Covariant Quantum Field Research Articles

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.

CHARGED SECTORS, SPIN AND STATISTICS IN QUANTUM FIELD THEORY ON CURVED SPACETIMES

The first part of this paper extends the Doplicher–Haag–Roberts theory of superselection sectors to quantum field theory on arbitrary globally hyperbolic spacetimes. The statistics of a superselection sector may be defined as in flat spacetime and each charge has a conjugate charge when the spacetime possesses non-compact Cauchy surfaces. In this case, the field net and the gauge group can be constructed as in Minkowski spacetime. The second part of this paper derives spin-statistics theorems on spacetimes with appropriate symmetries. Two situations are considered: First, if the spacetime has a bifurcate Killing horizon, as is the case in the presence of black holes, then restricting the observables to the Killing horizon together with "modular covariance" for the Killing flow yields a conformally covariant quantum field theory on the circle and a conformal spin-statistics theorem for charged sectors localizable on the Killing horizon. Secondly, if the spacetime has a rotation and PT symmetry like the Schwarzschild–Kruskal black holes, "geometric modular action" of the rotational symmetry leads to a spin-statistics theorem for charged covariant sectors where the spin is defined via the SU(2)-covering of the spatial rotation group SO(3).

CHERN–SIMONS GRAVITY, WILSON LINES AND LARGE-N DUAL GAUGE THEORIES

A five-dimensional Chern–Simons gravity theory based on the anti-de Sitter group SO (4, 2) is argued to be a useful model in which to understand the details of holography and the relationship between generally covariant and dual local quantum field theories. Defined on a manifold with boundary, conformal geometry arises naturally as a gauge invariance preserving boundary condition. By matching thermodynamic quantities for a particular background geometry, the dimensionless coupling constant of the Chern–Simons theory is directly related to the number of fields in a putative dual theory at high temperature. As a consistency check, the semiclassical factorization of Wilson line observables in the gravity theory is shown to induce a factorization in dual theory observables as expected by general arguments of large-N gauge theory.

Translocality and a duality principle in generally covariant quantum field theory

It is argued that the formal rules of correspondence between local observation procedures and observables do not exhaust the entire physical content of generally covariant quantum field theory. This result is obtained by expressing the distinguishing features of the local kinematical structure of quantum field theory in the generally covariant context in terms of a translocal structure which carries the totality of the non-local kinematic information in a local region. This gives rise to a duality principle at the dynamical level which emphasizes the significance of the underlying translocal structure for modelling a minimal algebra around a given point. We discuss the emergence of classical properties from this point of view.

Reeh-Schlieder Meets Newton-Wigner

The Reeh-Schlieder theorem asserts the vacuum and certain other states to be spacelike superentangled relative to local quantum fields. This motivates an inquiry into the physical status of various concepts of localization. It is argued that a covariant generalization of Newton-Wigner localization is a physically illuminating concept. When analyzed in terms of nonlocally covariant quantum fields, creating and annihilating quanta in Newton-Wigner localized states, the vacuum is seen to not possess the spacelike superentanglement that the Reeh-Schlieder theorem displays relative to local fields, and to be locally empty as well as globally empty. Newton-Wigner localization is then shown to be physically interpretable in terms of a covariant generalization of the center of energy, the two localizations being identical if the system has no internal angular momentum. Finally, some of the counterintuitive features of Newton-Wigner localization are shown to have close analogues in classical special relativity.

Algebraic Holography

A rigorous (and simple) proof is given that there is a one-to-one correspondence between causal anti-deSitter covariant quantum field theories on anti-deSitter space and causal conformally covariant quantum field theories on its conformal boundary. The correspondence is given by the explicit identification of observables localized in wedge regions in anti-deSitter space and observables localized in double-cone regions in its boundary. It takes vacuum states into vacuum states, and positive-energy representations into positive-energy representations.

Representation of the Resonance of a Relativistic Quantum Field Theoretical Lee-Friedrichs Model in Lax-Phillips Scattering Theory

The quantum mechanical description of the evolution of an unstable system defined initially as a state in a Hilbert space at a given time does not provide a semigroup (exponential) decay law. The Wigner-Weisskopf survival amplitude, describing reversible quantum transitions, may be dominated by exponential type decay in pole approximation at times not too short or too long, but, in the two channel case, for example, the pole residues are not orthogonal, and the evolution does not correspond to a semigroup (experiments on the decay of the neutral $K$-meson system strongly support the semigroup evolution postulated by Lee, Oehme and Yang, and Yang and Wu). The scattering theory of Lax and Phillips, originally developed for classical wave equations, has been recently extended to the description of the evolution of resonant states in the framework of quantum theory. The resulting evolution law of the unstable system is that of a semigroup, and the resonant state is a well-defined function in the Lax-Phillips Hilbert space. In this paper we apply this theory to a relativistically covariant quantum field theoretical form of the (soluble) Lee model. We construct the translation representations with the help of the wave operators, and show that the resulting Lax-Phillips $S$-matrix is an inner function (the Lax-Phillips theory is essentially a theory of translation invariant subspaces). In the special case that the $S$-matrix is a rational inner function, we obtain the resonant state explicitly and analyze its particle ($V, N, \theta$) content. If there is an exponential bound, the general case differs only by a so-called trivial inner factor, which does not change the complex spectrum, but may affect the wave function of the resonant state.

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.

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.

Covariant SPDEs and quantum field structures

Covariant stochastic partial differential equations (SPDEs) are studied in any dimension. A special class of such equations is selected and it is proved that the solutions can be analytically continued to Minkowski spacetime yielding tempered Wightman distributions which are covariant, obey the locality axiom and a weak form of the spectral axiom.

CONFORMAL INVARIANCE AND GRAVITATIONAL COUPLING IN QUANTUM FIELD THEORY

We study the quantum constraints of a conformalinvariant action for a scalar field. For this purpose webriefly present a reformulation of the duality principleadvanced earlier in the context of generally covariant quantum field theory, and apply it toexamine the finite structure of the quantum constraints.This structure is shown to admit a dimensional coupling(a coupling mediated by a dimensional coupling parameter) of states to gravity. Invariancebreaking is introduced by defining a preferredconfiguration of dynamical variables in terms of thelargescale characteristics of the universe. In thisconfiguration a close relationship between the quantumconstraints and the Einstein equations isestablished.

Problems of dynamics in generally covariant quantum field theory

Problems connected with the structural aspects of dynamics are addressed in the context of the algebraic approach to generally convariant quantum field theory. It is argued that the dynamical structure of observables in the generally convariant context becomes fundamentally state dependent. This makes it necessary to relate the entire dynamics to state-dependent automorphisms of the algebra of observables. The relevant states are highly correlated on large scales, so that we may not have exact accuracy for the identification of their observables in terms of a (quasi) local net of algebras. This feature is controlled by a scale fluctuation of the total observables around a point which is used to obtain a description of a one-parameter group of state-dependent automorphisms in terms of the modular group. In general, it is not clear whether the action of the latter group has a dynamical interpretation. We comment on a duality principle which could provide a straightforward means to obtain an “asymptotic” interpretation of the modular group on small scales.

Equivalence of light-front and covariant field theory.

In this paper we discuss the relation between the standard covariant quantum field theory and light-front field theory. We define covariant theory by its Feynman diagrams, whereas light-front field theory is defined in terms of light-cone time-ordered diagrams. A general algorithm is proposed that produces the latter from any Feynman diagram. The procedure is illustrated in several cases. Technical problems that occur in the light-front formulation and have no counterpart in the covariant formulation are identified and solved.

Outline of a generally covariant quantum field theory and a quantum theory of gravity

We study a tentative generally covariant quantum field theory, denoted the T theory, as a tool to investigate the consistency of quantum general relativity. The theory describes the gravitational field and a minimally coupled scalar field; it is based on the loop representation, and on a certain number of quantization choices. Four-dimensional diffeomorphism-invariant quantum transition probabilities can be computed from the theory. We discuss the choices on which the T theory relies, and the possibilities of modifying them.

Von Neumann algebra automorphisms and time-thermodynamics relation in generally covariant quantum theories

We consider the cluster of problems raised by the relation between the notion of time, gravitational theory, quantum theory and thermodynamics; in particular, we address the problem of relating the `timelessness' of the hypothetical, fundamental generally covariant quantum field theory with the `evidence' of the flow of time. By using the algebraic formulation of quantum theory, we propose a unifying perspective on these problems, based on the hypothesis that in a generally covariant quantum theory the physical time flow is not a universal property of the mechanical theory, but rather it is determined by the thermodynamical state of the system (`thermal time hypothesis'). We implement this hypothesis by using a key structural property of von Neumann algebras: the Tomita--Takesaki theorem, which allows us to derive a time flow, namely a one-parameter group of automorphisms of the observable algebra, from a generic thermal physical state. We study this time flow, its classical limit, and we relate it to various characteristic theoretical facts, such as the Unruh temperature and the Hawking radiation. We point out the existence of a state-independent notion of `time', given by the canonical one-parameter subgroup of outer automorphisms provided by the co-cycle Radon--Nikodym theorem.

A generally covariant quantum field theory and a prediction on quantum measurements of geometry

We define a generally covariant quantum field theory with an infinite number of degrees of freedom. This is obtained by combining the Husain-Kuchař model, the Loop Representation, the idea of defining diffeomorphism invariant observables in terms of material reference systems, and the Ashtekar-Isham C ∗-algebra representation theory. The theory can be seen as a c → 0 limit of general relativity coupled with material objects. The construction of the quantum theory can be completed to the point where physical expectation values can be computed, and quantitative physical predictions about gauge-invariant observables can be made within the model. The first physical result that we obtain is that the area of physical surfaces (defined by the matter coupled to the gravitational field) is quantized in units of A 0 = h ̵ G/2c 3 . This result was anticipated as a non-gauge-invariant result within quantum general relativity, but it becomes a definite result in the present model. It supports the prediction that the quantization of the area is a genuine physical effect. We discuss the meaning of this prediction in terms of quantum measurement theory, and the assumptions from which this prediction follows.

Unitarity condition in covariant quantum field theory with indefinite metric

Unitarity condition in covariant quantum field theory with indefinite metric

Topological quantum field theory

A twisted version of four dimensional supersymmetric gauge theory is formulated. The model, which refines a nonrelativistic treatment by Atiyah, appears to underlie many recent developments in topology of low dimensional manifolds; the Donaldson polynomial invariants of four manifolds and the Floer groups of three manifolds appear naturally. The model may also be interesting from a physical viewpoint; it is in a sense a generally covariant quantum field theory, albeit one in which general covariance is unbroken, there are no gravitons, and the only excitations are topological.

On generally covariant quantum field theory and generalized causal and dynamical structures

We give an example of a generally covariant quasilocal algebra associated with the massive free field. Maximal, two-sided ideals of this algebra are algebraic representatives of external metric fields. In some sense, this algebra may be regarded as a concrete realization of Ekstein's ideas of presymmetry in quantum field theory. Using ideas from our example and from usual algebraic quantum field theory, we discuss a generalized scheme, in which maximal ideals are viewed as algebraic representatives of dynamical equations or Lagrangians. The considered frame is no quantum gravity, but may lead to further insight into the relation between quantum theory and space-time geometry.

Generally covariant quantum field theory and scaling limits

The formulation of a generally covariant quantum field theory is described. It demands the elimination of global features and a characterization of the theory in terms of the allowed germs of families of states. A simple application is the computation of counting rates of accelerated idealized detectors. As a first orientation we discuss here the consequences of the assumption that the states have a short distance scaling limit. The scaling limit at a point gives a reduction of the theory to tangent space. It contains kinematical information but not the full dynamical laws. The reduced theory will, under rather general conditions, be invariant under translations and under a proper subgroup of the linear transformations in tangent space. One interesting possibility is that it is invariant under SLR(4). Then the macroscopic metric must evolve as a cooperative effect in finite size regions. The other natural possibility is that each family (coherent folium) of states defines a microscopic metric by the scaling limit and the tangent space theory reduces to a theory of free massless fields in a Minkowski space. Irrespective of the assumption of a scaling limit we show that the theory can be constructed from strictly local information.