Field In De Sitter Space Research Articles

Group theoretical approach to quantum fields in de Sitter space II. The complementary and discrete series

We use an algebraic approach based on representations of de Sitter group to construct covariant quantum fields in arbitrary dimensions. We study the complementary and the discrete series which correspond to light and massless fields and which lead new feature with respect to the massive principal series we previously studied (hep-th/0606119). When considering the complementary series, we make use of a non-trivial scalar product in order to get local expressions in the position representation. Based on these, we construct a family of covariant canonical fields parametrized by SU(1,1)/U(1). Each of these correspond to the dS invariant alpha-vacua. The behavior of the modes at asymptotic times brings another difficulty as it is incompatible with the usual definition of the in and out vacua. We propose a generalized notion of these vacua which reduces to the usual conformal vacuum in the conformally massless limit. When considering the massless discrete series we find that no covariant field obeys the canonical commutation relations. To further analyze this singular case, we consider the massless limit of the complementary scalar fields we previously found. We obtain canonical fields with a deformed representation by zero modes. The zero modes have a dS invariant vacuum with singular norm. We propose a regularization by a compactification of the scalar field and a dS invariant definition of the vertex operators. The resulting two-point functions are dS invariant and have a universal logarithmic infrared divergence.

Group theoretical approach to quantum fields in de Sitter space, I. The principal series

Using unitary irreducible representations of the de Sitter group, we construct the Fock space of a massive free scalar field. In this approach, the vacuum is the unique dS invariant state. The quantum field is a posteriori defined by an operator subject to covariant transformations under the dS isometry group. This insures that it obeys canonical commutation relations, up to an overall factor which should not vanish as it fixes the value of hbar. However, contrary to what is obtained for the Poincare group, the covariance condition leaves an arbitrariness in the definition of the field. This arbitrariness allows to recover the amplitudes governing spontaneous pair creation processes, as well as the class of alpha vacua obtained in the usual field theoretical approach. The two approaches can be formally related by introducing a squeezing operator which acts on the state in the field theoretical description and on the operator in the present treatment. The choice of the different dS invariant schemes (different alpha vacua) is here posed in very simple terms: it is related to a first order differential equation which is singular on the horizon and whose general solution is therefore characterized by the amplitude on either side of the horizon. Our algebraic approach offers a new method to define quantum field theory on some deformations of dS space.

EXCITED STATES OF DE SITTER SPACE SCALAR FIELDS: LATTICE SCHRÖDINGER PICTURE

In lattice Schrödinger picture, the wave functionals of a free real scalar field in de Sitter space for the Bunch–Davies vacuum and the corresponding excited states are given explicitly in momentum space for the following coupling cases: (i) massless conformal couplng, (ii) massless minimal coupling, and (iii) massive minimal coupling. Implications of these solutions for inflationary scenarios in cosmology are also discussed.

Localized particle states and dynamic gravitational effects

Scalar particles--i.e., scalar-field excitations--in de Sitter space exhibit behavior unlike either classical particles in expanding space or quantum particles in flat spacetime. Their energies oscillate forever, and their interactions are spread out in energy. Here it is shown that these features characterize not only normal-mode excitations spread out over all space, but localized particles or wave packets as well. Both one-particle and coherent states of a massive, minimally coupled scalar field in de Sitter space, associated with classical wave packets, are constructed explicitly. Their energy expectation values and corresponding Unruh-DeWitt detector response functions are calculated. Numerical evaluation of these quantities for a simple set of classical wave packets clearly displays these novel features. Hence, given the observed accelerating expansion of the Universe, it is possible that observation of an ultralow-mass scalar particle could yield direct confirmation of distinct predictions of quantum field theory in curved spacetime.

CPT groups for spinor field in de Sitter space

A group structure of the discrete transformations (parity, time reversal and charge conjugation) for spinor field in de Sitter space are studied in terms of extraspecial finite groups. Two CPT groups are introduced, the first group from an analysis of the de Sitter–Dirac wave equation for spinor field, and the second group from a purely algebraic approach based on the automorphism set of Clifford algebras. It is shown that both groups are isomorphic to each other.

Fields of accelerated sources: Born in de Sitter

This paper deals thoroughly with the scalar and electromagnetic fields of uniformly accelerated charges in de Sitter space–time. It gives details and makes various extensions of our Physical Review Letter from 2002. The basic properties of the classical Born solutions representing two uniformly accelerated charges in flat space–time are first summarized. The worldlines of uniformly accelerated particles in de Sitter universe are defined and described in a number of coordinate frames, some of them being of cosmological significance, the others are tied naturally to the particles. The scalar and electromagnetic fields due to the accelerated charges are constructed by using conformal relations between Minkowski and de Sitter space. The properties of the generalized “cosmological” Born solutions are analyzed and elucidated in various coordinate systems. In particular, a limiting procedure is demonstrated which brings the cosmological Born fields in de Sitter space back to the classical Born solutions in Minkowski space. In an extensive Appendix, which can be used independently of the main text, nine families of coordinate systems in de Sitter space–time are described analytically and illustrated graphically in a number of conformal diagrams.

Short distance and initial state effects in inflation: Stress tensor and decoherence

We present a consistent low energy effective field theory framework for parameterizing the effects of novel short distance physics in inflation, and their possible observational signatures in the Cosmic Microwave Background. We consider the class of general homogeneous, isotropic initial states for quantum scalar fields in Robertson-Walker (RW) spacetimes, subject to the requirement that their ultraviolet behavior be consistent with renormalizability of the covariantly conserved stress tensor which couples to gravity. In the functional Schr\"odinger picture such states are coherent, squeezed, mixed states characterized by a Gaussian density matrix. This Gaussian has parameters which approach those of the adiabatic vacuum at large wave number, and evolve in time according to an effective classical Hamiltonian. The one complex parameter family of $\alpha$ squeezed states in de Sitter spacetime does not fall into this UV allowed class, except for the special value of the parameter corresponding to the Bunch-Davies state. We determine the finite contributions to the inflationary power spectrum and stress tensor expectation value of general UV allowed adiabatic states, and obtain quantitative limits on the observability and backreaction effects of some recently proposed models of short distance modifications of the initial state of inflation. For all UV allowed states, the second order adiabatic basis provides a good description of particles created in the expanding RW universe. Due to the absence of particle creation for the massless, minimally coupled scalar field in de Sitter space, there is no phase decoherence in the simplest free field inflationary models. We apply adiabatic regularization to the renormalization of the decoherence functional in cosmology to corroborate this result.

On the stability of a self-similar spherical bubble of a scalar Higgs field in de Sitter space

An exact generalized discontinuous solution of the spherical-bubble type is obtained for a scalar Higgs field in de Sitter space. It is shown that the radius of such a generalized bubble evolves in accordance with one of the exact solutions to a dynamical problem considered previously for the bubble radius in the thin-wall approximation, where the bubble-wall thickness is negligible in relation to the bubble radius. Both the generalized solution and the self-similar bubble-type solution that was obtained earlier for the Higgs field in de Sitter space are studied for stability: it is shown that the former is stable, while the latter is unstable in this space. A physical interpretation of the reasons for the instability of the self-similar solution is given.

Discrete symmetries for spinor field in de Sitter space

Discrete symmetries, parity, time reversal, antipodal, and charge conjugation transformations for spinor field in de Sitter space, are presented in the ambient space notation, i.e., in a coordinate independent way. The PT and PCT transformations are also discussed in this notation. The five-current density is studied and their transformation under the discrete symmetries is discussed.

A NATURAL RENORMALIZATION OF THE ONE-LOOP EFFECTIVE ACTION FOR SCALAR FIELD IN CURVED SPACETIME

It has been shown that the negative norm states necessarily appear in a covariant quantization of the free minimally coupled scalar field in de Sitter space.1,2 In this process, ultraviolet and infrared divergences have been automatically eliminated.3 A natural renormalization of the one-loop interacting quantum field in Minkowski spacetime (λϕ4 theory) has been achieved through the consideration of the negative norm states.4 The one-loop effective action for scalar field in a general curved space-time has been calculated by this method, and a natural renormalization procedure in the one-loop approximation has been established.

High order correlation functions for a self-interacting scalar field in de Sitter space

We present the expressions of the three- and four-point correlation functions of a self-interacting light scalar field in a de Sitter spacetime at tree order, respectively, for a cubic and a quartic potential. Exact expressions are derived and their limiting behavior on superhorizon scales are presented. Their essential features are shown to be similar to those obtained in a classical approach.

NEGATIVE NORM STATES IN DE SITTER SPACE AND QFT WITHOUT RENORMALIZATION PROCEDURE

In recent papers,1,2 it has been shown that the presence of negative norm states or negative frequency solutions are indispensable for a fully covariant quantization of the minimally coupled scalar field in de Sitter space. Their presence, while leaving unchanged the physical content of the theory, offers the advantage of eliminating any ultraviolet divergence in the vacuum energy2 and infrared divergence in the two point function.3 We attempt here to extend this method to the interacting quantum field in Minkowski space-time. As an illustration of the procedure, we consider the λϕ4 theory in Minkowski space-time. The mathematical consequences of this method is the disappearance of the ultraviolet divergence to the one-loop approximation. This means, the effect of these auxiliary negative norm states is to allow an automatic renormalization of the theory in this approximation.

Initial Conditions for Inflation

Free scalar fields in de Sitter space have a one-parameter family of states invariant under the de Sitter group, including the standard thermal vacuum. We show that, except for the thermal vacuum, these states are unphysical when gravitational interactions are included. We apply these observations to the quantum state of the inflaton, and find that, at best, dramatic fine tuning is required for states other than the thermal vacuum to lead to observable features in the CMBR anisotropy.

COVARIANT TWO-POINT FUNCTION FOR MINIMALLY COUPLED SCALAR FIELD IN de SITTER SPACE–TIME

In a recent paper,1 it has been shown that negative norm states are indispensable for a fully covariant quantization of the minimally coupled scalar field in de Sitter space. Their presence, while leaving unchanged the physical content of the theory, offers an automatic and covariant renormalization of the vacuum energy divergence. This paper is a completion of our previous work. An explicit construction of the covariant two-point function of the "massless" minimally coupled scalar field in de Sitter space is given, which is free of any infrared divergence. The associated Schwinger commutator function and retarded Green's function are calculated in a fully gauge-invariant way, which also means coordinate independent.

Inflation with a Planck-scale frequency cutoff

The implementation of a Planck-scale high frequency and short wavelength cutoff in quantum theories on expanding backgrounds may have potentially nontrivial implications, such as the breaking of local Lorentz invariance and the existence of a yet unknown mechanism for the creation of vacuum modes. In scenarios where inflation begins close to the cutoff scale, these effects could have observable consequences as trans-Planckian modes are redshifted to cosmological scales. In close analogy with similar studies of Hawking radiation, a simple theory of a minimally coupled scalar field in de Sitter space is studied, with a high frequency cutoff imposed by a nonlinear dispersion relation. Under certain conditions the model predicts deviations from the standard inflationary scenario. We also comment on the difficulties in generalizing fluid models of Hawking radiation to cosmological space-times.

Gupta-Bleuler quantization for minimally coupled scalar fields in de Sitter space

We present a fully covariant quantization of the minimally coupled massless field on de Sitter space, thanks to a new representation of the canonical commutation relations. We thus obtain a formalism free of any infrared divergence. Our method is based on a rigorous group-theoretical approach combined with a suitable adaptation (Krein spaces) of the Wightman-Gärding axiomatic for massless fields (Gupta-Bleuler scheme). We make explicit the correspondence between unitary irreducible representations of the de Sitter group and the field theory on de Sitter spacetime. The minimally coupled massless field is associated with a representation which is the lowest term of the discrete series of unitary representations of the de Sitter group. In spite of the presence of negative-norm modes in the theory, no negative energy can be measured: expressions such as nk1nk2 ... |T00|nk1nk2 ... are always positive.

“Massive” vector field in de Sitter space

We present in this paper a covariant quantization of the “massive” vector field on de Sitter (dS) space based on analyticity in the complexified pseudo-Riemanian manifold. The correspondence between unitary irreducible representations of the de Sitter group and the field theory on de Sitter space–time is essential in our approach. We introduce the Wightman two-point function for the case of generalized free vector fields on de Sitter space. This function satisfies the conditions of (a) positiveness, (b) locality, (c) covariance, (d) normal analyticity, (e) transversality and (f) divergencelessness. The Hilbert space structure and the unsmeared field operators Kα(x) are also defined. This work is in the direct continuation of previous ones concerning the scalar and spinor cases.

An exact solution of a quark field coupled with a Yang - Mills field in de Sitter space

An exact solution of a quark field coupled to a Yang - Mills condensate in de Sitter space is derived. The colour helicity states have been introduced to facilitate calculation.

HOW HOT IS THE DE SITTER SPACE?

We show that the unique invariant locally Minkowskian state of quantum fields in de Sitter space M has for an observer moving along with a Killing vector field a temperature [Formula: see text] where R is the radius of M and a his acceleration. States with another temperature cannot be locally Minkowskian all over M.

Stochastic dynamics of large-scale inflation in de Sitter space.

In this paper we derive exact quantum Langevin equations for stochastic dynamics of large-scale inflation in de Sitter space. These quantum Langevin equations are the equivalent of the Wigner equation and are described by a system of stochastic differential equations. We present a formula for the calculation of the expectation value of a quantum operator whose Weyl symbol is a function of the large-scale inflation scalar field and its time derivative. The quantum expectation value is calculated as a mathematical expectation value over a stochastic process in an extended phase space, where the additional coordinate plays the role of a stochastic phase. The unique solution is obtained for the Cauchy problem for the Wigner equation for large-scale inflation. The stationary solution for the Wigner equation is found for an arbitrary potential. It is shown that the large-scale inflation scalar field in de Sitter space behaves as a quantum one-dimensional dissipative system, which supports the earlier results of Graziani and of Nakao, Nambu, and Sasaki. But the analogy with a one-dimensional model of the quantum linearly damped anharmonic oscillator is not complete: the difference arises from the new time-dependent commutation relation for the large-scale field and its time derivative. It is found that, for the large-scale inflation scalar field, the large time asymptotics is equal to the ``classical limit.'' For the large time limit the quantum Langevin equations are just the classical stochastic Langevin equations (only the stationary state is defined by the quantum field theory). \textcopyright{} 1996 The American Physical Society.