Mathematical Methods for Physicists: A concise introduction - Site Map

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Elements of group theory
Group theory did not ®nd a use in physics until the advent of modern quantum
mechanics in 1925. In recent years group theory has been applied to many
branches of physics and physical chemistry, notably to problems of molecules,
atoms and atomic nuclei. Mostly recently, group theory has been being applied in
the search for a pattern of `family' relationships between elementary particles.
Mathematicians are generally more interested in the abstract theory of groups,
but the representation theory of groups of direct use in a large variety of physical
problems is more useful to physicists. In this chapter, we shall give an elementary
introduction to the theory of groups, which will be needed for understanding the
representation theory.
De®nition of a group (group axioms)
A group is a set of distinct elements for which a law of `combination' is well
de®ned. Hence, before we give `group' a formal de®nition, we must ®rst de®ne
what kind of `elements' do we mean. Any collection of objects, quantities or
operators form a set, and each individual object, quantity or operator is called
an element of the set.
A group is a set of elements A, B, C; ...; ®nite or in®nite in number, with a rule
for combining any two of them to form a `product', subject to the following four
conditions:
(1) The product of any two group elements must be a group element; that is, if
A and B are members of the group, then so is the product AB.
(2) The law of composition of the group elements is associative; that is, if A, B,
and C are members of the group, then …AB†C ˆ A…BC†.
(3) There exists a unit group element E, called the identity, such that
EA ˆ AE ˆ A for every member of the group.
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