# Magnetic tuning of tunnel coupling between InAsP double quantum dots in InP
nanowires

Jason Phoenix  National Research Council Canada, Ottawa, Ontario, K1A0R6,
Canada  Department of Physics and Astronomy, University of Waterloo, Waterloo,
Ontario, Ontario, NL2 3G1, Canada  Marek Korkusinski  National Research
Council Canada, Ottawa, Ontario, K1A0R6, Canada  Dan Dalacu  National Research
Council Canada, Ottawa, Ontario, K1A0R6, Canada  Philip J. Poole  National
Research Council Canada, Ottawa, Ontario, K1A0R6, Canada  Piotr Zawadzki
National Research Council Canada, Ottawa, Ontario, K1A0R6, Canada  Sergei
Studenikin  National Research Council Canada, Ottawa, Ontario, K1A0R6, Canada
Department of Physics and Astronomy, University of Waterloo, Waterloo,
Ontario, Ontario, NL2 3G1, Canada  Robin L. Williams  National Research
Council Canada, Ottawa, Ontario, K1A0R6, Canada  Andrew S. Sachrajda  National
Research Council Canada, Ottawa, Ontario, K1A0R6, Canada  Louis Gaudreau
<EMAIL_ADDRESS>National Research Council Canada, Ottawa,
Ontario, K1A0R6, Canada

###### Abstract

We study experimentally and theoretically the in-plane magnetic field
dependence of the coupling between dots forming a vertically stacked double
dot molecule. The InAsP molecule is grown epitaxially in an InP nanowire and
interrogated optically at millikelvin temperatures. The strength of interdot
tunneling, leading to the formation of the bonding-antibonding pair of
molecular orbitals, is investigated by adjusting the sample geometry. For
specific geometries, we show that the interdot coupling can be controlled in-
situ using a magnetic field-mediated redistribution of interdot coupling
strengths. This is an important milestone in the development of qubits
required in future quantum information technologies.

Epitaxial quantum dots (QDs) embedded in nanowire waveguides are ideal sources
of single and entangled photons due to the high collection efficiency and
emission line purity which can be achieved with these devices [1, 2, 3, 4].
Additionally, this architecture has the potential to form the building blocks
of quantum information processors by coupling quantum dots in series within a
nanowire. Quantum dot molecules with clear signatures of molecular bonding and
antibonding states have been demonstrated, in which the carrier population can
be tuned utilizing the quantum-confined Stark effect [5, 6]. These optically
active quantum dots are also very promising candidates for quantum networking
units, because they can transfer the quantum information encoded in photonic
qubits to solid state qubits and process this information in coupled quantum
dot circuits [7, 8, 9].

Controlling the tunnel coupling between the dots is a key feature needed to
appropriately tune and perform quantum gates between qubits. In
electrostatically defined quantum dots, for example, inter-dot tunnel coupling
can be achieved via electrical gates designed for this purpose and linear
arrays of up to 9 qubits have been realized [10]. In epitaxial quantum dots,
tunnel coupling is determined by the distance separating the quantum dots,
which after the growth process can’t be changed. In this letter, we
demonstrate tunabilty of inter-dot coupling by applying a magnetic field
perpendicular to the dot stacking direction. We first perform optical
magnetospectroscopy of InAsP double quantum dots (DQDs) in InP nanowires and
identify an inverse power law which governs the energy difference between the
s-shell emission from each dot as a function of the inter-dot distance.
Emission energies are affected by differences in dot composition and strain,
while coupling between dots is determined at the growth stage by the barrier
thickness separating them. We will demonstrate, however, that we can tune, for
specific states, the emission energy difference on-demand over a range of
approximately 1 meV by applying a magnetic field parallel to the plane of the
quantum dots (i.e. Voigt geometry). As we will show, this energy shift is not
possible without quantum-mechanical coupling between dots, and we interpret
this result as an indication that magnetic field tuning of inter-dot tunnel
coupling is occurring due to the quantum analogue of the classical Lorentz
force.

The quantum dots used in this study are InAsP segments in a wurtzite crystal
phase InP nanowire, grown using selective area vapour-liquid-solid epitaxy in
a chemical beam epitaxy (CBE) system [3, 11]. The growth is catalysed by
position and size controlled gold nanoparticles on an InP (111)B substrate.
The size of the catalyst sets the diameter of the dot, $D$, and is defined
using electron-beam lithography. Diameters between 20 and 24 nm are used in
this study. The InAsP segments are grown by injecting AsH3 in the place of PH3
into the CBE chamber for 3 seconds during the InP nanowire growth producing
dots with thicknesses of $\sim$ 5 nm. For double dot samples (Fig. 1a),
different inter-dot spacers, $L$, are obtained in the same growth by virtue of
a nanowire diameter-dependent growth rate [11].

Figure 1: (a) Transmission electron microscopy image of a double dot nanowire
where the spacer was grown for 5 seconds. White arrows indicate dots, orange
arrows indicate inter-dot distance, $L$. (b) Normalized emission spectra of
three double-dot wires for three different inter-dot distances. The low-energy
s-shell peaks have been aligned to emphasize the relative energy of exciton
emissions for these three cases. The s-shell X- states are highlighted by
light blue boxes.

To enable efficient optical measurements the nanowires are further clad with a
$\sim$ 100 nm thick InP shell using growth conditions set to give a base
diameter of $\sim 220$ nm and a tip tapered to $\sim 20$ nm. The thicker shell
at the base supports the fundamental HE11 waveguide mode that provides optimal
coupling to the dot emission. The taper improves the coupling of the emitted
light to the lensed fibre used in the spectroscopy measurements [12, 1], i.e.
the taper controls the numerical aperture of the source.

A direct measurement of the inter-dot separation, $L$, using electron
microscopy techniques is not possible for clad nanowires. To determine $L$ we
instead use the diameter-dependent non-linear growth rate model developed in
Ref. 11. This approach only provides an estimate of the value of $L$ due to
($i$) a variation in nanowire diameters of $\pm 2$ nm for nominally identical
nanowires [13], ($ii$) a diameter-dependent incubation time to initiate the
nanowire growth, and ($iii$) a reservoir effect which delays the onset of the
spacer growth when switching from InAs to InP [14]. We assume a 5-second delay
to initiate the spacer growth and neglect the diameter-dependent incubation
time. The 3 double dot samples studied (DQD1, DQD2, DQD3) have dot diameters
$D_{1}=20$ nm, $D_{2}=22$ nm and $D_{3}=24$ nm. The first dot is incorporated
into the InP nanowire after growing the base for 905 seconds and the second
dot after growing a 15-second InP spacer on top of the first dot. Using the
assumptions above, we estimate spacer thicknesses of $L_{1}=9.5\pm 2$ nm,
$L_{2}=7\pm 2$ nm and $L_{3}=3\pm 2$ nm for the three samples.

Our experiment uses a 780 nm wavelength excitation laser coupled into the 1%
port of a 99-1 fibre beam splitter. The splitter’s input port is connected to
a 780HP single-mode fibre that runs inside a dilution refrigerator (base
temperature of 10 mK), where a lensed fibre focuses the light on the nanowire
sample. The lensed fibre also collects the emission from the chosen individual
nanowire, and the collected light is directed by the beam splitter to a
spectrometer for analysis. X, Y, Z piezo position controllers are used to
select each nanowire. The sample resides in the center of an 8 T
superconducting magnet in the Voigt configuration. Laser power levels at the
sample range from 1 nW to 1 $\mu$W in our experiments. See the Supplementary
Material for more information on this setup [15].

The high crystal purity of our nanowires results in extremely narrow
photoluminescence (PL) lines from the quantum dots, approaching the resolution
of the spectrometer ($\sim$0.024 nm). At the pumping levels used the samples
exhibit emission spectra characterised by the lowest-energy (s-shell) exciton
complexes: the neutral exciton (X), the negatively-charged exciton (X-) and
the neutral biexciton (XX). Emission lines are identified primarily through
power-dependence measurements [16, 2, 17, 3] and analysis of their behaviour
in a magnetic field [15]. Tests were initially conducted on single-dot samples
to observe the typical magnetic field dependence of emission lines associated
with each s-shell exciton complex [15]. These single-dot studies were then
used to establish a baseline for comparison with the double dots to determine
if they are coupled or not. Double-dot samples which display behaviour similar
to single dot structures can be assumed to have no inter-dot coupling, while
observed differences are assumed to arise from interactions between dots.

Figure 1(b) shows the s-shell emission spectra of three separate DQD samples
with different inter-dot distances, plotted relative to the lowest s-shell X-
peak. Two distinct sets of peaks are observed, each associated with one of the
quantum dots, and we refer to each set as either as the ‘low-energy’ or ‘high-
energy’ dot based on their relative s-shell emission energies. The figure
shows an increase in the energy difference between the ground state emission
of the two quantum dots (indicated by the light square boxes in each trace) as
$L$ is decreased. There are two mechanisms that result in a splitting of the
emission energies: a difference in the physical properties of the dots such as
composition and size, and a coupling between the dots. In this nanowire
system, we typically observe an energy difference between nominally identical
dots even for dots separated by $L>100$ nm where no coupling is expected. This
non-degeneracy is associated with a difference in As content between the two
dots[18] and in the limit of large $L$, we expect the emission energy
difference to be solely due to this difference in As content. With decreasing
$L$ we expect an increasing contribution due to tunnel coupling.

In coupled systems[19], the strengthening of the inter-dot coupling (the
increase in energy difference between the s-shell energies of the two dots,
$\Delta E_{S}$) can be described phenomenologically using a 1/$L^{3}$
dependence[20, 17]. For our data we obtain:

$\Delta E_{S}=\frac{33.0\times 10^{3}\text{
meV}\cdot\text{nm{\hbox{^3}}}}{(L+4.88\text{ nm})^{3}}+27.0\text{ meV}$ (1)

Since $L$ is measured from the edge of each dot, the offset in inter-dot
separation (4.88 nm) is a result of the finite thickness of the dots, which is
roughly 4.5 nm. The energy offset of 27 meV is included to account for the
difference in composition between the two dots within a nanowire which
produces an additional splitting as discussed above.

These data alone are not conclusive proof that inter-dot coupling is
responsible for the majority of the detuning as dot separation decreases. We
therefore provide more direct evidence of quantum mechanical coupling through
magneto-PL studies in the Voigt configuration. The magneto-PL of double dot
samples reveals the combined effects of the field strength and inter-dot
separation on the PL spectrum. Figure 2 compares the PL spectra of the high-
energy and low-energy dots as a function of magnetic field for samples DQD1
and DQD2 with inter-dot distances of $L_{1}=9.5$ nm (Fig. 2a) and $L_{2}=7$ nm
(Fig. 2b), respectively. DQD1 ($L_{1}=9.5$ nm) exhibits both Zeeman splitting
and diamagnetic shifts, and the spectrum of each dot is qualitatively similar
to that of a single-dot sample (see Supplementary Material [15]). The large
separation between the dots reduces the tunnel coupling between them to the
degree that they effectively behave as independent systems.

Figure 2: Magnetic field dependence of the emission spectra of (a) DQD1 with
inter-dot spacing of $L_{1}=9.5$ nm and (b) DQD2 with $L_{2}=7$ nm. Magnetic
decoupling of the high-energy (HE) and low-energy (LE) dots is observable in
(b) as a downward shift in the energy of the HE state.

Similarly, the low-energy QD in DQD2 ($L_{2}=7$ nm) shows a magnetic field
dependence similar to that of a single-dot nanowire. In contrast, the high-
energy QD in this sample reveals a downward shift in energy with increasing
B-field, a clear signature of a modification of the interdot coupling, which
is the main finding of this work. We interpret this energy shift as a quantum
analogue of the classical Lorentz force inducing an effective decoupling
between the QDs. In the Voigt configuration the magnetic field is applied
perpendicularly to the tunneling direction, and as carriers tunnel between
dots, the Lorentz force alters their trajectories, leading to a redistribution
of inter-dot coupling strengths. This concept is schematically shown in Fig.
3(a). The decoupling mechanism reduces the energy difference between the
emissions of the QDs from 47.5 meV to 46 meV which causes an effective
increase in inter-dot distance from $L_{2}=7$ nm to 10 nm.

We now perform a more quantitative analysis of the emission spectra of the
coupled quantum dot molecule by approximating the double-dot confinement with
a simple, separable three-dimensional double-well potential [15]. The
Hamiltonian of the system is written as
$\hat{H}=\hat{T}+\hat{V}_{l}(x,y)+\hat{V}_{v}(z)$, with
$\hat{T}=\hat{p}^{2}/2m$ being the kinetic energy operator of the particle
with mass $m$. The lateral confinement $\hat{V}_{l}(x,y)$ of either dot is
approximated by a two-dimensional parabolic potential with frequency
$\hbar\Omega_{e}=30$ meV for the electron, and $\hbar\Omega_{h}=15$ meV for
the hole [21]. Along the nanowire axis we deal with a one-dimensional double-
well potential $\hat{V}_{v}(z)$ with equal well widths (dot heights) $H=4.5$
nm (effectively accounting for interface effects). The difference in dot
composition, however, means the confinement depths are such that
$V_{1}>V_{2}$, where 1 (2) corresponds to the low (high) energy QD. We
calculate the single-particle energies and wave functions in this 3D
confinement numerically. Utilizing the effective masses for the electron,
$m_{e}=0.03$ $m_{0}$, and the hole, $m_{h}=0.06$ $m_{0}$, and assuming an
exciton binding energy of $25$ meV [21], we can find the well depths $V_{1}$
and $V_{2}$ by fitting the model positions of the emission maxima from either
dot to the measured results of uncoupled dots, as in Fig. 2(a) at the magnetic
field $B=0$ T. We obtain barrier height potentials $V_{1}^{e}=239$ meV and
$V_{2}^{e}=203$ meV for the electron, and $V_{1}^{h}=119.5$ meV and
$V_{2}^{h}=101.5$ meV for the hole, counting from the band edges of the InP
barrier material.

With the model fully parametrized, we now compute the energy levels of the
confined carriers as a function of the inter-dot distance. In Fig. 3(b) we
show the bonding (black lines) and antibonding (red lines) states of an
electron, including the three lowest shells, originating from the harmonic
lateral confinement. We detect a significant tunnel coupling for inter-dot
distances below $10$ nm, and a rich structure of level crossings arising from
the opposite shifts in levels of different vertical symmetry (bonding or
antibonding). Further, by performing the calculation for both types of
carriers, we may plot the energy difference between the emission maxima of
excitons built on the bonding and antibonding quantum molecular states (Fig.
3c). By taking the experimentally measured energy gaps and placing them on the
modelled curve in Fig. 3(c) we determine interdot distance of $9.5$ nm, $7$
nm, and $3$ nm corresponding to the samples DQD1, DQD2, and DQD3, as marked in
the figure. The model curve in Fig. 3(c) is in agreement with our empirical
fit of the data to a $1/L^{3}$ relation (Eq. 1), for which we used these
values of $L$.

Figure 3: (a) Schematic representation of the tunnelling direction of
electrons between dots for zero magnetic field, $t_{0}$ (solid line), and high
field, $t_{B}$ (dashed lines), in the Voigt configuration. Light blue shading
indicates the modified confinement region induced by the B-field. (b) Bonding
(black) and antibonding (red) quantum molecular states of an electron. The
blue arrows indicate the states whose coupling generates the emission line
configuration in panel (d) and Fig. 2(b). (c) The modelled energy gap between
s-shell emission lines of the high- and low-energy dots as a function of
inter-dot distance, $L$. The energy gaps from Fig. 1(b) are overlaid on the
model curve, allowing us to verify the values of $L$ for those samples. (d)
Emission line model showing the same shift in tunnel coupling strength as
observed in Fig. 1(b), without Zeeman splitting.

We then include the effects of the magnetic field in the Voigt configuration
into our model by introducing the generalized momentum operator for the
electron (hole) $\hat{\vec{\pi}}=\hat{\vec{p}}\pm\frac{e}{c}\vec{A}$, where
$e$ is the unit charge, $c$ is the speed of light, and $\vec{A}$ is the
magnetic vector potential [15]. With the choice $\vec{A}=B[0,0,y]$, $B$ being
the magnetic field magnitude, the kinetic energy expressed by this momentum
acquires two new terms: ($i$) a harmonic correction to the lateral confinement
in the $y$ direction, responsible for the diamagnetic shift of energy levels,
and ($ii$) an electron (hole) Hamiltonian term $\hat{H}_{yz}=\mp
i\hbar\Omega_{c}y\partial/\partial z$, where $\Omega_{c}=eB/mc$ is the
cyclotron energy. We include the two new terms in the basis of the zero-field
bonding and antibonding states. Owing to its bilinear form, the term
$\hat{H}_{yz}$ couples the vertical and horizontal wave function components
with opposite symmetry. The relevant coupling scheme is marked in Fig. 3(b)
with arrows. We find that the bonding $s$-shell orbital $B,s$ is coupled to
one of the antibonding $p$-shell orbitals $A,p$ (as marked by the green
double-arrow). On the other hand, the antibonding $s$-shell orbital $A,s$
couples with the bonding $p$-shell orbital $B,p$ (as marked by the blue
arrows). The strength of this coupling depends on the magnetic field, but also
on the energy gap between the relevant levels (large gap for the former pair,
small gap for the latter). As a result, the lowest-energy antibonding level,
$A,s$, will experience a strong shift towards the lower energies for inter-dot
distances larger than $\sim 7$ nm (i.e., past the crossing of the $A,s$ and
$B,p$ levels), but the level $B,s$ will not experience such a shift. This is
evident in Fig. 3(d), where we plot the calculated energies of the two lowest
orbitals as a function of the magnetic field. The lower-energy orbital
experiences only a diamagnetic shift, while the higher-energy excited state
exhibits both a downward shift arising from $\hat{H}_{yz}$ and an upward
diamagnetic shift. As these two levels form the two optically active excitonic
states, this non-monotonic shift is precisely what is seen in the experimental
spectra of Fig. 2(b).

We stress that the magnetic coupling mechanism does not renormalize the zero-
field tunneling elements directly. Indeed, these elements are relevant in the
creation of the bonding and antibonding orbitals of the same lateral symmetry
(e.g., $B,s$ and $A,s$). However, the appearance of the characteristic shift
as a function of $B$ guarantees that there is an overlap between the levels of
the two dots, and therefore constitutes a spectroscopic proof of the inter-dot
quantum-mechanical coupling. The decrease of the energy distance between the
peaks can be perceived as an effective renormalization of the bonding-
antibonding gap, however it is a result of a complex interaction of various
levels.

In summary, we have studied the magnetospectroscopy of epitaxial InAsP quantum
dots, demonstrating an inverse power relationship between the double-dot
energy difference and the inter-dot separation distance. Additionally, we have
provided conclusive proof of a technique for tuning the inter-dot tunnel
coupling. This technique does not require the fabrication of a series of
samples, but it instead relies on the magnetic field’s influence on the
electron and hole wave functions. In the sample DQD2 with $L_{2}=7$ nm, we
were able to achieve in-situ tuning of the tunnel coupling through the
application of a magnetic field in the Voigt configuration, as demonstrated by
a 1 meV reduction in emission energy detuning. Future experiments may involve
studies of the level structure of the dots by populating them with more
carriers, allowing access to higher excited states. The combination of the
Lorentz effect and the Voigt configuration have previously been used to map
out the wave functions of electrons confined in dots [22, 23], which we could
now observe as a function of the coupling strength. The small variations in
dot composition have important implications for experimental reproducibility
[21], so the difference in As content between dots should be addressed through
further refinements in the growth process. Overall, our experimental results
may have potential applications in quantum information technologies based on
epitaxial quantum dot qubits, especially where two-qubit gates need to be
performed and precise adjustment of the tunnel coupling is required. Moreover,
these nanowire DQDs can be used to generate entangled photon pairs for quantum
communication, and our results suggest a technique for adjusting the degree of
entanglement by tuning the inter-dot coupling.

##  .1 Acknowledgments

J.P. and S. S. thank the Natural Sciences and Engineering Research Council of
Canada for financial support.

##  .2 Conflict of interests

The authors declare they have no competing financial interests.

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# A look inside Feynman’s route to gravitation

M. Di Mauro  Dipartimento di Matematica, Università di Salerno, Via Giovanni
Paolo II,
Fisciano (SA), 84084, Italy
E-mail<EMAIL_ADDRESS>S. Esposito∗ and A. Naddeo∗∗ INFN, Sezione di
Napoli, C. U. Monte S. Angelo, Via Cinthia,
Napoli, 80125, Italy
∗E-mail<EMAIL_ADDRESS>
∗∗E-mail<EMAIL_ADDRESS>
###### Abstract

In this contribution we report about Feynman’s approach to gravitation,
starting from the records of his interventions at the Chapel Hill Conference
of 1957. As well known, Feynman was concerned about the relation of
gravitation with the rest of physics. Probably for this reason, he promoted an
unusual, field theoretical approach to general relativity, in which, after the
recognition that the interaction must be mediated by the quanta of a massless
spin-$2$ field, Einstein’s field equations should follow from the general
properties of Lorentz-invariant quantum field theory, plus self-consistency
requirements. Quantum corrections would then be included by considering loop
diagrams. These ideas were further developed by Feynman in his famous lectures
on gravitation, delivered at Caltech in 1962-63, and in a handful of published
papers, where he also introduced some field theoretical tools which were soon
recognized to be of general interest, such as ghosts and the tree theorem.
Some original pieces of Feynman’s work on gravity are also present in a set of
unpublished lectures delivered at Hughes Aircraft Company in 1966-67 and
devoted primarily to astrophysics and cosmology. Some comments on the relation
between Feynman’s approach to gravity and his ideas on the quantum foundations
of the fundamental interactions are included.

###### keywords:

Gravitation; Quantum field theory; Loop diagrams.

##  1 Introduction: a timeline

Among the many scientific interests that Feynman had in the 1950s and in the
1960s, a prominent place was taken by the understanding of the relation of
gravitation to the rest of physics,111“Next we shall discuss the possible
relation of gravitation to other forces. There is no explanation of
gravitation in terms of other forces at the present time. It is not an aspect
of electricity or anything like that, so we have no explanation. However,
gravitation and other forces are very similar, and it is interesting to note
analogies.” (Ref. 1, Vol. 1, Sec. 7-7); “My subject is the quantum theory of
gravitation. My interest in it is primarily in the relation of one part of
nature to another.” (Ref. 2, p. 697) and in particular the assessment of its
consistency with the uncertainty principle.222“…it would be important to see
whether Newton’s law modified to Einstein’s law can be further modified to be
consistent with the uncertainty principle. This last modification has not yet
been completed.”(Ref. 1, Vol. 1, Sec. 7-8) Feynman likely began to seriously
think about gravity in the early 1950s, just after completing his work on
quantum electrodynamics. This was attested by Murray Gell-Mann [3] in a paper
written for a Physics Today special issue devoted to Feynman’s
legacy,333Indeed here Gell-Mann remembered his visit to Caltech during the
Christmas vacation of 1954-55, the discussions with Feynman about quantum
gravity, and the fact that Feynman “had made considerable progress”. as well
as by Bryce S. DeWitt [4], in a letter to Agnew Bahnson written in November
1955 [5]. Feynman’s efforts began in a period in which general relativity,
after a stagnation which lasted about thirty years, was gradually emerging as
a mainstream research area, giving rise to a process known as the
_renaissance_ of general relativity [6].

A crucial event to consider, in order to reconstruct Feynman’s approach to
gravitation, is the 1957 Chapel Hill conference [4] on _The Role of
Gravitation in Physics_ , which was also pivotal to the renaissance of general
relativity. At that conference, the gravitational physics community delineated
the tracks along which subsequent work would develop in the subsequent
decades. The main threads were the following [7]: classical gravity, quantum
gravity, and the classical and quantum theory of measurement (as a bridge
between the previous two topics). In that conference Feynman’s work on
gravity, of which nothing had been published yet, was presented for the first
time, and put on paper in the records [4]. In fact these written records
testify that by the time of the conference he had already deeply thought, and
performed computations, about each of the above listed three topics, focusing
in particular on classical gravitational waves, on the arguments in favor of
quantum gravity from fundamental quantum mechanics, and finally on quantum
gravity itself. Thus, in this contribution, we take as starting point for our
reconstruction Feynman’s interventions at Chapel Hill and follow the
development of his work in the subsequent years, until the late sixties, where
he apparently lost interest in the subject.

The ultimate goal of Feynman’s work was the development of a quantum field
theory of gravitation, which led him to face deep conceptual as well as
mathematical issues, such as divergent integrals and the lack of unitarity
beyond the tree level approximation. This task accompanied Feynman for some
years, as stated in a letter [8] he wrote to Viktor F. Weisskopf in 1961 (“As
you know, I am studying the problem of quantization of Einstein’s General
Relativity. I am still working out the details of handling divergent integrals
which arise in problems in which some virtual momentum must be integrated
over”), as well as reported by William R. Frazer [9] in a short summary of the
talks given at the La Jolla conference, later in the same year.444The
International Conference on the Theory of Weak and Strong Interactions was
held in June 14-16, 1961, at the University of California, San Diego, in La
Jolla. Here, Geoffrey F. Chew gave his celebrated talk on the $S$-matrix [10],
while an afternoon session was devoted to the theory of gravitation, where
Feynman reported on his work on the renormalization of the gravitational
field, and recognized non-unitarity as the main difficulty, which was shared
also by Yang-Mills theory. A first comprehensive account of Feynman’s results
on quantum gravity can be found in the talk he gave in 1962 at the Warsaw
conference,555The International Conference on General Relativity and
Gravitation was held in Jabłonna and Warsaw in July 25-31, 1962, with Leopold
Infeld as the chairman of the local organizing committee. The discussion
focused on three main topics: the general properties of gravitational
radiation, the quantization of gravity and the exact solutions of the Einstein
field equations. whose written version was later published as a regular paper
in Acta Physica Polonica [2]. Further details were given by Feynman much
later, in a couple of papers [11, 12] published in 1972 in the Festschrift for
John A. Wheeler’s 60th birthday [13], which were written in a period in which
he was already deeply absorbed in the study of partons and strong
interactions.

Among the main sources which contribute to offer a clear account of Feynman’s
work on gravity issues, it is mandatory to include the famous Caltech Lectures
on Gravitation [14], delivered in 1962-63 and aimed to advanced graduate
students and postdocs. Finally, there is some unpublished material, included
in two sets of lectures, given in the 1960s at the Hughes Aircraft Company,
which only recently have been made available on the web.666See Ref. 15 for a
brief account of Feynman’s involvement in teaching at the Hughes Aircraft
Company. In particular, the 1966-67 set of lectures [16], which were devoted
to astronomy, astrophysics and cosmology, contains an introductory treatment
of general relativity, with an emphasis on applications to the main subjects.
While sharing many similarities with the above quoted Caltech lectures [14],
the Hughes treatment offers to the attentive reader several original points.
In those same years Feynman had succeeded in finding a new derivation of
Maxwell’s equations [17, 18], and a generalization of this approach to gravity
is suggested (but not pursued) in several places in the Hughes lectures on
astrophysics, as well as in the set of lectures given in the following year
and devoted to electromagnetism [19].

After outlining the main steps and sources which helped us to reconstruct the
full development of Feynman’s work on gravitation,777See Ref. 20 for a
comprehensive account of this work. in this contribution we focus on two key
issues: the formulation of quantum gravity as a quantum field theory of a
massless spin-$2$ field, the graviton, in whole analogy with quantum
electrodynamics, which is the content of Section 2, and the unitarity and
renormalization issues arising beyond the tree level approximation, presented
in Section 3. Our concluding remarks close the paper.

##  2 Gravity as a quantum field theory

Feynman’s strategy in approaching gravity was firstly outlined at Chapel Hill
conference in a series of critical comments (Ref. 4, pp. 272-276), in which a
non-geometric and field theoretical line of attack is put forward. His
starting point was an hypothetical, counterfactual situation, in which
scientists would discover the principles of Lorentzian quantum field theories
before general relativity.888In fact, in Ref. 4, p. 273, Feynman said:
“Instead of trying to explain the rest of physics in terms of gravity I
propose to reverse the problem by changing history. Suppose Einstein never
existed […]”. The main concern in such a situation would then be to include a
new force, the gravitational one, in the framework of quantum field theory.
This approach would be later completed in the first part of the Caltech
lectures [14]. Feynman’s reasoning was the following: on the basis of the
general principles of quantum field theory and of experimental results it is
possible to conclude that gravity, as any other force, has to be mediated by
exchanges of a virtual particle, which in this case is a massless neutral
spin-$2$ quantum, the graviton. Thus, by constructing a Lorentz invariant
quantum field theory of the graviton999The linear theory for a massless
spin-$2$ field and its massive counterpart was completely worked out by Markus
Fierz and Wolfgang E. Pauli in Ref. 21, on the basis of a previous work by
Paul A. M. Dirac [22], while iterative arguments similar to Feynman’s ones
were later put forward by Suraj N. Gupta [23] and Robert H. Kraichnan [24] in
an attempt to generate infinite nonlinear terms both in the Lagrangian and in
the stress-energy tensor. See for details Ref. 25. and by imposing certain
consistency requirements, full general relativity should be recovered.
Clearly, by following the same procedure for the spin-$1$ case Maxwell’s
equations are obtained (in this case it is much simpler, being the theory
linear). Such an approach testifies his ideas about fundamental interactions
as manifestations of underlying quantum theories [18], which were expressed by
him several times, for example in the Hughes Lectures[19]:

> I shall call conservative forces, those forces which can be deduced from
> quantum mechanics in the classical limit. As you know, Q.M. is the
> underpinning of Nature (Ref. 19, p.35).

Let us describe the steps in more detail. First of all, one has to establish
the spin of the mediating quantum. Both in the Caltech [14] and in the 1966-67
Hughes lectures [16], the choice of a spin-$2$ mediator was justified by the
observation that energy, which is the source of the gravitational force, grows
with the velocity. The same observation ruled out the possibility of a
spin-$0$ field, because the associated charge would decrease with the
velocity. This result can be traced back to an old argument by Einstein (never
published but recalled in Ref. 26, pp. 285-290), according to which the
vertical acceleration of a body would depend on its horizontal velocity, and
in particular would be zero for light, making light deflection impossible.
Once the spin of the graviton is established, one can easily construct the
linearized theory of the associated field, which is a massless spin-$2$ field.
Nonlinearity then comes into play because the graviton has to couple with
anything carrying energy-momentum, then also with itself, and this coupling
must be universal. The resulting nonlinear theory is general relativity.
General covariance and the geometric interpretation of general relativity are
finally recovered as a byproduct of the gauge invariance of the theory
101010See Ref. 14, p. 113.. For Feynman, this was only half of the whole
story. In fact, by pushing calculations beyond tree level, quantum gravity
effects would be taken into account. This was Feynman’s ultimate goal, i.e.
obtaining a quantum theory of gravity, which in this approach amounts to the
quantization of another field.

Let us describe how Feynman sketched the above procedure at the Chapel Hill
conference (Ref. 4, pp. 272-276). In whole analogy with electrodynamics, he
wrote down the following action:

$\displaystyle\int\left(\frac{\partial A_{\mu}}{\partial
x_{\nu}}-\frac{\partial A^{\nu}}{\partial x^{\mu}}\right)^{2}{\rm d}^{4}x+\int
A_{\mu}j^{\mu}{\rm d}^{4}x+\frac{m}{2}\int\dot{z}_{\mu}^{2}{\rm
d}s+\frac{1}{2}\int T_{\mu\nu}h^{\mu\nu}{\rm d}^{4}x$
$\displaystyle+\int(\textrm{second power of first derivatives of $h$}),$ (1)

where $h_{\mu\nu}$ is the new field associated with the graviton, i.e. a
symmetric second-order tensor field, which satisfies second order linear
equations of the kind:

$\displaystyle{h^{\mu\nu,\sigma}}_{,\sigma}-2{{\overline{h}^{\mu}}_{\sigma}}^{,\nu\sigma}=\overline{T}^{\mu\nu},$
(2)

where the bar operation is defined on a generic second rank tensor
$X_{\mu\nu}$ as:

$\displaystyle\overline{X}_{\mu\nu}=\frac{1}{2}\left(X_{\mu\nu}+X_{\nu\mu}\right)-\frac{1}{2}\eta_{\mu\nu}{X^{\sigma}}_{\sigma}.$
(3)

The equations of motion for particles also follow from the above action:

$\displaystyle
g_{\mu\nu}\ddot{z}^{\nu}=-\left[\rho\sigma,\mu\right]\dot{z}^{\rho}\dot{z}^{\sigma},$
(4)

where $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$, $\eta_{\mu\nu}$ is the Minkowski
metric, and $\left[\rho\sigma,\mu\right]$ are the Christoffel symbols of the
first kind.

However, when the field $h_{\mu\nu}$ is coupled to the matter according to
Eq.(4), the corresponding stress energy tensor $T_{\mu\nu}$ does not obey to a
conservation law, leading to a consistency problem. This happens at variance
with electromagnetism, where Maxwell equations guarantee conservation of the
current $j^{\mu}$. Instead here $T_{\mu\nu}$ does not take into account the
effect of gravity on itself, which requires nonlinearity.111111Feynman noticed
that nonlinearity was necessary in order to explain the precession of the
perihelion of Mercury (as discussed in Ref. 14, p. 75). Thus a suitable
$T_{\mu\nu}$ has to be found in order to satisfy the condition
$\partial_{\nu}T^{\mu\nu}=0$. The solution to this consistency issue121212See
for details Refs. 4, p. 274 and 14, pp. 78-79. can be obtained by adding to
the action a nonlinear third order term in $h_{\mu\nu}$, which leads to the
following equation for $T_{\mu\nu}$:

$\displaystyle
g_{\mu\lambda}T^{\mu\nu}_{\;\;\;\;\,,\nu}=-\left[\rho\nu,\lambda\right]T^{\rho\nu}.$
(5)

One can then go on to higher order approximations, until the procedure
converges. But finding the general solution of Eq. (5) is a really difficult
task in the absence of a standard procedure. Feynman’s idea was to look for an
expression for the $T_{\mu\nu}$, and hence for the action, that is invariant
under the following infinitesimal transformation of the whole tensor field
$g_{\mu\nu}$:

$\displaystyle g_{\mu\nu}^{\prime}=g_{\mu\nu}+g_{\mu\lambda}\frac{\partial
A^{\lambda}}{\partial x^{\nu}}+g_{\nu\lambda}\frac{\partial
A^{\lambda}}{\partial x^{\mu}}+A^{\lambda}\frac{\partial g_{\mu\nu}}{\partial
x^{\lambda}}.$ (6)

where the $4-$vector $A^{\lambda}$ is the generator. This is a geometric
transformation in a Riemannian manifold with metric, hence one can say that
geometry gives the metric $g_{\mu\nu}$. As such, in Feynman’s approach
geometry comes into play at the end and not at the beginning. By working out
calculations, the full nonlinear Einstein gravitational field equations are
obtained.

Lectures 3-6 of Feynman’s graduate course on gravitation (Ref. 14) contain all
the details of the above procedure. Interestingly, a proof is also given of
the ability of this field theory based formalism to reproduce key physical
effects of curved spacetime geometry. For instance, in Lecture 5 (Ref. 14, pp.
66-69) it is shown that, in the action of a scalar field, the time dilation
$t\rightarrow t^{\prime}=t\sqrt{1+\epsilon}$ exactly reproduces the effect of
a constant weak gravitational field described by the tensor
$g_{44}=1+\epsilon$, $g_{ii}=-1$, $i=1,2,3$. Incidentally such an effect plays
a pivotal role in producing the right result for the precession of Mercury’s
perihelion. Before moving to applications, Feynman devoted some lectures to
the discussion of the usual geometric approach to gravity and of its link with
the field theory based approach (Ref. 14, Lectures 7-10):

> Let us try to discuss what it is that we are learning in finding out that
> these various approaches give the same results (Ref. 14, p. 112).

Despite advocating one approach over another, Feynman was in fact intrigued by
the double nature of gravity, which has both a geometrical interpretation and
a field interpretation, and in Section 8.4 of Ref. 14 he recognized how an
explanation may be provided by gauge invariance. Indeed a viable procedure may
be established in order to obtain the invariance of the equations of physics
under space dependent variable displacements. This amounts to looking for a
more general Lagrangian, to be obtained by adding to the old one new terms,
involving a gravity field. The net result is a new picture of gravity, as the
field corresponding to a gauge invariance with respect to displacement
transformations.

Summing up, Feynman succeeded in obtaining the full nonlinear Einstein
equations by means of a consistency argument applied to a Lorentz invariant
quantum field theory of the graviton. According to John Preskill and Kip S.
Thorne [25], it is likely that he was completely unaware of the earlier work
by Gupta [23] and Kraichnan [24], which as mentioned had been developed along
a similar line of attack, but was still incomplete. So he would have developed
his approach independently, besides getting more complete results. A Lorentz-
invariant field theoretical approach to gravity would have been later pursued
also by Steven Weinberg [27, 28], albeit quite different from Feynman’s
one,131313Weinberg’s approach relied on the analyticity properties of
graviton-graviton scattering amplitudes, and it was quite more general, since
Weinberg actually proved that a quantum theory of a massless spin-2 field can
only be consistent if this field universally couples to the energy-momentum of
matter, hence the equivalence principle has to be obeyed. as well as by
Stanley Deser [29, 30, 31, 32]. Indeed Deser’s approach, while being similar
to Feynman’s one, was more elegant and general and led to completion the whole
program started with Kraichnan and Gupta. Finally, a rigorous and general
analysis of the relation between spin-$2$ theories and general covariance was
carried out by Robert M. Wald [33].

##  3 Fighting with loops: the renormalization of gravity

After obtaining the Einstein-Hilbert action and the full nonlinear Einstein
gravitational field equations, Feynman’s efforts were mainly directed towards
discussing quantum field theory issues beyond the tree level approximation,
i.e. loop diagrams, unitarity and renormalization.

To the best of our knowledge, these issues were publicly addressed for the
first time by Feynman in 1961 at the already mentioned La Jolla conference
[9], where nonlinearity was recognized as the very source of difficulty within
both gravitation and Yang-Mills theories. Indeed the sources of the
gravitational field are energy and momentum, and the gravitational field
carries energy and momentum itself. In the same way, the source of a Yang-
Mills field is the isotopic spin current, and the Yang-Mills field carries
isotopic spin itself. This means that both the gravitational field and the
Yang-Mills field are self-coupled, resulting in nonlinear field theories. A
difficulty that nonlinearity brings about is the fact that loop diagrams seem
to clash with unitarity.

As recalled in the introduction, Feynman’s results on quantum gravity can be
found in a report of the talk given at the 1962 Warsaw conference, later
published in Acta Physica Polonica [2],141414As pointed out by Trautman in
some recently published memories ([34], p. 406), the text of Feynman’s plenary
lecture became available too late to be included in the proceedings, therefore
it was published only in 1963 as a regular paper. with many details discussed
much later in the two 1972 Wheeler Festschrift papers [11, 12].

Also in this case Feynman followed his original strategy, leaving aside
quantization of space-time geometry, constructing a quantum field theory for
the graviton, and working out results at different perturbative orders. Since
the goal was the quantum theory, the Einstein equations and the corresponding
Lagrangian were assumed as a starting point, rather than derived from scratch.
The theory was coupled to a scalar field, and perturbative calculations up to
the next to leading order were pursued. This implied the inclusion of loop
diagrams, which make quantum corrections to enter the game. In Feynman’s own
words:

> I started with the Lagrangian of Einstein for the interacting field of
> gravity and I had to make some definition for the matter since I’m dealing
> with real bodies and make up my mind what the matter was made of; and then
> later I would check whether the results that I have depend on the specific
> choice or they are more powerful (Ref. 2, p. 698).

The metric was split in the following way:

$\displaystyle g_{\mu\nu}=\delta_{\mu\nu}+\kappa h_{\mu\nu};$ (7)

where the Minkowski metric is here denoted by $\delta_{\mu\nu}$ and $\kappa$
is a dimensionful coupling constant. Substituting (7) and expanding, the
Lagrangian for gravity coupled with a scalar field can be cast in the form:

$\displaystyle L$ $\displaystyle=$
$\displaystyle\int\left(h_{\mu\nu,\sigma}\overline{h}_{\mu\nu,\sigma}-2\overline{h}_{\mu\sigma,\sigma}\overline{h}_{\mu\sigma,\sigma}\right){\rm
d}\tau+\frac{1}{2}\int\left(\phi_{,\mu}^{2}-m^{2}\phi^{2}\right){\rm d}\tau$
$\displaystyle+\kappa\int\left(\overline{h}_{\mu\nu}\phi_{,\mu}\phi_{,\nu}-m^{2}\frac{1}{2}{h}_{\sigma\sigma}\phi^{2}\right){\rm
d}\tau+\kappa\int\left(hhh\right){\rm
d}\tau+\kappa^{2}\int\left(hh\phi\phi\right){\rm d}\tau+...$

where the bar operation has been defined in (3) and a schematic notation has
been adopted for the highly complex higher order terms. The first two terms
are simply the free Lagrangians of the gravitational field and of matter,
respectively. Before considering radiative corrections the classical solution
of the problem was worked out, which involved the variation of Eq. (3) with
respect to $h$ and, then, to $\phi$, giving rise to the following equations of
motion with a source term:

$\displaystyle
h_{\mu\nu,\sigma\sigma}-\overline{h}_{\sigma\nu,\sigma\mu}-\overline{h}_{\sigma\mu,\sigma\nu}$
$\displaystyle=$ $\displaystyle\overline{S}_{\mu\nu}\left(h,\phi\right),$ (9)
$\displaystyle\phi_{,\sigma\sigma}-m^{2}\phi$ $\displaystyle=$
$\displaystyle\chi\left(\phi,h\right).$ (10)

A close inspection revealed that Eq. (9) was singular, so that Feynman was
forced to resort to the invariance of the Lagrangian under the transformation:

$\displaystyle
h_{\mu\nu}^{{}^{\prime}}=h_{\mu\nu}+2\xi_{\mu,\nu}+2h_{\mu\sigma}\xi_{\sigma,\nu}+\xi_{\sigma}h_{\mu\nu,\sigma},$
(11)

$\xi_{\mu}$ being arbitrary. This meant that the source $S_{\mu\nu}$ had to be
divergenceless in order to make Eq. (9) consistent. Finally, by making the
gauge choice $\overline{h}_{\mu\sigma,\sigma}=0$, the law of the gravitational
interaction of two systems by means of the exchange of a virtual graviton was
obtained. Feynman then went on computing other processes, such as an
interaction vertex coupling two particles and a graviton and the gravitational
analog of gravitational Compton effect (i.e. with a graviton replacing the
photon).

After these preliminary calculations, Feynman went to the next to leading
order approximation, thus encountering diagrams with closed loops:

> However the next step is to take situations in which we have what we call
> closed loops, or rings, or circuits, in which not all momenta of the problem
> are defined (Ref. 2, pp. 703-704).

He realized that working out closed loop diagrams required the solution of a
number of conceptual issues, and he succeeded in showing that any diagram with
closed loops can be expressed in terms of sums of on shell tree diagrams,
which is the content of his celebrated tree theorem (which was treated in
detail in Ref. 11). Further details on the statement of the tree theorem and,
in particular, on the nature of the proof for the one-loop case were given by
Feynman in the discussion section (Ref. 2, pp. 714-717), while answering some
related questions by DeWitt. But the main problem to face in carrying out one-
loop calculations was the lack of unitarity, due to the presence of
contributions arising from the unphysical longitudinal polarization states of
the graviton, which did not cancel as they should. Following a suggestion by
Gell-Mann,151515“I suggested that he try the analogous problem in Yang-Mills
theory, a much simpler nonlinear gauge theory than Einsteinian gravitation.”
(Ref. 3, p. 53). Feynman considered the simpler Yang-Mills case (his results
in this case were summarized in Ref. 12) and found the same pathological
behavior:

> But this disease which I discovered here is a disease which exist in other
> theories. So at least there is one good thing: gravity isn’t alone in this
> difficulty. This observation that Yang-Mills was also in trouble was of very
> great advantage to me. […] the Yang-Mills theory is enormously easier to
> compute with than the gravity theory, and therefore I continued most of my
> investigations on the Yang-Mills theory, with the idea, if I ever cure that
> one, I’ll turn around and cure the other (Ref. 2, p. 707).

The solution to this issue was obtained by expressing each loop diagram as a
sum of trees and then computing the trees. This worked even if the process of
opening a loop by cutting a graviton line implies the replacement of a virtual
graviton with a real transverse one. Finally, in order to guarantee gauge
invariance the sum of the whole set of tree diagrams corresponding to a given
process has to be taken.

The same results, according to Feynman, could be obtained by direct
integration of the closed loop. In the last case a mass-like term has to be
added to the Lagrangian to avoid singularity but at the price of breaking
gauge invariance. At the same time a contribution has to be subtracted, which
is obtained by making a ghost particle (with spin-$1$ and Fermi statistics) to
go around the loop and artificially coupled to the external field. In this way
both unitarity and gauge invariance would be restored. This procedure was
worked out also for Yang-Mills theory, but in that case the ghost particle has
spin-$0$ [12].

Once successfully solved the one-loop case, Feynman’s efforts pointed toward a
further generalization of the above procedure to two or more loops:

> Now, the next question is, what happens when there are two or more loops?
> Since I only got this completely straightened out a week before I came here,
> I haven’t had time to investigate the case of 2 or more loops to my own
> satisfaction. The preliminary investigations that I have made do not
> indicate that it’s going to be possible so easily gather the things into the
> right barrels. It’s surprising, I can’t understand it; when you gather the
> trees into processes, there seems to be some loose trees, extra trees (Ref.
> 2, p. 710).

But, in Feynman’s words, preliminary attempts seem to suggest that novel
difficulties enter the game when dealing with two or more loops, as also
mentioned in the last of his published Lectures on Gravitation (Ref. 14,
Lecture 16, pp. 211-212). Here, once again, he recognized in the lack of
unitarity of some sums of diagrams the main source of the observed
pathological behavior and pointed out that a similar feature was shared also
by Yang-Mills theory. Finally, while hinting at the problem of finding ghost
rules for high order diagrams, he argued in favor of the non-renormalizability
of gravity as a consequence of these difficulties:

> I do not know whether it will be possible to develop a cure for treating the
> multi-ring diagrams. I suspect not – in other words, I suspect that the
> theory is not renormalizable. Whether it is a truly significant objection to
> a theory, to say that it is not renormalizable, I don’t know (Ref. 14,
> Lecture 16, pp. 211-212).

It is not clear whether Feynman was suggesting a link between non-unitarity
and non-renormalizability issues. But in any case Feynman’s results played a
prominent role in the development of gauge theory and quantum gravity.
Feynman’s rules for ghosts were later generalized to all orders by DeWitt [35,
36, 37], while Ludvig D. Faddeev and Viktor N. Popov [38] derived them in a
much simpler way, by means of functional integral quantization, setting the
standard for all subsequent work in the field. In particular, DeWitt proved
that Yang-Mills theory and quantum gravity are in fact unitary at two [35] and
arbitrarily many loops [36, 37]. However, while Yang-Mills theory was later
shown to be renormalizable (cf. Refs. 39-42), gravity presented divergences
which could not be renormalized (cf. Refs. 43-46), confirming Feynman’s
suspect. It should be mentioned that, in subsequent years, modified theories
of gravity, characterized by an action quadratic in the curvature, have been
put forward. Unlike ordinary general relativity, these theories are
renormalizable but not unitary [47].

It is worth mentioning that, unlike most of his contemporaries, Feynman did
not think about non-renormalizability as a signature of inconsistency of a
theory, as also recalled by Gell-Mann,161616“He was always very suspicious of
unrenormalizability as a criterion for rejecting theories” (Ref. 3, p. 53).
and claimed by Feynman himself in one of his last interviews, given in January
1988:

> The fact that the theory has infinities never bothered me quite so much as
> it bother others, because I always thought that it just meant that we’ve
> gone too far: that when we go to very short distances the world is very
> different; geometry, or whatever it is, is different, and it’s all very
> subtle ([48], p. 507).

In fact, within the modern view on quantum field theory, which was developed
in 1970s, non-renormalizability is considered only a signature of the fact
that the theory loses its validity at energies higher that a certain scale.
Nevertheless, one has an _effective field theory_ , which works and can be
useful to make predictions under that scale (interesting historical
discussions can be found in Refs. 49 and 50). This is true also for gravity
[51]. But, as remembered by John P. Preskill in a recent talk (Ref. 52, slide
37), although he anticipated this view, apparently Feynman was not really at
ease with it:

> I spoke to Feynman a number of times about renormalization theory during the
> mid-80s (I arrived at Caltech in 1981 and he died in 1988). I was surprised
> on a few occasions how the effective field theory viewpoint did not come
> naturally to him”. […] Feynman briefly discusses in his lectures on
> gravitation (1962) why there are no higher derivative terms in the Einstein
> action, saying this is the “simplest” theory, not mentioning that higher
> derivative terms would be suppressed by more powers of the Planck length.

As a further remark, let us notice that in the last years Feynman’s tree
theorem has spurred a renewed interest in researchers working in the context
of advanced perturbative calculations and generalized unitarity (cf. Refs. 53
\- 57).

##  4 Concluding remarks

In this paper we focused on Feynman’s contributions to the research in quantum
gravity, starting from his interventions at Chapel Hill conference in 1957 and
ending with the Wheeler festschrift papers. His approach was field theoretical
rather than geometric, reflecting his strong belief in the unity of Nature,
which is quantum at the deepest level. Quantization of gravity, according to
him, simply had to be considered as the quantization of another field, the
spin-$2$ graviton field. In this way full general relativity would be
recovered at leading order, while the inclusion of loop diagrams brought into
the picture a bunch of new difficulties. In this respect, Feynman’s struggle
against loops, while succeeding at one-loop order, failed with two- and
higher-loop diagrams. Nevertheless, his results triggered further efforts and
some tools he developed, such as the tree theorem, have recently become of
widespread use among people working on scattering amplitudes.

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