Parafermion theory from the q-deformed algebra with q a root of unity

2020 ◽  
Vol 17 (03) ◽  
pp. 2050045
Author(s):  
Won Sang Chung ◽  
Hassan Hassanabadi
Keyword(s):  
Partition Function ◽  
Specific Heat ◽  
Set Theory ◽  
Fock Space ◽  
Infinite Dimensional ◽  
Root Of Unity ◽  
Deformed Algebra ◽  
Bosonic Case ◽  
Fermionic Case ◽  
Bosonic Limit

In this paper, we find the [Formula: see text]-deformed algebra with the finite- and infinite-dimensional Fock space and both the fermionic limit and the bosonic limit. Using the cardinality of set theory, we propose the Hamiltonian interpolating bosonic case and fermionic case, which enables us to construct the proper partition function and internal energy. As examples, we discuss the specific heat of free [Formula: see text] parafermion gas model and [Formula: see text] parafermion star.

scholarly journals Semi-device-independent framework based on natural physical assumptions

Quantum ◽  
2017 ◽  
Vol 1 ◽  
pp. 33 ◽  
Author(s):  
Thomas Van Himbeeck ◽  
Erik Woodhead ◽  
Nicolas J. Cerf ◽  
Raúl García-Patrón ◽  
Stefano Pironio
Keyword(s):  
Fock Space ◽  
Outcome Measurement ◽  
Photon Number ◽  
Quantum Correlations ◽  
Mean Value ◽  
Laser Source ◽  
Optical Scheme ◽  
Infinite Dimensional ◽  
Energy Constrained ◽  
Quantum Optical

The semi-device-independent approach provides a framework for prepare-and-measure quantum protocols using devices whose behavior must not be characterized nor trusted, except for a single assumption on the dimension of the Hilbert space characterizing the quantum carriers. Here, we propose instead to constrain the quantum carriers through a bound on the mean value of a well-chosen observable. This modified assumption is physically better motivated than a dimension bound and closer to the description of actual experiments. In particular, we consider quantum optical schemes where the source emits quantum states described in an infinite-dimensional Fock space and model our assumption as an upper bound on the average photon number in the emitted states. We characterize the set of correlations that may be exhibited in the simplest possible scenario compatible with our new framework, based on two energy-constrained state preparations and a two-outcome measurement. Interestingly, we uncover the existence of quantum correlations exceeding the set of classical correlations that can be produced by devices behaving in a purely pre-determined fashion (possibly including shared randomness). This feature suggests immediate applications to certified randomness generation. Along this line, we analyze the achievable correlations in several prepare-and-measure optical schemes with a mean photon number constraint and demonstrate that they allow for the generation of certified randomness. Our simplest optical scheme works by the on-off keying of an attenuated laser source followed by photocounting. It opens the path to more sophisticated energy-constrained semi-device-independent quantum cryptography protocols, such as quantum key distribution.

scholarly journals White noise delta functions and continuous version theorem

1993 ◽  
Vol 129 ◽  
pp. 1-22
Author(s):  
Nobuaki Obata
Keyword(s):  
White Noise ◽  
Harmonic Analysis ◽  
Quantum Physics ◽  
Fock Space ◽  
Continuous Version ◽  
Infinite Dimensional ◽  
Schwartz Distribution ◽  
Delta Functions ◽  
Gelfand Triple ◽  
Gaussian Space

The recently developed Hida calculus of white noise [5] is an infinite dimensional analogue of Schwartz’ distribution theory besed on the Gelfand triple (E) ⊂ (L2) = L2 (E*, μ) ⊂ (E)*, where (E*, μ) is Gaussian space and (L2) is (a realization of) Fock space. It has been so far discussed aiming at an application to quantum physics, for instance [1], [3], and infinite dimensional harmonic analysis [7], [8], [13], [14], [15].

scholarly journals A Special Class of Infinite Dimensional Dirac Operators on the Abstract Boson-Fermion Fock Space

2014 ◽  
Vol 2014 ◽  
pp. 1-13
Author(s):  
Asao Arai
Keyword(s):  
Dirac Operator ◽  
Hilbert Spaces ◽  
Special Class ◽  
Spectral Properties ◽  
Fock Space ◽  
Dirac Operators ◽  
Fredholm Property ◽  
Perturbation Parameter ◽  
Coupling Constant ◽  
Infinite Dimensional

Spectral properties of a special class of infinite dimensional Dirac operatorsQ(α)on the abstract boson-fermion Fock spaceℱ(ℋ,𝒦)associated with the pair(ℋ,𝒦)of complex Hilbert spaces are investigated, whereα∈Cis a perturbation parameter (a coupling constant in the context of physics) and the unperturbed operatorQ(0)is taken to be a free infinite dimensional Dirac operator. A variety of the kernel ofQ(α)is shown. It is proved that there are cases where, for all sufficiently large|α|withα<0,Q(α)has infinitely many nonzero eigenvalues even ifQ(0)has no nonzero eigenvalues. Also Fredholm property ofQ(α)restricted to a subspace ofℱ(ℋ,𝒦)is discussed.

scholarly journals κ-DEFORMED STATISTICS AND CLASSICAL FOUR-MOMENTUM ADDITION LAW

2008 ◽  
Vol 23 (09) ◽  
pp. 653-665 ◽  
Author(s):  
MARCIN DASZKIEWICZ ◽  
JERZY LUKIERSKI ◽  
MARIUSZ WORONOWICZ
Keyword(s):  
Minkowski Space ◽  
Fock Space ◽  
Scalar Fields ◽  
Cross Product ◽  
Creation And Annihilation Operators ◽  
Multiplication Rule ◽  
Symmetry Properties ◽  
Deformed Algebra ◽  
Free Scalar

We consider κ-deformed relativistic symmetries described algebraically by modified Majid–Ruegg bi-cross-product basis and investigate the quantization of field oscillators for the κ-deformed free scalar fields on κ-Minkowski space. By modification of standard multiplication rule, we postulate the κ-deformed algebra of bosonic creation and annihilation operators. Our algebra permits one to define the n-particle states with classical addition law for the four-momentum in a way which is not in contradiction with the nonsymmetric quantum four-momentum co-product. We introduce κ-deformed Fock space generated by our κ-deformed oscillators which satisfy the standard algebraic relations with modified κ-multiplication rule. We show that such a κ-deformed bosonic Fock space is endowed with the conventional bosonic symmetry properties. Finally we discuss the role of κ-deformed algebra of oscillators in field-theoretic noncommutative framework.

scholarly journals Functional Representations for Fock Superalgebras

1998 ◽  
Vol 01 (02) ◽  
pp. 285-324 ◽  
Author(s):  
Joachim Kupsch ◽  
Oleg G. Smolyanov
Keyword(s):  
Reproducing Kernel ◽  
Fock Space ◽  
Function Spaces ◽  
Graded Algebras ◽  
Fock Representation ◽  
Infinite Dimensional ◽  
Gaussian Integration ◽  
Classical Function ◽  
Functional Representations ◽  
Wick Ordering

The Fock space of bosons and fermions and its underlying superalgebra are represented by algebras of functions on a superspace. We define Gaussian integration on infinite-dimensional superspaces, and construct super-analogs of the classical function spaces with a reproducing kernel — including the Bargmann–Fock representation — and of the Wiener–Segal representation. The latter representation requires the investigation of Wick ordering on Z2-graded algebras. As application we derive a Mehler formula for the Ornstein–Uhlenbeck semigroup on the Fock space.

scholarly journals Hardy-Type Space Associated with an Infinite-Dimensional Unitary Matrix Group

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Oleh Lopushansky
Keyword(s):  
Fock Space ◽  
Matrix Group ◽  
Unitary Matrix ◽  
Type Space ◽  
Infinite Dimensional ◽  
System A ◽  
Hardy Type ◽  
Complex Functions ◽  
Hardy Type Space ◽  
The Right

We investigate an orthogonal system of the homogenous Hilbert-Schmidt polynomials with respect to a probability measure which is invariant under the right action of an infinite-dimensional unitary matrix group. With the help of this system, a corresponding Hardy-type space of square-integrable complex functions is described. An antilinear isomorphism between the Hardy-type space and an associated symmetric Fock space is established.

THERMODYNAMICS OF q-MODIFIED BOSONS

1996 ◽  
Vol 10 (06) ◽  
pp. 683-699 ◽  
Author(s):  
P. NARAYANA SWAMY
Keyword(s):  
Mechanical Properties ◽  
Phase Transition ◽  
Equation Of State ◽  
Partition Function ◽  
Statistical Mechanics ◽  
Specific Heat ◽  
Thermodynamic Functions ◽  
Bose Condensation ◽  
Grand Partition Function ◽  
Statistical Mechanical

Based on a recent study of the statistical mechanical properties of the q-modified boson oscillators, we develop the statistical mechanics of the q-modified boson gas, in particular the Grand Partition Function. We derive the various thermodynamic functions for the q-boson gas including the entropy, pressure and specific heat. We demonstrate that the gas exhibits a phase transition analogous to ordinary bose condensation. We derive the equation of state and develop the virial expansion for the equation of state. Several interesting properties of the q-boson gas are derived and compared with those of the ordinary boson which may point to the physical relevance of such systems.

scholarly journals Entanglement in fermionic chains and bispectrality

2021 ◽  
pp. 2140001
Author(s):  
Nicolas Crampé ◽  
Rafael I. Nepomechie ◽  
Luc Vinet
Keyword(s):  
Signal Processing ◽  
Lie Algebras ◽  
Tridiagonal Matrix ◽  
Root Of Unity ◽  
Deformed Algebra ◽  
Bispectral Problem

Entanglement in finite and semi-infinite free Fermionic chains is studied. A parallel is drawn with the analysis of time and band limiting in signal processing. It is shown that a tridiagonal matrix commuting with the entanglement Hamiltonian can be found using the algebraic Heun operator construct in instances when there is an underlying bispectral problem. Cases corresponding to the Lie algebras [Formula: see text] and [Formula: see text] as well as to the q-deformed algebra [Formula: see text] at [Formula: see text] a root of unity are presented.

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