A remark on the boson-fermion correspondence

We introduce the space of skew-symmetric functions depending on an infinite number of variables and give a simple interpretation of the boson-fermion correspondence.     

Nonabelian orbifolds and the boson-fermion correspondence

For a positive integerl divisible by 8 there is a (bosonic) holomorphic vertex operator algebra (VOA) associated to the spin lattice _l . For a broad class of finite groupsG of automorphisms of we prove the existence and uniqueness of irreducibleg-twisted -modules and establish the modular-invariance of the partition functionsZ(g, h, ) for commuting elements inG. In particular, for any finite group there are infinitely many holomorphic VOAs admittingG for which these properties hold. The proof is facilitated by a boson-fermion correspondence which gives a VOA isomorphism between and a certain fermionic construction, and which extends work of Frenkel and others.Supported by NSA grant MDA904-92-H-3099.Supported by NSF grant DMS-9122030.     

Unitary representations of some infinite dimensional groups

We construct projective unitary representations of (a) Map(S ¹;G), the group of smooth maps from the circle into a compact Lie groupG, and (b) the group of diffeomorphisms of the circle. We show that a class of representations of Map(S ¹;T), whereT is a maximal torus ofG, can be extended to representations of Map(S ¹;G),     

Riemann surfaces,Clifford algebras and infinite dimensional groups

We consider functionals on one dimensional subshifts which have prescribed Randon-Nikodym derivative under transportation by conjugating homeomorphisms, and investigate their relation to Ruelle's transfer operator. In particular we show that two-sided functionals essentially are products of a functional which are supported on stable and unstable leaves. We also prove the meromorphicity of the Fourier transform of correlation functions for AxiomA follows in a more general setting.     

Riemann surfaces,Clifford algebras and infinite dimensional groups

We introduce a class of Riemann surfaces which possess a fixed point free involution and line bundles over these surfaces with which we can associate an infinite dimensional Clifford algebra. Acting by automorphisms of this algebra is a gauge group of meromorphic functions on the Riemann surface. There is a natural Fock representation of the Clifford algebra and an associated projective representation of this group of meromorphic functions in close analogy with the construction of the basic representation of Kac-Moody algebras via a Fock representation of the Fermion algebra. In the genus one case we find a form of vertex operator construction which allows us to prove a version of the Boson-Fermion correspondence. These results are motivated by the analysis of soliton solutions of the Landau-Lifshitz equation and are rather distinct from recent developments in quantum field theory on Riemann surfaces.     

A note on the dimensional crossover critical exponent

Letters in Mathematical Physics - We consider independent anisotropic bond percolation on $${\mathbb {Z}}^d\times {\mathbb {Z}}^s$$ where edges parallel to $${\mathbb {Z}}^d$$ are open with...     

Unitary representations of semi-direct product groups with infinite dimensional Abelian normal subgroup

We generalize Mackey's theory of induced representations to groups which are semidirect products of a locally compact group H by an infinite dimensional Abelian group A. A theorem on inducing in stages is proved and conditions ensuring conservation of the irreducibility of the representation through the inducing process are given. When A is a nuclear space or the Hilbert space A_O of a Gelfand triplet A_O ? A ? A′_O, a cylindrical ergodic measure on the dual space A′ of A (respectively A′_O) is associated to each irreducible representation of G, and it is proved that the representation is induced when the measure is concentrated on an orbit. In the last part, A is supposed to be a Hilbert space and sufficient conditions are given for the preceding ergodic measures to be concentrated on an orbit.     

Index formulas for generalized Wiener-Hopf operators and boson-fermion correspondence in 2N dimensions

The kernels of operators associated with special chiral gauge transformations (kinks) in the 2N-dimensional Dirac theory are explicitly determined. The result is used to obtain index formulas for Fredholm operators corresponding to continuous chiral gauge transformations. Moreover, the Fock space quadratic forms corresponding to the kinks are proved to converge to the Dirac field as the kink size goes to zero. It is also shown that forN 1, 2(mod 4) the Majorana field can be reached in a similar fashion.Work supported by the Netherlands Organisation for the Advancement of Research (NWO)     

Scattering and infinite dimensional symmetry schemes

We examine the recently proposed non-trivial binding of the Poincare group with internal symmetry groups, which leads to mass splitting without symmetry breaking, from the point of view of scattering. We give a very general realization of the corresponding infinite dimensional Lie algebra in the Fok space of asymptotic states built from two scalar nucleons. Assuming that the algebra describes the exact symmetry of theS-matrix we conclude that no elastic NN scattering exists.The author is indebted to Professor V. Votruba and Dr. J. Formánek for many stimulating discussions.     

A coriolis-like effect of the dynamical boson-fermion interaction in the interacting boson-fermion model

It is shown that in a particular basis the dynamical interaction of IBFM for a boson core coupled to a single-j particle in the SU(3) limit takes a tridiagonal form, as does the Coriolis interaction. Interference between this Coriolis-like effect in the boson-fermion dynamical interaction and the genuine Coriolis effect results in interesting features of IBFM: For a particular ratio of the interaction strengths, depending only on the angular momentum of the odd particle, the K = j band exactly follows the rotational energy rule, is completely uncoupled from the other bands and has exactly the same moment of inertia as a ground-state band of the core. A possible interpretation of this feature in terms of spectrum-generating supersymmetry is suggested.     

A note on embedding non-Abelian finite flavor groups in continuous groups

A non-Abelian finite flavor group G⊂SO(3)$G\subset \mathit{SO}(3)$ can have double covering G^′⊂SU(2)$G^{\prime }\subset \mathit{SU}(2)$ such that G⊄G^′$G\not\subset G^{\prime }$. This situation is not contradictory, but quite natural, and we give explicit examples such as G=Dn$G=D_n$, G^′=Q_2n$G^{\prime }=Q_{2n}$ and G=T$G=T$, G^′=T^′$G^{\prime }=T^{\prime }$. This observation can be crucial in particle theory model building.     

On the ascent of infinite dimensional Hamiltonian operators

In this paper, the ascent of 2 × 2 infinite dimensional Hamiltonian operators and a class of 4 × 4 infinite dimensional Hamiltonian operators are studied, and the conditions under which the ascent of 2 × 2 infinite dimensional Hamiltonian operator is 1 and the ascent of a class of 4 × 4 infinite dimensional Hamiltonian operators that arises in study of elasticity is2 are obtained. Concrete examples are given to illustrate the effectiveness of criterions.     

A note on the dimensional regularization method in the presence of infrared divergences

We indicate a practical way of applying the dimensional regularization scheme in the presence of infrared divergences, consisting of a finite subtraction procedure.     

A note on the ion sound theory in an infinite plasma with electric field

Stability of the electroacoustic wave in an infinite one dimensional plasma with a DC electric field is analysed. Residual charged particle-neutral collisions are modelled by a BGK term. With the ion collisions weak, the effect of ion acceleration in the DC electric field is taken into account. Conditions for unstable propagation are discussed.     

A note on the Monte Carlo simulation of one dimensional quantum spin systems

The one dimensional Ising model in a transverse field is treated by Monte Carlo methods in order to study the approach originally proposed by Suzuki, Miyashita and Kuroda.     

Ground state phase diagram of the infinite dimensional Hubbard model: A variational study

We use the Gutzwiller variational method to study the ground state phase diagram of the infinite dimensional Hubbard model, paying special attention to features associated with the Gaussian form of the tight binding band density of states. We only consider trial states for which the validity of the Gutzwiller approximation has been proven, i.e., the paramagnetic, ferromagnetic, and two-sublattice antiferromagnetic states. We map out the phase diagram numerically, and give approximate analytic arguments to explain the behaviour in the small-U, and large-U limits. We give two versions of the phase diagram: one restricted to homogeneous phases, and another when phase separation is allowed. In the latter case, a homogeneous antiferromagnetic state is found only at exact half-filling, where the state is also insulating. Off half-filling, four different regions are found: pure paramagnetic, and ferromagnetic states, as well as antiferromagnetic-paramagnetic, and antiferromagnetic-ferromagnetic mixed phases.Dedicated to Professor W. Brenig on the occasion of his 60th birthdayResearch performed within the program of the Sonderforschungsbereich 341 supported by the Deutsche Forschungsgemeinschaft     

Maximal violation of the Collins-Gisin-Linden-Massar-Popescu inequality for infinite dimensional states

We present a much simplified version of the Collins-Gisin-Linden-Massar-Popescu inequality for the 2x2xd Bell scenario. Numerical maximization of the violation of this inequality over all states and measurements suggests that the optimal state is far from maximally entangled, while the best measurements are the same as conjectured best measurements for the maximally entangled state. For very large values of d the inequality seems to reach its minimal value given by the probability constraints. This gives numerical evidence for a tight quantum Bell inequality (or generalized Csirelson inequality) for the 2x2xinfinity scenario.     

A note on blowup of smooth solutions for relativistic Euler equations with infinite initial energy

We study the singularity formation of smooth solutions of the relativistic Euler equations in (3+1)-dimensional spacetime for infinite initial energy. We prove that the smooth solution blows up in finite time provided that the radial component of the initial generalized momentum is sufficiently large without the conditions \(M(0)>0\) and \(s^{2}<\frac{1}{3}c^{2}\), which were two key constraints stated in Pan and Smoller (Commun Math Phys 262:729–755, 2006).