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Dynamical symmetry breakdown of a gauge invariant Lagrangian may fix physical coupling constants via a condition unrelated to the Gell-Mann-Low equation. This condition arises when the Lagrangian couples each separate gauge field to massless fermions of definite chirality.

The operators in a Wilson expansion are not in general multiplicatively renormalized in non-Abelian gauge theories. This is because of the renormalization of the gauge transformations themselves. Renormalized fields may be defined, which have the old gauge transformations. Alternatively, a special choice of gauge may be made, in which the gauge transformations are unchanged on renormalization. In any case, one gauge invariant factor appears in the renormalization of the Wilson operators.

The renormalization is performed in a manifestly covariant approach. The simplest form of the Ward identities z1=z2=…=zn=… is fulfilled automatically in every gauge. In the Yang-Mills theory the counter-terms are gauge-invariant and depend on the charge renormalization constant only. In pure gravitation the analysis of all divergences is reduced in the present approach to some special classification of the kth order scalar densities in a Riemannian space.

The divergent parts of vertex functions are calculated in the one-loop approximation of a unitary gauge theory. It is shown that finiteS-matrix elements may only be obtained provided one modifies the usual renormalization preseription.

By a generalization of canonical quantum electrodynamics with invariant one-parameter gauge families, a wide theory of neutral vector fields is formulated in a unified way that describes massive and massless fields on the same basis. Similarly to the case of quantum electrodynamics, there exist two invariant gauge families in the sense that any two gauges belonging to the same family are connected with each other by a generalized q-number gauge transformation. It is shown that the theory is manifestly renormalizable and the unitarity of the S-matrix is guaranteed by conservation of currents. Renormalization problems in this formalism are investigated.

A new gauge condition which leads directly to the 'multipole' hamiltonian for the interaction of molecules with the electromagnetic field is studied within the context of quantum electrodynamics ie when the molecules and field are considered as a closed dynamical system. The new formalism which maintains the advantages of the usual Coulomb gauge theory but avoids the difficulties associated with the presence of unphysical terms in the propagation function of the Coulomb gauge vector potential, is thus obtained without recourse to the unitary transformation of the usual Coulomb gauge formalism which has previously been used to achieve this result.

The anomalous dimensions of the twist-4 operators which determine the asymptotic behavior of radiative corrections are computed for non-Abelian gauge theories. In Lorentz gauges, the ghost problem appears to be highly non-trivial, and there is no simple set of operators which yield consistent results. However in the ghost-free axial gauges, we find a gauge invariant result for the expansion of two gauge invariant currents.

A comprehensive discussion is given of the renormalization of gauge theories, with or without spontaneous breakdown of gauge symmetry. The present discussion makes use of the Ward-Takahashi identities for proper vertices (as opposed to the identities for Green's functions) recently derived. The following features of the present discussion are significant: (1) The present discussion applies to a very wide class of gauge conditions. (2) The present discussion applies to any gauge group and any representation of the scalar fields. (3) The renormalized S matrix is shown to be gauge-independent. (4) Dependence of counterterms on the gauge chosen is discussed.

If the symmetry of the theory under global transformations generated by the charges is normal, the physical states of the system must be "color" singlets. (This is analogous to the physical states of two-dimensional quantum electrodynamics being neutral.) Consequently, the local color currents vanish in physical states. The (two-dimensional) inhomogeneous Lorentz invariance of the theory is also discussed.

Null-plane commutators of the currents in the spontaneously broken gauge theory of vector dominance are discussed, in comparison with those of the traditional field algebra model, using Dirac’s canonical formalism.

An alternative to the Higgs-Kibble mechanism for the formulation of spontaneously broken gauge theories has been recently proposed. It is shown how the new procedures can be generalised to the case when non-Abelian gauge groups are present.

Relations between gauge invariance and the conservation laws are discussed relative to the Korteweg‐deVries and related equations. The technique used here is conventional in field theory‐the canonical formalism related to the invariant variational problem. The function of constants of motion, when represented in terms of canonical variables, is to generate infinitesimal partial gauge transformations of any order, under which the variational problem is invariant. From this point of view, one example is presented together with the general theory.

An invariant functional formulation of the nonlinear chiral theories as well as of their generalizations is developed. To guarantee the manifest invariance of the generating functional under different choices of the local coordinates in the inner space (parametrization) new variables are utilized which are the left side of the classical equations of motions (with or without the source). In terms of the new variables an invariant regularization is introduced and an invariant perturbation theory is developed.

Ward -- Takahashi -- Slavnov identities were recently formulated by Lee in terms of generating functionals for non-Abelion gauge theories. These results are analyzed and extended to the most general gauge theories. (LBS)

Non-Abelian gauge theories involving massive spin-1/2 fermions and scalar fields without spontaneous symmetry breaking are considered in an arbitrary gauge. The renormalization group equation and its solution are derived in the one-loop approximation, using a massindependent dimensional regularization procedure, in terms of effective coupling constant, mass and gauge parameters. In the case of anSU2 theory, the various Ward-Taylor identities, and the gauge independence of physical quantities are confirmed in the course of studying some properties of the solutions of the renormalization group equation. Some extensions of results to larger groups and more complicated couplings are given.

Renormalizable field theories of locally coupled fermion fields can be formulated with subcanonical fields. The canonical formulation of theories with subcanonical fields is given. We study the behaviour of gauge fields which arise as bound states in these theories and propose a way to incorporate vacuum asymmetries (and as a consequence more degrees of freedom for low energies) like violation of scale invariance and isospin.

The advantages are pointed out of using a certain class of manifestly unitary gauges in theories of spontaneously broken gauge symmetry. These gauges are particularly useful in the recent theories of strong interactions.

Possible gauge structure associated with the q-number gauge transformation generated by a dipole gaugeon field is extensively studied in the framework of canonical quantum electrodynamics. The gauge structure is classified into two types, which will be referred to as Type I and Type II, respectively; A typical gauge structure in Type I is the one presented in a preceding paper. In Type I all possible one-parameter covariant gauges are divided into two gauge families complementary to each other, each of which is invariant with respect to the gauge transformation, while in Type II there does not occur such a division for gauges and a single gauge family covers the whole region for gauges. Circum­ stances of the gauge shift through renormalization and structure of binary dipole systems consisting of the longitudinal photon and the gaugeon fields are discussed for each of these· two types of gauge structure.

It is shown by means of a model that the renormalization and unitary gauges can be connected by a point transformation, and this fact is used to construct a formal proof of renormalization in the unitary gauge. The formal proof is then verified by demonstrating that for a fourth-order on-shell scattering process the S -matrix calculated directly in the unitary gauge is exactly equal to that calculated in the renormalization gauge. The calculation is refined to the point where it becomes purely graphical and this allows one to see by inspection how the cancellation of divergences occurs in the unitary gauge. The model considered here is Abelian, but it will be generalized to the non-Abelian case subsequently.

In a gauge theory of scalar mesons interacting with gauge vector particles, we show that scalar-meson four-point Green’s functions are free of infinities, provided that i) the scalar mesons do not couple to any fermions in the model and ii) there is no directλϕ4-term in the Lagrangian. The proof relies on a «gauge approximation» technique which systematically exploits the information provided by Ward-Takahashi identities. For non-Abelian gauge theories, the significance of this result is that a large class of scalar multiplets which can induce spontaneous symmetry breaking do not affect the issue of asymptotic freedom. In order to exploit the information of Ward-Takahashi identities we use the «axial gauge», in which there are no fictitious scalar particles and the identities assume their naive form, withZ1 =Z2 for all matter and gauge fields.

Canonical quantum electrodynamics in general one-parameter covariant gauges is for­ mulated by introducing two kinds of dipole-ghost fields. One corresponds to the longitudinal photon field, while the other is called a gaugeon field. None of these dipole ghosts are included in physical states which satisfy supplementary conditions. There are two invariant gauge families in the sense that two arbitrary gauges belonging to the same family are connected with each other by a q-number gauge transformation in a manifestly covariant way. The shift of gauge through renormalization is naturally derived from the gauge struc­ ture of the formalism.

Feynman's functional formulation of statistical mechanics is used to study general-relativistic quantum field theories at finite temperature. The techniques are then applied to gauge theories. The partition function $\mathrm{Tr}{e}^{\ensuremath{-}\ensuremath{\beta}H}$ is discovered to be a gauge-dependent quantity which is meaningless in most gauges. Instead, we define a physically meaningful partition function which is gauge-invariant and only equal to $\mathrm{Tr}{e}^{\ensuremath{-}\ensuremath{\beta}H}$ in certain "physical" gauges. Feynman rules for this partition function and for finite-temperature Green's functions are derived for a general gauge.

We construct natural generalizations of the gauge theory of hadrons (and leptons), now to include the B(1235) and possibly other mesons with CP = − 1. We are led to a concept of pseudospin multiplets including ϱ, ϱ′, A1, and B-type particles; this provides then a renormalizable approach to all known particle types with J ⩽ 1. A number of interesting mass relations and intriguing structure is found.

A generalized Valatin-Bogoliubov transformation is exhibited, which preserves the gauge invariance of the theory. Some consequences of this treatment of the BCS-model are analyzed in the framework of nonrelativistic axiomatic field theory.

Gauge fields together with non-linear field equations to govern them are introduced by requiring that the Lagrangian should be invariant under ·an extended translation in spacetime, i.e. a translation in which four parameters are replaced by four arbitrary coordinatedependent functions. A prescription is given to convert a non-invariant canonical ("pseudo") energy-momentum tensor into an invariant one. The symmetric part of these field equations is examined for the two cases: (1) under linear and non-relativistic approximation, it reduces to the classical gravitational-field equation, (2) for static and spherically symmetric field, its solution is shown to correspond to Schwarzschild's solution. The antisymmetric part has no classical analogues, for there are no sources of skew-symmetric energy-momentum tensors in the classical experiments. A reasonable method is proposed to eliminate this redundant field.

The method of using the dimension of space-time as a complex parameter introduced recently to regularize Feynman amplitudes is extended to an arbitrary Feynman graph. The method has promise of being particularly well-suited to gauge theories. It is shown how the renormalized amplitude, together with the Lagrangian counter-terms, may be extracted directly, following the method of analytic renormalization.

It is urged that the lesson of gauge invariance in quantum electrodynamics implies the irrelevance of «Schwinger term» difficulties in current algebra. The divergence equations of Veltman form the basis of a gauge-variation formalism in which these questions are avoided. Утверждается, что урок калибровочной инвариантности в квантовой злектродинамике указывает на то, что в алгебре токов трудности «выражения Швингера» не имеют отношения к делу. Расходящиеся уравнения Вельтмана образуют основу для калибровочно-вариационного формализма, в котором удается обойти эти вопросы.

The Gupta-Bleuler formulation of the subsidiary condition of quantum electrodynamics in the interaction picture is reviewed. This formalism is used to construct a simple proof of the invariance of the S-matrix under q-number gauge transformations A′μ = Aμ+δμF(□)δνA or, equivalently, under certain well-known modifications of the photon propagator.

This is the first of a series of papers addressed to the renormalizability question of spontaneously broken gauge theories. We give a brief outline of the motivation for such an investigation and describe the manner in which the renormalizability of such theories will be proved in the sequel. Put briefly, we will show that in an appropriate gauge, ultraviolet divergences of a spontaneously broken gauge theory are removed completely by the gauge-invariant counterterms in the Lagrangian which would make the Greon's functions of the corresponding unbroken gauge theory finite, that the $S$ matrix computed in this gauge is unitary, and that the $S$ matrix is independent of the gauge chosen. In this paper, the renormalizability question of the unbroken gauge theory is considered. We derive the Ward-Takahashi identities of the theory. We discuss several ways of regulating divergent Feynman integrals of the theory without destroying gauge invariance. Infrared divergences are avoided by the device of intermediate renormalization, wherein we choose as subtraction points some points where external momenta are Euclidean. This suffices to establish that the Bogoliubov-Parasiuk-Hepp renormalization will give renormalized Green's functions which satisfy the Ward-Takahashi identities. The existence of finite, renormalized Green's functions satisfying the Ward-Takahashi identities provides us with the means of proving the renormalizability of the spontaneously broken symmetry case. The Ward-Takahashi identities were previously derived for the gauge bosons by Slavnov. We present here a new derivation. The discussions on regularization methods and intermediate renormalization procedure and the renormalization conditions for matter fields, we believe, are new contributions of the present paper.

A general formal foundation of local theories using the Higgs-Kibble mechanism is given in the unitary gauge. The general term that is added to the Lagrangian owing to the presence of a gauge group is found both when the gauge group has and has not a non-trivial unbroken subgroup. The conjecture made by Weinberg about the form of this term when there is a non-trivial unbroken subgroup is found to be false in general. However when the unbroken subgroup is Abelian the conjecture is shown to be correct.

In two examples of spontaneously broken gauge theories in which electromagnetic isodoublet mass differences are finite, the sign of Δm|n−p for pointlike protons and neutrons is negative, although the general formula for mass differences in gauge theories allows both signs.

In the dual and fermion proposed by Neveu, Schwarz and Ramond, we find gauge identities involving vertex parts and also off-shell supergauge identities. When their models are translated into the secondary quantized field theory, we no longer can impose the gauge identities as operator ones, since they would lead to a contradiction of the second-quantization rules. We therefore require weak forms of the gauge identities just in an analogous way to the Lorentz condition in quantum electrodynamics. It is then shown that the weak forms of the identities are consistent with the second-quantization rules and also with field equations.

The ultraviolet behaviour of Green functions in the field theories with a local gauge symmetry is, to some extent, gauge dependent. This is due to the intervention of ghost-like gauge degrees of freedom. We argue that it may be possible, in the case of non-polynomial gauge theories, to find a class of gauges in which all ultraviolet divergences and their attendant ambiguities are suppressed.

This paper develops a non-local potential formalism for general gauge theories. With the help of this mathematical apparatus an argument for quantisation of the generalised charge is given, assuming that the Dirac monopoles are present.

An operator transformation is formulated which transforms electromagnetic potentials and spinor fields from one gauge to another. Commutation rules are transformed under this gauge transformation and, in particular, change from the commutation rules of the Lorentz gauge to those of the Coulomb gauge when the appropriate gauge transformation is made. The equations of motion are also correspondingly transformed. The transformation is used to generate commutation rules and equations of motion for the null-plane gauge.