The Operator Product Expansion Converges in Massless {\varphi_{4}^{4}}-Theory

It has been shown recently (Hollands and Kopper, Commun. Math. Phys. 313:257–290, 2012) that the mathematical status of the operator product expansion (OPE) is better than had previously been expected: namely considering massive Euclidean \({\varphi_4^4}\)-theory in the perturbative loop expansion, the OPE converges at any loop order when considering (as is usually done) composite operator insertions into correlation functions. In the present paper we prove the same result for the massless theory. While the short-distance properties of massive and massless theories may be expected to be similar on physical grounds, the proof in the massless case requires entirely new techniques, becausewe have to control with sufficient precision the exceptional momentum singularities of the massless correlation functions. Thebounds we state are organised in terms of weight factors associated to certain tree graphs (“tree dominance”). Our proof is again based on the flow equations of the renormalisation group, which we combine with such graph structures.     

The Operator Product Expansion Converges in Perturbative Field Theory

We show, within the framework of the massive Euclidean j⁴{\varphi^4} -quantum field theory in four dimensions, that the Wilson operator product expansion (OPE) is not only an asymptotic expansion at short distances as previously believed, but even converges at arbitrary finite distances. Our proof rests on a detailed estimation of the remainder term in the OPE, of an arbitrary product of composite fields, inserted as usual into a correlation function with further “spectator fields”. The estimates are obtained using a suitably adapted version of the method of renormalization group flow equations. Convergence follows because the remainder is seen to become arbitrarily small as the OPE is carried out to sufficiently high order, i.e. to operators of sufficiently high dimension. Our results hold for arbitrary, but finite, loop orders. As an interesting side-result of our estimates, we can also prove that the “gradient expansion” of the effective action is convergent.     

PCT Theorem for the Operator Product Expansion in Curved Spacetime

We consider the operator product expansion for quantum field theories on general analytic 4-dimensional curved spacetimes within an axiomatic framework. We prove under certain general, model-independent assumptions that such an expansion necessarily has to be invariant under a simultaneous reversal of parity, time, and charge (PCT) in the following sense: The coefficients in the expansion of a product of fields on a curved spacetime with a given choice of time and space orientation are equal (modulo complex conjugation) to the coefficients for the product of the corresponding charge conjugate fields on the spacetime with the opposite time and space orientation. We propose that this result should be viewed as a replacement of the usual PCT theorem in Minkowski spacetime, at least in as far as the algebraic structure of the quantum fields at short distances is concerned.     

Unified Free Quantum Fields for Massive and for Massless Particles

For any mass m ≧ 0 and arbitrary spin, free relativistic quantum fields are constructed using the same formulas for m > 0 and for m = 0. The transformation properties of these fields under P, C, T and some questions concerning super-selection rules are discussed.     

Quantum Theory of Massless Particles in Stationary Axially Symmetric Spacetimes

Quantum equations for massless particles of any spin are considered in stationary uncharged axially symmetric spacetimes. It is demonstrated that up to a normalization function, the angular wave function does not depend on the metric and practically is the same as in the Minkowskian case. The radial wave functions satisfy second order nonhomogeneous differential equations with three nonhomogeneous terms, which depend in a unique way on time and space curvatures. In agreement with the principle of equivalence, these terms vanish locally, and the radial equations reduce to the same homogeneous equations as in Minkowski spacetime.     

Asymptotic Completeness in a Class of Massless Relativistic Quantum Field Theories

This paper presents the first examples of massless relativistic quantum field theories which are interacting and asymptotically complete. These two-dimensional theories are obtained by an application of a deformation procedure, introduced recently by Grosse and Lechner, to chiral conformal quantum field theories. The resulting models may not be strictly local, but they contain observables localized in spacelike wedges. It is shown that the scattering theory for waves in two dimensions, due to Buchholz, is still valid under these weaker assumptions. The concepts of interaction and asymptotic completeness, provided by this theory, are adopted in the present investigation.     

Effective Gauge Theories of Composite W,Z and Massless Fermions

We have attempted to identify the circumstances under which a weakly coupled massive gauge theory such as the standard Glashow-Salam-Weinberg model, can emerge as a low energy effective Lagrangian of an ASF preon gauge theory. This involves many issues including the interplay between global and local symmetries and sum rules connecting long and short distance physics. The article puts together these issues and demonstrates that ASF preon models can be very tight and predictive frameworks. We make a systematic search of a minimal extension of the standard electro-weak theory, which could emerge as a low energy effective Lagrangian of an ASF preon gauge theory, in which the left-right symmetry is spontaneously broken by vacuum condensation.     

Dynamical Renormalization Group for Mode-Coupling Field Theories with Solenoidal Constraint

Journal of Statistical Physics - A correction to this paper has been published: https://doi.org/10.1007/s10955-021-02770-w     

Lorentz Covariant Canonical Formalism for Free Massive, Massless and Tachyonic Particles

We construct a consistent Lorentz-covariant canonical formalism for a free massive, massless and tachyonic particle in the framework of the absolute synchronization scheme of clocks. In the case of a massive particle our approach is canonically equivalent to the standard formulation which is not manifestly covariant. The absolute synchronization scheme seems to be the only one we can apply in the case of massless and tachyonic particles.     

Renormalization Group,Effective Action and Grand Unification Theories in Curved Space-Time

Recent developments in the renormalization group approach to grand unification theories (GUT's) in curved space-time are reviewed. The new kind of asymptotical conformal invariance in “finite” GUT's in curved space-time (with torsion) is derived. A discussion of asymptotically-finite GUT's in flat and curved space-time is presented. The modifications to the renormalization group in curved space with boundary are given. Some applications of the renormalization group are discussed as well as some prospects.     

On Superluminal Particles and the Extended Relativity Theories

Superluminal particles are studied within the framework of the Extended Relativity theory in Clifford spaces (C-spaces). In the simplest scenario, it is found that it is the contribution of the Clifford scalar component π of the poly-vector-valued momentum which is responsible for the superluminal behavior in ordinary spacetime due to the fact that the effective mass is imaginary (tachyonic). However, from the point of view of C-space, there is no superluminal (tachyonic) behavior because the true physical mass still obeys M ²>0. Therefore, there are no violations of the Clifford-extended Lorentz invariance and the extended Relativity principle in C-spaces. It is also explained why the charged muons (leptons) are subluminal while its chargeless neutrinos may admit superluminal propagation. A Born’s Reciprocal Relativity theory in Phase Spaces leads to modified dispersion relations involving both coordinates and momenta, and whose truncations furnish Lorentz-violating dispersion relations which appear in Finsler Geometry, rainbow-metrics models and Double (deformed) Special Relativity. These models also admit superluminal particles. A numerical analysis based on the recent OPERA experimental findings on alleged superluminal muon neutrinos is made. For the average muon neutrino energy of 17 GeV, we find a value for the magnitude that, coincidentally, is close to the mass of the muon m _μ =105.7 MeV.     

Renormalization of Path Ordered Phase Factors and Related Hadron Operators in Gauge Field Theories

We give a review of the renormalization and short distance properties of path ordered phase factors in nonabelian gauge field theories. It includes nonlocal gauge invariant meson, baryon and gluonium operators constructed with the help of such phase factors. Furthermore, the renormalization properties of functional derivatives of phase factors as they are needed in dynamical equations are considered. The discussion is based on an one dimensional auxiliary field formalism which enables the application of the usual language of local Green's functions.     

Universality of Operator Ordering in Kinetic Energy Operator for Particles Moving on two Dimensional Surfaces

When the motion of a particle is constrained on the two-dimensional surface, excess terms exist in usual kinetic energy 1/(2m)∑ p _i ² with hermitian form of Cartesian momentum p _i (i = 1,2,3), and the operator ordering should be taken into account in the kinetic energy which turns out to be 1/(2m)∑ (1/f _i )p _i f _i p _i where the functions f _i are dummy factors in classical mechanics and nontrivial in quantum mechanics. The existence of non-trivial f _i shows the universality of this constraint induced operator ordering in quantum kinetic energy operator for the constraint systems.     

Renormalizations and Operator Expansion in σ-Model

The operator expansion (OPE) is studied for the Green function n(0) n(x)) at x² → 0 (n(x) is the dynamical field of σ-model) in the framework of the two-dimensional σ-model with the O(N) symmetry group at large N. As a preliminary step we formulate the ronormalization scheme which permits introduction of an arbitrary intermediate scale μ² in the framework of 1/N expansion and discuss factorization (separation) of small (p < μ ) and large (p > μ ) momentum region. It is shown that definition of composite local operators and coefficient functions figuring in OPE is unambiguous only in the leading order in 1/N expansion when dominant are the solutions with exrtemum of action. Corrections of order f(μ²)/N (here /(μ²) is the effective interaction constant at the point μ²) in composite operators and coefficient functions essentially depend on factorization method of high and low momentum regions. It is shown also that contributions to the power corrections of order m²x²/(μ²)/N in the Green function (here m is the dynamical mass-scale factor in σ-model) arise simultaneously from two sources: from the mean vacuum value of the composite operator n ∂²n and from the hard particle contributions in the coefficient function of unite operator. Due to the analogy between σ-model and QCD the obtained result indicates theoretical limitations to the sum rule method in QCD.     

Quantum Double Actions on Operator Algebras and Orbifold Quantum Field Theories

Starting from a local quantum field theory with an unbroken compact symmetry group G in 1+1-dimensional spacetime we construct disorder fields implementing gauge transformations on the fields (order variables) localized in a wedge region. Enlarging the local algebras by these disorder fields we obtain a nonlocal field theory, the fixpoint algebras of which under the appropriately extended action of the group G are shown to satisfy Haag duality in every simple sector. The specifically 1+1 dimensional phenomenon of violation of Haag duality of fixpoint nets is thereby clarified. In the case of a finite group G the extended theory is acted upon in a completely canonical way by the quantum double D(G) and satisfies R-matrix commutation relations as well as a Verlinde algebra. Furthermore, our methods are suitable for a concise and transparent approach to bosonization. The main technical ingredient is a strengthened version of the split property which is expected to hold in all reasonable massive theories. In the appendices (part of) the results are extended to arbitrary locally compact groups and our methods are adapted to chiral theories on the circle. Received: 4 September 1996 / Accepted: 6 May 1997     

Theories of Variable Mass Particles and Low Energy Nuclear Phenomena

Variable particle masses have sometimes been invoked to explain observed anomalies in low energy nuclear reactions (LENR). Such behavior has never been observed directly, and is not considered possible in theoretical nuclear physics. Nevertheless, there are covariant off-mass-shell theories of relativistic particle dynamics, based on works by Fock, Stueckelberg, Feynman, Greenberger, Horwitz, and others. We review some of these and we also consider virtual particles that arise in conventional Feynman diagrams in relativistic field theories. Effective Lagrangian models incorporating variable mass particle theories might be useful in describing anomalous nuclear reactions by combining mass shifts together with resonant tunneling and other effects. A detailed model for resonant fusion in a deuterium molecule with off-shell deuterons and electrons is presented as an example. Experimental means of observing such off-shell behavior directly, if it exists, is proposed and described. Brief explanations for elemental transmutation and formation of micro-craters are also given, and an alternative mechanism for the mass shift in the Widom–Larsen theory is presented. If variable mass theories were to find experimental support from LENR, then they would undoubtedly have important implications for the foundations of quantum mechanics, and practical applications may arise.