Physical Degrees Of Freedom Research Articles

Kaluza-Klein-like analysis of the configuration space of gauge theories.

It is well known that Dirac quantization of gauge theories is not, in general, equivalent to reduced quantization. When both approaches are self-consistent some additional criterion must be found in order to decide which approach is more natural, or correct. Now, in most cases quantization on the physical degrees of freedom is properly curved-space quantization, with a curvature that is neither constant nor Ricci flat. In a series of two papers, this being the first, we show that, unlike reduced quantization, Dirac quantization (acting in the physical Hilbert space) corresponds in a natural way to such a curved-space quantization scheme, and has remarkable similarities with other curved-space quantization schemes proposed elsewhere. We begin here with an in-depth analysis of the geometry of the classical configuration space of gauge theories. In particular, the existence of a metric on the full configuration space establishes a Yang-Mills connection with respect to the orbits---we emphasize the importance of this connection, as well as the condition the gauge theory must satisfy in order to define it. We also discuss, in Kaluza-Klein-like fashion, three Levi-Civita connections, their associated Ricci tensors, and interrelationships among them.

Curved-space quantization: Toward a resolution of the Dirac versus reduced quantization question.

It is well known that Dirac quantization of gauge theories is not, in general, equivalent to reduced quantization. When both approaches are self-consistent some additional criterion must be found in order to decide which approach is more natural, or correct. Now, in many cases quantization on the physical degrees of freedom is properly curved-space quantization, with a highly nontrivial curvature: neither constant nor Ricci flat. On the other hand, the configuration space of the unreduced gauge theory is often (Ricci) flat, which makes Dirac quantization considerably simpler. We show that the natural minimal'' Dirac quantization scheme, together with certain restrictions we impose on the observables, is sufficient to make the quantum commutator of quadratic observables free of van Hove anomalies. This means the minimal'' Dirac quantization (acting in the physical Hilbert space) is actually a curved-space quantization scheme suitable for the type of curvature mentioned above, at least within a restricted (but still interesting) class of observables. In fact, we demonstrate that this curved-space quantization scheme, unlike minimal'' reduced quantization, has remarkable similarities with other curved-space quantization schemes proposed elsewhere. However, unlike these other schemes, it contains a piece which depends in an essential way on the gauge structure ofmore » the unreduced theory, and so could not have been guessed working strictly from within the classical reduced theory.« less

On the stability of the two-dimensional black hole

Abstract We reexamine the stability of the static black hole in two-dimensional string theory. The tachyonic mode ( t ) has a potential barrier near the black hole, while the graviton-dilaton mode ( h +φ) has a potential well near the black hole. ( h +φ) is not a physically propagating mode and nothing but a gauge artefact, although the potential well induces an exponentially growing mode with time. The stability should be based on the physical degrees of freedom. According to the analysis based on the physical mode (tachyonic mode), we find that the d =2 black hole is stable.

NONINTEGRABLE PHASE DYNAMICS IN TWO-DIMENSIONAL GAUGE MODELS

In two-dimensional gauge models the nonintegrable phases of the line Wilson integrals are the only physical gauge field degrees of freedom, providing the gauge field diminishes rather rapidly at spatial infinities. We construct a “physical” quantum Hamiltonian in terms of the physical degrees of freedom for the Schwinger model and the chiral Schwinger model. We show that in the latter case the physical quantum picture can be formulated in two equivalent ways: either in terms of the fermionic matter degrees of freedom moving in a linearly rising background electric field, or in terms of matter fields with exotic statistics.

Classical and quantum dynamics of the Faraday lines of force.

We study the vacuum Maxwell theory by expressing the electric field in terms of its Faraday lines of force. This representation allows us to capture the two physical degrees of freedom of the electric field by means of two scalar fields. The corresponding classical canonical theory is constructed in terms of four scalar fields, is fully gauge invariant, has an attractive kinematics, but a rather complicated dynamics. The corresponding quantum theory can be constructed in a well-defined functional representation, which we refer to as the Euler representation. This representation turns out to be related to the loop representation. The resulting quantization scheme is, perhaps, of relevance for non-Abelian theories and for gravity.

Canonical quantization of the string with dynamical geometry and anomaly free nontrivial string in two dimensions

The hamiltonian formulation of the string with a dynamical geometry and two-dimensional gravity with torsion is given. The canonical hamiltonian equals a linear combination of first-class constraints satisfying a closed algebra. It is the semidirect sum of the Virasoro algebra and the abelian subalgebra corresponding to the local Lorentz rotation. After making the canonical transformation the theory is quantized. It is proved that there exists a Fock space representation of pure two-dimensional gravity with torsion containing no central charge in the Virasoro algebra. Also constructed is a new Fock representation of the standard bosonic string. It is shown that two-dimensional string with dynamical geometry is anomaly free and describes two physical degrees of freedom.

Solution of the SU(2) Mandelstam constraints

It is shown how the Mandelstam constraints for an SU(2) pure lattice gauge theory with 3 N physical degrees of freedom may be solved completely in terms of 3 N Wilson and Polyakov loop variables and N − 1 gauge invariant discrete ±1 variables, thus enabling a manifestly gauge invariant formulation of the theory.

New way of the derivation of first-order Wess-Zumino terms

First-order Wess-Zumino terms, in the context of the linear self-dual constrained theory, are derived using the Batalin-Fradkin formalism with a suitable extension. Two Wess-Zumino terms are obtained: one does not modify the physical spectrum of the theory, while the other adds a physical degree of freedom to the spectrum of the theory.

Light-front QCD. III. Coupling constant renormalization.

In this third part in a series of papers on light-front QCD, we continue the study of regularization and renormalization in old-fashioned perturbative light-front formalism. We calculate lowest order radiative corrections to the quark-gluon coupling constant in light-front QCD. An attempt is made to understand the origin of antiscreening in a formulation of QCD with only physical degrees of freedom in terms of contributions from distinct Fock space sectors. The relevance of our results to bound state calculations in QCD with a truncated Fock space are discussed.

STRONG CP PROBLEM AND AN AXIAL U(1) GAUGE SYMMETRY

We extend the standard model by an axial U(1) gauge symmetry obtained from a rotation parametrized by a scalar field. In quantizing the system, the Peccei-Quinn symmetry is realized by introducing a BRST-exact term, which can provide a new approach to the strong CP problem without assuming a physical degree of freedom like the axion or the massless quark. We can find that the transition between different topological sectors of QCD via physical scattering processes is suppressed.

Gauge choices and physical variables in QED

In a guage theory like QED not all components of the gauge potentials are physical. Generally it proves necessary to remove some of the unphysical fields by a gauge-fixing. Clearly this should not, however, be done in such a way that physical degrees of freedom are removed. In this note it shall be heuristically shown that some widely used gauges can potentially do just that.

Immersion ideals and the causal structure of Ricci-flat geometries

A moving-frame analysis is given for the immersion of four-dimensional (pseudo-) Riemannian geometries in ten-dimensional (pseudo-) Euclidean space. The resulting Darboux bundles incorporate auxiliary connection forms that give concise quadratic expressions for the Riemann and Ricci tensors. Using Cartan-Kahler theory the authors calculate Cartan characters for the Ricci-flat case, which directly show how the three-dimensional foliation, constraints and causal evolution of dynamic geometry arise. Cartan character analyses are also reported for three-dimensional flat Riemannian geometry immersed in six-dimensional Euclidean space, for five-dimensional Ricci-flat geometry immersed in 15, and for some coupled and specialized fields. In all cases (m-1)-dimensional causal foliation naturally emerges from use of the auxiliary variables describing the immersions, and physical degrees of freedom are immediately discovered.

DIRAC'S QUANTIZATION OF THE CHIRAL SCHWINGER MODEL — PART II

We modify the Lagrangian of the chiral Schwinger model (CSM) by adding a nonlocal term. Then we quantize the modified model in the temporal gauge A0 = 0 by using Dirac's quantization procedure. It is shown that the "quantize first" approach leads to the nonlocal quantum theory of the nonmodified CSM. The "constrain first" approach gives us a quantum theory in which some of the physical degrees of freedom are represented by a quantized field with exotic statistics.

Comment on: Hamiltonian formulation of a theory with constraints [U. Kulshreshta, J. Math. Phys. 33, 633 (1992)

The Hamiltonian formulation of a constrained dynamical system, analyzed by Kulshreshta [J. Math. Phys. 33, 633 (1992)] is discussed. The theory possesses four coordinates and four primary second class constraints and thus two physical degrees of freedom. The correct equations of motion for the system are given. The reduced theory (q4=k) is shown to have three primary and one secondary second class constraints instead of three primary first class ones as claimed by Kulshreshta. The equations of motion are derived using the Dirac bracket.

The physical degrees of freedom in hot quenched QCD

We demonstrate the absence of a low-lying bosonic pole in the spectral function for the pseudoscalar meson channel for T>Tc. We also extract a set of effective four-Fermi couplings and show that this is small inthe vector channel for all T>Tc, but not in the pseudovector channel. A perturbative estimate is compatible with this result. The computations have been performed in quenched QCD on lattices with Nτ = 4.

CANONICAL DESCRIPTION OF A TWO-DIMENSIONAL GRAVITY

A two-dimensional gravity theory which was studied before within the Lagrangian methods, in the conformal gauge is investigated in terms of the Hamiltonian methods. Although the reparametrization invariant and the conformal gauge fixed Lagrangians lead to different number of physical degrees of freedom, it is shown that on mass-shell they are equivalent.

Spectral boundary conditions in one-loop quantum cosmology.

For fermionic fields on a compact Riemannian manifold with a boundary, one has a choice between local and nonlocal (spectral) boundary conditions. The one-loop prefactor in the Hartle-Hawking amplitude in quantum cosmology can then be studied using the generalized Riemann $\ensuremath{\zeta}$ function formed from the squared eigenvalues of the four-dimensional fermionic operators. For a massless Majorana spin-\textonehalf{} field, the spectral conditions involve setting to zero half of the fermionic field on the boundary, corresponding to harmonics of the intrinsic three-dimensional Dirac operator on the boundary with positive eigenvalues. Remarkably, a detailed calculation for the case of a flat background bounded by a three-sphere yields the same value $\ensuremath{\zeta}(0)=\frac{11}{360}$ as was found previously by the authors using local boundary conditions. A similar calculation for a spin-$\frac{3}{2}$ field, working only with physical degrees of freedom (and, hence, excluding gauge and ghost modes, which contribute to the full Becchi-Rouet-Stora-Tyutininvariant amplitude), again gives a value $\ensuremath{\zeta}(0)=\ensuremath{-}\frac{289}{360}$ equal to that for the natural local boundary conditions.

PHASE SPACE REDUCTION AND THE CHOICE OF PHYSICAL VARIABLES IN GAUGE THEORIES

The connection between the way of separation of physical variables and the form of the Hamiltonian path integral (HPI) is studied for the Yang-Mills quantum mechanics. It is shown that physical degrees of freedom are always described by curvilinear coordinates. It is also found that the ambiguity in determining physical variables follows from the reduction of the physical phase space. The latter leads to a modification of the standard HPI (HPI with gauge conditions).

Variations on QED in three dimensions

The quantization of three-dimensional electrodynamics in 2 + 2 De Sitter space is carried out in detail. In three dimensions we find five completely different types of quantum electrodynamics. The most interesting versions have spins +1 and/or −1. When both spins are combined as modes of a single vector potential with definite parity, then the field strength vanishes; nevertheless the theory has an infinite space of propagating physical modes. This version of three-dimensional QED is equivalent to three-dimensional singleton theory. The requirements of gauge invariance prohibit local interactions, in the classical theory and within the framework of canonical quantization, and interactions are thus limited to the boundary at spatial infinity, a torus. There are also two chiral theories, in which the physical modes have spin +1 only, or −1 only; here the field strength is not identically zero, though it vanishes on the physical subspace. The chiral theories do not have smooth limits as the curvature tends to zero. All these versions of electrodynamics are smooth limits of massive vector theories, and all can be regarded as quantized counterparts of classical Chern-Simons theory. The most standard form of three-dimensional QED has physical modes with spin zero; it is in a sense a limit of a massive theory with spin zero. In two dimensions the only physical degree of freedom is an excitation with vanishing energy and momentum, a topological mode.

Three-dimensional singletons

The three-dimensional analog of singleton gauge theory turns out to be related to the topological gauge theory of Schwartz and Witten. It is a fully-fledged gauge theory, though it involves only a single scalar field. Real, physical degrees of freedom propagate in 3-space, but they are ‘confined’ in the sense that they cannot be detected locally. The physical Hamiltonian density is not zero, but it is concentrated on the boundary at spatial infinity. This boundary surface, a torus, supports a two-dimensional conformal field theory.