Restoration of spontaneously broken continuous symmetries in de Sitter spacetime
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Abstract:
We formulate a functional approach to scalar quantum field theory in (n+1)-dimensional de Sitter spacetime and solve the functional Schr\"odinger equation for the conformally and minimally coupled scalar fields in both the k=0 and k=1 gauges. We show that there is a natural initial condition, the requirement that the field energy remain finite as the scale factor a becomes small, which specifies a unique, time-dependent, de Sitter vacuum state. This initial condition is closely related to Hawking's prescription of including in the functional integral only those field configurations which are regular on the Euclidean section. The Green's functions constructed using this initial condition are explicitly shown to be the analytic continuation of those derived using the Euclidean path-integral formalism and the regularity (boundary) condition. These Green's functions are used to study the Hawking effect and the restoration of continuous symmetries. In particular we study the restoration of a broken O(2) symmetry of a ${\ensuremath{\Phi}}^{4}$ theory. We argue that spontaneously broken continuous symmetries are always dynamically restored in de Sitter spacetime.Keywords:
Anti-de Sitter space
Scalar field theory
In 1917, de Sitter used the modified Einstein equation and proposed a model of the Universe without physical matter, but with a cosmological constant. De Sitter geometry, as well as Minkowski geometry, is maximally symmetrical. However, de Sitter geometry is better suited to describe gravitational fields. It is believed that the real Universe was described by the de Sitter model in the very early stages of expansion (inflationary model of the Universe). This article is devoted to the problem of classification of regular curves on the de Sitter space. As a model of the de Sitter plane, the upper half-plane on which the metric is given is chosen. For this purpose, an algebra of differential invariants of curves with respect to the motions of the de Sitter plane is constructed. As it turned out, this algebra is generated by one second-order differential invariant (we call it by de Sitter curvature) and two invariant differentiations. Thus, when passing to the next jets, the dimension of the algebra of differential invariants increases by one. The concept of regular curves is introduced. Namely, a curve is called regular if the restriction of de Sitter curvature to it can be considered as parameterization of the curve. A theorem on the equivalence of regular curves with respect to the motions of the de Sitter plane is proved. The singular orbits of the group of proper motions are described.
Anti-de Sitter space
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Following arXiv:1012.2107 we show that in global de Sitter space its isometry is broken by the loop IR divergences for any invariant vacuum state of the massive scalars. We derive kinetic equation in global de Sitter space. It follows from the Dyson-Schwinger equation of the Schwinger-Keldysh diagrammatic technique in IR limit and allows to understand the physical meaning and consequences of the loop IR divergences. In many respects the isometry breaking in global dS is similar to the one in the contracting Poincare patch of de Sitter space. Hence, as a warm up exercise we study the kinetic equation and properties of its solutions in the expanding and contracting Poincare patches of de Sitter space. Quite unexpectedly we find that under some initial conditions there is an explosive production of massive particles in the expanding Poincare patch.
Anti-de Sitter space
Isometry (Riemannian geometry)
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Semiclassical physics
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Exact solutions to the scalar wave equation are given on a de Sitter (dS) and anti-de Sitter (AdS) background with static metric, for arbitrary dimensions d. In particular in the case of dS space the boundary conditions at the horizon are in agreement with the Hawking effect (1975) and the solutions form a complete set of physical modes.
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Bulk reconstruction formulas similar to HKLL are obtained for de Sitter and anti-de Sitter spaces as the inverse Gel'fand Graev Radon transform. While these generalize our previous result on the Euclidean anti-de Sitter space, their validity in here is restricted only to odd dimensions in both instances. The exact Wightman function for the de Sitter space is then derived analytically. The GGR transform fixes the coefficient of the Wightman function. For the anti-de Sitter space it is shown that a reconstruction formula exists for the case of time-like boundary as well. The restriction on the domain of integration on the boundary is derived. As a special case, we point out that the formula is valid for the BTZ black hole as well.
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BTZ black hole
Radon transform
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The generic feature of non-conformal fields in Poincaré patch of de Sitter space is the presence of large IR loop corrections even for massive fields. Moreover, in global de Sitter there are loop IR divergences for the massive fields. Naive analytic continuation from de Sitter to Anti-de-Sitter might lead one to conclude that something similar should happen in the latter space as well. However, we show that there are no large IR effects in the one-loop two-point functions in the Poincaré patch of Anti-de-Sitter space even for the zero mass minimally coupled scalar fields. As well there are neither large IR effects nor IR divergences in global Anti-de-Sitter space even for the zero mass.
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Cosmological constant
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