Quantization of Forms on the Cotangent Bundle

We consider the following construction of quantization. For a Riemannian manifold $M$ the space of forms on T ^{⋆ M} is made into a space of (full) symbols of operators acting on forms on M. This gives rise to the composition of symbols, which is a deformation of the (“super”)commutative multiplication of forms. The symbol calculus is exact for differential operators and the symbols that are polynomial in momenta. We calculate the symbols of natural Laplacians. (Some nice Weitzenb?ck like identities appear here.) Formulae for the traces corresponding to natural gradings of Ω (T ^{⋆ M} ) are established. Using these formulae, we give a simple direct proof of the Gauss–Bonnet–Chern Theorem. We discuss these results in connection with a general question of the quantization of forms on a Poisson manifold. Received: 12 November 1998 / Accepted: 1 March 1999     

A Condensed Matter Interpretation of SM Fermions and Gauge Fields

We present the bundle (Aff(3)⊗ℂ⊗Λ)(ℝ³), with a geometric Dirac equation on it, as a three-dimensional geometric interpretation of the SM fermions. Each (ℂ⊗Λ)(ℝ³) describes an electroweak doublet. The Dirac equation has a doubler-free staggered spatial discretization on the lattice space (Aff(3)⊗ℂ)(ℤ³). This space allows a simple physical interpretation as a phase space of a lattice of cells. We find the SM SU(3) _c ×SU(2) _L ×U(1) _Y action on (Aff(3)⊗ℂ⊗Λ)(ℝ³) to be a maximal anomaly-free gauge action preserving E(3) symmetry and symplectic structure, which can be constructed using two simple types of gauge-like lattice fields: Wilson gauge fields and correction terms for lattice deformations. The lattice fermion fields we propose to quantize as low energy states of a canonical quantum theory with ℤ₂-degenerated vacuum state. We construct anticommuting fermion operators for the resulting ℤ₂-valued (spin) field theory. A metric theory of gravity compatible with this model is presented too.     

Relativistic fermion in a spherically symmetric potential well of finite depth in a two-dimensional space

The problem of existence of bounded relativistic fermion states in a spherically symmetric well of finite depth in a two-dimensional space is investigated. The well depth critical for the appearance of standard states with energies E = m, 0, and –m is determined; moreover, cases with zero and nonzero fermion momenta are considered. Approximate analytical expressions for the critical depths of narrow and wide wells are derived which are in good agreement with the results of numerical calculations. Approximate energies of levels located on the boundaries of the upper and lower continuums and determined analytically are in good agreement with the results of numerical calculations.     

Three-Point Functions for MN/SN Orbifolds¶with?= 4 Supersymmetry

The D1–D5 system is believed to have an “orbifold point” in its moduli space where its low energy theory is a ?=4 supersymmetric sigma model with target space M ^N /S _N , where M is T ⁴ or K3. We study correlation functions of chiral operators in CFTs arising from such a theory. We construct a basic class of chiral operators from twist fields of the symmetric group and the generators of the superconformal algebra. We find explicitly the 3-point functions for these chiral fields at large N; these expressions are “universal” in that they are independent of the choice of M. We observe that the result is a significantly simpler expression than the corresponding expression for the bosonic theory based on the same orbifold target space. Received: 29 March 2001 / Accepted: 20 January 2002     

SU(3) coupling schemes for odd-odd nuclei in the interacting boson – fermion – fermion model with both odd proton and odd neutron in natural parity orbits

The SU^BFF(3) dynamical symmetry limits of interacting boson – fermion – fermion model are identified and they are appropriate for heavy deformed odd – odd nuclei for configurations with both the odd proton and odd neutron occupying all the natural parity orbits in the corresponding valence shells. There are three symmetry limits and their correspondence with two quasi-particle (proton-neutron) Nilsson configurations is established; one of the limits mixes both Nilsson n_z's and Λ's and other two limits mix only Nilsson Λ's. The ¹⁹¹Ir (d,t) ¹⁹⁰Ir single nucleon transfer spectroscopic strengths are well described by one of the symmetry limits that mixes only Nilsson Λ's. Received: 22 June 1998     

The Girardeau's fermion-boson procedure in the light of the composite-boson many-body theory

We reconsider the procedure developed for atoms a few decades ago by Girardeau, in the light of the composite-boson many-body theory we recently proposed. The Girardeau's procedure makes use of a so called “unitary Fock-Tani operator” which in an exact way transforms one composite bound atom into one bosonic “ideal” atom. When used to transform the Hamiltonian of interacting atoms, this operator generates an extremely complex set of effective scatterings between ideal bosonic atoms and free fermions which makes the transformed Hamiltonian impossible to write explicitly, in this way forcing to some truncation. The scatterings restricted to the ideal-atom subspace are shown to read rather simply in terms of the two elementary scatterings of the composite-boson many-body theory, namely, the energy-like direct interaction scatterings — which describe fermion interactions without fermion exchange — and the dimensionless Pauli scatterings — which describe fermion exchanges without fermion interaction. We here show that, due to a fundamental difference in the scalar products of elementary and composite bosons, the Hamiltonian expectation value for N ground state atoms obtained by staying in the ideal-atom subspace and working with boson operators only, differ from the exact ones even for N = 2 and a mapping to the ideal-atom subspace performed, as advocated, from the fully antisymmetrical atomic state, i.e., the state which obeys the so-called “subsidiary condition”. This shows that, within this Girardeau's procedure too, we cannot completely forget the underlying fermionic components of the particles if we want to correctly describe their interactions.     

Modular conformal scaling group evidenced in lepton-quark mass

Lepton-quark mass may reflect a correspondence in spacetime structure described by a modular conformal scaling group. Stemming in part from a spacetime line element correspondenceds → (expλ _n)ds in which the eight quantitiesλ ₀,λ ₁, ...,λ ₇ constitute a closed set under a modular addition, the associated formula for lepton-quark mass (yielding values at the 1 GeV scale for the leptons and lighter quarks and at the physical pole for the top) is conjectured to bem=m _fQ²(exp −λ _n), wherem _f=10.245 TeV is the progenitor fermion mass,Q is the charge number of the lepton or quark, and the modular group parameterλ _n is indexed by a fermion principal quantum numbern that depends on three mutually independent projection operators.     

On ‘light’ fermions and proton stability in ‘big divisor’ D3/D7 large volume compactifications

Building on our earlier work (Misra and Shukla, Nucl. Phys. B 827:112, 2010; Phys. Lett. B 685:347–352, 2010), we show the possibility of generating “light” fermion mass scales of MeV–GeV range (possibly related to the first two generations of quarks/leptons) as well as eV (possibly related to first two generations of neutrinos) in type IIB string theory compactified on Swiss-Cheese orientifolds in the presence of a mobile space-time filling D3-brane restricted to (in principle) stacks of fluxed D7-branes wrapping the “big” divisor Σ _B . This part of the paper is an expanded version of the latter half of Sect. 3 of a published short invited review (Misra, Mod. Phys. Lett. A 26:1, 2011) written by one of the authors [AM]. Further, we also show that there are no SUSY GUT-type dimension-five operators corresponding to proton decay, and we estimate the proton lifetime from a SUSY GUT-type four-fermion dimension-six operator to be 10⁶¹ years. Based on GLSM calculations in (Misra and Shukla, Nucl. Phys. B 827:112, 2010) for obtaining the geometric K?hler potential for the “big divisor,” using further the Donaldson’s algorithm, we also briefly discuss in the first of the two appendices the metric for the Swiss-Cheese Calabi–Yau used, which we obtain and which becomes Ricci flat in the large-volume limit.     

On the operator equivalents and the crystal-field and spin Hamiltonian parameters

Operator equivalents used in magnetic resonance (electron paramagnetic resonance [EPR], electron-nuclear double resonance, nuclear magnetic resonance, etc.) are considered. The known non-Hermitian tensor operators O _±q ^(k) (0 ≤ q ≤ k) defined by D. Smith and J. H. M. Thornley (Proc. Phys. Soc. 89, 779–781, 1966) after H. A. Buckmaster (Can. J. Phys. 40, 1670–1677, 1962) are expressed in terms of Hermitian cosine and sine operators O _kq ^c and O _kq ^s. These new normalized operators are related to the nonnormalized extended Stevens operators O _k ^q and O _k ^−q ≡ Ω _k ^q defined by C. Rudowicz (J. Phys. C.: Solid State Phys. 18, 1415–1430, 1985), which are commonly used in EPR of transition ions. The relations between O _kq ^c,s and some other normalized operators occurring in the literature are also given. New crystal-field or spin Hamiltonian parameters B _kq ^c and B _kq ^s associated with the corresponding operators O _kq ^c and O _kq ^s are discussed.     

Warped Convolutions,Rieffel Deformations and the Construction of Quantum Field Theories

Warped convolutions of operators were recently introduced in the algebraic framework of quantum physics as a new constructive tool. It is shown here that these convolutions provide isometric representations of Rieffel’s strict deformations of C ^*–dynamical systems with automorphic actions of \mathbb Rⁿ{\mathbb R^n} , whenever the latter are presented in a covariant representation. Moreover, the device can be used for the deformation of relativistic quantum field theories by adjusting the convolutions to the geometry of Minkowski space. The resulting deformed theories still comply with pertinent physical principles and their Tomita–Takesaki modular data coincide with those of the undeformed theory; but they are in general inequivalent to the undeformed theory and exhibit different physical interpretations.     

Large-scale magnetic fields from inflation due to a <Emphasis Type="Italic">CPT</Emphasis>-even Chern–Simons-like term with Kalb–Ramond and scalar fields

We investigate the generation of large-scale magnetic fields due to the breaking of the conformal invariance in the electromagnetic field through the CPT-even dimension-six Chern–Simons-like effective interaction with a fermion current by taking account of the dynamical Kalb–Ramond and scalar fields in inflationary cosmology. It is explicitly demonstrated that magnetic fields on 1 Mpc scale with the field strength of ∼10⁻⁹ G at the present time can be induced.     

One-Parameter Homogeneous Differential Realization and Boson–Fermion Realization of the SPL(2,1) Superalgebra

One-parameter homogeneous differential realization of the SPL(2,1) superalgebra on the space of homogeneous polynomials and the corresponding boson–fermion realization are studied. The parameter α may be related to the interaction parameter U in one exactly solvable model for correlated electrons.The author was supported financially by Shenzhen Institute of Information Technology grant No. LG-060109. PACS: 12.60.Jv, 03.65.FD     

Experimental and computational study of the passage of an electron beam through the curvilinear element of the Drakon stellarator system in a tenuous plasma

Results are presented from the first stage of studies on the passage of an electron beam with energy 100–500 eV in a magnetic field of 300–700 Oe through the curvilinear solenoid of the KRéL unit, the latter being a prototype of the closing segment of the Drakon stellarator system, in the plasma-beam discharge regime. The ion density at the end of the curvilinear part of the chamber, n _i ≈8×10⁸–10¹⁰ cm⁻³, the electron temperature T _e ≈4–15 eV, and the positions at which the beam hits the target for different distances from it to the electron source are determined experimentally. The motion of the electron beam is computationally modeled with allowance for the space charge created by the beam and the secondary plasma. From a comparison of the experimentally measured trajectories and trajectories calculated for different values of the space charge, we have obtained an estimate for the unneutralized ion density of the order of 5×10⁷ cm⁻³. Zh. Tekh. Fiz. 69, 22–26 (February 1999)     

Correlation Functions for MN/SN Orbifolds

We develop a method for computing correlation functions of twist operators in the bosonic 2-d CFT arising from orbifolds M ^N /S _N , where M is an arbitrary manifold. The path integral with twist operators is replaced by a path integral on a covering space with no operator insertions. Thus, even though the CFT is defined on the sphere, the correlators are expressed in terms of partition functions on Riemann surfaces with a finite range of genus g. For large N, this genus expansion coincides with a 1/N expansion. The contribution from the covering space of genus zero is “universal” in the sense that it depends only on the central charge of the CFT. For 3-point functions we give an explicit form for the contribution from the sphere, and for the 4-point function we do an example which has genus zero and genus one contributions. The condition for the genus zero contribution to the 3-point functions to be non-vanishing is similar to the fusion rules for an SU(2) WZW model. We observe that the 3-point coupling becomes small compared to its large N limit when the orders of the twist operators become comparable to the square root of N – this is a manifestation of the stringy exclusion principle. Received: 20 July 2000 / Accepted: 17 December 2000     

Realisations of the representations of para-Fermi algebra in Fock space of Bose operators: part I

Using the matrix realisations of para-Fermi operators we find isomorphic mappings with respect to the Green product of the para-Fermi algebra into second-order polynomials of creation and annihilation para-Bose operators with arbitrary order of parastatistics. In the Fock space ℋ ₂ ¹ of two Bose operators all the irreducible representations of the para-Fermi algebra are realised. The spaces ofn-particle Bose statesn=1,2,..., from which ℋ ₂ ¹ is constructed as a direct sum, can be interpreted as spaces of para-Fermi states of para-statisticsn.     

A Revision on Classical Solutions to the Cauchy Boltzmann Problem for Soft Potentials

This short note complements the recent paper of the authors (Alonso, Gamba in J. Stat. Phys. 137(5–6):1147–1165, 2009). We revisit the results on propagation of regularity and stability using L ^p estimates for the gain and loss collision operators which had the exponent range misstated for the loss operator. We show here the correct range of exponents. We require a Lebesgue’s exponent α>1 in the angular part of the collision kernel in order to obtain finiteness in some constants involved in the regularity and stability estimates. As a consequence the L ^p regularity associated to the Cauchy problem of the space inhomogeneous Boltzmann equation holds for a finite range of p≥1 explicitly determined.     

Dirac Functional Determinants in Terms of the Eta Invariant and the Noncommutative Residue

 The zeta and eta–functions associated with massless and massive Dirac operators, in a D-dimensional (D odd or even) manifold without boundary, are rigorously constructed. Several mathematical subtleties involved in this process are stressed, as the intrinsic ambiguity present in the definition of the associated fermion functional determinant in the massless case and, also, the unavoidable presence (in some situations) of a multiplicative anomaly, that can be conveniently expressed in terms of the non-commutative residue. The ambiguity is here proven to disappear in the massive case, giving rise to a phase of the Dirac determinant – that agrees with very recent calculations which appeared in the mathematical literature – and to a multiplicative anomaly – also in agreement with other calculations, in the coinciding cases (in fact our results cover much more general situations). A number of physically relevant explicit examples are worked out. After explicit, nontrivial resummation of the mass series expansions, involving zeta and eta functions, our results are finally expressed in terms of quite simple formulae. Received: 7 October 1999 / Accepted: 27 January 2003 Published online: 5 May 2003 RID="*" ID="*" On leave from: ICE/CSIC and IEEC, Edifici Nexus 201, Gran Capità 2-4, 08034 Barcelona, Spain Communicated by R.H. Dijkgraaf     

Investigation of temperature-dependent electrical properties of p-VOPc/n-si heterojunction under dark conditions

An organic/inorganic heterojunction p-VOPc/n-Si was fabricated and its electrical properties were investigated. Temperature-dependent dark current–voltage (I–V) characteristics of the heterojunction exhibited rectification behaviour with a rectification ratio of 405 at ±1 V and room temperature. The current–voltage characteristics of the cell showed ohmic conduction at low voltages followed by a space charge-limited current (SCLC) conduction dominated by an exponential trap distribution at higher voltages. At room temperature, the series and shunt resistances were found to be approximately 1.4 and 100 kΩ, respectively. Diode ideality factor n was found to be 3.2 at room temperature and dropped to 1.9 at 363 K. Room temperature mobility of vanadyl phthalocyanine (VOPc) was extracted from the I–V characteristics in the SCLC region and was found approximately 15.5 × 10⁻³ cm² V⁻¹ s⁻¹. The effective barrier height, Ф_B, was estimated as 0.77 eV. The effect of temperature, on various heterojunction parameters was recorded under dark conditions and at temperatures ranging from 300 to 363 K.     

Probabilistic Averages of Jacobi Operators

I study the Lyapunov exponent and the integrated density of states for general Jacobi operators. The main result is that questions about these can be reduced to questions about ergodic Jacobi operators. I use this to show that for finite gap Jacobi operators, regularity implies that they are in the Cesàro–Nevai class, proving a conjecture of Barry Simon. Furthermore, I use this to study Jacobi operators with coefficients a(n) = 1 and b(n) = f(n ^ρ (mod 1)) for ρ > 0 not an integer.     

Coupled Fermion and Boson Production in a Strong Background Mean Field

 We derive quantum kinetic equations for fermion and boson production starting from a φ⁴ Lagrangean with minimal coupling to fermions. Decomposing the scalar field into a mean-field part and fluctuations we obtain spontaneous pair creation driven by a self-interacting strong background field. The produced fermion and boson pairs are self-consistently coupled. Consequently back reactions arise from fermion and boson currents determining the time-dependent self-interacting background mean field. We explore the numerical solution with cylindric boundary conditions for the time evolution of the mean field as well as for the number- and energy densities for fermions and bosons. We find that after a characteristic time all energy is converted from the background mean field to particle creation. Applying this general approach to the production of “quarks” and “gluons” a typical time scale for the collapse of the flux tube is 1.5 fm/c. Received February 14, 2002; accepted March 29, 2002 Published online June 24, 2002