Abstract
We derive the stochastic description of a massless, interacting scalar field in de Sitter space directly from the quantum theory. This is done by showing that the density matrix for the effective theory of the long wavelength fluctuations of the field obeys a quantum version of the Fokker-Planck equation. This equation has a simple connection with the standard Fokker-Planck equation of the classical stochastic theory, which can be generalised to any order in perturbation theory. We illustrate this formalism in detail for the theory of a massless scalar field with a quartic interaction.
Highlights
JHEP11(2017)065 expressions for the n-point functions in these theories, a simpler, finite behaviour should emerge that matches directly with the stochastic predictions, for example, in the late-time, static limit
We derive the stochastic description of a massless, interacting scalar field in de Sitter space directly from the quantum theory
This is done by showing that the density matrix for the effective theory of the long wavelength fluctuations of the field obeys a quantum version of the Fokker-Planck equation
Summary
In a theory of a massless, interacting scalar field, Φ(t, x), the simplest quantities that we could calculate are the n-point functions where all the fields are evaluated at exactly the same space-time point and at some suitably late time, Φn(t, x) ≡ lim Ω(t)|Φn(t, x)|Ω(t). When we differentiate ΦnL(t, x) with respect to the time and use the appropriate quantum form of the Fokker-Planck equation, we are led to the recursion relation9 This time we have written the result for the particular case of a quartic interaction, rather than for a general polynomial potential. We must first solve for the wave-functional and the corresponding density matrix of our full theory, which includes both the long and short wavelength parts of the field. For this purpose, the Schrodinger picture is the best suited, as we shall see. We turn next to the case of a scalar field theory with a quartic interaction
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