Abstract
There are eight possible Pin groups that can be used to describe the transformation behaviour 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.
Highlights
The space-time transformation behavior is governed by the Lorentz group Oð3; 1Þ, which comprises four connected components
We show that the consistent description of fermions in the presence of general relativity (GR) imposes severe restrictions on the choice of Pin group
We find that only two of the eight Pin groups are admissible: the group Pinþ 1⁄4 Pinþþ− and the group Pin− 1⁄4 Pin−−−. The source of these restrictions is the double cover of the frame bundle, which, in the context of GR, is needed in order to obtain an infinitesimal action of the space-time diffeomorphism group on the configuration space of fermions coupled to gravity
Summary
The space-time transformation behavior is governed by the Lorentz group Oð3; 1Þ, which comprises four connected components. We find that only two of the eight Pin groups are admissible: the group Pinþ 1⁄4 Pinþþ− and the group Pin− 1⁄4 Pin−−− The source of these restrictions is the double cover of the frame bundle, which, in the context of GR, is needed in order to obtain an infinitesimal action of the space-time diffeomorphism group on the configuration space of fermions coupled to gravity. Selecting the correct Pin groups is important from a fundamental point of view—it determines the transformation behavior of fermionic fields under reflections—and because the Pin group can affect observable quantities such as currents [13,14,15]. The two Pin groups Pinþ and Pin− that are compatible with GR are not the widely used Cliffordian Pin groups Pinð3; 1Þ and Pinð1; 3Þ
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