Conformal Spinning Quantum Particles in Complex Minkowski Space as Constrained Nonlinear Sigma Models in U(2,2) and Born's Reciprocity

We revise the use of 8-dimensional conformal, complex (Cartan) domains as a base for the construction of conformally invariant quantum (field) theory, either as phase or configuration spaces. We follow a gauge-invariant Lagrangian approach (of nonlinear sigma-model type) and use a generalized Dirac method for the quantization of constrained systems, which resembles in some aspects the standard approach to quantizing coadjoint orbits of a group G. Physical wave functions, Haar measures, orthonormal basis and reproducing (Bergman) kernels are explicitly calculated in and holomorphic picture in these Cartan domains for both scalar and spinning quantum particles. Similarities and differences with other results in the literature are also discussed and an extension of Schwinger's Master Theorem is commented in connection with closure relations. An adaptation of the Born's Reciprocity Principle (BRP) to the conformal relativity, the replacement of space-time by the 8-dimensional conformal domain at short distances and the existence of a maximal acceleration are also put forward.Comment: 33 pages, no figures, LaTe

On plane wave and vortex-like solutions of noncommutative Maxwell-Chern-Simons theory

We investigate the spectrum of the gauge theory with Chern-Simons term on the noncommutative plane, a modification of the description of the Quantum Hall fluid recently proposed by Susskind. We find a series of the noncommutative massive ``plane wave'' solutions with polarization dependent on the magnitude of the wave-vector. The mass of each branch is fixed by the quantization condition imposed on the coefficient of the noncommutative Chern-Simons term. For the radially symmetric ansatz a vortex-like solution is found and investigated. We derive a nonlinear difference equation describing these solutions and we find their asymptotic form. These excitations should be relevant in describing the Quantum Hall transitions between plateaus and the end transition to the Hall Insulator.Comment: 17 pages, LaTeX (JHEP), 1 figure, added references, version accepted to JHE

The magnetic mass of transverse gluon, the B-meson weak decay vertex and the triality symmetry of octonion

With an assumption that in the Yang-Mills Lagrangian, a left-handed fermion and a right-handed fermion both expressed as quaternion make an octonion which possesses the triality symmetry, I calculate the magnetic mass of the transverse self-dual gluon from three loop diagram, in which a heavy quark pair is created and two self-dual gluons are interchanged. The magnetic mass of the transverse gluon depends on the mass of the pair created quarks, and in the case of charmed quark pair creation, the magnetic mass $m_{mag}$ becomes approximately equal to $T_c$ at $T=T_c\sim 1.14\Lambda_{\bar{MS}}\sim 260$MeV. A possible time-like magnetic gluon mass from two self-dual gluon exchange is derived, and corrections in the B-meson weak decay vertices from the two self-dual gluon exchange are also evaluated.Comment: 22 pages, 9 figure

Gauge Theory on Fuzzy S^2 x S^2 and Regularization on Noncommutative R^4

We define U(n) gauge theory on fuzzy S^2_N x S^2_N as a multi-matrix model, which reduces to ordinary Yang-Mills theory on S^2 x S^2 in the commutative limit N -> infinity. The model can be used as a regularization of gauge theory on noncommutative R^4_\theta in a particular scaling limit, which is studied in detail. We also find topologically non-trivial U(1) solutions, which reduce to the known "fluxon" solutions in the limit of R^4_\theta, reproducing their full moduli space. Other solutions which can be interpreted as 2-dimensional branes are also found. The quantization of the model is defined non-perturbatively in terms of a path integral which is finite. A gauge-fixed BRST-invariant action is given as well. Fermions in the fundamental representation of the gauge group are included using a formulation based on SO(6), by defining a fuzzy Dirac operator which reduces to the standard Dirac operator on S^2 x S^2 in the commutative limit. The chirality operator and Weyl spinors are also introduced.Comment: 39 pages. V2-4: References added, typos fixe

Noncommutative gravity: fuzzy sphere and others

Gravity on noncommutative analogues of compact spaces can give a finite mode truncation of ordinary commutative gravity. We obtain the actions for gravity on the noncommutative two-sphere and on the noncommutative ${\bf CP}^2$ in terms of finite dimensional $(N\times N)$-matrices. The commutative large $N$ limit is also discussed.Comment: LaTeX, 13 pages, section on CP^2 added + minor change

Fuzzy sphere bimodule, ABS construction to the exact soliton solutions

In this paper, we set up the bi-module of the algebra ${\cal A}$ on fuzzy sphere. Based on the differential operators in moving frame, we generalize the ABS construction into fuzzy sphere case. The applications of ABS construction are investigated in various physical systems.Comment: Latex file without figure, 13 page

The Finite Temperature SU(2) Savvidy Model with a Non-trivial Polyakov Loop

We calculate the complete one-loop effective potential for SU(2) gauge bosons at temperature T as a function of two variables: phi, the angle associated with a non-trivial Polyakov loop, and H, a constant background chromomagnetic field. Using techniques broadly applicable to finite temperature field theories, we develop both low and high temperature expansions. At low temperatures, the real part of the effective potential V_R indicates a rich phase structure, with a discontinuous alternation between confined (phi=pi) and deconfined phases (phi=0). The background field H moves slowly upward from its zero-temperature value as T increases, in such a way that sqrt(gH)/(pi T) is approximately an integer. Beyond a certain temperature on the order of sqrt(gH), the deconfined phase is always preferred. At high temperatures, where asymptotic freedom applies, the deconfined phase phi=0 is always preferred, and sqrt(gH) is of order g^2(T)T. The imaginary part of the effective potential is non-zero at the global minimum of V_R for all temperatures. A non-perturbative magnetic screening mass of the form M_m = cg^2(T)T with a sufficiently large coefficient c removes this instability at high temperature, leading to a stable high-temperature phase with phi=0 and H=0, characteristic of a weakly-interacting gas of gauge particles. The value of M_m obtained is comparable with lattice estimates.Comment: 28 pages, 5 eps figures; RevTeX 3 with graphic

Star products made (somewhat) easier

We develop an approach to the deformation quantization on the real plane with an arbitrary Poisson structure which based on Weyl symmetrically ordered operator products. By using a polydifferential representation for deformed coordinates $\hat x^j$ we are able to formulate a simple and effective iterative procedure which allowed us to calculate the fourth order star product (and may be extended to the fifth order at the expense of tedious but otherwise straightforward calculations). Modulo some cohomology issues which we do not consider here, the method gives an explicit and physics-friendly description of the star products.Comment: 20 pages, v2, v3: comments and references adde

On One-Loop Gap Equations for the Magnetic Mass in d=3 Gauge Theory

Recently several workers have attempted determinations of the so-called magnetic mass of d=3 non-Abelian gauge theories through a one-loop gap equation, using a free massive propagator as input. Self-consistency is attained only on-shell, because the usual Feynman-graph construction is gauge-dependent off-shell. We examine two previous studies of the pinch technique proper self-energy, which is gauge-invariant at all momenta, using a free propagator as input, and show that it leads to inconsistent and unphysical result. In one case the residue of the pole has the wrong sign (necessarily implying the presence of a tachyonic pole); in the second case the residue is positive, but two orders of magnitude larger than the input residue, which shows that the residue is on the verge of becoming ghostlike. This happens because of the infrared instability of d=3 gauge theory. A possible alternative one-loop determination via the effective action also fails. The lesson is that gap equations must be considered at least at two-loop level.Comment: 21 pages, LaTex, 2 .eps figure

The Fuzzy Ginsparg-Wilson Algebra: A Solution of the Fermion Doubling Problem

The Ginsparg-Wilson algebra is the algebra underlying the Ginsparg-Wilson solution of the fermion doubling problem in lattice gauge theory. The Dirac operator of the fuzzy sphere is not afflicted with this problem. Previously we have indicated that there is a Ginsparg-Wilson operator underlying it as well in the absence of gauge fields and instantons. Here we develop this observation systematically and establish a Dirac operator theory for the fuzzy sphere with or without gauge fields, and always with the Ginsparg-Wilson algebra. There is no fermion doubling in this theory. The association of the Ginsparg-Wilson algebra with the fuzzy sphere is surprising as the latter is not designed with this algebra in mind. The theory reproduces the integrated U(1)_A anomaly and index theory correctly.Comment: references added, typos corrected, section 4.2 simplified. Report.no: SU-4252-769, DFUP-02-1