On locality in quantum general relativity and quantum gravity

The physical concept of locality is first analyzed in the special relativistic quantum regime, and compared with that of microcausality and the local commutativity of quantum fields. Its extrapolation to quantum general relativity on quantum bundles over curved spacetime is then described. It is shown that the resulting formulation of quantum-geometric locality based on the concept of local quantum frame incorporating a fundamental length embodies the key geometric and topological aspects of this concept. Taken in conjunction with the strong equivalence principle and the path-integral formulation of quantum propagation, quantum-geometric locality leads in a natural manner to the formulation of quantum-geometric propagation in curved spacetime. Its extrapolation to geometric quantum gravity formulated over quantum spacetime is described and analyzed.     

On the general covariance and strong equivalence principles in quantum general relativity

The various physical aspects of the general relativistic principles of covariance and strong equivalence are discussed, and their mathematical formulations are analyzed. All these aspects are shown to be present in classical general relativity, although no contemporary formulation of canonical or covariant quantum gravity has succeeded to incorporate them all. This has, in part, motivated the recent introduction of a geometro-stochastic framework for quantum general relativity, in which the classical frame bundles that underlie the formulation of parallel transport in classical general relativity are replaced by quantum frame bundles. It is shown that quantum frames can take over the role played by complete sets of observables in conventional quantum theory, so that they can mediate the natural transference of the general covariance and the strong equivalence principles from the classical to the quantum general relativistic regime. This results in a geometrostochastic mode of quantum propagation in general relativistic quantum bundles, which is mathematically implemented by path integration methods based on parallel transport along horizontal lifts of geodesics for the vacuum expectation values of a quantum gravitational field in a quantum spacetime supermanifold. The covariance features of this field are embedded in a quantum gravitational supergroup, which incorporates Poincaré as well as diffeomorphism invariance, and resolves the issue of time in quantum gravity.     

Covariant canonical quantization of fields and Bohmian mechanics

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 non-covariant 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-reparametrization invariant system and quantum gravity is discussed and compared with the conventional non-covariant Wheeler-DeWitt approach.Received: 11 October 2004, Published online: 6 July 2005PACS: 04.20.Fy, 04.60.Ds, 04.60.Gw, 04.60.-m     

On the Occurrence of Mass in Field Theory

This paper proves that it is possible to build a Lagrangian for quantum electrodynamics which makes it explicit that the photon mass is eventually set to zero in the physical part on observational ground. Gauge independence is achieved upon considering the joint effect of gauge-averaging term and ghost fields. It remains possible to obtain a counterterm Lagrangian where the only non-gauge-invariant term is proportional to the squared divergence of the potential, while the photon propagator in momentum space falls off like k ^–2 at large k which indeed agrees with perturbative renormalizability. The resulting radiative corrections to the Coulomb potential in QED are also shown to be gauge-independent. The experience acquired with quantum electrodynamics is used to investigate properties and problems of the extension of such ideas to non-Abelian gauge theories.     

Dimensional regularization of gauge theories in spherical spacetime: Free field trace anomalies

The O(n + 1) covariant formulation of massless quantum electrodynamics in spherical spacetime is further developed to allow a calculation of the energy-momentum tensor trace anomalies for the free Dirac, electromagnetic, and SU(2) gauge fields. The principal technical development is the construction of the Faddeev-Popov ghosts for electrodynamics and SU(2) Yang-Mills theory. This construction is unconventional first in that the gauge fixing term in the Lagrangian is not a perfect square, and second because it is necessary to remove radial as well as gauge degrees of freedom from the measure of the functional integral. The ghost fields are shown to satisfy a minimal scalar field equation. The free field effective action is found to be divergent in four dimensions, and is renormalized by the inclusion in the Lagrangian of a counterterm local in the gravitational fields. The energy-momentum tensor calculated from this renormalized effective action is shown to have a trace anomaly.     

Quantum mechanics of space and time

A formulation of relativistic quantum mechanics is presented independent of the theory of Hilbert space and also independent of the hypothesis of spacetime manifold. A hierarchy is established in the nondistributive lattice of physical ensembles, and it is shown that the projections relating different members of the hierarchy form a semigroup. It is shown how to develop a statistical theory based on the definition of a statistical operator. Involutions defined on the matrix representations of the semigroup are interpreted in terms ofCPT conjugations. The theory of particles of spin one-half and systems with higher spin is developed from first principles. Methods are also developed for defining energy, momentum, orbital angular momentum, and weighted spacetime coordinates without reference to a manifold.     

Polarized Spacetime Foam

An approximate model of a spacetime foam is presented. It is supposed that in the spacetime foam each quantum handle is like to an electric dipole and therefore the spacetime foam is similar to a dielectric. If we neglect of linear sizes of the quantum handle then it can be described with an operator containing a Grassman number and either a scalar or a spinor field. For both fields the Lagrangian is presented. For the scalar field it is the dilaton gravity + electrodynamics and the dilaton field is a dielectric permeability. The spherically symmetric solution in this case give us the screening of a bare electric charge surrounded by a polarized spacetime foam and the energy of the electric field becomes finite one. In the case of the spinor field the spherically symmetric solution give us a ball of the polarized spacetime foam filled with the confined electric field. It is shown that the full energy of the electric field in the ball can be very big.     

A Spin-Statistics Theorem for Quantum Fields¶on Curved Spacetime Manifolds¶in a Generally Covariant Framework

A model-independent, locally generally covariant formulation of quantum field theory over four-dimensional, globally hyperbolic spacetimes will be given which generalizes similar, previous approaches. Here, a generally covariant quantum field theory is an assignment of quantum fields to globally hyperbolic spacetimes with spin-structure where each quantum field propagates on the spacetime to which it is assigned. Imposing very natural conditions such as local general covariance, existence of a causal dynamical law, fixed spinor- or tensor type for all quantum fields of the theory, and that the quantum field on Minkowski spacetime satisfies the usual conditions, it will be shown that a spin-statistics theorem holds: If for some of the spacetimes the corresponding quantum field obeys the “wrong” connection between spin and statistics, then all quantum fields of the theory, on each spacetime, are trivial. Received: 1 March 2001 / Accepted: 28 May 2001     

Axiomatic Quantum Field Theory in Curved Spacetime

The usual formulations of quantum field theory in Minkowski spacetime make crucial use of features—such as Poincaré invariance and the existence of a preferred vacuum state—that are very special to Minkowski spacetime. In order to generalize the formulation of quantum field theory to arbitrary globally hyperbolic curved spacetimes, it is essential that the theory be formulated in an entirely local and covariant manner, without assuming the presence of a preferred state. We propose a new framework for quantum field theory, in which the existence of an Operator Product Expansion (OPE) is elevated to a fundamental status, and, in essence, all of the properties of the quantum field theory are determined by its OPE. We provide general axioms for the OPE coefficients of a quantum field theory. These include a local and covariance assumption (implying that the quantum field theory is constructed in a local and covariant manner from the spacetime metric and other background structure, such as time and space orientations), a microlocal spectrum condition, an “associativity” condition, and the requirement that the coefficient of the identity in the OPE of the product of a field with its adjoint have positive scaling degree. We prove curved spacetime versions of the spin-statistics theorem and the PCT theorem. Some potentially significant further implications of our new viewpoint on quantum field theory are discussed.     

Wavefront Sets in Algebraic Quantum Field Theory

The investigation of wavefront sets of n-point distributions in quantum field theory has recently acquired some attention stimulated by results obtained with the help of concepts from microlocal analysis in quantum field theory in curved spacetime. In the present paper, the notion of wavefront set of a distribution is generalized so as to be applicable to states and linear functionals on nets of operator algebras carrying a covariant action of the translation group in arbitrary dimension. In the case where one is given a quantum field theory in the operator algebraic framework, this generalized notion of wavefront set, called “asymptotic correlation spectrum”, is further investigated and several of its properties for physical states are derived. We also investigate the connection between the asymptotic correlation spectrum of a physical state and the wavefront sets of the corresponding Wightman distributions if there is a Wightman field affiliated to the local operator algebras. Finally we present a new result (generalizing known facts) which shows that certain spacetime points must be contained in the singular supports of the 2n-point distributions of a non-trivial Wightman field. Received: 27 July 1998 / Accepted: 3 March 1999     

Higher-Order Kinetic Term for Controlling Photon Mass in Off-Shell Electrodynamics

In relativistic classical and quantum mechanics with Poincaré-invariant parameter, particle worldlines are traced out by the evolution of spacetime events. The formulation of a covariant canonical framework for the evolving events leads to a dynamical theory in which mass conservation is demoted from a priori constraint to the status of conserved Noether current for a certain class of interactions. In pre-Maxwell electrodynamics—the local gauge theory associated with this framework —events induce five local off-shell fields, which mediate interactions between instantaneous events, not between the worldlines which represent entire particle histories. The fifth field, required to compensate for dependence of gauge transformations on the evolution parameter, enables the exchange of mass between particles and fields. In the equilibrium limit, these pre-Maxwell fields are pushed onto the zero-mass shell, but during interactions there is no mechanism regulating the mass that photons may acquire, even when event trajectories evolve far into the spacelike region. This feature of the off-shell formalism requires the application of some ad hoc mechanism for controlling the photon mass in two opposite physical domains: the low energy motion of a charged event in classical Coulomb scattering, and the renormalization of off-shell quantum electrodynamics. In this paper, we discuss a nonlocal, higher derivative correction to the photon kinetic term, which provides regulation of the photon mass in a manner which preserves the gauge invariance and Poincaré covariance of the original theory. We demonstrate that the inclusion of this term is equivalent to an earlier solution to the classical Coulomb problem, and that the resulting quantum field theory is renormalized.     

Noncommutativity and Lorentz violation in relativistic heavy ion collisions

We show that relativistic heavy ion collisions at LHC energies could be used as an experimental probe to detect fundamental properties of spacetime long speculated about. Our results rely on the recent proposal that magnetic fields of intensity much larger than that of magnetars should be produced at the beginning of the collisions and this could have an important impact on the experimental manifestation of a noncommutative spacetime. Indeed, in the noncommutative generalization of electrodynamics the interplay between a nonzero noncommutative parameter and an external magnetic field leads us to predict the production of lepton pairs of low invariant mass by free photons (an event forbidden by Lorentz invariant electrodynamics) in relativistic heavy ion collisions at present and future available energies. This unique channel can be clearly considered as a signature of noncommutativity. On the other hand, the search for such decays is worth anyway because their absence would ameliorate of three orders of magnitude the current bound on the noncommutative parameter.     

Quantum Theory of Path-Dependent Field Operator Based on Characteristics of Displacement Operators

The path-dependent operator formalism of quantum electrodynamics proposed by Mandelstam is reformulated through quantum field theory based on characteristics of displacement operators in Minkowski space. It is shown that total energy- and total angular-momentum operators can generate inhomogeneous Lorentz transformations on any local operator including path-independent bilinear forms constructed of path-dependent electron operator Ψ(x, P), but that generators for Ψ(x, P) itself are only their Ψ(x, P)-dependent parts. Such an unfamiliar feature is characteristic of the path-dependent operator formalism. The present approach possesses unique merits in making the logic of the formalism transparent as described in the following: i) Quantum electrodynamics can be formulated but for the help of potential operator even as a tool for calculation up to a final step. ii) Some restriction, which can be used to discuss propriety of gauge conditions, can be figured out. iii) By introducing a path-rearrangement operator, we can keep infinite variety of space-like pathes with the same end point throughout our formulation as they stand. iv) Several points which must be modified in the presence of magnetic monopole are closed up.     

Method of semiclassical representation in quantum electrodynamics

The operators of the classical amplitudes of an electromagnetic field are introduced and a method of transferring from quantum electrodynamics to the semiclassical approximation both in the case of a free field and in the case of the interaction of the field with a quantum system is given. The method considered enables one to set up solutions of quantum electrodynamics in the case of an intense field from the solutions of the semiclassical problem. An operator method of obtaining solutions of the equations of semiclassical electrodynamics is considered. The physical meaning of the quantum corrections to the semiclassical electrodynamics of an intense field is discussed.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 77–98, February, 1980.     

Stochastic electrodynamics. IV. Transitions in the perturbed harmonic oscillator-zero-point field system

In this fourth paper in a series on stochastic electrodynamics (SED), the harmonic oscillator-zero-point field system in the presence of an arbitrary applied classical radiation field is studied further. The exact closed-form expressions are found for the time-dependent probability that the oscillator is in the nth eigenstate of the unperturbed SED Hamiltonian H ₀ , the same H ₀ as that of ordinary quantum mechanics. It is shown that an eigenvalue of H ₀ is the average energy that the oscillator would have if its wave function could be just the corresponding eigenstate. The level shift for each unperturbed eigenvalue is found and shown to be unobservable for a different reason than in the corresponding QED treatment. Perturbation theory is applied to the SED Schrödinger equation to derive first-order transition rates for spontaneous emission and resonance absorption. The results agree with those of quantum electrodynamics, but the mathematics is strikingly different. It is shown that SED demands discarding the ideas of quantized energies, photons, and completeness of the Schrödinger equation, Finally, an intuitive physical SED model is suggested for the photoeffect and for Clauser's ⁽²⁾ coincidence experiment.     

Symmetry breaking in dynamical spacetimes

The behaviour of scalar electrodynamics under symmetry breaking by the Higgs mechanism is studied in a class of dynamical spacetimes—those having a Bianchi type I symmetry—which includes the spatially flat Robertson-Walker spacetimes. The Einstein equations are used to obtain the effective Lagrangian, from which it is shown thateither the gauge field does not become massive during symmetry breaking (in marked contrast with the case in which the background spacetime is static),or the symmetry breaking chooses not only a direction in the phase space of the Higgs field, but also a spatial direction in the spacetime sections.     

Quantum General Invariance and Loop Gravity

A quantum physical projector is proposed for generally covariant theories which are derivable from a Lagrangian. The projector is the quantum analogue of the integral over the generators of finite one-parameter subgroups of the gauge symmetry transformations which are connected to the identity. Gauge variables are retained in this formalism, thus permitting the construction of spacetime area and volume operators in a tentative spacetime loop formulation of quantum general relativity.     

Can Spacetime be a Condensate?

We explore further the proposal [Hu, B. L. (1996). General relativity as geometro-hydrodynamics. (Invited talk at the Second Sakharov Conference, Moscow, May 1996); gr-qc/9607070.] that general relativity is the hydrodynamic limit of some fundamental theories of the microscopic structure of spacetime and matter, i.e., spacetime described by a differentiable manifold is an emergent entity and the metric or connection forms are collective variables valid only at the low-energy, long-wavelength limit of such micro-theories. In this view it is more relevant to find ways to deduce the microscopic ingredients of spacetime and matter from their macroscopic attributes than to find ways to quantize general relativity because it would only give us the equivalent of phonon physics, not the equivalents of atoms or quantum electrodynamics.It may turn out that spacetime is merely a representation of certain collective state of matter in some limiting regime of interactions, which is the view expressed by Sakharov [Sakharov, A. D. (1968). Soviet Physics-Doklady 12, 1040–1041; Sakharov, A. D. (1967). Vacuum quantum fluctuations in curved space and the theory of gravitation. Doklady Akad. Nauk S.S.R. 177, 70; Adler, S. L. (1982). Reviews of Modern Physics 54, 729]. In this talk, working within the conceptual framework of geometro-hydrodynamics, we suggest a new way to look at the nature of spacetime inspired by Bose–Einstein condensate (BEC) physics. We ask the question whether spacetime could be a condensate, even without the knowledge of what the‘atom of spacetime’ is. We begin with a summary of the main themes for this new interpretation of cosmology and spacetime physics, and the ‘bottom-up’ approach to quantum gravity. We then describe the ‘Bosenova’ experiment of controlled collapse of a BEC and our cosmology-inspired interpretation of its results. We discuss the meaning of a condensate in different context. We explore how far this idea can sustain, its advantages and pitfalls, and its implications on the basic tenets of physics and existing programs of quantum gravity.