72
K. K. Leelamma
of the most important studies is the development of a quantum mechanical version
of the inverse scattering method used in the theory of integrable nonlinear evolution
equations like the Korteweg de Vries (KdV) equation. This method was developed
by Faddeev et al. [1–3] in formulating a quantum theory of solitons. Kulish and
Reshetikhin [4] showed that the quantum linear problem of the quantum sine-Gordon
equation was not associated with the Lie algebra sl(2) as in the classical case, but
with a deformation of this algebra. Sklyanin [5, 6] showed that deformations of Lie
algebraic structures were not special to the quantum sine-Gordon equation and that
it seemed to be part of a general theory. It was Drinfel’d who showed that a suitable
quantisation of Poisson Lie groups reproduced exactly the same deformed algebraic
structures encountered in the theory of quantum inverse scattering [7–9]. Almost at
the same time, Jimbo arrived at the same result [10, 11] from a slightly different
angle. In his work, the quantum algebras appeared in the context of the solution of
the Yang-Baxter Equation (YBE) which is a sufficient condition for solvability of
two-dimensional Ising model.
There is no universally accepted definition of a quantum group. There are several
approaches. In Drinfeld’s approach, the quantum group is defined as a deformation
of the universal enveloping algebra of a Lie algebra. This approach is similar to
the study of Lie groups via their Lie algebras. Jimbo also gave almost the same
definition. The new algebraic structures are called Quantised Universal Enveloping
Algebra (QUEA). In Manin’s work [12], quantum groups are defined as symmetries
of non-commutative or quantum spaces. We discuss this point in detail in Sect. 1.1.
Woronowicz [13–15] gave an entirely different approach based on non-commutative
C
∗ algebras. This is analogous to the classical theory of topological groups. He called
these groups, pseudo-groups. This approach is popular among mathematicians. The
theory of Faddeev and the Leningrad school [16] introduces quantum groups in terms
of R-matrices which are solutions of the Quantum Yang-Baxter Equation (QYBE).
This approach is directly connected to integrable quantum field theories and has no
classical analogue. In all the four approaches, quantum groups have the structure of
a Hopf algebra.
Quantum groups and quantum algebras have attracted much attention of physicists and mathematicians during the last decades of the twentieth century, especially
after the introduction of the q-harmonic oscillator. The fact that the energy levels of
the q-oscillator are not equally spaced and the success of the q-oscillator model in
accounting for the measurements on the infrared spectrum of a number of molecules,
indicated that q-deformation can take care of anharmonicity effects to some extent.
Motivated by these considerations, the Debye model of lattice heat capacity of crystals is reformulated [17], taking each mode as a q-oscillator. In the low temperature
limit, the model effectively coincides with the Debye model and in the high temperature limit, C v is found to be T -dependent, in very good agreement with the
experimental results obtained in the three cases studied. This model is discussed in
Sect. 2.
The problem of q-deformations of an anharmonic oscillator with quartic interaction [18] and its energy spectrum are studied. The energy values are found to
depend on the deformation parameter q. The various thermodynamic quantities such
K. K. Leelamma
of the most important studies is the development of a quantum mechanical version
of the inverse scattering method used in the theory of integrable nonlinear evolution
equations like the Korteweg de Vries (KdV) equation. This method was developed
by Faddeev et al. [1–3] in formulating a quantum theory of solitons. Kulish and
Reshetikhin [4] showed that the quantum linear problem of the quantum sine-Gordon
equation was not associated with the Lie algebra sl(2) as in the classical case, but
with a deformation of this algebra. Sklyanin [5, 6] showed that deformations of Lie
algebraic structures were not special to the quantum sine-Gordon equation and that
it seemed to be part of a general theory. It was Drinfel’d who showed that a suitable
quantisation of Poisson Lie groups reproduced exactly the same deformed algebraic
structures encountered in the theory of quantum inverse scattering [7–9]. Almost at
the same time, Jimbo arrived at the same result [10, 11] from a slightly different
angle. In his work, the quantum algebras appeared in the context of the solution of
the Yang-Baxter Equation (YBE) which is a sufficient condition for solvability of
two-dimensional Ising model.
There is no universally accepted definition of a quantum group. There are several
approaches. In Drinfeld’s approach, the quantum group is defined as a deformation
of the universal enveloping algebra of a Lie algebra. This approach is similar to
the study of Lie groups via their Lie algebras. Jimbo also gave almost the same
definition. The new algebraic structures are called Quantised Universal Enveloping
Algebra (QUEA). In Manin’s work [12], quantum groups are defined as symmetries
of non-commutative or quantum spaces. We discuss this point in detail in Sect. 1.1.
Woronowicz [13–15] gave an entirely different approach based on non-commutative
C
∗ algebras. This is analogous to the classical theory of topological groups. He called
these groups, pseudo-groups. This approach is popular among mathematicians. The
theory of Faddeev and the Leningrad school [16] introduces quantum groups in terms
of R-matrices which are solutions of the Quantum Yang-Baxter Equation (QYBE).
This approach is directly connected to integrable quantum field theories and has no
classical analogue. In all the four approaches, quantum groups have the structure of
a Hopf algebra.
Quantum groups and quantum algebras have attracted much attention of physicists and mathematicians during the last decades of the twentieth century, especially
after the introduction of the q-harmonic oscillator. The fact that the energy levels of
the q-oscillator are not equally spaced and the success of the q-oscillator model in
accounting for the measurements on the infrared spectrum of a number of molecules,
indicated that q-deformation can take care of anharmonicity effects to some extent.
Motivated by these considerations, the Debye model of lattice heat capacity of crystals is reformulated [17], taking each mode as a q-oscillator. In the low temperature
limit, the model effectively coincides with the Debye model and in the high temperature limit, C v is found to be T -dependent, in very good agreement with the
experimental results obtained in the three cases studied. This model is discussed in
Sect. 2.
The problem of q-deformations of an anharmonic oscillator with quartic interaction [18] and its energy spectrum are studied. The energy values are found to
depend on the deformation parameter q. The various thermodynamic quantities such