PHYSICAL REVIEW A, VOLUME 61, 012104

Free-motion time-of-arrival operator and probability distribution

I. L. Egusquiza1 and J. G. Muga21Fisika Teorikoaren Saila, Euskal Herriko Unibertsitatea, 644 P.K., 48080 Bilbao, Spain

2Departamento de Fı´sica Fundamental II, Universidad de La Laguna, La Laguna, Tenerife, Spain~Received 7 May 1999; published 9 December 1999!

We reappraise and clarify the contradictory statements found in the literature concerning the time-of-arrivaloperator introduced by Aharonov and Bohm in Phys. Rev.122, 1649 ~1961!. We use Naimark’s dilationtheorem to reproduce the generalized decomposition of unity~or positive-operator-valued measures! from anyself-adjoint extension of the operator, emphasizing a natural one, which arises from the analogy with themomentum operator on the half-line. General time operators are set within a unifying perspective. It is shownthat they are not in general related to the time of arrival, even though they may have the same form.

PACS number~s!: 03.65.2w

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I. INTRODUCTION

Even though there is no question that experimentalmeasure distributions of time of arrival for quantum systemthere has been a long debate about the capabilities of sdard quantum mechanics to address the very concept ofas an observable. In a nutshell, the problem arises becwe expect observables to be represented in the quantummalism by self-adjoint operators, whereas an old theoremPauli @1# states that for a semibounded self-adjoint operaH, no conjugate self-adjoint operatorT can exist, i.e., nooperator that is self-adjoint and satisfies the canonical cmutation relation@H,T#5 i\ over a dense domain. In othewords, there is no self-adjoint time operator if the Hamtonian is bounded from below, as we normally expect itbe. However, the theorem has not discouraged theoristsattempting to fit such an immediate classical concept intostandard framework~see @2# for a general discussion!. Inrelatively recent times a rather satisfactory answer has bgiven for the arrival time of freely moving states in ondimension, from the point of view of generalized spectfunctions, or positive-operator-valued measures~POVM!@3–6#. In short, the relevant statement is that self-adjointnis not strictly necessary for the formulation of probabilitiand the reality of measurements of observables, and thafact, any given observable is uniquely characterized~in aspecific sense! by the probability distributions of measurement results in the different states accessible to the sys~for a much more developed presentation, refer to@4,5#!.

This result has sparked a renewed interest in the anaof old and new proposals for time operators in general,in particular for the time-of-arrival operatorTAB ~see below!introduced by Aharonov and Bohm@7#, as in@6,8–21#.

However, a number of basic issues are still in disputein need of clarification. In particular, contradictory statments may be found about the domain of the time operTAB @9,14#, or about its eigenfunctions@16,14#; and the de-ficiency indices, which are important to determine its seadjoint extensions, appear to be, following different authoeither~0,1! or ~0,2! @10,14#, but only one of the possibilitiesmay be correct. Even though several self-adjoint ‘‘variantof TAB have been proposed@11,16,15#, none of them is a

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self-adjoint extension ofTAB in a proper sense. The framework to consider these extensions is Naimark’s dilation threm @4,5,22#, but an explicit simple construction of a seladjoint extension is lacking.~For a rather abstracpresentation of the theorem applied to the time-of-arriproblem, under the guise of ‘‘screen observables,’’ see@13#;also @4,6,5#.! In addition, the form of the time operators idifferent representations may also be, and has been, mising: the same operator has been presented in different fobut different operators have also been written in the saform.

It appears that in order to make further progress in tfield without being trapped by any of the numerous pitfamentioned, a detailed analysis of these topics is required,their clarification is the main objective of this paper. Evthough some of the arguments have to be rather technour aim is to provide an account readable by a general phcist. We shall in particular introduce a description of textension theorems as they apply to the problem at hand,develop a general analogy, sketched by Holevo@4#, betweenthe momentum operator in the half-line and the time opetor. We hope that it will provide a much better understandof the relevant issues. It will also allow us to join the strangiven by ~i! the initial proposal of the time-of-arrival operator of Aharonov and Bohm@7#, its modifications or variants@11,15–17#; ~ii ! the POVM idea as applied to quantum phyics and to the time operator in particular@5,14#; ~iii ! theaxiomatic approach of Kijowski to obtain good probabilidistributions for the time of arrival@11,12#; and ~iv! someapparently formal constructions in Allcock’s seminal pap@8#.

As is well known, the momentum operator in the half-linis self-adjoint. Moreover, it is maximally symmetric, becauits deficiency indices are (1,0), and it admits no self-adjoextension on the Hilbert space of functions defined onhalf-line ~for a fast and good review of deficiency indiceand the Neumann theory of extensions, with particular ephasis on the physical examples of momentum operatorLaplacian, see@23#—in addition, we will give a self-contained description in Sec. II!. On the other hand, there ia natural extension, namely, the momentum operator onfull line. It should be noticed that this is most definitely nthe unique extension, but just one of the infinitely many p

©1999 The American Physical Society04-1

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I. L. EGUSQUIZA AND J. G. MUGA PHYSICAL REVIEW A61 012104

sible extensions. Nonetheless, its ‘‘naturalness’’ is well jtified from a physical point of view. The question might thebe posed as to how we are to recover any information abthe distribution of probability of momentum in the half-linfrom the full line extension. This is actually the contentNaimark’s dilation theorem, together with the theorem thasserts the uniqueness of the POVM associated with a mmally symmetric operator, namely that theuniquedistribu-tion of probability associated with the momentum operaon the half-line is recovered from the spectral decomposiof the self-adjoint operator momentum on the full line,simple projection. Equivalently stated, we can reproducephysicalcontent of the momentum operator on the half-lifrom the standard, well-known techniques that apply to sadjoint operators.

We will show that the different alternatives that have beproposed to make sense of the time operator of Aharoand Bohm are actually consequences of this general rewhich we have phrased in terms of momentum operators,which can be extended to other cases as well. The generof the result also explains the coincidence of different cstructions, in particular the axiomatic approach of Kijows@11,12# and the attempts to create ex novo a self-adjointerator~@15# is a relevant example!.

We organize the paper as follows: in the next sectiongive a complete and very basic description of the momenoperator on the half-line and the problems that appear thWe also give the natural extension and describe more fNaimark’s dilation theorem in this context. In Sec. III westablish the parallel with the time-of-arrival operatorAharonov and Bohm, to be followed in the next section wthe general case of a time operator. Section V is devotediscussing and restating the different results known inliterature in the terms used in the previous sections.

II. THE MOMENTUM OPERATOR

Consider the momentum operatorp52 i\]x defined on adense domain of the Hilbert space of square integrable futions on the half-line,H.5L2(R1,dx). More precisely, it isdefined onD( p), the subspace of square integrable ablutely continuous functionsc whose derivative is also squarintegrable, and such thatc(0)50. Clearly, this operator issymmetric on its domain. Let us write down a set of genalized ~weak! eigenfunctions that provide us with a resoltion of the identity. The adequate set, which does not belto the Hilbert space and is parametrized by realp, is given bycp(x)5exp(ipx/\)/A2p\, as is only to be expected. Theform a complete basis, since*2`

` dp cp(x8 )cp(x)5d(x82x). However, they are not orthogonal~in the generalizedsense!:

E0

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dx cp8~x!cp~x!51

2d~p2p8!1

i

2pP

1

p2p8,

where P stands for principal part. This is evidently differefrom the usual constructions for self-adjoint operators.

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Let us now pass on to the position operatorx, defined onthose square integrable functions on the half-line,cPH. ,such that*0

`dx x2uc(x)u2,1`. This operator is self-adjoin

on its domain. Additionally, it is true that@ x,p#5 i\ on adense domain. However, in the case at hand the posoperator is bounded from below, and Pauli’s theorem thefore applies. It follows thatp is not a self-adjoint operatoand admits no self-adjoint extension inH. .

We shall now rephrase the statements of the precedtwo paragraphs in a different way. The domain of the opetor adjoint to p, i.e., p†, according to von Neumann’s formula, isD( p) % N( i ) % N(2 i ), whereN(6 i ) are the spacesof eigenvectors ofp† with eigenvalues6 i , respectively.This comes about because, even thoughp, being symmetric,has no imaginary eigenvalues, its adjoint does have eigvectors with imaginary eigenvalues. As a matter of fact,functions exp(ilx/\), with l a complex number with positiveimaginary part, are all eigenvectors ofp† of eigenvaluel.Notice that the dimension ofN(l), the space of eigenvectorwith eigenvaluel, is the same for alll with positive imagi-nary part. This is called the first deficiency index of the oeratorp, while the second one is the dimension of the spaof eigenvectors for any given eigenvalue of negative imanary part, in our case, 0. That is, the deficiency indices opare (1,0). But self-adjoint operators have deficiency indi(0,0), as do essentially self-adjoint operators, which,though not self-adjoint, have a unique self-adjoint extensinamely, their closure. If we were to build a self-adjoint etension ofp, we would need to include somehow those ements ofD( p†) that are not inD( p), but in such a way thatthe imaginary parts compensate. When the deficiency indare equal, this is achieved by the use of a unitary transmation fromN(l) to N(l) ~this is the von Neumann theorof self-adjoint extensions, whereby each self-adjoint extsion in the Hilbert space of definition of an operator ovedense domain, with equal defect indices, is given by a utary transformation between the deficiency subspaces!. Asthe deficiency indices in our case are different, it is not psible to extendp to include an action over the whole of thdomain ofD( p†) such that the imaginary parts compensaSince p is a closed operator, symmetric over its domawhich is dense inH. , and its deficiency indices are unequaone of them being 0, we say that it is a maximally symmetoperator, and the previous result is that it admits no sadjoint extension overH. .

Retaking the complete set of generalized eigenfuncticp(x), we can construct a positive-operator-valued meas~POVM! F. This is a map from intervals in the real line~as-algebra of subsets of a nonempty set! to the positive op-erators over a Hilbert space that satisfies three axiomswe will illustrate with the example at hand. The map for ocase is given by

^fuF~@a,b# !c&5Ea

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dxE0

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FREE-MOTION TIME-OF-ARRIVAL OPERATOR AND . . . PHYSICAL REVIEW A 61 012104

over allf,cPH. . This map sends intervals of the real linto positive operators acting onH. ~first axiom!, which addtogether when the intervals are disjoint~second axiom!, andwhich add to the identity operator when summed overreal line, because of the completeness proved above~thirdaxiom!. This differs from the usual decompositions for seadjoint operators in that it does not fulfill the property ththe positive operators be projectors, i.e., in our cF(@a,b#)2ÞF(@a,b#), because of the lack of orthogonalitshown above.

Now that we have the POVM associated withp, we canreconstruct the operator and the probability distribution ofvalues. The action of the operator is given by

~ pw!~x!5E2`

`

dpE0

`

dy pcp~x!cp~y !w~y!.

The probability distribution for its possible values overstatec is

Pc~p!5U E0

`

dx cp~x!c~x!U2

.

Notice that this indeed satisfies all the requirements for ibe a probability.

Let us see in more detail the relationship betweenprobability density and the POVM. We have written abothe expression for the action ofF(@a,b#). Symbolically, wecan writeF(dp), and its diagonal matrix element over a stac is

^cuF~dp!c&5E0

`

dxE0

`

dy c~x!eip(x2y)/\

2p\c~y! dp

5Pc~p!dp.

The expression just written defines in fact^fuF(dp)c&, andtherefore the whole POVM. Furthermore, the domain ofoperator associated with the POVM is also definedPc(p): by integration by parts it can readily be seen thatdomain of the operatorp is precisely the set of statesc forwhich the second moment of the probability distributiPc(p) is finite, and there it coincides withi pci2. Therefore,the probability density written above defines the POVM athe operator associated with it. Notice also that the probaity density need not be a continuous function. The onlyquirement is thatPc(p)dp be a good measure on the reline.

Another important property of the POVM is covarianunder displacements of momenta, which reflects the comtation relation@ x,p#5 i\. Namely, it is readily computedthat, for all realq,

^fueiqx/\F~@a,b# !e2 iqx/\c&5^fuF~@a1q,b1q# !c&.

In terms of the probability density, the statement is that

Pcq~p!5Pc~p1q!,

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wherecq5e2 iqx/\c is the shifted state.The probability density, written above for the case of pu

states, can be easily generalized to mixed states. Letr(y,x)be the matrix elements of the density matrixr in the positionrepresentation. The probability density associated withdensity matrix is then

Pr~p!5E0

`

dxE0

`

dyeip(x2y)/\

2p\r~y,x!.

In what follows we will use only pure states in the discusion, bearing in mind that the generalization to mixed stais straightforward.

It would seem that something akin to the spectral theorof self-adjoint operators has been achieved, and this isdeed the case. The difference, however, lies in the factthe diagonal matrix element of higher-order powers ofp on astatec, ^c,pnc&, does not necessarily coincide with the coresponding moment of the distributionPc(p), i.e.,*2`

` dp pnPc(p), for n>3. For the case at hand, for instance, this happens already forn53 if c8(0)Þ0. As a mat-ter of fact, in such a situation the diagonal matrix elementp3 over the statec has an imaginary part. Nonetheless, tmeasures of momenta can be readily associated withPc(p),which carries the relevant physical information.

Even so, we would normally like to understand the costructions above in terms of the more usual recipes for sadjoint operators. We shall now make essential use ofuniqueness theorem for POVMs of maximally symmetric oerators: given a maximally symmetric operatorA over a Hil-bert spaceH, there is a unique POVM,FA ~unique up toisomorphisms!, such that its first operator moment coincidwith the operator, and that the set of states over whichsecond moment exists is precisely the domain ofA @5,22#.This means that if we are to construct by whichever meanPOVM such that it fulfills these conditions, we will be obtaining again the same POVM.

Moreover, Naimark’s dilation theorem tells us that aPOVM associated with a symmetric operator defined odense subset ofH can be constructed from a self-adjoiextension of the operator to a larger spaceH, as follows: letE be the projection valued measure of the self-adjoint extsion „i.e., a POVM that satisfies the further requirement thE(@a,b#)25E(@a,b#)…, and P the projection operator fromH to H. ThenF(@a,b#)ªPE(@a,b#) is a POVM associatedwith the symmetric operator.

As a consequence, if we are to build a self-adjoint extsion of p in a larger space, we reproduce the unique POVand the whole of the physical content of the operator frthe usual analysis for the self-adjoint extension and a protion. Notice, however, that the possible extensions are inite. Not so the POVM, and that makes the freedomchoice of extension even more interesting.

Back to the case ofp52 i\]x defined on the half-line, wesee that there is a simple possibility: to extend the operatothe full line, i.e., 2 i\]x defined on a dense subset ofH

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I. L. EGUSQUIZA AND J. G. MUGA PHYSICAL REVIEW A61 012104

5L2(R,dx). Naturally enough, the action of this operatorthe elements ofD( p) is the same as that ofp, so it is anextension. This is not the only self-adjoint extension onwhole real line, of course. As a matter of fact, the deficienindices for the direct sum of the momentum operators onpositive and negative half-lines are (1,1), thus signaling ta one-dimensional continuum of alternative self-adjointtensions exists. They differ by the presence of a jump fution located onx50. However, most natural is the mometum on the full line, defined over absolutely continuofunctions, i.e., with no jump onx50. This is a self-adjointoperator, for which the standard spectral analysis is apcable.

More concretely, the projection valued measureE is givenby the expression

^fuE~@a,b# !c&

5Ea

b

dpE2`

`

dxE2`

`

dy cp~y !cp~x!f~x!c~y!,

with cp(x)5exp(ipx/\)/A2p\, as before, but now defineover the whole real line. The projectionP in our case issimply (Pc)(x)5Q(x)c(x), with Q being Heaviside’s stepfunction.

The probability distribution for the momentum operatover the whole real line for a statec is of course the modulussquared of the wave function in the momentum represetion, and its restriction to states that belong toD( p) is noneother than the probability distribution associated with tPOVM.

We see then that the reason we could not do the stananalysis for the momentum operator on the half-line is twe are being, in a way, far too restrictive in the behavior nx50 of the states on which it can act. There is a remindethe full line, seen for instance in the principal part that fobids orthogonality, or in the fact that the higher momentsthe probability distribution do not in general agree with tdiagonal matrix element of powers of the restricted momtum operator over the corresponding state. If our states wsuch that the functionc and all of its derivatives were zero ax50, there would be no problem with the higher momenand we would get no imaginary part for the powers ofp oversuch states. However, such a strong restriction on the alable wave functions would cut out many physically sensicases.

Let us now write slightly more formally the extensioprocedure we have performed to reobtain and interpretprobability density: we have started with a maximally symetric operator p52 i\]x acting on the dense domaiD( p),H.5L2(R1,dx), with deficiency indices (1,0). Wehave then considered an extension inL2(R,dx), making useof the isomorphism

L2~R,dx!5L2~R1,dx! % L2~R2,dx!.

The extension has been the natural one, i.e.,2 i\]x on thefull line @although, as we repeatedly stated, this is notonly possible extension; another easy choice would h

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been some extension ofp% (2 p), for instance#. The stan-dard spectral analysis for this operator produces the probaity density, which, when restricted toH. , givesPc(p), theprobability density out of which the POVM for the operatwe started from can be built.

III. THE TIME-OF-ARRIVAL OPERATOR

Within the long-running discussion on the concept of timin quantum mechanics, and in particular with respect tostatus of the time-energy uncertainty relations, AharonovBohm introduced in an important paper@7#, among otherquestions, ‘‘a clock’’ to measure time from the position amomentum of a freely moving test particle. The correspoing operator was obtained by a simple symmetrization ofclassical expressionmy/py , where y and py are, respec-tively, the position and momentum of the test particle. By tsame token, the operator obtained by symmetrizing the csical expression for the arrival time atx50 of a freely mov-ing particle having positionx and momentump, t52mx/p, is given by@2,20#

TABª2m

2~ xp211 p21x!.

Note the minus sign in comparison to the clock time in@7#.In spite of the somewhat subtle difference~in concept andsign! with the original time operator introduced in@7#, weshall refer to this operator as the Aharonov-Bohm~time-of-arrival! operator. Without regard for topological consideations, it is clear that it has the correct commutation relatwith the free-particle Hamiltonian on the line,H05 p2/2m. Itis thus a good candidate for a time operator, with the plsible physical interpretation, given by the corresponderule, that it is related to the time of arrival~notice that otherquantization rules might possibly give a different result,though this is the operator obtained not just by the symmtrization rule, but also by applying Weyl, Rivier, or BornJordan quantizations@24#!. However, it also follows, fromPauli’s theorem, that it cannot be a self-adjoint operator aing on a dense space.

Let us then examine first the question of the domainTAB . In order to do that, it is useful at this point to considthe Hilbert space of the free particle in the momentum rresentation,Hp :5L2(R,dp) ~on this space we know howp21 acts, thanks to the spectral theorem!. Formally, we thenobtain

TAB→i\m

2 S 1

p22

2

p

]

]pD .

There are other alternative expressions, such2 i\mp21/2]pp21/2, which would be valid forp.0, or forp,0 by analytic continuation, see, e.g.,@16#. At any rate,TAB understood as a differential operator presents a singpoint atp50, and this is the source of all the difficulties thhave appeared in the literature.

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FREE-MOTION TIME-OF-ARRIVAL OPERATOR AND . . . PHYSICAL REVIEW A 61 012104

The differential operator written above can only be aplied to absolutely continuous functions, but there are furtrequirements. One such is thatTABc belongs toHp , i.e., thatit be square integrable. This poses a restriction due tosingularity of the operator atp50. On computation, onefinds that the singularity is avoided if the functionc has oneof the following possible behaviors close top→0: eitherc(p);p1/2 or c(p)/p3/2→0. However, this is not enough tfix the domain of the operator, as Paul noticed long ago@9#.Given that, at least formally,TAB is symmetric, this shouldalso be a requirement on its domain. Integration by parts,demanding thatwuTABw& be equal to TABwuw& for all w inthe domain ofTAB , leads us to exclude the first possibilitthus defining the domain ofTAB , D(TAB), as the set of ab-solutely continuous square integrable functions ofp on thereal line, such thatc(p)/p3/2→0 as p→0 and iTABci2 isfinite. As for the alternative expression2 i\mp21/2]pp21/2,let us ask that@Q(p)2 iQ(2p)#c(p)/Aupu be absolutelycontinuous forc to be in its domain. This, together with thfurther requirement of symmetry, leads us to a domain tcoincides with that ofTAB . Since the respective actions alscoincide over this domain, we see that they are but equlent differential expressions for the operator, once theequate domain is taken into account.

TAB is closed over this domain, as can be checkedcomputing (TAB

† )† @there is no doubt thatD(TAB) is dense in

Hp , so the closure ofTAB must coincide with (TAB† )†]. Be-

fore that, though, let us examineTAB† and its domain.

In order to apply von Neumann’s formula, we havecheck whether there are statesc in Hp such that for allwPD(TAB) the following expression holds:

^cu~ TAB1 i !w&50,

since thenc is an eigenvector ofTAB† with eigenvaluei.

Analogously, we also have to study the case with eigenva2 i . By integration by parts, and application of the conditithat all functions inD(TAB) satisfy, namely, thatw(p)/p3/2

tends to zero asp tends to zero, it is found that there aretwoindependent eigenvectors with eigenvaluei, and none witheigenvalue2 i . The relevant eigenvectors are

c6~p!5Q~6p! A6p e2p2/2m\.

Notice that, contrary to the expectations of some auth@10,16,17#, both of them have to be taken into account: theare no requirements of derivability or continuity for the funtions in the domain of the adjoint.

The deficiency indices are therefore (2,0), and we havmaximally symmetric operator. As we have seen in the pceding section, this implies that no self-adjoint extensionexist, and we should redo for this case the analysis pformed for the momentum operator on the half-line. Hoever, the most convenient way of doing that is by passingthe energy representation, as was already pointed ouAllcock in his seminal work@8#, and emphasized again b

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Kijowski @11,12#. This change of representation is useffrom a mathematical point of view, because it implementtheorem@22# which states that a simple symmetric operawith deficiency indices (n,0) can be decomposed as a diresum ofn operators with deficiency indices (1,0). Since eaof these is in fact isomorphic to the momentum operatorthe half-line, we will be able to use directly the previouresults.

The change of representation corresponds to the decposition of the Hilbert spaceHp into the subspaces of postive and negative momentum, i.e.,

L2~R,dp!5L2~R1,dE! % L2~R1,dE!5H1 % H2 ,

where the first subspace is that of positive momenta, whethe second corresponds to negativep. The explicit isomor-phism is given by

c6~E!5~m/2E!1/4c~6A2mE!,

c~p!5S upum D 1/2FQ~p!c1S p2

2mD1Q~2p!c2S p2

2mD G ,where cPHp , c6PH6 , and the isomorphism relatec↔(c1 ,c2). The factorE21/4 is due to the change in thmeasure fromdp to dE and reciprocally for the factor(upu/m)1/2. The interesting point is that given this isomophism, the time operator of Aharonov and Bohm takesform 2 i\]E , as is well known. The domain of the operatoD(TAB), is sent by the isomorphism into the direct sumssquare integrable absolutely continuous functions in esubspace, such that for each subspace we have the restrthat ca(E)E21/2→0 asE→0. But this is exactly the caseconsidered above, since the requirementca(0)50 and thesquare integrability ofca8 (E) imply the restriction stated before. In other words, we have the isomorphism

TAB5~2 i\]E! % ~2 i\]E!5T1 % T2 ,

whereT6 are isomorphic to the momentum operator on thalf-line.

Therefore, the constructions carried out in the precedsection can be immediately translated to this situation,taking into account that the energy spectrum is degenewhereas the position spectrum is not. For instance, the cplete nonorthogonal set of generalized eigenfunctionscp isdoubled here into a set with a continuous parametert ~thenotation is intended to be suggestive! and a discrete onewith values1 or 2, as follows:

c1(t)~E!5S 1

A2p\eiEt/\,0D

and

c2(t)~E!5S 0,

1

A2p\eiEt/\D .

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I. L. EGUSQUIZA AND J. G. MUGA PHYSICAL REVIEW A61 012104

In what follows, we will not make the explicit distinctionbetween the element of the full Hilbert space and its comnent, if only one is zero.

These functions transform under the isomorphism to gthe following expressions:

ca(t)~p!5Q~ap!S ap

2pm\ D 1/2

eip2t/2m\.

It is straightforward to prove completeness, i.(a*2`

` dt ca(t)(p8)ca

(t)(p)5d(p2p8). Alternatively,(a*2`

` dt ca(t)(E8)ca

(t)(E)5d(E2E8)1, where1 is the twoby two identity matrix@d(E2E8)1 is the identity operatoron the full Hilbert spaceH1 % H2].

Nonorthogonality is also a direct translation:

E0

`

dE ca8(t8)

~E!ca(t)~E!5

1

2daa8S d~ t2t8!1

i

pP

1

t2t8D

or

E2`

`

dp ca8(t8)

~p!ca(t)~p!5

1

2daa8S d~ t2t8!1

i

pP

1

t2t8D .

It now behooves us to compute the POVM or, alterntively, the probability distribution for measured values of tTAB operator from which it can be readily recovered. Bdirect translation, we can write the probability density as

P (c1 ,c2)~ t !5U E0

`

dEe2 iEt/\

A2p\c1~E!U2

1U E0

`

dEe2 iEt/\

A2p\c2~E!U2

in the energy representation, or as

Pc~ t !5U E0

`

dp S p

2pm\ D 1/2

e2 ip2t/2m\c~p!U2

1U E2`

0

dp S 2p

2pm\ D 1/2

e2 ip2t/2m\c~p!U2

in the momentum representation. This is, of course, the sas Kijowski’s probability density. The essential propertycovariance under transformations generated by the Hatonian is also evident and a direct translation of the covance property signaled in the preceding section. Physicallmeans that the probability of arriving att for a given state isequal to the probability of arriving att2t for the same stateevolved for a timet. This is the reflection on the probabilitdensity of the canonical commutation relation@H,TAB#5 i\.

The domain ofTAB , now defined through the POVM, cabe characterized as the set of elements of the Hilbert spfor which *2`

` dt Pc(t)t2 is finite, and this quantity define

^TABcuTABc& ~thus realizing the minimum variance demanded by Kijowski and Werner@11,13#!.

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As before, we would like to understand all these constrtions in terms of a generalized self-adjoint extension, bying the uniqueness of the POVM associated with a mamally symmetric operator, and Naimark’s theorem. Fromstructure of the time operator of Aharonov and Bohnamely TAB5(2 i\]E) % (2 i\]E) on L2(R1,dE)% L2(R1,dE), it follows that anatural and simple extension~natural in the sense of following the natural extension inanalogy! is the operator (2 i\]E) % (2 i\]E) acting onL2(R,dE) % L2(R,dE), which is obviously self-adjoint. Inother words, we have introduced negative energies, resping the twofold degeneracy of the initial spectrum. On thspace the~doubled! Fourier transform acts as a unitary tranformation that provides us with thetimerepresentation of thestates, and the probability density for the time of arrival ova pure state in this extended space is nothing but the modsquared of the wave function in the time representation. Ntice that the time representation corresponds to the douspaceL2(R,dt) % L2(R,dt). We reobtain the by now usuaprobability density for time of arrival, by restriction to thinitial Hilbert space. The restrictions to the initial Hilbespace of the applications of the spectral theorem are thfore related toPc(t), as stated before.

The problems that the vicinity ofp50 ~alternativelyE50) pose for the analysis of the operator of Aharanov aBohm are thus seen to be due to the~physically imposed!restriction to the space of positive energies. There is aminder of the negative energies in aspects such as theorthogonality of the set of generalized eigenfunctions, simlarly to what happens in the paradigmatic example ofpreceding section.

IV. GENERAL TIME OPERATORS

In previous sections we have discussed the time of arrfor free motion, but in fact its mathematical form, andPOVM, can easily be generalized for arbitrary potentfunctions. The important point, however, is that in genethe resulting operators and POVMs cannot be physicallysociated with an arrival time.

Consider a self-adjoint Hamiltonian, bounded from blow, defined on a Hilbert spaceH, such that the spectradecomposition leads to the isomorphism

H5 % i 51N ~ % j 51

ni L2„@ai ,`!,dm i~E!…!,

where all the measuresm i are absolutely continuous withrespect to the Lebesgue measure. That is to say, the specof the Hamiltonian is absolutely continuous, and for agiven EPR the degeneracy is finite, the maximum degeeracy being( i 51

N ni . The Hamiltonian is realized in eacsubspace as the multiplication by the variableE. Therefore, anatural ~but most definitely not unique! candidate for thetime operator is again2 i\]E on each of these subspacerestricted to functions that vanish atai . Again each compo-nent of this time operator can be extended to the real lwhere it will be self-adjoint and admit a spectral decompsition, which, by projection, will give us again the POVMassociated with our candidate operator. The POVM, and

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FREE-MOTION TIME-OF-ARRIVAL OPERATOR AND . . . PHYSICAL REVIEW A 61 012104

associated probability density, will be unique for the givoperator if this is indeed maximally symmetric.

Notice, moreover, that if we do extend one of the Hilbsubspaces to be squarem-integrable functions on the realine, and demand that the covariance property of the restion of the time operator to that Hilbert subspace be mtained on the real line, then, under fairly general conditiothe extension of the time operator and HamiltonianL2

„R,dm i(E)… will be unitarily equivalent to the canonicallconjugate pair2 i\]E and multiplication byE. Therefore,modulo intervening unitary transformations, we have a wof constructing POVMs for time operators by Naimark diltions.

The usefulness or otherwise of the particular time opetor under consideration will then have more to do withactual properties and measured probability density rathan with mathematical difficulties that had riddled the woof many previous workers in the field.

To illustrate this point further, let us examine the standexample of a particle in a constant field. The HamiltonianHg5 p2/2m1mgq, clearly seen to be unbounded. Pauli’s agument is therefore not applicable, and a self-adjoint opetor canonically conjugate toHg does indeed exist, namelyTg5 p/mg. In the energy representation, this operator iscourse given by2 i\]E . It can be ascribed to the time oarrival to zero momentum, but in no possible way to the tiof arrival to a specified position. This example tells us than operator having the form2 i\]E in the energy representation, be it as a self-adjoint operator or associated witPOVM, does not necessarily mean that we have obtainetime-of-arrival operator, even though a time operator isdeed being considered.

At this point it is worthwhile mentioning the approachLeon et al. @25#, intended to generalizeTAB to the interactingcase of a potential barrier~under the hypothesis oasymptotic completeness and absence of bound states!. Theydo obtain good time operators, involving2 i\]E in the basisof outgoing or incoming scattering states, and their distritions, understood as POVMs. However, since the diagomatrix element of these operators is, over a given state,same as the expected time of arrival that the corresponasymptotic states~outgoing or incoming, respectively! wouldproduce in the case of no interaction, it is hard to understthem as bona fide time-of-arrival operators at an arbitrspatial point.

Finally, it is also interesting to examine the relation of toperatorTAB with older literature on the ‘‘time operator’’ byOlkhovski, Recami, and others@26#. The basic idea behindthis set of works is to extract a time operator from the retion that defines an average ‘‘presence time’’ atx50 by

^t&[

E2`

1`

dt uc~x50,t !u2t

E2`

1`

dt uc~x50,t !u2.

We shall discuss as in@26# the simple case of states withonegative momenta so that there is no need to consider

01210

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energy degeneracy in the following expressions. The eneFourier transform ofc(x50,t) is given by

h~E!5h21/2E2`

1`

c~x50,t !eiEt/\ dt

5h1/2^x50uE&^Euc~ t50!&.

In terms ofh we can write

^t&52 i\E0

`

dE h~E!*]h~E!

]E,

where h(E)[h(E)/@*dtuc(0,t)u2#1/2. Thus a ‘‘time opera-tor’’ may be again identified with2 i\]E , but now in thespace of functionsh(E). This operator is different fromTAB .

In terms of the position and momentum operators,average presence time becomes

^t&52m

2

^c~ t50!u p22x1 xp22uc~ t50!&

^c~ t50!u p21uc~ t50!&, ~1!

provided the integrals exist. This is to be contrasted withaverage time that can be defined using the current denJ(x50,t),

^t&J[

E2`

`

dt J~x50,t !t

E2`

`

dt J~x50,t !

5^c~ t50!uTABuc~ t50!&. ~2!

The different formal results are to be expected since, inclassical limit, ~1! is associated with an average presentime and~2! with an average passage time. Note, howevthat in both cases we can relate these measurements witformal operator2 i\]E , as has been pointed out all along

As stated above, no matter how one constructs gentime operators, one is inevitably led to2 i\]E ~plus, possi-bly, a function ofE), but the physical interpretation of thosoperators, self-adjoint or associated with a POVM, is quanother issue.

V. DISCUSSION

In the sections above, we have made reference to resand proposals of other workers in the subject. In this secthowever, we intend to restate some of those results andcussions in the light of our description. The best placestart is the initial paper of Aharonov and Bohm@7#, wherethey introduce the operatorTAB . They were not particularlyconcerned with the self-adjointness or otherwise of theerator, but they did signal the problem of the singularity fp50, which they brushed aside by stating that all compnents of the interesting wave functions would be of hienough momentum. Insofar as that is the case, then theindeed no problem in obtaining a probability distribution fmeasurements of the operator. Their real point of interes

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I. L. EGUSQUIZA AND J. G. MUGA PHYSICAL REVIEW A61 012104

introducingTAB was to show that there is no reason inherin the principles of quantum mechanics for the energy ofobserved system not to be observed in as short a time aspleases, where the time is measured in an observer sysThe topic of measurement limitations is revisited in mulater work of Aharonov and others@27#, where they make thedistinction between direct and indirect measurements of t

of arrival, indirect measurements being associated withTAB

~or with regularized variants! and direct measurements witcouplings of the particle with other degrees of freedom tin the classical limit would measure the time of arrival. Thdescribe a lower bound for the time uncertainty of diremeasurements inversely proportional to the typical energthe particle~not to the energy uncertainty!. In fact, the samedependence has been recently described, and traced bathe lower bound of the energy spectrum, for the indirmeasurement case@28#. But the time uncertainty refers to aensemble and does not preclude a precise determinatiothe time of arrival in an individual measurement; in othwords, this does not involve a limitation of the quantutheory to handle the arrival time concept and prescribein-trinsic arrival time distributions~formulated without explicitrecourse to a measuring apparatus or an additional ‘‘clodegree of freedom!. Numerical results have shown essentagreement between the~intrinsic! POVM distribution andoperational distributions given by phenomenological scrmodels, except in rather pathological cases@29,30#.

Going further back in time, we come to the workAllcock @8#. As we have signaled above, he advocateduse of the energy representation. Even more, he signaledneed for negative energies in a particular way. If we indewant to measure arrival times, we would have to prepwave packets that are localized on one side of the poinarrival (x5x0, say! for all times before a given one,t,t0.This directly entails that the temporal Fourier transformthe wave function will exhibit all positive and negative eergies, save for a set of null measure. In a way, it couldargued that the conceptual setup for measuring time ofrival requires by itself the introduction of negative energiFurthermore, the proof he proposes for the need for all ptive and negative energies is also illuminating: it goesway of the Paley-Wiener-Titchmarsh theorem anduniqueness theorem for analytic continuations. This is pcisely the same route taken to show the nonorthogonalitany set of functions that could be considered as~generalized!eigenfunctions of a time operator. In that proof it is clear ththe introduction of negative energies also eliminatesproblem of nonorthogonality. However, his interpretationthe nonorthogonality of~generalized! eigenfunctions for thetime of arrival as precluding any measurement thereof isodds with the operational approach within which the POVidea takes place.

A number of papers have also been published with a mfunctional analytical approach than those mentioned be@9,10,14#.

The axiomatic approach of Kijowski@11# has been mentioned all along, since its guiding idea, i.e., characterizingphysical content of a concept such as time of arrival in

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probability distribution, is crucial for our purposes, anclears the path for the introduction of the POVM as tphysically relevant object. Kijowski also emphasized tusefulness of the energy representation, and decomposeHilbert space of the free particle as the direct sum ofsubspaces of positive and negative momenta. He constr

both the operatorsT6 ~in his notation,t6Q , if Q is the point

x350) by comparison with the probability distribution, a

we do. He then points out thatT1 % T2 admits no self-adjoint extension, and goes no further that way. Howeveralso introduces a particularly interesting object, retaken mrecently by Delgado and Muga@15# ~see also@12#!, namely

T85T1 % (2T2). He then asserts that this operator is esstially self-adjoint. Let us examine this question from thpoint of view taken in this paper. As we have signaled, b

T1 andT2 are isomorphic to the momentum operator on thalf-line, and their deficiency indices are (1,0) for both

them. Therefore,2T2 is also a maximally symmetric opera

tor, with deficiency indices (0,1). The total operatorT1 %

(2T2), symmetric on a dense subspace, has deficiencydices (1,1), so it is not self-adjoint, although it admits seadjoint extensions. Contrary to what might be inferred frosome claims in@12#, there are infinite self-adjoint extensionparametrized byaP@0,2p). The extensionTa8 is defined on

D(Ta8 ), which is the set of elements ofH1 % H2 of the form

(c11le2E/\,c21leiae2E/\), where c6PD(T6) and lPC. The action on its domain is simply that of the coupledifferential operators (2 i\]E ,i\]E). The boundary condi-tions, however, are different. In terms of the momentum rresentation, the functions in the domain ofTa8 can havep1/2

behavior, for instance. At any rate, the probability distribtion deduced from these operators is not covariant under tshifts, so their physical interpretation is rather clouded unlthe state belongs to only one of the two subspaces~positiveor negative momenta! @20#.

Another interesting attempt to construct a self-adjoint oerator related to the time of arrival is that of Grot, Roveand Tate@16#. In essence, they give a realization of the ituition of Aharonov and Bohm@7# that for high momenta thep50 singularity of the differential operator is irrelevant,point later taken up by Paul@9# as well. However, this generalized self-adjoint operator also has some drawbacks fthe point of view of its interpretation@20,17#. From the pointof view taken in this paper, the problem with the proposalGrot, Rovelli, and Tate is that the strong operator limittheir deformed operator does not exist when their regulatomade to disappear~this can be seen, for example, computithe norm of the action by the regulated operator on aGaussian state, and then trying to eliminate the regulatothe norm blows up!. Therefore, the results obtained with thregulator cannot be at all extrapolated to the unregulacase.

Closest to the work carried out in this paper is the prestation by Busch, Grabowski, and Lahti@5,6#. However, theyrefer the idea of using Naimark’s dilation in this contextthe paper of Werner@13#, where he constructs ‘‘screen ob

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FREE-MOTION TIME-OF-ARRIVAL OPERATOR AND . . . PHYSICAL REVIEW A 61 012104

servables’’ using a version of imprimitivity adequate for tcase of POVMs. Our analysis is much more simplified aexplicit.

In conclusion, we have given a description of the POVMassociated with the momentum operator on the half-linethe time operator of Aharonov and Bohm in terms of Naimrk’s dilations, more concretely in terms of what for the mmentum operator is the natural dilation. We have clarifiseveral issues that have been a matter of debate or confuin the literature as of late, in particular the deficiency indicand constructions of self-adjoint extensions ofTAB , and wehave rederived the probability density for times of arrivderived axiomatically by Kijowski. We have also made

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proposal for the construction of the probability distributioof a wide class of time operators, for which it has beshown that they are not in general related to the arrival timin spite of the formal agreement with the arrival time opetor of the free motion case.

ACKNOWLEDGMENTS

We thank W. Amrein for very useful comments. Suppoby the CICYT under Project Nos. AEN-96-1668 and PB81482, by Gobierno de Canarias~Grant No. PB/95!, and bythe University of the Basque Country~Grant No.UPV063.310-EB187/98! is gratefully acknowledged.

vo

.G.

iza,

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