Fermion quantum field theory on curved and non-inertial backgrounds in standard-Minkowski form
Abstract
Quantum field theory on curved and non-inertial backgrounds contains background- and foliation-dependent quantities in the canonical Lagrangian, the hypersurface inner product and bilinear form, as well as in the equal-time anti-commutation relations. In this work, we determine a local fermion-field redefinition that brings these canonical structures into their standard-Minkowski forms, i. e., the forms they assume in Cartesian inertial coordinates on Minkowski spacetime, where the zeroth world coordinate is identified as the coordinate of time. Starting from the generally covariant Dirac action minimally coupled to a spin-1 gauge field, we derive the corresponding Lagrangian, fermionic inner product, and quantization rule in an Arnowitt-Deser-Misner decomposition, formulated in arbitrary world coordinates. We identify the generalized temporal gamma matrix as the common geometric factor governing the canonical temporal structure of all three quantities. Using a field redefinition, we transform this generalized temporal gamma matrix to its standard-Minkowski form, thereby mapping the fermionic inner product and the equal-time anti-commutation relation to their standard-Minkowski expressions, while transferring the explicit background and foliation dependence to the transformed Lagrangian and fermion-field operators. We show that such a field redefinition necessarily consists of a local rescaling and a fixing of the local Lorentz frame. This procedure restores the conventional canonical normalization from standard-Minkowski spacetime used for fermionic mode quantization and occupation-number operators. The transformed Lagrangian consequently assumes a generalized first-order Schrödinger form, leading to the familiar rest-energy term and spacetime-magnetic couplings, as well as to the leading non-relativistic limit, in which temporal derivatives are separated from spatial ones.
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BibTeX
@misc{pumpo2026fermion,
title = {{Fermion quantum field theory on curved and non-inertial backgrounds in standard-Minkowski form}},
author = {Fabio Di Pumpo and Tobias Asano and Alexander Friedrich and Gil Paz and Enno Giese},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11738},
eprint = {2609.11738},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11738},
}
RIS
TY - UNPB
TI - Fermion quantum field theory on curved and non-inertial backgrounds in standard-Minkowski form
AU - Fabio Di Pumpo
AU - Tobias Asano
AU - Alexander Friedrich
AU - Gil Paz
AU - Enno Giese
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.11738
UR - https://arxiv.org/abs/2609.11738
AB - Quantum field theory on curved and non-inertial backgrounds contains background- and foliation-dependent quantities in the canonical Lagrangian, the hypersurface inner product and bilinear form, as well as in the equal-time anti-commutation relations. In this work, we determine a local fermion-field redefinition that brings these canonical structures into their standard-Minkowski forms, i. e., the forms they assume in Cartesian inertial coordinates on Minkowski spacetime, where the zeroth world coordinate is identified as the coordinate of time. Starting from the generally covariant Dirac action minimally coupled to a spin-1 gauge field, we derive the corresponding Lagrangian, fermionic inner product, and quantization rule in an Arnowitt-Deser-Misner decomposition, formulated in arbitrary world coordinates. We identify the generalized temporal gamma matrix as the common geometric factor governing the canonical temporal structure of all three quantities. Using a field redefinition, we transform this generalized temporal gamma matrix to its standard-Minkowski form, thereby mapping the fermionic inner product and the equal-time anti-commutation relation to their standard-Minkowski expressions, while transferring the explicit background and foliation dependence to the transformed Lagrangian and fermion-field operators. We show that such a field redefinition necessarily consists of a local rescaling and a fixing of the local Lorentz frame. This procedure restores the conventional canonical normalization from standard-Minkowski spacetime used for fermionic mode quantization and occupation-number operators. The transformed Lagrangian consequently assumes a generalized first-order Schrödinger form, leading to the familiar rest-energy term and spacetime-magnetic couplings, as well as to the leading non-relativistic limit, in which temporal derivatives are separated from spatial ones.
ER -