BFV quantization on hermitian symmetric spaces

Gauge-invariant BFV approach to geometric quantization is applied to the case of hermitian symmetric spaces G/H. In particular, gauge invariant quantization on the Lobachevski plane and sphere is carried out. Due to the presence of symmetry, master equations for the first-class constraints, quantum observables and physical quantum states are exactly solvable. BFV-BRST operator defines a flat G-connection in the Fock bundle over G/H. Physical quantum states are covariantly constant sections with respect to this connection and are shown to coincide with the generalized coherent states for the group G. Vacuum expectation values of the quantum observables commuting with the quantum first-class constraints reduce to the covariant symbols of Berezin. The gauge-invariant approach to quantization on symplectic manifolds synthesizes geometric, deformation and Berezin quantization approaches.     

Arrival time in quantum field theory

Via the proper-time eigenstates (event states) instead of the proper-mass eigenstates (particle states), free-motion time-of-arrival theory for massive spin-1/2 particles is developed at the level of quantum field theory. The approach is based on a position-momentum dual formalism. Within the framework of field quantization, the total time-of-arrival is the sum of the single event-of-arrival contributions, and contains zero-point quantum fluctuations because the clocks under consideration follow the laws of quantum mechanics.     

Group actions on quantum bundles

Suppose we are given a group G acting through canonical transformations on a symplectic manifold (M, ω). If there is a quantum bundle over (M, ω), a carrier for wave functions in the geometric quantization theory, then G acts infinitesimally on the bundle in a natural way. We give a necessary and sufficient condition for the infinitesimal G-action to integrate up to a global G-action. This is used for an investigation how the choice of the quantum bundle over (M, ω) influences the integrability of the corresponding infinitesimal G-action. The relationship to group representations is briefly mentioned.     

Geometry of 2d spacetime and quantization of particle dynamics

We analyze classical and quantum dynamics of a relativistic particle in 2d spacetimes with constant curvature. We show that global symmetries of spacetime specify the symmetries of physical phase-space and the corresponding quantum theory. To quantize the systems we parametrize the physical phase-space by canonical coordinates. Canonical quantization leads to unitary irreducible representations of SO_↑(2.1) group.     

Quantum dynamics of a massless relativistic string (II)

The massless relativistic free string is studied in the gauge x₀ = τ. It is found that the classical solutions include transverse and longitudinal vibrations. The problem is treated both in the Lagrangian and Hamiltonian formalism. Different ways of quantizing the system are investigated. The path integral quantization leads to a Poincaré invariant quantum theory in any number of dimensions.     

Quantum groups as generalized gauge symmetries in WZNW models. Part II. The quantized model

This is the second part of a paper dealing with the “internal” (gauge) symmetry of the Wess–Zumino–Novikov–Witten (WZNW) model on a compact Lie group G. It contains a systematic exposition, for G = SU(n), of the canonical quantization based on the study of the classical model (performed in the first part) following the quantum group symmetric approach first advocated by L.D. Faddeev and collaborators. The internal symmetry of the quantized model is carried by the chiral WZNW zero modes satisfying quadratic exchange relations and an n-linear determinant condition. For generic values of the deformation parameter the Fock representation of the zero modes’ algebra gives rise to a model space of U _q (sl(n)). The relevant root of unity case is studied in detail for n = 2 when a “restricted” (finite dimensional) quotient quantum group is shown to appear in a natural way. The module structure of the zero modes’ Fock space provides a specific duality with the solutions of the Knizhnik–Zamolodchikov equation for the four point functions of primary fields suggesting the existence of an extended state space of logarithmic CFT type. Combining left and right zero modes (i.e., returning to the 2D model), the rational CFT structure shows up in a setting reminiscent to covariant quantization of gauge theories in which the restricted quantum group plays the role of a generalized gauge symmetry.     

On the renormalization of the quantum field theory of point-like monopoles and charges

The quantum field theory of point-like monopoles and charges is first formulated on a euclidean lattice for a convenient regularization. The regularization preserves the peculiar features of the theory, namely those related to the invariance and to the quantization condition. The partition function is expressed as a path integral over the particle's closed paths and the action is constructed in terms of arbitrary surfaces having those paths as boundaries. The possible divergences of the continuum limit are discussed, in particular the vacuum polarization ones. It is found that, although both the electric charge Q and the magnetic charge G are renormalized as Q = Z_QQ_R and G = Z_GG_R, the quantization condition is preserved by the renormalization i.e. Z_QZ_G = 1 so that QG = Q_RG_R = 2πn. Due to the dual symmetry of the theory, then, for Q = G we get Z_Q = Z_G = 1.     

Geometric quantization and quantum field theory in curved space-time

The Kostant-Souriau geometric quantization theory is applied to the problem of constructing a generally covariant quantum field theory. The occupation number formalism for a scalar field is introduced as a semiclassical approximation which is valid in low curvature regions of space-time and which depends on making a particular choice of polarization in the classical phase space of a single massive particle. The application of the formalism to particle creation problems is outlined.     

Control of tunnel coupling strength between InAs quantum dots and nanogap metallic electrodes through In-Ga intermixing

We have investigated the growth-temperature, T_G, dependence of the electronic properties of single self-assembled InAs quantum dots (QDs) coupled to nanogap metallic electrodes. The orbital quantization energies of QDs and the tunnel resistances exhibited strong T_G-dependence due to In-Ga intermixing during QD formation. It was found that the transparency of the tunnel junctions is controllable over a very wide range by simply changing the size and the growth temperature of QDs. By realizing strong QD-electrodes coupling, very high Kondo temperature T_K∼80 K was observed in our InAs QD system.     

On classical and quantum relativistic dynamics

A canonical formalism for the relativistic classical mechanics of many particles is proposed. The evolution equations for a charged particle in an electromagnetic field are obtained and the relativistic two-body problem with an invariant interaction is treated. Along the same line a quantum formalism for the spinless relativistic particle is obtained by means of imprimitivity systems according to Mackey theory. A quantum formalism for the spin-1/2 particle is constructed and a new definition of spin1/2 in relativity is proposed. An evolution equation for the spin-1/2 particle in an external electromagnetic field is given. The Bargmann Michel, and Telegdi equation follows from this formalism as a quasiclassical approximation. Finally, a new relativistic model for hydrogenlike atoms is proposed. The spectrum predicted is in agreement with Dirac's when radiative corrections have been added.     

The generator-coordinate-method with conjugate parameters and the unification of microscopic theories for large amplitude collective motion

A dynamic theory of large amplitude collective motion of many particle systems is presented which is relevant, for example, to nuclear fission. The theory is microscopic and makes use of a collective path, i.e. a suitably constructed set of distorted nonequilibrium Slater determinants. The approach is a generalization of the generator coordinate method (GCM) and improves its dynamic aspects by extending it to a pair of conjugate generator parameters q and p (DGCM). The problems connected with redundancy and superfluous degrees of freedom are solved by prediagonalizing the local oscillations about each point of the dynamic collective basis | q, p ～. For adiabatic large amplitude collective motion a Schrödinger equation is derived which appears to be nearly identical to the one obtained by a consistent quantization of semiclassical approaches as e.g. the adiabatic time dependent Hartree-Fock theory (ATDHF). In turn a collective path constructed by ATDHF proves to be particularly suited for being used in the present DGCM formalism. Altogether the formalism unifies two classes of microscopic approaches to collective motion, viz. the quantum mechanical GCM and the classical theories like cranking and ATDHF.     

Classical and quantum pumping in closed systems

Pumping of charge (Q) in a closed ring geometry is not quantized even in the strict adiabatic limit. The deviation form exact quantization can be related to the Thouless conductance. We use the Kubo formalism as a starting point for the calculation of both the dissipative and the adiabatic contributions to Q. As an application we bring examples for classical dissipative pumping, classical adiabatic pumping, and in particular we make an explicit calculation for quantum pumping in case of the simplest pumping device, which is a three site lattice model. We make a connection with the popular S-matrix formalism which has been used to calculate pumping in open systems.     

The QCD Hamiltonian and nonlinear quantum mechanics

We study the canonical quantization of SU(N) gauge theory in linear, noncovariant gauges. The canonical formalism is first discussed for the classical theory, with special attention to the features involving nonlinearity and the gauge degrees of freedom. The transition to the quantum theory is then performed for an arbitrary linear gauge, using the covariant quantization rules of nonlinear quantum mechanics. When the quantum Hamiltonian is written in the Weyl-ordered form appropriate for the application of the usual Dyson-Wick perturbative techniques, additional ordering terms appear with respects to the classical Hamiltonian. We discuss the relation of our results to those of previous authors, and the relevance of the ordering terms in field theory.     

On the structure of the irreducible representation basis for the exceptional group G2

Using the method of projection operators we have constructed the basis of the irreducible representation D

of the exceptional Lie group G₂ corresponding to the reduction of this group to the subgroup SU₃. The basis is nonorthogonal but convenient for calculations. The matrices of the generators of the group G₂ in this basis have been found. The problem of additional quantum number ω required for the complete labelling of the basis vectors is considered. For this purpose we introduce the operator ω which is cubic with respect to the generators of the group G₂ and scalar with respect to the subgroup SU₃. The matrix of this operator has been calculated in the nonorthogonal basis. This matrix has a nondegenerate spectrum of eigenvalues ω which can be used as the missing quantum number.     

Action-angle variables for dihedral systems on the circle

A nonrelativistic particle on a circle and subject to a cos−2(kφ) potential is related to the two-dimensional (dihedral) Coxeter system I₂(k), for k∈N. For such ‘dihedral systems’ we construct the action-angle variables and establish a local equivalence with a free particle on the circle. We perform the quantization of these systems in the action-angle variables and discuss the supersymmetric extension of this procedure. By allowing radial motion one obtains related two-dimensional systems, including A₂, BC₂ and G₂ three-particle rational Calogero models on R, which we also analyze.     

Quantum Mechanics in Curved Configuration Space

Different approaches are compared to formulation of quantum mechanics of a particle on the curved spaces. At first, the canonical, quasiclassical, and path integration formalisms are considered for quantization of geodesic motion on the Riemannian configuration spaces. A unique rule of ordering of operators in the canonical formalism and a unique definition of the path integral are established and, thus, a part of ambiguities in the quantum counterpart of geodesic motion is removed. A geometric interpretation is proposed for noninvariance of the quantum mechanics on coordinate transformations. An approach alternative to the quantization of geodesic motion is surveyed, which starts with the quantum theory of a neutral scalar field. Consequences of this alternative approach and the three formalisms of quantization are compared. In particular, the field theoretical approach generates a deformation of the canonical commutation relations between operators of coordinates and momenta of a particle. A cosmological consequence of the deformation is presented in short.