Singular Gauge Transformation Research Articles

Two Quantum Liquids

We describe a theory of the two dimensional Heisenberg-Hubbard model working in the spin and hole fermion representation based on the use of a singular gauge transformation on the spins. We are motivated by a new mean field theory for a system of strongly coupled fermions and bosons which is also described. In the latter case we obtain the bose condensed—fermi liquid phase; in the former we obtain equations governing generalized flux phases. We do not find superconducting solutions to these equations in the atomic limit, J=0. If solutions exist for finite J case then it is argued that an energy gap is present in that case.

Soft dilation theorem in covariant string field theory

The dilaton is studied in the context of covariant, cubic, open and closed string field theories. The string field configuration corresponding to a dilaton plane wave in flat space is constructed. Taking the zero-momentum limit, it is shown that the string coupling constant is correctly rescaled by a shift in the dilaton expectation value. The demonstration involves a subtle interplay between world-sheet cohomology and the algebra of string fields. This shift can be viewed as the result of a singular gauge transformation.

P-Form electrodynamics

A generalization of gauge theory in which the gauge potential1-form is replaced by a p-form is studied. Charged particles are then replaced by elementary extended objects of dimension p−1. It is shown that this extension is compatible with space-time locality only if the gauge group is U(1). A source which is a closed p−1 surface has zero total charge and corresponds to a particle-antiparticle pair. Its quantum rate of production in an external uniform field is evaluated semiclassically. The analog of the Dirac magnetic pole is constructed. It is another extended object, of dimension n−p−3, where n is the dimension of space-time. The “electric” and “magnetic” charges obey the Dirac quantization condition. This condition is derived in two different ways. One method makes use of local gauge patches and the other brings in singular gauge transformations. A topological mass term is introduced and it is shown that it can coexist with a magnetic pole when n=2p+1, provided the topological mass is quantized.

A Possible Step to Surfaces at Vanishing Bare Coupling in Quantumchromodynamics

Starting from a kind of half-dualized nonabelian action we show that for gbare → 0 it reduces to QCD. Integrating out the variables in the reversed order leads to a dual form of QCD. This form contains a constraint which can be solved in terms of surfaces with quark boundaries. Due to the nonabelian structure these surfaces cannot be moved in space-time by singular gauge transformations as the Dirac surface. It is conjectured that they become fully dynamical by quantum effects. The nontrivial structure of the dual theory at gbare → 0 is entirely due to it being nonabelian. The presence of the surfaces breaks self-duality at gbare → 0. A lattice version of the half-dualized action is briefly discussed.

Singular co-ordinate transformations in general relativity

In a manner quite similar to the Yang-Mills field strength which acquires additional two-dimensional σ-function terms under the singular gauge transformations, the Riemann-Christoffel curvature tensor in general relativity can also acquire such δ-function terms under a certain type of co-ordinate transformations on a certain class of space-times. As an illustration a Curzon metric with the conical-type singularities is adopted and the transformation properties of its energy-momentum tensor under singular co-ordinate transformations are discussed.

Comments on an exact rotating Julia-Zee dyon solution with the Kerr-Newman metric

We derive, à la Arafune et al., an exact rotating Julia-Zee dyon solution with the Kerr-Newman metric from a corresponding “abelian gauge” Schwinger dyon solution by applying a singular gauge transformation to it. The solution is characterized by four physical parameters, mass M, angular momentum S, electric charge Q, and magnetic charge Φ, and reduces to a rotating 't Hooft-Polyakov monopole in the case Q = 0.

Magnetic monopoles in SU(N) gauge theories

The potential A ( r) ≡ M( r ̂ × n ̂ )(r− r· n ̂ ) −1 is a static solution to the classical theory of non-abelian gauge fields coupled to a point magnetic source, for any matrix M in the Lie algebra of the gauge group G. This solution is rotationally invariant if the eigenvalues of M in the adjoint representation of G are quantized in half-integer units, but is stable to small perturbations only if all non-vanishing eigenvalues are ± 1 2 . In this paper, for the gauge groups G = SU( N), it is shown which sets of eigenvalues of M are consistent with the group structure, which consistent sets are gauge inequivalent, and which consistent gauge inequivalent sets correspond to stable monopoles. It is found that there are N inequivalent stable monopoles, including the trivial case M = 0 . Equivalence here is with respect to non-singular gauge transformations—the symmetry transformations of the classical theory. Singular gauge transformations are, in contrast, not symmetries but they are nevertheless useful for classifying solutions and for relating the above concept of local stability to the global, or topological, stability associated with the Dirac strings. In this context, it is shown that there are N distinct topological classes of monopoles, with the group structure of the center Z N≊Π 1( SU(N)/ Z N) of SU( N), that each class contains exactly one stable monopole, and that any other monopole in the same class has a strictly larger value of the magnetic charge magnitude tr M 2 . This leads to an interesting physical picture of local stability as a consequence of the minimization of magnetic energy. The paper concludes with some comments on related topics: the empirical absence of magnetic charge, `t Hooft's calculation of magnetic energy, magnetic confinement, and spontaneously broken theories.

Consequences of a singular gauge transformation

The fields and sources of the singular vector potential introduced by Bocchieri and Loinger, and discussed by Bohm and Hiley, are derived as distributions. Two generalized versions of Stokes’ theorem hold for these quantities.

Sufficient condition for confinement of static quarks by a vortex condensation mechanism

We derive a sufficient condition for confinement of static quarks by a vortex condensation mechanism. It admits vortices that are thick at all times at the cost of constraining them to a finite volume Γ i whose complement is not simply connected. The confining potential V( L) is estimated in terms of the change of free energy of a system enclosed in Γ i which is induced by a change in vorticity (= singular gauge transformation applied to boundary conditions on ϖΓ i. ) For Abelian gauge theories in 3 dimensions the confining Coulomb potential is reproduced as a lower bound.

Axially symmetric multi-instanton solutions generated by conformal mappings

It is shown that exact axially symmetric, regular self-dual multi-instanton solutions of the SU(2) Yang-Mills equations can be generated from the vacuum solution by means of singular conformal mappings of a complex plane followed by a singular gauge transformation. The regularizing gauge for the 't Hooft singular solution is found and the corresponding Yang-Mills gauge potential ${\mathit{A}}_{\ensuremath{\mu}}$ without any singularity and with the correct asymptotic behavior is given. The relations between the parameters in different forms of the solution are explored and the possible implications on the calculations of quantum effects are noted.

Dual transformations: An equivalence between the continuum and lattice versions of the two-dimensional Abelian Higgs model

The topological excitations in the two-dimensional Abelian Higgs model are studied in both the continuum and the Villain lattice version, and the results are compared. Using a singular gauge transformation, it is shown that there is an exact equivalence between the dual form of both versions. A fermion form of the theory is useful in studying its confinement properties beyond the dilute-gas limit. Also, the correct measure factor for the continuum version is derived.

Merons and meronlike configurations

We study successively for SU(2), SU(3), and SU(4) gauge groups and for Euclidean metric singular solutions of the equations of motions for "meronlike" Ans\"atze, namely for ${A}_{\ensuremath{\mu}}=(\frac{1}{2})i({\ensuremath{\partial}}_{\ensuremath{\mu}}U){U}^{\ensuremath{-}1}$, where $U$ is chosen among particular classes of gauge operators for the groups in question. For SU(3) and SU(4), apart from generalizations of logarithmically divergent SU(2) solutions, we construct explicity new classes of solutions which have meron-type topological index but a stronger than logarithmic divergence in the action. The study of a possible hierarchy of divergent solutions is thus initiated. But we start with a brief recapitulation of known SU(2) solutions mostly in order to study, carefully and in detail, the consequences of certain classes of singular gauge transformations. We show very explicitly that using them, solutions with a continuous spectrum of the topological index are obtained. Implications of our results are discussed.

Remarks on Zwanziger's local quantum field theory of electric and magnetic charge

Various aspects of Zwanziger's local Lagrangian formulation of a relativistic quantum field theory of electric and magnetic charge are investigated with a vew toward determining the consistency of the theory. A slight generalization of the classical particle theory is first studied and shown to be equivalent to the more-familiar nonlocal formulations due to Yan and Schwinger and to Dirac. The actions for each of these theories must first be altered so that the correct Lorentz force law results even if a charged-particle trajectory intersects a solenoidal ''string'' attached to each monopole. The resultant actions are then seen to be invariant under combined string rotations and (singular) gauge transformations. The duality invariance (i.e., invariance under interchange of electric and magnetic quantities) of these theories is also discussed. For the quantum field theory, it is shown how the boundary conditions are to be chosen so that one has rotational invariance about the string direction n, boost invariance along n, and duality invariance. The condition for full Lorentz invariance is discussed and it is argued that the Lorentz invariance of the classical and first-quantized theories suggests what is necessary in order that the quantum field theory is also Lorentz invariant. The Feynman rulesmore » for the theory are confirmed from the Faddeev-Popov ansatz, and their unitarity is explicitly verified. It is also shown that there are no unphysical intermediate states (if duality invariance is maintained), that the bare electric and magnetic charges are renormalized by the same factor, and that the theory is infrared-free.« less

The Singular Gauge Transformation in the Non-Abelian Gauge Theory

The Singular Gauge Transformation in the Non-Abelian Gauge Theory

Singular Gauge Transformations and Monopole-Vortex-Line Systems

Semi-infinite and finite monopole-vortex-line systems are constructed from the Nielscn­ Olesen SU(2) Higgs model. We clarify a connection between the Nambu model and the Nielsen-Olesen U(1) and SU(2) Higgs models in terms of singular gauge transformations. It is shown that under the singular gauge transformation Q the field tensor F' acquires an additional nonzero term - (i/e)SJ(a/J,-a,a)Q- 1 which leads to Dirac's strings. The behavior of the gauge potential and the Higgs scalars is discussed near the axis of symmetry and at large distances from the v.ortex line.

Regular N-instanton fields and singular gauge transformations

The singularities of 't Hooft's N-instanton solution obscure its topological properties. It is shown that these singularities may be displaced and in fact removed to infinity by suitable singular gauge transformations. In this new regular representation, the winding number of the field can be defined following the BPST prescription. The example of two instantons is discussed in detail.