q-Oscillators and a q-deformed Debye Model for Lattice Heat Capacity
73
as partition function, entropy and internal energy are also evaluated. It is discussed
in Sect. 3. These studies are expected to be of relevance in the context of lattice
dynamics.
The concept of q-deformation is also applied to investigate the magnetic properties
of ferromagnets [19]. The agreement between the linear spin-wave theory of ferromagnetism and experimental observations on ferromagnets is not satisfactory. The
q-deformed Holstein-Primakoff transformation is used to describe the spin variables
of a Heisenberg ferromagnet and the magnons are treated as q-bosons. The exchange
Hamiltonian in the nearest neighbour approximation is obtained for small values of
the deformation parameter when the excitation is low. The thermodynamic quantities in the low temperature region are also evaluated. It is found that the spontaneous
magnetisation and magnetic contribution to the heat capacity have q-dependent T
1
2
terms in addition to the well-known Bloch T
3
2 term. A comparison of the theoretical
results and experimental values in the cases of EuO and EuS, the simplest Heisenberg
ferromagnets, indicated that our model is an improvement over the linear spin-wave
theory.
1.1 Quantum Groups and non-commutative Spaces
The quantum algebras have been linked to geometries that have non-commutative
structures [20, 21]. The concept of space-time continuum has been fundamental to all
successful physical theories. However there are arguments that on a sub-microscopic
level, this concept has to be abandoned [22]. There is no experimental proof for
the assumption that space-time is smooth down to arbitrarily small distances. Perhaps it may be because of the idealisation of space-time concept that one comes
across tremendous problems in the unification of various interactions [23]. This motivates one to look for a new space-time concept. Quantum mechanical phase-space
is only partially non-commuting, only co-ordinates and momenta non-commute,
co-ordinates themselves are commuting. If at a sufficiently small length scale, coordinates become non-commuting operators, it will be impossible to measure the
position of a particle exactly. In this way, one may hope to remove the ultraviolet
divergences of conventional quantum field theory which are due to the possibility of
measuring field oscillations at one point. Thus non-commutativity is introduced as
a necessary condition in the generalised space-time concept. It has been argued that
physics at the Planck scale may be understood only with the help of non-commutative
geometry [23, 24].
In a non-commutative space with real co-ordinates (x, y, z), a unit of length along
the x-direction is defined as
x = (q − 1)x
(1)
or equivalently
x = (q − q
−1
)x
(2)
73
as partition function, entropy and internal energy are also evaluated. It is discussed
in Sect. 3. These studies are expected to be of relevance in the context of lattice
dynamics.
The concept of q-deformation is also applied to investigate the magnetic properties
of ferromagnets [19]. The agreement between the linear spin-wave theory of ferromagnetism and experimental observations on ferromagnets is not satisfactory. The
q-deformed Holstein-Primakoff transformation is used to describe the spin variables
of a Heisenberg ferromagnet and the magnons are treated as q-bosons. The exchange
Hamiltonian in the nearest neighbour approximation is obtained for small values of
the deformation parameter when the excitation is low. The thermodynamic quantities in the low temperature region are also evaluated. It is found that the spontaneous
magnetisation and magnetic contribution to the heat capacity have q-dependent T
1
2
terms in addition to the well-known Bloch T
3
2 term. A comparison of the theoretical
results and experimental values in the cases of EuO and EuS, the simplest Heisenberg
ferromagnets, indicated that our model is an improvement over the linear spin-wave
theory.
1.1 Quantum Groups and non-commutative Spaces
The quantum algebras have been linked to geometries that have non-commutative
structures [20, 21]. The concept of space-time continuum has been fundamental to all
successful physical theories. However there are arguments that on a sub-microscopic
level, this concept has to be abandoned [22]. There is no experimental proof for
the assumption that space-time is smooth down to arbitrarily small distances. Perhaps it may be because of the idealisation of space-time concept that one comes
across tremendous problems in the unification of various interactions [23]. This motivates one to look for a new space-time concept. Quantum mechanical phase-space
is only partially non-commuting, only co-ordinates and momenta non-commute,
co-ordinates themselves are commuting. If at a sufficiently small length scale, coordinates become non-commuting operators, it will be impossible to measure the
position of a particle exactly. In this way, one may hope to remove the ultraviolet
divergences of conventional quantum field theory which are due to the possibility of
measuring field oscillations at one point. Thus non-commutativity is introduced as
a necessary condition in the generalised space-time concept. It has been argued that
physics at the Planck scale may be understood only with the help of non-commutative
geometry [23, 24].
In a non-commutative space with real co-ordinates (x, y, z), a unit of length along
the x-direction is defined as
x = (q − 1)x
(1)
or equivalently
x = (q − q
−1
)x
(2)