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Where a preferred frame leaks into low-energy physics

Status

Proved: symmetry counts for compact groups, helicity statements under (B1) to (B3), supersymmetry statements for the stated algebra, absence of an invariant probability measure on frames, the one-loop coefficient for a scalar-line regulator, monomial supports of code Hamiltonians. Exact rational linear algebra: Table 1 and the dimension of the gauge-invariant forms. Calculation under hypotheses: code-distance amplification. Heuristic: the gauge-theory estimate.

Excludes

Under the stated hypotheses: equal graviton, photon and scalar speeds from a compact residual group or a vacuum without boosts; Lorentz invariance restored by a probability measure on frames; an unshifted one-loop fermion speed with a frame in the scalar-line regulator of a Yukawa model; equal speeds from stabiliser-code distance with single-qubit Pauli terms.

Revised

29 October 2026

Suppose a substrate selects a frame, or leaves only a compact subgroup of the Lorentz group unbroken. Does the residual symmetry then force gravitons, photons and scalar matter to share one limiting speed at low energy? Under the conditions of sections 1 and 2 it does not: for a compact residual group, and in a vacuum without boosts, the quadratic action of each species has a free speed of its own, with one exception, a supersymmetry algebra with a single speed constant, which fixes the common speed of the two-state multiplets (Proposition 5). A frame cannot be removed by averaging, because no probability measure on frames is Lorentz invariant (section 3). In a Yukawa model with the frame in the regulator of the scalar line, the fermion speed shifts at one loop by $g^2/48\pi^2$ when the scalar mass is small, for every regulator shape in a stated class and whatever the cutoff (section 4).

In Euclidean signature a group acting absolutely irreducibly, such as the hypercubic group, leaves a single coefficient for the scalar and graviton two-derivative actions, and for the Maxwell action when the group is the hypercubic one, but a real-time theory then requires reflection positivity (section 1). For a stabiliser code, a quantum error-correcting code defined by commuting Pauli operators, perturbed by single-qubit Pauli terms, a relative asymmetry between the couplings of two species is multiplied by the order of perturbation theory at which the logical coupling that sets the speed first appears, and that order is at least the code distance, the least number of qubits a non-trivial logical operator touches (section 5, hypotheses in Proposition 11).

The fine-tuning argument of Collins, Perez, Sudarsky, Urrutia and Vucetich [Collins et al.] is that a frame in a regulator feeds order-one Lorentz violation into the marginal operators of the low-energy theory; they estimate it with a Yukawa self-energy in which the regulator multiplies the fermion propagator. In Hořava–Lifshitz gravity [Hořava], Pospelov and Shang [Pospelov–Shang] find that when the Lorentz-violating sector couples to Lorentz-invariant matter only through Planck-suppressed operators, the transmitted violation is suppressed by the squared ratio of the two scales, provided the quadratic divergences from the instantaneous gravity modes are removed by higher-dimension terms. Coates, Melby-Thompson and Mukohyama [Coates et al.] obtain power-law divergences in the difference of the sound speeds of a gauge field and a scalar, which spoil that suppression unless the matter sector is also modified at high energy. For supersymmetric theories, Lorentz-violating operators in the supersymmetric Standard Model have dimension at least five [Groot Nibbelink–Pospelov], and in renormalisable Lorentz-violating supersymmetric gauge theories with linear supercharges the only violation sits in a constant $c$ of the supersymmetry algebra, which does not renormalise [Redigolo]. The independence of the one-loop coefficient from the shape of a regulator placed on the scalar line (Theorem 8) is new.

Exact Lorentz symmetry of a discrete substrate is the subject of quantum sprinklings and Lorentz-invariant point sets. Exact lattice translations together with a Zariski-dense discrete Lorentz group, a non-compact residual group, are treated in lattices with an arithmetic Lorentz group; the compact and boost-free groups below are a different case. A frame that is the velocity of a cluster of particles, and so a function of the state, is treated under a locality hypothesis in frames, Wilson terms and reflection positivity. A conserved lattice momentum for gravitons on a lattice, and the frame and photon-speed shift that follow when the momentum is made exact by band-limiting the vertices, are the subject of a conserved lattice momentum and graviton scattering.

1. What a residual symmetry group fixes

A free scalar with a frame has the action $\tfrac12 a\,\dot\phi^2-\tfrac12 b\,(\nabla\phi)^2$ and speed $c^2=b/a$. Lorentz invariance sets $a=b$. If only a subgroup $K$ of the Lorentz group survives, the allowed coefficient tensors are the $K$-invariant symmetric bilinear forms on momentum space. A compact $K$ always admits two independent ones, the Minkowski form $\eta$ and a positive-definite form obtained by averaging over $K$, so $b/a$ is a free parameter of the quadratic action of each scalar species. For spin 2, gauge invariance restricts the allowed forms, and the question is how many survive.

Write a real quadratic form in a symmetric tensor $h_{\mu\nu}$ and a momentum $k_\mu$ on $\mathbb R^4$ as a polynomial $Q(h,k)$ of degree 2 in the ten components of $h$ and degree 2 in $k$; this is a two-derivative action $\int h(-k)^{T}M(k)\,h(k)$ with $M$ quadratic in $k$. It is gauge invariant if

$$Q\bigl(h+k\otimes\xi+\xi\otimes k,\;k\bigr)=Q(h,k)\qquad\text{for all }\xi\in\mathbb R^4. \tag{1}$$

Let $\mathcal V$ be the space of gauge-invariant forms. A matrix $g\in GL(4,\mathbb R)$ acts by $k\mapsto gk$, $h\mapsto ghg^{T}$, and (1) is preserved, so $\mathcal V$ is a representation of $GL(4,\mathbb R)$. For a symmetric invertible matrix $G^{\mu\nu}$ define the Fierz–Pauli form with all contractions made with $G$:

$$Q_G(h,k)=-\tfrac12\,(k\!\cdot\!k)\,h_{\mu\nu}h^{\mu\nu}+k^{\mu}h_{\mu\nu}\,k_{\lambda}h^{\lambda\nu}-(k^{\mu}k^{\nu}h_{\mu\nu})\,h+\tfrac12\,(k\!\cdot\!k)\,h^{2}, \tag{2}$$

where indices are raised and contracted with $G$, $k\!\cdot\!k=G^{\mu\nu}k_\mu k_\nu$ and $h=G^{\mu\nu}h_{\mu\nu}$. For $G=\eta$ this is the linearised Einstein–Hilbert action, which satisfies (1). For a Lorentzian $G$ the form $Q_G$ is the Einstein–Hilbert form written in linear coordinates adapted to $G$, and a linear change of coordinates maps linearised gauge transformations to linearised gauge transformations. Both sides of (1) for $Q_G$ are polynomials in the entries of $G$, so the identity, valid on the open set of Lorentzian $G$, holds for every symmetric $G$. Each term of (2) contains three factors of $G$, so $Q_{tG}=t^{3}Q_G$ for a scalar $t$. In the eigenvalue basis $G=\mathrm{diag}(-1,c^2,c^2,c^2)$, a transverse-traceless mode with polarisation $h_{12}$ and momentum $k=(\omega,0,0,k_z)$ has $Q_G=c^{4}(\omega^{2}-c^{2}k_z^{2})$, so $Q_G$ describes helicity-2 waves with speed $c$.

Proposition 1 (gauge-invariant two-derivative forms). Let $\mathrm{Sym}^2$ be the space of symmetric $4\times4$ matrices, on which $O(4)$ acts by $s\mapsto g^{T}sg$. The map $\Phi:\mathrm{Sym}^2\to\mathcal V$, $\Phi(s)=\frac{d}{dt}Q_{1+ts}\Big\rvert_{t=0}$, is an isomorphism of $O(4)$-representations. In particular $\dim\mathcal V=10$, and for every subgroup $K\subset O(4)$ the number of $K$-invariant gauge-invariant forms is

$$n(K)=\dim\mathcal V^{K}=\dim(\mathrm{Sym}^2)^{K}. \tag{3}$$

Proof. The identity $Q_{G}(ghg^{T},gk)=Q_{g^{T}Gg}(h,k)$ holds because every contraction in (2) is a contraction with $G$ of vectors transformed by $g$. For $g\in O(4)$ and $G=1+ts$ it gives $\Phi(g^{T}sg)(h,k)=\Phi(s)(ghg^{T},gk)$, so $\Phi$ is equivariant.

Next, $\Phi$ is injective. The space $\mathrm{Sym}^2=\mathbb R\,1\oplus\mathrm{Sym}^2_0$ splits into the trace part and the nine-dimensional traceless part, which are irreducible and not isomorphic, so the kernel of $\Phi$ is a sum of a subspace of each. By homogeneity $\Phi(1)=3Q_1\neq0$. On the traceless part, label the components of $k$ by $(0,1,2,3)$, call the first one $\omega$ (the energy, as in section 2), and take the test mode above with the diagonal $G=1+ts$, $s=\mathrm{diag}(3,-1,-1,-1)$. Then $Q_G=-(G^{00}\omega^{2}+G^{33}k_z^{2})\,G^{11}G^{22}$ and $\frac{d}{dt}Q_G\Big\rvert_{t=0}=-(3\omega^{2}-k_z^{2})+2(\omega^{2}+k_z^{2})=3k_z^{2}-\omega^{2}\neq0$. Hence $\ker\Phi=0$, and $\dim\mathcal V\ge10$.

For the upper bound, expanding (1) in monomials of $(h,k,\xi)$ gives a homogeneous linear system with 1150 distinct equations for the $55\times10=550$ coefficients of $Q$ (55 monomials quadratic in the components of $h$, 10 quadratic in $k$). Its rank over $\mathbb Q$, computed in exact rational arithmetic, is 540, so the solution space has dimension $550-540=10$. Since $\Phi$ is injective and $\dim\mathrm{Sym}^2=10$, it is an isomorphism. $\square$

The lower bound $\dim\mathcal V\ge10$ follows from the injectivity of $\Phi$ and is proved by hand. The upper bound is the rank computation, which we do not reproduce by hand. For $O(4)$ the count can be done by hand: the $O(4)$-invariant forms of the required degree are spanned by $k^2\,h_{\mu\nu}h_{\mu\nu}$, $k^2h^2$, $k_\mu h_{\mu\nu}h_{\nu\lambda}k_\lambda$ and $(k_\mu h_{\mu\nu}k_\nu)\,h$, and (1) fixes their coefficients, up to a common factor, to $-\tfrac12,\tfrac12,1,-1$, which is (2) with $G=1$.

Table 1 lists $n(K)$ for the graviton and for a Maxwell field. For Maxwell, $Q(A,k)$ is quadratic in $A_\mu$ and in $k$ (100 coefficients), and the gauge transformation is $A\mapsto A+k\,\xi$ with scalar $\xi$. The columns “any form” count the invariants of the group without imposing gauge invariance. The groups are $O(4)$; the hypercubic group $H(4)$ of signed permutations of the four axes; $O(3)\times T$, spatial rotations together with time reversal $k_0\mapsto-k_0$ (with $h_{0i}\mapsto-h_{0i}$); and the cubic group of signed permutations of the three spatial axes times $T$. Every entry is the exact rank over $\mathbb Q$ of the linear conditions (generators of the Lie algebra for the continuous groups, generators of the group for the discrete ones). The gauge-invariant graviton entries agree with $\dim(\mathrm{Sym}^2)^K$ counted by hand: $1$ for $O(4)$ and $H(4)$; $2$ for $O(3)\times T$ and for the cubic group times $T$, the invariants being $\delta_{ij}$ and $u_\mu u_\nu$ with $u$ along the time axis. The Maxwell entry 20 for the trivial group is the number of quadratic forms in $F_{\mu\nu}=k_\mu A_\nu-k_\nu A_\mu$ (21) less the one proportional to $F\wedge F$, which vanishes for $F=k\wedge A$.

group	graviton, any form	graviton, gauge invariant	Maxwell, any form	Maxwell, gauge invariant
none	550	10	100	20
$O(4)$	4	1	2	1
$H(4)$	9	1	3	1
$O(3)\times T$	17	2	6	2
cubic $\times\,T$	26	2	7	2

Table 1. Dimensions of the spaces of two-derivative quadratic forms invariant under the group.

Corollary 2 (absolutely irreducible groups). Let $K\subset O(4)$ act on $\mathbb R^4$ absolutely irreducibly, as $H(4)$ does. Then $(\mathrm{Sym}^2)^K=\mathbb R\,1$, so every $K$-invariant gauge-invariant two-derivative graviton action is a multiple of $Q_1$, the Fierz–Pauli action with a single metric. The same holds for the scalar action $c^{\mu\nu}k_\mu k_\nu$, and, from Table 1, for a Maxwell field when $K=H(4)$.

Proof. A symmetric invariant bilinear form on an absolutely irreducible real representation is unique up to a factor, by Schur’s lemma applied to the complexification. For $H(4)$ the commutant of the signed permutations is scalar: commuting with all sign changes forces a diagonal matrix, and commuting with all permutations forces equal entries. Equation (3) gives the graviton statement, and for scalars the allowed coefficients are exactly $(\mathrm{Sym}^2)^K$. The Maxwell statement is the entry of Table 1. $\square$

For $H(4)$ the scalar, graviton and Maxwell actions then share the tensor $1$, so in Euclidean signature the relative weight of $\partial_0^2$ and $\partial_i^2$ is the same for each of them. Since $-1\in H(4)$, every odd-rank invariant tensor vanishes, which removes all operators with an odd number of indices; the only rank-two invariant is $\delta_{\mu\nu}$, so there is no dimension-four operator $c^{\mu\nu}\bar\psi\gamma_\mu\partial_\nu\psi$ with traceless $c^{\mu\nu}$. The first operators that distinguish $H(4)$ from $O(4)$ are built from the rank-four invariant $\sum_\mu e_\mu^{\otimes4}$: $\phi\sum_\mu\partial_\mu^4\phi$ for a scalar (four derivatives) and $\bar\psi\sum_\mu\gamma_\mu\partial_\mu^3\psi$ for a Dirac fermion (three). A common limiting speed in real time requires in addition that the Euclidean theory be reflection positive and that the linearised gauge invariance persist at interacting level. If the gauge invariance is only approximate, Table 1 shows what opens up: the $H(4)$-invariant graviton forms number nine, of which one is gauge invariant.

Corollary 3 (compact groups in real time). Let $K\subset O(3,1)$ be compact, acting on momentum space $\mathbb R^4$.

(a) $n(K)\ge2$ for the graviton, and the scalar and Maxwell quadratic actions likewise have at least two independent forms.

(b) The additional parameter is a speed. There is a $K$-invariant timelike line, spanned by a vector $x$ with $x^{T}\eta x=-1$, and with $\tilde u=\eta x$ the matrix $G_c=c^{2}\eta+(c^{2}-1)\,\tilde u\tilde u^{T}$ is $K$-invariant under $G\mapsto g^{T}Gg$ for every $c>0$. In a Lorentz frame with time axis $x$, $G_c=\mathrm{diag}(-1,c^2,c^2,c^2)$ and $G_1=\eta$.

The quadratic action of each species therefore has a parameter in addition to its normalisation, namely the speed $c$ relative to that frame, and nothing in the symmetry relates these parameters for different species.

Proof. (a) Averaging $g^{T}g$ over $K$ with Haar measure gives a $K$-invariant positive-definite form $\Omega$. In a basis in which $\Omega=1$ the group lies in $O(4)$, and the Minkowski form becomes a symmetric matrix $\eta'$ that is invariant under $K$ and indefinite, hence not proportional to $1$. Both $1$ and $\eta'$ lie in $(\mathrm{Sym}^2)^K$, so (3) gives $n(K)\ge2$; the number $n(K)$ does not depend on the basis, because $\mathcal V$ is stable under $GL(4,\mathbb R)$. For scalars the invariant coefficient tensors are the same two matrices. For Maxwell, rotate within $O(4)$ so that $\eta'=\mathrm{diag}(g_0,\dots,g_3)$; the forms $Q^{\mathrm{Max}}_{G}=-\tfrac14F_{\mu\nu}F_{\rho\sigma}G^{\mu\rho}G^{\nu\sigma}$, with $F_{\mu\nu}=k_\mu A_\nu-k_\nu A_\mu$, for $G=1$ and $G=\eta'$ are $K$-invariant and gauge invariant, and they are not proportional, because the coefficient of $F_{\mu\nu}^2$ in $Q^{\mathrm{Max}}_{\eta'}$ is $-\tfrac12g_\mu g_\nu$ and equal products $g_\mu g_\nu$ for all six pairs force all $g_\mu$ equal, which an indefinite $\eta'$ excludes.

(b) The forms $\Omega$ and $\eta$ are both $K$-invariant, so the matrix $\Omega^{-1}\eta$ commutes with $K$. It is similar to $\Omega^{-1/2}\eta\,\Omega^{-1/2}$, which is congruent to $\eta$, so it has exactly one negative eigenvalue, and the eigenvector $x$ of that eigenvalue satisfies $x^{T}\eta x<0$. Its line is $K$-invariant and timelike, so $gx=\pm x$ for $g\in K$; normalise $x^{T}\eta x=-1$ and let $\tilde u=\eta x$. Since $g^{T}\eta=\eta g^{-1}$, also $g^{T}\tilde u=\pm\tilde u$. Then $\eta$ and $\tilde u\tilde u^{T}$ are $K$-invariant under $G\mapsto g^{T}Gg$, and so is $G_c$. In a Lorentz frame with time axis $x$ one has $\eta=\mathrm{diag}(-1,1,1,1)$ and $\tilde u=-e_0$, so $G_c=\mathrm{diag}(-1,c^2,c^2,c^2)$. $\square$

By section 1, the graviton form $Q_{G_c}$, the Maxwell form $Q^{\mathrm{Max}}_{G_c}$ and the scalar form built from $G_c$ describe modes of speed $c$ relative to that frame, with $c$ free for each species; for $K=SO(3)$, with or without $T$, this is the setting of Proposition 4. Corollary 3 describes what the symmetry allows in the free action: the equality of the speeds of different species would be an input. A fermion kinetic term involves the coefficients $c^{\mu\nu}$ of $\bar\psi\gamma_\mu\partial_\nu\psi$; their counting for a compact $K$ is not carried out in sections 1 and 2, and section 4 treats a fermion in a model. A symmetry that mixes species, such as supersymmetry, is outside the corollary.

2. Vacua without boosts, and supersymmetry

A Hořava-type vacuum is invariant under time and space translations and spatial rotations, possibly with time reversal. Call this residual group $\Gamma$. We assume (B1) that the graviton has a linearised gauge invariance between foliation-preserving diffeomorphisms and the full linearised diffeomorphisms, (B2) that the photon has a $U(1)$ gauge invariance, and (B3) that the quadratic action has no cross term between $h$ and $A$, as follows for instance from charge conjugation $A\mapsto-A$.

Proposition 4 (speeds without boosts). Under (B1) to (B3):

(a) For every pair $c_T,c_\gamma>0$ the quadratic action $Q_{G_{c_T}}(h,k)+Q^{\mathrm{Max}}_{G_{c_\gamma}}(A,k)$, with $G_c=\mathrm{diag}(-1,c^2,c^2,c^2)$ and $Q^{\mathrm{Max}}_G$ the Maxwell form of section 1, is $\Gamma$-invariant and gauge invariant under the full linearised diffeomorphisms, hence under the foliation-preserving subgroup. Its helicity-2 modes propagate with speed $c_T$ and its helicity-1 modes with speed $c_\gamma$. The assumed symmetries therefore leave $c_T/c_\gamma$ unconstrained.

(b) Let $U$ be a unitary operator that commutes with the Hamiltonian $H$, the momentum $\vec P$ and the angular momentum $\vec J$. Then $U$ maps the one-particle states of momentum $\vec p$ and helicity $\lambda$ to states with the same $\vec p$ and $\lambda$. It therefore cannot map graviton states ($\lvert\lambda\rvert=2$) to photon states ($\lvert\lambda\rvert=1$): a symmetry that preserves helicity cannot intertwine the two sectors.

Proof. (a) Invariance under translations and rotations holds because $G_c$ is rotation invariant and constant, and $T$ acts trivially on it. Gauge invariance of $Q_G$ was shown in section 1, and the Maxwell form is invariant under $A\mapsto A+k\xi$ because it depends on $F$ only. The dispersion relation of transverse modes is $G^{\mu\nu}k_\mu k_\nu=0$, that is $\omega=c\lvert\vec k\rvert$, for each sector with its own $c$.

(b) The helicity of a one-particle state with momentum $\vec p$ is the eigenvalue of $\vec J\cdot\hat p$. If $U$ commutes with $\vec P$ and $\vec J$ it commutes with $\vec J\cdot\hat P$ as well, and with $H$ it preserves the energy. $\square$

In a Lorentz-invariant vacuum the relation between the two speeds comes from the boosts, which do not commute with $\vec J$; part (b) records that the symmetries remaining here cannot replace them. Equal speeds can still occur as a coincidence of the couplings; part (b) shows only that no such symmetry enforces them. A fermionic symmetry is an exception to the argument, because it can change the helicity by one half.

Exact supersymmetry is of this type. Suppose that a single algebra

$$\{\mathcal Q_\alpha,\mathcal Q^\dagger_\beta\}=2\bigl(H\,\delta_{\alpha\beta}+c\,\vec P\cdot\vec\sigma_{\alpha\beta}\bigr),\qquad \{\mathcal Q_\alpha,\mathcal Q_\beta\}=0, \tag{4}$$

acts on all fields, with supercharges $\mathcal Q_\alpha$ that commute with $H$ and $\vec P$, a real constant $c>0$ and Pauli matrices $\vec\sigma$. This is the supersymmetry algebra with the speed of light replaced by $c$, the form in which [Redigolo] finds the only Lorentz violation of supersymmetric gauge theories with linear supercharges. The second relation of (4), the absence of a central charge, is used below. For momentum along the $z$ axis we fix the convention $[J_z,\mathcal Q_1]=-\tfrac12\mathcal Q_1$ and $[J_z,\mathcal Q_2]=+\tfrac12\mathcal Q_2$; the opposite choice replaces $\lambda-\tfrac12$ by $\lambda+\tfrac12$ in the multiplets and does not affect the conclusion.

Proposition 5 (short multiplets have speed $c$). Assume (4), with supercharges commuting with $H$ and $\vec P$. On the sector of fixed energy $E$ and momentum $\vec p=p\hat z$ with $p>0$, the operators satisfy $\{\mathcal Q_1,\mathcal Q_1^\dagger\}=2(E+cp)$ and $\{\mathcal Q_2,\mathcal Q_2^\dagger\}=2(E-cp)$, and all other anticommutators among $\mathcal Q_1,\mathcal Q_2,\mathcal Q_1^\dagger,\mathcal Q_2^\dagger$ vanish. Hence every state has $E\ge cp$. An irreducible multiplet has two states, with helicities $(\lambda,\lambda-\tfrac12)$, if and only if $E=cp$, and four states otherwise. If the graviton and the photon each lie in two-state multiplets, with helicities $(2,\tfrac32)$ and $(1,\tfrac12)$, then both have the dispersion relation $E=c\lvert\vec p\rvert$, and $c_{\mathrm{GW}}=c_\gamma=c$.

Proof. With $\vec p$ along $z$, $\vec p\cdot\vec\sigma=p\,\sigma_z$ is diagonal, which gives the two anticommutators and shows that $\{\mathcal Q_1,\mathcal Q_2^\dagger\}=0$; the relations $\{\mathcal Q_\alpha,\mathcal Q_\beta\}=0$ are the second part of (4). In a positive-norm Hilbert space $\lVert \mathcal Q_2^\dagger\psi\rVert^2+\lVert \mathcal Q_2\psi\rVert^2=2(E-cp)\lVert\psi\rVert^2\ge0$, so $E\ge cp$. If $E=cp$ then $\mathcal Q_2=\mathcal Q_2^\dagger=0$ on the sector, and an irreducible multiplet is spanned by a state $\lvert\lambda\rangle$ annihilated by $\mathcal Q_1^\dagger$ and its image $\mathcal Q_1\lvert\lambda\rangle$, which is non-zero because $\lVert\mathcal Q_1\lvert\lambda\rangle\rVert^2=4cp\lVert\lambda\rVert^2$ and has helicity lower by one half: two states. If $E>cp$, the operators $\mathcal Q_1/\sqrt{2(E+cp)}$ and $\mathcal Q_2^\dagger/\sqrt{2(E-cp)}$ are the creation operators of two fermionic oscillators, since (4) gives the canonical anticommutation relations. An irreducible multiplet is spanned by a state $\lvert\lambda\rangle$ annihilated by $\mathcal Q_1^\dagger$ and $\mathcal Q_2$ together with its three images, with helicities $(\lambda,\lambda-\tfrac12,\lambda-\tfrac12,\lambda-1)$. A graviton in a two-state multiplet therefore has $E=cp$, and so does a photon. $\square$

The speed of every short multiplet is the constant $c$ of the algebra and does not depend on a coefficient in the action. Soft supersymmetry breaking leaves a residual difference of order $m_{\mathrm{soft}}^{2}/\Lambda_{\mathrm{LV}}^{2}$, where $\Lambda_{\mathrm{LV}}$ is the scale at which the Lorentz-violating terms appear; this estimate is quoted in [Coates et al.].

3. A frame cannot be averaged away

A construction on a point set that uses a unit timelike vector $u$ breaks Lorentz invariance unless its observables do not depend on $u$. One may try to restore invariance by drawing $u$ at random. No normalised random choice is Lorentz invariant.

Proposition 6 (no invariant measure on frames). Let $d\ge2$ and let $H^{d-1}$ be the set of future unit timelike vectors in $\mathbb R^{1,d-1}$. There is no $SO^{+}(1,d-1)$-invariant Borel probability measure on $H^{d-1}$.

Proof. For $u\in H^{d-1}$ we have $(u^{0})^{2}-(u^{1})^{2}=1+\lvert u_\perp\rvert^{2}>0$, so the rapidity $\theta(u)=\operatorname{artanh}(u^{1}/u^{0})$ is defined and continuous. A boost $B_\tau$ in the $(0,1)$ plane maps $(u^0,u^1)$ to $(\cosh \tau\,u^{0}+\sinh \tau\,u^{1},\ \sinh \tau\,u^{0}+\cosh \tau\,u^{1})$ and so shifts $\theta$ by $\tau$. If $\mu$ were invariant, the image measure $\nu=\theta_*\mu$ on $\mathbb R$ would satisfy $\nu(A+\tau)=\nu(A)$ for all $\tau$. Then $\nu([n,n+1))$ takes the same value $a$ for every integer $n$, and $\sum_n\nu([n,n+1))=1$ is impossible for a constant $a$, whether $a=0$ or $a>0$. $\square$

The invariant (Haar) measure on $H^{d-1}$ is the hyperbolic volume; it exists and has infinite total mass. An average over it is defined only for observables that are integrable over the rapidity, and the proposition says that no choice of normalisation turns it into a probability measure.

Corollary 7 (no equivariant frame from a Lorentz-invariant point process). Let $\mathcal P$ be a random locally finite subset of $\mathbb R^{1,d-1}$ whose law is invariant under $SO^{+}(1,d-1)$, for example a Poisson process. There is no measurable function $u(\mathcal P)\in H^{d-1}$ of the point set with $u(g\mathcal P)=g\,u(\mathcal P)$ almost surely for each $g$.

Proof. If there were one, the law $\mu$ of $u(\mathcal P)$ would satisfy $\mu(gA)=\Pr[g^{-1}u(\mathcal P)\in A]=\Pr[u(g^{-1}\mathcal P)\in A]=\mu(A)$, because $g^{-1}\mathcal P$ has the same law as $\mathcal P$. It would therefore be an invariant probability measure, which contradicts Proposition 6. $\square$

Corollary 7 is the argument of Bombelli, Henson and Sorkin [Bombelli–Henson–Sorkin], who prove related statements for sprinklings, in its simplest form. It gives a trichotomy for any matter construction on a Lorentz-invariant point set that uses a timelike axis. The axis may be drawn independently of the points; its law is then not invariant by Proposition 6, and the construction selects a frame. The axis may be a function of the points; Corollary 7 forbids this when the function is equivariant and measurable. Or the observables may be independent of the axis in each realisation. In Euclidean signature the corresponding group $SO(d)$ is compact and a Haar average over the axis exists, so the obstruction does not arise; whether such an averaged construction defines a Lorentz-invariant real-time theory is the question of reflection positivity. Fermions that see only the causal order shows that with one spinor space at every point of a sprinkling the order alone supplies no covariant kinetic term (its Theorem 1), and uses Proposition 6 to exclude a random global frame with an invariant joint law (its Proposition 3).

4. A frame in the matter regulator

The model is a Yukawa theory in which only the scalar line carries a frame-dependent regulator.

Work in Euclidean $\mathbb R^4$ with a unit vector $u$ along the time axis. A massless Dirac fermion $\psi$ with propagator $S(p)=1/(i(\gamma\!\cdot\!p))=-i(\gamma\!\cdot\!p)/p^{2}$ couples to a real scalar through $g\,\phi\bar\psi\psi$. The scalar propagator is replaced by

$$D(q)=\frac{f(\lvert q_\perp\rvert/\Lambda)}{q^{2}+M^{2}},\qquad q_\perp=q-(q\!\cdot\!u)u, \tag{5}$$

so that the frame enters only through the cutoff on the momentum orthogonal to $u$. This is a prescription for a propagator and not an action; it is all that a one-loop self-energy needs. We take $f(x)=F(x^{2})$ with $F$ continuously differentiable on $[0,\infty)$, $F(0)=1$, and $F(s)$, $sF'(s)$ bounded by $C_0(1+s)^{-\beta}$ for some $\beta>\tfrac12$. Examples are $e^{-x^{2}}$, $e^{-x^{4}}$ and $(1+x^{2})^{-2}$. The scalar has speed 1 for momenta much below $\Lambda$, and the question is the speed of the fermion.

The one-loop inverse propagator of $\psi$ is

$$W(p)=i(\gamma\!\cdot\!p)-g^{2}\!\int\!\frac{d^{4}q}{(2\pi)^{4}}\,S(p-q)\,D(q)=i(\gamma\!\cdot\!p)+ig^{2}\!\int\!\frac{d^{4}k}{(2\pi)^{4}}\,\frac{(\gamma\!\cdot\!k)}{k^{2}}\,D(p-k). \tag{6}$$

For $M>0$ and $\beta>\tfrac12$ the integral converges absolutely, because after the integration over the component along $u$ it is bounded by a multiple of the integral of $f(\lvert k_\perp\rvert/\Lambda)$ over $\lvert k_\perp\rvert$, and it is continuously differentiable in $p$. We need its derivative at $p=0$. Differentiating $D(p-k)$, using that $D$ is even, so that $(\partial_\nu D)(-k)=-(\partial_\nu D)(k)$, and integrating by parts (the boundary terms at $k=0$ and at infinity vanish) gives

$$\partial_{p_\nu}W\Big\rvert_{p=0}=i\gamma_\nu+ig^{2}\!\int\!\frac{d^{4}k}{(2\pi)^{4}}\,D(k)\Bigl(\frac{\gamma_\nu}{k^{2}}-\frac{2(\gamma\!\cdot\!k)\,k_\nu}{k^{4}}\Bigr). \tag{7}$$

Define $A=\int D/k^{2}$, $B_\parallel=\int D\,(k\!\cdot\!u)^{2}/k^{4}$ and $B_\perp=\tfrac13\int D\,k_\perp^{2}/k^{4}$, all with measure $d^{4}k/(2\pi)^{4}$. Rotation invariance about $u$ turns (7) into

$$W(p)=i\bigl[Z_\parallel\,(p\!\cdot\!u)\,(\gamma\!\cdot\!u)+Z_\perp\,(\gamma\!\cdot\!p_\perp)\bigr]+o(\lvert p\rvert),\qquad Z_\parallel=1+g^{2}(A-2B_\parallel),\quad Z_\perp=1+g^{2}(A-2B_\perp). \tag{8}$$

For $u$ the time axis, the pole of the propagator is at $Z_\parallel^{2}p_0^{2}+Z_\perp^{2}\vec p^{\,2}=0$, that is at $E=(Z_\perp/Z_\parallel)\lvert\vec p\rvert$ for $p_0=iE$. The fermion speed relative to the scalar is therefore

$$\delta c=\frac{Z_\perp}{Z_\parallel}-1=2g^{2}\,(B_\parallel-B_\perp)+O(g^{4}). \tag{9}$$

For an isotropic $D$ one has $B_\parallel=B_\perp=A/4$, so $\delta c=0$ and $Z=1+g^{2}A/2$, with $A=\frac{1}{16\pi^{2}}\ln(\Lambda^{2}/M^{2})$. This is the standard logarithmic wave-function factor $(g^{2}/32\pi^{2})\ln\Lambda^{2}$ of the Yukawa theory.

Theorem 8 (one-loop transfer from a frame in the scalar regulator). In the model (5), (6), let $f$ be in the class above and $M>0$. Then

$$\delta c=\frac{g^{2}}{16\pi^{2}}\;C\!\left(\tfrac{M}{\Lambda};f\right)+O(g^{4}),\qquad \lim_{M/\Lambda\to0}C\!\left(\tfrac{M}{\Lambda};f\right)=\frac13 . \tag{10}$$

Thus $\delta c\to g^{2}/48\pi^{2}$ for every such $f$. For the Gaussian $f=e^{-x^{2}}$ the correction is $O\bigl((M/\Lambda)^{2}\ln(\Lambda/M)\bigr)$. For a regulator that depends on the full four-momentum, $f=f(\lvert q\rvert/\Lambda)$, $C=0$.

Proof. We write (9) in four-dimensional polar coordinates about $u$. The regulator breaks isotropy only through the angle, and the shape of $f$ enters the radial integral through an additive constant that multiplies an angular average equal to zero.

Let $\chi$ be the angle between $k$ and $u$, so that $k\!\cdot\!u=\lvert k\rvert\cos\chi$, $\lvert k_\perp\rvert=\lvert k\rvert\sin\chi$ and $d^{4}k=4\pi\,\lvert k\rvert^{3}\sin^{2}\chi\,d\chi\,d\lvert k\rvert$. Then $B_\parallel-B_\perp=\int D\,(k\!\cdot\!u)^{2}/k^{4}-\tfrac13\int D\,k_\perp^{2}/k^{4}$ gives

$$\delta c=\frac{2g^{2}}{4\pi^{3}}\int_{0}^{\pi}\!d\chi\;w(\chi)\,R(\sin\chi),\qquad w(\chi)=\sin^{2}\chi\Bigl(\cos^{2}\chi-\tfrac13\sin^{2}\chi\Bigr), \tag{11}$$

with $R(\sin\chi)=\int_{0}^{\infty}\frac{y\,dy\,f(y\sin\chi/\Lambda)}{y^{2}+M^{2}}$. The weight $w$ is the quadrupole component of the angular distribution and has zero mean: $\int_0^\pi w\,d\chi=\pi/8-\tfrac13\cdot3\pi/8=0$.

In the radial integral put $x=y\sin\chi/\Lambda$ and $\varpi=M\sin\chi/\Lambda$, so that $R(\sin\chi)=\int_0^\infty x f(x)\,(x^{2}+\varpi^{2})^{-1}dx$. Split the range at $x=1$. On $(0,1)$ write $f=1+(f-1)$; since $f(x)-1=O(x^{2})$, the second term gives a convergent limit as $\varpi\to0$ by dominated convergence, and the first gives $\tfrac12\ln\bigl((1+\varpi^{2})/\varpi^{2}\bigr)$. On $(1,\infty)$ the integral converges to $\int_{1}^{\infty}f(x)\,dx/x$, finite because $f$ decays. Hence

$$R(\sin\chi)=\ln\frac{\Lambda}{M}-\ln\sin\chi+b_f+r(\varpi),\qquad b_f=\int_0^1\frac{f-1}{x}\,dx+\int_1^\infty\frac{f}{x}\,dx, \tag{12}$$

with $r(\varpi)\to0$ as $\varpi\to0$. Since $\varpi\le M/\Lambda$, the remainder is uniformly small in $\chi$ when $M/\Lambda$ is small. The constant $b_f$ and the term $\ln(\Lambda/M)$ do not depend on $\chi$.

Insert (12) in (11). The $\chi$-independent terms multiply $\int w\,d\chi=0$ and drop out. What remains is $-\int_0^\pi w(\chi)\ln\sin\chi\,d\chi$ plus a term that tends to zero. With $w=\sin^{2}\chi-\tfrac43\sin^{4}\chi$ and $I(m)=\int_0^\pi\sin^{m}\chi\,d\chi=\sqrt\pi\,\Gamma(\tfrac{m+1}{2})/\Gamma(\tfrac m2+1)$ one has $\int_0^\pi\sin^{m}\chi\ln\sin\chi\,d\chi=I'(m)$. Using the digamma values $\Psi(\tfrac32)=2-\gamma_{\mathrm E}-2\ln2$, $\Psi(2)=1-\gamma_{\mathrm E}$, $\Psi(\tfrac52)=\tfrac83-\gamma_{\mathrm E}-2\ln2$ and $\Psi(3)=\tfrac32-\gamma_{\mathrm E}$,

$$I'(2)=\frac\pi4\,(1-2\ln2),\qquad I'(4)=\frac{3\pi}{16}\Bigl(\frac76-2\ln2\Bigr),\qquad \int_0^\pi w\ln\sin\chi\,d\chi=I'(2)-\tfrac43I'(4)=-\frac{\pi}{24}.$$

Therefore $\delta c\to\frac{2g^{2}}{4\pi^{3}}\cdot\frac{\pi}{24}=\frac{g^{2}}{48\pi^{2}}$, which is (10) with $C=1/3$. For the Gaussian, $R(\sin\chi)=\tfrac12e^{\varpi^{2}}E_1(\varpi^{2})$ exactly, and its expansion gives $r=O(\varpi^{2}\ln\varpi)$. If $f$ depends on $\lvert k\rvert$ the radial integral does not depend on $\chi$ and (11) vanishes by $\int w=0$. $\square$

An isotropic regulator gives zero because the integrand of (9) is a quadrupole, whose angular average vanishes at each radius. A regulator that cuts off only the momentum orthogonal to $u$ removes more of the large-$\lvert q_\perp\rvert$ region at angles near $\pi/2$ than near the axis, and the induced angular dependence is the logarithm $-\ln\sin\chi$ of the infrared-sensitive radial integral. The shape of $f$ only changes the constant $b_f$, which the zero-mean weight removes.

Table 2 gives $C=\delta c/(g^2/16\pi^2)$ for three regulator shapes, computed from (11) by numerical quadrature.

$M/\Lambda$	$10^{-1}$	$10^{-2}$	$10^{-3}$	$10^{-4}$
$e^{-x^{2}}$	0.3256	0.3332	0.3333	0.3333
$e^{-x^{4}}$	0.3305	0.3333	0.3333	0.3333
$(1+x^{2})^{-2}$	0.3209	0.3331	0.3333	0.3333

Table 2. The coefficient $C(M/\Lambda;f)$.

The order of limits matters. With $M=0$ the double integral defining $B_\parallel-B_\perp$ is only conditionally convergent: integrating over the component along $u$ first at fixed $\lvert k_\perp\rvert$ gives zero, while the limit $M/\Lambda\to0$ of the absolutely convergent integral gives $1/3$. The mass must be kept until the end.

The result concerns a regulator on the scalar line only. The model of [Collins et al.], in which the regulator multiplies the fermion propagator and the quantity computed is the scalar self-energy, gives a Lorentz-violating coefficient $\xi=(g^2/6\pi^2)\bigl[1+2\int_0^\infty x f'(x)^2dx\bigr]$ that depends on the shape of $f$. Regulators placed on other lines, or on vertices, give other coefficients, and Theorem 8 gives universality in the shape of $f$ for this placement only.

For a gauge theory only an estimate is available. The coefficient multiplies a dimension-four operator, $c^{\mu\nu}\bar\psi\gamma_\mu\partial_\nu\psi$ with $c^{\mu\nu}\propto u^{\mu}u^{\nu}-\tfrac14\delta^{\mu\nu}$, and it does not depend on $\Lambda$, so a cutoff at the Planck scale gives no suppression. Replacing $g^{2}$ by $e^{2}$ gives $\delta c\sim\alpha/12\pi=1.9\times10^{-4}$ for $\alpha=1/137.036$. This is an order-of-magnitude replacement and not a computation: the factor from the gauge structure of the photon line is not included, and in a theory with several couplings the coefficients of different species need not coincide. A one-loop calculation of the photon speed in scalar electrodynamics with band-limited vertices, which select a frame, is in section 6.4 of a conserved lattice momentum and graviton scattering.

The data tables of [Kostelecký–Russell] list bounds on dimension-four coefficients of electrons, protons and neutrons that lie many orders of magnitude below $10^{-10}$, so a coefficient near $10^{-4}$ would need a cancellation by many orders of magnitude between independently renormalised parameters. The GW170817 observation bounds the difference between the speeds of gravity and light to between $-3\times10^{-15}$ and $+7\times10^{-16}$ of the speed of light [LIGO–Virgo et al.]. The Yukawa computation concerns a fermion and a scalar; applying it to the photon and the graviton would need the corresponding computation on those lines.

No internal symmetry forbids the operator, since it is a singlet. Exact Lorentz invariance of the regulator removes it; so does exact $H(4)$ invariance in Euclidean signature (section 1, where the only rank-two invariant is $\delta_{\mu\nu}$), and a supersymmetry with one algebra (4) fixes the speed within short multiplets (section 2).

5. Code distance and the ratio of speeds

Does the distance of a stabiliser code with Pauli perturbations equalise the speeds of two species? The reasoning that would make it so is that Lorentz-violating terms act as logical errors, which the code distance suppresses, while the kinetic terms do not. We pose the question for stabiliser codes. A species is pictured as a lattice field whose degrees of freedom are logical qubits of a code, with hopping amplitudes given by logical couplings; its speed is a function of those couplings, and the question is how the code distance enters that function. The first example below is a chain of this kind.

Let $\mathcal S$ be an abelian subgroup of the Pauli group on $n_{\mathrm q}$ qubits with $-1\notin\mathcal S$, with generators $S_s$, code space $\mathcal C$ (the joint $+1$ eigenspace) and projector $\Pi$. Phases are ignored throughout: two Pauli operators that differ by a phase count as equal, and $N(\mathcal S)/\langle\mathcal S\rangle$ is the set of logical classes, where $N(\mathcal S)$ is the normaliser of $\mathcal S$ in the Pauli group. The distance $d_{\mathrm c}$ is the least weight of an element of $N(\mathcal S)$ that is not, up to a phase, in $\mathcal S$. Let $H_0=-\Delta\sum_sS_s$ with $\Delta>0$ and $V=\sum_t\lambda_t P_t$ with $P_t$ Pauli strings and $\lambda_t$ arbitrary couplings. The low-energy Hamiltonian on $\mathcal C$ is the Schrieffer–Wolff series $H_{\mathrm{eff}}=\Pi e^{\mathcal T}(H_0+V)e^{-\mathcal T}\Pi$ [Bravyi–DiVincenzo–Loss], defined as a formal power series in the $\lambda_t$. We use only the following property of that series, shared by the Schrieffer–Wolff, Kato and Rayleigh–Schrödinger expansions: the term of order $N$ is a linear combination, with coefficients independent of the $\lambda_t$, of operators

$$\Pi\,P_{t_1}f_1(\mathbf S)\,P_{t_2}f_2(\mathbf S)\cdots f_{N-1}(\mathbf S)\,P_{t_N}\,\Pi, \tag{13}$$

where each $f_i$ is a function of the commuting stabilisers (energy denominators and projectors on syndrome sectors). Expand $H_{\mathrm{eff}}=\sum_La_L(\lambda)\,L$ in a basis of logical Pauli operators, one representative $L$ for each class in $N(\mathcal S)/\langle\mathcal S\rangle$.

Proposition 9 (disjoint monomial supports). Every monomial $\prod_t\lambda_t^{m_t}$ that appears in $a_L$ satisfies $[\prod_tP_t^{m_t}]=[L]$ in $N(\mathcal S)/\langle\mathcal S\rangle$. Hence $a_L$ and $a_{L'}$ for distinct classes have no monomial in common. If $V$ consists of single-qubit Paulis, every monomial in $a_L$ has degree at least the least weight in the class $[L]$, so at least $d_{\mathrm c}$ for $L\notin\langle\mathcal S\rangle$.

Proof. Between code projectors the stabilisers act as numbers, so only the Pauli product of an operator (13) matters, and a Pauli product determines its logical class.

A Pauli string either commutes or anticommutes with each stabiliser, so a function $f(\mathbf S)$ can be moved through $P_t$ at the price of flipping the arguments by signs. Pushing all $f_i$ to the right in (13) and acting on $\Pi$, where each stabiliser equals $+1$, turns (13) into a number times $\Pi R\Pi$ with $R=P_{t_1}\cdots P_{t_N}$, a Pauli string up to phase. If $R$ anticommutes with some $S_s$, then $\Pi R\Pi=0$. Otherwise $R\in N(\mathcal S)$ and $\Pi R\Pi=R\,\Pi=(\text{phase})\,L_R\Pi$, where $L_R$ is the logical representative of the class $[R]$. The product $R$ depends on the multiset $\{t_i\}$ only up to a phase, hence so does its class, and the monomial $\lambda_{t_1}\cdots\lambda_{t_N}$ feeds only $a_{L_R}$. For single-qubit $V$ the number of factors needed to build a string in a class is at least the weight of that class. $\square$

Proposition 9 gives a lower bound on the degree. Whether the coefficient at exactly that degree is non-zero depends on the code and on the perturbation, because sums over orderings with energy denominators can cancel; it is an assumption in Proposition 11 below. A species whose dynamics is carried by logical qubits has a speed determined by the coefficients $a_L$ of its logical Hamiltonian, and Proposition 9 says which microscopic couplings feed which $a_L$.

Corollary 10 (decoupled species). Suppose that the qubits split into two blocks, that $\mathcal S$ is generated by stabilisers each supported in one block, and that $V=V^{(1)}+V^{(2)}$ with the two terms acting on the respective blocks and containing disjoint sets of couplings $\lambda^{(1)},\lambda^{(2)}$. Then $H_{\mathrm{eff}}=H^{(1)}_{\mathrm{eff}}(\lambda^{(1)})+H^{(2)}_{\mathrm{eff}}(\lambda^{(2)})$, and the speeds are $c_\alpha=F_\alpha(\lambda^{(\alpha)})$. If $c_1=c_2$ holds on an open set of couplings, then both $F_1$ and $F_2$ are constant there. A relation between non-constant speeds is a restriction on the couplings, that is a tuning, or follows from a symmetry that maps the couplings of one species onto those of the other.

The splitting holds because $H_0$ and $V$ are sums of commuting block terms, so the Schrieffer–Wolff unitary is a product of unitaries each acting on its own block, and $\Pi$ is the product of the block projectors. An open set contains a product neighbourhood, which gives the statement about constants. The distance $d_{\mathrm c}$ enters only through the functions $F_\alpha$.

Proposition 11 (amplification). Let species 1 and 2 be as in Corollary 10, with $\epsilon$ the ratio of the perturbation to the gap $\Delta$. Assume that for each species the speed is set by the coefficient $a_{L_*}$ of one logical operator $L_*\notin\langle\mathcal S\rangle$, which has a non-zero homogeneous part $a_{L_*}^{(\ell)}$ of least degree $\ell$ in the couplings, in the form

$$c_\alpha=\kappa\,a^{(\ell)}_{L_*}(\lambda^{(\alpha)})\bigl(1+\rho(\lambda^{(\alpha)})\bigr),$$

where $\kappa$ is the same constant for both species and $\rho$ is a smooth function with $\rho=O(\epsilon^{2})$ and $\lambda\cdot\nabla\rho=O(\epsilon^{2})$. (The function $\rho$ collects the higher-degree parts of $a_{L_*}$ and the corrections to the speed law. The bound holds, for instance, when $a_{L_*}$ has no terms of degree $\ell+1$, which Proposition 9 does not exclude, and the corrections to the speed law are of relative order $\epsilon^2$, with $\lambda$-derivative of the same order, as in (15).) If the couplings of species 2 are those of species 1 multiplied by $1+\zeta$, then

$$\ln\frac{c_2}{c_1}=\ell\,\ln(1+\zeta)+O(\epsilon^{2}\zeta), \tag{14}$$

and $\ell\ge d_{\mathrm c}$ when $V$ is a sum of single-qubit terms.

Proof. The leading part $a_{L_*}^{(\ell)}$ is homogeneous of degree $\ell$ in the $\lambda$, hence is multiplied by $(1+\zeta)^{\ell}$. Along the ray $\lambda\mapsto(1+\zeta')\lambda$ the logarithm $\ln(1+\rho)$ has derivative $\lambda\cdot\nabla\rho/(1+\rho)=O(\epsilon^2)$ with respect to $\zeta'$, so it changes by $O(\epsilon^{2}\zeta)$ between $\lambda$ and $(1+\zeta)\lambda$. The bound $\ell\ge d_{\mathrm c}$ is Proposition 9. $\square$

So a 1% difference in the physical couplings of the two species produces a difference of about $\ell\%$ in their speeds. The distance reduces the absolute size of the logical couplings, here as $\epsilon^{d_{\mathrm c}}$, and does not reduce their ratios.

A repetition-code chain

Each species is a chain of blocks of $d_{\mathrm c}\ge3$ qubits with an intra-block coupling $-\Delta\,Z_iZ_{i+1}$ (the stabilisers), a field $-h\,X_i$ on every qubit, and a coupling $-J\,ZZ$ between the last qubit of one block and the first qubit of the next. The logical $\bar X=\prod X_i$ has weight $d_{\mathrm c}$ and is generated at order $d_{\mathrm c}$ in $h$, so the effective chain is a transverse-field Ising chain on logical qubits with field of order $h^{d_{\mathrm c}}/\Delta^{d_{\mathrm c}-1}$. The Jordan–Wigner transform gives a free Majorana chain whose quasiparticle energy is $2\varsigma(k)$, with $\varsigma(k)$ the smallest singular value of the $d_{\mathrm c}\times d_{\mathrm c}$ Bloch matrix $Y(k)$ with diagonal $h$, superdiagonal $-\Delta$ and corner entry $-Je^{ik}$. The chain is gapless when $J=h^{d_{\mathrm c}}/\Delta^{d_{\mathrm c}-1}$, which ties $J$ to the order-$d_{\mathrm c}$ coefficient of $\bar X$, and the speed per unit cell is $v=2\,d\varsigma/dk$ at $k=0$.

The speed follows in closed form. Put $\epsilon=h/\Delta$. At the critical $J$ the matrix $Y(0)$ has a one-dimensional right null space spanned by $\phi_j=\epsilon^{j-1}$ and a one-dimensional left null space spanned by $\bar\phi_j=\epsilon^{1-j}$, $j=1,\dots,d_{\mathrm c}$. First-order perturbation theory of the zero singular value gives $\varsigma(k)=\lvert\bar\phi^{T}Y'(0)\phi\rvert\,\lvert k\rvert/(\lVert\bar\phi\rVert\lVert\phi\rVert)$, where $Y'(0)$ has the single non-zero entry $-iJ$ in the corner. Since $J\bar\phi_{d_{\mathrm c}}\phi_1=h$ and $\lVert\phi\rVert\lVert\bar\phi\rVert=\epsilon^{1-d_{\mathrm c}}(1-\epsilon^{2d_{\mathrm c}})/(1-\epsilon^{2})$,

$$v=2\Delta\,\epsilon^{d_{\mathrm c}}\,\frac{1-\epsilon^{2}}{1-\epsilon^{2d_{\mathrm c}}}. \tag{15}$$

This is the speed law of Proposition 11 in the coupling $h$, with $\ell=d_{\mathrm c}$, $J$ fixed by criticality and $1+\rho=(1-\epsilon^{2})/(1-\epsilon^{2d_{\mathrm c}})$, and the proof of (14) applies to this one-parameter family. Let species 2 have $h$ larger by the factor $1+\zeta$, with $J$ re-tuned in both species. Then $d\ln v/d\ln h=d_{\mathrm c}-2\epsilon^{2}/(1-\epsilon^{2})+2d_{\mathrm c}\epsilon^{2d_{\mathrm c}}/(1-\epsilon^{2d_{\mathrm c}})$, which is $d_{\mathrm c}-0.20$ to two digits for $\epsilon=0.3$ and $d_{\mathrm c}\ge5$; the offset from $d_{\mathrm c}$ is the $O(\epsilon^{2}\zeta)$ term of (14). Hence $\ln(v_2/v_1)/\ln1.01$ is $4.80$ for $d_{\mathrm c}=5$ and $8.80$ for $d_{\mathrm c}=9$, so a 1% difference in $h$ becomes a 5% or a 9% difference in speed. If instead $h$, $\Delta$ and $J$ are all rescaled by $1+\zeta$, the speed rescales by $1+\zeta$ at every $d_{\mathrm c}$; the factor $d_{\mathrm c}$ is absent because the gap, against which the perturbative couplings are measured, scales with them.

A five-qubit code

For the $[[5,1,3]]$ code with $H_0=-\Delta\sum S_s$ ($\Delta=1$) and $V=\epsilon\sum_{q,a}\lambda_{q,a}\sigma^a_q$ with 15 couplings drawn once from a unit-variance Gaussian, numerical diagonalisation gives a splitting of the two lowest levels of order $\epsilon^{3}$ for $\epsilon$ between $10^{-3}$ and $1.6\times10^{-2}$. The coefficient of degree $d_{\mathrm c}=3$ is therefore non-zero for that draw and, being a polynomial in the couplings, for generic couplings. This is the non-vanishing hypothesis of Proposition 11 for this code at $d_{\mathrm c}=3$. No speed is computed for this code.

The proposition holds order by order in the series, for stabiliser codes with Pauli perturbations, and rests on the assumed speed law. The stability theorem for topologically ordered Hamiltonians [Bravyi–Hastings–Michalakis] protects quantities proportional to the identity on the code space, such as the degeneracy of the ground space. A speed is the ratio of two non-trivial logical couplings, which that theorem does not address. Holographic or approximate codes, non-Pauli perturbations and regimes in which the Schrieffer–Wolff series does not converge are outside the proposition.

6. Where a common limiting speed could survive

A common speed in real time from a group acting absolutely irreducibly on Euclidean momentum space, such as $H(4)$, would need a linearised diffeomorphism invariance that survives interactions, reflection positivity so that the continuation to real time exists, and no doublers; a Haar average over the frame in Euclidean signature (section 3) falls under the same conditions. Whether a substrate with these properties exists is open. For classical perturbative graviton theories on a Hamiltonian lattice in continuous time, a conserved lattice momentum shows (its Theorem 1) that a conserved lattice momentum with a real-analytic symbol is incompatible with a relativistic low-energy dispersion without three-wave resonances and non-vanishing low-energy graviton scattering. This obstructs exact diffeomorphism invariance there if that invariance supplies the conserved charge.

Whether a boost-free gravity theory has a supersymmetry of the form (4) acting on the graviton is open, and the size of the speed difference after soft breaking is known only through the estimate quoted in section 2.

A frame confined to the gravity sector, with no frame in the matter regulator, can have its transmitted violation suppressed as in [Pospelov–Shang]. The graviton speed $c_T$, in units of the matter speed, is then a free parameter of the gravity sector (Proposition 4), which must lie in the GW170817 window quoted in section 4. A tuned counterterm is the alternative, with one dimension-four coefficient per species and per order of perturbation theory; in the model of Theorem 8 it cancels the coefficient of (10).

Flows towards a common speed are analysed in other settings. Anber and Donoghue [Anber–Donoghue] give examples in which fields with different limiting speeds at high energy acquire a common one at low energy through interactions, with the differences normally vanishing only logarithmically and, in one model, as a power law. Iengo, Russo and Serone [Iengo–Russo–Serone] find in scalar Lifshitz-type models that a logarithmic running leaves Lorentz symmetry unrecovered to experimental accuracy unless the Lorentz-breaking terms are tuned to small values in the ultraviolet. Bednik, Pujolàs and Sibiryakov [Bednik–Pujolàs–Sibiryakov] find deviations suppressed by a power of the ratio of the infrared and ultraviolet scales in strongly coupled holographic examples.

The gauge-structure factor that converts the Yukawa coefficient $1/3$ into a photon–electron speed difference would settle whether the estimate of section 4 has the right size, and a two-loop calculation would show whether the one-loop coefficient survives. A calculation with regulators that act on all lines in a gauge-invariant way would show whether the independence of the shape of $f$ extends beyond a regulator on the scalar line.

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