Generally covariant dirac equation and associated boson fields

IL NUOVO CIMENTO VOL. XXVI, N. 5 1 o Dicembre 1962

Generally Covariant Dirac Equation and Associated Boson Fields (*).

H. LEUTWYLEI~

Institut ]iir Theoretische Physik der Universitdt Bern

(ricevuto il 10 Agosto 1962)

S u m m a r y . - - We reject the equation of the classical formalism of eova- riant spinor analysis wherein the field of the affine connection in spinor space, F~(x), is fixed by D~yv=O. By a closer analogy to tensor analysis in metric space we find only reality conditions imposed on F~(x), with considerable independent dynamical freedom remaining. This suggests an approach similar to the procedure of YANG and MILLS. Starting with a variational principle we obtain a nonlinear wave equation with nonvanishing rest mass for the affine field F~(x), whose structure is dictated by invariance requirements. As particular examples, we discuss MaxwelFs equations and the pseudoscalar meson-nucleon interaction, in addition to Heisenberg's nonlinear spinor equation.

1 . - I n t r o d u c t i o n .

The specia l - re la t iv is t ic Dirae e q u a t i o n is k n o w n to be i n v a r i a n t w i th res-

pect to a change of r e p r e s e n t a t i o n of the y -mat r ices

(1.1) y~ = S - I ~ S (# = 1, 2, 3, 4 ; x 4 = ct),.

if s i m u l t a n e o u s l y the wave f u n c t i o n ~v(x) is replaced b y

(1.2) ~ ' (x) = s l~v(x).

(') In partial fulfillment of the requirements lor Dr. Phil.

G E N E R A L L Y C O V A R I A N T D I R A C E Q U A T I O N AND A S S O C I A T E D BOSON F I E L D S 1067

I n a general ly relativist ic formal ism the defining relations for the matr ices ~,,

(1.3) {y., y~). = 2g.~(x) 1 , (i)

le~d necessarily to co-ordinate-dependent ma t r ix fields y,(x). Therefore, there is no reason to restr ict the group of similari ty t ransformat ions (1.1) to co- ordinate- independent matr ices S. Obviously, the special-relativistic Dirac equa- t ion is not inYariant with respect to co-ordinate-dependent t rans format ions S(x). I n analogy to the procedure known f rom general relat ivi ty , one m a y establish covariance by int roducing an affine connection F~(x) whose t rans- fo rmat ion propert ies insure tha t the quan t i ty

(1.4)

t ransforms as

(1.5) D' ' = S-1D~v 1~ ~)

This relat ion yields the t rans format ion law for the field F~,(x)

(1.6) 2x) - q (x)S - s - G s .

F~(x) is a mat r ix field, which we denote sometimes explicit ly by FzZ. = 1, 2, 3, 4 for an irreducible representa t ion of the matr ices y~AB).

no ta t ion eq. (1.4) takes the form

(A, B I n this

(1.4)'

The step f rom covariance under space- independent to space-dependen~

S- t ransformat ions m a y be considered as an analogue of the Y~ng-Mills method (~); in fact we will show in Section 4 tha t the isovectorfield b,(x) introduced by YANG and MILLS is contained in the field F,(x) if we app ly the co- va r i an t formal ism to reducible representa t ions ?~ where the isospin ro ta t ions m a y be in te rpre ted as a subgroup of the S- t ransformat ions (1.1).

The classical procedure (3) to determine the field / ' ( x ) involves an analogy between g,,, F,q. and 7~,, I~ respectively. Since in R iemann ian space the affine connection ~ is determined by the requi rement of conservat ion of

(1) Signature (+ -4- + - - ) ; 1: unit matrix in spinor space. (2) C. N. u a.nd R. L. MILLS: Phys. Rev., 96, 191 (1954). (~) E. WIGNER: Zeits. J. Phys., 53, 592 (t929); V. FOCK: Zeits. ]. Phys., 57, 261

(1929); H. TETRODE: Zeits. ]. Phys., 50, 336 (1928); E. SCI~I~0DI~GE~: Sitzber. Preuss. Ak. d. Wiss., 25, 105 (1932); V. BARGMANN: Sitzber. Preuss. Ak. d. Wiss., 25, 346 (1932).

1068 ~. LEUTWYLER

length under parallel displacement, i.e.,

(1.7)

one assumes the equat ion

(1.8) Cz - C m + ~ o

to establish an analogous connection between the fields F~~ and y,(x). At first sight the 16 mat r ix eqs. (1.8) seem to overdetermine the 4 mat r ix fields ]~O(x). t towever , this is not the case as was shown by Schr6dinger (*). Any given field ~z(x) satisfying the relations (1.3) gives rise to a field F ~ tha t fulfills (1.8). In irreducible representat ions Fff(x) is unique up to ~n addit ive multiple of the unit matr ix, which is in terpre ted as the eteetromag~etic potential.

In the present article we reject the analogy between eqs. (1.7) and (1.8). In fact the field y,(x) seems not to be the appropriate expression for the metr ic tensor in spinor space. The fundamenta l role in constructing invariants

(1.9) ~F ~ ~fCAyJ = (v~p)*

is played by the matr ix A whose definition is (5)

(1.10) y,r = - - A ( x ) y , ( x ) A - I ( x ) ; A(x ) =- A t (x ) .

Insert ing the t ransformat ion (1.1) one obtains the t ransformat ion law

(1.11) A ' - ~ S r

The relation between metric and afline connection m ay be in terpre ted in terms of basis vectors of the associated vector space. Here we are interested in the spinor space a t tached to each point x. Let eA(x) [A, B - - 1 , 2, 3, 4; spinor indices] denote the four l inearly independent basis vectors of the spinor space a t tached to the point x. Then the affine connection I s ( x ) furaishes a mapping of the vector space at the point x onto the vector space at the point x + S x . To each vector e~(x) is associated a mapping eA'(~) which m ay be expanded with respect to the basis e~(x+Sx) :

(1.12) e~'(x) = e~(z + ~x) + F .~(x)e~(x + ~x)~x" .

(4) E. SCHRSDISGER: Sitzber. Preuss. Ak. d. Wiss., 25, 105 (1932). (~) V. BARGMA)IN: 5itzber. Preuss. Ak. d. Wiss., 25, 346 (1932).

G E N E R A L L Y C O V A R I A N T D I R A C E Q U A T I O N A N D A S S O C I A T E D B O S O N F I E L D S 1 0 6 9

On the other hand the metric A ~B is defined by the scalar p roduc t of the basis

vectors :

( 1 . 1 3 ) A ~ ( x ) =- (e~(x), eB(x)) ---- (e~(x), e ~ ( x ) ) * = ( A ~ ( x ) ) * .

The dot indicates the ant i l inear i ty in the first factor of the scalar product , which leads to the t rans format ion p roper ty

(1.11) A]B ' = (Sc a)*Ad~SD ~ -- S ~A~'D S~)~ ~ (Sr ~ S , ~ .

Assuming the mapping (1.12) to conserve length, as it is done in t~iemannian

geometry , we find

: (x)) =

or equivalent ly

�9 a �9 ~ A~C (1.14) ( D , A ) ] ' ~ , A ~ n + ( F : ) A + F ~ J = 0

ins tead of (1.8). At first sight this equat ion seems to lead to a de terminat ion of I~,ff in te rms of A ]B quite analogous to the expression

(1.15)

obta ined f rom (1.7). In spite of this visual similari ty between (1.7) and (1.14) there is a vas t difference between the geometries described by g~, F~e~ and A ~B, F,~ n respectively. The reason is the following: The t rans format ion propert ies of gu~ involve derivat ives of the t rans format ion functions x " = X"(x) , while the matr ices S(x) in (1.11) m a y be given independent ly at each point of space. Therefore, in contradis t inct ion to g,,(x), A(x) may be brought to the pseudoeuclidean ]orm (fl in Dirac 's notat ion) simultaneously at every point x o/ space. On the other hand, the very fact tha t this is not the case for the field gz~(x) is responsible for its usefulness in describing curved spaces. The means for describing the geomet ry of spinor space is given direct ly b y the field F,(x); i t just does not possess a potent ia l f rom which i t could be derived (as was the case wi th the field F ~ ( x ) whose potent ia l was g,~(x)).

I n the fu ture we shall res t r ic t ourselves to co-ordinate sys tems in spiaor space where the field A ( x ) = - A is pseudoeuclidean everywhere. We cer ta inly can not obta in a more general geomet ry if we admi t co-ordinate sys tems where this is not the case. This would be analogous to a descript ion of special rela- t iv i ty in curvil inear co-ordinate systems. Wi th this restr ict ion the allowable t r ans format ions S(x) are to be un i t a ry relat ive to A:

(1.16) S t A S = A

,~ 68 - l l Nuovo ~imento.

'1~70 ]~I. LEUTWYLER

and the condition (1.14) then reads

(1.17) Fj, * = - - AFt, A - 1 .

This expression is mere ly a rea l i ty condition for the field %(x), which shows the fundamen ta l difference be tween (1.7) and (1.14).

Assuming the representa t ion y~ to be irreducible, we can develop F~ wi th respect to the basis genera ted by the 16 l inearly independent products

(1.18) r . = A.1 + B % + C7% ~ + D2~j~ +E.y~,

where 7,, and 7~ are defined by

1 (1.19) ~, = 1 ( ~ , ~ _ ~,~,) y5 - - 2 4 V ~ e~ '~} ' t~ ' r~ '~ "

The real i ty condit ion (1.17) then leads to

* ~* B "- ~* D ~ E * - - E . (1.20) A , = - - A , ; B , = , , Cu~*= C# ~ ; D~, = - - ~ ; i , =

Thus in our formal ism the geomet ry of a generally curved spinor space m a y be described b y the five affine-tensorial fields (1.20). This includes previous versions of spinor geomet ry as special eases:

The classical covar iant formal ism s tar t ing with (1.8) leads to an expression of these tensor fields in te rms of derivat ives of y,(x) (~), while the field A, (x ) is a rb i t r a ry and in te rpre ted as the e lectromagnet ic potent ia l . Other authors (7) replace (1.8) by D,y "~ = 0, which permits , in addi t ion to A ( x ) , an a rb i t r a ry field E~(x), while B~ ~, C~ ~ and D~ ~ are again given b y expressions containing der ivat ives of ?,(x). Final ly s tar t ing f rom a four-leg fo rmal i sm some authors (s) allow the field C,~(x) to play the role of a physical ly independent field. The purpose of this article is to include all these versions in a geometr ically satisfying manner and to show t h a t this formal i sm is capable of var ious physical applications.

(6) j . G. FLETCHER: NUOVO Cimento, 8, 451 (1958); B. E. LAURENT: Ark. ]. Fys., 16, 263 (1959).

(7) O. KLEIN: Ark. ]. Fys., 17, 517 (1960); B. E. LAURENT: Ark. ]. Fys., t6, 263 (1959); J. R. KLAUDER, H. LEUTWYLER and J. SCHAER: NUOVO Cimento, 23, 1099 (1962).

(s) T. KIMURA: Prog. Theor. Phys., 24, 386 (1960); A. M. BRODSKI, D. IVA- NE~KO and G. A. SOK�9 Soviet Physics JETP, 14, 930 (1962); V. I. RODICIIEV: Soviet Physics JETP, 13, 1029 (1962).

GENERALLY COVARIANT DIRAC EQUATION AND ASSOCIATED BOSON F I E L D S 1071

Although the field /'~(x) is not a spin-tensor, since i t t ransforms according to (1.6), we m a y - - i n analogy to the definition of the curva ture tensor in Rie- mann ian space--def ine a tensor q ~ by the relation

(1.21) (Dt, D ~ - - D , D , ) y J ~-- - - q ~ , W ; q~F,~ = ~F, y ~ - - ~ I ' ~ - - [F~, / '~]_.

Fo r ~ b an anMogous real i ty relat ion to (1.17) holds:

(1.22) ( D D - - D D I , ) A ~ q~ t A + A q 5 -~ O .

Another complete ly different way to obtain a spin-tensor f rom F , ( x ) is to subs t rac t the classical affine connection F~~ The difference of two affine connections is known to t ransform like a tensor, the addi t ional t e rms occurring in the t rans format ion law (1.6) cancelling each other. For this purpose, i t is useful to have a unique definition of the field F~~ This uniqueness m a y be achieved by adding to the relat ion (1.8) the equat ions (0)

(1.23)

= + 50 + A o-- o;

D.oB _---- + 5 ~ Bro_- o;

D oc =_ c o= o;

y t _ _ _ A y , A - 1 ; A t = A ,

yt~ ~" = B y t , B - 1 ; B r = - - B ,

y,* = C y C -1 ; C* = C -1 .

These relat ions fix the addi t ive mult iple of the unit ma t r ix t ha t is not determined b y (1.8).

This concludes the discussion of the (( geometr ical ~) notions to be used. I n order to introduce dynamics we proceed to formula te an action prin-

ciple including the fields gz~, FS~ , y,, F~., y~. Appendix A is devoted to the construct ion of the most general invar ian t t ha t can be constructed out of these fields, and their par t ia l derivatives, satisfying certain l inear i ty conditions. I n Section 2 a general discussion of the propert ies of the action principle resul t ing f rom this invar ian t is given, and a simple, i l lustrat ive example will be analysed in some detail. I n Section 3 we will, as a more physical application, derive t te isenbergs nonlinear spinor equat ion by requir ing s y m m e t r y of the Lagran- gian wi th respect to the Touschek t rans format ion and the Paul i -Giirsey group. The source of the nonlineari ty characteris t ic of Heisenberg 's equat ion is the field F,(x). Final ly in Section 4 we inves t igate a reducible 8 • 8-representat ion of the y-matr ices . I t is shown tha t as a special case the pseudoscalar isospin- invar ian t nucleon-pion sys tem is contained in our act ion principle, the meson field wi th nonvanishing res tmass being contained in FF,(x ).

(9) W. PAULI: A n n . P h y s . L p z . , 18, 305, 337 (1933).

1072 H. LEUTWYLER

2 . - T h e a c t i o n p r i n c i p l e .

~This section is concerned with the general features of the action principle based~on the Lagrangian constructed in Appendix A:

(2.1) 8f{L(R) + L (~ 2) + L(q~) + L~} V l ~ ddx = O.

The variat ion acts independent ly on the quantit ies ~e , , gt,~, F,, %0. According to the considerations in the in t roduct ion we don~t consider A as ~ field variable, bu t take it to be constant. The var ia t ion in ?,(x) h~s to be calculated f rom 3g.,,(x), while the var ia t ion in ~ ( x ) has to fulfill the real i ty condition (1.17).

A) General aspects o/ the equations o] motion.

a) V a r i a t i o n s w i t h r e s p e c t t o Fff~. In our me thod F~, occurs only in the first t e rm of the Lagr~ngian. Therefore the variat ions ~FzQ ~ lead in the wellknown method of Palat ini to (1.7).

Our formalism eliminates the following difficulty commonly encountered in the Palat inl method including a Dirac field: I f the affine connection of spinor space is not an independent dynamical variable bu t t aken to be determined by the equat ion D,y, = O, the expression for F, will depend on Fff~. Thus a var ia t ion in ~ leads to a change in F~, and thus Mso in L~. In such a case the action is no longer s ta t ionary if F ~ equals the Christoffel symbol, ~s required by (1.7).

b) T h e v a r i a t i o n s ~g.,~. To evaluate the variations of the metric we must also take into account the influence such a var ia t ion has on the matrices y~(x). This change in y, (x) is determined by eq. (1.3) up to an infinitesimal un i ta ry (10) t ransformat ion ~T:

(2.2)

Since the entire action is invar iant under a rb i t ra ry un i ta ry t ransformations, the var ia t ion

(2.3) ~1~ € ~ I F , = o ; ~%0--0

is equivalent to

(2.4) ~2r" = �89 ; ~ F , = - [aT, F , ] - ~ aT ; ~%0 = - - ~T%0.

(10) Unitary with respect to A ~ A being the same symbol as in (2.2).

G E N E R A L L Y C O V A R I A N T D I R A C E Q U A T I O N A N D A S S O C I A T E D B O S O N F I E L D S 1073

The action being s ta t ionary for a rb i t ra ry 3~ and all variations ~F~ which satisfy (1.17), the only contr ibut ing te rm in (2.4) is 3~ ' . Therefore we obtain the field equations for the tensor g~ considering the simultaneous variat ions

(2.5) ~g"~ ; ~7" = �89 �9

Wi thout explicit ly carrying out these variat ions in the terms L(~b~) and L( r we get (11) :

(2.6) al(R~,~-- 1 1 ~L(qb~) ~L(q~) _

1 <-- <-~ = ~ ~ {(n,r~ + D~,)(~ + ~i~0)-- (~1 + ~y 0 ) (~ , + r ,~)} V,

~L/~*g ~'~ denotes the variat ional derivat ive of L obtained by varying g~ and ~" simultaneously according to (2.5).

e) V a r i a t i o n s of ~f(x). Performing a part ial integrat ion we obtain the following covariant Dirac equation:

/z * ~r (~.7) (el + ~ ) ~ D ~ + D~{~ (~1 + %r~)~} + (e~ + ~,r~)~ = 0 .

In t roducing e l = p , + i q , ; e~=p~+iq~ we get

(2.8) (pl @ iq~y~)y~D~,v + Ty~ = O,

T = �89 + e~7~ ) + ~D~{ r ~ "(e 1. + e*r~) } .

d) V a r i a t i o n s of t h e f i e l d T'~(x). Now we turn to the field equations for F,(x). They consist of three par ts :

3L( r ~) 3L(~b) 3L~ (2.9) ~ - + - ~ +~F~ = o.

To evaluate the variat ional derivatives we m ay use the following formal de- vice (1~): 3F~ is a spin-tensor (in contradist inct ion to F, itself). Therefore D~ ~F~ is a covariant expression and one obtains, in complete analogy to the

(11) In the derivation of (2.6) use was made of 3Y5=0, which may be established with the help of ~he two relations

{7,. n)+ = o; ~,~ = - 1.

(12) B. E. L.~_URENT: Ark,. f. Fys., 16, 263 (1959).

1074 H. LEUTWYLER

Palat iui method in Rienmnnian space,

(2.1o)

With the help of this equat ion we find

(2.11) ~L(r

- - - - 2c~ D~ ,~" - - 2c2Dt,~ "~* =

2

(2.12) 3L,p ~ - = --~(e~ + e~r,)r'-7~(e * + e*~,,)~.

Generally speaking the t e rm 8L(~b~)/~F, contains second derivatives of F , ; we shall call it the (~ wave par t ~) of the eq. (2.9), in contrast to the (~ algebraic par t ~), 8L(r which contains F~ in an algebraic, l inear fashion. The na ture of the system of eqs. (2.9) depends ve ry strongly an the order of magni tude of [b I/] e l - - 1 ~, where I b I, I c ] represent typical coefficients in L(qb~) and in L(~5) respectively. This rat io has the dimension of the square of a length 1. One m a y classify the actions according to the value of an associated m o m e n t u m parameter roughly in three classes:

1) ~/1 is of the order of magni tude of the momenta occurring in the system. In this ease the algebraic par t in eq. (2.9) pl~ys the role of a mass- term. An example of this type will be considered in Section 4.

2) h/1 is ve ry small compared to the occurring momenta . I n this l imiting case one ma y neglect the algebraic part .

3) In the opposite limit, ]~/l being much larger than the occurring momenta, the field F,(x) wil l be determined algebraically.

However there is an interest ing pecul iar i ty in the l imiting case 3). In fact, 1 and ~5 commute with ~ " such tha t 1- and ~-components of 8L((~)/SF~ in (2.11), which is l inear in F, , vanish identically. This means, t ha t there does not exist a masslike t e rm for the fields A , and E, in (1.18)~ such tha t even in the limit 3) these fields will be determined by wave equations. An example of type 3) is t rea ted in Section 3.

B) Il lustrative example. - For the purpose of i l lustrat ion we consider an example where

(2.13) L(~b~) --~ b Tr ( ~ 5 r ; e2 : c2 --~ O.


The field equations which result from a var ia t ion of F~ then read

(2.14) D,(4b q5 ~'~ + 2c1~ ~''} = - - e ~ y " - - e * y ' ~ .

These equations may be developed with respect to the basis spanned by the y-matrices in order to obtain the field equations for the quantit ies A, , B~ ~,

D ~ Cff ~, ~, E in eq. (1.18). First let us consider the 1-component of (2.14):

(2.15) Tr (D u {4b~b "~ + 2GyU~}) -- 1 ~, {V I g lab Tr ~"=~ =

16b + ~ ~. {~/~(~"A ~- ~ . )}

I f we set

�9 P l e2 (2.16) A , = ze~,; 8-b = '

= - - 2pxvTy~0 .

then % denotes the electromagnetic potent ial [~0~----~0" according to (1.20)] and e the charge of the electron. Thus the 1-component of (2.14) reduces to Maxwell 's equations for ~(x) , the 1-component of I'f,(x). In order to simplify the equations for the other components let us assume tha t the deviat ion f rom the classical field F~,O(x) defined by eqs. (1.8) and (1.23) is small

(2.17)

c. = r ; + ~ r / + +rd+. . .

+.~= + ; + ~ + / + ++.~= +...

Denoting ordinary covariant derivatives by V~, we obtain

2s ~o - - 2 s ~y~'~o.

and two similar equations for the fields C, ~1 and D~ 1. For E~ 1 we get

45 ~7g {V/~E gl - - V,E/~I } = iql_2e Y'ySy-- ~ �9

As stated, we find masslike terms in the equations for B~ ~'~, CtJ ~ and Du~ but not in those for A, 1 and E, 1. Taking the limiting case 3)~ (Cl-+ c~)~ the fields B, ~1, C~ ~1, D~ 1 will be forced to vanish. The magni tude of the surviving field E , 1 compared to the magni tude of the electromagnetic potent ial will be determined by the ratio ql/P~.

1076 ~. LEUTWYLER

3. - Heisenberg's nonlinear spinor equation.

As a more phys ica l appl ica t ion (13) of the fo rmal i sm deve loped in Sect ions 1

and 2, we c o n s i d e r - - p u r e l y in a fo rmal w a y - - q u a n t i z e d fields ~0(x) and F , ( x )

and assume the L a g r a n g i a n to be a n t i s y m m e t r i z e d in y~, ~. W e impose t he fol lowing res t r ic t ions on our general va r i a t iona l pr inciple :

a) The L a g r a n g i a n is to be of t y p e 3), i .e. L ( r 2) negligible (b = 0).

b) W e assume s y m m e t r y wi th respect to the Tousehek t r a n s f o r m a t i o n (14),

(3.1) v 2 -~ ~ ' = e ~ ' ~ = (cos a + sin ~75)~,

as well as wi th respect to the Pau l i -Gi i r sey Group (~5)

(3.2) y ~ - ~ y / = a y ~ + b y s y ~ ~ Ial=+lbl== ]

W e = C-lW* .

Here the ma t r i x C is defined b y

(3.3) r f = c7 c- ; c * = ; A C �9 = - - C A T .

A n inves t iga t ion of the s y m m e t r y requ i rement s is f o u n d in A p p e n d i x B, where

i t is shown t h a t t h e y lead to the res t r ic t ions

(3.4) D,r = D , C -- 0 ,

(3.5) e~* = el ; e2* = e2 ; e3 = ea = 0 .

W i t h the help of the classical afline connec t ion (1.8) and (1.23) the cond i t ion

(3.4) res t r ic ts F , so t h a t

(3.6)

W e are now in a pos i t ion to discuss the ex t r ema l of the ac t ion func t iona l fo r va r ia t ions be longing to the class (3.6). A p p e n d i x C deals wi th the reso lu t ion

(is) Compare references (s), where it is shown that Hcisenberg's equation may be obtained with the help of covariant four-leg formalism, which amounts to consider only the fields A~ and C~ ~.

(14) B. TOUSCHEK: ~VUOVO Cimento, 5, 1281 (1957). (15) W. PAULI: Nuovo Cimento, 6, 204 (1957); F. GtiRS]~y: Nuovo Cime~to, 7, 411

(1958).

G E N E R A L L Y COVARIANT D I R A C E Q U A T I O N AND A S S O C I A T E D BOSON F I E L D S 1 0 7 7

of the field equations for ~ in the algebraic l imit ing case 3) in which we are in teres ted here. (For the purpose of this section we need only the expression for ~ since we do not consider the var ia t ions 8B~ and 8D~ ~ which lead to eqs. (C.6) and (C.8), as they are incompat ib le wi th the s y m m e t r y requirements b).) Specializing the general resolution (3.7) of Appendix C to e~*= e~

(ql = 0), e2* ~-- e2 (q2 = O) we find

(3.7) ~Q,uv c2p2- Cl])1 e2pl -~ Clp~.

- - 16(c 2 q- c2)2 {~v75Y'~vg ~ - ~75Y"~fg "~ - - 1-6(c 2 q_ c ~) ~ ' ~ ~'sy~YJ �9

Inser t ing this solution into the Dirac eq. (2.8), we obta in

(3.8) Y"D.~ ' --~(@'~,~vJ)y~y,,~, = o ,

C 2 3 ~(P2--P[) - - 2c~pxp~ 8 p,(c~ + c~)

This is in fact a covar iant formula t ion of Heisenberg 's nonlinear spinor equa- t ion (le), the nonl inear i ty in the present case being caused b y the presence of the field f~(x).

4. - Reducible representations: isospin-invariant meson-nucleon interaction.

I t is well known tha t the two states of the nucleon m a y be represented b y one e ight -component spinor

( 4 . 1 ) ~ = .

I n order to formula te the Dirae equat ion for this field we need an 8 • 8-representa t ion of the y-matrices. Le t us denote 8 • 8-matrices b y d~ in cont ras t to M which s tands for 4 • 4-matrices. A special representa t ion is given b y

I t is obvious t h a t such a representa t ion is reducible, decomposing into the direct sum of two equivalent irreducible representat ions. We therefore mus t

(is) H. P. DORR, W. ttEISENBER(L H. IRITTER, S. SCItLIEDER and K. YAMAZAKI: Zeits. ]. Nat. Forsch., 14 a, 441 (1959).

1078 n. LEUTWYLER

enlarge our formalism to include reducible representations, and revise the construct ion of the Lagrangian given in Appendix A. Since the analyt ical par t of the procedure does not make use of the irreducibil i ty of the ~ / representation, eq. (A.16) must still be true. Bu t to obtain (A.21) we needed the fact t ha t the matrices y~, generate a mat r ix basis in the representa t ion space. This is no longer t rue for reducible representations. In addit ions to the 16 linear ly independent products of the ~,'s, which we denote by Y(A) ( A = I , ..., 16), i t is convenient t o introduce the four matrices

(4.a) (::) (0il) (: :)

~ o = ' i ; ~ , = ; ~ , = ; r . i l O

The 64 linearly independent products

(4.4) "~r ~(A) ( r = 0 , 1 , 2 , 3 ; A = 1 , . . , 1 6 )

fo rm a ma t r ix basis. With the help of this basis the same kind of arguments used in Appendix A will carry through again. I t is shown in Appendix D tha t the only modification in the expression for the general invar iant is found in the t e rm L ( ~ ) . The three other terms, L(R) , L(qS), Lv, may be taken over direct ly f rom the irreducible case, simply by subst i tut ing ~,, ~,~ for y,, r respectively. The t e rm L(~b~) must be generalized to be of the form

(4.5) L & ) = 2: b:%A, Tr( ,A, A . A "

+ ~ b~# ~~ "JAA' Tr (Y(A) ~,,, 9(A,)~)o,,) " A , / l '

For irreducible representat ions the two terms in (4.5) are not essentially different, and each of them may be wri t ten in the form of the other. Inser t ing instead of ~(A) the quantit ies ~, ~ , ~ , YsY~, 95, the coefficients bm~a, again become invar iant tensors of the t ransformat ion group in l~iemannian space and m a y thus be expanded in a linear combination of the fundamenta l invariants g,~ and e m" and their products, as was arelady the case for irreducible representations.

Turning now to the physical in terpre ta t ion of the formalism, we first prove tha t any choice of the coefficients a, b, c, d, e in the action as generalized to reducible representat ions leads to a system of field equations tha t is invar iant with respect to rotat ions in isospin space. To this end, consider the subgroup of uni tary , unimodular 8 x 8-transformations S commuting with all four matrices ~ ,

(4.6) ~ = ~ ~; ~ t f i~ _ fi~ ; Det S = 1 .

G E N E R A L L Y C O V A R I A ] ~ T D I R A C E Q U A T I O N A N D A S S O C I A T E D B O S O N F I E L D S 1079

To get an explicit expression for these matr ices, we expand S into the basis

(4.4):

r ,A

Since the matr ices ?~ commute with 9,, the condition (4.6) reads

(4.s) Y ~.[9,, Z ~ ' ~ , ] = o . r A

The only l inear combinat ions of y-products commut ing with all four matr ices 9~ are just mult iples of 1. Thus we find

(4.9) ~ s~%A, = s ' i ; ~ = Z s ' , ~ . A r

Equiva len t ly we m a y write g as a direct product

(4.10) (a:) S = u • u = - - b * �9 ; u + u = l ; D e t u = l ,

where 1 denotes the 4 • 4 uni t ma t r ix and u is a uni tary , un imodular 2 • 2- mat r ix . The group of matr ices S satisfying (4.6) is thus isomorphic to SU2 which is the two-va lued representa t ion D�89 of the or thogonal group in three- dimensional euclidean space. This space is to be identified wi th isospin space. I n fact , if we t rans form the fields y~(x) and F,(x) according to

(4.H)

t hen the spinor fields !G, ~P. would suffer the changes

(4.12) t t ~ t ~ t

yJ~ = ay~ q- byJ. ; yJ = - - b y~ if- a y~, ,

which is just a ro ta t ion in isospin space. I f we take S to be independent of x, the field /~,(x) yields a representa t ion D�89174 D�89 = D~ (~ Do. This means t ha t _F, is composed of an isoveetor pa r t and an isoscalar par t , which m a y be seen immedia te ly by developing it wi th respect to the basis {4.4)

3 i l l ~ ^ (4.13) P,= 5rY; A,+ 5{55

A V=I A

I f we use the relations

(4.14) ~ - i ~ ~ = zq~ ; 2 c t ~ k = ~

1080 H . L : E U T W Y L E R

we find

Clearly under the transformation (4.11) the action remains invariant , since (4.11) may be interpreted as a special class of changes of the representation, which has the property of leaving ~, invariant, y~ = ~,. Thus we have prove4 the s ta tement tha t any variational principle start ing with an action built out of the fields g,,, Fff~, ~ , F, , y~, which is invariant under arbi t rary co-ordinate- dependent 8 • 8-transformations S(x), represents in particular an isospin-inva- r iant system. If we want to include ~ctions which are not isospin-invariant, we have to permit the matr ix fields (~7) ~, to occur explicitly in the Lagrangian.

Since our formalism is covariant with respect to space-dependent isospin rotations it must include the isovector field b, introduced by YA~G and MrLLS (2). Indeed, their field is contained in the vectorial part of P,(x):

(4.16) I~t,'~ b,'.

As a simple application of our reducible formalism we want to show tha t the matr ix field /~(x) contains also components which m a y be interpreted as the pseudoscalar mesonfield. In fact, if /~,(x) deviates from the classical field /~0(x) only by the particular term

(4.17) k ~ l

then the isovector field ~k(x) will interact with the spinor of the nucleon through the term (we put e2----0):

^ o e * ^ = - - ~ ~0 k - ^ ^ (4.18) k = l

Furthermore, according to (1.17), we have ~ * = ~ . For ~ we find

(4.19) k k

The expression L(~ 2) for reducible representations as given by eq. (4.5) now

(1~) The expression (4.3) for $i is correct only in this partioflar representation.


takes the form

(4.20) r (52 ) = b~ ~ ~ k ~,~k + b ~ o ~ . k

Although the first te rm is physically more important , we elaborate the second, more academic one too, since the to ta l is then the most general Lagrangian Z(~b~) allowed by our formalism for a field /~,(x) of the type (4.17).

The second te rm may be split up into

(4.21) L (R ~) + L(R, 5 " 5 ) + L{(5"5)~} �9

I n order to obtain the variations 3F, we do not need the most complicated part , L(R~). [Note tha t the variations ~F~ and ~gU" are t rea ted independent ly ; for the evaluat ion of ~gU~, which we do not consider here, q 3 is to be held fixed in (4.5).] For the second and third te rm in (4.21) we find

(4.22) L(R, 5 " 5 ) + L { ( 5 . 5 ) ~} =

= 16 5 " 5 bmJ"~t~:'~(R~,z,~o q- 2 5 " 5 (g~w g~ - - g~,o g~,)}"

Since b . , , "''~'~ is an invar iant tensor, i t may be wri t ten as a combinat ion of products of the fundamenta l invariants g~ and e ~ (~8). Using the sym- m e t r y properties of R~,~ we obtain

(4.23) L(R, 5 " 5 ) + L { ( 5 " 5 ) 2) : b H S " 5 ( R - - 2 4 5 " $ ) .

On the other hand we find for L(~)

(4.24) ~ ( $ ) = - c 5 " 5 .

Let us now evaluate the variations with respect to F~,(x) in the restr icted class (4.17). This amounts to variat ions of the field ~ ( x ) only, which lead to

(4.25) bT VgVgcp -? c 5 + b . (48 r - - R ) 5 = 8q~v~s~p.

For yJ(x) we find the equation

e 3 s . ^ (4.26) pxivUD~,~ q- -~ y~ = 4q~ 5 .'r y~y~.

(~) Compare Appendix A.

1 0 8 2 H. L E U T W Y L E R

Thus the formal ism contains in par t icular a field of the type of a pseudoscalar i sovector-meson with nonvanish ing restmass, the res tmass arising f rom tho Lagrangian t e r m L(~) .

The au thor great ly appreciates the guidance and cont inued interes t of Prof. J . 1~. KLAUDEIr and Prof. A. MERCmR in this work as well as the inva luable assistance given h im b y Prof. J . R. KLAUDEI~ in clarifying the presenta t ion of this paper.

A P P E N D I X A

Construction of the Lagrangian.

We assume the Lagrangian to consist of two par t s :

(A.~) L = Lr(g~, F.o~, ~ r . ~ , ~ , ~ , ~F~) +

+ L~(y~, ~, ~y~, ~(p, g~, F~,L, r~, F~).

This appendix is concerned with the construct ion of the mos t general expressions of type L r and L~, which are invar ian t with respect to a rb i t r a ry t ransfor- ma t ion S(x) as well as with respect to co-ordinate t rans format ions in Rieman- nian space, subject to the l ineari ty requi rements to be fo rmula ted luter~

a) The Lagrangian ]or the a]]ine connections Lr. - As a first step we wan t to show, tha t the fields F~,~ and ~F,% can occur in L r only in the combinat ion

(A.2)

while F, and ~zF, enter only in the form

(A.3)

This analyt ical pa r t of the construct ion m a y be obta ined by int roducing special co-ordinate systems.

F i rs t we per form the t r ans fo rmat ion

(A.4) x ; x / - ~ F r . . . . , = - x~ ) ( x " - x~)

which introduces a co-ordinate sys tem tha t is geodetic a t the point ~:

Q, o

(A.5) C gx) = 0 .

GENERALLY COVARIANT DIRAC EQUATION AND ASSOCIATED BOSON FIELDS 1 0 8 3

This condition is conserved under a t r ans fo rmat ion of th i rd order:

--c~ o _ o _ o

(A.6) x,~. = E, + i t , ~ ( x __ x ) ( ~ - - x~)(x r - x v) .

~o o This t r ans format ion changes ~aF~ ~(x) into

-- Q o ~,, o

(A.7) ~aF, ~(x) = + .

A suitable choice of te~v4 leads to

This equat ion m a y be shown to imply

Therefore, in the special co-ordinate sys tem introduced, Lv in fact depends on ly on the combinat ion (A.2). I n view of the tensorial t r ans format ion p roper t i es of Re~ the same is t rue in any co-ordinate system. Analogously the u n i t a r y t r ans fo rmat ion [in the sense (1.16)]

o ~ o/~

( a a 0 ) S(x) = 1 + IT,(x)(x - - x ) ,

leads to

(A.11) ' ~ C ( x ) = 0 .

This condition is again invar ian t under t ransformat ions of second order

(A.12) S(x) = 1 - - �89 - - x ' ) ( x ~ - x~) ,

which change ~aF/(x) into

(A.13) ~x.P.(a3) = ~xg(x) -? 0a..

A suitable choice of 0xz leads to

(A.14) a i /~ = �89 r

Because of (A.11) L v is thus a funct ion of ~ .~ . Since q).. is a spin-tensor, this mus t again be val id in any sys tem of co-ordina-

tes, such t ha t our assert ion is proved. L v mus t have the form

(A.15) L v = Lv(gl,~, 7~, I~al,~, ~i,~) �9

Iqow we impose the restriction, t ha t Lv be a t mos t l inear in / ~ e and quadra t ic in q 5 and does not contain a t e r m R • ~b. This corresponds to the postula tes used in general re la t iv i ty and electromagnet ism. F u r t h e r m o r e

1{)84 H. LEUTWYLER

we consider only irreducible representations of the m~trices ~ . The genera- lization to reducible matrices is straightforward and an example of a reducible 8 • 8-representation is t reated in Appendix D.

The linearity requirement tells us tha t L r must be of the form

(A.16)

With

(A.17)

L r = 1.(R) + L(r + L ( r .

L ( R ) = AI~'~R%.,~ + A~ ,

L( q~) = C'. ~ ~..ff .

Herein ~' B. ~. ~ denote spinor indices running from 1 to 4 and the coefficients A , B , C are to be constructed out of g,, and y,.

To evalute the invariance requirement under arbi t rary S-transformations, we may develop the matr ix C ~" in the space spanned by the 16 linearly independent y tp rodue t s :

(A.lS) Cop~ a - - p~'afl , Irma C ~ ' = c 1 " 1 + . y , ~ - c 3 ~,Z@ c4 y s y ~ @ c s " Y s .

The same may be done for the coefficient B by developing it in the space spanned by the direct product of two ~-matrices. The coefficients ci and their analogues in the expansion of B are not affected by S-transformations. Since under t ransformations of co-ordinates in Riemannian space they have to t ransform as tensors we may use a well known theorem of the theory of invariants (19) telling us tha t these coefficients have to be linear combinations of the tensors g,~ and e'~q" and their products. I n part icular there are no invariant tensors of odd rank. In this way we find e.g.

Because of the identity

(A.20) R Q ~ @ R~ @ R ~ . = 0 ,

the second term in (A.19) gives no contribution. Wi thou t explicitly giving the development into products of g,~ and s r've" (which gets ra ther lengthy), we obtain

L(R) =- a~ R~.~e(F~r)g "~ + a~ ,

A, ~ AA' Tr (y~A~ Of-) Tr (y~A') ~o), (A.21) L(q)') = x~ ~'~" A,A'

L(O) = el Tr (7"@,~) + c2 Tr (~'~*r

1

(A.~2) r"'*= �89 ~"~ ~'~176 Vlg t " ' ~ ,

(19) H. WEYL: The Classical Groups (Princeton, 1946).

G E N E R A L L Y C O V A R I A N T D I R A C E Q U A T I O N A N D A S S O C I A T E D B O S O N F I E L D S 1 0 8 5

[y(A); A = I , ..., 16 denotes the 16 l inearly independent products of y-matr ices used in the expansion (A.18).]

b) The Lagrangian ]or the spinor ]ield L~. - We consider only Lagrangians bil inear in % ~ and l inear in their der ivat ives of the general fo rm

(A.23) L,p = ~0~ t' O~jp + ~,~0~'~f + v~03yJ.

I f we again use a co-ordinate sys tem such tha t F,Q~(~)~-F,(~)=0, the matr ices 0~ have to be constructed with the help of g~,~(x) and ~,(x) alone. Developing these matr ices with respect to the basis genera ted by the matr ices y~ we obtain wi th the same kind of a rguments as used in a):

The ordinary der ivat ive ~ was replaced by D r in order to obta in an expression val id in any co-ordinate system. F r o m D,A =0 we obta in the real i ty conditions

( A . 2 5 ) e; - - e * ; e~' = - - e*" *" = e* 2 ~ e8 ~ e 3 ~ ea �9

A P P E N D I X B

Symmetry with respect to the groups of Touschek and Pauli-Giirsey.

a) Invariance with respect to the group o] Touschek. - Inser t ing the t ransfor- ma t ion (3.1) into Lv and making use of

(B.1) ys+= AysA-1; D~ exp [~Ys] = sin ~ D~y5,

we find the pa ramete r - inva r i an t s y m m e t r y condition

(B.2) (el+ e~ys)y'Dt, ysys+ D~ysysy (el + ~ y s ) - - 2 ( e 3 + e, ys) ---- 0 .

This is a restr ict ion on the field Fu(x ). The var ia t ions of the Lagrang ian have to be carried out with this restr ict ion in mind. I t is evident t ha t the conditions ment ioned in Section 3,

(B.3) D~y5 = e3 = e4 ~ 0

are sufficient, bu t one migh t believe the restr ict ion (B.3) to be s t ronger t han (B.2), i.e. t ha t we would loose a class of possible Touschek- invar ian t Lagrangians by imposing (B.3) instead of (B.2). Actual ly this is not the case.

69 - 1l N u o v o Cimento .

1086 H. LEUTWYLER

I t m a y be shown tha t in the algebraic l imit [3], in which we are in te res ted here, the Lagrang ian (1.24) together wi th the subsidiary condit ion (B.2) yields the same equations of mot ion for the spinor field ~v(x) as if we pu t e3~e4=0 and consider only var ia t ions ~F, tha t are subject to the subsidiary condit ion (B.3).

b) Invariance with respect to the group o] Pauli-Gi~rsey. - For a = 0 ; b = l the t rans format ion (3.2) is re lated to the charge conjugat ion (~0) by a Touschek t rans format ion with ~ = ~ / 2 . Since we require invar iance with respect to the full group of Touschek, s y m m e t r y wi th respect to the t ransformat ions of Pauli-Gfirsey implies in par t icular invar iance under

(B.4) ~ -~ ~v * = C-ly~ * .

Inser t ing this t ransformat ion in (A.23) we obtain the s y m m e t r y conditions

* D,C 0 (B.5) e * = el; e2= e2; = �9

Again these conditions are direct ly seen to be sufficient. As was the case in pa r t a) i t m a y again be shown t h a t they are necessary.

One does not get any other s y m m e t r y condition for different values of the pa ramete r s a and b in (3.2). The conditions (B.3) and (B.5) guarantee the invar iance of the Lagrangian with respect to the full groups of Touschek and Pauli-Gfirsey.

A P P E N D I X C

The algebraic limit [3] of the field equations for F,(x).

This appendix deMs with the resolution of eq. (2.11) with respect to the field F#(x) in the algebraic l imit (case [3] of Section 2). As a l ready pointed out in Section 2, the algebraic l imit is meaningful only for the fields B//, C~ ~ and D~ bu t not for A~ and E, . Wi th the help of the classical affine connection, defined b y eq. (1.8) and (].23), and the nota t ion

(20) Equation (B.4) represents charge conjugation only if DgC=O. In the general case is it given by

In particular, the transformation property of F~ implies

where 7~ denotes the electromagnetic potential.

GENERALLY COVARIANT DIRAC EQUATION AND ASSOCIATED BOSON FIELDS 1087

eq. (2.11) reads

(c.2) g* - �9 - - 4- 2 [Z~I, , C1~21'r4- C2y/~v*] ~)~(e I 4- e2y5)yv 4- y~'(e~ 4- 2y5)~/)'~/)

I f we insert the expansion (C.1) for A~ and neglect the wave par t , 8L(g~)/SY, then the components y~, 7~ and 75Y~ of eq. (C.2), which we are interested in, m a y be wri t ten

(C.3) 8c,(A~g ~'~- A ~'e) 4- 8c~e~'&-" AQ,, =

= {p~(1) 4- p2(ys)}g ~ - iql (7~) 4- iq2 ( y ~ * ) ,

= iql {g~yfye} _ ger(V~)} 4- p= {g~Y<y5 ye}--ger(ysy~)} 4- iq~ e~eyQ(y~)_p, e=O~e(y= r e} ,

In the above, ,~Mw has been abreviated by (M}. These three equations are to be solved for A~ , A ~ r and A ~ . Wi th the

help of the symmet ry properties of these equations it is possible to show th a t the solution is unique for any choice of the coefficients cl and c~. I t is given by

1 S 1 (C.6) GaS = ~ i go~fl -~- c~ -Jr- 402~ {~ 4- 2e2Tla~*} ,

(0.7) A ~'ev -- 1 2 (C 2 4- C 2) {(01VlY4- 02 w29)gcr (01V1/~4- c2 V2~)gO~y4- ECc/~Y0(Cl V2Q-- 02 wlq )} '

1 1 (C.8) A~fl= ~-o, S2go~fl . oY4- 4c~ {Cl T'a~ 202T2~} '

where we have employed the notat ion

(c.7)

JS11

$2

V1/*

V2/~

T1 ~v

Tu/~v

Pl P2 <~) + -~ <70,

iq2 iql 7- <1)--g- <~,5> ,

iql p~

iq~ Pl

iq, iq= ~- J r " ' ) - - 7 - <~''*) '

P2 Pl ~- <F ' ) + -g <y~'*) -

1088 H. LEUTWYLER

APPENDIX D

The Lagrangian for a reducible representation.

The same procedure as in Appendix A carries through for the analytical part of the construction. Thus eq. (A.17) is still valid with the difference that the spinor indices now run from 1 to 8. However, instead of the expansion (A.18) which is valid only for irreducible representations we have to use the basis (4.4):

A , V

With the same kind of arguments leading from (A.18) to L(~) in (A.21) we find

(D.2) L ( ~ ) : ~ c~ Tr (~ ~ ' ~ ) -~ ~ c~ Tr (~ ~'"*~,~). r

To find L(~ ") it is convenient to expand B"e"B* S in the space of the direct product (~,~(A))~a(~r,~(a,))B ~ [instead of taking the pairs (B, A); (C, D), as it was done in Appendix A].

In this way we obtain

A , A '

Finally we find

: - r ~ <-- * r

r r

+ ~ ~(e~" + e ; ~ ) ~ . r

~ow we have to investigate the invariance of these expressions with respect to arbitrary 8 • S-transformations. Since under these transformations the matrices ~ will not remain constant, but become x-dependent fields, which are not allowed to occur in the Lagrangian beside of g~,~, F,%, ~ , ft,, ~f, we must restrict the expressions (D.2), (D.3), (D.4) and find the most general invariants of these types which do not contain the fields $~(x) explicitly. Obviously the quantities

(D.5) L(52) = Z blW'OaAA' Tr (Y(A)~) Tr (~(A,)q~) + A.A '

~ - , f ) I I A , 4 ' v

(D.6) L(~) = c~ Tr (~"~,~) + c~ Tr (~'~*~,~),

G E N E R A L L Y " C O V A R I A N T D I E A C E Q U A T I O N A N D A S S O C I A T E D B O S O N F I E L D S 1089

are of this kind. We want to show now that these are in fact the only ones. I t suffices to consider invariance with respect to the special S-transformations introduced in Section 4 (4.6). Under these transformations the matrices

r r r

have to be numerically invariant [L depending not explicitly on ~] .

(D.8) S-~BInS = 2 ~ .

Since this must be true for any ~ given by (4.9), in part icular for S = ~ , we obtain, in view of the irreduciblity of the 2 x2 representation of the Panli algebra

which proves eq. (D.6) aud (D.7). To get (D.5) we first observe tha t the same argument goes through for the parts ( r = 0 ; r ' = l , 2, 3) and ( r = l , 2, 3; r ' = 0 ) in (D.3). The term ( r = 0 ; r ' = 0 ) i s obviously invariant and contained in (D.5) such tha t we need to consider only ( r = l , 2 , 3 ; r ' = l , 2 ,3) . With the help of the relations (4.14) we obtain the invariance requirement

(D.IO) bm'eaAA,i~ ~iz ~ t ~ ~ - b 'VeaAA,tm .

This means that the tensors b ~k have to be invariants of the three-dimensional orthogonal group. By the theorem mentioned in Appendix A the only invar iant quant i ty of this kind is the metric tensor 6 ~, snell tha t we must have

(D.11) bl~Vea4A'ik = b~'"o~ 5 ik �9

Insert ing this in (D.3) and expanding the tensor ~,~ with respect to the basis (4.4), the resulting" expressions may be shown to be of the form (D.5).

R I A S S U N T 0 (*)

Si respinge l'equazione del formalismo classico dell'analisi degli spinori covarianti in cui il campo delle connessioni affini nello spazio spinoriale, F~(x), 6 fissato d~ D~y~ = 0. Con una pifi stretta analogia all'analisi tensoriale nello spazio metrieo si trovano solo eondizioni di realitg imposte a F~,(x), mentre rimane una considerevole liberth dinamica indipendente. Questo suggerisce un aceostamento simile alla procedura di YAN(~ e ~r Partendo da un principio variazionale si ottiene un'equazione d'onda. non lineare con massa di quiete che non si annulla per il campo affine F~(x), la cui struttura ~ imposta dalle esigenze di invarianza. Come esempi particolari, si discutono le equazioni di Maxwell e l'interazione pseudosealare mesone-nucleone, oltre all'equazione spinoriale non lineare di Heisenberg.

(*) Traduzione a cura della Redazione.

Documents

Generally covariant dirac equation and associated boson fields