Comment on some theories of state reduction

PhysicsLetters A 167 (1992) 433434 North-Holland

PHYSICS LETTERS A

Comment on some theories of state reduction A.M. Jayannavar Institute of Physics, Sachivalaya Marg. Bhubaneswar 751005, India Received 9 March 1992; accepted for publication 15 June 1992 Communicated by J.P. Vigier

Recently several theories have been proposed to account for state reduction due to measurement. We show that the density matrix obtained in these theories is the same for the particle subjected to a classical white noise potential. We also point out classical analogs ofthese theories.

The relation between classical and quantum mechanics is one of the basic problems in the theory of measurement and interpretation of quantum theory. The linearity of the Schrodinger equation leads to

Diosi [4,51obtains the same equation (1), by taking into account the modification ofquantum dynamics by gravitational effects. In his case f(x, x’) = ~y(x—x’ )2• J 005 and Zeh have arrived at the same

superpositions of macroscopically distinguishable states and hence Schrödinger’s cat paradox. It is now understood that such paradoxes can be avoided if one treats the macroscopic body as a complex one interacting with an inevitable environment [1]. The environment surrounding a quantum system induces decoherence leading to classical behaviour [1,2]. Ghirardi et al. [3] have proposed a scheme wherein the wave function besides evolving through a Hamiltonian dynamics, is subjected to repeated collapse corresponding to localisation in space at random times. By this scheme one can suppress the linear superposition of states corresponding to the same macroscopic object being localized in far apart spatial regions. This will also lead to a description for macroscopic bodies in terms of trajectories. In mathematical language this procedure replaces the Schrödinger equation by a nonunitary master equation. For the simplest case of a free particle in one dimension they have obtained an equation for the density matrixp(x,x’, t) (~w*(xt)yi(x, t)) as [3]

master equation for systems interacting with an external environment [6]. We show in our paper that some features of these theories have classical analogs. One can obtain the same master equations proposed for state reduction by simply coupling a quantum particle to an effective white noise environment. We also show that some effects are not closely related to quantum mechanics at all. Recently Ballentine [7] has shown that these theories leading to eq. (1) are inconsistent as they violate the energy conservation and are incompatible with the existence ofequilibrium steady states. We also show that the violation of energy conservation and the existence of the equilibrium steady state can be easily seen in the classical counterpart of eq. (1). Most of the proceeding presentation is based on our work related to quantum transport in a dynamically disordered medium [8]. Consider a classical particle subjected to a random force2xby an environment, d

t

m

X

(1)

t3X

wheref(x, x’) is given by f(x, x’) =

1

—

exp [

—

~a (x—x’)2]

.

m~-j=f(t), (2) where f( t) is a Gaussian white noise. Equation (2) represents the well known Langevin equation [9] in the absence of a concomitant frictional force. One can easily verify that [81 the mean squared displace-

0375-9601/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.

433

Volume 167, number 5,6

PHYSICS LETTERS A

ment of the particle, , grows as t3. Since we have not provided the dissipative mechanism (i.e. absence offrictional force), the particle continues to absorb energy from the fluctuating force and accelerates indefinitely. In short, the particle heats up to an infinite temperature. This is also related to the fact that in the absence of friction there is no steady or equilibrium state. For equilibrium one must incorporate the fluctuation force concomitant with a frictional force (this is an essential part of the fluctuation—dissipation theorem) [9]. Now corresponding to the above classical equation (1) the Hamiltonian is given by H=p2/2m—xf(t). We will now quantize this system; the quantum Hamiltonian is H~

V2 —xf( t) (3) 2m wheref( t) is a Gaussian white noise, with = 0 and = 2 V~ô(I I’). The angular bracket represents averaging over all realizations of the random force. By using the Schrodinger equation one can easily write down the equation for the averaged —

—

density matrix p(x, x’, I) (~w*(x,t)W(x, I)). Following exactly the same procedure as that of ref. [8], we arrive at the same master equation (1), where f(x, x’) = (V~/h2)(x— x’) 2~ This is exactly the equation derived by Diosi [4,5], with a redefinition ofthe coefficient, namely ~y= V~/h2.We now talce the initial condition that the particle was prepared initially in a wave packet centered at the origin x= 0,

3 August 1992

Notice that in the asymptotic domain the width of the wave packet scales as t3. This is exactly the same dependence as obtained in the classical counterpart. The potential energy in eq. (3) is given by V(x, t) = xf( t), if we replace this potential by a space— time dependent Gaussian white noise with statistics given by =0 and =g(x—x’)ô(t—t’), we again arrive at eq. (1) for the averaged density matrix (see eq. (8) in ref. [81). If we assume the spatial part of the correlation function to be g(x— x’) = exp [ a (x— x’)2], we get the result derived by Ghirardi et al. [3]. Equation (17) in ref. [8] gives the exact result for the time evolution of the width of the wave packet. Ballentine has considered another interesting case [7] which corresponds to choosing a spatial conelation g(x—x’) = exp [ a (x—x’)4]. Such a correlation function conserves the energy of the quantum particle on the average [7]. This fact is also reflected in the fact that
—

—

one can easily verify that scales as t~in the asymptotic time domain. In conclusion, we have shown that the evolution equation obtained for the density matrix in several state reduction theories can be derived by a particle subjected to a classical Gaussian white noise potential. Some of the peculiar features of the state reduction theories such as the acceleration of particles and the incompatibility with the existence of equilibration have classical analogs.

i.e.

~(x, t=0)=

11 ~

—~

References exp(—x2/4a2)

[1] W.G. UnruhandW.H.Zurek, Phys. Rev.D40 (1989) 1071,

This ensures a correct normalization for the initial density matrix, p(x, x’, t=0)=W*(x, t=0)~t’(x,1 =0). This initial wave function has a width (or spatial spread) equal to a2. One can evaluate exactly the time development of the width [8] or the meansquared displacement; it is given by /1 2t2 3ma

434

V +

02 3m

(4)

and references therein. [2] A.O. Caldeira and AJ. Leggett, Phys. Rev. A 31(1985) 1059.

[3] G.C. Ghirardi, A. Rimini and T. Weber, Phys. Rev. D 34 (1986) [4] L. Diosi,470. Phys. Lett. A 120 (1987) 377. [5] L. Diosi, Phys. Rev. A 40 (1989) 1165. [6] E. Joos and H.D. Zeh, Z. Phys. B 59 (1985) 223. [7] L.E. Ballentine, Phys. Rev. A 43 (1991) 9. [81 A.M. Jayannavar and N. Kumar, Phys. Rev. Lett. 48 (1982) 553. [9] S. Chandrashekhar, in: Selected papers in stochastic processes, ed. N. Wax (Dover, New York, 1954).