The algebra of physical magnitudes

We define a physical magnitude as an equivalence class of measurement procedures and formulate sufficient restrictions on the equivalence relation to guarantee meaningful algebraic operations between magnitudes. These restrictions are not sufficient to let the Kochen and Specker argument go through. They are, however, stronger than mere statistical equivalence of measurement procedures and thus are relevant to the problem of the completeness of quantum mechanics. In fact, they give rise to a strong argument for the incompleteness of quantum mechanics.     

Universality in the algebra of vertex strengths as generated by bilocal currents

We review the derivation of the algebra of vertex strengths (AVS) from the algebra of current bilocals on the null plane in the deep Regge scaling region (DRSR). We discuss the difficulties and present a mathematical procedure yielding AVS through a smooth removal of the divergent ω^α dependences which differ on both sides of the infinite “ω-local” algebra of scaling functions. We suggest that the S_a charges in AVS (the 2⁺⁺ trajectories' vertex strengths) may be given by the antisymmetrized matrix elements of the vector bilocals.     

The algebra of physical observables in non-linearly realized gauge theories

We classify the physical observables in spontaneously broken non-linearly realized gauge theories in the recently proposed loopwise expansion governed by the Weak Power-Counting (WPC) and the Local Functional Equation. The latter controls the non-trivial quantum deformation of the classical non-linearly realized gauge symmetry, to all orders in the loop expansion. The Batalin–Vilkovisky (BV) formalism is used. We show that the dependence of the vertex functional on the Goldstone fields is obtained via a canonical transformation w.r.t. the BV bracket associated with the BRST symmetry of the model. We also compare the WPC with strict power-counting renormalizability in linearly realized gauge theories. In the case of the electroweak group we find that the tree-level Weinberg relation still holds if power-counting renormalizability is weakened to the WPC condition.     

Computer generated spatial filters for large object formats

A cheap computer generated hologram (CGH) contains 32 × 32 sampling cells, typically. When using a cheap CGH as spatial filter, the input object is also restricted to 32 × 32 sampling or resolution cells. However, practical objects are much larger. Sufficiently large filters would be too expensive in terms of computing and plotting time. We present a new method for producing CGH spatial filters. We were able to produce a 160 × 160 cell CGH filter with computing and plotting labour for only 32 × 32 sampling cells. We used that spatial filter as matched filter for pattern detection.     

On the physical representations of an infinite-dimensional Lie algebra

Irreducible Hermitian representations of the infinite-dimensional Lie algebra, presented by J. Formánek [Czech. J. Phys.B16 (1966), 1,281] and suitable for the classification of elementary particles, are characterized by assumptions I–III. All representations of this kind are found explicitly.     

Perturbative photon fluxes generated by high-frequency gravitational waves and their physical effects

We consider the electromagnetic (EM) perturbative effects produced by high-frequency gravitational waves (HFGWs) in the GHz band in a special EM resonance system, which consists of fractal membranes, a Gaussian beam (GB) passing through a static magnetic field. Under the synchro-resonance condition, coherence modulation of the HFGWs to the preexisting transverse components of the GB is predicted to produce the transverse perturbative photon flux (PPF), which has three novel and important properties. (1) The PPF has a maximum at a longitudinal symmetrical surface of the GB where the transverse background photon flux (BPF) vanishes; (2) the resonant effect will be high sensitive to the propagating directions of the HFGWs; (3) the PPF reflected or transmitted by the fractal membrane exhibits a very small decay to be compared with a very large decay of the much stronger BPF. Such properties might provide a new way to distinguish and display the perturbative effects produced by the HFGWs. We also discuss the high-frequency asymptotic behavior of the relic GWs in the microwave band and the positive definite issues of their energy-momentum pseudo-tensor. PACS  04.30.Nk; 04.25.Nx; 04.30.Db; 98.80.Cq     

The Virasoro algebra

Three theroems are proved. With suitable hypotheses in each case, characterizations are found for the Virasoro algebra, for some of its representations, and for the Ramond-Neveu-Schwarz superalgebra built around the Virasoro algebra.     

The blob algebra and the periodic Temperley-Lieb algebra

We determine the structure of two variations on the Temperley-Lieb algebra, both used for dealing with special kinds of boundary conditions in statistical mechanics models. The first is a new algebra, the blob algebra. We determine both the generic and all the exceptional structures for this two parameter algebra. The second is the periodic Temperley-Lieb algebra. The generic structure and part of the exceptional structure of this algebra have already been studied. We complete the analysis using results from the study of the blob algebra.     

Influence of elastic fields generated by point defects on the physical properties of matter

The influence of defects on the physical properties of matter in the vicinity of the points of structural phase transitions was investigated in the papers of A. P. Levanyuk et al. They took into account only one property of the defect: the capability of fixing the order parameter at the point of its occurrence. The electric fields generated by a point defect are, of course, not taken into account in such an approach. In the present study, the influence of the elastic fields generated by point defects on the behavior of the heat capacity in the vicinity of the second-order phase transition and on the absorption of sound are investigated. It is shown that consideration of the elastic fields leads to the same anomalies of thermodynamic characteristics as in the papers of Levanyuk.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 91–95, October, 1985.     

The BRS algebra of a free minimal differential algebra

We construct the Weil and the universal BRS algebras of theories that can have as a gauge symmetry a free minimal differential (Sullivan) algebra, the natural extension of Lie algebras allowing the definition of p-form gauge potentials (p > 1). The geometrical meaning of these p-form gauge potentials can be understood with the notion of a Quillen superconnection.     

Low-energy π N-scattering in the algebra of currents model with physical pions

In the algebra of currents model with physical pions the scattering S- and P-wavelengths and the energy dependence of N-scattering S- and P-phases is calculated. The results agree well with experimental data at low energies.Translated from Izvestiya VUZ. Fizika, No. 1, pp. 63–71, January, 1972.     

Soliton algebra by vortex-beam splitting

We experimentally demonstrate the possibility of breaking up intense vortex light beams into stable and controllable sets of parametric solitons. We report observations performed in seeded second-harmonic generation, but the scheme can be extended to all parametric processes. The number of generated solitons is shown to be determined by a robust arithmetic rule.     

The two-dimensional quantum Euclidean algebra

The algebra dual to Woronowicz's deformation of the two-dimensional Euclidean group is constructed. The same algebra is obtained from SU _q (2) via contraction on both the group and algebra levels.     

The classifying algebra for defects

We demonstrate that topological defects in a rational conformal field theory can be described by a classifying algebra for defects – a finite-dimensional semisimple unital commutative associative algebra whose irreducible representations give the defect transmission coefficients. We show in particular that the structure constants of the classifying algebra are traces of operators on spaces of conformal blocks and that the defect transmission coefficients determine the defect partition functions.     

Boson realizations of quantum algebras and some physical spaces with quantum algebra symmetry

The generators ofq-boson algebra are expressed in terms of those of boson algebra, and the relations among the representations of a quantum algebra onq-Fock space, on Fock space, and on coherent state space are discussed in a general way. Two examples are also given to present concrete physical spaces with quantum algebra symmetry. Finally, a new homomorphic mapping from a Lie algebra to boson algebra is presented.This work is supported by the National Foundation of Natural Science of China.     

q-Sigma-algebra generated by balls

The q--algebra, i.e., the system of sets closed under complementation, countable disjoint unions, and containing the empty set, generated by the system of open balls coincides with the -field of Borel sets inR sun forn=1,2, and 3. A first step to extend the proof forn=4, 5, 6, and 7 is indicated.     

The structure of theW ∞ algebra

A new definition of spectral data of a monopole is given for any compact Lie or Kac-Moody group. It is shown that the spectral data determines the irreducible monopole. In the case of maximal symmetry breaking the spectral data is shown to reduce to an earlier definition in terms of algebraic curves indexed by the nodes of the Dynkin diagram of the group. The structure of solutions to Nahm's equations corresponding to the monopole is discussed.Research supported in part by NSER C grant A8361 and FCAR grant EQ3518     

The operator algebra of orbifold models

We analyze the chiral properties of (orbifold) conformal field theories which are obtained from a given conformal field theory by modding out by a finite symmetry group. For a class of orbifolds, we derive the fusion rules by studying the modular transformation properties of the one-loop characters. The results are illustrated with explicit calculations of toroidal andc=1 models.     

The superconformal algebra in higher dimensions

The superconformal algebra for 4/4N-dimensional super-Minkowski space (d=4) can be identified with the simple superalgebra su (2,2/N). For even-dimension d=5,6 the superconformal algebra can be identified with a real form of the simple superalgebras F(4), D(4,1) respectively in Kac's classification. For even-dimension d>-7 it is impossible to define a superconformal algebra satisfying three natural conditions: (1) it acts as infinitesimal automorphisms on super-Minkowski space; (2) this action extends the natural action of the super-Poincaré algebra; (3) when the action of the even part of the superconformal algebra is reduced to an infinitesimal action on ordinary Minkowski space, it extends the natural action of the conformal algebra so (2, d).