Foundations of Physics

, Volume 38, Issue 1, pp 76–88 | Cite as

Is Classical Reality Completely Deterministic?

  • B. P. Kosyakov


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.


Cauchy Problem Find Phys World Line Classical Electrodynamic Fundamental String 
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  1. 1.
    Tabor, M.: Chaos and Integrability in Non-Linear Dynamics. Wiley, New York (1989) Google Scholar
  2. 2.
    de Vega, J.H., et al.: Classical splitting of fundamental strings. Phys. Rev. D 52, 4609 (1995) CrossRefADSMathSciNetGoogle Scholar
  3. 3.
    Blasone, M., Jizba, P., Kleinert, H.: Quantum behavior of deterministic systems with information loss: Path integral approach. Ann. Phys. (N.Y.) 320, 468 (2005) MATHCrossRefADSMathSciNetGoogle Scholar
  4. 4.
    Gozzi, E.: Hidden BRS invariance in classical mechanics. Phys. Lett. 201, 525 (1988) MathSciNetGoogle Scholar
  5. 5.
    Gozzi, E., Reuter, M., Thacker, W.D.: Hidden BRS invariance in classical mechanics, II. Phys. Rev. D 40, 3363 (1989) CrossRefADSMathSciNetGoogle Scholar
  6. 6.
    Abrikosov, A.A. Jr., Gozzi, E., Mauro, D.: Geometric dequantization. Ann. Phys. (N.Y.) 317, 24 (2005), quant-ph/0406028 MATHCrossRefADSMathSciNetGoogle Scholar
  7. 7.
    Macomber, H.K.: Time reversal in classical mechanics: a paradox. Am. J. Phys. 40, 1339 (1972) CrossRefADSGoogle Scholar
  8. 8.
    Norton, J.D.: Causation as folk science. Philos. Impr. 3(4) (2003),
  9. 9.
    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 MATHCrossRefGoogle Scholar
  10. 10.
    Kosyakov, B.: Introduction to the Classical Theory of Particles and Fields. Springer, Berlin (2007), pp. 374–376 MATHGoogle Scholar
  11. 11.
    Kosyakov, B.P.: Holography and the origin of anomalies. Phys. Lett. B 492, 349 (2000), hep-th/0009071 MATHCrossRefADSMathSciNetGoogle Scholar
  12. 12.
    Kleinert, H.: Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 4th edn. World Scientific, Singapore (2006), Sect. 18.13; kleinert/reb5/psfiles/pthic18.pdf MATHGoogle Scholar
  13. 13.
    Lanczos, C.: The Variational Principles of Mechanics. University of Toronto Press, Toronto (1949) MATHGoogle Scholar
  14. 14.
    Brown, J.D., York, J.W. Jr.: Jacobi’s action and the recovery of time in general relativity. Phys. Rev. D 40, 3312 (1989) CrossRefADSMathSciNetGoogle Scholar
  15. 15.
    Polyakov, A.M.: Gauge Fields and Strings. Harwood, Chur (1987), Sect. 9.1 Google Scholar
  16. 16.
    Mottola, E.: Functional integration over geometries. J. Math. Phys. 36, 2470 (1995) MATHCrossRefADSMathSciNetGoogle Scholar
  17. 17.
    Thacker, W.D.: New formulation of the classical path integral with reparametrization invariance. J. Math. Phys. 38, 2389 (1997) MATHCrossRefADSMathSciNetGoogle Scholar

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© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Russian Federal Nuclear CenterSarovRussia

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