Gupta-Bleuler Research Articles

A conformally covariant massless quantum field in 1 + 1 dimensions

In this paper, we construct a causal and conformally covariant massless free quantum field on the (1 + 1)-dimensional Minkowski spacetime. This field is of the Gupta–Bleuler type and does not suffer from infrared divergences. Although negative frequency states are used in its construction, the field has no unusual physical features: the energy–momentum tensor behaves reasonably when applied to physical states and it is also shown that particle detectors moving in the vacuum behave as expected.

Quantization of the electromagnetic field outside static black holes and its application to low-energy phenomena

We discuss the Gupta-Bleuler quantization of the free electromagnetic field outside static black holes in the Boulware vacuum. We use a gauge which reduces to the Feynman gauge in Minkowski spacetime. We also discuss its relation with gauges used previously. Then we apply the low-energy sector of this field theory to investigate some low-energy phenomena. First, we discuss the response rate of a static charge outside the Schwarzschild black hole in four dimensions. Next, motivated by string physics, we compute the absorption cross sections of low-energy plane waves for the Schwarzschild and extreme Reissner-Nordstrom black holes in arbitrary dimensions higher than three.

LOCAL QUANTUM CONSTRAINTS

We analyze the situation of a local quantum field theory with constraints, both indexed by the same set of space-time regions. In particular we find "weak" Haag–Kastler axioms which will ensure that the final constrained theory satisfies the usual Haag–Kastler axioms. Gupta–Bleuler electromagnetism is developed in detail as an example of a theory which satisfies the "weak" Haag–Kastler axioms but not the usual ones. This analysis is done by pure C*-algebraic means without employing any indefinite metric representations, and we obtain the same physical algebra and positive energy representation for it than by the usual means. The price for avoiding the indefinite metric, is the use of nonregular representations and complex valued test functions. We also exhibit the precise connection with the usual indefinite metric representation. We conclude the analysis by comparing the final physical algebra produced by a system of local constrainings with the one obtained from a single global constraining and also consider the issue of reduction by stages. For the usual spectral condition on the generators of the translation group, we also find a "weak" version, and show that the Gupta–Bleuler example satisfies it.

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.

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.

A modification of the 10D superparticle action inspired by the Gupta-Bleuler quantization scheme

We reconsider the issue of the existence of a complex structure in the Gupta-Bleuler quantization scheme. We prove an existence theorem for the complex structure associated with the d=10 Casalbuoni-Brink-Schwarz superparticle, based on an explicitly constructed Lagrangian that allows a holomorphic-antiholomorphic splitting of the fermionic constraints consistent with the vanishing of all first class constraints on the physical states.

Complex structure in the Gupta–Bleuler quantization method

We examine the general conditions for the existence of the complex structure intrinsic in the Gupta-Bleuler quantization method for the specific case of mixed first and second class fermionic constraints in an arbitrary space-time dimension. The cases d=3 and 10 are shown to be of prime importance. The explicit solution for d=10 is presented.

Two-dimensional boson and W-symmetry in the quantum Hall effect

We perform consistently the Gupta–Bleuler–Dirac quantization for a two-dimensional boson with parameter (α) on the circle, the boundary of the circular droplet. For α=1, we obtain the chiral (holomorphic) constraints. Using the representation of Bargmann–Fock space and the Schrodinger picture, we construct the holomorphic wave function. In order to interpret this function, we construct the coherent state representation by using the infinite-dimensional translation (W∞) symmetry for each Fourier (edge) mode. The α=1 chiral wave function explains the neutral edge states for integer quantum Hall effect very well. In the case of α=−1, we obtain a new wave function which may describe the higher modes (radial excitations) of edge states. The charged edge states are described by the |α|≠1 wave function. Finally, the application of our model to the fractional quantum Hall effect is discussed.

CANONICAL QUANTIZATION OF NONLOCAL FIELD EQUATIONS

We consistently quantize a class of relativistic nonlocal field equations characterized by a nonlocal kinetic term in the Lagrangian. We solve the classical nonlocal equations of motion for a scalar field and evaluate the on-shell Hamiltonian. The quantization is realized by imposing Heisenberg’s equation, which leads to the commutator algebra obeyed by the Fourier components of the field. We show that the field operator carries, in general, a reducible representation of the Poincaré group. We also consider the Gupta-Bleuler quantization of a nonlocal gauge theory and analyze the propagators and the physical modes of the gauge field.

Quantization of the electromagnetic field on static space–times

A quantization of the electromagnetic four-potential in the Lorentz gauge on static space–times with compact Cauchy surfaces is developed herein. For the classical problem, the field solution is obtained by use of a Hilbert space approach in which time evolution reduces to a unitary mapping of Cauchy data. For the quantum problem, a representation of the CCRs on a Fock space is constructed. The gauge condition is imposed as a constraint on state vectors and determines a physical subspace. The field equations are recovered on the physical Fock space (Gupta–Bleuler method).

On the Gupta-Bleuler Quantization of the Hamiltonian Systems with Anomalies

The Gupta-Bleuler quantization method of QED can be generalized to canonically quantized constrained systems with quantum second-class constraints. Such constraints may originate either from the second-class constraints, already presented in the classical description of the theory, or they may have their sources in quantum effects, in which case the theory is called anomalous. In this paper, I present a detailed description of how the Gupta-Bleuler ideas can be implemented in these cases and I argue that there are in principle no inconsistencies in quantum anomalous theories. Having quantized the anomalous theories canonically, I derive the path integral formulation of such theories and show that some new terms are necessarily present in this formulation. As an example, I show how the chiral Schwinger model can be quantized in the original fermionic formulation with no reference to the bosonized version used in the literature so far.

On the group theory of the polarization states of a massless field

It is shown that the theory of representations of the Poincaré group applied to the vector potentials of a massless field yields in a simple and direct way (without assuming that the field is a gauge field, and without detailed assumption on the form of the equation of motion), the structure of the polarizations, the Gupta–Bleuler condition, and gauge invariance of the theory. The method is shown to apply to the Maxwell field and to a five-dimensional generalization of the Maxwell field (which properly contains the Maxwell theory) associated with manifestly covariant dynamics.

CHIRAL BOSON, CHIRAL VACUUM AND EDGE STATES IN THE FRACTIONAL QUANTUM HALL EFFECT

By performing the Gupta–Bleuler quantization of a chiral boson, we obtain the chiral constraints, which correspond to the lowest Landau level conditions. From these, the chiral vacuum is defined as the vacuum of admixtures of many-harmonic oscillators. We construct the wave function for edge states of a droplet of incompressible quantum Hall fluid, by solving Schrödinger's equation on the basis of the chiral vacuum. This bosonic function can describe the collective edge modes, which are fundamentally a many-body effect of fermions at the lowest Landau level. In detail, the neutral edge state of FQHE is described by the α = 1 chiral boson theory. The charged edge states are described by the α ≠ 1 chiral boson theory.

THE CHIRAL BOSON THEORY AND EDGE STATES IN THE QUANTUM HALL EFFECT

We construct the wave functions for the edge states of a droplet of quantum Hall effect by performing the Gupta-Bleuler quantization of a chiral boson. These wave functions describe the chiral edge states of a many-electron system in the lowest Landau level. This demonstrates a crucial connection between the particle and condensed matter physics.

BRST formulation of the Gupta–Bleuler quantization method

In this paper it is shown how an algebra of mixed first- and second-class constraints can be transformed into an algebra of the Gupta–Bleuler type, consisting of holomorphic and antiholomorphic constraints. Its quantization is performed by BRST methods. A second-level BRST operator Ω is constructed by introducing a new ghost sector (the second-level ghosts), in addition to the ghosts of the standard BRST operator. An inner product in this ghost sector is found such that the operator Ω is Hermitean. The physical states, as defined by the non-Hermitean holomorphic constraints, are shown to be given in terms of the cohomology of this Hermitean BRST charge.

BRST cohomology and BRST gauge fixing

We discuss some general aspects of BRST cohomology. We investigate the BRST gauge-fixing procedure based on the harmonic gauges in the ghost-extended quantum mechanical Hilbert space. We obtain a general classification of states according to their BRST transformation properties. On the basis of these results we find necessary and sufficient conditions for the positivity of the physical state space and establish their equivalence with another set of conditions introduced by Spiegelglas. We apply the method to a system with a non-semisimple, solvable algebra of first-class constraints, which appears in the theory of massless higher spins. We find that both the Dirac and Gupta-Bleuler quantization procedures are contained within the BRST approach, but they correspond to different cohomology classes.

Asymptotic completeness in QED (II). Physical states

Physical states and fields in QED are defined as limits in the sense of Wightman functions of states and composite fields of the Gupta-Bleuler formalism. A formulation of asymptotic completeness proposed in an earlier publication for the Gupta-Bleuler case is transferred to the physical state space and shown to be valid in perturbation theory. An application to the calculation of inclusive cross sections is discussed.