Structure of Internal-Symmetry Groups

4 - I N T E R N A L - S Y M M E T R Y G R O U P S A N D T H E I R A U T O M O R P H I S M S 3637

3 6 ~ . Okubo and V . Mathur, Phys. Rev. D 2, 394 (1970). part icle to be the "spinor." This freedom i s not to be

37~hys i ca l ly , only continuous automorphisms a r e confused with the statement that there i s no "intrinsic"

relevant. difference between 3, and 3, . See also Ref. 25.

"M. Gell-Mann, R. Oakes, and B. Renner, Phys. Rev. 4 2 ~ . Okubo, Phys. Rev. D 2, 3005 (1970). See also

173, 2195 (1968). - K. Mahanthappa and L. Maiani, Phys. Letters m, 499

3 9 ~ . Glashow and S. Weinberg, Phys. Rev. Letters 3, (1 9 70).

224 (1968). 4 3 ~ e also thought this to be the case. (See Ref. 5.) "s. Weinberg, Phys. Rev. Letters 9, 616 (1966). 4 4 ~ . Dashen, Phys. Rev. D 3 ,1879 (1971). 4 1 ~ e may also do the reverse and assign the lighter

P I I Y S I C A L R E V I E W D V O L U M E 4 , N U M B E R 1 2 1 5 D E C E M B E R 1 9 7 1

Structure of Internal-Symmetry Groups* T. K. Kuo

Department of P h y s i c s , Purdue L'nioersity, La faye t t e , Indza?za 47907 (Received 3 March 1971)

It i s shown that, given SU(2) and T invariance, their outer automorphisms, which must themselves be symmetries, form a one-parameter gauge group. Thus, from isospin conservation and time-reversal invariance, one gets hypercharge conservation. Similarly, from angular momentum conservation and T invariance, one has fermion-number conservation. Further, the well-hown empirical relations ( - l)Y = ( - I ) ~ ' and ( - l)F = ( - a r e derived.

I. INTRODUCTION

Recently1 it was emphasized that the automorphisms of internal-symmetry groups a r e inherent conventions associated with their physical applica- tions. Since there i s a one-to-one correspondence between conventions and symmetries, this means that the outer automorphisms of internal-symmetry groups play an important role. In fact, according to the discussions in I, if one has an internal- symmetry group G containing a degenerate subgroup St then outer automorphisms of G which leave S invariant a r e themselves symmetries. These new symmetries a r e in general hidden symmetries. Indeed. in the numerous examples inves- tigated in I, we could not decide whether the derived symmetries were degenerate o r hidden. The identification can only be done by comparison with reality.

Let u s emphasize that there i s no fundamental difference between hidden and degenerate symmetr ies. (In this connection, we may remark that the familiar te rm "spontaneously broken" symmetry i s somewhat misleading.) If G is a symmetry group, and if S i s a subgroup of G, then hidden symmetries a r i s e when the physical s tates form irreducible representations, not of G, but of S. Indeed, the cosets of S in G constitute the hidden symmetries.

The physical significance of outer automorphisms may also be visualized in another way. When we have a (degenerate) symmetry group S (elements labeled a s Sa) and some physical s tates (labeled a s I a,)), we a r e accustomed to thinking that the labeling has already been given. Further, an element 9 i s understood to operate on the states and give r i s e to a reshuffling of them. Thus, / a i ) zlai,) and S 2 S, so that

where sb = (9)-1SSc. The symmetry of our system is reflected in the equivalence of the states I a,), which may be reshuffled. On the other hand, we could have regarded the elements SC to operate on S, and leave the physical s tates unchanged: / a ( ) 21 a(), S Z s b . In other words, the operation of SC gives r i s e to a reshuffling of elements in S. The symmetry of our system i s reflected, then, in the equivalence of the elements in S. Therefore, we had in the beginning the labeling of S and I a ) , the symmetry results a s a consequence of our freedom in reshuffling the states I a) o r the elements S4. The f i r s t view i s the usual one, while the second view corresponds to interpreting Sa a s elements of the inner automorphism of S. According to this second viewpoint, i t is quite c lear that outer automorphisms a r e just a s good a s the inner ones. Having adjoined these outer automorphisms to our

symmetry group, we have still to go back to the first, and "usual," viewpoint. It i s then necessary, a s was found in I, to interpret them a s hidden symmetries, in general. The two viewpoints a r e actually complementary. If S has neither a center nor an outer automorphism (except the trivial ones), then the two views a r e the same. If S has a center C, then to regard S a s inner automorphisms actually maps C to the identity., On the other hand, if S has some outer automorphisn~s, the "usual" viewpoint becomes rather cumbersome for visual- izing the existence of these symmetries.

In this work we wish to extend the previous considerations to include antiunitary symmetry oper- a tors - the t ime-reversal operator T. One of the most interesting propert ies of T i s that phase fac- t o r s of state vectors become nontrivial under T. Taking advantage of this property, we will show that the outer automorphisms of T and SU(2) (the isospin o r rotation group) a r e realized a s phase factors. Further, being just phase factors, these newly obtained symmetries a r e necessarily degen- e ra te symmetries. By explicit construction, we will in fact find that the outer automorphisms of T and SU(2) generate a gauge group U(1). Moreover, this U(1) group combines with the SU(2) group to form a U(2) group. Thus, given T and isospin, we derive the hypercharge conservation. Similarly, from T and the rotation group SU(2), one obtains the fermion-number conservation. Further. the well-known empirical relations (-I),' = (-1) and

= = a r e direct consequences of our construction.

In Sec. II, we give some general discussions on the questions of the structures of internal-symmetry groups. It seems obvious that these structures a r e rather nontrivial. Part icular emphasis will be laid on the conflicting nature of the various famil- i a r symmetry groups. Section ITI i s devoted to a discussion of some properties of the discrete C,

and T symmetries when they a r e adjoined with other internal-symmetry groups. The adjunction of T to an SU(2) group i s considered in Sec. IV. Finally, some concluding remarks a r e offered in Sec. V.

11. GENERAL REMARKS

It was recognized some time ago that the problem of combining several internal-symmetry groups into a la rger one is rather nontrivial. In fact, one is naturally led to studying the group Thus, Miche13 was able to show that, in combining the ~ o i n c a r 6 group to the gauge groups

r u Q ( l ) ? UB(l)> 9

a relation of the form

= (-1)€1Q + E ~ B + 6 3 L

must hold, where ti = O or 1. The different solutions correspond to different possible extensions of the gauge groups by the ~ o i n c a r g group. The "physical choice"

can only be decided by comparing with reality. Another familiar example has to do with the

problem of adjoining the t ime-reversal operator T to the internal-symmetry groups. For instance, i t i s well known that T and the isospin operators satisfy

These nontrivial commutation relations originate from the antiunitary nature of T and the requirement that T does not change the charge of a physical state.

In I, these considerations were car r ied one step further. It was argued that, by the very nature of the degenerate internal-symmetry groups, their outer automorphisms must themselves be symmetries. This requirement turns out to be very restrictive, a s was discussed in I.

As a further example along these lines. let u s begin with the chain of symmetries

(A chain Go C G, c G, C. . i s established if each Gi is a subgroup of G,,,.) What happens if we consider the outer automorphisms of the groups in Eq. ( I ) ? It turns out' that they a r e

C I UQ(1): Q - - Q ;

c2 c2 c2 U(2): Y - - Y , l2 -+I , , llS3 - - I1 , , ; (2)

c3 c3 SU(3) : F l , 3 , 4 , 6 , 8 - -F1,3,4,6,8. F Z , j , 7 "2.5.7 '

Since we obviously have the chain

C,C C2C C,, (3)

Eqs. (1)-(3) may be combined (with Ci2 = 1) into

where xs denotes the semi-direct product. Thus, given Eq. (I) , i t turns out that the outer automorphisms (the charge conjugation) have a natural generalization a s we enlarge the symmetry groups, so that Eq. (4) results .

The situation is completely different when we consider, in addition to U,(l), U(2), and SU(3), the parity operator, which forms a Z2 (P, P2 = 1) group. Instead of Eq. (I), we now have

4 - S T R U C T U R E O F I N T E R N A L - S Y M M E T R Y G R O U P S 3639

u,(1)xz2~u(2)xZ,CSu(3)xZ2. (5)

Besides C, ,, ,,, the introduction of parity gives r i s e to two new outer a u t o m ~ r p h i s m s , ~

U,(l)xZ,: P E (-I),P;

U(2)xZ2: P W ( - 1 ) 2 ' ~ . ( 6 )

[The automorphism V may be easily obtained with the methods discussed in I (further, V 2 = W2 a I).] Also, SU(3) X Z, has no outer automorphism except C, a s was given in Eq. (2). Our argument shows that U,(l) xZ, must be enlarged to [U,(l) X Z,] xsZ2, and U(2) XZ, to [U(2) x Z2]xSZ2, where the 2,'s behind the semi-direct product signs a r e formed out of V and W, respectively. However, Eq. (6) shows that V cannot be included in U(2) X Z, o r W, and U' not in SU(3) x 2,. We have

Thus, consideration of parity brings out the in- compatibility of the symmetry groups U,(l), U(2), and SU(3). We may in fact cal l UQ(l) XZ,, U(2) XZ,, and SU(3)xZ2 "conflicting symmetries." Even though they may be included in a chain a s in Eq. (5), their outer automorphisms cannot be so enlarged, a s in Eq. (7).

Physically this result is easy to understand. Given U,(l) X Z,, V corresponds to the ambiguity of the relative parity between states with even and odd charges. If we embed ~ ~ ( 1 ) in U(2), then V i s violated since in U(2) X Z, members in the same isospin multiplet must have the same parity. The same consideration applies to W when we embed ~ ( 2 ) in SU(3), a s was discussed in I, where we also showed that, in order to preserve Ur, we must enlarge U(2) to SU(3) x SU(3).

The conflict between U,(l) xZ, and U(2) x Z2 may be interpreted in the following way. The strong interaction is known to conserve isospin and parity, a s well a s the charge. However, one may ask whether the strong interaction only conserves U(2) approximately, in the sense that there may be a genuine isospin-breaking te rm in H,,. Equation (7) says that such a picture i s inconsistent, theoret- ically. Conversely, given that the symmetry group of He, i s U,(l) xZ2, i t i s also inconsistent to enlarge i t to groups with an SU(2) structure.

One has to be a little careful in dealing with the "conflicting symmetries" and the related problem of "broken symmetries." Our analysis was based on the assumption that one "knows" that H , pre- serves U(2) and H,, U,(l). Suppose one were to insist that U(2) XZ, be only a remnant symmetry of SU(3) XZ,; then indeed one might break SU(3) in such a way that the absolute parity of I = $ states

be fixed. (In reality, of course, the parity of I = $ states i s not absolute.) What we a r e empasizing i s that the reverse procedure i s more physical and consistent. We should s t a r t from U(2) x Z, and enlarge i t . In this case the absolute parity of I = ; states i s never measurable. Similarly, the symmetry propert ies of H, , and He, should be regarded a s having entirely different characteristics.

We should also emphasize that, even if we were to insist on starting from SU(3) X Z,, to break i t without preserving W, in a sense, yields a U(2) XZ, which is not "pure." For, what we get i s a ~ ( 2 ) x Z2 with an extra, exterior, constraint (that the I = + states have absolute parities). Finally, in assigning symmetries to H,,, we a r e not saying that SU(3) is in principle not possible. We have merely opted for U(2) a s a better choice.

Mathematically, these results seem to hinge on a number of "accidents." We may observe that, in physics, the enlargement of symmetry groups i s often "noncentral," in that for G , C Gi+,, the center of G, i s not in the center of G,,,. The enlargements of UQ(l) to ~ ( 2 ) and of SU(2) to SU(3) a r e all examples of noncentral enlargments. The conflict between SU(2) and SU(3), for instance, has i t s root in that (-I),', which i s essentially the identity in Su(2) (rotation of 2a), becomes just an undistin- guishing member in SU(3). Coupled with these facts a r e the existence of involutional (reflection- like) symmetries C, P, and T. The adjunction of these operators enhances the importance of elements of order two [such a s (-I),, (-1)2r] in the internal-symmetry groups. In a way SU(3), which has in i t s center only elements of order three, would be the "natural" symmetry group to consider if we had discrete symmetry operators which show "triality" rather than "duality" properties. All of these seem to suggest a rationale for the fundamental importance of the group SU(2), which happens to posses in i t s center a nontrivial element of or - der two. A further illustration will be given in Sec. IV, where we consider the problem of adjoining the t ime-reversal operator to SU(2).

111. INVOLUTIONAL SYMMETRIES

To facilitate our considerations later, we will now proceed to discuss some general propert ies of the involutional (reflection-like) symmetries C, P, and T in connection with their adjunction with the continuous internal-symmetry groups. Since their squares C2, P2, and T 2 represent the identity operation physically, i t seems safe to make the following ansatz:

Ansatz. Given any internal-symmetry group G, the operators C2, P2, and T2 must be in the cen- t e r of G.

3640 T . K . KUO 4 -

As an immediate consequence, we have the following theorem:

Theorem. If we exclude from our considerations the ~ o i n c a r ; group, then C2 = P2 = T2 = 1, the identity element in any internal-symmetry group G.

Several things need clarification. (1) G shall be understood to stand for any inter-

nal-symmetry group, whether i t be for the strong, the electromagnetic, o r the weak interactions. Here we a r e assuming that even though the "usual" C and P symmetries a r e violated by Hwk, the vio- lation is known and i t is meaningful to talk about C2 and P2, which, furthermore, a r e not violated

Hwk. (2) It i s not generally necessary to assume that

C2, P2, and T 2 be the identity in G. In fact, i t i s well known that T2 = SO that, for the Poin- car6 group, T 2 i s indeed not the identity.

(3) The gnsatz limits, to a certain extent, the freedom which the operators C, P, and T enjoy from the general viewpoints of group extensions. In such considerations, say extending G by P, i t is usually argued that gP, where g is any element in G, may be used a s the parity operator. Clearly, the ansatz dictates the choice of g to those for which (gP)2 i s in the center of G.

(4) The theorem follows immediately since in r e - ality the enlargments of symmetry groups a r e noncentral. Consider the strong-interaction symme- t r ies . If we believe in the possibility of the chain SU(2) C U(2) C SU(3) xSU(3) in the sense that i t makes sense to talk about an exact SU(3) xSU(3) limit, then C2 = P2 = T 2 = 1, since the identity is the intersection of the centers of SU(2), U(2), and SU(3) xSU(3). Alternatively, we may consider U(2) fo r H, and U,(1) for H,, the intersection of whose centers again yields the identity uniquely. (5) Note that in studying, for instance, the ad-

junction of P to SU(2), i t i s perfectly a l l right to have P2 = (-1)". However, if P has also to be adjoined to other groups, which do not have (-1)" in their centers, then we must give up the solution P2 = (-

(6) T2 = i s compatible with our discussions only because the rotation group is a symmetry common to a l l interactions. Note also that T2 = forbids any noncentral enlargements of the rotation group. (In fact, any "higher-symmetry" schemes which mix half-integral- with integral- spin states a r e forbidden.)

IV. ADJOINING THE TLME REVERSAL TO SU(2)

Consider the internal-symmetry group composed of SU(2) (isospin) and T (time reversal) , with T2 = 1 (see Sec. III). As we mentioned before, the antiunitary nature of T and the fact that T is not

supposed to a l te r the charge of a physical state imply the following relations:

l T , I , l = [ T , ~ , l = 0 ,

(T, I2}=0, (8)

T 2 = 1 . For our discussions it i s convenient to introduce7 a slight variant of T,

- T s e i " I z ~ = ~ ~ i - 1 2 (9)

which sat isf ies

Since 'T i s also antiunitary, we have

The symmetry T a n d SU(2) now form the direct- product group, [SU(2) x Z,]/Z2, where the factor group has to be taken since T2 must be identified with the center of SU(2).

From our discussions in I, i t follows immediately that the only outer automorphism of the group [ S U ( ~ ) x z,]/ 2, is

[go i s necessarily an outer automorphism since T commutes with SU(2).] Equation (12) is similar to our ear l ie r discussions of the W symmetry,

P W (- 1 ) 2 ' ~ . However, owing to the antiunitary nature of the physical consequences of go a r e entirely different from those of W.

In order to analyze the propert ies of go, we s t a r t from an I= $ state, say / a ) , which will be a s - sumed to be "elementary," in the sense that al l the states may be regarded a s composite states of / a), a s far a s their isospin propert ies a r e con- cerned. [Of course, on a composite state go is to operate in the usual way: go(/ a ) 1 a ) . . I a ) ) = (gal a ) ) . . (go/ a ) ) . ] Without loss of generality, then, we will only consider the state / a).

We now observe that the automorphism go may be realized on I a ) by the operation of multiplying by the phase ei'/2,

go/ a ) = ei */2 1.). (13)

Algebraically, this solution satisfies

g," = (-

goT= Tgoml,

[go, TI = 0 , where the second equation follows because T i s

4 - S T R U C T U R E O F I N T E R N A L - S Y M M E T R Y G R O U P S

antiunitary. Equation (12) is obviously an immediate consequence of Eq. (14).

The important point about Eq. (13) i s that go, when applied to I a ) , does not create a new state. Thus, the fact that T i s antiunitary has the re- markable consequence that g,, which induces an outer automorphism on is a degenerate symmetry operator.

Given that go is degenerate, we must now study the enlarged, still degenerate, symmetry group consisting of SU(2), and go. There a r e now two new outer automorphisms,

[f, ~ l = [ f , i l = 0 ,

and

[Note that (g,n(g,n =go(g,-lT)T= T 2 . ] The na- tures off and g, a r e completely different. Just a s before g, must again be a degenerate symmetry, being realized on / 0 ) a s a phase factor ei"14,

glI a ) =ei'" I a ) . (17)

Algebraically, we have

8: =go, - -

g lT= Tgl-', (18)

[gl, TI = lg,, go1 = 0 . On the other hand, for f we may define a state / &)

by

f l a ) = I Z ) . (19)

Then

g,,l (Y) =e-in/z 15). (20)

[Incidentally, Eq. (20) shows that / Z) must be different from I a ) . ] We have still to find the effect off ong,. Since g12=gm we have, from Eq. (15),

Summarizing, we find, from SU(2), T; and go, an additional degenerate symmetry g, (satisfying g,2 =go) a s well a s an (in general) hidden symmetry f, satisfying fg j-' =go-' and fg, f -' =gl-'.

Entirely similar reasoning leads then to the existence of another, degenerate, symmetry operator g,, with

-12 T -g lT , - -

gzT= Tgz-',

gzZ =g, ,

[gz, g o 1 = [gz, g,l= lg2,il = o , g21 a ) =ein181 a ) ,

fgzf -' =gz-' . Obviously, in this way, starting from T and

SU(2), we generate the symmetry f and a sequence of degenerate symmetries, go, g,, g2, . . . , with the properties

Further, i t i s clear that any product of the g's,

is also a degenerate symmetry operator. It follows that the g's actually generate an Abelian U(1) group (since any real number x, 0 s x 1, may be written a s x=x an/zn, for a, = O or 1). Let us define this U(1) group a s U ,(l) = {eieY}. Then

so that eieY can always be written a s a product of the gi' s.

Therefore, given T and SU(2), we have generated a one-parameter degenerate gauge group Uy(l) = {eteY} satisfying

and, by Eqs. (14) and (24),

[In other words, SU(2) and UJ1) generate U(2).] Needless to say, UY(l) i s nothing but the hypercharge gauge group. The other symmetry f [Eq. (15)] corresponds obviously to the usual G parity, satisfying Fqs. (15), (21), etc.] ,

Let us now pause for a moment, in order to get rid of a number of ambiguities neglected earlier. In Eq. (13), actually another solution exists,

go/ a ) =e-in'2/ a ) . (13')

This obviously corresponds to starting with ( Z) in-

KUO 4 -

stead of ( a ) , a s in Eq. (20). Next, in obtaining g,, g,, . . . [Eqs. (17), (22), etc.], one may take the solution (-1)2'gi instead of g,. However, since (-1)" - e i w Y - , al l these solutions a r e recovered when we have generated the U,(l) group. Lastly, in extending ,f from go to operate on gi, since we can only require fgi2f -' =g,,,-', we may, besides jg, f -' =g,-', get another solution ,fgi f -' = (-1)21gi-'. This ambiguity is removed, however, by considering fgiT,J-\ For, either solution [gi+,-l o r (-1)2rgi_l-1] gives fg i f -' =fgi+,Y-' =gi-l.

In summary, we find that hypercharge conservation follows from time reversa l invariance and SU(2) symmetry. Further, an "elementary" iso- spinor must have Y =*I . In general, for composite states, even and odd 1' values correspond to integral- and half-integral-isospin states, respectively.

If we s ta r t out from T and the rotation group [SU(2)], then, algebraically,

{T. 3 = 0 ,

T 2 = + * * *

(28) T e i e ' ~ =eic J T

It is clear that, by identical arguments a s above, we necessarily generate the fermion-number (F = B + L) gauge group UF(l) ={ei"} and the f e r - mion-number-conjugation (C,) operator, satisfying

[F, ? ] = [ F , T ] = 0 ,

V. CONCLUDING REMARKS

In this work we have found some rather nontrivial consequences in studying the outer automorphisms of symmetry groups. Specifically, we found that the outer automorphisms of the time r e - versa l and an SU(2) group generate unambiguously the degenerate symmetry group U(2). Thus, hypercharge conservation follows from T and isospin conservation. The fermion-number conservation i s a consequence of the angular momentum conservation. The well-known empirical relations (-1) = (-1)" and (-I), = (-1)'" were derived in the pro- cess .

In retrospect, these results a r e not a s surpr is - ing a s they might f i r s t appear. We may note that the "number laws" a r e realized a s phase factors on the physical s tates. Since the t ime-reversal operator i s intimately related to phase factors, i t i s perhaps not too surprising to find a cause and effect relation between them. Another important ingredient in our discussion i s the existence of spinors and isospinors, o r the empirical fact that the centers of SU(2) and a r e nontriv- ially represented. If there a r e only integral-spin and -isospin states, or , if SU(2)/Z2 instead of SU(2) were to be our symmetry group, then Eq. (12), which i s the starting point of a l l our discussions, i s trivial and nothing would have been learned from our considerations. The empirical relations (-1)'s and = in this connection, may be taken a s further clues to the intimate relationships between half-integral representations and the "number laws." Indeed, s ta r t - ing from these relations, what we showed in Sec. IV was that we may take "square roots" succes- sively. These "square roots" generate the gauge group U(1).

Our heavy reliance on 2' may lead to the following objection. Since we know that T i s violated (by the "superweak interaction"), but the fermion number i s not, is there not something wrong with our derivations? Actually this dilemma i s only an apparent one. We may s t a r t f rom the strong, the electromagnetic, and the weak interactions, for which T and SU(2) (rotation symmetry) a r e exact. We then obtain UF(l). Having obtained the larger group, we ask: How does the superweak interaction behave under this larger symmetry group? It is not hard to imagine that indeed the superweak interaction behaves so that i t violates T, but still conserves U(2). In a similar way, for T and isospin, we have obtained the hypercharge conservation, which i s a symmetry fo r H,, even though H , does not respect the isospin conservation.

Finally, i t i s not difficult to convince oneself that, had we started from T and U(2) o r SU(3) xSU(3), we would only have obtained inner automorphisms. This, again, testifies to the fundamental importance of the SU(2) group.

ACKNOWLEDGMENT

We would like to thank our colleagues for discussions.

S T R U C T U R E O F I N T E R N A L - S Y M M E T R Y G R O U P S 3643

*Supported in part by the U. S. Atomic Energy Com- mission.

'T. K. Kuo, preceding paper, Phys. Rev. D 4, 3620 (1971), hereafter referred to a s I.

2 ~ o m p a r e with the analogous situation in angular momentum conservation. If we have only SO(3) (the pure rotation group) instead of SU(2), then the center of SU(2) - ( - 1)'' -becomes trivial. The half-integral representations a r e obtained only if we enlarge our "geo- metrical picture" of rotation to include the extra element ( - 1 ) 2 ~ .

3 ~ . Michel, in Grozip Theoretzcul Concepts and Aletkods zn Elementary Particle Phys ics , edited by F . Giirsey (Gordon and Breach, New York, 1964); and in Particle Symmetr ies and Axlomutzc Field Theory, 1965 Brandeis Summer Institute in Theoretical Physics, edited by

hI. chr6tien and S. Deser (Gordon and Breach, New York, 1966).

4 ~ o n F. Kamber and N. Straumann, Helv. Phys. Acta. 37, 563 (1964). -

5 ~ . D. Lee and G. C. Wick, Phys. Rev. 148, 1385 (1966). 6 ~ . Zumino and D. Zwanziger, Phys. Rev. 164, 1959

(1967). he existence of T for SU(2) i s well known. We may

note that, if we adjoin T to an internal-symmetry group S which commutes with P , then the operator C T =T anticommutes with all the generators of S. Fo r SU(2), C =eiP'?; and hence we have Eqs. (9) and (10).

8 ~ o the extent that hadrons and leptons can be sepa- rated unambiguously, our arguments actually lead to sep- ara te baryon-number a d lepton-number conservation laws.

P H Y S I C A L R E V I E W D V O L U M E 4 , N U M B E R 1 2 1 5 D E C E M B E R 1 9 7 1

Motion of Charged Particles in a Homogeneous Magnetic Field*

Wu-yang ~ s a i ' and A s l m Ylldlz detferson Pkyszcul Luborutorj , Hari ard Cnz iersz t j , Cumbrzdge, AIlussutllusetts 02138

(Received 18 hlarch 1971)

A general and simple method is presented to calculate the eigenvalues of the eigenequation for the theory of a particle with any spin with anomalous-magnetic-moment coupling moving in a homogeneous magnetic field. The eigenvalues of the spin-: and spin-1 systems a r e oh- tained specifically. This method of approach does not require an explicit solution of the eigen- function equation. Different spin-1 theories (vector theory, multispinor theory, and F-component theory) a r e discussed, and in the case of no anomalous-magnetic-moment coupling terms, all these theories predict the same eigenvalues. Furthermore, by requiring that the energy eigenvalues be positive definite, we show that the vector spin-1 theory is consistent only when no anomalous-magnetic-moment coupling terms exist.

I . INTRODUCTION

The p rob lem of desc r ib ing the mot ion of c h a r g e d

p a r t i c l e s in a n e x t e r n a l e l ec t romagne t i c f ie ld h a s

been wi th u s f o r a long time.'.' Until now the only c a s e s d i s c u s s e d have been the spin-0 and spin- ; s y s t e m s , and the gene ra l ly accep ted me thod of a p - p r o a c h i s t o define the Lagrang ian equat ion of m o -

t ion and to u s e the d i f ferent ia l -equat ion technique to so lve f o r the e igenfunct ions and e igenva lues of

t he s y s t e m . T h i s app roach b e c o m e s inc reas ing ly compl i ca t ed when anomalous -magne t i c -momen t coupling t e r m s (a .m.m.c . t . ) are in t roducedZ and when we d i s c u s s t he h ighe r - sp in cases.

The pu rpose of t h i s p a p e r i s to p r e s e n t a s i m p l e , g e n e r a l me thod f o r ca lcula t ing the e igenva lues of p a r t i c l e s of any sp in with anomalous magne t i c m o -

m e n t moving in a homogeneous magne t i c f ield. T h e me thod i s i n t roduced in Sec. II, and the r e s u l t s of

T e r n o v et a1.' a r e ea s i ly r ep roduced . In Secs . III and IV, the e igenva lues of a spin-1 theo ry with

a.m.m.c.t . are ca lcula ted expl ic i t ly wi th in the v e c -

t o r theory of t he K e m m e r - P r o c a e q ~ a t i o n . ~ By ob - s e r v i n g the phys i ca l r e q u i r e m e n t tha t the s q u a r e s

of the e n e r g y e igenva lues be pos i t ive def in i te , we show that t he spin-1 theo ry i s cons i s t en t only when

the a .m .m.c . t . a r e not p re sen t . The p red ic t ions f r o m t h r e e d i f ferent sp in-1 t h e o r i e s - v e c t o r t heo -

r y , 3 mul t i sp ino r t heo ry ,4 and the 6-component theory5 -are d i s c u s s e d in Sec. V. We f ind tha t f o r t h e c a s e w h e r e the a .m.m.c . t . are a b s e n t , t h e s e

t h e o r i e s a l l p r e d i c t the s a m e r e s u l t s .

11. METHOD OF APPROACH

I n t h i s s ec t ion we i l l u s t r a t e t h e method of a p -

p r o a c h in the spin-; t heo ry with a . m . m . couplings.

O u r s t a r t i n g point i s t h e e igenvalue equation6

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Structure of Internal-Symmetry Groups