New Approach to Quantum Field Theory for Arbitrary Observers in Electromagnetic Backgrounds

Abstract

A reformulation of fermionic QFT in electromagnetic backgrounds is presented which uses methods analogous to those of conventional multiparticle quantum mechanics. Emphasis is placed on the (Schrödinger picture) states of the system, described in terms of Slater determinants of Dirac states, and not on the field operator $\hatψ(x)$ (which is superfluous in this approach). The vacuum state `at time $τ$' is defined as the Slater determinant of a basis for the span of the negative spectrum of the `first quantized' Hamiltonian $\hat{H}(τ)$, thus providing a concrete realisation of the Dirac Sea. The general S-matrix element of the theory is derived in terms of time-dependent Bogoliubov coefficients, demonstrating that the S-matrix follows directly from the definition of inner product between Slater determinants. The process of `Hermitian extension', inherited directly from conventional multiparticle quantum mechanics, allows second quantized operators to be defined without appealing to a complete set of orthonormal modes, and provides an extremely straightforward derivation of the general expectation value of the theory. The concept of `radar time', advocated by Bondi in his work on k-calculus, is used to generalise the particle interpretation to an arbitrarily moving observer. A definition of particle results, which depends {\it only} on the observer's motion and the background present, not on any choice of coordinates or gauge, or of the particle detector. We relate this approach to conventional methods by comparing and contrasting various derivations. Our particle definition can be viewed as a generalisation to arbitrary observers of Gibbons' approach.