Abstract Here, we point out that interactions with time delay can be described at the quantum level using a multi-time wave function , i.e. a wave function depending on one spacetime variable per particle. In particular, such a wave function makes it possible to implement direct interaction along light cones (not mediated by fields), as in the Wheeler–Feynman formulation of electrodynamics. Our results are as follows. (1) We derive a covariant two-particle integral equation and discuss it in detail. (2) It is shown how this integral equation (or equivalently, a system of two integro-differential equations) can be understood as defining the time evolution of ψ in a consistent way. (3) We demonstrate that the equation has strong analogies with Wheeler–Feynman electrodynamics and therefore suggests a possible new quantization of that theory. (4) We propose two natural ways how the two-particle equation can be extended to N particles. It is shown that exactly one of them leads to the usual Schrödinger equation with Coulomb-type pair potentials if time delay effects are neglected.
Abstract Here, we point out that interactions with time delay can be described at the quantum level using a multi-time wave function , i.e. a wave function depending on one spacetime variable per particle. In particular, such a wave function makes it possible to implement direct interaction along light cones (not mediated by fields), as in the Wheeler–Feynman formulation of electrodynamics. Our results are as follows. (1) We derive a covariant two-particle integral equation and discuss it in detail. (2) It is shown how this integral equation (or equivalently, a system of two integro-differential equations) can be understood as defining the time evolution of ψ in a consistent way. (3) We demonstrate that the equation has strong analogies with Wheeler–Feynman electrodynamics and therefore suggests a possible new quantization of that theory. (4) We propose two natural ways how the two-particle equation can be extended to N particles. It is shown that exactly one of them leads to the usual Schrödinger equation with Coulomb-type pair potentials if time delay effects are neglected.
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.
Landau and Peierls wrote down the Hamiltonian of a simplified version of quantum electrodynamics in the particle-position representation. We present a multi-time version of their Schrödinger equation, which bears several advantages over their original equation: the time evolution equations are simpler and more natural; they are more transparent with respect to choice of gauge; and, perhaps most importantly, they are manifestly Lorentz covariant. We discuss properties of the multi-time equations. Along the way, we also discuss the Lorentz covariant 3d Dirac delta distribution for spacelike surfaces and the inner product of photon wave functions on spacelike surfaces in an arbitrary gauge.
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.
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.
In relativistic quantum theory, one sometimes considers integral equations for a wave function ψ(x1,x2) depending on two space-time points for two particles. A serious issue with such equations is that, typically, the spatial integral over ∣ψ∣2 is not conserved in time -- which conflicts with the basic probabilistic interpretation of quantum theory. However, here it is shown that for a special class of integral equations with retarded interactions along light cones, the global probability integral is, indeed, conserved on all Cauchy surfaces. For another class of integral equations with more general interaction kernels, asymptotic probability conservation from t=−∞ to t=+∞ is shown to hold true. Moreover, a certain local conservation law is deduced from the first result.
The physics of special relativity is applied to a wave-packet model of a particle without the usual de Broglie-like assumption of Planck's rule, E = hv. The velocity of the particle is equated to the group-velocity of the wave-packet to associate the particle with its wave-packet. With such an identification it is demonstrated using special relativity that the momentum four-vector, P = [p, E/c2] of the particle is parallel and proportional to a central wave-number four-vector, K0 = [k0. ω0/c2], of the particle's wave-packet. The constant of proportionality, relating these two four-vectors, has the dimension of action and by diffraction experiments it has the value h/2π where h is Planck's constant. This refinement of the de Broglie wave-like nature of a particle suggests that special relativity underlies quantum physics more deeply than heretofore believed. The above quantum rules are used to find the wave-packet function in the one-dimensional case of a non-relativistic free particle and a particle in a conservative force field of potential V (x). This wave-packet function is shown to satisfy an integral time-evolution equation which relates the wave function at time t0 to the function at a slightly later time t. The resulting integral equation is equivalent to the propagator equation or path-integral hypothesised by Feynman. This fact also demonstrates, using the method of Feynman, that the wave-packet function satisfies Schrödinger's partial differential equation. The wave-packet function of a particle in a force field is directly shown to satisfy a linear integral equation which relates the wave-packet at one instant of time to its values at all later times. The kernel or propagator of this integral equation is also equivalent to Feynman's path integral. A computation demonstrates that the position of a moving particle in a simple quadratic potential well does not become more uncertain with the passage of time. This contrasts with the behaviour of a moving free particle which as a function of time becomes more and more difficult to accurately locate.