Multi-time wave functions for quantum field theory

Petrat, Sören; Tumulka, Roderich

doi:10.1016/j.aop.2014.03.004

Cited by 46 publications

(145 citation statements)

References 22 publications

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“…For a variable number of particles, a multi-time wave function becomes a so-called multitime Fock function (see [19] and also [5,6]). It can be represented as a sequence of n-particle wave functions ψ (n) , ψ = (ψ (0) , ψ (1) , ψ (2) , ψ (3) , ...).…”

Section: Review Of Important Conceptsmentioning

confidence: 99%

Multi-time formulation of particle creation and annihilation via interior-boundary conditions

Lienert

Nickel

2019

Rev. Math. Phys.

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Interior-boundary conditions (IBCs) have been suggested as a possibility to circumvent the problem of ultraviolet divergences in quantum field theories. In the IBC approach, particle creation and annihilation is described with the help of linear conditions that relate the wave functions of two sectors of Fock space: ψ (n) (p) at an interior point p and ψ (n+m) (q) at a boundary point q, typically a collision configuration. Here, we extend IBCs to the relativistic case. To do this, we make use of Dirac's concept of multi-time wave functions, i.e., wave functions ψ(x 1 , ..., x N ) depending on N space-time coordinates x i for N particles. This provides the manifestly covariant particle-position representation that is required in the IBC approach. In order to obtain rigorous results, we construct a model for Dirac particles in 1+1 dimensions that can create or annihilate each other when they meet. Our main results are an existence and uniqueness theorem for that model, and the identification of a class of IBCs ensuring local probability conservation on all Cauchy surfaces. Furthermore, we explain how these IBCs relate to the usual formulation with creation and annihilation operators. The Lorentz invariance is discussed and it is found that apart from a constant matrix (which is required to transform in a certain way) the model is manifestly Lorentz invariant. This makes it clear that the IBC approach can be made compatible with relativity.

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Section: Review Of Important Conceptsmentioning

confidence: 99%

Multi-time formulation of particle creation and annihilation via interior-boundary conditions

Lienert

Nickel

2019

Rev. Math. Phys.

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“…In recent years, there has been a renewed interest in constructing mathematically rigoros multi-time models, see [5] for an overview. Some of the current efforts to understand Dirac's multi-time models focus on the well-posedness of the corresponding initial value problems [6,7,8,9,10], other works also ask the question how the multi-time formalism could be exploited to avoid the infamous ultraviolet divergence of relativistic QFT and how a varying number of particles by means of creation and annihilation processes can be addressed [11,12,13]. Beside being candidate models for fundamental formulations of relativistic wave mechanics, a better mathematical understanding of such multi-time evolutions may also be beneficial regarding more technical discussions, such as the control of scattering estimates on vacuum expectation values of products of interacting field operators; see e.g.…”

Section: The Need For Multi-time Modelsmentioning

confidence: 99%

Multi-time dynamics of the Dirac-Fock-Podolsky model of QED

Deckert

Nickel

2019

Journal of Mathematical Physics

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Dirac, Fock, and Podolsky [1] devised a relativistic model in 1932 in which a fixed number of N Dirac electrons interact through a second-quantized electromagnetic field. It is formulated with the help of a multi-time wave function ψ(t 1 , x 1 , ..., t N , x N ) that generalizes the Schrödinger multi-particle wave function to allow for a manifestly relativistic formulation of wave mechanics. The dynamics is given in terms of N evolution equations that have to be solved simultaneously. Integrability imposes a rather strict constraint on the possible forms of interaction between the N particles and makes the rigorous construction of interacting dynamics a long-standing problem, also present in the modern formulation of quantum field theory. For a simplified version of the multi-time model, in our case describing N Dirac electrons that interact through a relativistic scalar field, we prove well-posedness of the corresponding multi-time initial value problem and discuss the mechanism and type of interaction between the charges. For the sake of mathematical rigor we are forced to employ an ultraviolet cut-off in the scalar field. Although this again breaks the desired relativistic invariance, this violation occurs only on the arbitrary small but finite length-scale of this cut-off. In view of recent progress in this field, the main mathematical challenges faced in this work are, on the one hand, the unboundedness from below of the free Dirac Hamiltonians and the unbounded, time-dependent interaction terms, and on the other hand, the necessity of pointwise control of the multi-time wave function.

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“…However, besides the strategy for the existence proof, nothing essential is expected to change for non-zero masses, since the boundary conditions are extracted from the tensor current j μν which stays the same in the massive case. With regard to the variable particle number case, the idea would be to choose the particle-position representation of QFT (see [5]) such that ψ = (ψ (0) , ψ (1) , ψ (2) , ...) where ψ (n) is an n-particle wave function. When compared to the N-particle case, additional possibilities for the boundary conditions appear, namely by relating ψ (n) with ψ (m) for n m. This corresponds to the recently introduced idea of "interior boundary conditions" which was used in [11,12] to avoid ultraviolet divergencies for a simple non-relativistic QFT.…”

Section: Discussionmentioning

confidence: 99%

“…Instead of going to quantum field theory (QFT) and trying to face the ultraviolet divergence problem there (see [5,8] for multi-time formulations of QFTs), we explore a different possibility to construct a rigorous model here: relativistic contact interactions.…”

Section: Introductionmentioning

confidence: 99%

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Interacting relativistic quantum dynamics for multi-time wave functions

Lienert

2016

EPJ Web Conf.

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Abstract. In this paper, we report on recent progress about a rigorous and manifestly covariant interacting model for two Dirac particles in 1+1 dimensions [9,10]. It is formulated using the multi-time formalism of Dirac, Tomonaga and Schwinger. The mechanism of interaction is a relativistic generalization of contact interactions, and it is achieved going beyond the usual functional-analytic Hamiltonian method.

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Multi-time wave functions for quantum field theory

Cited by 46 publications

References 22 publications

Multi-time formulation of particle creation and annihilation via interior-boundary conditions

Multi-time formulation of particle creation and annihilation via interior-boundary conditions

Multi-time dynamics of the Dirac-Fock-Podolsky model of QED

Interacting relativistic quantum dynamics for multi-time wave functions

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