Off-Shell Transport Dynamics

  • Tamás Sándor BiróEmail author
  • Antal Jakovác
Part of the SpringerBriefs in Physics book series (SpringerBriefs in Physics)


In Chap.  1 we discussed examples of quantum uncertainty, and also the energy variance and its resemblance to a temperature when special relativity is taken into account. We have demonstrated that the complex scalar free field theory, when viewed in terms of amplitude and phase variables, exhibits a coupling between the off-mass-shell relation of the classical four-momentum and the quantum scale variation of the magnitude of the quantum probability density. The classical dispersion relation \(P_iP^i-(mc)^2=0\) is no longer valid, unless one deals with plane waves of constant amplitude in space and time.


  1. 1.
    E. Wigner, On the quantum correction for thermodynamic equilibrium. Phys. Rev. 40, 749 (1932)ADSCrossRefGoogle Scholar
  2. 2.
    M. Hillery, R.F. O’Connell, M.O. Scully, E.P. Wigner, Distribution functions in physics: fundamentals. Phys. Rep. 106, 121 (1984)Google Scholar
  3. 3.
    W.B. Case, Wigner functions and Weyl transforms for pedestrians. Am. J. Phys. 76, 937 (2008)ADSCrossRefGoogle Scholar
  4. 4.
    W.P. Schleich, Quantum Optics in Phase Space (Wiley-VCH, Weinheim, 2001)CrossRefGoogle Scholar
  5. 5.
    S. Varro, P. Adam, T.S. Biro, G.G. Barnafoldi, P. Levai, Wigner 111. Colorful and deep; scientific symposium, in EPJ Web of Conferences, vol. 78 (2014), pp. 00001–08002Google Scholar
  6. 6.
    M.A. Manko, V.I. Manko, Probability description and entropy of classical and quantum systems. Found. Phys. 41, 330 (2011)Google Scholar
  7. 7.
    G. Baym, L.P. Kadanoff, Conservation laws and correlation functions. Phys. Rev. 124, 287 (1961)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    L.P. Kadanoff, G. Baym, Quantum Statistical Mechanics (Addison-Wesley, Boston, 1989)Google Scholar
  9. 9.
    K. Balzer, M. Bonitz, Nonequilibrium Green’s Functions Approach to Inhomogeneous Systems. Springer Lecture Notes in Physics, vol. 867 (Springer, Berlin, 2013)CrossRefGoogle Scholar
  10. 10.
    A.A. Vlasov, On vibrational properties of electron gas. J. Exp. Theor. Phys. 8, 291 (1938)Google Scholar
  11. 11.
    A.A. Vlasov, Many Particle Theory and Its Applications to Plasma (Gordon and Breach, New York, 1961)Google Scholar
  12. 12.
    L.V. Keldysh, Zh. Eksp. Teor. Fiz. 47, 1515 (1964) (in Russian); Sov. Phys. JETP 20, 1018 (1965)Google Scholar

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© The Author(s), under exclusive licence to Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.H.A.S. Wigner Research Centre for PhysicsBudapestHungary
  2. 2.Institute of PhysicsRoland Eötvös UniversityBudapestHungary

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