- Email: [email protected]
The undor equation of the meson field
Physica
VI,
no 9
THE UNCOR
October
EQUATION
1939
OF THE MESON FIELD
by F. J. BELINFANTE
Zusammenfassung Die Meson-Gleichung von P r o c a, Ii e m m c r und B h a b h a wircl mittcls Lincloren zweiter Stufc dargcstellt. Einc \‘erallgcmeinerung der Gleichung fiihrt zu ciner ncuen Meson-Gleichung, welchc im wcsentlichen aus ciner Kombination clcr Fallc (b) uncl (d) van I< e m m c r bcstcht. Die Ncutrerto-Glcichung wircl in ahnlicher Weise erwcitert. - Dns magnetischc Moment dcr Mesonen wird abgeleitct. Die ladungskonjugierten Wellenfunktionen geniigen einer Gleichung, in wclchcr die Vorzeichen aller Ladungen umgekehrt sind. Wenn man postuliert, class sic11 bci Beschreibung des physikalischen Geschehens mittels der ladungskonjugierten Grossen fiir alle physikalisch sinnvollen Grossen dieselben \Vcrtc ergcbon sollen, lasst sich folgern, dass Tcilchen mit ganzzahligem Spin der E i n s t c i n-B o s e-Statistik, und Teilchen mit halbzahligem Spin dem Ausschlicssungsprinzip geniigen miissen.
Resumo. La mezona ekvacio de P r o c a, Ii e m m e r kaj B h a b h a estas prezcntata per du&tufaj undoroj. Pligeneraligado cle la ekvacio donas novan mezonan ekvacion, kiu eefe konsistas el kombinajo de la lcazoj (b) kaj (d) de Ii e m m e r. La neutreta ekvacio estas plivastigata cn la sama maniero. La magneta momanto de la mezonoj estas kalkulata. La Sarge konjugitaj ondofunkcioj kontentigas ekvacion, en kiu la antatisignoj de ?iuj Sargoj estas inversaj. Se oni postulas ke EC prislrribo de la fizikaj okazajoj per la Sarge konjugitaj grandoj por eiuj observebloj devas rezulti la samaj valoroj, oni povas konkludi ke korpuskloj kun entjera spino clevas obei la statistikon de E i n s t e i II kaj B o s e, kaj ke korpuskloj kun entjcrplusduona spino devas obei la statistikon de 1; e r m i kaj D i r a c.
S 1. l‘hc P I’ o c n-Ii: c ~1 ~11c I’ meson cquntioe in wdov notation. The usual meson equations of I< e m m e r l), B h a h h a “) and Y u k a w a “) can be written in the following form: 4$LY - lo ?~,u) = DC, yvl = Dp yv - Dv s+ > +
+ g, v,) = Dp rwLv; (P,V = 0, 1,231. -
870 -
(1)
THE
UNDOR
EQUATION
OF THE
MESON
FIELD
f37L
Here x can be expressed in terms of the mass m of. the meson by x = m/h; UP” is an antisymmetrical tensor and V, is a four-vector giveri by *) :
By +N and +p we denote the wave-functions of the fields of neutrons and protons, so that after superquantization the expressions (2) represent operators which possess non-vanishing matrix elements for transitions of a proton into a neutron. The operators D, in (1) are defined by [U.C. (3)] :
We shall now write the P’r o c a-K e m m e r equations vector notation. For this purpose we put 7;,,” = co, = E,;
cbc = -
ccb = H,;
rp. = A,;
-
cpo= ‘PO= V;
(u, b, c being a cyclic permutation Further,
(1) in (4)
of 1, 2, 3).
we put
U oa =
u
ao =
e IO
-
%‘bc =
u,b
=
h,; v,,=a,,
vO=v,
(5)
v = $+#.
(6)
so that ; = -
++;p&,
Tt = -
Then, the equations x(f6 + fb2) = - ih x(?i + fb $j =
a = #+&,
++&J;
(1) read t) -
(a/cat) ;li: - (e/i%) (jib -
&),
rot-)A + (e/ihc) &x],
x(x + gb i$ = - rot ii + (a/cat) g x(v + gb v) = - div if-
(e/h)
VI (e/i&c) ([if, ii) + $i$,
(ii. i$.
In order to write these equations in undor notation we can make use of the K r a m e r s representation of undors [U.C. (28)], in *) For the notation used in the present paper we must refer to the preceding paper of the author on under calculus 4). References to formulae from that paper will be indicated by [U.C.]. In (2), (6) and (12) we have put % = 1 [U.C. (17)J. t) rot =-curl.
872
F. J. BEiINFANTE
which the D i r a c matrices ;;‘, p occurring
in (6) are given by
(8) and the charge-conjugated given by
tp = ep,
$” of a D i r a c wave-function
f =
0 0 0 -1 001 0 010 0’ -100 0
4 is
(9)
In this representation the components of the tensors represented by an undor of the second rank are related to the components of this undor by [U.C. (54), (58), (34n)]: F,-iF, -F,,-FF, 2X&,
=
F,--FF, -F/--F,,
L, - iL,
K,-iK, K”-
LO -I L,
- LO - L,
-
-LL,--iL,
-K”-KK, K,
-K,--iK,
G, + z’G,
-Go+G,
Go + Gz G, + iG,
ii=d+g,
KO=V+W,
-$ =i%-z.2,
F, =-S----Y,
&x-s,
L”=V-W,
&g+%,
G,=S--271.
; (10)
Here S is a scalar, 2, V a four-vector, z, d a six-vector, 2, W a pseudo-four-vector and Y a pseudo-scalar. In particular the undor of the second rank Xk,k,
represents, according --f
=
29f,
+k,
to (lo), the following
a = ‘ptZ+,
vzcptt$;
;=
ii- --&c+.
w=-+y&
y=qAy&;
- cpq3&,
(11)
tensors iL
- cpt p&;
-s=‘Ptp(CI.
(*2)
THE
UNDOR
EQUATION
OF THE
MESON
FIELD
873
Here we have put a, = -
cycl.; ys = - i&a+.
iga,,
From (10) we see that the P r o c a fieldx, sented by a symmetrical undor ‘%:k.
Writing
=
4 {‘rk,k,
+
yk,k,>
out the meson equations
=
(13)
V;%, g can be repre-
4 y{k,k,}.
(14
(7) with the help of (10) - (<2) in
terms of the undor components of \r@, and of sym X klk, = +%k, bk, + +%k, bk, = i&k, bk,J J and collecting the ten equations on account of (8) and (13) :
f@= 4 {(fb
+
gb)
(144
into one undor equation,
(fb -
+
gb)
we find,
Y!‘$i2’)
(16)
is a scalar operator which, operating on an undor of the second rank (lo), multiplies in this undor $, H’, Y and S by fband 2, V and B’, W by g,+ By an index in brackets we have here distinguished D ir ac matrices operating on each of the indices k, and k, of the undor. We remark that (15) and ( 16) are invariant by transformation to another representation of undors “). Putting
Dkz =
0
0
0 D&D,
0 -Do-D,
G---D,
-D,-
D,--iD, -Do--D, 0 iD,
0
we can write (15) at once in covariant [V.C. (821, (lb)]):
Do--D, -D&D, 0
=h
(17)
0
undor notation
“) (compare
2i$f;,k, + fk&,ld’ ‘E,J,~., + Dkll, Fk, + Dk,!, qk: = 0, where fk,k,l’l’are the matrix elements of the scalar interaction ator fop(16).
(18)
oper-
a74
F. J. BELINFANTE’
This interaction operator would take a particularly the constants fb and gb should happen to be equal: /b
=
simple form, if
gb (?I
Q9)
A preliminary interpretation 1) 5, of the binding energy of the deuteron in the (triplet) ground state and the attraction potential in the singlet state,, on which experimental data are available, seemed to indicate that 1 fb;lgb 1 did not differ much from unity indeed, but later investigations B), which accounted for the charge-independence of the nuclear forces’ between ‘heavy particles ‘), gave a different result. The simplification (19) seems, therefore, not to be allowed for the moment and we shall not make use of it in the following. 9 2. The generalized lneson equation and the neutretto equation. In the preceding section we have discussed the equation for the P r oc a - K e m m e r meson field: (case. (b) of K e m m e r 1)). This field SYm
was described by a symmetrical undor ‘I!R,R, and it interacted with the symmetrical part s?h,k, of the undor XR,k, = 24$,., +pk . It is plausible now to consider the generalization of the eqdatidns (15) and (18) by replacing these symmetrical undors sqk,k, and s??ksR,by the unsymmetrized undors Yh,L, and Xk,R,. The operator fopcan then be replaced by a more complicated scalar operator which, operating on an undor of the second rank (lo), multiplies S by f,,; x, V by g, ; ++ E, H by fb, g, W by gd, and Y by fd. The generalized meson equation 2i4%,k,
+ fopG%%&J~ + GT3 +
can then be written,
* o’, + (p"'
+
p"')
DO)
‘r,
k 18
=
0
(20)
according to (a)- (13), in vector notation: .
-++fos)
.=o;
x(X+gbz)= D,Z-[S,zij, x(V+av)= - (5.Z);(214 x(2 + fbi$ = -D&-V,
x(iTi+f$j
=[~~];
x(W+g,w)=-D,,Y,
~x(T&Jj
3Y; (214
x(Y+fdy)
=D,W+(T;.$.
THE
UNDOR
EQUATION
OF
THE
MESON
875
FIELD
Apart from the first equation (210), which defines the field component S in terms of the components of the field of heavy D i r a c particles, these equations represent exactly the cases (b) and (d) of K e m m e r, that is, the meson field suggested by M 0 11 e r and R o s e n f e 1 d “). The P r o c a field (b) describes mesons with a spin angular momentum A, whereas the “pseudo-scalar field” (d) describes spinless mesons 9). According to K e m m e r 6) the neutretto equation is obtained from the meson equation by changing +z #p into +(#z +p - $z +N) and Dp into V, ; and by postulating that the- tensors representing the neutretto field @‘k,R,shall be real. Thus (20) changes into 2M%,k,
+
fop
Mk,
with the additional
4JPk,
-
d&,
condition
4$l)>
(Yj? + Yj?) V” @k,k,= 0 (22)
+
(see [U.C. (64)]) that
CDk,k. = fit’) %(‘)(@k,k,)* = @f,k, = @f;,k, is a neutrettor of the second rank. The compatibility of the condition (23) must be shown. For this purpose we multiply of (22) by - P(l) G2) and find, on account [U.C. (81)] and 5?) Et21(@k,k,)* = @&, [U.C. 2iX{@fsk,
+
g(1)g(2)
or, interchanging 2ix
@k,
+
%
f$(@k,bks
-
$%k,+Nk,)*)
(23)
with the equation (22) the conjugate complex of - 9”) yt)* = yt) $9) (63a)]:
+
(YG”
+
y,!?)
vp@;xkl
=
0,
k, and k2: (@kl
bk,
-
%k,
hk,)?
+
(Yj?
+
$‘)
VP %k,
=
O,
(24)
where we have put (compare U.C. (40)): f$ = .gl) .jp
ff$ .jy* f(l)*
,
(25)
so that
(fopyk,k,)’
=
t$
yf,k.**
(26)
Now foponly multiplies the tensors represented by \rk,k, by the constant factors fo,&, fb,g, and fd,whereas, according to [U.C. (63), (63a), (64)], charge-conjugation of an undor of the second rank changes the tensors S ; A’, V; $, g; 5, W and Y, represented by it accordx*, V*; s*, g*; -s*, _ W* and - y*. ing to (lo), into -S*; We conclude therefore from (26) that & is the scalar operator multi-
876
F. J.‘BliLINFANTE
plying the tensors by the conjugate complex of the factors fO, g,, fbr g& fd. Now, asstiming that these constants are real
fo = f:,
gb =
k?>
fb =
fb*sg;l
=
$8
fd =
f%
(27)
we find
f$ = f*p If we make use of the fact that Jc$ &, is a neutrettor second rank [U.C. (63b)], the equation (24) turns into
(274
of the
+ fopN”p,,bk, -
@, hi k,)} + (y;’ + y!‘) VP (P&, = 0. (28) This equation for a& is identical with the original equation (22) so that the condition @k,k, = @Ek, (23) is indeed compatible for @k,k,r with (221, if the interaction constants (27) are real *). W@&,
§ 3. The charge-conjugated meson eqzcation. K r a m e r s 10) has shown that if 4 is a solution of the D i r a c equation for positive particles, then 4” is a solution of the equation for negative particles, that is, of the equation following from the D i r a c equation by changing e intd (- e). In the present section we shall show that the meson and the neutretto equations possess similar properties and that, if Y is a solution of the equation (20) for positive mesons (“theticons” t)), then ‘P [U.C. (63)] is a solution of the equation for negative mesons (“‘arneticons” t)) . For this purpose we proceed in a similar way as in the preceding section ; only this time we shall not interchange the indices k1 and &. In this way we derive from equation (20) : w3%,
+ 2Ep (@k, +Nk,)) + (Y:‘) + yjL2)) D*r yflk, = 0, (29)
where we have made use of g(‘) Ec2)(qfl +k,)* = Ec2)+;z . !$) 9:; = $2, (Pk,.
(30) Comparing (29) with’(20) we observe that \r&, satisfies an “arneticonic” equation (29) differing from the “theticonic” equation (20) for y&k, by the inversion of the sign of e (as D is replaced by D*) and by the change of
fOtJ(+8,,b$ into f$ (@k,hk,)’
(31)
*) If neutrettors of the second rank were defined by CD = W, instead of by @ = Qg, would have been necessary to take gb and fs real, but fO, g,j and /d purely imaginary. t) These names arc derivedlrom ~ETLXO~ = positive and b?vqTrxo; = negative.
it
THE
UNDOR
EQUATION
OF THE
MESON
FIELD
877
The electromagnetic potentials, being real, are not changed. We must now remember that the interaction of mesons with heavy particles, as described by the equations, should consist in the possibility of the absorption of an ameticon or the emission of a theticon by a proton which changes into a neutron, and vice versa, That is to say, the wave-functions in the equations (20) and (29) should be superquantized *) . If the wave-functions of anti-protons and neutrons (@ and.+N) are assumed to be a&commutative with each other (an assumption which simplifies the discussion of the canonical theory of quantized wave-fields and which enables us to introduce the formalism of the isotopic s#Gz in a natural way), we can express (31) by stating that, in order to change the equation for Y into that for ‘I?, not only should ‘I! be replaced by ‘l? and the electric charge e by eL = - e* > (32) but at the same time the “me.&” replaced by
charges
fo, & etc. should be
ft = - fo*,gt = - gb*,ft = - ff, etc.,
(324 and 4% by JrN and +p by @. There is no need to change the potentials of the M a x w e 11 i a n field occurring in the meson equation (20). This field can be described by a symmetrical neutrettor of the second rank, so that it is equal not only to its own charge-adjoint, but also to its charge-conjugated. In the same way the neutretto equation for Q, is changed into the equivalent equation for Q&,. This is seen at once by interchanging in (28) again k, and k, and by making use of the anticommutativity of 4% with Jllp and of $5 with dCN.Formally the infinities of the &functions from the commutation rules of protons and,of neutrons cancel each other. $4. The charge cwrelzt-density and the magnetic momelzt 01 mesons. Pro c au) and B h ab h a2) have derived the electric charge density and current of mesons with a spin A from a L a g r a n g i a n, which was chosen in such a way that the P r o c a equations and ‘the equations for the M a x w e 11 i a n field could both be deduced from it. *) The relativistic
question of the possibility invariance ofthe theory
of quaotization of the fields is maintained, is not discussed
in such a way that the in the present paper.
a28
I?. J. BEiINFANTE
In a similar way one can proceed for the generalized meson field “). If the field is normalized according to K e m m e r l) in such a way that x* , - z*, Y* and - W* are the canonical conjugates of?, 6, W and Y respectively, the expressions for the electric charge density and current take the following from “) la) : cp = (e/S) {(X* .S) - (5* . X) + Y* W - W* Y} = eY+popY, eT/c = (e/i%) {[ii*,
ii] + [ii*, Xj +
Here pop and&/C
are determined
(33)
by *)
or, in tensor notation,
These expressions are invariant by a transformation [U.C. (14), (14a. b)] from the K r a m e r s representation to any other representation of undors. The rnatrices.~,~ and TO& take the place here of the matrices 8 and 9: (that is, 8f3yp) in the case of the D i r a c electron. In analogy to [U.C. (20b)] it is convenient to write here (see [U.C. (ao)])
+ y(2)p Y* It,,= y+ y(‘)p 2h
(334
The main difference between the density matrices of electron and meson is that pop, being a singular matrix, cannot be made unity by transformation to any representation. A consequence of this singularity of popis that the meson equation (20) contains so called identities between the field components (differential equations not containing *) Compare L. d e I3 r o g 1 i e, lot. cit. Ia), page 22. The factor (J/21) is a consequence of I< e m m e r’s I) way of normalizing the meson wave-function and of our choice of the constants in (10). For instance, if the factor 2 in (10) is removed, the factor (1/2R) chariges into (l/88) I*).
THE
UNDOR
EQUATION
OF
TJ-LE
MESON
FIELD
879 .
derivatives with respect to the time), vii. the equations in the right hand column.of (21). : We can split up (20) into the proper equations of motion and the so-called identities by operating on it by (1 f p(1)p(2))/2. Abbreviating again by XI,R, = 2$$-h,$pRt and splitting up Yk,k, and Xk,k, according to :. %,k, = %,k, + %fk, 3 (35) \rz = 4 (1 + p(l)p@)) Y, Y” = Q( 1 - P(l)@(2))Y, we find the proper equations of motion of Y1: 2ix (Yr + fop XI) + (y(“fyiz,
.%) Yrr + ((8’) + (3t2’) D,Yr = 0, (36)
and the so-called identities: 2ix (YII + fop XII) + p-q) Comparing
. 5) YI = 0. .(36), (37) with (21) we observe without further
tion that YI represents the field components
(37) calcula-
2, 2, W and Y, and
YII the field components S, 6, z ands. Only of YI the components can be regarded as canonical .variables, whereas the components of Y” must be regarded as derived variables, defined by the equations (37) (like .I$ = rot % in quantum-electrodynamics), If for the present the interaction of mesons with heavy particles is neglected; the. meson equation (20) takes the form (2ix + r, DI”) Y = 0, putting we find
yi’-y
(I?, = yp + yf’).
1\2)= l?* and operating
(38)
on (38) by (1/2ix) . Pi-)@
l?i-) D” Y + (1/2ix) Pi-1 l?; DA Dp Y = 0.
(39)
From ytl y:;’ c yp y:“’ + y:“’ yj;” = _ &+
(40)
,we find
rip rpl = 0,
(41)
so that from (39) follows :l?ii) DA Y = (i/8x) . (IT-1 rp -
I?;-) -PA)(DA Dp -
Dr DA) Y.
(42)
880
I;.
From the definition
J. BELINFANTE
(3) of Dp follows
Dp D” = DLA Drl = (c/&c) . Vrx W‘l = (e/i8c).@,
D” Dr -
where $& denotes the M a x we 11 i a n field. The equation therefore be written in the following form: (c/~~&c).@~~ l?;x) rrl Y = (c/4mcZ).@
r;-) DAY=
(43) (42) can
FL+ rPY.
Adding (44) to (38) we find the following “equation of motion” from which aY/at can be solved by multiplication by p(i) : (ix + y;’ DA) Y = (c/8mc2) . $j+ I’;-’
(44) for Y,
rr Y.
(45)
The left hand member has the form of a “D i r a c equation” for the first index k, of the undor Y. If this equation is iterated like the ordinary D i r a c equation, we find (compare (40) and (43)) : (c/8mc2) . (- ix + y;) DP) (Q”c” r;-)
z
{x2 + 1 y(‘) 3
0
y(1) D{" P')
DP)
rp Y) = 1 (1) (1) D[h 3 YP YPl
+
= {(m”c”/tz’) - D, DA -
DPI}
4 id’),,,, . (e/W
\r
=
@-‘“>Y,
(46)
where we have put Y[A y,4 = - 2ki,. If only a magnetic
(47)
field is present @ = 0), we have
.$ ~1;;’ $j”P s G(“) . $j - i$?‘) .g
= ($. sr)).
(48)
Adding to (46) the corresponding equation with c$ (where in the left hand member l?l;-) occurs with the opposite sign), we find after multiplication by A2/2 y = {m2c2 + p,jpx - (ctz/2c) . (G . o”‘32’)) = (eA2/16mc2) .
r;-) Dp(gjAp q-j yr),
(49)
where = W) . DA is the operator of the kinetic momentum. In non-relativistic proximation we put cp”=mc2+T=cp,, PA
(50) ap-
(51) so that T is the operator of the non-relativistic kinetic energy. If, further, according to Y u k a w a 3), the right hand member
THE
UNDOR
EQUATION
OF THE
MESON
881
FIELD
I’(-) I’l;-) I’ DP(,@‘Y) . (eFz2/16mc2) of (49) is neglected in non-relat&&tic ap:roximation, this equation can be written as a S c h r 6d i n g e r equation (T Q mc2) : (Ey-“j
-
e%) Y? z TY =
= {&J2m so that the magnetic
-
moment
@A/2 mc) . $ (g(l)+J2)
.$)} Y,
(52)
of the meson
-+
pop = (c/2mc) . Q A (F+J2))
= (e/2mc) . Z&l
iw
is (e/2mc) times its spin angular momentum “) -+N S,, . Since the energy is given by “) E = JYt popE, Y, the non-relativistic x4) value of the magnetic moment is actually given by + P=/‘)r+PopSZopy. (54) As the value of the spin angular momentum is given in the same + way by “) S = $ Y+ pop3S, Y, the statement + p. = (e/2mc) .Z (55) holds for the values of these quantities as well as for the operators occurring in (53). 3 5. Charge-invariance and statistics. It is well-known that in the hole theory of electrons (superquantized theory of the D i r a c electron) there is an infinite c-number difference between the qnumber e++lC) (obtained by superquantization of the wave-function I# from the expression for the electric charge density e$tJr following in the usual way from the L a g r a n g i a n of unquantized wavemechanics) and the q-number representing the correct (observable) electric charge density. If the meson field is quantized, (33) must also be corrected by addition of infinite c-numbers. We have mentioned that to one description of D i r a c particles, mesons, neutrettos and the electromagnetic field by undor wavefunctions [U.C. (l)], (20), (22) there is an equivalent charge-conjugated description, in which some constants like e, f and g are replaced by cL, fL and gL’ (32), (32z), whereas every quantized undor *) is replaced by its charge-conjugated [U.C. (30)], (29). This suggests a kind of *) We assume that all fields (34)]) and not by “quasi-undors” Physica
VI
are described (reflection
by undon 4’ = f3#).
(reflection
$’ = ip#,
[U.C.(9), 56
.
a82
I:. J. BELINFANTE
symmetry between both ways-of describing physical situations *). By way of hypothesis one might assume that such a symmetry is a fznzria/~~c~ztal$roficdy of mturc. \\:e shall call this possible property the “ckn~Sc-ilzvn~iancc” of the physical world (not to be confused, however, with the principle of conservation of electric charge !). Therefore we shall postzrlntc that every physically significant quantity in quantum-mechanics (that is, every q-number cot~ctl~r representing the value of an obsr~ablc) is invariant by transition from one description of the fields of wave-functions to the chargeconjugated description, or, in shorter terms, is chaugc-ilt~)avirrllt. This postulate can serve to distinguish between wave-mechanical expressions, which after quantization cannot have a physical meaning any longer, and other analogous expressions, which may represent observables. For the present we shall leave this question out of consideration, but we shall show here that the postulate of chargeinvariance implies directly that photons and neutrettos must bc neutral, that D i r a c electrons must obey 17 e r m i-D i r a c statistics and that mesons ,mrst obey E i n s t e i n-B o s e statistics. The interesting fact is that this statistical behaviour of particles and quanta follows much more directly from the postulate of chargeinvariance then from postulates concerning the positive character of the total energy of free particles or quanta 7). From the L a g r a n g i a n of any kind of particles or quanta we can always deduce expressions for the electric charge density, the electric charge current, the total momentum ancl total energy of these corpuscles. The terms of the L a g r a n g i a n function depending on the derivatives of the field quantities ‘I” have always 3) the form of “) :I< Y+ B F Vp’\Ip. (56) If ‘I” is an undor **) ‘I!k,k~,,,,kSof ?ank AT, then [U.C. (12)] : !Y N B = B+ = II p”“; rP = C E,,y”“, (E,, = -c 1) ; B*I’* = l?BN; (57) II= I n= I so that, if we put .v \y‘” = & y* ) : zzz20” Z n p); s* .t: zzz1) (58) 11= I
THE
UNDOR
EQUATION
OF THE
MESON
FIELD
883
.
we have B d = (-
l)N E B*,
r % = - 32I?“.
(59) From (56) we find that the electric charge density, if it exists, is equal to ep = (eK/Ac) . ‘I? B PY, (- infinite c-number). (60) In the charge-conjugated description this expression is turned, on account of (32), into ,Lp= = (- eK/Ac) . ‘I? B I?’ \rj! (- infinite c-number), (61) therefore,
on account of (57)-((59)
:
eLpL = (- l)N. (eK/&c) . ‘Pl?‘B”Y*
(- infinite c-number).
(62)
If the expressions (60) and (62) for the electric charge density are postulated to be equal, the components of the wave-functions Y and ‘I!* occurring in (60) and (62) must be co~fimutative (apart from an infinite c-number term) if N is even, and must be anti-commzttative if N is odd. It is not true, of course, that the commutation rules follow rigovoatsly from ep = eLpL,
(63)
since in (63) the sum is .taken over the undor indices, and only‘the wave-functions ‘I! and Y* in one and the same point of space are multiplied with each other. In this case the a-function appearing in the commutation rules becomes infinite; its value corresponds formally to the sum or the difference of the two infinite c-numbers in (60) and (62). S ince the infinite c-number in (62) must be the chargeconjugated analogon of the infinite c-number in (60), this may be of some help in the “evaluating” of such infinite c-numbers. For photons and neutrett0.s it follows from (23) and from the symmetry of the operator pop ‘v&h respect to both undor indices, on which it operates, that p 3 YP+po,\r = ‘P+ popY” = v+ popY$ = pL.
On the other hand we find from (63) and (32) for any particles quanta pE -
pL.
(64)
or
(65) Comparing (64) with (65) we conclude that the electric charge density of the fields of neutrettos and photons must vanish, if it is
884
F. J. BELINFANTE
a charge-invariant expression. In a similar way we derive that by means of neutrettors of the first rank only neutral patricles 17) can be described. It does not follow from this, however, that neutral particles should necessarily be described by neutrettors ! For electrons we deduce from (65) that p = qJ+qJ -
c
166)
must be opposite equal to pL = ptge
-
c=
where CL takes the place of the infinite c-number conjugated description. From this we deduce ++9
+ +*+9*
= ++9
+ wAJ*
(664 C in the charge-
= F {(qJk)*+k + +k(w*>
=
= C + CL = c-number.
(67)
Similar relations can be deduced by postulating the chargeinvariance of the quantized expressions for electric charge current, total momentum and total energy. For instance, from
= i (C -
CL) = c-number.
(68)
It is obvious that relations like (67) and (68) are consistent with the anticommutativity relations of F e r m i-D i r a c statistics +k(x)* &&‘I
-I- +kr(x’) +)k(x)* = hr 8(x - x’),
(69)
but not with E i n s t e i n-B o s e statistics. In a similar way we find for mesons from p = yt P(I) & P12) Y-C
(70)
and pL =
yst
P(')
&
P2)
yf.
_
CL
(70n)
THE
UNDOR
EQUATION
OF THE
MESON
FIELD
885.
that y+ p(l) + pt2) y + yBt P(l) + P2) ys = c + CL* 2A 2F, Applying (57) -(59) we find yt
(71)
P(‘) + Pt2) \Y _ P(l) + PC’)y w \Y* = c + CL, (72) 28 2tz ) ( ) or, on account of (35), wt (Poti‘y’) - (POPY’ )m Yr* = C + CL = c-number. -V3) Again, in (73) the sum must be taken with respect to the undor indices, as in (67) and (68). It is obvious that (73) is consistent with E i n s t e i n-B o s e ‘commutativity relations between the components of Y’ and Y’*: Y:,k,(~)* YirlR,,@‘) - Y:,l,,,(x’) Yi,k,(~)* = c-number, (74) and not with F e r m i-D i r a c anticommutativity relations. The commutation rules for the components of Y” must be derived from those for Y’ by means of the so-called identities (37), so that it is not very alarming that we do not find any indication of them from (71)-(74). For neutral particles, indication of the commutation rules can be derived in this way from the expressions for the total momentum and the total energy, which are also obtained directly from the Lagr an g i a n. Generally we can postulate that the total L a g I a 1zg i a rt itself (integrated over space and time) shall be charge-invariant on account of the commutation rules of the field components. It is therefore not necessary to investigate the sign of the elzergy in order to derive the statistical behaviour of the corpuscles concerned l5) 16). It is true, however, that charge-invariance of the quantized expression for the total energy implies that by quantization according to the scheme of P auli and We is s k o p f 18) the so-called “states of negative energy” of free corpuscles (depending on the time by a factor e+lrri”l) can be interpreted, on account of the commutation relations (which do not need specification here !), as states of positive energy *) of corpuscles with opposite electric charge. We can understand this in the following way. By charge-conjugation of the quantized wave-function these states pass into charge-conjugated states of positive energy. If, now, the expression for the total energy is charge-invariant on account of the (un(
*) For the corpuscles
under
consideration
states
with
e-2nW
are of positive
energy.
886
THE
UNDOR
EQUATION
,OF THE
MESON
FIELD
specified) commutation rules of the q-number amplitudes a (J o rd a n-W i g n:e r or J o r d a n-K 1 e i n matrices), the terms in this expression arising from the so-called states of negative energy are automatically equal to the terms in the charge-conjugated expression arising there from states of positive energy of the charge-conjugated corpuscles (which are described with the help of the chargeconjugated q-number amplitudes b = a*). Using the latter (chargeconjugated) expression for the description of these terms in the total energy, the energy is given as a sum of only positive energies with amplitudes a*a or b*b. We observe that both the statistical behaviour of corpuscles and the possibility of describing so-called states of negative energy (of free corpuscles) as states of positive energy of charge-conjugated corpuscles follow directly from the postulate of charge-invariance of quantum-mechanical theories. The relation between the positive character of the energy of free corpuscles’ and the charge-invariance of energy seems to be still closer than that between charge-invariance and statistics. I wish to thank Prof. Received
July
15th
K r a m e r s for his interest in this work.
1939. REFERENCES
I ) N. 2) 3)
K e m m e r,
Proc.
roy.
Sot.
H. J. B h ab h a, Proc. roy. II. Y u k a w a, S. S a k a t a 20,319, 1938. 4) F. J. B e 1 i n f a n t e, Physica 5) H. F r 6 h 1 i c 11, W. H e i t I 1930. 6) N. K e m m e r, Proc. Cambr. 7) G. B r e i t, L. E. H o i s i n g
A. 1 GB, 127,
Sot. and
l’%iR.
A l(i(i, 501, 1938. M. T a k e t a n i,
B, 848, e r and
Proc.
phys.
math.
Sot.
Japan
1939, Zgr Ihe lert w/erred to by [U.C.;. N. K e m m e r. Proc. roy. Sot. A YGfl,
Phil. Sot. S4, 354, 1938. t on, S. S. S h are and
H. 1\1. T h a x t on,
Rev. 85, 1103, 1939. 8) C. Moller andL. Rosenfeld, NatureI43,241, 1939. 9) F. .J. Be 1 in f a n t e, Physica (i, 887. 1939. 10) 1-I. A. K r a m e r s, Proc. roy. Acad. Amsterdam 40, 8 14, 1937. 11) A. Pro c a, J. Phys. Radium 7, 347, 1936; D, 23, 1937. ‘12) F. .J. Be 1 infant e, Nature 143, 201, 1939. ,13) L. d e 1 r o g 1 i e, Actual. scient. industr. 41 I, 1936. 14) N. Kc m m e r, Proc. Roy. Sot. A, 1939. (In Ihe press). 15) H. A. K r a m e r s, .,Les nouvelles theories de la physique”, p. (Inst. int. d. Coop. intell., Paris 1939). -16) M. F i e r z, Hel. phys. Acta 12, 3, 1939. 17) E. AT a j oran a, Nuov. Cim. 14, 171, 1937. ’ 18) W. P a u 1 i and V. W e i s s k o p f, Hel. phys. Acta 7,709. 1134.
164-167,
154,
Phys.
1938.
VI,
no 9
THE UNCOR
October
EQUATION
1939
OF THE MESON FIELD
by F. J. BELINFANTE
Zusammenfassung Die Meson-Gleichung von P r o c a, Ii e m m c r und B h a b h a wircl mittcls Lincloren zweiter Stufc dargcstellt. Einc \‘erallgcmeinerung der Gleichung fiihrt zu ciner ncuen Meson-Gleichung, welchc im wcsentlichen aus ciner Kombination clcr Fallc (b) uncl (d) van I< e m m c r bcstcht. Die Ncutrerto-Glcichung wircl in ahnlicher Weise erwcitert. - Dns magnetischc Moment dcr Mesonen wird abgeleitct. Die ladungskonjugierten Wellenfunktionen geniigen einer Gleichung, in wclchcr die Vorzeichen aller Ladungen umgekehrt sind. Wenn man postuliert, class sic11 bci Beschreibung des physikalischen Geschehens mittels der ladungskonjugierten Grossen fiir alle physikalisch sinnvollen Grossen dieselben \Vcrtc ergcbon sollen, lasst sich folgern, dass Tcilchen mit ganzzahligem Spin der E i n s t c i n-B o s e-Statistik, und Teilchen mit halbzahligem Spin dem Ausschlicssungsprinzip geniigen miissen.
Resumo. La mezona ekvacio de P r o c a, Ii e m m e r kaj B h a b h a estas prezcntata per du&tufaj undoroj. Pligeneraligado cle la ekvacio donas novan mezonan ekvacion, kiu eefe konsistas el kombinajo de la lcazoj (b) kaj (d) de Ii e m m e r. La neutreta ekvacio estas plivastigata cn la sama maniero. La magneta momanto de la mezonoj estas kalkulata. La Sarge konjugitaj ondofunkcioj kontentigas ekvacion, en kiu la antatisignoj de ?iuj Sargoj estas inversaj. Se oni postulas ke EC prislrribo de la fizikaj okazajoj per la Sarge konjugitaj grandoj por eiuj observebloj devas rezulti la samaj valoroj, oni povas konkludi ke korpuskloj kun entjera spino clevas obei la statistikon de E i n s t e i II kaj B o s e, kaj ke korpuskloj kun entjcrplusduona spino devas obei la statistikon de 1; e r m i kaj D i r a c.
S 1. l‘hc P I’ o c n-Ii: c ~1 ~11c I’ meson cquntioe in wdov notation. The usual meson equations of I< e m m e r l), B h a h h a “) and Y u k a w a “) can be written in the following form: 4$LY - lo ?~,u) = DC, yvl = Dp yv - Dv s+ > +
+ g, v,) = Dp rwLv; (P,V = 0, 1,231. -
870 -
(1)
THE
UNDOR
EQUATION
OF THE
MESON
FIELD
f37L
Here x can be expressed in terms of the mass m of. the meson by x = m/h; UP” is an antisymmetrical tensor and V, is a four-vector giveri by *) :
By +N and +p we denote the wave-functions of the fields of neutrons and protons, so that after superquantization the expressions (2) represent operators which possess non-vanishing matrix elements for transitions of a proton into a neutron. The operators D, in (1) are defined by [U.C. (3)] :
We shall now write the P’r o c a-K e m m e r equations vector notation. For this purpose we put 7;,,” = co, = E,;
cbc = -
ccb = H,;
rp. = A,;
-
cpo= ‘PO= V;
(u, b, c being a cyclic permutation Further,
(1) in (4)
of 1, 2, 3).
we put
U oa =
u
ao =
e IO
-
%‘bc =
u,b
=
h,; v,,=a,,
vO=v,
(5)
v = $+#.
(6)
so that ; = -
++;p&,
Tt = -
Then, the equations x(f6 + fb2) = - ih x(?i + fb $j =
a = #+&,
++&J;
(1) read t) -
(a/cat) ;li: - (e/i%) (jib -
&),
rot-)A + (e/ihc) &x],
x(x + gb i$ = - rot ii + (a/cat) g x(v + gb v) = - div if-
(e/h)
VI (e/i&c) ([if, ii) + $i$,
(ii. i$.
In order to write these equations in undor notation we can make use of the K r a m e r s representation of undors [U.C. (28)], in *) For the notation used in the present paper we must refer to the preceding paper of the author on under calculus 4). References to formulae from that paper will be indicated by [U.C.]. In (2), (6) and (12) we have put % = 1 [U.C. (17)J. t) rot =-curl.
872
F. J. BEiINFANTE
which the D i r a c matrices ;;‘, p occurring
in (6) are given by
(8) and the charge-conjugated given by
tp = ep,
$” of a D i r a c wave-function
f =
0 0 0 -1 001 0 010 0’ -100 0
4 is
(9)
In this representation the components of the tensors represented by an undor of the second rank are related to the components of this undor by [U.C. (54), (58), (34n)]: F,-iF, -F,,-FF, 2X&,
=
F,--FF, -F/--F,,
L, - iL,
K,-iK, K”-
LO -I L,
- LO - L,
-
-LL,--iL,
-K”-KK, K,
-K,--iK,
G, + z’G,
-Go+G,
Go + Gz G, + iG,
ii=d+g,
KO=V+W,
-$ =i%-z.2,
F, =-S----Y,
&x-s,
L”=V-W,
&g+%,
G,=S--271.
; (10)
Here S is a scalar, 2, V a four-vector, z, d a six-vector, 2, W a pseudo-four-vector and Y a pseudo-scalar. In particular the undor of the second rank Xk,k,
represents, according --f
=
29f,
+k,
to (lo), the following
a = ‘ptZ+,
vzcptt$;
;=
ii- --&c+.
w=-+y&
y=qAy&;
- cpq3&,
(11)
tensors iL
- cpt p&;
-s=‘Ptp(CI.
(*2)
THE
UNDOR
EQUATION
OF THE
MESON
FIELD
873
Here we have put a, = -
cycl.; ys = - i&a+.
iga,,
From (10) we see that the P r o c a fieldx, sented by a symmetrical undor ‘%:k.
Writing
=
4 {‘rk,k,
+
yk,k,>
out the meson equations
=
(13)
V;%, g can be repre-
4 y{k,k,}.
(14
(7) with the help of (10) - (<2) in
terms of the undor components of \r@, and of sym X klk, = +%k, bk, + +%k, bk, = i&k, bk,J J and collecting the ten equations on account of (8) and (13) :
f@= 4 {(fb
+
gb)
(144
into one undor equation,
(fb -
+
gb)
we find,
Y!‘$i2’)
(16)
is a scalar operator which, operating on an undor of the second rank (lo), multiplies in this undor $, H’, Y and S by fband 2, V and B’, W by g,+ By an index in brackets we have here distinguished D ir ac matrices operating on each of the indices k, and k, of the undor. We remark that (15) and ( 16) are invariant by transformation to another representation of undors “). Putting
Dkz =
0
0
0 D&D,
0 -Do-D,
G---D,
-D,-
D,--iD, -Do--D, 0 iD,
0
we can write (15) at once in covariant [V.C. (821, (lb)]):
Do--D, -D&D, 0
=h
(17)
0
undor notation
“) (compare
2i$f;,k, + fk&,ld’ ‘E,J,~., + Dkll, Fk, + Dk,!, qk: = 0, where fk,k,l’l’are the matrix elements of the scalar interaction ator fop(16).
(18)
oper-
a74
F. J. BELINFANTE’
This interaction operator would take a particularly the constants fb and gb should happen to be equal: /b
=
simple form, if
gb (?I
Q9)
A preliminary interpretation 1) 5, of the binding energy of the deuteron in the (triplet) ground state and the attraction potential in the singlet state,, on which experimental data are available, seemed to indicate that 1 fb;lgb 1 did not differ much from unity indeed, but later investigations B), which accounted for the charge-independence of the nuclear forces’ between ‘heavy particles ‘), gave a different result. The simplification (19) seems, therefore, not to be allowed for the moment and we shall not make use of it in the following. 9 2. The generalized lneson equation and the neutretto equation. In the preceding section we have discussed the equation for the P r oc a - K e m m e r meson field: (case. (b) of K e m m e r 1)). This field SYm
was described by a symmetrical undor ‘I!R,R, and it interacted with the symmetrical part s?h,k, of the undor XR,k, = 24$,., +pk . It is plausible now to consider the generalization of the eqdatidns (15) and (18) by replacing these symmetrical undors sqk,k, and s??ksR,by the unsymmetrized undors Yh,L, and Xk,R,. The operator fopcan then be replaced by a more complicated scalar operator which, operating on an undor of the second rank (lo), multiplies S by f,,; x, V by g, ; ++ E, H by fb, g, W by gd, and Y by fd. The generalized meson equation 2i4%,k,
+ fopG%%&J~ + GT3 +
can then be written,
* o’, + (p"'
+
p"')
DO)
‘r,
k 18
=
0
(20)
according to (a)- (13), in vector notation: .
-++fos)
.=o;
x(X+gbz)= D,Z-[S,zij, x(V+av)= - (5.Z);(214 x(2 + fbi$ = -D&-V,
x(iTi+f$j
=[~~];
x(W+g,w)=-D,,Y,
~x(T&Jj
3Y; (214
x(Y+fdy)
=D,W+(T;.$.
THE
UNDOR
EQUATION
OF
THE
MESON
875
FIELD
Apart from the first equation (210), which defines the field component S in terms of the components of the field of heavy D i r a c particles, these equations represent exactly the cases (b) and (d) of K e m m e r, that is, the meson field suggested by M 0 11 e r and R o s e n f e 1 d “). The P r o c a field (b) describes mesons with a spin angular momentum A, whereas the “pseudo-scalar field” (d) describes spinless mesons 9). According to K e m m e r 6) the neutretto equation is obtained from the meson equation by changing +z #p into +(#z +p - $z +N) and Dp into V, ; and by postulating that the- tensors representing the neutretto field @‘k,R,shall be real. Thus (20) changes into 2M%,k,
+
fop
Mk,
with the additional
4JPk,
-
d&,
condition
4$l)>
(Yj? + Yj?) V” @k,k,= 0 (22)
+
(see [U.C. (64)]) that
CDk,k. = fit’) %(‘)(@k,k,)* = @f,k, = @f;,k, is a neutrettor of the second rank. The compatibility of the condition (23) must be shown. For this purpose we multiply of (22) by - P(l) G2) and find, on account [U.C. (81)] and 5?) Et21(@k,k,)* = @&, [U.C. 2iX{@fsk,
+
g(1)g(2)
or, interchanging 2ix
@k,
+
%
f$(@k,bks
-
$%k,+Nk,)*)
(23)
with the equation (22) the conjugate complex of - 9”) yt)* = yt) $9) (63a)]:
+
(YG”
+
y,!?)
vp@;xkl
=
0,
k, and k2: (@kl
bk,
-
%k,
hk,)?
+
(Yj?
+
$‘)
VP %k,
=
O,
(24)
where we have put (compare U.C. (40)): f$ = .gl) .jp
ff$ .jy* f(l)*
,
(25)
so that
(fopyk,k,)’
=
t$
yf,k.**
(26)
Now foponly multiplies the tensors represented by \rk,k, by the constant factors fo,&, fb,g, and fd,whereas, according to [U.C. (63), (63a), (64)], charge-conjugation of an undor of the second rank changes the tensors S ; A’, V; $, g; 5, W and Y, represented by it accordx*, V*; s*, g*; -s*, _ W* and - y*. ing to (lo), into -S*; We conclude therefore from (26) that & is the scalar operator multi-
876
F. J.‘BliLINFANTE
plying the tensors by the conjugate complex of the factors fO, g,, fbr g& fd. Now, asstiming that these constants are real
fo = f:,
gb =
k?>
fb =
fb*sg;l
=
$8
fd =
f%
(27)
we find
f$ = f*p If we make use of the fact that Jc$ &, is a neutrettor second rank [U.C. (63b)], the equation (24) turns into
(274
of the
+ fopN”p,,bk, -
@, hi k,)} + (y;’ + y!‘) VP (P&, = 0. (28) This equation for a& is identical with the original equation (22) so that the condition @k,k, = @Ek, (23) is indeed compatible for @k,k,r with (221, if the interaction constants (27) are real *). W@&,
§ 3. The charge-conjugated meson eqzcation. K r a m e r s 10) has shown that if 4 is a solution of the D i r a c equation for positive particles, then 4” is a solution of the equation for negative particles, that is, of the equation following from the D i r a c equation by changing e intd (- e). In the present section we shall show that the meson and the neutretto equations possess similar properties and that, if Y is a solution of the equation (20) for positive mesons (“theticons” t)), then ‘P [U.C. (63)] is a solution of the equation for negative mesons (“‘arneticons” t)) . For this purpose we proceed in a similar way as in the preceding section ; only this time we shall not interchange the indices k1 and &. In this way we derive from equation (20) : w3%,
+ 2Ep (@k, +Nk,)) + (Y:‘) + yjL2)) D*r yflk, = 0, (29)
where we have made use of g(‘) Ec2)(qfl +k,)* = Ec2)+;z . !$) 9:; = $2, (Pk,.
(30) Comparing (29) with’(20) we observe that \r&, satisfies an “arneticonic” equation (29) differing from the “theticonic” equation (20) for y&k, by the inversion of the sign of e (as D is replaced by D*) and by the change of
fOtJ(+8,,b$ into f$ (@k,hk,)’
(31)
*) If neutrettors of the second rank were defined by CD = W, instead of by @ = Qg, would have been necessary to take gb and fs real, but fO, g,j and /d purely imaginary. t) These names arc derivedlrom ~ETLXO~ = positive and b?vqTrxo; = negative.
it
THE
UNDOR
EQUATION
OF THE
MESON
FIELD
877
The electromagnetic potentials, being real, are not changed. We must now remember that the interaction of mesons with heavy particles, as described by the equations, should consist in the possibility of the absorption of an ameticon or the emission of a theticon by a proton which changes into a neutron, and vice versa, That is to say, the wave-functions in the equations (20) and (29) should be superquantized *) . If the wave-functions of anti-protons and neutrons (@ and.+N) are assumed to be a&commutative with each other (an assumption which simplifies the discussion of the canonical theory of quantized wave-fields and which enables us to introduce the formalism of the isotopic s#Gz in a natural way), we can express (31) by stating that, in order to change the equation for Y into that for ‘I?, not only should ‘I! be replaced by ‘l? and the electric charge e by eL = - e* > (32) but at the same time the “me.&” replaced by
charges
fo, & etc. should be
ft = - fo*,gt = - gb*,ft = - ff, etc.,
(324 and 4% by JrN and +p by @. There is no need to change the potentials of the M a x w e 11 i a n field occurring in the meson equation (20). This field can be described by a symmetrical neutrettor of the second rank, so that it is equal not only to its own charge-adjoint, but also to its charge-conjugated. In the same way the neutretto equation for Q, is changed into the equivalent equation for Q&,. This is seen at once by interchanging in (28) again k, and k, and by making use of the anticommutativity of 4% with Jllp and of $5 with dCN.Formally the infinities of the &functions from the commutation rules of protons and,of neutrons cancel each other. $4. The charge cwrelzt-density and the magnetic momelzt 01 mesons. Pro c au) and B h ab h a2) have derived the electric charge density and current of mesons with a spin A from a L a g r a n g i a n, which was chosen in such a way that the P r o c a equations and ‘the equations for the M a x w e 11 i a n field could both be deduced from it. *) The relativistic
question of the possibility invariance ofthe theory
of quaotization of the fields is maintained, is not discussed
in such a way that the in the present paper.
a28
I?. J. BEiINFANTE
In a similar way one can proceed for the generalized meson field “). If the field is normalized according to K e m m e r l) in such a way that x* , - z*, Y* and - W* are the canonical conjugates of?, 6, W and Y respectively, the expressions for the electric charge density and current take the following from “) la) : cp = (e/S) {(X* .S) - (5* . X) + Y* W - W* Y} = eY+popY, eT/c = (e/i%) {[ii*,
ii] + [ii*, Xj +
Here pop and&/C
are determined
(33)
by *)
or, in tensor notation,
These expressions are invariant by a transformation [U.C. (14), (14a. b)] from the K r a m e r s representation to any other representation of undors. The rnatrices.~,~ and TO& take the place here of the matrices 8 and 9: (that is, 8f3yp) in the case of the D i r a c electron. In analogy to [U.C. (20b)] it is convenient to write here (see [U.C. (ao)])
+ y(2)p Y* It,,= y+ y(‘)p 2h
(334
The main difference between the density matrices of electron and meson is that pop, being a singular matrix, cannot be made unity by transformation to any representation. A consequence of this singularity of popis that the meson equation (20) contains so called identities between the field components (differential equations not containing *) Compare L. d e I3 r o g 1 i e, lot. cit. Ia), page 22. The factor (J/21) is a consequence of I< e m m e r’s I) way of normalizing the meson wave-function and of our choice of the constants in (10). For instance, if the factor 2 in (10) is removed, the factor (1/2R) chariges into (l/88) I*).
THE
UNDOR
EQUATION
OF
TJ-LE
MESON
FIELD
879 .
derivatives with respect to the time), vii. the equations in the right hand column.of (21). : We can split up (20) into the proper equations of motion and the so-called identities by operating on it by (1 f p(1)p(2))/2. Abbreviating again by XI,R, = 2$$-h,$pRt and splitting up Yk,k, and Xk,k, according to :. %,k, = %,k, + %fk, 3 (35) \rz = 4 (1 + p(l)p@)) Y, Y” = Q( 1 - P(l)@(2))Y, we find the proper equations of motion of Y1: 2ix (Yr + fop XI) + (y(“fyiz,
.%) Yrr + ((8’) + (3t2’) D,Yr = 0, (36)
and the so-called identities: 2ix (YII + fop XII) + p-q) Comparing
. 5) YI = 0. .(36), (37) with (21) we observe without further
tion that YI represents the field components
(37) calcula-
2, 2, W and Y, and
YII the field components S, 6, z ands. Only of YI the components can be regarded as canonical .variables, whereas the components of Y” must be regarded as derived variables, defined by the equations (37) (like .I$ = rot % in quantum-electrodynamics), If for the present the interaction of mesons with heavy particles is neglected; the. meson equation (20) takes the form (2ix + r, DI”) Y = 0, putting we find
yi’-y
(I?, = yp + yf’).
1\2)= l?* and operating
(38)
on (38) by (1/2ix) . Pi-)@
l?i-) D” Y + (1/2ix) Pi-1 l?; DA Dp Y = 0.
(39)
From ytl y:;’ c yp y:“’ + y:“’ yj;” = _ &+
(40)
,we find
rip rpl = 0,
(41)
so that from (39) follows :l?ii) DA Y = (i/8x) . (IT-1 rp -
I?;-) -PA)(DA Dp -
Dr DA) Y.
(42)
880
I;.
From the definition
J. BELINFANTE
(3) of Dp follows
Dp D” = DLA Drl = (c/&c) . Vrx W‘l = (e/i8c).@,
D” Dr -
where $& denotes the M a x we 11 i a n field. The equation therefore be written in the following form: (c/~~&c).@~~ l?;x) rrl Y = (c/4mcZ).@
r;-) DAY=
(43) (42) can
FL+ rPY.
Adding (44) to (38) we find the following “equation of motion” from which aY/at can be solved by multiplication by p(i) : (ix + y;’ DA) Y = (c/8mc2) . $j+ I’;-’
(44) for Y,
rr Y.
(45)
The left hand member has the form of a “D i r a c equation” for the first index k, of the undor Y. If this equation is iterated like the ordinary D i r a c equation, we find (compare (40) and (43)) : (c/8mc2) . (- ix + y;) DP) (Q”c” r;-)
z
{x2 + 1 y(‘) 3
0
y(1) D{" P')
DP)
rp Y) = 1 (1) (1) D[h 3 YP YPl
+
= {(m”c”/tz’) - D, DA -
DPI}
4 id’),,,, . (e/W
\r
=
@-‘“>Y,
(46)
where we have put Y[A y,4 = - 2ki,. If only a magnetic
(47)
field is present @ = 0), we have
.$ ~1;;’ $j”P s G(“) . $j - i$?‘) .g
= ($. sr)).
(48)
Adding to (46) the corresponding equation with c$ (where in the left hand member l?l;-) occurs with the opposite sign), we find after multiplication by A2/2 y = {m2c2 + p,jpx - (ctz/2c) . (G . o”‘32’)) = (eA2/16mc2) .
r;-) Dp(gjAp q-j yr),
(49)
where = W) . DA is the operator of the kinetic momentum. In non-relativistic proximation we put cp”=mc2+T=cp,, PA
(50) ap-
(51) so that T is the operator of the non-relativistic kinetic energy. If, further, according to Y u k a w a 3), the right hand member
THE
UNDOR
EQUATION
OF THE
MESON
881
FIELD
I’(-) I’l;-) I’ DP(,@‘Y) . (eFz2/16mc2) of (49) is neglected in non-relat&&tic ap:roximation, this equation can be written as a S c h r 6d i n g e r equation (T Q mc2) : (Ey-“j
-
e%) Y? z TY =
= {&J2m so that the magnetic
-
moment
@A/2 mc) . $ (g(l)+J2)
.$)} Y,
(52)
of the meson
-+
pop = (c/2mc) . Q A (F+J2))
= (e/2mc) . Z&l
iw
is (e/2mc) times its spin angular momentum “) -+N S,, . Since the energy is given by “) E = JYt popE, Y, the non-relativistic x4) value of the magnetic moment is actually given by + P=/‘)r+PopSZopy. (54) As the value of the spin angular momentum is given in the same + way by “) S = $ Y+ pop3S, Y, the statement + p. = (e/2mc) .Z (55) holds for the values of these quantities as well as for the operators occurring in (53). 3 5. Charge-invariance and statistics. It is well-known that in the hole theory of electrons (superquantized theory of the D i r a c electron) there is an infinite c-number difference between the qnumber e++lC) (obtained by superquantization of the wave-function I# from the expression for the electric charge density e$tJr following in the usual way from the L a g r a n g i a n of unquantized wavemechanics) and the q-number representing the correct (observable) electric charge density. If the meson field is quantized, (33) must also be corrected by addition of infinite c-numbers. We have mentioned that to one description of D i r a c particles, mesons, neutrettos and the electromagnetic field by undor wavefunctions [U.C. (l)], (20), (22) there is an equivalent charge-conjugated description, in which some constants like e, f and g are replaced by cL, fL and gL’ (32), (32z), whereas every quantized undor *) is replaced by its charge-conjugated [U.C. (30)], (29). This suggests a kind of *) We assume that all fields (34)]) and not by “quasi-undors” Physica
VI
are described (reflection
by undon 4’ = f3#).
(reflection
$’ = ip#,
[U.C.(9), 56
.
a82
I:. J. BELINFANTE
symmetry between both ways-of describing physical situations *). By way of hypothesis one might assume that such a symmetry is a fznzria/~~c~ztal$roficdy of mturc. \\:e shall call this possible property the “ckn~Sc-ilzvn~iancc” of the physical world (not to be confused, however, with the principle of conservation of electric charge !). Therefore we shall postzrlntc that every physically significant quantity in quantum-mechanics (that is, every q-number cot~ctl~r representing the value of an obsr~ablc) is invariant by transition from one description of the fields of wave-functions to the chargeconjugated description, or, in shorter terms, is chaugc-ilt~)avirrllt. This postulate can serve to distinguish between wave-mechanical expressions, which after quantization cannot have a physical meaning any longer, and other analogous expressions, which may represent observables. For the present we shall leave this question out of consideration, but we shall show here that the postulate of chargeinvariance implies directly that photons and neutrettos must bc neutral, that D i r a c electrons must obey 17 e r m i-D i r a c statistics and that mesons ,mrst obey E i n s t e i n-B o s e statistics. The interesting fact is that this statistical behaviour of particles and quanta follows much more directly from the postulate of chargeinvariance then from postulates concerning the positive character of the total energy of free particles or quanta 7). From the L a g r a n g i a n of any kind of particles or quanta we can always deduce expressions for the electric charge density, the electric charge current, the total momentum ancl total energy of these corpuscles. The terms of the L a g r a n g i a n function depending on the derivatives of the field quantities ‘I” have always 3) the form of “) :I< Y+ B F Vp’\Ip. (56) If ‘I” is an undor **) ‘I!k,k~,,,,kSof ?ank AT, then [U.C. (12)] : !Y N B = B+ = II p”“; rP = C E,,y”“, (E,, = -c 1) ; B*I’* = l?BN; (57) II= I n= I so that, if we put .v \y‘” = & y* ) : zzz20” Z n p); s* .t: zzz1) (58) 11= I
THE
UNDOR
EQUATION
OF THE
MESON
FIELD
883
.
we have B d = (-
l)N E B*,
r % = - 32I?“.
(59) From (56) we find that the electric charge density, if it exists, is equal to ep = (eK/Ac) . ‘I? B PY, (- infinite c-number). (60) In the charge-conjugated description this expression is turned, on account of (32), into ,Lp= = (- eK/Ac) . ‘I? B I?’ \rj! (- infinite c-number), (61) therefore,
on account of (57)-((59)
:
eLpL = (- l)N. (eK/&c) . ‘Pl?‘B”Y*
(- infinite c-number).
(62)
If the expressions (60) and (62) for the electric charge density are postulated to be equal, the components of the wave-functions Y and ‘I!* occurring in (60) and (62) must be co~fimutative (apart from an infinite c-number term) if N is even, and must be anti-commzttative if N is odd. It is not true, of course, that the commutation rules follow rigovoatsly from ep = eLpL,
(63)
since in (63) the sum is .taken over the undor indices, and only‘the wave-functions ‘I! and Y* in one and the same point of space are multiplied with each other. In this case the a-function appearing in the commutation rules becomes infinite; its value corresponds formally to the sum or the difference of the two infinite c-numbers in (60) and (62). S ince the infinite c-number in (62) must be the chargeconjugated analogon of the infinite c-number in (60), this may be of some help in the “evaluating” of such infinite c-numbers. For photons and neutrett0.s it follows from (23) and from the symmetry of the operator pop ‘v&h respect to both undor indices, on which it operates, that p 3 YP+po,\r = ‘P+ popY” = v+ popY$ = pL.
On the other hand we find from (63) and (32) for any particles quanta pE -
pL.
(64)
or
(65) Comparing (64) with (65) we conclude that the electric charge density of the fields of neutrettos and photons must vanish, if it is
884
F. J. BELINFANTE
a charge-invariant expression. In a similar way we derive that by means of neutrettors of the first rank only neutral patricles 17) can be described. It does not follow from this, however, that neutral particles should necessarily be described by neutrettors ! For electrons we deduce from (65) that p = qJ+qJ -
c
166)
must be opposite equal to pL = ptge
-
c=
where CL takes the place of the infinite c-number conjugated description. From this we deduce ++9
+ +*+9*
= ++9
+ wAJ*
(664 C in the charge-
= F {(qJk)*+k + +k(w*>
=
= C + CL = c-number.
(67)
Similar relations can be deduced by postulating the chargeinvariance of the quantized expressions for electric charge current, total momentum and total energy. For instance, from
= i (C -
CL) = c-number.
(68)
It is obvious that relations like (67) and (68) are consistent with the anticommutativity relations of F e r m i-D i r a c statistics +k(x)* &&‘I
-I- +kr(x’) +)k(x)* = hr 8(x - x’),
(69)
but not with E i n s t e i n-B o s e statistics. In a similar way we find for mesons from p = yt P(I) & P12) Y-C
(70)
and pL =
yst
P(')
&
P2)
yf.
_
CL
(70n)
THE
UNDOR
EQUATION
OF THE
MESON
FIELD
885.
that y+ p(l) + pt2) y + yBt P(l) + P2) ys = c + CL* 2A 2F, Applying (57) -(59) we find yt
(71)
P(‘) + Pt2) \Y _ P(l) + PC’)y w \Y* = c + CL, (72) 28 2tz ) ( ) or, on account of (35), wt (Poti‘y’) - (POPY’ )m Yr* = C + CL = c-number. -V3) Again, in (73) the sum must be taken with respect to the undor indices, as in (67) and (68). It is obvious that (73) is consistent with E i n s t e i n-B o s e ‘commutativity relations between the components of Y’ and Y’*: Y:,k,(~)* YirlR,,@‘) - Y:,l,,,(x’) Yi,k,(~)* = c-number, (74) and not with F e r m i-D i r a c anticommutativity relations. The commutation rules for the components of Y” must be derived from those for Y’ by means of the so-called identities (37), so that it is not very alarming that we do not find any indication of them from (71)-(74). For neutral particles, indication of the commutation rules can be derived in this way from the expressions for the total momentum and the total energy, which are also obtained directly from the Lagr an g i a n. Generally we can postulate that the total L a g I a 1zg i a rt itself (integrated over space and time) shall be charge-invariant on account of the commutation rules of the field components. It is therefore not necessary to investigate the sign of the elzergy in order to derive the statistical behaviour of the corpuscles concerned l5) 16). It is true, however, that charge-invariance of the quantized expression for the total energy implies that by quantization according to the scheme of P auli and We is s k o p f 18) the so-called “states of negative energy” of free corpuscles (depending on the time by a factor e+lrri”l) can be interpreted, on account of the commutation relations (which do not need specification here !), as states of positive energy *) of corpuscles with opposite electric charge. We can understand this in the following way. By charge-conjugation of the quantized wave-function these states pass into charge-conjugated states of positive energy. If, now, the expression for the total energy is charge-invariant on account of the (un(
*) For the corpuscles
under
consideration
states
with
e-2nW
are of positive
energy.
886
THE
UNDOR
EQUATION
,OF THE
MESON
FIELD
specified) commutation rules of the q-number amplitudes a (J o rd a n-W i g n:e r or J o r d a n-K 1 e i n matrices), the terms in this expression arising from the so-called states of negative energy are automatically equal to the terms in the charge-conjugated expression arising there from states of positive energy of the charge-conjugated corpuscles (which are described with the help of the chargeconjugated q-number amplitudes b = a*). Using the latter (chargeconjugated) expression for the description of these terms in the total energy, the energy is given as a sum of only positive energies with amplitudes a*a or b*b. We observe that both the statistical behaviour of corpuscles and the possibility of describing so-called states of negative energy (of free corpuscles) as states of positive energy of charge-conjugated corpuscles follow directly from the postulate of charge-invariance of quantum-mechanical theories. The relation between the positive character of the energy of free corpuscles’ and the charge-invariance of energy seems to be still closer than that between charge-invariance and statistics. I wish to thank Prof. Received
July
15th
K r a m e r s for his interest in this work.
1939. REFERENCES
I ) N. 2) 3)
K e m m e r,
Proc.
roy.
Sot.
H. J. B h ab h a, Proc. roy. II. Y u k a w a, S. S a k a t a 20,319, 1938. 4) F. J. B e 1 i n f a n t e, Physica 5) H. F r 6 h 1 i c 11, W. H e i t I 1930. 6) N. K e m m e r, Proc. Cambr. 7) G. B r e i t, L. E. H o i s i n g
A. 1 GB, 127,
Sot. and
l’%iR.
A l(i(i, 501, 1938. M. T a k e t a n i,
B, 848, e r and
Proc.
phys.
math.
Sot.
Japan
1939, Zgr Ihe lert w/erred to by [U.C.;. N. K e m m e r. Proc. roy. Sot. A YGfl,
Phil. Sot. S4, 354, 1938. t on, S. S. S h are and
H. 1\1. T h a x t on,
Rev. 85, 1103, 1939. 8) C. Moller andL. Rosenfeld, NatureI43,241, 1939. 9) F. .J. Be 1 in f a n t e, Physica (i, 887. 1939. 10) 1-I. A. K r a m e r s, Proc. roy. Acad. Amsterdam 40, 8 14, 1937. 11) A. Pro c a, J. Phys. Radium 7, 347, 1936; D, 23, 1937. ‘12) F. .J. Be 1 infant e, Nature 143, 201, 1939. ,13) L. d e 1 r o g 1 i e, Actual. scient. industr. 41 I, 1936. 14) N. Kc m m e r, Proc. Roy. Sot. A, 1939. (In Ihe press). 15) H. A. K r a m e r s, .,Les nouvelles theories de la physique”, p. (Inst. int. d. Coop. intell., Paris 1939). -16) M. F i e r z, Hel. phys. Acta 12, 3, 1939. 17) E. AT a j oran a, Nuov. Cim. 14, 171, 1937. ’ 18) W. P a u 1 i and V. W e i s s k o p f, Hel. phys. Acta 7,709. 1134.
164-167,
154,
Phys.
1938.