Microscopic analysis of nuclear collective motions in terms of the boson expansion theory: (I). Formulation

A normal-ordered linked-cluster boson expansion theory, previously worked out by one of the authors (T.K.) and Tamura, has been developed further by reformulating it in a “physical” quasiparticle subspace which contains no spurious particle-number excitation modes. The expansion coefficients of the collective hamiltonian for low-lying quadrupole motions are determined starting from a microscopic fermion hamiltonian including self-consistent higher-order (many-body) interactions derived in our previous work. The contributions from the non-collective states with all possible non-collective one-boson excitations having I^π = 0⁺− 4⁺, which can directly couple to the collective states with one or two phonons, are taken into account in a systematic and compact way.     

Fokker-Planck approach to quantum lattice hamiltonians

Fokker-Planck equations have been applied in the past to field theory topics such as the stochastic quantization and the stabilization of bottomless action theories. In this paper we give another application of the FP-techniques in a way appropriate to the study of the ground state, the excited states and the critical behaviour of quantum lattice hamiltonians. Our approach is based on the choice of an exponential or Jastrow-like state which becomes the exact ground state of a discrete FP-hamiltonian. The “variational” parameters entering into the ansatz are fixed by forcing the FP-hamiltonian to coincide with the original hamiltonian except for terms not included in the ansatz.To illustrate the method we apply it to the Ising model in a transverse field (ITF). In one dimension we build up a FP-hamiltonian belonging to the same universality class as the standard ITF model. Likewise some considerations concerning the Potts model are outlined.     

Classical Electrodynamics of Point Charges

A simple mathematical procedure is introduced which allows redefining in an exact way divergent integrals and limits that appear in the basic equations of classical electrodynamics with point charges. In this way all divergences are at once removed without affecting the locality and the relativistic covariance of the theory, and with no need for mass renormalization. The procedure is first used to obtain a finite expression for the electromagnetic energy-momentum of the system. We show that the relativistic Lorentz-Dirac equation can be deduced from the conservation of this electromagnetic energy-momentum plus the usual mechanical term. Then we derive a finite lagrangian, which depends on the particle variables and on the actual electromagnetic potentials at a given time. From this lagrangian the equations of motion of both particles and fields can be derived via Hamilton's variational principle. The hamiltonian formulation of the theory can be obtained in a straightforward way. This leads to an interesting comparison between the resulting divergence-free expression of the hamiltonian functional and the standard renormalization rules for perturbative quantum electrodynamics.     

Superconformal algebras and their relation to integrable nonlinear systems

We relate the manifold of periodic functions on a circle with values in the Grassmann algebra to extended superconformal algebras. The graded Poisson brackets of these functions give the classical realization of the corresponding superconformal algebras and determine the hamiltonian structure for a class of integrable nonlinear equations. A super-generalization of the Korteweg-de Vries equation is found among these equations. In this way an important step in the program of the quantization of the Liouville equation is realized for the supersymmetric cases which are crucial in constructing a consistent quantum string theory. The construction of Miura transformations is outlined and the results for the N = 1,2 supersymmetric cases are presented.     

TDHF equations of motion for algebraic hamiltonians in a coherent state representation

Time-dependent Hartee-Fock (TDHF) equations are derived for nuclear systems with internal dynamical group U(r). The coordinates which appear in the TDHF equations are the coordinates which parameterize the U(r) coherent states. The TDHF orbits for the hamiltonian $\text{H}$ are identical with equations of motion for a classical system described by the hamiltonian function 〈$\text{H}$〉 obtained directly from the operator $\text{H}$. This quantum-classical correspondence facilitates interpretation of TDHF orbits. The phenomena of coexistence and critical elongation are discusses, as is the relation between the critical points of the function 〈$\text{H}$〉 and the spectral properties fo the operator $\text{H}$.     

Effective hamiltonians

Non-linear equations, generalizing those determining Bloch's transformation to an effective hamiltonian, are set up in a very simple way. It is shown how they immediately determine the results of Van Vleck, Bloch and Soliverez. Treating the Foldy-Wouthuysen transformation as a special case of the Van Vleck transformation, sixth-order expressions are obtained by means of the non-linear equations. The transformations of Soliverez, Roussy and Kvasnicka are shown to be strictly identical—because they all have a simple property which is proven to be unique for that of Soliverez. Kvasnicka relates his transformation to that of Primas'. We shall see that it is also very closely related to that of Van Vleck's. A certain non-hermitian effective hamiltonian also given by Roussy is proven to be identical to that of Bloch's. Primas' level shift transformation gives an effective hamiltonian for each level in zeroth order. This effective hamiltonian is seen to be identical to that of Soliverez's up to and including third order—and to all orders if one makes a simple change in ‘normalization’. Computer treatment of finite matrices is discussed. The sixth-order Van Vleck hamiltonian recently given by Jørgensen and Pedersen in terms of commutators is derived in a more simple way.     

A family of completely integrable multi-hamiltonian systems

We consider an N-component hierarchy of nonlinear evolution equations, previously known to be bi-hamiltonian and completely integrable. We show that there exist not just two, but (N + 1) compatible hamiltonian structures for this hierarchy. For the case N = 2 we relate our equations to a tri-hamiltonian hierarchy introduced by Kupershmidt.     

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.     

From the first to the second quantized string theory

Starting from the hamiltonian formulation of the bosonic string we calculate the Feynman transition amplitude D. Inverting D we get the kinetic operator and thus the free field action.     

New symmetries and constants of the motion from dynamical groups

It is shown that every dynamical group (sometimes also called spectrum generating group) gives rise to a proper Noether symmetry group of the action. For each generator there is a constant of the motion. Those which do not commute with the hamiltonian but connect states of different energy contain an explicit time dependence when expressed as a function of the Heisenberg variables p(t), q(t) which ensures their conservation. If the hamiltonian is in the Lie algebra, this time dependence is given by a simple “rotation” matrix in the adjoint representation. The statements are illustrated by exhibiting the conserved symmetry operators for the bound state problem with electric and magnetic charges.     

Stochastic processes in lattice (extended) supersymmetry

We show that ungauged N = 2 supersymetric models can be put on the (hamiltonian) lattice in such a way as to preserve a subalgebra of supersymmetry large enough to ensure the existence of the Nicolai mapping. The models can be interpreted as stochastic systems described by Langevin equations. We describe both Wilson and Susskind versions of the model.In two dimensions everything seems fine, but in 4D, one expects, on general grounds, that the continuum limit will be either trivial or non-Lorentz invariant.     

Spectra of quantum chains without the Yang-Baxter equation

We study one-dimensional reaction-diffusion models described by master equations and their associated two-state quantum hamiltonians. By choosing appropriate rates, the equations of motion decouple into certain subsets. We solve the first subset which has a close relation to the problem of lattice electrons in an electric field. In this way we obtain L(L − 1) + 1 energy levels of a quantum chain with L sites. The corresponding hamiltonian depends on seven parameters and does not look integrable using conventional methods. As an application, we compute the dynamical critical exponent of a new type of kinetic Ising model.     

Hidden hamiltonians of first-order equations

There are equations, like the KDV equation, of which the solutions behave like conservative systems although the equation is of first order in time. It is shown how equations of this kind can originate by a direct-product like process of fusion of two canonical conjugate variables. Conversely, for a class of dynamically well-behaved first-order equations a splitting of the independent variable into two conjugate parts and a corresponding hamiltonian functional can be found. It is shown how the action principle and the Noether theorem transform during this fusion or splitting process. A number of examples are discussed. It is shown how a KDV approximation can be derived directly from the hamiltonian of a second-order system without using the second-order wave equations.     

Exact excitation spectrum of the Zn + 1 × Zn + 1 generalized Heisenberg model

We describe the excitations of the generalization of the Heisenberg anisotropic hamiltonian to Z_{n + 1} spins. We analyse the Bethe ansatz equations without assuming the existence of strings. Excited states above an antiferromagnetic ground state are described by a finite number of parameters, which verify system of equations. We give the energy and momentum of the excitations. In a suitable limit we recover the spectrum of a relativistically invariant theory.     

Extending integrable hamiltonian systems from 2 to N dimensions

We describe a method of constructing N-dimensional integrable hamiltonian systems starting from two-dimensional ones. Several models are examined. Included are the two candidates for integrability discovered by Lakshmanan and Sahadevan for which we find the integrals of motion. Results for other N-dimensional integrable hamiltonian systems are also presented.     

Stochastic stability of a plasma torus

The hamiltonian equations for magnetic field lines in a toroidal plasma are derived from a variational principle; we find an equation for critical fluctuations

_n by assuming a marginal overlap of adjacent resonances throughout the small cross section for the torus. Comparison with the flux surfaces of JET as obtained from a static equilibrium code leads to critical fluctuations of the order of δB_crit ≲ B_px10⁻⁴ for n ≳ 10, where B_p is the poloidal equilibrium field. Finally, we solve the equations for magnetic field lines in the case of a set of nested flux surfaces perturbed by critical fluctuations; the picture of a completely destroyed torus has been verified with a stochasticity parameter √3, but no stable integration scheme for a stochastic magnetic line has been found.     

String length and covariant string field theory

String length differentials are shown to arise naturally in a BRST hamiltonian approach to the closed bosonic string. A gauge-covariant quadratic action is proposed which is shown to lead to the correct equations of motion. The gauge covariant action is extended to including a three-string interaction vertex, with exact invariance under an extended transformation on the string fields.     

Gauge covariant formulation of the generating operator. 2. Systems on homogeneous spaces

A gauge covariant approach to the operator Λ, generating the n-wave type equations on homogeneous spaces is proposed. The operator Λ̃ for the gauge equivalent equations is explicitly constructed. The main results (such as conservation laws, hierarchies of hamiltonian structures, etc.) for the n-wave type equations and their gauge equivalent ones are formulated in terms of Λ and Λ̃ respectively.     

IBFM form of SU(6/M) supergenerators for microscopic foundation

Canonical generators of the SU(6/M) supergroup are transformed to the IBFM form associated with the graded Schwinger realization which explicitly separates the SU(6)⊕SU(M) subalgebra. In the frame of this supersymmetry scheme we relate the IBFM hamiltonian to the microscopic quasiparticle-phonon hamiltonian.