Chapter 20
Theory of Continuous Groups
In Chap. 16, we classified the groups into finite groups and infinite groups. We
focused mostly on finite groups and their representations in the precedent chapters.
Of the infinite groups, continuous groups and their properties have been widely
investigated. In this chapter we think of the continuous groups as a collective term
that include Lie groups and topological groups. The continuous groups are also
viewed as the transformation group. Aside from the strict definition, we will see the
continuous group as a natural extension of the rotation group or SO(3) that we
studied briefly in Chap. 17. Here we reconstruct SO(3) on the basis of the notion of
infinitesimal rotation. We make the most of the exponential functions of matrices the
theory of which we have explored in Chap. 15. In this context, we study the basic
properties and representations of the special unitary group of SU(2) that has close
relevance to SO(3). Thus, we focus our attention on the representation theory of SU
(2) and SO(3). The results are associated with the (generalized) angular momenta
which we dealt with in Chap. 3. In particular, we show that the spherical surface
harmonics constitute the basis functions of the representation space of SO(3).
Finally, we study various important properties of SU(2) and SO(3) within a
framework of Lie groups and Lie algebras. The last sections comprise the description
of abstract ideas, but those are useful for us to comprehend the constitution of the
group theory from a higher perspective.
20.1 Introduction: Operators of Rotation and Infinitesimal
Rotation
To characterize the continuous transformation appropriately, we have to describe an
infinitesimal transformation of rotation. This is because a rotation with a finite angle
is attained by an infinite number of infinitesimal rotations.
© Springer Nature Singapore Pte Ltd. 2020
S. Hotta, Mathematical Physical Chemistry,
https://doi.org/10.1007/978-981-15-2225-3_20
799
Theory of Continuous Groups
In Chap. 16, we classified the groups into finite groups and infinite groups. We
focused mostly on finite groups and their representations in the precedent chapters.
Of the infinite groups, continuous groups and their properties have been widely
investigated. In this chapter we think of the continuous groups as a collective term
that include Lie groups and topological groups. The continuous groups are also
viewed as the transformation group. Aside from the strict definition, we will see the
continuous group as a natural extension of the rotation group or SO(3) that we
studied briefly in Chap. 17. Here we reconstruct SO(3) on the basis of the notion of
infinitesimal rotation. We make the most of the exponential functions of matrices the
theory of which we have explored in Chap. 15. In this context, we study the basic
properties and representations of the special unitary group of SU(2) that has close
relevance to SO(3). Thus, we focus our attention on the representation theory of SU
(2) and SO(3). The results are associated with the (generalized) angular momenta
which we dealt with in Chap. 3. In particular, we show that the spherical surface
harmonics constitute the basis functions of the representation space of SO(3).
Finally, we study various important properties of SU(2) and SO(3) within a
framework of Lie groups and Lie algebras. The last sections comprise the description
of abstract ideas, but those are useful for us to comprehend the constitution of the
group theory from a higher perspective.
20.1 Introduction: Operators of Rotation and Infinitesimal
Rotation
To characterize the continuous transformation appropriately, we have to describe an
infinitesimal transformation of rotation. This is because a rotation with a finite angle
is attained by an infinite number of infinitesimal rotations.
© Springer Nature Singapore Pte Ltd. 2020
S. Hotta, Mathematical Physical Chemistry,
https://doi.org/10.1007/978-981-15-2225-3_20
799