We present a covariant quantization of the free "massive" spin-3/2 fields in four-dimensional de Sitter space-time based on analyticity in the complexified pseudo-Riemannian manifold. The field equation is obtained as an eigenvalue equation of the Casimir operator of the de Sitter group. The solutions are calculated in terms of coordinate-independent de Sitter plane-waves in tube domains and the null curvature limit is discussed. We give the group theoretical content of the field equation. The Wightman two-point function $S^{i \bar j}_{αα'}(x,x')$ is calculated. We introduce the spinor-vector field operator $Ψ_α(f)$ and the Hilbert space structure. A coordinate-independent formula for the field operator $Ψ_α(x)$ is also presented.
We show that the induced representations of the de Sitter isometry group proposed many years ago by Nachtmann are equivalent to those derived from our general theory of external symmetry. These methods complete each other leading to a coherent theory of covariant fields with spin on the de Sitter spacetime. Some technical details of these representations are presented here for the first time.
The properties of the covariant quantum fields on de Sitter spacetimes are investigated focusing on the isometry generators and Casimir operators in order to establish the equivalence among the covariant representations and the unitary irreducible ones of the de Sitter isometry group. For the Dirac quantum field it is shown that the spinor covariant representation, transforming the Dirac field under de Sitter isometries, is equivalent with a direct sum of two unitary irreducible representations of the $Sp(2,2)$ group, transforming alike the particle and antiparticle field operators in momentum…
We construct a de Sitter invariant photon propagator in general covariant gauges. Our result is a natural generalization of the Allen-Jacobson photon propagator in Feynman gauge. Our propagator reproduces the correct response to a point static charge and the one-loop electromagnetic stress-energy tensor, strongly suggesting that it is suitable for perturbative calculations on de Sitter.
The Hamiltonian formalism of Einstein--Cartan (EC) gravity is a starting point for canonical quantum gravity. The existing formalisms are at most Lorentz covariant, or diffeomorphism covariant. Here we analyze the Hamiltonian EC gravity in a 5d covariant way, with the gauge group being the de Sitter (dS) group, which unifies the Lorentz transformations and translation in an elegant manner, and also coincides with the acceleration of the universe. We reformulate the EC equations into a dS-covariant form, then find out the dS-covariant constraints of the phase space, and make all the constraint …
A construction of the 2d and 4d fuzzy de Sitter hyperboloids is carried out by using a (vector) coherent state quantization. We get a natural discretization of the dS "time" axis based on the spectrum of Casimir operators of the respective maximal compact subgroups SO(2) and SO(4) of the de Sitter groups SO\_0(1,2) and SO\_0(1,4). The continuous limit at infinite spins is examined.
The covariant free fields of any spin on anti-de Sitter spacetimes are studied, pointing out that these transform under isometries according to covariant representations of the anti-de Sitter isometry group, induced by those of the Lorentz group. Applying the method of ladder operators it is shown that the covariant representations with unique spin are equivalent with discrete unitary irreducible representations of positive energy of the universal covering group of the isometry one. The action of the Casimir operators is studied finding how the weights of these representations may depend on th…
We derive covariant expressions for the one loop vacuum polarization induced by a charged scalar on de Sitter background. Two forms are employed: one in which two covariant derivatives act on de Sitter invariant basis tensors multiplied by scalar structure functions, and the other in which four covariant derivatives act on a single basis tensor times a structure function. The second representation permits the correction to dynamical photons to be expressed as a surface integral, which raises the important question of what sorts of effects can be absorbed into corrections of the initial state. …
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 commut…
In a recent paper (Phys. Rev. D78, 084031 (2008), arXiv:0808.0642, Ref. [1]) it was shown in examples that the covariant retarded Green's functions in particular gauges for electromagnetism and linearized gravity can be used to reproduce field configurations correctly in spite of the spacelike nature of past infinity in de Sitter spacetime. In this paper we extend the work of Ref. [1] concerning the electromagnetic field and show that the covariant retarded Green's function with an arbitrary value of the gauge parameter reproduces the electromagnetic field from two opposite charges at antipoda…
It is shown that Schwarzschild black hole and de Sitter solutions exist as exact solutions of a recently proposed relativistic covariant formulation of (power-counting) renormalizable gravity with a fluid. The formulation without a fluid is also presented here. The stability of the solutions is studied and their corresponding entropies are computed, by using the covariant Wald method. The area law is shown to hold both for the Schwarzschild and for the de Sitter solutions found, confirming that, for the $β=1$ case, one is dealing with a minimal modification of GR.
We construct the graviton propagator in $n$-dimensional anti-de Sitter spacetime in the most general covariant gauge. We then study the behaviour of this propagator for different values of the gauge parameters. We will show that in any gauge, apart from the Landau gauge, the graviton propagator in the AdS spacetime contains a complicated term involving the derivative of a hypergeometric function which can not be expressed in terms of elementary functions. We do our calculations in the Euclidean approach.
We consider the covariant graviton propagator in de Sitter spacetime in a gauge with two parameters, alpha and beta, in the Euclidean approach. We give an explicit form of the propagator with a particular choice of beta but with arbitrary value of alpha. We confirm that two-point functions of local gauge-invariant quantities do not increase as the separation of the two points becomes large.
In this article, we quantize the Maxwell ("massless spin one") de Sitter field in a conformally invariant gauge. This quantization is invariant under the SO$_0(2,4)$ group and consequently under the de Sitter group. We obtain a new de Sitter invariant two-points function which is very simple. Our method relies on the one hand on a geometrical point of view which uses the realization of Minkowski, de Sitter and anti-de Sitter spaces as intersections of the null cone in $\setR^6$ and a moving plane, and on the other hand on a canonical quantization scheme of the Gupta-Bleuler type.
Guided by idealized but soluble nonrenormalizable models, a nontraditional proposal for the quantization of covariant scalar field theories is advanced, which achieves a term-by-term, divergence-free perturbation analysis of interacting models expanded about a suitable pseudofree theory (differing from a free theory by an $\hbar$ term). This procedure not only provides acceptable solutions for models for which no acceptable solution currently exists, e.g., $ϕ^4_n$, for spacetime dimension $n\ge4$, but offers a new, divergence-free solution, for less-singular models as well, e.g., $ϕ^4_n$, for …
We examine the scalar sector of the covariant graviton two-point function in de Sitter spacetime. This sector consists of the pure-trace part and another part described by a scalar field. We show that it does not contribute to two-point functions of gauge-invariant quantities. We also demonstrate that the long-distance growth present in some gauges is absent in this sector for a wide range of gauge parameters.
We discuss the algebraic quantization of a real, massive scalar field in the Poincaré patch of the (d+1)-dimensional anti-de Sitter spacetime, with arbitrary boundary conditions. By using the functional formalism, we show that it is always possible to associate to such system an algebra of observables enjoying the standard properties of causality, time-slice axiom and F-locality. In addition, we characterize the wavefront set of the ground state associated to the system under investigation. As a consequence, we are able to generalize the definition of Hadamard states and construct a global alg…
The Wightman two-point function for the gravitational field in the linear approximation (the rank-2 ``massless'' tensor field) on de Sitter space has a pathological behaviour for large separated points (infrared divergence). This behaviour can be eliminated in the two-point function for the traceless part of this field if one chooses the Gupta-Bleuler vacuum. But it is not possible to do the same for the pure trace part (conformal sector). We briefly discuss the consequences of this pure trace behaviour for inflationary models.
We evaluate the one-graviton loop contribution to the vacuum polarization on de Sitter background in a 1-parameter family of exact, de Sitter invariant gauges. Our result is computed using dimensional regularization and fully renormalized with BPHZ counterterms, which must include a noninvariant owing to the time-ordered interactions. Because the graviton propagator engenders a physical breaking of de Sitter invariance two structure functions are needed to express the result. In addition to its relevance for the gauge issue this is the first time a covariant gauge graviton propagator has been …
I investigate a new idea of perturbation theory in covariant canonical quantization. I present preliminary results for a toy model of a harmonic oscillator with a quartic perturbation, and show that this method reproduces the quantized spectrum of standard quantum theory. This result indicates that when the exact solutions to classical equations are not known, covariant canonical quantization via perturbation theory could be a viable approximation scheme for finding observables, and suggests a physically interesting way of extending the scope of covariant canonical quantization in quantum grav…
The propagators of the Faddeev-Popov (FP) ghosts for Yang-Mills theories and perturbative quantum gravity in the covariant gauge are infrared (IR) divergent in de Sitter spacetime. We point out, however, that the modes responsible for these divergences will not contribute to loop diagrams in computations of time-ordered products in either Yang-Mills theories or perturbative quantum gravity. Therefore we propose that the IR divergent FP-ghost propagator should be regularized by a small mass term that is sent to zero in the end of any perturbative calculations. This proposal is equivalent to usi…
Using the adiabatic invariant action and applying Bohr-Sommerfeld quantization rule and first law of black hole thermodynamics a study of quantization of entropy and horizon area of Kerr-Newman-de Sitter black hole is carried out. The same entropy spectrum is obtained in two different coordinate systems. It is also observed that the spacing of the entropy spectrum is independent of the black hole parameters and also the corresponding quantum of horizon area is in agreement with the results of Bekenstein.
We give in this paper an explicit construction of the covariant quantization of the rank-two massless tensor field on de Sitter space (linear covariant quantum gravity on a de Sitter background). The main ingredient of the construction is an indecomposable representation of de Sitter group. We here make the choice of a specific simple gauge fixing. We show that our gauge fixing eliminates any infrared divergence in the two-point function for the traceless part of this field. But it is not possible to do the same for the pure trace part (conformal sector). We describe the related Krein space st…
In this paper, the two-point function of linearized gravitons on de Sitter (dS) space is presented. Technically, respecting the dS ambient space notation, the field equation is given by the coordinate-independent Casimir operators of the de Sitter group. Analogous to the quantization of the electromagnetic field in Minkowski space, the field equation admits gauge solutions. The notation allows us to exhibit the formalism of Gupta-Bleuler triplets for the present field in exactly the same manner as it occurs for the electromagnetic field. In this regard, centering on the spin-two part (the trac…
An algebra for covariant observers in de Sitter space
School of Physical Science and Technology, Ningbo University（宁波大学物理科学与技术学院）；School of Physics, Peking University（北京大学物理学院）；Center for High Energy Physics, Peking University（北京大学高能物理中心）
It is known that the covariant graviton two-point function in de Sitter spacetime is infrared divergent for some choices of gauge parameters. On the other hand it is also known that there are no infrared divergences requiring an infrared cutoff for the physical graviton two-point function for this spacetime in the transverse-traceless-synchronous gauge in the global coordinate system. We show in this paper that the covariant graviton Wightman two-point function with two gauge parameters is equivalent to the physical one in the global coordinate system in the sense that they produce the same tw…
The Becci-Rouet-Stora-Tyutin (BRST) operator quantization of a finite-dimensional gauge system featuring two quadratic super Hamiltonian and m linear supermomentum constraints is studied as a model for quantizing generally covariant gauge theories. The proposed model ``completely'' mimics the constraint algebra of General Relativity. The Dirac constraint operators are identified by realizing the BRST generator of the system as a Hermitian nilpotent operator, and a physical inner product is introduced to complete a consistent quantization procedure.
We present a proposal to relate the de Sitter Conjecture (dSC) to the Covariant Entropy Bound (CEB). By assuming an early phase of accelerated expansion where the CEB is satisfied, we take into account a contribution from extra-dimensions to the four-dimensional entropy which restricts the values of the usual slow-roll parameters. We show in this context that the dSC inequalities follow from the CEB -- including their mutual exclusion -- in both single and multi-field inflationary scenarios. We also observe that the order one constants, c and c' in the conjecture are given in terms of physical…
Covariant phase space quantization attempts to quantize the full space of classical solutions, leading to a quantum theory in which the usual time coordinate is missing. In this paper we explore how the time evolution of the quantum states of this system is restored. We compare covariant phase space methods for a nonrelativistic system (the harmonic oscillator) and three relativistic quantum cosmologies. We find that for the nonrelativistic case the preferred time coordinate emerges in a phase factor when one transforms among certain representations, giving exactly the phase that appears when …
This note derives the analogue of the Mukhanov-Sasaki variables both for scalar and tensor perturbations in the 1+3 covariant formalism. The possibility of generalizing them to non-flat Friedmann-Lemaitre universes is discussed.
In this paper, vacuum expectation value (VEV) of the energy-momentum tensor for a conformally coupled scalar field in de Sitter space-time is investigated through the Krein-Gupta-Bleuler construction. This construction has already been successfully applied to the de Sitter minimally coupled massless scalar field and massless spin-2 field to obtain a causal and fully covariant quantum field on the de Sitter background. We also consider the effects of boundary conditions. In this respect, Casimir energy-momentum tensor induced by Dirichlet boundary condition on a curved brane is evaluated.
A generally covariant system can be deparametrized by means of an ``extrinsic'' time, provided that the metric has a conformal ``temporal'' Killing vector and the potential exhibits a suitable behavior with respect to it. The quantization of the system is performed by giving the well ordered constraint operators which satisfy the algebra. The searching of these operators is enlightned by the methods of the BRST formalism.
Motivated by the success of the recently proposed method of anomaly cancellation to derive Hawking fluxes from black hole horizons of spacetimes in various dimensions, we have further extended the covariant anomaly cancellation method shortly simplified by Banerjee and Kulkarni to explore the Hawking radiation of the (3+1)-dimensional charged rotating black strings and their higher dimensional extensions in anti-de Sitter spacetimes, whose horizons are not spherical but can be toroidal, cylindrical or planar, according to their global identifications. It should be emphasized that our analysis …
We prove that, under the standard conformal scaling, a massless field in de Sitter spacetime admits an O(4)-invariant Fock quantization such that time evolution is unitarily implemented. This result disproves previous claims in the literature. We discuss the relationship between this quantization with unitary dynamics and the family of O(4)-invariant Hadamard states given by Allen and Folacci, as well as with the Bunch-Davies vacuum.
We propose a manifestly covariant canonical method of field quantization based on the classical De Donder-Weyl covariant canonical formulation of field theory. Owing to covariance, the space and time arguments of fields are treated on an equal footing. To achieve both covariance and consistency with standard noncovariant canonical quantization of fields in Minkowski spacetime, it is necessary to adopt a covariant Bohmian formulation of quantum field theory. A preferred foliation of spacetime emerges dynamically owing to a purely quantum effect. The application to a simple time-reparametrizatio…
We present a complete quantization of Lorentzian D=1+2 gravity with cosmological constant, coupled to a set of topological matter fields. The approach of Loop Quantum Gravity is used thanks to a partial gauge fixing leaving a residual gauge invariance under a compact semi-simple gauge group, namely Spin(4) = SU(2) x SU(2). A pair of quantum observables is constructed, which are non-trivial despite of being null at the classical level.
We present in this paper a fully covariant quantization of the minimally-coupled massless field on de Sitter space. We thus obtain a formalism free of any infrared (e.g logarithmic) 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 space-time. The minimally-coupled massless field is associated with a represent…
A formulation of Covariant Canonical Quantization is discussed, which works on an extended Hilbert space and reduces to conventional canonical quantization when constraining to the solution of the field equation a priori. From the formal point of view it may be seen as a formalism between the canonical operator and the functional integral approach. A covariant number operator and two symmetric vacua are constructed. By that means, certain well-known quantities like the LSZ formula are rederived via a projection limit. The time-ordering operator can be replaced by taking into account the mirror…
We solve perturbative constraints and eliminate gauge freedom for Ashtekar's gravity on de Sitter background. We show that the reduced phase space consists of transverse, traceless, symmetric fluctuations of the triad and of transverse, traceless, symmetric fluctuations of the connection. A part of gauge freedom corresponding to the conformal Killing vectors of the three-manifold can be fixed only by imposing conditions on Lagrange multiplier. The reduced phase space is equivalent to that of ADM gravity on the same background.
We present a new approach to the covariant canonical formulation of Einstein-Cartan gravity that preserves the full Lorentz group as the local gauge group. The method exploits lessons learned from gravity in 2+1 dimensions regarding the relation between gravity and a general gauge theory. The dynamical variables are simply the frame field and the spin-connection pulled-back to the hypersurface, thereby eliminating the need for simplicity constraints on the momenta. A consequence of this is a degenerate (pre)symplectic form, which appears to be a necessary feature of the Einstein-Cartan formula…
In our previous paper, we have presented a covariant BRST quantization of unimodular gravity which may account for the smallness of the cosmological constant, and have shown that the physical degrees of freedom in the theory are the same as general relativity. The formulation has been given by using rank-2 antisymmetric tensor fields for both ghosts and antighosts. Here we give an alternative formulation using a vector field for the antighost but keeping the same structure for the ghosts. This gives a significantly simpler covariant quantization with less ghosts and no tripole modes in the gho…
We canonically quantize closed string theory in the pp-wave background with a non-zero flux of the three-form field strength by using the covariant BRST operator formalism. In this canonical quantization, we completely construct new covariant free-mode representations, for which it is particularly important to take account of the commutation relations of the zero mode of the light-cone string coordinate X^{-} with other modes. All covariant string coordinates are composed of the free-modes. Moreover, employing these covariant string coordinates for the energy-momentum tensor, we calculate the …
In this paper, considering the linearized Einstein equation with a two-parameter family of linear covariant gauges in de Sitter spacetime, we examine possible vacuum states for the gravitons field with respect to invariance under the de Sitter group $SO_0(1,4)$. Our calculations explicitly reveal that there exists no natural de Sitter-invariant vacuum state (the Euclidean state) for the gravitons field. Indeed, on the foundation of a rigorous group theoretical reasoning, we prove that if one insists on full covariance as well as causality for the theory, has to give up the positivity requireme…
We present a coherent state quantization of the dynamics of a relativistic test particle on a one-sheet hyperboloid embedded in a three-dimensional Minkowski space. The group SO_0(1,2) is considered to be the symmetry group of the system. Our procedure relies on the choice of coherent states of the motion on a circle. The coherent state realization of the principal series representation of SO_0(1,2) seems to be a new result.
Unimodular gravity (UG) is an interesting theory that may explain why the cosmological constant is extremely small, in contrast to general relativity (GR). The theory has only the transverse diffeomorphism invariance and this causes lots of debate as to the equivalence of UG to GR in the covariant quantization. We study the covariant BRST quantization of UG by gauge fixing only the transverse diffeomorphism and show that the remaining physical degrees of freedom are two, the same number as GR. This is achieved by using antisymmetric tensor ghost fields which automatically satisfy the transvers…
We obtain a coordinate independent algorithm to determine the class of conformal Killing vectors of a locally conformally flat $n$-metric $γ$ of signature $(r,s)$ modulo conformal transformations of $γ$. This is done in terms of endomorphisms in the pseudo-orthogonal Lie algebra $\mathfrak{o}(r+1,s+1)$ up to conjugation of the its group $O(r+1,s+1)$. The explicit classification is worked out in full for the Riemannian $γ$ case ($r = 0, s = n$). As an application of this result, we prove that the set of five dimensional, $(Λ>0)$-vacuum, algebraically special metrics with non-degenerate optical …
Respecting that any consistent quantum field theory in curved space-time must include black hole radiation, in this paper, we examine the Krein-Gupta-Bleuler (KGB) formalism as an inevitable quantization scheme in order to follow the guideline of the covariance of minimally coupled massless scalar field and linear gravity on de Sitter (dS) background in the sense of Wightman-Gärding approach, by investigating thermodynamical aspects of black holes. The formalism is interestingly free of pathological large distance behavior. In this construction, also, no infinite term appears in the calculatio…
Canonical Hamiltonian field theory in curved spacetime is formulated in a manifestly covariant way. Second quantization is achieved invoking a correspondence principle between the Poisson bracket of classical fields and the commutator of the corresponding quantum operators. The Dirac theory is investigated and it is shown that, in contrast to the case of bosonic fields, in curved spacetime, the field momentum does not coincide with the generators of spacetime translations. The reason is traced back to the presence of second class constraints occurring in Dirac theory. Further, it is shown that…
In this paper, we generalize the simplified Dark Matter models with graviton mediator to the curved space-time, in particular to the de Sitter space. We obtain the generating functional of the Green's functions in the Euclidean de Sitter space for the covariant free gravitons. We determine the generating functional of the interacting theory between Dark Matter particles and the covariant gravitons. Also, we calculate explicitly the 2-point and 3-point interacting Green's functions for the symmetric traceless divergence-less covariant graviton. The results derived here are general in the sense …
The effect of string and quantum gravity inspired minimum-length deformed quantization on a free, massless scalar field is studied on de Sitter background at the level of second quantization. Analytic solution of a field operator is obtained to the first order in deformation parameter. Using this solution we then estimate the two point and four point correlation functions (with respect to the Bunch-Davies vacuum). The field operator shows up a non-linear dependence on creation and annihilation operators, therefore the perturbation spectrum proves to be non-Gaussian. The correction to the power…
We construct the graviton two-point function for a two-parameter family of linear covariant gauges in n-dimensional de Sitter space. The construction is performed via the mode-sum method in the Bunch-Davies vacuum in the Poincaré patch, and a Fierz-Pauli mass term is introduced to regularize the infrared (IR) divergences. The resulting two-point function is de Sitter-invariant, and free of IR divergences in the massless limit (for a certain range of parameters) though analytic continuation with respect to the mass for the pure-gauge sector of the two-point function is necessary for this result…
A Lorentz covariant quantization of membrane dynamics is defined, which also leaves unbroken the full three dimensional diffeomorphism invariance of the membrane. Among the applications studied are the reduction to string theory, which may be understood in terms of the phase space and constraints, and the interpretation of physical,zero-energy states. A matrix regularization is defined as in the light cone gauged fixed theory but there are difficulties implementing all the gauge symmetries. The problem involves the non-area-preserving diffeomorphisms which are realized non-linearly in the clas…
The BRST generator is realized as a Hermitian nilpotent operator for a finite-dimensional gauge system featuring a quadratic super-Hamiltonian and linear supermomentum constraints. As a result, the emerging ordering for the Hamiltonian constraint is not trivial, because the potential must enter the kinetic term in order to obtain a quantization invariant under scaling. Namely, BRST quantization does not lead to the curvature term used in the literature as a means to get that invariance. The inclusion of the potential in the kinetic term, far from being unnatural, is beautifully justified in li…
We compute the {\it exact} equation of state of circular strings in the (2+1) dimensional de Sitter (dS) and anti de Sitter (AdS) spacetimes, and analyze its properties for the different (oscillating, contracting and expanding) strings. The string equation of state has the perfect fluid form $P=(γ-1)E,$ with the pressure and energy expressed closely and completely in terms of elliptic functions, the instantaneous coefficient $γ$ depending on the elliptic modulus. We semi-classically quantize the oscillating circular strings. The string mass is $m=\sqrt{C}/(πHα'),\;C$ being the Casimir operator…
