Q-deformed Fermion Research Articles

BEHAVIOR OF q-DEFORMED QUANTUM PARTICLE STATISTICS ON THE HOLOGRAPHIC SCREEN

In this study, we propose an approach to investigate the relationship between MOND theory and the holographic principle by incorporating q-deformed theory. We first present a brief overview of Verlinde's entropic gravity assumption, which suggests that gravity can be interpreted as an entropic force arising from the statistical mechanics of quantum fields. Some thermo-statistical properties of q-deformed fermion gas model in two spatial dimensions are introduced. At the low-temperature limit, we derive the q-deformed thermal energy and analyze the impacts of fermionic q-deformation on MOND theory. Specifically, we consider the q-deformed Fermi-Dirac statistics of the bits on the holographic screen and examine MOND theory depending on q-deformed acceleration scale. Deformed Friedmann equation is studied by taking into account Friedmann–Robertson–Walker (FRW) universe. This equation shows a modified identification of the evolution of the universe that is compatible with both MOND theory and the holographic principle

Effects of bosonic and fermionic q-deformation on the entropic gravity

In this paper, we study thermodynamical contributions to the theory of gravity under the q-deformed boson and fermion gas models. According to Verlinde’s proposal, the law of gravity is not based on a fundamental interaction but it emerges as an entropic force from the changes of entropy associated with the information on the holographic screen. In addition, Strominger shows that the extremal quantum black holes obey neither boson nor fermion statistics, but they obey deformed statistic. Using these notions, we find q-deformed entropy and temperature functions. We also present the contributions that come from the q-deformed model to the Poisson equation, Newton’s law of gravity and Einstein’s field equations.

Truncated q-deformed fermion algebras and phase transition

In this paper, we apply the q-deformed fermion theory to the phase transition from the ordinary fermion into the truncated q-deformed fermion at the critical temperature $$ T_\mathrm{c}$$ .

Q-deformed Einstein equations from entropic force

In this study, we investigate the influences of fermionic q-deformation on the Einstein equations by taking into account of Verlinde’s entropic gravity approach and Strominger’s proposal on quantum black holes. According to Verlinde’s proposal, gravity is interpreted as an entropic force. Moreover, Strominger’s suggestion claims that extremal black holes obey deformed statistics instead of the standard Bose or Fermi statistics. Inspired by Verlinde’s and Strominger’s suggestions, we represent some thermostatistical functions of VPJC-type q-deformed fermion gas model for the high-temperature limit. Applying the Verlinde’s entropic gravity approach to the q-deformed entropy function, q-deformed Einstein equations with the effective cosmological constant are derived. The results obtained in this work are compared with the related works in the literature.

Q-deformed fermion in many-particle systems and its application to BCS theory

In recent decades, there have been increasing interests in quantum statistics beyond the standard Fermi–Dirac and Bose–Einstein statistics, such as the fractional statistics, quon statistics, anyon statistics and quantum groups, since they can provide new insights into the cosmology, nuclear physics and condensed matter. In this paper, we study the many-particle system formed by the q-deformed fermions (q-fermion), which is realized by deforming the quantum algebra of the anticommutation relations. From the standard perspective of the finite temperature field theory, we first construct the finite temperature Green’s function formalism for the free many-q-fermion system, then generalize it to the well known interacting fermionic system, the superconductor, and finally obtain a consistent q-deformed BCS (qBCS) theory. Interesting features can be implied from this model. For example, it predicts a Sarma-like ordered phase, which we call the q-deformed Sarma phase, at low temperature. It also presents a symmetric phase diagram in the parameter space and modified thermodynamic relations.

Thermosize effects in a q-deformed fermion gas model

We study the the high-temperature thermodynamic properties of the q-deformed fermion gas model by taking into account of the size effect of the gas particles. Starting from the logarithm of the grand partition function of the model, we calculate several thermodynamic functions of the model such as internal energy, entropy, and Helmholtz free energy by means of the deformation parameter q. Furthermore, the influences of the fermionic q-deformation on the thermosize effect in the confined deformed quantum gas systems such as the Seebeck-like and Peltier-like thermosize effects are discussed. Especially, we focus on the absorbed or released heat of the model in the Peltier-like thermosize effect. In the light of the results obtained in this work, we can conclude that the present q-deformed fermion model can be used to desing the new types of the micro-/nano-scaled quantum heat engines.

Fluctuations of a q-deformed fermion gas

The theory of q-deformed fermions is one of the theories of q-deformed oscillators. Within the framework of this theory and the traditional fluctuation theory, we investigate fluctuations of q-deformed fermion gas and obtain the expressions of fluctuations of the internal energy U, the particle number N and the correlation of fluctuations of the two physical quantities above. Further numerical calculation reveals that fluctuations of such a system have some interesting and particular features. We consider that this work may provide much insight into the theory of q fermions, and may also be helpful for the theory of q-deformed oscillators.

General thermostatistical properties of a q-deformed fermion gas in two dimensions

Starting with a deformed fermionic grand partition function, we study the high and low temperature thermostatistical properties of a special q-deformed fermion gas in two spatial dimensions. Many of the deformed thermostatistical functions such as the specific heat and the entropy are derived in terms of the real deformation parameter q for the range q < 1. For high temperatures, we specifically focus on the behavior of both the entropy function and the deformed virial coefficients in the equation of state for the q-fermion gas in two dimensions. Possible physical applications of the present q-fermion gas are briefly discussed.

Golden quantum oscillator and Binet–Fibonacci calculus

The Binet formula for Fibonacci numbers is treated as a q-number and a q-operator with Golden ratio bases q = φ and Q = −1/φ, and the corresponding Fibonacci or Golden calculus is developed. A quantum harmonic oscillator for this Golden calculus is derived so that its spectrum is given only by Fibonacci numbers. The ratio of successive energy levels is found to be the Golden sequence, and for asymptotic states in the limit n → ∞ it appears as the Golden ratio. We call this oscillator the Golden oscillator. Using double Golden bosons, the Golden angular momentum and its representation in terms of Fibonacci numbers and the Golden ratio are derived. Relations of Fibonacci calculus with a q-deformed fermion oscillator and entangled N-qubit states are indicated.

Thermodynamic geometry of deformed bosons and fermions

We construct the thermodynamic geometry of an ideal q-deformed boson and fermion gas. We investigate some thermodynamic properties such as the stability and statistical interaction. It will be shown that the statistical interaction of q-deformed boson gas is attractive, while it is repulsive for the q-deformed fermion one. Also, we will consider the singular point of the thermodynamic curvature to obtain some new results about the condensation of q-deformed bosons and show that there exists a finite critical phase transition temperature even in low dimensions. It is shown that the thermodynamic curvature of q-deformed boson and fermion quantum gases diverges as a power-law function with respect to temperature at zero-temperature limit.

Quasibosons composed of two q-fermions: realization by deformed oscillators

Composite bosons, here called quasibosons (e.g. mesons, excitons, etc), occur in various physical situations. Quasibosons differ from bosons or fermions as their creation and annihilation operators obey non-standard commutation relations, even for the ‘fermion+fermion’ composites. Our aim is to realize the operator algebra of quasibosons composed of two fermions or two q-fermions (q-deformed fermions) by the respective operators of deformed oscillators, the widely studied objects. For this, the restrictions on quasiboson creation/annihilation operators and on the deformed oscillator (deformed boson) algebra are obtained. Their resolving proves the uniqueness of the family of deformations and gives explicitly the deformation structure function (DSF) which provides the desired realization. In the case of two fermions as constituents, such realization is achieved when the DSF is a quadratic polynomial in the number operator. In the case of two q-fermions, q ≠ 1, the obtained DSF inherits the parameter q and does not continuously converge when q → 1 to the DSF of the first case.

A Comparative Study on q-Deformed Fermion Oscillators

In this paper, the algebras, representations, and thermostatistics of four types of fermionic q-oscillator models, called fermionic Newton (FN), Chaichian-Kulish-Ng (CKN), Parthasarathy-Viswanathan-Chaichian (PVC), Viswanathan-Parthasarathy-Jagannathan-Chaichian (VPJC), are discussed. Similarities and differences among the properties of these models are revealed. Particular emphasis is given to the VPJC-oscillators model so that its Fock space representation is analyzed in detail. Possible physical applications of these models are concisely pointed out.

Thermostatistics of Deformed Bosons and Fermions

Based on the q-deformed oscillator algebra, we study the behavior of the mean occupation number and its analogies with intermediate statistics and we obtain an expression in terms of an infinite continued fraction, thus clarifying successive approximations. In this framework, we study the thermostatistics of q-deformed bosons and fermions and show that thermodynamics can be built on the formalism of q-calculus. The entire structure of thermodynamics is preserved if ordinary derivatives are replaced by the use of an appropriate Jackson derivative and q-integral. Moreover, we derive the most important thermodynamic functions and we study the q-boson and q-fermion ideal gas in the thermodynamic limit.

DEFORMED QUANTUM STATISTICS IN TWO DIMENSIONS

It is known from the early work of May in 1964 that ideal Bose gas do not exhibit condensation phenomenon in two dimensions. On the other hand, it is also known that the thermostatistics arising from q-deformed oscillator algebra has no connection with the spatial dimensions of the system. Our recent work concerns the study of important thermodynamic functions such as the entropy, occupation number, internal energy and specific heat in ordinary three spatial dimensions, where we established that such thermostatistics is developed by consistently replacing the ordinary thermodynamic derivatives by the Jackson derivatives. The thermostatistics of q-deformed bosons and fermions in two spatial dimensions is an unresolved question that is the subject of this investigation. We study the principal thermodynamic functions of both bosons and fermions in the two-dimensional q-deformed formalism and we find that, different from the standard case, the specific heat of q-boson and q-fermion ideal gas, at fixed temperature and number of particles, are no longer identical.

Q-DEFORMED FERMIONS: ALGEBRA, FOCK SPACE AND THERMODYNAMICS

We investigate an algebra describing q-deformed Fermion oscillators. We establish the nature of the basic numbers which follow from this algebra, study the Fock space of these generalized Fermions and determine the Jackson Derivative appropriate for these Fermions. The statistical thermodynamics of these Fermions is examined and it is demonstrated that a consistent formulation preserving its full Legendre structure is accomplished by the use of Jackson Derivatives in place of the ordinary thermodynamic derivatives. We determine the important thermodynamic functions and illustrate the thermodynamic properties of q-Fermions.

REPLY TO THE COMMENT ON "NONCOMMUTATIVITY AS A POSSIBLE ORIGIN OF THE ULTRAHIGH-ENERGY COSMIC RAY AND THE TEV PHOTON PARADOXES"

A cogitative comment on Ref. 1 was given by the author of Ref. 2 arguing that the Lorentz-noninvariant modification of the kinematic dispersion law does not follow from deformed oscillator systems, and it cannot evade ultrahigh energy threshold anomalies by raising the respective thresholds. It is pointed out that the dispersion relations of a q-deformed boson and fermion are derived heedlessly and further study on noncommutative theories is necessary. However if such dispersion relations are assumed tenable, the tachyonic propagation and photon instability do not exist at all because the deviation from 1 in q must be imaginary.

Deformed Fermion Realization of sp (4) and Its Reductions

A specific q-deformation of the compact symplectic sp(4) algebra, one that is suitable for nuclear physics applications, is realized in terms of q-deformed fermion creation and annihilation operators of the shell-model. The generators of the algebra close on four distinct realizations of the u q (2) subalgebra. These reductions, which correspond to different pairing interactions, yield a complete classification of the basis states. An analysis of the role of the q-deformation is based on a comparison of the results for energies of the lowest isovector-paired 0+ states in the deformed and non-deformed cases.

Q-DEFORMED W-ALGEBRA FROM q-FERMION OSCILLATORS AND THE ALGEBRAIC STRUCTURE OF VERTEX OPERATORS IN 2D STRING THEORY

q-deformed fermion oscillators are used to construct q-deformed higher order Virasaro algebra in the Fock space representation. These facilitate the realization of q-deformed W∞-algebra. The vertex operators (positively dressed) in 2D string theory are interpreted in terms of q-fermionic states.

Q-Deformed structures and nonextensive statistics: a comparative study

The thermostatistics of q-deformed bosons and fermions can be built on the formalism of q-calculus. As in nonextensive statistics, the entire structure of thermodynamics is preserved but there are crucial differences between the two theories. We derive the most important thermodynamic functions of q-deformed fermions and bosons. We also study the semiclassical limit and the low-temperature behavior and point out the similarities and differences with the results of nonextensive statistics.

Generalized thermodynamics of q-deformed bosons and fermions.

We study the thermostatistics of q-deformed bosons and fermions obeying the symmetric (q<-->q(-1)) algebra and show that it can be built on the formalism of q calculus. The entire structure of thermodynamics is preserved if ordinary derivatives are replaced by an appropriate Jackson derivative. In this framework, we derive the most important thermodynamic functions describing the q-boson and q-fermion ideal gases in the thermodynamic limit. We also investigate the semiclassical limit and the low-temperature regime and demonstrate that the nature of the q deformation gives rise to pure quantum statistical effects stronger than undeformed boson and fermion particles.