Dynamical foundation of Very Special Relativity: a causal, unitary SIM(2)- or HOM(2)-invariant theory and the natural size of its coefficients

Very Special Relativity (VSR) proposes that the space-time symmetry group of nature is a proper subgroup of the Poincare group - in its spurion-inaccessible avatars, HOM(2) or SIM(2) together with translations - rather than the full Lorentz group. Cohen and Glashow showed that such symmetries reproduce most consequences of special relativity while evading many Lorentz-violation searches, and explicitly acknowledged four open directions: no complete causal, unitary dynamical VSR theory was constructed; the allowed nonlocal operator structures were not classified; quantitative phenomenology was deferred; and no microscopic mechanism fixes the size of the VSR coefficients. Twenty years of follow-up literature has delivered effective nonlocal VSR gauge theories that are renormalizable and unitary (Alfaro, Rivelles and collaborators), a rigorous free-field construction (Lee), Finsler-geometric, noncommutative and background-field realizations, and an extensive phenomenology. However, causality and microcausality of the genuinely nonlocal operators are not established anywhere in the literature, no UV completion or microscopic origin exists, a Thomas-precession incompatibility claim remains contested, and the naturalness of the tiny VSR coefficients (e.g. the $DISIM_b(2)$ deformation parameter bounded by $|b| < 10^{-26}$) is unexplained. The open problem is the construction of a single causal, unitary, microscopic VSR-invariant dynamical framework - or a proof of obstruction - together with a mechanism explaining the natural size of its coefficients.

Background

Special relativity (SR) identifies the space-time symmetry of nature with the Poincare group: space-time translations together with the full connected Lorentz group of rotations $J_i$ and boosts $K_i$. Cohen and Glashow (2006) asked whether the exact symmetry could instead be a proper subgroup of the Poincare group, which they called Very Special Relativity (VSR). Restricting to subgroups that contain translations and at least the two-parameter abelian group $T(2)$ generated by $T_1 = K_x + J_y$ and $T_2 = K_y - J_x$, there are four relevant avatars: $T(2)$ itself; $E(2)$ (adding $J_z$; isomorphic to the Euclidean motions of a plane); $HOM(2)$ (adding $K_z$; orientation-preserving similarities, or homotheties, of a plane); and $SIM(2)$ (adding both $J_z$ and $K_z$; the four-parameter similitude group). Their defining property is that adjoining any one of the discrete transformations $P$, $T$, or $CP$ enlarges the subgroup to the full Lorentz group. Consequently any theory conserving one of these discrete symmetries is automatically Lorentz invariant, and since CP-violating effects in nature are small, VSR departures from SR are expected to be small as well.

In perturbative approaches to Lorentz violation one inserts fixed background tensors, called spurions, into local operators. This works for $T(2)$ and $E(2)$, which admit invariant tensors (for $E(2)$ the null four-vector $n = (1,0,0,1)$). The avatars $HOM(2)$ and $SIM(2)$, however, admit no invariant local tensors at all - they are spurion-inaccessible. Lorentz-violating effects respecting $HOM(2)$ or $SIM(2)$ can therefore only be nonlocal: they are built from the non-invariant null vector $n$, which is preserved by $T_1$, $T_2$ and rotations about the $z$ axis and merely rescales under boosts along $z$, so that ratios such as $(p_1 \cdot n)/(p_2 \cdot n)$ are invariant. A key theorem of the source paper is that VSR implies SR in the context of local quantum field theory (or if CP is conserved); a VSR world therefore cannot be a precisely local quantum field theory, and its dynamics must be formulated with genuinely nonlocal operators.

Despite this, the kinematic content of SR survives almost intact in the $SIM(2)$ and $HOM(2)$ avatars: time dilation, the velocity-addition law, the existence of a center-of-mass frame, and a universal isotropic maximal attainable velocity all hold, and the little groups (the subgroups of the symmetry group that fix a standard momentum) of massive and massless particles have one-dimensional representations, unlike the Poincare case. VSR also offers an unusual origin for neutrino masses: lepton-number-conserving neutrino masses, while not Lorentz invariant, are VSR invariant and require no sterile states; an anisotropic charged-lepton electric dipole moment (EDM) of order $d_l \sim (m_\nu/2m_l)^2(e/2m_l)$ is then tied to the neutrino-mass scale and lies near experimental sensitivity for $m_\nu^2 \sim 10^{-4}\,\mathrm{eV}^2$. With suitable analyticity assumptions, CPT invariance can even be derived from $SIM(2)$ invariance.

The source paper explicitly acknowledged what it did not do: (1) no complete causal, unitary dynamical theory invariant under $SIM(2)$ or $HOM(2)$ was constructed - the authors state they expect that such a theory can be formulated, but do not formulate it; (2) only prototype nonlocal amplitudes were exhibited, not a systematic classification of the allowed VSR-preserving nonlocal structures; (3) quantitative phenomenology for the proposed probes (CP-violating meson decays such as $B^0 \to J/\psi K_s$, charged-lepton EDMs, and neutrino observables) was deferred to later work; (4) the origin and natural size of the VSR coefficients were not derived from any microscopic mechanism.

Subsequent literature built a substantial but incomplete structure. Gibbons, Gomis and Pope showed that general VSR is Finsler geometry (a geometry whose line element is homogeneous of degree one in displacements but anisotropic), that the inhomogeneous group $ISIM(2)$ ($SIM(2)$ together with space-time translations) admits only a one-parameter physically acceptable deformation $DISIM_b(2)$, and they emphasized the naturalness puzzle: the deformation parameter is bounded by $|b| < 10^{-26}$, raising the question of why such a dimensionless Lorentz-violating constant is so small. Alfaro, Rivelles and collaborators constructed nonlocal VSR-invariant gauge theories, Abelian and non-Abelian, that are renormalizable and unitary, including an electroweak model with the Standard Model particle content that generates neutrino masses; their loop calculations rely on Mandelstam-Leibbrandt-type regularization, a prescription for evaluating integrals containing light-like vectors that preserves gauge invariance and power counting, here requiring an auxiliary null vector whose removal must be taken as a careful limit to restore $SIM(2)$. Lee gave a rigorous construction of quantum fields with a preferred direction from the irreducible unitary representations of $ISIM(2)$, finding that free massive spin-1/2 and spin-1 fields are local and satisfy the Dirac and Proca equations. Realizations of the VSR subgroups were found on spacetimes with light-like noncommutativity (Sheikh-Jabbari and Tureanu) and as effective theories emerging from ordinary QED in a rapidly oscillating plane-wave background (Ilderton). Two consistency questions remain. Das and Mohanty argued that VSR predicts the wrong Thomas precession - the SR kinematic rotation produced by successive non-collinear boosts, observable in atomic spin-orbit splittings and in the Thomas-BMT equation governing spin precession of particles in storage rings - claiming a factor-of-two excess in spin-orbit coupling and a $10^3$ excess in ring precession, which would rule VSR out; this is contested by Ganjitabar and Shojai, who find no observable departure. And Asorey, Falceto and Marmo generalized the classical no-interaction theorems (which forbid interactions between point particles under a full Poincare-symmetry implementation) to Poincare subgroups, showing that VSR-type subgroups weaken these constraints but still restrict the admissible interactions.

Problem Statement

The problem is to give the Very Special Relativity proposal a dynamical foundation: construct a complete, causal, unitary quantum dynamical theory - an action or Hamiltonian, together with its state space and quantization - whose exact space-time symmetry group is a spurion-inaccessible VSR subgroup, SIM(2) or HOM(2) (with space-time translations), or else prove rigorously that no such theory can exist.

A dynamical foundation of VSR is a single framework that establishes: (1) Consistency: exact invariance under the full VSR group (not enlarged to the Lorentz group by quantization or radiative corrections), unitarity (a unitary S-matrix or probability-conserving evolution), and causal behavior - a suitable microcausality or analyticity condition holding despite the necessarily nonlocal n-dependent operators such as $(p \cdot n)/(q \cdot n)$ - together with a resolution of, or escape from, the disputed incompatibility with observed spin-precession (Thomas-precession) data. (2) Structure: the systematic determination of the VSR-preserving nonlocal operator content of the theory - which interactions are allowed, their gauge invariance, and their relation to the one-dimensional little-group representations. (3) Origin and size: a microscopic mechanism (a UV completion, emergent-symmetry dynamics, or comparable) from which the theory derives and which fixes the size of its VSR coefficients, explaining why they are small enough to have evaded precision Lorentz-invariance tests (for instance the bound $|b| < 10^{-26}$ on the $DISIM_b(2)$ deformation parameter) while being plausibly connected to the observed neutrino-mass scale; the framework must reproduce the established VSR phenomenology (neutrino-mass phenomenology, charged-lepton EDM and g-2 bounds, meson-decay and clock-comparison constraints) as its low-energy limits.

Alternatively, a rigorous proof that no such theory can exist resolves the problem. The completion boundary is a single framework that establishes all three components above - consistency, structure, and origin and size - or a proof that this is impossible; effective models with unproven causality, an unclassified VSR-preserving operator content, or unexplained coefficients do not by themselves complete it.

References

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