33. Schwinger-Keldysh: information carried by Keldysh Green’s functions (nonequilibrium diagrammatic methods 1)
DmitryLast time we introduced Keldysh Green’s functions and generating functional for non-equilibrium field theory of a single scalar field. Let us now discuss the information carried by them (and hopefully it will become clear why we need more than one Green’s function in the situation out of equilibrium).
For a free massive scalar field 
one has
(1)
(2)
(3)
Probably, the simplest way to see this is to use the WKB representation. For example, the limit 
of the 
Green’s function (neglecting the vacuum contribution) is given by
.
Recalling that 
we come to the expression (1).
After the Keldysh rotation one has
We conclude that the Keldysh Green function 
carries the information about distribution function 
, while the retarded and advanced Green functions define spectrum of particles (and are independent of the distribution function). This separation is only true for systems sufficiently close to the thermal equilibrium.
The reason why we need more than a single Green’s function is now clear. Away for equilibrium vacuum state gets excited, and the number of excitations in a given mode 
depends on time (how in particular? we will discuss it later). Advanced and retarded Green’s functions carry information only about the spectrum, so we need something else to describe dynamics of the system completely. This “something else” is the Keldysh Green’s function 
.
Let us discuss a bit the issue of the spectrum. In principle, a physical system is completely described by its Hamiltonian (and initial density matrix). The latter, generally speaking, contains information bout all the interactions between the relevant degrees of freedom in the system, and these interactions may be strong. How to understand which degrees of freedom are relevant for the dynamics? One needs to diagonalize the Hamiltonian, and this can be done only approximately in the general case. (Approximate) eigenstates of the Hamiltonian will correspond to relevant degrees of freedom - and that is what 
and 
- they show the basis of eigenstates of the diagonalized Hamiltonian (in the case of free field theory, as above, these states are Fourier harmonics).
Generally speaking, since diagonalization is not exact, the states may have some width (related to the strength of interactions). Only if their width is much smaller than the difference between the energies of nearest eigenstates, description in terms of separate degrees of freedom is possible.
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32. Eye on ArXiv: 30 Apr 2008 - Curvature perturbation from false vacuum inflation
DmitryJinn-Ouk Gong, Misao Sasaki, “Curvature perturbation spectrum from false vacuum inflation”
Another very nice and technical paper by Misao Sasaki and his collaborator. Let us recall how inflation developed in the old scenario introduced by Alan Guth back in 1981. Inflaton field was trapped in a local (false) minimum of its potential, so the Universe was exactly de Sitter-like during inflationary stage. Since the minimum was false (and true minimum corresponded to the Minkowski spacetime, very much in the spirit of KKLT or, better to say, KKLT was written very much in the spirit of old Guth’s scenario :-)), inflaton has to tunnel into the true minimum quantum mechanically (along classically fobidden Coleman-de Luccia trajectory). As we know now, in this scenario it is impossible to get a Universe with homogeneity and isotropy at the level we observe today, but let us forget about this issue for a moment…
If the inflaton is trapped in a local false minimum, how to calculate the spectrum of primordial perturbations which one observes in the end of de Sitter stage (after tunneling)? Naively, one has for the curvature perturbation
,
but inflaton does not move classically and 
. This is a well known situation in exact de Sitter space, where curvature perturbation 
is the pure gauge, since its effective action is proportional to the slow roll parameter 
and the latter is zero near the true minimum of the inflaton ponential. One may say that inflation is in the eternal regime of self-reproduction, which may end in some Hubble patches only by tunneling into Minkowski vacuum.
Let us model the end of de Sitter stage in different stage: by introducing a thermal correction to the inflaton mass: 
, where 
. At early stages 
is small and the effective mass is positive (inflaton is trapped near the minimum), while at late stages effective mass squared becomes negative, and the inflaton rapidly reaches the true minimum of the potential at 
.
The authors calculate exact QFT Green function of 
for such model and pair correlation function of Bardeen potential 
(from the corr. func. of energy-momentum tensor). Resulting power spectrum of the curvature perturbation turns out to be strongly blue: 
.
What is the main lesson of this paper? I guess, it is already learn by many: you generate curvature perturbation only if you explicitly breake down de Sitter invariance (for example, by introducing a time-dependent mass for the inflaton). In this case such things as 
deviate from zero (they are small w.r.t. a parameter which controls breakdown of de Sitter symmetry) and one can use old well known formulae to calculate curvature perturbation.
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