Q-DEFORMATION OF VIRASORO ALGEBRA AND LATTICE CONFORMAL THEORIES

1993 ◽  
Vol 08 (13) ◽  
pp. 1233-1242 ◽  
Author(s):  
ALEXANDER A. BELOV ◽  
KAREN D. CHALTIKIAN
Keyword(s):  
Virasoro Algebra ◽  
Poisson Brackets ◽  
Original Theory ◽  
Loop Algebras ◽  
Lattice Version ◽  
Definition Of

A natural definition of q-deformation of Virasoro and superconformal algebras is proposed. New Lie algebraic symmetries are shown to describe the lattice version of the original theory. On the classical (Poisson brackets) level these two-loop algebras are shown to be isomorphic to the Faddeev-Takhtadjan-Volkov lattice Virasoro algebra.

scholarly journals A NOTE ON SCALAR FIELD THEORY IN AdS3/CFT2

2010 ◽  
Vol 25 (14) ◽  
pp. 2815-2836
Author(s):  
PABLO MINCES
Keyword(s):  
Field Theory ◽  
Scalar Field ◽  
Central Charge ◽  
Virasoro Algebra ◽  
Radial Coordinate ◽  
Poisson Brackets ◽  
Scalar Field Theory ◽  
Classical Level ◽  
Conjugate Momentum ◽  
Conformal Fields

We consider a scalar field theory in AdS d+1, and introduce a formalism on surfaces at equal values of the radial coordinate. In particular, we define the corresponding conjugate momentum. We compute the Noether currents for isometries in the bulk, and perform the asymptotic limit on the corresponding charges. We then introduce Poisson brackets at the border, and show that the asymptotic values of the bulk scalar field and the conjugate momentum transform as conformal fields of scaling dimensions Δ- and Δ+, respectively, where Δ± are the standard parameters giving the asymptotic behavior of the scalar field in AdS. Then we consider the case d = 2, where we obtain two copies of the Virasoro algebra, with vanishing central charge at the classical level. An AdS3/CFT2 prescription, giving the commutators of the boundary CFT in terms of the Poisson brackets at the border, arises in a natural way. We find that the boundary CFT is similar to a generalized ghost system. We introduce two different ground states, and then compute the normal ordering constants and quantum central charges, which depend on the mass of the scalar field and the AdS radius. We discuss certain implications of the results.

Hamiltonian Systems, Pseudo-Poisson Brackets and Their Scattering Representation for Physical Systems

1999 ◽  
Author(s):  
B. M. J. Maschke ◽  
A. J. van der Schaft
Keyword(s):  
Hamiltonian Systems ◽  
Network Model ◽  
Geometric Structure ◽  
Symplectic Structure ◽  
Poisson Brackets ◽  
Physical Systems ◽  
Energy Conserving ◽  
Definition Of

Abstract This paper is concerned with the definition of the geometric structure of Hamiltonian systems associated with energy–conserving systems in relation with an interconnection topology of their network model. It is also presented how the symplectic structure of standard Hamiltonian systems has to be extended to pseudo–Poisson tensors in order to cope with invariants, equilibria and constraints. Finally a scattering representation of these pseudo–Poisson tensors is defined.

scholarly journals QUASI ROOT SYSTEMS AND VERTEX OPERATOR REALIZATIONS OF THE VIRASORO ALGEBRA

2013 ◽  
Vol 12 (07) ◽  
pp. 1350028
Author(s):  
BORIS NOYVERT
Keyword(s):  
Lie Algebras ◽  
Virasoro Algebra ◽  
Root Systems ◽  
Vertex Operators ◽  
Boson Fields ◽  
Dimension 2 ◽  
Definition Of ◽  
Extensive List ◽  
Scaling Dimension ◽  
Natural Way

A construction of the Virasoro algebra in terms of free massless two-dimensional boson fields is studied. The ansatz for the Virasoro field contains the most general unitary scaling dimension 2 expression built from vertex operators. The ansatz leads in a natural way to a concept of a quasi root systems. This is a new notion generalizing the notion of a root system in the theory of Lie algebras. We introduce a definition of a quasi root systems and provide an extensive list of examples. Explicit solutions of the ansatz are presented for a range of quasi root systems.

scholarly journals HOPF TERM, LOOP ALGEBRAS AND THREE-DIMENSIONAL NAVIER-STOKES EQUATION

1993 ◽  
Vol 08 (28) ◽  
pp. 2677-2686 ◽  
Author(s):  
YUTAKA MATSUO
Keyword(s):  
Current Algebra ◽  
Sigma Model ◽  
Three Dimensional ◽  
Virasoro Algebra ◽  
Navier Stokes ◽  
Nonlinear Sigma Model ◽  
Two Dimensional ◽  
Navier Stokes Equation ◽  
Loop Algebras ◽  
Extrinsic Geometry

The dynamics of the three-dimensional perfect fluid is equivalent to the motion of vortex filaments or "strings." We study the action principle and find that it is described by the Hopf term of the nonlinear sigma model. The Poisson bracket structure is described by the loop algebra, for example, the Virasoro algebra or the analog of O(3) current algebra. As a string theory, it is quite different from the standard Nambu-Goto string in its coupling to the extrinsic geometry. We also analyze briefly the two-dimensional case and give some emphasis on the w1+∞ structure.

CHIRAL BOSONIZED ACTION AND THE MINIMAL WESS-ZUMINO-WITTEN ACTION FOR CHIRAL TWO-DIMENSIONAL QCD FROM PATH-INTEGRAL METHOD

1994 ◽  
Vol 09 (18) ◽  
pp. 1643-1652
Author(s):  
JIAN-GE ZHOU ◽  
YAN-GANG MIAO ◽  
YAO-YANG LIU
Keyword(s):  
Invariant Measure ◽  
Path Integral ◽  
Integral Method ◽  
Poisson Brackets ◽  
Field Independent ◽  
Integration Measure ◽  
Path Integral Method ◽  
Definition Of ◽  
The Right ◽  
Constraint Surface

The path-integral method is applied to derive the chiral bosonized action and the minimal WZW action for chiral QCD2. An interesting feature in our model is that the Poisson brackets of the new set of constraints, which is a recombination of the original ones, are field-independent on the constraint surface. The correct contribution from det {Ωi, Ωj} or δ(Ωa) is considered, and it seems to affect the properties of the integration measure. Here Ωi are the original chiral constraints, Ωa are linear combination of Ωi. However, due to the determinants of Lab(ɸ) and [Formula: see text] (the definition of Lab(ɸ) and [Formula: see text] can be seen from Eqs. (22) and (41)) being ±1, the right-invariant measure is equivalent to the left-invariant measure, which is quite unlike the case of the gauge quasigroup. As a result, we can identify the integration measure in our model with the Haar measure.

The commutation laws of relativistic field theory

1952 ◽  
Vol 214 (1117) ◽  
pp. 143-157 ◽  
Keyword(s):  
Field Theory ◽  
Canonical Form ◽  
Action Principle ◽  
Poisson Brackets ◽  
Relativistic Field ◽  
Canonical Theory ◽  
Relativistic Field Theory ◽  
Dirac Particles ◽  
Canonical Variables ◽  
Definition Of

A definition of Poisson brackets is given which is related to the action principle, but does not require the introduction of canonical variables. This permits the laws for forming both the commutators of canonical theory and the anticommutators of Fermi-Dirac particles to be stated in a manifestly covariant way. Examples of the use of this method are given. The last section discusses tentatively the extension to the case of equations which cannot be written in canonical form.

Strong axioms of infinity in NFU

2001 ◽  
Vol 66 (1) ◽  
pp. 87-116 ◽  
Author(s):  
M. Randall Holmes
Keyword(s):  
Set Theory ◽  
Original Theory ◽  
Independence Result ◽  
Permutation Methods ◽  
New Foundations ◽  
Definition Of ◽  
The Continuum

This paper discusses a sequence of extensions of NFU, Jensen's improvement of Quine's set theory “New Foundations” (NF) of [16].The original theory NF of Quine continues to present difficulties. After 60 years of intermittent investigation, it is still not known to be consistent relative to any set theory in which we have confidence. Specker showed in [20] that NF disproves Choice (and so proves Infinity). Even if one assumes the consistency of NF, one is hampered by the lack of powerful methods for proofs of consistency and independence such as are available for use with ZFC; very clever work has been done with permutation methods, starting with [18] and [5], and exemplified more recently by [14], but permutation methods can only be applied to show the consistency or independence of unstratified sentences (see the definition of NFU below for a definition of stratification). For example, there is no method available to determine whether the assertion “the continuum can be well-ordered” is consistent with or independent of NF. There is one substantial independence result for an assertion with nontrivial stratified consequences, using metamathematical methods: this is Orey's proof of the independence of the Axiom of Counting from NF (see below for a statement of this axiom).We mention these difficulties only to reassure the reader of their irrelevance to the present work. Jensen's modification of “New Foundations” (in [13]), which was to restrict extensionality to sets, allowing many non-sets (urelements) with no elements, has almost magical effects.

GEOMETRY OF DIFFERENTIAL EQUATIONS AND PROJECTIVE REPRESENTATIONS OF THE WITT ALGEBRA

1992 ◽  
Vol 06 (11n12) ◽  
pp. 1969-2003 ◽  
Author(s):  
C. ITZYKSON
Keyword(s):  
Differential Equations ◽  
Virasoro Algebra ◽  
Precise Definition ◽  
Witt Algebra ◽  
Singular Vectors ◽  
Highest Weight ◽  
Projective Representations ◽  
Highest Weight Representations ◽  
Definition Of ◽  
Explicit Expressions

We give explicit expressions for the singular vectors in highest weight representations of the Virasoro algebra using a precise definition of fusion.

scholarly journals The mass-centre in the restricted theory of relativity and its connexion with the quantum theory of elementary particles

1948 ◽  
Vol 195 (1040) ◽  
pp. 62-81 ◽  
Keyword(s):  
Energy Momentum Tensor ◽  
Classical Mechanics ◽  
Momentum Tensor ◽  
Elementary Particles ◽  
The Other ◽  
Poisson Brackets ◽  
Theory Of Relativity ◽  
Mass Centre ◽  
General Systems ◽  
Definition Of

The Newtonian definition of the mass-centre can be generalized to the restricted theory of relativity in several ways. Three in particular lead to fairly simple expressions in terms of instantaneous variables for quite general systems. Of these only one is independent of the frame in which it is defined. It suffers from the disadvantage that its components do not commute (in classical mechanics, do not have zero Poisson brackets), and are therefore unsuitable as generalized co-ordinates in mechanics. Of the other two, one is particularly . simply defined, and the other has commuting co-ordinates. The Poisson brackets can be derived from quite general considerations because the various mass-centres are expressible in terms of integrals of the energy-momentum tensor which are directly connected with the infinitesimal operators of the group of Lorentz transformations. The definitions are readily applicable to a single particle in theories, such as are current for elementary particles, where a co-ordinate observable does not exist, but an energy-momentum tensor does, and furnish the nearest approach possible to such observables. They are applied to electrons, particles of spin 0 and ℏ (scalar- and vector-meson theories), and to photons.

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