Quantum Groups, q-Oscillators
and q-Deformed Quantum Mechanics
G. Vinod
Abstract The term quantum group was coined by the field medallist V. G. Drinfeld to describe a novel mathematical structure that made its first appearance in
the quantum inverse scattering method. Later it found applications in diverse areas
of mathematics and physics. In this article the mathematical structure of quantum
group is described from the point of view of physics. q-deformed oscillators, which
were introduced as the realisation of the quantum group su q (2), are discussed with
emphasis on the algebra related to the quon algebra. A q-deformed quantum mechanics is developed and a q-Schrödinger equation is proposed. Physical implications of
q-deformation are discussed and the applications of q-deformation to various fields
are mentioned.
Keywords Inverse scattering · q-formed oscillator · Quantum groups · Lie algebra
1 Quantum Groups
1.1 Introduction
Quantum group is a mathematical structure which became a major research area
in mathematics and theoretical physics in the last decade of the twentieth century.
Quantum group made its first appearance in the physics literature in connection with
the quantum inverse scattering method, a technique for studying integrable systems
in quantum field theory and statistical mechanics. This structure made its appearance
through the works of Kulish, Reshetikhin, Sklyanin Faddeev and Thakhatajan [1–4].
Faddeev observed that the Yang-Baxter Equation, which is a sufficient condition for
solvability of two-dimensional Ising model of statistical mechanics is to quantum
groups, just as the Jacobi identity is to classical Lie algebras. Drinfeld [4] realised
that the algebraic structure associated with quantum inverse scattering method can
G. Vinod (B)
Department of Physics, Sree Sankara College, Kalady, Ernakulam, Kerala, India
e-mail: vinodrohini@gmail.com
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021
K. S. Sreelatha and V. Jacob (eds.), Modern Perspectives in Theoretical Physics,
https://doi.org/10.1007/978-981-15-9313-0_7
91
and q-Deformed Quantum Mechanics
G. Vinod
Abstract The term quantum group was coined by the field medallist V. G. Drinfeld to describe a novel mathematical structure that made its first appearance in
the quantum inverse scattering method. Later it found applications in diverse areas
of mathematics and physics. In this article the mathematical structure of quantum
group is described from the point of view of physics. q-deformed oscillators, which
were introduced as the realisation of the quantum group su q (2), are discussed with
emphasis on the algebra related to the quon algebra. A q-deformed quantum mechanics is developed and a q-Schrödinger equation is proposed. Physical implications of
q-deformation are discussed and the applications of q-deformation to various fields
are mentioned.
Keywords Inverse scattering · q-formed oscillator · Quantum groups · Lie algebra
1 Quantum Groups
1.1 Introduction
Quantum group is a mathematical structure which became a major research area
in mathematics and theoretical physics in the last decade of the twentieth century.
Quantum group made its first appearance in the physics literature in connection with
the quantum inverse scattering method, a technique for studying integrable systems
in quantum field theory and statistical mechanics. This structure made its appearance
through the works of Kulish, Reshetikhin, Sklyanin Faddeev and Thakhatajan [1–4].
Faddeev observed that the Yang-Baxter Equation, which is a sufficient condition for
solvability of two-dimensional Ising model of statistical mechanics is to quantum
groups, just as the Jacobi identity is to classical Lie algebras. Drinfeld [4] realised
that the algebraic structure associated with quantum inverse scattering method can
G. Vinod (B)
Department of Physics, Sree Sankara College, Kalady, Ernakulam, Kerala, India
e-mail: vinodrohini@gmail.com
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021
K. S. Sreelatha and V. Jacob (eds.), Modern Perspectives in Theoretical Physics,
https://doi.org/10.1007/978-981-15-9313-0_7
91