The 'time of occurrence' in quantum mechanics
M D S R I N I V A S a n d R V I J A Y A L A K S H M I *
Department of Theoretical Physics, University of Madras, Guindy Campus, Madras 600 025, India
*E. D. P. Department, Bharat Petroleum Corporation Limited, Ballard Estate, Bombay 400 038, India
Abstract. Apart from serving as a parameter in describing the evolution of a system, time appears also as an observable property of a system in experiments where one measures 'the time of occurrence' of an event associated with the system. However, while the observables normally encountered in quantum theory (and characterized by self-adjoint operators or projection-valued measures) correspond to instantaneous measurements, a time of occurrence measurement involves continuous observations being performed on the system to monitor when the event occurs. It is argued that a time of occurrence observable should be represented by a positive-operator-valued measure on the interval over which the experiment is carried out. It is shown that while the requirement of time-translation invariance and the spectral condition rule out the possibility of a self-adjoint time operator (Pauli's theorem), they do allow for time of occurrence observables to be represented by suitable positive-operator-valued measures. It is also shown that the uncertainty in the time of occurrence of an event satisfies the time-energy uncertainty relation as a consequence o f the time-transla- tion invarianee, only if the time of occurrence experiment is performed on the entire time axis.
Keywords. Time operator; spectral condition; Pauli's theorem; time of occurrence of an event; continuous measurements; positive-operator-valued measures; time- energy uncertainty relation; time of arrival of photons.
1. Introduction
T h e n o t i o n o f t i m e a p p e a r s in a p h y s i c a l t h e o r y in t w o d i s t i n c t ways. F i r s t l y , t i m e a p p e a r s a s a p a r a m e t e r a g a i n s t w h i c h o n e c h a r t s t h e c o u r s e o f e v o l u t i o n o f t h e s t a t e o f a system, o r a s a p a r a m e t e r w h i c h specifies the i n s t a n t a t w h i c h a n ( i n s t a n t a n e o u s ) e x p e r i m e n t is c a r r i e d o u t . B a s e d o n t h i s n o t i o n o f t i m e i t is o f t e n a r g u e d t h a t i n q u a n t u m t h e o r y t h e r e is n o n e e d t o l o o k f o r a n o p e r a t o r t o r e p r e s e n t t i m e o r t o t a l k o f a s s o c i a t e d p r o b a b i l i t i e s , a s it is m e r e l y a p a r a m e t e r . H o w e v e r , t h e r e a l s o e x i s t s a n o t h e r m e a n i n g f u l n o t i o n o f t i m e a s a n o b s e r v a b l e p r o p e r t y o f a p h y s i c a l s y s t e m w h i c h a r i s e s i n e x p e r i m e n t s w h e r e o n e l o o k s f o r t h e t i m e a t w h i c h a p a r t i c l e d e c a y s , o r crosses a g i v e n s u r f a c e , o r is d e t e c t e d b y a d e t e c t o r , etc. T h e t i m e o f d e c a y , t i l n e o f a r r i v a l , t i m e o f d e t e c t i o n a r e c l e a r l y m e a s u r a b l e p r o p e r t i e s o f a s y s t e m (see A l l o c k 1969 f o r a n e x t e m i v ¢ d i s c u s s i o n ) , a n d c a n b e s u b s u m e d u n d e r t h e g e n e r a l n o t i o n o f 'the time o f occurrence o f an event associated with the system'.
W h e n w e a r e t a l k i n g , o f t i m e as a n o b s e r v a b l e p r o p e r t y o f a p h y s i c a l s y s t e m , i t s h o u l d b e q u i t e d e a r t h a t t h e r e is n o single u n i q u e t i m e o b s e r v a b l e , b u t a c t u a l l y a w h o l e c l a s s o f t i m e o f o c c u r r e n c e o b s e r v a b l e s - - o n e a s s o c i a t e d w i t h e a c h o b s e r v - a b l e e v e n t t h a t c o u l d o c c u r . I n f a c t s o m e o f t h e c o n f u s i o n o n t h e n o t i o n o f t i m e i n
173
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174 M D Srinivas and R Vijayalakshmi
quantum theory is mainly due to the fact that often it is not appreciated that there is nothing like the time observable, but a class o f time o f occurrence observables asso- ciated with each physical system. The main objective o f the theory should be to pro- vide a mathematical characterization of such time o f occurrence observables so that one arrives at a prescription for calculating say, the probability that an event occurs in a given interval o f time.
This paper addresses itself to an analysis o f the time o f occurrence observables in quantum mechanics. In the traditional approach to the problem o f time in quantum theory one looks for a time operator which is canonically conjugate to the Hamil- tonian operator. However, it was shown by Pauli (1933) at the early stages o f deve- lopment of quantum theory that the fact that the energy spectrum of every physical system is bounded from below automatically precludes the existence o f a self-adjoint time operator canonically conjugate to the Hamiltonian. In § 2 o f this paper we dis- cuss some o f the basic consequences this (spectral) condition that the Hamiltonian operator is bounded from below, as they are o f considerable importance to the rest of the paper. We shall also outline the various proofs o f Pauli's theorem which serve to clarify the basic assumptions involved.
Since a self-adjoint time operator is ruled out, there have been suggestions that time may be represented by a symmetric operator or even a nonsymmetric operator etc. It is our view that any mathematical characterization o f time in quantum theory should take into account the fundamental difference between observables usually encountered in quantum theory (and represented by self-adjoint operators) which correspond to measurements performed at a single instant, and the time o f occur- rence observables which involve continuous measurements performed over an ex- tended period o f time to observe when a given event occurs. In § 3 we show that the time o f occurrence observable associated with an experiment conducted over an interval (t 1, t z) to measure when an event e occurs, should be characterized by a positive-operator-valued (POV) measure
defined on the interval (tl, tz). This is to be contrasted with the mathematical charac- terization o f the observables usually encountered in quantum theory by spectral (or projection-valued) measures associated with self-adjoint operators via the spectral theorem. The actual construction o f the particular POV measure
A E (a)
to be associated with a given time of occurrence measurement, is to be based on the quantum theory of continuous measurements. In this paper we shall restrict our- selves to a study of some of the general properties common to all time of occurrence observables. In particular we shall formulate the basic condition imposed on a time o f occurrence observable by the requirement o f time-translation invariance o f the theory.
The time-translation invariance requirement takes a particularly simple form for time of occurrence measurements performed on the entire time axis which are now
associated w i t h POV measures o n the real line. In § 4 we shall prove a generalization o f Pauli's t h e o r e m which shows that the spectral condition and time-translation in- variance imply that in any time o f occurrence measurement p e r f o r m e d o n the entire real axis, the probability that the event occurs in any finite interval (tl, t~) or in a semi-infinite interval o f the form ( - - m , t) can never be unity. We shall also study some o f the other restrictions imposed on the time o f occurrence observables by the nature o f the spectrum o f the Hamiltonian.
In § 5 we discuss the celebrated time-energy uncertainty relation
A T A E >~ h/2, (1)
whose derivation and also the interpretation have remained generally controversial.
Especially in view o f its application to the relation between the life-time and energy- width o f an unstable state, AT in the above relation should be interpreted as the uncertainty in the time o f occurrence o f an event. We shall argue that the well- known formulations o f the above relation due to Mandelstam and T a m m (1945) and Wigner (1972) are not relevant to a discussion o f the uncertainty in the time o f decay or time o f arrival, etc., as they involve prescriptions for calculating A T in terms o f quantities clearly pertaining to instantaneous measurements only. In o u r approach, the uncertainty AT~q, q) in the time o f occurrence o f an event E as observed in an experiment performed over the interval (t 1, t2), can be c o m p u t e d in terms o f the associated POV measure
A --> T(q, q) (A). E
By employing the time-translation invariance requirement, we shall show that the time-energy uncertainty relation
AT[q, , , ) A E > / h / 2 ,
(2)
holds whenever (q, t z ) = ( - - 0 % oo) i.e., for time o f occurrence measurements performed on the entire time axis. However the relation (2) cannot be obtained for time o f occurrence measurements performed over finite or semi-infinite intervals o f time. In fact we show that in photon-counting experiments performed over the interval (0, oo), the uncertainty in the time of arrival o f the first p h o t o n is not constrained so as to satisfy the relation (2).
2. The spectral condition and Pauli's theorem
In this section we shall summarise some o f the well-known consequences o f the condition that the H a m i l t o n i a n operator H should have a spectrum which is b o u n d e d from below; in other words, there exists a number C > - - o o such that
H > c / . (3)
176 M D Srinivas and R Vijayalakahmi
Equation (3) constitutes one o f the basic assumptions o f both non-relativistic and relativistic q u a n t u m theories and is often referred to as the spectral condition. Phy- sically such a condition is necessary to ensure that the ground state of the system is stable.
The spectral condition (in a somewhat more general form) has played a crucial role in the various axiomatic formulations o f relativistic q u a n t u m field theories. In fact, m a n y o f the results that we cite in this section have been obtained in the recent studies of the problem o f causality and localizability in relativistic q u a n t u m mechanics (Hegerfeldt 1974; Skagerstam 1976; Perez a n d Wilde 1977; Hegerfeldt and Ruijsenaars 1980).
I f H be the self-adjoint Hamiltonian operator, let
...
denote the associated strongly continuous one-parameter group o f unitary operators.
I f H is bounded from below then one can employ Stone's theorem to show that the function
,
g(z) = (P, exp ( q - tl
is analytic for Im z > 0 and continuous for Im z >~0 for all vectors e, ff belonging to the Hilbcrt space 3/. For each real t, (9, Vt 4,) is a boundary value o f the above function. By a straightforward application o f the edge o f the wedge theorem (Streater and Wightman 1964) we can obtain the following result due to Perez and Wilde:
Lemma 1 (Perez and Wilde 1977): Let M be an open interval in the real line R and
~, 4, E 5~f. Then (~, Vt 4,) = 0 for t E M implies that (~, Vt ~b) ---- 0 for all t E R. In particular (4,, Vt ~b) = 0 for t E M implies that 4' = 0.
Now, let A be a bounded operator such that A Vt 4, = 0 for t ~ M. Then for any E ~ , we have (A t ~, Vt ~b) = 0 for t E M, where A t is the adjoint of A. We can then conclude from Lemma 1 that
( A t e , Vt ~) = (~, Al,'t 4,) = 0 ,
for all t E R. Since p was arbitrary to start with, we have the following result due to Hegerfeldt and Ruijsenaars.
Lemrna 2 (Hegerfeldt and Ruijsenaars 1980): Let A be a bounded operator such that A Vt ~b = 0 for t E M, where M c R is an open interval. Then A Vt ~b = 0 for all t E R .
F o r the sake o f completeness we shall also state the following result due to Borchers (1967) which can be employed to give a simple p r o o f o f Pauli's theorem.
Lemma 3 (Borchers
1967): Let E, F be projection operators such that F v, E = v, E v?*F,for
It I
where a > 0. ThenFE
= 0 implies that F re, E v 7 = 0for all
t E R .
It may be noted that the above results continue to hold when we replace V, by V~ -1 for, similar arguments can be employed for the case when the spectrum of H is bounded from above.
We shall now show how the spectral condition plays a crucial role in the result o f Pauli (1933) that a self-adjoint time operator does not exist. The main requirement placed on the time operator T is that it should be canonically conjugate to the Hamil- tonian H, which is normally expressed by means of the commutation relation
TH -- HT = -- iM, (5)
though it should be mentioned that there exists considerable confusion in the litera- ture (cf. Jammer 1974 where the opposite sign is chosen) regarding the sign on the right hand side o f the above equation. In any case the question as to what sign should be employed in the right hand side of (5) is not taken seriously as the operator T is shown not to exist. We shall later show that from fundamental considerations it follows that it is (5) which should be demanded of the time operator.
It is well-known (cf. Emch 1972) that the canonical commutation relation (CCR) (equation (5)) in the so-called Heisenberg form is implied (but not vice versa) by the CCR in the Weyl form
exp (~ Ha) exp (iT/3) = exp (-- ia/3) exp (iTfl) exp (~ Ha),
(6) for all reala, t,
where/3--> exp(iT~3)
is the strongly continuous one-parameter unitary group generated by the self-adjoint operator T. The Weyl-form CCR (6) can also be expressed equivalently in the following two forms:[ e x p ( ~ H e ) T e x p ( - - ~ H , O ] P
= ( T - - a l ) v, (7) for all real a, for all P in the domain .~ (T) of the operator T;exp (~ Ha) ET (A) exp ( --~ Ha) = ET (A + a),
(8)for all real a and Borel sets A C R, where A--> ET(A) is the unique spectral (or projec- tion-valued) measure associated with the self-adjoint operator T. We shall show in
178 M D Srinivas and R Vijayalakshmi
§ 3 that (6) (and hence, equivalently, equations (7) and (8)) is a consequence of the fundamental requirement o f time-translation invariance o f the theory.
The result o f Pauli can now be stated as follows:
Theorem 1 (Pauli 1933): I f the self-adjoint operator H is semibounded, then there does not exist a self-adjoint operator T such that the C C R (8) is satisfied.
There are several proofs of the above theorem. One can employ either the theorem o f von N e u m a n n (1931) for the representations o f C C R in the Weyt form (6), or the imprimitivity theorem of Mackey (1949), to conclude that if (8) is satisfied then the spectrum o f the Hamiltonian H will have to be absolutely continuous and range from -- ~ to + o% thus contradicting the hypothesis that H is semi-bounded. The same conclusion can also be reached by employing the following relation which is also equivalent to (6)
exp ( - - i
. ) , . . ) ] , ---- , . -- . .
(9)for all ~ E ~ (H). Equation (9) shows that the operators H and H--flI are unitarily equivalent and hence have the same spectrum for all real fl, from which it follows again that the spectrum of H ranges over the whole real line.
We shall employ the result of Borchers (Lemma 3) to give an alternate p r o o f o f Pauli's theorem, which is more in the spirit of the rest o f the results of this paper.
Proof of Theorem 1 : To start with, let A be a bounded Borel set and I A I denote its diameter. I f we now set for some a > I A I
F = ET(A),
and E = ET(A q- a),
then it follows from (8) and the basic properties of spectral measures that FE ~ O,
and FVrEVt -1 ---- Vt EV,-1F,
for all t. Hence from Lemma 3 it follows that , r v , Er,- -- o,
for all t ~ R. Since from (8) we have r =
v Evs),
we getF 2 = 0. In other words we have shown that
Er(A) = O, (10)
for each b o u n d e d Borel set A.
E T ( A ) , we obte]n
E T ( R ) = O, (11)
which contradicts the basic requirement E T ( R ) -= I that any spectral measure has to satisfy. This completes the p r o o f o f the theorem.
Since E T ( R ) can be obtained as a strong limit o f such
3. The 'time o f occurrence' observable
We saw in the last section that the result o f Pauli rules out the possibility o f associat- ing a self-adjoint o p e r a t o r with time. This has led to various suggestions that time should be represented by a symmetric operator (Engelmann and Fick 1959, 1964;
Paul 1962; Razavy 1967, 1969; Olkhovsky et al 1974) or even a non-symmetric operator ( H a b a and Novicki 1976) etc. However, in the absence o f a clear-cut pres- cription for calculating physically meaningful probabilities, any search for a n o n - self-adjoint time o p e r a t o r cannot be meaningful. It is therefore very i m p o r t a n t to first clarify the basic difference between the observables commonly e n c o u n t e r e d in q u a n t u m t h e o r y and time when it appears as an observable property o f a system.
The basic postulate o f quantum mechanics which associates each observable with a self-adjoint operator (von Neumann 1955) is designed mainly to answer questions o f the following type. ' I f p is the density operator characterizing the state o f a system (in the Heisenberg picture prior to the measurement o f A (t)) and if the observable A(t) is measured at time t, what is the probability Pr~l(t ) (A) that the o u t c o m e s lie in the Borel set A'. As an answer to this, we have the fundamental algorithm o f the theory
Pr~l(t) (A) z Tr (p E A (t) (A)), (12)
where A -+ E A (t) (A) is the unique spectral (or projection-valued) measure associated with the self-adjoint o p e r a t o r A (t). Thus each self-adjoint o p e r a t o r corresponds to a (instantaneous) measurement performed at some instant o f time. In the a b o v e situations, time itself is not an observable. It is merely a parameter which specifies when the observation is carried out.
The situations where we would like to speak o f time as an observable p r o p e r t y o f the system are very different from the ones encountered above. While it is mean- ingless to ask ' what is the time o f a system', it is perfectly legitimate to ask ' what is the time at which a particular event associated with the system occurs'. F o r ex- ample in the case o f a particle we can always device experiments to measure the time at which the particle say decays, or crosses a given surface, or is detected by a detector etc. Thus there are several observable times associated with a system such as the time o f decay or the t i m e o f arrival or the time o f detection, etc., all o f which c o m e u n d e r the general n o t i o n o f 'time o f occurrence o f an event ' - - w h e r e , it should be n o t e d that with each event one can associate a corresponding 'time o f occurrence observable'.
By providing an appropriate mathematical characterization o f such time o f occur- rence observables one would like to arrive at an algorithm for calculating probabili-
180 M D Srinivas and R Vijayalakshmi
ties such as Pr ((t x, t~); ~)--the probability that the event ~ occurs in the interval
(tl, t~).
The point we would like to emphasize is that the measurement o f the time o f occur- renee of any event involves performing continuous observations over an extended period of time to record when the event occurs. Thus the main difference between the observables commonly encountered in quantum theory and a time of occurrence observ- able is that while the former always refer to experiments performed at an instant o f time, the latter always refers to continuous measurements performed over an extended period of time. Hence there is no reason why a time of occurrence observable should be represented by a self-adjoint o p e r a t o r - o r , for that matter, why the associated probabilities should be given in terms of a spectral measure as in (12). Since we are here confronted with a situation where sequential measurements are performed, a mathematical characterization of time of occurrence observables should be based upon the quantum theory of measurement.
It is well known (see for example Benioff 1972; Srinivas 1975) that when a discrete sequence of instantaneous experiments (each characterized by a self-adjoint operator) are performed, the various probabilities associated with the composite experiment can no longer be characterized in terms of a spectral measure, but in terms of a so- called positive-operator-valued (POV) measure. In recent years several investigations have led to the development of a quantum theory of continuous measurements (see for example Davies 1969, I975, 1976; Srinivas and Davies 1981) mainly in connec- tion with the analysis of photon counting experiments in quantum optics and the various models of absorption and decay of systems. These investigations show that it is possible to compute the probability that an event occurs in a given interval of time in a situation where continuous measurements are performed. However these probabilities can be expressed in the form as in (12) only if we generalise the notion of an observable to include POV measures also.
A POV measure on a Borel space (X, B (X)), where B (X) is the g-algebra of Borel sets in X, is a mapping which assoicates with each Borel set A E B (X) a bounded positive operator T (A) such that the following properties are satisfied.
(i) 0 ~ T (A) ~ L for each A E B (X),
(13)
(ii) T(X) ~ 0,
(iii) If ~[A~]- is a sequence of mutually disjoint Borel sets, then r ( t j , a,) = r (A,),
(14)
(15)
where the right hand side converges in the strong operator topology. The POV measure A ~ T (A) is said to be normalized if
r
(x)
= i. (16)As it is easy to see, the notion of a POV measure is a generalization of that of a spectral (or projection-valued) measure wherein it is demanded that each T (A) be a projection operator.
Let X be the o u t c o m e space o f an experiment a n d let Prp (A) be the probability that the outcomes lie in the Borel set A E B (X) when an ensemble o f systems prepared in state p is subjected to the experiment. Then, regardless o f whether the experiment is such that it is performed at an instant o f time or over an extended (finite or infinite) interval o f time, there should exist a POV measure A--> T (A) on X such that
Prp (A) = Tr (p T (A)),
07)
if we demand that
(i) p ~
PrP
(A) should be additive in p under the formation o f mixtures(ii) p-~ PrP (A) should be continuous under the trace n o r m topology on the set o f (unnormalized) density operators.
It is thus clear that f r o m very general physical grounds it can be demanded t h a t time o f occurrence observables should be represented by POV measures. M o r e specifically let us consider the following situation where an ensemble o f systems prepared in state p at time t~ is subjected to continuous measurements over the time interval (t~, t~) to measure the time at which an event E occurs. Let us denote by
P~q,
,,~(A; E)
the probability that the event ~ occurs (h, t2) in the Borel subset A c (t 1, ta). In what follows we shall always assume that the event ~ is such that for each disjoint sequence ( A ~ - o f Borel subsets o f (tl, t~), • occurs in U l As if and only if ~ occurs in one and only one o f the subsets ~A~)---in other words, e occurs only once, if at all.
Then it is clear that there should exist a POV measure A-~ Tcq ' q~ (A) on (t 1, t~) such E
that
Pr~q, q, (A; ~) = T r (p Tt~, q, (A)). (18)
It m a y be noted that unlike the case o f observables represented by spectral measures, a time o f occurrence observable need not be normalised. In fact, the property
T~q, q~ ((t 1, t2)) = I, (19)
is satisfied if a n d only if the event ¢ is always certain to occur in the interval (tl, t~).
We shall refer to the POV measure A-~T,I,t~) (A) as the time o f occurrence measure on (t 1, t~) associated with the event ~. The actual construction o f such a time o f occurrence observable will have to be based upon a model o f continuous measure- ments performed over the interval (tl, t~) to monitor when the event ¢ occurs. W e shall not go into the theory o f such continuous measurements (see for example the references cited earlier) a n d confine ourselves only to the study o f some o f the general properties c o m m o n to all time o f occurrence observables. A particular example o f a time o f occurrence observable--the arrival time o f the first p h o t o n in a p h o t o n counting experiment--will be considered later in § 5.
182 M D Srinivas and R Vijayalakshmi
T h e general requirements that every time o f occurrence observable will have to satisfy are those arising f r o m causality and time-translation invariance o f the theory.
The requirement o f causality is that the probability
that the event E occurs in A C (t 1, t2) should n o t depend on whether or n o t the experiment is c o n t i n u e d beyond the time t2; i.e.
P ( q , is' (A; ~) = Pr~q, ,;)(A; 0, (20)
for all t 2 >~ t a. Therefore we have i
• ( a ) = " ,;, (A),
T ( t l , t~)
T(q,
(21)for A C (tl, t2) whenever t~ >~ t z. In particular
E E
T(q, t~) (A) = T(tx, ®) (A),
(22)
for all A c (t 1, t2). N o t e however that we c a n n o t d e m a n d o n the other h a n d t h a t
r~,,, ,8, (A) = T;,;,,,, (A),
for t~ < t,, f o r in q u a n t u m theory, the probability t h a t the event ~ occurs in the time period A C (t 1, t~) depends also on the entire sequence o f m e a s u r e m e n t s p e r f o r m e d earlier o n the system.
T h e r e q u i r e m e n t o f time-translation invariance is t h a t
Pr(Pq, t~) (A" E) = Pr p* , (,1+~, ,~+.) ( A + ~ ; O,
(23)
for all real ~- where A + r ~ ( t / t - - - r E A]- and p~ is the time-translate o f the state p.
i.e. pr is the state o f a n ensemble which is p r e p a r e d b y following a p r e p a r a t i o n p r o - cedure identical to t h a t employed for preparing systems in state p (at time t,), but carriedout at a t i m e z later (at time tl-k~" ). It is well k n o w n (see, for example, H o u t a p - pel et al 1965) t h a t
p~. ---- exp H ~r p exp H ~" , ( 2 4 )
where H is the H a m i l t o n i a n o f the system*. F r o m (23) a n d (24) it follows that the
*Note that the transformation p --* Pr of (24) is different from the time-evolution transformation p ---> V t p Vt where the latter denotes the state of the same ensemble of systems (which was origi- nally prepared in state p) at a time ~" later.
time-translation invariance imposes the following condition on a t i m e o f occurrence observable:
( ; ) " (-,.).
exp
H~" T,I,t~)(A )
exp 1I (tl+,r,t~+~.)(A+~-), (25) for all real -r, for all Borel sets A c (fi, t~). In particular for a time o f occurrence observable defined on the entire time axis (which corresponds to an experiment started at time - - o o a n d continued u p t o time + oo), the associated P O V m e a s u r e A---> T(' 0~, =) (A) will have to satisfy the conditionexp H -r T~_~, ® ) (A) exp H • = T(_®, ®) (A-+-~-). (26) We m a y n o t e here t h a t the earlier suggestions o f H e l s t r o m (1974) a n d H o l e v o (1978) that the t i m e observable in q u a n t u m theory m a y be represented by a P O V m e a s u r e on the real line satisfying a covariance condition such as (26) c a n n o w be clearly understood as being relevant to the case o f time of occurrence m e a s u r e m e n t s per- formed on the entire time axis.
It is easy to see t h a t the requirement (26) is the analogue for P O V m e a s u r e s o f the condition (8) that was imposed on the spectral measure A-~
E T
(A) associated with the (non-existent) self-adjoint time o p e r a t o r as a 'consequence' o f the C C R (5).O u r discussion has clearly established that the underlying physical principle behind the C C R (5) is the requirement o f time-translation invariance o f the theory*, which continues to lze applicable in the f o r m (25) a n d (26) even when there is no time- operator.
4. Generalization o f Pauli's theorem
I n the last section we saw that each time o f occurrence observable is associated with a P O V m e a s u r e A - + T ~ (tl, tg) (A) on the interval o f m e a s u r e m e n t
(tl, t2)
such t h a t the causality condition (21) a n d the time-translation invariance condition (25) are satisfied. W e shall n o w p r o v e a generalization of Pauli's t h e o r e m which shows w h a t restrictions are placed on a time o f occurrence measure o n the entire time axis by the spectral condition. F o r notational simplicity we shall hereafter write T('-oo, oo) (A) simply as T ~ (A). The time-translation invariance condition (26) c a n then be written as followsexp (~ H'r) TE (A) exp (~-~ H'r) = T" (A+~'),
(27)for all real ~, and Borel sets A C R. O u r basic result can now be stated as follows:
*It is of course well known (see for example Wightman 1962, Jauch 1968) that the CCR for position and momentum (which can also be expressed in the form (8)) is a consequence of the space- translation invariance of the theory.
184 M D Srinivas and R Vijayalakshmi
Theorem 2 (Generalization of Pauli's theorem): Let H be a semi bounded operator and let A ~ T'(A) be a POV measure on the real line which satisfies the covarianco condition (27). Then
(i) F o r each bounded Borel set A C R, r ~ (A) 4 = ~ ---~ 4 = 0.
(ii) F o r each finite t,
T ~ ((--oo, t]) 4 = 4 (iii) For each finite t,
T~ ((--oo, t]) 4 --- 0 Proof."
(28)
4 = 0, (29)
(R) 4 = 0. (30)
(i) Let [ A[ be the diameter of the bounded Borel set A, so that A I'1 (A + ~r) = for all ~- > [ A [. Henc~
T ¢ (A) q- 74 (A+~-) ~ I, (31)
for all 7 > [ A [. I f 4 be a vector such that
TE (A) 4 = 4, (32)
then (31) implies that
(4, rE ( a + , ) 4 ) = 0 - - ~ T , ( A + ~ ) 4 = 0,
for all ~- > [ A [. By applying (27) and Lemma 2 (due to Hegerfeldt and Ruijsenaars 1980) we conclude that
TE (A)exp ( ~ H~-)~b = 0 , (33)
for all real 9. Setting r = 0, we get 4 = T ¢ (A) 4 = 0 .
(ii) If 4 be a vector such that
TE ((--oo, t]) 4 = 4 , (34)
then it is easy to see that
.(I--r~ ((--o% t+~'])} 4 = 0, (35)
for all 1- > 0. Henc~ from (27) we obtain
~l--T~((--oo,
t])]-exp ( ~ H , 1 ~---O, for all ~r > 0. Applying Lemma 2 again, we get thato x p ( ~ H , ) { l - - T * ( ( - - o o , t])}exp(~ i
H , ) ~ = O (36)for all real ~.. In other words we have
rE ( ( - oo, t + d ) ~ = ~, (37)
for all real ~-. Taking the limit ~- ~ -- oo, we at once obtain ~ = 0 as required.
(iii) Let ~ be a vector such that
TE ((-- oo, t ] ) ~ = 0. 0 8 )
We th~a have
~ ((-- oo, t + ~ ] ) ~ = 0 ,
for all • < 0. Employing (27) we again get
TE ((--oo, t]) exp (-- i H ,) ~b = O,
(39)?1
for all T < 0. Again using Lcmma 2, we obtain
T ~ ( ( - ~ , t - ~ * ] ) ~ : 0 , (40)
for all real T. Taking the limit • --> oo, we obtain
r ~ (R) ¢ = 0, ( 4 0
as required. This completes the proof of the theorem.
It is obvious that the theorem of Pauli is subsum~l in the above result, for (29) implies that each of the operators T ~ ( ( - - o o , t]) will b¢ identically zero if assumed to be a projector, so that A --> T~(A) cannot be a proj~tion-valued measure.
We may also note that if the time of occurrence observable A--> T~(A) is normalized,
i.e.
T ~ (R) = I, (42)
186 M D Srinivas and R Vijayalakshmi then (30) implies that
T E ( ( - oo, t ] ) ¢ = 0 ~ = O. (43)
We shall now make a few remarks on the physical significance o f theorem 2. First o f all we should emphasize that the above result is applicable only for the case o f time o f occurrence measurements performed o n the entire time axis. N o w equations (28) to (30) are equivalent to the following:
(i) There does not exist any state # such that
Pr~_oo ' ~) (A; ~) : 1, if A is a b o u n d e d Borel set.
(ii) There does not exist any state p such that
Pr~_oo ' ~) ((-- oo, t] ; E) = 1, for any finite t.
(iii) P r ~ _ ~ , oo) ((-- oc, t]; ,) : 0 ~ Pr~_~, 6,) ((-- oc, ~ ) ; ,) : 0 (44) In somewhat metaphorical terms we can summarise the above as implying that
"nothing occurs with certainty in a finite interval o f time or p r i o r to any finite time instant, a n d if the occurrence o f an event p r i o r to some given instant can be ruled out with certainty then the event shall never occur'.
It should be emphasized that the generalization o f Pauli's theorem proved above does n o t in any sense preclude the existence o f time o f occurrence measures on the real line. In fact Holevo (1978) showed that if the Hamiltonian has absolutely c o n - ' tinuous spectrum ranging over the interval (a, b) where - - oo < a < b <~ o% then there exist several POV measures A + T'(A) satisfying the covariance condition (27). In the general case where the spectrum o f the H a m i l t o n i a n is not absolutely continuous, some further conditions will get imposed o n the time o f occurrence observable A--> T~(A) as we shall see below.
Let us denote by
= :~p.p (H) ~ 5t/~(H) ( ~ , - ~ ( H ) (45)
the decomposition o f the Hilbert space into the purely point, absolutely continuous and singularly continuous subspaces associated with the self adjoint operator H.
I f A ~ T~(A) is a POV measure on the real line, let us denote by ~ ( T E) a n d d ( T E) the following subsets o f 3 / .
~ ( T ' ) ---- U {Range T'(A) I A ~ B (R)}, (46)
= { I T'CR) = 0 ) . (47)
Therefore
e d ( T ' ) - - - 5 T'(A) = 0, (48)
for all A ~
B(R).
We have the following result:Theorem
3: Let A ~ TE(A) be a POV measure on the real line satisfying the covari- ante condition (27). Then(i) ~ ( T ~) c 57/.(H), (49)
(ii) Ytp.p (H) CjP(T~). (50)
Proof:
(i) This is a direct consequence o f the following result due to P u t n a m (1967) that if A is a bounded operator and for some real r we have
exp (~ H~.) A exp (--~ H~-) = A + D,
(51)where D > 0, then
Range D C= ~ a ( H ) . (52)
Now, from (27) and the fact that
Te((-- 0% t -j- ~-)) -- Te(( - ~ , t)) 7> 0, (53) for all real t and all'r ~> 0, we can conclude that
Range [Te(( - 0% t -b ~)) - - TE(( - 0% t))] C 3/a(H), (54) for all real t and ~- /> 0. Thus it follows that
~ ( T ' ) C g . ( H )
(ii) Let P ~ 3/p.p (H); that is P is a bound state o f H satisfying
H ~ = E ~ . (55)
N o w f r o m (27) and (55) it foUows that
[
(~-~ / H~-),. exp ( - ~ H . ) , ) (~, Te((--ao, t-}-z)) I~) ---- .exp r e ((--0% t))\
= ($, T E ((-- o% t)) P), for all real ~,. Taking the limit ~,-> - - ~ we get
T ' ((-- ~ , t)) ~ = 0,
(56)
(57)
188 M D Srinivas and R Vijayalakshmi
for all t from which it follows that
~p.p. (H) C J/' (T¢).
This completes the proof of the theorem.
From the above theorem it follows that if the Hamiltonian has a spectrum which is not purely absolutely continuous, then there does not exist any time of occurrence observable A -+ T~(A) on the entire axis which is normalised, for, if
T (R) = / , (55)
then we get
R (T ~) ---- ~ , (59)
and henc~ it follows from (49) that
( n ) = (60)
The other conclusion that can be drawn from the above theorem is that for any time of occurrence measurement performed on the entire time axis on a system in a bound state e of the Hamiltonian
Pr(_oo, (A; = 0, (61)
for all Borel sets A c R.
Finally we may emphasize again that the results of theorems 2 and 3 are obtained for the somewhat idealised situation of a time of occurrence measurement performed on the entire time axis. We have discussed this case in detail mainly because this would also have been the situation for which a self-adjoint time operator (which is again a spectral measure defined on the entire time axis) would have been relevant, in case such an operator existed. In the next section we shall present a simple example of a time of occurrence observable associated with measurements performed over a finite interval of time and show that many of the above results need not be valid for such observables.
5. The time-energy uncertainty relation
The uncertainty relation for time and energy
AT AE >1 ~ /1 (62)
was introduced by Heisenberg (1927) in the same paper in which he introduced the position momentum uncertainty relation
AQ AP i> ~- (63)
2"
is that the dispersions measured at the same systems in any state
¢,
from the way in which theory.
However the derivation as well as the interpretation of (62) has so far remained quite controversial (see for example Jammer 1974 for a review). By now the generally accepted interpretation o f (63) (see for example Gnanapragasam and Srinivas 1979) (or standard derivations) of position and momentum as time on two different but identically prepared ensembles of satisfy the relation (63). This interpretation directly follows (63) is derived from the fundamental principles of quantum The interpretation of (62) which Heisenberg himself advocated and which was later on discussed by Landau and Peirls (1931) and others (see for example Fock 1962, 1966) is that the energy of a system cannot be measured precisely and that in any measurement of energy which lasts for a time duration AT, there is bound to be an imprecision in the value of energy by an amount larger than or equal to lt/2AT. This interpretation has been generally found unacceptable (see for example Aharanov and Bohrn 1961, 1964; Allock 1969) firstly on the ground that quantum theory being a purely statistical theory, has nothing to say on individual measurements. It has also been shown (Aharnov and Bohm 1961, 1964; Kraus 1980) that the quantum theory of measurement does allow for measurements of energy of any arbitrary precision without there being any restrictions placed on the duration of measurement.
To us it appears natural that relation (62) should have a statistical interpretation similar to that of (63). In fact if AT has to be related to the uncertainty in the out- comes of an experiment to measure some observable time associated with a system, then it is most reasonable to interpret it as the dispersion in the times of occurrence of an event as measured in a time of occurrence experiment conducted over an ex- tended period of time. AE is to be interpreted as the dispersion in the values of energy as measured on a different but identically prepared ensemble of systems.
For the dispersion (AE) ~ of the energy values in state ¢, the standard rules of quantum mechanics yield the expression
(A.e)# = {(/¢¢, .n'¢) -- (¢,
(64)
for all ¢ in the domain ~ ( H ) of the Hamiltonian. Note that (AE) ¢~ is independent of tke time at which the (supposedly instantaneous) measurement of energy is carried out as long as no other measurement is carried out on the system prior to the measurement of energy.
The suggestion that AT should be viewed as the uncertainty in the time of occur- fence of some event goes back to Bohr (1928), who gave a heuristic proof of (62) where AT was taken to be the uncertainty in the time at which a particle executing one-dimensional motion crosses a given point. Another instance, where a similar interpretation has been found necessary, happens to be the case of an unstable system for which the following well-known relation
y = n , (65)
between the lifetime -r and the energy width y of the system, is often obtained as a consequence of the uncertainty relation (62). In recent times Allock (1969), Kijowski (1974), Helstrom (1974) and Holevo (1978) have also suggested that AT in relation (62) has to be viewed as the uncertainty in the time of occurrence of some event 1~.~2
190 M D Srinivas and R Vijayalakshmi
associated with the system. However, the crucial point here is that the uncertainty in the time of occurrence of an event depends also on the particular interval of time over which the experiment (to measure the time of occurrence) was carried out.
(A T(tl
uncertaintyIn fact we shall show in this section that if ,:~)ff denotes the
in the time of occurrence of an e v e n t , when an ensemble of systems prepared in state at time tl is subjected to continuous observations over the interval 01, t2), then the relation
(A T'(tx, ta)) ~r (AE) t~ /~ n/2,
can be obtained as a consequence of the fundamental principles of quantum theory (such as time-translation invariance) only when (tl, t2) = ( - - ~ , oo).
Before going into the formulation of a time-energy uncertainty relation along the lines indicated above, we shall first discuss the two well-known formulations of the time-energy uncertainty relation due to Mandelstam and Tamm (1945) and Wigner (1972). One of the main difficulties in deriving a relation of the form (62) from the basic principles of quantum theory has been that, in the absence of a self-adjoint time operator, one cannot employ the usual relations such as (64) to calculate the uncertainty AT. Mandelstam and Tamm circumvented this by associating with each observable A(t) (in the Heisenberg picture) 'time-uncertainty' [ATA(t)] ~ given by
(A TA(t)) ~ -~ (A A(t)) ~ (66)
(~, A(t) ~)
where (AA (t)) ~ is the usual uncertainty in the observable A(t) given by
(AA(t)) t~ = {(A(t) ~, A(t) ~ ) -- (~b, A(t) ~b)~} 1/2 (67) Now, by employing the Heisenberg equations of motion for A(t), it is easy to show that
(ATA(t)) $ (AE) ~ >~ hi2, (68)
where (AE) # is given by (64).
From (66) it is clear that (ATA(t)) ~ can be thought of as the time which would be required for the mean value of A(t) to vary by an amount equal to (AA(t))~bwthe uncertainty in the value of A(t) at time t. Here, experimentally (~, A(t) ~b) is obtained by carrying out a (instantaneous) measurement of A(t) at time t on an ensemble o f systems in state ~. d/dt (~h, A(t)~,) is obtained by measuring A(t) at times around t on different but identically prepared ensembles all in state ~b. Hence, the relation (68), though of considerable physical significance, has nothing to do with time of occur- rence measurements carried out over an extended period of time, and for no observable A(t) can the quantity (ATA(t)) ¢~ be related to the time of decay or the time of arrival, etc. of a particle.
Recently Wigner (1972) derived another relation of the form (62), where again the
uncertainty AT is defined without any explicit reference to a time operator. If $ is any fixed vector in the Hilbert space, then for each state ¢ define the function F,b(t ) as follows
F o ( t ) - - (~, exp ( - - ~ Ht)¢). (69)
Let us denote by Z~ the quantity
= {.j ,,,t}
whenever it exists, alid let /~$(E) be the Fourier transform of Y$(t). Now the 'un- certainties' (ATe°) $ and (AEg) ~ are defined as follows
(AT~)~ : { ft2 l Ftb(t) t ~ dt -- ( f t I F#(t) I~/z, tit)2} 1/2
z~ (71)
= { f (e)t d e - - ( f E (e)I lz,
Zff
(72)We then have the following uncertainty relation due to Wigner
(ATe) ¢ (AE~) # >~/1/2. (73)
Again, though (73) is a perfectly valid relation (barring questions of existence of the quantities (70) to (72)), our main objection is to the interpretation proposed by Wigner that if ~ is identified with the state of a resonance, then (73) can be viewed as the relation (65) between the uncertainty in the decay-time and the energy width of the state ~b. Firstly it should be noted (Bauer and Mello 1978) that (AE~) # given by (72) is quite different from the usual dispersion (AE) ff of energy values in the state ¢ as given by (64). More important is the fact that, according to the basic rules of quantum mechanics, the quantity
IF0 (t)l 2 = I (~, exp ( - - h H t ) ~ ) J2 (74) is to be interpreted as the probability that a system prepared in state ¢ say at time 0 is found to be in state ~ at time t hr an hzstantaneous measurement pelformed at t with no measurements being made in between times O, t. There is thus no a priori justification for interpreting I F# (t) 12 as the probability density that the system is found in state ~ around time t, in a situation where continuous measurements are made to check whether the system is in state ~ or not. We should however mention ihat under some limiting conditions the above interpretation
of IF o (t)t '
can be justified based on a suitable model of continuous measurements (Davies 1975,
192 M D Srinivas and R Vgayalakshmi
Friedman 1976; Misra and Sudarshan 1977) performed on the system. It may also be noted that similar expressions are sometimes considered in providing an algorithm for computing the time-delay in a scattering experiment (see for example Amrein et a1 1977).
It should thus be clear that the formulations of Mandelstam, Tamm and Wigner are not suitable for a general discussion of the uncertainty in the time of occurrence of an event in quantum theory. In our opinion such a discussion can only be based on the notion of a time of occurrence observable outlined in
5
3. Let us recall that if A -t T:& t2, (A) be such a time of occurrence measure on the interval (t,, t,), then the probability that the event e occurs in the Bore1 subset Ac
(t,, t,) is given byHence the uncertainty [A Thl, ,2,]4in the time of occurrence of event r in a measure- ment performed over the interval (t,, t,) (on an ensemble of systems prepared in state
$ at time t,) is given by
We shall now investigate whether an uncertainty relation of the form (62) can be derived for the quantity (A T:tl, t2,).B
We shall first consider the case of time of occurrence measurements performed on the entire time axis, and as in the last section write the associated time of occurr- ence measure on the real line as A -+ T' (A). The corresponding uncertainty in the time of occurrence shall be denoted as (A T E ) ~ and is given by
whenever the right side is well-defined. The uncertainty in the energy (A E)+ is of course as given by (64). Now, if the basic requirement of time-translation invariance
e x
(k
H a) Te (A) exp(-
H a) = Te (Afa) (78)is satisfied then Holevo (1978) has outlined a derivation of the relation
for suitable ¢. Since Holevo's proof does n o t indicate whether (or when) there do exist a large e n o u g h class o f states (say constituting a dense subset o f 3 / ) for which the relation (79) holds, we shall here present a proof o f (78) which also clarifies some o f the mathematical questions involved.
It is well-known that, unlike for a projection-valued measure, the dispersion (AT¢)#
o f (77) associated with a POV measure A --> Te (A) need n o t be well-defined on a dense subset of the Hilbert space 3 / . In_fact there do exist POV measures A --> T ~ (A) such that (A Tc)~ is n o t well-defined for any non-null vector ~b ~ 3 / (Akhiezer a n d Glaz- m a n 1961). However there is a class of POV measures on R which is in one-to-one correspondence with the set o f all maximal symmetric operators on ;Y£ and for which the dispersion (77) is always well-defined on a dense subset o f :~/. A linear operator r with dense domain ,~(r) is said to be symmetric if
(v, r ~) = (~ ~, ¢) (80)
for all ~, ~b ~ N (r). The symmetric operator r is said to be maximal symmetric if it has no symmetric extension defined on the same Hilbert space. It is well-known (Akhiezer and Glazman 1961) that with each maximal symmetric operator r, there is associated a unique POV measure A --> T (A) on the real line such that
(i) (V, r ~b) = f a d (~, T ( ( - - oo )q) $), for all ~ E : ~ a n d ~b E $ (r) ;
(81)
(ii) ~ ~ $ (r) i f f f 2, = d(~, T ( ( - - m , a]) ¢) <
in which case
( r ~b, r ~b) = f a = d (~b, T ( ( - - 0% a])~). (82) We thus see that if a POV measure A -+ T(A) is associated with a maximal symmetric operator r, then the associated dispersion (AT) '~ is given by
(AT) ~ ~- { f k 2 d (¢, T ((-- 0% A]) ~b) - (S a d (~,, T ( ( - - 0% a])
~))~)~'~
= {(~¢, ~-¢) - (¢, ~-.4,)~}- " (83)
for all ~b in the dense subset ~(-r). Another well-known fact that we shall employ (Cooper 1947) is that if A + T(A) is a POV measure associated with the maximal symmetric operator r, and for each real t if the operator W(t) is defined by
($, W(t) ~b) = f exp (/At) d(I °, T ( ( - - ~ , A] ) ~b), (84) then for either t ~> 0 or t ~<0, the set o f operators { W ( t ) } constitute a strongly
194
M D Srinivas and R Vijayalakshmi
continuous one-parameter semigroup of isometrics
(i.e. W(t) t W(t)= I)
whose generator is -r.We shall now present a proof of the time-energy uncertainty relation (79) for the class of arrival time measures on the entire time axis which are associated with maxi- mal symmetric operators.
Theorem 4
Let the POV measure A-~ T" (A) on the real line be associated with the maximal symmetric operator ~'.
If the covariance condition
e x p ( ~ H a ) T'(A) e x p ( - - ~ H a ) = T ' ( A + a ) , (85)
is satisfied, then the domain ~ of the operator (~-~ H - - H -r O is dense in ,3[ and
(T" H - - H 70 ~b = --i II ¢,
(86)
for all ¢ E ~. If the dispersions (A E)# and (A T')# are respectively given by (64) and (77) then
(A T 0 $ (A E)$ ~- ~ , (87)
for all ~b in the dense set ~ .
Proof."
From the condition (85) we get=fexp(iA/3) d(P,T'((--oo, A+a])
exp ( ; H a) ¢ )for all real/3 and ~, ~b E 3/. From this it follows that
e x p ( ~ H a )
W'(/3)=exp(--ia/3)W'(/3)
e x p ( ~ H a )(88)
for either a, 13 >~0 or a,/3 ~<0, where/3-+ W ~ (/3) is the strongly continuous one-para- meter semigroup of isometries generated by the maximal symmetric operator r *.
Since a-+ exp
(i/h) H a)
is also a strongly continuous semigroup of unitary operators for a~>0 or a~<0, we can employ the result of Foiaset al
(1960) (for two contraction semigroups obeying Weyl-like relations) to conclude that the generators 7" and H of these semigroups are such that the domain .,~ of the operator "r~H--Hz •
is densein 5 / / a n d that the C C R (86) is satisfied for all vectors in ~ . we see from (77) a n d (83) that
(A T,)~ = { ( ~ 4', ~, 4') - (4', ~, @ ) ~ / ~ ,
is well-defined for all 4' E ~ . Similarly, since ~ c ~ (H), we have that
Also as ~ C ~ (,rq
(89)
(AE)# = { ( H 4,, H V~) - - (4,, H V~)2} 1/2, (90) is well-defined for all 4' E ~. Now we can employ the C C R (86) a n d the standard procedure employed in deriving uncertainty relations to conclude
(A T ' ) ~ ( a E)~ i> ~, (91)
for all 4' ~ ~. This completes the p r o o f o f the theorem.
We have thus shown that the uncertainty in the time o f occurrence o f a n event, in an experiment carried out on the entire time axis, satisfies the time-energy uncer- tainty relation (87) as a consequence o f the time-translation invariance requirement (85). However, it is indeed very curious to note that such a relation cannot be derived for the uncertainties associated with time o f occurrence measurements conducted over a finite or a semi-infinite time interval. I f A ~ T{~, q} (A) is the associated time o f occurrence measure, it is easy to see that the time-translation invariance requirement
exp H a T(q, t~) (A) exp H a -- (q+a, q+~) ( A + a ) , (92) does not impose any serious restrictions on the POV measure A-+T(t1, t~ ) (A), unless tl = - - ~ , t~ = + ~ . In fact it could even happen that all the operators T ' (ti~ t 2 ) (A)} commute with the operators exp (i/h) H a), a n d still (92) is satisfied provided the time o f occurrence measurement is 'homogeneous in time'; i.e.
T E ( A ) : T E
('1, '2~ (tl+~, , 2 + ~ ( A + ~ ) . (93)
Obviously we cannot then expect to have any uncertainty relation between the dis- persions o f energy a n d the time o f occurrence.
The above point m a y be clarified by considering the example o f the time o f arrival of the first photon in photon counting experiments performed on a single-mode free field. These experiments have recently been analysed (Srinivas and Davies 1981) from the point o f view o f the quantum theory of continuous measurements. Now the Fock space o f a single-mode free field is the same as the Hilbert space associated with a harmonic oscillator and the Hamiltonian, which characterizes the evolution o f the system in the absence o f measurements, is given by
H = h w a t a, (94)
196 M D Srinivas and R Vijayalakshmi
where a, a t are respectively the annihilation and creation operators: if p is the state o f the field at time tl and the detector performs continuous measurements over the interval (q, t~), then the probability that the first photon is counted prior to the time ti-q-t is given by (see for example Mellow 1968, Scully and L a m b 1969, Srinivas and Davies 1981)
Pr~q, ,,~ [(t i, t i + t ) ; 1]
oo
= ~ n [1--exp (--~ t)]
n = l
exp [--;~ ( n - - l ) t] (n [Pl n), (95)
for all t < t2--ti, where [ n ) is the n-photon state and ;~ > 0 is a coupling parameter characterizing the measurement performed by the detector. It is easy to see that the probabilities (95) are associated with the time o f occurrence observables (q, ti+t)-+T~q, t~) [(ti, t t + t ) ] given by
O(3
Titq, q) [(t i, t i + t ) ] ---- ~ n [1--exp (--A t)]
n = l
exp [--;I ( n - - l ) t],
× In) ( n 1, (96)
for all t i < t 2 and 0 < t < ta--t t. The time o f occurrence observable (96) dearly satisfies the causality condition
T~q,i 'l' [(ti, t t + t ) ] = T~,l, oo~ [(ti, t i + t ) ] , (97) as also the 'homogeneity' property
i [(ti ' t i + t ) ] : T 1 [ ( t l + a ' t i + t + a ) ] '
T(ti, t~) ttxq ct, t2+a) (98)
for all real a. N o w from the fact that the Hamiltonian h w a ÷ a commutes with all its eigen projectors In) ( n I, it is easy to see that the time-translation invariance property
'_Ha)
: T i ( t l + G ~ ta+(Z) [ ( t i + a , t i + t + a ) ], (99) is trivially satisfied. Hence the POV measure (tz, t z + t ) ~ T~,I, ,~ [(tt, t z + t ) ] is a genuine time o f occurrence observable on the interval (ta, t~).
We can now evaluate the dispersion (ATe0 ' o~})$ in the arrival times of the first photon in a counting experiment performed from time 0 upto time oo. If we employ (76) and (96) we obtain
o o
n = l
n = l
(lOO)
whenever the right hand side is well defined. For the dispersion (AE)~ in the energy values we get from (64) and (94), the expression
oO oO
n = l n = l
(101)
for all ~b E ,~ (H). We thus see that for all n-photon states In) where n i> 2, (ATe0, oo))In> is finite while (AE) I n> is zero. Apart from this extreme case where the uncertainty product
(AT o, oo))¢ (AE)¢
vanishes, it should be clear that by suitable choice of h the uncertainty product
(A r 0, o ,)0 (ae)
can be made small for a large class of states $ ~ 5~t.
We are thus led to the conclusion that for time of occurrence measurements per- formed over finite or semi-infinite intervals, the time-translation invariance property does not lead to a time-energy uncertainty relation. This shows another basic conceptual difference between the position-momentum uncertainty relation (63) and the time-energy uncertainty relation (62) which is that while the former is always a consequence of a certain fundamental invariance property of the theory under space translations, the latter can be deduced from the property of time-translation in- variance only for the rather unrealistic case of time of occurrence measurements performed on the entire time axis. There does not as yet seem to be any funda- mental justification for assuming that the time-energy uncertainty relations hold also for time of occurrence measurements performed over a finite interval of time.
1 9 8 M D Srinivas and R V i j a y a l a k s h m i
Acknowledgement
W e a r e g r a t e f u l t o D r S D R i n d a n i f o r h e l p f u l d i s c u s s i o n s .
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