The Hamiltonian Formulation of Higher Order Dynamical Systems

Using Dirac's approach to constrained dynamics, the Hamiltonian formulation of regular higher order Lagrangians is developed. The conventional description of such systems due to Ostrogradsky is recovered. However, unlike the latter, the present analysis yields in a transparent manner the local structure of the associated phase space and its local sympletic geometry, and is of direct application to {\em constrained\/} higher order Lagrangian systems which are beyond the scope of Ostrogradsky's approach.Comment: 17 pages. Revised: references adde

Quantisation without Gauge Fixing: Avoiding Gribov Ambiguities through the Physical Projector

The quantisation of gauge invariant systems usually proceeds through some gauge fixing procedure of one type or another. Typically for most cases, such gauge fixings are plagued by Gribov ambiguities, while it is only for an admissible gauge fixing that the correct dynamical description of the system is represented, especially with regards to non perturbative phenomena. However, any gauge fixing procedure whatsoever may be avoided altogether, by using rather a recently proposed new approach based on the projection operator onto physical gauge invariant states only, which is necessarily free on any such issues. These different aspects of gauge invariant systems are explicitely analysed within a solvable U(1) gauge invariant quantum mechanical model related to the dimensional reduction of Yang-Mills theory.Comment: 22 pages, no figures, plain LaTeX fil

Topologically Massive Gauge Theories and their Dual Factorised Gauge Invariant Formulation

There exists a well-known duality between the Maxwell-Chern-Simons theory and the self-dual massive model in 2+1 dimensions. This dual description has been extended to topologically massive gauge theories (TMGT) in any dimension. This Letter introduces an unconventional approach to the construction of this type of duality through a reparametrisation of the master theory action. The dual action thereby obtained preserves the same gauge symmetry structure as the original theory. Furthermore, the dual action is factorised into a propagating sector of massive gauge invariant variables and a sector with gauge variant variables defining a pure topological field theory. Combining results obtained within the Lagrangian and Hamiltonian formulations, a new completed structure for a gauge invariant dual factorisation of TMGT is thus achieved.Comment: 1+7 pages, no figure

On Electric Fields in Low Temperature Superconductors

The manifestly Lorentz covariant Landau-Ginzburg equations coupled to Maxwell's equations are considered as a possible framework for the effective description of the interactions between low temperature superconductors and magnetic as well as electric fields. A specific experimental set-up, involving a nanoscopic superconductor and only static applied fields whose geometry is crucial however, is described, which should allow to confirm or invalidate the covariant model through the determination of the temperature dependency of the critical magnetic-electric field phase diagram and the identification of some distinctive features it should display.Comment: 14 pages (Latex) + 2 postscript figure

World-line Quantisation of a Reciprocally Invariant System

We present the world-line quantisation of a system invariant under the symmetries of reciprocal relativity (pseudo-unitary transformations on ``phase space coordinates" $(x^\mu(\tau),p^\mu(\tau))$ which preserve the Minkowski metric and the symplectic form, and global shifts in these coordinates, together with coordinate dependent transformations of an additional compact phase coordinate, $\theta(\tau)$). The action is that of free motion over the corresponding Weyl-Heisenberg group. Imposition of the first class constraint, the generator of local time reparametrisations, on physical states enforces identification of the world-line cosmological constant with a fixed value of the quadratic Casimir of the quaplectic symmetry group $Q(D-1,1)\cong U(D-1,1)\ltimes H(D)$, the semi-direct product of the pseudo-unitary group with the Weyl-Heisenberg group (the central extension of the global translation group, with central extension associated to the phase variable $\theta(\tau)$). The spacetime spectrum of physical states is identified. Even though for an appropriate range of values the restriction enforced by the cosmological constant projects out negative norm states from the physical spectrum, leaving over spin zero states only, the mass-squared spectrum is continuous over the entire real line and thus includes a tachyonic branch as well

Gauge Invariant Factorisation and Canonical Quantisation of Topologically Massive Gauge Theories in Any Dimension

Abelian topologically massive gauge theories (TMGT) provide a topological mechanism to generate mass for a bosonic p-tensor field in any spacetime dimension. These theories include the 2+1 dimensional Maxwell-Chern-Simons and 3+1 dimensional Cremmer-Scherk actions as particular cases. Within the Hamiltonian formulation, the embedded topological field theory (TFT) sector related to the topological mass term is not manifest in the original phase space. However through an appropriate canonical transformation, a gauge invariant factorisation of phase space into two orthogonal sectors is feasible. The first of these sectors includes canonically conjugate gauge invariant variables with free massive excitations. The second sector, which decouples from the total Hamiltonian, is equivalent to the phase space description of the associated non dynamical pure TFT. Within canonical quantisation, a likewise factorisation of quantum states thus arises for the full spectrum of TMGT in any dimension. This new factorisation scheme also enables a definition of the usual projection from TMGT onto topological quantum field theories in a most natural and transparent way. None of these results rely on any gauge fixing procedure whatsoever.Comment: 1+25 pages, no figure

The Physical Projector and Topological Quantum Field Theories: U(1) Chern-Simons Theory in 2+1 Dimensions

The recently proposed physical projector approach to the quantisation of gauge invariant systems is applied to the U(1) Chern-Simons theory in 2+1 dimensions as one of the simplest examples of a topological quantum field theory. The physical projector is explicitely demonstrated to be capable of effecting the required projection from the initially infinite number of degrees of freedom to the finite set of gauge invariant physical states whose properties are determined by the topology of the underlying manifold.Comment: 24 pages, no figures, plain LaTeX file; one more reference added. Final version to appear in Jour. Phys.

Spectrum of the non-commutative spherical well

We give precise meaning to piecewise constant potentials in non-commutative quantum mechanics. In particular we discuss the infinite and finite non-commutative spherical well in two dimensions. Using this, bound-states and scattering can be discussed unambiguously. Here we focus on the infinite well and solve for the eigenvalues and eigenfunctions. We find that time reversal symmetry is broken by the non-commutativity. We show that in the commutative and thermodynamic limits the eigenstates and eigenfunctions of the commutative spherical well are recovered and time reversal symmetry is restored