Relativistic time-of-arrival measurements predictions post-selection and causality problems Charis Anastopoulos

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Relativistic time-of-arrival measurements: predictions,

post-selection and causality problems

Charis Anastopoulos∗

Department of Physics, University of Patras, 26500 Greece

and

Maria-Electra Plakitsi †

The Moraitis School, Psichiko, 15452, Greece

October 12, 2022

Abstract

We analyze time-of-arrival probability distributions for relativistic particles in the con-

text of quantum ﬁeld theory (QFT). We show that QFT leads to a unique prediction,

modulo post-selection that incorporates properties of the apparatus into the initial state.

We also show that an experimental distinction of diﬀerent probability assigments is pos-

sible especially in near- ﬁeld measurements. We also analyze causality in relativistic

measurements. We consider a quantum state obtained by a spacetime-localized operation

on the vacuum, and we show that detection probabilities are typically characterized by

small transient non-causal terms. We explain that these terms originate from Feynman-

propagation of the initial operation, because the Feynman propagator does not vanish

outside the light-cone. We discuss possible ways to restore causality, and we argue that

this may not be possible in measurement models that involve switching the ﬁeld-apparatus

coupling on and oﬀ.

1 Introduction

Finding a consistent quantum description for time-of-arrival measurements is a classic problem

in the foundations of quantum theory. In the simplest time-of-arrival measurement, a particle

is prepared on an initial state |ψ0iwith positive momentum and localized around x= 0. A

particle detector is placed at x=L. The issue is to determine the probability P(t, L)dt that

the detector clicks at some moment between tand t+δt. Despite the apparent simplicity of

the problem, no unique time-of-arrival probability exists [1, 2]. Fundamentally, this is due to

the fact that time is not a quantum observable. There is no self-adjoint operator for time [3],

hence, we cannot rely on Born’s rule for a unique answer.

The time-of-arrival problem is exacerbated for relativistic particles, because it becomes

entangled with another foundational problem, the issue of localization. Particle localization

is essential to any description of measurements, because all particle-detection records are

∗anastop@physics.upatras.gr

†plma101803@moraitis.edu.gr

arXiv:2210.05591v1 [quant-ph] 11 Oct 2022

localized in space and in time. However, the existence of observables associated to spatial

localization is in conﬂict with the requirement of causality, as evidenced by several theorems

[4–6].

The most well known set-up where this conﬂict appears is Fermi’s two-atom problem.

Fermi studied information transmission through a quantum ﬁeld in a system of two localized

atoms at distance r[7]. He assumed that at time t= 0, atom A is in an excited state and

atom B is in the ground state. He asked when the inﬂuence of A will cause B to leave its

ground state. In accordance with the locality principle that spacelike separated events cannot

inﬂuence each other, he found that this happens only at time greater than r. However, Fermi’s

result was shown to be an artefact of an approximation [8]. Later studies of the problem came

to diﬀerent results that were heavily dependent on approximations. Eventually, Hegerfeldt

showed that the conﬂict of localization and causality is generic in relativistic systems [9,10],

it only requires energy positivity and the treatment of atoms as localized in disjoint spatial

regions——see, also the clariﬁcation of this result in [11].

In this paper, we analyze a broad class of probability distributions for the time of arrival of

relativistic particles. This class was identiﬁed in Ref. [12] through a analysis of measurements

in QFT. Part of our motivation is the possibility of experimentally distinguishing between

diﬀerent proposals for the time of arrival. We also analyse the structure of small apparently

super-luminal transient terms—analogous to the ones in the Fermi-atom problem—that ap-

pear in the time-of-arrival probability distribution. We identify their origin and examine their

implications for relativistic quantum theory of measurement.

An important reason for studying the time of arrival is that it forces us to re-conceptualize

the description of quantum measurements. Ever since von Neumann [13], measurements

have been described as almost instantaneous processes that occur at a single moment of

time t. In von Neumann measurements, the interaction of the apparatus with the measured

quantum system is switched on for a pre-determined time interval—the exact timing of the

measurement is determined by the shape of the switching function. Hence, the timing of the

measurement is an external parameter of the measurement scheme, and not a random variable

of the experiment. This logic has recently been extended to QFT measurements [14, 15]: the

measured ﬁeld and the apparatus are initially uncorrelated, and they interact only in a ﬁnite,

predetermined spacetime region.

Time-of-arrival measurements challenge this measurement paradigm. Actual particle de-

tectors (e.g., photographic plates, silicon strips) have a ﬁxed location in space and they are

made sensitive for a long time interval during which particles may be detected. Therefore,

the location of a detection event is a ﬁxed parameter of the experiment; the actual random

variable is the detection time. Von Neumann type measurements are fundamentally incapable

of describing time as a random variable, but it turns out that they can mimic some aspects of

time-of-arrival measurements. However, imitations have limitations: they work only to lowest

order in perturbation theory and that it eventually leads to causality problems.

Our results are the following.

First, we re-derive the relativistic time-of-arrival probabilities of Ref. [12] in a von Neu-

mann measurement scheme for quantum ﬁelds. The original derivation involved the Quantum

Temporal Probabilities (QTP) method [12, 16–19] that explicitly constructed a probability

density with respect to time. In the von Neumann measurement scheme, we use a switching

function that is localized in a compact spacetime region, and we reinterpret the probabilities

in order to deﬁne a probability density. This works only to leading order in perturbation

theory. We use this alternative derivation in order to identify the problems that persist in

this treatment of measurements.

Time-of-arrival probability distributions are post-selected, i.e., they refer only to the frac-

tion of particles that has been detected. They depend on the apparatus through a speciﬁc

operator ˆ

Sthat describes the localization of the detection records. We show that this operator

can be absorbed into a post-selection of the initial state. A unique time-of-arrival probabil-

ity measure ensues, modulo post-selection. This measure was ﬁrst derived by Leon [20] and

then rederived from a QFT analysis in [16]. In the non-relativistic limit, it coincides with

Kijowski’s time-of-arrival distribution [21].

The measured time of arrival probability distribution does depend on the properties of

the apparatus. We analyze the probability measure for an operationally meaningful class of

initial states that was recently proposed, in relation to experiments for distinguishing between

diﬀerent time-of-arrival proposals. We ﬁnd that the diﬀerent distributions are in principle

distinguishable in near-ﬁeld experiments.

Then, we analyse locality in the time-of-arrival probabilities. We consider an initial state

that is generated by a localized external source acting on the quantum ﬁeld vacuum. By

”localized” we mean that the source has support in a compact spacetime region. We ﬁnd that

the time-of-arrival probability has a small but non-zero contribution outside the light-cone of

the source’s support. We analyze the origin of this term, and we ﬁnd that it originates from

the fact that in QFT sources evolve with the Feynman propagator, and Feynman propagator

is non-zero outside the light-cone. We argue that von-Neumann type models, that rely on

switching on the interaction, may not be able to resolve this problem.

Finally, we brieﬂy revisit the QTP description of relativistic measurements, and present

possible strategies through which the super-luminal transient terms can be consistently re-

moved.

2 Relativistic time-of-arrival probabilities

In this section, we ﬁrst re-derive the time-of-arrival probabilities of Ref. [12] using a von

Neumann type of measurement. We show that these probabilities are unique modulo post-

selection, and that the eﬀect of the detector—as determined by its localization properties—can

be distinguished in near-ﬁeld measurements.

2.1 Detection probability from a von Neumann type measurement

We consider the measurement of a free scalar ﬁeld ˆ

φ(x) on a Hilbert space F, interacting with

an apparatus described by a Hilbert space H. As our focus is time-of-arrival measurements,

we will work only in one spatial dimension. The reason is that only particles that propagate

along the axis that connects the source with the detector contribute to the total probability.

The Hamiltonian of the total system is

H=ˆ

Hφ⊗ˆ

I+ˆ

I⊗ˆ

HA+ˆ

HI,(1)

where ˆ

Hφis the quantum ﬁeld Hamiltonian and ˆ

HAthe Hamiltonian of the apparatus. We

consider a von-Neumann type of measurement, in which the interaction is switched on for a

ﬁnite time. Then, we choose an interaction Hamiltonian

HI=ZdxF¯

t,¯x(t, x)ˆ

φ(x)⊗ˆ

J(x),(2)

where F¯

t,¯x(t, x) is a switching function centered around the spacetime point (¯

t, ¯x), and ˆ

J(x)

is a current operator on F∗.

We assume an initial state |ψifor the ﬁeld and an initial state |Ωifor the detector. It

is convenient to identify |Ωiwith the ground state of the Hamiltonian ˆ

HA,ˆ

HA|Ωi= 0, and

also to be annihilated by the generator of space translations ˆ

PAof the apparatus. Then, the

probability that the detector is found in an excited state, when measured after the interaction

has been switched oﬀ, is given by

Prob(¯

t, ¯x) = hψ0,Ω|ˆ

S†

t,¯x(ˆ

I⊗ˆ

I− |ΩihΩ|)ˆ

S¯

t,¯x|ψ0,Ωi,(3)

expressed in terms of the S-matrix ˆ

S¯

t,¯x=Texp h−iRdtdxF¯

t,¯x(t, x)ˆ

φ(t, x)⊗ˆ

J(t, x)i;ˆ

φ(t, x)

and ˆ

J(t, x) are Heisenberg-picture operators and Tstands for time-ordering.

To leading order in perturbation theory,

Prob(¯

t, ¯x) = Zdt1dx1dt2dx2F¯

t,¯x(t1, x1)F¯

t,¯x(t2, x2)G(tx, t1;t2, x2)hΩ|ˆ

J(t1, x1)ˆ

J(t2, x2)|Ωi,(4)

where

G(t1, x1;t2, x2) = hψ0|ˆ

φ(t1, x1)ˆ

φ(t2, x2)|ψ0i(5)

is a two-point correlation function for the ﬁeld.

We take a reference point x0= 0 on the apparatus, and we deﬁne |ωi=ˆ

J(0)|Ωi. Then,

we can write Ω|ˆ

J(t1, x1)ˆ

J(t2, x2)|Ωi=R(t2−t2, x2−x1), where

R(t, x) = hω|eiˆ

HAt−iˆ

PAx|ωi

is the detector kernel. The key property of R(t, x) is that by energy positivity, its Fourier

transform

R(E, K) := ZdtdxR(t, x)e−iEt+iKx = 2πhω|δ(E−ˆ

HA)δ(K−ˆ

PA) (6)

vanishes for E < 0.

In Eq. (4) ¯xand ¯

tappear as parameters, not as random variables. However, Eq. (4) has

a natural interpretation in terms of a probability density. Consider a homogeneous switching

function F¯

t,¯x(t, x) = f(t−¯

t, x −¯x), where f(t, x) = exp h−t2

2δ2

t−x2

2δ2

xifor space and time

spreads δxand δt, respectively. Gaussians satisfy the identity

f(t, x)f(t0, x0) = f2t+t0

2,x+x0

2pf(t−t0, x −x0).(7)

∗Certainly, the idea of a switching interaction is not realistic in QFT. Fields interact with the apparatuses

through terms deﬁned by the Standard Model of particle physics, and in a Poincar´e covariant theory, the

interaction terms are always present.

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摘要：

Relativistictime-of-arrivalmeasurements:predictions,post-selectionandcausalityproblemsCharisAnastopoulos*DepartmentofPhysics,UniversityofPatras,26500GreeceandMaria-ElectraPlakitsiTheMoraitisSchool,Psichiko,15452,GreeceOctober12,2022AbstractWeanalyzetime-of-arrivalprobabilitydistributionsforrelativi...

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Relativistic time-of-arrival measurements predictions post-selection and causality problems Charis Anastopoulos

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