Relativistic non-instantaneous action-at-a-distance interactions

Relativistic action-at-a-distance theories with interactions that propagate at the speed of light in vacuum are investigated. We consider the most general action depending on the velocities and relative positions of the particles. The Poincaré invariant parameters that label successive events along the world lines can be identified with the proper times of the particles provided that certain conditions are imposed on the interaction terms in the action. Further conditions on the interaction terms arise from the requirement that mass be a scalar. A generic class of theories with interactions that satisfy these conditions is found. The relativistic equations of motion for these theories are presented. We obtain exact circular orbits solutions of the relativistic one-body problem. The exact relativistic one-body Hamiltonian is also derived. The theory has three components: a linearly rising potential, a Coulomb-like interaction and a dynamical component to the Poincaré invariant mass. At the quantum level we obtain the generalized Klein–Gordon–Fock equation and the Dirac equation.     

Path integral representations in noncommutative quantum mechanics and noncommutative version of Berezin–Marinov action

It is known that the actions of field theories on a noncommutative space-time can be written as some modified (we call them θ-modified) classical actions already on the commutative space-time (introducing a star product). Then the quantization of such modified actions reproduces both space-time noncommutativity and the usual quantum mechanical features of the corresponding field theory. In the present article, we discuss the problem of constructing θ-modified actions for relativistic QM. We construct such actions for relativistic spinless and spinning particles. The key idea is to extract θ-modified actions of the relativistic particles from path-integral representations of the corresponding noncommutative field theory propagators. We consider the Klein–Gordon and Dirac equations for the causal propagators in such theories. Then we construct for the propagators path-integral representations. Effective actions in such representations we treat as θ-modified actions of the relativistic particles. To confirm the interpretation, we canonically quantize these actions. Thus, we obtain the Klein–Gordon and Dirac equations in the noncommutative field theories. The θ-modified action of the relativistic spinning particle is just a generalization of the Berezin–Marinov pseudoclassical action for the noncommutative case.     

Parity conservation and the Euclidean field formulation of relativistic quantum field theories

A method to construct Euclidean covariant fields corresponding to a relativistic quantum field theory with arbitrary spins is presented. The constructed fields act on a state space with an indefinite inner product, they commute (or anticommute) totally and (except for hermitian Fermion fields) adjoint relativistic fields correspond to adjoint Euclidean fields. The cases where this method can be applied include all Gårding-Wightman theories invariant under space inversion.     

Vertex calculation for a composite particle

The classical treatment and the quantization of composite relativistic systems is given a manifestly covariant formulation in presence of constraints. A particular formulation of Feynman's quantum mechanics is used to treat the scattering of composite relativistic systems. A covariant harmonic oscillator model is employed to calculate vertices of interactions: the results are similar to the corresponding ones in the usual field theories, but the presence of some convergence factors gives hope that a theory with composite particles may be finite.     

An algebraic <Emphasis Type="Italic">PT</Emphasis>-symmetric quantum theory with a maximal mass

In this paper, we draw attention to the fact that the studies by V.G. Kadyshevsky devoted to the creation of the geometric quantum field theory with a fundamental mass have had great development recently, as regards a non-Hermitian algebraic approach to construction of the quantum theory. The central idea of such theories is to construct a new scalar product in which the average values of non-Hermitian Hamiltonians are real. Many studies in this field include both purely mathematical ones and those containing the discussion of experimental results. We consider the development of an algebraic relativistic pseudo-Hermitian quantum theory with a maximal mass and discuss its experimentally important corollaries.     

Hyperfunction Quantum Field Theory: Localized Fields Without Localized Test Functions

This note addresses the problem of localization in quantum field theory; more specifically we contribute to the ongoing discussion about the most appropriate concept of localization which one should use in relativistic quantum field theory: through localized test functions or through the fields directly without localized test functions. In standard quantum field theory, i.e., in relativistic quantum field theory in terms of tempered distributions according to Gårding and Wightman, this is done through localized test functions. In hyperfunction quantum field theory (HFQFT), i.e., relativistic quantum field theory in terms of Fourier hyperfunctions this is done through the fields themselves. In support of the second approach we show here that it has a much wider range of applicability. It can even be applied to relativistic quantum field theories which do not admit compactly supported test functions at all. In our construction of explicit models we rely on basic results from the theory of quasi-analytic functions.     

A UV completion of scalar electrodynamics

In previous works, we constructed UV-finite and unitary scalar field theories with an infinite spectrum of propagating modes for arbitrary polynomial interactions. In this paper, we introduce infinitely many massive vector fields into a U(1) gauge theory to construct a theory with UV-finiteness and unitarity.     

An intuitive form of non-localism for quantum mechanics

The two-slit interference experiment provides a vivid demonstration of quantum mechanics. It is clear that one can describe the effects using standard probability theory if one makes a non-localism assumption. But many scientists dismiss non-localism as implying “action-at-a-distance”. This paper shows that it is possible to construct a non-local model which does not assume “action-at-a-distance”.     

What is the relativistic generalization of a linearly rising potential?

Recent progress in summing graphs of non-Abelian gauge theories focuses attention on certain manifestly Lorentz-invariant classical action-at-a-distance theories, whose solution allows for the construction of field-theoretic Green functions in the WKB approximation. The sum of graphs is of QED type, except that the gluon propagator is modified to incorporate the renormalization-group-invariant charge $\text{g}\text{(k)}$. The purely phenomenological choice $\text{g}{}^2\text{～ k}{}^{\mathrm{?2}}$ is equivalent non-relativistically to a linearly rising potential and yields a fully relativistic classical action-at-a-distance theory with exactly soluble circular orbits, whose Bohr quantization yields an asymptotic approximation to the poles of the Green function. One finds asymptotically linear Regge trajectories, but other phenomenological aspects are not as promising when only a linearly rising potential is used (in common with other phenomenological studies). As a redeeming feature, there do not appear to be any obvious pathologies of the sort familiar from string theories and ad hoc generalizations of linearly rising potentials.     

Comment Triton Binding Energy and Minimal Relativity

For relativistic three-body calculations, essentially two different approaches are in use: field theory and relativistic direct interactions. However, while results based upon relativistic field theory show an increase of the triton binding energy by about 0.3 MeV due to relativistic effects, calculations that claim to apply relativistic direct interactions obtain 0.3 MeV repulsion. In this paper, we discuss the origin of such a discrepancy. We show that the use of an invariant two-body amplitude increases the triton binding energy by about 0.3 MeV, consistent with the results from relativistic field theory. Furthermore, we point out that in calculations relying on the direct-interactions approach, indeed expansions are used, which may be a bad approximation and the reason for the discrepancy. Received November 4, 1996; revised January 15, 1998; accepted for publication January 19, 1998     

Action-at-a-distance electrodynamics in quasi-steady-state cosmology

Action-at-a-distance electrodynamics – alternative approach to field theory – can be extended to cosmological models using conformal symmetry. An advantage of this is that, the origin of arrow of time in electromagnetism can be attributed to the cosmological structure. Different cosmological models can be investigated, based on Wheeler–Feynman absorber theory, and only those models can be considered viable for our Universe which have net full retarded electromagnetic interactions, i.e., forward direction of time. This work evaluates the quasi-steady-state model and demonstrates that it admits full retarded and not advanced solution. Thus, quasi-steady-state cosmology (QSSC) satisfies this necessary condition for a correct cosmological model, based on action-at-a-distance formulation.     

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     

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.     

Perturbative symplectic field theory and Wigner function

We study relativistic quantum field theories in phase space, based on representations of the Poincaré group, using the Moyal product. We develop a perturbative theory for quantizing fields, with functional methods in phase space. The two-point function is related to relativistic Wigner functions for bosons and fermions. As an example we analyze the complex scalar field with quartic self-interaction.     

The pion propagator in relativistic quantum field theories of the nuclear many-body problem

Pion interactions in the nuclear medium are studied using renormalizable relativistic quantum field theories. Previous studies using pseudoscalar πN coupling encountered difficulties due to the large strength of the πNN vertex. We therefore formulate renormalizable field theories with pseudovector πN coupling using techniques introduced by Weinberg and Schwinger. Calculations are performed for two specific models: the scalar-vector theory of Walecka, extended to include π and ρ mesons in a non-chiral fashion, and the linear σ-model with an additional neutral vector meson. Both models qualitatively reproduce low-energy πN phenomenology and lead to nuclear matter saturation in the relativistic Hartree formalism, which includes baryon vacuum fluctuations. The pion propagator is evaluated in the onenucleon-loop approximation, which corresponds to a relativistic random-phase approximation built on the Hartree ground state. Virtual $\text{N}\text{N}$ loops are included, and suitable renormalization techniques are illustrated. The local-density approximation is used to compare the threshold pion self-energy to the s-wave pion-nucleus optical potential. In the non-chiral model, s-wave pion-nucleus scattering is too large in both pseudoscalar and pseudovector calculations, indicating that additional constraints must be imposed on the lagrangian. In the chiral model, the threshold self-energy vanishes automatically in the pseudovector case, but does so for pseudoscalar coupling only if the baryon effective mass is chosen self-consistently. Since extrapolation from free space to nuclear density can lead to large effects, pion propagation in the medium can determine which πN coupling is more suitable for the relativistic nuclear many-body problem. Conversely, pion interactions constrain the model lagrangian and the nuclear matter equation of state. An approximately chiral model with pseudovector coupling is favored. The techniques developed here allow for a consistent treatment of these models using renormalizable relativistic quantum field theores.     

Newtonian versus relativistic nonlinear cosmology

Both for the background world model and its linear perturbations Newtonian cosmology coincides with the zero-pressure limits of relativistic cosmology. However, such successes in Newtonian cosmology are not purely based on Newton's gravity, but are rather guided ones by previously known results in Einstein's theory. The action-at-a-distance nature of Newton's gravity requires further verification from Einstein's theory for its use in the large-scale nonlinear regimes. We study the domain of validity of the Newtonian cosmology by investigating weakly nonlinear regimes in relativistic cosmology assuming a zero-pressure and irrotational fluid. We show that, first, if we ignore the coupling with gravitational waves the Newtonian cosmology is exactly valid even to the second order in perturbation. Second, the pure relativistic correction terms start appearing from the third order. Third, the correction terms are independent of the horizon scale and are quite small in the large-scale near the horizon. These conclusions are based on our special (and proper) choice of variables and gauge conditions. In a complementary situation where the system is weakly relativistic but fully nonlinear (thus, far inside the horizon) we can employ the post-Newtonian approximation. We also show that in the large-scale structures the post-Newtonian effects are quite small. As a consequence, now we can rely on the Newtonian gravity in analyzing the evolution of nonlinear large-scale structures even near the horizon volume.     

Superfield Formulation for Non-Relativistic Chern–Simons-Matter Theory

We construct a superfield formulation for non-relativistic Chern–Simons-matter theories with manifest dynamical supersymmetry. By eliminating all the auxiliary fields, we show that the simple action reduces to the one obtained by taking non-relativistic limit from the relativistic Chern–Simons-matter theory proposed in the literature. As a further application, we give a manifestly supersymmetric derivation of the non-relativistic ABJM theory.      

Construction of gauge invariant effective nucleonic theories: Functional approach

Starting from relativistic quantum field theories, describing interacting nucleons and pions coupled to the dynamical electromagnetic field, the pion degrees of freedom are eliminated by means of functional integration. Apart from taking into account some operators perturbatively in e, e.g. the vacuum polarization, this procedure is exact, giving effective theories for nucleons and photons. The subsequent nonrelativistic reduction yields the corresponding nonrelativistic quantum field.theory. The latter is unique, irrespective of the precise form of the original nucleon-pion interaction. Nucleonic potentials and electromagnetic interactions are mutually consitent. Local gauge invariance is satisfied at any stage of the formal development.     

Self-Consistent Approximations in Relativistic Plasmas: Quasiparticle Analysis of the Thermodynamic Properties

We generalize the concept of conserving, Φ-derivable, approximations to relativistic field theories. Treating the interaction field as a dynamical degree of freedom, we derive the thermodynamic potential in terms of fully dressed propagators, an approach which allows us to resolve the entropy of a relativistic plasma into contributions from its interacting elementary excitations. We illustrate the derivation for a hot relativistic system governed by electromagnetic interactions.     

Point-form quantum field theory and meson form factors

We comment on canonical quantization of relativistic field theories on a Lorentz-invariant surface of the form x ² = τ². By this choice of the quantization surface all components of the four-momentum operator become interaction dependent, whereas the generators of Lorentz transformations stay free of interactions – a feature characteristic for Dirac’s “point-form” of relativistic dynamics. In the sequel we demonstrate how field theoretical concepts may enter the framework of relativistic quantum mechanics. To this aim we employ a Poincaré-invariant approximation scheme, which allows to reduce a field theoretical many-body problem to a multichannel problem for a Bakamjian-Thomas-type mass operator. As an application of this multichannel formalism we will discuss the scattering of an electron by a (confined) quark-antiquark pair. It will be sketched how an electromagnetic meson form factor can be extracted from the one-photon exchange optical potential.