Symmetries induced by conserved vector currents in the theory of a scalar field

Let us consider a quantum theory of one scalar, real, local, Poincaré covariant fieldA(x) with the restricted spectrum condition (massive one particle states and a unique vacuum). The asymptotic fieldsA _{in out} (x) are assumed to be irreducible. Our conjecture is that under some technical assumptions the charge of every real, hermitean, locally conserved, Poincaré covariant quantum (pseudo) vector fieldj (x) relatively local toA(x), appearing in this theory-vanishes. This means that in a theory of one scalar, real field with a massive particle one can not expect to get symmetry groups induced by conserved (pseudo) vector currents, only by global, selfadjoint, Poincaré invariant generators.Our arguments can be easily extended to a theory of one complex scalar field, in this case the only symmetry transformation induced by a current can be the gauge transformation.We prove also that under very weak assumptions two fields related to each other by a unitary (or similarity) transformation are equal barring some patological cases.     

The symmetry,inferable from Bogoliubov transformation,between processes induced by a mirror in two-dimensional and a charge in four-dimensional space-time

We consider the symmetry between creation of pairs of massless bosons or fermions by an accelerated mirror in (1+1)-dimensional space and emission of single photons or scalar quanta by an electric or scalar charge in (3+1)-dimensional space. The relation of Bogoliubov coefficients describing the processes generated by a mirror to Fourier components of the current or charge density implies that the spin of any disturbances bilinear in the scalar or spinor field coincides with the spin of quanta emitted by the electric or scalar charge. The mass and invariant momentum transfer of these disturbances are essential for the relation of Bogoliubov coefficients to invariant singular solutions and the Green functions of wave equations for both (1+1)-and (3+1)-dimensional spaces, and especially for the integral relations between these solutions. One of these relations leads to the coincidence of the self-action changes and vacuum-vacuum amplitudes for an accelerated mirror in two-dimensional space-time and a charge in four-dimensional space-time. Both invariants of the Lorentz group, spin and mass, play an essential role in the established symmetry. The symmetry embraces not only the processes of real quanta radiation, but also the processes of the mirror and charge interactions with fields carrying spacelike momenta. These fields accompany their sources and determine the Bogoliubov matrix coefficients α _ω′ω ^{B, F} . It is shown that the Lorentz-invariant traces ±trα^B,F describe the vector and scalar interactions of the accelerated mirror with a uniformly moving detector. This interpretation rests essentially on the relation between propagators of the waves with spacelike momenta in two-and four-dimensional spaces. The traces ±trα^{B, F} coincide with the products of the mass shift Δm_{1, 0} of the accelerated electric or scalar charge and the proper time of the shift formation. The symmetry fixes the value of the bare fine structure constant α₀=1/4π.     

BRST quantization of a scalar particle in a curved background

The BRST formalism is employed to quantize a scalar particle and interactions with an external scalar field (x ) and vector gauge fieldA (x ) in the background of an arbitrary gravitational field. The second-quantized actions are obtained.     

Electrodynamics of a Scalar Particle in a Plane Electromagnetic Wave Field

In the present paper, compact expressions are derived for the probability of photon emission by a scalar particle and for the probability of creating pairs of scalar particles in an arbitrary plane electromagnetic wave field. Based on these general expressions, the amplitude of elastic scattering of a scalar particle and the amplitude of elastic scattering of a photon are derived by the method of dispersion relations (in the first-order approximation for the fine-structure constant ₀ = e ²/4). The real components of these amplitudes determine the radiative corrections for particle masses in the examined fields. Some particular cases of the plane wave field are examined. In particular, the above-indicated amplitudes in the external electromagnetic field being a superposition of a constant crossed field and a plane elliptically polarized electromagnetic wave propagating along the direction orthogonal to the magnetic and electric components of the constant crossed field are investigated. The amplitude of elastic scattering of a scalar particle in an arbitrary plane electromagnetic wave field is also obtained by direct calculations of the corresponding mass operator of the scalar particle in this field.     

Self-Similar Static Spherically Symmetric Scalar Field Models

Dynamical systems techniques are used to study the class of self-similar static spherically symmetric models with two non-interacting scalar fields with exponential potentials. The global dynamics depends on the scalar self-interaction potential parameters k ₁ and k ₂. For all values of k ₁, k ₂, there always exists (a subset of) expanding massless scalar field models that are early-time attractors and (a subset of) contracting massless scalar field models that are late-time attractors. When k ₁ 1/ and k ₂ 1/ , in general the solutions evolve from an expanding massless scalar fields model and then recollapse to a contracting massless scalar fields model. When k ₁ < 1/ or k ₂ < 1/ , the solutions generically evolve away from an expanding massless scalar fields model or an expanding single scalar field model and thereafter asymptote towards a contracting massless scalar fields model or a contracting single scalar field model. It is interesting that in this case a single scalar field model can represent the early-time or late-time asymptotic dynamical state of the models. The dynamics in the physical invariant set which constitutes a part of the boundary of the five-dimensional timelike self-similar physical region are discussed in more detail.     

The unitary bogoliubov transformation for a quantized scalar field in a friedman space-time

The time-dependent creation and annihilation operators for a complex scalar field, in a Friedmann space-time, defining particle states with respect to which the Hamiltonian is diagonal, are related by a Bogoliubov transformation to the creation and annihilation operators defined in strict analogy with the procedure carried out in Minkowski space. The Bogoliubov transformation is here written in terms of a unitary operator,U, and an expression for that operator is found via the generating functionF=i InU. The properties of the representation obtained by makingU act upon the state vector , to give a new state _U, are discussed. It is shown that the particle-number operator remains constant in such a picture so that the evolution of the system with time is clearly seen to depend upon the energy _k on the one hand, and upon the state vector _U on the other. Also, it is pointed out that this new representation permits the in and out states to be defined unambiguously.On leave of absence from Istituto de Fisica G. Galilei (Padova) and Istituto Nazionale di Fisica Nucleare (Sezione di Padova).     

Quantized fields in external field

This is the second part of an article devoted to the study of quantized fields interacting with a smooth classical external field with fast space time decrease. The case of a charged scalar field is considered first. The existence of the corresponding Green's functions is proved. For weak fields, as well as pure electric or scalar external fields, the BogoliubovS-operator defined in Part I of this work is shown to be unitary, covariant, causal up-to-a-phase. Its perturbation expansion is shown to converge on a dense set in Fock space. These results are generalised to a class of higher spin quantized fields, nicely coupled to external fields, which includes the Dirac theory, and, in the case of minimal and magnetic dipole coupling, the spin one Petiau-Duffin-Kemmer theory. It is not known whether this class contains examples of physical interest involving quantized fields carrying spins larger than one.     

The topological structure of the unitary and automorphism groups of a factor

It is proved that a large class ofII ₁ factors have unitary group which is contractible in the strong operator topology, but whose fundamental group in the norm topology is isomorphic to the additive real numbers as proven by Araki-Smith-Smith [1]. The class includes the approximately finite dimensional factor of typeII ₁ and the group factor associated with the free group on infinitely many generators. This contractibility is used to prove the contractibility of the automorphism group of the approximately finite dimensional factor of typeII ₁ and typeII . It is further shown that the fundamental group of the automorphism group of the approximately finite dimensional factor of typeIII , 0<<1, is isomorphic to the integer group .Dedicated to Huzihiro ArakiThis research is supported in part by NSF Grant DMS-9206984     

Relationship between (2 + 1) and (3 + 1)-Friedmann–Robertson–Walker cosmologies; linear,non-linear,and polytropic state equations

It is shown that Friedmann–Robertson–Walker (FRW) cosmological models coupled to a single scalar field and to a perfect fluid fitting a wide class of matter perfect fluid state equations, determined in (3+1) dimensional gravity can be related to their (2+1) cosmological counterparts, and vice-versa, by using simple algebraic rules relating gravitational constants, state parameters, perfect fluid and scalar field characteristics. It should be pointed out that the demonstration of these relations for the scalar fields and potentials does not require the fulfilment of any state equation for the scalar field energy density and pressure. As far as to the perfect fluid is concerned, one has to demand the fulfilment of state equations of the form p+ = f(). If the considered cosmologies contain the inflation field alone, then any (3+1) scalar field cosmology possesses a (2+1) counterpart, and vice-versa. Various families of solutions are derived, and we exhibited their correspondence; for instance, solutions for pure matter perfect fluids and single scalar field fulfilling linear state equations, solutions for scalar fields coupled to matter perfect fluids, a general class of solutions for scalar fields subjected to a state equation of the form p + = are reported, in particular Barrow–Saich, and Barrow–Burd–Lancaster–Madsen solutions are exhibited explicitly, and finally perfect fluid solutions for polytropic state equations are given.     

Spontaneous Symmetry Breaking and Higgs' Mechanism in the Nonsymmetric Jordan-Thiry Theory

In the paper we construct the nonsymmetric Jordan-Thiry theory unifying Moffat's theory of gravitation, the Yang-Mills' field, the Higgs' fields and scalar forces in a geometric manner. We discuss spontaneous symmetry breaking, the Higgs' mechanism and mass generation in the theory. The scalar field Ψ (as in classical Jordan-Thiry theory) is connected to the effective gravitational constant. This field is massive and has Yukawa-type behavior. We discuss the relation between R₊ invariance and U(1)_F from G. U. T. within Einstein λ-transformation, and fermion number conservation. In this way we connect W _μ-field from nonsymmetric theory of gravitation with a gauge field A from G. U. T. We derive the equation of motion for a test particle from conservation laws in the hydrodynamic limit.     

A symmetry relating certain processes in 2-and 4-dimensional space-times and the value α0 = 1/4π of the bare fine structure constant

The symmetry manifests itself in exact relations between the Bogoliubov coefficients for processes induced by an accelerated point mirror in 1 + 1 dimensional space and the current (charge) densities for the processes caused by an accelerated point charge in 3 + 1 dimensional space. The spectra of pairs of Bose (Fermi) massless quanta emitted by the mirror coincide with the spectra of photons (scalar quanta) emitted by the electric (scalar) charge up to the factor e ²/ħc. The integral relation between the propagator of a pair of oppositely directed massless particles in 1 + 1 dimensional space and the propagator of a single particle in 3 + 1 dimensional space leads to the equality of the vacuum-vacuum amplitudes for the charge and the mirror if the mean number of created particles is small and the charge e = √ħc. Due to the symmetry, the mass shifts of electric and scalar charges (the sources of Bose fields with spin 1 and 0 in 3 + 1 dimensional space) for the trajectories with a subluminal relative velocity β₁₂ of the ends and the maximum proper acceleration w ₀ are expressed in terms of the heat capacity (or energy) spectral densities of Bose and Fermi gases of massless particles with the temperature w ₀/2π in 1 + 1 dimensional space. Thus, the acceleration excites 1-dimensional oscillation in the proper field of a charge, and the energy of oscillation is partly deexcited in the form of real quanta and partly remains in the field. As a result, the mass shift of an accelerated electric charge is nonzero and negative, while that of a scalar charge is zero. The symmetry is extended to the mirror and charge interactions with the fields carrying spacelike momenta and defining the Bogoliubov coefficients α^B,F. The traces trα^B,F, which describe the vector and scalar interactions of the accelerated mirror with a uniformly moving detector, were found in analytic form for two mirror trajectories with subluminal velocities of the ends. The symmetry predicts one and the same value e ₀ = √ħc for the electric and scalar charges in 3 + 1 dimensional space. Arguments are adduced in favor of the conclusion that this value and the corresponding value α₀ = 1/4π of the fine structure constant are the bare, nonrenormalized values. The text was submitted by the author in English.     

Scalar particle with polarizability in a uniform external magnetic field

The model of a scalar structured particle is considered, which possesses polarizability in an external electromagnetic field. The expression for the 4-dimensional current density is found. The exact solution of the equations describing a scalar particle with polarizability in a uniform external magnetic field is obtained. Up to the terms of order O(H²), the energy spectrum can be formally obtained by the substitution of the particle mass in the expression for a pointlike scalar particle: mm–H²/2, where is the magnetic polarizability of the particle. It is shown that the rms radius of a trajectory can be obtained by the substitution of the charge in the well-known formula for a structureless scalar particle: ee(1{-H²/m)^1/2 (where is the electric polarizability).Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 91–94, January, 1991.I thank A. I. L'vov for discussions.     

Automorphisms of the affineSU(3) fusion rules

We classify the automorphisms of the (chiral) level-k affineSU(3) fusion rules, for any value ofk, by looking for all permutations that commute with the modular matricesS andT. This can be done by using the arithmetic of the cyclotomic extensions where the problem is naturally posed. Whenk is divisible by 3, the automorphism group (Z ₂) is generated by the charge conjugationC. Ifk is not divisible by 3, the automorphism group (Z ₂×Z ₂) is generated byC and the Altschüler-Lacki-Zaugg automorphism. Although the combinatorial analysis can become more involved, the techniques used here forSU(3) can be applied to other algebras.     

On the uniqueness of the equilibrium state for Ising spin systems

We show that for an Ising spin system of arbitrary spin with a ferromagnetic pair interaction and a periodic external magnetic field there is a unique equilibrium state if and only if the magnetization is continuous with respect to a uniform change in the external field. Hence, if the critical temperatureT _c is defined as the temperature where the spontaneous magnetization (which is a non-increasing function of the temperature) becomes positive, then the equilibrium state is unique forT>T _c and is non-unique forT<T _c (when the external field is zero). This implies that the correlation functions have a cluster property forT>T _c .We also show that for an anti-ferromagnet consisting of two sublattices there is a unique equilibrium state if and only if the staggered magnetization is continuous with respect to a change in the staggered field.Supported in part by U.S.A.F.O.S.R. under contract F 44620-71-C-0013, P001.     

Nuclearity and Thermal States in Conformal Field Theory

We introduce a new type of spectral density condition, that we call L ²- nuclearity. One formulation concerns lowest weight unitary representations of and turns out to be equivalent to the existence of characters. A second formulation concerns inclusions of local observable von Neumann algebras in Quantum Field Theory. We show the two formulations to agree in chiral Conformal QFT and, starting from the trace class condition for the conformal Hamiltonian L ₀, we infer and naturally estimate the Buchholz-Wichmann nuclearity condition and the (distal) split property. As a corollary, if L ₀ is log-elliptic, the Buchholz-Junglas set up is realized and so there exists a β-KMS state for the translation dynamics on the net of C^*-algebras for every inverse temperature β > 0. We include further discussions on higher dimensional spacetimes. In particular, we verify that L ²-nuclearity is satisfied for the scalar, massless Klein-Gordon field. Dedicated to László Zsidó on the occasion of his sixtieth birthday Supported by MIUR, GNAMPA-INDAM and EU network “Quantum Spaces–Non Commutative Geometry” HPRN-CT-2002-00280     

Bianchi type IX asymptotical behaviours with a massive scalar field: chaos strikes back

We use numerical integrations to study the asymptotical behaviour of a homogeneous but anisotropic Bianchi type IX model in General Relativity with a massive scalar field. As it is well known, for a Brans-Dicke theory, the asymptotical behaviour of the metric functions is ruled only by the Brans-Dicke coupling constant ₀ with respect to the value –3/2. In this paper we examine if such a condition still exists with a massive scalar field. We also show that, contrary to what occurs for a massless scalar field, the singularity oscillatory approach may exist in the presence of a massive scalar field having a positive energy density.     

On the Relation Between Orthogonal,Symplectic and Unitary Matrix Ensembles

For the unitary ensembles of N×N Hermitian matrices associated with a weight function w there is a kernel, expressible in terms of the polynomials orthogonal with respect to the weight function, which plays an important role. For the orthogonal and symplectic ensembles of Hermitian matrices there are 2×2 matrix kernels, usually constructed using skew-orthogonal polynomials, which play an analogous role. These matrix kernels are determined by their upper left-hand entries. We derive formulas expressing these entries in terms of the scalar kernel for the corresponding unitary ensembles. We also show that whenever w/w is a rational function the entries are equal to the scalar kernel plus some extra terms whose number equals the order of w/w. General formulas are obtained for these extra terms. We do not use skew-orthogonal polynomials in the derivations     

Hilbert space cocycles as representations of (3 + 1)-d current algebras

It is proposed that instead of normal representations, one should look at cocycles of group extensions valued in certain groups of unitary operators acting in a Hilbert space (e.g. the Fock space of chiral fermions), when dealing with groups associated to current algebras in gauge theories in 3 + 1 spacetime dimensions. The appropriate cocycle is evaluated in the case of the group of smooth maps from the physical three-space to a compact Lie group.The cocyclic representation of a componentX of the current is obtained through two regularizations, (1) a conjugation by a background potential dependent unitary operatorh _A, (2) by a subtraction-h _A ^-1 _xhA, where _x is a derivative along a gauge orbit. It is only the total operatorh _A ^-1 Xh _A -h _A ^-1 _xhA which is quantizable in the Fock space using the usual normal ordering subtraction.Supported by the Alexander von Humboldt Foundation     

The big bang as a result of the first-order phase transition driven by a change of the scalar curvature in an expanding early Universe: The “hyperinflation” scenario

We suggest that the Big Bang could be a result of the first-order phase transition driven by a change in the scalar curvature of the 4D spacetime in an expanding cold Universe filled with a nonlinear scalar field φ and neutral matter with an equation of state p = νε (where p and ε are the pressure and energy density of the matter, respectively). We consider the Lagrangian of a scalar field with nonlinearity φ⁴ in a curved spacetime that, along with the term–ξR|φ|² quadratic in φ (where ξ is the interaction constant between the scalar and gravitational fields and R is the scalar curvature), contains the term ξRφ₀(φ + φ⁺) linear in φ, where φ₀ is the vacuum mean of the scalar field amplitude. As a consequence, the condition for the existence of extrema of the scalar-field potential energy is reduced to an equation cubic in φ. Provided that ν > 1/3, the scalar curvature R = [κ(3ν–1)ε–4Λ] (where κ and Λ are Einstein’s gravitational and cosmological constants, respectively) decreases with decreasing ε as the Universe expands, and a first-order phase transition in variable “external field” parameter proportional to R occurs at some critical value R _c < 0. Under certain conditions, the critical radius of the early Universe at the point of the first-order phase transition can reach an arbitrary large value, so that this scenario of unrestricted “inflation” of the Universe may be called “hyperinflation.” After the passage through the phase-transition point, the scalar-field potential energy should be rapidly released, which must lead to strong heating of the Universe, playing the role of the Big Bang.     

Mirror Symmetry on Kummer Type <Emphasis Type="Italic">K</Emphasis>3 Surfaces

We investigate both geometric and conformal field theoretic aspects of mirror symmetry on N=(4,4) superconformal field theories with central charge c=6. Our approach enables us to determine the action of mirror symmetry on (non-stable) singular fibers in elliptic fibrations of _N orbifold limits of K3. The resulting map gives an automorphism of order 4,8, or 12, respectively, on the smooth universal covering space of the moduli space. We explicitly derive the geometric counterparts of the twist fields in our orbifold conformal field theories. The classical McKay correspondence allows for a natural interpretation of our results.