Massless Conformal Scalar Field Research Articles

Entanglement dynamics in de Sitter spacetime

We apply a master equation approximation with dynamical coarse graining to a pair of detectors interacting with a scalar field. By solving the master equation numerically, we investigate the evolution of negativity between comoving detectors in de Sitter space. For a massless conformal scalar field with conformal vacuum, it is found that a pair of detectors can perceive entanglement beyond the Hubble horizon scale if the initial separation of detectors is sufficiently small. At the same time, violation of the Bell–Clauser–Horne–Shimony–Holt inequality on the super-horizon scale is also detected. For a massless minimal scalar field with Bunch–Davies vacuum, on the other hand, the entanglement decays within Hubble time scale, owing to the quantum noise caused by particle creations in de Sitter space, and the entanglement on the super-horizon scale cannot be detected.

Massless conformal fields, AdS(d + 1)/CFTd higher spin algebras and their deformations

We extend our earlier work on the minimal unitary representation of SO(d,2) and its deformations for d=4,5 and 6 to arbitrary dimensions d. We show that there is a one-to-one correspondence between the minrep of SO(d,2) and its deformations and massless conformal fields in Minkowskian spacetimes in d dimensions. The minrep describes a massless conformal scalar field, and its deformations describe massless conformal fields of higher spin. The generators of Joseph ideal vanish identically as operators for the quasiconformal realization of the minrep, and its enveloping algebra yields directly the standard bosonic AdS(d+1)/CFTd higher spin algebra. For deformed minreps the generators of certain deformations of Joseph ideal vanish as operators and their enveloping algebras lead to deformations of the standard bosonic higher spin algebra. In odd dimensions there is a unique deformation of the higher spin algebra corresponding to the spinor singleton. In even dimensions one finds infinitely many deformations of the higher spin algebra labelled by the eigenvalues of Casimir operator of the little group SO(d−2) for massless representations.

Deformed twistors and higher spin conformal (super-)algebras in four dimensions

Massless conformal scalar field in six dimensions corresponds to the minimal unitary representation (minrep) of the conformal group SO(6,2). This minrep admits a family of deformations labelled by the spin t of an SU(2)_T group, which is the 6d analog of helicity in four dimensions. These deformations of the minrep of SO(6,2) describe massless conformal fields that are symmetric tensors in the spinorial representation of the 6d Lorentz group. The minrep and its deformations were obtained by quantization of the nonlinear realization of SO(6,2) as a quasiconformal group in arXiv:1005.3580. We give a novel reformulation of the generators of SO(6,2) for these representations as bilinears of deformed twistorial oscillators which transform nonlinearly under the Lorentz group SO(5,1) and apply them to define higher spin algebras and superalgebras in AdS_7. The higher spin (HS) algebra of Fradkin-Vasiliev type in AdS_7 is simply the enveloping algebra of SO(6,2) quotiented by a two-sided ideal (Joseph ideal) which annihilates the minrep. We show that the Joseph ideal vanishes identically for the quasiconformal realization of the minrep and its enveloping algebra leads directly to the HS algebra in AdS_7. Furthermore, the enveloping algebras of the deformations of the minrep define a discrete infinite family of HS algebras in AdS_7 for which certain 6d Lorentz covariant deformations of the Joseph ideal vanish identically. These results extend to superconformal algebras OSp(8*|2N) and we find a discrete infinite family of HS superalgebras as enveloping algebras of the minimal unitary supermultiplet and its deformations. Our results suggest the existence of a discrete family of (supersymmetric) HS theories in AdS_7 which are dual to free (super)conformal field theories (CFTs) or to interacting but integrable (supersymmetric) CFTs in 6d.

Minimal unitary representation of 5d superconformal algebra [formula omitted] and [formula omitted] higher spin (super)-algebras

We study the minimal unitary representation (minrep) of SO(5,2), obtained by quantization of its geometric quasiconformal action, its deformations and supersymmetric extensions. The minrep of SO(5,2) describes a massless conformal scalar field in five dimensions and admits a unique “deformation” which describes a massless conformal spinor. Scalar and spinor minreps of SO(5,2) are the 5d analogs of Dirac's singletons of SO(3,2). We then construct the minimal unitary representation of the unique 5d superconformal algebra F(4) with the even subalgebra SO(5,2)×SU(2). The minrep of F(4) describes a massless conformal supermultiplet consisting of two scalar and one spinor fields. We then extend our results to the construction of higher spin AdS6/CFT5 (super)-algebras. The Joseph ideal of the minrep of SO(5,2) vanishes identically as operators and hence its enveloping algebra yields the AdS6/CFT5 bosonic higher spin algebra directly. The enveloping algebra of the spinor minrep defines a “deformed” higher spin algebra for which a deformed Joseph ideal vanishes identically as operators. These results are then extended to the construction of the unique higher spin AdS6/CFT5 superalgebra as the enveloping algebra of the minimal unitary realization of F(4) obtained by the quasiconformal methods.

Deformed twistors and higher spin conformal (super-)algebras in six dimensions

Massless conformal scalar field in six dimensions corresponds to the minimal unitary representation (minrep) of the conformal group SO(6,2). This minrep admits a family of labelled by the spin t of an SU(2)T group, which is the 6d analog of helicity in four dimensions. These deformations of the minrep of SO(6,2) describe massless conformal fields that are symmetric tensors in the spinorial representation of the 6d Lorentz group. The minrep and its deformations were obtained by quantization of the nonlinear realization of SO(6,2) as a quasiconformal group in arXiv:1005.3580. We give a novel reformulation of the generators of SO(6,2) for these representations as bilinears of deformed twistorial oscillators which transform nonlinearly under the Lorentz group SO(5,1) and apply them to define higher spin algebras and superalgebras in AdS7. The higher spin (HS) algebra of Fradkin-Vasiliev type in AdS7 is simply the enveloping algebra of SO(6,2) quotiented by a two-sided ideal (Joseph ideal) which annihilates the minrep. We show that the Joseph ideal vanishes identically for the quasiconformal realization of the minrep and its enveloping algebra leads directly to the HS algebra in AdS7. Furthermore, the enveloping algebras of the deformations of the minrep define a discrete infinite family of HS algebras in AdS7 for which certain 6d Lorentz covariant deformations of the Joseph ideal vanish identically. These results extend to superconformal algebras OSp(8 ∗ |2N) and we find a discrete infinite family of HS superalgebras as enveloping algebras of the minimal unitary supermultiplet and its deformations. Our results suggest the existence of a discrete family of (supersymmetric) HS theories in AdS7 which are dual to free (super)conformal field theories (CFTs) or to interacting but integrable (supersymmetric) CFTs in 6d.

High-temperature expansion of the one-loop free energy of a scalar field on a curved background

The complete form of the high-temperature expansion of the one-loop contribution to the free energy of a scalar field on a stationary gravitational background is derived. The explicit expressions for the divergent and finite parts of the high-temperature expansion in a three-dimensional space without boundaries are obtained. These formulas generalize the known one for the stationary spacetime. In particular, we confirm that for a massless conformal scalar field the leading correction to the Planck law proportional to the temperature squared turns out to be nonzero due to non-static nature of the metric. The explicit expression for the so-called energy-time anomaly is found. The interrelation between this anomaly and the conformal (trace) anomaly is established. The natural simplest Lagrangian for the "Killing vector field" is given.

Minimal unitary representation of [formula omitted] and its [formula omitted] deformations as massless 6 D conformal fields and their supersymmetric extensions

We study the minimal unitary representation (minrep) of SO ( 6 , 2 ) over an Hilbert space of functions of five variables, obtained by quantizing its quasiconformal realization. The minrep of SO ( 6 , 2 ) , which coincides with the minrep of SO ∗ ( 8 ) similarly constructed, corresponds to a massless conformal scalar field in six spacetime dimensions. There exists a family of “deformations” of the minrep of SO ∗ ( 8 ) labeled by the spin t of an SU ( 2 ) T subgroup of the little group SO ( 4 ) of lightlike vectors. These deformations labeled by t are positive energy unitary irreducible representations of SO ∗ ( 8 ) that describe massless conformal fields in six dimensions. The SU ( 2 ) T spin t is the six-dimensional counterpart of U ( 1 ) deformations of the minrep of 4D conformal group SU ( 2 , 2 ) labeled by helicity. We also construct the supersymmetric extensions of the minimal unitary representation of SO ∗ ( 8 ) to minimal unitary representations of OSp ( 8 ∗ | 2 N ) that describe massless six-dimensional conformal supermultiplets. The minimal unitary supermultiplet of OSp ( 8 ∗ | 4 ) is the massless supermultiplet of ( 2 , 0 ) conformal field theory that is believed to be dual to M-theory on AdS 7 × S 4 .

Cyclic inflation

We present an inflationary model that is geodesically complete and does not suffer from the trans-Planckian problem. In most inflationary models, massless (conformal) scalar field fluctuations in a de Sitter background gives rise to a scale-invariant spectrum. In this work, we realize scale-invariant perturbations from thermal fluctuations in (conformal) radiation during a radiation dominated contraction era prior to inflation. As the modes exit the Hubble radius during the contraction phase, scale-invariant fluctuations are generated. After many cycles, we enter into a power-law inflationary phase, that stretches the modes produced in the previous contraction phase to scales that we observe today.

Casimir energy density for spherical universes in n-dimensional spacetime

We consider the Casimir effect for the massless conformal scalar field in an n-dimensional, closed, static universe. We calculate the renormalized vacuum energy density using the covariant point-splitting method, the mode-sum regularization and the renormalized vacuum energy with the zeta-function regularization. We observe that all odd spacetime dimensions give us the zero renormalized vacuum energy density. For even spacetime dimensions the renormalized vacuum energy density oscillates in sign. The result agrees with three regularization techniques. The Casimir energy density for spherical universes in n-dimensional spacetime is regarded as interesting both to understand the correspondence between the sign of the effect and the dimension of manifold in topology and as a key to confirming the Casimir energy for half spherical universes (manifold with boundary) in n-dimensional spacetime.

Casimir energy of the massless conformal scalar field on S-2 by the point-splitting method

We calculate the Casimir energy of the massless conformal scalar field on the surface (S-2) of a 3 dimensional Riemann sphere by using the point-splitting, mode sum and the ζ-function renormalization methods. We also consider the half space case with both the Dirichlet and the Neumann boundary conditions. This problem is interesting since the Casimir energy could be calculated analytically by various methods, thus allowing us to compare different regularization schemes.

Back reaction of a conformal field on a three-dimensional black hole

The first order corrections to the geometry of the (2+1)-dimensional black hole due to back-reaction of a massless conformal scalar field are computed. The renormalized stress energy tensor used as the source of Einstein equations is computed with the Green function for the black-hole background with transparent boundary conditions. This tensor has the same functional form as the one found in the nonperturbative case which can be exactly solved. Thus, a static, circularly symmetric and asymptotically anti-de Sitter black hole solution of the semiclassical equations is found. The corrections to the thermodynamic quantities are also computed.

The energy localization problem and the renormalized vacuum energy in static Robertson-Walker universes

We calculate the renormalized quantum vacuum energy inside a spherical boundary for the massless conformal scalar field in curved background Robertson-Walker geometry. We use the mode sum method with an exponential cutoff. In our calculations we do not make assumptions about the exterior geometry or the global topology of the universe.

Casimir energy in a curved background with a spherical boundary: An exactly solvable case.

We reconsider the Casimir effect for the massless conformal scalar field in a ``half Einstein universe.'' This case is interesting in that it could be solved analytically by several different methods. Thus, it allows us to compare and understand effects of different approaches. We first calculate the renormalized vacuum energy by using the mode sum method and study the effects of different cutoff functions. Next, we calculate the renormalized vacuum energy-momentum tensor by using the covariant point-splitting method. We construct the Green's functions by using the eigenfunctions, which are obtained by solving the wave equation with the appropriate boundary conditions (Dirichlet and Neumann). We discuss further properties of this case as well as its relation to previous calculations.

Inflation inR 2 cosmology with massless conformal scalar field and adiabatic regularization

A cosmology described by a Robertson-Walker metric with Lagrangian √-g(@#@R+σR2) and a massless conformal scalar field as matter content is considered. It is shown that, when the adiabatic regularization is performed, there is, in the spatially flat case, an inflationary solution which is not stable in the future, at first perturbative order, if the condition σ>t p 2 /2160π (tp Planck time) is fulfilled and stable in the opposite case.

Temperature in Friedmann thermodynamics and its generalization to arbitrary space-times

In recent articles we have introduced Friedmann thermodynamics, where certain geometric parameters in Friedmann models were treated like their thermodynamic counterparts (temperature, entropy, Gibbs potential, etc.). This model has the advantage of allowing us to determine the geometry of the universe by thermodynamic stability arguments. In this paper, in search for evidence for the definition of “gravitational” temperature, we will investigate a massless conformal scalar field in an Einstein universe in detail. We will argue that the “gravitational” temperature of the Einstein universe is given asTg=1/2π) (ħc/k) (1/R0), where R0 is the radius of the Einstein universe. This is in accord with our definition of “gravitational” temperature in Friedmann thermodynamics and determines the dimensionless constant as 1/2π. We discuss the limitations of the model we are using. We also suggest a method to generalize our “gravitational” temperature to arbitrary space-times granted that they are sufficiently smooth.

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.

Notes on the Hawking effect in de Sitter space. II.

The Hawking effect for a quantized massless conformal scalar field in two- and four-dimensional de Sitter space is studied. Two different charts, on which the mode functions are defined, are the spatially flat chart and the static one. The Bogoliubov coefficients are evaluated without any approximation. In both two and four dimensions, the particle number distributions reveal the exact thermal spectrum.

Notes on the Hawking effect in de Sitter space. I.

We consider the Hawking effect for a quantized massless conformal scalar field in two- and four-dimensional de Sitter space. The relation among Bogoliubov coefficients is investigated without explicit integration of Klein-Gordon products. Our method presents a clear view of the property of Bogoliubov coefficients. In the two-dimensional case the thermal distribution is exactly derived. The application to the four-dimensional case is not straightforward, but we can derive the same result with some techniques.

Large Scale Quantum Fluctuations in the Inflationary Universe

Quantum fluctuations of an inflation-driving scalar field are evaluated in a way manifestly independent of the choice of coordinate gauge conditions. It is found that the dynamical degree of freedom of the fluctuating field is represented in terms of a nearly massless conformal scalar field in the unperturbed de Sitter background. Implications of the result are discussed. In particular, it is argued that classical cosmological density perturbations may not be generated in the sense as discussed in the literature.

Calculation of the trace anomaly of the conformal energy-momentum tensor in Kasner spacetime by adiabatic regularization

The anomalous trace of the energy-momentum tensor of a quantized massless conformal scalar field in a homogeneous anisotropic, spatially flat background is calculated by the method of adiabatic regularization. This method yields the same result as obtained earlier by other formal methods for a general spacetime, but has the advantage of simplicity in computation and clarity in physical interpretation. Explicit expressions are also obtained for the regularized energy density and pressure, which are useful in formulating semiclassical theories of gravity and in calculating the back reaction of quantized fields in relativistic cosmology.