brauer, weyl - spinors in n dimensions

Spinors in n DimensionsAuthor(s): Richard Brauer and Hermann WeylSource: American Journal of Mathematics, Vol. 57, No. 2 (Apr., 1935), pp. 425-449Published by: The Johns Hopkins University PressStable URL: http://www.jstor.org/stable/2371218 .Accessed: 16/02/2011 06:19

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SPINORS IN n DIMENSIONS.

By RICHARD BRAUER AND HERMANN WEYL.

Introduction and Summary. Let by, be the group of orthogonal transformations o:

n (I) xt~~X E o (ic) x,* (i I,1~ 2, .. n)

k=l

of the n-dimensional space, and b?n+ the subgroup of proper transformations, having determinant + 1 and not - 1. We shall first operate withiii the contiiiuum of all complex numbers, whereas the particular conditions pre- vailing under restriction to real variables will be studied only at the end of the paper (?? 8 and 10). A given representation P: o -> G(o) of degree N defines a certain kind of "covariant quantities": a quantity characterized by N numbers a1, , aN relative to an arbitrary Cartesian coo6rdinate system in the underlying n-dimensional Euclidean space will be called a quamntity of k1ind r, provided the components aK experience the linear transformation G(o) under the influence of the co,6rdinate transformation o. The quantity is called primitive if the representation is irreducible. The proposition that every representation breaks up into irreducible parts, states that the most general kind of quantities is obtained by juxtaposition of several independent primitive quantities.

By a tensor of rcank f we shall mean here what usually is called a skew- symmmetric tensor: a skew-symmetric function a(i1 if) of f indices ranging independently from 1 to n which transforms according to the law

n - if)--) Y o(i1k1) . . . o(ifkf) * a0i kr ) kl,. . ., kf = 1

uncder the influence of the rotation o. The tensors of rank f form the sub- stratum of a representation rf of degree ( ).

We often have to distinguish between even and odd dimensionality, and we shall accordingly put n = 2v or n 2v + 1. Let us use the notation v <n> and in passing notice the congruence

In (n 1) <n> (mod 2).

E. Cartan developed a general method of constructing irreducible representations of b.n (or anv other semi-simple group) by considering the in-

425

42 6 RICHARD BRAUER AND HERMANN WEYL.

finitesinmal operatiolns, alnd he foundl t as the building stolnes of the whole edifice the tenasor represenatationis Pf together with one further double-vallued representation, A : o -> S(o) of dlegr-ee 2". The quanatities of kinad A are called spim,ors. In the four-dimenasional world this kinid of quanltities has come to its due holnors by Dirac's theory of the spilnning electroln. Cartani, accordling to his stanadpoinit, states the tranisformationi law S(o) of spiniors onaly for the inafilnitesimal rotatiolns o. Here we shall give a simple filnite descriptionl of the represenitationi A alnd shall derive from it by the simplest algebraic mealls the maini properties of the spilnors. Onie will be able to judge by this theory to what extelnt recelnt ilnvestigationis about spinaor calculus reveal those essenatial features that stay unichanaged for higher dimenisiolns. Onae of the chief results will be that Dirac's equatiolns of the motioln of aln electroln alnd the expression for the electric currenit are ulniquely determilned eveln in the case of arbitrary dimelnsiolnality.

Our ilnvestigatioln will be arralnged as follows: we start (? 2) with a certaini associative algebra II of order 22v which proves to be a complete matrix algebra in. 2Pc dimenisionas, alnd leads to the desired defilnitioll of A (? 3). We shall first get A as a collineationa representatioln such that onlly the ratios of the spinaor comlponaelnts have a mealning. In the case of eveln dimensionality it 2v we shall prove (? 3) that the product A X A of A by the contragredient representatioln A splits up according to the equivalelnce:

AX roF+Fr+Pr2 + * rn.

whereas in the odd case

AXFro+r2?+F4+' + Eni

(? 5). The collineationa representatioln A can be normalized so as to give an orclilnary, though double-valued represelntation A satisfyinag the equivalenace A ~ A (?? 4, 5). If olne restricts olneself to the proper orthogonial transformations in a space of eveni dimensionaality, A splits up into two representa- tionis A+ and A- each of degree 2'-1 (? 6). The four products of the type A X :s will be determinied incdividually for A = Al or A-, ancd so will the equivalences of type A .- A. The trlansitioln from our finite to Cartan's infilnitesimal description can be easily performed (? 7). In considerinag real transformations only, the differences of the inertial index lhave to be taken into account (? 8); it will be proved that A is equivalent to A again -but for a sign the determiniation of which is of peculiar interest alnd closely related

t Bulletin Soci6te Math6matique de France, vol. 41 (1913), p. 53. Coim1pare also Weyl, Mathematisehe Zeitschrift, vol. 24 (1926), p. 342.

SPINORS IN It DIMIENSIONS. 42 7

to the inertial index. IrreducibilityT and equivalence of the occurriing representations will be ascertained in ? 9, and the relation to physics will be clis- cussed in ? 10. In parts of the inavestigationa we must have recourse to the law of dualityT of tenasors anad tenasor representations rf as formulated in the preliminary ? 1. The last sectiona (? 11) is devoted to the clemonastrationi of a well-knaowji fuinclaniental propositioni conacerninig the automorphisnis of the coniplete matrix algebra, a propositioni inadispenisable for the definitioll of A.

1. Duality of tensors. rP is the representation of degree 1 of the full rotatioln group b,, associatilng the signiature a (o) with the rotatioll o o (o) + 1 for the proper, o-(o) - 1 for the imiiproper rotatiolns. AnyT representation r : -> G (o) gives rise to another represelntation ar : o (o) G (o), coil- cidling with r ulnder restrictioln to b.1+.

The equation (2) * (ill. ) = (l . . . if)

in which i1 if i'1 . i'f denotes anyT eveni perniutation of the figures from 1 to II, associates a telnsor a* of rank ii - f with every telnsor a of ralnk f. This relatioln is ilnvarialnt with respect to proper orthogolnal tralnsformatiolls. Thus the la,w of duality rF,f -rf prevails for the telnsor representations rf of b,,+. Wheln takilng the improper orthogorial tranisformatiolns ilnto colnsidera- tioln it is to be replaced by

rt, .f r--"rf.

Ina the case of aln eveln niumber of dimelnsiolns it 2v', the represelntation, rF cleserves particular attelntion. It satisfies the equivalelnce -rv, rv. (2) or rather

(3) a(il . ilv) .l .( .

iv)

now establishes a transformation a -> , of the space of the tensors of rank v upo,n itself. We added the factor il in order to make this transforniation invTolutorial: -* a; for if il iv i'l . 'v is an even perniutation, i*l i'vil iv has the character (- 1)". We may distinguish between positive andl negative tensors of rank v according as a*- a or a - a. Any tensor of rank v can be decomposed in a unique manner into a positive ancd a negative part:

a, =-( + a,* + t( -2 t

Hence, as a representation of the group b+2v, rv splits up into two representations rv+ + rv- of half the degree.

2. The algebrca H. Our procedure is exactly the same as followed by

428 RICHARD BRAUER AND HERMANN WEYL.

Dirac in his classical paper on the spinning electron.t We introduce n quantities pi which turn the fundamental quadratic form into the square of a lilnear form: (4) X12 + Xn. 2 (plXl+ * + PnXn) 2.

For this purpose we must have

(5)) op2 aL , pkpi -Pipk (k 7+ i). The quantities pi engender an algebra consisting of all linear combinations of the 2n units

(6) ea,... =. p " p. . . (cal, , an integers mod 2).

The recipe for multiplication of the units reads, according to (5):

ea,. . . an 'efil.. On ( ) eyl . ..Yn; 75 (Xi + /3in I ( a,k.

One easily convinces oneself that this rule of multiplication is associative. One may write the most general quantity a of our algebra in the form

(7) a= + (1/f!) , a,(il if)pi, p, + (f==O, 1, ,n), (il,. . if) f

splitting a into parts according to the number f of the different factors p. Since the product of f different p's like Pi, * pif is skew-symmetric with respect to the indices i1 .f, one will choose the coefficients ac (i1 if) in (7) also skew-symmetric; one is then allowed to extend the sum > in (7) over the indices i1,- , f independently from 1 to n. Consequently the quantity a is equivalent to a "tensor set " consisting of m + 1 tensors, one of each of the ranks 0, 1, , f, * *, n. The addition of two tensor sets and the multiplication of a set by a number has the trivial significance within the algebra H. But how are we to express the multiplication of two tensor sets a and b ? It suffilces to describe the case of an a containing merely one tensor ac of ralnk f, and a b containing merely one tensor 8 of rank g (whereas the other parts vanish). The product splits into different parts according to the number r of coincidences among the indices of oc and ,8. As

Pil Pf r- Pli . i7 PIl Plr Pk Pkg-r n(e g s o)r(r-1)/2p the "cont tpio" Pk- Pkg_

one gets as part r- of the product essentially the " contraction"

tProceedings of the Royal Society (A), vol. 117 (1927), p. 610; vol. 118 (1928), p. 351.

SPINORS IN It DIMIENSIONS. 429

(8) l t- k * kg-r ) a (i...ifr(t *1. * Ir) * 1* * 1? k, **g 7)

This process, however, has to be followed by "alternation," i. e. alternating summation over all permutations of the f + g - 2r indices in -y. Since -y is already skew-symmetric with respect to the f - r indices i and the g - r indices k, it is sufficient to extend an alternating sum over all " mixtures " of the indices i1 if-r with the indices k.. 7g-r. This will be indicated by the symbol M. By taking into consideration the factor 1/f! attached to the f-th term in (7) and the several distributions of the r equal indices 11 Ir among the indices of ac and /3, one gets finally the result: The " product " of the two tensors ac and 83 is a tensor set in which only tensors of rank f + g -2 appear; the integer X is limited by the bounds

X _O, 2r f +g-n, X f, xig.

The part r is given by

( )<">(I/,-!) -My (ii .if-r 471 kg-r)

where y denotes the contraction (8).-We are not so much interested in the exact description of this process of multiplication as in the fact that it is orthogonally invariant.

3. Spinors in a space of even dimenisionzality. In this section we suppose n = 2v to be even. The algebra H is known to the quantum theorist from the process of " superquantizing" that allows the passage from the theory of a single particle to the theory of an undetermined number of equal particles subjected to the Fermi statistics. This connection at once yields a definite representation pi -> PF by matrices Pi of order 2". Into its description enter the two-rowed matrices

1= 10 1gI 0 01_ Q0o1i I 01' I 0 -1' p~_ 1 0'

The two rows and columns will be distinguished from each other by the signs + and -. 1', P, Q anticommute with each other; their squares are- 1. Besides pl,- , p2V we sometimes use the notation p1, , pl, q1, , qv. The representation then is given by

pa 3Pa, 1 X ...X I' X P X I X . X.1, (9 a Q. 1 X... X 1 X Q X I X 1. . . X V)

On the right side we have v factors; the factors P, Q respectively, occur at tlle ac-th place. The rows and columns of our matrices or the coo6rdinates XA in


the 2v-dimnensional representation space, according to the notation introduced, are distinguished from each other by a combination of signs (.-,, 02, , UV), (7a = ) One verifies at once that the desired rules prevail:

(10) p2_ PkUP, -P,Pk (i' =7& 1)

In this manner we have established a definite representatlion x -> X of degree 2v for the algebra I. We maintain that all miiatrices X appealr here as imtages of elemtents x of the algebi-a. As the algebra II is of the same order 22 -_ (2v) 2as the algebra colnsisting of all matrices in the 2v-cdimensional space, the relation x ?> X is a one-to-one isomorphic mapping of H upon the complete matrix algebra of the 2-cdimensional " spin space ": the algebra H is isonmorphic to the complete matrix algebra in spin space. In order to prove our statement, let us compute the matrix Ua representing Ua iPaqa

(11) Ua iPaQa I X . .

*.X I X If X I X .

.* X1

and then (11') Ul . . .Ua-lPa I X X 1 X PX1X X 1

together with U1 UaiQa. (The factors different from 1 occur at the o-th place.) Thus the following elements

2( + Ua) =- Za++n Ul . . .

Ua-1(pa - iga) Za

UiM . . .

Ua-1(p?a + iqa) Za , 2(1Ua) = Z.a-

are represented by products similar to (11) but containing one of the matrices

1 0 0 1 0 0 0 0 0 0 ll0 0 ' 1 0 ll0 1,

at the ac-th place. Colnsequelntly the image of the element fI (Za6ara) is the a=1

matrix containing a term different from 0, lnamely 1, only at the crossing poilnt of the row u1 UV with the column T1 7*v (O-a= ?4 Ta = +)-

We are now in a position to establish the colnnectioln with the rotations o 11 o (ik) 11 in the n-dimelnsiolnal space (Metlhod A). We chalnge, by means of the orthogolnal matrix o(ih)

k=1 k=1

and we observe at olnce that the new P*i, like the old oiies, satisfy the relationis (10). Consequently pi -> P*1i defines a new representation of our algebra H. Sin?ce the full miiatr7'ix algebra, hoowever, allows on?.ly in ?er autom o rphismis,t

t See the pioof in ? 11.


this representation has to be equivalent to the original one; that is, there exists a non-singular matrix S(o) such that

(13) P*- ~S (o)Pi S (o) - (it -~ ,2. , n).

S (o) is determined by this equation but for a numerical factor, the "gauge factor": S (o) is to be interpreted in the "homogeneous" sense, not as an affile transformationi of the 2kc-dimensional vector space, but as a collineation of the projective space conlsistinig of its rays. After fixinig the gauge factors for two rotations o, o' and their product o'o in an arbitrary manner, we necessarily have a relationi like

(14) S(o'o)- c S(o') S(o).

Consequently we are dealing with a collineationi representtation of degree 2V of the rotationi group, the so-called spin r7epr'esentation A : o > S(o).

The same conniection can be described as follows (Method B). Or- thogonal transformation of the tenasors of an arbitrary tensor set defines an autoniorphic mappinig x -> x of the algebra II of the tensor sets upon itself. Such a mapping however, in the representation x-> X of the tensor sets by matrices X of order 2v, is necessarily displayed in the form

XI -- X SXS-' (S independent of x).

Let us write down this equation in components: X= = XJE 11; it then reads

X*JK - 51R SKT 1R XT f,T

S= 1 SJR jj is the matrix contragredienit to S. Hence the components XJK experience the transformation S X S and this proves the reductioni

(15) AX ro +al+ r?P - PO+ rl+ +r,v-+ +} ?(rv -orv).

The quantities {j1A) and {0.4} of the kind A, A shall be called covar7ian-t and contravairiant spinior-s respectively. Let us write the components VIA of a covariant spinor as a column and the componients OA of a contravarianit spinor as a row. Our last equation tells us that one is able to formii by linear combination of the (2") 2products O4A B: otie scalar, one vector, onTe tensor of rank 2, etc. The scalar is, of course,

++ = E Oq,A A

The vector has the components 4Pq f. Inideed, in carrying out the trans- folrllation O** 5I, s* - S-1, one gets,

432 RICHARD BRAUER AND HERAMANN WEYL.

p*P. oii* Pi= - o(ik) cS-PP*k S& = o(ik) pPk qp. k=1 k=1

The tensor of rank 2 has the components cp(PjPk) > [i y k] ; etc. In this manner we are able to carry out the reduction (15) explicitly.

4. Connectionz between covar-iantt and contravariant spinors. Let n be even as before. We propose to show that the representation i is equivalent to the representation A. For this purpose we observe that the relations (10) characteristic for the matrices Pi hold at the same time for the transposed matrices P'i. According to the proposition on the automorphisms of our matrix algebra II we already have had occasion to use, there must exist a defilite non-singular matrix C such that

(16) P', CPiC-

for all i. It is easy to write down C explicitly. For we have

Pa Pa, QcLa - Q- ( xQa,

But the product pi pv commutes with the pa and anticommutes with the qa, if v is odd; if v is even the situation is reversed. Hence one can take

c -p.. .. pv or q . qv

according as v is odd or even. In this way one finds in both cases:

(17) c== 0 1 10 i X 01 X . (V factors) 1 0 -i 0 1 0

and one verifies at once the relations (16). Along with (12) we have

P1 ->J*, P 0 (ki)P',k. k

This transition is expressed on the one hand in the form

P-i > S'(o)-1 P' S'(o) = S(o)P' S(o)1 .

On the other hand the transformation of P'" = CPqC-1 is obviously performed by means of CS (o) CW1. Hence an equation like

CS (o) C-1 =-p (O) *S (O)

must hold where p (o) is a numerical factor dependent on o. On multiplication of S(o) by X, S(o) is multiplied by 1/X and p is thus changed into pA2.

SPINORS IN n DIMIENSIONS. 433

Hence we may dispose of the arbitrary gauge factor in S in such a way that p becomes = 1:

(18) S(O) = CS(O)C-1. This has the effect that (19) (det S)2 = 1.

S(o) is now uniquely determined but for the Sign. After normalizing this sign for two rotations o, o' and the compound o'o in an arbitrary manner, the composition factor c in (14) becomes - I; for the matrices X t S(o'o) and X A S (o') S (o) both satisfy the normalizing condition

C=CxC-1.

A now is an ordinary, though double-valued representation instead of a collineation representation.

Equation (18) gives the explicit relation between the covariant and contravariant spinors: if C is the matrix 1J CAB 11 the substitution

( = E CAB qB B

changes the covariant spinor ;, into a contravariant spinor p. The " square " of the double-valued representation A is single-valued and

is decomposed, according to formula

into the tensor representations rf.

5. Odd number of dimensions. n - 2i' + 1. To our quantities pi,, **, p2V

a further one p2v+1 has to be added, p22v+1 = 1, which anticommutes with the previous pi. The representation pi -* Pi (i 1, , 2v) can be extended by establishing the correspondence

P- >Pn=-- 1, X 1, X ***X 1, (n 2v+ 1).

Let t be = 1 or i according as v is evren or odd. The product

(20) u tplp2 . . . Pn

commutes with all quantities of the algebra and satisfies the equation u2_ 1. In the representation just described u is represented by the matrix 1. There exists a second representation of the algebra:

(21) p1>-Pt(i=1,2,i * *,n)

in which u 1 and which thus proves to be inequivalent to the first one.

434 RICIIARD BRAUER AND IIERAIANN WEYL.

The order 2 (2v)2 of the algebra H this time is twice as large as the order of the algebra of all matrices X in the 2v-cdimensional spin-space. Our isomorphic mapping x -> X therefore becomes a one-to-one correspondence only after reducing 11 modulo (1 - u) ; this is accomplished by adding the coniditioni u 1 to the defining equations (5). This new algebra may be realized as a subalgebra in II in different manniers; for inistanice, as the algebra of the quantities x satisfyilng the condition x =ux. It is more convenient to consider the even quantities in Il. Their basis coinsists of the products of an even number of p; in (6) one has to add the restriction a, + + , =-0 (mod 2); the corresponiding tensor sets conitain tensors of even rank only. Any odd quantity may be written in the form utx where x is even. The arbitrary quantity x + ux' of the algebra rI (x and x' even) is represented by the same matrix as the even quantity x + x'. Hence the correspondence x -> X is a one-to-one correspondence within the algebra Hl, of the even quantities. The second representation (21) coincides with the first for the even quantities.

The procedure is now as above (MIethod A). Let 11 0(ilc) 11 be a proper orthogonal transformation. Then (12) yields a new representation of H. By multiplication we get

U*-tPl P=_ cldet [o(RiU)] U =U.

Hence this representation like the original one associates the matrix + 1 (and not 1) with u; by means of Pi --P* we thus map the algebra H reduced mnodulo (1 - u) isomorphically upon itself, and consequently an equation like

P*. - SPiS-l

holds. The representation a o -> S (o) may1 be extenaded to the improper rotations by making the matrix + 1 or 1 correspond to the reflection xi > xi that commutes with all rotations. (Wlhether one chooses + 1 or -1 does not make any difference here since the represenitation A is double- vTalued.)

(MIethod B). The orthogonal transformation o is an isomorphic mapping of the manifold of all even tensor sets upon itself. After representing this manifold by the algebra of all matrices X in 2v dimensions in the manner described above, o appears as an automorphism X --> X* of the complete matrix algebra: X- SXVS-1. One gets S(o) here at the same time for all proper ancd improper rotations o. Furthermore, we obtain the decomposition

(22) Ax A ro+r2+ + 2Vro+ur,1+ + ur3+ ,


the last sumi concluding with the term rv or 'Orv. Consequently there is con- tailned in A X A a proper scalar, an imiproper vector, a proper tensor of rank 2, etc.

The nt = (2i' + 1)-dimensional oroup of rotationis b, comprises the (I? - 1)-dimensional one b,,-, by subjecting the variables x1, ..X2V to ain

orthogonal transformation alnd leavinog

x,v+l unchanged. This restriction to a subgroup carries the representation A of b1, as here defined, over into the representation A of the (n - 1) -dimelnsional group of rotations which we defined in ? 3. The same restriction splits a tenisor of rank f in the n-dimensional space into two tenisors of ranik f and f - 1 respectively in the (Il - 1) - dimensional space. And thus the decomposition (22) goes over into the decomposition (15).

The matrix C, (17), which satisfied the equations P'i = CPi.C- (for i - 1, 2, , 2v) fulfills the condition

CP C-1 = (- 1 ) 1P

f or P? = P2V+1. Hence it can be used here for the same purpose as in ? 4 only if v even. In the opposite case one must replace C by CUP,,:

0i1 0 1 1 0 iK, 1 -? 0 X 1 0 X ? 0,

and one then has CPIC-1 P'i (for all i). Under both circumstalnces the equation (18) obtains for the C determined in this manner and after ani appropriate normalization of the gauge factor in S (o). Here again we have A ~ A and we are able to express explicitly the transfornmatioln C which chaniges covariant spinors into contravariant onies.

6. Splitting of A under restriction to pr'oper rotations. In the case of odd dimensionality it makes no differenice whether one considers the group bll or bl+ sinice the reflection commuting with all rotations is an improper rotation. If, however, a == 2i' is even, restrictioln to b,,+ effects a splitting of the spin representationi A into tw,o ilnequivalent representations A+ and A- of degree 2-1, and one will have to distinguish between " positive " anid " nega- tire " spinlors accordingly. This comes about as follows.

Again we form

(23) U tp .pv > U 1 X 1 X . X 1.

We separate the eveen comnbinations of signs (C,. ,uv) as characterized by rl *V = + 1 fromn the odd ones. Accoridinig to such an arrangement U appears- in the form

436 RICHARD BRAUER AND HERAIANN WEYL.

1 0 (24) =- s

0 -1

As a consequence of equations (12) one has for the proper rotations o: U -- U* - U. As P*i - SP,S-1 implies U* = SUS-1 the matrix S commutes with (24) and thus breaks up into an "even " and an " odd " part:

S+ O

o S

The matrices S+(o) and S-(o) in the two representations A+ and A- of degree 2v-1 are uniquely determined but for a common sign. Hence the fact that the reflection is associated with the matrix + 1 in A+, with the matrix - 1 in A-, means an actual inequivalence.

What is the significance of the partition of X into four squares for the corresponding quantities x of the algebra II or for the tensor sets? (1) We see from the equation UP - Pi U that the even quantities commute with U and that the odd ones anticommute. Even and odd quantities are consequently represented by matrices of the following shape respectively:

x x (25) (26)

x x

(the squares not marked by a cross are occupied by zeros). (2) The in- volutorial operation

a ->a* au, A4->A* AU

leavTes the two front squares in

A

unichanged while it reverses the signs in the two back squares. Let us agree to ascribe the signature + or - to a quantity a according as a* a or a*- a. These quantities then are represented by matrices of the form (27), (28) respectively:

SPINORS IN fn DIMIENSIONS. 437

x. x (27) - - (28)

x x

Every quantity may be uniquely written as the sum of two quantities of signatures + alid -. (Besides the operation a -> a * one could of course also consider the following one: a -* at = ua. But the crossing of both signatures is carried out in a more convenient way by crossing the signature here applied with the division into even and odd quantities. For we have at = a* for even quantities and at a* for odd ones.) Thus we finally get this scheme:

x x

x x

even odd odd even + + : signature.

The question as to how our star operation is expressed in terms of tensor sets is answered by the equation:

Pi . . pf* U ( 1 )<f> tpf+1 . .

P

showing that the transition from a {a} to a* -{a*} is defined b+

a* (ifl * . . . fn-f) (1) <f*0 a(t, * i f)

(where i1 * i if'1 * it f is any even permutation). The factor ( 1) <0>* equals i".

Hence, taking into consideration the splitting of rv into rv+ + rv- as explained in ? 1, we get the following reductions:

(29) A+XA+ro+Pr2+A+ x A- ri+r3+ A- X + r + r3 +

- -X a-ro +r2 + .

Of the two sums in the first column, one breaks off with rv-1, the other with rv+, whereas the sums of the second column end with rv- and rv-J respectively.

From (16) we obtain by multiplication

(- 1) U' = CUC-1 or CU ( 1) UC.

This shows that C is of form (25) or (26) according as v is even or odd. With C, C2 being the partial matrices of C, we thus have

SPINORS IN n DIMIENSIONS. 437

x x (27) (28)

x x Every quantity may be uniquely written as the sum of two quantities of signatures + and -. (Besides the operation a -- a* one could of course also consider the following one: a -- at = ua. But the crossing of both signatures is carried out in a more convenient way by crossing the signature here applied with the division into even and odd quantities. For we have at = a* for even quantities and at-= a* for odd ones.) Thus we finally get this scheme:

x x

even odd odd even + - + - : signature.

The question as to how our star operation is expressed in terms of tensor sets is answered by the equation:

Pi . . . pf * U= (-)f*pf+1 . . .

pn,

showing that the transition from a= {a} to a* = {a,*} is defined by

0Z* (ifl *. i'n- f ) == (- 1 ) <f> t * aZ (ij . . . if )

(where i1* if. i'l . in-f is any even permutation). The factor (- 1) <v>* equals i".

Hence, taking into consideration the splitting of rv into rv+ + rV- as explained in ? 1, we get the following reductions:

(29) Ax +roP+Pr+ XA-X 'r0+Pr3+ ? A- x r + r3 + A- X A-X --ro +r2 + .

Of the two sums in the first column, one breaks off with rv_, the other with rv+, whereas the sums of the second column end with rv- and rv-, respectively.

From (16) we obtain by multiplication

(- 1)U=-- CUC-1 or CU= ( 1)vUC.

This shows that C is of form (25) or (26) according as v is even or odd. With Cl, C, being the partial matrices of C, we thus have


for X = S,6 or Tfl (3# (x), but dSa = Sa, dTa= -Ta.

(b) X8X [S.aTfT, XI = for all S and T

except for X = S6 alnd Ta for which we have:

SSO = Sa, STa Tfl.

This is readily seeni from the expression

[SaTP, X] = Sa (TflX + XTfl) (XS.a + SaX) Tp.

In this way we have arrived at Cartan's infinitesimal descriptioni of the spill representationi.

Nothinig essenitial has to be added in the case of odd dimnensioctality. It is theni most colnveniienit to assume the funidamenital quadratic form in the shape

(x0) 2+ 2(XIyI+ + xvyv)

(31) shows that A is double-valued and not single-valued. For in accordance with this equation the rotation o:

xl e-> xi, y1 --> c y1 (all other variables ulnchaniged)

is associated with the operationi S (o) multiplyilng the variable xc1 . a, in the spin space by e"i0L (offa - 1).

8. Conditions of reality. For the real orthogonal transformations the question arises whether the conijugate complex representation : o S(o) is equivalelnt to \. The Pi beilng Hermitian matrices, P5 equals P'i. Further- more, the equationis:

P*, - o(ki)Pe imply P*. fi o(ki)P; k k

provided the o (ik) are real. This leads at onice to the result

S(o) =p(o)S(o).

Hence the Hermitian unit form YXAXA in spin space goes over, by means of the substitution S, into p fold the unit form. So p must be positive and

detSJ2 2

But on account of our normalization of S causinig (det S)2 to be = 1 we find p = 1,

17

440 RICHARD BRAUER AND HERMTANN WEYL.

S(O) S(O), A=A;

i. e. the r epr esenitationi A of the r eal orthogonial gr oup is uniitary. When restricting oneself to real variables one must be aware of the possi-

bility that the fundamental quadratic form

(32) EaxkX

may have an inertial intdex t different fromi 0. This is of particular import for physics as, accordilng to relativity theory, t - 1 for the four-dimelnsiolnal world. One now has to subject the determinilng pi of the algebra II to the equation

(plxl + + p.kxb) 2 Na kxiXk or 2(pjp7 + pep ) a

One will get the new pi from the old ones by means of the transformation H' if the fundamental form (32) arises from the normal form with aik = 8k by means of the transformation H.

But here againi it is convenient to base a more detailed investigation upon the real normal form

(33) (xl)2 . . . -(x)4 + (xt+1)2 + (x8)2 = c, (xI) . i

(Without alny loss of genierality we miiay, suppose 2t n.) In accordalnce with physics, let us call the first t variables xi the temporal, the last i - t the spatial co6rdilnates. The subject of our conisideratioln is the group b,7 of Lorentz tranisformationis; that is, of all real linlear tranisformationls o carryinlg the funidamenital form (33) inlto itself.

Pt+i, . ., P., keep their previous significance, while P1, , Pt assurme the factor i V- 1. We thus have

Pi P'i for (i =1, ,t); Pi P'i for (i=t+1,* n).

The Hermitian conjugate A' of a matrix A may be denoted by A. The Pi as well as the P'i satisfy the fundamental rules of commutation. Both sets of matrices must be changed one into the other by means of a certain transformation B. It is easy enough to write down B explicitly:

(34) B =--V-<t> . Pi . . . Pt. To be exact, we have

f To be quite definite: the variables xi are subjected to the Lorentz-transformation o: xi .> o (ik) xk. The p, (or Pi) then undergo the contragredient

k transformation; but in raising the index by means of pi = eip, one may introduce quantities pi transforming cogrediently with the variables xi.


(35) P', S=BP B-1 or -P'i - BP B-1

according as t is even or odd. The factor i-<t> has been added in order to make B ilermitian: B B. The transposed matrix B' coincides with B but for the sign, namely B' = (-1)<t>B. In the case of an even n the matrix B is of form (25) or (26) according as t is even or odd. All these properties could be fairly easily derived from general considerations; it is not worth the trouble, however, as one may read them at once from the explicit expression (34).

One obtains from (35) the relation

(36) BS(o)B-' = p(o )S(o)

or after multiplication by S'(o) on the left:

S'BS = pB:

the Hermitian form B goes over, by means of the transformation S, into the multiple p of itself. In consequence p is real and one infers, in the same manner as in the definite case, the equation

p(o) =? 1.

As to its dependence on o, p (o) satisfies the condition

p(ofo) - p(o')p(o).

A new consideration, however, is required for determining this sign p. In a Lorentz transformation o (ik) 1j the temporal minor of the whole determinant:

o(ll, * *,o(it)

(37) is either 1 Or<~

o(1) * (tt)

We shall put or (o) + 1 or 1 according as the first or the second case prevails, and call o- (o) the temporal signature; it is a character, i. e.

cr_(O O) == r_ (0 ) * af (0).

We need not trouble to prove this here directly, because we shall see in the course of our further investigations that the p(o) in (36) coincides with ar (o). In the same manner one may introduce a spatial signature or+(o) by nieans of the spatial minor of the matrix 11 o(ie) 11. The latter, though, is = cr(o) *Q;

442 RICHARD BRAUER AND HERAMANN WEYL.

hence the character cr(o) distinguishing the proper and improper transformations equals r,a_. Of the Lorentz transformations having u- 1 one may say that they reverse the sense of time whereas those having u+ 1 reverse the spatial sense. The group of Lorentz transformations falls apart into four pieces not connected with each other and distinguished from each other by the values of the two signatures r and cr.

To prove (37) let us introduce the two vectors

oil' - {o (tl ) ,. * * 0 (itt) }, oi" { o ( , t + 1 ), . (illq)

in the realms of the temporal and spatial co6rdinates respectively. The scalar product (a' Ii') in these two partial spaces has its usual significance a'ib+ + + a'tb't. The relations characteristic for the Lorentz transformation then read:

(?* ?') 8ik + (O*"Ok") ( 1, 5 2,**, t). From these we derive

= 1/+(/ ! :(0o + (1/2 !) l" 2 (1 " 2 ")1 ( ' )I+

(otIot')1- (ot "o1"), (o1"/I02"1), (o +"ot"1 t1)

All terms on the right side are > 0;. hence the whole determinant on the left is ? 1. This determinant however is the square of Q.

The fact that the sigmw p imz (36) equals o- is proved in the following manner. In accordance with

n P*i - ( o(lti)Pk

k=1 we find

o(11) * o(t) (38) P*. e P*t . . . .Pt + *

o(tl) . o(tt)

But a product like Pi,. Pi, * P, * * Pt where i1 . . .t are different indices always has the trace 0 except if il . . .t is a permutation of 1 . . . t; whereas

tr(Pw . . . Pt P1 * * Pt) = (_ 1) )<t>tr(Pi2 P . . pt2) (- )t-<t> 2v

Hence on multiplying equation (38) by P1 . . . Pt to the right and forming the trace, one is led to this value of the determinant (2:

SPINORS IN n DIMENSIONS. 443

Using the definitions of S: P*i --- SPiS-1, and of B, one readily obtains:

2S === tr(SBS-1 * B) - tr(B - SBS-1). According to (36)

S-i PB'- tSB' PB-'1B.

Replacement of B' by B is allowed as B' coincides with B but for a numerical factor. So one finally gets, with T = BS j1 tJK 11

21a = p * tr(BSSB) = p * tr(BS SB) - p - tr(TJF T) p I tJK 12, J,K

and this equation shows p to have the sign of Q. Any representation r : o -> G (o) of the Lorentz group gives rise to

another one cr-r : o -- (o) G (o). Equation (36) or

S (o) (r_- (o) B-1S (o) B

then proves the equivalence: (39) A

The transformation B changes the conjugate of a covariant spinor V/ into a contravariant spinor 4: q)' ==- B (in so far as we confine ourselves to Lorentz's transformations of temporal signature u- - 1). (39) yields, on account of (15), (22), the decompositions

(40) A X A -r-ro { +r-]? ++ urv- + + (r-rv +--'r+,Pv) [n=t 2v]; 07+ro + o?Pr + + u+rv-1

AX cr-ro + +r, + -r2 +** ' [m 2v+ 1].

The latter series breaks off with oJ'rv or cr+rv. In the case n- 2v we have the splitting of A into A+ and A-, when

restricting ourselves to the group b,+ of proper Lorentz transformations [o (o) = 1]. This restriction wipes out the difference between the two signatures cr and o-+. As we mentioned before, B is of form (25) or (26) according as t is even or odd. Hence one has

for even t: A~-/uA, A-'o(A-; for odd t: Q+~c u& A-uA/+.

9. IrrXeducibility. Irreducibility of rf is granted a for-tiori if one is able to prove that there does not exist any homogeneous linear relation with con- stant coefficients (independent of o) among the minors of order f of the matrix of an arbitrary rotation 11 o (ik) 11. This can be shown without using


any other rotations than permutations of the coordinate axes combined with changes of signs. For let us assume that we have such a non-trivial relation R in which a definite minor A (k:: kf ) occurs with a coefficient different from 0. By suitable exchange we can place this minor in the left upper corner of the matrix. We will now take into account the changes of signs only:

the matrices of which have only their chief minors A (i, * if) different from 0. The linear relation R will contain, apart from A (1 2 * f), at least one more term A (1' 2' fi') with a coefficient different from zero. At least one of the indices 1' 2' f', let us say 1, is different from 1, 2, * * *, f. By changing the sign of the one variable xi, the relation R is carried over into a new one R' in which A (1 2 . * f) occurs with the same, A (1' 2' * * f') however with the opposite coefficient. Hence the sum 2 (R + R') certainly is shorter than R, that is, contains less terms than R; but A (1 2 * f) occurs in it with the same coefficient different from 0 as before. The procedure of shortening may be continued until the presupposed linear relation XI 0 leads to the im- possible equation A(1 2 * f) 0.

These considerations were based upon the comnplete group b?n. If one allows proper rotations only, b,+ one may have to combine the permutation in the first step with a change of sign of one variable. The second step can be performed in the same manner provided 2f < n, for then one may choose I as above: as one of the indices 1', 2', * * , f' different from 1, 2, . ,f, furthermore choose m as an index that does not occur in the row 1, 2, . , f, 1', 2', * * *, f', and then change the signs of both variables xi and xma simul- taneously. Even when w = 2v, f = v the procedure of shortening will work as long as the relation R still contains a term A (1' 2'- * v') the indices of which are not just the complement v + 1, . . *, n of 1, . . , v. Thus one will be led in this case finally to a relation of the form:

(41) cA(1,2< * ,v) +c'A(v+ 1, ,n) 0.

Such a relation obtains indeed:

A(v+ 1, * , i) A(1,2, *,v)

SPINORS IN It DIMENSIONS. 445

but there exists of course no other one of the type (41). From this we learn not only that the two representations rJ' and rv- are irreducible, but at the same time that they are inequivalcnt; for it proves that there does not hold any linear relation with fixed coefficients between the components of the two matrices associated with the same arbitrary rotation o in these representations. For the components of these two matrices are

[B Q - 'v) +4 iPBQ(kl k ' v) ]

with

Vki . .. kv J 2 L Vk:L . .. kv)J k'l . .. k' ,

i:L * * , * ,.'1 * i'V and k1 * kv, k1'l* . .k'v are even permutations of the figures 1, 2, * , n. The reasoning above shows that there exists no universal linear relation between the quantities B (Q::v )

The inequivalence of two such rf the ranks f of which do not give the sum n, is granted by their having different degrees.

This whole argument was based upon the comnplex orthogonal group. But nothing is to be modified when one confines oneself to the real orthogonal transformations. Furthermore one sees, by formulating the result in an infinitesimal manner, that it cannot be effected by the inertial index. The infinitesimal transformation

(42) dxi= Xk, dxk xi (i=/=1k)

(all other incremelnts being 0; this transformation engenders the permutation xi ->xk, xk --> - xi as well as the change of sign xi x->--Xi, Xk---> --xk) has to be replaced, if the fundamental quadratic form contains terms with the minus sign, for couples (xi, Xk) consisting of a temporal andl a spatial variable by

dxzi Xk, dxk xi

while it has to be kept unchanged for couples of variables (Xi, Xk) both temporal or both spatial. The statement of irreducibility under all transformations (42) in the definite case is identical with the statement of irreducibility under the transformations replacing them in the indefinite case; one only needs to replace the temporal variables Xk by V-1 Xk.

The product r X r of a representation r with its contragredient r contains the identity r17 at least jz times when r reduces into ,u parts. If we are allowed to make use of the general and elementary theorem that the irreducible


parts of a representation are uniquely determined t (in the sense of equivalence and except for their arrangement), then the formulae (15), (22), (29) show at once the irreducibility of i\ or A+ and A- respectively and the inequivalence of the latter. Another direct proof runs as follows:

Take the full group bit in the even case n = 2v. Using the fundamental quadratic form in the shape (30), let us consider the " diagonal" infinitesimal rotations (43) dx, = if aXa, dya i0aYa (c , * 1, * , v)

(0cP, independent parameters). It is associated in A with the diagonal transformation

dxgL .. .V (i/2) (1 + ** + Covv) xul ... UP (a=+).

Given a partial space P' of the total spin space P, different from 0 and invariant under A., one chooses a non-vanishing vector z:

Z =i zAeA = {ZA} [A = (.a,, , * v)] A

occurring in P'. By performing the substitutioli (43) repeatedly one is able to isolate each term ZAeA, as these parts are of different " weights " (i/2) (aioi + * * + uvcv). Therefore at least one of the fundamental vectors eA occurs in P'. But eA = el ... up goes over into any other fundamental vector e1. .. TP by exchanging Xa -> ?Ya, Ya > X.a those couples (Xa, ya) for which the signs o-a and Ta do not coincide. P' is therefore identical with the total P.-Irreducibility of A for odd n = 2v + 1 is an immediate consequence of the irreducibility for even n, we just proved; one has to restrict oneself merely to the subgroup b.,-1 within bs, n n 2v + 1. One sees in the same manner that the two parts A+., A- are irreducible and inequivalent for the group bit n = 2v.

10. Dirac's theory. Let us suppose we are dealing with a spinor field .VA (Xl' - X4) in an n-dimensional " world " with the fundamental metric form (33). The most essential feature of Dirac's theor-y is that one should be able to form a vector by linear combination of the products qAqB. If n is even, one sees from equation (40) that exactly one such vector si exists-that behaves like a vector at least for all Lorentz transformations not reversing the sense of time; and one such vector for all Lorentz transformations not reversing the spatial sense. In the case n odd, one vector of the second, and

t Compare e. g. Weyl, Theory of Groups and Quantum Mechanics (London, 1931), p. 136.

SPINORS IN It DIMENSIONS. 447

no vector of the first kind exists. Only the first type can be used when one believes in the equivalence of right and left, but is prepared to abandon the equivalence of past and future. n has then to be even and the vector is

si ==_ qBPi f.

From this vector one can derive the scalar field:

(44) , ,B(a/x)( pi = iPi). i

One needs a scalar that arises from linear combination of the products OA

. O/B/aXi in Dirac's theory as the main part of the actioni quantity which accounts for the fundamental features of the whole quantum theory. There is no ambiguity: for (A X a) X IX contains the identity rP or rather the representation o-rA' just once if decomposed inlto its irreducible parts. That is shown by equation (40) when one takes into account the fundamental lemma of the theory of representations asserting that the product, r X P1 contains the identity rP once, or not at all, according as the two irreducible representations r, rP of the same group are equivalent or not. Dirac's quantity of action contains, apart from (44), a second term which is a linear combination of the undifferentiated products qAbB; it is multiplied by the mass, and accounts for the inertia of matter. There exists just one such scalar, namely VB+, in the case of an even as well as an odd n.

Furthermore one may consider as essential the fact that the time component of the electric current is positive-definite in Dirac's theory, namely proportional to the "probability density" 2 t/AfA; this grants the atomistic

A structure of electric charge. If the fundamental form (33) is of inertial index t, this property however is not possessed by the vector contained in A X A but by the tensor of rank t with the components

t= i . Pitt (i*, it different),

the "temporal " component, S12 ... t, of which is = (but for a numerical factor). It seems to be required by the scheme of Maxwell's equations that electric current should be a vector; this requirement, together with the postu- late of the atomic structure of electricity, compels us to assume the inertial index t to be = 1I

11. Appendix. Automorphisms of the complete matriix algebra. A one-


to-one correspondence X T>X* of the ring of all n-rowed matrices upon itself is isomorphic wheni satisfyitng the conditions

(X + Y)* eX* + Ye, (AX)* X - *, (XY)* X*Y*

(X an a,rbitrary number). The only such automoorphism is " si,milarity":

X* = AXA--,

A being a fixed non-singular matrix.

Proof. The equation GX y-X has a solution X 7 0 only if y is an eigen-value of the matrix G; for the columns of the matrix X must be eigen- vectors belonging to the eigen-value y. The eigen-values of G thus are characterized in a manner invariant with respect to the given automorphism. Consequently G* has the same eigen-values as G. Thus we are led to proceed as follows. Let us choose n fixed different numbers -y, * y ,y and with them form the diagonal matrix

'Yl

As G* has the same eigen-values as G, a non-singular matrix A can be determined such that G* = AGA-'. Let us replace every X* by X** A-AX*A and now consider the automorphism X -> X** that leaves G unchanged. The matrix Ei7 containing an element different from 0, namely 1, only at the crossing point of the i-th row with the kc-th column is determined by the properties

GEik = yiEik EikG ykEk

except for a numerical factor. Hence we have

(45) Eik -> EiP*= c'kEik-

The equation E E2, =E,, furnishes a =o, a = 1. After putting 2=i , ai (Xlk the relation

leads to a k os,fk. On account of i= 1 one therefore has 3 = /ci and =ik ca/ak. Hence in accordance with (45) an arbitrary matrix X= Xik

and its image X88 = J* 1x are linked by the relation

SPINORS IN n DIMENSIONS. 449

xi*c a$xik/ak or X88 AoXAo-'

where Ao is the diagonal matrix with the terms a,, - *, ca. This demonstration furnishes a method for constructing a spinor from a

given tensor set g. The method will be used preferably in the case where g consists of only one tensor of definite rank. Our representation of degree 2" of the algebra II associates with g a matrix G. Let us assume that G has the (simple) eigen-value y and let . be the corresponding eigen-vector in spin space: Gq = y -Ar. The rotation o carries g into a set g (o) represented by the matrix G(o). y is a (simple) eigen-value of G(o) as well as of G, and the solution VI (o) of the equation

G (o) +(o) yl +()

arises from f by the transformation S ( o) corresponding to o in the spin representation.

THE INSTITUTE FOR ADVANCED STUDY,

PRINCETON, NEW JERSEY.

brauer, weyl - spinors in n dimensions

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