Hamiltonian treatment of self-dual antisymmetric tensors

Self-dual antisymmetric tensors occur in certain (4n + 2)-dimensional supergravity models, in particular the one connected to type IIB superstrings. So far they have been quantized only using light front methods. Here we show that the existing covariant action for such fields leads to a consistent hamiltonian system also for ordinary “timelike” dynamics, although rather complicated second class constraints are present.     

Hamiltonian BRST quantization of the interacting antisymmetric tensor field

We study the hamiltonian BRST quantization of the non-abelian antisymmetric tensor field. We find the constrained system which arises from the standard action by Dirac's procedure, and eliminate the second-class constraints by introducing Dirac brackets. Having isolated the underlying first-class constrained system, we quantize it using the hamiltonian BRST techniques of Batalin and Fradkin. We study the Lorentz covariant gauge fixing of this system, and discuss the relationship between our results and other recent studies of the interacting antisymmetric tensor field.     

Stability of critical behaviour,critical-exponent renormalization and first-order transitions

The critical behaviour of systems with short-range interactions in which the free energy has divergent second derivatives is shown to be unstable under small perturbations. The perturbations can arise from additional terms in the hamiltonian with a long-range nature, but also from hidden variables, subjected to constraints. In general there will be either critical-exponent renormalization or first-order transitions; in special cases one can also have more complicated multicritical behaviour. In the analysis use is made of Legendre transformations and homogeneity properties of the short-range system. The results are rather general and independent of specific properties of operators in the hamiltonian, or the nature and number of hidden variables.     

Hamiltonian formulation of string field theory

Witten's string field theory is quantized in the hamiltonian formalism. The constraints are solved and the hamiltonian is expressed in terms of only physical degrees of freedom. Thus, no Faddeev-Popov ghosts are introduced. Instead, the action contains terms of arbitrarily high order in the string functionals. Agreement with the standard results is demonstrated by an explicit calculation of the residues of the first few poles of the four-tachyon tree amplitude.     

The Hamiltonian formulation of the Euler equation and subsequent constraints on the properties of randomly stirred fluids

It is shown that the existence of a new integral of the motion for inviscid flow, evident in the hamiltonian formulation of the Euler equation, results in some important constraints on the properties of randomly stirred fluids.     

Polaron formalism applied to donor-acceptor pairs in semiconductors

The effective hamiltonian for the electronic states of a donor-acceptor pair in a polar semiconductor is formulated as a Fröhlich hamiltonian generalized to an electron and a positive hole interacting in the field of the positive donor and negative acceptor cores. The polarization of the lattice by the ion cores is simplified by a Platzman transformation; that by the electron and positive hole, by a further unitary transformation in the adiabatic approximation. The total effective hamiltonian including the lattice polarization energy is then considered for application of the variational principle to determine the parameters of the electron and positive hole effective mass functions. The Inglis-Williams analysis of the zero-phonon luminescent spactra of pairs is reconciled with the present theory by determining what single effective hamiltonian for the excited electronic state is consistent with their theoretical spactra. Within the approximations of their analysis, which are clarified, it is found to be the total effective hamiltonian. Finally, a more general and rigorously-derived effective hamiltonian is presented for the direct theoretical determination of the zero-phonon radiative transition energies of donor-acceptor pairs in compound semiconductors.     

How the Jones polynomial gives rise to physical states of quantum general relativity

Solutions to both the diffeomorphism and the hamiltonian constraint of quantum gravity have been found in the loop representation, which is based on Ashtekar's new variables. While the diffeomorphism constraint is easily solved by considering loop functionals which are knot invariants, there remains the puzzle why several of the known knot invariants are also solutions to the hamiltonian constraint. We show how the Jones polynomial gives rise to an infinite set of solutions to all the constraints of quantum gravity thereby illuminating the structure of the space of solutions and suggesting the existance of a deep connection between quantum gravity and knot theory at a dynamical level.This essay received the third award from the Gravity Research Foundation, 1992-Ed     

Hamiltonian formulation of the supermembrane

The hamiltonian formulation of the supermembrane theory in eleven dimensions is given. The covariant split of the first and second class constraints is exhibited, and their Dirac brackets are computed. Gauge conditions are imposed in such a way that the reparametrizations of the membrane with divergence-free 2-vectors are unfixed.     

Phase space action for minimal surfaces of any dimension in curved spacetime

The phase space action, depending on coordinates, momenta and Lagrange multipliers (which turn out to be components of the induced metric), has been written down for a world sheet of arbitary dimension (a string generalized to membranes) in a curved embedding spacetime. Canonical and hamiltonian formalisms have been formulated in a covariant and general way with the true dynamical variables being separated from the redundant ones. The membrane constraints follow directly from the variation of our action; their suitable superposition gives a hamiltonian from which we derive the equations of motion for a membrane via the Poisson brackets. The same hamiltonian we obtain also in a different way from a variation of the action. For n = 2 all equations coincide with those of strings.     

Magnetic multipole moments

Literature definitions of magnetic multipole moment operators are shown to be at variance, and new definitions are formulated which are consistent with a general multipole interaction hamiltonian and with the radiation field of a dynamic charge distribution. The applicability of traceless multipole moments is examined.

The multipole hamiltonian is used to derive expressions for some magnetic quadrupole distortion tensors. For those describing the quadrupole moment induced by a magnetic field and by a field gradient the number of independent components for various molecular symmetries is evaluated.     

IBM-2 description of M1 properties in deformed nuclei

Results of schematic calculations are presented in which various terms breaking F-spin symmetry are considered in the hamiltonian of the neutron-proton interacting boson model (IBM-2). Specific attention is paid to the effect of F-spin symmetry breaking on γ → ground and γ → γ M1 transitions in deformed nuclei. A comparison with available M1 data in the rare-earth nuclei is presented. The constraints implied by these data on the form of the IBM-2 hamiltonian in well-deformed nuclei are discussed.     

Dispersion relations for light propagation through a two-level system

Interaction of radiation with two-level system is considered using the Dicke hamiltonian. By making use of the Schwinger representation, the hamiltonian is written in a trilinear form. The hamiltonian is made bilinear by replacing one set of boson operators by C-numbers. A Bogoliubov transformation is then used to reduce the hamiltonian to diagonal form. The density matrix separates into a product of density matrices for separate modes. The energies of the photon-like modes are found to follow dispersion relations which depend on the given set of initial conditions.     

Hamiltonian formulation of QED in the superaxial gauge

We present a hamiltonian formulation of QED in a fully fixed axial gauge. The equal-time commutators for all field variables are computed and are shown to lead to the correct equations of motion. The constraints and gauge conditions hold as strong operator relations.     

Yang-Mills theory on a cylinder

Yang-Mills theory on a two dimensional cylinder is studied in the hamiltonian formalism, without using gauge conditions. Since the only gauge invariant variable is the Wilson loop (holonomy) this system is equivalent to a finite dimensional system. The eigenstates and eigenvalues of the hamiltonian are found exactly.     

Constrained systems,OSP(1,1|2) and string theory

The underlying OSP invariance of the Fradkin-Vilkovisky formalism is discussed. Ghost degrees of freedom are interpreted as negative dimensional phase space variables that eliminate unphysical degrees of freedom by the Parisi-Sourlas mechanism, ensuring manifest covariance. The formalism makes use of subsidiary constraints, extending the usual algebra of constraints. A relations between abelian and nonabelian constraint algebras is established, and exploited to construct a nonabelian representation of the OSP generators. For theories based entirely on constraints such as string theories, the natural Fradkin-Vilkovisky hamiltonian is a manifestly OSP invariant squared length of a graded phase space vector. As an application, the OSP covariant formulation of bosonic strings is discussed.     

The anomalous chiral Schwinger model and its quantization

We solve the anomalous chiral Schwinger model in the hamiltonian formulation implementing the Faddeev proposal. We diagonalize the hamiltonian by functional techniques and construct the eigenfunctionals of the quantized system. The spectrum is non Lorentz invariant since the Poincarè algebra is not closed. We point out that, relinquishing the Faddeev condition, there would be an infinite degeneration of the hamiltonian. Among the degenerate 1-particle eigenfunctionals it is possible to find a state with relativistic spectrum. We also examine the meaning of a non local modification of the hamiltonian which restores Lorentz invariance.     

Stability in bihamiltonian systems and multidimensional rigid body

The presence of two compatible hamiltonian structures is known to be one of the main, and the most natural, mechanisms of integrability. For every pair of hamiltonian structures, there are associated conservation laws (first integrals). Another approach is to consider the second hamiltonian structure on its own as a tensor conservation law. The latter is more intrinsic as compared to scalar conservation laws derived from it and, as a rule, it is “simpler”. Thus it is natural to ask: can the dynamics of a bihamiltonian system be understood by studying its hamiltonian pair, without studying the associated first integrals?     

Doubly-occupied orbital MCSCF methods

A new computational method for a previously suggested Multi-Configuration SCF molecular wave function is developed and applied to some simple molecules. The method involves the use of a single Fock-like hamiltonian matrix for all the MCSCF orbitals and does not involve orthogonality constraints or coupling operators. Application to lithium hydride shows that this method is capable of recovering a large part of the correlation energy and compares very favourably with conventional lengthy CI methods.     

Spectral singularities of the quartic anharmonic oscillator

The singularities of the eigenvalues of the quartic anharmonic oscillator hamiltonian are studied numerically as a function of the quartic coupling parameter. Their location is found to be consistent with previous WKB calculations but the identification of which levels are singular is found to differ.