Initial state propagators

It is possible to define a general initial state for a quantum field by introducing a contribution to the action defined at an initial-time boundary. The propagator for this theory is composed of two parts, one associated with the free propagation of fields and another produced by the operators of this initial action. The derivation of this propagator is shown for the case of a translationally and rotationally invariant initial state. In addition to being able to treat more general states, these techniques can also be applied to effective field theories that start from an initial time. The eigenstates of a theory with interacting heavy and light fields are different from the eigenstates of the theory in the limit where the interactions vanish. Therefore, a product of states of the noninteracting heavy and light theories will usually contain excitations of the heavier state once the interactions are included. Such excitations appear as nonlocal effects in the effective theory, which are suppressed by powers of the mass of the heavy field. By appropriately choosing the initial action, these excitations can be excised from the state leaving just effects that would be produced by a local action of the lighter fields. QUANTUM field theory is typically used for systems in simple quantum states. For describing scattering processes this is appropriate. To an excellent approximation the particles that participate in or result from a scattering process can be treated as the free-particle states of a noninteracting theory when looking long before or long after a collision has occurred. Propagation is always made in reference to vacuum states defined in a far past and a distant future. Being able to use these states brings many boons. Subtleties such as which vacuum to use—the ground state of the free or of the interacting theory—largely do not matter, diagrammatic calculations permit many short-cuts such as the amputation of external legs, and the propagator which is the basis of the perturbative description of processes is of the simplest possible form. However, in other physical settings being limited to only the asymptotic vacuum and the free particle states is a bit too restrictive. It might not be practical, or even theoretically sound, in a particular system to define a state in an infinitely distant past. Avoiding general excited states also means missing out on the possibility of describing other interesting dynamics—for example, systems which are not in an equilibrium state or the behaviour of quantum fields in the very early universe. Given the central role of scattering processes in the development of quantum field theory, it is not surprising that the treatment of more complicated quantum states, other than the thermal state, have received far less attention. But the importance of quantum fields in cosmology has encouraged an interest in a broader understandingof fields in more general †Electronic address: [email protected]