On the fundamental spinor fields representing massless fermions in d dimensions

Volume 197, number 3

PHYSICS LETTERS B

ON THE FUNDAMENTAL SPINOR FIELDS REPRESENTING MASSLESS FERMIONS IN d DIMENSIONS M. D A N I E L l CERN, CH-1211 Geneva 23; Switzerland

Received 11 August 1987

We establish a relation between chirality and "helicity" in higher dimensions. This is a generalization of the well-known result that in four dimensions a spinor field of positive chirality can only be associated with a massless fermion of helicity - ½.We give also the transformation properties of the fundamental spinor fields, representing massless fermions, in d dimensions under the discrete symmetries of charge conjugation, parity, and time reversal. It is shown that self-charge-conjugatefundamental spinor fields can only exist in d= 1, 2 (mod 8) dimensions.

It is well known that in four dimensions a spinor field of positive chirality can be associated with a massless fermion o f helicity - ½ but not with one o f helicity ½. This is indeed the case with a massless neutrino, where a positive chirality Weyl spinor is used to describe such a particle. This is an example of a more general result pointed out by Weinberg [ 1 ] some time ago. Given that the general irreducible representation o f the Lorentz group in four dimensions is labelled by (A, B), where 2A and 2B are integers, Weinberg has shown that a massless particle annihilation operator, a(/~, 2), of helicity 2 can only be used to construct fields which transform under the Lorentz group according to a representation (A, B) such that B-A=2.

(1)

The limitation imposed by (1) is solely due to the non-semi-simple nature o f the little group for massless particles. It is clear now that massless fermions with helicity 2 = ½, - ½ can be associated with spinor fields transforming as (0, ½) and ( l , 0) respectively. These are Weyl spinors of chirality -1 and 1. Consequently, there is a correlation between helicity and chirality in four dimensions. In this note we ask what (if any) is the correlation Permanent address: The King's school, Canterbury, CT2 7NF, UK.

between "helicity" and chirality in d dimensions. First, however, we need to clarify what we mean by "helicity" in higher dimensions. In four dimensions, the helicity 2 is a real number, which labels the onedimensional physically permissible irreducible representations of the little group [2,1 ]. For global reasons the value of 2 is restricted to be a positive or negative integer or half odd integer. In higher dimensions the "helicity" index 2 describes the transformation properties of the massless particle states under the little group - for the massive case see Weinberg [ 3]. In d dimensions the physically relevant little group is S O ( d - 2 ) . When d is even the spinor representations o f S O ( d - 2) and those o f the Lorentz group SO(I, d - 1 ) are doubled. I f a massless particle rotates with one o f the spinor representations o f S O ( d - 2 ) , it is necessary to ask which spinor representation o f SO (1, d - 1 ) may be associated with. We show that the non-semi-simple nature of the little group restricts this choice in a manner analogous to the four-dimensional case. Indeed we find that there is always a definite correlation between "helicity" [ i.e. a spinor representation o f SO ( d - 2 ) ] and chirality in d dimensions. We may remark here that, if one is only interested in counting degrees o f freedom, one does not need to know which spinor representation of S O ( d - 2 ) is associated with a given spinor representation o f the Lorentz group. On the other hand, if one is inter-

0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

363

Volume 197, number 3

PHYSICS LETTERSB

ested in finding how the spinor representation of the little group induces a spinor representation of the Lorentz group (as in the case of the work of Wigner [2]), then one must be careful to take into account the constraints imposed by the non-semi-simple nature of the little group as the analysis of Weinberg [ 1 ] shows in four dimensions. This is also necessary, if one wants to study how massless spinor fields transform under the discrete symmetries of charge conjugation (ga), parity ( # ) , and time reversal (Y). In the past the analysis of ~, ~ , and ~ t r a n s f o r mations in d dimensions has been carried out by making use ofbispinors and the Dirac equation [4,5 ]. For a discussion of the (¢ operation see also ref. [ 6]. In the case of massless fermions the fundamental spinor fields are not bispinors, and indeed in a parity-violating theory bispinors are not always available. In such a case, one has to make use of fundamental spinor fields and the correct Weyl equations, which should embody the correct correlation between chirality and "helicity". To carry out the analysis we need a bit of a formalism about the spinor representations of (pseudo)orthogonal groups. In an even-dimensional space, d = 2v, let N and N be the fundamental spinor representations of SO(l, 2 v - 1 ) and let gN and ~ur~ be the spinors transforming according to these representations. We denote also by r and ~ the fundamental spinor representations of S O ( 2 v - 2 ) . Let yJ") j = 1, ..., 2v be the 2")< 2" matrices generating the Clifford algebra C ( R (~,2"- ~)). The set of matrices Zj,= -~[y)"), 7 / ) ] form a reducible representation of the algebra of SO(l, 2 v - 1). In the reduced basis [ 7 ], where y ) " ) = y ) " - ' ) ® a 2 j = l , ..., 2 v - 1 and 7(2~)=1("-~)®O'1 with y),-l) being appropriate 2"- ~®2"- ~ matrices, the generators Z~, are diagonalized. I I Zij =

-

with, ref. [ 8],

z ~ ={z~, zi~.} = l . f~ r1j~,(v-- 1) ~,(.-~'k

364

__ ,(.-- t 2f l r~ j, ( v - - l ) ,rk

~ , ( " - - ')}, 1), _ i l2/s

(2)

1. { . - - 1 ) }, 1), i ~rj

(3)

j, k = l , 2 ..... 2 v - 1 . In this basis Y~,")+I=Yt")...Y~ ) ocl("-~)®a3 and the spinors ~,n and gr~ have chiralities XN--1 and Zn = - 1. The generators for the r and ~ representations o f S O ( 2 v - 2 ) = SO(l, 2 v - 1 ) are given by formulae analogous to (2) and (3) namely zr__fl~,(.-2).,(.-2)

Z r

!~,(.-2) , 2/'j

- - I. 2 / ' j

gk

=

~,~.-2),

y!~,(v--2)

~J

(4)

},

-- I ~,(v--2)

1 s,

2rj

(5)

with y),-2) j = 1.... , 2 v - 3 appropriate 2~-2X2 "-2 matrices [ 8 ]. We are now ready to discuss the little group and its implications. For massless particles the algebra of the little group in d dimensions is identified with that of all translations and rotations in ( d - 2 ) dimensions. The generators corresponding to translations are given by ref. [9] La=Za,2v_ 1 2 ¢ - Z a , 2 .

,

a=l,...,2v-2,

(6)

and it is easy to verify that they form an invariant abelian subalgebra

[La, Lb] =0.

(7)

In the N and Iq representations, of course, we have

taN =~Za 1~'("--1)7~ iZ,l

1~,(.--1

L~ = 2 r a

l)

1) , - i ~1l~(,-a l~(v--l)

)Y~-I ) + i 2Za

•

(8)

(9)

On setting these generators to zero the little group algebra reduces to the algebra of S O ( 2 v - 2 ) . This procedure may appear rather ad hoc, but unless this is done, the massless particle states (the "helicity" states which form representations of the little group) will not form a finite set. From the above discussion we have S O ( 2 v - 2 ) c little group c SO(l, 2 v - 1) and consequently the fundamental spinor representations N and l~l of SO(I, 2 v - 1 ) decompose as N=r~L

+ s I

29 October 1987

lq= r ~ L

(10)

For massless particles there is no rest frame so instead a "standard" light-like d-momentum KA= (k, 0 ..... k) is introduced and the states, u(k, 2), are labelled by k and the running "helicity" index 2. These states provide the numerical coefficients u~ (k, 2), which satisfy

PHYSICS

Volume 197, number 3 [exp(~"'Jk"(~-~)'~-~)~] ,,Nxtt2) ~t.~ Ij Ik lJnm~m k '~' = u.N'~(k, p) [exp(*2~ .... b.(.-:)7~.-2)+~

+

½o~o.:.-:/.- ~))]~,

(11 )

[,~,,.,.{'-.,,Jk~,(~-')~,t~-~))].mUNm,+(k, 2) "~1 J k 2 '~ i'j = u N'~(k, #) t,-^mr°~+~.1 . . . . b., 2 (v) 2) r ~b . (av 1 ..... - - ~ t.,.,

(12)

2v-- 2 . , (/.'-- 2) ~tI l a

l J p2 "

The coefficients u ~'~ and u ~'r satisfy relations identical to (11 ) and (12). As we have stated already in order that the states u(k, 2) form a finite set, it is necessary to represent the translations of the little group by zero. In the N representation we must have N N,r (La),mUm (k,2)

=0,

N N,I (La),mUm (k, 2) =0,

a = 1..... 2 u - 2 .

(13, 14)

A similar set of equations holds true with N replaced by lq. Using the matrix representations for L N and L ~ given by (8) and (9) respectively we immediately deduce that (1 +iy~_-I >) uN'r(k, 2) =0,

(15)

(1 +iyi~,- 1)) uN'r(k, 2) =0,

(16)

(1 - i y ~ S ~ >) u~'~(k, 2) =0,

(17)

u~'r(k, 2 ) = 0 .

(18)

(1 - i y ~ z I ))

We shall now obtain another set of equations for the numeric coefficients u(k, 2). To do this we consider (11 ) and (12) and choose a special rotation made out of infinitesimal rotations in the planes (1, 2 ), ( 3, 4), ..., ( 2/, - 3, 2u - 2 ). Comparing now equal powers of infinitesimal parameters on both sides of (11 ) and (12) we obtain, after some algebra, the following set of equations: (1 +iy~Z P) uN'r(k, 2) =0,

(19) (20)

(1 -iy~.": I ))

ur
(1 - i 7 ~ - - P)

u~'r( k, 2) =0.

(22)

(1 + i y ~ - I ))

(21)

Solving eqs. (16)-(22) we obtain

uN'~(k, 2) =0, uN'r(k,2) = 0

(23)

LETTERS

29 October 1987

B

with uN'r(k, 2) and uSS(k, 2) in general different from zero. The same analysis goes through with numeric coefficients vN'r(k, 2), vr<~(k,2), vN'~(k, 2) and v~S(k, 2) which satisfy equations similar to (11 ) and (12) but with the right-hand side transforming according to the complex conjugate representations r* and t*, respectively. The result is that all the numeric coefficients of the type vN,~ and v~'r vanish, and only vNx and v~s are in general different from zero. The vanishing of all the numeric coefficients U nN,r, U nr~,r, ~.r, ~,r in d dimensions is the analogue of (1), the constraint satisfied by the helicity 2, pointed out by Weinberg in four dimensions. We can now follow Weinberg [ 1 ] and boost the non-vanishing numerical coefficients uN'~(k, 2), ur<~(k, 2), vN'r(k, 2) and v~'r(k, 2) of the "standard" reference frame to a frame defined by an arbitrary light-like momentum pA = { IPl, P}. First we consider the boost B( IPl ) along the ( d - 1 )-direction given by B A B(

Ipl )

coshO

I

[sinhO

0 ... sinhO 1 1 ... 0 0 ... coshO

where O = ln llPl/kl, which takes kA= {k, 0 ..... k} into { IP I, 0 .... , Ipl }- Let R (p) be the rotation which takes the ( d - 1 )-axis into the direction of the unit vector =P/IP I. Then the boosted coefficients are given by UnN,r (P, 2) =

(DN[R(p)]DN[B( IP[ )]),,mUNm'r(k, 2). (24)

The numeric coefficients u~S(p,2), u N ' r ( f f , ~,) and v. ~ , 2) are constructed in a similar manner. The general relativistic fields in d dimensions, {aM}, transform under various irreducible representations, M, of the Lorentz group SO(l, d - 1 ). It is necessary, however, to label the fields with both M and a label s which specifies the transformation properties under the "little" group SO ( d - 2); there is in general no one-to-one connection between M and s [ 3 ]. In the case of spinors ~/N and g~ our analysis shows that the only fields that can be constructed are 365

Volume 197, number 3

PHYSICS LETTERS B

29 October 1987

Table 1 d=2v=4k, v even.

~//N'r ( X )

= j d/t (p) ~ [exp( - ip.x) uN'r(p, 2) a(p, 2, r) + e x p ( i p . x ) vN'r(p, 2) a*(p, ,~, ~)],

(25)

Operation

Transformation

(~

( ~ / N . r ( x ) 66; I =

__2~ylzZN.rflN,re~V-l)-I

(~//lq,~(X))*

~'Nx(t, X) ~-~=~Ur~'~(t,--X)

and

-- N.rt~N.r~'N . . . . . .

I I/"N'r(--t, x )

~,,, (x) = J d/z(p) ~ t e x p ( - i p . x )

u~"(p, 2) a(p, 2, ?)

+ e x p ( i p . x ) vN'r(R, 2) a*(p, 2, ~)],

(26)

and no other ones. Thus, the annihilation operator a(p, 2, r) is associated with the spinor field V Nx of chirality ZN = + 1, and a(p, 2, r) is associated with the spinor field ~tr~,r of chirality Zr~ = - 1. For example, it is inconsistent to use a(p, 2, r) to construct a spinor field g/rq.~which has negative chirality. This is the generalization of Weinberg's result in higher dimensions. The previous analysis was confined to even dimensions. In odd dimensions, d = 2 v + 1, there is a unique spinor representation, M, of S O ( l , 2v), which decomposes under the "little" group SO(l, 2 t , - 1 ) as M = r ~ r , where r is the unique spinor rep= resentation of S O ( v - 1 ) . In this case we can write uniquely ~jM'r ( X ) =~

dt/£(p) ~

[exp(--ip.x)

~/M.r(,, 2)a(p,2,

+ e x p ( i p . x ) v~X(p, 2) a*(p, ;t, r)],

r) (27)

To complete this note we give also the transformations of the spinor fields vN'~(X), g/~'r(X) and g/M'r(x) under if, ~ , and ~. This is of interest in even-dimensional spaces, because one can see directly whether the chirality is maintained or not after ~g, .~, and ~, Moreover, once the transformation under ~g is known, it is easy to find out what values of d can accomodate massless self-charge-conjugate fundamental spinor fields. We summarize our results in tables 1, 2, and 3 ~t. In table 1 ZN,r, aN, r and flN,r a r e generally complex phases, although it is possible to take 0tfN,r~---ZN0gr with a~ real, and "~These g~, ~, and ~-transformations were also derived independently by S. Chadha. 366

2~ = ( -- 1 )us+s(~- 1)/2+~<,+1)/2 The transformation properties for ~'~'r(x) are obtained from table 1 by substitution N--,/~I, r ~ and taking into account the relations Z~,~=ZN,r, /~,r=--flN,~ and a ~ s = - - o ~ * ~ . In table 2 the properties o f ~ ' r ( x ) are obtained again by substitution and taking into account the relation /~r~.~= flN,r(aN* ~) 2. With the choice aN.~=Zr~a~ and a~ real the latter gives fl~,r =/]N,~In tables 1 and 2 P~_- [) is a 2 ~- l × 2 . - l matrix [ 8 ] satisfying y ~ , - t > . = ( _ 1 ):P[~z~)yjP[~:I >-' , j = 1, ..., 2 v - 1.

(28)

Moreover, P ~2v-' - ~1) P ~2P-' - ~1) * -- P- a2.v-~ - I ) *IP ~ - I ) ---2 " --~- 2V-- 1 • It is clear also that P ~ , - I ) is the matrix that brings about the equivalence N*~/q,

for v even,

N* ~ N,

for v odd.

(29)

It is of interest to consider the case d = 4 k + 2 and define a new field @~.~(x) := ~#~/N'~(X) ~g- ~. In this case one can show that ~ga(p, 2, r) ~g-~=a(p, 2, r) and Ta* (p, 2, ~') ~ - ~ --2 ~-12~-2,,~*+'w-,2, r). Hence, the field VSS(x) can be constructed in terms of creation and annihilation operators as follows:

Table 2 d= 2v= 2k+ 2, v odd. Operation

Transformation ~,N'r(x) ~ '=,~;_'2ZN,r~N,rP~=p -I (~'N'r(x))*

~uN'r(t,x) ~-1 =~U~S(t,--X) ~'~

~'-~N.r(/,X ) ~---1 ~---Z*N,rt~N,rOOPN,r ~ t~* pO,-l),,,gr.r[ 2~--1 W ~,--t,f ¢.~ av2

Volume 197, number 3

PHYSICS LETTERS B

Table 3 d=2v+ 1. Operation Transformation (~

C~//M,r(x ) (~ - I = 2 ~ _ I I ZM,r~M,r P ~ ) - i ( ~//M,r ( X ) ) *

.~

~ l g M ' r ( t , X~_, X2v ) ~ - I = ~ )

ff

~

~/

M,r

~'--I

(I,X) ~

v

~ e ) i/t M,r ( t, - - X ± , X2v ) *

v)

--~2v~M,r]~M,rOlM,rP~v ~/

M,r

(--t,x)

~,~.'~(x) u~'~(p, 2) a(p,2, ~)

=f d#(p) ~ [exp(-ip.x) +2~-t2~-2 e x p ( i p . x )

v~,f(p,

2)

a*(p, 2, r ) ] .

Now 2v-2 [~,=2k+, = ( - - 1 ) k =1, =-1,

k=2q,

d=8q+2,

k=2q+l,

d=8q+6.

W i t h 2 ~-2.-2'= 1, ~ s ' ~ ( x ) can be identified with ~uN,r a n d it is, therefore, clear that self-charge-conjugate massless fermions can exist in d = 2 ( m o d 8). We now turn o u r attention to the o d d - d i m e n s i o n a l case. It is well known that, i f parity in o d d d i m e n sions is defined in the usual way o f the reflection o f all the spatial coordinates, it is simply equivalent to a p r o p e r Lorentz transformation. So one can arbitrarily define parity as the reflection o f all spatial coordinates x~ except x:~ [4]. One could equally define parity as the i m p r o p e r t r a n s f o r m a t i o n o b t a i n e d by reflecting any odd n u m b e r o f spatial coordinates, but all these reflections should be equivalent to the above choice by p r o p e r Lorentz transformations. In table 3 P~g) is a 2 ~ × 2 ~ matrix satisfying ~,j(.u) . =

29 October 1987

Following the procedure outlined for the evend i m e n s i o n a l case one can show in the o d d - d i m e n sional case that is possible to have self-charge-conjugate massless spinor fields in d = 1 ( m o d 8). Finally, we would like to r e m a r k that, i f we combine ¢'N'f(X) a n d ¢/~'~(x) to form a b i s p i n o r field

~(x)-\¢,N,~(x) ) , then it is easy to check that the results o f tables 1 a n d 2 are in agreement with ref. [4]. The results for selfcharge-conjugate spinors also agree with those o f Wetterich a n d van N i e u w e n h u i z e n [6]. I a m grateful to the Theoretical Physics D i v i s i o n o f C E R N for its hospitality during the course o f this work a n d to J.S. Bell for reading critically the m a n uscript. I would like to thank also S. Chadha for m a n y helpful discussions on <¢, 2 , a n d ~.

References [ 1] S. Weinberg, Phys. Rev. 134 (1964) 882. [2] E.P. Wigner, Ann. Math. 40 (1939) 149. [3] S. Weinberg, Phys. Lett. B 143 (1984) 97. [4] M. Belen Gavela and R.I. Nepomechie, Class. Quant. Grav. 1 (1984) 21. [ 5 ] C. Wenerich, Nucl. Phys. B 234 (1984) 413. [6] F. Gliozzi, J. Scherk and D. Olive, Nucl. Phys. B 122 (1977) 253; P. van Nieuwenhuizen, in: Supergravity 1981, eds. S. Ferrara and J.G. Taylor (Cambridge U.P., Cambridge, 1982); C. Wetterich, Nucl. Phys. B 211 (1983) 177; P. van Nieuwenhuizen, in: Groups and topology I! (Les Houches, 1983) eds. B. De Witt and R. Stora (North-Holland, Amsterdam, 1984). [ 7 ] M. Gonrdin, Basics of Lie groups (Editions Fronti6res, Gifsur-Yvette, France, 1982). [8] S. Chadha and M. Daniel, Rutherford-Appleton preprint RAL-85-040 (1985). [9] N. Ohta, Phys. Rev. D 31 (1985) 442.

( _ 1 V'P~Y) ~(v)p~v) - 1 ," = A p [ j = Z P •

367