Gupta Bleuler Triplets Research Articles

Super-gauge field in de Sitter universe

The Gupta-Bleuler triplet for vector-spinor gauge field is presented in de Sitter ambient space formalism. The invariant space of field equation solutions is obtained with respect to an indecomposable representation of the de Sitter group. By using the general solution of the massless spin-$\frac{3}{2}$ field equation, the vector-spinor quantum field operator and its corresponding Fock space is constructed. The quantum field operator can be written in terms of the vector-spinor polarization states and a quantum conformally coupled massless scalar field, which is constructed on Bunch-Davies vacuum state. The two-point function is also presented, which is de Sitter covariant and analytic.

Covariant and infrared-free graviton two-point function in de Sitter spacetime

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 traceless part, ${\cal{K}}^t$), the field solution is written as a product of a generalized polarization tensor and a minimally coupled massless scalar field. Then, admitting a de Sitter-invariant vacuum through the so-called "Krein space quantization", the de Sitter fully covariant two-point function is calculated. This function is interestingly free of pathological large distance behavior (infrared divergence). Moreover, the spin-zero part (the pure-trace part; ${\cal{K}}^{pt}$) of the field is discussed in this paper. It is shown that the implications of the dS group unitary irreducible representations restrict the gauge-fixing parameter to the optimal value, which remarkably results in the pure-trace part be written in terms of a conformally coupled massless scalar field.

Massless qft on (1 + 1)-de Sitter space-time

It is known that a massless free field on a (1 + 1)-dimensional de Sitter space-time does not allow a de Sitter invariant Fock vacuum on a Hilbert space. We show that this result no longer holds on an indefinite inner product space. To this end we use the Gupta-Bleuler formalism to construct a causal, de Sitter and conformally covariant massless free quantum field on the (1 + 1)-dimensional de Sitter space-time admitting a de Sitter invariant vacuum in an indefinite inner product space. The field is defined rigorously as an operator-valued distribution and is covariant in the usual strong sense: Vg−1ϑ(x)Vg = ϑ(g · x) for any g in the de Sitter group, where V is a unitary representation of the de Sitter group on the space of states. The formalism of Gupta-Bleuler triplets also allows for an explicit description of the gauge degree of freedom. We show that, although the field itself is not an observable (it is gauge dependent), the stress tensor is. The component t00(x) is positive in all physical states, and vanishes in the vacuum. In addition, since the field is conformally covariant, the model does not exhibit a conformal anomaly in the trace of the energy-momentum tensor.

Singleton Field Theory and Flato–Frønsdal Dipole Equation

We study solutions of the equations (△ -λ )φ = 0 and (△ -λ )2φ = 0 in global coordinates on the covering space CAdSd of the d-dimensional Anti de-Sitter space subject to various boundary conditions and their connection to the unitary irreducible representations of \(\overline {{\text{SO}}} _0 \)(d-1,2). The ‘vanishing flux’ boundary conditions at spatial infinity lead to the standard quantization scheme for CAdSd in which solutions of the second- and the fourth-order equations are equivalent. To include fields realizing the singleton unitary representation in the bulk of CAdS d one has to relax the boundary conditions thus allowing for the nontrivial space of solutions of the dipole equation known as the Gupta–Bleuler triplet. We obtain explicit expressions for the modes of the Gupta–Bleuler triplet and the corresponding two-point function. To avoid negative-energy states one must also introduce an additional constraint in the space of solutions of the dipole equation.

Deformations of massless Gupta–Bleuler triplets in 3+2 De Sitter space

We study deformations of massless indecomposable representations (Gupta–Bleuler triplets) for the De Sitter group SO(3,2), showing that there is no deformation of such representations toward a kind of massive Gupta–Bleuler triplets which would have corresponded to an analog of a Higgs–Kibble mechanism in anti De Sitter space.

Some Properties of Massless Particles in Arbitrary Dimensions

Various properties of two kinds of massless representations of the n-conformal (or (n+1)-De Sitter) group [Formula: see text] are investigated for n≥2. It is found that, for space-time dimensions n≥3, the situation is quite similar to the one of the n=4 case for Sn-massless representations of the n-De Sitter group [Formula: see text]. These representations are the restrictions of the singletons of [Formula: see text]. The main difference is that they are not contained in the tensor product of two UIRs with the same sign of energy when n>4, whereas it is the case for another kind of massless representations. Finally some examples of Gupta–Bleuler triplets are given for arbitrary spin and n≥3.

Massless Gupta-Bleuler vacuum on the (1+1)-dimensional de Sitter space-time

We construct a causal, de Sitter, and conformally covariant massless free quantum field on the (1+1)-dimensional de Sitter space-time admitting a de Sitter invariant vacuum in an indefinite inner product space. The field is defined rigorously as an operator-valued distribution and is covariant in the usual strong sense: $V{}_{g}^{\ensuremath{-}1}\ensuremath{\varphi}(x)V{}_{g}=\ensuremath{\varphi}(g\ensuremath{\cdot}x)$ for any $g$ in the de Sitter group, where $V$ is a unitary representation of the de Sitter group on the space of states. We use the formalism of Gupta-Bleuler triplets which also allows for an explicit description of the gauge degree of freedom. As a consequence the model does not suffer from infrared divergences, contrary to what happened in previous treatments of this problem. The causality and the covariance of the theory are assured thanks to a suitable choice of the space of solutions of the classical field equation. We show that, although the field itself is not observable (it is gauge dependent), the stress tensor and the energy-momentum vector are. The energy operator ${P}_{0}$ is positive in all physical states, and vanishes in the vacuum. In addition, the field is conformally covariant and the model does not exhibit a conformal anomaly in the trace of the energy-momentum tensor.

Some remarks on the Gupta–Bleuler triplet

Indecomposable representations of the Poincaré group associated to infrared singular field theory models are discussed in the framework of the general theory of the Gupta–Bleuler triplet formulated by Araki. It is shown that the definition of maximal Hilbert space structures, related to the infrared properties of the states of the models, can be exploited to construct representation spaces for the Gupta–Bleuler triplet. The examples of the two-dimensional massless scalar field and of the electromagnetic field in the Landau gauge are discussed. In particular, in the first example, the relation between the Gupta–Bleuler triplet and the algebraic treatment of the massless scalar field is investigated. In the case of electromagnetism, the structure of the representation of the Poincaré group in the Landau gauge is clarified. The explicit form of the corresponding Gupta–Bleuler triplet for the one-particle space of the electromagnetic field is exhibited.

Bargmann–Wigner equations in de Sitter space

Bargmann–Wigner equations in (3,2)-de Sitter space are found for all spins s≥1 and all masses. The massless fields have gauge freedom; they can be extended to indecomposable representations of the form of Gupta–Bleuler triplets.

Invariant bilinear forms on 3+2 de Sitter space

Massive elementary systems in 3+2 de Sitter space are associated with unitary irreducible representations of the kinematic group SO0(3,2). On the other hand, massless elementary systems are connected to certain non-decomposable representations of this group. As usual they display gauge invariance and conformal invariance properties. The authors give in this work the corresponding invariant bilinear forms. They are indefinite in the second case, positive-definiteness being restored on (physical) quotient spaces. In particular they present a table of scalar products involving lowest-energy states cyclic for the two Gupta-Bleuler triplets of de Sitter QED and make precise some of its conformal invariance aspects.

The singleton dipole

The space of solutions of the dipole equation $$(\square - \frac{5}{4}\varrho )^2 \phi (x) = 0,$$ in a 3+2 de Sitter space with curvature constant ϱ, contains a complete Gupta-Bleuler triplet, consisting of pure gauge modes, physical modes, and “scalar” or “auxiliary” modes. Indefinite metric quantization is carried out precisely as in more conventional gauge theories. The associated Lagrangian and Hamiltonian field theory formulations reveal an interesting interplay between fields on the de Sitter manifold and their boundary values at spatial infinity.

On pseudo-unitary representations

The structure of representations of a group on a finite dimensional complex vector space with a non-degenerate invariant (indefinite) inner product is analysed in terms of layers of Gupta-Bleuler triplets.

Linear de Sitter gravity

It is shown that there are two equivalent field potentials for massless spin-2 fields in (3,2)–de Sitter space; a symmetric two-tensor and a three-tensor of mixed symmetry. Each theory carries two inequivalent Gupta–Bleuler triplets, which correspond to the two helicities. They differ in their behavior at spatial infinity. All four fields appear when writing linear conformal gravity in de Sitter space. Two of them describe the unitary conformal gravitons, and two describe part of the conformal ghost.

Conformal quantum Yang–Mills

Conformally covariant quantization of non-Abelian gauge theory is presented, and the invariant propagators needed for perturbative calculations are found. The vector potential acquires a richer gauge structure displayed in the larger Gupta–Bleuler triplet whose center is occupied by conformal QED. Path integral formulation and BRS invariance are shown on a formal level in one covariant gauge.

Gauge fixing and Gupta–Bleuler triplets in de Sitter QED

Binegar et al. recently developed de Sitter QED within the frame of a particular gauge fixing c= (2)/(3) . Such a choice leads to the simplest possible structure of the Gupta–Bleuler triplets, with the SO(3,2) representations D(3,0) and D(1,1) appearing to describe special gauge solutions. The present work examines the general case c≠ (2)/(3) by considering explicitly the corresponding homogeneous propagators and some fundamental carrier states for the related indecomposable representations of SO(3,2). The latter expressions involve ‘‘logarithmic’’ gauge states which independently carry extensions of the D(3,0) and D(1,1) Lie algebra representations.

Gauge fixing and massless, integral-spin fields in de sitter space

The classical field theories of massless fields of arbitrary, fixed, integral spins reach their simplest Gupta-Bleuler triplet structure when the gauge-fixing parameter c is put equal to zero (‘the Feynman Gauge’). This value of c is peculiar to the flat space theory. In a De Sitter massless field theory with spin s, the ‘best value’ of c is shown to be equal to 2/2s+1.