Linear bosonic and fermionic quantum gauge theories on curved spacetimes

Abstract

We develop a general setting for the quantization of linear bosonic and fermionic field theories subject to local gauge invariance and show how standard examples such as linearised Yang-Mills theory and linearised general relativity fit into this framework. Our construction always leads to a well-defined and gauge-invariant quantum field algebra, the centre and representations of this algebra, however, have to be analysed on a case-by-case basis. We discuss an example of a fermionic gauge field theory where the necessary conditions for the existence of Hilbert space representations are not met on any spacetime. On the other hand, we prove that these conditions are met for the Rarita-Schwinger gauge field in linearised pure $N=1$ supergravity on certain spacetimes, including asymptotically flat spacetimes and classes of spacetimes with compact Cauchy surfaces. We also present an explicit example of a supergravity background on which the Rarita-Schwinger gauge field can not be consistently quantized.

Appendix A: Spinor and gamma-matrix conventions

We review some aspects of spinors in higher dimensions following [35], being mainly interested in properties of Majorana spinors. Let $D \text{ mod } 8=2,3,4$ and we denote by $\eta ^{ab} = \text{ diag }\left( -,+,+,\dots ,+\right) ^{ab}$ the $D$-dimensional Minkowski metric. The $\gamma$-matrices $\gamma ^a$, $a=0,\dots ,D-1$, are complex $2^{\lfloor D/2\rfloor }\times 2^{\lfloor D/2\rfloor }$-matrices satisfying the Clifford algebra relations $\{\gamma ^a,\gamma ^b\}=2\,\eta ^{ab}$. We take the timelike $\gamma$-matrix to be antihermitian ${\gamma ^0}^\dagger =-\gamma ^0$ and the spatial $\gamma$-matrices hermitian ${\gamma ^i}^\dagger = \gamma ^i$, for all $i=1,\dots ,D-1$. We further fix $\beta := i\gamma ^0$ which satisfies $\beta ^\dagger =\beta$. There exists a charge conjugation matrix $C$, which is antisymmetric, i.e. $C^{\mathrm{T}}=-C$, in the dimensions we are considering, see Table 1 in [35]. Further properties are $C^\dagger = C^{-1}$ and, for all $a=0,\dots ,D-1$,

$$\begin{aligned} {\gamma ^a}^{\mathrm{T}} = - C\gamma ^a C^{-1}. \end{aligned}$$

(7.1)

We define the charge conjugation operation on spinors $\chi \in \mathbb C ^{2^{\lfloor D/2\rfloor }}$ by

$$\begin{aligned} \chi ^c := -\beta \,C^*\, \chi ^*, \end{aligned}$$

(7.2)

where $*$ denotes component-wise complex conjugation. This operation squares to the identity, ${\chi ^c}^c =\chi$, for all $\chi$. A Majorana spinor is defined by the reality condition $\chi ^c =\chi$ and the space of Majorana spinors is a real vector space of dimension $2^{\lfloor D/2\rfloor }$. For every Majorana spinor $\chi$ the Dirac adjoint equals the Majorana adjoint, $\overline{\chi } := \chi ^\dagger \beta = \chi ^{\mathrm{T}} C$, and thus the hermitian structure $\overline{\chi }\lambda$ on Dirac spinors equivalently reads for Majorana spinors

$$\begin{aligned} \overline{\chi }\lambda = \chi ^{\mathrm{T}}C\lambda = -\lambda ^{\mathrm{T}} C\chi , \end{aligned}$$

(7.3)

where in the last equality we have used that $C^{\mathrm{T}} = -C$. We thus have a non-degenerate $\mathbb R$-bilinear antisymmetric map $\chi ^{\mathrm{T}}C\lambda$ on the space of Majorana spinors. However, this map takes values in the purely imaginary numbers $i\mathbb R$ and therefore should be rescaled by the imaginary unit in order to take values in the reals $\mathbb R$.