Abstract
Path integral expressions are given for the wave function and for the density matrix in a theory which describes state vector reduction. The magnitude of the contribution of each classical path to the propagator is not unity, as in ordinary quantum theory, but depends upon the path.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References and Remarks
D. Bohm and J. Bub,Rev. Mod. Phys. 38, 453 (1966).
P. Pearle,Phys. Rev. D 13, 857 (1976).
Recent reviews appear in P. Pearle, inQuantum Concepts in Space and Time, R. Penrose and C. J. Isham, eds. (Clarendon Press, Oxford, 1986), p. 84, andNew Techniques and Ideas in Quantum Measurement Theory, D. M. Greenberger, ed.,Ann. N.Y. Acad. Sci., 1986, p. 539.
P. Pearle,Phys. Rev. D 29, 235 (1984).
A. Zeilinger, inQuantum Concepts in Space and Time, R. Penrose and C. J. Isham, eds. (Clarendon Press, Oxford, 1986), p. 16.
R. Penrose,ibid., p. 129.
F. Karolyhazy, A. Frenkel, and B. Lukacs,ibid., p. 109.
L. Diosi,Phys. Lett. 120A, 377 (1987).
N. Gisin,Phys. Rev. Lett. 52, 1657 (1984).
G. C. Ghirardi, A. Rimini, and T. Weber,Phys. Rev. D 34, 470 (1986); for a recent review seeFound. Phys. 18, 1 (1988).
P. Pearle, “Combining stochastic dynamical statevector reduction with spontaneous localization,” to be published in Phys. Rev. A, March, 1989.
G. C. Ghirardi, P. Pearle, and A. Rimini, preprint in preparation, 1989.
In Refs. 11 and 12, the Ito stochastic differential equations that are equivalent to the Stratonovich Eqs. (1), (3), and (16) were given. The Brownian motion functionw(x, t) given there is related to the white noise functionW(x, t) given here byW =∂w/∂t. The Ito form (Eq. (1) multiplied bydt withW dt replaced bydw and the last bracketed term multiplied by 1/2) was used there, because it is most easy to employ in calculating the statistical behavior of the ensemble of solutions. The Stratonovich form is used here because it may be manipulated as ifW were just an ordinary function of time approximating white noise.
The Ito form of Eq. (3) is [13]dψ(x, t) = {−iH 0 +W(x, t) − (1/2)⋋}ψ(x, t). Since according to the Ito calculus,d|ψ(x, t)|2 =ψ*dψ +ψdψ* + 〈dψ*dψ〉 = −i[ψ*(H 0 ψ) − (H 0 ψ*)ψ] + 2|ψ|2 dw, it follows thatd〈N 2〉 =d〈∫dx|ψ(x, t)|2〉 = 0.
R. P. Feynman and A. R. Hibbs,Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965), Chap. 4.
L. S. Schulman,Techniques and Applications of Path Integration (Wiley, New York, 1981), Chap. 4.
Expressions similar in form to Eq. (9) arise in ordinary quantum theory when the path integral for the density matrix of a system in interaction with an environment is traced over the environmental variables: See R. Feynman and F. Vernon, Jr.,Ann. Phys. (N. Y.) 24, 118 (1963); Ref. 15, Chap. 12; A. Schmid,J. Low Temp. Phys. 49, 609 (1982); A. O. Caldeira and A. J. Leggett,Physica 121A, 587 (1983).
In the CSL theory, as in the SL theory which inspired it, the reduction of a wave function for a macroscopic (many-particle) object proceeds at a faster rate than for a single particle. (This is at the heart of the resolution of the trigger problem.) The many-particle theory has other interesting features not encountered when dealing with single particle behavior. We do not discuss these features here; they are treated in Ref. 12. Our aim here has been to provide enough information to facilitate understanding of these and other problems in the CSL theory via path integrals.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Pearle, P., Soucek, J. Path integrals for the continuous spontaneous localization theory. Found Phys Lett 2, 287–296 (1989). https://doi.org/10.1007/BF00692673
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00692673