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)).

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 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 Deformation of the Wheeler–DeWitt equation

2014 ◽  
Vol 29 (20) ◽  
pp. 1450106 ◽  
Author(s):  
Mir Faizal
Keyword(s):  
Commutation Relation ◽  
Canonical Commutation Relation ◽  
Big Bang ◽  
Minimum Length ◽  
Second Quantization ◽  
Canonical Commutation Relations ◽  
Commutation Relations ◽  
The Big Bang ◽  
Big Bang Singularity ◽  
Modified Theory

In this paper, we will analyze the consequences of deforming the canonical commutation relations consistent with the existence of a minimum length and a maximum momentum. We first generalize the deformation of first quantized canonical commutation relation to second quantized canonical commutation relation. Thus, we arrive at a modified version of second quantization. A modified Wheeler–DeWitt equation will be constructed by using this deformed second quantized canonical commutation relation. Finally, we demonstrate that in this modified theory the big bang singularity gets naturally avoided.

scholarly journals SUPERSYMMETRIC CANONICAL COMMUTATION RELATIONS

2006 ◽  
Vol 21 (13n14) ◽  
pp. 2937-2951 ◽  
Author(s):  
FLORIN CONSTANTINESCU
Keyword(s):  
Vector Fields ◽  
Canonical Quantization ◽  
Canonical Commutation Relations ◽  
Commutation Relations

We discuss the unitarily-represented supersymmetric canonical commutation relations which are subsequently used to canonically quantize massive and massless chiral, antichiral and vector fields. The canonical quantization shows some new facets which do not appear in the nonsupersymmetric case. Our tool is the supersymmetric positivity generating the Hilbert–Krein structure of the N = 1 superspace.

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