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Undor calculus and charge-conjugation
Ehysica
VI No 9
UNDOR
October
CALCULUS
1939 .
AND CHARGE-CONJUGATION
by F. J. BELINFANTE Instituut
voor Theoretische Natuurkunde
der Rijks-Universitrit,
Leiden
Zusammenfassung Grossen, welche sich transformieren wie Produkte 4-komponentiger D i r a c scher Wellenfunktionen, werden ,,Undoren” genannt. Zu jedem Undor I# kann durch Linearkombination der Komponenten seines komplex Konjugierten $* ein neuer ,,ladungskonjugierter” Undor 4s gebildet werden. Wenn + = $*, wird die Wellenfunktion + ein ,,Neutrettor” genannt. Undoren zweiter Stufe entsprechen gewisse Tensoren, Neutrettoren zweiter Stufe entsprechen reelle Tensoren. Mit Hilfe eines ,,metrischen” Undors zweiter Stufe werden ,,kontravariante” Undoren definiert, welche sich kontragredient zu den gewohnlichen (,,kovarianten”) Undoren transformieren. Endlich wird der ,,Gradient-Neutrettor” eingaftihrt. Resumo Kvantojn, kiuj transformigas kiel produtoj de kvar-komponantaj D i r a c’aj ondofunkcioj, ni nomas ,,undoroj”. El ?iu undoro 4 per unuagrada kombinado de la komponantoj de @a komplekse konjugito $* nova ,,.$.arge konjugita” undoro $” povas esti konstruata. Se + = 4”. ni nomas la ondofunkcion + ,,neiitretoro”. Duadtufaj undoroj reprezentas certajn tensorojn, dua$tufaj netitretoroj reprezentas realajn tensorojn. Per ,,metrika” undoro duagtufa ,,kontra?ivariantaj” undoroj estas difinataj, kiuj transformigas kontrafipase al la ordinaraj (,,kunvariantaj”) undoroj. Fine la ,,gradiento-neiitretoro” estas prezentata.
3 1. Irvtroduction. It is well known that - with respect to the restricted L o r e n t z group, excluding spatial reflections through
the origin l tensors can be expressed in terms of spinors l) “). As soon as spatial reflections are taken into account, however, it is necessary to consider @airs of spinors transforming one into the other by a reflection through the origin. An example of such a pair of spinors is the wave function of the D i r a c electron. In the following we shall investigate the properties of quantities transforming like products of such D i r a c wave-functions “). Such I’hysica
VI
849 54
850
F. J;.BELJNFANTE
quantities we shall call “undors ” *). They form a generalization of D i r a c wave-functions in the same sense as tensors form a generalization of vectors. Just as the representations of the L o r e n t z group by the transformations of most tensors are not azlsredzlzierf, so the representations by most undors are likewise reducible. In particular we shall discuss, in the following, the relation between undorsof the second rank and tensors, and the analogon in undorcalculus to real tensors : “neutrettors” 4). Finally we shall deduce the metrical undor and define the gradient undor. The whole set of mathematical relations will be built up in such a form, that we shall be able to apply it later on to the theory of mesons and neutrettos 5) 6) 7) “) 9). ‘.‘i i - _ 5 2. The D i Y a c wave-function (under of the first mnk). The D ir a c-equation of a positive particle (a positon or a proton according to the value of x = me/h) can be written in the following form:
if we put
(3)
Here e is the elementary charge (e > 0) and%, 8 is the potential four-vector of the electromagnetic field. As in (1) the interaction with heavy.quanta is neglected, this equation does not account for the anomalous magnetic moment of the proton lo) “) 6) 11). We shall call quantities transforming like the four-component D i r a c wave-function 4, four-spinors or ztndors of the first rank. In the following we shall often use a representation, of them, whi,ch is ausredzcziert with respect to the group of restricted L or en t z transformations, and in which the first two components of a fourspinor transform like the two-component quantity called a covariant conjugatedspinorby Van der Waerdenl),, Laporte and L?hlenbeck2), andcalledaregularspinorby Kramers12), whereas the last two components transform like the spin-conjugated 1) *) Drrivcd t) See.II.
from ttndn = wave. A. K r a m c r 8, lot.
cit. I*), pap
263.
UNDOR
CALCULUS
AND
851
CHARGE-CONJUGATION
of such a quantity, called a contravariant regular spinor by V a n d e r W a e r d e n 1) and others 2). Explicitly: \ x’ = x
h=h
y’ = y cos 8 + z sin 8
cos- ; + $2isin-
+; = t/q i sin7 e
z’ = - y sin 8 + z cos 3 t’ &t
+j = #,cos-
I
i
+ $,cos,
8
2
3.
~ .w 8 + cCdi.sin 2 2
8 +;=$3isinT++4cos$
z’ =z t’ =t x’ = x Y’ = Y
where
If we require that (1) is L o r e n t z-invariant, we find that the D i r a c matrices are give? in the representation (4) by (. + a = p.2 (5) and (6)
P = (A + 34&a if we denote by Pxt
PY,
Pz and
z{qx,
ay, 4
852
F. J,. BELINFANTE
the P a u 1 i matrices la) operating s of the four-spinor
From p2 = 1 we deduce A2 -
on the discrete arguments
I’ and
B2 = 1, therefore,
IO p =
OAO 0 OOA I/A 0 00 0 IplOO
(8)
with A = A + B = I/(A - B). Apart from a numerical factor the transformation-matrix of + for a spatial reflection through the origin 14) must be equal to p :
6’(- x, - Y, - 2, t) = iP #(x3 Y, z, t)* As a double reflection should not change the geometrical of the four-spinor, j must be a square root of f 1: p=
f
(9) meaning
1.
(10) By (4) a representation of the complete L o r e n t z group by transformations of 4 is not yet uniquely given. If we complete (4) by making a definite choice for the representation of reflections, the matrix p will be fixed by (9), (10) apart from a square root of f 1. On the other hand we may choose A = 1, that is, P = Pxi (11) the representation by + of the L o r e n t z group including reflections is then given by (9), (10) together with (4), apart from the same factor j in the reflection. In the following we shall denote the conjugate complex of a matrix by an asterisk *; by a cross t the adjoint (“Hermitian conjugate”) of a matrix. For instance
UNDOR
Further,
CALCULUS
AND
853
CHARGE-CONJUGATION
we put
for instance :
We introduce the nonrzalimtio~z- or density-matrix defined apart from a real numerical factor by p+s = ap,
-3s
==a:,
9, which
at =a.
is
(13)
Further, we shall postulate that under a linear transformation by a non-singular matrix S to another representation: 0’ z .y(J, clg’ = +t.q, p’ = spy,
;f
(14)
= S&l,
(14n)
the real expression 4Jtw4J shall be invariant,
so that 9’
=1
g-1
3, s-1.
(144
The definitions (13) are indeed invariant under these transformations (14n, b). In our particular representation (4), in which + q+ =;> (15) we have on account of (13), apart from a numerical 0
0
factor,
01
(16) after the choice (11) of p we have, therefore, 9 = 1.
(17) from
The density operator will remain unity so long as, starting the representation (4), (5), (1 I), (17), we admit only mitnvy tvansfowzatiom, for which sst= 1. (18)
854
F. J: BELJNFANTE
In the particular representation (5), (11) it is easily verified with regard to the complete L o r e n t z group, 4J+w+ is a scalar, and that -? 1 Ic = tJ1+4& together form a four-vector nuity equation
(19)
with p = #t&d,
(likeTwith
that,
(20)
ct), which satisfies the conti-
. . aP = 0 div +1 + at as a consequence of (13) and the D i r a c equation
(21)
(1) :
= ix($+p+th$ - ++q3+)+ $ #+{(22 - 23)s- a$ .: - %)}JI= 0. It is, therefore, ‘possible to regardTand p as the probability current-density and we shall normalize c-number solutions of (1) in one-particle wave mechanics by jfj
dx dy dz (+O$) = 1.
(22)
In the following we shall confine ourselves mainly to those representations, in which (17) is fulfilled, and we shall therefore drop S practically throughout. As regards the representation (4), this restriction (17) means taking A = A*, B = - B* in (6) and IhI = 1 in (a), so that P+ = P
(174
becomes a H e r m i t i a n matrix. As regards transformations by (14) to other representations, it means restriction to unitary transformations ( 18).
UNDOR
CALCULUS
AND
3 3. Charge-con.jugation of four-spinors. A matrix apart from a complex unity factor eic by *) &=-s;*,
855.
CHARGE-CONJUGATION
p;E=-sdr:p*,
S is defined
$X*=1;
(23)
so where sx = - iayaz, so that in the representation
In this representation
the restriction
‘(24)
(5), (6) +’ + 5 = a.
(25)
(5), (8) this matrix
b 0 2 = eic 0 -ll/hl therefore
cycl.,
0
aEis equal to
0 -pL(
0 IhI II/Al0 0 0
0 0’ 0
(26)
(17) makes it equal to di = 8 . pyay
(27) in the representation (4), (5), (8). In the following the particular representation (4), (5), (1 l), (17), (27) with eic = 1 will be called the K r a m e r s 15) representation: + ;E=pyfxy, 9=1; ;==: a = ha,-+ p=px, (28) By means of the matrix ;Ewe construct plex (+*) of the four-spinor wave-function (I), another four-component quantity
from the conjugate com(4) of a positive particle
#” = Is+*.
(29) From ( l), (3) and (23) we can easily deduce that 4” satisfies the equation 15) {ix + (;. 5)
+ f3D*o}tp = 0,
(30)
that is, the wave equation for a negative particle (a negaton or a hystaton t) according to the value of x). For this reason +G is called *) This matrix P is identical with the matrix C* introduced 71 Hystaton (antiproton) is derived from 6a-;ctro& = last;
by P a u 1 i, lot. proton fromnporoq
cit. Id). = first.
a56
F. J. BELINFANTE
the &urge-cortjtigated 15) of 9. From (30) we can conclude 14) that with respect to restricted L o r e n t z transformations - the chargeconjugated of an undor of the first rank is again an undor 15). As regards reflection: from (30) we can conclude only that the transformation of C/J~must be again of the form (9). But the representations of the complete L o r e n t z group by the transformations of 4 and by those of v might be different with respect to the sign of i. In order to examine this, we compare the charge-conjugated four-spinor in the reflected system of co-ordinates, which is defined bY $p E $1” = go’* = j*gp*+*; (31) with the charge-conjugated four-spinor 4” in the original system of co-ordinates. This last, (29), will be transformed into (31) by (9) with perhaps a different value of j, - say i@) : Jr”’ = jw p 0” = p p -g+** (34 From (31), (32) and (23) we find 1w = _ j**
(33) The charge-conjugated of an undor of the first rank is therefore itself an undor with respect to the com@ete Lo r e n t z group (jt8) = 11, if i= - j* (sothatj= fionaccountof (10)). (34) We can also prove directly that +” is an undor with the choice (34) of i, without making use of the equation (30). Let A be the linear operator of some Lo ren t z transformation of the undor 4. Then v transforms like an.undor if q” = A(& (35) or sn* +* = fqJ*, (354 that is, if ne = %A*. (36) Indeed this condition is satisfied for the restricted transformations on account of (23n). For the reflection fied, if j (3% = ;Ej* f3*, or, on account of (23), if i = - i*.
L or en t z (9) it is satis(37) (34)
UNDOR
CALCULUS
AND
857
CHARGE-CONJUGATION
This result of RI a j o r a n a “) and R a c a h 16) means that it serves a useful purpose to define the reflection of a D i r a c wavefunction in such a way that a double spatial reflection through the origin inverts its sign. In the following we shall see that this same definition enables us to describe the P r o c a field (that is, a field consisting
of a four-vectorx,
V and an antisymmetrical
tensor of
the second rank s, q by means of a sy;rmetrical undor. Since according to (34) $” can now be regarded as a regular undor, it is natural to postulate that by a transformation (14) to another representation of undors, the undor $” shall be transformed in the same way as all other undors $. This means that (If’ E 0’” = S’ q* must be obtained
= g’s* +*
(380)
from GC(29) by a transformation $f’ zzzsy
This holds independently
(14) :
= s&+*
W)
of the choice of $, if S’S* = S ;E,or ;E’ z.z sss*-‘.
(14c)
Under the transformation (14n, c) the definitions (23) are indeed invariant, that is to say, if ;Eis defined in one representation in accordance with (23) and is then transformed to another representation according to (14c), the relations (23) will hold again between the transformed charge-conlugator g’ and the transformed D i r ac matrices
p’, 2, etc. In the same way the relation 85 = ;f”8*
= (a-&)“,
(234
which is valid in Ii r a m e r s’ representation, is invariant by a transformation ( 14b, c) and in consequence holds in every representation of undors. It follows from (27) that ;E = S” (17b) holds in the representation (4), (17) ; and from (14~) we deduce that this relation (17b) holds in all rc#wesentations (17), for which tlac density matrix is unity, because (17b) is invariant by a unitary transformation (St = S-l, S” = S*-I) : P=
=
g-1
SWS”
=
sgs*-1
=
g
858
F. 1. BELINFANTE
If F is an operator, which operates on four-spinors, a charge-transformed operator F& by
we can define
(F+)E = F”+“,
(39)
so F” = fF*F.
(40)
On the other hand, if F depends on the electric elementary c, we can define the charge-inverse FL of F by {F(e)}= = F(-
Then we can summarize the connection
e).
charge (41)
between (1) or
and (30) or H$J8 = &O&8 = -
&,+”
(304
by stating that H&., = i
H$,,
&,L,,= -
&$.
(42)
K r a m e r s 15) has pointed out that a description of electrons by means of 4 and Hop, and a charge-conjugated descrifition by means of 9” and HoLp, should be equivalent. It can be shown that the meson theory shows a similar kind of charge-invariance. 5 4. Neutrettors of the first rank.
We shall call self-charge-conjugated
four-spinors v=+
(43)
neutrettors of the Jirst rank *). These quantities are adequate for the description of the neutral particles of the theory of M a j o r a n a “). It can easily be shown that M a j o r a n a ‘s “real” D i r a c wavefunctions are exactly what we call neutrettors. Proceeding by a transformation (14) with
s = St-1 = (1 + i&csY)/dZ *) The name ‘“neutrimws” Editor in Nature a)), seems described by these quantities. described.
(44)
(in analogy to spimws proposed by me in a Letter to the to be less adequate, since it suggests that neutrinos can be By neutrettors, however; only photons and neutrettos are
UNDOR
CALCULUS
from the Kramer representation
AND
CHARGE-CON
a59
JU‘GATION
s representation
(28) to a Ma j or an a
;EMAJ
=
we deduce from (23) : --f
-
1,
+
8MAJ
(45)
1,
+
(45Q) This is indeed the characteristic of the representation discussed by Majorana*). Four-spinors, which are self-charge-conjugated, are on account of (29) and (45) real in a A/r u j o Y n n a ~efiresentahow. According to M a j o r a n a “) neutrettovs can describe neutral particles t). For the purpose of building up a canonical theory of M a j o r an a particles the K r a m e r s representation is very convenient. Denoting in this representation (28) the first two components of d, as a Kramer s spinor by P%AJ
=-PMAJ,
Y%AJ’-YMAJ,
and the last two components K r a m e r s spinor v by
aitfAj=:MAJB
ZAJ=-G*AJ.
as the spin-conjugaled
01 of another
(464 we can write (29) in the following
form:
(48) This four-spinor
is a neutrettor,
II
if
’ rS. = J1 = +” = / t, / , that means, if u E v. *) The
representation actually used by by taking S = tdi.(&(l -a~) t) Note added in proofs. The Ma j or transformation properties of !Z have also Rev. 64, 56, 1938.
(44), but
(49)
M a j o r a n a is obtained from (28) not by + ~~(1 + uy)} in (14). an a theory of neutral particles and the been investigated by W. H. Furry, Phys.
860
F. J. BELINFANTE
In other words, a Ize&rettor of the first rank consists of a two-qbinor and its spin-conjugated and can be written as
in a Ii In a should theory; theory
r a m e r s representation. canonical theory of neutral particles only the two-spinor u be regarded as a canonical variable like + in D i r a c ‘s U* takes a place comparable with that of $* in D i r a c’s and %s can be expressed in terms of u*.
$5. Undors of the second rank. We shall call a sixteen-component quantity Y W, = ys,*,,s,rs (A,, k, = 1, 2, 324; SI,Y~, ~2, ~2 = + &, - 4) (51) transforming like the product dlR,+;, = +s,,r,+:,,r, of two fourspinors an zrndor of the second rank. With respect to the undor-indices we regard it as a matrix with oozecolumn and 16 rows. Still, we shall wyite it as a square matrix with 4 x 4 elements:
yk,k,
‘r,,
‘r,,
‘r23
‘Y;4
y31
y32
y33
‘r,4
‘r,,
‘r42
‘r,,
‘yk4
(54
= -
Matrices like &‘), #, &I, &, p’“‘, +, $0, $1, gh), gw, p,, etc. are assumed to operate on the argument k, of \rk&. Taking these operators as unity matrices with respect to the index, on which they do not operate, we can regard them as matrices with 16 rows and 16 colums. With respect to the restricted L o r e n t z group an undor of the second rank represents one regular spinor, one conjugate complex spinor and two mixed spinors- of the second rank; it represents, therefore “) i2), two four-vectors 7’K, K” andT, Lo (transforming ;;l’ct), two scaiars F, and Go, one regular complex three-vector Kennzahlen of an antisymmetrical self-dual tensor 2))
like (the
UNDOR
CALCULUS
and one conjugate
AND
861
CHARGE-CONJUGATION
complex three-vector z’=
22 + is*,
W)
where z,, g, and &,, gz form two antisymmetrical tensors of the second rank. Generally all these quantities are complex. In the K r a m e r s representation (28) we can write: Yll
Yl2
Yl3
y14
‘r,,
y22
‘r,,
‘r,4
‘r,,
‘r32 ‘r,,
‘r,
‘r,,
‘r,,
‘r,
‘r,
=
4
F,- iF, Fo--Fz - F,--iF, -F&-F, L,- iL, LO-L, -LO-L, -L,--iL,
K,--iK,,
-K”-K,
-F;2y
-Fs”G” . (54) 0 I -G:, + G, G, + iG,
In an arbitrary representation of undors obtainable from (28) by a transformation (14) we should replace the left hand member Y of (54 by’ cp-~ S(W y = y (55) By spatial reflection according to (9) by
through the origin the undor Y will transform y’ = j’p(l) (j(2)y*
(56)
In the representation (28) used by us we have f3(1)p(2) = eg)&?), so that this transformation (56) can be written, on account of (54), as
i? = (2 + i&/i,
K” = (V + W)i/i,
z = (X -i&/j,
LO = (V -
z = (ii - i&/i,
F,=(-
G’= (ii + i&j,
Go = (S - iY)i/j,
w)i/j, (58)
the new quantitieix, respect to restricted
S - iY)j/i,
V; $, W; s, g; S and Y are still tensors with L o r e n t z transformations (two four-vectors,
862
F. J. BELINFANTE
an anti-symmetrical tensor and two scalars), whereas we can ‘now write (57) in the following form: ~~-~, g
V’=-j-v;
= + g, w’ = -
w;
j&-s,
ii%+&
y’ = -y;
S’ = + s.
(59)
With respect to the complete L o r e n t z group including reflections, an undor of the second rank consists therefore of a regular scalar S ; a regular four-vector X, V; an antisymmetrical tensor of the second rank%, G (which can be regarded as a pseudo-tensor of the second rank g, g) ; a pseudo-four-vector (that is, an antisymmetrical tensor of the third rank) 2, W, and a pseudo-scalar (anti-symmetrical tensor of the fourth rank) Y. It is often assumed “) that the meson field can be regarded as a P r o c a field 17)) that is, a field consisting of a four-vector x V and a six-vector z, 2 only (case (b) of K e m m e r “)) . Such a field can be described by an undor of the second rank satisfying the relations Y,, = - j2Ys1, Y,,=
- j2Y4,,
‘r,=
- i2Y32, Yz4= - i2Y4,.
(60)
The most symmetrical method to achieve this and at the same time the one and only possibility to achieve (60) in a way which is independent of the representation of undors, that is, invariant under the transformation ‘r’
= 9,Smy
(55a)
.’ .
is to postulate
that the undor describing the P r o c a field must be with respect to its two indices, and at the same time that for undors i2 in (9), (10) must be equal to .minus unity: symmetrical
y kd, = Yk,k,, i2i-1. Indeed, in general a symhetricnl sents : _
(61) -
(610)
undor of the second rank repre-
(b). a regular 4-vector 2, V and a-6-vector Ey H:if i2 = -
1; (62)
(c). a pseudo-4-vector.g,
Wand a 6-vector c I$ if jz = + 1;
UNDOR
CALCULUS
whereas an antisymnzetrical
AND
863
CHARGE-CONJUGATION
undor of the second rank represents:
(d) . a pseudo-$-vectors,
W, a scalar S and’a pseudo-scalar Y, jfj2 = - 1;
(a). a regular 4-vectorx,
V, a scalar S and a pseudo-scalar Y, if i2 = + 1.
(621)
Here (a), (b), (c), (d) refer to the tensors composing the field in the four cases considered by K e m m e r 6). Now we define the charge-conjqated of an undor of the second rank by and its charge-adjoint
YGM, = 22’) L!z2)(Yk,k,)* by
(63)
YEk, = ‘ezk, = it?‘)s(2) (\rk,k,)*.
(634
Then WI = w-M1! (69 is a self-charge-adjoint undor of the second rank. Now we can express the tensors represented by Ys in terms of those represented according to (54), (58) by ‘I!‘. In this way we find from (28), (54) and (58) :
-;jr! =&F ) V”=V*;
&=E*
‘i2
yg = y*; S” = s*.
=s*,
WC = w*;
> gL$b
,
(64)
We observe that by the choice of constants made in (58) we have achieved that the tensors represented according to (54), (58) by the charge adjoint Ys of an undor of the second rank Y, are the conjugate complex of the tensors represented by the undor Y itself *). If now a neukettor of the second rank is defined as a self-chavgeadjoint undor of the second rank, it represents by (54) and (58) according to (64) a set of real tensors. Such neutrettors are therefore adequate for the description of the M a x w e 11 i a n field “) and of K e m m e r’s neutretto field 7). A specimen of a neutrettor of the second rank is given by +f C&(63b). +
+++
*) The constants in the definition by (54), (58) of the tensors S; A, V; E, II; U, \V and Y in terms of the components of the undor \r are uniquely determined by the conditions (59) and (04) apart from arbitrary real numerical factors to these tensors, which are all chosen equal to unity in (58)
864
F. J. BELiNFANTE
Taking i=i
(344
in the following, in accordance with (34) and (6 la), all factors i/i and i/j vanish from (58). 5 6. Covardant u&or calcdus. The fact that the linear combinations (29) (and, apart from linear combinations differing from (29) by a numerical factor, only these) of the components of a conjugate complex undor Jr: form again the components of a regular undor dCR, enables us to define a metrical z&or &, that is, an undor of the second rank, which connects contravariant and covariant undors with each other. Since, on account of (19), (23) and (13), the expression ++%f+$ = (it P+fb+ = ,“, 4; (~+%)ik+k = Fk (t%“”P+c’J)k’4; (,k = = c (tb*p*)”
(Jf $k = r, x” $A
(65)
i,k
is a scalar, we can regard x” = f (a+* f3*)k~~~
(66)
as a regular contravariant undor. We shall connect with this y,” an ordinary covariartt undor ~1by xl
=
T
f&k xk
=
Fk f&k
(s*
p*)
“4:
.
(67)
Since x, shah be a regular undor and its components are linear combinations of those of +X, the undor a must be equal to (v)l apart from some numerical factor, for which we shall choose unity : x‘=
$5
As this result should be independent T glk (8*p*)k” l
(68)
of $, we find
= SF, or gtb*p*
= it
! 69)
We conclude : g = g(j*B*-‘.
iv
UNDOR
CALCULUS
AND
CHARGE-CON
In Kramer-s’ representation metrical undor g,, takes the following
(S(3 = pX, E = p,+) form :
0 -1
g=that
ip,a, =
865 .
JUdATION
.the
00
1 0
0
00
0
01
0
0 -1
0
is, +1
=
-
02,
=
$2
qJ1,
$3
=
+4,
94
=
,-
q.J3.
(7Ia)
We observe that in this representation the metrical undor is antisymmetrical just as the metrical spinor is in spinor calculus and unlike the metrical tensor in tensor calculus (which is symmetrical) : lr = - g kkl = - l&k). (72) This property of the metrical undor is invariant under transformations (14) to other representations. This follows from (70), (14a, b, c) : g;Q, = S’ p’*a’*--l
= se p*a*+
S” = sgs-
A consequence of this antisymmetry %‘$k
=
-
= z Sj’l g,, S&k. (144
of g is (73) undors are con-
i?Jd#‘*
Conjugate complex contravariant and covariant nected by the coqkgafe comfilex metrical w&or g;i
=
(f&k)*
=
F
(74)
Efk(p-l)k!s
This undor is in K r a m e r s’ representation 0 -1 00 1 0 00 0 0 01 0 0 -1 0
g;& =
and transforms
to another
The contravariant
representation g*’ = s* g*st.
metrical u&or f
given by
by (144
g’” is determined
filkgkm
=
(744
by (75)
v
or gkm
Physica
VI
3
g-1;
gdi
s
gtd.
VW 55
866
F. J. BELINFANTE
In K r a m e r s’ representation 010 -100 000-l 001
gknr =
0 0
we find from (71) and (740) 010 -100 000-l 001
g *iA _-
0
0 0 (76) 0
Transformation to another representation changes the contravariant metrical undors gkm and g*kd according to (14d, e) and (75~2). Co- and contravariant undors are now connected by
Here the summation must always be carried out with respect to the last index of the metrical undor. From (65), (68), (73) it follows that ++ 9 p t,, = T (fk +k = -
: +??(,“.
(19fi)
In the same way we derive $+ap+-q;(J~*”
=-q)*“!.p;.
(19b)
For the current-density four-vector (20) we can derive expressions, for instance *) -+ i/c = c ,q #I) p = z# tjJfk p; (JI Inserting tion
similar
P@)
(46a, b) into (71a) we find in K r a m e r s’ representa!+LUi,
therefore, following putting z’s 3 w:
$2=-q,
(/3=-+2,
!+4=7)5’,
(78)
V a n d e r W a e r d e n’s notation
1) and
(784
*) Compare
P. A. M.
D i r a c, lot.
cit. I*), and W.
Go r d o n, lot. cit.
10).
UNDOR
CALCULUS
AND
863.
CHARGE-CONJUGATION
The difference between our undor calculus and Van de r W a e r d e n’s spinor calculus is, that we have taken care from the beginning that the transformations of “contravariant” undors should indeed be contragredient to those of “covariant” undors by all transformations of the complete L o r e n t z group iuzchdiwg spatial reflections.
’
We might have derived (14e) and (76) in a simpler way. Making use of our knowledge of V a n d e r W a e r d e n’s metrical spinor, we conclude that the contravariant metrical undor gkn must have the form OaO 0 -a00
0 U-64
000-b
OOb ;o I in K r a m e r s’ representation, in which the first two components of a four-spinor behave like the components of a conjugate complex covariant V a n d e r W a e r d e n two-spinor and the last two components like a regular contravariant V an d er W aer d en spinor. The ratio (a/b) is now determined by the condition that, if ‘p and 4 are two arbitrary regular covariant undors, the expression (Pk gk
+I
=
‘pw
g-’
(79)
+
shall be a scalar with respect to the spatial reflection (9), (10) (as well as it was, on account of (76a), a scalar by restricted L o r e n t z transformations). In K r a m e r s’ representation we find from (76a), (9) and p = pX (28): 4%
lJl2 -
‘p2 41)
-
b(cp3
$4
-
94 G3) =
=
44
a ?(ff3
= $1
-
9;
44
$4
-
(~4 $3)
-
WP;
-
b iVf9
44
$2
‘pi 44) -
7
92 44,
therefore, a/b=-12=-
111~.
(76b)
Choosing j according to (34) we find a = b and (76a) becomes identical to (76) for K r a m e r s’ representation (28). If, now, we fiostdate that the scalar (79) shall be an invariant under
a68
F. J; BELINFANTE
transformation
(14) to another representation,
W’,’
=
p-1
g-1
we find
p-1, (14il
in accordance with ( 144. A still shorter way of deriving (71) is by making use of (54) and (58). Since the metrical undor is supposed not to change its form by L o r e n t z transformations and spatial reflections, it must represent a scalar. Therefore it must have, according to (54), (58), the following form in K r a m e r s’ representation : 0 -s gkl
=
4
0 0
s
0
00
0
0
OS’
0
o-s
-
0
Takings = 2wefind (71). We remark that g is a nezltrettor of the second rank on account of the special definition (63~) given for the charge-adjoint of an undor of the second rank. In the literature use is often made of an abbreviated notation for the contravariant charge-conjugated of a regular undor & or yk,k,:
(80) Further,
we shall denote by ap and yp (p = 0, 1,2,3)
h
the matrices
a0 = - a0 G 12
a2, a3) = {al, a2, a3} =Z,
(24 {Yl,
Y2,
Y3)
The probability then given by
The relations
=
{y’,
y”,
density
y”>
=r;=
PZ
and current
Y0 =
-
ypS = -
Ey’**,
apt3
=
P.
of a II i r a c electron
(13), (23) and (23~~)can now be written pts = 3 p,
Yo
are
as
=4;ap,
a@% = SaF*.
(81)
In undor caIculus the gradient four-vector VP is, according to (54), (58), (60), (64), represented by a symmetricalneutrettor. In K r a-
UNDOR
CALCULUS
m e r s’ representation form :
AND
this gradient 0
neutrettor
v, -
VZ
c at Vkl
---- ia c at
- v,-
has the following
--ia Y-.Vi c at
iv,
---_la
0
869
CHARGE-CONJUGATION
-v,-
iv, =
v,
0
0
iv,
0
0
Vlk
(82) .
The D i r a c equation of a free electron can now be written in covariant undor-notation : {ix + y’ VP}+ = 0 -+
i&
+ vkl+’ = 0.
I am much indebted to Prof. Kramer s for many on the questions treated in the present paper. Received
July
14th,
(14 discussions
1939.
REFERENCES 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 9)
B. L. v a n d e r W a e r d e n, Giittinger Nachr. 1008, 100. 0. Laporte andG.E. Uhlenbeck, Phys.Rev.37,1380,1931. F. J. B e 1 i n fan t e, Nature 143, 201, 1939. E. Majorana, Nuov.Cim.14, 171, 1937. H. Y u k a w a, S. S a k a t a and M. T a k c t an i, Proc. phys. math. Sot. Japan 30,319, 1938. N. K e m m e r, Proc. roy. Sot. A 106, 127, 1938. N. K e m m e r, Proc. Cambr. Phil. Sot. 34, 3.54, 1938. H. Yukawa, S. Sakata, 111. Kobayasi and M. Taketani, Proc. phys. math. Sot. Japan 80, 720, 1938. C. Meller andL. Rosenfeld, Nature143,241, 1939. H. F r 8 h 1 i c h and W. H e i t 1 e r, Nature 141, 37, 1938. H. FrGhlich, W. Heitler and N. Kernmer, Proc. roy. Sot. .4 186, 154 193R. H. ‘1. K r a m e r s, Hand- u. Jahrb. d. cbem. Phys. I, 5 61-64. W. Pa uli, 2. Phys. 43, 601, 1927. W. P a u 1 i, Ann. Inst. H. PoincarC 6, 130, 1936. H. A. K r a m e r s, Proc. roy. Acad. -4msterdam 40, 814, 1937. G. R a c a h, Nuov. Cim. 14, 322, 1937. A. P r o c a, J. Phys. Radium 7,347, 1936; 8, 23, 1937. P.A.M. Dirac, Proc.roy.Soc.A118,351,1928~ -W. G o r d o n, 2. Phys. SO, 630, 1928.
VI No 9
UNDOR
October
CALCULUS
1939 .
AND CHARGE-CONJUGATION
by F. J. BELINFANTE Instituut
voor Theoretische Natuurkunde
der Rijks-Universitrit,
Leiden
Zusammenfassung Grossen, welche sich transformieren wie Produkte 4-komponentiger D i r a c scher Wellenfunktionen, werden ,,Undoren” genannt. Zu jedem Undor I# kann durch Linearkombination der Komponenten seines komplex Konjugierten $* ein neuer ,,ladungskonjugierter” Undor 4s gebildet werden. Wenn + = $*, wird die Wellenfunktion + ein ,,Neutrettor” genannt. Undoren zweiter Stufe entsprechen gewisse Tensoren, Neutrettoren zweiter Stufe entsprechen reelle Tensoren. Mit Hilfe eines ,,metrischen” Undors zweiter Stufe werden ,,kontravariante” Undoren definiert, welche sich kontragredient zu den gewohnlichen (,,kovarianten”) Undoren transformieren. Endlich wird der ,,Gradient-Neutrettor” eingaftihrt. Resumo Kvantojn, kiuj transformigas kiel produtoj de kvar-komponantaj D i r a c’aj ondofunkcioj, ni nomas ,,undoroj”. El ?iu undoro 4 per unuagrada kombinado de la komponantoj de @a komplekse konjugito $* nova ,,.$.arge konjugita” undoro $” povas esti konstruata. Se + = 4”. ni nomas la ondofunkcion + ,,neiitretoro”. Duadtufaj undoroj reprezentas certajn tensorojn, dua$tufaj netitretoroj reprezentas realajn tensorojn. Per ,,metrika” undoro duagtufa ,,kontra?ivariantaj” undoroj estas difinataj, kiuj transformigas kontrafipase al la ordinaraj (,,kunvariantaj”) undoroj. Fine la ,,gradiento-neiitretoro” estas prezentata.
3 1. Irvtroduction. It is well known that - with respect to the restricted L o r e n t z group, excluding spatial reflections through
the origin l tensors can be expressed in terms of spinors l) “). As soon as spatial reflections are taken into account, however, it is necessary to consider @airs of spinors transforming one into the other by a reflection through the origin. An example of such a pair of spinors is the wave function of the D i r a c electron. In the following we shall investigate the properties of quantities transforming like products of such D i r a c wave-functions “). Such I’hysica
VI
849 54
850
F. J;.BELJNFANTE
quantities we shall call “undors ” *). They form a generalization of D i r a c wave-functions in the same sense as tensors form a generalization of vectors. Just as the representations of the L o r e n t z group by the transformations of most tensors are not azlsredzlzierf, so the representations by most undors are likewise reducible. In particular we shall discuss, in the following, the relation between undorsof the second rank and tensors, and the analogon in undorcalculus to real tensors : “neutrettors” 4). Finally we shall deduce the metrical undor and define the gradient undor. The whole set of mathematical relations will be built up in such a form, that we shall be able to apply it later on to the theory of mesons and neutrettos 5) 6) 7) “) 9). ‘.‘i i - _ 5 2. The D i Y a c wave-function (under of the first mnk). The D ir a c-equation of a positive particle (a positon or a proton according to the value of x = me/h) can be written in the following form:
if we put
(3)
Here e is the elementary charge (e > 0) and%, 8 is the potential four-vector of the electromagnetic field. As in (1) the interaction with heavy.quanta is neglected, this equation does not account for the anomalous magnetic moment of the proton lo) “) 6) 11). We shall call quantities transforming like the four-component D i r a c wave-function 4, four-spinors or ztndors of the first rank. In the following we shall often use a representation, of them, whi,ch is ausredzcziert with respect to the group of restricted L or en t z transformations, and in which the first two components of a fourspinor transform like the two-component quantity called a covariant conjugatedspinorby Van der Waerdenl),, Laporte and L?hlenbeck2), andcalledaregularspinorby Kramers12), whereas the last two components transform like the spin-conjugated 1) *) Drrivcd t) See.II.
from ttndn = wave. A. K r a m c r 8, lot.
cit. I*), pap
263.
UNDOR
CALCULUS
AND
851
CHARGE-CONJUGATION
of such a quantity, called a contravariant regular spinor by V a n d e r W a e r d e n 1) and others 2). Explicitly: \ x’ = x
h=h
y’ = y cos 8 + z sin 8
cos- ; + $2isin-
+; = t/q i sin7 e
z’ = - y sin 8 + z cos 3 t’ &t
+j = #,cos-
I
i
+ $,cos,
8
2
3.
~ .w 8 + cCdi.sin 2 2
8 +;=$3isinT++4cos$
z’ =z t’ =t x’ = x Y’ = Y
where
If we require that (1) is L o r e n t z-invariant, we find that the D i r a c matrices are give? in the representation (4) by (. + a = p.2 (5) and (6)
P = (A + 34&a if we denote by Pxt
PY,
Pz and
z{qx,
ay, 4
852
F. J,. BELINFANTE
the P a u 1 i matrices la) operating s of the four-spinor
From p2 = 1 we deduce A2 -
on the discrete arguments
I’ and
B2 = 1, therefore,
IO p =
OAO 0 OOA I/A 0 00 0 IplOO
(8)
with A = A + B = I/(A - B). Apart from a numerical factor the transformation-matrix of + for a spatial reflection through the origin 14) must be equal to p :
6’(- x, - Y, - 2, t) = iP #(x3 Y, z, t)* As a double reflection should not change the geometrical of the four-spinor, j must be a square root of f 1: p=
f
(9) meaning
1.
(10) By (4) a representation of the complete L o r e n t z group by transformations of 4 is not yet uniquely given. If we complete (4) by making a definite choice for the representation of reflections, the matrix p will be fixed by (9), (10) apart from a square root of f 1. On the other hand we may choose A = 1, that is, P = Pxi (11) the representation by + of the L o r e n t z group including reflections is then given by (9), (10) together with (4), apart from the same factor j in the reflection. In the following we shall denote the conjugate complex of a matrix by an asterisk *; by a cross t the adjoint (“Hermitian conjugate”) of a matrix. For instance
UNDOR
Further,
CALCULUS
AND
853
CHARGE-CONJUGATION
we put
for instance :
We introduce the nonrzalimtio~z- or density-matrix defined apart from a real numerical factor by p+s = ap,
-3s
==a:,
9, which
at =a.
is
(13)
Further, we shall postulate that under a linear transformation by a non-singular matrix S to another representation: 0’ z .y(J, clg’ = +t.q, p’ = spy,
;f
(14)
= S&l,
(14n)
the real expression 4Jtw4J shall be invariant,
so that 9’
=1
g-1
3, s-1.
(144
The definitions (13) are indeed invariant under these transformations (14n, b). In our particular representation (4), in which + q+ =;> (15) we have on account of (13), apart from a numerical 0
0
factor,
01
(16) after the choice (11) of p we have, therefore, 9 = 1.
(17) from
The density operator will remain unity so long as, starting the representation (4), (5), (1 I), (17), we admit only mitnvy tvansfowzatiom, for which sst= 1. (18)
854
F. J: BELJNFANTE
In the particular representation (5), (11) it is easily verified with regard to the complete L o r e n t z group, 4J+w+ is a scalar, and that -? 1 Ic = tJ1+4& together form a four-vector nuity equation
(19)
with p = #t&d,
(likeTwith
that,
(20)
ct), which satisfies the conti-
. . aP = 0 div +1 + at as a consequence of (13) and the D i r a c equation
(21)
(1) :
= ix($+p+th$ - ++q3+)+ $ #+{(22 - 23)s- a$ .: - %)}JI= 0. It is, therefore, ‘possible to regardTand p as the probability current-density and we shall normalize c-number solutions of (1) in one-particle wave mechanics by jfj
dx dy dz (+O$) = 1.
(22)
In the following we shall confine ourselves mainly to those representations, in which (17) is fulfilled, and we shall therefore drop S practically throughout. As regards the representation (4), this restriction (17) means taking A = A*, B = - B* in (6) and IhI = 1 in (a), so that P+ = P
(174
becomes a H e r m i t i a n matrix. As regards transformations by (14) to other representations, it means restriction to unitary transformations ( 18).
UNDOR
CALCULUS
AND
3 3. Charge-con.jugation of four-spinors. A matrix apart from a complex unity factor eic by *) &=-s;*,
855.
CHARGE-CONJUGATION
p;E=-sdr:p*,
S is defined
$X*=1;
(23)
so where sx = - iayaz, so that in the representation
In this representation
the restriction
‘(24)
(5), (6) +’ + 5 = a.
(25)
(5), (8) this matrix
b 0 2 = eic 0 -ll/hl therefore
cycl.,
0
aEis equal to
0 -pL(
0 IhI II/Al0 0 0
0 0’ 0
(26)
(17) makes it equal to di = 8 . pyay
(27) in the representation (4), (5), (8). In the following the particular representation (4), (5), (1 l), (17), (27) with eic = 1 will be called the K r a m e r s 15) representation: + ;E=pyfxy, 9=1; ;==: a = ha,-+ p=px, (28) By means of the matrix ;Ewe construct plex (+*) of the four-spinor wave-function (I), another four-component quantity
from the conjugate com(4) of a positive particle
#” = Is+*.
(29) From ( l), (3) and (23) we can easily deduce that 4” satisfies the equation 15) {ix + (;. 5)
+ f3D*o}tp = 0,
(30)
that is, the wave equation for a negative particle (a negaton or a hystaton t) according to the value of x). For this reason +G is called *) This matrix P is identical with the matrix C* introduced 71 Hystaton (antiproton) is derived from 6a-;ctro& = last;
by P a u 1 i, lot. proton fromnporoq
cit. Id). = first.
a56
F. J. BELINFANTE
the &urge-cortjtigated 15) of 9. From (30) we can conclude 14) that with respect to restricted L o r e n t z transformations - the chargeconjugated of an undor of the first rank is again an undor 15). As regards reflection: from (30) we can conclude only that the transformation of C/J~must be again of the form (9). But the representations of the complete L o r e n t z group by the transformations of 4 and by those of v might be different with respect to the sign of i. In order to examine this, we compare the charge-conjugated four-spinor in the reflected system of co-ordinates, which is defined bY $p E $1” = go’* = j*gp*+*; (31) with the charge-conjugated four-spinor 4” in the original system of co-ordinates. This last, (29), will be transformed into (31) by (9) with perhaps a different value of j, - say i@) : Jr”’ = jw p 0” = p p -g+** (34 From (31), (32) and (23) we find 1w = _ j**
(33) The charge-conjugated of an undor of the first rank is therefore itself an undor with respect to the com@ete Lo r e n t z group (jt8) = 11, if i= - j* (sothatj= fionaccountof (10)). (34) We can also prove directly that +” is an undor with the choice (34) of i, without making use of the equation (30). Let A be the linear operator of some Lo ren t z transformation of the undor 4. Then v transforms like an.undor if q” = A(& (35) or sn* +* = fqJ*, (354 that is, if ne = %A*. (36) Indeed this condition is satisfied for the restricted transformations on account of (23n). For the reflection fied, if j (3% = ;Ej* f3*, or, on account of (23), if i = - i*.
L or en t z (9) it is satis(37) (34)
UNDOR
CALCULUS
AND
857
CHARGE-CONJUGATION
This result of RI a j o r a n a “) and R a c a h 16) means that it serves a useful purpose to define the reflection of a D i r a c wavefunction in such a way that a double spatial reflection through the origin inverts its sign. In the following we shall see that this same definition enables us to describe the P r o c a field (that is, a field consisting
of a four-vectorx,
V and an antisymmetrical
tensor of
the second rank s, q by means of a sy;rmetrical undor. Since according to (34) $” can now be regarded as a regular undor, it is natural to postulate that by a transformation (14) to another representation of undors, the undor $” shall be transformed in the same way as all other undors $. This means that (If’ E 0’” = S’ q* must be obtained
= g’s* +*
(380)
from GC(29) by a transformation $f’ zzzsy
This holds independently
(14) :
= s&+*
W)
of the choice of $, if S’S* = S ;E,or ;E’ z.z sss*-‘.
(14c)
Under the transformation (14n, c) the definitions (23) are indeed invariant, that is to say, if ;Eis defined in one representation in accordance with (23) and is then transformed to another representation according to (14c), the relations (23) will hold again between the transformed charge-conlugator g’ and the transformed D i r ac matrices
p’, 2, etc. In the same way the relation 85 = ;f”8*
= (a-&)“,
(234
which is valid in Ii r a m e r s’ representation, is invariant by a transformation ( 14b, c) and in consequence holds in every representation of undors. It follows from (27) that ;E = S” (17b) holds in the representation (4), (17) ; and from (14~) we deduce that this relation (17b) holds in all rc#wesentations (17), for which tlac density matrix is unity, because (17b) is invariant by a unitary transformation (St = S-l, S” = S*-I) : P=
=
g-1
SWS”
=
sgs*-1
=
g
858
F. 1. BELINFANTE
If F is an operator, which operates on four-spinors, a charge-transformed operator F& by
we can define
(F+)E = F”+“,
(39)
so F” = fF*F.
(40)
On the other hand, if F depends on the electric elementary c, we can define the charge-inverse FL of F by {F(e)}= = F(-
Then we can summarize the connection
e).
charge (41)
between (1) or
and (30) or H$J8 = &O&8 = -
&,+”
(304
by stating that H&., = i
H$,,
&,L,,= -
&$.
(42)
K r a m e r s 15) has pointed out that a description of electrons by means of 4 and Hop, and a charge-conjugated descrifition by means of 9” and HoLp, should be equivalent. It can be shown that the meson theory shows a similar kind of charge-invariance. 5 4. Neutrettors of the first rank.
We shall call self-charge-conjugated
four-spinors v=+
(43)
neutrettors of the Jirst rank *). These quantities are adequate for the description of the neutral particles of the theory of M a j o r a n a “). It can easily be shown that M a j o r a n a ‘s “real” D i r a c wavefunctions are exactly what we call neutrettors. Proceeding by a transformation (14) with
s = St-1 = (1 + i&csY)/dZ *) The name ‘“neutrimws” Editor in Nature a)), seems described by these quantities. described.
(44)
(in analogy to spimws proposed by me in a Letter to the to be less adequate, since it suggests that neutrinos can be By neutrettors, however; only photons and neutrettos are
UNDOR
CALCULUS
from the Kramer representation
AND
CHARGE-CON
a59
JU‘GATION
s representation
(28) to a Ma j or an a
;EMAJ
=
we deduce from (23) : --f
-
1,
+
8MAJ
(45)
1,
+
(45Q) This is indeed the characteristic of the representation discussed by Majorana*). Four-spinors, which are self-charge-conjugated, are on account of (29) and (45) real in a A/r u j o Y n n a ~efiresentahow. According to M a j o r a n a “) neutrettovs can describe neutral particles t). For the purpose of building up a canonical theory of M a j o r an a particles the K r a m e r s representation is very convenient. Denoting in this representation (28) the first two components of d, as a Kramer s spinor by P%AJ
=-PMAJ,
Y%AJ’-YMAJ,
and the last two components K r a m e r s spinor v by
aitfAj=:MAJB
ZAJ=-G*AJ.
as the spin-conjugaled
01 of another
(464 we can write (29) in the following
form:
(48) This four-spinor
is a neutrettor,
II
if
’ rS. = J1 = +” = / t, / , that means, if u E v. *) The
representation actually used by by taking S = tdi.(&(l -a~) t) Note added in proofs. The Ma j or transformation properties of !Z have also Rev. 64, 56, 1938.
(44), but
(49)
M a j o r a n a is obtained from (28) not by + ~~(1 + uy)} in (14). an a theory of neutral particles and the been investigated by W. H. Furry, Phys.
860
F. J. BELINFANTE
In other words, a Ize&rettor of the first rank consists of a two-qbinor and its spin-conjugated and can be written as
in a Ii In a should theory; theory
r a m e r s representation. canonical theory of neutral particles only the two-spinor u be regarded as a canonical variable like + in D i r a c ‘s U* takes a place comparable with that of $* in D i r a c’s and %s can be expressed in terms of u*.
$5. Undors of the second rank. We shall call a sixteen-component quantity Y W, = ys,*,,s,rs (A,, k, = 1, 2, 324; SI,Y~, ~2, ~2 = + &, - 4) (51) transforming like the product dlR,+;, = +s,,r,+:,,r, of two fourspinors an zrndor of the second rank. With respect to the undor-indices we regard it as a matrix with oozecolumn and 16 rows. Still, we shall wyite it as a square matrix with 4 x 4 elements:
yk,k,
‘r,,
‘r,,
‘r23
‘Y;4
y31
y32
y33
‘r,4
‘r,,
‘r42
‘r,,
‘yk4
(54
= -
Matrices like &‘), #, &I, &, p’“‘, +, $0, $1, gh), gw, p,, etc. are assumed to operate on the argument k, of \rk&. Taking these operators as unity matrices with respect to the index, on which they do not operate, we can regard them as matrices with 16 rows and 16 colums. With respect to the restricted L o r e n t z group an undor of the second rank represents one regular spinor, one conjugate complex spinor and two mixed spinors- of the second rank; it represents, therefore “) i2), two four-vectors 7’K, K” andT, Lo (transforming ;;l’ct), two scaiars F, and Go, one regular complex three-vector Kennzahlen of an antisymmetrical self-dual tensor 2))
like (the
UNDOR
CALCULUS
and one conjugate
AND
861
CHARGE-CONJUGATION
complex three-vector z’=
22 + is*,
W)
where z,, g, and &,, gz form two antisymmetrical tensors of the second rank. Generally all these quantities are complex. In the K r a m e r s representation (28) we can write: Yll
Yl2
Yl3
y14
‘r,,
y22
‘r,,
‘r,4
‘r,,
‘r32 ‘r,,
‘r,
‘r,,
‘r,,
‘r,
‘r,
=
4
F,- iF, Fo--Fz - F,--iF, -F&-F, L,- iL, LO-L, -LO-L, -L,--iL,
K,--iK,,
-K”-K,
-F;2y
-Fs”G” . (54) 0 I -G:, + G, G, + iG,
In an arbitrary representation of undors obtainable from (28) by a transformation (14) we should replace the left hand member Y of (54 by’ cp-~ S(W y = y (55) By spatial reflection according to (9) by
through the origin the undor Y will transform y’ = j’p(l) (j(2)y*
(56)
In the representation (28) used by us we have f3(1)p(2) = eg)&?), so that this transformation (56) can be written, on account of (54), as
i? = (2 + i&/i,
K” = (V + W)i/i,
z = (X -i&/j,
LO = (V -
z = (ii - i&/i,
F,=(-
G’= (ii + i&j,
Go = (S - iY)i/j,
w)i/j, (58)
the new quantitieix, respect to restricted
S - iY)j/i,
V; $, W; s, g; S and Y are still tensors with L o r e n t z transformations (two four-vectors,
862
F. J. BELINFANTE
an anti-symmetrical tensor and two scalars), whereas we can ‘now write (57) in the following form: ~~-~, g
V’=-j-v;
= + g, w’ = -
w;
j&-s,
ii%+&
y’ = -y;
S’ = + s.
(59)
With respect to the complete L o r e n t z group including reflections, an undor of the second rank consists therefore of a regular scalar S ; a regular four-vector X, V; an antisymmetrical tensor of the second rank%, G (which can be regarded as a pseudo-tensor of the second rank g, g) ; a pseudo-four-vector (that is, an antisymmetrical tensor of the third rank) 2, W, and a pseudo-scalar (anti-symmetrical tensor of the fourth rank) Y. It is often assumed “) that the meson field can be regarded as a P r o c a field 17)) that is, a field consisting of a four-vector x V and a six-vector z, 2 only (case (b) of K e m m e r “)) . Such a field can be described by an undor of the second rank satisfying the relations Y,, = - j2Ys1, Y,,=
- j2Y4,,
‘r,=
- i2Y32, Yz4= - i2Y4,.
(60)
The most symmetrical method to achieve this and at the same time the one and only possibility to achieve (60) in a way which is independent of the representation of undors, that is, invariant under the transformation ‘r’
= 9,Smy
(55a)
.’ .
is to postulate
that the undor describing the P r o c a field must be with respect to its two indices, and at the same time that for undors i2 in (9), (10) must be equal to .minus unity: symmetrical
y kd, = Yk,k,, i2i-1. Indeed, in general a symhetricnl sents : _
(61) -
(610)
undor of the second rank repre-
(b). a regular 4-vector 2, V and a-6-vector Ey H:if i2 = -
1; (62)
(c). a pseudo-4-vector.g,
Wand a 6-vector c I$ if jz = + 1;
UNDOR
CALCULUS
whereas an antisymnzetrical
AND
863
CHARGE-CONJUGATION
undor of the second rank represents:
(d) . a pseudo-$-vectors,
W, a scalar S and’a pseudo-scalar Y, jfj2 = - 1;
(a). a regular 4-vectorx,
V, a scalar S and a pseudo-scalar Y, if i2 = + 1.
(621)
Here (a), (b), (c), (d) refer to the tensors composing the field in the four cases considered by K e m m e r 6). Now we define the charge-conjqated of an undor of the second rank by and its charge-adjoint
YGM, = 22’) L!z2)(Yk,k,)* by
(63)
YEk, = ‘ezk, = it?‘)s(2) (\rk,k,)*.
(634
Then WI = w-M1! (69 is a self-charge-adjoint undor of the second rank. Now we can express the tensors represented by Ys in terms of those represented according to (54), (58) by ‘I!‘. In this way we find from (28), (54) and (58) :
-;jr! =&F ) V”=V*;
&=E*
‘i2
yg = y*; S” = s*.
=s*,
WC = w*;
> gL$b
,
(64)
We observe that by the choice of constants made in (58) we have achieved that the tensors represented according to (54), (58) by the charge adjoint Ys of an undor of the second rank Y, are the conjugate complex of the tensors represented by the undor Y itself *). If now a neukettor of the second rank is defined as a self-chavgeadjoint undor of the second rank, it represents by (54) and (58) according to (64) a set of real tensors. Such neutrettors are therefore adequate for the description of the M a x w e 11 i a n field “) and of K e m m e r’s neutretto field 7). A specimen of a neutrettor of the second rank is given by +f C&(63b). +
+++
*) The constants in the definition by (54), (58) of the tensors S; A, V; E, II; U, \V and Y in terms of the components of the undor \r are uniquely determined by the conditions (59) and (04) apart from arbitrary real numerical factors to these tensors, which are all chosen equal to unity in (58)
864
F. J. BELiNFANTE
Taking i=i
(344
in the following, in accordance with (34) and (6 la), all factors i/i and i/j vanish from (58). 5 6. Covardant u&or calcdus. The fact that the linear combinations (29) (and, apart from linear combinations differing from (29) by a numerical factor, only these) of the components of a conjugate complex undor Jr: form again the components of a regular undor dCR, enables us to define a metrical z&or &, that is, an undor of the second rank, which connects contravariant and covariant undors with each other. Since, on account of (19), (23) and (13), the expression ++%f+$ = (it P+fb+ = ,“, 4; (~+%)ik+k = Fk (t%“”P+c’J)k’4; (,k = = c (tb*p*)”
(Jf $k = r, x” $A
(65)
i,k
is a scalar, we can regard x” = f (a+* f3*)k~~~
(66)
as a regular contravariant undor. We shall connect with this y,” an ordinary covariartt undor ~1by xl
=
T
f&k xk
=
Fk f&k
(s*
p*)
“4:
.
(67)
Since x, shah be a regular undor and its components are linear combinations of those of +X, the undor a must be equal to (v)l apart from some numerical factor, for which we shall choose unity : x‘=
$5
As this result should be independent T glk (8*p*)k” l
(68)
of $, we find
= SF, or gtb*p*
= it
! 69)
We conclude : g = g(j*B*-‘.
iv
UNDOR
CALCULUS
AND
CHARGE-CON
In Kramer-s’ representation metrical undor g,, takes the following
(S(3 = pX, E = p,+) form :
0 -1
g=that
ip,a, =
865 .
JUdATION
.the
00
1 0
0
00
0
01
0
0 -1
0
is, +1
=
-
02,
=
$2
qJ1,
$3
=
+4,
94
=
,-
q.J3.
(7Ia)
We observe that in this representation the metrical undor is antisymmetrical just as the metrical spinor is in spinor calculus and unlike the metrical tensor in tensor calculus (which is symmetrical) : lr = - g kkl = - l&k). (72) This property of the metrical undor is invariant under transformations (14) to other representations. This follows from (70), (14a, b, c) : g;Q, = S’ p’*a’*--l
= se p*a*+
S” = sgs-
A consequence of this antisymmetry %‘$k
=
-
= z Sj’l g,, S&k. (144
of g is (73) undors are con-
i?Jd#‘*
Conjugate complex contravariant and covariant nected by the coqkgafe comfilex metrical w&or g;i
=
(f&k)*
=
F
(74)
Efk(p-l)k!s
This undor is in K r a m e r s’ representation 0 -1 00 1 0 00 0 0 01 0 0 -1 0
g;& =
and transforms
to another
The contravariant
representation g*’ = s* g*st.
metrical u&or f
given by
by (144
g’” is determined
filkgkm
=
(744
by (75)
v
or gkm
Physica
VI
3
g-1;
gdi
s
gtd.
VW 55
866
F. J. BELINFANTE
In K r a m e r s’ representation 010 -100 000-l 001
gknr =
0 0
we find from (71) and (740) 010 -100 000-l 001
g *iA _-
0
0 0 (76) 0
Transformation to another representation changes the contravariant metrical undors gkm and g*kd according to (14d, e) and (75~2). Co- and contravariant undors are now connected by
Here the summation must always be carried out with respect to the last index of the metrical undor. From (65), (68), (73) it follows that ++ 9 p t,, = T (fk +k = -
: +??(,“.
(19fi)
In the same way we derive $+ap+-q;(J~*”
=-q)*“!.p;.
(19b)
For the current-density four-vector (20) we can derive expressions, for instance *) -+ i/c = c ,q #I) p = z# tjJfk p; (JI Inserting tion
similar
P@)
(46a, b) into (71a) we find in K r a m e r s’ representa!+LUi,
therefore, following putting z’s 3 w:
$2=-q,
(/3=-+2,
!+4=7)5’,
(78)
V a n d e r W a e r d e n’s notation
1) and
(784
*) Compare
P. A. M.
D i r a c, lot.
cit. I*), and W.
Go r d o n, lot. cit.
10).
UNDOR
CALCULUS
AND
863.
CHARGE-CONJUGATION
The difference between our undor calculus and Van de r W a e r d e n’s spinor calculus is, that we have taken care from the beginning that the transformations of “contravariant” undors should indeed be contragredient to those of “covariant” undors by all transformations of the complete L o r e n t z group iuzchdiwg spatial reflections.
’
We might have derived (14e) and (76) in a simpler way. Making use of our knowledge of V a n d e r W a e r d e n’s metrical spinor, we conclude that the contravariant metrical undor gkn must have the form OaO 0 -a00
0 U-64
000-b
OOb ;o I in K r a m e r s’ representation, in which the first two components of a four-spinor behave like the components of a conjugate complex covariant V a n d e r W a e r d e n two-spinor and the last two components like a regular contravariant V an d er W aer d en spinor. The ratio (a/b) is now determined by the condition that, if ‘p and 4 are two arbitrary regular covariant undors, the expression (Pk gk
+I
=
‘pw
g-’
(79)
+
shall be a scalar with respect to the spatial reflection (9), (10) (as well as it was, on account of (76a), a scalar by restricted L o r e n t z transformations). In K r a m e r s’ representation we find from (76a), (9) and p = pX (28): 4%
lJl2 -
‘p2 41)
-
b(cp3
$4
-
94 G3) =
=
44
a ?(ff3
= $1
-
9;
44
$4
-
(~4 $3)
-
WP;
-
b iVf9
44
$2
‘pi 44) -
7
92 44,
therefore, a/b=-12=-
111~.
(76b)
Choosing j according to (34) we find a = b and (76a) becomes identical to (76) for K r a m e r s’ representation (28). If, now, we fiostdate that the scalar (79) shall be an invariant under
a68
F. J; BELINFANTE
transformation
(14) to another representation,
W’,’
=
p-1
g-1
we find
p-1, (14il
in accordance with ( 144. A still shorter way of deriving (71) is by making use of (54) and (58). Since the metrical undor is supposed not to change its form by L o r e n t z transformations and spatial reflections, it must represent a scalar. Therefore it must have, according to (54), (58), the following form in K r a m e r s’ representation : 0 -s gkl
=
4
0 0
s
0
00
0
0
OS’
0
o-s
-
0
Takings = 2wefind (71). We remark that g is a nezltrettor of the second rank on account of the special definition (63~) given for the charge-adjoint of an undor of the second rank. In the literature use is often made of an abbreviated notation for the contravariant charge-conjugated of a regular undor & or yk,k,:
(80) Further,
we shall denote by ap and yp (p = 0, 1,2,3)
h
the matrices
a0 = - a0 G 12
a2, a3) = {al, a2, a3} =Z,
(24 {Yl,
Y2,
Y3)
The probability then given by
The relations
=
{y’,
y”,
density
y”>
=r;=
PZ
and current
Y0 =
-
ypS = -
Ey’**,
apt3
=
P.
of a II i r a c electron
(13), (23) and (23~~)can now be written pts = 3 p,
Yo
are
as
=4;ap,
a@% = SaF*.
(81)
In undor caIculus the gradient four-vector VP is, according to (54), (58), (60), (64), represented by a symmetricalneutrettor. In K r a-
UNDOR
CALCULUS
m e r s’ representation form :
AND
this gradient 0
neutrettor
v, -
VZ
c at Vkl
---- ia c at
- v,-
has the following
--ia Y-.Vi c at
iv,
---_la
0
869
CHARGE-CONJUGATION
-v,-
iv, =
v,
0
0
iv,
0
0
Vlk
(82) .
The D i r a c equation of a free electron can now be written in covariant undor-notation : {ix + y’ VP}+ = 0 -+
i&
+ vkl+’ = 0.
I am much indebted to Prof. Kramer s for many on the questions treated in the present paper. Received
July
14th,
(14 discussions
1939.
REFERENCES 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 9)
B. L. v a n d e r W a e r d e n, Giittinger Nachr. 1008, 100. 0. Laporte andG.E. Uhlenbeck, Phys.Rev.37,1380,1931. F. J. B e 1 i n fan t e, Nature 143, 201, 1939. E. Majorana, Nuov.Cim.14, 171, 1937. H. Y u k a w a, S. S a k a t a and M. T a k c t an i, Proc. phys. math. Sot. Japan 30,319, 1938. N. K e m m e r, Proc. roy. Sot. A 106, 127, 1938. N. K e m m e r, Proc. Cambr. Phil. Sot. 34, 3.54, 1938. H. Yukawa, S. Sakata, 111. Kobayasi and M. Taketani, Proc. phys. math. Sot. Japan 80, 720, 1938. C. Meller andL. Rosenfeld, Nature143,241, 1939. H. F r 8 h 1 i c h and W. H e i t 1 e r, Nature 141, 37, 1938. H. FrGhlich, W. Heitler and N. Kernmer, Proc. roy. Sot. .4 186, 154 193R. H. ‘1. K r a m e r s, Hand- u. Jahrb. d. cbem. Phys. I, 5 61-64. W. Pa uli, 2. Phys. 43, 601, 1927. W. P a u 1 i, Ann. Inst. H. PoincarC 6, 130, 1936. H. A. K r a m e r s, Proc. roy. Acad. -4msterdam 40, 814, 1937. G. R a c a h, Nuov. Cim. 14, 322, 1937. A. P r o c a, J. Phys. Radium 7,347, 1936; 8, 23, 1937. P.A.M. Dirac, Proc.roy.Soc.A118,351,1928~ -W. G o r d o n, 2. Phys. SO, 630, 1928.