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    Symmetry: An Introduction to Group Theory and Its Applications
    Symmetry: An Introduction to Group Theory and Its Applications
    Symmetry: An Introduction to Group Theory and Its Applications
    Ebook456 pages3 hours

    Symmetry: An Introduction to Group Theory and Its Applications

    By Roy McWeeny

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    The crucial significance of symmetry to the development of group theory and in the fields of physics and chemistry cannot be overstated, and this well-organized volume provides an excellent introduction to the topic.
    The text develops the elementary ideas of both group theory and representation theory in a progressive and thorough fashion, leading students to a point from which they can proceed easily to more elaborate applications. The finite groups describing the symmetry of regular polyhedral and of repeating patterns are emphasized, and geometric illustrations of all main processes appear here — including more than 100 fully worked examples.
    Designed to be read at a variety of levels and to allow students to focus on any of the main fields of application, this volume is geared toward advanced undergraduate and graduate physics and chemistry students with the requisite mathematical background.

    LanguageEnglish
    PublisherDover Publications
    Release dateMay 23, 2012
    ISBN9780486138800
    Symmetry: An Introduction to Group Theory and Its Applications

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      Symmetry - Roy McWeeny

      INDEX

      PREFACE

      THE OPERATIONAL principles underlying the construction of symmetrical patterns were certainly known to the Egyptians. They were formulated symbolically during the nineteenth century and played their part in the development of the theory of groups. But only during the past thirty-five years has the immense importance of symmetry in physics and chemistry been fully recognized.

      The value of group theoretical methods is now generally accepted. In chemical physics alone, the symmetries of atoms, molecules and crystals are sufficient to determine the basic selection and intensity rules of atomic spectra, and of the electronic, infra-red and Raman spectra of molecules and crystals. The vector model for coupling angular momenta, the properties of spin, the Zeeman and Stark effects, the splitting of energy levels by a crystalline environment, and even the nature of the periodic table may all be traced back to symmetry of one kind or another.

      Many excellent textbooks on group theory and its applications are already available. They range from rigorous but formal presentations to highly condensed accounts which deal with particular applications and give only a superficial treatment of underlying concepts. The first kind of approach is unpalatable, except to the professional mathematician; the second provides a useful vocabulary and a set of working rules—but no real understanding of group theory. The present book is intended primarily for physics and chemistry graduates who possess a fair amount of mathematical skill but lack the formal equipment demanded by the standard texts (by Wigner, Weyl and others). Accordingly, the elementary ideas of both group theory and representation theory (which, incidentally, provides the basic mathematical tools of quantum mechanics) are developed in a leisurely but reasonably thorough way, to a point at which the reader should be able to proceed easily to more elaborate applications. For this purpose, emphasis is placed upon the finite groups which describe the symmetry of regular polyhedra and of repeating patterns. By restricting the scope in this way, it is possible to include geometrical illustrations of all the main processes. In fact, over a hundred fully worked examples have been incorporated into the text.

      A discussion of the permutation group and of the rotation groups would have defeated the aim of introducing all the basic ideas in a limited space. However, there is much to be said for dealing with special groups in the context in which they occur. A study of the rotation group, for example, is essential to any full discussion of the quantum mechanical central field problem. And such applications present no real difficulty once the basic principles have been grasped.

      The book is constructed so that it may be read at various levels and with emphasis on any one of the main fields of application. Chapter 1 is concerned mainly with elementary concepts and definitions; Chapter 2 with the necessary theory of vector spaces (though sections 2.8–10 may be omitted in a first reading). Chapters 3 and 4 are complementary to 1 and 2: they provide the reader with an opportunity of actually working with groups and representations, respectively, until the ideas already introduced are fully assimilated. The more formal theory of irreducible representations is confined to Chapter 5. In a first reading it would be sufficient to grasp the main ideas behind the orthogonality relations (5.3.1), passing then to section 5.4 for the properties of group characters, and finally to the construction of irreducible basic vectors according to (5.7.8) and the examples which follow. The rest of the book deals with applications of the theory. Chapter 6 is concerned largely with quadratic forms, illustrated by applications to crystal properties and to molecular vibrations. Chapter 7 deals with the symmetry properties of functions, with special emphasis on the eigenvalue equation in quantum mechanics. Chapter 8 covers more advanced applications, including the detailed analysis of tensor properties and tensor operators.

      Much of the material in this book has been presented to final year undergraduates and to graduate students in the Departments of Mathematics, Physics and Chemistry at the University of Keele, and the exposition has often been guided by their response. It is a pleasure to acknowledge this enjoyable and valuable contact, and also the many conversations with colleagues both here and at the University of Uppsala. Finally, anyone writing a book in this field must be deeply conscious of his debt to those who pioneered the subject, over thirty years ago; I should like to acknowledge, in particular, that the books by Wigner and Weyl have been a constant influence.

      R. MCWEENY

      CHAPTER 1

      GROUPS

      1.1. Symbols and the Group Property

      A large part of mathematics involves the translation of everyday experience into symbols which are then combined and manipulated, according to determinable rules, in order to yield useful conclusions. In counting, the symbols we use stand for numbers and we make such statements as 2 + 3 = 5 without giving much thought to the meaning of either the symbols themselves or the signs + (which indicates some kind of combination) and = (which indicates some kind of equivalence). In group theory we use symbols in a much wider sense. They may, for instance, stand for geometrical operations such as rotations of a rigid body; and the notions of combination and equivalence must then be defined operationally before we can start translating our observations into symbols. We do arithmetic without much thought. only because we are so familiar with the operational definitions, which are far from trivial, which we learnt as children. But it is worth reminding ourselves how we began to use symbols.

      How did we learn to count? Perhaps we took sets of beads, as in Fig. 1.1, giving each set a name 1, 2, 3, . . . (the whole numbers ). A set of cows, for example, can then be given the name 3, or said to contain 3 cows, if its members can be put in one-to-one correspondence with the beads of the set named 3 (a bead for each cow, no cows or beads left over). The same number is associated with different sets if, and only if, their members can be put in one-to-one correspondence : in this case the numbers of objects in the different sets are equal. If x objects in one set can be related in this way to y objects in another set we write x = y. If the numbers of fingers on my two hands are x and y, I can say x = y because I can put them into one-to-one correspondence: and I can say x = y = 5 because I can put the members of either set in one-to-one correspondence with those in the set named 5 in Fig. 1.1. This provides an operational definition of the symbol =. We observe that the sets in Fig. 1.1 have been given distinct names because none can be put in one-to-one correspondence with any other: 1 ≠ 2 ≠ 3 ≠ 4 . . . . Numbers may be combined under addition (or added ), for which we usually use the symbol +, by putting together different sets to make a new set. If we put together a set of 4 objects and a set of 1 object the resultant set is said to contain (4+1) objects: but there is another name for the number of objects in this set because it can be put in one-to-one correspondence with the set of 5 objects. Hence the different collections contain equal numbers of objects and we write 4 + 1 = 5. The whole numbers are conveniently arranged in the ordered sequence (Fig. 1.1) such that 1 + 1 = 2, 2 + 1 = 3, 3 + 1 = 4, etc., so that sets associated with successive numbers are related by the addition of 1 object. Generally, we say that if the members of sets containing x and y objects, when put together to form a new set, can be put into one-to-one correspondence with those of a set of z objects, then x + y = z. The operational meaning of the law of combination (indicated by the + ) and of the equivalence ( = ) is now absolutely clear. But, of course, the terminology is quite arbitrary: instead of 2 + 3 = 5 we could just as well write

      2 combined with 3 gives 5 or 2 ! 3 : 5

      What matters is that we agree upon (i) what the symbols stand for, (ii) what we shall understand by saying two of them are equal, or equivalent, and (iii) what we shall understand by combining them.

      FIG. 1.1. Sets of objects representing the whole numbers.

      In group theory we deal with collections of symbols, A, B, . . . , R, . . . , which do not necessarily stand for numbers and which are accordingly set in distinctive type (gill sans—instead of the usual italic letters). We refer to the members of the collection as elements and often denote the whole collection by showing one or more typical elements in braces, {R} or {A, B, . . . , R, . . . }. The elements may, for example, represent geometrical operations such as rotations of a rigid body, and the law of combination is then non-arithmetic. Nevertheless rotations can be compounded in the sense that one (R) followed by a second (S) gives the same final result as a third (T): the italic phrases describe, respectively, the law of combination (sequential performance) and the nature of the equivalence. In order not to restrict the law of combination we shall normally leave the question completely open, simply writing side by side the two symbols to be combined and, purely formally, calling the result of combination their product. We adopt the = sign to denote equivalence in whatever sense may be appropriate. In dealing with any particular collection of elements, the meaning of the term product and the significance of the = sign will be agreed upon at the outset.

      When the elements of a collection are the natural numbers and the law of combination is addition one important property is at once evident. Three numbers may be combined, by applying the law of combination twice, in either of two ways: with the general terminology these are

      (i)  (abc)1 = a(bc)    or (ii)  (abc)2 = (ab)c

      and the results are always identical. To give an example

      (i) (234)1 = 2(34) = 27 = 9   (ii) (234)2 = (23)4 = 54 = 9

      The order in which the terms of a product are dealt with is therefore irrelevant and there is no need to bracket together the pairs to show how a triple product must be interpreted. The law of combination is said to be associative. In this case there is no ambiguity and an ordered product of any number of elements is uniquely equivalent to a certain single element. This is a fundamental property of all the collections we shall study and is assumed in defining the group property:

      (1.1.1)

      The condition that the combination of two elements should yield an element of the same kind (i.e. also in the collection) is itself non-trivial: the collection is then closed and we refer to the closure property. It should be noticed that the law of combination refers to an ordered pair, so that the possibility RS ≠ SR is admitted from the start. It must also be stressed that a collection with the group property is not a group unless further conditions are fulfilled. Before introducing these we give examples of various collections with the group property:

      EXAMPLE 1.1. We first consider the natural numbers under addition. This collection has already been mentioned in defining the group property—which it evidently possesses. In dealing with such collections, in which two laws of combination are recognized (addition and multiplication), it is of course usual to employ the special symbol + to indicate combination by addition. The associative property a(bc) = (ab)c may be written without ambiguity as

      a + (b + c) = (a + b) + c

      A second important property is ab = ba or, with the + notation,

      a + b = b + a

      The elements of the collection are in this case said to commute and the law of combination to be commutative.

      EXAMPLE 1.2. Let us now take the natural numbers under multiplication. The product under multiplication is in this case the simple arithmetic product of two numbers. Again, to avoid ambiguity, the law of combination may be indicated explicitly by the × sign. The law is again associative [e.g. 3 × (4 × 2) = (3 × 4) × 2] and a(bc) = (ab)c may in this case be written

      a × (b × c) = (a × b) × c

      The collection is clearly commutative under multiplication also: ab = ba or, with the × notation,

      a × b = b × a

      In both the preceding examples the order in which the elements are combined is irrelevant. Collections in which the law of combination is commutative are termed Abelian: but many of those we shall meet are non-Abelian and the reference to an ordered pair must be carefully observed. The following example introduces a non-commutative law of combination:

      EXAMPLE 1.3. Let us consider operations which bring the lamina of Fig. 1.2 into a position indistinguishable from that which it originally occupied. Such an operation (which need not be mechanically feasible) is said to bring the system into self-coincidence and is called a symmetry operation. If we single out one point (+) on the lamina, and identify different operations by observing what happens to this point, it is clear that the lamina is brought into self-coincidence by the six operations indicated in Fig. 1.2(a–f). So long as we ignore the possibility of turning the lamina over, this is the full set of symmetry operations. The symbols beneath the six results are merely conventional names for the operations which produce them. We include the so-called identity operation E stand for rotations through 120° anti-clockwise (+ve) and clockwise (–ve) while σ(1), σ(2) and σ(3) stand for reflections which send all points on one side of a broken line (including the +) into their images on the other side. It is not necessary to define positive rotation through 240° as another . We note also that if the final positions of the + signs in Fig. 1.2 are superimposed in a single triangle they form a pattern (Fig.1.2(g)) or set of equivalent points which has the full symmetry of the lamina itself. The number of equivalent points in such a set is the number of symmetry operations in the collection (including E).

      FIG. 1.2. Symmetry operations. (a)–(f) indicate operations bringing the triangular lamina into self-coincidence. (g) indicates six equivalent points.

      We now define a law of combination by agreeing that RS (R and S denoting any two symmetry operations) shall stand for operation S followed by operation R: symmetry operations may be combined by sequential performance. The convention of putting the first operation on the right operates first. Once introduced, it must be carefully observed in all that follows. Since the combined effect of R and S is, by definition, to send the lamina into self-coincidence, it follows that RS = T where T is one of the six possible symmetry operations, and the = implies equivalence of effect (i.e. RS and T each bring the + into the same final position). To give an example, σ(1)C3 represents the operation C3 followed by reflection σ(1) : but σ(1) operating on the result shown in Fig. 1.1(b) gives that shown in Fig. 1.1(e)—which is also the result of the single operation σ(2). We therefore write

      σ(1)C3 = σ(2)

      It is again stressed that writing the symbols side by side indicates sequential performance of the operations which they represent, and that the = sign indicates equivalence, as regards final effect. It is at once clear that the symbols must now, in general, be regarded as non-commutative: for reversal of order shows that C3σ(1) = σ(3) ≠ σ(1)C3.

      1.2. Definition of a Group

      We now complete the formal definition of a group. A collection of elements {A, B, . . . , R, . . . } is a group G if

      (1.2.1)

      In a non-Abelian group it is clearly important to know whether ER is the same as RE: if E is a right unit is it also a left unit? A simple argument shows that this is so: for if we let ER = S, then E = E(RR−1) = SR−1; but if R−1 is the unique right inverse of R this means S = R and hence ER = R. A similar argument shows that R−1R = E : a right inverse is also a left inverse.

      We now re-examine the collections discussed in section 1.1.

      EXAMPLE 1.4. The collection of natural numbers, combined under addition, satisfies (a) but does not form a group because there is no unit—no number x such that n + x = x + n = n. We may remedy this omission by introducing a unit under addition which we call 0 with this property: n + 0 = 0 + n = n. But the collection is still not a group because n has no inverse, xn say, such that n + xn = xn + n = 0. To remove this defect we must include the negative integers, written − 1, − 2, etc. and defined by n + (− n) = (− n) + n = 0. The resultant collection { . . . − n, . . . , − 2, − 1, 0, 1, 2, . . . , n, . . . } is then a group under addition.

      EXAMPLE 1.5. The collection of natural numbers, combined under multiplication, satisfies conditions (a) and (b), 1 being the unit under multiplication because n × 1 = 1 × n = n. But the collection is not a group unless we admit also the inverses written n−1 or 1/n such that n(n−1) = (n−1)n = 1. And if we admit n−1 we must, according to (a), admit all numbers of the form mn−1 (or m/n). The collection of all positive rational numbers (excluding zero) is thus a group under multiplication.

      EXAMPLE 1.6. The collection of symmetry operations of an equilateral triangle (example 1.3) obviously satisfies both (a) and (b). It satisfies (c) because for every operation we can find an operation which restores the lamina to its original position ; for every element there is an inverse. It should be noted that an element and its inverse need not be different members of the collection: each of the reflections, for example, is its own inverse. The collection {E, C, σ(1), σ(2), σ(3)} is therefore a group. it is the symmetry group usually denoted by C3ν and treated more fully in chapter 3.

      1.3. The Multiplication Table

      If a group G contains g elements, g is called its order. We shall be concerned mainly with groups of finite order—finite groups—and in this case the properties of the group are conveniently summarized in a multiplication table which sets out systematically the products of all g² pairs of elements.

      EXAMPLE 1.7. Let us consider two finite groups, with ordinary multiplication as the law of combination:

      G1 = {1, − 1},  G2 = {1, − 1, i, − i}

      The multiplication tables of G1 and G2 are

      G1 :

      G2:

      The rows and columns of such tables are labelled by the group elements while the body of the table contains their g² possible products: the element equivalent to RS (in that order) is placed at the intersection of the R-row and the S-column.

      Generally the multiplication table is the array each entry being written as the single element to which it is equivalent. Such a table exhibits a certain structure which is sufficient to specify an abstract group.

      EXAMPLE 1.8. Consider the group G2 in example 1.7 and give the elements new names, A, B, C, D. The table then reads

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