Probability conservation for multi-time integral equations Matthias Lienert November 20 2022

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Probability conservation for multi-time integral equations

Matthias Lienert∗

November 20, 2022

Abstract

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|2is 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.

Keywords: multi-time wave functions, Bethe-Salpeter equation, probability conservation, relativistic

quantum theory, integral equations.

This paper is dedicated to the memory of Detlef Dürr,

a wonderful person, scientist and mentor.

1 Introduction

1.1 Motivation

An elegant but little-known approach to relativistic quantum theory involves wave functions ψ(x1, ..., xN)

depending on many space-time variables xk∈R4for many particles k= 1,2, ..., N. In addition to other

applications, such as by Detlef Dürr and coauthors [1,2] in the foundations of relativistic quantum theory,

these multi-time wave functions (see [3] for an introduction) make it possible write down closed integral

equations which describe a ﬁxed number of relativistic, interacting particles in a manifestly covariant way.

The best-known example is the Bethe-Salpeter (BS) equation [4] which has been used to describe bound

states in quantum ﬁeld theory (QFT). At the time of its discovery, it was hoped that the BS equation

represented a fully relativistic – and interacting – generalization of the Schrödinger equation, at least for

processes where fermion creation and annihilation are not relevant (such as bound state problems).

The formulation of a completely relativistic wave equation for two-body systems has, in a

certain sense, solved a long-standing problem of quantum mechanics. The natural and simple

way in which relativistic invariance is achieved is, of course, very real progress, which may lead

one to hope that the main features of the equation are more permanent than the solidity of its

present ﬁeld theoretic foundation might suggest. Furthermore, it is hardly necessary to recall

that the usefulness of the equation has been amply demonstrated in several high-precision

calculations of energy levels. – Wick, 1954 [5]

However, there is a serious issue with such integral equations. Due to their non-locality in conﬁguration

space (in the sense of PDEs), they typically do not imply (local) continuity equations, nor do they conserve

the (global) probability integral. In the context of the Bethe-Salpeter equation, it has been said:

[...] The absence of a positive-deﬁnite norm for the wave function and of any orthogonality

theorem. – Wick, 1954, listing problems of the BS equation [5]

Nakanishi (1965) explicitly calculated the normalization integrals in some special cases of the

equal-mass Wick-Cutkosky model, and discovered that certain B-S amplitudes have negative

or zero norm. – Nakanishi, 1969 [6]

∗Marvel Fusion GmbH, Theresienhöhe 12, 80339 Munich, Germany. E-mail: lienertmat@gmail.com

arXiv:2210.05759v2 [quant-ph] 20 Nov 2022

Of course, these quotes mean nothing else than that the quantities proposed as a norm do not actu-

ally constitute one. They cannot have the physical meaning of a probability integral. Considering that

quantum physics is based on the notion of probability, this seems rather problematic for the physical

justiﬁcation of the Bethe-Salpeter equation.

The motivation for integral equations for a multi-time wave function can also be approached from a

second angle – one that was dear to Detlef Dürr: the quantization of Wheeler-Feynman (WF) electro-

dynamics [7–11]; see [12–14] for some of Detlef Dürr’s works on the topic. This theory pursues the idea

that the ultraviolet divergence problem of classical Maxwell-Lorenz electrodynamics can be avoided by

“integrating out” the ﬁelds. The result is a dynamics where interactions between particles occur directly

and exactly when particle world-lines are light-like separated.

The discussions with Detlef Dürr about ﬁnding a suitable quantum version of that theory sparked

my personal curiosity about the subject. After studying previous proposals for quantizations of WF

electrodynamics [15–18], [19, chap. 8], which all encounter their own diﬃculties, it seemed to me that

integral equations for a multi-time wave function might be a more promising way forward. In [20], I laid

out how these types of integral equations make it possible to transfer the principle of direct interactions

along light cones, that forms the core of WF electrodynamics, to the quantum level. This was done

in a way that retains the Dirac-Schrödinger equation with a spin-dependent Coulomb potential as the

non-relativistic limit, thus staying close to empirically successful models. In a series of papers [21–24],

my co-authors and me were able to prove that multi-time integral equations provide a fully relativistic

and interacting quantum dynamics which does not suﬀer from the ultraviolet divergence problem, even

for singular light-cone interactions [24].

However, the question of probability conservation was left open and, as we have seen for the BS

equation, there is reason for concern. Equations with interaction terms which are non-local in the

conﬁguration space of quantum mechanics do typically not imply local conservation laws. It is then at

best unclear whether global probability conservation holds true. Historically, Feynman himself saw the

problem of probability conservation as one of the central obstacles to quantizing WF electrodynamics, as

he reports in his Nobel lecture:

I found that if one generalized the action from the nice Lagrangian forms [...] to these forms

[...] then the quantities which I deﬁned as energy, and so on, would be complex. The energy

values of stationary states wouldn’t be real and probabilities of events wouldn’t add up to

100%. – Feynman, 1965 [25]

I don’t think we have a completely satisfactory relativistic quantum-mechanical model, even

one that doesn’t agree with nature, but, at least, agrees with the logic that the sum of

probability of all alternatives has to be 100%. – Feynman, 1965 [25]

In view of these diﬃculties, if one is not ready to dismiss multi-time integral equations altogether, one

may conclude that the equations are not exactly the right ones and that some modiﬁcation is in order.

From the point of view that the interaction term in the BS equation which quantum electrodynamics

suggests (an inﬁnite series of Feynman diagrams in need of renormalization) is not the most simple and

natural, such a modiﬁcation seems easy to accept. In addition, the argument that non-local interaction

terms usually preclude local conservation laws does not apply to global conservation laws, leaving room

for logical possibilities which may not have been suﬃciently explored. I am going to adopt these positions

here. This makes it possible to prove that for certain classes of integral equations the global probability

integral is, in fact, conserved.

2 The integral equation

For simplicity, we focus on the case of N= 2 Dirac particles. Moreover, we set c=1=~. Then the class

of integral equations we shall study reads:

ψ(x1, x2) = ψfree(x1, x2) + iZd4x0

1d4x0

2G1(x1−x0

1)G2(x2−x0

2)K(x0

1, x0

2)ψ(x0

1, x0

2).(1)

Here ψ:R4×R4→C4⊗C4is a multi-time wave function with 16 complex components for two particles.1

ψfree(x1, x2)is a solution of the free Dirac equation in x1, x2, i.e.:

(−iγµ

k∂xµ

k+mk)ψfree(x1, x2)=0, k = 1,2.(2)

1One can also study (1) on a sub-domain of R4×R4, e.g., on the set Sof space-like conﬁgurations.

The space-time integral in (1) extends over R4×R4, the entire conﬁguration space-time. G1and G2are

Green’s functions of the Dirac equations of particles 1 and 2. We use the convention of [26, Appendix E]:

(−iγµ

k∂xµ

k+mk)Gk(xk−x0

k) = δ(4)(xk−x0

k).(3)

Here and in the following, particle indices in γ-matrices, Green’s functions and propagators indicate on

which spin index their matrix structure acts.

K(x1, x2)is the so-called interaction kernel, a covariant, matrix-valued distribution. We require the

following symmetry condition with respect to its matrix structure:

K†(x1, x2) = γ0

1γ0

2K(x1, x2)γ0

1γ0

2.(4)

As explained in [20], direct interactions along light cones in the spirit of Wheeler-Feynman electromag-

netism can be expressed by the interaction kernel

Ksym(x1, x2) = λ γµ

1γ2,µ δ((x1−x2)2)(5)

where λ∈Ris a coupling constant and (x1−x2)2= (x0

1−x0

2)2−|x1−x2|2denotes the Minkowski square.

Note that (5) contains both retarded and advanced interaction terms, as can be seen by decomposing the

delta distribution. The retarded part is given by:

Kret(x1, x2) = λ γµ

1γ2,µ

2|x1−x2|δ(x0

1−x0

2− |x1−x2|).(6)

With these conventions, the factor iin the interaction term in (1) is required to obtain the correct

non-retarded limit, i.e., a Schrödinger equation with spin-dependent Coulomb potential [20].

Relation to the Bethe-Salpeter equation. The BS equation is contained in the class of equations

(1) for the case that Gkare Feynman propagators SF

kfor the two particles k= 1,2, and for the case

that K(x1, x2)is given by an inﬁnite series of Feynman diagrams. In the so-called ladder-approximation

of the BS equation, only a certain sub-class of these Feynman diagrams (consisting of those exchanging

only one virtual photon at a time) is considered. Then Ksimpliﬁes to

KBSL(x1, x2) = λ γµ

1γν

2DF

µν (x1, x2)(7)

where DF

µν is the Feynman propagator of a photon (see [27, p. 331]). In Lorenz gauge:

µν (x1, x2) = ηµν DF(x1, x2)(8)

where DFis the Feynman propagator of the wave eq. and ηµν the Minkowski metric. As both DFand

4πδ((x1−x2)2)are Green’s functions of the wave equation, (7) closely resembles (5). However, a crucial

diﬀerence is that only (5) is supported on the light cone; (7) also has support outside.

Role of the dynamics. As discussed in [23], one can best understand the dynamics deﬁned by (1) in

the case of Gk=Sret

k, the retarded Green’s function of the Dirac equation for particle k. Then, for each

incoming free wave function ψfree, the integral equation deﬁnes a unique interacting solution ψwhich

agrees with ψfree in the inﬁnite past. Thus Eq. (1) can be viewed as a machinery which takes an incoming

free solution and computes an interacting correction to it.

Notes on retarded Green’s functions. We now collect useful properties of retarded Green’s functions

which are rooted in their simple relation to the propagator of the Dirac equation. These will play a crucial

role in the upcoming arguments. Namely, we have:

Sret(x−x0) = θ(x0−x00)S(x−x0)(9)

where θis the Heaviside function and Sthe propagator of the Dirac equation. Scan be used to time-evolve

every free solution of the Dirac equation from one Cauchy surface Σto another:

ψfree(x) = −iZΣ

dσµ(x0)S(x−x0)γµψfree(x0).(10)

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Probabilityconservationformulti-timeintegralequationsMatthiasLienert*November20,2022AbstractInrelativisticquantumtheory,onesometimesconsidersintegralequationsforawavefunction(x1;x2)dependingontwospace-timepointsfortwoparticles.Aseriousissuewithsuchequationsisthat,typically,thespatialintegraloverjj2i...

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