Discrete symmetries and non-Lorentzian tangent-space groups: the open directions of Weinberg's causality-based spin-statistics program in higher spacetime dimensions

Weinberg (1984) showed that for massive particles in any spacetime dimension $d \geq 4$ with Lorentz tangent-space group $O(d-1,1)$, microcausality (vanishing of field (anti)commutators at spacelike separation) forces the standard spin-statistics connection, an essentially unique field construction from particle-annihilation and antiparticle-creation operators, dimension-dependent antiparticle little-group transformation rules, and dimensional obstructions to self-conjugate particles. The paper left three extensions open: the discrete symmetries T, C, P; massless particles; and tangent-space groups beyond $O(d-1,1)$. Of these, the massless case was settled by Ohta (1985); the discrete symmetries and the tangent-space groups beyond $O(d-1,1)$ remain open. The open problem is to complete the causality-based program: derive the causality-consistent action of T, C, P on the causal field construction and its particle-antiparticle phase choices for all field types and dimensions, and extend the entire construction (field structure, statistics, antiparticle rules, self-conjugacy constraints) to local frame groups $G_T$ other than $O(d-1,1)$, reducing to the known results in the Lorentz limit.

Background

In relativistic quantum field theory, one-particle states are classified by the method of Wigner (1939): each state of a given mass carries a unitary irreducible representation of the Poincare group (the group of translations and Lorentz transformations of flat spacetime), and its spin content is governed by the little group, the subgroup of the Lorentz group $O(d-1,1)$ that leaves a standard reference momentum invariant. For a massive particle in $d$ spacetime dimensions the little group is the rotation group $O(d-1)$; for a massless particle it is the Euclidean group $E(d-2)$, whose Abelian translation subgroup must act trivially on physical states (otherwise one obtains continuous-spin representations carrying an infinite tower of helicity-like labels), leaving effectively $O(d-2)$.

A quantum field is an operator-valued distribution transforming covariantly under a finite-dimensional, generally non-unitary matrix representation of the Lorentz group. Microcausality (also called locality) is the requirement that two fields either commute or anticommute at spacelike separated points; it is necessary for the Lorentz invariance of scattering amplitudes. The spin-statistics connection asserts that this choice is not free: fields whose one-particle states transform as tensors (integer spin) must be quantized with commutators (bosons), while spinor-tensors (half-integer spin) require anticommutators (fermions).

Weinberg (1984) asked which of these structural facts follow from microcausality alone for massive particles in any spacetime dimension $d \geq 4$. Writing the field for a particle in an irreducible Lorentz representation as a linear combination of particle-annihilation and antiparticle-creation operators with coefficient functions fixed by covariance, he found that this construction is essentially unique for each Lorentz type; that microcausality forces the usual spin-statistics connection in every dimension; that antiparticles transform under the little group in the same way as the particles, except for some bosons and all fermions in $d = 5 + 4n$ dimensions, where the antiparticle realizes the inequivalent 'doubled' representation (in odd spacetime dimensions $d$, where the rotation group $O(d-1)$ acts on an even number of spatial dimensions, irreducible representations related by reversal of the invariant epsilon tensor, the outer automorphism induced by a spatial reflection, can be inequivalent, a phenomenon called representation doubling); and that these constraints explain why, in certain dimensions, some kinds of massive bosons or fermions cannot be their own antiparticles. A particle that is its own antiparticle corresponds to a real, or Majorana-type, field; which Majorana, Weyl (chiral), and Majorana-Weyl spinor types can exist at all is governed by $d$ mod 8 (Wetterich 1983).

The analysis presumes that physics in locally inertial frames is invariant under the Lorentz group $O(d-1,1)$ acting on orthonormal local frames; this is the tangent-space group (the structure group acting on local frames) of standard general relativity. In a companion paper, Weinberg (1984) studied 'quasi-Riemannian' theories of gravitation in $d > 4$ dimensions, in which the local invariance group $G_T$ of inertial frames is not the Lorentz group but is required to contain a subgroup acting as $O(3,1)$ on four of the coordinates and leaving the others invariant; such theories were of interest because they can yield chiral fermions in four dimensions without non-gravitational gauge fields.

The 1984 spin-statistics paper left three directions open: (i) the discrete symmetries, namely time reversal T (an antiunitary inversion of the time coordinate), charge conjugation C (particle-antiparticle interchange), and parity P (inversion of spatial coordinates), in particular their interplay with the causality-fixed relative phases between the particle and antiparticle terms of the fields; (ii) massless particles, whose little group differs; (iii) tangent-space groups other than $O(d-1,1)$.

Direction (ii) has since been resolved: Ohta (1985) showed that microcausality forces the usual spin-statistics connection for massless particles in any number of dimensions, with an essentially unique field construction, antiparticles transforming like particles under $O(d-2)$, and, unlike the massive case, no causality obstruction to self-conjugacy in any dimension; he also showed that the reversed statistics of Faddeev-Popov ghosts (the auxiliary fields of statistics opposite to the naive assignment, introduced when quantizing gauge theories) does not by itself violate causality. The complementary question of which $O(d-2)$ representations a massless particle of a given field type can furnish was analyzed by Weinberg (2020) and settled in full generality by the proof of his 'decapitation conjecture' (that the little-group representation is obtained from the field's Lorentz representation by deleting its highest row, in Young-tableau language) by Distler (2020). Direction (i) has received only short contemporaneous treatments, none of which supplies the operator- and phase-level T, C, P action on the causal fields of general field type: a general CPT theorem for Minkowski spacetimes of any dimension, valid for local Lorentz-invariant Lagrangians (Gavela and Nepomechie 1984; CPT denotes the combined operation of charge conjugation, parity, and time reversal), a study of the C, P, and T transformations of the massive spin-1/2 field in higher dimensions (Shimizu 1985), a study of the causality constraints on the T, C, P transformations in higher-dimensional spacetime (Chadha and Daniel 1986), and an analysis of the C, P, T transformation laws of fundamental massless spinor fields (Daniel 1987). Later studies of discrete symmetries in higher-dimensional and Kaluza-Klein settings (Wetterich 2011; Mankoč Borštnik and Nielsen 2014; Chakraborty et al. 2025) use other frameworks and do not address this causality-based T, C, P construction. Direction (iii) has only brief comments by Ohta (1985) and no systematic treatment: for a general local frame group $G_T$, the little group of massive and massless particles, the possible doubling of its representations, the causal field construction, the statistics assignment, and the self-conjugacy constraints all remain undetermined.

Problem Statement

The open core of the question left open by Weinberg's 1984 causality-based analysis of quantum fields in higher spacetime dimensions, after the massless case was resolved by later work, is the following.

(1) Discrete symmetries. For fields of massive and massless particles in $d \geq 4$ spacetime dimensions constructed from microcausality as in Weinberg (1984) and Ohta (1985), determine the transformations under time reversal T, charge conjugation C, and parity P that are consistent with causality: the action of T, C, and P on the field operators and on the one-particle states; the constraints these symmetries impose on the relative phases between the particle and antiparticle terms of the causal fields; the fate of the antiparticle little-group transformation rules (including the doubling in $d = 5 + 4n$ dimensions) under each symmetry; and the combined CPT structure in each dimension, determined at the operator and phase level for the causal fields. This must be done for arbitrary tensor and spinor-tensor field types, massive and massless, in all $d \geq 4$, completing and verifying the partial treatments of Shimizu (1985), Chadha and Daniel (1986), and Daniel (1987).

(2) General tangent-space groups. Extend the causality-based construction to spacetime theories whose local frame (tangent-space) group $G_T$ is not $O(d-1,1)$. The general class of such groups is the target; it includes the quasi-Riemannian tangent groups considered by Weinberg (1984), which contain a subgroup acting as $O(3,1)$ on four coordinates and leaving the others invariant. Parts (1) and (2) are independent sub-problems. For each class of groups $G_T$ addressed (with the causality structure defined by the invariant light cone of the Lorentz subgroup), determine: the little group of massive and massless particles and whether representation doubling occurs; the operator structure of causal fields and its uniqueness; the spin-statistics connection; the antiparticle transformation rules; and the dimensional constraints on self-conjugate (Majorana-type) particles. The results must reduce to those of Weinberg (1984) for massive fields and Ohta (1985) for massless fields when $G_T = O(d-1,1)$.

References

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