Yang-Baxter algebras of monodromy matrices in integrable quantum field theories

We consider a large class of two-dimensional integrable quantum field theories with non-abelian internal symmetry and classical scale invariance. We present a general procedure to determine explicitly the conserved quantum monodromy operator generating infinitely many non-local charges. The main features of our method are a factorization principle and the use of $\text{P}$, $\text{T}$, and internal symmetries. The monodromy operator is shown to satisfy a Yang-Baxter algebra, the structure constants (i.e. the quantum R-matrix) of which are determined by two-particle S-matrix of the theory. We apply the method to the chiral SU(N) and the O(2N) Gross-Neveu models.     

Splitting the degeneracy of harmonically-bound identical spinless bosons: General derivation

The Gartenhaus-Schwartz model of N particles interacting via pairwise harmonic potentials $\text{V}{}_{\mathrm{ij}}\text{=}\text{1}\text{2}\text{kR}{}_{\mathrm{ij}}{}^2$ is extended by addition of an arbitrary pairwise interaction small enough to be treated by degenerate perturbation theory. The theory is worked out for a rarefied fluid of N spinless bosons, to give the splittings of the degeneracies of the U(3N?3) symmetry inherent in the Gartenhaus-Schwartz model. Following Moshinsky and Kramer the external field is formally introduced in order to reduce the zeroth-order problem to the 3N-dimensional harmonic oscillator. For systems with more particles than quanta, a recipe is given to construct states in which all the excitation is in internal degrees of freedom and none is in translation. Analytical solutions are obtained for states with no more than two quanta in excitation of relative motion of any one pair of particles.     

On Kaluza-Klein theory

Assuming the compactification of 4 + K-dimensional space-time implied in Kaluza-Kleintype theories, we consider the case in which the internal manifold is a quotient space, $\text{G}\text{H}$. We develop normal mode expansions on the internal manifold and show that the conventional gravitational plus Yang-Mills theory (realizing local G symmetry) is obtained in the leading approximation. The higher terms in the expansions give rise to field theories of massive particles. In particular, for the original Kaluza-Klein 4 + 1-dimensional theory, the higher excitations describe massive, charged, purely spin-2 particles. These belong to infinite dimensional representations of an O(1,2).     

Current algebra theorems for the U(1) “Goldstone boson”

The U(1) problem is reconsidered from the point of view of the $\text{1}\text{N}$ expansion. It is argued that various heuristic ideas about the η′ are valid from this point of view. Current algebra theorems, similar to soft π theorems, are derived for the η′. They are valid to lowest order in $\text{1}\text{N}$.     

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.     

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.     

Manifest chiral symmetry preservation in the 1N-expanded SU(N) Thirring model

By utilizing manifestly chiral-invariant auxiliary field operators, it is demonstrated that continuous chiral symmetry is preserved explicitly in the $\text{1}\text{N}$ expansion of 2-dimensional theories whose fermions have a Gross-Neveu type of potential. The effective lagrangian derived in the $\text{1}\text{N}$ limit describes a massless scalar field whose derivative coupling to the fermions vanishes as N → ∞, and a decoupled massive scalar field.     

Search for the symmetry of the Fokker-Planck equation

It is shown that the Fokker-Planck equations with linear drift γx and nonlinear drift $\text{γx +}\text{δ}\text{x}$, i.e. the Rayleigh process, have exact symmetries on their own solution manifolds, the sl(2, R) ? Weyl algebra and the sl(2, R) ? I, respectively. The symmetries of these two types of processes for 1 + 2-dimensional cases are explicitly found out too.     

New superalgebra and goldstone spin-32 particle

A new supersymmetry group is introduced, whose algebra contains a spinor-vector generator. The non-linear realizations of this group lead to a Goldstone particle of spin-$\text{3}\text{2}$. The introduction of local gauge symmetry (Higgs effect) is discussed.     

Conjugacy and stability theorems in connection with contraction of lie algebras

Let $\text{a ? Г}{}_{\mathrm{n}}\text{(}\text{g}\text{?}\text{)}$, the n-dimensional Grassmannian variety, and ? be a Lie algebra of the dimension m?n. We study some properties of the set $\text{B <}\text{G(a)}$, where $\text{G(a)}$ is the Zariski closure of the orbit of $\text{G=GL(}\text{g}\text{?}\text{)}$ at the point a. The group GL(?) is the group of automorphisms of ?. The set B is shown to be the set of Lie algebras which are contractions of ā. ā is a subalgebra of ?. The main results are formulated in theorems on the conjugacy of contracted algebras and stability of the Lie algebra ā under contraction. The conjugacy theorem relates the algebras in the set B.     

The strong topology on the algebra of polynomials

It is shown that under a quite general condition on the operator T (unbounded, symmetric) and on the domain $\text{D}$ for the representation x → T of the algebra $\text{P}$(x) on $\text{D}$ in $\text{P}$(T) the strongest locally convex topology τ_∞ coincides with the strong topology σ^$\text{D}$.     

N = 1 supergravity in the light-cone gauge

N = 1 supergravity is studied in a light-cone gauge. The non-linear transformations and the hamiltonian are constructed to order K in the coupling constant as a realization of the super-Poicaré algebra. The only input except the algebraic structure is the helicity content ($\text{±2, ±}\text{3}\text{2}$) and the dimensionality of the coupling constant.     

Integral equations for the scattering of N identical particles

Integral equations are obtained for the scattering of N identical particles using a form of the N-particle scattering equations derived previously. The equations couple together only transition operators between physical two cluster channels, the breakup amplitudes being expressed in terms of quadratures over two-cluster amplitudes. The kernel of the equations becomes connected after a single iteration. The number of coupled equations for identical particles is $\text{1}\text{2}\text{N}\text{or}\text{1}\text{2}\text{(N?1)}$ when N is even or odd respectively.     

Current algebra of quarks confined to a cavity

Equal-time commutators of fields with charges are calculated in a cavity approximation to the MIT bag model, with N flavours of non-interacting quarks confined to a rigid spherical cavity and SU(N) symmetry arbitrarily broken by mass terms. It is proved that inside the cavity the algebra is identical with that of free field theory, whilst on the boundary quark fields commute with axial charges. Vector divergences and sigma commutators belong to a $\text{(N,}\text{N}\text{) + (}\text{N}\text{, N)}$ multiplet of chiral SU(N) × SU(N). Axial divergences contain additional surface terms which do not contribute to sigma commutators. A non-strange quark mass in the range 20–44 MeV is required to give a value 30–70 MeV for the nucleon matrix element of the sigma commutator relevant to pion-nucleon scattering.     

Supersymmetric pure spinor theory

A theory based solely on a 2×(N=2)-component Weyl spinor field χ(x) of subcanonical dimension $\text{1}\text{2}$ allows the local construction (without derivatives) of effective fermi and bose fields with spins up to N=2. It is demonstrated that the lagrangian $\text{～:}\text{det}\text{xx}{}^1\text{: (x)}$ studied earlier is invariant under a global N=2 supersymmetry transformation and can be cast into a form $\text{～(}\text{det DD}{}^1\text{):ππ}{}^1\text{:}$ involving the scalar chiral superfield $\text{π=}\text{exp}\text{[θ}{}_{\mathrm{x}}\text{(x + iθσθ}{}^1\text{)]}$ the components of which are finite part products of the basic field. The theory can be generalized to an N-supersymmetric theory in a 2N-dimensional space-time yielding the Thirring model as special case for N=1.