scholarly journals Singular Bogoliubov transformations and inequivalent representations of canonical commutation relations

2019 ◽  
Vol 31 (08) ◽  
pp. 1950026 ◽  
Author(s):  
Asao Arai
Keyword(s):  
Fock Space ◽  
Sufficient Conditions ◽  
Bogoliubov Transformation ◽  
Fock Representation ◽  
Canonical Commutation Relations ◽  
Commutation Relations ◽  
Direct Sum ◽  
Boson Fock Space ◽  
Inequivalent Representations ◽  
Bogoliubov Transformations

We introduce a concept of singular Bogoliubov transformation on the abstract boson Fock space and construct a representation of canonical commutation relations (CCRs) which is inequivalent to any direct sum of the Fock representation. Sufficient conditions for the representation to be irreducible are formulated. Moreover, an example of such representations of CCRs is given.

scholarly journals A family of inequivalent Weyl representations of canonical commutation relations with applications to quantum field theory

2016 ◽  
Vol 28 (04) ◽  
pp. 1650007 ◽  
Author(s):  
Asao Arai
Keyword(s):  
Field Theory ◽  
Scalar Field ◽  
Fock Space ◽  
Complex Hilbert Space ◽  
Space Region ◽  
Zero Field ◽  
Canonical Commutation Relations ◽  
Commutation Relations ◽  
Time Zero ◽  
Conjugate Momentum

We consider a family of irreducible Weyl representations of canonical commutation relations with infinite degrees of freedom on the abstract boson Fock space over a complex Hilbert space. Theorems on equivalence or inequivalence of the representations are established. As a simple application of one of these theorems, the well-known inequivalence of the time-zero field and conjugate momentum for different masses in a quantum scalar field theory is rederived with space dimension [Formula: see text] arbitrary. Also a generalization of representations of the time-zero field and conjugate momentum is presented. Comparison is made with a quantum scalar field in a bounded region in [Formula: see text]. It is shown that, in the case of a bounded space region with [Formula: see text], the representations for different masses turn out to be mutually equivalent.

L2(E*, μ)-WEYL REPRESENTATIONS

2002 ◽  
Vol 05 (04) ◽  
pp. 581-592
Author(s):  
XIAOSHAN HU ◽  
ZHIYUAN HUANG ◽  
XIANGJUN WANG
Keyword(s):  
Infinite Dimensions ◽  
Direct Construction ◽  
Fock Representation ◽  
Canonical Commutation Relations ◽  
Commutation Relations

For the canonical commutation relations in infinite dimensions, we offer an explicit direct construction of Weyl representations Wϕ generated from the Fock representation by any ϕ ∈ L2(E*, μ, R) over the Q-space (E*, μ). Moreover, we obtain that, for any ϕ, ψ ∈ L2(E*, μ,R), Wϕ+ψ and Wϕ are unitarily equivalent, proving a conjecture posed by Robinson in Ref. 2. Our construction employs Wiener–Itô decomposition of the space L2(E*, μ, R) (respectively L2(E*, μ, C)).

On Some Markov Processes Arising from the Eyre-Hudson Super Lie Algebra Representations

1998 ◽  
Vol 01 (03) ◽  
pp. 485-498
Author(s):  
K. R. Parthasarathy
Keyword(s):  
Brownian Motion ◽  
Markov Processes ◽  
Fock Space ◽  
Linear Manifold ◽  
Free Field ◽  
Natural Manner ◽  
Commutation Relations ◽  
Boson Fock Space ◽  
Lie Algebra Representations ◽  
Super Lie Algebra

It is well-known3,5 that Brownian motion and Poisson process arise naturally from the canonical commutation relations (CCR) of free field operators in a boson Fock space. Eyre and Hudson2 have recently shown how to construct fields of operators in a boson Fock space obeying super Lie commutation relations. We establish the essential self-adjointness of their real and imaginary parts on the domain ∊, the linear manifold generated by all the exponential (coherent) vectors and determine a family of Markov processes which they give rise to in a natural manner. These Markov processes yield examples of Evans–Hudson flows3,5 and Azéma-like martingales.1,4,6

scholarly journals GROUND STATE OF THE MASSLESS NELSON MODEL WITHOUT INFRARED CUTOFF IN A NON-FOCK REPRESENTATION

2001 ◽  
Vol 13 (09) ◽  
pp. 1075-1094 ◽  
Author(s):  
ASAO ARAI
Keyword(s):  
Scalar Field ◽  
Ground State ◽  
Coupling Constant ◽  
Fock Representation ◽  
Nelson Model ◽  
Canonical Commutation Relations ◽  
Commutation Relations ◽  
Quantum Particles ◽  
Infrared Cutoff ◽  
Massless Quantum

We consider a model of quantum particles coupled to a massless quantum scalar field, called the massless Nelson model, in a non-Fock representation of the time-zero fields which satisfy the canonical commutation relations. We show that the model has a ground state for all values of the coupling constant even in the case where no infrared cutoff is made. The non-Fock representation used is inequivalent to the Fock one if no infrared cutoff is made.

scholarly journals Noisy effects in interferometric quantum gravity tests

2017 ◽  
Vol 15 (08) ◽  
pp. 1740014 ◽  
Author(s):  
F. Benatti ◽  
R. Floreanini ◽  
S. Olivares ◽  
E. Sindici
Keyword(s):  
Quantum Gravity ◽  
Weak Coupling ◽  
Scale Effects ◽  
External Environment ◽  
Planck Scale ◽  
Canonical Commutation Relations ◽  
Commutation Relations ◽  
Photon Propagation ◽  
Gravity Theories ◽  
Planck Scale Effects

Quantum-enhanced metrology is boosting interferometer sensitivities to extraordinary levels, up to the point where table-top experiments have been proposed to measure Planck-scale effects predicted by quantum gravity theories. In setups involving multiple photon interferometers, as those for measuring the so-called holographic fluctuations, entanglement provides substantial improvements in sensitivity. Entanglement is however a fragile resource and may be endangered by decoherence phenomena. We analyze how noisy effects arising either from the weak coupling to an external environment or from the modification of the canonical commutation relations in photon propagation may affect this entanglement-enhanced gain in sensitivity.

scholarly journals Canonical transformations for fermions in superanalysis

2014 ◽  
Vol 26 (06) ◽  
pp. 1450009
Author(s):  
Joachim Kupsch
Keyword(s):  
Infinite Number ◽  
Orthogonal Group ◽  
Degrees Of Freedom ◽  
Fock Space ◽  
Continuous Representation ◽  
Canonical Transformations ◽  
Module Extension ◽  
Bogoliubov Transformations

Canonical transformations (Bogoliubov transformations) for fermions with an infinite number of degrees of freedom are studied within a calculus of superanalysis. A continuous representation of the orthogonal group is constructed on a Grassmann module extension of the Fock space. The pull-back of these operators to the Fock space yields a unitary ray representation of the group that implements the Bogoliubov transformations.

scholarly journals MATRIX THEORY, HILBERT SCHEME AND INTEGRABLE SYSTEM

1998 ◽  
Vol 13 (34) ◽  
pp. 2731-2742 ◽  
Author(s):  
YUTAKA MATSUO
Keyword(s):  
Hilbert Scheme ◽  
Matrix Theory ◽  
Fock Space ◽  
Homology Class ◽  
Orthogonal Basis ◽  
Inner Product ◽  
Virasoro Symmetry ◽  
Hilbert Scheme Of Points ◽  
The Matrix ◽  
Boson Fock Space

We give a reinterpretation of the matrix theory discussed by Moore, Nekrasov and Shatashivili (MNS) in terms of the second quantized operators which describes the homology class of the Hilbert scheme of points on surfaces. It naturally relates the contribution from each pole to the inner product of orthogonal basis of free boson Fock space. These bases can be related to the eigenfunctions of Calogero–Sutherland (CS) equation and the deformation parameter of MNS is identified with coupling of CS system. We discuss the structure of Virasoro symmetry in this model.

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