Mathematical Methods for Physics and Engineering - Matematica.NET

GROUP THEORYif matrices are involved. In the notation in which G = {G 1 ,G 2 ,...,G n } the orderof the group is clearly n.As we have noted, for the integers under addition zero is the identity. Forthe group of rotations and reflections, the operation of doing nothing, i.e. thenull operation, plays this role. This latter identification may seem artificial, butit is an operation, albeit trivial, which does leave the system in a physicallyindistinguishable state, and needs to be included. One might add that without itthe set of operations would not form a group and none of the powerful resultswe will derive later in this and the next chapter could be justifiably applied togive deductions of physical significance.In the examples of rotations and reflections mentioned earlier, • has been takento mean that the left-hand operation is carried out on the system that resultsfrom application of the right-hand operation. ThusZ = X • Y (28.4)means that the effect on the system of carrying out Z isthesameaswouldbe obtained by first carrying out Y and then carrying out X. The order of theoperations should be noted; it is arbitrary in the first instance but, once chosen,must be adhered to. The choice we have made is dictated by the fact that mostof our applications involve the effect of rotations and reflections on functions ofspace coordinates, and it is usual, and our practice in the rest of this book, towrite operators acting on functions to the left of the functions.It will be apparent that for the above-mentioned group, integers under ordinaryaddition, it is true thatY • X = X • Y (28.5)for all pairs of integers X, Y . If any two particular elements of a group satisfy(28.5), they are said to commute under the operation •; if all pairs of elements ina group satisfy (28.5), then the group is said to be Abelian. Thesetofallintegersforms an infinite Abelian group under (ordinary) addition.As we show below, requirements (iii) and (iv) of the definition of a groupare over-demanding (but self-consistent), since in each of equations (28.2) and(28.3) the second equality can be deduced from the first by using the associativityrequired by (28.1). The mathematical steps in the following arguments are allvery simple, but care has to be taken to make sure that nothing that has notyet been proved is used to justify a step. For this reason, and to act as a modelin logical deduction, a reference in Roman numerals to the previous result,or to the group definition used, is given over each equality sign. Such explicitdetailed referencing soon becomes tiresome, but it should always be available ifneeded.1044