Real-Time Observables

  • Mikko Laine
  • Aleksi Vuorinen
Part of the Lecture Notes in Physics book series (LNP, volume 925)


Various real-time correlation functions are defined (Wightman, retarded, advanced, time-ordered, spectral). Their analytic properties are discussed, and general relations between them are worked out for the case of a system in thermal equilibrium. Examples are given for free scalar and fermion fields. A physically relevant spectral function related to a composite operator is analyzed in detail. The so-called real-time formalism is introduced, and it is shown how it can be used to compute the same spectral function that was previously determined with the imaginary-time formalism. The need for resummations in order to systematically determine spectral functions in weakly coupled systems is stated. The concept of Hard Thermal Loops (HTLs), which implement a particular resummation, is introduced. HTL-resummed gauge field and fermion propagators are derived. The main plasma physics phenomena that the HTL resummation captures are pointed out. A warning is issued that although necessary HTL resummation is in general not sufficient for obtaining a systematic weak-coupling expansion.


Wick rotation Time ordering Heisenberg operator Wightman function Retarded and advanced correlators Kubo-Martin-Schwinger relation Spectral representation Sum rule Analytic continuation Density matrix Schwinger-Keldysh formalism Hard Thermal Loops Landau damping Plasmon Plasmino Dispersion relation 


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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Mikko Laine
    • 1
  • Aleksi Vuorinen
    • 2
  1. 1.AEC, Institute for Theoretical PhysicsUniversity of BernBernSwitzerland
  2. 2.Department of PhysicsUniversity of HelsinkiHelsinkiFinland

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