Twistors and geometric quantisation theory

The basic geometry of twistors is developed as an application of geometric quantisation theory to the conformal group. It is found, however, that the Kähler form is not positive and that the quantised Hilbert space is trivial. This serves both to highlight difficulties in the quantisation theory for semi-simple Lie groups and to point out some of the obstacles in the way of developing a rigorous theory of twistors. It also suggests some areas in which the interplay between the two theories may be helpful in clarifying issues.     

Geometry and 2-Hilbert space for nonassociative magnetic translations

Letters in Mathematical Physics - We suggest a geometric approach to quantisation of the twisted Poisson structure underlying the dynamics of charged particles in fields of generic smooth...     

Pre-quantisation for an arbitrary Abelian structure group

A geometric study of canonical quantisation generalising the pre-quantisation technique of Kostant is presented. The concept of a quantising fibre bundle with an arbitrary abelian structure group arises naturally in this framework. It is demonstrated that quantising fibre bundles induce vector-field representations. Necessary and sufficient conditions for the quantisability of symplectic manifolds are derived and a proof for the existence of two-dimensional non-reducible quantising toral bundles is given.     

On Fermi-like quantisation of classical mechanics

In this work we generalise some previously obtained results concerning the quantisation of classical finite models according to the symmetric (Fermi-like) scheme of quantisation. We consider models whose dynamics is defined through some non-singular Lie bracket and show that we can make the dynamics with any prescribed bracket relations, as defined by a certain type ofnon-singular symmetric brackets, coexist. The quantisation scheme established is: (a) defined up to an arbitrary factor and, (b) sensitive to the addition of total time derivatives to the corresponding Lagrangian. Both unconstrained and constrained models are considered.     

Hypercomplex Representations of the Heisenberg Group and Mechanics

In the spirit of geometric quantisation we consider representations of the Heisenberg(–Weyl) group induced by hypercomplex characters of its centre. This allows to gather under the same framework, called p-mechanics, the three principal cases: quantum mechanics (elliptic character), hyperbolic mechanics and classical mechanics (parabolic character). In each case we recover the corresponding dynamic equation as well as rules for addition of probabilities. Notably, we are able to obtain whole classical mechanics without any kind of semiclassical limit ħ→0.     

Quantum Structures in Einstein General Relativity

We introduce the notion of a quantum structure on an Einstein general relativistic classical spacetime M. It consists of a line bundle over M equipped with a connection fulfilling certain conditions. We give a necessary and sufficient condition for the existence of quantum structures and classify them. The existence and classification results are analogous to those of geometric quantisation (Kostant and Souriau), but they involve the topology of spacetime, rather than the topology of the configuration space. We provide physically relevant examples, such as the Dirac monopole, the Aharonov–Bohm effect and the Kerr–Newman spacetime. Our formulation is carried out by analogy with the geometric approach to quantum mechanics on a spacetime with absolute time, given by Jadczyk and Modugno.     

Some remarks on the dynamical origin of charge and space-time quantisation

It is argued here that the concept of dynamical origin of charge as formulated in a previous paper requires the quantisation of space-time. Indeed, in this scheme, it is pointed out that the quantisation of electric charge in unit ofe is a direct consequence of this space-time quantisation.     

Schwinger’s action principle,supersymmetry and path integrals

We use Schwinger’s action principle in quantum mechanics to obtain the quantisation from Lagrangian for the fermionic variables, as well as when it contains auxiliary coordinates. We illustrate this with a supersymmetric Lagrangian which naturally includes auxiliary variables. We further show that the action principle also leads to Feynman’s path integral quantisation, which is aesthetically pleasing.     

Semiclassical Theory of Quasiparticles in the Superconducting State

We have developed a semiclassical approach to solving the Bogoliubov-de Gennes equations for superconductors. It is based on the study of classical orbits governed by an effective Hamiltonian corresponding to the quasiparticles in the superconducting state and includes an account of the Bohr-Sommerfeld quantisation rule, the Maslov index, torus quantisation, topological phases arising from lines of phase singularities (vortices), and semiclassical wave functions for multidimensional systems. The method is illustrated by studying the problem of an SNS junction and a single vortex.     

Review of AdS/CFT Integrability,Chapter II.2: Quantum Strings in AdS<Subscript>5</Subscript> × S<Superscript>5</Superscript>

We review the semiclassical analysis of strings in AdS₅ × S⁵ with a focus on the relationship to the underlying integrable structures. We discuss the perturbative calculation of energies for strings with large charges, using the folded string spinning in AdS₃ ? AdS₅ as our main example. Furthermore, we review the perturbative light-cone quantisation of the string theory and the calculation of the worldsheet S-matrix.     

A theory of extended quantisation

A new theory of quantisation is presented. After arguments are given indicating that mass-energy in the universe is quantised, this quantisation is mathematically related to the lifespan and maximum size of the universe. Various consequences are then deduced, such as the existence of a minimum force.     

On structure preserving quantisations

Maps of functions on classical phase space to quantum operators do not preserve the algebraic structure. After locating the algebraic reasons for it, the problem of quantisation is redefined and the Moyal bracket is discussed for its structure preservation. This quantisation entails the inclusion of Schwartz distributions to the space of classical functions.     

Magnetic levels in superlattices

A magnetic field parallel to the layers of a GaAs---GaAlAs superlattice leads to a quantisation of the subband dispersion relation. The discrete energy levels are calculated with a semiclassical quantisation scheme and it is shown that within the energy width of the subbands, closed orbits, and in the superlattice minigap open orbits, are formed. Experimentally this behaviour is observed as sharp peaks in the interband Landau level absorption for energies within the subband width (closed orbits) and the disappearance of these peaks at higher energies (where no closed orbits exist).     

Propagators and relaxation for stochastically quantised U(1) in the temporal gauge

We show that stochastic quantisation can be used to provide a simple derivation of the propagator for a maximal axially fixed gauge field. The relaxation and correlation properties of systems both unfixed and axially fixed are examined and compared. A comparison of the two methods of gauge fixing used in stochastic quantisation is made.     

Semiclassical quantisation rules for the Dirac and Pauli equations

We derive explicit semiclassical quantisation conditions for the Dirac and Pauli equations. We show that the spin degree of freedom yields a contribution which is of the same order of magnitude as the Maslov correction in Einstein-Brillouin-Keller quantisation. In order to obtain this result a generalisation of the notion of integrability for a certain skew product flow of classical translational dynamics and classical spin precession has to be derived. Among the examples discussed is the relativistic Kepler problem with Thomas precession, whose treatment sheds some light on the amazing success of Sommerfeld’s theory of fine structure [Ann. Phys. (Leipzig) 51 (1916) 1].     

A Note on the Faddeev–Popov Determinant and Chern–Simmons Perturbation Theory

A refined expression for the Faddeev–Popov determinant is derived for gauge theories quantised around a reducible classical solution. We apply this result to Chern–Simons perturbation theory on compact spacetime 3-manifolds with quantisation around an arbitrary flat gauge field isolated up to gauge transformations, pointing out that previous results on the finiteness and formal metric-independence of perturbative expansions of the partition function continue to hold.     

Stochastic quantisation of a gauge field in the spatial axial gauge

The stochastic quantisation method of Nelson can be applied to quantise an Abelian gauge field in the spatial axial gauge.     

A unified semiclassical description of elastic scattering by a central field

The diffraction-integral formulation of the semiclassical limit of the quantal wavefunction, as proposed in an earlier paper, is applied to the treatment of elastic scattering by a general central-field potential. Angular-momentum quantisation is shown to be a natural consequence of the theory and partial-wave series representations of the s-matrix and asymptotic wavefunction are derived. In addition, the method and results establish a connection between refractive, diffractive, dynamical and uniform-approximation theories of semiclassical scattering.