Index of subfactors and statistics of quantum fields. I

We identify the statistical dimension of a superselection sector in a local quantum field theory with the square root of the index of a localized endomorphism of the quasi-local C^*-algebra that represents the sector. As a consequence in a two-dimensional theory the possible values of the statistical dimension below 2 are restricted to a given discrete set.Supported in part by Ministero della Pubblica Istruzione and CNR-GNAFA     

Fock representations of quantum fields with generalized statistics

We develop a rigorous framework for constructing Fock representations of quantum fields obeying generalized statistics. The main features of these representations are investigated. Various aspects of the underlying mathematical structure are illustrated by means of explicit examples.     

Approximate approach to the quantum statistics of the superposition of coherent and chaotic fields

Formulae recently derived for the integrated intensity distribution, the photon-counting distribution and its factorial moments in the statistics of the superposition of multimode coherent and chaotic fields are analyzed in greater detail and their validity as approximate formulae for light of arbitrary spectrum is investigated. It is shown by explicit calculation of the third factorial moment of the photon-counting distribution for the superposition of a one-mode coherent field with a Gaussian Lorentzian field that the proposed formulae hold with very good accuracy over a wide range of conditions.The authors thank Dr. Z. Braunerová and M T. Kojecký of the Computer Center of Palacký University for their help with calculations.     

A transition in the spectral statistics of quantum optical model by different electromagnetic fields

In this paper, we have considered the effects of different quantized electromagnetic fields on the spectral statistics of two-level atoms. The Berry-Robnik distribution and the maximum likelihood estimation technique are used to analyze the effect of the mean photon numbers, the two level atoms numbers and also the quantum number of considered states on the fluctuation properties of different systems which are described by different sets of the Dicke Hamiltonian’s parameters. Our results describe the obvious effect of mean photon number on the spectral statistics and show more regular dynamics when this quantity reaches 700. Also, we observed universality in the spectral statistics of considered systems when the number of two level atoms approaches an unrealistic limit (N _A ~ 200) and there are some suggestions about the effect of the quantum number of selected levels and the atom-field coupling constant on level statistics.     

Spin and statistics of quantum kinks

Apart from some brief and inconclusive remarks concerning the problem of spin and statistics of quantum kinks in space-time dimension D > 2, we give a detailed discussion of the D = 2 situation. Our main results is that two-dimensional quantum kinks are statistical “schizons”; they exist in the same Hilbert space either as bosons or as fermions. In those cases where one can introduce local kink-sector generating operators as in the sine-Gordon model, the Bose and Fermi fields are strictly local fields, which are relatively non-local with respect to each other.     

Renormalizations in quantum statistics

A simple proof is given that quantum statistics is multiplicatively renormalizable (if the corresponding quantum field theory is), that the renormalization constants are independent of the temperature and the parameters in the theory with the dimensions of mass, and that the renormalization constants (of the counterterms) are equal to the renormalization constants of the corresponding quantum field theory.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 26–30, January, 1981.I should like to thank B. L. Voronov, O. K. Kalashnikov, A. D. Linde, E. S. Fradkin, and A. E. Shabad for helpful discussions.     

Generalized statistics of macroscopic fields

We define the notion of generalized statistics and give some examples. In particular, we consider the relationsa _i a _j ^* -q _ij a _j ^* a _i = _ij for - 1 q _ij =q _ji + 1 and we prove the existence of a Fock space representation of these relations.This work has been supported by the Deutsche Forschungsgemeinschaft (SFB 123).     

New forms of quantum statistics

We propose a new two-parameter deformation of the algebra of creation and destruction operators, which allows the construction of a new family of Hillbert spaces with positive definite inner product. This provides a continuous interpolation between two new forms of statistics named orthofermi and orthobose statistics. Positivity of the inner product over the two-parameter region is discussed.     

Boltzmann statistics of quantum friction

In this paper we investigate the thermodynamic properties of simple quantum-mechanical systems in the presence of friction. Using the propagators for these simple models we calculate the response functions in Boltzmann statics. In the low temperature region the response functions exhibit singular behaviour.     

Lie algebraic aspects of quantum statistics. parafermi statistics

We study in detail the Lie-algebraical properties of the quantization condition for spinor fields

$\text{[a}{}^{\mathrm{+}}{}_{\mathrm{i}}\text{,a}{}^{\mathrm{?}}{}_{\mathrm{j}}\text{],a}{}^{\mathrm{\pm }}{}_{\mathrm{i}}\text{= ±2δ}{}_{\mathrm{ij}}\text{a}{}^{\mathrm{\pm }}{}_{\mathrm{j}}$

.It turns out that the parafermi statistics is one particular solution of this relation: it is the minimal rank simple Lie algebra generated by the operators a^±_i entering into the above relation. We point out some other solutions.     

On fermi quantum fields constructed from bose quantum fields and their applications

In former papers a representation of the quantum Fermi and para-Fermi fields was proposed. This representation is such that the only basic quantum entities are Bose quantum fields. In this paper we show several possibilities of application: (i) to lower the number of elementary particles; (ii) to describe as separate states of a fundamental particle other particles that presently are considered as different, and to induce an ordering among them; (iii) to obtain relations among the quantum numbers of those particles; (iv) to obtain a physical picture of some unstable particles. This article is concerned with the physical interpretation of the formalism, and some of the statements that are contained here have a conjectural character.     

Mathematical and physical foundations of quantum statistics

It is shown that the peculiar properties of quantum statistics can be explained as a consequence of the destroyed isotropy of the subquantum space.     

The quantum geometry of spin and statistics

Both, spin and statistics of a quantum system can be seen to arise from underlying (quantum) group symmetries. We show that the spin–statistics theorem is equivalent to a unification of these symmetries. Besides covering the Bose–Fermi case we classify the corresponding possibilities for anyonic spin and statistics. We incorporate the underlying extended concept of symmetry into quantum field theory in a generalised path integral formulation capable of handling general braid statistics. For bosons and fermions the different path integrals and Feynman rules naturally emerge without introducing Grassmann variables. We also consider the anyonic example of quons and obtain the path integral counterpart to the usual canonical approach.     

Search for elementary quantum statistics

Bose-Einstein and Fermi-Dirac are the main quantum statistics. Therefore, it is likely that if truly elementary building blocks of Nature exist, they are either bosons or fermions, so that it is also likely that one, and only one, of the following possibilities, concerning those elementary building blocks, is correct: (i) all of them are fermions; (ii) some of them are bosons, others fermions; (iii) all of them are bosons; (iv) the distinction between these cases is methodological, not physical. Since tensors can be constructed from spinors, most physicists support one of the first two points of view. However, by starting from the fact that now it is known that bosonization makes sense, and developing a former research by Penney, we defend the point of view that, at least in a finite model of the Universe, the third point of view is the more likely. To avoid confusion we state that we are not concerned with the whole set of the so-called “elementary particles” since most physicists believe by now that, e.g., hadrons are built from quarks, nor concerned with quarks since many physicists suspect they are also composite objects. This research concerns the true elementary building blocks of Nature, assuming that such set exists, whatever those building blocks are. Finally, we extend this research to general finite associative algebras, enlarging the physical applicability of our point of view concerning the role of bosons in Nature.     

Produced entropy in quantum statistics

Hg-films were condensed with admixtures of HgBr₂ or HgCl₂ onto a substrate at 4.2 °K. The superconducting transition temperature is measured as a function of the impurity concentration. The transition temperature increases from 3.96 °K of the pure quenched film to 4.06 °K at a content of 0.3 mol.% HgBr₂. Higher impurity concentrations lower the transition temperature and superconductivity vanishes at about 16 mol.% HgBr₂ or 23 mol.% HgCl₂ respectively. Above these concentrations the electrical resistivity shows a negative temperature coefficient. By means of Debye-Scherrer diagrams there is found a diminishing of the crystallite size with increasing impurity concentration.     

Entanglement concentration using quantum statistics

We propose an entanglement concentration scheme which uses only the effects of quantum statistics of indistinguishable particles. This establishes the fact that useful quantum information processing can be accomplished by quantum statistics alone. Because of the basis independence of statistical effects, our protocol requires less knowledge of the initial state than most entanglement concentration schemes. Moreover, no explicit controlled operation is required at any stage.     

Eigenfunction statistics on quantum graphs

We investigate the spatial statistics of the energy eigenfunctions on large quantum graphs. It has previously been conjectured that these should be described by a Gaussian Random Wave Model, by analogy with quantum chaotic systems, for which such a model was proposed by Berry in 1977. The autocorrelation functions we calculate for an individual quantum graph exhibit a universal component, which completely determines a Gaussian Random Wave Model, and a system-dependent deviation. This deviation depends on the graph only through its underlying classical dynamics. Classical criteria for quantum universality to be met asymptotically in the large graph limit (i.e. for the non-universal deviation to vanish) are then extracted. We use an exact field theoretic expression in terms of a variant of a supersymmetric σ model. A saddle-point analysis of this expression leads to the estimates. In particular, intensity correlations are used to discuss the possible equidistribution of the energy eigenfunctions in the large graph limit. When equidistribution is asymptotically realized, our theory predicts a rate of convergence that is a significant refinement of previous estimates. The universal and system-dependent components of intensity correlation functions are recovered by means of an exact trace formula which we analyse in the diagonal approximation, drawing in this way a parallel between the field theory and semiclassics. Our results provide the first instance where an asymptotic Gaussian Random Wave Model has been established microscopically for eigenfunctions in a system with no disorder.