Covariant Quantum Field Research Articles

Crossed products, conditional expectations and constraint quantization

Recent work has highlighted the importance of crossed products in correctly elucidating the operator algebraic approach to quantum field theories. In the gravitational context, the crossed product simultaneously promotes von Neumann algebras associated with subregions in diffeomorphism covariant quantum field theories from type III to type II, and provides the necessary ingredients to gravitationally dress operators, thereby enforcing the constraints of the theory. In this note we enhance the crossed product construction to the context of general gauge theories with arbitrary combinations of internal and spacetime local symmetries. This is done by leveraging the correspondence between the crossed product and the extended phase space. We then undertake a detailed study of constraint quantization from the perspective of a generic crossed product algebra. We study and compare four distinct approaches to constraint quantization from this point of view: refined algebraic quantization, BRST quantization, path integral quantization, and the commutation theorem for crossed products. Far from simply reproducing existing analyses, the operator algebraic viewpoint sheds new light on old problems by reformulating the dressing of operators in terms of conditional expectations and other closely related projection maps. We conclude by applying our approach to the constraint quantization of three distinct gauge theories including a discussion of gravity on null hypersurfaces.

Covariant quantum field theory of tachyons

Covariant quantum field theory of tachyons

Analytic constraints on the energy-momentum tensor in conformal field theories

In this work we investigate the matrix elements of the energy-momentum tensor for massless on-shell states in four-dimensional unitary, local, and Poincar\'e covariant quantum field theories. We demonstrate that these matrix elements can be parametrised in terms of covariant multipoles of the Lorentz generators, and that this gives rise to a form factor decomposition in which the helicity dependence of the states is factorised. Using this decomposition we go on to explore some of the consequences for conformal field theories, deriving the explicit analytic conditions imposed by conformal symmetry, and using examples to illustrate that they uniquely fix the form of the matrix elements. We also provide new insights into the constraints imposed by the existence of massless particles, showing in particular that massless free theories are necessarily conformal.

Construction of p-Adic Covariant Quantum Fields in the Framework of White Noise Analysis

In this article we construct a large class of interacting Euclidean quantum field theories, over a p-adic space time, by using white noise calculus. We introduce p-adic versions of the Kondratiev and Hida spaces in order to use the Wick calculus on the Kondratiev spaces. The quantum fields introduced here fulfill all the Osterwalder-Schrader axioms, except the reflection positivity.

On Wick Polynomials of Boson Fields in Locally Covariant Algebraic QFT

This work presents some results about Wick polynomials of a vector field renormalization in locally covariant algebraic quantum field theory in curved spacetime. General vector fields are pictured as sections of natural vector bundles over globally hyperbolic spacetimes and quantized through the known functorial machinery in terms of local $^*$-algebras. These quantized fields may be defined on spacetimes with given classical background fields, also sections of natural vector bundles, in addition to the Lorentzian metric. The mass and the coupling constants are in particular viewed as background fields. Wick powers of the quantized vector field are axiomatically defined imposing in particular local covariance, scaling properties and smooth dependence on smooth perturbation of the background fields. A general classification theorem is established for finite renormalization terms (or counterterms) arising when comparing different solutions satisfying the defining axioms of Wick powers. The result is specialized to the case of general tensor fields. In particular, the case of a vector Klein-Gordon field and the case of a scalar field renormalized together with its derivatives are discussed as examples. In each case, a more precise statement about the structure of the counterterms is proved. The finite renormalization terms turn out to be finite-order polynomials tensorially and locally constructed with the backgrounds fields and their covariant derivatives whose coefficients are locally smooth functions of polynomial scalar invariants constructed from the so-called marginal subset of the background fields. The notion of local smooth dependence on polynomial scalar invariants is made precise in the text.

A small non-vanishing cosmological constant from the Krein–Gupta–Bleuler vacuum

We point out a potential relevance between the Krein–Gupta–Bleuler (KGB) vacuum leading to a fully covariant quantum field theory for gravity in de Sitter (dS) spacetime and the observable smallness of the cosmological constant. This may provide a formulation of linear quantum gravity in a framework amenable to developing a more complete theory determining the value of the cosmological constant.

Composite graviton self-interactions in a model of emergent gravity

We consider a theory of scalars minimally coupled to an auxiliary background metric. The theory is generally covariant and subject to the constraint of vanishing energy-momentum tensor. Eliminating the auxiliary metric leads to a reparametrization invariant, non-polynomial, metric-independent action for the scalar fields. Working in the limit of a large number of physical scalars, a composite massless spin-2 state, the graviton, was identified in previous work, in a two-into-two scalar scattering process. Here, we further explore the possibility that dynamical emergent gravity is a natural feature of generally covariant quantum field theories, by studying the self-interactions of the emergent composite graviton. We show that the fine-tuning previously imposed to ensure the vanishing of the cosmological constant, as well as the existence of the massless spin-2 state, also assures that the emergent graviton's cubic self-interactions are consistent with those of Einstein's general relativity, up to higher-derivative corrections. We also demonstrate in a theory with more than one type of scalar that the composite graviton coupling is universal.

Casimir energy–momentum tensor for curved boundaries in de Sitter space-time

In the present paper, we study Casimir effect for a conformally coupled scalar field propagating on background of $$4+1$$ -dimensional de Sitter (dS) space-time. The field is supposed to obey Dirichlet boundary condition on two 4-dimensional curved boundaries. Technically, we accomplish our calculations using an indefinite field quantization scheme, which already has been successfully applied to the dS minimally coupled massless scalar field and the dS linear gravity to obtain a causal and fully covariant quantum field on dS space-time.

Canonical quantization of the covariant fields on de Sitter space–times

The properties of the covariant quantum fields on de Sitter space–times 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 [Formula: see text] group, transforming alike the particle and antiparticle field operators in momentum representation. Their basis generators and Casimir operators are written down finding that the covariant representations are equivalent with unitary irreducible ones from the principal series whose canonical weights are determined by the fermion mass and spin.

Vacuum states for gravitons field in de Sitter space

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 requirement of the inner product. However, one may still look for states with as much symmetry as possible, more precisely, a restrictive version of covariance by considering the gravitons field and the associated vacuum state which are, respectively, covariant and invariant with respect to some maximal subgroup of the full de Sitter group. In this regard, we treat $SO(4)$ case, and find a family of $SO(4)$-invariant states. The associated $SO(4)$-covariant quantum field is given, as well.

Quantum Field Theories on Categories Fibered in Groupoids

We introduce an abstract concept of quantum field theory on categories fibered in groupoids over the category of spacetimes. This provides us with a general and flexible framework to study quantum field theories defined on spacetimes with extra geometric structures such as bundles, connections and spin structures. Using right Kan extensions, we can assign to any such theory an ordinary quantum field theory defined on the category of spacetimes and we shall clarify under which conditions it satisfies the axioms of locally covariant quantum field theory. The same constructions can be performed in a homotopy theoretic framework by using homotopy right Kan extensions, which allows us to obtain first toy-models of homotopical quantum field theories resembling some aspects of gauge theories.

Hadamard States for Quantum Abelian Duality

Abelian duality is realized naturally by combining differential cohomology and locally covariant quantum field theory. This leads to a C$^*$-algebra of observables, which encompasses the simultaneous discretization of both magnetic and electric fluxes. We discuss the assignment of physically well-behaved states to such algebra and the properties of the associated GNS triple. We show that the algebra of observables factorizes as a suitable tensor product of three C$^*$-algebras: the first factor encodes dynamical information, while the other two capture topological data corresponding to electric and magnetic fluxes. On the former factor we exhibit a state whose two-point correlation function has the same singular structure of a Hadamard state. Specifying suitable counterparts also on the topological factors we obtain a state for the full theory, providing ultimately a unitary implementation of Abelian duality.

Galilean covariance, Casimir effect and Stefan–Boltzmann law at finite temperature

The Galilean covariance, formulated in 5-dimensions space, describes the nonrelativistic physics in a way similar to a Lorentz covariant quantum field theory being considered for relativistic physics. Using a nonrelativistic approach the Stefan–Boltzmann law and the Casimir effect at finite temperature for a particle with spin zero and 1/2 are calculated. The thermo field dynamics is used to include the finite temperature effects.

Differential cohomology and locally covariant quantum field theory

We study differential cohomology on categories of globally hyperbolic Lorentzian manifolds. The Lorentzian metric allows us to define a natural transformation whose kernel generalizes Maxwell's equations and fits into a restriction of the fundamental exact sequences of differential cohomology. We consider smooth Pontryagin duals of differential cohomology groups, which are subgroups of the character groups. We prove that these groups fit into smooth duals of the fundamental exact sequences of differential cohomology and equip them with a natural presymplectic structure derived from a generalized Maxwell Lagrangian. The resulting presymplectic Abelian groups are quantized using the [Formula: see text]-functor, which yields a covariant functor from our categories of globally hyperbolic Lorentzian manifolds to the category of [Formula: see text]-algebras. We prove that this functor satisfies the causality and time-slice axioms of locally covariant quantum field theory, but that it violates the locality axiom. We show that this violation is precisely due to the fact that our functor has topological subfunctors describing the Pontryagin duals of certain singular cohomology groups. As a byproduct, we develop a Fréchet–Lie group structure on differential cohomology groups.

Revisiting the equivalence of light-front and covariant QED in the light-cone gauge

We discuss the equivalence between light-front time-ordered-perturbation theory and covariant quantum field theory in light-front quantization, in the case of quantum electrodynamics at one-loop level. In particular, we review the one-loop calculation of the vertex correction, fermion self-energy and vacuum polarization. We apply the procedure of integration by residue over the light-front energy in the loop to show how the perturbative expansion in covariant terms can be reduced to a sum of propagating and instantaneous diagrams of light-front time-ordered perturbation theory. The detailed proof of equivalence between the two formulations of the theory resolves the controversial question on which form should be used for the gauge-field propagator in the light-cone gauge in the covariant approach.

Casimir energy-momentum tensor for a quantized bulk scalar field in the geometry of two curved branes on Friedmann-Robertson-Walker background

In a previous work [S. Rahbardehghan et al. in Phys. Lett. B 750, 627 (2015)], we considered a simple brane-world model; a single $4$-dimensional brane embedded in a $5$-dimensional de Sitter (dS) space-time. Then, by including a conformally coupled scalar field in the bulk, we studied the induced Casimir energy-momentum tensor. Technically, the Krein-Gupta-Bleuler (KGB) quantization scheme as a covariant and renormalizable quantum field theory in dS space was used to perform the calculations. In the present paper, we generalize this study to a less idealized, but physically motivated, scenario, namely we consider Friedmann-Robertson-Walker (FRW) space-time which behaves asymptotically as a dS space-time. More precisely, we evaluate Casimir energy-momentum tensor for a system with two $D$-dimensional curved branes on background of $D+1$-dimensional FRW space-time with negative spatial curvature and a conformally coupled bulk scalar field that satisfies Dirichlet boundary condition on the branes.

Quantum Gravity from the Point of View of Locally Covariant Quantum Field Theory

We construct perturbative quantum gravity in a generally covariant way. In particular our construction is background independent. It is based on the locally covariant approach to quantum field theory and the renormalized Batalin–Vilkovisky formalism. We do not touch the problem of nonrenormalizability and interpret the theory as an effective theory at large length scales.

Abelian Duality on Globally Hyperbolic Spacetimes

We study generalized electric/magnetic duality in Abelian gauge theory by combining techniques from locally covariant quantum field theory and Cheeger-Simons differential cohomology on the category of globally hyperbolic Lorentzian manifolds. Our approach generalizes previous treatments using the Hamiltonian formalism in a manifestly covariant way and without the assumption of compact Cauchy surfaces. We construct semi-classical configuration spaces and corresponding presymplectic Abelian groups of observables, which are quantized by the CCR-functor to the category of $C^*$-algebras. We demonstrate explicitly how duality is implemented as a natural isomorphism between quantum field theories. We apply this formalism to develop a fully covariant quantum theory of self-dual fields.

Locally covariant quantum field theory and the spin–statistics connection

The framework of locally covariant quantum field theory (QFT), an axiomatic approach to QFT in curved spacetime (CST), is reviewed. As a specific focus, the connection between spin and statistics is examined in this context. A new approach is given, which allows for a more operational description of theories with spin and for the derivation of a more general version of the spin–statistics connection in CSTs than previously available. This part of the text is based on [C. J. Fewster, arXiv:1503.05797.] and a forthcoming publication; the emphasis here is on the fundamental ideas and motivation.

Examining a covariant and renormalizable quantum field theory in de Sitter space by studying “black hole radiation”

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 calculation of expectation values of the energy–momentum tensor (we have an automatic and covariant renormalization) which results in the vacuum energy of the free field to vanish. However, the existence of an effective potential barrier, intrinsically created by black holes gravitational field, gives a Casimir-type contribution to the vacuum expectation value of the energy–momentum tensor. On this basis, by evaluating the Casimir energy–momentum tensor for a conformally coupled massless scalar field in the vicinity of a nonrotating black hole event horizon through the KGB quantization, in this work, we explicitly prove that the hole produces black-body radiation which its temperature exactly coincides with the result obtained by Hawking for black hole radiation.