Path integral approach to space-time probabilities: a theory without pitfalls but with strict rules

0.18 MB·English

by  Dmitri Sokolovski

Path integral approach to space-time probabilities: a theory without pitfalls but with
strict rules
D. Sokolovski1,2
1
Departmento de Qu´ımica-F´ısica, Universidad del Pa´ıs Vasco, UPV/EHU, Leioa, Spain
2
IKERBASQUE, Basque Foundation for Science, E-48011 Bilbao, Spain
(Dated: December 11, 2013)
Followingtherenewedinterestinthetopic[1],werevisittheproblemofassigningprobabilitiesto
classes of Feynman paths passing through specified space-time regions. We show that by assigning
of probabilities to interfering alternatives, one already makes an assumption that the interference
hasbeendestroyedthroughinteractionwithanenvironment,orameter. Includingtheeffectsofthe
meterallowstoconstructaconsistenttheory,freeoflogical’pitfalls’identifiedinRef.[1]. Wherevera
3
metercannotbeconstructed,orcannotbesettoeffectthedesireddecoherence,formallyconstructed
1
probabilitieshavenoclearphysicalmeaning,andcanviolatethenecessarysumrules. Weillustrate
0
theabove approach by analysing thethreeexamples considered in [1].
2

n PACSnumbers: PACSnumbers: 03.65.-w,03.75.lm,02.50.-r
a
J

8 I. INTRODUCTION that the chosen restriction automatically prescribes the
1 type of the meter required to destroy interference. In

Sect. V we emphasise the distinction between the finite-
] Recently Halliwell and Yearsley [1] revisited the prob-
h time and continuous quantum measurements, and pro-
lem of assigning probabilities to amplitudes obtained by
p ceedtoanalysetheformertype. InSect. VIweformulate
restricting Feynman paths to space-time regions. They
- three questions which define a quantum measurement.
t emphasised that the method suffers from serious defi-
n Section VII describes the mixed state of the system af-
a ciencies: ”seemingly obvious notionof ’restricting paths’ terinterferencebetweentheclassesofFeynmanpathshas
u leads to the quantum Zeno effect and hence to unphysi-
beendestroyed. InSectionVIIIweproveageneralresult
q cal results” This is the sense in which we would say that
concerning the emergence of a kind of the Zeno effect in
[ path integrals constructions may suffer from pitfalls’[1].
high-accuracy ’ideal’ measurements. In Sections IX, X,
2 Their criticism extends to the path integral analysis of XI we analyse the examples used in Ref.[1] to illustrate
v the quantum traversal time [2]-[8]. In order to mitigate
’unreasonable properties’ of path integrals amplitudes.
4 the Zeno effect, the authors of [1] follow Alonso et al
Section XII contains our conclusions.
4 [9] in suggesting ’temporal coarse graining’ whereby the
2
observed system is controlled only at discrete times, be-
1
tween which it is allowed to evolve freely. The problem
. II. PATH DECOMPOSITION OF A
1 is closely related to the more general question concern-
TRANSITION AMPLITUDE
0 ingtheoriginofquantumprobabilities(see,forexample,
3
[10]-[15]).
1 For a system governed by a Hamiltonian Hˆ, consider
: We disagree with the conclusions of Ref.[1] with re-
v the transition amplitude
gards to the ’pitfalls’ of the path integral approach to
i
X quantum probabilities, and would like to restore, in as K(F,I,t)= ψ exp( iHˆt)ψ (1)
F I
r much as possible, the good name of quantum traversal h | − | i
a
time. The purpose of this paper is to show that the ap- between some initial and final states ψ and ψ . In
I F
| i | i
proach is indeed a consistent theory, with its own strict itsHilbertspace,chooseacompeteorthonormalbasis x
| i
rules,yetfreefrom’unphysical’features. Webaseourar- wherexmaytakediscreteorcontinuosvalues,sothat(I
gument on a simple considerationwhich, it appears,was is the unity)
not taken into account in Ref.[1]. Assigning probability
tointerferingalternatives,onemustassumethatthesaid x x =I. (2)
interferencehassomehowbeendestroyed. Destructionof Xx | ih |
interferenceisaphysicalprocess,whichmustresultfrom
interaction with another physical system, e.g., a meter. Writing exp( iHˆt) as K exp[ iHˆ(tk/K)], inserting
− k=1 −
We demonstrate that once the inevitable effects of the (2) between the exponeQntials and also before and after
meter are properly taken into account, one has a theory ψ and ψ , and sending K to infinity yields the cel-
I F
| i | i
free from logical contradictions. The rest of the paper ebrated Feynman path integral [16] (or, as the case may
is organised as follows. In Section II we briefly describe be, path sum),
path decomposition of the propagator. In Section III we
choose a variable, and construct its amplitude distribu- K(F,I,t)= A[path]. (3)
tion by restricting Feynman paths. In Sect. IV we show pXaths
2
Hereapathisdefinedbythesequenceofthex’slabelling IV. PATH RESTRICTION VS. DYNAMICAL
the states through which the system passes at the inter- INTERACTION
′
mediate times t =0,ǫ,2ǫ,...,t, ǫ t/K. In the continu-
≡ ′
ous limit K we will denote it x(t) The functional The task of converting the amplitudes (4) into
→ ∞
A[path] limK→∞ ψF xK xK exp( iHˆǫ)xK−1 probabilities requires some additional care. Our re-
≡ h | ih | − | i
...exp( iHˆǫ)x x ψ is the amplitude the path arrangement of the Feynman histories into classes was
0 0 I
− | ih| | i
contributestoK(F,I,t). WithFeynmanpathsandtheir purely cosmetic. Like the paths themselves, the classes
contributions A[path] known, the task of evaluation of remain interfering alternatives. To assign the probabil-
the transition amplitudes reduces, at least formally, to ities one must first destroy the interference. This must
addition of complex numbers. Conceptual simplicity be done by bringing the system in contact with another
of the Feynman quantum mechanics is matched by the quantum system, or systems, and we must decide what
difficulty of actually performing the path sum (3). It type ofanadditionalsystem(a meter)is suitablefor the
does, however, make a convenient starting point for a task.
discussion of quantum measurements. Conveniently, the answer is already contained in our
choiceof the functional F. Consider the equationof mo-
tion satisfied by the restricted path integral (4). If this
equation in maps onto a Schroedinger equation describ-
III. MEASURABLE PROPERTIES OF A
ing the originalsystem plus another system, we immedi-
QUANTUM SYSTEM
ately obtaina recipe for constructingthe meter with the
desiredproperties. If, onthe otherhand,the equationof
Next we may wish to enquire about the system’s be- motion does not look like such a Schroedinger equation,
haviour in the time interval between its preparation in we must stop and admit that the interference cannot be
the state ψI and its subsequent detection in ψF . We destroyed by any means available to us. The question
| i | i
may not be interested in all the details, but just in the whether further progress can be made in this latter case
values of some quantity F. For the answer to exist, the is beyond the scope of this paper.
value of F must be defined for each of the possible his- Thetypeoftheequationsatisfiedbytherestrictedprop-
tories,i.e., shouldbe a functional F[path]defined onthe agator depends on the choice of the functional F, and
Feynman paths. It is reasonable to re-arrange the paths mustbe establishedineachindividualcase. One general
according to the value of F, and define the probability result was proven in Refs.[17]: let the functional be of
amplitude to have this value equal to f as (δ(z) is the the form
Dirac delta)
t
′ ′ ′
F[path]= β(t)a(x(t))dt (8)
Φ(F,I,tf)= A[path]δ(F[path] f). (4) Z0
| −
pXaths where β(t) is a known function of time and a(x) is some
functionofx. Thentheprobabilityamplitudeforthesys-
Equation (4) can be also written in an operator form
tem to arriveata ’location’ x by taking only the paths
| i
with the properties F[path] = f, Φ(x,tf) (we drop the
Φ(F,I,t|f)=hψF|Uˆ(t|f)|ψIi (5) implied dependence in the initial state||ψIi to simplify
notations), satisfies the Schroedinger equation (SE)
where the restricted evolution operator propagates the
initial state only along the paths which have the prop- ∂ Φ=[Hˆ i∂ β(t)Aˆ]Φ (9)
t f
−
erty F[path] = f. If there are several quantities F , F ,
1 2
...F ,theprobabilityamplitudeforthemtohave,jointly, with the initial condition
N
the values f ,f ...f , K(F,I,tf ,f ,..,f ), can be con-
1 2 N 1 2 N
| Φ(x,0f)= xψ δ(f) (10)
structed in a similar manner. With the part of the path | h | Ii
summation (3) already performed, the full propagator is Here the operator Aˆ, diagonal in the chosen representa-
given by an ordinary quadrature,
tion, is defined by the choice of a(x) in Eq.(8),
K(F,I,t)= df1 df2... dfNΦ(F,I,tf1,f2,..,fN).(6) Aˆ dxx a(x) x. (11)
Z Z Z | ≡Z | i h |
It is tempting to define the probability to have the Equation (9) is, of course, a SE describing the system
property F[path]=f simply as interactingwithavonNeumannmeter[18],withpointer
position f, designed to measure the operator Aˆ. Unlike
P(F,I,tf) Φ(F,I,tf)2. (7) in the original von Neumann’s approach, the pointer re-
| →| | |
mains coupled to the system for a finite time, since the
Next we will show that this temptation should so far be quantity F in Eq.(8) refers to the whole of the time in-
resisted. terval [0,t].
3
V. FINITE-TIME MEASUREMENTS VS. THEIR is a suitable meter can be found. For a functional of the
CONTINUOUS COUNTERPARTS type (8) the answer is ’yes’. One should then prepare
the system in the desired initial state, set the pointer to
Before proceeding we must emphasise an important zero,turnontheinteraction,andaccuratelymeasurethe
distinctionbetweentwokindsofmeasurements. Suppose pointer’s position at the time t.
we have only one meter, and in the spirit of [1], [9], ac- The third question follows from the second. The am-
tivate it through a sequence of sharp strong pulses, that plitude Φ in Eq.(9) is not normalisable due to the pres-
is we choose enceofthe δ(f)in(10), andcannotbe usedto construct
physical probabilities. To obtain a physical amplitude
N
′ Ψ(x,0f), one should prepare the pointer in a physical
β(t)= δ(t−tn), tn =nt/(N +1) (12) state G|(f), df G(f)2 = 1, and replace the initial con-
nX=1 | |
dition (10) wRith
Assume, for simplicity, that the operator Aˆ in Eq. (11)
Ψ(x,0f)= xψ G(f). (14)
I
is a projector onto a part of the Hilbert space Ω. Then | h | i
the pointer wouldmove one notch to the righteachtime The result can be written in an equivalent form [17]
the system is in Ω at t=t , or otherwise remains where
n
′ ′
it is. At the time t we look at the pointer once, and, Ψ(x,tf)= dfG(f f )Φ(x,tf ). (15)
| Z − |
having found it shifted by m notches, conclude that the
system was in Ω m times out of all N trials. We can- which has a simple interpretation. The function G(f
′ −
not, however, say exactly when, as the outcome of the f ) plays the role of a filter, selecting a limited range of
measurement is a single number, yielding the value of a the values of F[x(t)]’s which contribute to the pointer’s
functional on otherwise unspecified virtual trajectories. advancementtopositionf. Foranaccuratemeasurement
Since theobservationtookt secondsto complete,wecall oneshouldchooseG(f)sharplypeakedaroundzero,with
it a finite-time measurement. a small yet finite width ∆f. Now
SupposethatinsteadwehaveN identicalmeters,only
P(x,tf)= Ψ(x,tf)2 (16)
one of which is briefly activated at each t = tn, that is, | | | |
we choose yields the probability the find the system in x , and the
| i
pointeratalocationf. Itisalsotheprobabilitythatthe
′
β (t)=δ(t t ), (13)
n − n observed system arrives at x and F has the value in the
As a result, at t we have N readings, all either 0 or 1, interval [f ∆f,f +f ∆f]. Accordingly,
− −
and the exact knowledge at which of the t the system
n
was found in Ω. Accordingly, there is a set of probabil- P(tf)= dxP(x,tf) (17)
| Z |
ities P(x ,x ,..x ), x = 1. This is a prototype of
1 2 N n
a continuous measurement [1±9], whose outcome is a real yields the probability to find the pointer at x without
trajectoryfollowedby the observedsystem. Like the au- askingabout the state of the system. It is alsothe prob-
thorsof[1]weareinterestedinfinite-timemeasurements, ability that, for the observed system, F has the value
which we will consider through the rest of the paper. within [f ∆f,f +f ∆f] regardless of where it ends
− −
up once the measurement is finished. The correct nor-
malisation of P(x,tf),
|
VI. THREE QUESTIONS TO A QUANTUM
MEASUREMENT dxdfP(x,tf)=1 (18)
Z |
We see now that to describe a quantum measurement, is guaranteed, since the evolution according to the SE
onemustprovideclearanswerstoatleastthreefollowing (9) is unitary. We note that the limitation on the accu-
questions: racy of a measurement is of purely quantum nature: the
a) What is being measured? values inside the interval [f ∆f,f +f ∆f] cannot
− −
bedistinguishedsinceinterferencebetweenthemhasnot
b) How it is being measured?
c) To what accuracy is it being measured? been destroyed.
To answer the first question, we need to specify a vari-
able which characterises the system in the absence of
VII. STATE OF THE SYSTEM AFTER A
a meter. In our case it is the functional in (8), whose
MEASUREMENT
physicalmeaning depends on the choice of the switching
′ ′
function β(t). Thus, for β(t) = 1/t = const it repre-
sents the time average of the of the quantity Aˆ, and for Wenotethatrestrictingitsevolutiontopathswiththe
β(t′)=δ(t′ t ),theinstantaneousvalueofAˆ. Choosing property F[path] = f, we effectively ’chop’ the state of
0
′ ′− ′ the system into sub-states
β(t) =δ(t t ) δ(t t ) allows to measure the sum
1 2
− ± −
or the difference of A’s values at t and t , and so forth.
1 2
Φ(tf) = dxx Φ(x,tf). (19)
Toanswerthe secondquestiononemustaskfirstifthere | | i Z | i |
4
′ ′
whichaddup to the Schroedingerstate ofa freely evolv- where C = G(f )df . In the high-accuracy (ideal mea-
ing system, surement)liRmitα 0,ther.h.s. ofEq.(24)vanishes,and
→
withitvanishestheprobabilitytofindthepointeratany
df Φ(x,tf) =exp( iHˆt)ψI . (20) location f, P(tf)= b Ψ(x,tf)2dx, which is of course
Z | | i − | i | a | | |
wrong. The way arouRnd this difficulty is to assume that
This is the particular property of the interaction in in addition to its smooth part Φ˜(x,tf), Φ(x,tf) has a
Eq.(9). It can be seen as the quantum analog of the | |
number of δ-singularities,
conditionthataclassicalmeter shouldmonitorthe mea-
sured system without affecting its evolution (for details Φ(x,tf)=Φ˜(x,tf)+ c (x)δ(f f ). (25)
k k
see Refs.[20]). A meter of a finite accuracy ∆f, uses lin- | | −
X
ear combinations of the fine-grained sub-states (19)
Now as the accuracy improves, the probability P(x,tf)
|
becomes
′ ′
Ψ(tf) = df G(f f )Φ(tf) , (21)
| | i Z − | | i
limα→∞P(x,tf)= (26)
and then destroys the coherence between these ’coarse- |
grained’ states. The final mixed state of the system ρˆ ck(x)2δ(f fk)+α−1 C 2 Φ˜(x,tf)2.
| | − | | | | |
after measurement, therefore, is Xk
Suppose now that one is conducting an experiment on
ρˆ(t)= df Ψ(tf) Ψ(tf), (22)
Z | | ih | | a large number of identical systems, all post-selected in
the state x , and receives a signal proportional to the
with trρˆ(t)=1, as follows from Eq.(18). | i
number of cases in which the pointer is found in f. As
theaccuracyimproves,thesignalisdominatedbystrong
peaks at f = f . In addition, there is a smaller sig-
VIII. POSSIBILITY OF A ’ZENO EFFECT’ k
nal revealing more and more details of the structure of
Φ˜(x,tf)2, and eventually fading altogether. The po-
The convolution formula (10) has the advantage that | | |
sitions of the peaks f , and the coefficients c must be
if the main properties of the fine-grained amplitude k k
determined for each particular case.
Φ(x,tf)areknown,wecanqualitativelyestimatethere-
| Forexample,earlierwe haveshown[22]thatfor asys-
sult of smearingit with the function G(f). For example,
tem in a finite-dimensional Hilbert space at attempt to
suppose that a transitionis classically allowed. Than, in
determine precisely the value of the time average of an
the semiclassical limit, rapidly oscillating Φ(x,tf) has
| operatorAˆinevitablyleads to aZeno effecttrapping the
a narrow stationary region around the classical value of
system in the eigenstates (eigen sub-spaces of Aˆ). An-
F[x(t)], f . Then a not-too-accurate meter will always
cl ′ other example of such behaviour was given in Ref.[23]
return the value f , since if G(f f ) is centred at any
cl − which analysed a measurement of qubit’s residence time
f = f the integral (10) will vanish, destroyed by the
6 cl withaslightlymoresophisticatedvariantofthevonNeu-
oscillations [21].
mann meter.
Itispossibletomakeanothergeneralstatementabout
Forasysteminaninfinitevolume,e.g.,for <x< ,
the properties of Φ(x,tf). For a system confined to a −∞ ∞
| there exists another possibility. While Ψ(x,tf)2 must
finite interval (volume) a x b, the amplitude dis- | | |
≤ ≤ vanish as Aˆ , the range of x’ s involved in the inte-
tribution Φ(x,tf) cannot be a smooth function for all →∞
| gral (17) may increase proportionally, so that the prob-
x. Rather, to ensure conservation of probability in the
ability is conserved without Φ(x,tf) acquiring singular
high accuracy limit ∆f 0, it must have singularities, |
→ terms. Physically this would mean that the meter scat-
typically, of the Dirac delta type.
ters the observed system into a wide range of its final
Theproofisbasedonconservationofprobability. Sup-
positions.
pose we have some reference function G(f) and want to
For the following, it suffices to note that, as it fol-
improve the accuracy by making it narrower. This can
lows from Eq.(26), an accurate determination of the
be achieved by simple scaling,
value of any quantity of the type (8) may suppress some
G(f) Gα(f)=α1/2G(αf), (23) of the transitions and lead to sharply defined values
→
F[path] = f in those transitions which survive. With
where we recalled that G(f) is also the initial state of k
some hesitation we follow the authors of Ref.[1] in call-
the pointer, and as such must be normalised to unity,
df G (f)2 =1. Suppose nowthatΦ(x,tf)is smooth. ingita’Zenoeffect’alsointhecasethemeasuredsystem
α
| | | hasacontinuousspectrum. InaconventionalZenoeffect
RThen, as α , the width of Gα ∆f/α 0, an we
→ ∞ ∼ → [27]-[28], frequent observations trap the system in one of
should be able to take Φ outside the integral (10),
the eigenstates of the measured quantity. Here, the ac-
′ ′ tionof he meter restricts the systemto a particulartype
Ψ(x,tf)= dfG (f f )Φ(x,tf ) (24)
α
| Z − | ≈ ofevolution,withoutfreezingitaltogether. Withthiswe
are ready to analyse the cases discussed by the authors
Φ(x,tf) G (f′)df′ =α−1/2CΦ(x,tf),
| Z α | of [1].
5
IX. THE PROBABILITY NOT TO ENTER THE solarge,thatthe transmittedandreflectedwavepackets
RIGHT HALF-SPACE are well separated and do not overlap. With the help
of Eq.(28) the amplitude distribution for the duration τ
Westartwiththetaskofdefiningtheprobabilitythat, spent by a free particle in the right half-space becomes
in one dimension, a free particle of a mass M does not [cf. Eq.(29)]
enter the region Ω 0 x < [1]. Equivalently, one
≡ ≤ ∞ (33)
canask: ’whatistheprobabilitythattheparticlespends
in Ω precisely a zero duration, τ = 0?’ The functional Φ˜(x,tτ) for x>0;
Φ(x,tτ)= |
yielding the duration a Feynman paths spends in Ω is | (cid:26) δ(τ)ψ0( x,t)+Φ˜( x,tτ) for x<0.
− − − |
well known [2],
Hereψ (x,t) x exp[ iHˆt)ψ isjustthe freelyprop-
0 I
t ′ ′ agating wave≡pahck|et, an−d Φ˜(x|,tτi) is a smooth function,
tΩ(t,[x(·)])=Z θΩ(x(t ))dt, (27) involvingthe Fouriertransform|ofT(k,V)atallrelevant
0
energies. The last two terms clearly correspond to the
where θ (z) = 1 for z inside Ω, and zero otherwise.
Ω particle being reflected, for example, we have
The operator in Eq.(10) is just the projector onto the
right half-space, Aˆ = Pˆ = ∞dxx x. The coupling ψ0( x,t) ψ∞(x,t), (34)
Ω 0 | ih | − − ≡
−i∂τθΩ(x) allows to identifyRthe meter as a continuous whereψ∞(x,t)isthewavepacketreflectedbyaninfinite
version of the Larmor clock [24]-[26], a large magnetic
potential wall, V = . This may again seem strange,
moment which precesses in a magnetic field for as long ∞
since everything said so far referred to a free particle.
as particle remains inside Ω. The solution to Eq.(9) is
Thefollowingpointiscentraltoouranalysis: assignment
given by the Fourier integral [6]
ofprobabilities tointerferingclasses ofquantumhistories
∞ cannotbeconsideredoutsidethecontextofmeasurements
Φ(x,tτ)=(2π)−1 dV exp(iVτ)ψV(x,t) (28) of the property of interest.
| Z−∞ So far we have only contemplated a classification of
ψV(x,t) x exp[ iHˆVt)ψI , Feynman paths according to the duration spent in Ω,
≡h | − | i
yet the result already contains a reference to the re-
where [we use θ(x) for θ[0,∞)] flection which would be caused by a Larmor clock,
Hˆ =p2/2m+Vθ(x). (29) should we decide to employ one. Moreover, the fine-
V grained amplitude Φ(x,tτ) contains information about
|
In other words, to find the traversaltime amplitude dis- all measurement scenarios, which are, in turn, speci-
tribution one needs to know the results of evolving the fied by the choice of the filter G in Eq.(21). Suppose,
initialstate forallpotentialsteps addedinthe right-half for example, we decide to make no measurement at all
′
space - even though we are discussing the properties of by making G(f f ) in Eq.(10) so broad that it can
−
a free particle. The transmission (T) and reflection (R) be replaced by a constant. It is useful to note that
amplitudesforsuchastepatanenergyEareeasilyfound (since (2π)−1 dτ dVexp(iVτ)T(k,V) = T(k,0) = 1)
to be (k =√2ME) Φ˜(x,tτ) add uRp toRthe freely propagating pulse,
|
T(k,V)=2/[1+(1 V/E)1/2], R=T 1, (30) t
− − Φ˜(x,tτ)dτ =ψ0(x,t). (35)
Z |
Now if the initial state is a wave packet, initially in the 0
left half space, and moving from left to right, Thus with no measurement make the last two terms in
(33) cancel, leaving us, as it should, with only the free
xψ = dkA(k)exp[ikx iE(k)t], (31) wave packet travelling to the right of the origin x=0.
I
h | i Z − Or we may want to know what is the probability for
xψI 0 for x 0 spending precisely a duration τ in the right half-space.
h | i≡ ≥
Again,we cannotavoidemployinga highly accurateme-
it evolves into
ter. The result of our attempt is known from Sect. VII:
we can neglect all but the singular in τ terms, thus ob-
ψ (x,t)= dkT(k,V)A(k)exp[ikx iE(k)t] (32)
V Z − taining
dkA(k)exp[ ikx iE(k)t] P(x,tτ)= ψ∞(x,t)2δ(τ), (36)
−Z − − | | |
which corresponds to the wave packet reflected as if by
+ dkT(k,V)A(k)exp[ ikx iE(k)t]. an infinite wall, at the meter with only zero readings. In
Z − −
thelightofwhatwassaidabove,thisishardlysurprising.
Here the first term is the transmitted part, the second Classically,onecanarrangeameterwhichwouldnotper-
term describes the reflection from an infinite step, V = turbthemeasuredsystem. Onecanalsosetthemeterin
, and the third reflected term accounts for the fact suchawaythatitwouldactonthesystem,andthencor-
∞
that the step isn’t, after all, infinite. We will consider t rectly measure the variable for the system perturbed by
6
′ ′
the very measurement[20]. Quantally,the first option is exp iS [x(t)] Ut [x(t)] , with S denoting the free-
0 Ω 0
{ − }
notavailablesince itis necessaryto destroyinterference. particle action. The amplitude to arrive in x by travel-
′
Our ideal measurement of is, in this sense, correct: in ling along all paths satisfying t [x(t)] = τ is, therefore,
Ω
order to measure τ to a greataccuracy,the meter would Ψ(x,tτ)exp( Uτ),andtheamplitude to arrivethereat
| −
needtoexertaforcewhichwouldn’tlettheparticleenter all is
the region,andthen confirmthe zeroresult. In asimilar t
′ ′ ′
way, we can analyse the ’softer’ measurements involving ψ (x,t)= exp( Uτ )Φ(x,tτ )dτ , (39)
U
Z − |
different shapes and widths of the function G. For ex- 0
ample a detailed analysis of Aharonov’s ’weak measure- where Φ(x,tτ′) is given by Eq.(33). This is very simi-
ments’ [30]-[32] can be found in [33]. lar to the am|plitude for obtaining a zero reading if one
Equallyimportantaretherestrictionswhichtheabove measures the duration spent in the right half space by
principle puts on what one can use the fine-grained am- a meter whose initial state G(τ) is θ(τ)exp( Uτ). We,
plitudesfor. Forexample,simplysummingΦ˜(x,tτ)over therefore,alreadyknowwhatwillhappenifon−eincreases
|
a certain range of τ, squaring the modulus of the result, absorption U in order to eliminate the particles which
and declaringit the probability to have τ within a range have entered the right half-space, and then managed to
isdangerous. Theauthorsof[1]havetriedseparatingthe surviveuntilt. Since dτ′θ(τ′)exp( Uτ′)=1/U,which
−
amplitudesintojusttwoclasses,thoseforwhichτ iszero, vanishes as U ,Rthe contributions from the smooth
and those for which it is not, so that the two amplitudes part of Φ(x,tτ→′)∞will vanish, leaving us again with the
|
are given by A(x,tτ = 0) = ψ0(x,t) and A(x,tτ = particle fully reflected from the origin,
| − | 6
0) = dτ[Φ˜(x,tτ)+Φ˜( x,tτ)] = ψ (x,t)+ψ ( x,t).
Now tRhe probab|ility to h−ave e|ntered t0he right ha0lf−-space ψU(x,t)≈ψ∞(x,t), as U →∞. (40)
appears to have the value This result is equivalent to two complimentary state-
ments. (a) Restricting an evolution to the Feynman
P(tτ =0) dxA(x,tτ)2 = (37) pathswhichdonotenteraspacialregionleadstoperfect
| 6 ≡Z | | |
reflectionfromthe region’sboundary. (b)Sucharestric-
dxψ∞(x,tτ)2+ dxψ0(x,tτ)2 =2, tion can be achieved by introducing a large absorbing
Z | | | Z | | | potential, the physical cause of the reflection.
which, as the authors of [1] pointed out, must be wrong.
To find a reasonfor this discrepancy we revisit the mea-
XI. THE TIME OF CROSSING INTO THE
surementasdefined byEqs.(9) and(21). Apparently, no RIGHT HALF-SPACE FOR THE FIRST TIME
initialstate ofthe meter G(f) effects the coarse-graining
ofthefine-grainedamplitudesintothesetwoclasses. We,
Thethirdcaseconsideredin[1]involvescontrollingthe
therefore,no longer haveEq.(18), itself a consequenceof
first time a system enters a given region of space Ω or,
unitarity of the system-meter evolution. Moreover, non-
more generally, a certain sub-space of its Hilbert space.
conservation of the number of particles in Eq.(37) sug-
ForaparticleofamassM inonedimension,weconstruct
geststhattheprobabilitiesP(tτ =0)andP(tτ =0)are
| 6 | a functional whose value gives the time a Feynman path
not measurable by any other scheme, short of injecting
enters Ω for the first time.
more particles into the system.
t
′ ′
ΘΩ[x()]=limγ→∞ dt exp γtΩ(t,[x()]) (41)
· Z {− · }
0
X. THE PROBABILITY NOT TO BE
′
wheret ([x(t),t)]isthetraversaltimefunctionaldefined
ABSORBED IN THE RIGHT HALF-SPACE Ω
earlierinEq.(27). Thefunctionaladdsupdt’sforaslong
asexp γt [x(t)]isnotzero,i.e.,foraslongasthepath
Ω
Another case discussed in the Ref. [1] is absorptionof {−
had made no incursion into Ω, and marks the moment
a particle by an optical potential confined to the right
the border of Ω is crossed for the first time. If the path
half space,
originates from inside Ω at t = 0, the value of Θ [x(t)]
Ω
is set to zero. If a path has not yet visited Ω by the
U(x)=iUθ(x). (38)
time t, the value is set to t. In this way, every Feynman
As before, we consider a wave packet incident on the path is labeled by its first crossing time, and we can re-
absorbing potential from the left, and a time t so large arrange the paths into the classes, as was done in Sect..
that a free pulse would be fully contained to the right Forthe amplitude tofirstcrossintoΩ=[0, )atatime
∞
of the origin x = 0. Following [1] we wish to approx- τ, 0 τ t we have
≤ ≤
imate the probability not to enter the right half space
′ ′
with the probability not to be absorbed by U(x). The Φ(x,tτ)= dx x ψI (42)
| Z h | i
problem is easily solved by the technique of the pre-
x(t)=x
vious Section. The amplitude not to be absorbed af-
Dxexp iS [x() δ(Θ [x()] τ)
′ 0 Ω
ter travelling along a Feynman path x(t) is obviously Zx(0)=x′ { · } · −
7
where the path integration is over the paths starting at alternatives. Destructionof interference [e.g., conversion
′
t = 0 in x and ending in x at the time t. The first of a pure state (20) into a statistical mixture (22)] is
crossing time expansion has been derived by many au- a physical process, and must be executed by a physical
thors [34]-[37], and in the Appendix we offer yetanother agent, which we call a meter. This puts serious restric-
one based on the direct evaluation of the restricted path tions on the probabilities we may construct.
integral in (42). For a wave packet (31) approaching the Firstly,suchametermustexist. Precisetypeoftheinter-
origin from the left, from Eq.(53) we have action,requiredtodestroythe interference,is prescribed
by the propertyof Feynmanpaths we wishto control. It
Φ(x,t|τ)=ψ∞(x,t)δ(τ −t) (43) may happen that it does not correspond to a coupling
+(i/2M)θ(τ)θ(t τ)K(x,0,t τ)∂xψ∞(0,τ), which is physically acceptable.
− −
Secondly, even if the meter exists, it must be capable of
where, as before, ψ∞(x,t) is the wave packet scattered
effecting the desired separation of Feynman paths into
by an infinite wall at x = 0. We see that initially [0−
classes. This requires finding an acceptable initial state
stands for limǫ→0(0−ǫ)] for the meter. This may not always be possible.
Φ(x,0− τ)=ψI(x)δ(τ), (44) Thirdly, a measurement must be classified according to
|
its accuracy, determined by the width and shape of the
and, using Eq.(52), find the equation of motion,
initialmeter’sstate. Therangeofpossiblemeasurements
i∂ Φ(x,tτ)=HˆΦ(x,tτ) (45) stretches for highly inaccurate ’weak’ measurements to
t
| | highly accurate ’ideal’ ones.
−i∂τ[δ(τ −t)ψ∞(x,t)θ(t)]. A simple analysisof Sect. VIII shows that anideal mea-
Integrating Eq.(45) over τ shows that surement of the type discussed in [1] may lead to a kind
of a Zeno effect. This is neither an ’unphysical’ result,
dτΦ(x,tτ) =ψ(x,t), (46) nor a ’pitfall’ of a theory, but a general quantum me-
Z | chanicalrule. To evencontemplate the accurate value of
a variable F, one must assume that the interference has
which, together with Eq.(43) gives the standard first
crossing time expansion, used for example in [36]. This been destroyed to the required degree, i.e., consider also
the effects of an external meter. The meter would then
completes the first task outlined in Sect. VI, that is,
defining the quantity of interest. perturb the system, and yield a sharply defined value of
F, which correctly describes this perturbed motion.
We do, however, fail in the second task, in specifying
a meter for the first crossing time. Indeed, the Eq.(45) This ’Zeno effect’ cannot be avoided completely. It can,
however, be mitigated, e.g., by requesting less informa-
does not look like a SE describing interaction between
two quantum systems. It is non-homogenous, with the tion about F, and reducing the accuracy of the mea-
surement ∆f. The authors of Refs. [1] and [9] chose to
source term fully determined by the evolution in the left
half-space with an infinite wall at the origin. We cannot consideranidealmeasurementofaquantityobtainedby
even guarantee that it conserves the probability. Indeed replacingthe integral(8)withadiscretesum(12). With
constricting with the help of Eq.(10) a square integrable this, the system is allowed to evolve freely between tn
solution to represent the particle and a potential meter, and tn−1, and is not reduced to the Zeno-like evolution
if ǫ is kept sufficiently large. In general, one is lead to
Ψ(x,0− τ)=ψI(x)G(τ), Ψ(x,0− τ)2dτdx=1,(47) consider finite-time measurements of different quantities
| Z | | | to different accuracies, varying ∆f and ǫ, to achieve a
desired ratio between the information obtained and the
andusingEq.(45)toevaluatetherateofchangeofP(t)=
Ψ(x,tτ)2dτdx, we find perturbation incurred.
| | |
R We illustrate the above with a brief review of the ex-
dP(t)
= (48) amples considered in Ref.[1]
dt
(a) The probability for a free particle not to enter the
∞ right half-space equals the probability to spend there a
∗
2Re[ dτ∂τG(τ t) dxψ∞(x,t)Ψ (x,tτ)]. zero net duration. This duration is represented by the
Z − Z |
0 traversal time functional, the relevant meter exists as a
It is unlikely that the r.h.s. of Eq.(48) vanishes identi- continuous version of the Larmor clock []. An accurate
cally, and we abandon our attempts to find probabilities clock prevents the particle from entering the regionand,
forthefirstcrossingtime(41),justaswepromisedinwe under the circumstances, the probability not to enter is
would, in the second passage of Section ... unity.
(b) The probability amplitude to enter the right half-
space cannot be defined as the net probability to spend
XII. CONCLUSIONS AND DISCUSSION thereanydurationotherthanzero. Eventhoughtheme-
ter(Larmorclock)exists,itcannotbe setupto separate
In summary, assigning probabilities to interfering al- the Feynman paths into these two classes. One may be
ternatives implies destruction of coherence between the confident that this cannot be done by any other means
8
either, since thus defined probabilities do not add up to (2π)−1/2 exp(iλz)dλ w can rewrite it as
unity. R
(c) The probability not to be absorbed in the right half- K(x,x′,tτ)=(2π)−1/2 dλexp(iλτ) (49)
| Z ×
space is similar to the probability not to enter it, and
tends to unity as the magnitude of the absorbing poten- x(t)=x t t′
′ ′′ ′′
Dxexp i dt[L λexp( γ θ (x(t )dt ]
tial increases. This is a different way of saying that an Zx(0)=x′ { Z0 0− − Z0 Ω }
infiniteabsorbingpotentialmustreflectallincomingpar-
ticles. where L(x,x˙) is the particle’s Lagrangian. Expanding
(d)Wefoundnomeaningfulprobabilitiesassociatedwith the second exponential we find the amplitude for a path
′
thefirstcrossingtimeamplitudes(43),asnophysicalme- x(t) to be
ter of the type discussed here can be realised. To iden- t ∞
tify the time of the first crossing, one needs the record A′[x(t′)]=exp(i Ldt′) 1+ ( iλ)n (50)
Z { − ×
the particle’s past. For this reason, even classically, we 0 nX=1
cannot construct a single pointer which would stop once t t1 tn−1 n tm
′ ′
the particle first crosses into the region of interest. dt1 dt2.. dtnexp[ γ θΩ(x(t))dt .
Z Z Z − Z }
Finally,wenotethatinallabovecaseswehavefailedto 0 0 0 mX=1 0
arriveattheclassicallimit. Thismaycausesomeconcern The first term in the curly brackets corresponds to free
[1]Whereameterforourfinite-timemeasurementexists, motion. The integrand of the n-th term corresponds to
the remedy is simple [20]. We do not improve accuracy theparticlemovinginatimedependedopticalpotential.
indefinitely, but stop while ∆f exceeds the width of the For 0 t′ < tn−1 we have nγθω(x), for tn−1 t′ < tn−2
≤ ≤
stationary region of the fine-grained distribution in Eq. the potential is reduced to (n 1)γθ , and so on. For
ω(x)
(9). This region occurs around the classical value of F, t t′ < t the absorbing po−tential is turned off. As
1
≤
f , and is very narrow if the system is nearly classical. we send γ , the distinction between, say, the terms
cl
→ ∞
Withtheaccuracychosenhighyetfinite,onlythisregion containing nγ and (n 1)γ disappears, and the particle
−
contributes to the integral (10) Thus the pointer always moves in an infinite absorbing potential until t , after
n
points at fcl, and we can replace the SE by the classical which it is switched off. Integrals 0t1dt2... 0tn−1dtn can
equations of motion. now be evaluated to yield tn−1/(nR 1)!, aRnd summing
n −
Where no meter exists, the situation appears to be over n we have
moredifficult. Classically,onecanalwaysdefinethe first
t
crossing time, seemingly with no reference to meters or A′[x(t′)]=exp(i Ldt′) (51)
Z −
measurements. This is not quite so, as one always has 0
at his disposal a classical trajectory x¯(t) from which all t t ′ t1 ′
otherquantities, including the firstcrossingtime, canbe iλZ dt1exp(−iλt1)exp(iZ Ldt +Z L∞dt),
0 t1 0
derived. From quantum mechanical point of view, x¯(t)
comprisestheresultsofmeasuringtheparticle’spositions where L∞ =limγ→∞L γθΩ(x) is the Lagrangianwith
−
an infinite absorbing potential introduced in the right
at all times. This suggests that the first crossing time
half-space. InsertingEq.(51)into(49)yieldsthe netam-
shouldbe determinedinacontinuousquantummeasure-
′
plitudeonallpathsconnectingx andx,andfirstcrossing
ment,yieldingasequenceofparticle’spositionsx(t),and
the origin x=0 at the time τ
then evaluating the functional (41) on this ’real’, rather
thanvirtual,trajectory. Amoredetailedanalysiswillbe K(x,x′,tτ)=K(x,x′,t)δ(τ) (52)
given in our future work [38]. | −
′′ ′′ ′′ ′ ′′
∂τ dx K(x,x ,t τ)K∞(x ,x,τ)dx
− Z −
where K∞ and K are the propagators with and with-
ACKNOWLEDGMENTS out the infinite absorbing potential in the right half
space, i.e., K(x,x′,t τ) θ(t τ) x exp( iHˆt)x′
We acknowledge support of the Basque Government and K∞(x,x′,τ) θ(−τ) x e≡xp( i−Hˆτ)hx′|, whe−re Hˆ∞| i
≡ h | − | i ≡
(Grant No. IT-472-10),and the Ministry of Science and limγ→∞Hˆ γθ(x) . Note that the infinite absorbing
−
Innovation of Spain (Grant No. FIS2009-12773-C02-01). potential is equivalent to an infinite potential wall intro-
IamalsogratefultoJ.N.L.Connorforbringingtheissue duced at x=0, so that
to my attention.
′ ′
K∞(x,x,t) 0 for x,x 0.
≡ ≥
Differentiating the product in (52) and noting that
XIII. APPENDIX Hˆθ(x)−θ(x)Hˆ∞ =−(1/2M)[∂x2,θ(x)]=(1/2M)δ(x)∂x,
we rewrite Eq.(52) as
′ ′ ′
Consider the path integral in Eq.(42) keeping, for K(x,x,tτ)=K(x,x,t)δ(τ)θ(x)+K∞(x,x,t)δ(τ t)+
| −
′′ ′
the moment γ finite. Using the identity δ(z) = (i/2M)K(x,0,t τ)∂x′′K∞(x =0,x,τ), (53)
−
9
which is the standardform of the first crossingtime am-
′
plitudeforaparticletravellingbetweenx andx[34]-[37].
[1] J. J. Halliwell and J. M. Yearsley, Phys. Rev. D 86, drecht, 2000).
024016 (20012). [20] D. Sokolovski, Phys. Rev.A 66, 032101 (2002)
[2] D.Sokolovskiand L.M.Baskin, Phys.Rev.A,36,4604- [21] D. Sokolovski, Phys. Rev.A 59, 1003 (1999)
11 (1987) [22] D. Sokolovski, Phys. Rev.A 84, 062117 (2011)
[3] D.SokolovskiandJ.N.LConnor,SolidStateComm.,36, [23] D. Sokolovski, Phys. Rev. Lett. 102, 230405 (2009)
475-481 (1994) (2009)
[4] .D.Sokolovski,S.BrouardandJ.N.LConnor,Phys.Rev. [24] A.I.Baz’,Yad.Fiz.4,252(1966)[Sov.J.Nucl.Phys.4,
A,50, 1240-1256 (1994) 182(1967)]; 5,229(1967)[5,161(1967)]; V.F.Rybachenko,
[5] D.Sokolovski, Phys. Rev.A, 52,1, pp.R5-R8(1995) ibid 5, 895 (1965) [ibid. 5, 635 (1967)]
[6] D.Sokolovski, Phys. Rev.Lett., 25, 4946 (1997) [25] M. Biittiker, Phys. Rev.B 27, 6178 (1983)
[7] D.Sokolovski, Proc. Roy.Soc., 460,1505-1517 (2003) [26] D. Sokolovski and J. N. L. Connor, Phys. Rev. A 47,
[8] D.Sokolovski, in Time in Quantum Mechanics, ed. by 4677 (1993)
J.G.Muga,R.SalaMayatoandI.L.Egusquiza,(Springer, [27] D.Home and M.A.B. Whitaker, Ann. Phys., 258, 237
2008), pp.195-234 (1997).
[9] D. Alonso, R. Sala Mayato and J. G. Muga, Phys. Rev. [28] P.Facchi, H. Nakazato, and S.Pascazio, Phys.Rev.Lett.,
A 67, 032105 (2012). 86, 2699 (2002)
[10] R.B.Griffiths,J.Stat.Phys.36,219(1984);Am.J.Phys. [29] P.Facchi and S.Pascazio, Phys.Rev.Lett., 89, 080401
55, 11(1987); Phys. Rev.Lett. 70, 2201(1993). (2002)
[11] M.Gell-MannandJ.B.Hartle,Phys.Rev.A76,022104 [30] Y.Aharonov,D.AlbertandL.Vaidman,Phys.Rev.Lett,
(2007) 60, 1351 (1988)
[12] S. Goldstein and D. N. Page, Phys. Rev. Lett. 74, 3715 [31] Y. Aharonov, J. Anandan, S. Popescu and L. Vaidman,
(1995) Phys.Rev.Lett, 64, 2965 (1990)
[13] J. J. Halliwell, Phys.Rev.D 60, 105031 (1999) [32] Y. Aharonov and L. Vaidman in Time in Quantum Me-
[14] L. Diosi, Phys.Rev.Lett., 92, 170401 (2004) chanics. Vol.1,SecondEdition,ed.by.G.Muga,R.Sala
[15] J. B. Hartle, Phys.Rev.A 78, 012108(2008) Mayato and I. Egusquiza, (Springer, Berlin Heidelberg,
[16] R. P. Feynman and A. R. Hibbs, Quantum Mechanics 2008)
and Path Integrals (New York: McGraw-Hill,, 1965). [33] D. Sokolovski, Phys. Rev.A 76, 042125 (2007)
[17] D. Sokolovski and R. Sala Mayato, Phys. Rev. A 71, [34] T. E. Feuchtwang, Phys.Rev.B 10, 4121,(1974).
042101(2005); Phys. Rev. A 73, 052115 (2006). 76, [35] A. Auerbach and S. Kivelson. Nucl. Phys. B 257 799,
039903(E) (2006). (1985).
[18] J.vonNeumann,Mathematical FoundationsofQuantum [36] J. J. Halliwell, Phys.Lett. A, 207, 237 (1995).
Mechanics(PrincetonUniversityPress,Princeton,1955), [37] J. J. Halliwell and J. M. Yearsley, Phys.Rev. A 79,
pp.183217. 0621101 (2009).
[19] M. B. Mensky, Quantum Measurements and Path Inte- [38] D. Sokolovski, (to bepublished)
grals (IOP, Bristol, 1993); Quantum Measurements and
Decoherence: Models and Phenomenology (Kluwer, Dor-