Cylindrical Spacetime Research Articles

STRING HAMILTONIAN FROM GENERALIZED YANG–MILLS GAUGE THEORY IN TWO DIMENSIONS

Two-dimensional SU (N) Yang–Mills theory is known to be equivalent to a string theory, as found by Gross in the large N limit, using the 1/N expansion. Later it was found that even a generalized YM theory leads to a string theory of the Gross type. In the standard YM theory case, Douglas and others found the string Hamiltonian describing the propagation and the interactions of states made of strings winding on a cylindrical space–time. We address the problem of finding a similar Hamiltonian for the generalized YM theory. As in the standard case we start by writing the theory as a theory of free fermions. Performing a bosonization, we express the Hamiltonian in terms of the modes of a bosonic field, that are interpreted as in the standard case as creation and destruction operators for states of strings winding around the cylindrical space–time. The result is similar to the standard Hamiltonian, but with new kinds of interaction vertices.

Unpolarized radiative cylindrical spacetimes: trapped surfaces and quasilocal energy

We consider the most general vacuum cylindrical spacetimes, which are defined by two global, spacelike, commuting, non-hypersurface-orthogonal Killing vector fields. The cylindrical waves in such spacetimes contain both + and × polarizations, and are thus said to be unpolarized. We show that there are no trapped cylinders in the spacetime, and present a formal derivation of Thorne's C-energy, based on a Hamiltonian reduction approach. Using the Brown–York quasilocal energy prescription, we compute the actual physical energy (per unit Killing length) of the system, which corresponds to the value of the Hamiltonian that generates unit proper-time translations orthogonal to a given fixed spatial boundary. The C-energy turns out to be a monotonic non-polynomial function of the Brown–York quasilocal energy. Finally, we show that the Brown–York energy at spatial infinity is related to an asymptotic deficit angle in exactly the same manner as the specific mass of a straight cosmic string is to the former.

Gauge fixing and the Hamiltonian for cylindrical spacetimes

We introduce a complete gauge fixing for cylindrical spacetimes in vacuo that, in principle, do not contain the axis of symmetry. By cylindrically symmetric we understand spacetimes that possess two commuting spacelike Killing vectors, one of them rotational and the other one translational. The result of our gauge fixing is a constraint-free model whose phase space has four field-like degrees of freedom and that depends on three constant parameters. Two of these constants determine the global angular momentum and the linear momentum in the axis direction, while the third parameter is related with the behavior of the metric around the axis. We derive the explicit expression of the metric in terms of the physical degrees of freedom, calculate the reduced equations of motion and obtain the Hamiltonian that generates the reduced dynamics. We also find upper and lower bounds for this reduced Hamiltonian that provides the energy per unit length contained in the system. In addition, we show that the reduced formalism constructed is well defined and consistent at least when the linear momentum in the axis direction vanishes. Furthermore, in that case we prove that there exists an infinite number of solutions in which all physical fields are constant both in the surroundings of the axis and at sufficiently large distances from it. If the global angular momentum is different from zero, the isometry group of these solutions is generally not orthogonally transitive. Such solutions generalize the metric of a spinning cosmic string in the region where no closed timelike curves are present.

Dynamics of Rotating Cylindrical Shells in General Relativity

Cylindrical spacetimes with rotation are studied using the Newmann-Penrose formulas. By studying null geodesic deviations the physical meaning of each component of the Riemann tensor is given. These spacetimes are further extended to include rotating dynamic shells, and the general expression of the surface energy-momentum tensor of the shells is given in terms of the discontinuation of the first derivatives of the metric coefficients. As an application of the developed formulas, a stationary shell that generates the Lewis solutions, which represent the most general vacuum cylindrical solutions of the Einstein field equations with rotation, is studied by assuming that the spacetime inside the shell is flat. It is shown that the shell can satisfy all the energy conditions by properly choosing the parameters appearing in the model, provided that $ 0 \le \sigma \le 1$, where $\sigma$ is related to the mass per unit length of the shell.

Twist fields and broken supersymmetry

A twist field on a cylindrical space–time has the defining property that translation about a spatial circle results in multiplying the field by a phase. In this paper we investigate how such multivalued twist fields fit into the framework of constructive quantum field theory. Twisted theories have an interest in their own right; the twists also serve as infrared regulators that partially preserve the underlying symmetries of the Hamiltonian. The main focus of this paper is to investigate the extent that boson–fermion twist-field systems are compatible with the Lie symmetry and with the N=2 supersymmetry that one expects in the same examples without twists. We consider free systems and nonlinear boson–fermion interactions that arise from a holomorphic, quasihomogeneous, polynomial superpotential. We choose the twisting angles to lie on a chosen line in twist parameter space (leaving one free twist parameter). Doing this, we can obtain Lie symmetry and half the number of supersymmetry generators that one expects in our examples without the twists. We also show that the Hamiltonians for scalar twist fields yield twisted, positive-temperature expectations with the “twist-positivity” property. This is important because it justifies the existence of a functional integral representation for twisted, positive-temperature trace functionals. We regularize these systems in a way that preserves symmetry to the maximal extent. We pursue elsewhere other aspects and applications of this method, including bounding the extent of supersymmetry breaking.

Geodesic completeness of orthogonally transitive cylindrical space–times

In this paper a theorem is derived in order to provide a wide sufficient condition for an orthogonally transitive cylindrical space–time to be singularity free. The applicability of the theorem is tested on examples provided by the literature that are known to have regular curvature invariants.

Would a topology change allow Ms. Bright to travel backward in time?

Many who have studied special relativity have pondered the question, can a person traveling faster than light take a round-trip journey and return earlier than they set off? The answer could go either way, but for any given journey all observers will agree on whether the return was earlier or later than the departure. In the usual space–time, the person must at some point change direction to return to their starting point, and the event of the return is unambiguously timelike separated from the event of the departure. If space is circular rather than linear, however, a superluminal traveler can return home without changing direction, and the issue of whether the return is unambiguously later or earlier than the departure needs to be reconsidered. The device of avoiding a direction change by considering a circular space has been considered by other authors in the context of the twin paradox. In this article I will discuss the twin paradox and superluminal travel on a cylindrical space–time with particular attention to the role of coordinate patchings and their physical interpretation.

SYMPLECTIC STRUCTURE OF LINEAR GRAVITY COUPLED TO A MASSIVE PARTICLE

Jackiw-Teitelboim model of linear gravity coupled to a point particle is studied in a curved cylindrical spacetime. The classical solutions for the metric, the particle trajectory, and the scalar field are explicitly given. We studied the symplectic structure on the phase space of the classical solutions.

Combining Cauchy and characteristic codes. II. The interface problem for vacuum cylindrical symmetry.

This paper continues a series investigating methods of combining Cauchy and characteristic techniques in numerical relativity. We present a technique for passing gravitational wave information between an inner Cauchy region and an outer, compactified, characteristic region via a numerical interface. In this developmental work the method is applied to vacuum cylindrical spacetime with two degrees of gravitational freedom. A comparison of our numerial results with an analytic solution, and a number of numerical experiments involving the propagation of Gaussian wave packets, indicate that the techniques presented here can produce very satisfactory results. \textcopyright{} 1995 The American Physical Society.

Three dimensional black holes and cylindrical general relativity

A black string solution of the four dimensional Einstein-Hilbert action, S = 1 16πG ∫ d 4X −g (R−2Λ) , in a cylindrical spacetime is found. The three dimensional dilaton-gravity theory obtained through dimensional reduction, with action S = 1 2π ∫ d 3x −ge (R − 2Θ) , contains the corresponding three dimensional black hole solution. In this three dimensional theory one is able to define mass and angular momentum for the black hole using the Hamiltonian formulation. The causal structure is analyzed and the temperature of the black hole is computed.

A new approach to instanton calculations in the O(3) nonlinear σ-model

We construct all instantons in the 0(3) σ -model on a cylindrical space-time, known not to exist on a finite time interval. The scale parameter, ρ , is related to the boundary condition in time. This may cure the ρ → 0 divergent instanton gas, through a proper inclusion of in and out states in the path integral.

CANONICAL QUANTIZATION OF TWO-DIMENSIONAL GAUGE FIELDS

SU (N) gauge fields on a cylindrical space-time are canonically quantized via two routes revealing almost equivalent but different quantizations. After removal of all continuous gauge degrees of freedom, the canonical coordinate Aµ (in the Cartan subalgebra [Formula: see text]) is quantized. The compact route, as in lattice gauge theory, quantizes the Wilson loop W, projecting out gauge-invariant wave functions on the group manifold G. After a Casimir energy related to the curvature of SU (N) is added to the compact spectrum, it is seen to be a subset of the noncompact spectrum. States of the two quantizations with corresponding energy are shifted relative to each other, such that the ground state on G, χ0(W), is the first excited state Ψ1(Aµ) on [Formula: see text]. The ground state Ψ0(Aµ) does not appear in the character spectrum, as its lift is not globally defined on G. Implications for lattice gauge theory and the sum-over-maps representation of two-dimensional QCD are discussed.

Representations of the SU(N) T-algebra and the loop representation in 1 + 1 dimensions

We consider the Yang--Mills phase space on a cylindrical spacetime () and the associated algebra of gauge-invariant functions, the T-variables. We solve the Mandelstam identities both classically and quantum mechanically by considering the T-variables as functions of the eigenvalues of the holonomy and their associated momenta. It is shown that there are two inequivalent representations of the quantum T-algebra. Then we compare this reduced phase-space approach to Dirac quantization and find it gives essentially equivalent results. We proceed to define a loop representation in each of these two cases. One of these loop representations (for N=2) is more or less equivalent to the usual loop representation.

Bosonization as duality

We show how to systematically derive the rules for bosonization in two dimensions as a particular case of a duality transformation. The duality process amounts to gauging the global symmetry of the original (fermionic) theory, and constraining the corresponding field strength F μν to vanish. Integration over the Lagrange multiplier, Λ, for this constraint then reproduces the original theory, and integration over the gauge fields generates the dual theory with Λ as the new (bosonized) variable. We work through the bosonization of the Dirac fermion, the massive and massless Thirring models, and a fermion on a cylindrical spacetime as illustrative examples.

FREE FIELD REPRESENTATION OF TODA FIELD THEORIES

We study the following problem: Can a classical sl n Toda field theory be represented by means of free bosonic oscillators through a Drinfeld-Sokolov construction? We answer affirmatively in the case of a cylindrical space-time and for real hyperbolic solutions of the Toda field equations. We establish in fact a one-to-one correspondence between such solutions and the space of free left and right bosonic oscillators with coincident zero modes. We discuss the same problem for real singular solutions with nonhyperbolic monodromy.

Phase space structure and the path integral for gauge theories on a cylinder

The physical phase space of gauge field theories on a cylindrical spacetime with an arbitrary compact simple gauge group is shown to be the quotient R 2r W A , r a rank of the gauge group, W A the affine Weyl group. The path integral (PI) formula resulting from Dirac's operator method contains a symmetrization with respect to W A rather than the integration domain reduction. It gives a natural solution to Gribov's problem. Some features of fermion quantum dynamics caused by the nontrivial phase space geometry are briefly discussed.

Euclidean Majorana fermions, fermionic integrals, and relative Pfaffians

The construction of Euclidean Majorana fermions on a two-dimensional cylindrical space–time is discussed, based on a given real-time theory. After a review of the basics of defining Euclidean fermions in the free theory, a functional integral formulation for the Euclidean theory is introduced. This functional integral connects the expectation values of the Euclidean fields with the theory of relative Pfaffians. A proof, which draws on these ideas, is given for the Feynman–Kac formula of a general interacting theory.

Gauge fixing and Gribov copies in pure Yang-Mills on a circle

In order to understand how gauge fixing can be affected on the lattice, we first study a simple model of pure Yang-mills theory on a cylindrical spacetime [ SU (N) on S 1 × R ] where the gauge fixed space is explicitly displayed. On the way, we find that different gauge fixing procedures lead to different Hamiltonians and spectra, which however coincide under a shift of states. The lattice version of the model is then compared.

Supersymmetry and the spectral condition on a cylinder

We study the spectral condition for supersymmetric Wess-Zumino field theories on a cylindrical spacetime. This condition is preserved under certain ultraviolet cutoff procedures and also leads to analyticity of HKR regularized field operators in a complex neighborhood of spacetime.

Temperature Dependence of theCPN-1 Model and the Analogy with Quantum Chromodynamics

The two-dimensional CPN-1 model — a simple field-theoretic analogue of four-dimensional quantum chromodynamics (QCD) — is analysed and reviewed. The major themes are the temperature dependence of the CPN-1 model, and the analogy between CPN-1 and QCD. A detailed treatment of the 1/N approximation of the CPN-1 model is given. The main results emerging from this approximation are discussed at length. These are: asymptotic freedom, dimensional transmutation, confinement and topological charge nonquantization at zero temperature T = O, screening and topological charge quantization at finite temperature T. The analogy with QCD is explained in detail. A new, qualitative, analysis of the CPN-1 model at finite temperature is introduced. This approach exploits the conformal invariance of the model to “heat” an arbitrary CPN-1 field from T = O to finite temperature. This is achieved by conformal-transforming the flat Euclidean space-time of the T = O theory to the cylindrical space-time of the finite temperature theory.