Path Integrals

  • Roberto Mauri
Part of the Soft and Biological Matter book series (SOBIMA)


In the previous chapters we saw that stochastic processes can be described using two equivalent approaches, one Lagrangian, leading to the Langevin equation, the other Eulerian, exemplified in the Fokker-Planck equation. Both descriptions allow to determine the stochastic properties of a system, provided that these properties are known at an earlier time. In addition, though, a third approach exists, where the evolution of a system in time is formulated by writing down the probability of observing a trajectory, or “path”, of its macroscopic variables. The first successful attempt to define path integration is due to Norbert Wiener, who in 1921 replaced the classical notion of a single, unique trajectory of a Brownian particle with a sum, or functional integral, over an infinity of possible trajectories, to compute the probability distribution describing the diffusion process (Wiener, Proc. Natl. Acad. Sci. USA 7:253, 294, 1921). In a second development, this concept was applied to quantum mechanics, first by Paul Dirac (Phys. Z. Sowjetunion 3, 1932) and then by Richard Feynman (Rev. Mod. Phys. 20:367, 1948), expressing the propagator of the Schrödinger equation in terms of a complex-valued path integral.


  1. 1.
    Chalchian, M., Demichev, A.P., Demichev, A.: Path Integrals in Physics: Stochastic Processes and Quantum Mechanics. Taylor & Francis, New York (2001) Google Scholar
  2. 2.
    Dirac, P.A.M.: Phys. Z. Sowjetunion 3 (1932) Google Scholar
  3. 3.
    Feynman, R.P.: Rev. Mod. Phys. 20, 367 (1948) MathSciNetADSCrossRefGoogle Scholar
  4. 4.
    Feynman, R.P., Hibbs, A.R.: Quantum Mechanics and Path Integrals. McGraw-Hill, New York (1965) zbMATHGoogle Scholar
  5. 5.
    Gelfand, I.M., Yaglom, A.M.: J. Math. Phys. 1, 48 (1960) ADSCrossRefGoogle Scholar
  6. 6.
    Graham, R.: Phys. Rev. Lett. 38, 51 (1977) MathSciNetADSCrossRefGoogle Scholar
  7. 7.
    Onsager, L., Machlup, S.: Phys. Rev. 91, 1505 (1953) MathSciNetADSzbMATHCrossRefGoogle Scholar
  8. 8.
    Schulman, L.S.: Techniques and Applications of Path Integration. Wiley, New York (1981) zbMATHGoogle Scholar
  9. 9.
    Wiegel, F.W.: Introduction to Path-Integral Methods in Physics and Polymer Science. Word Scientific, Singapore (1986) CrossRefGoogle Scholar
  10. 10.
    Wiener, N.: Proc. Natl. Acad. Sci. USA 7, 253 (1921), also see p. 294 ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Roberto Mauri
    • 1
  1. 1.Department of Chemical Engineering, Industrial Chemistry and Material ScienceUniversity of PisaPisaItaly

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