# Crystalline Vacuum as the Background of the Standard Model

L. M. Pismen  Technion – Israel Institute of Technology, Haifa 32000, Israel

###### Abstract

A multiscale expansion procedure reveals a crystalline vacuum state arising as
a result of resonant interactions among weak Planck-scale waves on spinor and
cospinor manifolds. Quarks are presented as elastic modulations of the vacuum
crystal. The complementing modes are identified with colors, and distinctions
between flavors are attributed to asymmetries of the vacuum lattice.
Repercussion of the theory include insights into the nature of quantum
uncertainty and dark matter and energy.

Introduction. The simple and beautiful structure of the Standard Model (SM)
suggests some relation to the structure of vacuum on the Planck scale, which
is inaccessible experimentally and can only be probed by its effects on the
observable fields on a long phenomenological scale. This letter explores a
mechanism of presenting SM fields as long-scale (“elastic”) modulations of a
Planck-scale crystalline structure.

The essential feature of long-scale equations describing envelopes of
underlying structures, common in macroscopic nonlinear physics ch93 ; lp06 ,
is their universality, which renders them independent of details of underlying
physics and totally determined by the type of the transition and applicable
symmetries. Although, contrary to a macroscopic situation, Planck-scale
physics is unknown, it is further assumed that the Einstein equations continue
to be valid on this scale, adjusted to a topology compatible with the SM
structure. The above mentioned resilience of long-scale expansions suggests
that the results might be insensitive to deviations from known physics on
short scales. The principal tool of the analysis is the multiscale expansion
called forth by the huge “desert” between the Planck and SM scales.

It has been long hoped, starting from the original (4+1)D Kaluza–Klein
scenario kk1 , that extra dimensions hold the key to the structure of
phenomenological fields localized on the 4D world brane. Compact extra-
dimensional manifolds inevitably suffered from transverse instabilities, which
were quelled to some extent in models of “large” extra dimensions large ;
Arkani ; Randall . Some theories interpreted the brane as a domain wall
separating distinct extra-dimensional vacua wall , a structure explored in
(4+1)D lattice computations Kaplan ; Blum ; others extended to multiple branes
and even to a crystalline structure of branes Kaloper . A single extra
dimension, originally meant to unify gravity with the electromagnetic field
has turned insufficient to lodge the growing panoply of particles and fields.
This prompted Pauli Pauli  to explore two extra dimensions accommodating SU(2)
gauge fields an a work that remained unpublished, as he had not arrived at
desired results. Lattice methods have been also applied to a (4+2)D model 6d ,
where the 4D world is interpreted as a vortex defect in 6D. This topology has
been explored in connection with SM phenomenology (Libanov  and later works by
the same group). The number of extra dimensions swells to six Scherk  or seven
Witten  in string theories.

Waves in spin-spaces. The mechanism to be presented here aspires to avoid
extra dimensions, but endows hidden 2D manifolds that _merge_ into the 4D
world with dynamics of their own. The prototype of such manifolds are _spin-
spaces_ , which feature as an efficient analytical tool in the Newman–Penrose
(NP) formalism NP ; PR . The common way of applying this formalism to weak
deviations from vacuum is to go down from a 4D configuration to spinor
equations, but the opposite direction will be followed here.

The starting point is the Minkowski null tetrad expressed in a standard way
through base spinors $o^{A},\,\iota^{A},\,o^{A^{\prime}},\,\iota^{A^{\prime}}$
defined on the two spinor manifolds. According to the NP formalism, the
components $R_{ABC^{\prime}D^{\prime}}$ of the Ricci and Weil spinors can be
expressed by solving a set of 18 equations having a common structure that
involves the intrinsic derivatives in the tetrad directions and spin
coefficients. Denoting the Minkowski spin bases as

$\eta_{(0)}^{A}=o^{A},\;\;\eta_{(1)}^{A}=\iota^{A},\;\;\eta_{(0)A}=o_{A},\;\;\eta_{(1)A}=-\iota_{A}.$
(1)

and the respective primed (cospinor) bases
$\eta{}_{(1)}^{A^{\prime}}=o^{A^{\prime}}$ etc., the intrinsic derivatives can
be rewritten in the compact form
$\nabla_{(ab^{\prime})}=\eta_{(a)}^{A}\eta_{(b^{\prime})}^{B^{\prime}}\nabla_{AB^{\prime}}$,
so that

$D=\nabla_{(00^{\prime})},\;\;\delta=\nabla_{(01^{\prime})},\;\;\delta^{\prime}=\nabla_{(10^{\prime})},\;\;D^{\prime}=\nabla_{(11^{\prime})}.$
(2)

Accordingly, the spin coefficients are denoted as
$\gamma_{(ab^{\prime}cd)}=\eta^{A}_{(c)}\nabla_{(ab^{\prime})}\eta_{A(d)}$,
which corresponds PR  (omitting parentheses) to

$\displaystyle\boldsymbol{\kappa}=\gamma_{00^{\prime}00},\;\;\boldsymbol{\varepsilon}=\gamma_{00^{\prime}10},\;\;\boldsymbol{\gamma^{\prime}}=\gamma_{00^{\prime}01},\;\;\boldsymbol{\tau^{\prime}}=\gamma_{00^{\prime}11}$
(3)
$\displaystyle\boldsymbol{\sigma}=\gamma_{01^{\prime}00},\;\;\boldsymbol{\beta}=\gamma_{01^{\prime}10},\;\;\boldsymbol{\alpha^{\prime}}=\gamma_{01^{\prime}01},\;\;\boldsymbol{\rho^{\prime}}=\gamma_{01^{\prime}11}$
(4)
$\displaystyle\boldsymbol{\rho}=\gamma_{10^{\prime}00},\;\;\boldsymbol{\alpha}=\gamma_{10^{\prime}10},\;\;\boldsymbol{\beta^{\prime}}=\gamma_{10^{\prime}01},\;\;\boldsymbol{\sigma^{\prime}}=\gamma_{10^{\prime}11}$
(5)
$\displaystyle\boldsymbol{\tau}=\gamma_{11^{\prime}00},\;\;\boldsymbol{\gamma}=\gamma_{11^{\prime}10},\;\;\boldsymbol{\varepsilon^{\prime}}=\gamma_{11^{\prime}01},\;\;\boldsymbol{\kappa^{\prime}}=\gamma_{11^{\prime}11}.$
(6)

The priming signs corresponds to the interchange of all 0 and 1 indices.
Complex conjugation is equivalent to priming/unpriming of all indices, so that
the respective spin coefficients will be further specified as
$\gamma_{(ab^{\prime}c^{\prime}d^{\prime})}=\eta^{A^{\prime}}_{(c^{\prime})}\nabla_{(ab^{\prime})}\eta_{A^{\prime}(d^{\prime})}$.
For example, the first equation of the NP system reads in this notation as

$\displaystyle\nabla_{00^{\prime}}\gamma_{10^{\prime}00}-\nabla_{10^{\prime}}\gamma_{00^{\prime}00}=|\gamma_{10^{\prime}00}|^{2}+\gamma_{01^{\prime}00}\gamma_{01^{\prime}00^{\prime}}$
(7)
$\displaystyle-\gamma_{00^{\prime}00}(\gamma_{00^{\prime}11}+2\gamma_{10^{\prime}10}+\gamma_{0^{\prime}1^{\prime}1^{\prime}0^{\prime}}-\gamma_{10^{\prime}01})$
(8)
$\displaystyle+\gamma_{10^{\prime}00}(\gamma_{00^{\prime}10}+\gamma_{0^{\prime}01^{\prime}0^{\prime}})+R_{000^{\prime}0^{\prime}}.$
(9)

The flat vacuum state (1) will be further perturbed by weak gravitational
waves with momenta $\mathbf{k}_{a}=\kappa_{A}\kappa_{A^{\prime}}$ measured on
the Planck scale. An antisymmetric perturbation metric is trivial, as it
amounts to rescaling the spinor basis, and the operators
$\nabla_{(ab^{\prime})}$ give a non-zero outcome only when acting on
expressions containing both spinors and cospinors. Therefore a perturbation of
the spinor base should involve cospinors and vice versa. Thus, the spin bases
are perturbed as

$\displaystyle\qquad\eta_{(a)}^{A}\to\eta_{(a)}^{A}+\epsilon\sum_{f}\zeta^{A}_{(af)}\mathcal{E}_{(f)},$
(10) $\displaystyle\mathcal{E}_{(f)}=\exp\left({\rm
i}\kappa_{F}\kappa_{F^{\prime}}S^{FF^{\prime}}\right),\;\;S^{FF^{\prime}}=\sum_{q}\eta_{(q)}^{F}\eta_{(q)}^{F^{\prime}}.$
(11)

The primed (cospinor) bases are perturbed in a similar way.

Perturbations of the derivative operators (2) originate in perturbations of
associated spin bases:

$\displaystyle\nabla^{[1]}_{(ab^{\prime})}$ $\displaystyle=$
$\displaystyle\epsilon\sum_{f,g}\left(\eta_{(a)}^{A}\zeta_{(b^{\prime}f)}\mathcal{E}_{(f)}+\eta_{(b^{\prime})}^{B^{\prime}}\zeta_{(ag)}^{A}\mathcal{E}_{(g)}\right)\nabla_{AB^{\prime}},$
(12) $\displaystyle\nabla^{[2]}_{(ab^{\prime})}$ $\displaystyle=$
$\displaystyle\epsilon^{2}\sum_{f,g}\zeta_{(af)}^{A}\zeta_{(b^{\prime}g)}^{B^{\prime}}\mathcal{E}_{(f)}\mathcal{E}_{(g)}\nabla_{AB^{\prime}}.$
(13)

As the exponent in Eq. (11) contains the sum of all coordinates and all spin
frames may oscillate with the same set of momenta, the action of
$\nabla_{(ab^{\prime})}$ is the same for all $a,b^{\prime}$ and just reduces
to multiplication by $-\mathbf{k}_{f}$, so that the indices can be omitted.

Since $\nabla_{(ab^{\prime})}$ act nontrivially only on perturbed bases, all
spin coefficients are at most of $\mathcal{O}(\epsilon)$. In the first order,

$\gamma^{[1]}_{(cd)}=-\epsilon\eta^{A}_{(c)}\sum_{f}\mathbf{k}_{f}\zeta_{A(df)}\mathcal{E}_{(f)}.$
(14)

In the second order, there are two distinct expressions:

$\displaystyle\gamma^{[2]}_{(cd)}=\nabla\gamma^{[1]}_{(cd)}=\epsilon^{2}\sum_{f,g}\zeta^{A}_{(cf)}\zeta_{A(dg)}\mathbf{k}_{f}\cdot\mathbf{k}_{g}\mathcal{E}_{(g)}\mathcal{E}_{(f)},$
(15)
$\displaystyle\gamma^{[2]}_{(ab^{\prime}cd)}=\eta_{(c)}^{A}\nabla^{[1]}_{ab^{\prime}}\eta_{A(d)}=$
(16)
$\displaystyle\epsilon^{2}\eta^{A}_{(c)}\nabla_{AB^{\prime}}\sum_{f}\left(\eta_{(a)}^{C}\zeta_{(b^{\prime}f)}\mathcal{E}_{(f)}+\eta_{(b^{\prime})}^{B^{\prime}}\zeta_{(af)}^{C}\mathcal{E}_{(f)}\right).$
(17)

The linear $\mathcal{O}(\epsilon)$ terms in the left-hand side of Eq. (9)
contain intrinsic derivatives acting upon the perturbations of bases denoted
by the last number in the definition of the respective spin coefficient.
Taking Eq. (9), we see that the last two numbers in the indices of the two
spin coefficients are identical, while the indices of the intrinsic derivative
acting on one spin coefficients is identical to the first two indices of
another one, so that their combination vanishes.The same applies to pairs of
linear terms in the rest of NP equation system.

Crystalline Vacuum. A wave with a Planck-scale momentum is a _ghost_ that
averages to zero on phenomenological scales. Only constant terms, which may be
subject to long-scale modulation are relevant for observable phenomena. Such
terms may arise in $\mathcal{O}(\epsilon^{2})$ only as _standing waves_. The
linear terms in the $\mathcal{O}(\epsilon^{2})$ HP equations contain
unperturbed directional derivatives $\nabla$ acting upon $\gamma^{[2]}_{(cd)}$
and $\nabla^{[1]}_{ab^{\prime}}$ directional derivatives acting upon
$\gamma^{[1]}_{(cd)}$. The former are equivalent to $\gamma^{[2]}_{(cd)}$ in
Eq. (15) and generate a standing wave when $\mathbf{k}_{f}=-\mathbf{k}_{g}$.
Returning to Eq. (9), we see that since this expression is independent of
$(ab^{\prime})$ both linear terms cancels, as they do in
$\mathcal{O}(\epsilon)$. However, standing waves can be generated by nonlinear
terms combining terms of $\gamma^{[1]}_{(cd)}$ with momenta of opposite signs.

In the third order, Planck-scale oscillations are suppressed in triplet
combinations forming a resonant triangle. Relevant solutions containing only a
restricted set of wavenumbers $k_{(a)}$ should include triplets of waves in
the spinor and cospinor manifolds satisfying

$\kappa_{F(f)}+\kappa_{F(g)}+\kappa_{F(h)}=0,\quad\kappa_{F^{\prime}(f^{\prime})}=-\kappa_{F(f)}.$
(18)

A resonant triangle does not need to be equilateral, which contrasts a common
situation in macroscopic symmetry breaking phenomena ch93 ; lp06 ; alx . A
vacuum crystal would naturally arise if second-order interactions involving
standing waves reduce the vacuum energy, which is further reduced due to
triplet resonances in the third-order of the expansion of the NP equations.
The three modes of the resonant triangle are interpreted as _colors_. The
“white” state combining all three colors is necessary for suppressing third-
order oscillations. Assigning a particular color to constituent modes of a
resonant triangle is arbitrary, and it is not necessary to attribute color
exchange and “color confinement” to interactions meditated by gluons.

The four amplitudes of perturbations of spinor and cospinor bases
$\zeta^{A}_{(af)},\,\zeta^{A^{\prime}}_{(a^{\prime}f)}$ are interpreted as
_quarks_ of different _flavors_ , each coming in three colors. Antiquarks
relate to reverse waves, which are always present, as they are necessary for
second-order interactions. The “color symmetry” ensures satisfying the
resonance condition in triplet interactions involving any set of quarks, and
therefore quark triplets combined in _baryons_ are non-separable. _Mesons_
combining a quark and an antiquark (and therefore colorless) are recognized as
standing spinor waves in the second order of the above expansion underlying
the emergence of the vacuum crystal. In this framework, there is no need to
attribute quark confinement to a vague and counterintuitive notion of
interaction strength increasing with separation.

The problematic moment is that so far we have identified only _four_ quark
flavors. This calls for a speculative suggestion. An additional spinor
manifold is invoked, assuming a 6D manifold that comprises _three_ spin-
spaces, one of which remaining flat, but its bases acquiring a dependence on
the coordinates of the crystallized 4D manifold to generate colored quarks of
two additional flavors.

As it has been briefly discussed in the introduction, higher-dimensional
theories have a long history but have never found either experimental evidence
or even an unequivocal theoretical justification. Six-dimensional spinors have
been invoked both as a device for unification of Standard Model fields and as
the background of supersymmetric theories 6d . It is an attractive proposition
that “the dynamics of Einstein’s curved space geometry runs its course in
superspace as the dynamics of a particle unfolds in spacetime” wh64 . It
allows to envisage the “Big Bang” (a misnomer coined by Fred Hoyle as a jibe)
as an event that had happen in a primordial higher-dimensional space. In the
framework of the current proposal, this is a crystallization event in two
coupled spinor manifolds leaving the remaining uncoupled manifold flat. The
symmetry should be restored within black holes, as crystalline units dissolve
when space contracts beyond the limit allowing for a multiscale analysis.

Modulated waves. Observable quark masses are determined by modulations of the
modes forming a resonant triangle on the phenomenological scale far exceeding
the intrinsic scale of the crystalline vacuum. The energy of a modulated state
depends on the direction of a modulation wave with respect to the waves
forming the vacuum crystal. The macroscopic analogue is found in preferred
modulations normal to the waves forming ubiquitous hexagonal patterns in 2D
knp . Applying a similar approach to the spin-space metric while assuming
tentatively that the ratio of the modulation wavelength to the scale of the
vacuum crystal is of the same $\mathcal{O}(\epsilon)$, we compute lowest-order
corrections of spin coefficients induced by $\mathcal{O}(\epsilon)$ long-scale
modulations due to perturbations of spinor space momenta
$\kappa_{F}\to\kappa_{F}+\epsilon\widetilde{\kappa}_{F}$ and similar for the
cospinor space. This modulates $\mathcal{E}_{(f)}$ as

$\mathcal{E}_{(f)}\to\mathcal{E}_{(f)}\left[1+{\rm i}\epsilon
Q_{[1]}-\epsilon^{2}Q_{[2]}+\mathcal{O}(\epsilon^{3})\right],$ (19)
$\displaystyle Q_{[1]}$
$\displaystyle=\left(\widetilde{\kappa}_{F}\kappa_{F^{\prime}}+\widetilde{\kappa}_{F^{\prime}}\kappa_{F}\right)S^{FF^{\prime}}+\kappa_{F}\kappa_{F^{\prime}}S^{FF^{\prime}}_{[1]},$
(20) $\displaystyle Q_{[2]}$
$\displaystyle=\left(\widetilde{\kappa}_{F}\widetilde{\kappa}_{F^{\prime}}S^{FF^{\prime}}\right)^{2}+\left(\widetilde{\kappa}_{F}\kappa_{F^{\prime}}+\widetilde{\kappa}\right)S^{FF^{\prime}}_{[1]}$
(21) $\displaystyle+\kappa_{F}\kappa_{F^{\prime}}S^{FF^{\prime}}_{[2]},$ (22)
$\displaystyle S^{FF^{\prime}}_{[1]}$
$\displaystyle=\sum_{q,f}\left(\eta_{(q)}^{F}\zeta_{(qg)}^{F^{\prime}}\mathcal{E}_{(g)}+\eta_{(q)}^{F^{\prime}}\zeta_{q,h}^{F}\mathcal{E}_{(h)}\right),$
(23) $\displaystyle S^{FF^{\prime}}_{[2]}$
$\displaystyle=\sum_{q,g,h}\zeta_{(qg)}^{F}\zeta_{(qh)}^{F^{\prime}}\mathcal{E}_{(f)}\mathcal{E}_{(g)}.$
(24)

The respective lowest-order modulations of spin coefficients are

$\displaystyle\widetilde{\gamma}^{[1]}_{(cd)}=-{\rm
i}\epsilon^{2}\eta^{A}_{(c)}\sum_{f}\mathbf{k}_{f}\zeta_{A(df)}\mathcal{E}_{(f)}\times$
(25)
$\displaystyle\left[\left(\widetilde{\kappa}_{F}\kappa_{F^{\prime}}+\widetilde{\kappa}_{F^{\prime}}\kappa_{F}\right)S^{FF^{\prime}}+\kappa_{F}\kappa_{F^{\prime}}S^{FF^{\prime}}_{[1]}\right].$
(26)

The only second-order contribution of long-scale modulation comes from linear
terms or the NP system, but both terms cancel as before, so that the lowest-
order effect of modulation is delayed to the third order. Further expansion to
$\widetilde{\gamma}^{[2]}_{(cd)}$ or
$\gamma^{[2]}_{(cd)},\,\gamma^{[2]}_{(ab^{\prime}cd)}$ defined by Eq. (17)
does not prevent the cancellation of linear terms, so that modulation affects
only nonlinear terms of the type
$\epsilon^{3}\gamma_{(ab)}\widetilde{\gamma}^{[1]}_{(cd)}$. This expression
includes, besides terms dependent on modulation momenta
$\widetilde{\kappa}_{F},\,\widetilde{\kappa}_{F^{\prime}}$ and three “colored”
exponents belonging to the resonant triangle, also additional terms coming
from the expansion of $S^{FF^{\prime}}$, which include an extra color and
therefore should average to zero unless complemented by an anticolored wave.

Modulations of this extra wave, appearing in higher orders of the expansion,
can be associated with _leptons_ , which may be involved only in _weak_
interactions appearing in the fourth order of the expansion. The three
distinct leptons and the respective neutrinos should be associated with the
three generations corresponding to the respective flavors.

Discussion. The key point of the current proposal is an ultimate unification
presenting matter fields as nothing more than modulated Planck-scale
gravitational waves. Although the theory is purely “classical”, it does not
call for conventional quantization but rather for a reinterpretation of
quantum mechanics attributing the source of quantum uncertainty to the
envelope character of phenomenological fields.

It is pointless to speculate on the existence of a “metaphysics” below the
Planck scale, that would determine resonant momenta. The proposed theory does
not predict parameters of SM dependent on the parameters of the underlying
crystal, which should have been selected at “Big Bang”. It is likely that
different resonances are responsible for crystalline structures in different
universes that may arise at unconnected locations within the 6D spinor
manifold, and the resonant momenta may be inferred from observed quark masses
rather than the other way around.

In principle, it might be possible to compute the energy gain due to
crystallization _ab initio_ by solving the NP equation system to the third
order in order to compute the Ricci tensor that determines the energy of
crystals based on a specific resonances. Still, there is no warranty that the
“optimal” crystalline structure would be chosen. The anthropic principle may
be evoked to estimate values appropriate to a long-lived universe and fitting
the observed phenomenology. Still less feasible would be computing quark
masses _ab initio_ by incorporating Eq. (26) in the multiscale expansion, but
distinctions between flavors can be naturally attributed to an asymmetry of
the vacuum lattice.

The inferred “inflation” period corresponds to a rapid growth of a fluctuation
in a primordial flat space developing into a sustainable crystal. A small
assembly of cells, likely disordered, is not a crystal yet. A three-wave
resonance leading to the genesis of quarks emerges at the size of
$\mathcal{O}(\epsilon^{-3})$ Planck units. Identifying this with the “grand
unification” epoch dominated by quark–gluon plasma at $10^{6}$ seconds ($t\sim
10^{39}$ Planck units) after the Big Bang suggests the scaling estimate
$\epsilon\sim 10^{-13}$.

The crystalline scale remains invariant as a universe expands: more
crystalline units are just added to fill the available space in the 6D
manifold. The energy gain due to crystallization can be identified with the
“dark energy” driving the expansion of a universe. The expansion would
accelerate as the boundary of a universe in the 6D manifold extends, similar
to surface tension effects leading to Ostwald ripening ost  in macroscopic
two-phase systems. “Dark matter” may be built up by macroscopic fields that
are unrelated to the Planck-scale crystal and therefore do not interact with
its modulations. In an unlikely case of a collision between two universes
based on different resonant momenta, the two modulating fields may not
interact remaining mutually dark, unless, by chance, a new resonance is
formed, leading to a more complicated, and likely disastrous, phenomenology.

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