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We apply generalized Bogoliubov transformations to the transfer matrix of relativistic field theories regularized on a lattice. We derive the conditions these transformations must satisfy to factorize the transfer matrix into two terms which propagate fermions and antifermions separately, and we solve the relative equations under some conditions. We relate these equations to the saddle point approximation of a recent bosonization method and to the Foldy-Wouthuysen transformations which separate positive from negative energy states in the Dirac Hamiltonian., 25 pages

In the absence of interaction it is well known that the Kogut-Susskind regularizations of fermions in the spin and flavor basis are equivalent to each other. In this paper we clarify the difference between the two formulations in the presence of interaction with gauge fields. We then derive an explicit expression of the transfer matrix in the spin basis by a unitary transformation on that one in the flavor basis which is known. The essential key ingredient is the explicit construction of the fermion Fock space for variables which live on blocks. Therefore the transfer matrix generates time translations of two lattice units., 16 pages

We assume that the most important quark correlations are pairwise at all baryon densities. We introduce correlated pairs by means of Bogoliubov transformations which are functions of time and spatial gauge fields, in the formalism of the transfer matrix with lattice regularization. The dependence on time and gauge fields allows us to enforce gauge invariance and other symmetries term by term in the transformed quantities. The resulting action should be suitable for the description of multiquark mesons and baryons as states of a quark and a diquark. We derive the quark contribution to the free energy at finite chemical potential in a certain approximation. Its expression cannot be evaluated analytically, but it has a definite sign., Comment: 36 pages

We construct renormalizable models of gauge theories in four dimensions, with gauge fields composite of bosonic or fermionic constituents. They are obtained by a regularization of $\mathrm{C}{\mathrm{P}}^{n\ensuremath{-}1}$ models where the constraint on constituent fields is replaced by a constraint on expectation values.

We apply to lattice QCD a bosonization method previously developed in which dynamical bosons are generated by time-dependent Bogoliubov transformations. The transformed action can be studied by an expansion in the inverse of the nilpotency index, which is the number of fermionic states in the structure function of composite bosons. When this number diverges the model is solved by the saddle point method which has a variational interpretation. We give a stationary covariant solution for a background matter field whose fluctuations describe mesons. In the saddle point approximations live fermionic quasiparticles with quark quantum numbers which are confined, in the sense that they propagate only in pointlike color singlets. Conditions for chiral symmetry breaking are determined, to be studied numerically, and a derivation of mesons-nucleons action is outlined., Comment: 33 pages

A noncompact formulation of gauge theories on the lattice recently proposed makes use of auxiliary fields. We show that in such a formulation the representation of the gauge group in the case of SU(2) is completely reducible, one of the irreducible multiplets containing the Yang-Mills fields plus one auxiliary field. Elimination of this field by a gauge-invariant constraint leads to Wilson's formulation.

We evaluate the Faddeev-Popov determinant in a complete axial gauge obtained by modifying the gauge A 3 =0 on a torus. The Gribov horizon does not enclose a compact volume.

Recently a new bosonization method has been used to derive, at zero fermion density, an effective action for relativistic field theories whose partition function is dominated by fermionic composites, chiral mesons in the case of QCD. This approach shares two important features with variational methods: the restriction to the subspace of the composites, and the determination of their structure functions by a variational calculation. But unlike standard variational methods it treats excited states on the same footing as the ground state. I show that this bosonization method is an approximation of an exact procedure in which composites are introduced in the presence of fermionic states with the quantum numbers of the constituents (quasiparticles). This procedure consists of an independent Bogoliubov transformation at each time slice. The time-dependent parameters of the transformation are then associated with composite fields. In this way states of nonvanishing fermion (baryon) number (neglected in the bosonization approach) are retained. By the exact procedure I derive an effective action for QCD at finite temperature and baryon density. I test the result on a four-fermion interaction model.

It is shown that starting from one and the same transfer matrix formulation of QCD on a lattice, it is possible to obtain both the action of Hasenfratz and Karsch as well as an action where the chemical potential is not coupled to the temporal links.

We present a new method of bosonization of fermion systems applicable when the partition function is dominated by composite bosons. Restricting the partition function to such states we get an euclidean bosonic action from which we derive the Hamiltonian. Such a procedure respects all the fermion symmetries, in particular fermion number conservation, and provides a boson mapping of all fermion operators., Comment: 12 pages

As a first step to derive the IBM from a microscopic nuclear hamiltonian, we bosonize the pairing hamiltonian in the framework of the path integral formalism respecting both the particle number conservation and the Pauli principle. Special attention is payed to the role of the Goldstone bosons. We construct the saddle point expansion which reproduces the sector of the spectrum associated to the addition or removal of nucleon pairs., 23 pages

We define nonlinear changes of variables in Berezin integrals as new integration variables of homogeneous polynomials of the defining elements in the Grassmann algebra. We perform such a change of variables in the partition function of QCD by introducing as integration variables cubic and quadratic polynomials of the quark field, with the quantum numbers of the nucleon and the meson respectively, and we suggest a perturbative scheme using these these functions as free states.

Recently it has been found that in a noncompact lattice regularization of the SU(2) gauge theory the physical volume is larger than in the Wilson theory with the same number of sites. In its original formulation the noncompact regularization is directly applicable to U(n) theories for any n and to SU(n) theories for n=2. In this paper we extend it to SU(n) for any n and investigate some of its properties., 14 pages

We define the chemical potential as the Lagrange multiplier of the baryon charge operator in the transfer matrix formalism of QCD on a lattice. Transforming the partition function into an euclidean path integral we get the Hasenfratz-Karsh action both for Wilson and Kogut-Susskind fermions. In the latter case the chemical potential in the spin-diagonal basis is half that in the flavour basis. Some open problems in the spin-diagonal basis are pointed out., Comment: 16 pages

The pairing interaction among identical nucleons in a single-particle level is treated in the hamiltonian formalism using even Grassmann variables. A minimal (irreducible) basis having a remarkable symmetry property is set up using composite, commuting variables with a finite index of nilpotency. Eigenvalues and eigenfunctions of given energy, seniority and zero third component of the angular momentum of the pairing hamiltonian are then found. The eigenvectors, which cannot be cast solely in terms of composite bosons with angular momentum zero and two, are expanded in the minimal basis with coefficients analytically expressed in terms of a generalized hypergeometric function., 15 pages, LateX, 1 Postscript figure

An action is constructed with noncompact gauge fields which is gauge invariant on the lattice. It requires the introduction of auxiliary fields which disappear in the continuum limit, with the possible exception of an abelian field.

A quadratic action in even Grassmann variables with the quantum numbers of the pions has been studied. It includes the $\sigma$-field in order to be invariant under SU(2)_L times SU(2)_R transformations over the quarks. This action exhibits the Goldstone phenomenon reducing its symmetry to the O(3) isospin invariance. The model has been investigated in the Stratonovitch-Hubbard representation, in which form it is reminiscent of the Gell-Mann-Levy model. By the saddle point method a renormalizable expansion in inverse powers of the index of nilpotency of the mesonic fields (which is 24), is generated. The way it might be used in a new perturbative approach to QCD is outlined., Comment: Talk presented at Lattice '97 (Theoretical Developments), Edinburgh (UK), July 1997. LaTeX2e, 3 pages, uses espcrc2

We show that in the hamiltonian formalism of compact QED on the lattice gauge transformations appear as representations of translations on a circle or on a line according to the boundary conditions on the wave functions. When gauge invariance is enforced, one gets at strong coupling a linear potential in the first case and a Coulombic potential in the second.

The mass formula summation/sub J/(- 1)/sup 2J/(2J + 1) m/sub J/ /sup 2/ = 0 is derived for a very general class of interactions with spontaneously broken supersymmetry. It shows the vanishing of the graded trace of the square of the mass operator, with m/sub J/ the mass associated with a (real) field of spin J. This mass relation is shown to be true even in the presence of explicit breaking, provided it fulfills suitable requirements.

It is pointed out that in the hamiltonian formulation of gauge theories on a lattice consistency of the Gauss constraint requires that the gauge group be compact. As a consequence the electric charge is quantized irrespective of the compactness of the transverse gauge field.

We study the strong coupling limit of noncompact Yang-Mills theories on a lattice. We find that quarks are superconfined namely single quarks cannot be separated even by one lattice spacing. At large but finite values of the coupling constant superconfinement is relaxed, and the quark-quark wave function decays at large distance not faster than exponentially, a behaviour incompatible with a linear effective potential.

The generators of gauge field and matter field gauge transformations for the SU(2) gauge group can be written as orbital and spin angular momenta respectively. The Gauss law requires that the sum of these momenta vanish. This is only possible if the state functionals are restricted to be single-valued or double-valued with respect to the gauge fields and the matter fields gauge variables at the same time. Such a restriction can be imposed on scalar matter fields by introducing a polar representation.

The asymptotic behaviour of the form factor for non-relativistic bound systems is derived for local and nonlocal two- and three-body potentials, taking into account the Pauli principle. The effect of the Pauli principle is very important, giving rise to the extra factor q−2Kmin, where Kmin is the minimum degree in the homogeneous polynomials expansion, which grows like A 4 3 for large particle number A.

We derive the general formula for the supertrace of the quartic mass matrix in a general supersymmetric gauge theory, with arbitrary representations for the chiral multiplets. This formula clarifies the non-renormalization theorems in presence of gauge interactions and gives “extended renormalization theorems” for N = 2 and N = 4 supersymmetric Yang-Mills theories. In particular we find the known result that g ren = g bare for the N = 4 theory and the new result m ren = m bare for the N = 2 gauge interactions of massive hypermultiplets. We give arguments to the extent that the latter non-renormalization theorem persists to all orders in perturbation theory.