Remarks about Notations
Occasionally, we use for explanations of coordinate-invariant expression with three-dimensional tensors an index form including the Levi-Civita or fully antisymmetric symbols
defined by
(1)
Our most important notations deviating from an essential part of English literature are the following very rational ones for scalar product, vector product and dyadic product of two vectors
and
(2)
and for the volume product of three vectors
and
(3)
They agree fully or partially with many weighty sources, in particular, most completely with Rosenfel’d [1] but also widely with, e.g., [2] [3] [4] [5] [6] and many weighty others.
The action of operators
onto vectors
from the left and co-vectors
from the right we denote by
(4)
A bilinear form
in vectors
and co-vectors
is then
(5)
Operators in arbitrary n-dimensional spaces obey a Hamilton-Cayley identity which in cases of
takes on the forms (e.g., [6] [7] )
(6)
where
is the corresponding identity operator in considered dimension. Unspecifically for dimension, we introduced for the first three invariants with respect to similarity transformation
notations by special symbols according to (other notations, e.g., Fyodorov [6] [7] and Lagally [8] )
(7)
The second invariant in the form (7) is at once the determinant of two-dimensional operators. By introduction of new higher operator invariants this series may be continued to higher dimension but in this case it is difficult to invent for them new bracket symbols (but
for determinant seems to be acceptable to add for 4D-case and
becomes then third invariant and inversely
vanishes in 3D-case) and an index numbering for invariants is a possible solution. The Hamilton-Cayley identities allow to introduce specifically for dimension complementary operators
to
and with their help the corresponding inverse operators
in the cases
are
(8)
Symmetric and antisymmetric parts of an operators
can be determined in spaces with a given invariant symmetric scalar product (Euclidean or pseudo-Euclidean ones with a nonsingular metric tensor
,
) from
in coordinate form
with
,
or in index form (upper index
means transposition
)
(9)
Specifically, three-dimensional antisymmetric second-rank tensors
can be mapped in unique way onto (axial) vectors
according to
(10)
with
built from vector
as alternative notation for the antisymmetric operator
with correspondences
(11)
In considered spaces we can relate in unique way covariant vectors
with contravariant vectors
, in coordinate form by
, and may define quadratic forms
according to
(12)
The explained notations are convenient for two- and three-dimensional coordinate-invariant calculations1.
1. Introduction
The aim of present article is to review and to continue to develop mathematical aspects for the classical and quantum-mechanical description of the polarization of two-mode quasiplane quasimonochromatic light beams by means of the Lie group
. In a possible continuation it is intended to apply this to the investigation of the transformation of polarized light beams in reflection and refraction problems and to discuss the quantum-mechanical conditions for unpolarized light beams. Classically and for vacuum which we only consider, it is mostly sufficient to investigate a two-dimensional polarization matrix composed from the electric field and with components perpendicular to beam propagation from which can be determined the Stokes parameters which vanish for unpolarized light beams. Quantum-statistically this can be done from a density operator for, at least, two mutually orthogonal boson modes and to determine their expectation values for the polarization matrix.
There are two well-known realizations of the Lie group
, first, the translation of the classical formula for the angular momentum into a vector operator according to the rules of transition to canonical quantum mechanics and, second, the two-mode realization by the two-dimensional fundamental representation of
acting onto two quantum-mechanical states which correspond to two orthogonal polarization vectors in the operator representation of the electric field.
The quantum-statistical description of two-mode light polarization rests on the Jordan-Schwinger realization of the Lie algebra operators to
by quadratic combinations of the boson annihilation and creation operators of the two modes (Jordan 1935 [9] , and of Schwinger 1952 [10] , the last republished in [11] ) in the specific application to polarization. This description was developed in short form by Jauch and Rohrlich [12] in 1955. The monograph of Peřina [13] from 1971 contains a chapter about the polarization properties of light in which the polarization matrix (called coherence matrix) is quantum-statistically defined. In the same year 1971, Prakash and Chandra [14] determined the general form of two-mode unpolarized light beams by its definition of
invariance and Agarwal [15] determined the quasiprobabilities of such light beams. After some time of stagnation the rigorous quantum-statistical description of light polarization was further developed by many authors and author groups and was reviewed in the article of Luis and Sánchez-Soto [16] . V. Peřinová, A. Lukš and J. Peřina [17] consider and refer the application of
operators to atomic coherent states. The best-known realization of
not considered here is the application to quantum-mechanical angular momentum (e.g., [3] [18] [19] ).
With coordinate-invariant methods initiated in second half of last century mainly by F.I. Fyodorov [4] [5] [6] we develop some mathematical aspects of the groups
and for
in parametrization by the three-dimensional vector
instead of the Euler angles
that is rarely to find compared with the huge “group” literature from which we cite the early works of Lyubarski [20] , Gürsey, [21] , Behrends, Dreitlein, Fronsdal and Lee, [22] , Wybourne [23] , Gilmore [24] , Barut and Rączka [25] and Hamermesh [26] . The group
can be with advantage also parameterized by a three-dimensional vector
which simplifies mainly the composition law for two and more transformations and was developed with a few predecessors also mainly by Fyodorov. This is discussed here in details and applied for the calculation of the invariant (Haar) measure in both mentioned parametrization (Section 16). In Section 5 it is shown how one can determine representation matrices if one knows a complete set of basis operators, in our case for the fundamental two-dimensional representation of
with the vector parameter
. Some important decompositions (disentanglements) of these matrices are calculated in Section 10 and Section 11 considers shortly the representation of
by quaternions. A comparison of the two mentioned parametrizations by vector
and Euler angles
is made in Appendix A.
2. Short Quantum Description of Two-Mode Light Polarization of Beams by
Transformations
The group
can be applied in quantum optics for the theory of polarized and unpolarized light beams and for the lossless beamsplitter, in each case when two amplitudes can be transformed into each other or into two other amplitudes.
First we consider a light beam in an isotropic medium (here vacuum) which may be composed of two partial beams with two possible polarizations described by normalized, in general, complex-valued polarization vectors
and
. The electric field
of such a beam can be represented in the following way
(2.1)
The mean value of the wave vector of the beam is denoted by
and its mean frequency by
. Both are assumed here to be real-valued. The complex-valued amplitudes to two possible (mean) polarizations
and
are denoted by
and
and they depend slowly from position and time
that we do not explicitly write since it does not play a role in our further considerations. In quantum-mechanical context the electric field
becomes an operator and, correspondingly, the amplitudes
too proportional to pairs of boson annihilation creation operators
. The polarization vectors are orthogonal to the wave vector
and should satisfy the orthonormalization conditions
(2.2)
The description of the beam polarization by vectors
and
is not obligatory and we can use also other two polarization vectors
and
which are connected with the primary ones and satisfying the relations (2.2) by
(2.3)
The general transformation
of this kind proves to be a two-dimensional unitary unimodular transformation. Then the new amplitudes
and
to polarization vectors
and
in the beam are connected with the primary ones by
(2.4)
and the primary amplitudes are connected with the new amplitudes by the same matrix
but in the way
(2.5)
The two complex polarization vectors
contain 4 real components and by conditions (2.2) they are restricted to 3 independent real components. The general state of polarization of a light beam is a partially polarized state. An unpolarized state is a state which remains invariant with respect to all transformations
of the
-group. From Section 5 on we consider a parametrization of the unitary unimodular matrices
by a three-dimensional vector
.
Another device where the
transformations may be applied with advantage is a beamsplitter between two isotropic media. In this case we have an incident wave
from medium “1” and (or) an incident wave
from medium “2” of coupled modes from which result a reflected or refracted wave
and
in media “1” and “2” in dependence from which side we see the incident wave. In this case it is interesting to consider, in general, polarized incident waves and to calculate from them the reflected and refracted waves. The transformations of the beam are here actively made but inner details of the action of beamsplitter (e.g., a layer) are not involved. The correspondences to a singular beam with two possible polarizations are here
(2.6)
One may, however, consider in case of a beamsplitter only one incident wave from medium “1” or medium “2” with two possible polarizations but in this case we have a reflected or refracted partially polarized wave in both media that means, at least, 6 possible polarizations (or 8 in case of incident waves from both sides) and the consideration are not in full analogy to a simple beam.
3. The Group
Embedded into the Symplectic Group
of All Quadratic Combinations of Two Pairs of Boson Operators
Lie algebras were created as the tool to describe the local properties of Lie groups in the neighborhood of the identity element. It seems that pairs of boson annihilation and creation operators in quadratic combinations can be considered as elementary building stones of series of Lie algebras, in particular
,
,
,
(e.g., [23] [24] [25] and many others). For a more general insight into the Lie group of polarization transformations of two modes which leads to a realization of the two-dimensional unitary unimodular group
it is favorable to embed them into the group of all possible transformations made by quadratic combinations of two pairs of annihilation and creation operators of a two-mode system.
Pairs of boson annihilation boson annihilation and creation operators
comprise 2n operators which obey the following commutation relations (I identity operator in (Hilbert) representation space)
(3.1)
These boson annihilation and creation operators result from (Hermitean) canonical operators
and
by the definitions
(3.2)
with the commutation relations
(3.3)
The relations (3.1), (3.2) and (3.3) form the Heisenberg-Weyl algebra of an n-mode system. As important partial set of quadratic combinations of n pairs of boson operators may be considered the number operators
defined by
(3.4)
The operator N is the total number operator and is the sum of all partial number operators.
Due to our main interest here for the description of polarized and unpolarized light beams we consider now two pairs of boson annihilation and creation operators
and
. We introduce three Hermitean operators
by the definitions (e.g., Schwinger [10] , p. 545, Jauch and Rohrlich [12] , pp. 40-49, Messiah [19] , Chap. XIII)
(3.5)
They satisfy the abstract commutation relations for a Lie algebra
(3.6)
where
is the Levi-Civita symbol (or Levi-Civita pseudo-tensor) and here ad once they are the structure coefficients of the Lie algebra
. From (3.6) follows
(3.7)
and therefore (the operator
is analogous to the fully antisymmetric volume product)
(3.8)
In addition to (3.5) we introduce raising and lowering operators
and
as usual by
(3.9)
which satisfy the commutation relations
(3.10)
The operator C defined by (see also (3.8))
(3.11)
commutes with all operators of the Lie algebra
(3.12)
and is usually taken as the Casimir operator to the Lie algebra
. It can be represented after substitution of
according to (3.5) using the number operators defined in (3.4) in the following way
(3.13)
For the finite-dimensional irreducible representations which are parameterized by a discrete parameter j it is proportional to the identity operator I of the corresponding representation space of dimension
and for these representations it can be specialized to
(3.14)
The Casimir operator C does not belong to operators of the Lie algebra
. The number operator
also commutes with all operators of the Lie algebra
(3.15)
and can be taken in addition to the operators of
forming the Lie algebra
to unitary transformations of the group
.
With the 4 annihilation and creation operators
one may form 16 quadratic combination from which only 10 are linearly independent. These are the 4 squared operators (
,
,
,
and the remaining 12 squared operators reduce to 6 independent operators due to the mutual commutation relations that means totally to 10 independent operators.
From the quadratic combinations we may separate the Lie algebra of the two-mode squeezing group determined by the following three basic Hermitean operators
(3.16)
They satisfy the abstract commutation relations for a Lie algebra
(3.17)
In addition to the three operators
we introduce in analogy to
the operators
and
by linear combinations
(3.18)
which satisfy the commutation relations
(3.19)
By the formal substitutions
(3.20)
they make the transition to the commutation relations (3.6) for a Lie algebra
. The operator
defined by
(3.21)
commutes with all operators
of the Lie algebra
(3.22)
and is usually taken as the Casimir operator of the Lie algebra
. For irreducible representations it is proportional to the identity operator I of the representation space and is signified by a parameter k (Bargmann index) according to
(3.23)
In the realization of
by (3.16) the operator
takes on the form
(3.24)
which using (3.5) can be also represented
(3.25)
In comparison, the Casimir operator C of
in (3.13) can be represented by
(3.26)
The operator
in the realization (3.5) commutes with all operators
of the Lie algebra
in the realizations (3.16) and the operator
with all operators
of the Lie algebra
and we have
(3.27)
The two sets of 4 operators
and
form the extended Lie algebras
and
of only unitary but not necessarily unimodular operators, respectively.
The 6 operators
are not closed as a Lie algebra since the commutators between the
operators
and the
operators
lead to new operators according to
(3.28)
which cannot be represented as linear combinations of these 6 operators. Closing them to a new Lie algebra can be obtained by adding two new groups of operators
, formed from the pairs of annihilation and creation operators
,
(3.29)
From (3.29) we combine the operators
and
to new operators
and
according to (
)
(3.30)
leading explicitly to
(3.31)
They satisfy the commutation relations of an
-algebra in analogy to (3.19)
(3.32)
For completeness there remains to calculate the commutation relation of
with the operators
and with
for which we find
(3.33)
and
(3.34)
The operators
and
are connected with
and
by
(3.35)
Thus the operators
and
are already contained as linear combinations of the operators
and
or
and
and must not separately be taken into account for closing the Lie algebra of the above mentioned 6 operators. The operators
form two bases of two Lie algebras
of squeezing operators of the two modes with indices “1” and “2” separately.
The 10 operators
form a possible basis of the 10-parameter Lie algebra
with
independent commutation relations. An informative overview about the structure of the Lie algebra with its commutation relations give the root diagrams that we investigate in the next Section.
4. The Root Diagrams for the Homogeneous and Inhomogeneous Symplectic Group
of Two Pairs of Boson Operators
For a first overview about the structure of a Lie group it is very useful to consider the root diagram (e.g., [21] [22] [23] [24] [25] ) of its Lie algebra which describes the neighborhood of the identity operator I of the Lie group. We suppose that the compact part of the Lie group in the neighborhood of the identity element can be parameterized by vectors
with d independent components where d is called the dimension of the Lie group and we require that the vector
describes the identity element. We denote the group operators by
. In the neighborhood of the identity element
an arbitrary group operator
can be expanded in a Taylor series according to
(4.1)
The operators
are infinitesimal operators of the Lie group and the set of all possible linear combinations of these operators forms the Lie algebra of dimension d. We give here some definitions and a few basic results to the general theory of Lie algebras and discuss later the root diagrams from Equation (4.16) (refrootdef) on.
The commutator
of two arbitrary elements X and Y takes on the role of multiplication in the Lie algebra and the result Z has to be an element of the Lie algebra for closing it (
denotes the trace of an operator A)
(4.2)
From the definition of the commutator follows immediately
(4.3)
which is called the Jacobi identity. It takes on the role of the associative law for, e.g., the algebra of real and complex numbers or the algebra of matrices. Now we choose a basis of d linearly independent operators
in such way that arbitrary operators of the Lie algebra can be represented as
(4.4)
where
are vector components (in analogy to vectors
and later with operators
to
). Then from (4.2) we find for the commutator
(4.5)
where we have introduced coefficients
by definition
(4.6)
which are called the structure coefficients of the Lie algebra with respect to the chosen basis
. From (4.6) follows
(4.7)
From the Jacobi identity (4.3) follows then for arbitrary three basis operators
follows then
(4.8)
and therefore
(4.9)
By contraction over the indices
and then interchanging the free indices
and
follow the two equations
(4.10)
where we used that from the antisymmetry of the structure coefficients in the lower indices follows for the sum terms in (4.10)
(4.11)
Forming the difference of the Equations (4.10) and using the symmetry (4.11) the first two sum terms cancel and from the third sum terms using again (4.11) results
(4.12)
and each of the equations (10) simplifies to
(4.13)
from which follows
(4.14)
The symmetric tensor
is called the Killing form and with its help one may define a bilinear symmetric scalar product of two operators
and
written
as follows
(4.15)
The second-rank symmetric tensor
is a kind of metric tensor for the Lie algebra and plays an important role for the distinction of different kinds of Lie algebras (e.g., Levy-Maltsev theorem, [23] [25] ).
Now comes into play the Cartan subalgebra of the Lie algebra which is the linear space of the maximum of commuting operators of the Lie algebra with operators usually denoted by
. The number r of independent operators of the Cartan subalgebra is called the rank of the Lie algebra. From linear combinations of the Lie-algebra operators
one may select by linear combinations d operators
which are eigenvectors of the operator of the Cartan subalgebra in the sense
(4.16)
The vectorial eigenvalues
are called the root vectors of the Lie algebra and their dimensionality is equal to the rank of the Lie algebra or the dimension r of the Cartan subalgebra. Only the vectorial eigenvalue
is r-fold degenerate and their root vectors are linear combinations of the operators
. The root diagram represents the
root vectors in the r-dimensional space plus the r operators
of the Cartan subalgebra in the center. The basis operators of the Cartan subalgebra are not uniquely determined and can be defined in different variants ways of the theory. The commutating operators
of the Cartan subalgebra plus the root vectors determine already a certain amount of all commutation relations. For the remaining commutation relations of the periphery the theory of Lie algebras derives relations from the Jacobi identity which restrict their possibilities. This is only a minimum of the many well-known relations for Lie algebras (see, e.g., [23] [24] [25] and many others).
For some generalizations we extend the mainly here considered Lie group
. In Figure 1 we represent the two-dimensional root vectors of the Lie algebra
to the symplectic group
as arrows in two bases of the Cartan subalgebra
and
. For example, the arrow in the left-hand partial picture to the operator
means the commutator relation
and the right-hand partial picture the commutator relation
.
Each pair of boson annihilation and creation vectors
or corresponding canonical operators
with the commutation relations
or
forms also a Lie algebra called Heisenberg-Weyl algebra which is of dimension zero and reduces to a point. However, if we take in addition to pairs of boson operators of the Heisenberg-Weyl algebra the corresponding number operators
then due to commutation relations
Figure 1. Root diagrams of Lie algebra to the homogeneous symplectic group
in different bases. It is 10-dimensional and therefore a basis possesses 10 operators. In first basis of the Cartan subalgebra we have
and in the second basis
where
and
are the number operators to the two modes. The identity operator I in the Cartan subalgebra in the center of the diagram does not play a role since it commutes with all operators of the Lie group and does not provide a contribution to the roots.
,
,
the Heisenberg-Weyl algebras taken together with the number operators also form Lie algebras of corresponding rank. The same is, for example, by combination of Lie algebras
or
representable by pairs of annihilation and creation operators we find new algebras which we call inhomogeneous Lie algebras
or
, respectively.
In Figure 2 we represent the root diagrams of
in different basis systems. Besides the operators of the Cartan subalgebra in the center they contain there the identity operator I for closing them. In Figure 3 we represent the root diagram of a subalgebra of
in Figure 2 and make from it the transition to the root diagram of the Lie algebra
. In Figure 4 the root diagram for
of the right-hand Figure 3 is stretched in direction of the ordinate to the canonical form in a way that it takes on the maximal symmetry of a regular hexagon. One may check that operator
,
,
,
, commutes with all operators
to the Lie algebra
constructed from the inhomogeneous group
in described way and illustrated in Figure 4
(4.17)
Therefore one may add to the operator
a multiple of
Figure 2. Root diagram of Lie algebra to the inhomogeneous symplectic group
. These root diagrams contain in addition to the homogeneous symplectic group
the pairs of operators
and
with commutation relations
. Therefore, the identity operator I belongs to the Lie algebra and since it commutes with all operators it belongs to the Cartan subalgebra in the center of the diagrams. The diagram is quasi two-dimensional since no root operator possesses a component in direction of I perpendicular to the paper plane.
Figure 3. I somorphism of root diagram to Lie subalgebra of
to the Lie algebra
of homogeneous unitary group
. The root diagram of this Lie algebra on the left-hand side is part of the root diagram in Figure 2. The operators in the center are defined by
,
with
,
. Since the operators
and
commute with the operators
both root diagrams are equivalent.
and for the same reason a multiple of the identity operator I without changing the root diagram of
that is represented on the right-hand picture of Figure 4. If we add to this scheme now the annihilation and creation operators
which do not commute with the operator
then we may obtain the root diagram for the Lie
Figure 4. Root diagram of Lie algebra to the unitary unimodular group
in two equivalent bases. The operators in the center are defined by
,
with
,
and additionally
. The two schemes are equivalent because the number operator
commutes with all operators of the Lie algebra to
in considered realization and thus the operators of the Cartan subalgebra in the center can be substituted using the given identity. These diagrams correspond to the decontorted right-hand diagram of Figure 3 in perpendicular direction which leads to highest symmetry of the root diagram for
that becomes clear if we take into account
,
,
,
,
,
.
algebra to the inhomogeneous group
represented in Figure 5. It contains 15 operators, 3 of the Cartan subalgebra in the center and 12 in the periphery and is very similar to the root diagram of the Lie algebra
to the exceptional Lie group
with 14 basis operators (e.g., [23] ). The root scheme to the next unitary unimodular group
is three-dimensional and thus of rank 3. It possesses 3 operators in the center belonging to the Cartan subalgebra and 12 in the periphery which with their tips form in the most symmetric way the 12 corners of a cuboctahedron which is a semi-regular polyhedron.
In cases when the identity operator I belongs to the Cartan subalgebra in addition to 2n annihilation and creation operators (inhomogeneous groups) the root scheme, nevertheless, remains quasi of rank n since operator I commutes with all operators and does not provide a contribution to a higher rank
.
A conclusion is that by transition from known series of Lie groups to their inhomogeneous partner groups some problems emerge with their root diagrams and algebraic properties (traces, see also Appendix B).
5. Fundamental Representation
of
in Parametrization by a Three-Dimensional Vector
We now derive the fundamental representation of
in the basis of the
Figure 5. Root scheme of Lie algebra of inhomogeneous group
in realization by 3 pairs of boson annihilation and creation operators. The 3 operators (H) of the Cartan subalgebra are defined by
with
,
,
. The distance from the center to the tips of the star is equal to 1. The diagram is very similar to that of the exceptional group
and becomes the same if we omit the operator I in the center since it does not contribute to a third dimension. On the other side all commutators should belong to the diagram.
operators
and clarify in this way the transformation properties of the involved basic quantities such as boson operators and polarization vectors. For this purpose, we use a real three-dimensional vector parameter
and represent the general element x of the Lie algebra
in the following way
(5.1)
In Appendix A, we establish the connection to the Euler angles as parameters. The transition from the Lie algebra
to the Lie group
and its inversion is made by the exponential mapping
(5.2)
The construction of the fundamental representation of
in the basis of the operators
requires to consider a part of the Lie algebra of the inhomogeneous unitary unimodular group
with the operators
and their commutation relations as a possible set of basis operators. Instead of writing down all commutation relations, we use the necessary ones in the following mapping of
onto matrices
in the two-dimensional fundamental representation of
(5.3)
The matrices
are essentially (multiplied by factor 2) the Pauli spin matrices
which possess the properties
(5.4)
The most direct relation to the Pauli spin matrices is one reason that we construct the fundamental representation of the Lie algebra
in the basis
and not in the adjoint basis
.
From (5.3) follows for the representation of the Lie algebra operators
(5.5)
This is the mapping
of the operators x on two-dimensional matrices
according to
(5.6)
By means of the well-known operator expansion (e.g., [19] )
(5.7)
and using the Hamilton-Cayley identity for two-dimensional operators
in (5.6) we find the corresponding mapping
into the Lie group (note the difference between
and x)
(5.8)
In described way we obtain from (5.3) the two-dimensional fundamental representation
of
by unitary unimodular matrices
in the basis of creation operators
(5.9)
In analogy we find from (5.9) for the transformation of annihilation operators
(5.10)
According to
(5.11)
these transformations possess the total number operator
as basic invariant.
The form of the matrix
is
(5.12)
and is explicitly found in described way using the Hamilton-Cayley identity and with abbreviations
and
(compare, e.g., Gilmore [24] , p. 150, Equation (6.41))
(5.13)
The relations of unitarity
and of unimodularity
in addition are more explicitly (
denotes determinant of two-dimensional matrix or operator
)
(5.14)
The character of the representation or trace of the representation matrices is (
denotes trace of a matrix or operator
)
(5.15)
with
expressed by the matrix elements using the unimodularity and unitarity (5.14) of
(5.16)
We mention still that using
(5.17)
and additionally using the identity (5.7) the matrix (5.13) together with the chosen basis
can be straightforwardly derived also as follows
(5.18)
that is identical with (5.13). The components of
and their relations to annihilation and creation operators are explained in (3.5) and (3.9) and furthermore
is used.
Other bases, for example,
, lead only to a reordering with changing signs in the matrices in (5.13). In explained way one may calculate also other irreducible and reducible representations even with other dimension if one possesses a suited basis.
6. Determination of a Basic Range of Vector Parameter
for
We now determine a basic range for the vector parameter
. Two vector parameters
and
which lead to the same explicitly given matrix
in (5.13) are called equivalent and this is denoted by
. We show that the transformations of the parameter
according to
(6.1)
leave the operators
in (5.13) unchanged. From (6.1) follows for modulus and direction of
(6.2)
with correlated signs. In connection with
and
we see that the transformation (6.1) preserves the matrices
(5.13). The minimal difference of two different equivalent points is obtained if we set
in (6.1) and it is
(6.3)
Such equivalent points possess opposite directions considered from the center. Therefore, one may choose as basic range of inequivalent parameters
a three-dimensional ball2 of radius
(6.4)
Inner points of this three-dimensional ball possess equivalent points only outside it. The whole surface (sphere
) of this three-dimensional ball
, independently on the direction of
, corresponds to the negatively taken identity matrix
that means topologically to one point of the group manifold.
We consider now the transformation
(6.5)
It is only necessary to investigate the case
since the remaining part is identical with the already considered transformation (6.1). From
(6.6)
follows
(6.7)
Thus the transformation (6.5) is the transition from the matrices (5.13) to their negative matrices.
The inversion of the matrices is made by the transformation
(6.8)
In applications the vector parameter
does not possess the same weight (or Haar measure) independently on the modulus
and therefore not the same topology as a usual three-dimensional sphere with equal weight for all
and with volume
for radius R. As a second possible fundamental range of inequivalent parameters
one may also choose
(6.9)
Therefore, the invariant measure for the described basic ranges of parameters
should be vanishing for all
. We come back to this important problem in Section 16 when we discuss the derivation of an invariant measure over the group
. This invariant measure has to become vanishing for
and
such as for the center
.
7. Inversion of the Mapping
From the matrix
one may determine the parameter
up to the equivalence
given in (6.1). First from (5.15) follows
(7.1)
Therefore
(7.2)
or for
and
instead of
and
(7.3)
Thus from the matrix (5.13) the vector parameter
can be determined with the indeterminacy of
described by (6.1).
8. Eigenvalues and Eigenvectors of
in the Two-Dimensional Fundamental Representation with Vector Parameter
In this Section we determine the right-hand and left-hand eigenvectors (spinors)
and
of the matrix
in (5.13) to the well-known eigenvalues
according to
(8.1)
We consider
and
as column vectors and
and
as row vectors. From scalar multiplication of the first equation with
and of the third equation with
and forming the difference of the obtained equations follows for
the orthogonality of
and
and similarly of
and
. This leads to the following representation of
(8.2)
The operators
and
are one-dimensional projection operators with the properties
(8.3)
Since
are unitary unimodular matrices their eigenvalues
are complex numbers on the unit circle in the complex plane. The unimodularity of the matrix
makes the product of its eigenvalues equal to 1 and thus complex conjugate in two-dimensional case. Concretely, one finds from the eigenvalue equation
(8.4)
the following well-known solutions for the eigenvalues
(8.5)
as already mentioned. Inserting these eigenvalues into (8.1) in the concrete representation (5.13) one obtains equations with the following solutions for the two-dimensional eigenvectors in a non-normalized form
(8.6)
In general, they are complex vectors. There are possibilities to represent these eigenvector in another way and to choose other proportionality factors. As a more symmetrical and normalized form of the eigenvectors (8.6) one may choose
(8.7)
which leads to
(8.8)
with the orthonormality relations
(8.9)
In degenerate case
one finds
and thus
and in degenerate case
one has
corresponding to
that means only to one group transformation.
9. Composition Law for Vector Parameters
Corresponding to Products of
Transformations
The group
is described by the vector parameter
, for example, in the fundamental representation by the matrix
given in (5.13) with respect to a basis discussed in Section 5. We now consider the composition of two such matrices
and
with the vector parameters
and
to the product matrix
and ask how the vector parameter
to the matrix
is connected with the vector parameters
and
according to the correspondences
(9.1)
From the multiplication of two such matrices of the form (5.13) and reorganization of the obtained terms one easily finds the following scalar and vector equation (
)
(9.2)
where
denotes the vector product of the vectors
and
. These two equations can be resolved with respect to the vector
in unique way that provides the following formula (notation
see below)
(9.3)
One may call this formula the composition law for the chosen vector parameter of the group
. The factor on first line of the right-hand gives the direction of the new vector
which as we have seen in Section 4 possesses another meaning for
in comparison to the rotation group and the factor on the second line the modulus of
. For the new parameter
we find
(9.4)
The above composition formulae are somehow similar (but not identical) to formulae for spherical trigonometry of the surface (sphere) of a three-dimensional ball but here play a role also the inner points. In addition, in applications our three-dimensional ball possesses different weights of its points in dependence on the modulus
and thus another topology as a usual three-dimensional ball with equal weight measure for all its inner points. For example, as discussed the whole surface (sphere) of our three-dimensional ball with
corresponds to the operator
that means to only one point and therefore the (Haar) measure has to vanish for
. In Section 14-15, we consider another parametrization where the composition law for
takes on a simpler form.
We mention that Fyodorov [5] (§. 29, p. 447) and [6] (§. 3, p. 18) introduced a special notation
for the composition of parameters corresponding to the product of two group elements, in our case of
and
to
according to
(9.5)
For the composition of three parameters holds an associative law
(9.6)
corresponding to the associative law for the multiplication of group elements. Therefore, one may omit the inner brackets. The introduced symbols are very convenient and hardly come into conflict with other generally used symbols (only Hermitean scalar products are sometimes denoted by such brackets).
10. Decompositions of
Matrices or Disentanglement Relations for the Group
Beside the composition it is sometimes useful to decompose the general matrix
in (5.13) into products of simpler matrices which is called disentanglement. The obtained decompositions are then true for arbitrary irreducible representations. With the two-dimensional fundamental representation of
in representation by the vector parameter
in Section 5 we have developed at once the mathematical means for the disentanglement of
group operators that we present here. The method is the same as used, for example, in [27] [28] for
. We have to decompose the general matrices
of the fundamental representation derived in (5.13) into products of simpler matrices and have to look for the corresponding decompositions of the general group operators
into products of special group operators. Another method is the derivation of differential equations for the exponents in the decomposition formulae by introduction of an additional parameter and differentiation with respect to this parameter and then to solve the obtained differential equations that is made in a paper of Ban [29] based on Lie algebra methods (see also, e.g., [30] [31] ).
The following more special matrices of (5.13) are mainly of interest
(10.1)
The first two are oppositely triangular matrices and the third is a diagonal matrix.
If we take into account the most interesting decompositions of the general operator of
into products of special operators with
and
separately in the arguments of the exponentials, we can make the following 6 product decompositions of the 2D matrix
using its unimodularity
(10.2)
Evidently, these decompositions are not specific for
and are applicable for all two-dimensional unimodular matrices of the Special linear group
, for example, also similar for the two-dimensional non-unitary fundamental representation of
and decompositions of matrices including triangular matrices were already known to Gauss [32] . The obtained disentanglement relations in the corresponding 6 considered orderings of group operators of
are
(10.3)
with the explicit form of the elements of the two-dimensional matrix
(see (5.13))
(10.4)
In the special case
corresponding to
(10.5)
we obtain from (10.3) and (10.4) the disentanglement relations
(10.6)
where on the right-hand side the operator
appears although it is not on the left-hand side. The matrices
in (10.5) themselves do not form a group.
These formulae are important, for example, for the derivation of
group-coherent states in the sense of Perelomov [32] and for their representation.
We consider now the following decompositions of the unimodular 2D matrix
into products of two unimodular matrices
(10.7)
They correspond to the following disentanglement of group operators with
and with
(10.8)
with the relations
(10.9)
From these relations follows
(10.10)
The stable part in the decompositions (10.8) is the factor
. The inversion of (10.9) and (10.10) can be found using
(10.11)
which follows from combination of the relation for
in (10.9) with the relation for
in (10.10).
11. Parametrization of
by Quaternions
For some completeness we will shortly consider the parametrization of
by quaternions which were introduced by W.R. Hamilton in the middle of the 19th century after searching for more general number systems than complex numbers (e.g. [33] [34] [35] ).
A quaternion
consists of a scalar part
and of a vectorial part
which both (by definition) are real in case of real quaternions. The associative but not commutative multiplication law in the quaternion algebra
for two quaternions
and
is
(11.1)
where
denotes the scalar product and
the vector product of two vectors
and
. From (11.1) follows
(11.2)
The quaternion
is called the conjugate quaternion to
and the product
is proportional to the identical quaternion
and therefore
is the reciprocal quaternion to r. The nonnegative number
is the modulus or norm of a quaternion. The multiplication law (11.1) can be realized by matrix multiplication, in lowest-dimensional case by special 2D matrices, for example, by the following correspondence
(11.3)
with the additive decomposition of
as follows3
(11.4)
where
are the three Pauli spin matrices
explicitly given in (5.3). By comparison with (5.13), one finds the following correspondences between
matrices
and real quaternions
(11.5)
This means that the squared modulus of the quaternion is the determinant
of the two-dimensional matrix
which due to unimodularity is equal to 1 and the scalar part is half the trace
of the matrix
. Therefore,
matrices correspond to real unit quaternions (modulus equal to 1 by definition) with 3 independent real parameters. The multiplication of matrices (5.13) or the quaternion multiplication (11.1) allow to establish the composition law of two
transformations. Clearly, the noncommutative matrix multiplication is the more generally applicable operation in comparison to quaternion multiplication.
12. Regular Representation
of
as Basic Representation of
In this Section we construct the regular (or adjoint) representation of
which provides the group of inner automorphisms of
and thus the transformation of the vector operator
. It uses the operators of the abstract Lie algebra themselves as a basis and, therefore, is three-dimensional. If we use
as basis, we can construct the three-dimensional representation matrices to
from the commutators
in analogy to (5.3) that we do not write down explicitly. From this realization we find in analogy to (5.12) the regular representation of
in the form
(12.1)
with the following three-dimensional matrix
with particularly simple structure
(12.2)
where
denotes the trace,
the second invariant and
the determinant of the operator
(see also below) and where we took into account
. It is a special complex unimodular matrix which is equivalent to a real orthogonal matrix as we will now show. Inserting (5.13) for
, we obtain an explicit form of this matrix which we do not write down. It is now straightforward to get the matrix
of the mapping
in the basis of the operators
instead of
that can be represented in the vector form4
(12.3)
and explicitly in representation by vector components (sum convention)
(12.4)
or written in coordinate-invariant form
(12.5)
The matrix
in the representation
corresponding to the basis
is a three-dimensional real orthogonal matrix with determinant equal to +1 (i.e. a proper rotation) that means
(12.6)
The rotation axis is described by the unit axial vector
and the rotation angle is
or, equivalently, by the unit vector
and the rotation angle
. In the form
it describes by convention an anti-clockwise rotation of the vector
and in the form
a clockwise rotation of the vector
about an angle
if the vector
is perpendicular to the clock plane.
Rotations with the parameters
and
lead to the same matrix
. We consider the following transformations of the vector parameter
(12.7)
from which follows for modulus and direction of
(12.8)
Inserted in (12.5) it leaves the rotation
unchanged.
Two other special cases are, in particular, the special case
in (6.1) for which follows
(12.9)
In comparison, for the transformation
we find
(12.10)
This allows to restrict a fundamental region of the parameters
to the three-dimensional sphere
with identification of opposite points on the surface of the two-dimensional sphere
with
as its boundary. The very direct relation to covariant quantities of
is an advantage of using the vector parameter
in comparison to the Euler angles
.
We can look to relation (12.3) also in another way. For this purpose we take the vector parameter
for an arbitrary element
of the Lie algebra to
and consider its transformation
(12.11)
from which follows
(12.12)
This means that all elements of
with vector parameters
where
is obtained from
by an arbitrary rotation of the three-dimensional rotation group
are equivalent and form one class within
. This transformation changes only the axis direction
and all elements of
with the same
are equivalent.
The relation between the three-dimensional matrix
and the two-dimensional matrix
and its inversion in covariant form is (e.g., see [18] (Equations (2.16) and (2.32)), [25] (pp. 42-43) and [36] (Equation (2.2.15))5)
(12.13)
where
and
denote here the trace and determinant of two-dimensional matrices, respectively. Matrices
and
lead to the same
. Thus we have constructed the known 2-1 homomorphism of
to
where
is the center of
consisting of two elements
and
. We can look to Equations (1)-(6) as to the inner automorphisms of the unitary unimodular group
that means to the inner transformations of the operators
which leave unchanged the commutation relations. This is important for the coordinate-invariant interpretation.
13. Parametrization of Rotation Group
by Vector
in Coordinate-Invariant Description
The three-dimensional rotation operator
can be represented in the following exponential form
(13.1)
which defines a real three-dimensional antisymmetric operator
. Three-dimensional anti-symmetric operators
can be mapped onto three-dimensional axial vectors
according to
(13.2)
from which follows
(13.3)
Only the second invariant
is in general non-vanishing whereas trace and determinant vanish and the Hamilton-Cayley identity for
reduces to
(13.4)
with the consequence that all powers of
reduce to powers of
and
multiplied by factors
(13.5)
From the Taylor series
and analogously for
by substitution
follows
(13.6)
Due to relation
for an exponential operator
we check for determinant
(13.7)
Using
with consequence
and in addition
according to (13.3) taking into account
follows explicitly from (13.6) for the trace and the second invariant of
(13.8)
The equality
of first and second invariant of
is due to
combined with the general identity
for general three-dimensional operators
.
Our next problem is the transition from the antisymmetric operator
in the above formulae to a representation by the vector parameter
by means of the formulae (13.2). First we make the transition of
to a representation by
(13.9)
which in coordinate-invariant representation is
(13.10)
Inserting (13.10) into (13.6) for
follows
(13.11)
The abbreviation
is a unit vector in direction of the rotation axis and
the rotation angle counter-clockwise taken. The application of
onto an arbitrary vector
leads to
(13.12)
By comparison of
with the general form
of group elements in representations of
in (5.2) which in specialization to the vector basis becomes
we find that the operators
are represented in the three-dimensional regular representation by the following matrices closely related to the Levi-Civita symbol
(13.13)
One may check that this three-dimensional matrix representation of the operators
satisfies the commutation relations (3.10) and may be taken as alternative starting point for the construction of the three-dimensional representation of
.
14. Cayley-Gibbs-Fyodorov Parametrization of Rotation Group
by Vector Parameter
in Coordinate-Invariant Description
There is yet another very interesting parametrization of the three-dimensional rotation group obtained by specialization from the Cayley representation (e.g., [7] ) of proper orthogonal operators
of arbitrary dimension (i.e., operators satisfying
with determinant
) by antisymmetric operators
which is possible in general n-dimensional case as follows6
(14.1)
where the second relation of antisymmetry of
follows from
(14.2)
We now consider specific properties of the transformations in three-dimensional case. After expansion of
in (14.1) in a Taylor series of powers of
according to
(14.3)
and reduction of powers of
higher than or equal 3 by the Hamilton-Cayley identity (6) (third equation) to powers lower than 3 taking into account the antisymmetry of
with the consequence
, we find the following reduced relations between
and
(14.4)
with the invariants
(14.5)
They do not contain powers of
and
higher than quadratic ones. Furthermore using (13.1) and (14.1) we have
(14.6)
with the invariants
(14.7)
Independently on the choice of the chosen sign of
and
the relations between
and
are true since they are involved only in the combinations
and
, respectively.
A further specifics of the three-dimensional case of the Cayley representation is that a real antisymmetric tensor
(or operator in Euclidean space) with its 3 independent components can be mapped
onto a three-dimensional real vector
which in representation by vector indices in analogy to (13.1) takes on the form (see also ((10) and (13.1));
)
(14.8)
from which follows, for example
(14.9)
For the squared operator
we find (
is a symmetric operator)
(14.10)
From this using (14.6) one finds the following relations between the vector parameters
and
(14.11)
with the consequence
(14.12)
The direct relation to the trigonometric functions in the rotation matrix
given in (12.5) by the vector parameter
is
(14.13)
and this rotation matrix
is therefore uniquely represented by vector parameter
as follows
(14.14)
The modulus
of the vector parameter
with real components
is stretched in comparison to the modulus
of the vector parameter
with real components
.
All rotations of
about an angle
correspond to the parameter
and in the whole region
we have the correspondences of
to
as the rotation axis. The region to angles
can be reduced by transition
to angles in the region
with simultaneous transition to the opposite rotation axis
. This is not possible for the fundamental representation of
and we cannot find equivalent angles within the region
which are equivalent by changing the direction
of the rotation axis.
The relation (14.11) between
and
maps all vectors of the three-dimensional ball of vectors
with
onto the three-dimensional Euclidean space of vectors
where opposite points
of the boundary
correspond to single points of
. For
the boundary corresponds to
where opposite directions have to be identified. For
the parameter
does not uniquely determine a transformation
of the form (5.13).
The two-dimensional matrix
of the fundamental representation of
in (5.13) takes on the following very simple form in representation by the vector parameter
in components
(14.15)
The minus sign in front of
appears because
changes its sign in the region
. The special case
corresponds to the parametrization of the special element
of
by Perelomov [32] (Section 4.1) where the complex parameter
there corresponds to our
. Furthermore, by comparison of (5.13) and (14.15) with (11.5) we find that the Cayley-Gibbs-Fyodorov parametrization possesses the following relation to the quaternion representation of
matrices
(14.16)
This means that the Cayley-Gibbs-Fyodorov parametrization and the quaternion representation of
are connected by a simple relation but the parametrization of the group
by the parameter
is not unique.
The parametrization of elements of
and of the three-dimensional rotation group
by the three-dimensional vector parameter
which is dual to the antisymmetric operator
of the Cayley representation was introduced by Gibbs (according to [6] ) and was used by Fyodorov as the fundament of his approach to the representation theory of this group in coordinate-invariant representation and, furthermore, was extended by him to the Lorentz group [6] . This parametrization is advantageous, in particular, for the coordinate-invariant representation of the composition law of two, in general, non-commuting operators in these groups. We consider this in next Section.
15. Composition Law for the Vector Parameters
and
Corresponding to Products of Rotations of
As in the case of
(Section 9) we derive now the composition law of the two vector parameters
and
to a new vector parameter
for the product of two rotations
and
according to
(15.1)
This composition law can be obtained from the representation (13.11) of the rotation operator
and possesses the explicit form
(15.2)
It seems to be also possible to find it from the composition law for
using the formulae (9.2) and (9.3).
We calculate now the antisymmetric operator
for the product of two three-dimensional rotations
(i.e.,
,
). If we substitute
in the formula for
in last of Equations (14.4) and express
and
by the antisymmetric operators
and
we find
(15.3)
where we used the specific three-dimensional identities for general antisymmetric operators
and
(15.4)
which can be directly checked. From the first of these identities and after multiplication of it with
and then forming the traces we find scalar identities for general antisymmetric operators
and
as follows
(15.5)
Furthermore, starting from first of Equations (14.4) with
and using the identities (15.5) we find
(15.6)
Inserting the relation (15.6) into (15.3) this provides the composition law for two rotations in the following final operator form
(15.7)
with the only non-vanishing invariant
of the antisymmetric operator
(15.8)
From (15.7) using the definition (14.8) of the vector parameter
by the antisymmetric operator
follows the composition law expressed by the vector parameter
in agreement with [6] [37] 7
(15.9)
and is near to a formula of Rodrigues cited, e.g., in [33] and [35] (p. 17). The appearance of the vector product
in this relation is the consequence that the order of two rotations
does not commute. A linearization is obtained by stretching
(Appendix C). From (15.9) follows for the modulus
(15.10)
Our derivation of the composition law (15.9) of two vector parameters
and
distinguishes from the more directly obtained of Fyodorov by using specific three-dimensional identities for antisymmetric operators. The composition of two vector parameters
and
to a new vector parameter
for the product of two rotations
can be obtained from (12.5) but this is also possible using (9.2)-(9.4). The composition law (9.3) for the vector parameter
is more complicated than that for the vector parameter
and can be also obtained from (15.9) using (14.8) and (14.6). This is an advantage of using the vector parameter
which, however, is nonlinear in
.
Following Fyodorov ( [6] ) we introduced in (9.5) and used in (15.9) the convenient symbol
for the composition of two vector parameters
and
corresponding to the product
or
. It allows to formulate some composition rules in a simple way. In general, due to non-commutativity but associativity of the group multiplication one has
(15.11)
meaning that one may omit the inner brackets in the composition of three transformations. Explicitly, we calculate in considered case (
denotes the volume product of three vectors
)
(15.12)
An important special case of (15.12) corresponds to
of
where
and
which provide all conjugate elements
. Since in both cases the transition to the inverse element means the transition
or
, respectively, we have to calculate
for which we find from (15.12)
(15.13)
This means that the vector parameters
to conjugate elements of the groups
possess the same modulus (length)
and therefore correspond to rotations about the same angle but, in general, to different rotation axes
and
. If the rotation axes of
and of
are parallel that means
then one finds
(15.14)
The formulae (15.13) and (15.14) show that rotations around arbitrary axes but with the same rotation angle belong to the same class of conjugate elements and that with different rotation angles to different classes of conjugate elements.
16. Invariant Integration over Group
and
The s-dimensional irreducible representations
of a finite group G with N elements
possess for any fixed
the following orthogonality relations in index notation for operators, e.g., [20] [26]
(16.1)
In transition to a Lie group one has to substitute
by the chosen parameter, in our case of
, by the vector parameter
or by the vector parameter
and the summation over the discrete elements g by integration over the chosen parameter, where
must be substituted by the corresponding parameter for the inverse element, in our case by
that means in our first considered case and h by a fixed parameter