The direction of time in quantum field theory

The algebra of observables associated with a quantum field theory is invariant under the connected component of the Lorentz group and under parity reversal, but it is not invariant under time reversal. If we take general covariance seriously as a long-term goal, the algebra of observables should be time-reversal invariant, and any breaking of time-reversal symmetry will have to be described by the state over the algebra. In consequence, the modified algebra of observables is a presentation of a classical continuous random field. First some mathematical preliminaries are necessary. Quantum field theory is presented at an elementary level in terms of an operator-valued distribution, φ̂(x). That this is a distribution reflects the fact that φ̂(x) is not itself an operator that we can associate with a measurement; for an operator, we have to smooth the quantum field by averaging, to obtain φ̂f = ∫ f(x)φ̂(x)d4x, where the test function f(x) is generally taken to be a Schwartz space function, which is zero at infinity and smooth both in real space and, as f̃(k), in Fourier space. There are notational, conceptual, and mathematical advantages to working with the smeared operators φ̂f instead of with the operator-valued distribution φ̂(x), and we can always get back to operator-valued distributions, albeit improperly, by using Dirac delta functions. Routinely, φ̂f is expressed as the sum of non-observable creation and annihilation operators a†f and af , φ̂f = af + a † f∗ , where af and a † f∗ are both complex linear in f to ensure that φ̂f is complex linear. The quantized Klein-Gordon field, for example (because it is the most elementary non-interacting quantum field), can be straightforwardly presented in terms of commutation relations between creation and annihilation operators[1],