Multitime Schrödinger equations cannot contain interaction potentials
Abstract
Multitime wave functions are wave functions that have a time variable for every particle, such as ϕ(t{sub 1},x{sub 1},...,t{sub N},x{sub N}). They arise as a relativistic analog of the wave functions of quantum mechanics but can be applied also in quantum field theory. The evolution of a wave function with N time variables is governed by N Schrödinger equations, one for each time variable. These Schrödinger equations can be inconsistent with each other, i.e., they can fail to possess a joint solution for every initial condition; in fact, the N Hamiltonians need to satisfy a certain commutator condition in order to be consistent. While this condition is automatically satisfied for noninteracting particles, it is a challenge to set up consistent multitime equations with interaction. We prove for a wide class of multitime Schrödinger equations that the presence of interaction potentials (given by multiplication operators) leads to inconsistency. We conclude that interaction has to be implemented instead by creation and annihilation of particles, which, in fact, can be done consistently [S. Petrat and R. Tumulka, “Multitime wave functions for quantum field theory,” Ann. Physics (to be published)]. We also prove the following result: When a cutoff length δ > 0 ismore »
 Authors:
 Mathematisches Institut, LudwigMaximiliansUniversität, Theresienstr. 39, 80333 München (Germany)
 Department of Mathematics, Rutgers University, 110 Frelinghuysen Road, Piscataway, New Jersey 088548019 (United States)
 Publication Date:
 OSTI Identifier:
 22250893
 Resource Type:
 Journal Article
 Resource Relation:
 Journal Name: Journal of Mathematical Physics; Journal Volume: 55; Journal Issue: 3; Other Information: (c) 2014 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; ANNIHILATION; COMMUTATORS; EQUATIONS; HAMILTONIANS; LORENTZ INVARIANCE; MATHEMATICAL SOLUTIONS; POTENTIALS; QUANTUM FIELD THEORY; QUANTUM MECHANICS; RELATIVISTIC RANGE; WAVE FUNCTIONS
Citation Formats
Petrat, Sören, Email: petrat@math.lmu.de, and Tumulka, Roderich, Email: tumulka@math.rutgers.edu. Multitime Schrödinger equations cannot contain interaction potentials. United States: N. p., 2014.
Web. doi:10.1063/1.4867524.
Petrat, Sören, Email: petrat@math.lmu.de, & Tumulka, Roderich, Email: tumulka@math.rutgers.edu. Multitime Schrödinger equations cannot contain interaction potentials. United States. doi:10.1063/1.4867524.
Petrat, Sören, Email: petrat@math.lmu.de, and Tumulka, Roderich, Email: tumulka@math.rutgers.edu. 2014.
"Multitime Schrödinger equations cannot contain interaction potentials". United States.
doi:10.1063/1.4867524.
@article{osti_22250893,
title = {Multitime Schrödinger equations cannot contain interaction potentials},
author = {Petrat, Sören, Email: petrat@math.lmu.de and Tumulka, Roderich, Email: tumulka@math.rutgers.edu},
abstractNote = {Multitime wave functions are wave functions that have a time variable for every particle, such as ϕ(t{sub 1},x{sub 1},...,t{sub N},x{sub N}). They arise as a relativistic analog of the wave functions of quantum mechanics but can be applied also in quantum field theory. The evolution of a wave function with N time variables is governed by N Schrödinger equations, one for each time variable. These Schrödinger equations can be inconsistent with each other, i.e., they can fail to possess a joint solution for every initial condition; in fact, the N Hamiltonians need to satisfy a certain commutator condition in order to be consistent. While this condition is automatically satisfied for noninteracting particles, it is a challenge to set up consistent multitime equations with interaction. We prove for a wide class of multitime Schrödinger equations that the presence of interaction potentials (given by multiplication operators) leads to inconsistency. We conclude that interaction has to be implemented instead by creation and annihilation of particles, which, in fact, can be done consistently [S. Petrat and R. Tumulka, “Multitime wave functions for quantum field theory,” Ann. Physics (to be published)]. We also prove the following result: When a cutoff length δ > 0 is introduced (in the sense that the multitime wave function is defined only on a certain set of spacelike configurations, thereby breaking Lorentz invariance), then the multitime Schrödinger equations with interaction potentials of range δ are consistent; however, in the desired limit δ → 0 of removing the cutoff, the resulting multitime equations are interactionfree, which supports the conclusion expressed in the title.},
doi = {10.1063/1.4867524},
journal = {Journal of Mathematical Physics},
number = 3,
volume = 55,
place = {United States},
year = 2014,
month = 3
}

Multitime wave functions are wave functions that have a time variable for every particle, such as ϕ(t{sub 1},x{sub 1},...,t{sub N},x{sub N}). They arise as a relativistic analog of the wave functions of quantum mechanics but can be applied also in quantum field theory. The evolution of a wave function with N time variables is governed by N Schrödinger equations, one for each time variable. These Schrödinger equations can be inconsistent with each other, i.e., they can fail to possess a joint solution for every initial condition; in fact, the N Hamiltonians need to satisfy a certain commutator condition in ordermore »

Consistency of multitime Dirac equations with general interaction potentials
In 1932, Dirac proposed a formulation in terms of multitime wave functions as candidate for relativistic manyparticle quantum mechanics. A wellknown consistency condition that is necessary for existence of solutions strongly restricts the possible interaction types between the particles. It was conjectured by Petrat and Tumulka that interactions described by multiplication operators are generally excluded by this condition, and they gave a proof of this claim for potentials without spincoupling. Under suitable assumptions on the differentiability of possible solutions, we show that there are potentials which are admissible, give an explicit example, however, show that none of them fulfills themore » 
Infinitely many solutions for the nonlinear Schrödinger equations with magnetic potentials in R{sup N}
In this paper, we study a nonlinear Schrödinger equations with magnetic potentials in R{sup N} involving subcritical growth. Under some decaying and weak symmetry conditions of both electric and magnetic potentials, we prove that the equation has infinitely many nonradial complexvalued solutions by applying the finite reduction method. 
Standing waves for coupled nonlinear Schrödinger equations with decaying potentials
We study the following singularly perturbed problem for a coupled nonlinear Schrödinger system which arises in BoseEinstein condensate: −ε{sup 2}Δu + a(x)u = μ{sub 1}u{sup 3} + βuv{sup 2} and −ε{sup 2}Δv + b(x)v = μ{sub 2}v{sup 3} + βu{sup 2}v in R{sup 3} with u, v > 0 and u(x), v(x) → 0 as x → ∞. Here, a, b are nonnegative continuous potentials, and μ{sub 1}, μ{sub 2} > 0. We consider the case where the coupling constant β > 0 is relatively large. Then for sufficiently small ε > 0, we obtain positive solutions of this systemmore » 
Groupinvariant solutions of semilinear Schrödinger equations in multidimensions
Symmetry group methods are applied to obtain all explicit groupinvariant radial solutions to a class of semilinear Schrödinger equations in dimensions n ≠ 1. Both focusing and defocusing cases of a power nonlinearity are considered, including the special case of the pseudoconformal power p = 4/n relevant for critical dynamics. The methods involve, first, reduction of the Schrödinger equations to groupinvariant semilinear complex 2nd order ordinary differential equations (ODEs) with respect to an optimal set of onedimensional point symmetry groups, and second, use of inherited symmetries, hidden symmetries, and conditional symmetries to solve each ODE by quadratures. Through Noether's theorem,more »