Gauge invariance in fractional field theories

The principle of local gauge invariance is applied to fractional wave equations and the interaction term is determined up to order in the coupling constant . Based on the Riemann-Liouville fractional derivative definition, the fractional Zeeman effect is used to reproduce the baryon spectrum accurately. The transformation properties of the non-relativistic fractional Schrödinger-equation under spatial rotations are investigated and an internal fractional spin is deduced.     

Supersymmetric gauge theories in quantum mechanics

A general construction for supersymmetric U(1) gauge invariant Hamiltonians in quantum mechanics is given. For a given number of fermionic and bosonic degrees of freedom it is shown that for four supercharges the interactions are determined uniquely, and coincide with the dimensionally reduced N = 1, d = 3 + 1 supersymmetric electrodynamics. With two supercharges one gets models which cannot be obtained through dimensional reduction. For two special choices of a parameter one recovers the dimensionally reduced d = 1 + 1 Weyl supersymmetric and Majorana supersymmetric electrodynamics.     

Gauge invariance properties of transition amplitudes in gauge theories. I

The functional approach developed earlier for scattering theory in quantum field theory makes it possible to make an explicit and complete study of the gauge invariance properties oftransition amplitudes (not just of the gauge transformations of Green's functions) in covariant and noncovariant gauges. This paper is devoted to the Abelian gauge theory of quantum electrodynamics. Using the powerful technique of functional differentiation and starting from the Coulomb gauge, the gauge invariance property of transition amplitudes,up to gauge-dependent scaling factors, isexplicitly established in arbitrary gauges. The key ingredients in the analysis are the derived exact expression for the vacuum-to-vacuum transition amplitude, introducing in the process arbitrary gauges, and the idea of stimulated emissions by external sources studied earlier.     

The background field method and the S-matrix

We prove that the S-matrix can be correctly obtained from the gauge-invariant effective action in the background field approach to gauge theories. In addition, we present a computation of the two-loop fermionic contributions to the Yang-Mills β-function.     

Differentiable probabilities: A new viewpoint on spin,gauge invariance,gauge fields,and relativistic quantum mechanics

A new approach to developing formulisms of physics based solely on laws of mathematics is presented. From simple, classical statistical definitions for the observed space-time position and proper velocity of a particle having a discrete spectrum of internal states we derive u generalized Schrödinger equation on the space-time manifold. This governs the evolution of an N component wave function with each component square integrable over this manifold and is structured like that for a charged particle in an electromagnetic field but also includes SU(N) gauge field couplings. This construction reveals a new hasis for gauge invariance and new insight into the appearance of spin and other such properties in relativistic quantum mechanics and suggests a new charged particle model.     

Gauge dependence of world lines and invariance of theS-matrix in relativistic classical mechanics

The notion of world lines is studied in the constraint Hamiltonian formulation of relativistic point particle dynamics. The particle world lines are shown to depend in general (in the presence of interaction) on the choice of the equal-time hyperplane (the only exception being the elastic scattering of rigid balls). However, the relative motion of a two-particle system and the (classical)S-matrix are independent of this choice.A preliminary version of this paper was circulated as ICTP, Trieste, Internal Report IC/79/59On leave of absence from Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Sofia 1184, Bulgaria     

Isospectral transformations in relativistic and nonrelativistic mechanics

The modified Korteweg de-Vries hierarchy of partial differential equations generating transformations of the one-dimensional Dirac equation, is shown to reduce in the limitc to the Korteweg de-Vries hierarchy, generating isospectral transformations of the Schrödinger equation. The former hierarchy reduces into relativistic and the latter into nonrelativistic isoperiodic transformation in the limit0.     

Thermodynamic calculations in relativistic finite-temperature quantum field theories

A Minkowski space formalism of finite-temperature quantum field theory is used to compute static thermodynamic quantities in the one- and two-loop approximation in an elegant and straightforward way using a generalization of Weinberg's tadpole method of calculating effective potentials. Systematic diagrammatic techniques for low- and high-temperature expansions are developed. Renormalizability by zero-temperature symmetric counterterms is proven for all orders in the loop expansion and demonstrated explicitly to two loops. Many useful computational techniques applicable to general finite temperature calculations are explained.     

Exact factorized S-matrix of the chiral field in two dimensions

The exact factorized S-matrix and the spectrum for the two-dimensional SU(N)-principal chiral field are presented.     

Yang-Mills instantons and the S-matrix

We present a scheme for calculating gauge-invariant S-matrix elements in the presence of instantons. We exploit the conformal invariance of the zero-mass field equations. The asymptotic in and out states are defined by their values on null infinity $\text{J}$. We use this method to calculate to lowest-order S-matrix elements for scalar particles and fermions in a dilute gas of SU(2) instantons and anti-instantons. The scalar particles acquire an effective mass and an effective interaction of the form $\text{exp}\text{(?(?}{}^2\text{/16π) ?}\text{?}\text{)}$, where ? is the scale of the instanton, plus other interactions which cannot be presented by a local effective lagrangian. The fermions acquire the effective lagrangian obtained by 't Hooft. In the case of a single flavour of fermions, this corresponds to a mass term.     

Subsidiary conditions and physical S-matrix unitarity in indefinite-metric quantum gravitation theory

The quantum theory of gravitational fields is formulated in a manifestly Lorentz covariant manner in the framework of indefinite-metric quantum field theory. The physical state subspace is defined by the two subsidiary conditions Q_B|phys〉 = Q_c|phys〉 = 0, where the conserved charges Q_B and Q_c are the generators of the BRS transformation ond of the Faddeev-Popov ghost scale transformation, respectively. By clarifying the metric structures of the Fock space of asymptotic fields with help of the Ward-Takahashi identities, the physical S-matrix unitarity is established in just the same way as in the canonical theory of Yang-Mills fields.     

Renormalization character and quantum S-matrix for a classically integrable theory

The quantum theory of a N-component generalization of the sine-Gordon model is investigated. We find at the one-loop order that the model is renormalizable only when the corresponding classical theory is completely integrable: N = 1 (sine-Gordon model) and N = 2 (reduced O(4) σ-model). Moreover the coupling constant does nor renormalize in these two cases. Although the S-matrix for N = 2 is factorizable at the tree level, an anomaly appears at the one-loop order. Its effect is like a local quartic coupling.     

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.     

Incompatibility of gauge invariance and nonrelativistic locality in path integral formulation

It is shown that a modification of the usual gauge transformations is essential to the path integral formulation of nonrelativistic quantum mechanics as a consequence of defining the locality condition as follows: The contribution from each path comes entirely from the points on the path. Arguments are based on the similarity between Wiener and Feynman functional (path) integrals.     

Unitarity and the energy averaged S-matrix

A unitarity relation between mean width $\text{γ}$, level distance D and the absorptive part of the energy averaged S-matrix is proved and discussed. Consequences for the determination of $\text{γ}\text{D}$ are pointed out.