Infinite Statistics Research Articles

INFINITE STATISTICS, SYMMETRY BREAKING AND COMBINATORIAL HIERARCHY

The physics of symmetry breaking in theories with strongly interacting quanta obeying infinite (quantum Boltzmann) statistics known as quons is discussed. The picture of Bose/Fermi particles as low energy excitations over nontrivial quon condensate is advocated. Using induced gravity arguments, it is demonstrated that the Planck mass in such low energy effective theory can be factorially (in number of degrees of freedom) larger than its true ultraviolet cutoff. Thus, the assumption that statistics of relevant high energy excitations is neither Bose nor Fermi but infinite can remove the hierarchy problem without necessity to introduce any artificially large numbers. Quantum mechanical model illustrating this scenario is presented.

Using kinematic boundary lines for particle mass measurements and disambiguation in SUSY-like events with missing energy

We revisit the method of kinematical endpoints for particle mass determination, applied to the popular SUSY decay chain ? 02 ? ? 01. We analyze the uniqueness of the solutions for the mass spectrum in terms of the measured endpoints in the observable invariant mass distributions. We provide simple analytical inversion formulas for the masses in terms of the measured endpoints. We show that in a sizable portion of the SUSY mass parameter space the solutions always suffer from a two-fold ambiguity, due to the fact that the original relations between the masses and the endpoints are piecewise-defined functions. The ambiguity persists even in the ideal case of a perfect detector and infinite statistics. We delineate the corresponding dangerous regions of parameter space and identify the sets of ``twin'' mass spectra. In order to resolve the ambiguity, we propose a generalization of the endpoint method, from single-variable distributions to two-variable distributions. In particular, we study analytically the boundaries of the {mj?(lo),mj?(hi)} and {m??,mj??} distributions and prove that their shapes are in principle sufficient to resolve the ambiguity in the mass determination. We identify several additional independent measurements which can be obtained from the boundary lines of these bivariate distributions. The purely kinematical nature of our method makes it generally applicable to any model that exhibits a SUSY-like cascade decay.

Origin of holographic dark energy models

We investigate the origin of holographic dark energy models which were recently proposed to explain the dark energy-dominated universe. For this purpose, we introduce the spacetime foam uncertainty of δ l ⩾ l p α l α − 1 . It was argued that the case of α = 2 / 3 could describe the dark energy with infinite statistics, while the case of α = 1 / 2 can describe the ordinary matter with Bose–Fermi statistics. However, two cases may lead to the holographic energy density if the latter recovers from the geometric mean of UV and IR scales. Hence the dark energy with infinite statistics based on the entropy bound is not an ingredient for deriving the holographic dark energy model. Furthermore, it is shown that the agegraphic dark energy models are the holographic dark energy model with different IR length scales.

Spacetime Foam: From Entropy and Holography to Infinite Statistics and Nonlocality

Due to quantum fluctuations, spacetime is foamy on small scales. The degree of foaminess is found to be consistent with holography, a principle prefigured in the physics of black hole entropy. It has bearing on the ultimate accuracies of clocks and measurements and the physics of quantum computation. Consistent with existing archived data on active galactic nuclei from the Hubble Space Telescope, the application of the holographic spacetime foam model to cosmology requires the existence of dark energy which, we argue, is composed of an enormous number of inert “particles” of extremely long wavelength. We suggest that these “particles” obey infinite statistics in which all representations of the particle permutation group can occur, and that the nonlocality present in systems obeying infinite statistics may be related to the nonlocality present in holographic theories. We also propose to detect spacetime foam by looking for halos in the images of distant quasars, and argue that it does not modify the GZK cutoff in the ultra-high energy cosmic ray spectrum and its contributions to time-offlight differences of high energy gamma rays from distant GRB are too small to be detectable.

Representation of orthofermion spin operators

The algebraic expressions for the total spin operators expressed in terms of orthofermion creation and annihilation operators are combined into a single equation. The terms in this expressions are rearranged to provide representations of local spin operators. This task is essential for modelling a system of orthofermions in the presence of a magnetic field. By factorizing the orthofermion annihilation (creation) operators into charge and spin dependent parts, it is shown that the latter part suffices to represent spin number, raising and lowering operators. Finally a connection is provided between the spin systems and Greenberg’s infinite statistics.

Holographic foam, dark energy and infinite statistics

Quantum fluctuations of spacetime give rise to quantum foam, and black hole physics dictates that the foam is of holographic type. Applied to cosmology, the holographic model requires the existence of dark energy which, we argue, is composed of an enormous number of inert “particles” of extremely long wavelength. These “particles” necessarily obey infinite statistics in which all representations of the particle permutation group can occur. For every boson or fermion in the present observable universe there could be ∼1031 such “particles”. We also discuss the compatibility between the holographic principle and infinite statistics.

A FORMULA FOR THE SENSITIVITY TO sin2 2θ13 IN REACTOR EXPERIMENTS WITH A SPECTRAL ANALYSIS

Using an analytical approach, the sensitivity to sin 2 2θ13 with infinite statistics in a spectral analysis is investigated in reactor neutrino oscillation experiments with one reactor and two identical detectors. We derive an useful formula for the sensitivity which depends only on two parameters [Formula: see text] (the uncorrelated bin-to-bin systematic error over the square-root of the number of bins) and σ dB (the bin-to-bin correlated detector specific systematic error) if they are smaller than other errors. It is pointed out that the spectrum analysis is advantageous over the rate analysis in search of neutrino oscillations for [Formula: see text].

Holographic foam, dark energy and infinite statistics

Holographic foam, dark energy and infinite statistics

Neutrino statistics and non-standard commutation relations

Recently it was suggested that the neutrino may violate the Pauli exclusion principle (PEP). This renews interest in the systematic search for bilinear commutation relations that could describe deviations from PEP. In the context of this search we prove a no-go theorem which forbids a finite occupancy limit for an arbitrary system with a bilinear commutation relation. In other words, either the upper limit on the occupancy number is 1 (the ordinary fermionic case) or there is no upper limit at all. Some examples of the latter class include the usual Bose statistics, as well as non-standard quon statistics and infinite statistics.

Infinite statistics and theSU(1, 1) phase operator

A few years ago, Agarwal (1991 Phys. Rev. A 44 8398) showed that the Susskind–Glogower phase operators, expressible in terms of Bose operators, provide a realization of the algebra for particles obeying infinite statistics. In this paper we show that the SU(1, 1) phase operators, constructed in terms of the elements of the su(1, 1) Lie algebra, also provide a realization of the algebra for infinite statistics. There are many realizations of the su(1, 1) algebra in terms of single or multimode bose operators, three of which are discussed along with their corresponding phase states. The Susskind–Glogower phase operator is a special case of the SU(1, 1) phase operator associated with the Holstein–Primakoff realization of su(1, 1).

Orthofermion statistics and its application to the infinite U Hubbard model

We present an algebra of the creation and annihilation operators for spin 1 2 fermions which avoid double occupancy of an orbital state. These fermions, called orthofermions, obey the quantum statistics in which the state vector is antisymmetric only for the exchange of the orbital indices. Because of this peculiar property, representation of the number and the spin operators in terms of creation and annihilation operators is much more complex compared to the usual representation of spin 1 2 fermions. We have found that this representation is very similar to the representation of the number operator in the infinite statistics of Greenberg. As an application of this statistics, we study the thermodynamics of the infinite U Hubbard model and obtain the known exact results in one dimension. Since our approach is valid in any dimension, it can be employed to ascertain the accuracy of the approximate solutions of the Hubbard model.

The energy operator for a model with a multiparametric infinite statistics

In this paper we consider an energy operator (a free Hamiltonian), in the second-quantized approach, for the multiparameter quon algebras: aia†j − qija†jai = δij, i, j ∊ I with (qij)i,j∊I any Hermitian matrix of deformation parameters. We obtain an elegant formula for normally ordered (sometimes called Wick-ordered) series expansions of number operators (which determine a free Hamiltonian). As a main result (see theorem 1) we prove that the number operators are given, with respect to a basis formed by 'generalized Lie elements', by certain normally ordered quadratic expressions with coefficients given precisely by the entries of the inverses of Gram matrices of multiparticle weight spaces. (This settles a conjecture by Meljanac S and Perica A (1994 J. Phys. A: Math. Gen. 27 4737–44).) These Gram matrices are Hermitian generalizations of the Varchenko matrices, associated with a quantum (symmetric) bilinear form of diagonal arrangements of hyperplanes. The solution of the inversion problem of such matrices in Meljanac S and Svrtan D (1996 Math. Commun. 1 1–24 (theorem 2.2.17)), leads to an effective formula for the number operators studied in this paper. The one-parameter case, in the monomial basis, was studied by Zagier, Stanciu and Møller.

Superselection Theory for Subsystems

An inclusion of observable nets satisfying duality induces an inclusion of canonical field nets. Any Bose net intermediate between the observable net and the field net and satisfying duality is the fixed-point net of the field net under a compact group. This compact group is its canonical gauge group if the occurrence of sectors with infinite statistics can be ruled out for the observable net and its vacuum Hilbert space is separable.

Non-standard Neutral Kaon Dynamics from Infinite Statistics

The neutral kaon system can be effectively described by non-unitary, dissipative, completely positive dynamics that extend the usual treatment. In the framework of open quantum systems, we show how the origin of these non-standard time evolutions can be traced to the interaction of the kaon system with a large environment. We find that a heat bath of quanta obeying infnite statistics could constitute a realistic example of such an environment.

Generalized Fock spaces, new forms of quantum statistics and their algebras

We formulate a theory of generalized Fock spaces which underlies the different forms of quantum statistics such as ``infinite'', Bose-Einstein and Fermi-Dirac statistics. Single-indexed systems as well as multi-indexed systems that cannot be mapped into single-indexed systems are studied. Our theory is based on a three-tiered structure consisting of Fock space, statistics and algebra. This general formalism not only unifies the various forms of statistics and algebras, but also allows us to construct many new forms of quantum statistics as well as many algebras of creation and destruction operators. Some of these are : new algebras for infinite statistics, q-statistics and its many avatars, a consistent algebra for fractional statistics, null statistics or statistics of frozen order, ``doubly-infinite'' statistics, many representations of orthostatistics, Hubbard statistics and its variations.

NEW INCARNATIONS OF INFINITE STATISTICS

In this paper, we present two new realizations of infinite statistics and construct their respective number operators in a simple manner. One of the realizations assumes the individual oscillators to be bosonic but the mutually commutation relationship (CR) to be of the quonic form. Interestingly, the system of mutually commuting oscillators satisfying individual relations of the type [Formula: see text] (no sum over i) is shown to describe a collection of bosons. This demonstrates that for these examples the mutual CRs between the oscillators decide the statistics.

On dynamics of pseudorapidity fluctuations in central CCu collisions at 4.5 A GeV/ c

Results on dynamical fluctuations of charged particles in the pseudorapidity space of central CCu interactions at 4.5 A GeV/ c are performed in the transformed variables and using higher order scaled factorial moments modified to remove the bias of infinite statistics in the normalization. The intermittency behavior is found up to eighth order of the moments increasing with the order and leading to the pronounced multifractality. Two differed intermittent-like rises are obtained, one indicating an occurrence of the non-thermal phase transition, and no critical behavior is found to be reached in another case. The observations may be treated to show different regimes of particle production during the cascade. Comparison with some conventional model approximations notes the multiparticle character of the fluctuations. The results presented can be effective in sense of sensitivity of intermittency to the hadronization phase.

Remarks on the number operator for generalized quons

We construct the number operator for particles obeying infinite statistics, defined by a generalized q-deformation of the Heisenberg algebra, and prove the positivity of the norm of linearly independent state vectors.

A simulation of emission and transmission noise propagation in cardiac SPECT imaging with nonuniform attenuation correction

Transmission computed tomography provides information needed for nonuniform attenuation correction of cardiac single photon emission computed tomography (SPECT). Nonuniform attenuation correction is accomplished using an iterative ML-EM algorithm and a projection-backprojection operation that incorporates attenuation factors measured from the reconstructed transmission map. The precision and accuracy of the attenuation corrected emission reconstruction is a function of emission and transmission statistics. This paper presents an error propagation analysis that uses a mathematical cardiac chest phantom to simulate various combinations of total emission counts C and transmission flux I0 under ideal imaging conditions (without geometric response distortion and without scatter). The spatial average, spatial variance, and accuracy measures for a 4 x 4 pixel region in the heart are tabulated after 30 iterations of the ML-EM algorithm. The confidence intervals for these measures were determined from 1000 realizations of reconstructions from projections randomly generated with the same transmission and emission statistics. It can be shown empirically from the simulation results that the spatial %rms uncertainty for the simulated cardiac region has a simple expression: %rms2 = K1/C+K2/I0(2)+B2 where K1 and K2 are least-square estimates based on the simulation results, and B is the measured spatial %rms uncertainty for the simulation at infinite statistics. For a transmission incident flux of 1500 events per projection bin of 0.712 cm and typical clinical emission events totaling 1 x 10(5), the spatial %rms uncertainty is approximately 14%. At clinical transmission and emission statistics, the statistical noise in the simulated attenuation-corrected reconstructions are dominated by the emission statistics.

“Infinite statistics” and its two-parameter q-deformation

We generalize Greenberg's q-mutator algebra for infinite statistics to the most general form invariant under unitary transformations in the indices and study its consequences. Constraints on the parameter space arising from the positivity of the metric in the underlying Fock space are obtained.