More about superconformal field theory. Hamiltonian formalism

The Hamiltonian formalism for theN=1,d=4 superconformal system is given. The first-order formalism is found by starting from the canonical covariant one. As the conformal supergravity is a higher-derivative theory, to analyze the second-order Hamiltonian formalism the Ostrogradski transformation is introduced to define canonical momenta.     

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.     

Hamiltonian formalism,quantization and S matrix for supergravity

The canonical formalism for supergravity is constructed. The algebra of canonical constraints is found. The correct expression for the S matrix is obtained. Usual “covariant methods” lead to an incorrect S matrix in supergravity, since a new four-particle interaction of ghostfields survives in the Lagrangian expression of the S matrix.     

Hamiltonian Dynamics for Einstein’s Action in <Emphasis Type="Italic">G</Emphasis>→0 Limit

The Hamiltonian analysis for the Einstein’s action in G→0 limit is performed. Considering the original configuration space without involve the usual ADM variables we show that the version G→0 for Einstein’s action is devoid of physical degrees of freedom. In addition, we will identify the relevant symmetries of the theory such as the extended action, the extended Hamiltonian, the gauge transformations and the algebra of the constraints. As complement part of this work, we develop the covariant canonical formalism where will be constructed a closed and gauge invariant symplectic form. In particular, using the geometric form we will obtain by means of other way the same symmetries that we found using the Hamiltonian analysis.     

Hamilton's principle,covariant Hamiltonian,and canonical formalism in four and five dimensions

A discussion of the 1950s and 1960s on the existence of an explicit covariant canonical formalism is renewed. A new point of view is introduced where Hamilton's principle, based on the existence of a Hamiltonian, is postulated independently from the Lagrange formalism. The Hamiltonian is determined by transformation properties and dimensional considerations. The variation of the action without constraints leads to an explicit covariant canonical formalism and correct equations of motion. The introduction of the charge as a fifth momentum gives rise to a reformulation of classical relativistic point mechanics as a five-dimensionalU(1) gauge theory with a theoretically invisible extra dimension. A generalization to other gauge groups is given. The inversion of the proper time is introduced as a new particle-antiparticle symmetry that allows one to show that in the five-dimensional classical theory all particles have positive energy.     

Continuously infinite commensurate-incommensurate phase transition of a two-dimensional competing Ising model

We consider the critical behavior of a two-dimensional competing axial Ising model including interactions up to third nearest neighbors in one direction. On the basis of a low-temperature analysis relating the transfer matrix of this model with the Hamiltonian of theS = 1/2XXZ chain, it is shown that the usual square root singularity dominating commensurate-incommensurate phase transitions of two-dimensional systems merges into a continuously infinite transition for certain relations among the coupling parameters. The conjectured equivalence between the maximum eigenstate of the transfer matrix associated with this model and the ground state of theXXZ chain is tested numerically for lattice widths up to 18 sites.     

CANONICAL QUANTIZATION OF GAUGE FIELDS (I)——THE YANG-MILLS FIELD

The Yang-Mills field is quantized within the canonical formalism in covariant gauges. The interaction Lagrangian of X and X', i.e. the unphysical components of A_μ, is studied. In this Lagramgian there is only one term contributing to the S matrix elements between physical states. It is the source of the breaking of the unitarity of the physical S matrix. We get the gauge compensating term by solving a simple functional differential equation. If the gauge compensating term is added to the action, the S matrix in the physical state vector space can be expressed in a form which has no couplings of physical and unphysical particles, and so the physical S matrix is gauge independent and unitary.     

Generalized Hamiltonian formalism of (2+1)-dimensional non-linear \sigma-model in polynomial formulation

We investigate the canonical structure of the (2+1)-dimensional non-linear model in a polynomial formulation. A current density defined in the non-linear model is a vector field, which satisfies a formal flatness (or pure gauge) condition. It is the polynomial formulation in which the vector field is regarded as a dynamic variable on which the flatness condition is imposed as a constraint condition by introducing a Lagrange multiplier field. The model so formulated has gauge symmetry under a transformation of the Lagrange multiplier field. We construct the generalized Hamiltonian formalism of the model explicitly by using the Dirac method for constrained systems. We derive three types of the pre-gauge-fixing Hamiltonian systems: In the first system the current algebra is realized as the fundamental Dirac Brackets. The second one manifests the similar canonical structure as the Chern-Simons or BF theories. In the last one there appears an interesting interaction as the dynamic variables are coupled to their conjugate momenta via the covariant derivative. Received: 29 September 1998 / Published online: 14 January 1999     

Quantization of non-local field theory

The scheme of quantisation of non-local field theory is formulated. An intermediate regularisation is introduced into the non-local Lagrangian of the classical scalar field in such a way that the procedure of the canonical quantisation leads to the appearance of additional ghost states with indefinite metrics. The ghost states disappear when the regularisation is removed but the propagator of the scalar particle becomes non-local and theS-matrix is finite, unitary, causal and covariant in each perturbation order.     

Gauge independence for noncovariant supergauges

Previous work on noncovariant gauge choices within the superfield formalism was restricted to renormalization problems. Here we show the gauge independence and (global) supersymmetry for theS-matrix with suitably defined on-shell physical sources.     

Canonical transformations in a higher-derivative field theory

It has been suggested that the chiral symmetry can be implemented only in classical Lagrangians containing higher covariant derivatives of odd order. Contrary to this belief, it is shown that one can construct an exactly soluble two-dimensional higher-derivative fermionic quantum field theory containing only derivatives of even order whose classical Lagrangian exhibits chiralgauge invariance. The original field solution is expressed in terms of usual Dirac spinors through a canonical transformation, whose generating function allows the determination of the new Hamiltonian. It is emphasized that the original and transformed Hamiltonians are different because the mapping from the old to the new canonical variables depends explicitly on time. The violation of cluster decomposition is discussed and the general Wightman functions satisfying the positive-definiteness condition are obtained.     

Second-Order Five-Dimensional Chern–Simons Gravity

Continuing our previous discussion of the canonical covariant formalism (Zandron, O. S. (in press). International Journal of Theoretical Physics), the second-order canonical fünfbein formalism of the topological five-dimensional Chern–Simons gravity is constructed. Since this gravity model naturally contains a Gauss–Bonnet term quadratic in curvature, the second-order formalism requires the implementation of the Ostrogradski transformation in order to introduce canonical momenta. This is due to the presence of second time-derivatives of the fünfbein field. By performing the space–time decomposition of the manifold M ⁵, the set of first-class constraints that determines all the Hamiltonian gauge symmetries can be found. The total Hamiltonian as generator of time evolution is constructed, and the apparent gauge degrees of freedom are unambiguously removed, leaving only the physical ones.     

Quantization in Quantum System with Time-Dependent Boundary Condition

An approach to quantize a quantum mechanical system with time-dependent boundary condition is proposed in the framework of canonical quantization. It can be achieved by introducing the time-dependent boundary condition into the usual Lagrangian. We set up the effective Hamiltonian formalism and believe that this formalism can provide a generalized method to calculate the boundary effects.     

Topological Five-Dimensional Chern–Simons Gravity Theory in the Canonical Covariant Formalism

The canonical covariant formalism (CCF) of the topological five-dimensional Chern–Simons gravity is constructed. Because this gravity model naturally contains a Gauss–Bonnet term, the extended CCF valid for higher curvature gravity must be used. In this framework, the primary constraint and the total Hamiltonian are found. By using the equations of the CCF, it is shown that the bosonic five-form which defines the total Hamiltonian is a first-class dynamical quantity strongly conserved. In this context the equations of motion are also analyzed. To determine the effective interactions of the model, the toroidal dimensional reduction of the five-dimensional Chern–Simons gravity is carried out. Finally the first-order CCF and the usual canonical vierbein formalism (CVF) are related and the Hamiltonian as generator of time evolution is constructed in terms of the first-class constraints of the coupled system.     

Covariant Hamiltonian Field Theory: Path Integral Quantization

The Hamiltonian counterpart of classical Lagrangian field theory is covariant Hamiltonian field theory where momenta correspond to derivatives of fields with respect to all world coordinates. In particular, classical Lagrangian and covariant Hamiltonian field theories are equivalent in the case of a hyperregular Lagrangian, and they are quasi-equivalent if a Lagrangian is almost-regular. In order to quantize covariant Hamiltonian field theory, one usually attempts to construct and quantize a multisymplectic generalization of the Poisson bracket. In the present work, the path integral quantization of covariant Hamiltonian field theory is suggested. We use the fact that a covariant Hamiltonian field system is equivalent to a certain Lagrangian system on a phase space which is quantized in the framework of perturbative quantum field theory. We show that, in the case of almost-regular quadratic Lagrangians, path integral quantizations of associated Lagrangian and Hamiltonian field systems are equivalent.     

Canonical quantization and chromodynamics in a spherical cavity

The canonical quantization formalism is applied to the Lagrange density of chromodynamics, which includes gauge fixing and Faddeev-Popov ghost terms in a general covariant gauge. We develop the quantum theory of the interacting fields in the Dirac picture, based on the Gell-Mann and Low theorem and the Dyson expansion of the time evolution operator. The physical states are characterized by their invariance under Becchi-Rouet-Stora transformations. Subsequently, confinement is introduced phenomenologically by imposing, on the quark, gluon, and ghost field operators, the linear boundary conditions of the MIT bag model at the surface of a spherically symmetric and static cavity. Based on this formalism, we calculate, in the Feynman gauge, all nondivergent Feynman diagrams of second order in the strong coupling constantg. Explicit values of the matrix elements are given for low-lying quark and gluon cavity modes.     

Mesonic degrees of freedom in nuclei and the definition of meson-exchange currents

We describe the nucleus by a meson-nucleon system. Starting from a covariant field theoretical Hamiltonian we derive an effective Schrödinger equation for the nucleonic components. The meson-exchange currents are then defined unarbitrarily by an effective operator (current) in the space of the nucleonic components. The advantage over theS-matrix method [1] is discussed. In the nonrelativistic limit the meson-current as well as the seagull (pair) current agrees with theS-matrix result. Recoil and wavefunction orthogonalization cancels completely in this limit.     

Gravity and the frame field

The covariant derivative of a single massive fermion field on a Riemannian manifold is defined. The standard method of defining free bosonic Lagrangians from the fermion covariant derivative does not give the usual Lagrangian density for the free gravitational field. We express the fermion Lagrangian mass term as a frame field term added to the covariant derivative; this extended covariant derivative defines a gravitational Lagrangian density proportional to the usual scalar curvatureR, plus a term quadratic in the curvature components. The quadratic term is expected to be negligible at distances much greater than the fermion Compton wavelength, and is of a general form widely studied in recent years. The frame field term used to derive this gravitational Lagrangian is essentially the same as that used previously to derive the electroweak interaction boson mass matrix without using the Higgs-Kibble mechanism.     

Some fundamental difficulties with quantum mechanical collision theory

When quantum scattering theory is applied strictly from the point of view that the state of a system is completely described by the density matrix, whether pure or mixed, it is not possible to assume that colliding particles are at all times individually in pure states. Exact results are significantly different from conventionally accepted approximations. In particular, it turns out that the cross section as ordinarily defined in theS-matrix formalism is an adequate parameter for deciding the outcome of interactions only when the particles are carefully prepared in matching pure states. In general the use of the cross section in studying pair collisions in a real gas is shown to be analogous to a repeated “collapse of the wave function” after each collision, and involves arbitrary removal of nondiagonal elements of the density matrix, thus violating the basic laws of quantum dynamical evolution.