scholarly journals Observations on Noncommuting Coordinates and on Fields Depending on Them

2003 ◽  
Vol 4 (S2) ◽  
pp. 913-919 ◽  
Author(s):  
R. Jackiw
Keyword(s):  
Noncommuting Coordinates

LAGRANGIAN AND HAMILTONIAN FORMALISM ON THE h-DEFORMED QUANTUM PLANE

2000 ◽  
Vol 12 (05) ◽  
pp. 711-724
Author(s):  
SUNGGOO CHO ◽  
SANG-JUN KANG ◽  
KWANG SUNG PARK
Keyword(s):  
Hamiltonian Formalism ◽  
Quantum Plane ◽  
Noether’S Theorem ◽  
Noether's Theorem ◽  
Commutation Relations ◽  
Quantum Deformations ◽  
Initial Choice ◽  
Derivatives Of ◽  
Noncommuting Coordinates

It is known that there are only two quantum planes which are covariant under the quantum deformations of GL(2) admitting a central determinant. Contrary to the q-deformed quantum plane, the h-deformed quantum plane has a structure suitable for defining time derivatives and variations as closely as in the ordinary plane. From these we derive differential calculi including the skew-derivatives of Wess–Zumino as well as variational calculi on the quantum plane. These calculi enable us to generalize the Lagrangian and Hamiltonian formalism on the ordinary plane to the quantum plane. In particular, we construct commutation relations between noncommuting coordinates and momenta which do not depend on the initial choice of Lagrangian. We also discuss the symmetry of a Lagrangian and Noether's theorem.

scholarly journals Vacuum Expectation Value of the Higgs Field and Dyon Charge Quantisation from Spacetime Dependent Lagrangians

2003 ◽  
Vol 18 (31) ◽  
pp. 2207-2216
Author(s):  
Rajsekhar Bhattacharyya ◽  
Debashis Gangopadhyay
Keyword(s):  
Classical Solution ◽  
Electric Charge ◽  
Higgs Field ◽  
Vacuum Expectation ◽  
Vacuum Expectation Value ◽  
Lagrangian Formalism ◽  
Expectation Value ◽  
Yang Mills ◽  
Noncommuting Coordinates

The spacetime dependent Lagrangian formalism of Refs. 1 and 2 is used to obtain a classical solution of Yang–Mills theory. This is then used to obtain an estimate of the vacuum expectation value of the Higgs field viz. ϕa = A/e, where A is a constant and e is the Yang–Mills coupling (related to the usual electric charge). The solution can also accommodate noncommuting coordinates on the boundary of the theory which may be used to construct D-brane actions. The formalism is also used to obtain the Deser–Gomberoff–Henneaux–Teitelboim results10 for dyon charge quantisation in Abelian p-form theories in dimensions D = 2(p+1) for both even and odd p.

scholarly journals SPACE–TIME DEPENDENT LAGRANGIANS AND WEAK–STRONG DUALITY: SINE-GORDON AND MASSIVE THIRRING MODELS

2002 ◽  
Vol 17 (12) ◽  
pp. 729-738 ◽  
Author(s):  
RAJSEKHAR BHATTACHARYYA ◽  
DEBASHIS GANGOPADHYAY
Keyword(s):  
Quantum Gravity ◽  
Strong Duality ◽  
Space Time ◽  
Holographic Principle ◽  
Time Dependent ◽  
Length Scales ◽  
Noncommuting Coordinates

The formalism of space–time dependent Lagrangians developed in Ref. 1 is applied to the sine-Gordon and massive Thirring models. It is shown that the well-known equivalence of these models (in the context of weak–strong duality) can be understood in this approach from the same considerations as described in Ref. 1 for electromagnetic duality. A further new result is that all these can naturally be linked to the fact that the holographic principle has analogues at length scales much larger than quantum gravity. There is also the possibility of noncommuting coordinates residing on the boundaries.

HOPF ALGEBRA STRUCTURE OF (2+1)-DIMENSIONAL QUANTUM SUPERSPACE, ITS DUAL AND ITS DIFFERENTIAL CALCULUS

2010 ◽  
Vol 25 (26) ◽  
pp. 2241-2253 ◽  
Author(s):  
MUTTALIP OZAVSAR
Keyword(s):  
Hopf Algebra ◽  
Differential Calculus ◽  
Lie Superalgebra ◽  
Algebra Structure ◽  
Hopf Superalgebra ◽  
Hopf Algebra Structure ◽  
Quantum Supergroup ◽  
Noncommuting Coordinates

We consider a (2+1)-dimensional quantum superspace which has noncommuting coordinates in Manin sense and it was shown that this space has a Hopf algebra structure, i.e. the quantum supergroup, when it is extended by the inverse of the bosonic variable. Differential structures on this space were given by constructing the differential calculus in the sense of Woronowicz. Thus, we deduce that the corresponding quantum Lie superalgebra which as a commutation superalgebra appears classical, and as Hopf structure is non-cocommutative q-deformed. Finally, dual Hopf superalgebra was given.

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