Field In De Sitter Space Research Articles

Creation of spin-1/2 particles by an electric field in de Sitter space.

In the present article we solve the Dirac equation in a de Sitter universe when a constant electric field is present. Using the Bogoliubov transformations, we compute the rate of spin 1/2 created particles by the electric field. We compare our results with the scalar case. We also analyze the behavior of the density of particles created in the limit H=0, when de Sitter background reduces to a flat space-time.

Hawking radiation in de Sitter space of arbitrary dimension

The Hawking radiation for a quantised massless conformal scalar field in de Sitter space of arbitrary dimension is studied. Two cases are considered, the global chart and the spatially flat chart. In each case, the author discussed the thermal properties of the Bogoliubov transformation to the theory on the static chart without any approximation. It is proved that the vacuum states defined on two charts have the particles which exhibit the exact thermal distribution with respect to that on the static chart.

Excited-state spectra of de Sitter-space scalar fields.

To illustrate gravitational effects on the dynamics of a quantized field, the spectrum of excited states available to a linear scalar field in de Sitter space is examined in detail. Explicit Schr\"odinger-picture wave functionals are obtained for the excitation-number eigenstates of the familiar Fockspace description. The field energies of these states are calculated from expectation values of the appropriate Hamiltonian. The Euclidean vacuum state is seen to be the lowest-energy de Sitter-invariant state, although for any massive or nonconformally coupled field instantaneous Hamiltonian diagonalization, breaking de Sitter invariance, yields states of lower energy. All other de Sitter-invariant vacua are characterized by uniform excitation, relative to the Euclidean vacuum, in all field modes. Associated with any vacuum state is a Fock-space basis of excited states. These have field energies in integral increments above the vacuum; i.e., they represent the quantized excitations of the field's normal modes. The energy increments---the (renormalization-independent) energies of individual "particles"---differ markedly from the classical normal-mode frequencies of the field. For fields with combined mass and curvature coupling above a certain threshold, "particle" energies oscillate at late and early times rather than approach a fixed limit; for fields with strong curvature coupling these energies can even become negative. They show exponential rather than oscillatory time dependence if the combined mass and curvature coupling is below the threshold. Excitation energies of a massless, conformally coupled field have the same time dependence as the energy of a classical relativistic particle, but for other fields with the same mass-curvature-coupling sum, "particle" energies have an amplified component, which grows as the radius of the space. The interaction between a field in an excited state and a monopole "detector" is also calculated: Each excitation gives, above the vacuum signal, a finite response with a finite width in energy. These effects on "particle" energies and interactions are due to modulation of the field normal modes in a time-dependent spacetime metric, so similar effects should occur in general spacetimes.

Making inflation work: Damping of density perturbations due to Planck energy cutoff.

In this paper we propose an alternative method for the computation of classical density perturbations from a quantum field in an inflationary scenario. We compute the power spectrum of density perturbations directly from vacuum fluctuations of the ``time-time'' component of the energy-momentum tensor. We compute the inhomogeneous part of the correlation function 〈0\ensuremath{\Vert}${T}_{0}^{0}$(x,t) ${T}_{0}^{0}$(y,t)\ensuremath{\Vert}0〉 for a massless minimally coupled scalar field in de Sitter space. The Fourier transform of this two-point function leads to the scale-invariant spectrum of perturbations, but is ultraviolet divergent. This expression can be made finite by introducing an (ad hoc) small-distance cutoff in the proper length. We argue that this cutoff should be of the order of the Planck length, and show that, in such a case, the density fluctuations have the acceptable magnitude (\ensuremath{\sim}${10}^{\mathrm{\ensuremath{-}}4}$) for the case of primordial inflation. Thus the inflationary scenario can be made to work without any fine-tuning.

Stochastic inflation as a time-dependent random walk.

This paper exploits the ``stochastic inflation'' paradigm introduced by Starobinskii to study the evolution of long-wavelength modes for a free scalar field \ensuremath{\Phi} in an inflationary Universe. By relaxing the assumption of a ``slow roll,'' it becomes obvious that the well-known late-time infrared divergence of the vacuum for a massless field in de Sitter space may be viewed as a consequence of the fluctuation-dissipation theorem. This stochastic model is also extended to allow for nonvacuum states and power-law inflation, situations where the fluctuation-dissipation theorem no longer holds. One recovers the correct late-time form for the expectation value 〈${\ensuremath{\Phi}}^{2}$〉 in these cases as well, corroborating thereby the intuitive picture that, quite generally, the long-wavelength modes of the field evolve in a thermal ``bath'' provided by the shorter-wavelength modes.

Massless fields in the static de Sitter space: Exact solutions and choice of the vacuum states

Investigations of quantum fields in the de Sitter space have a long history (a bibliography of studies made up to 1981 can be found in the monograph of [i]). Being a curved sPace with maximal symmetry, the de Sitter space makes it possible to construct explicitly a quantum theory of free fields and also some models of interacting fields and investigate effects associated with the influence of the gravitational field on quantum processes. The de Sitter space is an example of a manifold that possesses an event horizon -the cosmological horizon. In the behavior of quantum fields in this space there is an organic intertwining of thermal effects (due to the finiteness of the Hawking temperature [2] associated with the cosmological horizon) with the effects of acceleration and curvature. It is no surprise that problems of quantum field theory in de Sitter space, which are of fundamental interest, have long attracted attention. In recent years a new stimulus to such investigations has been given by the appearance of the theory of an inflationary universe [3,4].

Green's functions of scalar field in de Sitter space: Discrete lattice formalism.—I

We derive, in discrete lattice formalism, the Green's function of free scalar field in (1+D)-dimensional de Sitter space. ForD=1 andD=3 cases our results agree with previous work of Bunch and Davies by conventional field-theoretical approach. We describe how our treatment may be extended to deal with systems with self-interaction and internal symmetry.

Quantum probability distributions in the early Universe. III. A geometric representation of stochastic systems.

We present a geometric interpretation of the dynamics of quantum stochastic systems obeying the Wigner equation. We apply this representation to the quantum fluctuations of the homogeneous long-range free scalar field in de Sitter space (m<, , , ...) space. Up to initial conditions, each trajectory uniquely characterizes the evolution of the system. The stationary states of the system act like fixed points or attractors in the moment space. We present the evolution of the scalar field as a flow in the moment space and show that the Bunch-Davies vacuum is the fixed point.

Non-gaussian density perturbations in inflationary cosmologies

The primordial mass density fluctuations may have arisen from quantum fluctuations in a (massless) scalar field that occurred during an inflationary era. We show that it is possible for primordial mass density fluctuations, which arose in this way, to be highly non-gaussian. We also show that the “bad” infrared properties of the propagator for a massless scalar field in de Sitter space can translate itself into a power spectrum, for the two-point spatial correlation of objects that do not trace the mass, which behaves like k−3, at small wave numbers k.

Massless minimally coupled scalar field in de Sitter space.

We quantize the massless minimally coupled scalar field in de Sitter space, and find a one-complex-parameter family of O(4)-invariant Hadamard Fock vacua which break de Sitter invariance. The different Fock spaces corresponding to the different choices of vacuum are subspaces of a single space of states. We make some remarks about the existence of E(3)- and O(1,3)-invariant Fock vacuum states.

Flat-space singletons.

Singletons exist, as particles and as local fields, only in 3+2 de Sitter space. Their kinematical properties make them natural candidates for constituents of massless fields, and perhaps for quarks. It is interesting to find out how to describe this type of compositeness in flat space. A theory of interacting singleton fields in de Sitter space is now available, and in this paper we study the flat-space limit of the Green's functions of that theory. The flat-space limit is an autonomous theory of Green's functions, but is not an operator field theory. The three-point function is calculated and its flat-space limit is found to reveal glimpses of a physical interpretation. Causal and spectral properties are in accord with the tenets of axiomatic field theory. The theory is a generalization of local field theory, in which photons appear as composite objects although the physical S matrix is the same as in conventional QED.

Quantum instabilities and the cosmological constant

Several recent studies have found that the stress-energy of quantum fields in de Sitter space will take the form of a growing effective cosmological constant (Λ) with sign opposite to that of the background spacetime. This leads, in self-consistent scheme, to the spontaneous decay of the effective value of Λ, and has been proposed as a possible solution to the “problem of the cosmological constant”. By modeling the back-reaction of the spacetime to the quantum-stress-energy, it is shown that it is unlikely that such quantum instabilities can lower the value of Λ by a large factor and yield a universe even remotely like our own.

Vacuum states in de Sitter space.

We examine possible vacuum states for scalar fields in de Sitter space, concentrating on those states (1) invariant under the de Sitter group O(1,4) or (2) invariant under one of its maximal subgroups E(3). For massive fields there is a one-complex-parameter family of de Sitter-invariant states, which includes the ``Euclidean'' vacuum state as a special case. We show these states are generated from the Euclidean vacuum by a frequency-independent Bogoliubov transformation, and obtain formulas for the symmetric, antisymmetric, and Feynman functions. In the massless minimally coupled case we prove that there exists no de Sitter-invariant Fock vacuum state. However one can find Fock states which are E(3) invariant. These states include the Bunch-Davies and Ottewill-Najmi vacua as special cases.

The de Sitter vacuum

We show that the vacuum for a scalar field in de Sitter space is not unique, but rather forms a one-parameter family. We examine the response of a particle detector in the general vacuum.

Quantum fluctuations in the new inflationary universe

The evolution of quantum fluctuations of a scalar field in de Sitter space is analyzed in the context of the new inflationary scenario. The duration of the inflationary phase is estimated and the problem of density perturbations resulting from quantum fluctuations of the Higgs field is discussed.

Massless, half-integer-spon fields in de Sitter space

Gauge-invariant wave equations are obtained for massless fields in de Sitter space, with arbitrary, half-integer spin. Massless quanta with spins $s=\frac{1}{2}, \frac{3}{2}, \dots{}$ carry a series of unitary, irreducible representations of the de Sitter group. The special gauge fields carry another series of unitary, irreducible representations, hence ghost fields as well as the Goldstone and the associated special gauge field are actually massive in de Sitter space. Fields (properly: potentials) do not admit chirality for spins other than \textonehalf{}, but the field strengths associated with spin $\frac{3}{2}$ do admit duality.

Elementary particles in a curved space

The Fierz-Pauli program is carried out for spin-2 fields in-de Sitter space. A spin-s field is associated with an irreducible representation D(E 0, s) of the universal covering group of SO(3, 2), with extremal weight (L 0 5, L1 2) → (E 0, s). (L 0 5 is the time translation operator.) The particular cases D(s+1, s) (s=1, 2,...) are characterized by invariance of the field equations under a gauge group and is for this and other reasons called ‘the massless case’. The Fierz-Pauli program is carried out for s=2, E 0>3; it fails in the special case E 0=3 only. The limit, as E 0 → 3, of the Fierz-Pauli field equation agrees with the linearized form of Einstein's gravitational field equation, with cosmological constant λ=-3ρ. Here ρ is the curvature constant of de Sitter space and the linearization referred to is defined relative to the de Sitter metric.

Quantum field theory in de Sitter space: renormalization by point-splitting

We examine the modes of a scalar field in de Sitter space and construct quantum two-point functions. These are then used to compute a finite stress tensor by the technique of covariant point-splitting. We propose a renormalization ansatz based on the DeWitt-Schwinger expansion, and show that this removes all am biguities previously present in pointsplitting regularization. The results agree in detail with previous work by dimensional regularization, and give rise to an anomalous trace with the conventional coefficient. We describe how’ our treatment may be extended to more general situations.

Quantised fields over de Sitter space

The theory of quantised fields in de Sitter space is investigated. It is demonstrated that de Sitter invariant fields transform like Minkowskian fields under a subgroup of the conformal group, if the de Sitter space is parametrised by horospherical coordinates. Special attention is thus focused on the resolution of the apparent causality conflict. For fields of spin less than or equal to one the field equations are derived; the case of arbitrary spin is dealt with by means of Weinberg-type fields. Furthermore, the author gives an improved version of the renormalisability proof for the model of a quantised scalar particle in the classical de Sitter background field.