Times of Arrival and Gauge Invariance Siddhant Das∗ and Markus N¨othy Mathematisches Institut, Ludwig-Maximilians-Universitat M¨unchen,Theresienstr. 39, D-80333 M¨unchen,Germany (Dated: December 18, 2020) We revisit the arguments underlying two well-known arrival-time distributions in quantum me- chanics, viz., the Aharonov-Bohm and Kijowski (ABK) distribution, applicable for freely moving particles, and the quantum flux (QF) distribution. An inconsistency in the original axiomatic derivation of Kijowski's result is pointed out, along with an inescapable consequence of the \nega- tive arrival times" inherent to this proposal (and generalizations thereof). The ABK free-particle restriction is lifted in a discussion of an explicit arrival-time setup featuring a charged particle mov- ing in a constant magnetic field. A natural generalization of the ABK distribution is in this case shown to be critically gauge-dependent. A direct comparison to the QF distribution, which does not exhibit this flaw, is drawn (its acknowledged drawback concerning the quantum backflow effect notwithstanding). I. INTRODUCTION unexpected consequence of the \negative arrival times", characteristic of the ABK proposal and generalizations The distribution of arrival (or detection) times of a thereof, is discussed, which casts doubts on the associ- quantum particle amenable to laboratory time-of-flight ated time-energy uncertainty relation. The QF distri- (TOF) experiments is far from settled, as evidenced by bution is motivated as a special case of the arrival-time the multitude of inequivalent theoretical predictions sug- distribution in Bohmian mechanics (de Broglie-Bohm or gested in the literature [1{4]. In a typical theoretical pilot-wave theory) but has also been arrived at from other discussion of a TOF experiment, one considers a parti- angles in specific instances. Focusing next on the ar- rival times of particles subject to external potentials, we cle of mass m with a well-localized wave function 0(x) at time zero, propagating either freely or in specified ex- are led to the question of gauge invariance. A natu- ternal potentials. Critical to any such discussion is the ral generalization of the ABK formula to scalar poten- probability Π(τ) dτ that the particle's time-of-arrival on tials, the so-called \standard arrival-time distribution", a given surface Q is between times τ and τ + dτ, subject and its direct extension to vector potentials are intro- to the condition duced. For concreteness, we study the arrival-time dis- Z 1 tributions of a charged particle in a constant magnetic dτ Π(τ) = 1: (1) field (Sec. III), discovering that the said extension is not 0 gauge-invariant, while the QF distribution is. The former For completeness, one could add a \non-detection prob- even ceases to be a meaningful probability distribution ability" P (1) to the left-hand side of (1), accounting for in some gauges. We conclude in Sec.IV, drawing lessons the fraction of experimental runs in which the particle for future work. In what follows, we take as the units does not intercept Q even as t ! 1. of measurement of mass, length, and time, m, σ (the 2 Most approaches lead to an ideal (or intrinsic) TOF width of the wave packet 0), and σ m=~, respectively; distribution Π(τ), i.e., an apparatus-independent theo- formally, this amounts to setting ~ = m = σ = 1 in every retical prediction, given by some functional of the initial equation. quantum state j 0i and the surface Q. In the category of ideal TOF distributions | insofar as such a descrip- tion may be warranted | the Aharonov-Bohm-Kijowski (ABK), and the quantum flux (QF) distributions are ex- arXiv:2102.02661v1 [quant-ph] 3 Feb 2021 emplary. These distributions have been arrived at from different theoretical viewpoints, making them important II. THEORETICAL VIEWPOINTS benchmarks for comparing with experiments. However, it has long been observed that both predictions practi- Work by Aharonov-Bohm [5] and Paul [6] in the 1960s cally coincide in the presently accessible far-field or scat- suggested a TOF distribution starting with the classical tering regime, making an experimental test distinguish- arrival-time formula ing the two proposals very challenging. We provide a critical review of these developments in Sec. II, pointing out an inconsistency in the origi- L − z τ = (2) nal axiomatic derivation of Kijowski's distribution. An p ∗ [email protected] describing a freely moving particle that at t = 0 had po- y [email protected] sition z and momentum p, and arrived after time τ at a 2 distant point L on a line1. They sought the symmetric −∞ < τ < 1, as opposed to Eq. (1). A typical nar- quantization [3, Sec. 5] rative accompanying these \negative arrival times" char- acteristic of (5-6) and generalizations thereof invokes the 1 τ^ = Lp^− 1 − p^− 1 z^ +z ^p^− 1; (3) notion of a quantum state prepared in the infinite past 2 that evolves into 0 at time zero (see [21, p. 155], [12, p. 4679], or [3, p. 4337] for details). The crux of these promoting (2) to the Hilbert space operatorτ ^, osten- arguments is that the negative arrival-time probability sibly a quantum observable, wherez ^ andp ^ = − i@=@z should be interpreted as the non-detection probability; denote the usual position and momentum operators, re- accordingly, spectively, of quantum mechanics. Formally, (3) is canon- ically conjugate to the free-particle Hamiltonian, i.e., Z 1 PAB(1) = dτ ΠAB(−τ): (7) p^2; τ^ = 2i1; (4) 0 This then precludes the application of usual quantum althoughτ ^ is not a self-adjoint operator (i.e., a quantum formulas for expectation values that inevitably include observable in the sense of Dirac and von Neumann). Nev- negative eigenvalue contributions. In particular, taking ertheless, its (generalized) eigenfunctions constitute an h jτ^j i as the mean arrival-time and following the work over-complete set [7], defining a positive operator-valued in [22] generalized to account for L 6= 0; we find that measure (or POVM)2, and in turn the arrival-time dis- tribution [9, 10] i Z h jτ^j i = dz dz0 2L − z − z0 sgnz − z0 Z 1 4 2 1 X p ~ R ΠAB(τ) = dp θ(αp) jpj 0(p) ∗ 0 2π × 0 z 0 z ; (8) α = ± −∞ 2 hence hτi vanishes for any real (z) sgn(·) is odd, im- iτ 2 0 × exp − p + ipL : (5) 2 plying h jτ^j i = − h jτ^j i , leading to the absurd con- clusion that every arrival is instantaneous in these cases. ~ Here, 0(p) = hpj 0i is the momentum representation of In view of Eq. (4), one could scrutinize some version the initial state j 0i and θ(·) is Heaviside's step function. of the time-energy uncertainty relation [1, Ch. 3], the A reformulation of (5) due to Leavens [11], main concern of Aharonov-Bohm [5] and Kijowski [19] among others. However, the distribution (5-6) decays too Z 1 3 1 X 1 + iαsgn(z − L) slowly to have finite first and second moments , unless ΠAB(τ) = dz 32π jz − Lj3=2 j 0i satisfies α = ± −∞ 2 − 3=2 ~ lim p 0(p) = 0: (9) × τ (z) − τ (L) ; (6) p!0 As a result, one typically restricts the domain of the op- involves only the time-dependent position space wave eratorτ ^ to the subspace of wave functions fulfilling this function. Several authors have arrived at (or endorsed) condition [7, 10]. But this leaves out, e.g., Gaussian ΠAB [12{16], most notably Allcock [17, 18] and Kijowski wave packets / exp− αz2 + iβz that violate (9) pro- [19] (see below). Allcock deduced only the α = + contri- ducing an infinite ∆τ, being otherwise perfectly reason- bution of (5), starting from a phenomenological detector able states amenable to present-day experiments [23{25]. model based on a complex absorbing potential (see also Secondly, even when the moments are finite, they may no [20]). longer satisfy the desired (Robertson-Schr¨odinger)uncer- Due to the negative eigenvalues ofτ ^ (perhaps at- tainty inequality, once the contribution of the \negative tributable to the incomplete classical formula, see arrival times" is discarded from expectation value inte- footnote1), Π AB(τ) is normalized over the interval grals [26, p. 190]. Therefore, substantiating the time- energy uncertainty relation via arrival-time operators ful- filling (4), of which (3) is but one example [12, 22, 27{33], seems academic at best. This does little to generate en- 1 The correct classical TOF formula valid for an arbitrary initial thusiasm for cutting-edge TOF experiments. Quite to point (z; p) in phase space is the contrary. ( (L − z)=p; sgnp = sgn(L − z); In 1974 Kijowski [19] rediscovered (5) (rather, a three- τ = 1; otherwise; dimensional version of it applicable to arrivals on a plane) quantizing which seems nothing short of impossible. 2 Quantum observables defined by self-adjoint operators cannot be found for many other experiments as well. The notion of an observable was thus generalized to POVMs [8, pp. 480{484], 3 This by no means invalidates a TOF distribution so long as it and there are various (inequivalent) suggestions for arrival-time is normalizable `ala Eq. (1), which merely forces Π(τ) to decay POVMs. faster than 1/τ as τ ! 1. 3 p p by imposing a set of plausible axioms he believed a TOF pz ! −pz with the momentum integration being per- distribution should satisfy. He initially considered freely formed over pz < 0. For generic wave functions that are moving particles, for which neither \left" nor \right moving", he proposed 1 1 X Z Z ~ ~ it 2 t(p) = 0(p) exp − p ; (10) F0( ) = dpx dpy dpz θ(αpz) 2π 2 2 α = ± R −∞ 2 prepared in special initial states that | in momentum p ~ × jpzj (p) : (15) representation | vanish for pz ≤ 0, the so-called \right moving wave functions".