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Pin Groups in General Relativity PHYSICAL REVIEW D 101, 021702(R) (2020) Rapid Communications Pin groups in general relativity Bas Janssens Institute of Applied Mathematics, Delft University of Technology, 2628 XE Delft, Netherlands (Received 16 September 2019; published 28 January 2020) There are eight possible Pin groups that can be used to describe the transformation behavior of fermions under parity and time reversal. We show that only two of these are compatible with general relativity, in the sense that the configuration space of fermions coupled to gravity transforms appropriately under the space- time diffeomorphism group. DOI: 10.1103/PhysRevD.101.021702 I. INTRODUCTION We derive these restrictions in the “universal spinor ” For bosons, the space-time transformation behavior is bundle approach for fermions coupled to gravity, as – – governed by the Lorentz group Oð3; 1Þ, which comprises developed in [3 5] for the Riemannian and in [6 9] for four connected components. Rotations and boosts are the Lorentzian case. However, our results remain valid in contained in the connected component of unity, the proper other formulations that are covariant under infinitesimal “ ” orthochronous Lorentz group SO↑ 3; 1 . Parity (P) and time space-time diffeomorphisms, such as the global approach ð Þ – reversal (T) are encoded in the other three connected of [2,10 12]. To underline this point, we highlight the role components of the Lorentz group, the translates of of the space-time diffeomorphism group in restricting the ↑ admissible Pin groups. SO ð3; 1Þ by P, T and PT. For fermions, the space-time transformation behavior is Selecting the correct Pin groups is important from a 3 1 fundamental point of view—it determines the transforma- governed by a double cover of Oð ; Þ. Rotations and — boosts are described by the unique simply connected tion behavior of fermionic fields under reflections but also ↑ ↑ because the Pin group can affect observable quantities such double cover of SO ð3; 1Þ, the spin group Spin ð3; 1Þ. as currents [13–15]. Due to their transparent definition in However, in order to account for parity and time reversal, ↑ terms of Clifford algebras, the “Cliffordian” Pin groups one needs to extend this cover from SO ð3; 1Þ to the full − − Pinð3; 1Þ¼Pinþ þ and Pinð1; 3Þ¼Pin þþ have attracted Lorentz group Oð3; 1Þ. much attention [13,16–19]. Remarkably, the two Pin groups This extension is by no means unique. There are no less − Pinþ and Pin that are compatible with GR are not the than eight distinct double covers of Oð3; 1Þ that agree widely used Cliffordian Pin groups Pinð3; 1Þ and Pinð1; 3Þ. with Spin↑ð3; 1Þ over SO↑ð3; 1Þ. They are the Pin groups abc Pin , characterized by the property that the elements ΛP Λ Λ2 − Λ2 II. THE LORENTZIAN METRIC and T covering P and T satisfy P ¼ a, T ¼ b and 2 ðΛPΛTÞ ¼ −c, where a, b and c are either 1 or −1 (cf. [1,2]). In order to establish notation, we briefly recall the frame In this paper, we show that the consistent description of or vierbein formalism for a Lorentzian metric g on a four- fermions in the presence of general relativity (GR) imposes dimensional space-time manifold M. μ severe restrictions on the choice of Pin group. In fact, we A frame ex based at x is a basis ea∂μ of the tangent space find that only two of the eight Pin groups are admissible: T M, with basis vectors labeled by a ¼ 0, 1, 2, 3. The space − − −−− x the group Pinþ ¼ Pinþþ and the group Pin ¼ Pin . FðMÞ of all frames (with arbitrary x) is called the frame The source of these restrictions is the double cover of the bundle, and we denote by FxðMÞ the set of frames with frame bundle, which, in the context of GR, is needed in base point x. Note that the group Glð4; RÞ of invertible order to obtain an infinitesimal action of the space-time 4 4 a × matrices Ab acts from the right on FxðMÞ, sending ex diffeomorphism group on the configuration space of 0 0μ μ b to the frame ex ¼ exA with e a ¼ ebAa. This action is free fermions coupled to gravity. 0 and transitive; any two frames ex and ex over the same point 0 a x are related by ex ¼ exA for a unique matrix Ab. For a given Lorentzian metric g, the orthonormal frame Published by the American Physical Society under the terms of bundle OgðMÞ ⊂ FðMÞ is the space of all orthonormal the Creative Commons Attribution 4.0 International license. μ μ ν frames e , satisfying gμνe e η . Since two orthonormal Further distribution of this work must maintain attribution to a a b ¼ ab 0 the author(s) and the published article’s title, journal citation, frames ex and ex over the same point x differ by a Lorentz 3 0 and DOI. Funded by SCOAP . transformation Λ, ex ¼ exΛ, the Lorentz group Oð3; 1Þ acts 2470-0010=2020=101(2)=021702(6) 021702-1 Published by the American Physical Society BAS JANSSENS PHYS. REV. D 101, 021702 (2020) g ψ∶ → g freely and transitively on the set OxðMÞ ⊂ FxðMÞ of by smooth maps M S that assign to each space-time orthonormal frames based at x. point x a spinor ψ x based at x. The configuration space for Specifying a metric g at x is equivalent to specifying the the fermionic fields at a fixed metric g is thus the space g g Γ g g set OxðMÞ of orthonormal frames. Since OxðMÞ ⊂ FxðMÞ ðS Þ of sections of the spinor bundle S . is an orbit under the action of the Lorentz group Oð3; 1Þ on FxðMÞ, specifying the metric at x is equivalent to picking a IV. FERMIONIC FIELDS COUPLED TO GR point in the orbit space RxðMÞ¼FxðMÞ=Oð3; 1Þ. This is We now wish to describe the configuration space for the set of equivalence classes ½ex of frames at x, where two 0 fermionic fields coupled to gravity. This is not simply the frames ex and ex are deemed equivalent if they differ by a 0 product of the configuration space of general relativity and Lorentz transformation Λ, ex ¼ exΛ. We denote the bundle that of a fermionic field; the main difficulty here is that the of all equivalence classes ½ex (with arbitrary x)byRðMÞ. g To describe fermions in the presence of GR, it will be very space S where the spinor field ψ takes values depends convenient to view a metric g on M as a section of RðMÞ;a on the metric g. A solution to this problem was proposed smooth map g∶M → RðMÞ that takes a point x to an in [3,4] for the Riemannian case, and in [6–9] for metrics of Lorentzian signature. In order to handle reflections, we equivalence class ½ex of frames at x. The configuration space [20] of general relativity can thus be seen as the space need to adapt this procedure as follows. ΓðRðMÞÞ of sections of the bundle RðMÞ. First, we choose a twofold cover of Glð4; RÞ that agrees ˜ 4 R 4 R with the universal cover Glþð ; Þ over Glþð ; Þ.In III. FERMIONIC FIELDS Sec. V we show that there are only two such covers, which, þ − IN A FIXED BACKGROUND for want of a better name, we will call Gin and Gin . Having made our choice of GinÆ, we choose what one may We start by describing fermionic fields on M in the call a Gin structure; a twofold cover u∶ Qˆ → FðMÞ with a presence of a fixed background metric g. In order to do this, GinÆ-action that is compatible with the Glð4; RÞ-action on a number of choices have to be made, especially if we wish FðMÞ. Corresponding to every (not necessarily orthogonal) to keep track of the transformation behavior of spinors frame e , there are thus two gin frames qˆ and qˆ 0 .If under parity and time reversal. x x x ˜ ∈ Æ ∈ 4 R The local transformation behavior is fixed by choosing A Gin covers A Glð ; Þ, then the two gin frames ˆ ˜ ˆ 0 ˜ one out of the eight possible Pin groups Pinabc, together corresponding to exA are qxA and qxA. Æ 3 1 with a (not necessarily C-linear) representation V that We denote by Pin the twofold cover of Oð ; Þ inside Æ ˆ Æ extends the spinor representation of Spin↑ð3; 1Þ ⊂ Pinabc. Gin . Choosing a Gin structure Q for the group Gin is g Æ For example, V consists of n copies of C4 in the case of n equivalent to choosing a Pin structure Q for the group Pin . ˆ ˆ Dirac fermions, and it consists of m copies of C2 in the case Indeed, for every GinÆ structure Q, the preimage Qg ⊂ Q of of m Majorana fermions [21]. OgðMÞ ⊂ FðMÞ under the map u∶ Qˆ → FðMÞ is a PinÆ- Once a Pin group has been selected, the second choice structure, since the restriction ug∶ Qg → OgðMÞ of u to Qg one has to make is a choice of Pin structure. A Pin structure intertwines the PinÆ-action on Qg with the action of the is a twofold cover u∶ Qg → OgðMÞ of the orthonormal Lorentz group Oð3; 1Þ on OgðMÞ. Conversely, every PinÆ- frame bundle, equipped with a Pinabc-action that is com- structure u∶ Qg → OgðMÞ gives rise to the associated GinÆ- g patible with the action of the Lorentz group on O ðMÞ. The structure Qˆ ¼ðQg ×GinÆÞ=PinÆ. This is the space of ˜ abc compatibility entails that if Λ ∈ Pin covers Λ ∈ Oð3; 1Þ, equivalence classes ½q ; A˜ , where ðq Λ˜ ; A˜ Þ is identified ˜ x x then uðq ΛÞ¼uðq ÞΛ for all pin frames q in Qg.
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