IL NUOVO CIMENTO VOL. XXX, N. 3 1 o Novembre 1963

Spin Connection in Terms of Dirac Operators.

H. G. Loos

G i a n n i n i S c i e n t i / i c Corpora t ion - S a n t a A n n , Cal.

(rieevuto il 6 Giugno 1963)

S u m m a r y . - - A derivation is given for the expression of the spin con- nection in terms of the Dirac operators, using the Hamilton-Cayley theorem, modified for singular operators.

I n any analyt ic theory of spinor fields one has to introduce a spin-covariant

der ivat ive of spinors,

(1)

s

V~ ~ - - 8~V - - F ~ ,

s

v~ r Cr + e f t , s

v y ~ , I - [F~, ~]

where ~0 and ~ respect ively are any four-component spinor and its adjoint

and M is any linear operator in four-dimensionM spin space. The transfor- ma t ion propert ies of the spin-connection matr ices F~ are fixed by the require-

men t tha t the spin-covariant differential of a quan t i ty wi th spinor character

is a quan t i ty of the same kind.

The field of 1)irac operators y~(x ~) is introduced, as usual, by decomposit ion

of the metr ic tensor,

(2) g ~ f f := y ~ y ~ ,

s

and in order tha t the energy m o m e n t u m operators iV, and the Dirac oper- ators y~. refer to different degrees of freedom, these operators mus t corn-

902 H.G. ZOOS

mute, i.e.,

(3) o = v ~ = v ~ 7 ~ - [ r ~ , 7~],

where ~7 denotes the ordinary covariant derivative. For given ya(x'), (3) is a restr ict ion for F~(x~). In fact, the traceless parts of the matr ices/ '~(x ~) are determined uniquely by the mat r ix fields y;.(x'); an explicit expression for F - - } ( T r I ' ~ ) I in terms of the y~. has been derived by FLETCm~R (~), for space of any number of dimensions.

The purpose of this note is to give a more e lementary derivation of Fletcheffs result, for the case of four dimensions.

Transvect ion of (3) with y x to the right and use of (2) gives

(4) }(V~7~)7 ~ = I ~ - - I T ~ F ~ / ,

which equation may be wri t ten as

(5) }(V~7~)7 ~ = LG,

where L is a linear operator acting on 4 • 4 complex matrices M:

(6)

From (5) we have formally

(7)

L M = M - - �88 ~.

F z = 5 -1{ 1 (~7xy~)~} ,

where L -1 is the inverse of the operator L defined by (6). Using (2) i t can be shown by straightforward calculation tha t the operator L has the following

eigenmatrices and eigenvalues.

eigenvalue

0

1_ 2

(8) 1

_3 2

2

eigenmatrices

I

YE~Ya

7~

YE~Y~.Y~,Y~1

These eigenmatrices span the space of all 4 • complex matrices, since any

4 • complex m~trix M has a Clifford expansion,

(9) M = mI $ m~7~ § m~YE~Y~j $ m~TJ~Y~ § m~7~7~7~7~ �9

SPIN CONNECTION IN TERMS OF DIRAC OPERATORS 903;

I t follows tha t for any I :<4 ma t r ix M,

(lO) L ( L - - ~ ) ( L - - 1) (L- - ~ ) (L- - 2)M 0 .

This can be verified by substituting' in (10) for M its Clifford expansion; by

commuta t ion of the factors on the lef t -hand side of (10) one can always bring

to bear on any t e rm of the ('lifford expansion a factor which annihilates tha t term.

Since the opera tor L has an eigenwflue 0, the inverse operat ion L 1 is not

defined for ~ll 4 >~ 4 matrices, i n order to find out for which matr ices M the expression L 1,'1/ is meaningful , we consider the eigenvalue equat ion

(11 ) Lml~.j - - I~mk, ~ ,

where m~,~ are eigenmatrices of L belon~ing to the eigenvalue l~. Multiplication on the left by L -~ oqves

1 ( 1 2 ) L - l t t ~ l , , J 7 m z d ,

unless /~ wmishes. Since the eigenmatrices m,,j span the space of all 4 • matrices, we can expand any 4 • ma t r ix M in eigenmatrices m~,j:

(13) M = ~ c~.~ m~,~, /c,3

and find by appl icat ion of (12)

(*]~, j

k,j k,j

unless one of the lk vanishes. Suppose lo is the only vanishing eigenvMue of L. Then, (141 is only defined if co,~ v~mishes, i .e . , if the expansion of M in te rms of eigemm~trices does not contain the eigenm~trices too,j, belonging to the

eigenvalue 10. Also, for this (.ase one m a y add to (141 an a rb i t r a ry linear (:om- binat ion of eigemnatrices mo,~:

(151

since Lmo,~ O. (8) shows tha t in the present case mo,~ is the ident i ty m~trix. Hence, the expression L- ' M is defined, up to a t e rm a I , for all l x-1 matr ices

9 0 4 H . G . LOOS

M' whose Clifford expansions have no te rm I , i.e., for all 4 X4 matrices M' of vanishing trace. For such matrices, the factor L in (10) m~y be dropped,

~nd we have

(16) ( L - - � 8 9 ~) (L- - 2 ) M ' = 0 .

B y writing

(17) L = E - A ,

where E is the ident i ty operator in the space of all 4 X 4 matrices, (16) becomes

(18) (4A 4 -~ 4A 3 - - A 2 - - A ) M ' = 0 .

We proceed with the ansatz

(19) L - I M ' = ~ c ~ A ' M ' , r=O

where the c~ are complex numbers. Multiplying (19) by L = E - - A we obtain

R

(20) M ' = ~ ( c r A ' - - crA~+I)M ' �9

Equa t ion (20) is identically satisfied on account of (18) if

(21)

where 2 is a number.

C o = l ,

Hence, (19) becomes

(22)

where

(23)

( ~ 0 = 1 ,

C 1 - - C o = - - 2 ,

C 2 - - C 1 = - - ~

C a - C 2 = 4 ~ ,

c3= 42,

The solution to this set of equations is

7 __4 2 Ci=6, V 2 = 3 , V3 = 3""

4 2 ~-Aa~ M ' L - ~ M ' - - - ( I ~- ~ A -4- ~ A + 3 ~ J ,

A M ' = �88 ~ ,

SPIN CONNE(~TION 1N TERMS (SP I)IRAC OPERATORS ,Q()5

on account of (6) and (17). Equa t ion (22) is valid for any 4 • complex ma t r ix with vanishing trace. Applicat ion to (7) is permissible because (V~ya)y a has indeed a zero trace, as follows b y taking the t race of (4), together with Ya Ya= 4I, which results fronl (2). Remember ing t ha t an a rb i t r a ry t e r m G I inay t)e added to L-1M ' we have finally

4A,, 2 4 (24) F I(I I L-I -t ~ + .~- ~)(V~y~)~+ GI

which agrees with Flet(,her's result (*) for the spin connection i l t a four-

dimensional Riemann space. I t follows fi 'om (23) tha t the t race of AM' is

the same as the tra(,e of M'. Sim,e (V~ya)y a has zero trace, the first t e rm in (24)~

N

(25) I~ = 14(1 q- ~-A + J-A ~ .j- ~ )(V,,ya) y ,

o

has zero trace. Hence, the traeeless p{~rt F~ of the spin connection P~ is

uniquely de termined by the fiehl of I) irac operators y~, and we have

o

(26) F~ ~ F , + a~I ,

where a~ is an a rb i t r a ry vector.

(1) ,7. ( ] . F L E T C I I E R : NIr Cdmento, 8 , 4 5 1 ( 1 9 5 8 ) .

R I A S S U N T O

Viene data. una derivazione per l'espressione della connessione di spin in termini degli opera.tori di Dira.e nsando il teorema, di ttamilton-Cayley modifieato per operatori singoli.

5S - ll N,mmo Cim, e~do.