Relationships in infrared intensity theories,in particular the Mayants-Averbukh theory

A brief survey is given of the Mayants-Averbukh infrared intensity theory in relation to the more well-known but equivalent polar tensor theory. In addition, the appearances of the symmetry invariant parameter matrices D_n⁰ of the Mayants-Averbukh theory were derived and tabulated for various symmetries about the midpoint of bond n. The use of these matrices and a single bond coordinate system will offer a convenient alternative to the Mayants-Averbukh treatment of a central, symmetric bond. The rotational mode equations of the Mayants-Averbukh and polar tensor theories have been investigated to elucidate the constrainsts which they impose on infrared intensity theories based on the bond dipole moment model and the atomic point charge model. It was found that the valence-optical theory is in full conformity with the rotational modes only if all electrooptical parameters $\text{?μ}{}_{\mathrm{n}}\text{?γ}{}_{\mathrm{i}}$ are neglected, where γ_i is the ith internal angular coordinate. The constraints imposed on the equilibrium charge-charge flux theory correspond to neglect of all charge flux parameters. The generalized valence-optical theory was found to be incompatible with the rotational mode equations of the Mayants-Averbukh theory. However, its basic dipole moment equation was found useful for suggesting a unique interpretation of a set of d parameters (elements of D_n⁰) in terms of bond dipole moment components.     

All-loop group-theory constraints for color-ordered SU(N) gauge-theory amplitudes

We derive constraints on the color-ordered amplitudes of the L-loop four-point function in SU(N) gauge theories that arise solely from the structure of the gauge group. These constraints generalize well-known group theory relations, such as U(1) decoupling identities, to all loop orders.     

Relationships in Raman intensity theories,in particular the Mayants-Averbukh theory

Brief surveys are given of the Mayants-Averbukh Raman intensity theory, and of the polar tensor Raman intensity theory recently presented by Bogaard and Haines. It was found that these intensity theories in essence are equivalent. In addition, the appearances of the symmetry invariant parameter matrices F_n⁰ of the Mayants-Averbukh theory were derived and tabulated for various symmetries of bond n. These matrices, and a single bond coordinate system, can be used as a convenient alternative to the Mayants-Averbukh treatment of bonds which have some kind of symmetry with respect to the midpoint of the bond. A modification of the Mayants-Averbukh treatment is also suggested. The rotational mode equations of the Mayants-Averbukh theory have been investigated to elecudate the constraints which they impose on Raman intensity theories based on the bond polarizability model. It was found that the valenceoptical theory is in conformity with the rotational modes only if all electrooptical parameters $\text{?α}{}_{\mathrm{ii}}{}^{\mathrm{(n)}}\text{?γ}{}_{\mathrm{p}}$ are neglected, where α_ii⁽ⁿ⁾ (i = 1, 2, 3) are the diagonal components of the polarizability α⁽ⁿ⁾ of bond n, and γ_p is the pth internal angular coordinate. Furthermore, the valence-optical theory was found to be strictly applicable only for cylindrical bond symmetry, C_mv (m ≥ 4). A generalized valence-optical Raman intensity theory, allowing also for non-zero off-diagonal components α_ij⁽ⁿ⁾, was found to be incompatible with the rotational mode equations of the Mayants-Averbukh theory. However, its basic polarizability equation was useful for suggesting a unique interpretation of a set of f parameters (elements of F_n⁰) in terms of components of the anisotropic part of a symmetric bond polarizability.     

Universal Covering Group of U(n) and Projective Representations

Using fiber bundle theory, we construct the universal covering group of U(n),U(n), and show that U(n) is isomorphic to the semidirect product SU(n) ∝.We give a bijection between the set of projective representations of U(n) and theset of equivalence classes of certain unitary representations of SU(n) ∝.Applying Bargmann's theorem, we give explicit expressions for the liftings ofprojective representations of U(n) to unitary representations of SU(n) ∝. Forcompleteness, we discuss the topological and group theoretic relations betweenU(n), SU(n), U(t), and Z _n .     

A connection between ν-dimensional Yang-Mills theory and (ν−1)-dimensional,non-linear σ-models

We study non-linear σ-models and Yang-Mills theory. Yang-Mills theory on the ν-dimensional lattice ? ^v can be obtained as an integral of a product over all values of one coordinate of non-linear σ-models on ? ^v?1 in random external gauge fields. This exhibits two possible mechanisms for confinement of static quarks one of which is that clustering of certain two-point functions of those σ-models implies confinement of static quarks in the corresponding Yang-Mills theory. Clustering is proven for all one-dimensional σ-models, for theU(n) ×U(n) σ-models,n=1, 2, 3, ..., in two dimensions, and for the SU(2) × SU(2) σ-models for a large range of couplingsg ² ? O(ν). Arguments pertinent to the construction of the continuum limit are discussed. A representation of the expectation of Wilson loops in terms of expectations of random surfaces bounded by the loops is derived when the gauge group is SU(2),U(n) or O(n),n=1, 2, 3, ..., and connections to the theory of dual strings are sketched.     

Invariant length scale in relativistic kinematics: lessons from Dirichlet branes

Dirac–Born–Infeld theory is shown to possess a hidden invariance associated with its maximal electric field strength. The local Lorentz symmetry O(1,n) on a Dirichlet-n-brane is thereby enhanced to an O(1,n)×O(1,n) gauge group, encoding both an invariant velocity and acceleration (or length) scale. The presence of this enlarged gauge group predicts consequences for the kinematics of observers on Dirichlet branes, with admissible accelerations being bounded from above. An important lesson is that the introduction of a fundamental length scale into relativistic kinematics does not enforce a deformation of Lorentz boosts, as one might assume naively. The exhibited structures further show that Moffat's non-symmetric gravitational theory qualifies as a candidate for a consistent Born–Infeld type gravity with regulated solutions.     

Rational Generalised Moonshine from abelian orbifoldings of the Moonshine Module

We consider orbifoldings of the Moonshine Module with respect to the abelian group generated by a pair of commuting Monster group elements with one of prime order p=2,3,5,7 and the other of order pk for k=1 or k prime. We show that constraints arising from meromorphic orbifold conformal field theory allow us to demonstrate that each orbifold partition function with rational coefficients is either constant or is a hauptmodul for an explicitly found modular fixing group of genus zero. We thus confirm in the cases considered the Generalised Moonshine conjectures for all rational modular functions for the Monster centralisers related to the Baby Monster, Fischer, Harada-Norton and Held sporadic simple groups. We also derive non-trivial constraints on the possible Monster conjugacy classes to which the elements of the orbifolding abelian group may belong.     

In Support of n-Correlation

In this paper we examine n-correlation for either the eigenvalues of a unitary group of random matrices or for the zeros of a unitary family of L-functions in the important situation when the correlations are detected via test functions whose Fourier transforms have limited support. This problem first came to light in the work of Rudnick and Sarnak in their study of the n-correlation of zeros of a fairly general automorphic L-function. They solved the simplest instance of this problem when the total support was most severely limited, but had to work extremely hard to show their result matched random matrix theory in the appropriate limit. This is because they were comparing their result to the familiar determinantal expressions for n-correlation that arise naturally in random matrix theory. In this paper we deal with arbitrary support and show that there is another expression for the n-correlation of eigenvalues that translates easily into the number theory case and allows for immediate identification of which terms survive the restrictions placed on the support of the test function.     

Z′ mass limits,masses and coupling of higgs bosons,and Z′ decays in an E6 superstring based model

We demonstrate that current phenomenological constraints on Z-Z′ mixing for an E₆ grand unified group with low energy gauge group SU(2)_L×U(1)_Y×U(1)_Y, allow only a narrow range of Higgs vacuum expectation values consistent with possibilities favored by renormalization group expectations. Modest improvements in bounds on this mixing will lead to substantial bounds on the Z′ mass if alternative renormalization group solutions are not found. We then explore the constraints upon relations between Higgs masses in this model. In addition we explore the couplings of these Higgs to the gauge particles of the theory and emphasize the associated implications for Higgs detection in decays of the Z′.     

Operator formalism on general algebraic curves

The usual Laurent expansion of the analytic tensors on the complex plane is generalized to any closed and orientable Riemann surface represented as an affine algebraic curve. As an application, the operator formalism for the b − c systems is developed. The physical states are expressed by means of creation and annihilation operators as in the complex plane and the correlation functions are evaluated starting from simple normal ordering rules. The Hilbert space of the theory exhibits an interesting internal structure, being splitted into n (n is the number of branches of the curve) independent Hilbert spaces. In this way we are able to realize new kinds of conformal field theories at genus zero with symmetry group Virⁿ ⊗ G, Vir being the Virasoro group and G denoting a discrete and nonabelian crystallographic group. Exploiting the operator formalism a large collection of explicit formulas of string theory is derived. Finally, we develop as an important byproduct new methods in order to handle differential equations related to monodromy, like the Riemann monodromy problem.     

Constraints on the currents in principal models giving rise to integrable nonlinear systems

Quadratic constraints on the current in the principal SU(2n) model stand at the origin of an extended reduction mechanism. We derive local conservation laws both for the reduced model and for a class of solutions of the principal SU(2n) model.     

Collective hamiltonians with Kac-Moody algebraic conditions

We describe the general framework for constructing collective-theory hamiltonians whose hermiticity requirements imply a Kac-Moody algebra of constraints on the associated jacobian. We give explicit examples for the algebras sl(2)_k and sl(3)_k. The reduction to W_n-constraints, relevant to n-matrix models, is described for the jacobians.     

The absence of cut-off effects for the fixed-point action in one-loop perturbation theory

In order to support the formal renormalization group arguments that the fixed-point action of an asymptotically free model gives cut-off independent physical predictions in one-loop perturbation theory, we calculate the finite-volume mass gap m(L) in the non-linear σ-model. No cut-off effect of the type g⁴ (a/L)ⁿ is seen for any n. The results are compared with those of the standard and tree level improved symanzik actions.     

Constraints for anomalous dimensions of local light-cone operators in [ϕ 3]6 theory

Using the MS scheme, we derive in [? ³]₆ theory the collinear conformal Ward identity for the Green's functions of local light-cone operators of leading twist. The Ward identity for special collinear conformal transformations and renormalization group invariance give constraints for the off-diagonal part of the anomalous dimension matrix for the general case of β#0. We compute the anomaly of special conformal transformation in lowest loop order and obtain from the constraints the off-diagonal part of the anomalous dimension in 2-loop order.     

A new method of constructing symmetry-adapted linear combinations for a finite group: I. Salc's of equivalent invariant functions

A theory of representation for a finite group G has been presented based on the basis functions defined on a set of equivalent points S(ⁿ) with respect to the symmetry operations of G. The general expressions are given for the symmetry-adapted linear combinations of equivalent invariant functions and the σ-bonding hybrid atomic orbitals.     

On the periodic orbits of a strongly chaotic system

A point particle sliding freely on a two-dimensional surface of constant negative curvature (Hadamard-Gutzwiller model) exemplifies the simplest chaotic Hamiltonian system. Exploiting the close connection between hyperbolic geometry and the group SU(1,1)/⦅±1⦆, we construct an algorithm (symboliv dynamics), which generates the periodic orbits of the system. For the simplest compact Riemann surface having as its fundamental group the “octagon group”, we present an enumeration of more than 206 million periodic orbits. For the length of the nth primitive periodic orbit we find a simple expression in terms of algebraic numbers of the form m + √2n (m, nϵN are governed by a particular Beatty sequence), which reveals a strange arithmetical structure of chaos. Knowledge of the length spectrum is crucial for quantization via the Selberg trace formula (periodic orbit theory), which in turn is expected to unravel the mystery of quantum chaos.     

Two-dimensionalR n-gravitation

A system with constraints is considered: a string theory whose Lagrangian is thenth power of the Gauss curvature of a space-time manifold (n∈N,n>1). The problem is solved exactly because after the constraints are utilized we deal with a variational problem with a trivial Lagrangian, i.e., its Euler-Lagrange equations are satisfied identically. One can say that the constraints “swallow” all dynamical degrees of freedom of the field theory. The investigation is a continuation of the 1989 work of Burlankov and Pavlov, who solved the problem of two-dimensionalR ²-gravitation under the gauge γ=1.     

Hamiltonian analysis of 4-dimensional spacetime in Bondi-like coordinates

We discuss the Hamiltonian formulation of gravity in four-dimensional spacetime under Bondi-like coordinates {v,r,x~a,a=2,3}. In Bondi-like coordinates, the three-dimensional hypersurface is a null hypersurface, and the evolution direction is the advanced time v. The internal symmetry group SO(1,3) of the four-dimensional spacetime is decomposed into SO(1,1), SO(2), and T~±(2), whose Lie algebra so(1,3) is decomposed into so(1,1), so(2), and t~±(2) correspondingly. The SO(1,1) symmetry is very obvious in this type of decomposition, which is very useful in so(1,1) BF theory. General relativity can be reformulated as the four-dimensional coframe(e_μ~I) and connection(ω_μ~(IJ))dynamics of gravity based on this type of decomposition in the Bondi-like coordinate system. The coframe consists of two null 1-forms e~-, e~+and two spacelike 1-forms e~2, e~3. The Palatini action is used. The Hamiltonian analysis is conducted by Dirac's methods. The consistency analysis of constraints has been done completely. Among the constraints, there are two scalar constraints and one two-dimensional vector constraint. The torsion-free conditions are acquired from the consistency conditions of the primary constraints about π_(IJ)~μ. The consistency conditions of the primary constraints π_(IJ)~0=0 can be reformulated as Gauss constraints. The conditions of the Lagrange multipliers have been acquired. The Poisson brackets among the constraints have been calculated. There are 46 constraints including 6 first-class constraints π_(IJ)~0=0 and 40 second-class constraints. The local physical degrees of freedom is 2.The integrability conditions of Lagrange multipliers n_0, l_0, and e_0~A are Ricci identities. The equations of motion of the canonical variables have also been shown.     

Quantization of relativistic systems with constraints

The general dynamical system with constraints is quantized, and the S-matrix is constructed in the most general class of gauges including relativistic ones. In the case when constraints do not form a group a new type of additional diagrams arises securing unitarity of the theory: the four-fermion interaction of ghost fields.     

Gradient properties of the renormalisation group equations in multicomponent systems

The existence of a metric, which enables the renormalisation group β functions of a multicomponent field theory to be written as a gradient, has very important implications for the asymptotic behavior of the renormalisation group equations. It is shown that a very simple metric exists in a field theory with n-component Bose fields and arbitrary φ⁴ interaction, when the β functions are calculated perturbatively up to and including the 2-loop diagrams. This same metric is shown to work for all irreducible diagrams, but it must and can be modified to accommodate reducible 3-loop contributions. The prospects and outlook of this aspect of the renormalisation group are discussed.