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Lie Groups
Chapter 3
Lie Groups It is now time to formulate a definition of a Lie group and to describe some of the major properties of such groups. Readers whose interests lie only in the applications to solid state physics (where only finite groups appear) may safely omit this chapter.
Definition
of a linear Lie group
A Lie group embodies three different forms of mathematical structure. Firstly, it satisfies the group axioms of Chapter 1 and so has the group structure described in Chapter 2. Secondly, the elements of the group also form a "topological space", so that it may be described as being a special case of a "topological group". Finally, the elements also constitute an "analytic manifold'. Consequently a Lie group can be defined in several different (but equivalent) ways, depending on the degree of emphasis that is being accorded to the various aspects. In particular, it can be defined as a topological group with certain additional analytic properties (Pontrjagin 1946, 1986) or, alternatively, as an analytic manifold with additional group properties (Chevalley 1946, Adams 1969, Varadarajan 1974, Warner 1971). Both of these formulations involve the introduction of a series of ancillary concepts of a rather abstract nature. Very fortunately, every Lie group that is important in physical problems is of a type, known as a "linear Lie group", for which a relatively straightforward definition can be given. As will be seen, this definition is both precise and simple, in that it involves only familiar concrete objects such as matrices and contains no mention of topological spaces or analytic manifolds. (Readers who are interested in the general definition of a Lie group in terms of analytic manifolds may, for example, find this formulation in Appendix J of Cornwell (1984).) The basic feature of any Lie group is that it has a non-countable number of elements lying in a region "near" its identity and that the structure of this region both very largely determines the structure of the whole group and 35
GROUP T H E O R Y IN P H Y S I C S
36
is itself determined by its corresponding real Lie algebra. To ensure that this is so, the elements in this region must be parametrized in a particular analytic way. Of course, to say that certain elements are "near" the identity means that a notion of "distance" has to be composed, and it is here that the complications of the general treatment start. However, all the Lie groups of physical interest are "linear", in the sense that they have at least one faithful finite-dimensional representation. This representation can be used to provide the necessary precise formulation of distance and to ensure that all the other topological requirements are automatically observed. Definition Linear Lie group of dimension n A group ~ is a linear Lie group of dimension n if it satisfies the following conditions (A), (B), (C) and (D): (A) G must possess at least one faithful finite-dimensional representation r. Suppose that this representation has dimension m. Then the "distance" d(T, T') between two elements T and T' of G may be defined by m
d(T, T') = + ( E
m
~" I r(T)jk - r(T')jk 12}~/2.
j=lk=l
(This distance function d(T, T') will be called the "metric".) Then (i) d(T', T) - d(T, T'); (ii) d(T, T) - O; (iii) d(T, T') > O if T ~= T'; (iv) if T, T' and T" are any three elements of G,
d(T, T") < d(T, T') + d(T', T"), all of which are essential for the interpretation of d(T, T ~) as a distance. (The choice of this metric implies that the group is being endowed with the topology of the m2-dimensional complex Euclidean space C m2 (see Example II of Appendix B, Section 2).) The set of elements T of G such that d(T, E) < 6, where ti is positive real number, is then said to "lie in a sphere of radius 5 centred on the identity E", which will be denoted by M~. Such a sphere will be sometimes referred to as a "small neighbourhood" of E. (B) There must exist a ~ > 0 such sphere M~ of radius 6 centred n real parameters Xl,X2,...,Xn sponding to the same element T by x l = x2 . . . . - xn = O.
that every element T of G lying in the on the identity can be parametrized by (no two such sets of parameters correof G), the identity E being parametrized
LIE GROUPS
37
Thus every element of M~ corresponds to one and only one point in an n-dimensional real Euclidean space ]Rn, the identity E corresponding to the origin (0, 0 , . . . , 0) of IRn. Moreover, no point in ] a n corresponds to more than one element T in M6. (C) There must exist a ~7 > 0 such that every point in IRn for which n
E xj2 < ~72
(3.2)
j=l
corresponds to some element T in M~.
The set of point elements T so obtained will be denoted by Rv. Thus Rv is a subset of M~, and there is a one-to-one correspondence between elements of G in Rv, and points in I ~ n satisfying Condition (3.2). The final set of conditions ensures that in terms of this parametrization the group multiplication operation is expressible in terms of analytic functions. Let T ( x l , x 2 , . . . , X n ) denote the element of G corresponding to a point satisfying Condition (3.2) and define F ( x l , x 2 , . . . ,xn) by r(x~,x2,...,x~) = r ( T ( x ~ , x 2 , . . . , x ~ ) ) for all (Xl,X2,...,Xn) satisfying Condition (3.2). (D) Each of the matrix elements of F(Xl,X2,...,Xn) must be an analytic function of X l , X 2 , . . . , X n for all ( x l , x 2 , . . . , x n ) satisfying Condition
(~.2). The term "analytic" here means that each of the matrix elements Fjk must be expressible as a power series in Xl - x ~ x~ X n - Xn0 for all (x ~ 1 7 6 ~ satisfying Condition (3.2). This implies that all the derivatives OFjk/OXp, 02Fjk/OXPOx q etc. must exist for all j , k - 1 , 2 , . . . , m at all points satisfying Condition (3.2), including in particular the point (0,0,... ,0) (Fleming 1977). In particular one can define the n m • m matrices al, a 2 , . . . , an by (ap)~k =
(orjk/OxP)~=~
.....
~=o
.
(3.3)
These conditions together imply the following very important theorem. T h e o r e m I The matrices a l , a 2 , . . . ,an defined by Equation (3.3) form the basis for a n-dimensional real vector space. Proof See, for example, Chapter 3, Section 1, of Cornwell (1984).
It should be noted that, although a l , a 2 , . . . ,an form the basis of a real vector space, there is no requirement that the matrix elements of these matrices need be real. (This point is demonstrated explicitly in Example III.)
GROUP T H E O R Y IN PHYSICS
38
It will be shown in Chapter 8 that the matrices ai, a 2 , . . . , an actually form the basis of a "real Lie algebra", a vital observation on which most of the subsequent theory is founded. However, the rest of the present chapter will be devoted to "group theoretical" aspects of linear Lie groups. The above definition requires a parametrization only of the group elements belonging to a small neighbourhood of the identity element. In some cases this parametrization by a single set of n parameters x i, x 2 , . . . , xn is valid over a large part of the group or even over the whole group, but this is not essential. In Section 2 it will be shown that the whole of the "connected" subgroup of a linear Lie group of dimension n can be given a parametrization in terms of a single set of n real numbers which will be denoted by yi, y 2 , . . . , yn. However, this latter parametrization is not required to satisfy all the conditions of the above definition, and so need bear little relation to the parametrization by Xi~
X2~
9
9 X 9n .~
The following examples have been chosen because they illustrate all the essential points of the definition without involving any heavy algebra. E x a m p l e I The multiplicative group of real numbers As in Example I of Chapter 1, Section 1, let g be the group of real numbers t (t ~- 0) with ordinary multiplication as the group multiplication operation, the identity E being the number 1. g has the obvious one-dimensional faithful representation F(t) = [t], so condition (A) is satisfied and the metric d of Equation (3.1) is given by d(t,t') = I t - t'l. In particular, d(t, 1) = I t - 1 I. Let 5 = ~i so that 89< t < 2for all t in M~. A convenient parametrization for t E M~ is then t - exp xi.
(3.4)
As required in (B), the identity 1 corresponds to x i = 0. Condition (C) is obeyed with ~ = log 3, as x 2 < (log 3)2 implies 2 < exp x i < 3. By Equation (3.4) F(xi) = e x p x i , which is certainly analytic, so that condition (D) is satisfied. Thus g is a linear Lie group of dimension 1. It should be noted that Equation (3.3) implies that ai = [1], thereby confirming the first theorem above. It is significant that the parametrization in Equation (3.4) extends to all t > 0 (with - o c < x i < +oc) and that this set forms a subgroup of g. Moreover, every group element t such that t < 0 can be written in the form t = ( - 1 ) e x p x i for some xi. E x a m p l e I I The groups 0(2) and SO(2) 0(2) is the group of all real orthogonal 2 • 2 matrices A, SO(2) being the subgroup for which det A = +1. If A E 0(2), F ( A ) = A provides a faithful finite-dimensional representation. The orthogonality conditions A A = A A = 1 require that (All)2 d- (Ai2) 2
--
(Aii)2 + (A2i)2 = (A2i)2 + (A22)2
-
(Ai2) 2 + (A22) 2 - i
(3.5)
LIE GROUPS and
39
AliA21 + A22A12
= AliA12 + A22A21 = 0.
(3.6)
Equations (3.5) imply that (All) 2 = (A22) 2 and (A12) 2 = (A21) 2, so that there are only two sets of solutions of Equations (3.6), namely: (i) All = A22 and A12 = -A21. Equations (3.5) imply that det A = +1, i.e. A e SO(2). Moreover, from Equations (3.5), d(A, 1) = 2(1 - All) 1/2. (ii) All - -A22 and A12 = A21. In this case det A = - 1 and d(A, 1) = 2. With the choice fi = v/2, condition (B) requires the parametrization of part of set (i) but it is not necessary to include set (ii), as it is completely outside M~. A convenient parametrization is [ t
= r(A)
cosxl -sinxl
=
sinxl I cosxl "
(3.7)
Clearly x l = 0 corresponds to the group identity 1 and the dimension n is 1. Every point of IR 1 such that x~ < (7~/3) 2 gives a matrix A in M~, so condition (C) is satisfied. In fact the parametrization of Equation (3.7) extends to the whole of the set (i) with - ~ <_ x 1 < 7T, that is, to the whole of SO(2). Condition (D) is obviously obeyed, so 0(2) and SO(2) are both linear Lie groups of dimension 1. Further, Equation (3.3) gives
[Ol
al
-1
--
0
'
again confirming the first theorem above. Although the parametrization of Equation (3.7) extends to the whole of SO(2), it cannot apply to the set (ii). However, every n of set (ii) can be written as A-
[ 0 11 [ c~ -1
0
-sinxl
sinxl ]--[--sinxl
cosxl
cosxl
c~
sinxl
]
(3.8)
for some xl such that -zr < xl _< lr. E x a m p l e I I I The group SU(2) SU(2) is the group of 2 • 2 unitary matrices u with det u = 1. A faithful finitedimensional representation is provided by F ( u ) = u. The defining conditions imply that every u E SU(2) has the form u=
-/F*
a*
'
(3.9)
where a a n d / 3 are two complex numbers such that [a[ 2 + [/312 = 1. With a = al + ia2, ~ = ~1 + i/32 (al,a2,/31,f12 being real), this latter condition becomes a l2 + a 2 +/312 +/322 = 1. An appropriate parametrization is then OL2 -'- X 3 / 2 , /~1 = X2/2, ~2 -- Xl/2, ~1 --- +{1 - (1/4)(x 2 + x 2 + x 2 ) } 1/2,
GROUP T H E O R Y IN PHYSICS
40
for then xl - x2 - x3 ----0 corresponds to the identity 1, and d(u, 1) = 211 - {1 - (1/4)(x 2 § x 2 + x2)}1/2] 1/2, so that d(u, 1) < ~ if and only if x 2 + x 2 + x 2 < {2~ 2 -Z~i 1 4 } 1/2 . Thus, with < 2v/2 and r / < 2~ 2 - ~1~4 , conditions (S) and (C) are satisfied and M6 and R n coincide. Condition (D) is clearly true, so SU(2) is a linear Lie group of dimension 3. Incidentally, Equation (3.3) gives
al=~
1[0
i
0
,a2=~
if 0 1] -1
0
,a3=~
1[/ 0] 0
-i
'
/310/ "
so that the first theorem above is yet again confirmed. Although this parametrization is the most convenient for establishing that SU(2) is a linear Lie group, it is not the most useful for some practical calculations. Indeed only the matrices u with c~1 >_ 0 can be parametrized this way, whereas it will be shown in Example III of Section 2 that there exist parametrizations of the whole of SU(2). There is no difficulty in principle in generalizing the arguments used in Examples II and III to show that for all g >_ 2, o ( g ) , SO(N), V(N) and 1) g 2 and SU(N) are linear Lie groups of dimensions 51 N ( g - 1) , 8 9 N 2 - 1 respectively, but the detailed algebra is rather more lengthy. (U(1) is a special case that is very easy to treat along the lines of Example I, because u = [expixl], -Tr < XI __< 71", is a parametrization.) Finally, a Lie subgroup can be defined in the obvious way. D e f i n i t i o n Lie subgroup of a linear Lie group A subgroup G' of a linear Lie group G that is itself a linear Lie group is called a "Lie subgroup" of G.
2
The connected group
components
o f a l i n e a r Lie
D e f i n i t i o n Connected component of a linear Lie group A maximal set of elements T of G that can be obtained from each other by continuously varying one or more of the matrix elements F(T)jk of the faithful finite-dimensional representation r is said to form a "connected component" of G. (It can be shown that the concept of connectedness as defined for a general topological space (Simmons 1963) is equivalent, for the type of space being considered here, to that implied by the above definition.) E x a m p l e I The multiplicative group of real numbers This group was considered in Example I of Section 1. The set t > 0 forms
41
LIE GROUPS
one connected component (which actually constitutes a subgroup) and the set t < 0 forms another connected component. As t = 0 is excluded from the group, one set cannot be obtained continuously from the other. E x a m p l e I I The groups 0(2) and SO(2) In the group 0(2) that was examined in Example II of Section 1, every matrix A of SO(2) can be parametrized by Equation (3.7) with -Tr <_ x~ _< 7r, whereas if A is a member of the set (ii) (i.e. if det A = - 1 ) , A can be written in the form of Equation (3.8). Thus SO(2) constitutes one connected component and the set (ii) is another connected component. It is obvious that these two sets cannot be connected to each other, because in a connected component det F(T) must vary continuously with T (if it varies at all), but det A cannot take any values between +1 and - 1 for A E 0(2). These examples suggest the following general theorem. T h e o r e m I The connected component o f a linear Lie group G that contains the identity E is an invariant subgroup of G. This component is often referred to as "the connected subgroup of ~". Moreover, each connected component of a linear Lie group 6 is a right coset of the connected subgroup. Proof See, for example, Chapter 3, Section 2, of Cornwell (1984).
In principle G may have a countably infinite number of connected components, but in all cases of physical interest this number is finite. The axioms imply that the connected subgroup is always a linear Lie group. D e f i n i t i o n Connected linear Lie group A linear Lie group is said to be "connected" if it possesses only one connected component. Thus the whole of a connected linear Lie group of dimension n can be parametrized by n real numbers Yl,Y2,-..,Yn which form a connected set in IRn in such a way that all the matrix elements F(T)jk are continuous functions of the parameters. There is no requirement that these functions be analytic nor that they provide a one-to-one mapping. Consequently this parametrization does not necessarily satisfy all the conditions appearing in the definition of a linear Lie group. As the sets x l , x 2 , . . . , x n and y l , y 2 , . . . , Y n are required for different purposes, they need not be interchangeable. The parametrizations do coincide in Examples I and II above, but Example III below reflects the general situation. E x a m p l e I I I The group SU(2) Every pair of complex numbers a a n d / 3 of Equation (3.9) that satisfy the condition lal 2 + 1/312 = 1 can be written as a = cosy1 exp(iy2), /3 -- sin Yl exp(iy3),
GROUP THEORY IN PHYSICS
42 where
(3.11)
0 ~ Yl _~ 7r/2, 0 ~ Y2 _< 2zr, 0 < Y3 _< 2zr. Thus u = r(u) =
cos yl exp(iy2) - sin yl exp(-iy3)
sinylexp(iy3)
cos y i exp ( - i y2 )
I
'
(312)
whose matrix elements axe obviously continuous functions of Yl, Y2 and Y3. This is therefore a parametrization of the whole of SU(2). (This parametrization fails to satisfy the conditions involved in the definition of a linear Lie group because it does not provide a one-to-one mapping of the appropriate regions, for the identity corresponds to the whole set of points yl - 0, Y2 -- 0, 0 _~ Y3 __ 27r. Consequently or/Oy3 = 0 at yl = y2 = y3 = 0.) Similar arguments show that SO(N) and SU(N) are connected linear Lie groups for all N > 2, as is U(N) for all N >_ 1. The relationship between a connected linear Lie group and its corresponding real Lie algebra will be studied in some detail in Chapter 8, where it will be shown that the Lie algebra very largely determines the structure of the group. Indeed, it is for this purpose that the parametrization in terms of x l, x 2 , . . . , xn is required. However, the rest of this chapter is devoted to certain "global" properties of linear Lie groups, and for these it is the parametrization in terms of Yl, y 2 , . . . , yn that is relevant.
3
T h e c o n c e p t of c o m p a c t n e s s for linear Lie groups
Although the concept of a "compact" set in a general topological space has a curiously elusive quality, the following theorem, often referred to as the "Heine-Borel Theorem", provides a very straightforward characterization of such sets in finite-dimensional real and complex Euclidean spaces. As this will suffice to distinguish a compact linear Lie group from a non-compact linear Lie group, no attempt will be made to give a detailed account of compactness, nor even a definition of the notion. (A lucid account of this and other general topological ideas may be found in the book of Simmons (1963).) T h e o r e m ! A subset of points of a real or complex finite-dimensional Euclidean space is "compact" if and only if it is closed and bounded. As mentioned in Section 1, by introducing the faithful m-dimensional representation F, the Lie group has been endowed with the topology of C m2. However, it is often helpful to invoke the continuous parametrization of the connected subgroup by yl, y 2 , . . . , Yn introduced in Section 2. As the continuous image of a compact set is always another compact set (Simmons 1963), it
LIE GROUPS
43
follows that if the linear Lie group has only a finite number of connected components and the parameters yl, Y2,..., Yn range over a closed and bounded set in IR ~, then the group is compact. A "bounded" set of a real or complex Euclidean space is merely a set that can be contained in a finite "sphere" of the space. The term "closed" implies something more involved, so perhaps a few words of explanation may be needed. Although the specification of a general closed set can be fairly difficult, the only subsets of IRn that are relevant here are connected, and for these the characterization is straightforward. Indeed, in ]R 1 every connected closed set is of the form a l <_ yl _ bl. Similarly, the set aj ~_ yj ~_ bj, j -- 1, 2 , . . . , n, of IR~ is closed, but if any of the end points aj and bj are not attained the set is not closed. This set is bounded if and only if all the aj and by are finite. These considerations imply the following identification.
C h a r a c t e r i z a t i o n Compact linear Lie group of dimension n A linear Lie group of dimension n with a finite number of connected components is compact if the parameters Yl, Y2,..., Yn range over the closed finite intervals aj ~_ yj ~_ bj, j - 1, 2 , . . . , n. The Lie groups of physical interest that are non-compact usually fail to be compact by virtue of giving an unbounded set of matrix elements in @m2 . As the sets of matrix elements F(T)jk of a linear Lie group G are bounded if and only if there exists a finite real number M such that d(T, E) < M for all T E G, such groups are very easy to recognize in practice. If a Lie group ~ is compact, then every Lie subgroup $ of G must also be compact (except in the rare case when S has a "non-closed" parametrization). On the other hand, if G is non-compact, S may easily possess compact Lie subgroups. For semi-simple Lie groups there exists a criterion for compactness that is expressed purely in Lie algebraic terms, as will be shown in Chapter 11, Section 10. The real importance of the distinction between compact and non-compact groups lies in the fact that the representation theory of compact Lie groups is very largely the same as that for finite groups, whereas for non-compact groups the theory is entirely different. E x a m p l e I The multiplicative group of real numbers As noted in Example I of Section 1, a faithful one-dimensional representation of this group is provided by F(t) - It]. Obviously this set is unbounded in C 1, so the group is non-compact. E x a m p l e I I The groups O(N) and SO(N) For 0(2) and SO(2), Examples II of Sections 1 and 2 imply that the range of the only parameter y l ( = Xl) is - l r <_ yl _< lr. Similar statements are true for O(N) and SO(N) for N _> 3, and consequently o ( g ) and SO(N) are compact
GROUP THEORY IN PHYSICS
44 for all N > 2.
E x a m p l e I I I The groups U(N) and SU(N) As all the intervals in Conditions (3.11) are closed and finite, SU (2) is compact. The same is true of SU(N) for all N > 2, and of U(N) for all N > 1.
4
Invariant integration
If to each element T of a group g a complex number f(T) is assigned, then f(T) is said to be a "complex-valued function defined on g". One example that has been met already is the set of matrix elements F(T)jk (for j, k fixed) of a matrix representation F of g. For a finite group sums of the form ETEg f (T) are frequently encountered, particularly in representation theory. Because the Rearrangement Theorem shows that the set {T'T; T E g} has exactly the same members as G, it follows that for any T' E
E f(T'T)= E TEg
f(T),
TEg
and the sum is said to be "left-invariant". Similarly
E f(TT') = E f(T), T6g
TEg
so such sums are also "right-invariant". Moreover, with f(T) = 1 for all T E G, the sum is finite in the sense that ~-'~Teg 1 -- g, the order of G. In generalizing to a connected linear Lie group, it is natural to make the hypothesis that the sum can be replaced by an integral with respect to the parameters Yl,y2,...,yn. However, questions immediately arise about the left-invariance, right-invariance and finiteness of such integrals. For general topological groups these become problems in measure theory. Using this theory Haar (1933) showed that for a very large class of topological groups, which includes the linear Lie groups, there always exists a left-invariant integral and there always exists a right-invariant integral. (Accounts of these developments, including proofs of the theorems that follow, may be found in the books of nalmos (1950), Loomis (1953) and Hewitt and Ross (1963).) Let
f(T) diT -
dyl ... 1
and
f(T) d~T -
dyn f(T(yl,..., Yn))az(yl,..., Yn)
(3.13)
n
/bl /abn dyl..,
1
dyn f(T(yl,..., yn))a~(yl,..., Yn)
(3.14)
n
be the left- and right-invariant integrals of a linear Lie group G, so that
fJg
=
fJg I(T)d
(3.15)
45
LIE GROUPS
(3.16)
fG f ( T T ' ) d ~ T = ff~ f ( T ) d ~ T
for any T ~ E G and any function f ( T ) for which the integrals are well defined. Here a z ( y l , . . . , y ~ ) and a ~ ( y l , . . . , y n ) are left- and right-invariant "weight functions", which are each unique up to multiplication by arbitrary constants. The left- and right-invariant integrals may be said to be f i n i t e if ~
~
b
l
L
b
d t T =-
n
dyl . . .
dyn az (Yl , . . . , Yn )
1
n
and d~T -
dyn ar (Yl,
dyl . . . 1
9 9Yn) 9
n
are finite. If the multiplicative constants can be chosen so that al (Yl,..., Yn) and a t ( y 1 , . . . ,y~) are equal, so that the integrals are both left- and rightinvariant, then G is said to be "unimodular', and one may write dIT = d~T = dT
and (71(Yl,
. . . , Yn)
--
6rr(Yl,...,Yn)
--
o'(yx,...,yn).
If G has more than one connected component, the integrals in Equations (3.13) and (3.14) can be generalized in the obvious way to include a sum over the components. The significance of the distinction between compact and non-compact Lie groups lies in the first two of the following theorems, the first of which was originally proved by Peter and Weyl (1927). They imply that compact Lie groups have many of the properties of finite groups, summation over a finite group merely being replaced by an invariant integral over the compact Lie groups, whereas for non-compact groups the situation is completely different. T h e o r e m I If G is a compact Lie group, then G is u n i m o d u l a r and the invariant integral I(T) dT -
dye...
exists and is finite for every continuous function f ( T ) . be chosen so that dT -
dyn a ( y l , . . . , Yn) '- 1.
dyl . . . 1
Thus a ( y l , . . . , Yn) can
n
(A function f ( T ) is continuous if and only if f ( T ( y l , . . . , function of y l , . . . , yn.)
y n ) ) is a continuous
T h e o r e m I I If ~ is a n o n - c o m p a c t Lie group then the left- and rightinvariant integrals are both infinite.
46
GROUP T H E O R Y IN PHYSICS
For non-compact groups the question of when G is unimodular is partially answered by the following theorem. T h e o r e m III
If G is Abelian or semi-simple then G is unimodular.
The definition of a semi-simple Lie group is given in Chapter 11, Section 2. The other non-compact linear Lie groups have to be investigated individually. In practice, explicit expressions for weight functions are seldom needed. Indeed, in dealing with the compact Lie groups all that is usually required is the knowledge (embodied in the first theorem above) that finite left- and right-invariant integrals always exist.
Lie Groups It is now time to formulate a definition of a Lie group and to describe some of the major properties of such groups. Readers whose interests lie only in the applications to solid state physics (where only finite groups appear) may safely omit this chapter.
Definition
of a linear Lie group
A Lie group embodies three different forms of mathematical structure. Firstly, it satisfies the group axioms of Chapter 1 and so has the group structure described in Chapter 2. Secondly, the elements of the group also form a "topological space", so that it may be described as being a special case of a "topological group". Finally, the elements also constitute an "analytic manifold'. Consequently a Lie group can be defined in several different (but equivalent) ways, depending on the degree of emphasis that is being accorded to the various aspects. In particular, it can be defined as a topological group with certain additional analytic properties (Pontrjagin 1946, 1986) or, alternatively, as an analytic manifold with additional group properties (Chevalley 1946, Adams 1969, Varadarajan 1974, Warner 1971). Both of these formulations involve the introduction of a series of ancillary concepts of a rather abstract nature. Very fortunately, every Lie group that is important in physical problems is of a type, known as a "linear Lie group", for which a relatively straightforward definition can be given. As will be seen, this definition is both precise and simple, in that it involves only familiar concrete objects such as matrices and contains no mention of topological spaces or analytic manifolds. (Readers who are interested in the general definition of a Lie group in terms of analytic manifolds may, for example, find this formulation in Appendix J of Cornwell (1984).) The basic feature of any Lie group is that it has a non-countable number of elements lying in a region "near" its identity and that the structure of this region both very largely determines the structure of the whole group and 35
GROUP T H E O R Y IN P H Y S I C S
36
is itself determined by its corresponding real Lie algebra. To ensure that this is so, the elements in this region must be parametrized in a particular analytic way. Of course, to say that certain elements are "near" the identity means that a notion of "distance" has to be composed, and it is here that the complications of the general treatment start. However, all the Lie groups of physical interest are "linear", in the sense that they have at least one faithful finite-dimensional representation. This representation can be used to provide the necessary precise formulation of distance and to ensure that all the other topological requirements are automatically observed. Definition Linear Lie group of dimension n A group ~ is a linear Lie group of dimension n if it satisfies the following conditions (A), (B), (C) and (D): (A) G must possess at least one faithful finite-dimensional representation r. Suppose that this representation has dimension m. Then the "distance" d(T, T') between two elements T and T' of G may be defined by m
d(T, T') = + ( E
m
~" I r(T)jk - r(T')jk 12}~/2.
j=lk=l
(This distance function d(T, T') will be called the "metric".) Then (i) d(T', T) - d(T, T'); (ii) d(T, T) - O; (iii) d(T, T') > O if T ~= T'; (iv) if T, T' and T" are any three elements of G,
d(T, T") < d(T, T') + d(T', T"), all of which are essential for the interpretation of d(T, T ~) as a distance. (The choice of this metric implies that the group is being endowed with the topology of the m2-dimensional complex Euclidean space C m2 (see Example II of Appendix B, Section 2).) The set of elements T of G such that d(T, E) < 6, where ti is positive real number, is then said to "lie in a sphere of radius 5 centred on the identity E", which will be denoted by M~. Such a sphere will be sometimes referred to as a "small neighbourhood" of E. (B) There must exist a ~ > 0 such sphere M~ of radius 6 centred n real parameters Xl,X2,...,Xn sponding to the same element T by x l = x2 . . . . - xn = O.
that every element T of G lying in the on the identity can be parametrized by (no two such sets of parameters correof G), the identity E being parametrized
LIE GROUPS
37
Thus every element of M~ corresponds to one and only one point in an n-dimensional real Euclidean space ]Rn, the identity E corresponding to the origin (0, 0 , . . . , 0) of IRn. Moreover, no point in ] a n corresponds to more than one element T in M6. (C) There must exist a ~7 > 0 such that every point in IRn for which n
E xj2 < ~72
(3.2)
j=l
corresponds to some element T in M~.
The set of point elements T so obtained will be denoted by Rv. Thus Rv is a subset of M~, and there is a one-to-one correspondence between elements of G in Rv, and points in I ~ n satisfying Condition (3.2). The final set of conditions ensures that in terms of this parametrization the group multiplication operation is expressible in terms of analytic functions. Let T ( x l , x 2 , . . . , X n ) denote the element of G corresponding to a point satisfying Condition (3.2) and define F ( x l , x 2 , . . . ,xn) by r(x~,x2,...,x~) = r ( T ( x ~ , x 2 , . . . , x ~ ) ) for all (Xl,X2,...,Xn) satisfying Condition (3.2). (D) Each of the matrix elements of F(Xl,X2,...,Xn) must be an analytic function of X l , X 2 , . . . , X n for all ( x l , x 2 , . . . , x n ) satisfying Condition
(~.2). The term "analytic" here means that each of the matrix elements Fjk must be expressible as a power series in Xl - x ~ x~ X n - Xn0 for all (x ~ 1 7 6 ~ satisfying Condition (3.2). This implies that all the derivatives OFjk/OXp, 02Fjk/OXPOx q etc. must exist for all j , k - 1 , 2 , . . . , m at all points satisfying Condition (3.2), including in particular the point (0,0,... ,0) (Fleming 1977). In particular one can define the n m • m matrices al, a 2 , . . . , an by (ap)~k =
(orjk/OxP)~=~
.....
~=o
.
(3.3)
These conditions together imply the following very important theorem. T h e o r e m I The matrices a l , a 2 , . . . ,an defined by Equation (3.3) form the basis for a n-dimensional real vector space. Proof See, for example, Chapter 3, Section 1, of Cornwell (1984).
It should be noted that, although a l , a 2 , . . . ,an form the basis of a real vector space, there is no requirement that the matrix elements of these matrices need be real. (This point is demonstrated explicitly in Example III.)
GROUP T H E O R Y IN PHYSICS
38
It will be shown in Chapter 8 that the matrices ai, a 2 , . . . , an actually form the basis of a "real Lie algebra", a vital observation on which most of the subsequent theory is founded. However, the rest of the present chapter will be devoted to "group theoretical" aspects of linear Lie groups. The above definition requires a parametrization only of the group elements belonging to a small neighbourhood of the identity element. In some cases this parametrization by a single set of n parameters x i, x 2 , . . . , xn is valid over a large part of the group or even over the whole group, but this is not essential. In Section 2 it will be shown that the whole of the "connected" subgroup of a linear Lie group of dimension n can be given a parametrization in terms of a single set of n real numbers which will be denoted by yi, y 2 , . . . , yn. However, this latter parametrization is not required to satisfy all the conditions of the above definition, and so need bear little relation to the parametrization by Xi~
X2~
9
9 X 9n .~
The following examples have been chosen because they illustrate all the essential points of the definition without involving any heavy algebra. E x a m p l e I The multiplicative group of real numbers As in Example I of Chapter 1, Section 1, let g be the group of real numbers t (t ~- 0) with ordinary multiplication as the group multiplication operation, the identity E being the number 1. g has the obvious one-dimensional faithful representation F(t) = [t], so condition (A) is satisfied and the metric d of Equation (3.1) is given by d(t,t') = I t - t'l. In particular, d(t, 1) = I t - 1 I. Let 5 = ~i so that 89< t < 2for all t in M~. A convenient parametrization for t E M~ is then t - exp xi.
(3.4)
As required in (B), the identity 1 corresponds to x i = 0. Condition (C) is obeyed with ~ = log 3, as x 2 < (log 3)2 implies 2 < exp x i < 3. By Equation (3.4) F(xi) = e x p x i , which is certainly analytic, so that condition (D) is satisfied. Thus g is a linear Lie group of dimension 1. It should be noted that Equation (3.3) implies that ai = [1], thereby confirming the first theorem above. It is significant that the parametrization in Equation (3.4) extends to all t > 0 (with - o c < x i < +oc) and that this set forms a subgroup of g. Moreover, every group element t such that t < 0 can be written in the form t = ( - 1 ) e x p x i for some xi. E x a m p l e I I The groups 0(2) and SO(2) 0(2) is the group of all real orthogonal 2 • 2 matrices A, SO(2) being the subgroup for which det A = +1. If A E 0(2), F ( A ) = A provides a faithful finite-dimensional representation. The orthogonality conditions A A = A A = 1 require that (All)2 d- (Ai2) 2
--
(Aii)2 + (A2i)2 = (A2i)2 + (A22)2
-
(Ai2) 2 + (A22) 2 - i
(3.5)
LIE GROUPS and
39
AliA21 + A22A12
= AliA12 + A22A21 = 0.
(3.6)
Equations (3.5) imply that (All) 2 = (A22) 2 and (A12) 2 = (A21) 2, so that there are only two sets of solutions of Equations (3.6), namely: (i) All = A22 and A12 = -A21. Equations (3.5) imply that det A = +1, i.e. A e SO(2). Moreover, from Equations (3.5), d(A, 1) = 2(1 - All) 1/2. (ii) All - -A22 and A12 = A21. In this case det A = - 1 and d(A, 1) = 2. With the choice fi = v/2, condition (B) requires the parametrization of part of set (i) but it is not necessary to include set (ii), as it is completely outside M~. A convenient parametrization is [ t
= r(A)
cosxl -sinxl
=
sinxl I cosxl "
(3.7)
Clearly x l = 0 corresponds to the group identity 1 and the dimension n is 1. Every point of IR 1 such that x~ < (7~/3) 2 gives a matrix A in M~, so condition (C) is satisfied. In fact the parametrization of Equation (3.7) extends to the whole of the set (i) with - ~ <_ x 1 < 7T, that is, to the whole of SO(2). Condition (D) is obviously obeyed, so 0(2) and SO(2) are both linear Lie groups of dimension 1. Further, Equation (3.3) gives
[Ol
al
-1
--
0
'
again confirming the first theorem above. Although the parametrization of Equation (3.7) extends to the whole of SO(2), it cannot apply to the set (ii). However, every n of set (ii) can be written as A-
[ 0 11 [ c~ -1
0
-sinxl
sinxl ]--[--sinxl
cosxl
cosxl
c~
sinxl
]
(3.8)
for some xl such that -zr < xl _< lr. E x a m p l e I I I The group SU(2) SU(2) is the group of 2 • 2 unitary matrices u with det u = 1. A faithful finitedimensional representation is provided by F ( u ) = u. The defining conditions imply that every u E SU(2) has the form u=
-/F*
a*
'
(3.9)
where a a n d / 3 are two complex numbers such that [a[ 2 + [/312 = 1. With a = al + ia2, ~ = ~1 + i/32 (al,a2,/31,f12 being real), this latter condition becomes a l2 + a 2 +/312 +/322 = 1. An appropriate parametrization is then OL2 -'- X 3 / 2 , /~1 = X2/2, ~2 -- Xl/2, ~1 --- +{1 - (1/4)(x 2 + x 2 + x 2 ) } 1/2,
GROUP T H E O R Y IN PHYSICS
40
for then xl - x2 - x3 ----0 corresponds to the identity 1, and d(u, 1) = 211 - {1 - (1/4)(x 2 § x 2 + x2)}1/2] 1/2, so that d(u, 1) < ~ if and only if x 2 + x 2 + x 2 < {2~ 2 -Z~i 1 4 } 1/2 . Thus, with < 2v/2 and r / < 2~ 2 - ~1~4 , conditions (S) and (C) are satisfied and M6 and R n coincide. Condition (D) is clearly true, so SU(2) is a linear Lie group of dimension 3. Incidentally, Equation (3.3) gives
al=~
1[0
i
0
,a2=~
if 0 1] -1
0
,a3=~
1[/ 0] 0
-i
'
/310/ "
so that the first theorem above is yet again confirmed. Although this parametrization is the most convenient for establishing that SU(2) is a linear Lie group, it is not the most useful for some practical calculations. Indeed only the matrices u with c~1 >_ 0 can be parametrized this way, whereas it will be shown in Example III of Section 2 that there exist parametrizations of the whole of SU(2). There is no difficulty in principle in generalizing the arguments used in Examples II and III to show that for all g >_ 2, o ( g ) , SO(N), V(N) and 1) g 2 and SU(N) are linear Lie groups of dimensions 51 N ( g - 1) , 8 9 N 2 - 1 respectively, but the detailed algebra is rather more lengthy. (U(1) is a special case that is very easy to treat along the lines of Example I, because u = [expixl], -Tr < XI __< 71", is a parametrization.) Finally, a Lie subgroup can be defined in the obvious way. D e f i n i t i o n Lie subgroup of a linear Lie group A subgroup G' of a linear Lie group G that is itself a linear Lie group is called a "Lie subgroup" of G.
2
The connected group
components
o f a l i n e a r Lie
D e f i n i t i o n Connected component of a linear Lie group A maximal set of elements T of G that can be obtained from each other by continuously varying one or more of the matrix elements F(T)jk of the faithful finite-dimensional representation r is said to form a "connected component" of G. (It can be shown that the concept of connectedness as defined for a general topological space (Simmons 1963) is equivalent, for the type of space being considered here, to that implied by the above definition.) E x a m p l e I The multiplicative group of real numbers This group was considered in Example I of Section 1. The set t > 0 forms
41
LIE GROUPS
one connected component (which actually constitutes a subgroup) and the set t < 0 forms another connected component. As t = 0 is excluded from the group, one set cannot be obtained continuously from the other. E x a m p l e I I The groups 0(2) and SO(2) In the group 0(2) that was examined in Example II of Section 1, every matrix A of SO(2) can be parametrized by Equation (3.7) with -Tr <_ x~ _< 7r, whereas if A is a member of the set (ii) (i.e. if det A = - 1 ) , A can be written in the form of Equation (3.8). Thus SO(2) constitutes one connected component and the set (ii) is another connected component. It is obvious that these two sets cannot be connected to each other, because in a connected component det F(T) must vary continuously with T (if it varies at all), but det A cannot take any values between +1 and - 1 for A E 0(2). These examples suggest the following general theorem. T h e o r e m I The connected component o f a linear Lie group G that contains the identity E is an invariant subgroup of G. This component is often referred to as "the connected subgroup of ~". Moreover, each connected component of a linear Lie group 6 is a right coset of the connected subgroup. Proof See, for example, Chapter 3, Section 2, of Cornwell (1984).
In principle G may have a countably infinite number of connected components, but in all cases of physical interest this number is finite. The axioms imply that the connected subgroup is always a linear Lie group. D e f i n i t i o n Connected linear Lie group A linear Lie group is said to be "connected" if it possesses only one connected component. Thus the whole of a connected linear Lie group of dimension n can be parametrized by n real numbers Yl,Y2,-..,Yn which form a connected set in IRn in such a way that all the matrix elements F(T)jk are continuous functions of the parameters. There is no requirement that these functions be analytic nor that they provide a one-to-one mapping. Consequently this parametrization does not necessarily satisfy all the conditions appearing in the definition of a linear Lie group. As the sets x l , x 2 , . . . , x n and y l , y 2 , . . . , Y n are required for different purposes, they need not be interchangeable. The parametrizations do coincide in Examples I and II above, but Example III below reflects the general situation. E x a m p l e I I I The group SU(2) Every pair of complex numbers a a n d / 3 of Equation (3.9) that satisfy the condition lal 2 + 1/312 = 1 can be written as a = cosy1 exp(iy2), /3 -- sin Yl exp(iy3),
GROUP THEORY IN PHYSICS
42 where
(3.11)
0 ~ Yl _~ 7r/2, 0 ~ Y2 _< 2zr, 0 < Y3 _< 2zr. Thus u = r(u) =
cos yl exp(iy2) - sin yl exp(-iy3)
sinylexp(iy3)
cos y i exp ( - i y2 )
I
'
(312)
whose matrix elements axe obviously continuous functions of Yl, Y2 and Y3. This is therefore a parametrization of the whole of SU(2). (This parametrization fails to satisfy the conditions involved in the definition of a linear Lie group because it does not provide a one-to-one mapping of the appropriate regions, for the identity corresponds to the whole set of points yl - 0, Y2 -- 0, 0 _~ Y3 __ 27r. Consequently or/Oy3 = 0 at yl = y2 = y3 = 0.) Similar arguments show that SO(N) and SU(N) are connected linear Lie groups for all N > 2, as is U(N) for all N >_ 1. The relationship between a connected linear Lie group and its corresponding real Lie algebra will be studied in some detail in Chapter 8, where it will be shown that the Lie algebra very largely determines the structure of the group. Indeed, it is for this purpose that the parametrization in terms of x l, x 2 , . . . , xn is required. However, the rest of this chapter is devoted to certain "global" properties of linear Lie groups, and for these it is the parametrization in terms of Yl, y 2 , . . . , yn that is relevant.
3
T h e c o n c e p t of c o m p a c t n e s s for linear Lie groups
Although the concept of a "compact" set in a general topological space has a curiously elusive quality, the following theorem, often referred to as the "Heine-Borel Theorem", provides a very straightforward characterization of such sets in finite-dimensional real and complex Euclidean spaces. As this will suffice to distinguish a compact linear Lie group from a non-compact linear Lie group, no attempt will be made to give a detailed account of compactness, nor even a definition of the notion. (A lucid account of this and other general topological ideas may be found in the book of Simmons (1963).) T h e o r e m ! A subset of points of a real or complex finite-dimensional Euclidean space is "compact" if and only if it is closed and bounded. As mentioned in Section 1, by introducing the faithful m-dimensional representation F, the Lie group has been endowed with the topology of C m2. However, it is often helpful to invoke the continuous parametrization of the connected subgroup by yl, y 2 , . . . , Yn introduced in Section 2. As the continuous image of a compact set is always another compact set (Simmons 1963), it
LIE GROUPS
43
follows that if the linear Lie group has only a finite number of connected components and the parameters yl, Y2,..., Yn range over a closed and bounded set in IR ~, then the group is compact. A "bounded" set of a real or complex Euclidean space is merely a set that can be contained in a finite "sphere" of the space. The term "closed" implies something more involved, so perhaps a few words of explanation may be needed. Although the specification of a general closed set can be fairly difficult, the only subsets of IRn that are relevant here are connected, and for these the characterization is straightforward. Indeed, in ]R 1 every connected closed set is of the form a l <_ yl _ bl. Similarly, the set aj ~_ yj ~_ bj, j -- 1, 2 , . . . , n, of IR~ is closed, but if any of the end points aj and bj are not attained the set is not closed. This set is bounded if and only if all the aj and by are finite. These considerations imply the following identification.
C h a r a c t e r i z a t i o n Compact linear Lie group of dimension n A linear Lie group of dimension n with a finite number of connected components is compact if the parameters Yl, Y2,..., Yn range over the closed finite intervals aj ~_ yj ~_ bj, j - 1, 2 , . . . , n. The Lie groups of physical interest that are non-compact usually fail to be compact by virtue of giving an unbounded set of matrix elements in @m2 . As the sets of matrix elements F(T)jk of a linear Lie group G are bounded if and only if there exists a finite real number M such that d(T, E) < M for all T E G, such groups are very easy to recognize in practice. If a Lie group ~ is compact, then every Lie subgroup $ of G must also be compact (except in the rare case when S has a "non-closed" parametrization). On the other hand, if G is non-compact, S may easily possess compact Lie subgroups. For semi-simple Lie groups there exists a criterion for compactness that is expressed purely in Lie algebraic terms, as will be shown in Chapter 11, Section 10. The real importance of the distinction between compact and non-compact groups lies in the fact that the representation theory of compact Lie groups is very largely the same as that for finite groups, whereas for non-compact groups the theory is entirely different. E x a m p l e I The multiplicative group of real numbers As noted in Example I of Section 1, a faithful one-dimensional representation of this group is provided by F(t) - It]. Obviously this set is unbounded in C 1, so the group is non-compact. E x a m p l e I I The groups O(N) and SO(N) For 0(2) and SO(2), Examples II of Sections 1 and 2 imply that the range of the only parameter y l ( = Xl) is - l r <_ yl _< lr. Similar statements are true for O(N) and SO(N) for N _> 3, and consequently o ( g ) and SO(N) are compact
GROUP THEORY IN PHYSICS
44 for all N > 2.
E x a m p l e I I I The groups U(N) and SU(N) As all the intervals in Conditions (3.11) are closed and finite, SU (2) is compact. The same is true of SU(N) for all N > 2, and of U(N) for all N > 1.
4
Invariant integration
If to each element T of a group g a complex number f(T) is assigned, then f(T) is said to be a "complex-valued function defined on g". One example that has been met already is the set of matrix elements F(T)jk (for j, k fixed) of a matrix representation F of g. For a finite group sums of the form ETEg f (T) are frequently encountered, particularly in representation theory. Because the Rearrangement Theorem shows that the set {T'T; T E g} has exactly the same members as G, it follows that for any T' E
E f(T'T)= E TEg
f(T),
TEg
and the sum is said to be "left-invariant". Similarly
E f(TT') = E f(T), T6g
TEg
so such sums are also "right-invariant". Moreover, with f(T) = 1 for all T E G, the sum is finite in the sense that ~-'~Teg 1 -- g, the order of G. In generalizing to a connected linear Lie group, it is natural to make the hypothesis that the sum can be replaced by an integral with respect to the parameters Yl,y2,...,yn. However, questions immediately arise about the left-invariance, right-invariance and finiteness of such integrals. For general topological groups these become problems in measure theory. Using this theory Haar (1933) showed that for a very large class of topological groups, which includes the linear Lie groups, there always exists a left-invariant integral and there always exists a right-invariant integral. (Accounts of these developments, including proofs of the theorems that follow, may be found in the books of nalmos (1950), Loomis (1953) and Hewitt and Ross (1963).) Let
f(T) diT -
dyl ... 1
and
f(T) d~T -
dyn f(T(yl,..., Yn))az(yl,..., Yn)
(3.13)
n
/bl /abn dyl..,
1
dyn f(T(yl,..., yn))a~(yl,..., Yn)
(3.14)
n
be the left- and right-invariant integrals of a linear Lie group G, so that
fJg
=
fJg I(T)d
(3.15)
45
LIE GROUPS
(3.16)
fG f ( T T ' ) d ~ T = ff~ f ( T ) d ~ T
for any T ~ E G and any function f ( T ) for which the integrals are well defined. Here a z ( y l , . . . , y ~ ) and a ~ ( y l , . . . , y n ) are left- and right-invariant "weight functions", which are each unique up to multiplication by arbitrary constants. The left- and right-invariant integrals may be said to be f i n i t e if ~
~
b
l
L
b
d t T =-
n
dyl . . .
dyn az (Yl , . . . , Yn )
1
n
and d~T -
dyn ar (Yl,
dyl . . . 1
9 9Yn) 9
n
are finite. If the multiplicative constants can be chosen so that al (Yl,..., Yn) and a t ( y 1 , . . . ,y~) are equal, so that the integrals are both left- and rightinvariant, then G is said to be "unimodular', and one may write dIT = d~T = dT
and (71(Yl,
. . . , Yn)
--
6rr(Yl,...,Yn)
--
o'(yx,...,yn).
If G has more than one connected component, the integrals in Equations (3.13) and (3.14) can be generalized in the obvious way to include a sum over the components. The significance of the distinction between compact and non-compact Lie groups lies in the first two of the following theorems, the first of which was originally proved by Peter and Weyl (1927). They imply that compact Lie groups have many of the properties of finite groups, summation over a finite group merely being replaced by an invariant integral over the compact Lie groups, whereas for non-compact groups the situation is completely different. T h e o r e m I If G is a compact Lie group, then G is u n i m o d u l a r and the invariant integral I(T) dT -
dye...
exists and is finite for every continuous function f ( T ) . be chosen so that dT -
dyn a ( y l , . . . , Yn) '- 1.
dyl . . . 1
Thus a ( y l , . . . , Yn) can
n
(A function f ( T ) is continuous if and only if f ( T ( y l , . . . , function of y l , . . . , yn.)
y n ) ) is a continuous
T h e o r e m I I If ~ is a n o n - c o m p a c t Lie group then the left- and rightinvariant integrals are both infinite.
46
GROUP T H E O R Y IN PHYSICS
For non-compact groups the question of when G is unimodular is partially answered by the following theorem. T h e o r e m III
If G is Abelian or semi-simple then G is unimodular.
The definition of a semi-simple Lie group is given in Chapter 11, Section 2. The other non-compact linear Lie groups have to be investigated individually. In practice, explicit expressions for weight functions are seldom needed. Indeed, in dealing with the compact Lie groups all that is usually required is the knowledge (embodied in the first theorem above) that finite left- and right-invariant integrals always exist.