Abstract
We interpret the concept of determinism for a classical system as the requirement that the solution to the Cauchy problem for the equations of motion governing this system be unique. This requirement is generally believed to hold for all autonomous classical systems. Our analysis of classical electrodynamics in a world with one temporal and one spatial dimension provides counterexamples of this belief. Given the initial conditions of a particular type, the Cauchy problem may have an infinite set of solutions. Therefore, random behavior of closed classical systems is indeed possible. With this finding, we give a qualitative explanation of how classical strings can split. We propose a modified path integral formulation of classical mechanics to include indeterministic systems.
Similar content being viewed by others
References
Tabor, M.: Chaos and Integrability in Non-Linear Dynamics. Wiley, New York (1989)
de Vega, J.H., et al.: Classical splitting of fundamental strings. Phys. Rev. D 52, 4609 (1995)
Blasone, M., Jizba, P., Kleinert, H.: Quantum behavior of deterministic systems with information loss: Path integral approach. Ann. Phys. (N.Y.) 320, 468 (2005)
Gozzi, E.: Hidden BRS invariance in classical mechanics. Phys. Lett. 201, 525 (1988)
Gozzi, E., Reuter, M., Thacker, W.D.: Hidden BRS invariance in classical mechanics, II. Phys. Rev. D 40, 3363 (1989)
Abrikosov, A.A. Jr., Gozzi, E., Mauro, D.: Geometric dequantization. Ann. Phys. (N.Y.) 317, 24 (2005), quant-ph/0406028
Macomber, H.K.: Time reversal in classical mechanics: a paradox. Am. J. Phys. 40, 1339 (1972)
Norton, J.D.: Causation as folk science. Philos. Impr. 3(4) (2003), http://www.philosophersimprint.org/003004/
Kosyakov, B.P.: Exact solutions of classical electrodynamics and the Yang–Mills–Wong theory in even-dimensional spacetime. Theor. Math. Phys. 119, 493 (1999), hep-th/0207217
Kosyakov, B.: Introduction to the Classical Theory of Particles and Fields. Springer, Berlin (2007), pp. 374–376
Kosyakov, B.P.: Holography and the origin of anomalies. Phys. Lett. B 492, 349 (2000), hep-th/0009071
Kleinert, H.: Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 4th edn. World Scientific, Singapore (2006), Sect. 18.13; http://www.physik.fu-berlin.de/ kleinert/reb5/psfiles/pthic18.pdf
Lanczos, C.: The Variational Principles of Mechanics. University of Toronto Press, Toronto (1949)
Brown, J.D., York, J.W. Jr.: Jacobi’s action and the recovery of time in general relativity. Phys. Rev. D 40, 3312 (1989)
Polyakov, A.M.: Gauge Fields and Strings. Harwood, Chur (1987), Sect. 9.1
Mottola, E.: Functional integration over geometries. J. Math. Phys. 36, 2470 (1995)
Thacker, W.D.: New formulation of the classical path integral with reparametrization invariance. J. Math. Phys. 38, 2389 (1997)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kosyakov, B.P. Is Classical Reality Completely Deterministic?. Found Phys 38, 76–88 (2008). https://doi.org/10.1007/s10701-007-9185-x
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10701-007-9185-x