IL NUOVO CIMENTO VOL. 107 B, N. 1 Gennaio 1992

Lagrang ian F o r m u l a t i o n o f Bohr ' s M e a s u r e m e n t Theory (*).

A. WIDOM and Y. SRIVASTAVA Physics Department, Northeastern University - Boston, M A 02115 Dipartimento di Fisica and I N F N , Universit4 di Perugia - Perugia, Italia

(ricevuto il 17 Gennaio 1991; approvato il 13 Marzo 1991)

Summary. -- In the conventional formulation of quantum mechanics, a measure- ment is an interaction between a quantum object and a classical apparatus. The quantum motions of the object dynamically induce the apparatus into a classical path (e.g., into a reading). The rules for computing the induced apparatus Lagrangians are here discussed.

PACS 03.65 - Quantum theory; quantum mechanics.

1 . - I n t r o d u c t i o n .

It has been over half a century since those who formulated (and profoundly ques- tioned) quantum mechanics agreed on a more or less well-defined set of rules for in- terpreting physical measurements [1]. Over the objections of Einstein that the foun- dations were incomplete[2], and over the objections of Landau that for relativistic fields the rules yield divergent answers, the interpretation of Bohr eventually be- came the convention, and here this convention will be merely assumed to be true.

A measurement, according to Bohr, is any interaction between a quantum object and an apparatus obeying classical mechanics to a sufficient degree of accuracy. The ((measurement)~ is the ,,interactiom~, and requires no human observer. This has been stressed by Landau and Lifshitz [3]. During this interaction, the state of the quantum object should be expanded as a superposition of amplitudes over a basis which de- pends on the nature of the interaction

(1) I~) = E a~ Jn). n

If the quantum object were in state In), then the apparatus would respond with a characteristic classical path (i.e. the apparatus reading). In what follows, this charac- teristic classical path will be viewed as arising from an effective Lagrangian. Each

(*) The authors of this paper have agreed to not receive the proofs for correction.

71

72 A. WIDOM and Y. SRIVASTAVA

state of the quantum object will ,(grow, on to the apparatus a characteristic Lagran- gian contribution Ln,

(2) (measurement): In} ---> L n .

For the superposition of amplitudes in eq. (1), the effective Lagrangian Ln ((pushes)) the apparatus into a reading with probability lan 12, so that the actual reading is stochastic.

The purpose of this work is to discuss a technical problem associated with the Bohr view. For a given measurement, how does one calculate the measurement basis, and the induced measurement Lagrangians in eq. (2)? For very general classes of sys- tems, the above technical problem will here be solved [4]. The deeper issues involved in the completeness of the theory will not be discussed in detail [1-3].

2. - C l a s s i c a l a p p a r a t u s p a t h s .

An apparatus, which obeys classical mechanics to a sufficient degree of accuracy will, in what follows, be described via the usual rules of classical analytical mechan- ics. The generalized coordinates will be denoted by

(3) X = (X 1 , x 2, . . . , x n ) ,

and the generalized velocities will be denoted by

(4) v = (v I , v2 , . . . , vn),

(5) v j = (dxJ / dt) .

Given a Lagrangian for the apparatus

(6) L = L(x, v),

the reading of the apparatus obeys

(7) (d / dt)(~L / av j) = ( ~L / ~xJ) .

Following Bohr's dictates, the quantum object is never directly observed. One may only note the path of the apparatus. To decipher any information concerning the state of the quantum object, the key step is in eq. (2), i.e. one must know how the state In} induces the apparatus reading, via the measurement Lagrangians Ln (x, v) here being employed.

3. - M a t r i x q u a n t u m o b j e c t s .

Suppose that the quantum object were described by a Hamiltonian matrix which depended upon the apparatus coordinates

(8) H(x) = IIHab II"

Let S(x) be a unitary matrix

(9) S § (x) = S -1 (x)

LAGRANGIAN FORMULATION OF BOHR'S MEASUREMENT THEORY 73

which brings the Hamiltonian matrix into diagonal form

(10a) W(x) = S + (x) H(x) S(x) ,

(10b) W(x) = II~ab Wa II.

For a given apparatus path, one seeks a solution to the Schradinger equation for the column vector u,

(11) ih(au/at ) = H(x) u ,

having the form

(12) u = S(x) ~'.

With

(13) p~. (x) = ih S + (x) aS (x ) / ax j

denoting the generators of the unitary matrix function S(x), eqs. (11) and (12) read

(14) ih(aw/at) = [W(x) - vJpj (x)] w,

where eqs. (5) and (10) have been employed. We note (in passing) that, from a mathematical viewpoint, quantum states here

are exhibited as fiber bundles over the apparatus manifold, while the Cartan connec- tion generators have zero non-Abelian curvature

(15) FiN (x) = (aP~ (x) / ax j - aPj (x) / ax i) + (i / h)[Pi (x), Pj ( x ) ] = 0.

From the viewpoint of physics, the quantum state In} will induce (in the apparatus) a Lagrangian contribution L, (x, v) by virtue of the interaction (i.e. measurement). The eigenvalue problem determining the above is simply

(16) [W(x) - - V i Pi (x)] ln) = - L~ (x, v ) l n ) ,

as is evident from eq. (14).

4. - C a n o n i c a l s y s t e m s .

In sect. 3, the Heisenberg matrix formalism was used to show how a quantum ob- ject induces a classical interaction reading in the apparatus. At first, let us consider two systems interacting on the completely classical level, the total Lagrangian will be denoted by

(17) L = L(Q, Q, x, ~c).

The canonical momenta of the Q-degrees of freedom are defined by

(18) Pa = (aL /aQa) ,

74 A. WIDOM and Y. SRIVASTAVA

and the Routhian is defined by

(19) R = Q a P a - L = R(P , Q, x, 5c).

We recall that in classical mechanics the Routhian is the ,,Hamiltoniam~ with respect to the Q-degrees of freedom

(20a)

(20b)

- P a = (DR/DQa),

Qa = (DR / D P a ) ,

while remaining ~,Lagrangian~ with respect to the x-degrees of freedom

(21) (d/dt ) (~R /~5c i) = (aR /ax i ) .

Having recollected the classical mechanics formalism, let us now suppose that the Q- degrees of freedom describe a quantum object. Then, through the Routhian, one may construct the Hamiltonian operator for the quantum object

(22) H(x, x) = R ( - i h ~ / ~Q, Q, x, 5r

while still retaining a Lagrangian view of the classical apparatus reading. For the canonical systems here being discussed, the eigenvalue problem for meas-

urements is given by

(23) H(x, v) ~F~ (Q) = - L~ (x, v) ~ (Q).

Equation (23) theoretically describes Bohr's notion that a superposition of states for the quantum object

(24a) 7~(Q) = ~ an ~Vn (q) n

yields an apparatus reading

(24b) (d / dt)(3Ln/ 3V i) = (3L / 3x~ ) ,

with probability

(24c) P~ = la~ I ~ �9

Equations (22) and (23) are the central results of this work, which has (as its modest aim) furnished some computational rules with which to implement Bohr's proce- dure.

5 . - C o n c l u s i o n s .

The motivation for considering the computational rules required to implement Bohr's measurement theory is that by now laboratory systems of (,macroscopic size, exist which exhibit coherent superposition of amplitudes on a (,macroscopic scale, [5]. (If we are allowed to speculate, we submit that more humane biological experiments may even bring (,SchrSdinger's cat~ into a more clear focus.)

Continued studies of superposition of amplitudes on a macroscopic scale will no doubt force us into those considerations laid dormant (prematurely) regarding Ein-

LAGRANGIAN FORMULATION OF BOHR'S MEASUREMENT THEORY 75

stein's objections as to the completeness of the standard quantum-mechanical formu- lation. Until that point, however, it is important to have agreed upon rules of compu- tation with which to push the Bohr view to its limit. I t is the authors' hope that the considerations of this work would have clarified some technical points regarding im- plementation of the Bohr view.

This work was supported by the Department of Energy in the US and by INFN in Italy.

R E F E R E N C E S

[1] See, for example, Quantum Theory and Measurement, edited by J. A. WHEELER and W. H. ZUREK (Princeton University Press, Princeton, N.J., 1983).

[2] For N. Bohr's discussions with A. Einstein on epistemological problems in atomic physics, see Chapt. 7 of Albert Einstein: Philosopher Scientist, Vol. 7 of the Library of Living Philosophers, edited by P. A. SCHLIPP (Evanston, Ill., 1949) (reprinted by Open Court, La Salle, Ill., 1970).

[3] L. LANDAU and E. M. LIFSHITZ: Statistical Physics, second edition (Addison Wesley Pub- lishing Co., Reading, Mass., 1969), p. 28.

[4] The quantum dynamics of ferromagnetic grains illustrating some of the general formalism presented here can be found in a recent work: Quantum electrodynamic circuit theory of measurements on ferromagnetic grains by A. WIDOM and Y. S RIVASTAVA: Nuovo Cimento B, 193, 185 (1989).

[5] For a recent review, see Y. SRIVASTAVA and A. WIDOM: Phys. Rep., 148, 1 (1987).