Multi-time wave functions for quantum field theory

Sören Petrat, Roderich Tumulka

In a recent work (Petrat and Tumulka 2013 arXiv:1309.0802), we have described a formulation of a model quantum field theory in terms of a multi-time wave function and proposed a suitable system of multi-time Schrödinger equations governing the evolution of that wave function. Here, we provide further evidence that multi-time wave functions provide a viable formulation of relevant quantum field theories by describing a multi-time formulation, analogous to the one in Petrat and Tumulka (2013 arXiv:1309.0802), of another model quantum field theory. This model involves three species of particles, say x -particles, anti- x -particles, and y -particles, and postulates that a y -particle can decay into a pair consisting of an x and an anti- x particle, and that an x –anti- x pair, when they meet, annihilate each other creating a y -particle. (Alternatively, the model can also be interpreted as representing beta decay.) The wave function is a multi-time version of a time-dependent state vector in Fock space (or rather, the appropriate product of Fock spaces) in the particle-position representation. We write down multi-time Schrödinger equations and verify that they are consistent, provided that an even number of the three particle species involved are fermionic.

Matthias Lienert, Sören Petrat, Roderich Tumulka

Roderich Tumulka

In a recent work (Petrat and Tumulka 2013 arXiv:1309.0802), we have described a formulation of a model quantum field theory in terms of a multi-time wave function and proposed a suitable system of multi-time Schrödinger equations governing the evolution of that wave function. Here, we provide further evidence that multi-time wave functions provide a viable formulation of relevant quantum field theories by describing a multi-time formulation, analogous to the one in Petrat and Tumulka (2013 arXiv:1309.0802), of another model quantum field theory. This model involves three species of particles, say x-particles, anti-x-particles, and y-particles, and postulates that a y-particle can decay into a pair consisting of an x and an anti-x particle, and that an x–anti-x pair, when they meet, annihilate each other creating a y-particle. (Alternatively, the model can also be interpreted as representing beta decay.) The wave function is a multi-time version of a time-dependent state vector in Fock space (or rather, the appropriate product of Fock spaces) in the particle-position representation. We write down multi-time Schrödinger equations and verify that they are consistent, provided that an even number of the three particle species involved are fermionic.

Matthias Lienert, Sören Petrat, Roderich Tumulka

Matthias Lienert, Sören Petrat, Roderich Tumulka

In non-relativistic quantum mechanics of N particles in three spatial dimensions, the wave function ψ ( q 1 , ..., q N , t ) is a function of 3 N position coordinates and one time coordinate. It is an obvious idea that in a relativistic setting, such functions should be replaced by ϕ (( t 1 , q 1 ), ..., ( t N , q N )), a function of N space-time points called a multi-time wave function because it involves N time variables. Its evolution is determined by N Schrödinger equations, one for each time variable; to ensure that simultaneous solutions to these N equations exist, the N Hamiltonians need to satisfy a consistency condition. This condition is automatically satisfied for non-interacting particles, but it is not obvious how to set up consistent multi-time equations with interaction. For example, interaction potentials (such as the Coulomb potential) make the equations inconsistent, except in very special cases. However, there have been recent successes in setting up consistent multi-time equations involving interaction, in two ways: either involving zero-range ( δ potential) interaction or involving particle creation and annihilation. The latter equations provide a multi-time formulation of a quantum field theory. The wave function in these equations is a multi-time Fock function, i.e., a family of functions consisting of, for every n = 0, 1, 2, ..., an n -particle wave function with n time variables. These wave functions are related to the Tomonaga–Schwinger approach and to quantum field operators, but, as we point out, they have several advantages.

Wayne Nicholas Polyzou, Tracie Michlin, Fatih Bulut