- Email: [email protected]
Quantization of two-component higher order spinor equations
ANNALS
OF PHYSICS:
20, 184-202
Quantization
of Two-Component Spinor Equations* A.
Ph,ysics
(1062)
0.
Department,
BARUT~
Syracuse
G. H.
ASD
University,
Higher
Order
&ICLLEN$
Syracuse,
New
York
A two-component fermion field is quantized using a Lagrangian formalism containing higher order derivatives of the field variables and an indefinite metric in Hilbert space. A consistent quantization necessitates beside a massive spin one-half field, obeying the Feyrlniarl-(:ell-Mann equation, the introduction of a massless spin one-half field, obeying the nentrino equation. Gauge invariant electromagnetic and vector interactions are introduced in a manner which depends on the transformation properties of the field variables. The physical rest,rictions placed on the S-matrix by the use of an indefinite metric eliminates the massless field from the “composit,e” field for electrodynamics and pseudo-vector mesons intera&ions and implies parity invariance but not for four-field weak interactions. The results obtained for quantum elect’rodynamics are equivalent, to the usual fourxomponent t,heory. I. INTRODUCTIOS
It has been generally recognized that the description of spin one-half particles hy four-component spinors is redundant and t’mo-component, spinors should be su&ientS. The use of t’wo-component, spinors offer many advantages, bot,h of fundament,al and practical nature. Peynman and Gell-Mann (1) were the first to give a two-component second order spinor equation for a spin one-half particle interacting with an ext’ernal electromagnet.ic field and established t’he vect,or and axial vector charact’er of weak interactions. Brown (2) has reduced this equat#ion to two coupled linear two-component spinor equations and showed its equivalence to the Dirac eyuatjion. The coupled linear equations have heen subsequently quantized (3, 4). The question arose early whet,her one could quantize the Feynman-Gell-Mann equation (or similar higher order spinor equations) directly from a Lagrangian formalism without, linearizing it,. It, is clear that. the simplest Lagrangian which * Supported in part by the U. S. Air Force Office of Scientific Research. i Now at Lawrence Radiation Laboratory, University of California, California. $ Now on a Resident Research Associateship at the U. 8. Naval Ordnance White Oak, Silver Spring, Md. 184
Berkeley, Laboratory,
HIGHER
ORDER
SPINOR
EQTTATIONS
185
gives a second order equation is of third order. And higher order Lagrangian theories necessit’ate the use of an indefinit#e metric in Hilbcrt space (5). Iiibblc and Polkinghorne (6) have replaced in the Lagrangian the two-component spinors by four-component ones and linearized the resultant, equations. They then use a subsidiary condition to climinak the superfluous components. It, has been also suggested that t(hese superfluous components of liihhle and Polkinghornc might be used to int,roduw an isoduhlet, of fermions ( 7 ) . The purpose of this paper is tJo present, a complet’e Lagrangian cluantization of the higher order two-component spinor equation in it,s own right. We follow t’o the end the consequences of higher order Lagrangians and bhc use of an indefinite metric and formulate a consistent, second ctuant,ized theory of t,hese equat~ions. The indefinit,e metric in quant#ized field bheories has been discussed again and again since it was first proposed by Dirac in I I143 (8, 9) to overcome various difficulties of quantum field theory (10) and recently again for the purpose of consistency of t,he field theories (11). Whereas, in concret#cexamples, some of t,hese discussions deal wit,h simple models using indefinite metric, we give a complete physical case of such a t,heory. It is known, for example, in the case of elect,rodynamics, that a local theory wit’h an indefinit,e metric is equivalent to a nonlocad t,hcory with a positive definite metric. If this fact is genera~lly t,rue, then the consistent’ quant,izatJion of the higher order two-component spinor cquat,ions presented here may be an indiration of t’heir nonlocal rharacter. The main starting point of the present work is t,he introduction of an auxiliary field which is a sum of t,he Fcynman-Gell-Mann field 4 and a masslesst,tvo component, field 7. The necessity of doing so and the general propert,ies of the Lagrangians and the equations are discussedin the nest section. In S&ion III we quantize the fire fields, and in t,he following sect~iont’he interact)ing fields. The resultjs imply that,, in our formalism, all massive spin one-half particles must be accompanied by a masslessspin one-half parkle obeying t#heneutrino equation. The physical interpretation of the A-matrix in the presence of an indefinite mrt,rie eliminates the masslessfield in t,he vase of strong and clectromagnet,ic interact,ions, and, furthermore, implies parity int-ariance, whereas the original t,heory is only C:P-invariance. The application of t,hr t.heory to the weak intcractions is also briefly discussed. We use natural tmit,s wit,h fi = c = 1 and a metric2 ( +, -? -, - ). IT. LAGRANGIAPT
We start from the Feynmatr-Gell-Mann
6’0RMAI,ISRI
equation
1
+(A-) = 0
(1)
186 with
F,, = A,,, - A,,,; The simplest free particle tive of the field variable
a’” = -0”’ =
Lsgrangian
for Eq. ( 1) is of third order in the deriva-
c = c#B+id”Z,(0’) + my$,
(2)
where S” = (I, u,)
Z” = (1, -u;)
(3)
with
Equation (2) is invariant under the nonunitary two-dimensional irreducible representation of the proper Lorentz group. It can he shown that no metric of the form (4, I’4) exists for this t,wo-dimensional representation (121 (i.e., there is no scalar of the form 4’4). However, a Lorentz vector of the form 4’Y4 does exist as can be seen by not’ing that under infinitesimal Lorentx transformations the t’wo-component spinor 4 transforms as &n:‘)
= S(L)4(.Z)
= (1 + ; ,,“C)
4(s).
(5)
To quantize Eq. (1) by conventional techniques, we have t,o linearize the relation (2). In this linearization we cannot obtain t,he proper commutation relations without t#he use of an indefinite metric. To see this, we define iKZ,g5
= m.$,
(6)
whose inverse is ia”Z,( = m4 + CC,
(7)
where GYz,< = 0,
( iI2 + m’)< = 0,
(8)
and c is a new constant paramet,er entering into t,he theory. In Eq. (7) we made use of the relation (ia”z,j(--ia”Z,)
= (-ia’Z,j(ia’z,)
= 0’.
(9)
HIGHER
ORDER
SPINOR
187
EQITATIONH
E:quat,ion (2) for the free part,icle case t,hen t,akes the form 6: =
++itP&$
+
(+i#E
p( -
m.$+@ -
mc$i,t (10)
+
c(m<+(
+
ml+<
-
<+id”S,~
-
~+itE,~).
A calculat’ion leads to nonzero anticommut8at8ion relat,ions, which depend on the parameter 6, between the massless field { and the massive field c+. This can hc seen easily when wc writ’c Eq. ( 10 ) in the Kcmmer form (11) where
and evaluate [P(z),
*(CC’)]+
=
i6”(.r
-
(,13 )
x’),
with n” = -it&/+/j”. For spacelike surfaces
(13’)
(p = 0), Eq. (13) is equal to
[c$(.r),(b+(k)]+ = sRcx- x’),
[((x), t+(.r’))+ = 2(x - x’),
[4(x),
[‘(.r’)],
= -e-‘63cX
- x’).
(W)
The quantizat,ion of Eq. i 10 ) hy Brown (.2), Tonin (3)) and Theis (4) amounts to assuming c = 0. This is equivalent, to t#he Dirac equat)ion written in terms of two-component spinors as seen explicit,ly hy sett,ing 6 = 0 in 1%~. ( 11). This scheme eliminates t’hc massless l-field and hence eliminates the use of an indefinitr metric. Their results for quantum electrodynamics are equivalent, to thr four-component, Dirac theory. We now consider t,he quantization of Eq. (2 ) as it stands which can be carried out, using a higher order Lagrangian t,hcory.’ Thcrc are several advantages for considering two-component spinors instead of the usual four-component, spinors. The theory based on two-component spinors is simpler, more re&ictive, and srparat’cs csharge and spin degrees of freedom. From the relativist,ic relation bet,ween mass and momentum, we would expert all fields to obey the KleinGordon equation ( 0’ 1 A qumtizntion
of the coupled
+
equations
m’)&(x) (6) and
=
0
(7) is also possible.
(14)
(m 2 0 and u = spin and ( or ) tensor index ) X spin zero particle would be described by a one-romponentj field while a spin one-half particlc by a twocomponent field due to the twofold spin degeneracy. For the spin one-half cast t#he wave function for the free part#icle would he 1’ esp (ip.x - iEt) where v is a two-component’ spinor. The sign of the energy in the exponent t’ells us whet,her we are describing an electron or a position. It is, indeed, this reasoning that led Feynman-Gell-Mann t,o t’heir unique form ( 1,’ - Ai ) for weak interactions. E’or further reference we note the relat,ion between t’hc Dirac spinor appearing in their (I: - -4) theory and the t,wo-component spinor appearing in Eq. ( I ‘)
Equat,ion ( 1) ran also he obt’ainrd from t’he second order Dirac equation ( 1.9)) a procedure which is valid even when m = 0, whereas the Feynman-Gell-Mann procedure breaks down for nl = 0. Another method, st’arting from Eq. ( 14) and stressing the transformation properties, is given in Xppendix I. The procedure by which we quantize the higher order Lagrangian and cstahlish contact with Eq. ( 1 ) is the following. A variat#ion of Eq. (2) yields the equation of motion ( 0” + m’jia”8,~(xj
= 0.
(16)
This result is more genera1 than we want. To avoid the use of a subsidiary dition, we define a new field # related t.o 4 by ia”Z,#
=
S(L)
=
(
(17)
+.
Since 4 transforms under Lorent’z t8ransformat,ions transforms as J/‘( x’) = S( L)+(s) with
con-
by S(h)
given by (5) , J/
1 + ; EpvrJfiVi- = s’t-1. )
In turn we may define a 3”‘( S = )~L,,$“‘) related to T’” in the same way as S is to S. We now choose a Hrrmit8ian Lagrangian for t’he +-field d: = ; [(o”~+z”a,~ which
- a,~+Z”o”~)
+ m2(#+8”a,lj
- a&+.2”+)],
(19)
leads to the desired result (O*+
m’)(id’Z,+)
= (02 + m”j+ = 0.
(‘LO)
The inverse of (23) is given by (21)
HIGHER
ORDER
SPIA-OR
189
EQUATIONS
where (22)
idPZp?f = 0
and (l/O’) is symbolic in that when it acts on the plane wave decomposit,ion, it brings down a factor ( -1/X2). It, should he noted that the fields defined by Eqs. (17), (21 j, and (22) do not’ obey t#he same equations of mot,ion as the fields defined by Eels. (G)-(8). The fields of Eqs. ((j-18) are so defined that the linearized form (Ekl. (‘ICI j of Eq. (2) is the Dirac Lagrangian written in terms of two-component spinors piur a term depending on the mnssless field it). The fields of Eqs. (17), (21), and (22) are defined so that’ l3q. (2) can he replaced by Ekl. (19). In t,his manner Eki. (19) leads t’o the correct free prat,icle equation of mot,ion (i.e., Eq. (20)) for the Feynman-Gell-Mann field whereas 15~1.(2) does not. Just as in t’he linearized form of (a), n-e must also introduce a massless field obeying the nrutrino equation ( 33). Here too the quantizat,ion of t’he #-field leads to the use of an indefinite met.ric because of the massless v-field. III.
QUAPCTIZ4TIOT
OF
FREE
FIELD
two-component spinor +-field, it was In order to quant’ize the “composite” necessary t,o generalize the Schwinger action prinriple t.o include higher order derivat,ives of the spinor field. The details of this procedure are given in the suhsequent, paper (I,$). We note that the main difficult~ies encountered were the precise definitions of the canonical variables and the int’erpretat8ion of the energy density. It t,urns out t,hat the energy densit,y is the 7”’ component] of t,he “energymomentum” tensor plm a term which is an integral over a total tangential divergence. This integral term can he shown to tend to zero for most physical pr0CCSSeS.
The canonical “coordinates” are the normal derivatives of the field variables, JCk’4( x ) , where 8 = n,dP is the normal derivative. Here ( k ) means the derivative t,aken X: limes. The conjugat,e canonical “momentlum” is given by fiCk” = nw ncklr with II”;”
= ‘Z
(-1
)(ri+n) (3
where N is the highest, order derivative
ny$y(,--k)
of C$appearing
py in the Lagrangian
VI
(23) and
is the binomial coeflicient, n !i[m!( n - m) !]. Since there exists one less pair 0 1v2 of conjugate variables than the highest order derivat’ive, we have t,hat, 0 5 I,, 5 N - 1. Further 2, is t,he t,angential derivat,ive so defined that n,a” = O(i.e. 3, = d, Finally,
II’“(“)
is defined as
nb)
and
_a,(1) = _a,,cj,, . . . _a”( .
190
BhRUT
AKD
MULLES
and
L &m(nh(“I) E---
+
ad: + all perm. of p, Y’“‘, XCm’ . (25) a4,cn)X(m)p
1
With these definitions of canonical conjugate variables, the Hamiltonian equations of motion are equivalent t,o the Euler-Lagrange equations, obtained from varying the Lagrangian, i.e.,
go (--)rLa,cnjLfl(n)= 0. The spacelike commutation [8’?$&), are derivable
relations,
fip(;r’)]*
(26)
which are given by
= i&bPG,(.c
(s - .$
- .r’),
< 0
(27)
from the generator F(a)
= /“da,
In Eq. (28) the $-variation
II:k)P&%&)
[g
-
T”“&r,
1 .
(28)
is the total variation
h#Bpcn,= b&n)
- cp,+)&rs
defined such that the energy-moment’um N--l
- ~gYs&+(n,
(29)
tensor T’” is symmetric (30)
where f( ,),AV E ,g[ppp4
+ pplp4
+ nwA~YPj(y#]~
(31)
relations with the field The definitions of P”, J”“, Q and their commutation variables are the same as in the case of first order Lagrangians. In the present case the Lagrangian ( 19) is of order N = 2; hence X- takes the values 0, 1. Using the “natural” coordinate system (,n,, = ( I, 0, 0, 0) ), we have for our canonical conjugates
++-O=-a,++hP
ad: ad: 5. [(o" + d)q+ + aOa"~+%l, a, a(aca,+) = 2 maa (3'2) = ___ad: = .$ #++S,. a(aoaoti)
However, the commut#ation relations (27) for the variables (32) do not lead to Euler-Lagrange equations equivalent to the Hamilton equations. This apparent discrepancy is not due to an inconsistency in our theory. It, arises because our Lagrangian, when written in t,erms of canonical variables, is symmetric under
HIGHER
ORDER
SPISOR
191
EQUATIONS
the interchange @ J+ II^ “)’ but not symmetric under # 4 Ij[‘“)T. This symmetry affects the field generat’ors in such a way that t,he correct commutation relations 116) are given by [&Jr),
rI,“(x’)]+
= &6(S
- 2);
[a”~&>,
lI,““(.r’)],
= ; &bl3(.L - .c’).
(33)
A completely con&ent quantizat~ion of our q-field rccluires physically interpretable annihilation and creat’ion operators and a positive definit’e espectat)ion value for t’he observables. To t’his end we first, derive the commutjation relation between I++and 4’ for any two space-time points using Green’s functions. Let [$~~i.r), &+(x’)]+ where G(.c - .r’) is the homogeneous ( 0’
From
ow
= iGnb(.r - zr’), Green’s function
+ m2)id”S,G
(34)
satisfying
= 0.
relation *=
- 722, T-4+$,
v-e are led to G(J~‘) = ?$?!
A(sx’) + ;< F(r.c’).
(35)
A( I’ - s’) is bhe usual Klein-Gordon Green’s function and F(.r - .r’) is the one satisfying #Z,F = 0. E.quation (35 ) leads t,o t,he result [&(.r), +b+(.r’)]+ = d”SvnbA(l..r’); [TJ,i.r), ?&A]+
= itnYFab(.r.r’).
(36)
An explicit8 representat’ion for the Green’s functions tells us that, for spacelike poink, (x - s’)~ < 0, we have A(sr’)
= 0,
II’l.r.r’) = -d”(x G(xn’) = 0,
&A( .w’) = 2(x - x’); - x’),
d,,F(ss’) = 0;
a,,G(.r.r’) = 0,
f30doG(.rx') = -iS”(x
i:37) - x').
Using the standard methods, we t#hen obtain t,he following plane wax’e deromposit#ionsfor our fields
192
BARUT
where the U’s are eigenspinors
AND
MULLEN
of the spin projection P ‘f
= I+(1 f
operators
rJ.$,.
Now pY = a1 + TWOwith w1 = (p’ + ~n’)“~, CO?= (p2)liz, and r = f. (33) yields finally the commutation relations
[G’(P), a&-Q+ = b+(pj, br~(P’)l+= b&p’ ; [C(+)+(p), CC+) (p’)]+ = [d’-‘+(p), P(p’)]+
= -s,,
Equation
(39) .
The minus sign in the last equation leads to an indefinite metric. To seethis we calculate the energy using T”” = 71.o$,o + T”“$,o”++ h.c.
w0)
and the ‘(charge” (X) of the system. The net results are H = c wl(u:purp + t&b,,) - c W&$+)+C;+) + p
cg’),
P (41)
Q
=
; P-'
X(a;t,a,
-
bfpbrpj
-
C
h(~$+)+~j+~
-
dJ(-)
&I).
P
Although the energy for each field is positive definite, t,he total energy is not. To insure the positive definite charact’er of the expectation value of H, we introduce an indefinite metric ( I’) in the following way. Let
H = c
ciHi
c2 = -1,
Cl = +1,
i=1,2
such that [a:
p+,
7
cGp’]+ = [bSp, b,,p,]+ = ($-‘1,
= [Q’,
&)I+
Cd&p
= C&’
,
(43)
.
The indefinite metric (I’) has the following properties
r: = ri = r;l.
Vl ) r21- = 0,
I? = rlrz)
(44)
We now define the metric conjugation operation by the following relations tij'
=
riqjri
=
+bi
fii'
i #j;
=
riGiri
=
c,+~
(4,s)
where the ci’s can always be set, equal t’o f 1 by a suit’able renormalization of the $;‘s. We say t,hat if ci is equal to plus (minus) 1 then fiZ has positive (negative) metric parity. By metric invariance of, say, the Lagrangian, we mean the following : c'=
rcr
=
c.
(46)
HIGHER
ORDER
SPINOR
193
EQVATIONS
The expectation value of an operator is 0 = (~1 I’0 la). In particular with !Z j 0) = j 0), the eigenvalues of bhe I’i’s, which are diagonal in a representation labeled by the ith number operator Ni:r~i, are given by ni: (c~)‘~~. Wit,h these definitions t’he (t) notation is really the adjoint, defined with respect, to the indefinite metric, and is related to th.e Hermitian conjugate (h.c.), defined wit,h respect to the definite metric by cl+ =
rah,c.r
= C+
ah.r. =
b+
,
=
rCh.c.r
rbh.c.
=
r
=
b1r.c.
-Ch.c.,
&
=
3 rdh.c.r
=
-#l.c..
(47)
Consider symbolically the expectation value of H for a state of one “a” part(icle and one “c” part.icle. Then A is positive definite, for
A = (01 acrclHlatct IO) + (01 acrc2H2atctjO) '-
~(01
dlr2~+~~+c+
-'
&r2(0
[ 0)
lo) +
c1"c26(o
+
~~(01
j O}
2
dxktcutc+
10)
0.
We also obtain t,he desired result for the charge operator. The plane wave amplitudes have the usual interpreCation of annihilabion and creation operat,ors. This completes the consistent quantization of the free field case. IV.
INTERACTIOn’
STRUCTURE
Equation (A.1 ) gave the prescription for adding electromagnetic interactions to the free part#icle equation of motion for the +-field. r\Tow, however, our quantization deals wit)h t,he G-field, so we must generalize it. Whereas Eq. (A.1 j used T*", we now expect t,he new procedure t’o contain 3”“. We introduce a vector quantit#y B, which stands for eit,her t)he eleckomagnetic field (eA,) or a pseudovector meson field (i@,, j . The prescription becomes &a”*
-+ 5YD”n,,lF/,
i43)
where
CD,,= d, + iB,* and e and g must be real. Further,
D, = d, + iB,,
the t,erms linear in t,he derivative
(fJ VP + - La D,~,n”“J, Cd $
E Q,sp- #,
transform
as (49)
where the ( -?+) is inserted for normalization. It is important to not#e that 3” must stand next to t.he #-field so thaO the Lagrangian will remain covariant. The reason for this is that only odd order tensors exist with %natrices standing next to a field variable +ty-pyy
. . . 8”$ = Lorentz tensor,
194
BARUT
ASI)
MULLEN
and also due to the relation 3’” = gp” + igpYt 3 z”~:“. We can now write
down the total Lagrangian
which leads to the Euler-Lagrange
equation
ad: Dx a(Dk*q)
--ad: av
as
+ Dva>,
(note that our general action principle t’he third term). The final result is ( ~pD”T’”
ad: a(q*D”*@)
(51)
= O
leads to taking D”D,
in reverse order in
+ m”)iDh .f&S = 0.
( 5”)
We wrote T’” instead of ( Yvt)-’ so that Eq. ( 52) becomes identical man-Gell-Mann equation (A.5)) if we identify iOx?‘+
5 4
iDhzxv
= 0.
t,o t.he Feyn(53)
or inversely
with”
For B, = eA, we obtain [
Eq.
(.I)
and for B, = ig@, we obtain
d 1
(a, + g+J(a" - g@) + d + 8 gd"sp" cp= 0,
where 5,” = (@“.,
(ci.5)
- (a,,,) - a(o,a, - +pa,).
One might ask why Eq. (54) was select,ed over some other form of interaction for either a pseudo-scalar or pseudo-vector meson. To answer t.his question, we 2 Here
7 appears
-DprD”‘T~“/na2 $1 = obtain
D$‘+/PL~,
Eq.
to have
charge,
and P1 = [DII’D”‘T*”
but
if we introduce
+ m2J/m2
with
D,’
$2 = ~/,n” so that the field $J in the (1) and a,%$ = 0 instead of (53’).
the
projection
operat.ors
= ~3, + y (1 + r3)Ap Lagrangian
(50)
Pi
=
and the fields
is + = $1 +
#z R-e
HIGHER
ORDER
SPINOR
derive Eq. (5-l) from the conventional gy5@, in [y”(P, Multjiplying
in on t’he left by [~“(p, [
Dirac
- igr5sP)
formalism
by replacing
ie14, by
- m]ys = 0.
- igy’+,,)
(p, + igr”@,) (p” - ig~w)
195
EQUaTIONS
-
(56)
+ m], we obtain
HZ2 - ; gy5ypysl””
1
# = 0.
(57)
We chose this form for the multiplier since it is the one that leads to a second order equation having ~‘9 as a solution. This implies that Eq. (57) can be written in two-component form of Eq. (54). Had we t,ried to develop a theory with pseudo-scalar mesons, we would have been led to a two-component form having the meson field variable appearing in the denominator in the equation of motion. Hence, for simplicity of form, we adopt, a pseudovector theory of meson interartion. A.
QUAN'~~M
ELECTRODYSAMICS
We now have a t(otal Lagrangian given by Eq. (50) for electromagnetic processes which can be written as Co + Cr. In turn sr can be broken down int#o t hrcc t)erms + Cr,(dv),
(58)
where we have split up J& into 4 and 7. The fields Cam contains bilinear t,erms of t’hat type. We perform plicit,ly that t,erms C1, and C I2 are met,ric invariant we mean that, cI,,, are invariant’ under tii ---) fi;’ = by Eqs. (121-i-14). This result places restrict)ions on &‘-matrix. These restrictions may be stated as: (a) Only transitions between st,ates of the same = CjCj(#i, #j) # 0 only for C; = Cj (b) All physical observables commut’e with t,he
in parentheses mean t’hat’ this splitting to show eswhile CJ, is not. By this r,tiir’i = cili, as defined a physically interpret,able
CI,(M)
+ Cr?(llv)
metric
are allowed:
metric
operat’or:
( tii’,
#j')
0’
=
r()h.c.r
=
0h.c.
(c) r commutes with S-mat’rix (St = Stl.r. ) (insures unitarity of S-matrix). In addition to the t.hree restrict,ions, we note t’hat if the generator is bilinear
then
SO = i[O, F]for any operator
0 remains unchanged
when an indefinite
metric is introduced.
+
FIG. 1. The solid line is a massive spin the dashed line is a massless spin one-half
one-half particle.
particle,
the
wavy
line
is a photon,
and
For 0 = Ntotnl this means t,hat the norm of a state is preserved and the Heisenberg equations of motion remain valid. Turning now to our Lagrangian (SO) or (.58), we see that to first order in e or g we may neglect ce13 due to condition (c). Further, for the scattering of massive spin one-half particles d: 12 would not enter as it cont#ains only massless fields. We are left with just, the Feynman-Gell-Mann field, 4. Sow to second order s (2) - P[:c,(x,) ::&,(x,):] - P[:cI,(i) ::c1,(2j: +
:Cr,(l)::C1,(2):
+
:c,,(l)::c,(2j:
+ :d31Z(l)::S11(2): (59)
+ :cI,(l) + CI,(l)
::~e,(2): ::c1,(2):
+ :~e,L(l) ::~c,,(2): + :c,(l)
+ :Cr,(lj
::JZr,(2):],
where : . .. : denot,es the normal product ( IO’). Metric terms 3, 6, 7, 8 and for scattering of massive particles, 4, .j leaving s (2) -
P[:e,,(i)
::cI,(2):
::s1,(2j:
The second t,erm above, after contraction would lead to diagrams in which there is (Fig. 1). If we normalize the metric parity to +l and the unphysical photons to not contribute to massive scattering. Continuing this procedure we see that’
+ :s6,,(i) of t,he a single of the 1, then
invariance eliminates we can eliminate 2,
::cI,(2):].
(60)
time ordered normal product, massless intermediate particle physical (transverse) phot,ons the second term in (60 ) does
up to any order, the q-field does not
HIGHFlR
contribute. as
ORDER
SPIA-OR
Thus, we need only consider &(#x$) b:=
w*j”*Tp’.
197
EQUATIOXS
[D,“D,*++S’S”BXDxqb
+ JZ~(+C$) which -
dD!D,*++S”4].
can be written r6l)
For elect,romagnetic processes we have Q, = D, and symbolically, ( I/D,DyT’“) acting on 4 bring down a term ( -l/m’). If we define Dp*++S” = d, D,*X” = ---in’+‘, and D,Z’ = iII-‘, (61) becomes
which is nothing more than the Dirac equation writ,ten in terms of tn-o-component spinors. This Lagrangian was used by Brown (2) and Tonin (3) to calculate quantum electrodynamical processes. Therefore, the restrictions placed on our more general theory by the indefinit,e met.& are equivalent to taking t’he limit 6 = 0 in Eq. (10). To summarize, in order to define a physically interpretable S-matrix in a Hilhert space of indefinite metric, we are led to certain restrict’ions. In essence, these conditions &ate that a linear operator 0 in the total Hilbert space is an observable corresponding to the observable 0’ if it acts on t)he vectors of the subspace of positive definit,e metric as a Hermitian operator. Although it2 was the q-field which forced us to enlarge t#he Hilbert space, it must be emphasized that one rannot’ just ignore t>his field (set t = 0) from the start. The reason is that from (58) t,here are terms containing the v-field which are metric invariant. The metric invariance of the S-matrix insures its unitarity and we would t#hus expect t’hat the q-dependent part would make a contribution to physical processes. However, we must also sat,isfy condition (a ), which states that transiGons between st!at,es of opposite met#ric parity are forbidden. This is t,he reason that the q-dependent term in (58) does not contribute. The need for all three conditions points out tb.e import*ant fact’ that the Hermiticit)y of an operator in the subspace of posit’ive definite metric does not insure its ohservahility. As Tonin has shown (S), the Lagrangian (6%) leads to results equivalent to t,he usual four-component case in the realm of quantum elec~~rodynamics. The equakn (61 ) also exhibits the full parity invariance as required. This is due t.o t,he symmetry between the interchange of Q w + in calculating matrix elements (i.e., Lmder I’:PD iv 4 and I%$ - a). Due t,o the general form for our total Lagrangian, the same results are obtained for pseudo-vector meson theory (eA, + ig+P). That is, the skong interactions are P-invariant and t,he v-field does not, enter into t.he results. We have thus the t,heorem t#hat t#he two-component, spinor theory with vector int,eract,ions
198
BAIZUT
L4PiD MULLEN
(@P-invariance ) and metric invariance implies P-invariance.3 If the q-field is associated with t#heneutrino, then its nonexistence in elect,romagnetic and strong interactions is explained, or, conversely, supports our hypotheses. B.
WEAK
INTERACTIONS
The second order t,wo-component spinor equation (1) was used by FeynmanGell-Mann to just#ify the V - A theory of the four field Fermi interaction. The assumption was that if all fermions are described by a two-component spinor then the weak interactions should be expressed in terms of them. The result, in terms of four-component spinors, was
JL - (WC 1 + r”Nd ($Yr(1 + YW
(67)
or equivalently -
WC1 - r”h”(l
+ r”MWl
- Y5hrU
+ r5M.
Using relation (15)) we may also write this as 6, -
G#J+w (4+%#4,
(69)
where of course C#I is the two-component spinor field obeying Eq. (1). In this approach the neutrino (m, e = 0) obeys the equation p,24+4 = p,qpx%w)
= 0
with phzh+( V) = 0. In terms of our #-field we can write (69) for massive particles as Go -
(aA~+S”z”Z”a”~) (aA#+?Z”z,Z’a,#),
(70)
since &XX+ = +. This is the simplest structure containing only the +-field. However, it is just a choice and does not depend upon any new symmetry rules other than metric invariance. Again, if the v-field is the neutrino, then (70) is good for nonleptonic decays. For c,, to be metric invariant when it, contains the v-field, it must contain 11in pairs. This implies transitions of the form + -+ + + 11+ 4, which may be able to explain pure leptonic decays. The structure of the Lagrangian is arbitrary for such a process so it remains to find addit,ional symmetry principles. There also remains the problem of finding the correct C,,, for leptonic strangeness conserving and violating decays which contain only one neutrino. IV.
SUMMARY
AND
CONCLUSIONS
We have quantized a single “composite” field given by Eq. (31) using a higher order Lagrangian ( 19). We chose this approach because (a) we had to plus
3 There is a counterpart of this theorem charge symmetry implies P-invariance.
in ordinary
Yukawa
interactions.
CP-invariance
HIGHER
ORDER
SPINOR
EQUaTIOKS
199
deal with only one field instead of three; (b) it led to a method of introducing electromagnetic and strong interactions which are equivalent in form and valid for all values of spin; and (c) it implies the concept that all massive spin onehalf particles are coupled to a neutrino particle. Whether the neutrino coupled to each massive particle is the same remains to he seen when we can uniquely write down the weak interaction Lagrangian. But for sbrong and electromagnetic interactions tbe resbrict’ions placed on the S-matrix by t,he indefinit,e metric uniquely determine the int,eractions and imply P-invariance. Thus we cannot get the answer from the electromagnetic and strong interactions since the physical parts of the Lagrangians in t’hese cases are not a function of the s-field. The significance of t!hese restrictions is that it is a necessary but not a sufficient condition that an observable operator in the total Hilhert space act as a Hermitian operator on the vectors of t,he suhspare of positive definite metric*. -4lt’hough this insures bhe diagonalizahility of t’he operat’or, we must. also include the conditions t$hat only t)ransitions between states of the same metric parity are allowed and that the S-mat#rix is unit,ary and not pseudo-unitary. Finally, we add that the successful quantization of the FeynmanP6ell-Mann field, performed here, suggests several int’eresting dire&ions of investigation. One, as already stated, is to find a unique weak int(eraction structure. Another is a more detailed investigation of t,he concept of metric conjugation and metric parit’y. The success of our theory seems t)o put, t’he role of the indefinite metric in yuantSum field t,heory on a firmer footing. It’ may thus prove fruihful to work out, a formal theory of scattering using an indefinite met’ric. We finally remark t,hat although the usual proof of the TCP theorem depends on the spin-statist,ics connection, which in turn depends on the positive definite metric, the spinstat#istirs theorem can be proved in most cases witIhout’ the assumption of a positive definit.e metric from t’he action principle il7), or within the framework of S-matrix theory (18). APPENDIX
I
In this appendix we discussbriefly t(he invariance properties of Ey. ( 1). We consider first the Lorentz invariance of this equat)ion. Equation (,1) must have t,he same form in every frame, hence
(AS) Lorentz invariance demands that (A.2)
200
RARUT
Ah-D
MULLEN
Using the fact’ that
(AA) relation
iA.2)
reduces to [LP, uPv] = gavaPa _ g”“uPP + goPuav _ gwuBY*
A solut,ion of (A.4) exists under the proper Lorentz We now show that Eq. separately. Consider first, x’ = -x,
(A.4)
with 2s”’ = CJ~’ and therefore Eq. (1) is invariant group. (1) is invariant under PC and T but not P and C’ the PC transformation defined by
P’ = -P,
db’)
= w%*(
-xl, (A.5)
A,’ = -g,,A’(
-x),
Flj = -Fij(
FL = Fed-x),
-x),
which gives [(p, + eA,,)2 - rn2 + em”OiFo;- ea”jFij](PC)+*(-x) For Eq. (1) to be PC invariant, [(p,
+ eA,)”
-
i.e., (A.6)
= 0.
(A.61
equal to
,m” + moi*Foi + ecrij*Fi&*( -x)
= 0,
(A.7)
we must have that (pc)-‘,““(p~~)
= aoi*
and - (Pc)-$TiqPC)
= (Tij*.
These last two relations reduce to [(PC),
n31+
=
0,
[(PC), u,]- = 0.
(A.81
A solution for (A.8) is PC = CQ. Therefore Eq. (1) is PC invariant with &x’)
= u&*(-x,
x0).
An analogous procedure shows us that Eq. (1) is also T-invariant &z’)
= u2+*(x, -20).
(A.91 with (A.10)
We next postulate that the gauge and Lorentz invariant way of introducing electromagnetic interactions into Eq. (14) for a particle of arbitrary spin is given by (19) a,#’ + D,D,T”*
(A.ll)
HIGHER
ORDER
SPINOR
EQVATIONS
201
with D, = 13~+ ieA, . T’” depends on the transformation properties of the $-field manner under infinitesimal Lorent, transformations &‘(S’) This is equivalent
= ,l~‘&“T:,‘(bb(.ll).
in the following (A.12)
to the usual notation
&iv = gav + +A” Finally
we may writ’e (D,D,T;;:
+ WL%,,&~(.L.) = 0.
For a one-component field (scalar), This leads to the usual Klein-Gordon [(a, + id,)”
(A.15)
we have S”’ = 0 so that TiR = gp”sab . eyuabion for spin zero particles, + m$(.r)
= 0.
(A.16)
For the two-component case, as was said previously, t’he proper Lorentz group admits a two-dimensional irreducible representation but not the full Lorent’z group. This implies parity noninvariance of the equations of motion and is in keeping with weak interact’ions. In this case, we found above t,hat 2S’” = /” and therefore D,D,T”” = D,” - (~/‘3)d“‘F~~ . Using Pa = iD, we obtain Eq. (1). We might just point out that had we considered t#he four-component case, we would be led to the second order Dirac equation which is invariant under the full Lorentz group. This in turn implies P-invariance. I~ECEIVED:
February
28, 1962 REFEREKCES
1. R. FEYNMAN AND M. GELL-MANN, Phys. Rev. 2. L. M. BROIVN, Phys. Rev. 111, 957 (1958). 3. M. TONIN, Nuovo cinlento 14, 1108 (1959).
4.
W. R. THEIS, Portschr. Physik 7, 559 5. A. PAIS AND G. E. UHLENBECK, I’hys. 6. J. W. KIBBLE AND J. C. POLKINGHORNE, 7. G. MARX, Nuclear Phys. 9, 337 (1958). 8. P. A. M. DIRAC, Conamu~~. Dublin Inst. 9. W. PAULI, Revs. Modern Phys. 16, 175 (1950); R. ASCOLI AND E. MINARDI, J. PLEBANSKI, Nuovo cimento 18, 884
109, 192 (1958).
(1959). Rev. 79, 145 (1950). Nuouo cimento 8, 74 (1958). Advanced (1943); Nucleav
(1960).
Studies 8. N.
Phys.
A, No. 1 (1943). Proc. Phys. Sot. 63, 681 9, 242 (1958); H. M. FRIED AND GUPTA,
202
BARUT
AND
MULLEN
10. See, for example, the review paper by K. NAGY, ;V~touo cimento S~cppl. 17, 92 (1960). 11. E.C.G. SUDARSHAN, Phys. Rev. 123,2183 (1961); E. C.G.SUDARSHAN AND H.J. SCHNITZER, Phys. Ret>. 123, 2193 (1961); R. SPITZER, preprint. 12. A. 0. BARVT, to he published. 13. A. 0. BARUT, Ann. Phys. (N. k’.) 6,95 (1958). 14. A. 0. BARUT AND G. H. MULLEN, Ann. Phys. (X. Y.) 20.203-218 (1962). 15. For a more detailed discussion of this point see J. SCHWINGER, Phil. Mug. 44, 1171 (1953).
16. W. THIRRING, “Principles York,
of Quantum
Electrodynamics,”
1958.
17. R. ARNOWITT AND S. DESER, (to be published). f8. H. P. STAPP, Phys. Rev. (to be published). 19. C.G.BOLLINI,NUOUO cimento 19,560 (1959).
p. 51. Academic
Press.
Sew
OF PHYSICS:
20, 184-202
Quantization
of Two-Component Spinor Equations* A.
Ph,ysics
(1062)
0.
Department,
BARUT~
Syracuse
G. H.
ASD
University,
Higher
Order
&ICLLEN$
Syracuse,
New
York
A two-component fermion field is quantized using a Lagrangian formalism containing higher order derivatives of the field variables and an indefinite metric in Hilbert space. A consistent quantization necessitates beside a massive spin one-half field, obeying the Feyrlniarl-(:ell-Mann equation, the introduction of a massless spin one-half field, obeying the nentrino equation. Gauge invariant electromagnetic and vector interactions are introduced in a manner which depends on the transformation properties of the field variables. The physical rest,rictions placed on the S-matrix by the use of an indefinite metric eliminates the massless field from the “composit,e” field for electrodynamics and pseudo-vector mesons intera&ions and implies parity invariance but not for four-field weak interactions. The results obtained for quantum elect’rodynamics are equivalent, to the usual fourxomponent t,heory. I. INTRODUCTIOS
It has been generally recognized that the description of spin one-half particles hy four-component spinors is redundant and t’mo-component, spinors should be su&ientS. The use of t’wo-component, spinors offer many advantages, bot,h of fundament,al and practical nature. Peynman and Gell-Mann (1) were the first to give a two-component second order spinor equation for a spin one-half particle interacting with an ext’ernal electromagnet.ic field and established t’he vect,or and axial vector charact’er of weak interactions. Brown (2) has reduced this equat#ion to two coupled linear two-component spinor equations and showed its equivalence to the Dirac eyuatjion. The coupled linear equations have heen subsequently quantized (3, 4). The question arose early whet,her one could quantize the Feynman-Gell-Mann equation (or similar higher order spinor equations) directly from a Lagrangian formalism without, linearizing it,. It, is clear that. the simplest Lagrangian which * Supported in part by the U. S. Air Force Office of Scientific Research. i Now at Lawrence Radiation Laboratory, University of California, California. $ Now on a Resident Research Associateship at the U. 8. Naval Ordnance White Oak, Silver Spring, Md. 184
Berkeley, Laboratory,
HIGHER
ORDER
SPINOR
EQTTATIONS
185
gives a second order equation is of third order. And higher order Lagrangian theories necessit’ate the use of an indefinit#e metric in Hilbcrt space (5). Iiibblc and Polkinghorne (6) have replaced in the Lagrangian the two-component spinors by four-component ones and linearized the resultant, equations. They then use a subsidiary condition to climinak the superfluous components. It, has been also suggested that t(hese superfluous components of liihhle and Polkinghornc might be used to int,roduw an isoduhlet, of fermions ( 7 ) . The purpose of this paper is tJo present, a complet’e Lagrangian cluantization of the higher order two-component spinor equation in it,s own right. We follow t’o the end the consequences of higher order Lagrangians and bhc use of an indefinite metric and formulate a consistent, second ctuant,ized theory of t,hese equat~ions. The indefinit,e metric in quant#ized field bheories has been discussed again and again since it was first proposed by Dirac in I I143 (8, 9) to overcome various difficulties of quantum field theory (10) and recently again for the purpose of consistency of t,he field theories (11). Whereas, in concret#cexamples, some of t,hese discussions deal wit,h simple models using indefinite metric, we give a complete physical case of such a t,heory. It is known, for example, in the case of elect,rodynamics, that a local theory wit’h an indefinit,e metric is equivalent to a nonlocad t,hcory with a positive definite metric. If this fact is genera~lly t,rue, then the consistent’ quant,izatJion of the higher order two-component spinor cquat,ions presented here may be an indiration of t’heir nonlocal rharacter. The main starting point of the present work is t,he introduction of an auxiliary field which is a sum of t,he Fcynman-Gell-Mann field 4 and a masslesst,tvo component, field 7. The necessity of doing so and the general propert,ies of the Lagrangians and the equations are discussedin the nest section. In S&ion III we quantize the fire fields, and in t,he following sect~iont’he interact)ing fields. The resultjs imply that,, in our formalism, all massive spin one-half particles must be accompanied by a masslessspin one-half parkle obeying t#heneutrino equation. The physical interpretation of the A-matrix in the presence of an indefinite mrt,rie eliminates the masslessfield in t,he vase of strong and clectromagnet,ic interact,ions, and, furthermore, implies parity int-ariance, whereas the original t,heory is only C:P-invariance. The application of t,hr t.heory to the weak intcractions is also briefly discussed. We use natural tmit,s wit,h fi = c = 1 and a metric2 ( +, -? -, - ). IT. LAGRANGIAPT
We start from the Feynmatr-Gell-Mann
6’0RMAI,ISRI
equation
1
+(A-) = 0
(1)
186 with
F,, = A,,, - A,,,; The simplest free particle tive of the field variable
a’” = -0”’ =
Lsgrangian
for Eq. ( 1) is of third order in the deriva-
c = c#B+id”Z,(0’) + my$,
(2)
where S” = (I, u,)
Z” = (1, -u;)
(3)
with
Equation (2) is invariant under the nonunitary two-dimensional irreducible representation of the proper Lorentz group. It can he shown that no metric of the form (4, I’4) exists for this t,wo-dimensional representation (121 (i.e., there is no scalar of the form 4’4). However, a Lorentz vector of the form 4’Y4 does exist as can be seen by not’ing that under infinitesimal Lorentx transformations the t’wo-component spinor 4 transforms as &n:‘)
= S(L)4(.Z)
= (1 + ; ,,“C)
4(s).
(5)
To quantize Eq. (1) by conventional techniques, we have t,o linearize the relation (2). In this linearization we cannot obtain t,he proper commutation relations without t#he use of an indefinite metric. To see this, we define iKZ,g5
= m.$,
(6)
whose inverse is ia”Z,( = m4 + CC,
(7)
where GYz,< = 0,
( iI2 + m’)< = 0,
(8)
and c is a new constant paramet,er entering into t,he theory. In Eq. (7) we made use of the relation (ia”z,j(--ia”Z,)
= (-ia’Z,j(ia’z,)
= 0’.
(9)
HIGHER
ORDER
SPINOR
187
EQITATIONH
E:quat,ion (2) for the free part,icle case t,hen t,akes the form 6: =
++itP&$
+
(+i#E
p( -
m.$+@ -
mc$i,t (10)
+
c(m<+(
+
ml+<
-
<+id”S,~
-
~+itE,~).
A calculat’ion leads to nonzero anticommut8at8ion relat,ions, which depend on the parameter 6, between the massless field { and the massive field c+. This can hc seen easily when wc writ’c Eq. ( 10 ) in the Kcmmer form (11) where
and evaluate [P(z),
*(CC’)]+
=
i6”(.r
-
(,13 )
x’),
with n” = -it&/+/j”. For spacelike surfaces
(13’)
(p = 0), Eq. (13) is equal to
[c$(.r),(b+(k)]+ = sRcx- x’),
[((x), t+(.r’))+ = 2(x - x’),
[4(x),
[‘(.r’)],
= -e-‘63cX
- x’).
(W)
The quantizat,ion of Eq. i 10 ) hy Brown (.2), Tonin (3)) and Theis (4) amounts to assuming c = 0. This is equivalent, to t#he Dirac equat)ion written in terms of two-component spinors as seen explicit,ly hy sett,ing 6 = 0 in 1%~. ( 11). This scheme eliminates t’hc massless l-field and hence eliminates the use of an indefinitr metric. Their results for quantum electrodynamics are equivalent, to thr four-component, Dirac theory. We now consider t,he quantization of Eq. (2 ) as it stands which can be carried out, using a higher order Lagrangian t,hcory.’ Thcrc are several advantages for considering two-component spinors instead of the usual four-component, spinors. The theory based on two-component spinors is simpler, more re&ictive, and srparat’cs csharge and spin degrees of freedom. From the relativist,ic relation bet,ween mass and momentum, we would expert all fields to obey the KleinGordon equation ( 0’ 1 A qumtizntion
of the coupled
+
equations
m’)&(x) (6) and
=
0
(7) is also possible.
(14)
(m 2 0 and u = spin and ( or ) tensor index ) X spin zero particle would be described by a one-romponentj field while a spin one-half particlc by a twocomponent field due to the twofold spin degeneracy. For the spin one-half cast t#he wave function for the free part#icle would he 1’ esp (ip.x - iEt) where v is a two-component’ spinor. The sign of the energy in the exponent t’ells us whet,her we are describing an electron or a position. It is, indeed, this reasoning that led Feynman-Gell-Mann t,o t’heir unique form ( 1,’ - Ai ) for weak interactions. E’or further reference we note the relat,ion between t’hc Dirac spinor appearing in their (I: - -4) theory and the t,wo-component spinor appearing in Eq. ( I ‘)
Equat,ion ( 1) ran also he obt’ainrd from t’he second order Dirac equation ( 1.9)) a procedure which is valid even when m = 0, whereas the Feynman-Gell-Mann procedure breaks down for nl = 0. Another method, st’arting from Eq. ( 14) and stressing the transformation properties, is given in Xppendix I. The procedure by which we quantize the higher order Lagrangian and cstahlish contact with Eq. ( 1 ) is the following. A variat#ion of Eq. (2) yields the equation of motion ( 0” + m’jia”8,~(xj
= 0.
(16)
This result is more genera1 than we want. To avoid the use of a subsidiary dition, we define a new field # related t.o 4 by ia”Z,#
=
S(L)
=
(
(17)
+.
Since 4 transforms under Lorent’z t8ransformat,ions transforms as J/‘( x’) = S( L)+(s) with
con-
by S(h)
given by (5) , J/
1 + ; EpvrJfiVi- = s’t-1. )
In turn we may define a 3”‘( S = )~L,,$“‘) related to T’” in the same way as S is to S. We now choose a Hrrmit8ian Lagrangian for t’he +-field d: = ; [(o”~+z”a,~ which
- a,~+Z”o”~)
+ m2(#+8”a,lj
- a&+.2”+)],
(19)
leads to the desired result (O*+
m’)(id’Z,+)
= (02 + m”j+ = 0.
(‘LO)
The inverse of (23) is given by (21)
HIGHER
ORDER
SPIA-OR
189
EQUATIONS
where (22)
idPZp?f = 0
and (l/O’) is symbolic in that when it acts on the plane wave decomposit,ion, it brings down a factor ( -1/X2). It, should he noted that the fields defined by Eqs. (17), (21 j, and (22) do not’ obey t#he same equations of mot,ion as the fields defined by Eels. (G)-(8). The fields of Eqs. ((j-18) are so defined that the linearized form (Ekl. (‘ICI j of Eq. (2) is the Dirac Lagrangian written in terms of two-component spinors piur a term depending on the mnssless field it). The fields of Eqs. (17), (21), and (22) are defined so that’ l3q. (2) can he replaced by Ekl. (19). In t,his manner Eki. (19) leads t’o the correct free prat,icle equation of mot,ion (i.e., Eq. (20)) for the Feynman-Gell-Mann field whereas 15~1.(2) does not. Just as in t’he linearized form of (a), n-e must also introduce a massless field obeying the nrutrino equation ( 33). Here too the quantizat,ion of t’he #-field leads to the use of an indefinite met.ric because of the massless v-field. III.
QUAPCTIZ4TIOT
OF
FREE
FIELD
two-component spinor +-field, it was In order to quant’ize the “composite” necessary t,o generalize the Schwinger action prinriple t.o include higher order derivat,ives of the spinor field. The details of this procedure are given in the suhsequent, paper (I,$). We note that the main difficult~ies encountered were the precise definitions of the canonical variables and the int’erpretat8ion of the energy density. It t,urns out t,hat the energy densit,y is the 7”’ component] of t,he “energymomentum” tensor plm a term which is an integral over a total tangential divergence. This integral term can he shown to tend to zero for most physical pr0CCSSeS.
The canonical “coordinates” are the normal derivatives of the field variables, JCk’4( x ) , where 8 = n,dP is the normal derivative. Here ( k ) means the derivative t,aken X: limes. The conjugat,e canonical “momentlum” is given by fiCk” = nw ncklr with II”;”
= ‘Z
(-1
)(ri+n) (3
where N is the highest, order derivative
ny$y(,--k)
of C$appearing
py in the Lagrangian
VI
(23) and
is the binomial coeflicient, n !i[m!( n - m) !]. Since there exists one less pair 0 1v2 of conjugate variables than the highest order derivat’ive, we have t,hat, 0 5 I,, 5 N - 1. Further 2, is t,he t,angential derivat,ive so defined that n,a” = O(i.e. 3, = d, Finally,
II’“(“)
is defined as
nb)
and
_a,(1) = _a,,cj,, . . . _a”( .
190
BhRUT
AKD
MULLES
and
L &m(nh(“I) E---
+
ad: + all perm. of p, Y’“‘, XCm’ . (25) a4,cn)X(m)p
1
With these definitions of canonical conjugate variables, the Hamiltonian equations of motion are equivalent t,o the Euler-Lagrange equations, obtained from varying the Lagrangian, i.e.,
go (--)rLa,cnjLfl(n)= 0. The spacelike commutation [8’?$&), are derivable
relations,
fip(;r’)]*
(26)
which are given by
= i&bPG,(.c
(s - .$
- .r’),
< 0
(27)
from the generator F(a)
= /“da,
In Eq. (28) the $-variation
II:k)P&%&)
[g
-
T”“&r,
1 .
(28)
is the total variation
h#Bpcn,= b&n)
- cp,+)&rs
defined such that the energy-moment’um N--l
- ~gYs&+(n,
(29)
tensor T’” is symmetric (30)
where f( ,),AV E ,g[ppp4
+ pplp4
+ nwA~YPj(y#]~
(31)
relations with the field The definitions of P”, J”“, Q and their commutation variables are the same as in the case of first order Lagrangians. In the present case the Lagrangian ( 19) is of order N = 2; hence X- takes the values 0, 1. Using the “natural” coordinate system (,n,, = ( I, 0, 0, 0) ), we have for our canonical conjugates
++-O=-a,++hP
ad: ad: 5. [(o" + d)q+ + aOa"~+%l, a, a(aca,+) = 2 maa (3'2) = ___ad: = .$ #++S,. a(aoaoti)
However, the commut#ation relations (27) for the variables (32) do not lead to Euler-Lagrange equations equivalent to the Hamilton equations. This apparent discrepancy is not due to an inconsistency in our theory. It, arises because our Lagrangian, when written in t,erms of canonical variables, is symmetric under
HIGHER
ORDER
SPISOR
191
EQUATIONS
the interchange @ J+ II^ “)’ but not symmetric under # 4 Ij[‘“)T. This symmetry affects the field generat’ors in such a way that t,he correct commutation relations 116) are given by [&Jr),
rI,“(x’)]+
= &6(S
- 2);
[a”~&>,
lI,““(.r’)],
= ; &bl3(.L - .c’).
(33)
A completely con&ent quantizat~ion of our q-field rccluires physically interpretable annihilation and creat’ion operators and a positive definit’e espectat)ion value for t’he observables. To t’his end we first, derive the commutjation relation between I++and 4’ for any two space-time points using Green’s functions. Let [$~~i.r), &+(x’)]+ where G(.c - .r’) is the homogeneous ( 0’
From
ow
= iGnb(.r - zr’), Green’s function
+ m2)id”S,G
(34)
satisfying
= 0.
relation *=
- 722, T-4+$,
v-e are led to G(J~‘) = ?$?!
A(sx’) + ;< F(r.c’).
(35)
A( I’ - s’) is bhe usual Klein-Gordon Green’s function and F(.r - .r’) is the one satisfying #Z,F = 0. E.quation (35 ) leads t,o t,he result [&(.r), +b+(.r’)]+ = d”SvnbA(l..r’); [TJ,i.r), ?&A]+
= itnYFab(.r.r’).
(36)
An explicit8 representat’ion for the Green’s functions tells us that, for spacelike poink, (x - s’)~ < 0, we have A(sr’)
= 0,
II’l.r.r’) = -d”(x G(xn’) = 0,
&A( .w’) = 2(x - x’); - x’),
d,,F(ss’) = 0;
a,,G(.r.r’) = 0,
f30doG(.rx') = -iS”(x
i:37) - x').
Using the standard methods, we t#hen obtain t,he following plane wax’e deromposit#ionsfor our fields
192
BARUT
where the U’s are eigenspinors
AND
MULLEN
of the spin projection P ‘f
= I+(1 f
operators
rJ.$,.
Now pY = a1 + TWOwith w1 = (p’ + ~n’)“~, CO?= (p2)liz, and r = f. (33) yields finally the commutation relations
[G’(P), a&-Q+ = b+(pj, br~(P’)l+= b&p’ ; [C(+)+(p), CC+) (p’)]+ = [d’-‘+(p), P(p’)]+
= -s,,
Equation
(39) .
The minus sign in the last equation leads to an indefinite metric. To seethis we calculate the energy using T”” = 71.o$,o + T”“$,o”++ h.c.
w0)
and the ‘(charge” (X) of the system. The net results are H = c wl(u:purp + t&b,,) - c W&$+)+C;+) + p
cg’),
P (41)
Q
=
; P-'
X(a;t,a,
-
bfpbrpj
-
C
h(~$+)+~j+~
-
dJ(-)
&I).
P
Although the energy for each field is positive definite, t,he total energy is not. To insure the positive definite charact’er of the expectation value of H, we introduce an indefinite metric ( I’) in the following way. Let
H = c
ciHi
c2 = -1,
Cl = +1,
i=1,2
such that [a:
p+,
7
cGp’]+ = [bSp, b,,p,]+ = ($-‘1,
= [Q’,
&)I+
Cd&p
= C&’
,
(43)
.
The indefinite metric (I’) has the following properties
r: = ri = r;l.
Vl ) r21- = 0,
I? = rlrz)
(44)
We now define the metric conjugation operation by the following relations tij'
=
riqjri
=
+bi
fii'
i #j;
=
riGiri
=
c,+~
(4,s)
where the ci’s can always be set, equal t’o f 1 by a suit’able renormalization of the $;‘s. We say t,hat if ci is equal to plus (minus) 1 then fiZ has positive (negative) metric parity. By metric invariance of, say, the Lagrangian, we mean the following : c'=
rcr
=
c.
(46)
HIGHER
ORDER
SPINOR
193
EQVATIONS
The expectation value of an operator is 0 = (~1 I’0 la). In particular with !Z j 0) = j 0), the eigenvalues of bhe I’i’s, which are diagonal in a representation labeled by the ith number operator Ni:r~i, are given by ni: (c~)‘~~. Wit,h these definitions t’he (t) notation is really the adjoint, defined with respect, to the indefinite metric, and is related to th.e Hermitian conjugate (h.c.), defined wit,h respect to the definite metric by cl+ =
rah,c.r
= C+
ah.r. =
b+
,
=
rCh.c.r
rbh.c.
=
r
=
b1r.c.
-Ch.c.,
&
=
3 rdh.c.r
=
-#l.c..
(47)
Consider symbolically the expectation value of H for a state of one “a” part(icle and one “c” part.icle. Then A is positive definite, for
A = (01 acrclHlatct IO) + (01 acrc2H2atctjO) '-
~(01
dlr2~+~~+c+
-'
&r2(0
[ 0)
lo) +
c1"c26(o
+
~~(01
j O}
2
dxktcutc+
10)
0.
We also obtain t,he desired result for the charge operator. The plane wave amplitudes have the usual interpreCation of annihilabion and creation operat,ors. This completes the consistent quantization of the free field case. IV.
INTERACTIOn’
STRUCTURE
Equation (A.1 ) gave the prescription for adding electromagnetic interactions to the free part#icle equation of motion for the +-field. r\Tow, however, our quantization deals wit)h t,he G-field, so we must generalize it. Whereas Eq. (A.1 j used T*", we now expect t,he new procedure t’o contain 3”“. We introduce a vector quantit#y B, which stands for eit,her t)he eleckomagnetic field (eA,) or a pseudovector meson field (i@,, j . The prescription becomes &a”*
-+ 5YD”n,,lF/,
i43)
where
CD,,= d, + iB,* and e and g must be real. Further,
D, = d, + iB,,
the t,erms linear in t,he derivative
(fJ VP + - La D,~,n”“J, Cd $
E Q,sp- #,
transform
as (49)
where the ( -?+) is inserted for normalization. It is important to not#e that 3” must stand next to t.he #-field so thaO the Lagrangian will remain covariant. The reason for this is that only odd order tensors exist with %natrices standing next to a field variable +ty-pyy
. . . 8”$ = Lorentz tensor,
194
BARUT
ASI)
MULLEN
and also due to the relation 3’” = gp” + igpYt 3 z”~:“. We can now write
down the total Lagrangian
which leads to the Euler-Lagrange
equation
ad: Dx a(Dk*q)
--ad: av
as
+ Dva>,
(note that our general action principle t’he third term). The final result is ( ~pD”T’”
ad: a(q*D”*@)
(51)
= O
leads to taking D”D,
in reverse order in
+ m”)iDh .f&S = 0.
( 5”)
We wrote T’” instead of ( Yvt)-’ so that Eq. ( 52) becomes identical man-Gell-Mann equation (A.5)) if we identify iOx?‘+
5 4
iDhzxv
= 0.
t,o t.he Feyn(53)
or inversely
with”
For B, = eA, we obtain [
Eq.
(.I)
and for B, = ig@, we obtain
d 1
(a, + g+J(a" - g@) + d + 8 gd"sp" cp= 0,
where 5,” = (@“.,
(ci.5)
- (a,,,) - a(o,a, - +pa,).
One might ask why Eq. (54) was select,ed over some other form of interaction for either a pseudo-scalar or pseudo-vector meson. To answer t.his question, we 2 Here
7 appears
-DprD”‘T~“/na2 $1 = obtain
D$‘+/PL~,
Eq.
to have
charge,
and P1 = [DII’D”‘T*”
but
if we introduce
+ m2J/m2
with
D,’
$2 = ~/,n” so that the field $J in the (1) and a,%$ = 0 instead of (53’).
the
projection
operat.ors
= ~3, + y (1 + r3)Ap Lagrangian
(50)
Pi
=
and the fields
is + = $1 +
#z R-e
HIGHER
ORDER
SPINOR
derive Eq. (5-l) from the conventional gy5@, in [y”(P, Multjiplying
in on t’he left by [~“(p, [
Dirac
- igr5sP)
formalism
by replacing
ie14, by
- m]ys = 0.
- igy’+,,)
(p, + igr”@,) (p” - ig~w)
195
EQUaTIONS
-
(56)
+ m], we obtain
HZ2 - ; gy5ypysl””
1
# = 0.
(57)
We chose this form for the multiplier since it is the one that leads to a second order equation having ~‘9 as a solution. This implies that Eq. (57) can be written in two-component form of Eq. (54). Had we t,ried to develop a theory with pseudo-scalar mesons, we would have been led to a two-component form having the meson field variable appearing in the denominator in the equation of motion. Hence, for simplicity of form, we adopt, a pseudovector theory of meson interartion. A.
QUAN'~~M
ELECTRODYSAMICS
We now have a t(otal Lagrangian given by Eq. (50) for electromagnetic processes which can be written as Co + Cr. In turn sr can be broken down int#o t hrcc t)erms + Cr,(dv),
(58)
where we have split up J& into 4 and 7. The fields Cam contains bilinear t,erms of t’hat type. We perform plicit,ly that t,erms C1, and C I2 are met,ric invariant we mean that, cI,,, are invariant’ under tii ---) fi;’ = by Eqs. (121-i-14). This result places restrict)ions on &‘-matrix. These restrictions may be stated as: (a) Only transitions between st,ates of the same = CjCj(#i, #j) # 0 only for C; = Cj (b) All physical observables commut’e with t,he
in parentheses mean t’hat’ this splitting to show eswhile CJ, is not. By this r,tiir’i = cili, as defined a physically interpret,able
CI,(M)
+ Cr?(llv)
metric
are allowed:
metric
operat’or:
( tii’,
#j')
0’
=
r()h.c.r
=
0h.c.
(c) r commutes with S-mat’rix (St = Stl.r. ) (insures unitarity of S-matrix). In addition to the t.hree restrict,ions, we note t’hat if the generator is bilinear
then
SO = i[O, F]for any operator
0 remains unchanged
when an indefinite
metric is introduced.
+
FIG. 1. The solid line is a massive spin the dashed line is a massless spin one-half
one-half particle.
particle,
the
wavy
line
is a photon,
and
For 0 = Ntotnl this means t,hat the norm of a state is preserved and the Heisenberg equations of motion remain valid. Turning now to our Lagrangian (SO) or (.58), we see that to first order in e or g we may neglect ce13 due to condition (c). Further, for the scattering of massive spin one-half particles d: 12 would not enter as it cont#ains only massless fields. We are left with just, the Feynman-Gell-Mann field, 4. Sow to second order s (2) - P[:c,(x,) ::&,(x,):] - P[:cI,(i) ::c1,(2j: +
:Cr,(l)::C1,(2):
+
:c,,(l)::c,(2j:
+ :d31Z(l)::S11(2): (59)
+ :cI,(l) + CI,(l)
::~e,(2): ::c1,(2):
+ :~e,L(l) ::~c,,(2): + :c,(l)
+ :Cr,(lj
::JZr,(2):],
where : . .. : denot,es the normal product ( IO’). Metric terms 3, 6, 7, 8 and for scattering of massive particles, 4, .j leaving s (2) -
P[:e,,(i)
::cI,(2):
::s1,(2j:
The second t,erm above, after contraction would lead to diagrams in which there is (Fig. 1). If we normalize the metric parity to +l and the unphysical photons to not contribute to massive scattering. Continuing this procedure we see that’
+ :s6,,(i) of t,he a single of the 1, then
invariance eliminates we can eliminate 2,
::cI,(2):].
(60)
time ordered normal product, massless intermediate particle physical (transverse) phot,ons the second term in (60 ) does
up to any order, the q-field does not
HIGHFlR
contribute. as
ORDER
SPIA-OR
Thus, we need only consider &(#x$) b:=
w*j”*Tp’.
197
EQUATIOXS
[D,“D,*++S’S”BXDxqb
+ JZ~(+C$) which -
dD!D,*++S”4].
can be written r6l)
For elect,romagnetic processes we have Q, = D, and symbolically, ( I/D,DyT’“) acting on 4 bring down a term ( -l/m’). If we define Dp*++S” = d, D,*X” = ---in’+‘, and D,Z’ = iII-‘, (61) becomes
which is nothing more than the Dirac equation writ,ten in terms of tn-o-component spinors. This Lagrangian was used by Brown (2) and Tonin (3) to calculate quantum electrodynamical processes. Therefore, the restrictions placed on our more general theory by the indefinit,e met.& are equivalent to taking t’he limit 6 = 0 in Eq. (10). To summarize, in order to define a physically interpretable S-matrix in a Hilhert space of indefinite metric, we are led to certain restrict’ions. In essence, these conditions &ate that a linear operator 0 in the total Hilbert space is an observable corresponding to the observable 0’ if it acts on t)he vectors of the subspace of positive definit,e metric as a Hermitian operator. Although it2 was the q-field which forced us to enlarge t#he Hilbert space, it must be emphasized that one rannot’ just ignore t>his field (set t = 0) from the start. The reason is that from (58) t,here are terms containing the v-field which are metric invariant. The metric invariance of the S-matrix insures its unitarity and we would t#hus expect t’hat the q-dependent part would make a contribution to physical processes. However, we must also sat,isfy condition (a ), which states that transiGons between st!at,es of opposite met#ric parity are forbidden. This is t,he reason that the q-dependent term in (58) does not contribute. The need for all three conditions points out tb.e import*ant fact’ that the Hermiticit)y of an operator in the subspace of posit’ive definite metric does not insure its ohservahility. As Tonin has shown (S), the Lagrangian (6%) leads to results equivalent to t,he usual four-component case in the realm of quantum elec~~rodynamics. The equakn (61 ) also exhibits the full parity invariance as required. This is due t.o t,he symmetry between the interchange of Q w + in calculating matrix elements (i.e., Lmder I’:PD iv 4 and I%$ - a). Due t,o the general form for our total Lagrangian, the same results are obtained for pseudo-vector meson theory (eA, + ig+P). That is, the skong interactions are P-invariant and t,he v-field does not, enter into t.he results. We have thus the t,heorem t#hat t#he two-component, spinor theory with vector int,eract,ions
198
BAIZUT
L4PiD MULLEN
(@P-invariance ) and metric invariance implies P-invariance.3 If the q-field is associated with t#heneutrino, then its nonexistence in elect,romagnetic and strong interactions is explained, or, conversely, supports our hypotheses. B.
WEAK
INTERACTIONS
The second order t,wo-component spinor equation (1) was used by FeynmanGell-Mann to just#ify the V - A theory of the four field Fermi interaction. The assumption was that if all fermions are described by a two-component spinor then the weak interactions should be expressed in terms of them. The result, in terms of four-component spinors, was
JL - (WC 1 + r”Nd ($Yr(1 + YW
(67)
or equivalently -
WC1 - r”h”(l
+ r”MWl
- Y5hrU
+ r5M.
Using relation (15)) we may also write this as 6, -
G#J+w (4+%#4,
(69)
where of course C#I is the two-component spinor field obeying Eq. (1). In this approach the neutrino (m, e = 0) obeys the equation p,24+4 = p,qpx%w)
= 0
with phzh+( V) = 0. In terms of our #-field we can write (69) for massive particles as Go -
(aA~+S”z”Z”a”~) (aA#+?Z”z,Z’a,#),
(70)
since &XX+ = +. This is the simplest structure containing only the +-field. However, it is just a choice and does not depend upon any new symmetry rules other than metric invariance. Again, if the v-field is the neutrino, then (70) is good for nonleptonic decays. For c,, to be metric invariant when it, contains the v-field, it must contain 11in pairs. This implies transitions of the form + -+ + + 11+ 4, which may be able to explain pure leptonic decays. The structure of the Lagrangian is arbitrary for such a process so it remains to find addit,ional symmetry principles. There also remains the problem of finding the correct C,,, for leptonic strangeness conserving and violating decays which contain only one neutrino. IV.
SUMMARY
AND
CONCLUSIONS
We have quantized a single “composite” field given by Eq. (31) using a higher order Lagrangian ( 19). We chose this approach because (a) we had to plus
3 There is a counterpart of this theorem charge symmetry implies P-invariance.
in ordinary
Yukawa
interactions.
CP-invariance
HIGHER
ORDER
SPINOR
EQUaTIOKS
199
deal with only one field instead of three; (b) it led to a method of introducing electromagnetic and strong interactions which are equivalent in form and valid for all values of spin; and (c) it implies the concept that all massive spin onehalf particles are coupled to a neutrino particle. Whether the neutrino coupled to each massive particle is the same remains to he seen when we can uniquely write down the weak interaction Lagrangian. But for sbrong and electromagnetic interactions tbe resbrict’ions placed on the S-matrix by t,he indefinit,e metric uniquely determine the int,eractions and imply P-invariance. Thus we cannot get the answer from the electromagnetic and strong interactions since the physical parts of the Lagrangians in t’hese cases are not a function of the s-field. The significance of t!hese restrictions is that it is a necessary but not a sufficient condition that an observable operator in the total Hilhert space act as a Hermitian operator on the vectors of t,he suhspare of positive definite metric*. -4lt’hough this insures bhe diagonalizahility of t’he operat’or, we must. also include the conditions t$hat only t)ransitions between states of the same metric parity are allowed and that the S-mat#rix is unit,ary and not pseudo-unitary. Finally, we add that the successful quantization of the FeynmanP6ell-Mann field, performed here, suggests several int’eresting dire&ions of investigation. One, as already stated, is to find a unique weak int(eraction structure. Another is a more detailed investigation of t,he concept of metric conjugation and metric parit’y. The success of our theory seems t)o put, t’he role of the indefinite metric in yuantSum field t,heory on a firmer footing. It’ may thus prove fruihful to work out, a formal theory of scattering using an indefinite met’ric. We finally remark t,hat although the usual proof of the TCP theorem depends on the spin-statist,ics connection, which in turn depends on the positive definite metric, the spinstat#istirs theorem can be proved in most cases witIhout’ the assumption of a positive definit.e metric from t’he action principle il7), or within the framework of S-matrix theory (18). APPENDIX
I
In this appendix we discussbriefly t(he invariance properties of Ey. ( 1). We consider first the Lorentz invariance of this equat)ion. Equation (,1) must have t,he same form in every frame, hence
(AS) Lorentz invariance demands that (A.2)
200
RARUT
Ah-D
MULLEN
Using the fact’ that
(AA) relation
iA.2)
reduces to [LP, uPv] = gavaPa _ g”“uPP + goPuav _ gwuBY*
A solut,ion of (A.4) exists under the proper Lorentz We now show that Eq. separately. Consider first, x’ = -x,
(A.4)
with 2s”’ = CJ~’ and therefore Eq. (1) is invariant group. (1) is invariant under PC and T but not P and C’ the PC transformation defined by
P’ = -P,
db’)
= w%*(
-xl, (A.5)
A,’ = -g,,A’(
-x),
Flj = -Fij(
FL = Fed-x),
-x),
which gives [(p, + eA,,)2 - rn2 + em”OiFo;- ea”jFij](PC)+*(-x) For Eq. (1) to be PC invariant, [(p,
+ eA,)”
-
i.e., (A.6)
= 0.
(A.61
equal to
,m” + moi*Foi + ecrij*Fi&*( -x)
= 0,
(A.7)
we must have that (pc)-‘,““(p~~)
= aoi*
and - (Pc)-$TiqPC)
= (Tij*.
These last two relations reduce to [(PC),
n31+
=
0,
[(PC), u,]- = 0.
(A.81
A solution for (A.8) is PC = CQ. Therefore Eq. (1) is PC invariant with &x’)
= u&*(-x,
x0).
An analogous procedure shows us that Eq. (1) is also T-invariant &z’)
= u2+*(x, -20).
(A.91 with (A.10)
We next postulate that the gauge and Lorentz invariant way of introducing electromagnetic interactions into Eq. (14) for a particle of arbitrary spin is given by (19) a,#’ + D,D,T”*
(A.ll)
HIGHER
ORDER
SPINOR
EQVATIONS
201
with D, = 13~+ ieA, . T’” depends on the transformation properties of the $-field manner under infinitesimal Lorent, transformations &‘(S’) This is equivalent
= ,l~‘&“T:,‘(bb(.ll).
in the following (A.12)
to the usual notation
&iv = gav + +A” Finally
we may writ’e (D,D,T;;:
+ WL%,,&~(.L.) = 0.
For a one-component field (scalar), This leads to the usual Klein-Gordon [(a, + id,)”
(A.15)
we have S”’ = 0 so that TiR = gp”sab . eyuabion for spin zero particles, + m$(.r)
= 0.
(A.16)
For the two-component case, as was said previously, t’he proper Lorentz group admits a two-dimensional irreducible representation but not the full Lorent’z group. This implies parity noninvariance of the equations of motion and is in keeping with weak interact’ions. In this case, we found above t,hat 2S’” = /” and therefore D,D,T”” = D,” - (~/‘3)d“‘F~~ . Using Pa = iD, we obtain Eq. (1). We might just point out that had we considered t#he four-component case, we would be led to the second order Dirac equation which is invariant under the full Lorentz group. This in turn implies P-invariance. I~ECEIVED:
February
28, 1962 REFEREKCES
1. R. FEYNMAN AND M. GELL-MANN, Phys. Rev. 2. L. M. BROIVN, Phys. Rev. 111, 957 (1958). 3. M. TONIN, Nuovo cinlento 14, 1108 (1959).
4.
W. R. THEIS, Portschr. Physik 7, 559 5. A. PAIS AND G. E. UHLENBECK, I’hys. 6. J. W. KIBBLE AND J. C. POLKINGHORNE, 7. G. MARX, Nuclear Phys. 9, 337 (1958). 8. P. A. M. DIRAC, Conamu~~. Dublin Inst. 9. W. PAULI, Revs. Modern Phys. 16, 175 (1950); R. ASCOLI AND E. MINARDI, J. PLEBANSKI, Nuovo cimento 18, 884
109, 192 (1958).
(1959). Rev. 79, 145 (1950). Nuouo cimento 8, 74 (1958). Advanced (1943); Nucleav
(1960).
Studies 8. N.
Phys.
A, No. 1 (1943). Proc. Phys. Sot. 63, 681 9, 242 (1958); H. M. FRIED AND GUPTA,
202
BARUT
AND
MULLEN
10. See, for example, the review paper by K. NAGY, ;V~touo cimento S~cppl. 17, 92 (1960). 11. E.C.G. SUDARSHAN, Phys. Rev. 123,2183 (1961); E. C.G.SUDARSHAN AND H.J. SCHNITZER, Phys. Ret>. 123, 2193 (1961); R. SPITZER, preprint. 12. A. 0. BARVT, to he published. 13. A. 0. BARUT, Ann. Phys. (N. k’.) 6,95 (1958). 14. A. 0. BARUT AND G. H. MULLEN, Ann. Phys. (X. Y.) 20.203-218 (1962). 15. For a more detailed discussion of this point see J. SCHWINGER, Phil. Mug. 44, 1171 (1953).
16. W. THIRRING, “Principles York,
of Quantum
Electrodynamics,”
1958.
17. R. ARNOWITT AND S. DESER, (to be published). f8. H. P. STAPP, Phys. Rev. (to be published). 19. C.G.BOLLINI,NUOUO cimento 19,560 (1959).
p. 51. Academic
Press.
Sew