A Connection between Lie Algebra Roots and Weights and the Fock Space Construction

1994 ◽  
pp. 265-284
Author(s):  
George W. Mackey
Keyword(s):  
Lie Algebra ◽  
Fock Space ◽  
Space Construction

scholarly journals A Quantization Procedure of Fields Based on Geometric Langlands Correspondence

2009 ◽  
Vol 2009 ◽  
pp. 1-14
Author(s):  
Do Ngoc Diep
Keyword(s):  
Symmetry Group ◽  
Lie Algebra ◽  
Vector Bundles ◽  
Sigma Model ◽  
Fock Space ◽  
Loop Algebra ◽  
Target Space ◽  
Automorphic Representations ◽  
Langlands Correspondence ◽  
Geometric Langlands

We expose a new procedure of quantization of fields, based on the Geometric Langlands Correspondence. Starting from fields in the target space, we first reduce them to the case of fields on one-complex-variable target space, at the same time increasing the possible symmetry groupGL. Use the sigma model and momentum maps, we reduce the problem to a problem of quantization of trivial vector bundles with connection over the space dual to the Lie algebra of the symmetry groupGL. After that we quantize the vector bundles with connection over the coadjoint orbits of the symmetry groupGL. Use the electric-magnetic duality to pass to the Langlands dual Lie groupG. Therefore, we have some affine Kac-Moody loop algebra of meromorphic functions with values in Lie algebra=Lie(G). Use the construction of Fock space reprsentations to have representations of such affine loop algebra. And finally, we have the automorphic representations of the corresponding Langlands-dual Lie groupsG.

scholarly journals PARAFERMIONS, PARABOSONS AND REPRESENTATIONS OF 𝔰𝔬(∞) AND 𝔬𝔰𝔭(1|∞)

2009 ◽  
Vol 20 (06) ◽  
pp. 693-715 ◽  
Author(s):  
N. I. STOILOVA ◽  
J. VAN DER JEUGT
Keyword(s):  
Lie Algebra ◽  
Fock Space ◽  
Complete Solution ◽  
Lie Superalgebra ◽  
Explicit Construction ◽  
Fock Spaces ◽  
Infinite Rank ◽  
Infinite Set ◽  
Basis Vectors

The goal of this paper is to give an explicit construction of the Fock spaces of the parafermion and the paraboson algebra, for an infinite set of generators. This is equivalent to constructing certain unitary irreducible lowest weight representations of the (infinite rank) Lie algebra 𝔰𝔬(∞) and of the Lie superalgebra 𝔬𝔰𝔭(1|∞). A complete solution to the problem is presented, in which the Fock spaces have basis vectors labelled by certain infinite but stable Gelfand–Zetlin patterns, and the transformation of the basis is given explicitly. We also present expressions for the character of the Fock space representations.

The Vacuum Distributions of the Truncated Virasoro Fields are Products of Gamma Distributions

2017 ◽  
Vol 24 (01) ◽  
pp. 1750004 ◽  
Author(s):  
Luigi Accardi ◽  
Andreas Boukas ◽  
Yun-Gang Lu
Keyword(s):  
Fourier Transform ◽  
Characteristic Function ◽  
Heisenberg Group ◽  
Lie Algebra ◽  
Fock Space ◽  
Random Variables ◽  
Virasoro Algebra ◽  
Commutation Relations ◽  
Gamma Distributions

In a recent paper, using a splitting formula for the multi-dimensional Heisenberg group, we derived a formula for the vacuum characteristic function (Fourier transform) of quantum random variables defined as self-adjoint sums of Fock space operators satisfying the multidimensional Heisenberg Lie algebra commutation relations. In this paper we use that formula to compute the characteristic function of quantum random variables defined as suitably truncated sums of the Virasoro algebra generators. By relating the structure of the Virasoro fields to the quadratic quantization program and using techniques developed in that context we prove that the vacuum distributions of the truncated Virasoro fields are products of independent, but not identically distributed, shifted Gamma-random variables.

Multiphoton Laguerre polynomial state as minimum uncertainty states of intensity-dependent squeezing

2014 ◽  
Vol 92 (9) ◽  
pp. 1016-1020
Author(s):  
Qian-Fan Chen ◽  
Hong-Yi Fan
Keyword(s):  
Coherent State ◽  
Lie Algebra ◽  
Explicit Form ◽  
Coherent States ◽  
Fock Space ◽  
Laguerre Polynomial ◽  
Annihilation Operator ◽  
Nonlinear Realization ◽  
Generalized Coherent States ◽  
Minimum Uncertainty

For a general multiphoton annihilation operator, F = f(N)ap, where N = a†a, we find the explicit form of an operator, G†, which satisfies [F, G†] = 1. Based on the nonlinear realization of the SU(1,1) Lie algebra whose generators are [Formula: see text], [Formula: see text], and R0 = [N/p] + 1/2. We introduce the concept of intensity-dependent multiphoton squeezing and find that the state Lm(–yR†)|j⟩, 0 ≤ j ≤ p, where Lm(x) is a Laguerre polynomial, is a minimum uncertainty state for intensity-dependent multiphoton squeezing. We also construct the phase states for multiphoton operator, which turn out to be SU(1,1) generalized coherent states. Additionally, we show that the photon-added coherent state |α, m⟩ (m is a non-negative integer), which can be interpreted as a nonlinear coherent state, can be expressed as [Formula: see text] in the whole Fock space.

scholarly journals AUXILIARY REPRESENTATIONS OF LIE ALGEBRAS AND THE BRST CONSTRUCTIONS

2000 ◽  
Vol 15 (04) ◽  
pp. 281-291 ◽  
Author(s):  
ČESTMÍR BURDÍK ◽  
A. PASHNEV ◽  
M. TSULAIA
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Scalar Product ◽  
Fock Space ◽  
Kernel Operator ◽  
Brst Charge ◽  
Representations Of Lie Algebras ◽  
Definition Of

The method of constructing auxiliary representations for a given Lie algebra is discussed in the framework of the BRST approach. The corresponding BRST charge turns out to be non-hermitian. This problem is solved by the introduction of the additional kernel operator in the definition of the scalar product in the Fock space. The existence of the kernel operator is proven for any Lie algebra.

KMS states for reduced groups, theta functions and the Powers–Størmer construction

1989 ◽  
Vol 105 (2) ◽  
pp. 397-410 ◽  
Author(s):  
K. C. Hannabuss
Keyword(s):  
Fock Space ◽  
Theta Functions ◽  
Loop Groups ◽  
Vector Group ◽  
Kms States ◽  
Twisted Convolution ◽  
Fock States ◽  
Convolution Algebra ◽  
Schwartz Functions ◽  
Space Construction

AbstractKMS states of a twisted convolution algebra of Schwartz functions on a vector group are classified and related to KMS states of twisted L1-algebras for certain subquotients. The KMS states for the subquotient algebras are also related to Fock states of vector groups. In the particular case of the subquotient Tn × ℤn of ℚ2n this links the Fock space construction of the theta functions with their appearance in KMS states of loop groups and in the Kac character formula.

Fock space construction of the massless dipole field

1983 ◽  
Vol 147 (1) ◽  
pp. 140-170 ◽  
Author(s):  
Anton Z Capri ◽  
Gebhard Grübl ◽  
Randy Kobes
Keyword(s):  
Fock Space ◽  
Dipole Field ◽  
Space Construction
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