Gauge transformations and quantum mechanics I. Gauge invariant interpretation of quantum mechanics

Quantum mechanical operators are interpreted according to their equations of motion. Operators representing physical quantities which have classical analog are constructed by requiring that the quantum and the classical (i.e., Newtonian) equations of motion have a term by term correspondence. Of special importance to the interpretation of quantum mechanics is the particle's energy operator. In the presence of a time-varying electric fieldE, the particle's energy operator is constructed so that its time derivative is the power operatorJ · E (J being the current operator). This interpretation of operators, such as the particle's energy operator, is gauge invariant despite the possible explicit dependence on electromagnetic potentials of the operators concerned. A gauge invariant interpretation of quantum mechanics is obtained by expanding the wave-function (in an arbitrary gauge) in the orthonormal set of eigenfunctions of the particle's energy operator (in the same gauge) and by interpreting the resulting expansion coefficients as probability amplitudes. This formulation possesses all the traditional gauge freedom and contains no gauge ambiguity. (Here, by gauge invariance we also mean that the dependence on paths in the DeWitt-Mandelstam formalism and on the procedures for path averaging in the Belinfante-Rohrlich-Strocchi formalism does not occur.) In particular, probability amplitudes and transition matrix elements are gauge invariant, and the transition matrix elements between states of different energies are proportional to the corresponding matrix elements of J · E, rather than J_μA_μ. Lamb found experimental evidence that led to the conclusion that “the usual interpretation of probability amplitudes” was gauge dependent and was correct only in the gauge in which the interaction Hamiltonian was of the form of the electric dipole interaction −er · E(0, t), instead of the usual −eA(0, t) ·p/mc. It is shown here that the gauge invariant formulation for bound systems derives the electric dipole interaction in any arbitrary gauge as the result of the long wavelength and lowest order approximation of fields. For a quantum system interacting with a precessing magnetic field, the Güttinger-Schwinger procedure of quantizing the system along the instantaneous magnetic field has been known to yield the correct transition probabilities during the interaction. This quantization procedure follows directly from the gauge invariant formulation. The electric and the magnetic multipole interactions appearing in the gauge invariant formulation directly correspond to terms in the classical Poynting theorem. The gauge invariant magnetic multipole interactions differ from their counterparts in the conventional formalism. For example, the gauge invariant magnetic dipole interaction involves the time derivative of the magnetic field. This result is shown to be consistent with the Poynting theorem. Although the gauge invariant interpretive scheme proposed here is formulated for a nonrelativistic, spinless charged particle, the extension to the Dirac equation is straightforward.     

Gauge transformations and quantum mechanics II. Physical interpretation of classical gauge transformations

New gauges are introduced. The potentials, vector and scalar, in these gauges are obtained in closed forms by the Green's function method. These closed form solutions are explicity expressed only in terms of the charge and current densities. The physical interpretation is on how potentials propagate from the charge and current densities. The Coulomb gauge and the Lorentz gauge are special cases of a new gauge defined in this paper. It is called the complete α-Lorentz gauge. The scalar potential propagates at speed αc from the charge density for any positive α. When α is one, the usual solutions for the Lorentz gauge are recovered. When α is not one, our results show that, in order to satisfy the requirement that electromagnetic fields be gauge invariant and in order to conform to Maxwell's interpretation that electromagnetic fields propagate at speed c from the charge and current densities (we only consider the vacuum), the vector potential must contain two mathematically and physically independent gradient components. Furthermore, one such component must propagate at speed αc while the other must at speed c from charge and current densities. Our discussions on the Coulomb gauge are based on the results obtained by letting α go to (positive) infinity. Guided by Maxwell's interpretation, we introduce a new decomposition of the vector potential in the Lorentz gauge into a longitudinal and a transverse component. For an arbitrary charge and current distribution, it is shown that the transverse component will generate all the fields only in the radiation zone. However, for a point charged particle, the transverse component only generates the “free fields”everywhere in the instantaneous rest frame of the charged particle.     

Gauge generators,transformations and identities on a noncommutative space

By abstracting a connection between gauge symmetry and gauge identity on a noncommutative space, we analyse star (deformed) gauge transformations with the usual Leibniz rule as well as undeformed gauge transformations with a twisted Leibniz rule. Explicit structures of the gauge generators in either case are computed. It is shown that, in the former case, the relation mapping the generator with the gauge identity is a star deformation of the commutative space result. In the latter case, on the other hand, this relation gets twisted to yield the desired map.     

Slow motion approximations in general relativity II. Gauge transformations

When the field equations of general relativity are expanded in powers of a small parameter, the general covariance of the exact theory implies a corresponding gauge invariance of the equations obtained in the expansion. In a slow motion expansion, the derivation of this gauge transformation is complicated by the fact that the time coordinate is singled out for special treatment. In a previous paper, a new (3 + 1)-dimensional decomposition of the field equations was obtained which is particularly suitable as a starting point for slow motion approximations. The present paper gives a systematic method, again using covariant techniques throughout, for obtaining the corresponding gauge transformations to arbitrarily high accuracy. The calculations are explicitly carried out as far as is required in the 2 1/2-post-Newtonian approximation.     

Gauge Symmetry and Transverse Symmetry Transformations in Gauge Theories

The transverse symmetry transformations associated with the normal symmetry transformations are proposed to build the transverse constraints on the basic vertices in gauge theories. I show that, while the BRST symmetry in non-Abelian gauge theory QCD (Quantum Chromodynamics) leads to the Slavnov-Taylor identity for the quark-gluon vertex which constrains the longitudinal part of thevertex, the transverse symmetry transformation associated with the BRST symmetry enables to derive the transverse Slavnov-Taylor identity for the quark-gluon vertex, which constrains the transverse part of the quark-gluon vertex from the gauge symmetry of QCD.     

Spin-2 Quantum Gauge Theories and Perturbative Gauge Invariance

In the framework of causal perturbation theory we analyze the gauge structure of a massless self-interacting quantum tensor field. We look at this theory from a pure field theoretical point of view without assuming any geometrical aspect from general relativity. To first order in the perturbation expansion of the S-matrix we derive necessary and sufficient conditions for such a theory to be gauge invariant, by which we mean that the gauge variation of the self-coupling with respect to the gauge charge operator Q is a divergence in the sense of vector analysis. The most general trilinear self-coupling of the graviton field turns out to be the one derived from the Einstein–Hilbert action plus divergences and coboundaries.     

Gauge Symmetry and Transverse Symmetry Transformations in Gauge Theories

The transverse symmetry transformations associated with the normal symmetry transformations are proposed to build the transverse constraints on the basic vertices in gauge theories. I show that, while the BRST symmetry in non-Abelian gauge theory QCD （Quantum Chromodynamics） leads to the Slavnov-Taylor identity for the quark-gluon vertex which constrains the longitudinal part of the vertex, the transverse symmetry transformation associated with the BRST symmetry enables to derive the transverse Slavnov-Taylor identity for the quark-gluon vertex, which constrains the transverse part of the quark-gluon vertex from the gauge symmetry of QCD.     

Norm and Gauge

汉语的规、范常用来翻译gauge和norm。Norm (normalization,renormalization),gauge 是近代数学和物理的重要概念。规范理论和重整化密切相关。     

Poincaré transformations and Galilei transformations

Poincaré and Galilei transformations are seen to be contained in each other in one space dimension more.     

Gauge algebra and quantization

Quantization of a general gauge theory in the lagrangian approach is accomplished in closed form. The generating equation is found, containing all the relations of the open gauge algebra. A new class of diagrams is revealed, required by BRS-symmetry, but completely definable only from the requirement of unitarity.     

Gauge theory and bags

The surface dependent phase factor $\text{P}\text{(}\text{exp}\text{∫}\text{F}{}_{\mathrm{\mu \nu }}{}^1\text{d}\text{σ}{}^{\mathrm{\mu \nu }}$) is introduced and its variational derivatives are investigated. Under certain assumptions it is shown to satisfy a differential equation which coincides with the membrane equation.     

Gauge and Higgs bosons

The Review summarizes much of particle physics and cosmology.Using data from previous editions,plus 3,062 new measurements from 721 papers,we list,evaluate,and average measured properties of gauge bosons and the recently discovered Higgs boson,leptons,quarks,mesons,and baryons.We summarize searches for hypothetical particles such as supersymmetric particles,heavy bosons,axions,dark photons,etc.All the particle properties and search limits are listed in Summary Tables.We also give numerous tables,figures,formulae,and reviews of topics such as Higgs Boson Physics,Supersymmetry,Grand Unified Theories,Neutrino Mixing,Dark Energy,Dark Matter,Cosmology,Particle Detectors,Colliders,Probability and Statistics.Among the 117 reviews are many that are new or heavily revised,including new reviews on Pentaquarks and Inflation.     

Gauge invariance and renormalization

The renormalization of Abelian and non-Abelian local gauge theories is discussed. It is recalled that whereas Abelian gauge theories are invariant to local c-number gauge transformations δA_μ(x) = ?_μ,…, with □Λ = 0, and to the operator gauge transformation δA_μ(x) = ?_μφ(x), …, δφ(x) = α^?1?·A(x), with □φ = 0, non-Abelian gauge theories are invariant only to the operator gauge transformations $\text{δA}{}_{\mathrm{\mu }}\text{(x) ～}\textcolor{red}{}{}_{\mathrm{\mu }}\text{C(x)}$, …, introduced by Becchi, Rouet and Stora, where

_μ is the covariant derivative matrix and C is the vector of ghost fields. The renormalization of these gauge transformation is discussed in a formal way, assuming that a gauge-invariant regularization is present. The naive renormalized local non-Abelian c-number gauge transformation $\text{δA}{}_{\mathrm{\mu }}\text{(x) = (Z}{}_1\text{/Z}{}_3\text{)gA}{}_{\mathrm{\mu }}\text{(x) ×}\text{Λ}\text{(x)+?}{}_{\mathrm{\mu }}\text{Λ}\text{(x)}$, …, is never a symmetry transformation and is never finite in perturbation theory. Only for $\text{Λ}\text{(x) = (Z}{}_3\text{/Z}{}_1\text{)L}$ with L finite constants or for $\text{Λ}\text{(x) = Ω}\text{z}\text{?}{}_3\text{C(x)}$ with Ω a finite constant does it become a finite symmetry transformation, where $\text{z}\text{?}{}_3$ is the ghost field renormalization constant. The renormalized non-Abelian Ward-Takahashi (Slavnov-Taylor) identities are consequences of the invariance of the renormalized gauge theory to this formation. It is also shown how the symmetry generators are renormalized, how photons appear as Goldstone bosons, how the (non-multiplicatively renormalizable) composite operator A_μ × C is renormalized, and how an Abelian c-number gauge symmetry may be reinstated in the exact solution of many asymptotically fr ee non-Abelian gauge theories.     

Action Principle and Algebraic Approach to Gauge Transformations in Gauge Theories

The action principle is used to derive, by an entirely algebraic approach, gauge transformations of the full vacuum-to-vacuum transition amplitude (generating functional) from the Coulomb gauge to arbitrary covariant gauges and in turn to the celebrated Fock–Schwinger (FS) gauge for the Abelian (QED) gauge theory without recourse to path integrals or to commutation rules and without making use of delta functionals. The interest in the FS gauge, in particular, is that it leads to Faddeev–Popov ghosts-free non-Abelian gauge theories. This method is expected to be applicable to non-Abelian gauge theories including supersymmetric ones.     

Gauge invariance and daisies

Perturbative calculation of effective potentials based on fine-tuning of coupling constants must be carefully done in order to preserve its gauge invariant contents.