Canonical formulation of the spherically symmetric einstein-Yang-Mills-Higgs system for a general gauge group

The dynamics of the spherically symmetric system of gravitation interacting with scalar and Yang-Mills fields is presented in the context of the canonical formalism. The gauge group considered is a general (compact and semisimple) N parameter group. The scalar (Higgs) field transforms according to an unspecified M-dimensional orthogonal representation of the gauge group. The canonical formalism is based on Dirac's techniques for dealing with constrained hamiltonian systems. First the condition that the scalar and Yang-Mills fields and their conjugate momenta be spherically symmetric up to a gauge is formulated and solved for global gauge transformations, finding, in a general gauge, the explicit angular dependence of the fields and conjugate momenta. It is shown that if the gauge group does not admit a subgroup (locally) isomorphic to the rotation group, then the dynamical variables can only be manifestly spherically symmetric. If the opposite is the case, then the number of allowed degrees of freedom is connected to the angular momentum content of the adjoint representation of the gauge group. Once the suitable variables with explicit angular dependence have been obtained, a reduced action is derived by integrating away the angular coordinates. The canonical formulation of the problem is now based on dynamical variables depending only on an arbitrary radial coordinate r and an arbitrary time coordinate t. Besides the gravitational variables, the formalism now contains two pairs of N-vector variables (R, π_r), (Θ, π_Θ), corresponding to the allowed Yang-Mills degrees of freedom and one pair of M-vector variables, (h, π_h), associated with the original scalar field. The reduced Hamiltonian is invariant under a group of r-dependent gauge transformations such that R plays the role of the gauge field (transforming in the typically inhomogeneous way) and in terms of which the gauge covariant derivatives of Θ and h naturally appear. No derivatives of R appear in the Hamiltonian and the gauge freedom allows us to define a gauge in which R is zero. Also the r and t coordinates are fixed in a way consistent with the equations of motion. Some nontrivial static solutions are found. One of these solutions is given in closed form; it is singular and corresponds to a generalization of the singular solution found in the literature with different degrees of generality and the geometry is described by the Reissner-Nordström metric. The other solution is defined through its asymptotic behavior. It generalizes to curved space the finite energy solution discyssed by Julia and Zee in flat space.     

A gravitating SO(3, 1) gauge field

In this article, we postulate SO(3, 1) as a local symmetry of any relativistic theory. This is equivalent to assuming the existence of a gauge field associated with this noncompact group. This SO(3, 1) gauge field is the spinorial affinity which usually appears when we deal with weighting spinors, which, as is well known, cannot be coupled to the metric tensor field. Furthermore, according to the integral approach to gauge fields proposed by Yang, it is also recognized that in order to obtain models of gravity we have to introduce ordinary affinities as the gauge field associated with GL(4) (the local symmetry determined by the parallel transport). Thus if we assume both L(4) and SO(3, 1) as local independent symmetries we are led to analyze the dynamical gauge system constituted by the Einstein field interacting with the SO(3, 1) Weyl-Yang gauge field. We think this system is a possible model of strong gravity. Once we give the first-order action for this Einstein-Weyl-Yang system we study whether the SO(3, 1) gauge field could have a tetrad associated with it. It is also shown that both fields propagate along a unique characteristic cone. Algebraic and differential constraints are solved when the system evolves along a null coordinate. The unconstrained expression for the action of the system is found working in the Bondi gauge. That allows us to exhibit an explicit expression of the dynamical generator of the system. Its signature turns out to be nondefinite, due to the nondefinite contribution of the Weyl-Yang field, which has the typical spinorial behavior. A conjecture is made that such an unpleasant feature could be overcome in the quantized version of this model.     

Characteristic numbers of U1-valued lattice gauge fields

If a U₁-valued latice gauge field u defined on a periodic, 2-dimensional lattice satisfies the generic continuity condition uuuu ≠ − 1, it can be used to construct a principal U₁-bundle over the torus and in that bundle a connection such that parallel transport along bonds is given by u. In higher dimensions this construction can only be carried out if u is monopole-free (otherwise no such bundle can exist). The characteristic classes and numbers of this bundle can then be calculated from u in a straightforward way. Examples are given of u's with maximum possible characteristic numbers, along with a discussion of the behavior of u and of its topology under an action-decreasing deformation.     

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.     

Fundamental length hypothesis and new concept of gauge vector field

Through an analysis of quantum field theory with “fundamental length” l[1–10], a new concept of gauge vector field is determined. The electromagnetic field is considered in detail. The new electromagnetic potential turns out to be a 5-vector associated with the De Sitter group SO(4,1). The extra fifth component, called τ-photon, similar to the scalar and longitudinal photons, does not correspond to an independent dynamical degree of freedom. Gauge-invariant equations of motion for all components of the electromagnetic 5-potential are found. Though the new gauge group remains Abelian, it is nevertheless larger than the conventional gauge group. In particular, the new gauge transformations intrinsically depend on the fundamental length l. Therefore one can consider them as a base for modification of QED at small distances (?l) in a profound way. The underlying physics becomes much richer due to the appearance of new interactions mediated by the τ-photons [14].     

Compactification of d = 11 supergravity on S4 (or 11 = 7 + 4, too)

Eleven-dimensional supergravity can compactify spontaneously on either S⁷ or S⁴. We study the latter case, with attention to its connection with a possible gauged N = 4 supergravity in d = 7. We derive the linearized field equations and supersymmetry transformation rules for the effective d = 7 supergravity multiplet. There are five third rank antisymmetric tensors in this multiplet which are in the 5 representation of the gauge group USp(4) and they are propagated with a self-duality condition in 7 dimensions. There is also a 14 of scalar fields and they are found to propagate with a non-conformal wave operator.     

Spin gauge fields: From Berry phase to topological spin transport and Hall effects

The paper examines the emergence of gauge fields during the evolution of a particle with a spin that is described by a matrix Hamiltonian with n different eigenvalues. It is shown that by introducing a spin gauge field a particle with a spin can be described as a spin multiplet of scalar particles situated in a non-Abelian pure gauge (forceless) field U (n). As the result, one can create a theory of particle evolution that is gauge-invariant with regards to the group Uⁿ (1). Due to this, in the adiabatic (Abelian) approximation the spin gauge field is an analogue of n electromagnetic fields U (1) on the extended phase space of the particle. These fields are force ones, and the forces of their action enter the particle motion equations that are derived in the paper in the general form. The motion equations describe the topological spin transport, pumping, and splitting. The Berry phase is represented in this theory analogously to the Dirac phase of a particle in an electromagnetic field. Due to the analogy with the electromagnetic field, the theory becomes natural in the four-dimensional form. Besides the general theory, the article considers a number of important particular examples, both known and new.     

Stringy differential geometry for double field theory, beyond Riemann

While the fundamental object in Riemannian geometry is a metric, closed string theories call for us to put a two-form gauge field and a scalar dilaton on an equal footing with the metric. Here we propose a novel differential geometry which treats the three objects in a unified manner, manifests not only diffeomorphism and one-form gauge symmetry but also O(D, D) T-duality, and enables us to rewrite the known low energy effective action of them as a single term. We comment that the notion of cosmological constant naturally changes.     

Monopoles and topological field theory

We present a topological quantum field theory for magnetic monopoles in an SU(N) Yang-Mills-Higgs model. This field theory is obtained by gauge fixing the topological action defining the monopole charge. This work extends to the three-dimensional case the quantization of invariant polynomials in four dimensions. We choose the Bogomolny self-duality equations as gauge conditions for the magnetic monopole topological field theory. In this way the geometrical equation discussed e.g. in Atiyah and Hitchin's work are recovered as ghost equations of motion. We give the cocycles of the corresponding topological symmetry. In the N→∞ limit interesting phenomena occur. The functional integration is forced to cover only the moduli space and the role of the ghosts stemming from the gauge fixing process is to provide a smooth semiclassical approximation.     

Higgs and Wilson lines in field and string theory

This is a short review on basics of the use of the Wilson line to break gauge symmetry in theories with compact extra dimensions. We show how the computation of the one-loop effective field theory leads to a finite result. We then explain the realization of this breaking and the effective potential computation in an open string theory framework with D-branes. To cite this article: K. Benakli, C. R. Physique 8 (2007).     

How chiral are type-II superstrings?

We study the type-II superstrings in four dimensions by studying vacua where massless chiral multiplets transform as complex representations of the non-abelian gauge group. We show that the gauge group can only be SU(3) and that such fields transform as 3 of SU(3). However, attempts to obtain the theory with N=1 supergravity fail. It turns out that the “different” constructions via asymmetric orbifolds give the same massless spectrum with necessarily N=2 supergravity.     

Strings on pp-waves and massive two-dimensional field theories

We find a general class of pp-wave solutions of type IIB string theory such that the light cone gauge worldsheet Lagrangian is that of an interacting massive field theory. When the light cone Lagrangian has (2,2) supersymmetry we can find backgrounds that lead to arbitrary superpotentials on the worldsheet. We consider situations with both flat and curved transverse spaces. We describe in some detail the background giving rise to the N=2 sine Gordon theory on the worldsheet. Massive mirror symmetry relates it to the deformed CP¹ model (or sausage model) which seems to elude a purely supergravity target space interpretation. To cite this article: J. Maldacena, L. Maoz, C. R. Physique 4 (2003).     

Composite electrodynamics

This paper solidifies the foundations for a singleton theory of light, first proposed two years ago. This theory is based on a pure gauge coupling of the scalar singleton field to the electromagnetic current. Like quarks, singletons are essentially unobservable. The field operators are not local observables and therefore need not commute for spacelike separation. This opens up possibilities for generalized statistics, just as is the case for quarks. It then turns out that a pure gauge coupling, in which ∂μφ(x) couples to the conserved current j^μ(x), generates real interactions— the effective theory is precisely ordinary electrodynamics in de Sitter space. Here we improve our theory and explain it in much more detail than before, adding two new results. (1) The concept of normal ordering in a theory with unconventional statistics is worked out in detail. (2) We have discovered the natural way of including both photon helicities. Quantization, it may be noted, is a study in representation theory of certain infinite-dimensional, nilpotent Lie algebras, of which the Heisenberg algebra is the prototype.     

Higher gauge theory and a non-Abelian generalization of 2-form electrodynamics

In conventional gauge theory, a charged point particle is described by a representation of the gauge group. If we propagate the particle along some path, the parallel transport of the gauge connection acts on this representation. The Lagrangian density of the gauge field depends on the curvature of the connection which can be calculated from the holonomy around (infinitesimal) loops. For Abelian symmetry groups, say G=U(1), there exists a generalization, known as p-form electrodynamics, in which (p−1)-dimensional charged objects can be propagated along p-surfaces and in which the Lagrangian depends on a generalized curvature associated with (infinitesimal) closed p-surfaces. In this article, we use Lie 2-groups and ideas from higher category theory in order to formulate a discrete gauge theory which generalizes these models at the level p=2 to possibly non-Abelian symmetry groups. An important feature of our model is that it involves both parallel transports along paths and generalized transports along surfaces with a non-trivial interplay of these two types of variables. Our main result is the geometric picture, namely the assignment of non-Abelian quantities to geometrical objects in a coordinate free way. We construct the precise assignment of variables to the curves and surfaces, the generalized local symmetries and gauge invariant actions and we clarify which structures can be non-Abelian and which others are always Abelian. A discrete version of connections on non-Abelian gerbes is a special case of our construction. Even though the motivation sketched so far suggests applications mainly in string theory, the model presented here is also related to spin foam models of quantum gravity and may in addition provide some insight into the role of centre monopoles and vortices in lattice QCD.     

Isolation and quantization of the two degrees of freedom of the electromagnetic potential for any gauge

The true dynamical degrees of freedom (TDDF) of the electromagnetic potential are found for any gauge. They are the components of the Fourier transform of the electromagnetic potential on a two-dimensional spacelike plane orthogonal to the lightlike momentum vector for k² = 0 and vanish for k² ≠ 0. Gauge invariance is related to the (two-parameter) indeterminacy of this spacelike plane and the arbitrariness of the component of the electromagnetic potential along the momentum vector. By direct quantization of the TDDF for any gauge (compatible with the equations of motion), some of the well-known problems of the usual treatments are avoided. For instance, the constraint div E = 0 is a c-number (agrees with the commutation relations) without choosing a gauge, there appears no need for an indefinite metric in the space of state amplitudes, the commutators for creation and annihilation operators of every component of the electromagnetic potential (timelike, longitudinal, and transverse) have the same sign, and the energy of the electromagnetic field is positive for any gauge. When gauges are chosen, the results of the literature are recovered. In our treatment, gauge fixation and quantization commute.     

On the vacuum structure of an SU(2) Yang-Mills theory

By using a compactification of the spatial part R³ of Minkowski-space different from the one-point compactification to S³, we get a new classification of the vacua for an SU(2) gauge theory. It contains, besides the vacua arising in the S³ compactification, the Gribov vacua as new classes. We discuss the role of pseudoparticle solutions within this framework and comment on the problem of the Coulomb gauge degeneracy.     

Polynomial invariants for SU(2) monopoles

We present an explicit expression for the topological invariants associated to SU(2) monopoles in the fundamental representation on spin four-manifolds. The computation of these invariants is based on the analysis of their corresponding topological quantum field theory, and it turns out that they can be expressed in terms of Seiberg-Witten invariants. In this analysis we use recent exact results on the moduli space of vacua of the untwisted N = 1 and N = 2 supersymmetric counterparts of the topological quantum field theory under consideration, as well as on electric-magnetic duality for N = 2 supersymmetric gauge theories.     

Hypersymmetry field rotation angle

The paper determines a limit energy under which hypersymmetry (HySy) is broken. According to gauge theories, interaction mediating spin-0 bosons must be massless. The theory of HySy predicted massive intermediate bosons. Hypersymmetry field rotation, described in this paper, justifies the mass of the HySy mediating boson. The mass of intermediate bosons must arise from dynamical spontaneous breaking of the group of HySy. HySy rotation is performed in the velocity-dependent D field. The derived rotation of the field is defined by the spontaneous symmetry breaking and precession of the velocity v around its third projection in the D field (that produced the mass of the field's bosons). The latter represents the real- and effective velocities of a boson-emitting particle in the direction towards a target particle. The mass of the discussed (fictitious) Goldstone bosons can be removed by the unitarity gauge condition through Higgs (BEH) mechanism. According to the simultaneous presence of a Standard Model (SM) interaction's symmetry group and the (beyond SM) HySy group, their bosons should be transformed together. Spontaneous breakdown of HySy may allow performing a transformation that does not influence the SM physical state of the investigated system. The paper describes a field transformation that eliminates the mass of the intermediate bosons, rotates the SM- and HySy bosons’ masses together while leaving the SM bosons intact. The result is an angle that characterises the HySy by a precession mechanism of the velocity that generates the field. In contrast to the known SM intermediate bosons, the HySy intermediate bosons have no fixed mass. The mass of the HySy intermediate bosons (that appear as quanta of a velocity-dependent gauge field D) depends on the relative velocity of the particles whose interaction they mediate. So, the derived precession angle is a function of that velocity.