The CPT theorem states that a unitary and Lorentzinvariant theory must also be invariant under a discrete symmetry mathbfCRT which reverses charge time and one spatial direction. In this article we study a mathbbZ2 times mathbbZ2 symmetry group in which two of the nontrivial symmetries Reflection Reality and a 180 degree rotation are implied by Unitarity and Lorentz Invariance respectively while the third is mathbfCRT. In cosmology Scale Invariance plays the role of Lorentz Invariance. This naturally leads to converses of the CPT theorem as any two of the discrete mathbbZ2 symmetries will imply the third one. Furthermore in many field theories the Reflection Reality mathbbZ2 symmetry is actually sufficient to imply the theory is fully unitary over a generic range of couplings. Building upon previous work on the Cosmological Optical Theorem we derive nonperturbative reality conditions associated with bulk Reflection Reality in all flat FLRW models and mathbfCRT in de Sitter spacetime in arbitrary dimensions. Remarkably this mathbfCRT constraint suffices to fix the phase of all wavefunction coefficients at future infinity up to a real sign  without requiring any analytic continuation or comparison to past infinity  although extra care is required in cases where the bulk theory has logarithmic UV or IR divergences. This result has significant implications for de Sitter holography as it allows us to determine the phases of arbitrary npoint functions in the dual CFT.
['High Energy Physics - Theory', 'Astrophysics - Cosmology and Nongalactic Astrophysics', 'General Relativity and Quantum Cosmology']
The Cosmological CPT Theorem
{'The Cosmological CPT Theorem': 'Harry Goodhew 1 ,a,b , Ayngaran Thavanesan 2 ,a , and Aron C. Wall 3 ,a,c  \n- a Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, UK.\n- b Leung Center for Cosmology and Particle Astrophysics, National Taiwan University, Taipei 10617, Taiwan.\n- c School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540 USA.', 'Abstract': "The CPT theorem states that a unitary and Lorentz-invariant theory must also be invariant under a discrete symmetry CRT which reverses charge, time, and one spatial direction. In this article, we study a Z 2 × Z 2 symmetry group, in which two of the nontrivial symmetries ('Reflection Reality' and a 180 degree rotation) are implied by Unitarity and Lorentz Invariance respectively, while the third is CRT . (In cosmology, Scale Invariance plays the role of Lorentz Invariance.) This naturally leads to converses of the CPT theorem, as any two of the discrete Z 2 symmetries will imply the third one. Furthermore, in many field theories, the Reflection Reality Z 2 symmetry is actually sufficient to imply the theory is fully unitary, over a generic range of couplings. Building upon previous work on the Cosmological Optical Theorem, we derive non-perturbative reality conditions associated with bulk Reflection Reality (in all flat FLRW models) and CRT (in de Sitter spacetime), in arbitrary dimensions. Remarkably, this CRT constraint suffices to fix the phase of all wavefunction coefficients at future infinity (up to a real sign) - without requiring any analytic continuation, or comparison to past infinity - although extra care is required in cases where the bulk theory has logarithmic UV or IR divergences. This result has significant implications for de Sitter holography, as it allows us to determine the phases of arbitrary n -point functions in the dual CFT.  \n2  \nat735@cantab.ac.uk", '1 Introduction': "In quantum field theory, symmetries and other physical principles impose fundamental constraints on dynamical systems. This philosophy of 'bootstrapping' has proven to be powerful in removing the need for repetitive computations, since typical Lagrangian methods lead to a huge amount of redundancy when going from theory to observables. In particle physics, the role of fundamental principles is hard to observe when calculating scattering amplitudes from Lagrangians in perturbation theory with many redundancies, for example different Lagrangians can be related to each other by gauge transformations or field redefinitions. As a result, the on-shell method was developed to describe all physically consistent S -matrices [1-6]. Recently, this approach has been used to tackle the difficult task of computing late-time correlators which live at the boundary at the end of inflation through the program known as the Cosmological Bootstrap (see e.g. [7-20] as well as a recent Snowmass white paper [21]). The goal of the Cosmological Bootstrap program is to circumvent the need for standard perturbation theory techniques to compute cosmological correlators, which, even at tree-level, require nested time integrals to be performed, and instead use symmetries and physical principles to directly infer the form and properties of the cosmological correlators.  \nHowever, this approach raises a new question: what are the symmetries that underpin the physical processes within the universe as well as our universe as a whole? One of the most universal and important symmetries in flat spacetime is CRT (the simultaneous reversal of time, T , reflection across a plane, R , and charge conjugation, C ). This symmetry ensures, for example, that particles and antiparticles have the same mass. Famously, the CPT theorem ensures that this symmetry is always valid in any QFT which is Lorentz invariant and unitary.  \nWhat are the implications of this symmetry for our universe as a whole? At first sight one might think that CRT is simply no longer a useful symmetry in cosmology (at least over distances longer than the Hubble scale), because the arrow of expansion breaks time reversal symmetry. Applying CRT would relate an expanding universe to a (hypothetical) contracting universe, and would therefore (seemingly) not place any interesting constraints on the expanding cosmology alone. On the other hand, in global de Sitter, which has a both a contracting phase and an expanding phase, CRT symmetry maps operators at future infinity ( I + ) to operators past infinity ( I -). But this formulation of CRT also does not appear to be very useful, if we only have access to observables at I + , as is typical for applications to inflationary cosmology, or dS/CFT. 1 If this were the only implication of CRT , it would not be of much use in cosmology.  \nSuppose we have a spacetime which can be well approximated by the Bunch-Davies vacuum state in de Sitter spacetime (e.g. during slow roll inflation). In this setup, we will show that by doing a suitable analytic continuation in the conformal time parameter η , we can access the past dS-Poincar'e patch, and hence place nontrivial constraints on correlators and wavefunction coefficients in the future Poincar'e patch. At future infinity ( η = 0) the constraints on the wavefunction coefficient become particularly simple, as the η continuation becomes trivial for IR finite correlators. In this case, we will be able to determine the phase of wavefunction coefficients without any analytic continuation.  \nSimilarly, in any FLRW cosmology with spatially flat time slices, we will show that one can also continue to a contracting cosmology by continuing the spatial coordinates y (or equivalently the momenta k ). This constraint on the wavefunction coefficients, which comes from Unitarity rather than CRT , gives a nonperturbative form of the Cosmological Optical Theorem [13-15], which agrees with the existing tree level results. (Here, we are referring to the hermitian analyticity reality conditions that apply to single diagrams, not to the 'cutting rules' which relate different diagrams.)  \nThe reason we can obtain these stronger constraints, is that we are using the additional conditions  \nthat (i) we are considering the vacuum state only, and (ii) the dynamics of the theory come from a local Lagrangian that has CRT symmetry around every point. This combination of discrete symmetries with locality turns out to give results much more powerful than those which can be obtained by simply thinking of discrete symmetries as acting globally on correlators, without considering the etiology of the correlators, i.e. the manner in which they arise from interactions, starting from a Bunch-Davies boundary condition.", '1.1 The CPT Theorem and its Converses': "In this paper we also hope to clarify the structure of the CPT theorem in flat spacetime. The usual CPT theorem proves the following implication:  \nLorentz Invariance + Unitarity = ⇒ CRT invariance .  \nIn this paper we derive a new formulation of the CPT theorem for integer spin fields in which we consider two discrete transformations, Reflection Reality ( RR ) and a 180 · rotation ( R π ), that are implied by Unitarity and Lorentz Invariance respectively. These form a Z 2 × Z 2 symmetry group with CRT and the identity. 2 This allows us to weaken the premises of the CPT theorem to:  \n180 · rotation + Reflection Reality = ⇒ CRT invariance .  \nThe existence of a 180 · rotation follows from an analytic continuation of the existence of a single SO + (1 , 1) Lorentz Invariance symmetry, which is itself a subgroup of the full Lorentz group SO + ( d, 1). On the other hand, Reflection Reality ( RR ) is a discrete symmetry which holds for all theories with real (but not necessarily positive) norm on states, and hence all unitary theories have to satisfy non-perturbatively. 3  \nThis symmetry group structure enables us to make converse statements to the usual CPT theorem, as any two of the transformations implies the third. So we also have:  \nCRT invariance + 180 · rotation = ⇒ Reflection Reality ;  \nCRT invariance + Reflection Reality = ⇒ 180 · rotation .  \nIt is important to note that a reflection real theory is not necessarily unitary, e.g. a theory with negative norm states or spin-statistics violating fields. However, this situation typically arises only in cases where one makes a bad discrete choice as to the matter content of a theory. In many cases (e.g. where you start with a unitary theory and continuously deform the coupling constants in a way that doesn't change the number of degrees of freedom), RR is enough to accidentally imply Unitarity. We will argue that this happens for a generic (codimension 0) set of couplings in Section 5.1.  \nWhile this fact, that RR is often enough to imply Unitarity, is not commonly discussed, it plays a structural role in many theoretical physics constructions. For example, it explains why field theorists working in Euclidean signature are very careful to ensure their theories are covariant, but don't spend much time worrying about Unitarity - this is because the discrete symmetries in a covariant Euclidean signature theory are usually sufficient to automatically ensure Unitarity upon continuation to Minkowski  \nsignature. Another example of this phenomenon was studied in [25], where the covariant and canonical formulations of the Maxwell field were compared in curved spacetime. In this case, covariance suffices to imply Unitarity, but not vice versa. In our current language, this is because Unitarity accidentally follows from CRT together with a discrete subgroup of covariance.", '1.2 Applications to Cosmology': "Furthermore, we identify an isomorphic Z 2 × Z 2 group structure in de Sitter, relating discrete symmetries arising from Scale Invariance, Unitarity and CRT in global de Sitter.  \nUsing the global structure of de Sitter as a guide, we are able to also identify the set of discrete symmetries pertaining to the Poincar'e patch of de Sitter through analytic continuations and use this to derive an analogous theorem in cosmology. In global de Sitter, the analogue of a Lorentz boost is an SO + (1 , 1) subgroup of the full de Sitter group SO + ( d +1 , 1). Any such SO + (1 , 1) boost defines a future Poincar'e patch, in which it acts as a scaling symmetry D λ on the conformal coordinates: ( η, y ) → ( λη, λ y ), for λ > 0. 4 The 180 · rotation thus becomes an analytic continuation of the scaling symmetry that sends ( η, y ) → ( -η, -y ), which we call Discrete Scale Invariance ( D -1 ). This is particularly interesting in the context of inflation where we often break de Sitter boosts (these act as special conformal transformations at the boundary) and so we lose the full set of de Sitter isometries. Here we need only to invoke an analytic continuation of Dilatations. This symmetry should therefore be valid in realistic models of slowroll inflation, up to terms proportional to the tilt n s -1.  \nThe Cosmological CPT theorem now says that:  \nScale Invariance + Unitarity = ⇒ CRT invariance.  \nwhile the weakened form and converse results are:  \nDiscrete Scale Invariance + Reflection Reality = ⇒ CRT invariance;  \nCRT invariance + Discrete Scale Invariance = ⇒ Reflection Reality;  \nCRT invariance + Reflection Reality = ⇒ Discrete Scale Invariance.  \nWe remind the reader that RR is a discrete symmetry which all unitary theories have to satisfy nonperturbatively, as well as to all orders in perturbation theory. This will become important when we discuss the constraints of these discrete symmetries on cosmological observables. 5 Besides the spacetime manifold itself, we will study the action of these symmetries on two basic types of objects: cosmological correlators in the in-in formalism, and wavefunction coefficients.  \nThe first structure are cosmological n -pt correlators B ( η 1 . . . η n ; y 1 . . . y n ). Here, we find in section 3.4 the following symmetry transformations for equal time correlators (the cosmological correlators are all on an equal time at the end of inflation) in the Poincar'e patch:  \nCRT : B n ( η ; y ) = B ∗ n ( -η ; -y ) ; (1.1)  \nD ± -1 : B n ( η ; y ) = B n ( -η ; -y ) ; (1.2)  \nRR : B n ( η ; y ) = B ∗ n ( η ; y ) , (1.3)  \n̸  \ne.g. RR implies the simple constraint that equal-time correlators are real. However, we will find that we can derive a more powerful constraint when working with the wavefunction coefficients considered in the cosmology literature.  \nNon-perturbative constraints on wavefunction coefficients. Secondly, we also study the application of these symmetries to the coefficients of the Wavefunction of the Universe (WFU) Ψ (see e.g. [27-36]). Note that the WFU Ψ we work with here is the QFT in curved spacetime limit of the Wheeler-DeWitt states considered in the canonical quantum gravity literature (see e.g. [24, 37-53]). By Taylor expanding Ψ, one obtains the wavefunction coefficients ψ n , which have proved useful in understanding Unitarity in the inflationary context through the Cosmological Optical Theorem [13-15], although until now these constraints have only been able to be expressed in a perturbative way. However, as a result of our approach through the Cosmological CPT theorem which has no reliance on perturbation theory, we are, for the first time, able to make non-perturbative statements of bulk Unitarity.  \nIt is important to realise that we cannot simply directly extend the equations (1.1)-(1.3) from B n to ψ n , as this method fails to produce interesting constraints on the phase of the wavefunction in the future Poincar'e patch. (The problem is that ψ n is generated by a single sheet of the path integral, instead of two sheets like B n .) Instead, we have to re-analyse the problem by directly considering the effect of the symmetries on the local Lagrangian, as stated above. This also identifies the reason for why the authors of [54] found some examples of non-local interactions which did not satisfy (1.9) at tree-level.  \n(In our analytic continuation of possible wavefunction coefficients ψ n , we find that we can no longer assume a priori that a 360 · complex rotation of the scaling acts trivially on ψ n . As often happens with complex functions, there is a possible monodromy when going around the singularity at the origin. Thus, we must distinguish between the two signs of the 180 · rotation, which we call D ± -1 . This extends the Z 2 × Z 2 to a larger group isomorphic to the automorphisms of the integers: Aut( Z ). 6 )  \nIn section 6.1 we find that the local Lagrangian symmetries enforce the following relationships between bulk wavefunction coefficients, ψ n :  \nCRT : [ ψ n ( η ; y ; Ω)] ∗ = ψ n ( e -iπ η ; -y ; e -iπ Ω); (1.4)  \nD ± -1 : ψ n ( η ; y ; Ω) = ψ n ( e ± iπ η ; e ± iπ y ; Ω) ; (1.5)  \nRR : [ ψ n ( η ; y ; Ω)] ∗ = ψ n ( η ; -e iπ y ; e -iπ Ω) . (1.6)  \nIn the above expressions, the factors of e ± iπ indicate the direction of analytic continuation in the complex plane. Ω is the Weyl factor of the bulk metric, which ends up being rotated by the non-perturbative analytic continuations of η or y .  \nPerturbative constraints. The above results can be expanded to arbitrary order in perturbation theory. Importantly, the rotation of the Weyl factor Ω leads to a dependence on the number of loops L in the Feynman-Witten diagram. However, the effects of this Ω rotation end up being trivial in d = odd spatial dimensions, for all tree level diagrams in the bulk (and more generally, loop diagrams that are free from divergences).  \nIn sections 6.2 and 7.2 we find the following perturbative constraints on the bare (UV-finite) wavefunc-  \ntion coefficients, valid for general real d :  \nCRT : [ ψ ( L ) n ( η 0 ; k ) ] ∗ = e iπ ( d +1)( L -1) ψ ( L ) n ( e -iπ η 0 ; k ) , (1.7)  \nD ± -1 : ψ ( L ) n ( η 0 ; k ) = e ± iπd ψ ( L ) n ( e ± iπ η 0 ; e ∓ iπ k ) , (1.8)  \nRR : [ ψ ( L ) n ( η 0 ; k ) ] ∗ = e iπ (( d +1) L -1) ψ ( L ) n ( η 0 ; e -iπ k ) . (1.9)  \nHere we have taken the Fourier Transform to enable a more direct comparison to the Cosmology literature (our results are given in both position and momentum space in the body of the paper). The last symmetry, RR , holds for correlators in any flat FLRW spacetime and has been previously observed for both contact [13, 14] and exchange [54] tree-level diagrams. The corrections for UV divergent amplitudes are discussed in Sections 6.2 and 7.5.  \nPhase formula. One particularly powerful application of (1.7) is to push the wavefunction coefficients to the future boundary where, for light fields, the time dependence factorises straightforwardly. The resulting boundary wavefunction coefficient, ψ ( L ) n , no longer depends on η , and so (in Section 7.4) we find it is possible to deduce the phase of an arbitrary coefficient:  \narg( ψ ( L ) n ) = -π 2 ( ( d +1)( L -1) + dn -∑ α ∆ α ) + π N , (1.10)  \nwhere the last term indicates that CRT can only fix the phase up to an overall ± sign. ∆ is the conformal dimension of the n operators ϕ + used to differentiate the boundary wavefunction Ψ[ ϕ -], which is written in the basis of the conjugate operators ϕ -with dimension d -∆. 7 This result holds for any boundary wavefunction coefficients that UV and IR-finite (i.e. free from logarithmic divergences in η or the UV cutoff ϵ ), as well as the coefficient in front of the leading log( η ) divergence for any IR-divergent wavefunction coefficients, or the leading log( ϵ ) divergence for any UV-divergent diagram. 8  \nIf we restrict attention to the case where all particles have ∆ = d , as for the stress-tensor and massless scalar fields, we find the phase is independent of n :  \narg( ψ ( L ) n ) = -π 2 ( d +1)( L -1) + π N . (1.11)  \nThis then tells us that massless scalar fields and gravitons have real boundary wavefunction coefficients to all loop order in even spacetime dimension provided they don't diverge in the UV. When UV divergences are present one finds in dimensional regularisation (dim-reg) a generic complex phase proportional to the dim-reg parameter δ , i.e. arg( ψ ( L ) n ) ∝ πδ . For tree-level, IR-finite and scale-invariant Feynman-Witten diagrams in cosmology, this was recently found independently [54] where the authors proved this reality condition by direct analysis of the bulk time integrals. 9 The implications of this for cosmology in terms of no-go theorems for parity violations has previously been explored for the specific case of the tree-level scalar trispectrum in [65,66], and will be treated for more general tree-level and loop diagrams in [67]. This  \n8 There is also an extra correction for diagrams with spinor propagators, see footnote 39.  \n9 AT would like to express his gratitude to David Stefanyszyn, Xi Tong and Yuhang Zhu for various extensive and helpful discussions on reality in cosmology.  \nis also consistent with what is found in flat-space amplitudes which can be shown to be real at tree-level, see e.g. [68] 10", '1.3 Implications for dS/CFT': "These constraints not only have implications for cosmology [67] but may also illuminate the role of Unitarity in de Sitter quantum gravity (dS/CFT), where previous attempts to identify the imprints of bulk unitary time evolution in the boundary theory had proven to be elusive (see e.g. [22,23]). Given the close relationship between CRT and Unitarity, and the fact that CRT simplifies at the future boundary I + , our work gives some reason to think that bulk Unitarity corresponds to some equally natural restriction on the boundary CFT side. At the very least, if somebody claims to have an example of dS/CFT holography, the phase formula would give a simple way to rule out examples where the bulk theory violates Reflection Reality ( RR ), without any need to do calculations that probe the bulk interior. Then, to ensure that the bulk theory is fully unitary, the remaining task is to check by hand that that the bulk Lagrangian has no negative norm fields. We expect that success at this task would not require any additional fine-tuning of continuous parameters, although it might require making good discrete choices. 11  \nA very simple check of the phase formula is to calculate the required number of degrees of freedom c dS of the holographic CFT in de Sitter. There are a number of possible definitions of c dS , including:  \n- 1. various coefficients of the trace anomaly ⟨ T i i ⟩ in ψ ( L ) 1 , when d = even, or\n- 2. the coefficient of the 2-point stress tensor correlator ⟨ T ij ( k ) T lm ( -k ) ⟩ in ψ ( L ) 2 for general d ,  \nwhere L is the loop order in the bulk theory, not the boundary CFT. These definitions coincide with the Virasoro central charge c when d = 2, but differ in arbitrary d . Happily, the phase formula agrees for all these indicators, 12 so regardless of the definition of c dS we find that  \narg( c dS ) = arg( ψ ( L ) n ) = -π 2 ( d +1)( L -1) + π N . (1.12)  \nFrom (1.12) we can infer that for d = 2 , 4 , . . . , we expect the central charge to be imaginary at tree level (but real at odd-loop order), while for d = 1 , 3 , 5 . . . it is real at all orders (section 7.5). This matches what has been found in the literature:  \n- · For d = 2 at tree-level, c = i 3 ℓ 2 G N (e.g. from applying Friedel's analysis to dS [41]); while for the case of pure gravity at 1-loop order we have c = i 3 ℓ 2 G N +13 [70]; 13\n- · For d = 3 at tree-level, c dS ∝ -ℓ 2 G N (e.g. in higher-spin dS/CFT [24] or Maldacena [27]);\n- · For d = 4 at tree level, c dS ∝ -i ℓ 3 G N (e.g. Maldacena [27]).\n- · For d = 5 we have c dS ∝ + ℓ 4 G N , the same sign as a normal unitary CFT [71].  \n13 We thank Jordan Cotler and Kristan Jensen for making us aware of their previous result for the 1-loop contribution to the central charge in dS 3 as well as upcoming work for loop calculations in dS 2 and dS 4 . The more observant reader will notice that the real 1-loop contribution is positive in the convention chosen by the authors and the same sign as the tree-level contribution as predicted from the combination of the normalisability constraint and the phase formula.  \nAlthough our phase formula does not determine the real sign of c dS , this can be fixed by the stability requirement that G N > 0, and one finds that the sign of the tree-level c dS has a repeating pattern in d mod 4 [71], as represented by the four cases above. This compatibility with previous results, as well as the results from more non-trivial explicit checks (forthcoming in [72] and summarised in Section 7.4 below) confirm the validity of the phase formula (1.10).", '1.4 Plan of Paper': "The outline of the paper is as follows:  \nWe begin in Section 2 by reviewing the CPT theorem in flat space. We identify three discrete symmetries which form a Z 2 × Z 2 group and are implied by the usual ingredients of the CPT theorem, namely Lorentz Invariance, Unitarity and CRT . We highlight the constraints these discrete symmetries impose on the observables and use the power of the group structure to provide a proof of the CPT theorem as well as converse statements. Then, in Section 3, we use the embedding formalism to extend these symmetries to de Sitter space, in both global and Poincar'e coordinates. We describe how the Z 2 × Z 2 group acts on cosmological correlators, with the 180 · rotation taking the form of a Discrete Scale Invariance symmetry. We also define the wavefunction coefficients, leaving their symmetry transformations to later sections.  \nTo lay the groundwork for further results, in Section 4, we discuss the important distinction between symmetries of the spacetime, and symmetries of the local Lagrangian, two concepts which do not quite overlap with each other. (Only the latter is useful for constraining wavefunction coefficients in a single Poincar'e patch.) In Section 5 we show that, in many generic scenarios, the Reflection Reality symmetry RR suffices to imply that the theory is fully Unitary. (This greatly increases the importance of the converse CPT theorem that concludes to RR .)  \nIn Section 6, we sketch how to implement discrete symmetries in cosmology, by analytically continuing either the η coordinate (to implement the local Lagrangian form of CRT ) or the k coordinate (to implement the local Lagrangian form of RR ), with these two transformations related by a discrete scale transform (180 · rotation) of the spacetime. This result can be expressed in both non-perturbative and perturbative forms. In Section 7 we provide a more concrete and detailed derivation of the perturbative conditions. By taking the limit as η → 0, we derive the phase formula for the wavefunction coefficients at future infinity, and we check the formula by comparing to calculations of specific Feynman-Witten amplitudes in the literature. Finally, we conclude in Section 8 with a discussion of the implications of these results for dS/CFT and inflation, as well as some directions for possible future work.", '2 CPT in Flat space': 'The CPT theorem in flat space states that a Unitary and Lorentz-invariant theory will also be CRT invariant. However, it is not possible to make converse statements. In this section we will present a new perspective on the proof of the CPT theorem, highlighting the group structure of a set of discrete symmetries that are consequences of CRT , Unitarity and Lorentz Invariance. This group structure necessitates the existence of converse statements and thus highlights which aspects of Unitarity or Lorentz Invariance are missed in these converse statements; however in Section 5 we will discuss how in many field theories, the full principle of Unitarity is accidentally implied up to some discrete choices.', '2.1 Lorentz Invariance': 'A common confusion in the literature is interchangeable labelling of the proper orthochronous Lorentz group (also known as the restricted Lorentz group) SO + ( d, 1) and the proper Lorentz group SO ( D, 1). The precise statement of Lorentz Invariance is invariance under the proper orthochronous Lorentz group SO + ( d, 1) which consists of those Lorentz transformations that preserve both the orientation of space and the direction of time. Table 1 provides a classifcation of all the subgroups of the Lorentz group which contain at least one of the four connected components { 1 , P, T, P T } . 14 The collection of the four interconnected parts can be organised into a group as the quotient group O ( d, 1) /SO + ( d, 1) which is also isomorphic to the Z 2 × Z 2 group. Any Lorentz transformation can be defined through a correct, proper orthochronous Lorentz transformation combined with two additional pieces of data, which distinguish one of the four interconnected parts. This structure is characteristic of finite-dimensional Lie groups.  \nTable 1: Subgroups of the Lorentz group which contain at least one of the four connected components { 1 , P, T, P T } , where d is the number of spatial dimensions. In applications to de Sitter, one should replace d with D = d +1.', '2.2 Outline of CPT Theorem': 'The CPT theorem states that any QFT that is:  \n- 1. Unitary,\n- 2. Lorentz invariant, and has\n- 3. Energy bounded from below,  \nmust also be invariant under the combined CRT symmetry. We will now dissect each assumption one by one.  \nAssumption 3 allows us to analytically continue to Euclidean signature, using the fact that e -H ∆ τ is suppressed for positive Euclidean times ∆ τ > 0. Let us adopt coordinates in Euclidean signature where τ = it and x I , I = 1 ... d are the spatial coordinates. We highlight here (as it will become important in the cosmological case) that the R in CRT reflects a single spatial coordinate, say x 1 →-x 1 .  \nThe SO + ( d, 1) Lorentz Invariance in d + 1-dimensional Minkowski spacetime analytically continues to SO ( d + 1) rotational invariance in Euclidean signature. On the other hand, Unitarity implies that Euclidean correlators satisfy Reflection Positivity, which we will define in the following section. However, it is not necessary to enforce either of these relationships in their entirety to ensure CRT symmetry. We do not require full SO + ( d, 1) Lorentz Invariance; we simply need the subgroup SO + (1 , 1) boost (this will  \nbecome important when studying the implications of the CPT theorem for cosmology). In fact, all that is required is a discrete, Z 2 symmetry from each of them:  \nwhere in each row, the first arrow represents the Wick rotation to Euclidean Signature and the second arrow is an implication. The CPT theorem is then simply the statement that the combination of these two symmetries is a CRT transformation. Therefore, requiring the symmetry of correlators under both Reflection Reality and this 180 · rotation enforces their invariance under the CRT transformation.  \nSuppose, for concreteness, we start by considering vacuum correlators of the form  \nB n = 〈 ϕ 1 ( τ, x ) ϕ 2 ( τ, x ) . . . ϕ n ( τ, x ) 〉 , (2.1)  \nwhere ϕ is some generic, real, scalar field and we have suppressed the dependence on the other spatial coordinates that we do not transform. Without loss of generality, we will choose to work in a real basis for all fields, so that the transformation CT just becomes T and we have an RT theorem (this will become important when discussing heavy fields in cosmology). 15 Then the Z 2 transformations of interest act in the following way on Euclidean field theory correlators:  \n- · As we will show in Sections 2.3 and 2.4, Euclidean Reflection Positivity implies a weaker principle know as Reflection Reality. We can enforce this by insisting that our correlators are invariant under the transformation:  \nRR : B ( τ, x ) → B ∗ ( -τ, x ) , (2.2)  \nRR ( B n ) = 〈 ϕ 1 ( -τ, x ) ϕ 2 ( -τ, x ) . . . ϕ n ( -τ, x ) 〉 ∗ ; (2.3)  \n- · A 180 · (= π ) rotation in the τ -x plane corresponds to the transformation: 16  \nR π : B ( τ, x ) → B ( -τ, -x ) , (2.4)  \nR π ( B n ) = 〈 ϕ 1 ( -τ, -x ) ϕ 2 ( -τ, -x ) . . . ϕ n ( -τ, -x ) 〉 ; (2.5)  \n- · Finally, CRT is the transformation:  \nCRT : B ( τ, x ) → B ∗ ( τ, -x ) , (2.6)  \nCRT ( B n ) = 〈 ϕ 1 ( τ, -x ) ϕ 2 ( τ, -x ) . . . ϕ n ( τ, -x ) 〉 ∗ . (2.7)  \nIt might seem surprising that there is no transformation of the τ coordinate here. However, this is simply a reflection of the fact that these correlators are a function of Euclidean time, τ = it , so simultaneously complex conjugating and sending t →-t leaves τ unchanged.  \nIt is straightforward to see that combining any two of these transformations produces the third and therefore (together with the identity) they form a group which is isomorphic to Z 2 × Z 2 (or the Klein 4-group). The consequence of this is that a theory respecting any two of these discrete symmetries will automatically respect the third. 17  \nIf we exploit this group structure to derive CRT from invariance under the other two transformations we recover the standard CPT theorem. However, there also exist two converse directions, where we use  \nCRT + RR = ⇒ R π ,  \nor else  \nCRT + R π = ⇒ RR .  \nThis last implication is particularly interesting as, in many field theories, Reflection Reality is strong enough to accidentally imply the full principle of Unitarity, despite being weaker that it. This will be discussed further in Section 5.  \nTo generalise to correlators of operators that carry spin, it is of course also necessary to specify how the fields transform under the coordinate transformations above, e.g. by defining them as tensor fields. In the case of a tensor field with an odd number of τ indices, the transformation τ → -τ will reverse its sign; this is to ensure that e.g. the time component of a real vector field V t remains self adjoint: ( V t ) † = V t = ⇒ ( V τ ) † = -V τ . (The spatial polarisations are unaffected by this transformation.)  \nFor theories involving spinor fields, it is also necessary to consider projective representations of Z 2 × Z 2 . In this paragraph, let us assume for simplicity that the theory is fully rotationally covariant. Then a 360 · rotation gives a factor of ( -1) 2 s (where s is the spin); this is of course the same as the fermion parity ( -1) F for fields that satisfy spin-statistics. Hence, the 180 · rotation R π has order 4 when acting on half-integer spin fields, whilst Unitarity requires that RR 2 = +1 and hence in a rotationally covariant theory we also need CRT 2 = +1. This is only possible if the group anticommutes in the sense that [ A,B ] := ABA -1 B -1 = -1 when A,B are any two distinct elements of { RR , CRT , R π } . One thus finds that the appropriate double cover of Z 2 × Z 2 for spinors is the dihedral group D 4 . 18 We will focus on fields with integer spin representations and not consider half-integer spin fields in detail this paper, but a similar formalism should also provide a much easier approach to obtain a Unitarity constraint for correlators involving spinors in flat space.', '2.3 Reflection Positivity': "In this section we review how the Reflection Positivity of correlators follows from the assumption of Unitarity, as it is an important element in the proof of the CPT theorem presented in the previous section, and may be unfamiliar to a reader.  \nConsider a ket state defined on the τ = 0 slice of a Euclidean field theory. This may be defined by (an arbitrary sum/integral of) a time ordered Euclidean correlator, in which all operators are inserted on the  \n18 With the addition of one additional magical fact that Unitarity requires CRT 2 = ( -1) 2 s ( -1) F , which requires a close inspection of the transformation properties of 2-pt correlators to prove, the considerations in this paragraph provide all the necessary ingredients to prove the spin-statistics theorem. But spin-statistics requires at least covariance under 90 · rotations, not just 180 · rotations, so it is less robust than the CPT theorem.  \nside τ < 0:  \n∣ ∣ ψ 〉 = ∑ ∫ f O n ( τ n ) . . . O 2 ( τ 2 ) O 1 ( τ 1 ) ∣ ∣ 0 〉 (2.8)  \nwith τ 1 ≤ τ 2 ≤ . . . ≤ τ n < 0. We have suppressed writing the spatial coordinates for notational conciseness as they are left invariant by this transformation. Note that f has an explicit dependence on τ 19 and is an arbitrary distribution over the spatial coordinates and choices of fields, such that the resulting n -point correlator is finite. Similarly, the corresponding bra is 20  \n〈 ψ ∣ ∣ = ∑ ∫ f ∗ τ →-τ 〈 0 ∣ ∣ O † 1 ( -τ 1 ) O † 2 ( -τ 2 ) . . . O † n ( -τ n ) . (2.9)  \nThe positivity of the inner product ⟨ ψ | ψ ⟩ ≥ 0 therefore implies the principle of Reflection Positivity:  \n( ∑ ∫ f ∗ τ →-τ 〈 0 ∣ ∣ O † 1 ( -τ 1 ) O † 2 ( -τ 2 ) . . . O † n ( -τ n ) )( ∑ ∫ f O n ( τ n ) . . . O 2 ( τ 2 ) O 1 ( τ 1 ) ∣ ∣ 0 〉 ) ≥ 0 , (2.10)  \na principle which is much simpler in spirit than the number of symbols it takes to write it out. Note that even in a unitary theory, it is possible for the RHS to vanish if the correlator vanishes, for example if we insert the equation of motion in a free theory, or a pure gauge state in a gauge theory. Such null states may be projected out of the theory, leaving a Hilbert space with a positive-definite inner product.  \nFor a fixed number, N , of insertions on each side, we would obtain the positivity of some 2 N -point function. However, it is also allowed to consider states coming from quantum superpositions of different values of N . Because the sums are taken independently on both sides of the τ = 0 surface, Reflection Positivity also places constraints on the correlations with odd numbers of points. It is not, however, allowed for the operators which define the ket to stray 'offside' onto the wrong side of the τ = 0 surface, even if you include the corresponding reflected operator.  \nIn Lorentzian signature, Unitarity consists of two statements: (i) there exists a positive norm on the state space, and (ii) this norm is preserved under time evolution. Strictly speaking, Reflection Positivity is the Wick rotation of (i), not (ii). 21 But (ii) follows if we additionally use the τ -translation invariance of the Euclidean path integral. This implies that if we insert an empty strip of width ∆ τ between two correlators, it doesn't matter which side we insert it on:  \n| χ ⟩ † ( e -H ∆ τ | ψ ⟩ ) = ( e -H ∆ τ | χ ⟩ ) † | ψ ⟩ . (2.11)  \nfrom which it follows that H = H † , guaranteeing that the Lorentzian evolution e iHt preserves the norm. (More generally, if one assumes the Euclidean partition function is a covariant functional of the metric, one could deduce that Lorentzian evolution with respect to arbitrary lapse and shift functions will also be unitary.) Some constraints from only having states with positive norm were explored in [76, 77], and explained in a particularly clear way in [78].  \n̸", '2.4 Reflection Reality': 'Since every positive number is real, Reflection Positivity obviously implies the weaker principle that:  \n( ∑ ∫ f ∗ τ →-τ 〈 0 ∣ ∣ O † 1 ( -τ 1 ) O † 2 ( -τ 2 ) . . . O † n ( -τ n ) )( ∑ ∫ f O n ( τ n ) . . . O 2 ( τ 2 ) O 1 ( τ 1 ) ∣ ∣ 0 〉 ) ∈ R . (2.12)  \nThis is simply the assertion that the inner product is real (but not necessarily positive). This is the analogue of (i) in the previous section. The further statement corresponding to (ii), that this (possibly indefinite) real product is preserved by Lorentzian time evolution, follows from the same argument in (2.11).  \nEq. (2.12) does not yet manifestly take the form of a Z 2 symmetry, because of the annoying restrictions that (a) the amplitude must be a product of two conjugate correlators, (b) each of which is restricted to one side of τ = 0. However, it turns out that when we only care about deriving a reality condition, neither of these restrictions are actually required. To see this, simply consider a state of the form  \n∣ ∣ β 〉 = ∣ ∣ 0 〉 + β ∣ ∣ ψ 〉 , (2.13)  \nwhere β is an arbitrary complex number. Now Eq. (2.12) tells us that  \n〈 β ∣ ∣ β 〉 = 〈 0 | 0 〉 + β 〈 0 | ψ 〉 + β ∗ 〈 ψ | 0 〉 + | β | 2 〈 ψ | ψ 〉 ∈ R . (2.14)  \nBut the first and last terms are also real by Eq. (2.12). And the only way the sum of the middle terms can be real for all values of β , is if  \n〈 ψ | 0 〉 = 〈 0 | ψ 〉 ∗ , (2.15)  \nfrom which it follows that  \n( ∑ ∫ f ∗ τ →-τ 〈 0 ∣ ∣ O † 1 ( -τ 1 ) O † 2 ( -τ 2 ) . . . O † n ( -τ n ) ) ∣ ∣ 0 〉 = 〈 0 ∣ ∣ ( ∑ ∫ f O n ( τ n ) . . . O 2 ( τ 2 ) O 1 ( τ 1 ) ∣ ∣ 0 〉 ) ∗ . (2.16)  \nSimilarly, at this point the restriction that the correlators remain on one side of the τ = 0 surface can be lifted, either by using translation invariance, or by analyticity of the Euclidean position space correlators. It follows that an arbitrary correlator satisfies the following Reflection Reality ( RR ) condition:  \n〈 O 1 ( τ 1 ; x 1 ) O 2 ( τ 2 ; x 2 ) . . . O n ( τ n ; x n ) 〉 = 〈 O † 1 ( -τ 1 ; x 1 ) O † 2 ( -τ 2 ; x 2 ) . . . O † n ( -τ n ; x n ) 〉 ∗ (2.17)  \nwhich now takes the form of a Z 2 symmetry that reverses the sign of Euclidean time τ and also complex conjugates the amplitude. This symmetry is therefore required to hold in any unitary field theory. But it is somewhat weaker than Unitarity as it is compatible with the existence of states with negative-norm.', '2.5 Lorentzian Correlators': 'In Lorentzian signature, RR is simply a hermitian conjugation, and implies that self-adjoint fields have real expectation values. As ϕ = ϕ † , the action of taking the adjoint simply reverses the order of operators, and (in operator ordering) RR implies:  \n〈 ϕ 1 ( t 1 ; x 1 ) ϕ 2 ( t 2 ; x 2 ) . . . ϕ n ( t n ; x n ) 〉 = 〈 ϕ n ( t n ; x n ) . . . ϕ 2 ( t 2 ; x 2 ) ϕ 1 ( t 1 ; x 1 ) 〉 ∗ (2.18)  \nbut without any positivity conditions. We also now introduce time ordered correlators as the object of consideration in Lorentzian signature,  \nG n = 〈 T ϕ 1 ( t 1 ; x 1 ) ϕ 2 ( t 2 ; x 2 ) . . . ϕ n ( t n ; x n ) 〉 . (2.19)  \nThe same group structure then emerges from the transformations:  \n- · Reflection Reality is the statement that:  \nRR : G ( t ; x ) → G ∗ ( t ; x ) , (2.20)  \nRR ( G n ) = 〈 T ϕ 1 ( t 1 ; x 1 ) ϕ 2 ( t 2 ; x 2 ) . . . ϕ n ( t n ; x n ) 〉 ∗ , (2.21)  \nwhere the bar indicates that time ordering has been exchanged with anti-time ordering (in other words, the order would have been reversed if we had been using operator ordering). For equal time correlation functions this relationship therefore enforces that these correlators are real. In other words, RR implies the existence of a real norm on the Hilbert space (which need not be positive). As we shall see later, this norm is also preserved under Lorentzian time evolution.  \n- · The 180 · rotation for time-ordered correlators becomes:  \nR π : G ( t ; x ) → G ( -t ; -x ) , (2.22)  \nR π ( G n ) = 〈 T ϕ 1 ( -t 1 ; -x 1 ) ϕ 2 ( -t 2 ; -x 2 ) . . . ϕ n ( -t n ; -x n ) 〉 , (2.23)  \nwhere time ordering has been preserved notwithstanding the reversal of the sign of t .  \n- · Finally, CRT is the transformation:  \nCRT : G ( t ; x ) → G ∗ ( -t ; -x ) , (2.24)  \nCRT ( G n ) = 〈 T ϕ 1 ( -t 1 ; -x 1 ) ϕ 2 ( -t 2 ; -x 2 ) . . . ϕ n ( -t n ; -x n ) 〉 ∗ . (2.25)', '3 CPT in de Sitter': 'We previously showed that in flat space the CPT theorem in Euclidean signature can be understood through a Z 2 × Z 2 symmetry group that relates the discrete symmetries of Reflection Reality RR , the 180 · Euclidean rotation R π , and CRT . The first two of these symmetries are guaranteed by the Lorentzian properties of Unitarity and SO + (1 , 1) invariance.  \nAnalogous properties also exist in de Sitter, leading to an isomorphic Z 2 × Z 2 group. In this section we will determine the effect of the three discrete symmetries RR , R π , and CRT on the coordinates by considering the embedding of de Sitter in a D +1-dimensional Minkwoski spacetime.', '3.1 Global de Sitter': "Similar to how a Lorentz invariant theory in a D + 1-dimensional (where D = d + 1) Minkowski spacetime is invariant under the restricted Lorentz group SO + ( D +1 , 1), a de Sitter invariant theory in a d +1-dimensional de Sitter spacetime is invariant under SO + ( d +1 , 1), which consists of those de Sitter transformations that preserve both the orientation of space and the direction of time.  \nIn Euclidean signature, the SO + ( d +1 , 1) de Sitter invariance in d +1-dimensional de Sitter spacetime becomes SO ( d +2) rotational invariance of a S d +1 sphere of radius ℓ embedded in R D +1 :  \nδ AB x A x B = τ 2 + x I x I = ℓ 2 , (3.1)  \nand hence its metric is given by,  \nd s 2 = dΘ 2 + ℓ 2 cos 2 ( Θ ℓ ) ( dΦ 2 d +sin 2 (Φ d ) dΩ 2 d -1 ) . (3.2)  \nThe embedding space and the coordinates of the Euclidean dS sphere are related by  \nτ = ℓ sin ( Θ ℓ ) , (3.3)  \nx I = ℓ cos ( Θ ℓ ) sin(Φ d ) z i , (3.4)  \nx D = ℓ cos ( Θ ℓ ) cos(Φ d ) , (3.5)  \nwhere the z i describe a unit S d -1 satisfying z i z i = 1 and are expressible in terms of d -1 angular variables, Φ i .  \nWe now act with the same flat space transformations described in Section 2, but now in the τ -x D plane of the embedding space. In the embedding space the CRT transformation and rotations act on these coordinates as  \nCRT  \n: ( τ, x 1 , . . . , x d , x D ) → ( τ, x 1 , . . . , x d , -x D ) , (3.6)  \nR θ : ( τ, x 1 , . . . , x d , x D ) → ( τ cos( θ ) + ix D sin( θ ) , x 1 , . . . , x d , x D cos( θ ) + iτ sin( θ )) . (3.7)  \nThe reason these are identical for θ = π , despite representing different transformations when acting on correlators, is that CRT additionally includes complex conjugation, which does not affect the coordinates. We also note that in this formulation this rotation can be understood as an analytic continuation of the Lorentz boost to imaginary values of the rapidity, θ = iξ . The action of the Z 2 transformations from Section 2 on the Euclidean dS correlators is the following:  \n- · Reflection Reality requires invariance under the transformation:  \nRR : B (Θ; Φ) → B ∗ ( -Θ;Φ) , (3.8)  \nRR ( B n ) = 〈 ϕ 1 ( -Θ 1 ; Φ 1 ) ϕ 2 ( -Θ 2 ; Φ 2 ) . . . ϕ n ( -Θ n ; Φ n ) 〉 ∗ , (3.9)  \nwhere Reflection Reality maps an insertion on the S d +1 across the Θ = 0 plane and a global complex conjugation acting on both complex fields and amplitudes/correlators;  \n- · A 180 · rotation corresponds to the transformation:  \nR π : B (Θ; Φ) → B ( -Θ; π -Φ) , (3.10)  \nR π ( B n ) = 〈 ϕ 1 ( -Θ 1 ; π -Φ 1 ) ϕ 2 ( -Θ 2 ; π -Φ 2 ) . . . ϕ n ( -Θ n ; π -Φ n ) 〉 , (3.11)  \nwhere the rotation amounts to a reflection from a pole on one axis of the sphere to the opposite pole;  \n- · Finally, CRT is the transformation:  \nCRT : B (Θ; Φ) → B ∗ (Θ; π -Φ) , (3.12)  \nCRT ( B n ) = 〈 ϕ 1 (Θ 1 ; π -Φ 1 ) ϕ 2 (Θ 2 ; π -Φ 2 ) . . . ϕ n (Θ n ; π -Φ n ) 〉 ∗ , (3.13)  \nwhere the transformation is a reflection from a pole on one axis of the sphere to the opposite pole as well as a global complex conjugation acting on both complex fields and amplitudes/correlators.  \nWhen we do a Wick rotation τ = ix 0 to Lorentzian signature, this converts the embedding Euclidean space into a Minkowski space:  \nd s 2 = η µν d x µ d x ν = -(d x 0 ) 2 +d x I d x I , I = 1 , . . . , D . (3.14)  \nDe Sitter spacetime then corresponds to a hyperboloid in these coordinates [79] satisfying,  \nη µν x µ x ν = -( x 0 ) 2 + x I x I = ℓ 2 = 1 H 2 = d ( d -1) 2Λ , (3.15)  \nwhere ℓ is the de Sitter radius which is inversely related to the usual Hubble constant H parameterising the expansion rate set by the positive cosmological constant Λ. This hyperboloid can be sliced up in many ways; the slicings we will consider are the closed slicing of global de Sitter (this section) and the Poincar'e flat slicing (Section 3.2). 22 The flat slicing coordinates of interest to inflationary cosmology cover only the region with x 0 + x D > 0, whilst the closed slicing of global de Sitter covers the whole hyperboloid.  \nIn Lorentzian global de Sitter coordinates (also referred to as closed slicing), the metric w.r.t. the proper time coordinate T is given by  \nd s 2 = -d T 2 + ℓ 2 cosh 2 ( T ℓ ) ( dΦ 2 d +sin 2 (Φ d ) dΩ 2 d -1 ) , (3.16)  \nwhere T ∈ R and dΩ 2 d -1 is the round metric on the unit d -1-sphere. We can see from the metric that constant T surfaces are d -spheres which shrink from the past asymptotic boundary I -at T = -∞ to T = 0 and grow from T = 0 to the future asymptotic boundary I + at T = + ∞ . The embedding space and global de Sitter coordinates are related as  \nx 0 = ℓ sinh ( T ℓ ) , (3.17)  \nx i = ℓ cosh ( T ℓ ) sin(Φ d ) z i , (3.18)  \nx D = ℓ cosh ( T ℓ ) cos(Φ d ) . (3.19)  \nThe CRT transformation and rotations act on the embedding space coordinates as  \nCRT : ( x 0 , x 1 , . . . , x d , x D ) → ( -x 0 , x 1 , . . . , x d , -x D ) , (3.20)  \nR θ : ( x 0 , x 1 , . . . , x d , x D ) → ( x 0 cos( θ ) + ix D sin( θ ) , x 1 , . . . , x d , x D cos( θ ) + ix 0 sin( θ )) . (3.21)  \nThe CRT transformation in Lorentzian global de Sitter is then straightforwardly given by  \nCRT : ( T, Φ 1 , . . . , Φ d ) → ( -T, Φ 1 , . . . , π -Φ d ) , (3.22)  \nCRT : G ( T ; Φ d ) → G ∗ ( -T ; π -Φ d ) , (3.23)  \nCRT ( G n ) = 〈 T ϕ 1 ( -T 1 ; π -Φ d 1 ) ϕ 2 ( -T 2 ; π -Φ d 2 ) . . . ϕ n ( -T n ; π -Φ dn ) 〉 ∗ . (3.24)  \nUnfortunately, an arbitrary rotation doesn't have such a simple expression; the reason for this is that this rotation in the embedding space formalism misaligns the axis x 0 from the axis of the hyperboloid, so that the slices are now ellipsoids not spheres and attempting to parameterise the resulting slices using spherical coordinates fails. However, for the specific 180 · rotation in Section 2.5 we have  \nR π : ( T, Φ 1 , . . . , Φ d ) → ( -T, Φ 1 , . . . , π -Φ d ) , (3.25)  \nR π : G ( T ; Φ d ) → G ( -T ; π -Φ d ) , (3.26)  \nR π ( G n ) = 〈 T ϕ 1 ( -T 1 ; π -Φ d 1 ) ϕ 2 ( -T 2 ; π -Φ d 2 ) . . . ϕ n ( -T n ; π -Φ dn ) 〉 , (3.27)  \nwhich as before, acts on the coordinates in the same way as the CRT transformation, with the difference between the two reappearing when we consider a complex object. Finally we can infer that the RR  \ntransformation in these coordinates is given by  \nRR : ( T, Φ 1 , . . . , Φ d ) → ( T, Φ 1 , . . . , Φ d ) , (3.28)  \nRR : G ( T ; Φ d ) → G ∗ ( T ; Φ d ) , (3.29)  \nRR ( G n ) = 〈 T ϕ 1 ( T 1 ; Φ d 1 ) ϕ 2 ( T 2 ; Φ d 2 ) . . . ϕ n ( T n ; Φ dn ) 〉 ∗ , (3.30)  \nwhere, once again, we see that for equal time correlation functions this relationship enforces that these correlators are real, i.e. RR implies the existence of a real norm on the Hilbert space, which need not be positive. 23", "3.2 Poincar'e de Sitter": "Having shown in Section 3.1 that global de Sitter inherits the three discrete symmetries of interest in the CPT theorem from the embedding space, we can now consider these transformations in the Poincar'e patch, where we perform inflationary calculations. However, things will become much less trivial here, since a single Poincar'e patch explicitly chooses a specific orientation of time from the perspective of the global time coordinate. Thus, the discrete transformations will each require an analytic continuation outside of the physical domain of the original Poincar'e patch. In this section, we will establish the behaviour of this analytic continuation at the level of the coordinates and in Sections 3.3-3.5 we will discuss the implications for the inflationary observables known as the cosmological correlators and the Wavefunction of the Universe.  \nIn the flat slicing of the Poincar'e patch we have the metric,  \nd s 2 = -d t 2 + e 2 Ht d y i d y i , i = 1 , . . . , d , (3.31)  \nwhere t ∈ R is generally referred to as cosmological time in the literature. These coordinates are related to the embedding space coordinates by  \nx 0 = 1 H sinh ( Ht ) + H 2 y i y i e Ht , (3.32)  \nx D = 1 H cosh ( Ht ) -H 2 y i y i e Ht , (3.33)  \nx i = e Ht y i , 1 ≤ i ≤ d, (3.34)  \nThe transformation t → t + iπ H recovers both parts of the CRT transform in these coordinates but it naturally reflects all spatial coordinates rather than just one. 24 It is possible to recover just the transformation in (3.20) by additionally transforming all y i →-y i . Unfortunately, analytically continuing the time variable is not easily interpreted as a time reversal. However, if we go to conformal time 25 , where the metric becomes  \nd s 2 = ( -1 Hη ) 2 ( -d η 2 +d y i d y i ) , i = 1 , . . . , d , (3.36)  \n25 In cosmology conformal time and cosmological time are related by d t = a d η where for de Sitter we have a ( t ) = e Ht and thus  \n∫ η -∞ d η ' = ∫ t -∞ e -Ht ' d t ' = ⇒ η = -1 H e -Ht . (3.35)  \nthen our new coordinates are related to those in the embedding space by  \nx 0 = 1 H sinh ( Ht ) + H 2 y i y i e Ht = H 2 η 2 -1 -H 2 y i y i 2 H 2 η = η 2 -1 -y i y i 2 Hη , (3.37)  \nx D = 1 H cosh ( Ht ) -H 2 y i y i e Ht = H 2 y i y i -1 -H 2 η 2 2 H 2 η = y i y i -1 -η 2 2 Hη , (3.38)  \nx i = e Ht y i = -y i Hη , 1 ≤ i ≤ d , (3.39)  \nη = y 0 = -1 H ( x 0 + x D ) , y i = x i x 0 + x D . (3.40)  \nIn the final equality for (3.37) and (3.38) we have used the fact that we can rescale η → η/H and y i → y i /H without loss of generality. In order to be consistent with the cosmology literature we will work with the parametrisation in which η ≤ 0 (with η = 0 -being the same future asymptotic boundary I + in global de Sitter) and y i ∈ R , i.e. we will pick the Poincar'e patch covering x 0 + x D ≥ 0. Such foliations only cover half of de Sitter space; however CRT or the 180 · rotation will take you from one Poincar'e patch to the other.  \nWe see that the CRT transformation acts on the coordinates as:  \nCRT : ( η, y 1 , . . . , y d ) →-( η, y 1 , . . . , y d ) = ( -η, -y 1 , . . . , -y d ) . (3.41)  \nTherefore, in order to recover a CRT transformation in a single spatial and time coordinate in the embedding space, or equivalently in global de Sitter slicing, it is necessary to transform all spatial coordinates in the Poincar'e patch in addition to analytically continuing our conformal time variable. 26 The analytic continuation in conformal time η is just an artefact of the fact that T (or CRT ) is a transformation within the full global de Sitter which is not captured by a single Poincar'e patch. Thus CRT involves an analytic continuation since the observables are defined for one Poincar'e patch where η ≤ 0. We will discuss in Sections 3.3-3.5 how this analytic continuation manifests itself at the level of observables.  \nWe can similarly consider how the Lorentzian SO + (1 , 1) boost in the embedding space manifests in the Poincar'e patch, since it is this boost which, when analytically continued, becomes SO (2) in Euclidean signature and gives us the necessary R π rotation in the Z 2 × Z 2 group. From the perspective of the Poincar'e patch the generator of the SO + (1 , 1) boost becomes:  \nL 0 D ≡ x 0 ∂ ∂x D -x D ∂ ∂x 0 (3.42)  \n= η ∂ ∂η + y i ∂ ∂y i ≡ D, (3.43)  \nwhere we have used the fact that  \n∂ ∂x 0 ≡ ∂y µ ∂x 0 ∂ ∂y µ = Hη 2 ∂ ∂η + Hηy i ∂ ∂y i (3.44)  \n∂ ∂x D ≡ ∂y µ ∂x D ∂ ∂y µ = Hη 2 ∂ ∂η + Hηy i ∂ ∂y i , (3.45)  \nHence we find that the SO + (1 , 1) boost acts as spacetime dilatations (which we will refer to as just Dilatations for the sake of simplicity) on the Poincar'e patch, which in the cosmology literature is generally  \nreferred to as bulk Scale Invariance. 27 The action of the rotation operator on the coordinates in embedding space is:  \n( x 0 , x 1 , . . . , x d , x D ) = ( -1 H 2 η + η -y i y i 2 η , -y 1 Hη , . . . , -y d Hη , -1 H 2 η -η + y i y i 2 η ) , (3.46)  \nR θ : ( x 0 , x 1 , . . . , x d , x D ) → ( x 0 cos( θ ) + ix D sin( θ ) , x 1 , . . . , x D cos( θ ) + ix 0 sin( θ )) (3.47)  \n= ( -e iθ H 2 η + η e iθ -y i y i 2 ηe iθ , -y 1 Hη , . . . , -y d Hη , -e iθ H 2 η -η e iθ + y i y i 2 ηe iθ ) . (3.48)  \nFrom this we can read off that this rotation is equivalent to the Poincar'e patch coordinate transformation,  \nR θ : ( η, y 1 , . . . , y d ) → e -iθ ( η, y 1 , . . . , y d ) . (3.49)  \nJust as the flat space transformation was an analytic continuation of Lorentz boosts to imaginary rapidity, this rotation has become the analytic continuation of the Dilatation operator to complex scales. In particular the 180 · rotation acts on the coordinates as a negative Dilatation:  \nD -1 : ( η, y 1 , . . . , y d ) →-( η, y 1 , . . . , y d ) = ( -η, -y 1 , . . . , -y d ) . (3.50)  \nNote that the action on coordinates is the same as for CRT . This is expected as the same thing happens in flat space; the key differences for the 180 · rotation are i) it does not involve any complex conjugation, and ii) the correlator continues to be time-ordered. This is particularly interesting in the context of inflation where we often break de Sitter boosts (these act as special conformal transformations at the boundary) and so we lose the full set of de Sitter isometries. Here we need only to invoke an analytic continuation of Dilatations, which comes naturally from the embedding space formalism.  \nIn momentum space we would have almost the same relations:  \nCRT  \n: ( η, k 1 , . . . , k d ) →-( η, -k 1 , . . . , -k d ) = ( -η, k 1 , . . . , k d ) , (3.51)  \nD -1 : ( η, y 1 , . . . , y d ) →-( η, k 1 , . . . , k d ) = ( -η, -k 1 , . . . , -k d ) . (3.52)  \napart from the fact that the complex conjugation in CRT in position space reverses the sign of k , due to the Fourier transform, cancelling out the reflection of each spatial coordinate.  \nWe will now move on to discussing the constraints these discrete symmetries impose on the inflationary observables known as cosmological correlators.", '3.3 Inflationary Observables': "In the cosmological context the purpose of studying de Sitter spacetime is to understand the behaviour of quantum fluctuations at the end of inflation. These perturbations then grow over the history of the universe to produce all the structure that we see. Due to the randomness inherent to the quantum nature of these fluctuations and the single universe that we have to observe we study them through their correlation functions. In this section we focus on cosmological correlators calculated using the in-in formalism, since constraints for them only require the spacetime symmetries considered in Section 3.2. In this paper, we will focus only on observables involving fields with integer spin representations and will not consider spinor  \nfields in detail. However, a similar formalism should provide a much easier approach to deriving symmetry and Unitarity constraints for observables involving half-integer spin fields in cosmology (see e.g. [65,82-94] for discussions of spinors in de Sitter space and cosmology), which has yet to be done in the cosmology literature.  \n(Later, we will revisit these same observables using the Wavefunction of the Universe (WFU) which introduces a new type of n -pt functions known as the wavefunction coefficients. However, as we will discuss in Section 3.5 this requires consideration of a different type of symmetry, namely the local Lagrangian symmetries discussed in Section 4, as it is those which will impose non-trivial constraints on the WFU and wavefunction coefficients.)  \nThe correlators of interest in cosmology are equal time correlation functions evaluated on some time slice, usually taken to be the asymptotic future of de Sitter or some (quasi-)de Sitter inflationary spacetime,  \n⟨O ( η ) ⟩ = ⟨ Ψ( η ) |O ( η ) | Ψ( η ) ⟩ . (3.53)  \nWe do not (a priori) know the state | Ψ( η ) ⟩ . However, we assume that the universe was in a vacuum of the full theory, | Ψ ⟩ = | Ω ⟩ . Just like in flat space it will prove very helpful to move to the interaction picture where operators evolve according to the free Hamiltonian whilst the evolution of states is dictated by the interacting Hamiltonian. Then these correlators are  \n⟨O ( η ) ⟩ = ⟨ Ω( η i ) | U † I ( η i , η ) O I ( η ) U I ( η i , η ) | Ω( η i ) ⟩ , (3.54)  \nwhich are known as 'in-in' correlators due to the fact that both vacuum states are defined in the past at η i . This is in opposition to flat space amplitudes where the ket state corresponds to the future 'out' vacuum.  \nWe could perform a Dyson expansion for this expression, using the fact that this time evolution operator  \nis  \nU ( η i , η ) = T exp [ -i ∫ η η i d ηH int [ ϕ, ϕ ' ] ] . (3.55)  \nHowever, | Ω( η i ) ⟩ is the in vacuum of the full theory (despite being expressed in the interaction picture). This is relevant as the operators, O , will be constructed from the interaction picture creation and annihilation operators, which act straightforwardly on the free (Bunch-Davies) vacuum, | 0 ⟩ . By taking η i to the infinite past we can turn off the interactions adiabatically so that the free and interacting vacua coincide. We will discuss the procedure by which this is done more extensively in Section 7.1.", '3.4 Symmetries of the Cosmological Correlators': "In this section we explore how the symmetry transformations CRT and D -1 from Section 3.2 act on inin cosmological correlators and thus infer the constraint from Reflection Reality ( RR ). The cosmological correlators can be expressed as  \nB n ( η 0 ; y 1 . . . y n ) = ⟨ Ω( η 0 ) | ϕ 1 ( η 0 ; y 1 ) . . . ϕ n ( η 0 ; y n ) | Ω( η 0 ) ⟩ (3.56)  \n= ⟨ Ω( η i ) | U † ( η i , η 0 ) ϕ 1 ( η 0 ; y 1 ) . . . ϕ n ( η 0 ; y n ) U ( η i , η 0 ) | Ω( η i ) ⟩ , (3.57)  \nwhere η i is some time at which we specify the initial conditions at the start of inflation and η 0 is the reheating surface at the end.  \nImportantly, for a correlator to be invariant under a transformation, we require several different properties of the theory and background in which it lives. To illustrate this, let us consider the action of each  \nof the symmetries individually. For CRT we find  \nCRT : B n ( η ; y 1 . . . y n ) → B ∗ n ( -η ; -y 1 , . . . , -y n ) (3.58) = CRT ( ⟨ Ω( η i ) | ) U † ( η i , η ) ϕ ∗ 1 ( -η ; -y 1 ) . . . ϕ ∗ n ( -η ; -y n ) U ( η i , η ) CRT ( | Ω( η i ) ⟩ ) , (3.59)  \nwhere we have used the fact that the time evolution operator defined in (3.55) is invariant under CRT . Thus we recover our original correlator if:  \n- · The state is CRT -invariant, CRT ( | Ω( η i ) ⟩ ) = | Ω( η i ) ⟩ , which is true for the Bunch-Davies vacuum in which the in-in correlators are usually defined;\n- · The theory is CRT -invariant, CRT : H → H which ensures that U ( η i , η ) is also CRT 0invariant;\n- · The fields are CRT -invariant, CRT ( ϕ ( η ; y )) ≡ ϕ ∗ ( -η ; -y ) = ϕ ( η ; y ). This is actually ensured by the fact that their equation of motion and the initial Bunch-Davies vacuum state is CRT -invariant. However we will see in Section 7.4 that at the boundary the CRT transformation acts more nontrivially on the fields.  \nFor the Dilatation operator, we will use the intuition from Section 3.2 to assume we can access λ = -1 despite the fact that it usually defined with λ ∈ R ≥ 0 and then address this analytic continuation in more detail in Section 6,  \nD λ : B n ( η ; y 1 , . . . , y n ) → B n ( λη ; λ y 1 , . . . , λ y n ) (3.60) = ⟨ Ω( λη i ) | U † ( λη i , λη ) ϕ 1 ( λη ; λ y 1 ) . . . ϕ n ( λη ; λ y n ) U ( λη i , λη ) | Ω( λη i ) ⟩ . (3.61)  \nThus we recover our original correlator if:  \n- · The state is scale-invariant, | Ω( λη i ) ⟩ = | Ω( η i ) ⟩ , which is true for the Bunch-Davies vacuum in which the in-in correlators are usually defined, since the Bunch-Davies vacuum (as well as any other α -vacua) is invariant under all the de Sitter isometries;\n- · The theory is scale-invariant, D λ : H → H which ensures that U ( λη i , λη ) = U ( η i , η );\n- · The fields are scale-invariant, D λ ( ϕ ( η ; y )) ≡ ϕ ( λη ; λ y ) = ϕ ( η ; y ). This is actually ensured by their scaling properties, if they were not scale invariant themselves then this transformation would also involve transforming the fields as we will see in Section 7.4.  \nWe can then use CRT and D -1 to infer RR , and thus find in analogy with Sections 2.4 and 2.5, that RR constrains B n to be real by ensuring that the time evolution operator is unitary, U † U = 1 . Hence we are justified in labelling this property and its associated transformation RR .  \nWe are now equipped to construct a symmetry group from the action of these transformations,  \nCRT : B n ( η ; y ) = B ∗ n ( -η ; -y ) ; (3.62)  \nD ± -1 : B n ( η ; y ) = B n ( -η ; -y ) ; (3.63)  \nRR : B n ( η ; y ) = B ∗ n ( η ; y ) , (3.64)  \nJust as in flat space these three transformations form a closed group and therefore, any two of them imply the last.  \nThus we are not asserting that these are guaranteed to be symmetries of an inflationary theory, just that one of them will come for free if the other two are satisfied. Similar to how a theory can be invariant under  \nthe discrete 180 · rotation R π but not invariant under SO + (1 , 1) flat space, it is relatively easy to come up with theories in de Sitter that accidentally satisfy this discrete ( λ = -1) scaling transformation without actually respecting continuous scale transformations ( λ > 0). For example, consider the interaction ga 2 ϕ 4 in D = 3 + 1-spacetime dimensions, where g is a real coupling. This is clearly not scale invariant (as a scale-invariant non-derivative interaction would contain four powers of a rather than two) but is both Unitary (since it has a real coupling) and CRT invariant as it does have Discrete Scale Invariance due to the even number of extra scale factors.  \nThe focus here has been on equal time correlators due to their importance to cosmology. However, Unitarity also has implications for unequal time correlators. In this case the result is almost identical to the flat space results expressed in Lorentzian signature:  \nCRT : T B n ( η ; y ) = T B ∗ n ( -η ; -y ) ; (3.65)  \nD ± -1 : T B n ( η ; y ) = T B n ( -η ; -y ) ; (3.66)  \nRR : T B n ( η ; y ) = T B ∗ n ( η ; y ) , (3.67)  \nwhere the bar indicates that time ordering has been exchanged with anti-time ordering. The main difference between flat space and here being that it is all the spatial coordinates that are inverted rather than just one. This is the case in any dimension due to the way the global de Sitter transformation is projected onto the Poincar'e patch.", '3.5 Wavefunction of the Universe': "The same cosmological correlation functions discussed in Sections 3.3 and 3.4 can be calculated by using a basis of ϕ field eigenstates for both the bra and the ket:  \n⟨O ( η ) ⟩ = ∫ D ϕ D ˜ ϕ ⟨ Ψ( η ) | ϕ ; η ⟩⟨ ϕ ; η |O ( η ) | ˜ ϕ ; η ⟩⟨ ˜ ϕ ; η | Ψ( η ) ⟩  \n(3.68) = ∫ D ϕ D ˜ ϕ Ψ ∗ [ ϕ ; η ]Ψ[ ˜ ϕ ; η ] ⟨ ϕ ; η |O ( η ) | ˜ ϕ, η ⟩ (3.69) = ∫ D ϕ Ψ ∗ [ ϕ ; η ]Ψ[ ϕ ; η ] O ( ϕ ; η ) , (3.70)  \nwhere O ( ϕ ; η ) is obtained by replacing instances of ˆ ϕ with the corresponding eigenvalue, ϕ , of the state | ϕ ; η ⟩ , and the Wavefunction of the Universe (WFU) is defined as 28  \nΨ[ ϕ ; η ] ≡ ⟨ ϕ ; η | Ψ( η ) ⟩ = ⟨ ϕ ; η | Ω ⟩ . (3.71)  \nFrom (3.71) it alreay becomes apparent that the same analysis used for the cosmological correlators in Section 3.4 will not apply to the WFU, since the WFU is not an overlap of two vacuum states | Ω ⟩ . We will explain how to deal with this problem in Section 4.  \nThe WFU satisfies the Schrodinger equation,  \ni ℏ d d η Ψ[ ϕ ; η ] -H ( ϕ ( η ; x ); η )Ψ[ ϕ ; η ] = D Ψ[ ϕ ; η ] = 0 , (3.72)  \nsubject to the Bunch-Davies initial condition and can be parameterised as  \nΨ[ ϕ ; η ] ≡ ⟨ ϕ ; η | Ω ⟩ = exp { -∞ ∑ n =2 ∫ n ∏ a =1 d d k a (2 π ) d ϕ k a ψ n ( η ; k 1 , . . . , k n ) δ d ( ∑ k a ) } . (3.73)  \nDespite not being directly observable themselves, it has become customary to discuss the role of symmetries and fundamental principles on the terms in this expansion, ψ n , the so called coefficients of the WFU. The relations between the cosmological correlators B n and ψ n can be found in various parts of the literature, see e.g. [17,34,96]. In particular we have developed our understanding of Unitarity within the cosmological context in terms of its implications on these wavefunction coefficients. This is due to the discovery of the Cosmological Optical Theorem [13-15] where Unitarity of the time evolution operator was used to derive a constraint on contact wavefunction coefficients (Feynman-Witten amplitudes with only external lines):  \nψ n ( η 0 ; | k | , k ) + ψ ∗ n ( η 0 ; -| k | , -k ) = 0 , (3.74)  \nwhere this Unitarity constraint involves an analytic continuation of the magnitude of the external momenta (we will sometimes refer to the magnitudes as 'energies') from the positive real axis through the lower half of the complex plane. However, for more complicated diagrams with internal lines the right hand side picks up a combination of lower point functions, i.e. in analogy with the Cutkosky cutting rules in flat space Unitarity in cosmology relates different orders in perturbation theory [15, 97]. It was later shown in [54] that if one also analytically continues the internal energies (3.74) would hold for all tree-level contact and exchange diagrams with some caveats for certain types of non-local interactions. Unfortunately, these results are understood only perturbatively [15,97], whereas the symmetry perspective that we present here is valid non-perturbatively and, as such provides new insights on these objects.  \nBy considering the local Lagrangian symmetries , in Section 4, we will show that a similar relationship holds for diagrams of arbitrary loop order when we analytically continue both external and internal energies . (Although we do not, in this paper, consider cutting rules, only constraints on individual Feynman-Witten diagrams.) This also provides insight into why certain types of non-local interactions may not satisfy (3.74). Through analogy with RR in flat space, we will interpret this as a limited consequence of Unitarity.", '4 Local Lagrangian Symmetries': "In this section we describe the notion of a local Lagrangian symmetry, and explain why this type of symmetry is needed to obtain a useful constraint on wavefunction coefficients in a single Poincar'e patch.", '4.1 Distinction from Spacetime Symmetries': "In order to proceed, we must now make a distinction between two classes of symmetries:  \n- 1. A spacetime symmetry , which acts on the entire manifold M by moving points around by means of some map m : x → y . (In addition to some local transformation of the fields ϕ ( x ), and possibly a complex conjugation of the amplitude.)\n- 2. A local Lagrangian symmetry , which is an assertion that the Lagrangian L at a point p ∈ M respects some symmetry. Here, the symmetry acts on the tangent space T p ( M ) (in addition to transforming the fields ϕ ( p ) and possibly complex conjugating).  \nThe relation between these kinds of symmetry is somewhat subtle.  \nAspacetime symmetry can sometimes give rise to a local Lagrangian symmetry. This happens whenever there is a fixed point of the map m , that is a point p for which m : p → p . In this case there will be an induced map on the tangent space T p , and this will normally 29 give rise to a symmetry of the local Lagrangian.  \nAs translations in Minkowksi do not have fixed points, they do not give rise to any local Lagrangian symmetry by themselves. But they do, of course, ensure that if there is a local Lagrangian symmetry at one point p , it can be translated to any other point q .  \nThere can also exist local Lagrangian symmetries which do not arise from any spacetime symmetry of M . A classic example of this, is the standard assumption that a field theory has local Lorentz Invariance at each point p , even on a spacetime manifold which has no Killing symmetry. (This example also shows that the definition of 'local' has nothing to do with gauge invariance, as we typically assume local Lorentz Invariance even when doing field theory on a fixed, non-gravitational background.) Another simple example would be a time-varying Hamiltonian H ( t ), in a case where at each moment of time t the Hamiltonian is time-reversal invariant, but there exists no time t 0 for which H ( t 0 + t ) = H ( t 0 -t ).  \nLet us now consider the implications for the discrete symmetries in our Z 2 × Z 2 group. In the case of Reflection Reality ( RR ), the fixed points include the entire real part of the Lorentzian manifold M . Hence, if we are working in Lorentzian signature, we are automatically entitled to extend RR to a symmetry of the local Lagrangian at each point p .  \nThe same is not true of an individual CRT symmetry, since in this case t →-t and not all points are fixed by a given CRT . But in a manifold with time translation symmetry (e.g. Minkowski) we can always translate the story to find a CRT symmetry at any given point. Alternatively, if we have local Lorentz Invariance, this can be Wick rotated to obtain a local 180 · rotation, and then RR (which is always local by the argument above) implies local CRT .", '4.2 Flat Space Wavefunction': "Below, we will be interested in using discrete symmetries to constrain vacuum wavefunction coefficients. But, as we have alluded to, it will turn out that the local Lagrangian versions of this symmetry are much more useful than the spacetime versions.  \nLet us consider the spacetime version of the Z 2 × Z 2 group, starting in flat spacetime. Recall that the vacuum wavefunction Ψ at τ = it = 0, is given by a path integral in the lower half of Euclidean space, that is it is a one-sided path integral over a spacetime with τ < 0:  \nΨ[ ϕ ( x )] ≡ 〈 ϕ ( x ) | Ψ 〉 = ∫ τ< 0 D ϕe -I [ ϕ ( x,τ )] (4.1)  \n(This might be followed by some amount of Lorentzian time evolution, which will not affect the basic points below.)  \nThere is no problem with identifying the constraint that CRT places on the flat space wavefunction. Assuming ϕ is real, and at fixed t , it is simply:  \nCRT : Ψ[ ϕ ( x )] = Ψ ∗ [ ϕ ( -x )] (4.2)  \nwhere x is the single spatial coordinate that gets flipped. Nor does it matter much if we think of this as the spacetime or the local form of CRT , as long as it holds fixed the t -slice on which the wavefunction is evaluated. (Assuming time translation symmetry, Ψ is independent of t .)  \nBut what is the implication for RR on the wavefunction? Naively, there isn't one. RR tells you that correlators satisfy hermitian conjugation (2.18):  \nRR : 〈 Ψ | ϕ 1 ϕ 2 . . . ϕ n | Ψ 〉 = 〈 Ψ | ϕ n . . . ϕ 2 ϕ 1 | Ψ 〉 ∗ , (4.3)  \nbut this holds for every possible value of Ψ (that lives in a state space which admits a real inner product) regardless of the functional form of Ψ[ ϕ ]. What went wrong? The problem is that the one-sided path  \nintegral is done over negative τ values only. Hence, a spacetime symmetry which flips τ → -τ does not place a meaningful constraint on Ψ. Because it reverses the order of operators, it maps a ket state | Ψ ⟩ to a bra state ⟨ Ψ | , rather than placing any constraint on the ket itself.  \nA similar problem occurs for the 180 · rotation, assuming we attempt to apply it in isolation and not as part of a bigger group of symmetries. 30  \nThe story above also applies, without any essential change, to global de Sitter where CRT allows us to write a concise non-perturbative constraint for the full de Sitter wavefunction at equalT :  \nΨ[ T ; ϕ (Φ d )] = CRT (Ψ[ T ; ϕ (Φ d )]) ≡ Ψ ∗ [ -T ; ϕ ( π -Φ d )] , (4.4)  \nwhere we have used the fact that (as discussed in Section 3.1) for equal-time quantities, such as the de Sitter wavefunction, CRT simplifies to a complex conjugation, a flip in the global time coordinate and a rotation on the d -1-sphere. Note that at T = 0, CRT acts to constrain the wavefunction to be its own conjugate after a spatial reflection.", "4.3 Poincar'e Wavefunction of the Universe": "In Poincar'e de Sitter, things naively get even worse. As the Poincar'e patch explicitly breaks timereversal invariance, it seems that we can't even constrain the wavefunction using CRT . This is because the standard Poincar'e CRT (described in Section 3.2) maps the future patch P + to the past patch P -. But then it seems to place no constraints on P + considered in isolation.  \nHowever, this is too quick. In fact, as we shall see, discrete symmetries can place quite strong constraints on the wavefunction Ψ. This is so for two basic reasons:  \n- · We can use the local Lagrangian form of discrete symmetries such as CRT and RR , and\n- · We can make use of the fact that Ψ, the Bunch Davies state, satisfies various analytic continuation properties.  \nRegarding the first point, let us consider a much more powerful form of local Lagrangian CRT :  \nLL CRT : For every point p ∈ P + , there exists a CRT symmetry of its tangent space.  \nThis is a much more powerful principle. In fact though, we can obtain it from Poincar'e CRT if we also assume invariance under the full de Sitter group.  \nPoincar'e CRT + SO + ( d +1 , 1) = ⇒ LL CRT .  \nConversely, if we stipulate that all local Lagrangian symmetries must arise from spacetime symmetries, and also spatial translations, we find that the full de Sitter group is implied:  \nLL CRT from spacetime + spatial translations = ⇒ SO + ( d +1 , 1)  \nbecause the conjugation of translations by the local CRT symmetries can get you special conformals, and from there one generates the whole group. Of course, the Poincar'e version of CRT is an easy corollary.  \nWe therefore find, that in the context of homogeneous cosmology, local Lagrangian CRT can arise from spacetime symmetries only in the case of de Sitter. These wavefunction constraints will be considered in Sections 6.1-6.2 and (in more specific detail) in Section 7.  \nOn the other hand, local Lagrangian RR , which does not reflect a Lorentzian time direction, can be meaningfully applied in any real FLRW cosmology, regardless of the existence of any Killing symmetry. In the case where space is flat, we can use this to prove a more general version of the Cosmological Optical Theorem (in Section 6.3).  \nBut before this, in the next Section 5, we will examine the close relationship between the local Lagrangian version of RR , and Unitarity in Lorentzian signature.", '5 Accidental Unitarity': 'While RR is weaker than full Unitarity, it turns out it gets you a lot closer to full Unitarity than you might think! In this section, we will argue that, among the set of QFTs satisfying RR , a codimension 0 subset are also fully Unitary.', '5.1 Perturbative Equivalence of Reflection Reality and Unitarity': "Suppose we start with a unitary QFT, and consider deforming it by some small perturbation δL to the Lagrangian density L . 31 Let us assume that this small perturbation does not change the number of configuration degrees of freedom in the theory. 32 It should then be possible to view it as a change to the Hamiltonian δH . If we consider all possible terms in the Hamiltonian (satisfying some set of symmetries) up to some degree of irrelevance, this should give us a finite-dimensional space of perturbations to the couplings, so H ∈ C N for some finite dimension N . Now the subspace of perturbations that preserve Unitarity should be those satisfying:  \nδH = δH † , (5.1)  \nwhere † is defined using the positive norm of the original theory's state space. This can be regarded as imposing a real structure on the space of couplings, i.e. it restricts you to a real subspace δH ∈ R N ⊂ C N such that if a vector v is in the space R N , iv is not in the R N . In other words all of its basis vectors should be (complex)-linearly independent. Since Unitary implies Reflection Positivity and hence Reflection Reality, for any v ∈ R N , RR ( v ) = v . Since RR is antilinear this means that RR ( iv ) = -iv . We now know how RR acts on the entire space, and it is clear that the space of RR symmetric perturbations is no larger than the space of unitary ones. Hence, for small perturbations of this sort, RR and Unitarity are equivalent conditions.  \nFor a bosonic field theory in Lorentzian signature, such perturbations are in one-to-one correspondence with real contributions to δL . 33 If however, we consider perturbations that involve derivative couplings, in Lorentzian signature there can also be (typically divergent) measure factors required to restore Unitarity, that are tantamount to fixing certain imaginary terms in the action. However, upon passing to Euclidean signature, such terms are fixed by RR alone, without using any additional feature of Unitarity. This is one way in which Euclidean field theory is nicer than Lorentzian field theory.", '5.2 How to get Theories with Negative Norm States': 'Does this mean that all RR satisfying theories are also unitary? Clearly not, since there exist theories with negative norm states. But to get them you have to break one of the assumptions in the argument above. Since the argument above refers only to continuous variations of the Lagrangian, one way to obtain such theories is to make a bad discrete choice of field content, which is incompatible with Unitarity. One easy example of this is a negative-norm scalar field. Here we start with a field with negative propagator term, e.g:  \nL = -1 2 ∂ µ ϕ∂ µ ϕ + V ( ϕ ) . (5.2)  \nAlthough this action looks unbounded below in Euclidean signature, this problem can be resolved if we define a new field χ = iϕ and do our functional integrals along the real χ direction. But then χ † = -χ , so any terms in ˆ V ( χ ) = V ( iϕ ) with odd powers of χ will have an unexpected imaginary sign, relative to the requirements of Unitarity. This is because the states with an odd number of χ quanta have negative norm. 34 This is an example of a bad discrete choice which leads to a RR but not Unitary theory. Another example of such a bad choice is a spin-statistics violating theory, e.g:  \nL = i∂ µ ψ∂ µ ψ, (5.3)  \nwhere ψ and ψ are scalar fermions. In this case the 1-particle state space has indefinite signature and there is no way to extend RR to full Unitarity.  \nFinally we can also consider choices which change the number of degrees of freedom. An illuminating example is the Proca Lagrangian:  \nL = 1 4 F µν F µν + 1 2 m 2 A µ A µ . (5.4)  \n̸  \nIn D spacetime dimensions, the number of propagating degrees of freedom is D -1 when m 2 = 0, but only D -2 when m 2 = 0. The latter is, of course, due to the usual gauge symmetry of a massless photon field. Since the gauge theory has a bunch of null sates, which have 0 inner product with any other vector, it can be a boundary between a unitary theory, and a theory containing negative norm states. And in fact, in the tachyonic case where m 2 < 0, there is a propagating temporal mode which has negative norm. If then, you have a unitary theory with a gauge symmetry, and you want to deform it to another unitary theory, the safest thing to do is to restrict attention to only gauge-invariant perturbations to the Lagrangian.  \nAlthough the above counterexamples show that RR does not imply Unitarity, it is encouraging that we had to work hard to find such counterexamples-generic perturbations to a Lagrangian cannot do the trick. Put another way, once we impose RR , we have a decent chance of accidentally ending up with a unitary theory by pure luck. There is not going to be any need to fine-tune further parameters with infinite precision to end up in the Unitary case.', '5.3 Unitarity from CRT': "It is similarly possible to argue that Unitarity can accidentally follow from a local CRT invariance of the action.  \nIf we restrict to locally Lorentz invariant bulk Lagrangians, all such covariant terms have a 180 · rotation symmetry, 35 and hence CRT implies RR implies (for small perturbations as above) Unitarity. Thus CRT is also sufficient to accidentally imply Unitarity, in the absence of bad discrete choices for the field content.  \nIt should be noted that the above argument refers to local CRT invariance of the Lagrangian L at each point. In Section 3, we discussed a global CRT transformation that flips a future Poincar'e-dS to the past one. This global CRT symmetry will not, by itself, suffice to prove local CRT invariance of the Lagrangian, as it only flips CRT around one particular static patch bifurcation surface. But, global CRT together with the full rotational symmetry SO + ( d +1 , 1) implies local CRT at any point.  \nThis helps to motivate the crucial importance of CRT in de Sitter as a check on holographic ideas. Suppose somebody hands you a claimed instance of dS/CFT and you have already checked that it is conformally invariant. But you want to know if it is dual to a bulk unitary theory. According to this argument, checking CRT invariance gets you nearly all the way to deciding Unitarity. As long as the low energy bulk fields don't violate spin-statistics or have negative-norm states, it is then reasonably likely that you have a Unitary theory on your hands. On the other hand, if you slap together some partition function without paying any regard to discrete symmetries, it would be infinitely unlikely that you will end up with a Unitary bulk theory. We will now explore how the symmetries described in Sections 3-4 impose constraints on the coefficients of the Wavefunction of the Universe.", '6 Analytic Continuation in Cosmology': 'Similar to how the physical domain of Mandelstam variables in amplitudes is defined for s, t ≥ 0 and crossing symmetry involves analytically continuing to s, t ∈ C , we will now discuss how our discrete transformations in Section 3.2 will involve analytically continuing the conformal time outside of the physical domain of η ≤ 0 and in Fourier space will also involve analytically continuing the magnitude of the momenta k = √ k · k ≥ 0.', '6.1 Nonperturbative Analytic Continuation': "To perform the analytic continuation explained in Section 3.2 more carefully, let us do QFT on a fixed spacetime Poincar'e patch metric. 36 We take the metric to have the slightly more general form:  \nd s 2 = ( A d ζ ) 2 +( Be Hζ d y i ) 2 = A 2 d ζ 2 + B 2 e 2 Hζ d y i d y i , (6.1)  \nwhere A and B are constants. It should be noted that because a typical matter action has a factor of √ g out front, the sign of A and B matters, not just A 2 and B 2 . Some special cases of this metric are:  \n- · A = B = +1: Normal Euclidean hyperbolic space H 0 , with ζ = τ ;\n- · A = + i , B = +1: Future dS-Poincar'e P + , with ζ = t ;\n- · A = -i , B = +1: Past dS-Poincar'e P -, with ζ = -t ;\n- · A = B = + i : Funky Euclidean hyperbolic space H + with all minus signature.  \nThe funky hyperbolic space H + does not in general have the same properties as a normal field theory: massive fields become tachyons, and (depending on the theory) it need not satisfy Reflection Reality.  \nCorrect path via Wick rotation of g tt : From this it can be seen that a correct way to analytically continue from P + to P -is to hold B fixed and take A along the path:  \nA = i 1 -2 q = e (1 -2 q ) iπ/ 2 , q ∈ [0 , 1] (6.2)  \nwhere q = 0 corresponds to H + and q = 1 corresponds to H -. The halfway point, q = 1 2 , corresponds to normal Euclidean hyperbolic space H 0 . The advantage of this path is that a well-behaved QFT (e.g. an arbitrary p -form field) has an action whose real part is bounded below everywhere in the interior of the q -interval. As usual the Lorentzian path integral, being oscillatory, is on the edge of convergence. (If we define the initial condition by taking A to have a small positive part at early times, we obtain the Bunch-Davies state; hence, the initial condition is independent of the choice of q ∈ [0 , 1].)  \nGeometric path via η continuation: However, the correct path does not have any obvious geometric interpretation in terms of analytically continuing the time coordinate; that is, the path does not lie on the complexification of P + . If we choose instead to rotate the η coordinate, this corresponds to starting in H + and shifting Ht → Ht + iπ . This time we are holding A fixed and taking B along the path:  \nB = i 2 q = e iπq , q ∈ [0 , 1] (6.3)  \nAt q = 1 2 , this contour passes through funky hyperbolic space H + where the QFT need not converge. Worse still, at q = 1, it ends up at A = + i , B = -1 which is not the same as P -!  \nHowever, the two paths are very similar. For each value of q , they differ only by an overall rescaling of the metric. That is, the geometrical path can be converted back into the correct path if, for each value of q ∈ [0 , 1], we additionally rotate the metric by  \ng ab → Ω 2 g ab , Ω = ( -i ) 2 q = e -iπq (6.4)  \nThis looks very similar to an imaginary Weyl transformation, although we do not rescale any of the matter fields. At q = 1 2 , this puts us back to normal hyperbolic space H 0 , and at q = 1, it yields P -, after applying Ω = -1. Then, Ω 2 = +1 and we have simply rotated the metric g ab a full cycle in the complex plane. However, in general the amplitudes become singular as g ab → 0, and hence we have rotated around a singularity. So we are on a new Riemann sheet, and we do not necessarily get the same physics as if we had not done the Weyl rescaling!  \nIf we write our amplitudes as an explicit function of Ω, we can thus conclude that we can continue from Bunch-Davies in P + (evolving forwards in time) to Bunch-Davies in P -(evolving backwards in time) by doing the transformation:  \nA -( η ; y 1 , . . . , y d ; Ω) = A + ( e -iπ η ; y 1 , . . . , y d ; e -iπ Ω) , (6.5)  \nwhere in this expression, the use of e iπ rather than -1 is a mnemonic indicating in which direction one should rotate in the complex plane to reach -1 (all such rotations are to be done simultaneously, in this case holding the ratio of η and Ω fixed). But P + and P -are also related by CRT symmetry, which tells us that P + and P -are also related by sending y i →-y i and complex conjugating the amplitude. Hence, invariance under CRT also requires P + and P -to also be related by the transformation:  \nA -( η ; y 1 , . . . , y d ; Ω) = A + ( -η ; -y 1 , . . . , -y d ; Ω) ∗ . (6.6)  \nwhere a minus sign that is written out explicitly, represents a mere reflection of the coordinate (no analytic continuation). Combining these together, it follows that, in a CRT invariant theory, the wavefunction coefficients in P + must obey:  \nCRT : ψ ∗ n ( η ; y 1 , . . . , y d ; Ω) = ψ n ( e -iπ η ; -y 1 , . . . , -y d ; e iπ Ω) . (6.7)  \nIn momentum space we would have almost the same relation:  \nCRT : ψ ∗ n ( η ; k 1 , . . . , k d ; Ω) = ψ n ( e -iπ η ; k 1 , . . . , k d ; e -iπ Ω) , (6.8)  \napart from the fact that complex conjugation in position space reverses the sign of k , cancelling out the reflection of each spatial coordinate.  \nGeometric path via y continuation: There is another way to achieve a similar result, by analytically continuing the y variables rather than the η variable. If we analytically continue y i → e -iπq y i (for all of the y coordinates at once, and for q ∈ [0 , 1] as before), then to hold the metric fixed, we end up doing the exact same rotation of B as in (6.3). As before, to convert this to the correct contour for the A parameter, we will need to rotate Ω to obtain equivalence to the correct path. Also, as η keeps the same sign, we will need to invoke some discrete symmetry which remains within P + . The obvious choice here is RR .  \nThe simplest way to write down a correct constraint on the wavefunction coefficient is to start with CRT in the form (6.7), and also act with a 180 · rotation. As the latter is an analytic continuation of the SO + (1 , 1) scaling symmetry, it takes the form:  \nD ± -1 : ψ n ( η ; y 1 , . . . , y d ; Ω) = ψ n ( e ± iπ η ; e ± iπ y 1 , . . . , e ± iπ y d ; Ω) , (6.9)  \nwhere the ± sign depends on the direction of the 180 · rotation. When considering wavefunction coefficients, it is no longer an a priori truth that the 360 · rotation is a trivial transformation. This enhances the symmetry group to Aut( Z ), see the discussion surrounding Table 2 in Section 7.2. Hence, the combined statement (which should follow from Reflection Reality) implies the following result for the y -continued wavefunction coefficient:  \nRR : ψ ∗ n ( η ; y 1 , . . . , y d ; Ω) = ψ n ( η ; -e iπ y 1 , . . . , -e iπ y d ; e -iπ Ω) , (6.10)  \nwhere, similarly to previous expressions, multiplication by -e iπ is shorthand for an analytic continuation followed by a reflection, and cannot validly be replaced with +1!  \nIn momentum space, these relations become:  \nD ± -1 : ψ n ( η ; k 1 , . . . , k d ; Ω) = ( -1) ± d ψ n ( e ± iπ η ; e ∓ iπ k 1 , . . . , e ∓ iπ k d ; Ω) , (6.11)  \nand  \nRR : ψ ∗ n ( η ; k 1 , . . . , k d ; Ω) = ( -1) d ψ n ( η ; e -iπ k 1 , . . . , e -iπ k d ; e iπ Ω) . (6.12)  \nwhere the prefactor of ( -1) d arises because of the scaling of the overall delta function δ d ( ∑ n a k a ) due to momentum conservation, which transforms like k -d , and which we (purely conventionally) exclude from our definition of ψ n . (We would not need to explicitly include this factor if we were manipulating ψ ' n , which is defined to include the delta function.) In cases involving fractional dimensions (e.g. dim. reg.), this prefactor becomes e iπd in the case of RR and e ± iπd in the case of D ± -1 , transforming in the opposite direction as the k argument.  \nTo summarise, we obtain the following non-perturbative results for the wavefunction coefficients in position space:  \nCRT : ψ ∗ n ( η ; y 1 , . . . , y d ; Ω) = ψ n ( e -iπ η ; -y 1 , . . . , -y d ; e -iπ Ω); (6.13)  \nD ± -1 : ψ n ( η ; y 1 , . . . , y d ; Ω) = ψ n ( e ± iπ η ; e ± iπ y 1 , . . . , e -± iπ y d ; Ω) ; (6.14)  \nRR : ψ ∗ n ( η ; y 1 , . . . , y d ; Ω) = ψ n ( η ; -e iπ y 1 , . . . , -e iπ y d ; e -iπ Ω) . (6.15)  \nand in momentum space:  \nCRT : ψ ∗ n ( η ; k 1 , . . . , k d ; Ω) = ψ n ( e -iπ η ; k 1 , . . . , k d ; e -iπ Ω); (6.16)  \nD ± -1 : ψ n ( η ; k 1 , . . . , k d ; Ω) = e ± iπd ψ n ( e ± iπ η ; e ∓ iπ k 1 , . . . , e ∓ iπ k d ; Ω) ; (6.17)  \nRR : ψ ∗ n ( η ; k 1 , . . . , k d ; Ω) = e iπd ψ n ( η ; e -iπ k 1 , . . . , e -iπ k d ; e -iπ Ω) . (6.18)  \nTo avoid confusion, in the expressions above that involve complex conjugation, the arguments η, k, Ω on the LHS should be taken to be real, so that analytic continuation occurs only on the RHS. 37  \nSimilar to the constraint (4.4) on the global de Sitter wavefunction, CRT also allows us to write a concise non-perturbative constraint for the full de Sitter wavefunction in the Poincar'e patch:  \nΨ[ η ; Ω; ϕ ( y )] = CRT (Ψ[ η ; Ω; ϕ ( y )]) ≡ Ψ[ e -iπ η ; e -iπ Ω; ϕ ( -y )] ∗ , (6.19)  \nΨ[ η ; Ω; ϕ ( k a )] = CRT (Ψ[ η ; Ω; ϕ ( k a )]) ≡ Ψ[ e -iπ η ; e -iπ Ω; ϕ ( k a )] ∗ , (6.20)  \nwhere η , Ω and ϕ are real, and it is important that the complex conjugation on the RHS is the final step , after evaluating the rest of the expression. Note that at η = 0, CRT acts to constrain the wavefunction to be its own conjugate after the spatial reflection and Weyl rotation.  \nAs all of the above analytic continuations are equivalent to the correct Wick rotation, which should work for any fixed background QFT, Eqs. (6.7)-(6.12) will always be valid non-perturbatively , at least in the non-gravitational coupling constants. 38 In the next section, we will see how to rewrite these results perturbatively. In that context, the RR conditions (6.10) and (6.12) will become the statement of Hermitian analyticity, and (as expected from past work [13,14,54]) they are actually valid for all FLRW spacetimes, to all orders in perturbation theory (and for Bunch-Davies). This makes sense because unlike the other two transformations, RR does not require the existence of a map from P + to P -.", '6.2 Perturbative Discrete Symmetries': "The transformations in Section 6.1 are somewhat obscure, due to their explicit dependence on the Weyl factor Ω. If, however, we are doing unrenormalised perturbation theory around a free, covariant, parity-even bosonic theory, a remarkable simplification appears. This is because the action takes the form:  \nI = ∫ d D x √ g L 0 [ ϕ, g ab ] , (6.21)  \nwhere, apart from the overall factor of √ g , only integer powers of the metric appear in L . 39 In this section, we are defining I with the conventions of the Euclidean action, so that the amplitude always goes like e -I , even in Lorentzian signature; thus the Lorentzian action is imaginary.  \n38  \nIn the case of gravity, at least two additional problems present themselves: (1) the conformal mode problem, which means  \nthat the path integral is not convergent for any value of  \nq  \n, and (2) we cannot simply write  \ng  \nab  \n=  \ng  \n0  \nab  \n+  \nh  \nab  \nand treat  \nh  \nany other field, because the gauge conditions of the graviton no longer make sense unless we allow Ω to act on  \nh  \nFurthermore, as  \nH  \n0  \nab  \nab  \nlike  \nas well.  \nreverses the sign of Λ, there is reason to expect that the result here cannot make sense nonperturbatively,  \nas in many approaches to quantum gravity (e.g. the string landscape), Λ is not a freely adjustable parameter. That said,  \nwhen working perturbatively in the gravitational coupling, in all the cases we have checked we get the same phases as in the  \ncase of a scalar field.  \n39 For spinor fields, the propagator term contains a factor of γ µ which scales like the square-root of a metric. So in theories with fermions, we expect an additional power of ( -1) in CRT for each fermion propagator, in any dimension D .  \nIt follows that when we analytically continue Ω → -Ω (regardless of which direction we go around Ω = 0), the sole effect is to multiply the √ g factor by a minus sign for each spacetime dimension: ( -1) D . This is then equivalent to acting on the Lagrangian with the transformation:  \nL ( -Ω) = ( -1) D L (Ω) . (6.22)  \nSo when D = d +1 is even, there is actually no difference between the correct path and the paths where we analytically continue η or y i . We can just (in unrenormalised perturbation theory) ignore the Ω →-Ω instruction.  \nOn the other hand, when D = d +1 is odd, the Lagrangian changes sign: L → -L . In any FeynmanWitten diagram, this provides an extra minus sign for every vertex V , and also every internal edge I (because the sign of the propagator term also reverses). In any connected diagram, the Euler formula V -I = 1 -L allows us to write the change of any unrenormalised Feynman-Witten amplitude A in terms of the number of loops L :  \nA ( L ) ( -Ω) = e iπ ( d +1)( L -1) A ( L ) (Ω) , (6.23)  \nwhere A ( L ) denotes a perturbative amplitude computed at some loop order L (e.g. L = 0 corresponds to tree-level and L = 1 corresponds to 1-loop order), and we allow d to be non-integer. The reason we keep saying 'unrenormalised', is that this pattern does not continue to hold when doing renormalisation theory. The reason is that the UV cutoff ϵ has units of length, and so it implicitly depends on the value of g ab and hence Ω. If, for example, one has a log divergence of the form C log( ϵ ), then when Ω → e -iπ Ω, this will also send ϵ → e -iπ ϵ , thus producing the shift:  \nC log ( e iπ ϵ ) = C [ log( ϵ ) -iπ ] , (6.24)  \nso that any log-divergent term produces an imaginary shift in the corresponding finite term (this shift has no explicit dependence on D ). Of course, this effect does not matter for tree-level diagrams, as these are UV-finite and thus do not require renormalisation. Nor does it affect any loop diagrams that happen to be finite. 40 Also, the coefficient of the log divergence itself still satisfies the phase rule (except when considering a diagram with higher powers of the log( ϵ ), in which case this remark would apply to the leading order log divergence). 41  \nA conceptually similar effect comes if we are doing bulk perturbation theory, not around a free theory, but around an interacting CFT where various operators have non-integer anomalous dimensions. In this case, the conformal perturbation theory can have essentially arbitrary non-integer powers of the metric determinant g , and so Ω →-Ω can introduce strange phases even when D = d +1 is even. To say it as a slogan, the CRT transformation takes the simple form (6.23) only for amplitudes that involve no bulk quantum anomalies.  \nIn cases without such quantum anomalies, the only extra factor to worry about is ( -1) D (1 -L ) , and so, combining our results with those of the previous section, we therefore obtain the following perturbative results for the wavefunction coefficients in position space:  \nCRT : ψ ( L ) ∗ n ( η ; y 1 , . . . , y d ) = e iπ ( d +1)( L -1) ψ ( L ) n ( e -iπ η ; -y 1 , . . . , -y d ) ; (6.25)  \nD ± -1 : ψ ( L ) n ( η ; y 1 , . . . , y d ) = ψ ( L ) n ( e ± iπ η ; e ± iπ y 1 , . . . , e ± iπ y d , ) ; (6.26)  \nRR : ψ ( L ) ∗ n ( η ; y 1 , . . . , y d ) = e iπ ( d +1)( L -1) ψ ( L ) n ( η ; -e iπ y 1 , . . . , -e iπ y d ) , (6.27)  \nwhere ψ ( L ) n denotes a perturbative wavefunction coefficient computed at some loop order L . Having these constraints in position space may prove to be quite powerful in a position-space cosmological bootstrap program, which is yet to be explored in the literature but could be more well-motivated in terms of connecting to cosmological observations, since imprints of physical principles, such as locality, in the late-time non-Gaussianity is obscured in Fourier space [100]. 42 In momentum space we obtain:  \nCRT : ψ ( L ) ∗ n ( η ; k 1 , . . . , k d ) = e iπ ( d +1)( L -1) ψ ( L ) n ( e -iπ η ; k 1 , . . . , k d ) ; (6.28)  \nD ± -1 : ψ ( L ) n ( η ; k 1 , . . . , k d ) = e ± iπd ψ ( L ) n ( e ± iπ η ; e ∓ iπ k 1 , . . . , e ∓ iπ k d , ) ; (6.29)  \nRR : ψ ( L ) ∗ n ( η ; k 1 , . . . , k d ) = e iπ (( d +1) L -1) ψ ( L ) n ( η ; e -iπ k 1 , . . . , e -iπ k d ) . (6.30)  \nAs this section considers only the case of a parity-even Lagrangian, technically the arguments in this section don't care about whether we reflect the y -coordinates in CRT or RR . However, we have put the signs in the correct place for parity-odd theories as well, as can be deduced in Section 6.3 where we do not restrict to parity-even theories.  \nThe analytic continuation of the conformal time coordinate η may result in an interesting connection to the recent work in [101], where the in-in correlators were related to an in-out prescription involving evolution through a pair of Poincar'e patches stitched together by their η = 0 surfaces. The authors of [101] refer to the transformation on the Feynman-Witten diagrams used in this construction as a time reversal which is not quite correct as it is really a CRT transformation.", '6.3 Perturbative Cosmological Optical Theorem for FLRW': "Next, we show how to prove (6.27) (and thus (6.30)) perturbatively, for a general spatially-flat FLRW cosmology with arbitrary scale factor a ( ζ ):  \nd s 2 = ( A d ζ ) 2 +( Ba ( ζ ) d y i ) 2 , (6.31)  \nand we continue to define P ± as before, even though in the general case there is no way to go from P + to P -by an extension of the real spacetime manifold. So here, P -is more of an abstraction representing what would happen if you solve the same QFT but on a contracting spacetime instead of an expanding one.  \nWe now drop the assumptions that the Lagrangian is Lorentz-covariant, or that it is invariant with respect to time or space reversal. This may be done by allowing the action to depend covariantly on the 1-index purely temporal and d -index purely spatial permutation symbols:  \nϵ t , ϵ i 1 i 2 ...i d , (6.32)  \nwhere we define each permutation symbol so that it is invariant under all real coordinate transforms in (time / space) except those for which det Jacobian = -1, in which case the (time / space) permutation symbol flips sign. 43 However, for any kind of analytic continuation, the ϵ 's have to remain the same sign throughout, as the analytic continuation of 1 is 1. Now let a given amplitude be labelled T = ± 1 according to whether it depends on ϵ t an (even/odd) number of times, and let it be labelled X = ± 1 according to whether it depends on ϵ i 1 i 2 ...i d an (even/odd) number of times.  \nReflection Reality ( RR ) may now be stated as the principle that on the Euclidean cosmology, each term in the Lagrangian satisfies L = T L ∗ , i.e. the τ -reversal even terms are real and the τ -reversal odd  \nterms are imaginary. From this it follows that, if we do the y -rotation described in the previous section, we obtain:  \nψ ∗ n ( η ; y 1 , . . . , y d ; Ω) = Tψ n ( η ; e iπ y 1 , . . . , e iπ y d ; e -iπ Ω) , (6.33)  \nbut note that we are not yet reflecting the y -coordinates here. Doing that reflection in each coordinate gives us an additional factor when d and the parity are both odd: 44  \nψ n ( η ; -y 1 , . . . , -y d ; Ω) = X d ψ n ( η ; y 1 , . . . , y d ; Ω) . (6.34)  \nFurthermore, since ϵ t substitutes for a factor of √ g tt in the Lagrangian L , and ϵ i 1 i 2 ...i d substitutes for a factor of √ g ( d ) in L -this is equivalent saying you don't need the metric to integrate a p -form in p dimensions-the Ω continuation is also modified from (6.22) and becomes: 45  \nψ ( L ) n ( η ; y 1 , . . . , y d ; e iπ Ω) = TX d ( -1) D ( L -1) ψ ( L ) n ( η ; y 1 , . . . , y d ; Ω) = TX d e iπD ( L -1) ψ ( L ) n ( η ; y 1 , . . . , y d ; Ω) . (6.35)  \nBy combining together Eqs. (6.33)-(6.35), we observe that the dependence on T and X cancels out, and so we verify the relation from the previous section:  \nRR : ψ ( L ) ∗ n ( η ; y 1 , . . . , y d ) = e iπ ( d +1)( L -1) ψ ( L ) n ( η ; -e iπ y 1 , . . . , -e iπ y d ) . (6.36)  \nIn momentum space we thus recover (6.30). This is the same as the Hermitian Analyticity component of the Cosmological Optical Theorem (COT), see e.g [54], i.e. the part of the COT which does not involve cutting rules.", '7 The Cosmological CPT Theorem': "Our 'Cosmological CPT theorem' makes the following three claims:  \nDiscrete Scale Invariance + RR invariance = ⇒ CRT invariance ;  \nCRT invariance + Discrete Scale Invariance = ⇒ RR invariance ;  \nCRT invariance + RR invariance = ⇒ Discrete Scale Invariance ,  \nwhere we remind the reader that Reflection Reality is a discrete symmetry which ALL unitary theories have to satisfy non-perturbatively, as well as to all orders in perturbation theory.  \nIn Section 6, we already outlined the necessary analytic continuations needed to define these three symmetries { CRT , RR , D } . In Sections 7.1-7.3, we will rederive the perturbative form of these symmetries, from a more concrete and detailed perspective. In Section 7.4 we will take the limit as η → 0 in order to derive the form of the symmetries at the boundary, including the phase formula. Section 7.5 will then check this result by comparing it to specific calculations in the literature.", '7.1 Unitarity and the Bunch-Davies Vacuum': "In this section we discuss our approach to enforcing the Bunch-Davies vacuum in a way that is consistent with Unitarity. Establishing such consistency is particularly important here as it enforces the direction in which we must rotate the spacetime coordinates through the complex plane. This extends the discussion in Section 3.3 where we defined the correlation functions in terms of the vacuum of the full theory, | Ω ⟩ . In the presence of interactions perturbation theory requires that we rewrite this state in terms of the free (Bunch-Davies) vacuum. It is standard to equate these two states in the early time limit by introducing a small imaginary contribution to the time, η i = -∞ (1 -iε ), rotating the contour off the negative real axis and turning off the interactions. To see this consider the free vacuum expanded in the basis of energy eigenstates of the full theory (see [102] for a more pedagogical derivation),  \n| 0 ⟩ = ∑ N | N ⟩⟨ N | 0 ⟩ . (7.1)  \nWe then consider the evolution of this state in the full theory,  \ne -iH ( η -η i ) | 0 ⟩ = ∑ n e -iE n ( η -η i ) | n ⟩⟨ n | 0 ⟩ (7.2)  \n= e -iE Ω ( η -η i ) | Ω ⟩⟨ Ω | 0 ⟩ + ∑ N =Ω e -iE N ( η -η i ) | N ⟩⟨ N | 0 ⟩ (7.3)  \n̸  \n= e -iE Ω ( η -η i )   | Ω ⟩⟨ Ω | 0 ⟩ + ∑ N =Ω e -i ( E N -E Ω )( η -η i ) | N ⟩⟨ N | 0 ⟩   . (7.4)  \n̸  \nThe vacuum is, by definition, the lowest energy state in the theory and so E N -E Ω > 0. Therefore, in the limit η i →-∞ (1 -iε ) only the first term remains and  \ne -iH ( η -η i ) | 0 ⟩ = e -iH ( η -η i ) | Ω ⟩⟨ Ω | 0 ⟩ . (7.5)  \nThis is just the Hamiltonian time evolution operator and so  \n⟨O ( η ) ⟩ = ⟨ 0 | U † I ( η ) O I ( η ) U I ( η ) | 0 ⟩ (7.6)  \nHowever, as was noted in [103,104], this approach spoils the Unitarity of the time evolution operator,  \n̸  \nU -1 I ( η ) = T exp [ -i ∫ η -∞ (1 -iε ) d ηH int [ ϕ, ϕ ' ] ] = U † I ( η ) = T exp [ -i ∫ η -∞ (1+ iε ) d ηH int [ ϕ, ϕ ' ] ] . (7.7)  \nTherefore, following [103], we instead introduce the shifted Hamiltonian,  \nH ε = He εη , (7.8)  \nwhich was shown to produce the same results perturbatively as the more standard rotation. Furthermore, it has the same effect of turning off the interactions in the infinite past so that interacting and free vacua agree thus we remain in the Bunch-Davies vacuum state. The reason for this is that, in both cases the ε -prescription contributes a negative real part to the exponent of the Hamiltonian in the infinite past thereby ensuring that it vanishes,  \nlim η →-∞ H ε ∼ lim η →-∞ e ik T η + εη = lim η →-∞ e i ( k T -iε ) η = 0 . (7.9)  \nFrom (7.9) we can see that this ε -prescription is equivalent to introducing a small negative imaginary part to the energies. In terms of this new Hamiltonian the expectation value is  \n⟨O ( η ) ⟩ = ⟨ 0 | U † ε ( η ) O I ( η ) U ε ( η ) | 0 ⟩ , U ε ( η ) = T exp [ -i ∫ η -∞ d ηH ε int [ ϕ, ϕ ' ] ] , (7.10)  \nwhich is now a well defined, unitary operator.  \nWhen we act with each of our three transformations it is important to ensure that we remain in the same vacuum state (else they cannot be symmetries of the theory). This will require that we rotate each of our coordinates in a particular direction through the complex plane when moving from the positive to the negative real axis (or vice versa). To see this we first consider RR , which involves a rotation of the momenta through the negative half of the complex plane,  \nk → e -iθ k . (7.11)  \nFrom these momenta we generate a set of energies 46 , k = √ k · k , which are the variables that are typically used to express both correlation functions and wavefunction coefficients. Under this rotation the energy transforms as √ √  \nk → k · k e -2 iθ = √ k · k e -2 iθ , (7.12)  \nwhere the final line follows as k is real. To ensure the single-valuedness of k during this rotation we need to continuously follow a Riemann sheet of the square root (i.e. we will not add any factors of 2 π to the exponent). Therefore, we have that  \nk → e -iθ k . (7.13)  \nIn order for the interactions to vanish in the early time limit and thus to remain in the Bunch-Davies vacuum we must ensure that  \nRe( ikη + εη ) < 0 ⇒ lim η →-∞ e ikη + εη = 0 . (7.14)  \nUnder this rotation in the energies this condition becomes,  \nRe( ike -iθ η + εη ) = k sin( θ ) η + εη > 0 (7.15)  \nwhich is true for positive θ (thus retroactively justifying our decision to rotate into the lower half of the complex plane). The direction of this rotation (as well as the fact that the factor ε is left unchanged) was noticed in the discussion of the Cosmological Optical Theorem [13] where the iε prescription was included on the energy, k → k -iε . In that case it was necessary to complex conjugate the energy when sending k → -k ∗ to keep the sign of ε constant. Although here the ε is treated independently of k the same behaviour is observed, k is reflected and ε remains unchanged.  \nNext, we consider the CRT transformation acting on this Hamiltonian. In this case the momentum is unchanged and we just rotate the time coordinate, η → ηe -iθ so that,  \nRe( ikηe -iθ + εηe -iθ ) = k sin( θ ) η + εη cos( θ ) . (7.16)  \nThis is a linear combination of sines and cosines and so could equivalently be represented as a shifted sine function which is non-zero at θ = 0. Therefore, this function is guaranteed to change sign over the range 0 ≤ θ ≤ π (or -π ≤ θ ≤ 0) and we must also rotate ε alongside η , ε → εe iθ , so that the exponent is  \nRe( ikηe -iθ + εη ) = k sin( θ ) η + εη (7.17)  \nwhich is negative just as for the rotation in k .  \nFinally, we must consider the simultaneous rotation of k and η this comes from the analytic continuation of the scaling transformation and so when we send η → λη we must simultaneously send k → λ -1 k . Therefore, the two rotate in opposite directions through the complex plane such that the combination kη  \nis unchanged. As a result of this the exponent in the infinite time limit of the Hamiltonian has a real part given by,  \nRe( ikη + εηe iθ ) = εη cos( θ ) . (7.18)  \nJust as before this will change sign and so we must rotate ε in the same direction as k . This doesn't enforce a particular direction to the rotation of k or η however.", '7.2 The Loop Momentum Integral': "In Section 6.2 we established the transformation properties of the wavefunction coefficients from the analytic continuation of the coordinates and metric. Here we present an alternative perspective to reach the same conclusions by looking at the explicit calculation of the wavefunction coefficients.  \nIn order to achieve this goal we first break up the wavefunction into distinct parts that are easier to tackle on their own. As explained in [35] the wavefunction coefficients can be calculated by performing a series of nested time integrals plus, potentially, some loop integrals,  \nψ ' ( L ) n ( η 0 ; k ; ε ) = i V ∫ η 0 V ∏ v a d +1 v d η v e εη v L ∏ l d d P l n ∏ a K k a ( η ) I ∏ i G p i ( η, η ' ) δ d ( n ∑ a k a ) , (7.19)  \nwhere the prime in ψ ' ( L ) n is used to explicitly highlight that these wavefunction coefficients contain the δ -function whilst those without the δ -function will be denoted by ψ ( L ) n . The external energies are indicated by k a = | k a | whilst the p i are the internal energies which are built from the k a plus potentially some of the loop momenta, P l . Note that here we have made the ε prescription explicit and that there are no derivatives acting on any of our propagators. This is purely for the sake of notational simplicity and adding any derivatives will not add any technical complications.  \nWe do not explicitly consider UV divergences in this section (which can lead to some additional corrections with a relative i sign, as discussed in Section 6.2). However, as our results are valid in general dimensions, it is straightforward to renormalise these divergences by dimensional regularisation and this will be discussed explicitly in Section 7.5.  \nThe first component we will look at is the time integral which runs from -∞ to η 0 when we have a transformation involving the time coordinate it is these limits that pick up a minus sign. To remove this we redefine the variables inside the integral with η v → -η v 47 The effect of this on the measure factor is then straightforward as everything is just some power of η .  \nThe loop integral is slightly more involved. To treat this we first change variables so that rather than doing the loop integral over some d dimensional vectors we instead perform it over some set of loop energies. This requires a change of basis which was elucidated for arbitrary loops in [105]. For our purposes, the most important fact about this determinant is that it is related to the volume of a simplex with edges that are the lengths of squares of energies. Therefore, its zeros and functional form are unchanged under a transformation which sends k →-k . Explicitly, for a single loop, the integration measure becomes  \n∫ d d P l = N √ Vol 2 ( s ) ∫ Γ I l ∏ i 2 p i d p i [ Vol 2 ( p, s ) Vol 2 ( s ) ] d -I l -1 2 , (7.20)  \nwhere I l is the number of internal lines in the loop, l ; P l is the momentum running through this loop; and N accounts for the integral over the d -I L sphere plus an extra, unimportant, proportionality constant.  \nUnder the rescalling p → λp and s → λs this transforms as  \nλ d ∫ d d P l = Nλ 2 I l √ λ 2 I l -2 √ Vol 2 ( s ) ∫ Γ I l ∏ i 2 p i d p i [ λ 2 I l Vol 2 ( p, s ) λ 2 I l -2 Vol 2 ( s ) ] d -I l -1 2 . (7.21)  \nFor positive λ we can see that the scaling on each side agrees. Naively, on the left hand side continuing λ →-λ would appear to have no impact. However, from the right hand side we can see how our analytic continuation in the energies alters this conclusion, instead recovering a factor of ( -1) d . To see this, we first note that the region, Γ, depends only on the squares of the energies and so is unchanged. However, each of our square roots of energies does pick up a factor of -1. As we now have a potentially non-integer exponent in the dimension of the integral d we will need to be careful with e iπd vs e -iπd . For RR , in accordance with the discussion in Section 7.1, all rotations in the energy must be taken in the lower half plane and so λ = λe -iπ . Therefore we have,  \n( -λ ) d ∫ d d P l = Ne -iπd λ d √ Vol 2 ( s ) ∫ Γ I l ∏ i p i d p i [ Vol 2 ( p, s ) Vol 2 ( s ) ] d -I l -1 2 . (7.22)  \nWenext explore the transformations of the propagators. To do this we consider the differential equation,  \nO k ( η ) = a d -1 ∂ 2 ∂η 2 +( d -1) a ' a d -2 ∂ ∂η + m 2 a d +1 + a d -1 k 2 ⇒ { O k ( η ) G k ( η, η ' ) = -iδ ( η -η ' ) , O k ( η ) K k ( η ) = 0 . (7.23)  \nThe propagators additionally satisfy the boundary conditions  \nlim η → η 0 K k ( η ) = 1 , lim η →-∞ K k ( η, η ' ) ∝ e ikη , (7.24)  \nlim η,η ' → η 0 G k ( η, η ' ) = 0 , lim η,η ' →-∞ G k ( η, η ' ) ∝ e ikη . (7.25)  \nThese boundary conditions at early time will ensure that our iε prescription will cause the early time integrand to vanish. Let us first consider three relationships that our differential operator satisfies. These will later be linked to symmetries of the wavefunction and are labeled accordingly,  \nRR : O ∗ k ( η ) = O e -iπ k ( η ) , (7.26)  \nCRT : O ∗ k ( η ) = e -iπ ( d +1) O k ( e -iπ η ) , (7.27)  \nD ± -1 : O k ( η ) = e ± iπ ( d +1) O e ∓ iπ k ( e ± iπ η ) . (7.28)  \nHere, the direction of the rotations in k and η are dictated by the asymptotic time behaviour, preserving the Bunch-Davies vacuum discussed in Section 7.1 and the ± sign in D ± -1 represents the ambiguity in the direction of rotation of η → e ± iπ η 48 .  \nWe start by considering Reflection Reality,  \nO ∗ k ( η ) K ∗ k ( η ) = 0 , lim η → η 0 K ∗ k ( η ) = 1 , lim η →-∞ K ∗ k ( η ) ∝ e -ikη , (7.29)  \nO e -iπ k ( η ) K e -iπ k ( η ) = 0 , lim η → η 0 K e -iπ k ( η ) = 1 , lim η →-∞ K e -iπ k ( η ) ∝ e -ikη . (7.30)  \nIn light of (7.26) these two differential equations are identical to each other as are the boundary conditions therefore,  \nRR : K ∗ k ( η ) = K e -iπ k ( η ) . (7.31)  \nSimilarly, the bulk-bulk propagator satisfies  \n∗ ∗ ' ' ∗ ' ∗ ' -ikη  \n-O k ( η ) G k ( η, η ) = -iδ ( η -η ) , lim η → η 0 -G k ( η, η ) = 0 , lim η →-∞ -G k ( η, η ) ∝ e , (7.32) O e -iπ k ( η ) G e -iπ k ( η, η ' ) = -iδ ( η -η ' ) , lim η → η 0 G e -iπ k ( η, η ' ) = 0 , lim η →-∞ G e -iπ k ( η, η ' ) ∝ e -ikη , (7.33)  \nwhich gives a very similar expression for this propagator,  \nRR : G ∗ k ( η, η ' ) = -G e -iπ k ( η, η ' ) . (7.34)  \nNotice, that we are free to introduce a minus sign in the early time limit in (7.32), this is because the boundary condition when we introduce ε is that the propagator vanishes, so it is just the sign of the imaginary exponent that needs to be fixed.  \nThe CRT transformation acts similarly,  \nO ∗ k ( η ) K ∗ k ( η ) = 0 , lim η → η 0 K ∗ k ( η ) = 1 , lim η →-∞ K ∗ k ( η ) ∝ e -ikη , (7.35) O k ( e -iπ η ) K k ( e -iπ η ) = 0 , lim η → η 0 K k ( e -iπ η ) = 1 , lim η →-∞ K k ( e -iπ η ) ∝ e -ikη . (7.36)  \nThe additional phase introduced to the differential operator can simply be dropped when considering the bulk-boundary propagator as the right hand side of the differential equation is zero and so,  \nCRT : K ∗ k ( η ) = K k ( e -iπ η ) . (7.37)  \nFor the Green's function this is not the case, but the boundary conditions can have an arbitrary phase introduced,  \nO ∗ ( η ) G ∗ ( η, η ' ) = iδ ( η -η ' ) , O k ( e -iπ η ) G k ( e -iπ η, e -iπ η ' ) = iδ ( η -η ' ) ,  \nlim η →-∞ G ∗ k ( η, η ' ) ∝ e -ikη , lim η →-∞ e iπ ( d +1) G k ( e -iπ η, e -iπ η ' ) ∝ e -ikη ,  \nk k (7.38) lim η → η 0 G ∗ k ( η, η ' ) = 0 , lim η → η 0 e iπ ( d +1) G k ( e -iπ η, e -iπ η ' ) = 0 , (7.39) (7.40)  \nwhere the time reversal has introduced an extra minus sign into the delta function due to its action on the limits of the integral. To see this consider,  \n∫ η ' -∞ d η δ ( η -η ' ) → ∫ -η ' ∞ d η δ ( η + η ' ) = -∫ η ' -∞ d η δ ( -η + η ' ) . (7.41)  \nOnce again, we can see that (7.27) allows us to extract a relationship between the complex conjugate of the bulk-bulk propagator and its analytic continuation,  \nCRT : G ∗ k ( η, η ' ) = e iπ ( d +1) G k ( e -iπ η, e -iπ η ' ) . (7.42)  \nNext we consider dilatations. This symmetry simply introduces a phase to both equations whilst leaving the boundary conditions unchanged. Therefore,  \nD ± -1 : K k ( η ) = K e ∓ iπ k ( e ± iπ η ) , and G k ( η, η ' ) = e ∓ iπd G e ∓ iπ k ( e ± iπ η, e ± iπ η ' ) . (7.43)  \nThus, we see that the bulk-boundary propagators pick up very simple relationships from each of our symmetries. The bulk-bulk propagators have similar relationships but additionally introduce a dimension dependent phase.  \nFinally, we must consider the transformation of the delta function. This will behave similarly to the right hand side of the differential equation for the bulk-bulk propagator (7.41),  \nδ d ( e iθ k ∑ a k a ) = e -iθd δ d ( k ∑ a k a ) . (7.44)  \nThis is unlike what we expect for the delta function which, for real numbers, transforms as  \nδ ( ax ) = 1 | a | δ ( x ) . (7.45)  \nHowever, just as for the loop and time integrals, the analytic continuation that we perform will also rotate the integrals over the external momenta by e iθd . This is then matched by the analytic continuation of the delta function.  \nWe now have all the ingredients to understand how our three transformations work on a wavefunction coefficient. The first symmetry we will consider is Reflection Reality which relates the complex conjugate to the rotation of all momenta through the lower half of the complex plane,  \nRR : [ ψ ' ( L ) n ( η 0 ; k ; ε ) ] ∗ (7.46)  \n= ( -i ) V ∫ η 0 V ∏ v a d +1 v d η v e εη v L ∏ l d d P l n ∏ a K ∗ k a ( η ) I ∏ i G ∗ p i ( η, η ' ) δ d ( n ∑ a k a ) (7.47)  \n= ( -i ) V ∫ η 0 V ∏ v a d +1 v d η v e εη v L ∏ l d d P l n ∏ a K e -iπ k a ( η ) I ∏ i -G e -iπ p i ( η, η ' ) δ d ( n ∑ a k a ) (7.48)  \n= e iπ ( I -V + d ( L -1)) i V ∫ η 0 V ∏ v a d +1 v d η v e εη v L ∏ l d d ( e -iπ P l ) n ∏ a K e -iπ k a ( η ) I ∏ i G e -iπ p i ( η, η ' ) δ d ( e -iπ n ∑ a k a ) (7.49)  \n= e iπ (1+ d )( L -1) ψ ' ( L ) n ( η 0 ; e -iπ k ; ε ) . (7.50)  \nwhere in the penultimate line we have employed the Feynman-Euler relationship, V -I + L = 1 and the rotation on k applies to all internal and external energies. Next we investigate Scale Invariance which rotates all momenta magnitudes as well as the conformal time,  \nD ± -1 : ψ ' ( L ) n ( η 0 ; k ; ε ) = i V e ± iπd ( V + L -I -1) ∫ η 0 V ∏ v e ∓ iπd a d +1 v d η v e εη v L ∏ l d d ( e ∓ iπ P l ) (7.51)  \n× n ∏ a K e ∓ iπ k a ( e ± iπ η ) I ∏ i G e ∓ iπ p i ( e ± iπ η, e ± iπ η ' ) δ d ( e ∓ iπ n ∑ a k a ) (7.52)  \n= i V ∫ e ± iπ η 0 V ∏ v a d +1 v d η v e e ∓ iπ εη v L ∏ l d d ( e ∓ iπ P l ) n ∏ a K e ∓ iπ k a ( η ) I ∏ i G e ∓ iπ p i ( η, η ' ) δ d ( e ∓ iπ n ∑ a k a ) (7.53)  \n= ψ ' ( L ) n ( e ± iπ η 0 ; e ∓ iπ k ; e ∓ iπ ε ) . (7.54)  \nThe final transformation we will consider is CRT which flips the conformal time with an overall complex  \nconjugation of the wavefunction coefficient,  \nCRT : [ ψ ' ( L ) n ( η 0 ; k ; ε ) ] ∗ (7.55)  \n= ( -i ) V ∫ η 0 V ∏ v a d +1 v d η v e εη v L ∏ l d d P l n ∏ a K ∗ k a ( η ) I ∏ i G ∗ p i ( η, η ' ) δ d ( n ∑ a k a ) (7.56)  \n= ( -i ) V e -iπdV ∫ η 0 V ∏ v e iπd a d +1 v d η v e εη v L ∏ l d d P l n ∏ a K k a ( e -iπ η ) I ∏ i e iπ ( d +1) G p i ( e -iπ η, e -iπ η ' ) δ d ( n ∑ a k a ) (7.57)  \ne iπ ( I -V )( d +1) i V ∫ -η 0 V ∏ v a d +1 v d η v e e -iπ εη v L ∏ l d d P l n ∏ a K k a ( η ) I ∏ i G p i ( η, η ' ) δ d ( n ∑ a k a )  \n= (7.58) = e iπ (1+ d )( L -1) ψ ' ( L ) n ( e -iπ η 0 ; k ; e iπ ε ) . (7.59)  \nTable 2: Table showing how the transformations combine for ψ ( L ) n . In this table the first transformation is in the column heading and then the row heading second. We see that CRT and RR each square to 1. Furthermore, they combine to give D ± -1 . Since a priori (before imposing dilation symmetry) the operations D + -1 and D --1 act differently, we no longer have the Z 2 × Z 2 structure as the dilatation operator doesn't square to itself. Instead this group has the same structure as the group Aut( Z ) of automorphisms of the integers.  \nAn interesting observation one can make is that for ψ ' ( L ) n , CRT and RR introduce equal phases, and in even spacetime dimensions all three transformations leave any real ψ ' ( L ) n invariant at any loop order, provided ψ ' ( L ) n is UV-finite. This also highlights the fact that in the original contact Cosmological Optical Theorem [13,14] the minus sign picked up between ψ ( L ) n and its reflected counterpart can be understood as an artefact of the fact that the δ -function has been stripped off. Indeed, when we consider ψ ( L ) n without this delta function stripped off we find  \nCRT : [ ψ ( L ) n ( η 0 ; k ; ε ) ] ∗ = e iπ ( d +1)( L -1) ψ ( L ) n ( e -iπ η 0 ; k ; e iπ ε ) , (7.60)  \nD ± -1 : ψ ( L ) n ( η 0 ; k ; ε ) = e ± iπd ψ ( L ) n ( e ± iπ η 0 ; e ∓ iπ k ; e ∓ iπ ε ) , (7.61)  \nRR : [ ψ ( L ) n ( η 0 ; k ; ε ) ] ∗ = e iπ (( d +1) L -1) ψ ( L ) n ( η 0 ; e -iπ k ; ε ) . (7.62)  \nThese relationships each imply an operator that leaves the wavefunction coefficient unchanged up to an overall phase in the presence of the corresponding symmetry,  \nCRT ( ψ ( L ) n ) = [ ψ ( L ) n ( e -iπ η 0 , k , e iπ ε ) ] ∗ = e iπ ( d +1)( L -1) ψ n ( η 0 ; k ; ε ) , (7.63)  \nD ± -1 ( ψ ( L ) n ) = ψ ( L ) n ( e ± iπ η 0 ; e ∓ iπ k ; e ∓ iπ ε ) = e ∓ iπd ψ n ( η 0 ; k ; ε ) , (7.64)  \nRR ( ψ ( L ) n ) = [ ψ ( L ) n ( η 0 ; e -iπ k ; ε ) ] ∗ = e iπ (( d +1) L -1) ψ n ( η 0 ; k ; ε ) , (7.65)  \nwhere, for clarity, we have kept the ε -dependence to explicitly show how it rotates. Although, in the final wavefunction coefficient ψ ( L ) n we take ε → 0 so that this rotation, ultimately, will not influence  \nthese transformations 49 . These symmetry transformations can then be combined, as presented in Table 2. Just as in the flat space case the combination of CRT and RR generates the other transformations and they square to themselves. However, here we do not recover the Z 2 × Z 2 group but instead the group Aut( Z ) of automorphisms of the integers. It is important to note that we need both D ± -1 for this group to close as (in arbitrary dimensions) they are not generically self inverse. This is not an obstacle as both transformations are implied by the Scale Invariance of the theory due to the fact that the Bunch-Davies condition is preserved by both rotations (this condition is what restricts us from considering rotations in the other directions for either CRT or RR ). Interestingly, the ambiguity in this rotation direction appears absent from ψ ' ( L ) n where D ± -1 doesn't introduce a phase. However, this is merely an artifact of the dilatation symmetry, without imposing it the two transformations act differently. Furthermore, to connect RR to CRT we still need to be able to rotate in both directions using our dilatation operator. To see this consider how D + -1 acts on an already transformed wavefunction coefficient,  \nD + -1 ( RR ( ψ ( L ) n )) = D + -1 ( e iπ ( -1+( d +1) L ) ψ ( L ) n ) = e iπ ( d +1)( L -1) ψ ( L ) n = CRT ( ψ ( L ) n ) , (7.66)  \nD  \n+ -1 ( CRT ( ψ ( L ) n )) = D + -1 ( e iπ ( d +1)( L -1) ψ ( L ) n ) = e iπ (( d +1) L -2 d -1) ψ ( L ) n = RR ( ψ ( L ) n ) . (7.67)  \n̸  \nWe can see that this agrees with the group structure in Table 2,  \n̸  \nCRT · RR = D + -1 ⇒ { D + -1 · RR = CRT , D + -1 · CRT = CRT · RR · CRT = RR , (7.68)  \nwhere the second implication follows from the fact that the CRT transformation squares to one.  \nWe will now move on to discuss how these transformations can be used to identify constraints of the I + boundary correlators where η = 0 -which will be of particular importance for dS/CFT!", '7.3 Spinning Fields': "The analysis above also applies to spinning fields. To see this it is convenient to use the following notations and conventions. In d +1-dimensional spacetime, for traceless 50 , integer spin fields we use the free action developed in [106] and discussed in the context of the cosmological bootstrap in [15,107]:  \nS = ∫ d d +1 x [ a ( η )] d -1 1 2 s ! [ ( σ ' i 1 ...i s ) 2 -c 2 s ( ∂ j σ i 1 ...i s ) 2 -δc 2 s ( ∂ j σ ji 2 ...i s ) 2 -m 2 a 2 ( σ i 1 ...i s ) 2 ] . (7.69)  \nThe totally-symmetric, traceless tensor σ i 1 ...i s has spatial indices i 1 = 1 , . . . , d which span the d -dimensional spacelike hypersurface orthogonal to the η coordinate. 51 σ i 1 ...i s has (2 s +1) components, which each create states ('particles') with helicities 0 , ± 1 , . . . , ± s , and we have enforced invariance under dilatations by including inverse factors of the scale factor for each coordinate derivative. Following [15, 107] we Fourier transform and diagonalise this using the helicity modes, σ h , defined by:  \nσ i 1 ...i s ( η ; x ) = ∫ k e i k · x S ∑ h = -S e h i 1 ...i s ( k ) σ h ( η ; k ) , (7.70)  \nThese helicity tensors are defined as an outer product of helicity vectors,  \ne h i 1 ...i s = e h 1 i 1 . . . e h s i s , (7.71)  \nwhich satisfy the following relations:  \ne h i ( k ) [ e h ' i ( k ) ] ∗ -4 δ hh ' = 0 (orthogonality and normalisation) ,  \n(7.72) [ e h i ( k ) ] ∗ -e h i ( -k ) = 0 ( σ i 1 ...i s ( x ) is real) . (7.73)  \nNote that these fields are not assumed to be transverse, h is allowed to take d different values including 0 where e 0 is proportional to the momentum. The contributions from the other helicity modes are therefore transverse by the orthogonality condition.  \nWe parameterise the WFU, Ψ, at conformal time η 0 in terms of the helicities of the integer spin field as  \nΨ[ η 0 ; σ ( k )] = exp [ -∞ ∑ n =2 1 n ! ∑ h i = ± ∫ k 1 ,..., k n ψ h 1 ...h n n ( η 0 ; k )(2 π ) d δ d ( ∑ k a ) σ h 1 ( η 0 ; k 1 ) . . . σ h n ( η 0 ; k n ) ] . (7.74)  \nSpatial translations and spatial rotations ensure that wavefunction coefficients can be written as a product of a helicity factor , which is an SO ( d ) invariant function of helicity vectors and spatial momenta, multiplied by a trimmed wavefunction coefficient which is only a function of the magnitudes of the momenta in the literature:  \nψ h 1 ...h n n = (tensor structure) × (trimmed wavefunction coefficient) . (7.75)  \nWe take all coefficients appearing in the tensor structure to be real and therefore include any factors of i that might appear when converting to momentum space, or simply as part of the Feynman rules, in the trimmed part which we will denote as ψ n for brevity. For a general ψ h 1 ...h n n , we have  \nψ h 1 ...h n n = [ e h 1 ( k 1 ) . . . e h n ( k n ) k α 1 1 . . . k α n n ] ψ n , (7.76)  \nfor some integer α i . Note that we can choose for both parity-even and parity-odd tensor structures to be invariant with respect to the discrete symmetries of CRT , RR and D which we discuss in this paper. 52 Hence, all our results extend directly from the scalar ψ n discussed in Section 7.2 to the tensor case.", '7.4 Constraints on Boundary Wavefunction Coefficients': "All of the results that we have presented in this section so far rely on an analytic continuation, either in momenta or time. However, by considering the wavefunction of the universe at the future boundary of dS, where the time dependence often trivialises, we can derive direct constraints on the wavefunction coefficients. This future boundary is where the reheating surface is understood to live in the inflationary paradigm. Therefore, such constraints are of particular cosmological relevance.  \nIn the late time limit the bulk fields take the form  \nlim η → 0 -ϕ ( η, y ) = ϕ + ( y ) η ∆ + + ϕ -( y ) η ∆ -, (7.77)  \n= ϕ + ( y ) η ∆ + ϕ -( y ) η d -∆ , (7.78)  \nwhere for the case of scalars, the mass m 2 /H 2 = m 2 ℓ 2 = ∆( d -∆) is related to the conformal dimension in the usual way ∆ = d/ 2 + ν , ν = √ d 2 / 4 -m 2 /H 2 . 53 For massive spins fields the mass and spin are related to the conformal dimension by ∆ = d/ 2 + µ , µ = √ ( d +2 s -4) 2 / 4 -m 2 /H 2 . We define ϕ + with ∆ + ≡ ∆ and ϕ -(with ∆ -= d -∆). For heavy scalar fields ( m > dH/ 2), ∆ is complex and so these boundary operators do not represent a self adjoint basis 54 . We will therefore restrict our discussion to light fields and leave a more comprehensive examination, including heavy fields, to future work. In this case, ∆ is real and positive so the ϕ -term dominates. Starting from the expansion of the wavefunction in terms of its coefficients, 3.71, we can similarly define a 'boundary' wavefunction 55 which is a functional of these boundary fields ϕ -and has no explicit η -dependence  \nΨ[ ϕ -( k a )] = exp { -∞ ∑ n =2 ∫ [ n ∏ a =1 d d k a (2 π ) d ϕ -( k a ) ] ψ n ( k ) δ d ( ∑ k a ) } , (7.79)  \nwhere we have introduced boundary wavefunction coefficients 56 denoted by ψ n to distinguish them from the bulk wavefunction coefficients ψ n . 57 The choice to Taylor expand the wavefunction in the field operator with dimension d -∆ rather than using the seemingly more sensible choice of labeling this operator's dimension ∆ is purely conventional. It is standard to refer to ∆ + as the dimension of a field in cosmology (for example a massless field has dimension ∆ = d ) but it turns out that this is not the part of the field that survives on the boundary.  \nLet us now count the scalings of each term in the exponent one by one to see how ψ n ( k a ) scales under k → λ k which in turn will scale the internal and external energies k → λk . The d -dimensional d d k measures each scale with volume and thus they will scale with an overall factor of λ nd . This cancels with an equivalent scaling of the Fourier transformed ϕ -( k a ) which give an overall scaling of λ -nd . 58 Each boundary field must contribute an extra scaling like η ∆ -d to ensure the Scale Invariance of the bulk field (7.78). The result of this is that in addition to the factor coming from the Fourier Transform the fields contribute a further λ dn -∑ α ∆ α . Finally, the δ d ( k 1 + . . . + k n ) scales with inverse volume λ -d . 59 We know that the exponent of (7.79) must be dimensionless, thus by dimensional analysis, any IR-finite boundary wavefunction coefficient will scale as  \nψ n ( λ k ) = λ d λ -nd λ ∑ α ∆ α ψ n ( k ) = λ d (1 -n )+ ∑ α ∆ α ψ n ( k ) . (7.81)  \nWe will now look at the constraints from CRT , RR and D -1 on the boundary wavefunction coefficients ψ n , where this dimensional analysis exercise will prove to be useful. Let us look at each transformation individually.  \n56 These can be interpreted as the correlators of the operators of some conjectured CFT living at the boundary, i.e. dS/CFT. 57 To obtain the boundary wavefunction coefficient we differentiate the boundary wavefunction Ψ with respect to the sources ϕ -which have dimension d -∆  \nψ n ( k ) ≡ lim ϕ -→ 0 δ n δ n ϕ -Ψ[ ϕ -( k a )] . (7.80)  \n58 This is in the exact same way that in (3.73) the original bulk field ϕ k a scaled as λ d resulting in the bulk wavefunction coefficient scaling inversely to the δ d ( ∑ k a ).  \nRR does not care about the time at which the wavefunction is evaluated and so the resulting constraint is unchanged: 60  \nRR : [ ψ ( L ) n ( k ) ] ∗ = e iπ [( d +1) L -1] ψ ( L ) n ( e -iπ k ) . (7.82)  \nScale Invariance in the bulk requires that  \nψ ( L ) n ( η/λ ; λ k ) = λ d ψ ( L ) n ( η ; k ) . (7.83)  \nAs was derived in (7.81), Scale Invariance of the boundary theory at I + where η = 0 -tells us that the wavefunction coefficients must scale as  \nψ ( L ) n ( λ k ) = λ d (1 -n )+ ∑ α ∆ α ψ ( L ) n ( k ) , (7.84)  \nwhere ∆ α are the dimensions of the external fields. The Cosmological CPT theorem tells us that the SO + (1 , 1) boost which is the important continuous symmetry in the flat space CPT theorem acts in the Poincar'e patch as dilatations D , i.e. Scale Invariance λ ∈ R + , which we can analytically continue to SO (2), provided we have a Hamiltonian bounded from below. Under analytic continuation of λ ∈ R + through the complex plane λ ∈ C , 61 we can then access λ = e -iπ , to conclude that:  \nψ ( L ) n ( e -iπ k ) = ( -1) d (1 -n )+ ∑ α ∆ α ψ ( L ) n ( k ) , (7.85)  \nwhere for cases with fractional d and ∆ we have  \nD ± -1 : ψ ( L ) n ( e ∓ iπ k ) = e ∓ iπ [ d (1 -n )+ ∑ α ∆ α ] ψ ( L ) n ( k ) . (7.86)  \nWe need to take into account that ψ ( L ) n is analytic through the lower half-plane C -i . Given that the RR (7.82) transformation rotates ψ ( L ) n counter-clockwise, we must use the D + -1 in (7.86) to land on  \nD + -1 · RR : [ ψ ( L ) n ( k ) ] ∗ = e iπ [( d +1) L -1 -d (1 -n ) -∑ α ∆ α ] ψ ( L ) n ( k ) . (7.87)  \nWe can also derive the same expression independently using CRT , which tells us directly that  \n[ ψ ( L ) n ( η ; k ) ] ∗ = e iπ ( d +1)( L -1) ψ ( L ) n ( e -iπ η ; k ) , (7.88)  \n[ ψ ( L ) n ( k ) ] ∗ = e iπ [( d +1)( L -1)+ dn -∑ α ∆ α ] ψ ( L ) n ( k ) , (7.89)  \nwhere for (7.88) we have to continue 1 /η ∝ k ≡ | k | in the lower-half plane C -i , and as previously explained the factor of η ∑ α d -∆ α comes from the fields scaling as η ∆ -d ∝ λ d -∆ at the boundary. We have now identified the discrete symmetries which the boundary wavefunction coefficients ψ ( L ) n satisfy:  \nCRT : ψ ( L ) n ( k ) = e -iπ [( d +1)( L -1)+ dn -∑ α ∆ α ] [ ψ ( L ) n ( k ) ] ∗ , (7.90)  \nD ± -1 : ψ ( L ) n ( k ) = e ± iπ [ d (1 -n )+ ∑ α ∆ α ] ψ ( L ) n ( e ∓ iπ k ) , (7.91)  \nRR : ψ ( L ) n ( k ) = e -iπ [( d +1) L -1] [ ψ ( L ) n ( e -iπ k ) ] ∗ , (7.92)  \nwhere the last condition from RR holds for boundary correlators in any flat FLRW spacetime with a future conformal boundary. We can use (7.90) to solve for the phase of ψ ( L ) n directly, obtaining the following phase formula for the boundary wavefunction coefficients:  \ne i arg( ψ ( L ) n ) ≡ ψ ( L ) n ( k ) | ψ ( L ) n ( k ) | = ± √ √ √ √ ψ ( L ) n ( k ) ψ ( L ) ∗ n ( k ) = ± ( -i ) ( d +1)( L -1)+ dn -∑ α ∆ α , (7.93)  \nwhere there is a ± out front because CRT cannot determine the overall real sign, because obtaining the phase involves taking a square root. 62 Hence, we obtain the result (1.10) quoted in Section 1.2:  \narg( ψ ( L ) n ) = -π 2 ( ( d +1)( L -1) + dn -∑ α ∆ α ) + π N . (7.94)  \nRemarkably (7.93) and (7.94) will hold for any ψ ( L ) n computed in cosmology provided that:  \n- · spacetime is de Sitter (possibly with boost-breaking terms);\n- · the Lagrangian is locally CRT -invariant;\n- · the amplitude is UV and IR finite;\n- · and involves fields in representations with integer spin and real ∆ (no spinors or principal series).  \nFurthermore, the phase of ψ ( L ) n has no dependence on the details of the bulk interactions, e.g. derivative couplings will have the same phase, provided the Feynman-Witten diagrams have the same external legs with the same ∆'s.  \nThe steps leading up to this phase formula implicitly assumed that the amplitudes were IR and UVfinite. We have already discussed the -iπ shift due to UV log divergences in Section 6.2; we now make the corresponding remark for IR divergences. For a typical IR-divergent amplitude, there is a term like C log( -η ) which, upon rotating η from 0 -to 0 + , becomes C log ( -e -iπ η ) This provides an extra -iπC shift that adjusts the IR-finite piece of the amplitude, which does not conform to (7.94). But, the leading order IR-divergence will continue to satisfy (7.94). An example of such a calculation will be quoted in Section 7.5 below.", '7.5 Explicit Checks of the Phase Formula': 'We will now provide a list of non-trivial checks of the phase formula (7.93) and (7.94) for various specific Feynman-Witten diagrams in cosmology.  \nThe first five calculations i -v below all involve conformally coupled scalars (denoted by φ ), while the remaining two vi and vii involve massless fields (denoted by ϕ ). For the calculations in iii -vii , we work in non-integer dimensions for the purposes of dimensional regularisation (dim-reg) using the prescription described in Appendix C of [109]. In this approach, the mass of the field is also renormalised to keep the order of the Hankel function ν fixed, which ensures that the integrals can be computed analytically in the dim-reg parameter δ . 63', 'i. Conformal tree-level IR-finite coefficient ψ tree 3 ,φφ ˜ σ': 'The phase of the contact three-point wavefunction coefficient ψ tree 3 ,φφ ˜ σ , generated via a simple cubic interaction φφ ˜ σ in 3 + 1-dimensions, involving two conformally-coupled scalars (with ∆ = 2) and a massive scalar ˜ σ with mass m < √ 2 H (corresponding to 2 < ∆ ≤ 3), in order for the time integral to converge in the IR (i.e. when η = 0 -), is found in Appendix B of [13] to be  \narg( ψ tree 3 ,φφ ˜ σ ) = π 2 ( ν + 1 2 ) . (7.95)  \nThis agrees with (7.93) and (7.94) as it correctly predicts the generically complex phase of a wavefunction coefficient involving a light field with a mass m< √ 2 H .', 'ii. Conformal tree-level IR-divergent coefficient ψ tree 3 ,φφφ': 'The contact three-point wavefunction coefficient ψ tree 3 ,φφφ for the interaction involving three conformallycoupled scalars, generated via a simple cubic interaction φφφ in 3 + 1-dimensions, where the time integral diverges in the IR (i.e. when η = 0 -), is found to be  \nψ tree 3 ,φφφ = i (log( -i ( k 1 + k 2 + k 3 ) η 0 ) + γ ) ( η 0 ) 3 H 4 + O ( 1 η 2 0 ) , (7.96)  \ni.e. it has a log( -η ) IR-divergence. However, as we previously explained (7.93) and (7.94) continue to hold for the phase of the coefficient of the leading log( -η ) IR-divergence and thus we find correctly that there is an i in front of the log( -η ) term. This is another non-trivial check of the phase formula as it correctly predicts the imaginary phase of the log( -η ) in the IR-divergent wavefunction coefficient.', 'iii. Conformal tree-level IR-finite coefficient in non-integer dimension d = 3 + δ , ψ tree 6 ,φ 6': 'The phase of the contact six-point wavefunction coefficient ψ tree 6 ,φ 6 for the interaction involving six conformally-coupled scalars, generated via a simple 6th-order interaction φ 6 in non-integer d = 3 + δ dimensions (which rescales the conformal dimension of the conformally coupled scalar ∆ = d/ 2 + 1 / 2 = (4 + δ ) / 2), was first computed in [111]. However, an incorrect phase was found, which will be corrected in [72]. For this particular Feynman-Witten diagram one finds  \nψ tree 6 ,φ 6 = -e -iπδ Γ(3 + 2 δ ) H 4+ δ ( k (6) T ) 3+2 δ . (7.97)  \nOnce again (7.93) and (7.94) correctly predict the generically complex phase of the wavefunction coefficient in non-integer dimensions d = 3 + δ .', 'iv. Conformal 1 -loop UV-finite coefficient in non-integer dimension d = 3 + δ , ψ ( L =1) 4 ,φ 6': 'The phase of the 1-site 1-loop four-point wavefunction coefficient ψ ( L =1) 4 ,φ 6 for the interaction involving four conformally-coupled scalars, generated via a 6th-order interaction φ 6 in non-integer d = 3 + δ -dimensions (which, as previously explained, rescales the conformal dimension of the conformally coupled scalar ∆ = d/ 2 + 1 / 2 = (4 + δ ) / 2), was first computed in [111]. However, an incorrect phase was found, which will be corrected in [72]. For this particular Feynman-Witten diagram one finds  \nψ ( L =1) 4 ,φ 6 = λe -iπδ 2 5+ δ π 4+ δ 2 H 2 ( k (4) T ) 1+ δ Γ(1 + δ )Γ ( 2 + δ 2 ) . (7.98)  \nSince ψ ( L =1) 4 ,φ 6 comes from a UV-finite diagram we see that is is analytic in the k T -energy pole as δ → 0. As predicted we find lim δ → 0 ψ ( L =1) 4 ,φ 6 ∈ R . Additionally (7.93) and (7.94) correctly predicts the generically complex phase of ψ ( L =1) 4 ,φ 6 when δ is kept finite. This will be important when considering the next examples, which are UV-divergent.', 'v. Conformal 1 -loop UV-divergent coefficient in non-integer dimension d = 3 + δ , ψ ( L =1) n,φ ( n +4) / 2': 'The 2-site, 1-loop de Sitter wavefunction coefficient for arbitrary number n of conformally coupled fields in external legs in non-integer d = 3+ δ -dimensions will be computed in upcoming work [112] 64 where the author generalises the recently developed differential equation approach [113,114] to computing flat FLRW cosmological wavefunction coefficients to loop diagrams. The result for the phase of the UV-divergent term is  \narg ( ψ ( L =1) n,φ ( n +4) / 2 ) = -nπ (2 + δ ) 4 + π N , (7.99)  \nwhich is again compatible with our phase formula. Note that the phase for n = 4 matches the phase of (7.98) because the phase formula depends only on the external legs and not the internal structure of the Feynman-Witten diagram.', "vi. Massless 1 -loop coefficient in non-integer dimension d = 3 + δ , ψ ( L =1) 2 ,ϕ ' 3": "As previously discussed, in cases with a logarithmic UV-divergence, one can renormalise these using dim-reg. In dim-reg, the logarithmic UV-divergence turns into a 1 /δ divergence (for IR-finite 1-loop diagrams), which can be canceled by expanding the phase using the simple fact that any generic complex number A can be expressed as A = | A | e i arg( A ) . For IR-finite 1-loop diagrams we thus find  \nlim δ → 0 [ | ψ ( L =1) | e i arg( ψ ( L =1) ) ] ∼ 1 δ (1 ± iπδ + O ( δ 2 )) = 1 δ ± iπ + O ( δ ) . (7.100)  \nThis phenomenon found in [111], where the authors computed a parity-odd contribution to the scalar trispectrum using the in-in formalism. We have demonstrated that this is in fact a generic feature of UV-divergent loop diagrams and will be explored further in [67].  \nIn [97], the 1-loop four-point wavefunction coefficient ψ ( L =1) 2 ,ϕ ' 3 for the interaction involving three massless scalars, generated via a boost-breaking derivative interaction ϕ ' 3 in non-integer d = 3+ δ -dimensions (which rescales the conformal dimension of the massless scalar ∆ = d/ 2 + 3 / 2 = (6 + δ ) / 2), was computed with an incorrect phase that will be corrected in [72]. For this particular Feynman-Witten diagram one finds  \nψ ( L =1) 2 ,ϕ ' 3 = H 2 S 1+ δ (2 π ) 3+ δ k 3+ δ ( -iH ) δ I ( δ ) , (7.101)  \nwhere S d -2 is the surface area of the ( d -2)-dimensional unit sphere (i.e. S 1 = 2 π ) and I ( δ ) is a dimensionless quantity with no contribution to the phase defined in [97]. As we stated in Sections 7.2 and 7.4 the phase formula still holds for the case of derivative interactions; the phase is also important for the consistency of the cutting rules derived in [97].", 'vii. Graviton 2-pt function in integer dimension ψ ( L ) 2 = ⟨ T ij ( k ) T lm ( -k ) ⟩': 'This graviton 2-point function in cosmology is equivalent to calculating the 2-point function of the stress tensor in dS/CFT. Our phase rule also gives the correct prediction in this case. The answer depends on the dimension d and loop order L as follows:  \n- · In even d +1-spacetime dimensions, ⟨ T ij ( k ) T lm ( -k ) ⟩ ∈ R to ALL loop order;\n- · In odd d + 1-spacetime dimensions, ⟨ T ij ( k ) T lm ( -k ) ⟩ ∈ i R at tree level and even loop order (i.e. when L ∈ 2 Z );\n- · In odd d +1-spacetime dimensions, ⟨ T ij ( k ) T lm ( -k ) ⟩ ∈ R at odd loop order (i.e. when L ∈ 2 Z +1).  \nHence for the case of d = 2 we expect the central charge, which is equivalent to the coefficient of the 2-point stress tensor correlator, to be imaginary at tree level and real at 1-loop order and so on. This matches what has been found in the literature, for a list of previous results see Section 1.3.', '8 Discussion': "In this paper we have provided a new perspective on the CPT theorem, which naturally enables us to make converse statements. Our analysis highlights the profound interplay between Unitarity, Lorentz Invariance, and the discrete symmetries. By demonstrating that any two of the Z 2 symmetries in the set { RR , CRT , 180 · } imply the third, we have established a more unified and fundamental understanding of these fundamental principles, in both flat and de Sitter spacetimes.  \nImportantly, as shown in Section 5, the RR symmetry alone is often enough to ensure full Unitarity over a codimension-0 domain of generic couplings in many field theories. This means that assuming CRT together with covariance is often sufficient to obtain a physically realistic QFT. This is very convenient from a practical point of view, since the detailed statement of Unitarity in Lorentzian spacetimes is quite finicky - for example it often requires fine-tuning the measure factors in the path integral - if you try to impose it directly. This perspective is even more valuable in cosmology, where the dynamical nature of the spacetime makes it tricky to use the (manifestly unitary but not manifestly covariant) Hamiltonian formalism.  \nIn the cosmological context, we used discrete symmetries to derive non-perturbative constraints on both cosmological correlators, and wavefunction coefficients. This allowed us to extend the tree-level results for Hermitian Analyticity [13, 14, 54]. Our non-perturbative RR result requires analytically continuing the momenta k and the bulk Weyl factor Ω. We also derived a somewhat more explicit perturbative statement of the result, at arbitrary loop order, which like [13,14,54] involves an analytic continuation of the momenta k alone. This result holds for all flat FLRW cosmologies.  \nIn the more restrictive context of Poincar'e de Sitter, one can instead invoke CRT by analytically continuing the η coordinate. These results reveal that the CRT constraint is powerful enough to determine the phase of wavefunction coefficients at future infinity. Surprisingly, this does not require any analytic continuation or comparison to past infinity; the constraints can just be read off directly at the level of the wavefunction coefficients.  \nIt should be noted, however, that in d = even spatial dimensions, the phase does depend on whether the number of bulk loops is even or odd. Also, in any dimension, when the bulk theory has a logarithmic UV or IR divergence, the associated finite term gets an imaginary shift relative to the coefficient of the log divergence. Most of our equations have been written for general real dimension d , making it easy to apply our results to calculations using dimensional regularisation.  \nOur phase rule has important implications for the dS/CFT correspondence, providing a new tool to determine the phases of correlators in the holographic dual conformal field theory, thereby advancing our understanding of quantum gravity in cosmological settings. As dS/CFT theories are non-unitary-in the sense that the boundary theory, thought of as a Euclidean field theory, does not itself satisfy a boundaryRR , they do not satisfy the reality conditions of normal unitary theories. It is therefore critical to identify  \nthe (presumably equally constraining) properties that a dS/CFT must satisfy to be dual to a unitary bulk. This article provides substantial progress towards this goal, but further work is needed to clarify what this constraint means from the perspective of the boundary field theory. It would also be useful if this reality constraint can be promoted to a more restrictive positivity constraint on dS/CFT, similar to how Reflection Positivity is more restrictive than Reflection Reality in unitary theories. Put another way, our reality conditions are necessary for bulk unitarity, but it would be even nicer to identify a necessary and sufficient condition for bulk unitarity.  \nOur CRT results also have some applicability to inflation cosmology, due to the slow roll approximation which makes the spacetime approximately described by Poincar'e de Sitter. The constraints from CRT are compatible with a breaking of de Sitter isometries (namely the breaking of de Sitter boosts but preservation of Discrete Scale Invariance) during inflation, so long as CRT still manifests as a symmetry of the local Lagrangian, as discussed in Section 4. However, in general the existence of a nonzero spectral tilt n s -1 (related to the slow-roll parameter) implies that this form of CRT is slightly broken, which should lead to small corrections to our CRT result for wavefunction coefficients.  \nThere are a number of other directions in which this work can be usefully extended. Our results lead fairly directly to a no-go theorem for certain specific cosmological correlators of bosonic fields [67] derived from the phase formula in this paper. An interesting direction which should certainly be explored is using a similar formalism to constrain spinors using discrete symmetries, which is yet to be done in the cosmology literature. To derive such constraints it will be necessary for us to use projective representations of the group Z 2 × Z 2 with the appropriate double cover of the group for half-integer spin fields being the dihedral group D 4 . Further work should also certainly be done on extending the analysis of the boundary wavefunction coefficients to the case of heavy fields in the principal series (discussed by [55-64]). In this case, the two dimensions ∆ = d/ 2 ± iν correspond to equally-leading and oscillatory modes as η → 0 -, leaving to subtle questions about the best way to define the boundary conditions at I + . Presumably, the same analytic continuation in η which led to the phase rule, will now lead to an enhancement of one mode over the other by an amount exponential in ν .  \nAnother interesting direction would be to see the implications of our discrete symmetry constraints and the phase rule for the recently proposed differential equation method for computing wavefunction coefficients in general flat FLRW cosmologies [113,114]. It might also be worth seeing if there is a phase rule for the de Sitter S-matrix, constructed along the lines of [115,116], or to the in-out formalism of [101]. It could also be that our phase rule provides useful constraints on flat space amplitudes, via the fact that going on the total 'energy' pole k T → 0 (which is equivalent to going to the infinite past η →-∞ ), i.e. the 'flat-space' limit, the Bunch-Davies vacuum looks like the flat-space vacuum.  \nFinally, it should be noted that in the derivation of our phase rule, we have implicitly assumed that the de Sitter vacuum is stable, and hence not subject to decay. In the string theory community, it is generally believed that de Sitter vacua in the string landscape will always be metastable. The probability that no vacuum decay event occurs anywhere, should fall off as an exponential of the total elapsed spacetime volume at late times. Although this correction looks badly divergent, it seems to have the right form to be renormalised by a d -dimensional cosmological constant counterterm in log Z CFT . However, this counterterm has a sign which is real, unlike the usual holographic counterterms in dS/CFT which are imaginary [44, 48, 71, 117]. As vacuum decay would lead to branches of the wavefunction disappearing into other possibilities, the assumption that vacuum decay does not occur is effectively a nonunitary postselection rule. Thus, we do not expect that our phase rule is necessarily valid for such processes. It would be interesting to study such vacuum transitions in more detail.", 'Acknowledgements': "It is a pleasure to thank Santiago Agu'ı Salcedo, Gon¸calo Araujo Regado, Dionysios Anninos, Tarek Anous, Nima Arkani-Hamed, Paolo Benincasa, Calvin Chen, Chandramouli Chowdhury, Nick Cooper, Mang Hei Gordon Lee, Hofie Sigridar Hannesdottir, Daniel Harlow, Ted Jacobson, Rohit Kalloor, Rifath Khan, Juan Maldacena, Ciaran McCulloch, Prahar Mitra, Sebastian Mizera, Enrico Pajer, Guilherme Pimentel, Bharathkumar Radhakrishnan, Adel Rahman, Muthusamy Rajaguru, Alan Rios Fukelman, Vladimir Schaub, Vasudev Shyam, David Stefanyszyn, Leonard Susskind, Marija Tomaˇsevi'c, Xi Tong, Erik Verlinde, Tom Westerdijk, Edward Witten, and Yuhang Zhu for useful discussions. AT would like to especially emphasise the importance of the initial collaboration with Carlos Duaso Pueyo, who was the first to encourage him to continue pursuing this research program.  \nHG was supported jointly by the Science and Technology Facilities Council through a postgraduate studentship and the Cambridge Trust Vice Chancellor's Award. HG is supported by a Postdoctoral Fellowship at National Taiwan University funded by the Ministry of Education (MOE) NTU-112L40001. AT is supported by the Bell Burnell Graduate Scholarship Fund and the Cavendish (University of Cambridge). AT and AW are supported by the AFOSR grant FA9550-19-1-0260 'Tensor Networks and Holographic Spacetime'. AW was also supported by the STFC grant ST/P000681/1 'Particles, Fields and Extended Objects', an Isaac Newton Trust Early Career grant, and NSF PHY-2309135 to the Kavli Institute for Theoretical Physics (KITP). 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