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

Abstract

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.

Another Proof of Born’s Rule on Arbitrary Cauchy Surfaces

“…2.2; they arise especially from multi-time wave functions [9,11,25,33]; see [23] for an introduction and overview. While certain ways of implementing an ultraviolet cutoff [7,26] lead to multi-time wave functions that cannot be evaluated on arbitrary Cauchy surfaces, models without cutoff define a hypersurface evolution, either on the non-rigorous [28,29] or on the rigorous level [6,[19][20][21][22]. As a consequence, our result proves in particular a Born rule for multi-time wave functions, thereby generalizing a result of Bloch [4] (see also Remark 4 in [24]).…”

“…with outcome s k = 1 if a particle was found and s k = 0 otherwise. 7 We say that the outcome matrix s is compatible with L (denoted s : L) whenever…”

“…meaning that the symmetric difference between the two sets is a set of measure 0 in Γ(Υ). This is the case because, as described in Footnote 7, the configurations in the symmetric difference have at least one particle in the 2d set ∂Δ k for some k. 7 We could also have defined B k by Δ k ∩ B instead of (41), but that would have caused a bit of trouble because these sets would not have been disjoint. Our choice (41), on the other hand, has the consequence, which may at first seem like a drawback, that ∪ k B k = B because we have removed the points on the 2d triangles where two tetrahedra meet.…”

Another Proof of Born's Rule on Arbitrary Cauchy Surfaces

“…Some examples are described in [24]; they arise especially from multi-time wave functions [9,11,33,25]; see [23] for an introduction and overview. While certain ways of implementing an ultraviolet cutoff [7,26] lead to multi-time wave functions that cannot be evaluated on arbitrary Cauchy surfaces, models without cutoff define a hypersurface evolution, either on the non-rigorous [28,29] or on the rigorous level [20,21,6,22,19]. As a consequence, our result proves in particular a Born rule for multi-time wave functions, thereby generalizing a result of Bloch [4] (see also Remark 4 in [24]).…”

“…It is well known that for a sequence P n of projections, weak convergence to the projection P (i.e., xΨ|P n |Ψy Ñ xΨ|P |Ψy for every Ψ) implies strong convergence (i.e., P n Ψ Ñ P Ψ for every Ψ). 7 Set P n " U Σ Υn P Υn pM Bn pLqqU Υn Σ and P " P Σ pM P pLqq. Then Theorem 1 provides the weak convergence, and the strong convergence was what we claimed.…”