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The celebrated Aharonov-Bohm effect is perhaps the first example in which the the interplay between classical topology and quantum theory was explored. This connection has continued to shed light on diverse areas of physics like quantum statistics, anomalies, condensed matter physics, and gauge theories. Several attempts were made to generalize the Aharonov-Bohm effect by modifying the simple solenoidal current distribution used by them to the case of multiple solenoids, and a toroidal solenoid, for example. A particularly ambitious task is to confine the magnetic flux to the interior of a knotted solenoid. While it is to be expected that a non-trivial phase factor will be picked up by the wave function of a charged particle travelling in the complement of the knot in three-dimensional space, the lack of symmetry defied attempts to explicitly solve the associated scattering problem. In this paper we report on a way to make progress towards this problem based on multipole expansions. The vector potential produced by a knot is obtained by making a multipole expansion, which is then used to calculate the S-matrix for the scattering of the charged particle, in the Born approximation. It is found that the S-matrix carries an imprint of the knottedness at the octopole order. For the case of a torus knot, a curious factorization property is seen to hold., Comment: 26 pages

The time evolution of an open quantum system is governed by the Gorini-Kossakowski-Sudarshan-Lindlad equation for the reduced density operator of the system. This operator is obtained from the full density operator of the composite system involving the system itself, the bath, and the interactions between them, by performing a partial trace over the bath degrees of freedom. The entanglement between the system and the bath leads to a generalized Liouville evolution that involves, amongst other things, dissipation and decoherence of the system. In a similar fashion, the time evolution of a physical observable in a classically constrained dynamical system is governed by a generalization of the Liouville equation, in which the usual Poisson bracket is replaced by the so-called Dirac bracket. The generalization takes into account the reduction in the phase space of the system because of constraints, which arise either because they are introduced by hand, or because of some underlying gauge invariance. We derive an intriguing, but precise classical-quantum correspondence between the aforementioned situations which connects the Lindblad operators to the constraints. The correspondence is illustrated in a system of coupled simple harmonic oscillators studied earlier in the context of the area law of black holes by Bombelli, Koul, Lee, and Sorkin, and independently by Srednicki., Comment: v2: 23 pages, typos fixed and title slightly changed

Studying quantum entanglement in systems of indistinguishable particles, in particular anyons, poses subtle challenges. Here, we investigate a model of one-dimensional anyons defined by a generalized algebra. This algebra has the special property that fermions in this model are composites of anyons. A Hubbad-like Hamiltonian is considered that allows hopping between nearest neighbour sites not just for the fundamental anyons, but for the fermionic anyon composites. Some interesting results regarding the quantum entanglement of these particles are obtained.

Issues related to quantum entanglement in systems of indistinguishable particles, as discussed in the information theoretic approach, are extended to anyonic statistics. Local and non-local measurements discussed in this framework are carefully analysed in the two-site anyonic Hubbard model which provides a concrete case-study. The von Neumann entropy, the single-particle density matrix, the pair correlation function, and the pseudo-momentum distribution function are worked out paying special attention to the dependence on the statistics parameter.

Anyons in one spatial dimension can be defined by correctly identifying the configuration space of indistinguishable particles and imposing Robin boundary conditions. This allows an interpolation between the bosonic and fermionic limits. In this paper, we study the quantum entanglement between two one-dimensional anyons on a real line as a function of their statistics., Comment: 26 pages, 2 figures

A general method for deriving exact expressions for vector potentials produced by arbitrarily knotted solenoids is presented. It consists of using simple physics ideas from magnetostatics to evaluate the magnetic field in a surrogate problem. The latter is obtained by modelling the knot with wire segments carrying steady currents on a cubical lattice. The expressions for a 31 (trefoil) and a 41 (figure-eight) knot are explicitly worked out. The results are of some importance in the study of the Aharonov-Bohm effect generalised to a situation in which charged particles moving through force-free regions are scattered by fluxes confined to the interior of knotted impenetrable tubes., Comment: 7 pages, 1 figure

A free particle is constrained to move on a knot obtained by winding around a putative torus. The classical equations of motion for this system are solved in a closed form. The exact energy eigenspectrum, in the thin torus limit, is obtained by mapping the time-independent Schrodinger equation to the Mathieu equation. In the general case, the eigenvalue problem is described by the Hill equation. Finite-thickness corrections are incorporated perturbatively by truncating the Hill equation. Comparisons and contrasts between this problem and the well-studied problem of a particle on a circle (planar rigid rotor) are performed throughout., Comment: 9 pages

Simple physics ideas are used to derive an exact expression for a flat connection on the complement of a torus knot. The result is of some mathematical importance in the context of constructing representations of the knot group -- a topological invariant of the knot. It is also a step forward in the direction of obtaining a generalisation of the Aharonov-Bohm effect, in which charged particles moving through force-free regions are scattered by impenetrable, knotted solenoids., Comment: 7 pages, 1 figure

The origin of thermal and quantum entanglement in a class of three-dimensional spin models, at low momenta, is traced to purely topological reasons. The establishment of the result is facilitated by the gauge principle which, when used in conjunction with the duality mapping of the spin models, enables us to recast them as lattice Chern-Simons gauge theories. The thermal and quantum entanglement measures are expressed in terms of the expectation values of Wilson lines, loops, and their generalisations. For continuous spins, these are known to yield the topological invariants of knots and links. For Ising-like models, they are expressible in terms of the topological invariants of three-manifolds obtained from finite group cohomology -- the so-called Dijkgraaf-Witten invariants., Comment: RevTex4, 6 pages

It has recently been shown that the maximal kinematical invariance group of polytropic fluids, for smooth subsonic flows, is the semidirect product of SL(2,R) and the static Galilei group G. This result purports to offer a theoretical explanation for an intriguing similarity, that was recently observed, between a supernova explosion and a plasma implosion. In this paper we extend this result to discuss the symmetries of discontinuous flows, which further validates the explanation by taking into account shock waves, which are the driving force behind both the explosion and implosion. This is accomplished by constructing a new set of Rankine-Hugoniot conditions, which follow from Noether's conservation laws. The new set is dual to the standard Rankine-Hugoniot conditions and is related to them through the SL(2,R) transformations. The entropy condition, that the shock needs to satisfy for physical reasons, is also seen to remain invariant under the transformations., Comment: 14 pages, 1 figure

The maximal invariance group of Newton's equations for a free nonrelativistic point particle is shown to be larger than the Galilei group. It is a semi-direct product of the static (nine-parameter) Galilei group and an $SL(2,R)$ group containing time-translations, dilations and a one-parameter group of time-dependent scalings called {\it expansions}. This group was first discovered by Niederer in the context of the free Schr\"odinger equation. We also provide a road map from the free nonrelativistic point particle to the equations of fluid mechanics to which the symmetry carries over. The hitherto unnoticed $SL(2, R)$ part of the symmetry group for fluid mechanics gives a theoretical explanation for an observed similarity between numerical simulations of supernova explosions and numerical simulations of experiments involving laser-induced implosions in inertial confinement plasmas. We also give examples of interacting many body systems of point particles which have this symmetry group., Comment: Plain TeX File: 15 Pages

It has recently been found that supernova explosions can be simulated in the laboratory by implosions induced in a plasma by intense lasers. A theoretical explanation is that the inversion transformation, ($\Sigma: t \to -1/t,~ {\bf x}\to {\bf x}/t$), leaves the Euler equations of fluid dynamics, with standard polytropic exponent, invariant. This implies that the kinematical invariance group of the Euler equations is larger than the Galilei group. In this paper we determine, in a systematic manner, the maximal invariance group ${\cal G}$ of general fluid dynamics and show that it is a semi-direct product ${\cal G} = SL(2,R) \wedge G$, where the $SL(2,R)$ group contains the time-translations, dilations and the inversion $\Sigma$, and $G$ is the static (nine-parameter) Galilei group. A subtle aspect of the inclusion of viscosity fields is discussed and it is shown that the Navier-Stokes assumption of constant viscosity breaks the $SL(2, R)$ group to a two-parameter group of time translations and dilations in a tensorial way. The 12-parameter group ${\cal G}$ is also known to be the maximal invariance group of the free Schr\"odinger equation. It originates in the free Hamilton-Jacobi equation which is central to both fluid dynamics and the Schr\"odinger equation., Comment: Plain TeX File: 19 Pages

It is shown that in the two-exponential version of Liouville theory the coefficients of the three-point functions of vertex operators can be determined uniquely using the translational invariance of the path integral measure and the self-consistency of the two-point functions. The result agrees with that obtained using conformal bootstrap methods. Reflection symmetry and a previously conjectured relationship between the dimensional parameters of the theory and the overall scale are derived., Comment: Plain TeX File; 15 Pages

The origin of the rather mysterious duality symmetry found in quantum Liouville theory is investigated by considering the Liouville theory as the reduction of a WZW-like theory in which the form of the potential for the Cartan field is not fixed a priori. It is shown that in the classical theory conformal invariance places no condition on the form of the potential, but the conformal invariance of the classical reduction requires that it be an exponential. In contrast, the quantum theory requires that, even before reduction, the potential be a sum of two exponentials. The duality of these two exponentials is the fore-runner of the Liouville duality. An interpretation for the reflection symmetry found in quantum Liouville theory is also obtained along similar lines., Comment: Plain TeX file; 9 pages

The quantisation of the two-dimensional Liouville field theory is investigated using the path integral, on the sphere, in the large radius limit. The general form of the $N$-point functions of vertex operators is found and the three-point function is derived explicitly. In previous work it was inferred that the three-point function should possess a two-dimensional lattice of poles in the parameter space (as opposed to a one-dimensional lattice one would expect from the standard Liouville potential). Here we argue that the two-dimensionality of the lattice has its origin in the duality of the quantum mechanical Liouville states and we incorporate this duality into the path integral by using a two-exponential potential. Contrary to what one might expect, this does not violate conformal invariance; and has the great advantage of producing the two-dimensional lattice in a natural way., Comment: Plain TeX File; 36 pages

The phase space path integral Wess-Zumino-Witten $\to$ Toda reductions are formulated in a manifestly conformally invariant way. For this purpose, the method of Batalin, Fradkin, and Vilkovisky, adapted to conformal field theories, with chiral constraints, on compact two dimensional Riemannian manifolds, is used. It is shown that the zero modes of the Lagrange multipliers produce the Toda potential and the gradients produce the WZW anomaly. This anomaly is crucial for proving the Fradkin-Vilkovisky theorem concerning gauge invariance., Comment: Plain TeX file, 27 pages

The quantum Wess-Zumino-Witten $\to$ Liouville reduction is formulated using the phase space path integral method of Batalin, Fradkin, and Vilkovisky, adapted to theories on compact two dimensional manifolds. The importance of the zero modes of the Lagrange multipliers in producing the Liouville potential and the WZW anomaly, and in proving gauge invariance, is emphasised. A previous problem concerning the gauge dependence of the Virasoro centre is solved., Comment: Plain TeX file, 15 pages

The path integral description of the Wess-Zumino-Witten $\to$ Liouville reduction is formulated in a manner that exhibits the conformal invariance explicitly at each stage of the reduction process. The description requires a conformally invariant generalization of the phase space path integral methods of Batalin, Fradkin, and Vilkovisky for systems with first class constraints. The conformal anomaly is incorporated in a natural way and a generalization of the Fradkin-Vilkovisky theorem regarding gauge independence is proved. This generalised formalism should apply to all conformally invariant reductions in all dimensions. A previous problem concerning the gauge dependence of the centre of the Virasoro algebra of the reduced theory is solved., Comment: Plain TeX file; 28 Pages

We study the infra-red limit of non-abelian Chern-Simons gauge theory perturbed by a non-topological, albeit gauge invariant, mass term. It is shown that, in this limit, we may construct an infinite class of integrable quantum mechanical models which, for the case of SU(2) group, are labelled by the angular momentum eigenvalue. The first non-trivial example in this class is obtained for the triplet representation and it physically describes the gauge invariant coupling of a non-abelian Chern-Simons particle with a particle moving on $S^3$ - the SU(2) group manifold. In addition to this, the model has a fascinating resemblance to the Landau problem and may be regarded as a non-abelian and a non-linear generalisation of the same defined on the three-sphere with the uniform magnetic field replaced by an angular momentum field. We explicitly solve for some eigenstates of this model in a closed form in terms of some generalised orthogonal polynomials. In the process, we unravel some startling connections with Anderson's chain models which are important in the study of disordered systems in condensed matter physics. We also sketch a method which allows us, in principle, to find the energy eigenvalues corresponding to the above eigenstates of the theory if the Lyapunov exponents of the transfer matrix of the infinite chain model involved are known., Comment: Plain TeX File, 23 Pages

We investigate a modification of the 2+1 dimensional abelian Chern-Simons theory, obtained by adding a Proca mass term to the gauge field. We are particularly interested in the infrared limit, which can be described by two {\it a priori} different "topological" quantum mechanical models. We apply methods of equivariant cohomology and the ensuing supersymmetry to analyze the partition functions of these quantum mechanical models. In particular, we find that a previously discussed phase-space reductive limiting procedure which relates these two models can be seen as a direct consequence of our supersymmetry., Comment: 12 pages, UU-ITP 13/94 and HU-TFT-94-24