FOCK SPACE FOR GENERALIZED STATISTICS AND BOSON-FERMION SUPERSELECTION RULE

1996 ◽  
Vol 10 (21) ◽  
pp. 1035-1041
Author(s):  
L. DE FALCO ◽  
R. MIGNANI ◽  
R. SCIPIONI
Keyword(s):  
Quantum Gravity ◽  
Density Matrix ◽  
Tensor Product ◽  
Fock Space ◽  
Superselection Rule ◽  
Expectation Values ◽  
Ladder Operators ◽  
Von Neumann ◽  
Von Neumann Equation ◽  
Natural Way

We introduce a generalized Fock space for a recently proposed operatorial deformation of the Heisenberg-Weyl (HW) algebra, aimed at describing statistics different from the Bose or Fermi ones. The new Fock space is obtained by the tensor product of the usual Fock space and the space spanned by the eigenstates of the deformation operator ĝ. We prove a “statistical Ehrenfest-like theorem”, stating that the expectation values of the ladder operators of the generalized HW algebra — taken in the ĝ-subspace — are creation and annihilation operators defined in the usual Fock space and obeying the ordinary statistics, according to the ĝ-eigenvalues. Moreover, such a “statistics” operator ĝ can be regarded as the generator of a boson-fermion superselection rule. As a consequence, the generalized Fock space decomposes into incoherent sectors, and therefore one gets a density matrix diagonal in the ĝ eigenstates. This leads, under suitable conditions, to the possibility of continuously interpolating between different statistics. In particular, it is necessary to assume a nonstandard Liouville-Von Neumann equation for the density matrix, of the type already considered e.g. in the framework of quantum gravity. It is also preliminarily shown that our formalism leads in a natural way — due to the very properties of the operator ĝ — to a grading of the HW algebra, and therefore to a supersymmetrical scheme.

MONOTONIC INDEPENDENCE ON THE WEAKLY MONOTONE FOCK SPACE AND RELATED POISSON TYPE THEOREM

2005 ◽  
Vol 08 (02) ◽  
pp. 259-275 ◽  
Author(s):  
JANUSZ WYSOCZAŃSKI
Keyword(s):  
Limit Theorem ◽  
Fock Space ◽  
Type Theorem ◽  
Random Variables ◽  
Direct Proof ◽  
Binomial Coefficients ◽  
Second Quantization ◽  
The Central Limit Theorem ◽  
Natural Way ◽  
Poisson Type

We show how the construction of t-transformation can be applied to the construction of a sequence of monotonically independent noncommutative random variables. We introduce the weakly monotone Fock space, on which these operators act. This space can be derived in a natural way from the papers by Pusz and Woronowicz on twisted second quantization. It was observed by Bożejko that, by taking μ = 0, for the μ-CAR relations one obtains the Muraki's monotone Fock space, while for the μ-CCR relations one obtains the weakly monotone Fock space. We show that the direct proof of the central limit theorem for these operators provides an interesting recurrence for the highest binomial coefficients. Moreover, we show the Poisson type theorem for these noncommutative random variables.

UNIFIED VARIATIONAL PRINCIPLES FOR THE VON NEUMANN, MASTER AND BOLTZMANN-BLOCH EQUATIONS AND THE T-MATRIX

1995 ◽  
Vol 09 (10) ◽  
pp. 1227-1242
Author(s):  
MASUMI HATTORI ◽  
HUZIO NAKANO
Keyword(s):  
Density Matrix ◽  
Variational Principles ◽  
Matrix Theory ◽  
Bloch Equation ◽  
Irreversible Processes ◽  
Collision Term ◽  
Maximum Condition ◽  
Von Neumann ◽  
Maximum Problem ◽  
T Matrix

The variational principle of irreversible processes, which was previously presented for the von Neumann equation as a stationarity problem and then converted into a maximum problem by contracting the density matrix perturbatively, is reinvestigated w.r.t. the contraction of the density matrix. The present contraction relies on the T-matrix theory of scattering, where no perturbational consideration enters. By taking the electron transport in solids as a typical example, the contraction is performed in two steps: the even component of the density matrix as to time reversal is eliminated first and then the off-diagonal elements in the scheme of diagonalizing the unperturbed Hamiltonian. The maximum problem thus obtained is for the diagonal elements of the odd component of the density matrix. The maximum condition gives the master equation, which is reduced to the Boltzmann-Bloch equation in the scheme of one-body picture. It is noticeable in this equation that the collision term is given in terms of the T-matrix in scattering theory.

Extension of normal functionals on W*-tensor products

1975 ◽  
Vol 78 (2) ◽  
pp. 301-307 ◽  
Author(s):  
Simon Wassermann
Keyword(s):  
Representation Theory ◽  
Tensor Product ◽  
Hilbert Spaces ◽  
General Result ◽  
Von Neumann Algebras ◽  
Tensor Products ◽  
Von Neumann ◽  
Hilbert Algebras ◽  
Special Cases

A deep result in the theory of W*-tensor products, the Commutation theorem, states that if M and N are W*-algebras faithfully represented as von Neumann algebras on the Hilbert spaces H and K, respectively, then the commutant in L(H ⊗ K) of the W*-tensor product of M and N coincides with the W*-tensor product of M′ and N′. Although special cases of this theorem were established successively by Misonou (2) and Sakai (3), the validity of the general result remained conjectural until the advent of the Tomita-Takesaki theory of Modular Hilbert algebras (6). As formulated, the Commutation theorem is a spatial result; that is, the W*-algebras in its statement are taken to act on specific Hilbert spaces. Not surprisingly, therefore, known proofs rely heavily on techniques of representation theory.

scholarly journals Semi-classical approximations based on Bohmian mechanics

2020 ◽  
Vol 35 (14) ◽  
pp. 2050070 ◽  
Author(s):  
Ward Struyve
Keyword(s):  
Wave Function ◽  
Quantum Theory ◽  
Quantum Gravity ◽  
Quantum Electrodynamics ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Bohmian Mechanics ◽  
Expectation Values ◽  
Usual Approach ◽  
Classical Approximation

Semi-classical theories are approximations to quantum theory that treat some degrees of freedom classically and others quantum mechanically. In the usual approach, the quantum degrees of freedom are described by a wave function which evolves according to some Schrödinger equation with a Hamiltonian that depends on the classical degrees of freedom. The classical degrees of freedom satisfy classical equations that depend on the expectation values of quantum operators. In this paper, we study an alternative approach based on Bohmian mechanics. In Bohmian mechanics the quantum system is not only described by the wave function, but also with additional variables such as particle positions or fields. By letting the classical equations of motion depend on these variables, rather than the quantum expectation values, a semi-classical approximation is obtained that is closer to the exact quantum results than the usual approach. We discuss the Bohmian semi-classical approximation in various contexts, such as nonrelativistic quantum mechanics, quantum electrodynamics and quantum gravity. The main motivation comes from quantum gravity. The quest for a quantum theory for gravity is still going on. Therefore a semi-classical approach where gravity is treated classically may be an approximation that already captures some quantum gravitational aspects. The Bohmian semi-classical theories will be derived from the full Bohmian theories. In the case there are gauge symmetries, like in quantum electrodynamics or quantum gravity, special care is required. In order to derive a consistent semi-classical theory it will be necessary to isolate gauge-independent dependent degrees of freedom from gauge degrees of freedom and consider the approximation where some of the former are considered classical.

scholarly journals On the central Haagerup tensor product

1994 ◽  
Vol 37 (1) ◽  
pp. 161-174 ◽  
Author(s):  
Pere Ara ◽  
Martin Mathieu
Keyword(s):  
Tensor Product ◽  
Large Class ◽  
Von Neumann Algebras ◽  
Bounded Operators ◽  
Multiplier Algebra ◽  
Von Neumann ◽  
Isometric Representation

For a large class of C*-algebras including all von Neumann algebras, the central Haagerup tensor product of the multiplier algebra with itself has an isometric representation as completely bounded operators.

scholarly journals Quantum Weak Invariants: Dynamical Evolution of Fluctuations and Correlations

Entropy ◽  
2020 ◽  
Vol 22 (11) ◽  
pp. 1219
Author(s):  
Zeyi Shi ◽  
Sumiyoshi Abe
Keyword(s):  
Time Evolution ◽  
Open Quantum System ◽  
Dynamical Evolution ◽  
Time Dependent ◽  
Von Neumann Entropy ◽  
Completely Positive Map ◽  
Expectation Values ◽  
Completely Positive ◽  
Temporal Asymmetry ◽  
Von Neumann

Weak invariants are time-dependent observables with conserved expectation values. Their fluctuations, however, do not remain constant in time. On the assumption that time evolution of the state of an open quantum system is given in terms of a completely positive map, the fluctuations monotonically grow even if the map is not unital, in contrast to the fact that monotonic increases of both the von Neumann entropy and Rényi entropy require the map to be unital. In this way, the weak invariants describe temporal asymmetry in a manner different from the entropies. A formula is presented for time evolution of the covariance matrix associated with the weak invariants in cases where the system density matrix obeys the Gorini–Kossakowski–Lindblad–Sudarshan equation.

scholarly journals REPRESENTATIONS OF GENERALIZED Ar STATISTICS AND EIGENSTATES OF JACOBSON GENERATORS

2006 ◽  
Vol 21 (21) ◽  
pp. 1691-1700 ◽  
Author(s):  
M. DAOUD
Keyword(s):  
Analytic Function ◽  
Coherent States ◽  
Uncertainty Relation ◽  
Fock Space ◽  
Complete Set ◽  
Natural Way

We investigate a generalization of Ar statistics discussed recently in the literature. The explicit complete set of state vectors for the Ar statistics system is given. We consider a Bargmann or an analytic function description of the Fock space corresponding to Ar statistics of bosonic kind. This brings, in a natural way, the so-called Gazeau–Klauder coherent states defined as eigenstates of the Jacobson annihilation operators. The minimization of Robertson uncertainty relation is also considered.

scholarly journals ENTANGLEMENT DYNAMICS IN FINITE QUDIT CHAIN IN CONSISTENT MAGNETIC FIELD

2012 ◽  
Vol 10 (06) ◽  
pp. 1250068 ◽  
Author(s):  
E. A. IVANCHENKO
Keyword(s):  
Magnetic Field ◽  
Entanglement Dynamics ◽  
Von Neumann ◽  
Magnetic Field Modulation ◽  
Entanglement Measures ◽  
Analytical Formulae ◽  
Initial States ◽  
Von Neumann Equation ◽  
Field Modulation ◽  
The Magnetic Field

Based on the Liouville–von Neumann equation, we obtain a closed system of equations for the description of a qutrit or coupled qutrits in an arbitrary, time-dependent, external magnetic field. The dependence of the dynamics on the initial states and the magnetic field modulation is studied analytically and numerically. We compare the relative entanglement measure's dynamics in bi-qudits with permutation particle symmetry. We find the magnetic field modulation which retains the entanglement in the system of two coupled qutrits. Analytical formulae for the entanglement measures in finite chains from two to six qutrits or three quartits are presented.

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