Quasi-multiplication and 1-algebras

Let $\text{B}$ be a ¹-algebra with identity. With $\text{B}$ we associate a quasi-algebra Q($\text{B}$) consisting of sequences whose entries are elements of $\text{B}$. For A,B in $\text{B}$ we give general expansions pertaining to (A+B)ⁿ, e^A+B and e^Ae^B. We also discuss the case where $\text{B}$ is the ¹-algebra generate creation and annihilation operators. Another example deals with the ¹-algebra of field operators.     

On semipositive definiteness of 2n-degree forms

The aim of this paper is to present necessary and sufficient conditions for the semipositive definiteness of 2n-degree forms. These conditions allow to verify whether a given map Λ: $\text{A}$ → $\text{A}$ (where $\text{A}$ is the algebra of linear operators on the n-dimensional Hilbert space $\text{H}$) is a semipositive map.     

Bosonic construction of the Lie algebras of some non-compact groups appearing in supergravity theories and their oscillator-like unitary representations

We give a construction of the Lie algebras of the non-compact groups appearing in four dimensional supergravity theories in terms of boson operators. Our construction parallels very closely their emergence in supergravity and is an extension of the well-known construction of the Lie algebras of the non-compact groups $\text{SP}\text{(2n,}\text{R}$ and $\text{SO}\text{(2n)}{}^1$ from boson operators transforming like a fundamental representation of their maximal compact subgroup U(n). However this extension is non-trivial only for n?4 and stops at n = 8 leading to the Lei algebras of SU(4) × SU(1, 1), SU(1, 1), SU(5, 1), SO(12)¹ and E₇(7). We then give a general construction of an infinite class of unitary irreducible representations of the respective non-compact groups (except for E₇₍₇₎ and SO(12)¹ obtained from the extended construction). We illustrate our construction with the examples of SU(5, 1) and SO(12)¹.     

The U(612) supersymmetric limit of the interacting Boson-Fermion Model

The $\text{U(}\text{6}\text{12}\text{)}$ dynamical supersymmetry of the Interacting Boson-Fermion Model, appropriate for nuclei where the odd nucleon can occupy the single-particle orbits $\text{J =}\text{1}\text{2}\text{,}\text{3}\text{2}\text{,}\text{and}\text{5}\text{2}$ is analyzed. The subgroup structure of $\text{U(}\text{6}\text{12}\text{)}$ is discussed and the generators and Casimir operators of these subgroups are given. In each of the three dynamical symmetries of $\text{U(}\text{6}\text{12}\text{)}$ the energy formulas, reduction rules, and wave functions are obtained and closed expressions for E2 transition rates and one-nucleon transfer intensities are derived. Examples of $\text{U(}\text{6}\text{12}\text{)}$ symmetries are presented.     

An effective method of investigation of positive maps on the set of positive definite operators

Let $\text{A}$₁ be the algebra of linear operators on the n-dimensional Hilbert space $\text{H}$₁, and let $\text{A}$₂ be the algebra of linear operators of the m-dimensional Hilbert space $\text{H}$₂. Let $\text{L}$($\text{A}$₁, $\text{A}$₂) denote the complex space of linear maps from $\text{A}$₁ to $\text{A}$₂. By a positive map we mean an element of the space $\text{L}$($\text{A}$₁, $\text{A}$₂) (superoperator with respect to $\text{H}$₁) which maps positive definite operators in $\text{A}$₁ into positive definite operators in $\text{A}$₂. The aim of this paper is to present an effective method which allows to verify whether a given superoperator Λ∈$\text{L}$($\text{A}$₁, $\text{A}$₂) is a positive map. Besides that necessary and sufficient conditions for the positive definiteness of even-degree forms in many variables are given.     

Spinor-inverted solution to thirring model and its generalization to U(n) symmetry

Using the properties of massless free Fermi fields in (1-1) dimensions, it is shown that the spinor inverted form of Klaiber's operator solution to Thirring model is also a scale-invariant solution of the model. But unlike the former it admits a nonvanishing SU(n) current coupling in the generalization of the model to include U(n) symmetry. The value of this coupling constant is fixed and equals Dashen-Frishman number $\text{?4π}\text{(n + 1)}$. The general form of the 2m-point function is given and operates product expansions are exhibited in terms of composite local operators. Scale dimensions of all the bilinear and quadrilinear local operators with U(n) symmetry are computed and are found to depend on n. However, different parts of a composite local operator belonging to different irreducible U(n) representations have the same dimension.     

Integrable quantum field theories and Bogoliubov transformations

The Federbush, massless Thirring and continuum Ising models and related integrable relativistic quantum field theories are studied. It is shown that local and covariant classical field operators exist that generate Bogoliubov transformations of the annihilation and creation operators on the Fock spaces of the respective models. The quantum fields of these models are closely related or equal to quadratic forms implementing these transformations, and hence formally inherit the covariance and locality of the underlying classical field operators. It is proved that the Federbush and massless Thirring fields on the physical sector do not satisfy the equation of motion. Closely related fields are defined that do satisfy it, and which lead to the same S-matrix, but these fields are presumably non-local. Bethe transforms are constructed for the various models, and on the unphysical sector the relation with the field theory approach is established.     

Thirring model with U(n) symmetry: Scale invariant only for fixed values of a coupling constant

Generalizations of the Thirring model to Fermi fields with U(n) symmetry are treated. When interactions quadratic in the SU(n) currents are introduced, scale invariance (with anomalous dimensions) is maintained only for values of the coupling g_v=0 or $\text{g}{}_{\mathrm{<mtext>v</mtext>}}\text{=}\text{4φ}\text{(n+1)}$.     

Accidental degeneracies in the zeeman effect and the symmetry groups

It is well known that the problem of a charged particle in a general central potential, subject to an external megnetic field, has no degeneracy. However, if the central potential has a symmetry group larger that O(3), as happens for the oscillator (U(3)), Coulomb (O(4)) and free particle (E(3)), even in the presence of a magnetic field a residual degeneracy remains. In the present paper the symmetry Lie algebras are found which are responsible for the residual degeneracies in the problems mentioned, first in the case of weak magnetic fields when the Hamiltonian has only linear terms in the magnetic field intensity $\text{H}$ and, in the last section, for strong magnetic fields when the term in $\text{H}$² is also included. Furthermore in the classical limit the canonical transformations are found that take the Hamiltonians of the problems mentioned in the presence of a magnetic field into other Hamiltonians in which the nature of the symmetry Lie algebra becomes obvious.     

Hydrodynamics of an n-component phonon system

The dynamic properties of an n-component phonon system in d dimensions, which serves as a model for a structural phase transition of second order, are investigated. The symmetry group of the hamiltonian is the group of orthogonal transformations $\text{O}$(n). For n ≥ 2 a continuous symmetry is broken for T<T_c, where T_c is the transition temperature. We derive the hydrodynamic equations for the generators of this group, the $\text{1}\text{2}\text{n (n ? 1)}$ angular-momentum variables. Besides the usual hydrodynamics of a phonon system, there are $\text{1}\text{2}\text{n (n ? 1)}$ additional independent diffusive modes for T > T_c. In the ordered phase we find 2 (n ? 1) propagating modes with linear dispersion and quadratic damping. Formally the hydrodynamics is similar in the isotropic Heisenberg ferromagnet (n = 2) or the isotropic antiferromagnet (n ≥ 3). The relaxing modes for T < T_c require special care. We study the critical dynamics by means of the dynamical scaling hypothesis and by a mode-coupling calculation, both of which give the critical dynamical exponent $\text{z =}\text{1}\text{2}\text{d}$. The results are compared with the 1/n expansion. It is shown that for large n there is a non-asymptotic region characterized by an effective exponent $\text{z}\text{?}\text{= φ/2ν}$, where φ is the crossover exponent for a uniaxial perturbation, and ν the critical exponent of the correlation length.     

A classification of compactifying solutions for d =11 supergravity

Supergravity in eleven dimensions is known to have classical solutions of the type (anti-de Sitter space-time) × (7-dimensional Einstein space). We give a list of all homogeneous 7-manifolds which admit an Einstein metric. Known solutions are reviewed, with some emphasis on the SU(3) × SU(2) × U(1) compactifications. Their topology is discussed in detail.The list includes three new solutions, with symmetry groups SU(3) × SU(2), SO(5) and SO(5) × U(1). The first solution has no supersymmetry, while the second and third yield respectively N = 1 and N = 2 supersymmetry in four dimensions. The last two solutions may be extended to solutions with nonzero internal photon curl, breaking all supersymmetry.The existence of a spin structure on homogeneous manifolds $\text{G}\text{H}$ is discussed and related to topological properties of $\text{G}\text{H}$. As an illustration, we treat the coset spaces SU(m + 1) × SU(n + 1)/SU(m) × SU(n) × U(1), which include the spaces with SU(3) × SU(2) × U(1) symmetry.     

On duality in e+e- annihilation

With the help of a relativistic generalization of a formula for the leptonic width of quarkonium in the quasiclassical approximation it is shown that the sum of a sufficiently large number of total cross sections for the production of the quarkonium state (Q$\text{Q}$)_n in e⁺e^- annihilation is equal to the total cross section for production of a quasi-free quark- antiquark pair Q$\text{Q}$.     

The uniform and the strong topology on realizations of the algebra of polynomials

It is shown that under quite general assumptions on the operators A₁,…,A_n (unbounded, symmetric) and on the domain $\text{D}$ on the realization $\text{P}\text{(A}{}_1\text{,…,A}{}_{\mathrm{n}}\text{)}$ of the algebra of polynomials $\text{P}\text{(x}{}_1\text{,…,x}{}_{\mathrm{n}}\text{)}$, the strongest locally convex topology τ_st coincides with the uniform topology τ_$\text{D}$ as well as with the strong operator topology τ_s. In the case n = 2 some conditions are given, under which these general assumptions are fulfilled.     

Quantizing a classically ergodic system: Sinai's billiard and the KKR method

Sinai's “billiards on a torus,” i.e., free motion of a particle in a plane amongst reflecting discs of radius R centred on points of the unit square lattice, is a classically ergodic system with two freedoms, parametrized by R. Quantal energy levels E_n are given by the vanishing of the Korringa-Kohn-Rostoker (KKR) determinant of solid state theory. This gives a rapid computational scheme for computing E_n as functions of R. Except for the integrable case R = 0, no degeneracies are found, illustrating the theorem that two parameters, not one, are required to make levels cross in a generic system. The same theorem leads to the prediction that the probability distribution of the spacings S of neighbouring levels is $\text{O}$(S) as S → 0, in good agreement with computation. The KKR determinant is transformed analytically to give the level density d(E) semiclassically (i.e., as ? → 0) as the sum of a steady contribution $\text{d}\text{?}\text{(E)}$ and an oscillatory contribution d_osc(E). $\text{d}\text{?}$ is $\text{O}$(?^?2) and is given by the Weyl “area” formula plus “edge,” “corner” and “curvature” corrections, in excellent agreement with computation. d_osc is given by a sum over classical closed orbits (all unstable). Nonisolated closed orbits (not hitting discs) contribute terms with $\text{O}$$\text{(?}{}^{\mathrm{<mtext>?3</mtext><mtext>2</mtext><mtext>)</mtext>}}$ to d_osc, while isolated closed orbits (bouncing between discs) contribute terms with $\text{O}$(?^?1) to d_osc. The isolated orbits are vastly more numerous than the nonisolated orbits and their contributions cannot be neglected. As a means of calculating the individual E_n (rather than the smoothed spectrum), the KKR method is much more efficient than the classical path sum.     

Spatial theory for algebras of unbounded operators

In this paper, the spatial theory of ¹-automorphisms is investigated in the context of algebras of unbounded operators. In particular, it is shown that ¹-automorphisms, satisfying some order relation, of the Op¹-algebra generated by the position and momentum operators q_j, p_j(j=1,...., n) on the Schwartz space $\text{I}\text{(}\text{R}{}^{\mathrm{n}}\text{)}$ are unitarily implemented.     

The p-approximations and inverse problems of expansion and contraction of n-fermion operators

The orthogonal projectors from the space $\text{L}$(n) of n-fermion operators onto its sub-spaces $\text{L}$_p(n) consisting of p-reducible elements of $\text{L}$(n>), as well as those from $\text{L}$(n) onto $\text{L}$_p(n)?$\text{L}$_p?1(n) are constructed. Using the above projectors the inverse problems of contraction and expansion are solved.     

Self-consistent Hartree-Fock calculation of exchange effects in a weakly inhomogeneous electron gas

An exact calculation is given for the function B_x(n) which enters the Hohenberg-Kohn-Sham expansion, in powers of $\text{?n(}\text{r}\text{)}$, of the echange energy of an inhomogeneous eletron gas. The calculation is made for arbitrary inter-particle interaction. It is shown that the gradient expansion does not exist, in the case of Coulomb interaction, unless correlations are taken into account.     

A free-algebraic solution for the planar approximation

An explicit solution for the generating functional of n-point functions in the planar approximation is given in terms of two sets of free-algebraic annihilation and creation operators.     

Magnetic monopoles in SU(N) gauge theories

The potential $\text{A}\text{(}\text{r}\text{) ≡}\text{M}\text{(}\text{r}\text{?}\text{×}\text{n}\text{?}\text{)(r?}\text{r}\text{·}\text{n}\text{?}\text{)}{}^{\mathrm{?1}}$ is a static solution to the classical theory of non-abelian gauge fields coupled to a point magnetic source, for any matrix $\text{M}$ in the Lie algebra of the gauge group G. This solution is rotationally invariant if the eigenvalues of $\text{M}$ in the adjoint representation of G are quantized in half-integer units, but is stable to small perturbations only if all non-vanishing eigenvalues are $\text{±}\text{1}\text{2}$. In this paper, for the gauge groups G = SU(N), it is shown which sets of eigenvalues of $\text{M}$ are consistent with the group structure, which consistent sets are gauge inequivalent, and which consistent gauge inequivalent sets correspond to stable monopoles. It is found that there are N inequivalent stable monopoles, including the trivial case $\text{M}\text{=}\text{0}$. Equivalence here is with respect to non-singular gauge transformations—the symmetry transformations of the classical theory. Singular gauge transformations are, in contrast, not symmetries but they are nevertheless useful for classifying solutions and for relating the above concept of local stability to the global, or topological, stability associated with the Dirac strings. In this context, it is shown that there are N distinct topological classes of monopoles, with the group structure of the center $\text{Z}{}_{\mathrm{N}}\text{?Π}{}_1\text{(}\text{SU}\text{(N)/}\text{Z}{}_{\mathrm{N}}\text{)}$ of SU(N), that each class contains exactly one stable monopole, and that any other monopole in the same class has a strictly larger value of the magnetic charge magnitude $\text{tr}\text{M}{}^2$. This leads to an interesting physical picture of local stability as a consequence of the minimization of magnetic energy. The paper concludes with some comments on related topics: the empirical absence of magnetic charge, `t Hooft's calculation of magnetic energy, magnetic confinement, and spontaneously broken theories.