Exploring the meaning of an individual Feynman path

Volume 126, number 4

PHYSICS LETTERS A

4 January 1988

EXPLORING THE MEANING OF AN INDIVIDUAL FEYNMAN PATH Debashis GANGOPADHYAY’ Department of Physics, College of Textile Technology, Berhampur, Murshidabad, West Bengal, India

and Dipankar HOME Department of Physics, Bose Institute, Calcutta 700009, India Received 4 September 1987; accepted for publication 3 November 1987 Communicated by J.P. Vigier

We suggest a plausible procedure for classifying individual Feynman trajectories and ascribing physical meaning to them within the standard path integral formalism. The treatment pertains to the case of a free particle and calls for further generalisation.

In recent years there has been a renewed interest [1—4]in the relationship between the quantum measurement problem and the concept of a trajectory in Feynman’s path integral formalism, stemming from the central role played by a trajectory q( 1) in the path integral formulation of quantum mechanics. This is in contrast to the usual canonical quantum mechanics where position eigenvalues are usually avoided owingto a pathological problem of normalisation. On the other hand, there is an inherent conceptual difficulty in setting up a general scheme for the classification of individual Feynman trajectories. In Feynman’s own words [51: It is to be expected that the postulates can be generalised by the replacement of the idea of ‘paths in a region of space—time R’ to ‘paths of class R’ or ‘paths having property R’. But which properties correspond to which physical measurements has not been formulated in a general way.” Aharonov and Vardi [1] made a significant attempt to develop an operational meaning associated with an individual “Feynman path”. In their scheme of dense measurements within the Schrodinger framework they showed that: (a) a particle follows with probability unity the trajectory that is being observed; (b) the familiar weight (phase) factor exp(iSTh) is obtained for a state; (c) a theoretical scheme is possible that allows for measurement of the relative phase between any two trajectories. The motivation underlying this Letter is to probe (i) whether the interesting result (a) can be incorporated in the general path integral approach withcrut any conceptual difficulty, and (ii) whether the problem of classification of trajectories embodied in (c) may also be studied within Feynman’s formalism. As an initial step in this direction, we study the case of a free particle. Let us begin by recalling some basic results. The Feynman path integral for the transition amplitude of a particle having mass m is given by “...

= lim

$...f~ii’[dqj (2~th)1’2] exp(~dtL)~ $ 2~q(t)exp[(i/h)S],

(1)

N&1 fixed

Present address: Institute of Physics, Sachivalaya Marg, Bhubaneswar 751005, India.

0375-9601/88/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

219

Volume 126, number 4

PHYSICS LETTERS A

4 January 1988

where q~=q0 is the initial position at time t=0, q~=q~=q~ is the final position at time the action for the lagrangian L and J ~q( t) signifies functional integration. For any functional F[q(t)], the average ofF, viz. is defined as [6] ~=$~q(t)F[q(t)] exp{(i/h)S[q(t)]}

t= T,

S=J~dtLis

(2)

and it can be shown that [6] /öF\ ____

i/ 8S\ ~ \F6q(t)/

—

(3)

with ö /~qdenoting functional differentiation. It is known [6] that for a free particle, choosing F[q(t)] q0 <1>, = q~<1> one obtains ={q~+ [(qT—qO)/T]t}< 1>

=qciassicai<

1>

=

1 and the boundary conditions =

,

(4)

so that the average value of the trajectory is indeed the classical path (i.e. a straight line). Again, considering the most general expression for q( t): q(l)=q~1(t)+y(t)

(5)

,

where <~(t)> = 0 along with the boundary conditions y(0) = y( T) = 0, one may account for deviations from the classical path q~1(t).y(t) denotes fluctuations from the classical trajectory q~1(t)and on an average the fluctuations tend to mutually cancel i.e., <~(t) > = 0. One usually stipulates the trial function for y( t) as y(t)= ~ a,, sin(nx/T).

(6)

Note that the form for y( t) in (6) satisfies the requirements just mentioned. We stress that this form for y( t) is usually regarded as an ad-hoc input such that it satisfies the boundary conditions; since the average of a harmonic function over fundamental periods vanishes, the requirement =0 is automatically met. We shall show that eq. (6) can be derived from the action principle (by computing up to the relevant order in i/h) as a natural consequence provided we are prepared to identify the different paths by an index n and treated each trajectory dynamically. Let us write the action S[q(t)] =S[q~1(t)+y(t)] =S~1+

J

2=S~+~[y(t)],

(7)

dt ~m~

using the vanishing of the linear term in j’ from the boundary conditions y( 0) = y( T) = 0. At this stage we note that ~[y] = f~dt~mj’2 is an action-like quantity. Therefore the relation (3), which is a perfectly general result, can be also applied to .S~,whence we write /8F\ \&y(t

‘~F 6~5’\ 1)/~~\

~y(t1)/~

(8)

Choosing F[y]= exp(_

~$

we obtain from (8): 220

dl

~~2y2)

(9)

Volume 126, number 4

PHYSICS LETTERS A

4 January 1988

_~([l+O(i/h)] Jdt~=0. So to O( i/h) we can write —

~(

Jdt(9+w2y))=0.

(10)

The solution for y( I) satisfying eq. (10) and the given boundary conditions is obviously (with w = nit/fl y(t) =

~

y,,(t)

=

~ a,, sin[(nx/T)t]

.

(11)

The following subtleties regarding the derivation of (11) are worth noting: (1) The sum is not just a mathematical artifact ofFourier decomposition but it also implies classification of trajectories, denoted by the index, n, as each trajectory y,, ( I) has a corresponding dynamical equation of motion j~,,+ &y,, = 0 with solution y,,( t) = a,, sin( nit tIT). This is an interesting insight provided by the above treatment. (2) As fluctuations need not be small (in keeping with Feynman’s spirit) our result nowhere demands the requirement of small fluctuations. One need only calculate consistently up to the relevant orders in i/h. It is logical to expect that as the source of fluctuations is essentially quantum in nature, what should matter is the relevant order in i/h and not the absolute values of the fluctuations. This is what is precisely borne out by our result. (3) Choosing a similar form for F[q(t)] as (9) and using eq. (3), one would have been led to a similar equation as (10), but the correct result for q~1( t) could not have been obtained since no solution for q( t) of ~,,

the form (11) can satisfy the boundary conditions =q0, For

‘Q,(t)=

J~q(t)ö(q(l)_Q~(t))exp[(i/h)S].

(12)

Q,,( t) = q~1(t) we have

Igci(t) A exp[(i/2h)mv T] where A is the normalisation factor, v= (q7--- q0)/T is the classical velocity. Hence we obtain ,

P~1=IIq~iI=A.

(13)

(14)

Similarly for Q,,(t) =y~(t)= a,, sin(nitt/T) we have P~=I

IQxI~4

,

(15)

which implies that the probability for following any particular path is the same. Now, suppose a particle is observed to follow a particular trajectory. Since the probabilities are the same for all individual paths, subsequently there should be no preference for the particle to switch over to any other trajectory. This is in essence Aharonov’s result (a) which can, thus, be accommodated within the standard path integral formalism. The 221

Volume 126, number 4

PHYSICS LETTERS A

4 January 1988

above result has its analogue in the Schrodinger formulation of quantum mechanics. This is the quantum Zeno paradox which stems from the observation that the survival probability of a given quantum system (i.e., the probability for finding the system in the intitial state after being left to itself for a certain period oftime) tends to unity in the limit of a continuous series of observations to find out whether the system is in the original state or not [7]. To sum up, the index, n, associated with each path via the relation y~(t)=a,, sin(nxt/T), provides a crucial handle to classify individual Feynman paths. Each path y,, (1) now satisfies a dynamical equation of motion in the form of eq. (10) with the boundary conditions y,,( 0) = y,, ( T) = 0. The classical path satisfies a different equation of motion, m~~1=0, with the boundary conditions q~1(0)=q0,q~I(T)=qT.Of course, what we have illustrated in this paper pertains only to the case of a free particle. The generalisation of this treatment, incorporating potential fields, seems worth studying and is presently under investigation. One of us (DH) thanks Professor B.B. Biswas and Professor S.C. Roy (Bose Institute, Calcutta) for their encouragement. DH is also grateful to Professor R.A. Brown (Macquarie University, Sydney) for instructive discussions concerning the path integral formalism.

References [1] [2] [3] [4] [5] [6]

Y. Aharonov and M. Vardi. Phys. Rev. D 21(1980) 2235. Y. Nambu, in: Proc. 1st tnt. Symp. on Foundations of quantum mechanics, Physical Society ofJapan, Tokyo (1983) p. 363. C.M. Caves. Phys. Rev. D 33 (1986) 1643. C.M. Caves, Phys. Rev. D 35 (1987) 1815. R.P. Feynman, Rev. Mod. Phys. 20 (1948) 384. R.P. Feynman and A.R. Hibbs, Quantum mechanics and path integrals (McGraw-Hill, New York, 1965). [7] D. Home and M.A.B. Whitaker, J. Phys. A 19 (1986) 1847, and references therein.

222