Background Field Method and Initial-Time Singularity for Coherent States

Abstract

The background field method is adopted for studying the dynamics of coherent states within an interacting scalar field theory. Focusing on a coherent state that corresponds to the homogeneous condensate, the quantum depletion of the expectation value of the field-operator is demonstrated to be due to the annihilation of the condensate constituents into relativistic quanta. Moreover, due to the fact that the initial field acceleration and energy for the nonsqueezed coherent states are determined in terms of bare coupling constant, instead of the renormalized one, the appearance of perturbative singularities is shown to be inevitable. In other words, consistency of these states requires the finiteness of the bare coupling constant, through the resummation.

De Sitter space as BRST invariant coherent state of gravitons

“…Next we are interested in computing the leading quantum corrections to the dynamics of 1-point expectation values for the gauge sector, resulting from the quantum pair creation of scalar particles. As it was demonstrated explicitly in [21], the relevant dynamics follows from ( 26)-( 29) (albeit with additional classical source) by retaining only the expectation values of bilinear operators in S.…”

“…They can be built around the vacuum of the theory by the action of the field displacement operator (see e.g. [20,21]), written in terms of canonical degrees of freedom without invoking their asymptotic representation. In particular, the coherent state for a complex scalar field takes the following form…”

“…It must be stressed that the order in which the exponentials appear in this definition of the coherent state is necessary for reproducing (21). The consistency of this state requires the transversality of E c j , thus (22) will be maintained; the only modification being the generation of the nonvanishing expectation value for µ .…”

“…In the absence of S 0 and S j these are classical equations of motions for the theory at hand. The computation of these quantum terms requires the knowledge of 2-point and 3-point functions, however we could instead follow [21] evaluating them explicitly in the coherent state up to a desirable order in and g. The explicit form of the equation for the scalar field will be useful when discussing the particle production and will be given when relevant.…”

“…In other words, even though the dynamics is given by the background dependent equation of motion, the initial conditions for the mode functions are the ones in Minkowski vacuum. See [20,21] for the discussion of this point and implications for perturbative dynamics.…”