The Pin Groups in Physics: C, P and T

Abstract

A simple, but not widely known, mathematical fact concerning the coverings of the full Lorentz group sheds light on parity and time reversal transformations of fermions. Whereas there is, up to an isomorphism, only one Spin group which double covers the orientation preserving Lorentz group, there are two essentially different groups, called Pin groups, which cover the full Lorentz group. Pin(1,3) is to O (1,3) what Spin(1,3) is to SO (1,3). The existence of two Pin groups offers a classification of fermions based on their properties under space or time reversal finer than the classification based on their properties under orientation preserving Lorentz transformations -provided one can design experiments that distinguish the two types of fermions. Many promising experimental setups give, for one reason or another, identical results for both types of fermions. These negative results are reported here because they are instructive. Two notable positive results show that the existence of two Pin groups is relevant to physics:• In a neutrinoless double beta decay, the neutrino emitted and reabsorbed in the course of the interaction can only be described in terms of Pin(3,1).• If a space is topologically nontrivial, the vacuum expectation values of Fermi currents defined on this space can be totally different when described in terms of Pin(1,3) and Pin(3,1).Possibly more important than the two above predictions, the Pin groups provide a simple framework for the study of fermions; it makes possible clear definitions of intrinsic parities and time reversal; it clarifies colloquial, but literally meaningless, statements. Given the difference between the Pin group and the Spin group it is useful to distinguish their representations, as groups of transformations on "pinors" and "spinors", respectively.The Pin(1,3) and Pin(3,1) fermions are twin-like particles whose behaviors differ only under space or time reversal.A section on Pin groups in arbitrary spacetime dimensions is included.

Global Symmetries, Counterterms, and Duality in Chern-Simons Matter Theories with Orthogonal Gauge Groups

“…By substituting the value for the background B 2 = 1 4 dA we find the mixed anomaly 1 8π dAdA between the time-reversal symmetry and the U (1) symmetry. 40 This is the same parity anomaly as that of one massless Dirac fermion. The lines can carry fractional U (1) charges determined by their charges under the one-form symmetry.…”

Universal coverings of orthogonal groups

“…The Pin and Spin groups (Clifford-Lipschitz groups) widely used in algebraic topology [12,44,4,49,50,51], in the definition of pinor and spinor structures on the riemannian manifolds [59,41,47,82,28,29,56,31,27,1,17,18,2], spinor bundles [67,68,37,36,38], and also have great importance in the theory of the Dirac operator on manifolds [8,7,77,35,3]. The Clifford-Lipschitz groups also intensively used in theoretical physics [23,32,34,69,33,10,19].…”

Space and time dimensions of algebras with application to Lorentzian noncommutative geometry and quantum electrodynamics

“…Indeed, in Minkowski space time we want plane wave solutions of the (massive) Dirac equation: ψ(x) = ue ikµx µ . Compatibility with the dispersion relation k 2 = m 2 e , where m e is the fermion mass, implies the metric (+, −, −, −) for the West-coast convention and (−, +, +, +) for the East-coast one [22].…”

“…Chargeconjugation symmetry requires JD = DJ and the reality of the fermionic Lagrangian implies that D is self-adjoint with respect to the indefinite inner product. Thus, the West-coast convention corresponds to J = J − and η = η + while the East-coast one to J = J + and η = η − [22]. This is related to the signature of the metric.…”