Relativistic Nonlocal Quantum Field Theory

A theory is defined to be relativistic if its Hamiltonian, total momenta, and boost's generators satisfy commutation relations of the Poincaré group. Field theories with usual local interactions are known to be relativistic. A simple example of a relativistic nonlocal theory is found. However, it has divergences. Some conditions are obtained which are necessary in order that a nonlocal theory be relativistic and divergenceless.     

Relativistic Quantum Mechanics and Field Theory

The problems which arise for a relativistic quantum mechanics are reviewed and critically examined in connection with the foundations of quantum field theory. The conflict between the quantum mechanical Hilbert space structure, the locality property and the gauge invariance encoded in the Gauss' law is discussed in connection with the various quantization choices for gauge fields.     

Remarks on Causality in Relativistic Quantum Field Theory

It is shown that the correlations predicted by relativistic quantum field theory in locally normal states between projections in local von Neumann algebras (V ₁),(V ₂) associated with spacelike separated spacetime regions V ₁,V ₂ have a (Reichenbachian) common cause located in the union of the backward light cones of V ₁ and V ₂. Further comments on causality and independence in quantum field theory are made. Originally published in International Journal of Theoretical Physics, Vol. 44, No. 7, 2005,Due to a publishing error, authorship of the article was credited incorrectly. The corrected article is reprinted in its entirety here. The online version of the original article can be found at     

Propagation of Relativistic Quantum Particles in Spacetime in Quantum Field Theory

The paper deals with a recent systematic study of the propagation of relativistic quantum particles in spacetime. This study was a reaction to the overwhelming number of experiments dealing with the localization of not only massive but also of photons by detectors. The method of study is based on a configuration unitarity expansion of the vacuum-to-vacuum transition amplitudes as, massive and massless, particles propagate between emitters and detectors. Topics treated are the amplitudes of propagation from one time-space coordinate to another, limiting velocities of particles and their reconciliations with relativity, emergence of particles into cones in detection regions versus the direction of their moments, stimulated emissions by external sources in spacetime, scattering theory in quantum field theory in configuration space, and finally a spacetime for mulation of closed-time path for multi-particle states.     

Spin in Relativistic Quantum Theory

We discuss the role of spin in Poincaré invariant formulations of quantum mechanics.     

Steps Towards the Axiomatic Foundations of the Relativistic Quantum Field Theory: Spin-Statistics, Commutation Relations, and CPT Theorems

A realistic physical axiomatic approach of the relativistic quantum field theory is presented. Following the action principle of Schwinger, a covariant and general formulation is obtained. The correspondence principle is not invoked and the commutation relations are not postulated but deduced. The most important theorems such as spin-statistics, and CPT are proved. The theory is constructed form the notion of basic field and system of basic fields. In comparison with others formulations, in our realistic approach fields are regarded as real things with symmetry properties. Finally, the general structure is contrasted with other formulations.     

On Hot Bangs and the Arrow of Time in Relativistic Quantum Field Theory

 A recently proposed method for the characterization and analysis of local equilibrium states in relativistic quantum field theory is applied to a simple model. Within this model states are identified which are locally (but not globally) in thermal equilibrium and it is shown that their local thermal properties evolve according to macroscopic equations. The largest space–time regions in which local equilibrium states can exist are timelike cones. Thus, although the model does not describe dissipative effects, such states fix in a natural manner a time direction. Moreover, generically they determine a distinguished space–time point where a singularity in the temperature (a hot bang) must have occurred if local equilibrium prevailed thereafter. The results illustrate how the breaking of the time reflection symmetry at macroscopic scales manifests itself in a microscopic setting. Received: 17 January 2003 / Accepted: 5 March 2003 Published online: 17 April 2003 Communicated by H. Araki and K. Fredenhagen     

Relativistic Quantum Field Inertia and Vacuum Field Noise Spectra

The equivalence statements for quantum scalar field vacuum states that havebeen used for the thermal-like Hawking effect and Unruh effect are surveyed.An important ingredient in this framework is the concept of a vacuum field noisespectrum, by which one can obtain information about the curvature invariants ofclassical worldlines (relativistic classical trajectories). It is argued, in the spiritof the free-fall-type universality, that the preferred quantum field vacua withrespect to accelerated worldlines should be chosen from the class of all thosepossessing stationary spectra for their quantum fluctuations. For scalar quantumfield vacua there are six stationary cases, as shown by Letaw some time ago,and reviewed here. However, nonstationary vacuum noises can be treated by afew mathematical methods that are mentioned as well. Since the informationabout the kinematical curvature invariants of the worldlines is of radiometricorigin, suggestions are given on the more useful application of such an academicformalism to radiation and beam radiometric standards for high-energyaccelerators and in astrophysics. We conclude with a look at related axiomaticquantum field topics and some other recent work.     

Four-Body Bound-States Systems in Relativistic Scalar Quantum Field Theory: Variational Basis-State Approach

The variational method within the Hamiltonian formalism of quantum field theory has been used in order to investigate the effect of virtual pairs for four-body scalar systems consisting of two particles and two antiparticles of the same mass. The scalar particles and antiparticles interact via a massive or massless mediating scalar field. The ground state energy solutions of Fock-space variational trial states (\(|N{\bar {N}}N{\bar {N}}{ \rangle }+|N{\bar {N}}N{\bar {N}}N{\bar {N}\rangle }\)) of the relativistic wave equations have been studied. We have compared these results with the previous work of four-body system (variational trial states of the form \(|N{ \bar {N}}N{\bar {N}}{\rangle }\)) and we have shown that the inclusion of virtual pairs has a noticeable effect at low coupling and at high coupling the energy of the system is changed by an important amount. In other words, the calculations show that the inclusion of virtual pairs augments the binding energy of the four-body system by a substantial amount at strong coupling. This study can pave the way for some new ideas in order to investigate the effect of virtual pairs, for example, for a bound-states quark-antiquark or tetraquark systems in future.     

Stability of Covariant Relativistic Quantum Theory

In this paper we study the relativistic quantum-mechanical interpretation of the solution of the inhomogeneous Euclidean Bethe-Salpeter equation. Our goal is to determine conditions on the input to the Euclidean Bethe-Salpeter equation so the solution can be used to construct a model Hilbert space and a dynamical unitary representation of the Poincaré group. We prove three theorems that relate the stability of this construction to properties of the kernel and driving term of the Bethe-Salpeter equation. The most interesting result is that the positivity of the Hilbert space norm in the non-interacting theory is not stable with respect to Euclidean covariant perturbations defined by Bethe-Salpeter kernels. The long-term goal of this work is to understand which model Euclidean Green functions preserve the underlying relativistic quantum theory of the original field theory. Understanding the constraints imposed on the Green functions by the existence of an underlying relativistic quantum theory is an important consideration for formulating field-theory motivated relativistic quantum models.This work supported in part by the U.S. Department of Energy, under contract DE-FG02-86ER40286     

Non-perturbative Methods in Relativistic Field Theory

This talk reviews relativistic methods used to compute bound and low energy scattering states in field theory, with emphasis on approaches that John Tjon and I discussed (and argued about) together. I compare the Bethe–Salpeter and Covariant Spectator equations, show some applications, and then report on some of the things we have learned from the beautiful Feynman–Schwinger technique for calculating the exact sum of all ladder and crossed ladder diagrams in field theory.     

Finsler Geometry and Relativistic Field Theory

Finsler geometry on the tangent bundle appears to be applicable to relativistic field theory, particularly, unified field theories. The physical motivation for Finsler structure is conveniently developed by the use of gauge transformations on the tangent space. In this context a remarkable correspondence of metrics, connections, and curvatures to, respectively, gauge potentials, fields, and energy-momentum emerges. Specific relativistic electromagnetic metrics such as Randers, Beil, and Weyl can be compared.     

Necessary and sufficient conditions for current conservation in the framework of Relativistic Quantum Field Theory

Necessary and sufficient conditions for the conservation of vector currents covariant under translations and of the Conformal currents, are formulated and proven. Only the weak spectrum condition is assumed, thus the theorems serve as generalizations of a well-known theorem due to Coleman. This is done under the axioms of Relativistic Quantum Field Theory due to Wightman.     

General Requirements for a Relativistic Quantum Theory

The equations of a relativistic quantum theory for two or more particles should satisfy at least the following criteria. (1) They should be Poincaré invariant. (2) The cluster property should hold. (3) Causality should not be violated over distances much larger than the Compton wavelengths of the particles involved. (4) The electromagnetic interaction between charged particles should be formulated in a gauge-invariant way. (5) If, for a two-particle system, one of the masses becomes infinitely large, the equations should reduce to the relevant relativistic equation for the other particle. (6) In the nonrelativistic limit the equation should reduce to the Schr?dinger equation. In this paper it will be shown how a quasi-potential theory, which was introduced many years ago [1] and which was applied to a number of systems [2–12], meets all these requirements. Received March 25, 1997; revised July 15, 1997; accepted for publication March 18, 1998