Resurgent deformation quantisation

We construct a version of the complex Heisenberg algebra based on the idea of endless analytic continuation. The algebra would be large enough to capture quantum effects that escape ordinary formal deformation quantisation.     

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.     

Circle actions in geometric quantisation

The aim of this article is to present unifying proofs for results in geometric quantisation with real polarisations by exploring the existence of symplectic circle actions. It provides an extension of Rawnsley’s results on the Kostant complex, and gives a partial result for the focus–focus contribution to geometric quantisation; as well as, an alternative proof for theorems of Śniatycki and Hamilton.     

The theory of rest-mass quantisation

The general relativity concept of density-dependent space-curvature and the mass-energy relation of special relativity indicate a rest-mass quantisation rule which makes it possible to account for the interconversion of mass and energy in a simple manner and obtain the known quantum postulates as corollaries, thereby throwing new light on the nature of matter and radiation, the uncertainty principle, and the structure of elementary particles.     

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.     

Canonical BRST quantisation of worldsheet gravities

We reformulate the BRST quantisation of chiral Virasoro and W₃ worldsheet gravities. Our approach follows directly the classic BRST formulation of Yang-Mills theory in employing a derivative gauge condition instead of the conventional conformal gauge condition, supplemented by an introduction of momenta in order to put the ghost action back into first-order form. The consequence of these simple changes is a considerable simplification of the BRST formulation, the evaluation of anomalies and the expression of Wess-Zumino consistency conditions. In particular, the transformation rules of all fields now constitute a canonical transformation generated by the BRST operator Q, and we obtain in this reformulation a new result that the anomaly in the BRST Ward identity is obtained by application of the anomalous operator Q², calculated using operator products, to the gauge fermion.     

Canonical quantisation in non-Abelian background fields

The canonical quantisation of Yang-Mills theory in a local background gauge is presented. The theory is shown to possess a BRS-supersymmetry, and the associated conserved charge is used to define a subsidiary condition on physical states. The Furry approximation is introduced, and an explicit example of the method in the case of a uniform external field strength is presented. The particle production rate is calculated, and the connection with the proper time formalism established. The results do not agree with those obtained by symmation of the corresponding perturbation series.     

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.     

Derivations of Lie brackets and canonical quantisation

An extensive analysis of the Dirac problem of canonical quantisation is reported. In this a known solution [1] has been found to be unique to within a canonical transformation under a certain prescribed condition. This proves a conjecture due to Streater [2]. A further canonically inequivalent solution is obtained by relaxing this condition. The results obtained are discussed in terms of the derivation algebras pertaining to the Classical and Quantum Lie brackets. Applications to the study of higher symmetries and to realisations of Lie algebras as polynomial functions of canonical operators are pointed out.     

On the quantisation of one-dimensional bags

We discuss the quantisation of one-dimensional MIT bags by expanding the fields as a sum of classical modes and truncating the series after the first term. We obtain the lowest states of a bag in a world containing two scalar quark fields. Problems associated with the zero-point oscillations of the field are discussed.     

A concise method for kinetic energy quantisation

We present a straightforward method for obtaining exact classical and quantum molecular Hamiltonians in terms of arbitrary coordinates. As compared to other approaches the resulting expressions are rather compact, the physical meaning of each quantity is quite transparent, and in some cases the calculation effort will be greatly reduced. We also investigate systems with constraints to find the suggested method to be applicable in contrast to most conventional approaches to kinetic energy operators which cannot directly be applied to constrained systems. Two examples are discussed in detail, the monohydrated hydroxide anion and the protonated ammonia dimer.     

Exact quantisation of the relativistic Hopfield model

We investigate the quantisation in the Heisenberg representation of a relativistically covariant version of the Hopfield model for dielectric media, which entails the interaction of the quantum electromagnetic field with the matter dipole fields, represented by a mesoscopic polarisation field. A full quantisation of the model is provided in a covariant gauge, with the aim of maintaining explicit relativistic covariance. Breaking of the Lorentz invariance due to the intrinsic presence in the model of a preferred reference frame is also taken into account. Relativistic covariance forces us to deal with the unphysical (scalar and longitudinal) components of the fields, furthermore it introduces, in a more tricky form, the well-known dipole ghost of standard QED in a covariant gauge. In order to correctly dispose of this contribution, we implement a generalised Lautrup trick. Furthermore, causality and the relation of the model with the Wightman axioms are also discussed.     

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].     

Batalin-Vilkovisky field-antifield quantisation of fluctuations around classical field configurations

The Lagrangian field-antifield formalism of Batalin and Vilkovisky (BV) is used to investigate the application of the collective coordinate method to soliton quantisation. In field theories with soliton solutions, the Gaussian fluctuation operator has zero modes due to the breakdown of global symmetries of the Lagrangian in the soliton solutions. It is shown how Noether identities and local symmetries of the Lagrangian arise when collective coordinates are introduced in order to avoid divergences related to these zero modes. This transformation to collective and fluctuation degrees of freedom is interpreted as a canonical transformation in the symplectic field-antifield space which induces a time-local gauge symmetry. Separating the corresponding Lagrangian path integral of the BV scheme in lowest order into harmonic quantum fluctuations and a free motion of the collective coordinate with the classical mass of the soliton, we show how the BV approach clarifies the relation between zero modes, collective coordinates, gauge invariance and the center-of-mass motion of classical solutions in quantum fields. Finally, we apply the procedure to the reduced nonlinear O(3) σ-model.