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Dec 1993

Volume 34, Issue 12, pp. 5413-6059

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Diffeomorphism cohomology in Beltrami parametrization

Giuseppe Bandelloni and Serge Lazzarini

J. Math. Phys. 34, 5413 (1993); http://dx.doi.org/10.1063/1.530313 (28 pages) | Cited 6 times

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Considering local conformal field theories on a Riemann surface by coupling conformal matter fields with a complex structure parametrized by a Beltrami differential, the local diffeomorphism cohomology modulo d of the Becchi–Rouet–Stora operator is computed directly by means of the spectral sequences method. Then, thanks to both power counting and locality principles of the Feynman algorithm, the local theory is analyzed. On the one hand, in the ghost number one sector, consistent anomalies turn out to be exactly, after elimination of a trace anomaly involving matter fields, the well‐known holomorphically split diffeomorphism anomaly due to the vacuum. On the other hand, in the ghost number zero sector, local observables of the theory, namely, the vertex operators, are generically calculated, as well as all possible classical actions for Lagrangian conformal models.
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11.10.Gh Renormalization
03.70.+k Theory of quantized fields

Robustness in quantum measurements

Thomas Breuer, Anton Amann, and Nicolaas P. Landsman

J. Math. Phys. 34, 5441 (1993); http://dx.doi.org/10.1063/1.530314 (10 pages) | Cited 2 times

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Conditions are formulated under which a representation of an intrinsic C∗‐ algebra of (often quasilocal) observables of an infinite system is appropriate to describe measurement‐type processes: such a representation should allow for the description of robust experiments, it should be separable, and the pointer observable should be in its weak closure. If the pointer values are discrete the existence of such a measurement representation can be proven. If the pointer can take continuously many values, then the existence can only be proven under the additional assumptions of having an asymptotically Abelian system or dealing with type I representations. In the constructed measurement representations the pointer observable turns out to be classical. The structure of the representation suggests that spontaneous symmetry breaking might be a physical explanation of the emergence of the classical pointer.
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03.65.Ta Foundations of quantum mechanics; measurement theory

Spectral analysis and stability properties of a relativistic deformation of the harmonic oscillator

E. Caliceti and A. M. Cherubini

J. Math. Phys. 34, 5451 (1993); http://dx.doi.org/10.1063/1.530315 (17 pages)

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The spectral properties are established and the stability of the eigenvalues under the singular perturbation representing the relativistic corrections are proven for a family of ordinary differential operators describing a relativistic deformation of the quantum harmonic oscillator.
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02.30.Tb Operator theory
03.65.-w Quantum mechanics

Spectral properties of observables and convex mappings in quantum mechanics

Gianni Cassinelli and Pekka J. Lahti

J. Math. Phys. 34, 5468 (1993); http://dx.doi.org/10.1063/1.530316 (8 pages) | Cited 5 times

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Observables of a physical system can be identified with convex mappings sending states into probability measures. The properties of these mappings determine the spectral properties of the observables. The mutually exclusive cases of injective and surjective mappings are characterized, and the convex structure of their ranges are investigated. In particular, the validity of the Krein–Milman property for these convex sets is studied.  
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03.65.Db Functional analytical methods

Quantum deformation of Bose parastatistics

Ludmil K. Hadjiivanov

J. Math. Phys. 34, 5476 (1993); http://dx.doi.org/10.1063/1.530317 (17 pages) | Cited 14 times

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A q‐deformation of the transformation of the Chevalley basis to an odd basis of generators of the universal enveloping of the Lie superalgebra G(n)≡osp(1‖2n) is presented. It is shown that one thus obtains a reasonable quantum deformation of the algebra of n para‐Bose oscillators.
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03.65.Fd Algebraic methods
11.30.Pb Supersymmetry

Symmetries of two‐mode squeezed states

D. Han, Y. S. Kim, Marilyn E. Noz, and Leehwa Yeh

J. Math. Phys. 34, 5493 (1993); http://dx.doi.org/10.1063/1.530318 (16 pages) | Cited 10 times

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It is known that the symmetry of two‐mode squeezed states is governed by the group Sp(4) which is locally isomorphic to the O(3,2) de Sitter group. It is shown that this complicated ten‐parameter group can be regarded as a product of two three‐parameter Sp(2) groups. It is shown also that two coupled harmonic oscillators serve as a physical basis for the symmetry decomposition. It is shown further that the concept of entropy is needed when one of the two modes is not observed. The entropy is zero when the system is uncoupled. The system reaches thermal equilibrium when the entropy becomes maximal.
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02.20.-a Group theory
03.65.Ta Foundations of quantum mechanics; measurement theory
11.30.Cp Lorentz and Poincaré invariance
42.50.Ar Photon statistics and coherence theory

Commutator expansion of the self‐energy operator for an electron in an external potential

Levere Hostler

J. Math. Phys. 34, 5509 (1993); http://dx.doi.org/10.1063/1.530319 (24 pages) | Cited 1 time

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Techniques of Erickson and Yennie [Ann. Phys. N.Y. 35, 271 (1965)] are applied in an investigation of a simple Taylor’s series which involves products of commutators and nested commutators and which represents the self‐energy operator Σ of an electron in an external potential. A technique of ‘‘moving the parameter integrals to the mass’’ leads to a relatively simple representation of all terms as a one‐dimensional integral over a common virtual electron mass squared parameter. Although not a true expansion in powers of the external field, the method provides a window on the higher‐order terms in the self‐energy operator. Mass eigenfunction expansion concepts are shown to be quite useful in a discussion of infrared divergences.
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12.20.Ds Specific calculations
11.10.St Bound and unstable states; Bethe-Salpeter equations
31.30.J- Relativistic and quantum electrodynamic (QED) effects in atoms, molecules, and ions
02.90.+p Other topics in mathematical methods in physics (restricted to new topics in section 02)

Even and odd symplectic and Kählerian structures on projective superspaces

O. M. Khudaverdian and A. P. Nersessian

J. Math. Phys. 34, 5533 (1993); http://dx.doi.org/10.1063/1.530320 (16 pages) | Cited 5 times

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Supergeneralization of CP(N) provided with even and odd Kählerian structures from Hamiltonian reduction are constructed. Operator Δ, which is used in Batalin–Vilkovisky quantization formalism and mechanics, which are bi‐Hamiltonian under corresponding even and odd Poisson brackets, are considered.
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11.30.Pb Supersymmetry
11.10.Ef Lagrangian and Hamiltonian approach

Rigorous calculation of the collective excitation spectrum in a mean field model

B. Momont

J. Math. Phys. 34, 5549 (1993); http://dx.doi.org/10.1063/1.530269 (11 pages) | Cited 1 time

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The dynamics of macroscopic density fluctuations is studied in the Overhauser model. The spectral decomposition of the time derivative on the Hilbert space of macroscopic density fluctuations is completely determined. The spectrum of the collective excitations is rigorously calculated.
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05.30.Fk Fermion systems and electron gas
03.65.Fd Algebraic methods
72.15.Nj Collective modes (e.g., in one-dimensional conductors)

Non‐Abelian solitons in two‐dimensional lattice field theories

V. F. Müller

J. Math. Phys. 34, 5560 (1993); http://dx.doi.org/10.1063/1.530270 (29 pages)

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For a class of self‐interacting multicomponent scalar field theories with a global discrete non‐Abelian symmetry group, mixed order–disorder correlation functions are defined in terms of Euclidean functional integrals. These correlation functions satisfy Osterwalder–Schrader positivity. From a representation of the correlation functions in terms of the transfer matrix, the dual algebra at fixed time is derived. This algebra implies parafermion operators showing non‐Abelian braid group statistics. In a pure phase of spontaneous symmetry breaking for a related class of order–disorder correlation functions a convergent polymer representation is developed, emerging from a combined low‐ and high‐temperature‐type expansion. The infinite volume correlation functions of this class show exponential clustering in the disorder fields.
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03.70.+k Theory of quantized fields
05.50.+q Lattice theory and statistics (Ising, Potts, etc.)
11.30.Qc Spontaneous and radiative symmetry breaking

On a model of a relativistic particle with curvature and torsion

V. V. Nesterenko

J. Math. Phys. 34, 5589 (1993); http://dx.doi.org/10.1063/1.530271 (7 pages) | Cited 3 times

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A new version of the model of a relativistic particle with curvature and torsion considered in V. V. Nesterenko [J. Math. Phys. 32, 3315 (1991)] is investigated. Two integrals along the world trajectory of its curvature and torsion are added to the standard action for the pointlike spinless relativistic particle. Since here the three‐dimensional space–time is considered at the beginning, the torsion of the world curve is defined with a sign in contrast to the previous consideration. Upon obtaining a complete set of constraints in the phase space a generalized Hamiltonian description of a new version of the model is constructed. This enables one to quantize the model canonically and to derive exactly the relation between the spin and mass of the states.
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11.10.-z Field theory
11.90.+t Other topics in general theory of fields and particles (restricted to new topics in section 11)
14.80.-j Other particles (including hypothetical)

Canonical approximate quantum measurements

Masanao Ozawa

J. Math. Phys. 34, 5596 (1993); http://dx.doi.org/10.1063/1.530272 (29 pages) | Cited 13 times

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In order to generalize the Wigner formula for the joint probability distribution of successive canonical measurements of discrete observables to continuous observables, a mathematical model is investigated of a canonical approximate measurement of an arbitrary observable, which generalizes von Neumann’s model of measuring interaction and Davies’s covariant instrument for approximate position measurement. Calculations of two types of root‐mean‐square errors show that this model clears certain requirements for good approximate measurements. A modified Wigner formula for successive canonical approximate measurements of arbitrary observables is established. It is also shown that the canonical approximate measurement of a discrete observable is reduced to the conventional measurement satisfying the von Neumann–Lüders collapse postulate by a minor modification in the process of measurement.
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03.65.Ta Foundations of quantum mechanics; measurement theory
03.65.Db Functional analytical methods
02.50.-r Probability theory, stochastic processes, and statistics

The functional integral for fields in a cavity

Luciano Vanzo

J. Math. Phys. 34, 5625 (1993); http://dx.doi.org/10.1063/1.530273 (14 pages) | Cited 4 times

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The functional integral for a scalar field confined in a cavity and subjected to linear boundary conditions is discussed herein. It is shown how the functional measure can be conveniently dealt with by modifying the classical action with boundary corrections. The nonuniqueness of the boundary actions is described with a three‐parameter family of them giving identical boundary conditions. In some cases, the corresponding Green’s function will define a kind of generalized Gaussian measure on function space. The vacuum energy is discussed, paying due attention to its anomalous scale dependence, and the physical issues involved are considered.
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03.70.+k Theory of quantized fields
02.30.-f Function theory, analysis

Quantum transformation theory in fermion Fock space

Yong‐de Zhang and Zhong Tang

J. Math. Phys. 34, 5639 (1993); http://dx.doi.org/10.1063/1.530274 (7 pages) | Cited 17 times

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In this article a general linear quantum transformation U for the following fermion system with n modes, (b′+,math′)=U(b+,math)U−1=(b+,math)(B, A, mathD), is studied. All above transformations in Fock space are proved to form a ray representation of a group which is isomorphic with O(2n,C). The explicit expressions of operator U in terms of the normal ordered product and non‐normal ordered product are obtained. A series of auxiliary identities are given. It has been mentioned that the results are applicable to the case of fermion fields with interaction.
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12.90.+b Miscellaneous theoretical ideas and models (restricted to new topics in section 12)
03.65.Fd Algebraic methods

Gauge invariance of systems with first‐class constraints

Alejandro Cabo, Masud Chaichian, and Domingo Louis Martinez

J. Math. Phys. 34, 5646 (1993); http://dx.doi.org/10.1063/1.530275 (13 pages) | Cited 7 times

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The infinitesimal canonical transformations which map solutions of the total Hamiltonian equations of motion into each other are investigated. For that, the generating function Ψ of such transformations should satisfy certain conditions. In general, Ψ is a function which depends on the coordinates, the momenta, and the Lagrange multipliers λ. However, it is shown that the requirement of independence of the function Ψ on the Lagrange multipliers is sufficient for the existence of gauge invariant transformations in the Lagrange formalism. It is shown that the condition that the Poisson brackets between Ψ and all the primary first‐class constraints are a linear combination of the latter ones provides the λ independence of the function Ψ. The existence of such a λ‐independent function Ψ is proven for some systems. In particular, this is proven for the relevant case of systems having primary and secondary first‐class constraints. The authors suggest the possibility that for some specific systems a λ‐independent generating function Ψ cannot be constructed. This conclusion concerns the systems with more than primary and secondary constraints.
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03.65.Ca Formalism
03.70.+k Theory of quantized fields
11.10.Ef Lagrangian and Hamiltonian approach

Solvable (nonrelativistic, classical) n‐body problems on the line. I

F. Calogero and Ji Xiaoda

J. Math. Phys. 34, 5659 (1993); http://dx.doi.org/10.1063/1.530276 (12 pages) | Cited 5 times

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Several solvable n‐body problems on the line are exhibited. They are characterized by equations of motion of Newtonian type, mjxj=Fj, j=1,...,n, with the ‘‘forces’’ Fj given as explicit functions of the ‘‘particle coordinates’’ xk and their velocities mathk, k=1,...,n; the forces Fj generally also depend on some free parameters, so that in each case there is actually a class of solvable models. In this paper the emphasis is on translation‐invariant models, and on values of n sufficiently small (‘‘few‐body problems’’) to allow the display of the solution of the initial‐value problem in completely explicit form.
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45.05.+x General theory of classical mechanics of discrete systems

The planar isosceles problem for Maneff’s gravitational law

Florin N. Diacu

J. Math. Phys. 34, 5671 (1993); http://dx.doi.org/10.1063/1.530277 (20 pages) | Cited 7 times

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Maneff’s gravitational law explains, with a very good approximation, the perihelion advance of the inner planets as well as the orbit of the Moon. Here the invariant set of planar isosceles solutions of the three‐body problem for Maneff’s model is studied. The application of Maneff’s law in atomic physics provides, in the case of the isosceles problem, a model with relativistic correction for the helium atom. It is shown that every solution leads to a collision singularity and consequently has no periodic orbits. Using McGehee’s technique the triple‐collision singularity is blown up and the binary‐collision solutions are regularized. The flow on the collision manifold is shown to be nongradientlike and the set of collision/ejection solutions is described. The center manifold and the block‐regularization problems are analyzed. The network of homoclinic and heteroclinic orbits is further discussed. Finally an anisotropic model having the property that the flow on the collision manifold changes drastically when the mass parameter is varied is studied, giving rise to a subcritical pitchfork bifurcation of the equilibria.
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95.10.Ce Celestial mechanics (including n-body problems)
45.05.+x General theory of classical mechanics of discrete systems
34.10.+x General theories and models of atomic and molecular collisions and interactions (including statistical theories, transition state, stochastic and trajectory models, etc.)

Poisson structure of dynamical systems with three degrees of freedom

Hasan Gümral and Yavuz Nutku

J. Math. Phys. 34, 5691 (1993); http://dx.doi.org/10.1063/1.530278 (33 pages) | Cited 33 times

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It is shown that the Poisson structure of dynamical systems with three degrees of freedom can be defined in terms of an integrable one‐form in three dimensions. Advantage is taken of this fact and the theory of foliations is used in discussing the geometrical structure underlying complete and partial integrability. Techniques for finding Poisson structures are presented and applied to various examples such as the Halphen system which has been studied as the two‐monopole problem by Atiyah and Hitchin. It is shown that the Halphen system can be formulated in terms of a flat SL(2,R)‐valued connection and belongs to a nontrivial Godbillon–Vey class. On the other hand, for the Euler top and a special case of three‐species Lotka–Volterra equations which are contained in the Halphen system as limiting cases, this structure degenerates into the form of globally integrable bi‐Hamiltonian structures. The globally integrable bi‐Hamiltonian case is a linear and the SL(2,R) structure is a quadratic unfolding of an integrable one‐form in 3+1 dimensions. It is shown that the existence of a vector field compatible with the flow is a powerful tool in the investigation of Poisson structure and some new techniques for incorporating arbitrary constants into the Poisson one‐form are presented herein. This leads to some extensions, analogous to q extensions, of Poisson structure. The Kermack–McKendrick model and some of its generalizations describing the spread of epidemics, as well as the integrable cases of the Lorenz, Lotka–Volterra, May–Leonard, and Maxwell–Bloch systems admit globally integrable bi‐Hamiltonian structure.
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02.20.-a Group theory
02.30.Hq Ordinary differential equations

Analysis of the Green’s function approach to one‐dimensional inverse problems. I. One parameter reconstruction

Sailing He, Sergei I. Kabanikhin, Vladimir G. Romanov, and Staffan Ström

J. Math. Phys. 34, 5724 (1993); http://dx.doi.org/10.1063/1.530279 (23 pages) | Cited 2 times

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An analysis of a Green’s function approach (based on wave splitting) to the one‐dimensional electromagnetic inverse problem is given. The Green’s functions refer to split components of the fundamental solution. The linear system of equations for the Green’s functions is shown to be well‐posed for the inverse problem and stability estimates, theorems on existence, uniqueness, local correctness, and convergence are given.
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03.50.De Classical electromagnetism, Maxwell equations
11.80.-m Relativistic scattering theory
41.20.Jb Electromagnetic wave propagation; radiowave propagation

Boundary values as Hamiltonian variables. I. New Poisson brackets

Vladimir O. Soloviev

J. Math. Phys. 34, 5747 (1993); http://dx.doi.org/10.1063/1.530280 (23 pages) | Cited 12 times

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The ordinary Poisson brackets in field theory do not fulfill the Jacobi identity if boundary values are not reasonably fixed by special boundary conditions. It is shown that these brackets can be modified by adding some surface terms to lift this restriction. The new brackets generalize a canonical bracket considered by Lewis, Marsden, Montgomery, and Ratiu for the free boundary problem in hydrodynamics. The definition of Poisson brackets used herein permits the treating of to treat boundary values of a field on equal footing with its internal values and the direct estimation of estimate the brackets between both surface and volume integrals. This construction is applied to any local form of Poisson brackets.
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03.50.-z Classical field theories
46.05.+b General theory of continuum mechanics of solids

The critical surface for the tridimensional massless Gaussian fixed point: An unstable manifold with two relevant directions

Emmanuel A. Pereira

J. Math. Phys. 34, 5770 (1993); http://dx.doi.org/10.1063/1.530281 (11 pages) | Cited 1 time

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The domain of attraction is determined in a small neigborhood of the tridimensional massless Gaussian fixed point, obtaining a critical surface in a manifold with one marginal and two relevant directions (besides the marginal direction due to the field strengh renormalization). The results are proven within the context of a hierarchical approximation.
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11.10.-z Field theory
05.20.-y Classical statistical mechanics

The Hausdorff moments in statistical mechanics

E. Scalas and G. A. Viano

J. Math. Phys. 34, 5781 (1993); http://dx.doi.org/10.1063/1.530282 (20 pages) | Cited 3 times

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A new method for solving the Hausdorff moment problem is presented which makes use of Pollaczek polynomials. This problem is severely ill posed; a regularized solution is obtained without any use of prior knowledge. When the problem is treated in the L2 space and the moments are finite in number and affected by noise or round‐off errors, the approximation converges asymptotically in the L2 norm. The method is applied to various questions of statistical mechanics and in particular to the determination of the density of states. Concerning this latter problem the method is extended to include distribution valued densities. Computing the Laplace transform of the expansion a new series representation of the partition function Z(β) (β=1/kBT) is obtained which coincides with a Watson resummation of the high‐temperature series for Z(β).
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02.90.+p Other topics in mathematical methods in physics (restricted to new topics in section 02)
02.20.-a Group theory

On an old article of Tzitzeica and the inverse scattering method

A. Yu. Boldin, S. S. Safin, and R. A. Sharipov

J. Math. Phys. 34, 5801 (1993); http://dx.doi.org/10.1063/1.530283 (9 pages) | Cited 8 times

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Tzitzeica’s class of surfaces in R3 is considered in connection with the inverse scattering method for the associated equation uxy=eu−e−2u. The Bäcklund transformation for this equation is derived from Tzitzeica’s Darboux transformation for it.  
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02.40.-k Geometry, differential geometry, and topology
03.65.Pm Relativistic wave equations
04.20.Jb Exact solutions

C‐integrable nonlinear partial differential equations. III

F. Calogero and Ji Xiaoda

J. Math. Phys. 34, 5810 (1993); http://dx.doi.org/10.1063/1.530284 (22 pages) | Cited 12 times

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A technique based on a change of dependent variables, used in a previous paper to generate C‐integrable nonlinear partial differential equations (PDEs) (i.e., nonlinear PDEs linearizable by an appropriate Change of variables) in 1+1 dimensions (one time and one space variables), is extended to the case of more than one space dimension. Several examples of evolution C‐integrable PDEs in 1+2 dimensions (one time and two space variables) are exhibited.
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02.30.Jr Partial differential equations

A hierarchic array of integrable models

Peter G. O. Freund and Anton V. Zabrodin

J. Math. Phys. 34, 5832 (1993); http://dx.doi.org/10.1063/1.530285 (11 pages) | Cited 3 times

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Motivated by Harish–Chandra theory, starting from a simple CDD‐pole S matrix, a hierarchy of new S matrices involving ever ‘‘higher’’ (in the sense of Barnes) gamma functions are constructed. These new S matrices correspond to scattering of excitations in ever more complex integrable models. From each of these models, new ones are obtained either by ‘‘q deformation,’’ or by considering the Selberg‐type Euler products of which they represent the ‘‘infinite place.’’ A hierarchic array of integrable models is thus obtained. A remarkable diagonal link in this array is established. Though many entries in this array correspond to familiar integrable models, the array also leads to new models. Setting up this array led to new results on the q‐gamma function and on the q‐deformed Bloch–Wigner function.
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75.10.Jm Quantized spin models, including quantum spin frustration
02.90.+p Other topics in mathematical methods in physics (restricted to new topics in section 02)
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