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On the statistical behaviour of known and unknown elementary particles
Physica
VII,
no 3
:
Maart
1’940
ON THE STATISTICAL BEHAVIOUR OF KNOWN AND UNKNOWN ELEMENTARY PARTICLES by W. PAUL1 I’hysikalischcs Institut Instituut voor
and F. J. BELINFANTE
dcr EidgenGssischen Technischen Hochschule, Theoretische Natuurkunde der Rijks-Universiteit,
Zurich (Schwciz) Leiden
Zusammenfassung Es wird untersucht, in wie, weit Spekulationen iiber die Statistik willktirlicher, hypothetischer Teilchen in einer relativistisch invarianten Theorie moglich sind, werin alle oder ein Teil der folgenden drei Postulate vorausgesetzt werden: (I) die Energie ist positiv; (II) Observable an verschiedenen Raum-Zeitpunkten mit raumartiger Verbindungslinie sind kommutativ; (III) es gibt zwei Hquivalente Beschreibungen der Natur, in welchen die Elementarladungen entgegengesetzte Vorzeichen haben und in welchen einander entsprechende Feldgrossen sich bei L o r e n t zTransformationen in gleicher Weise transformieren. Der eine von uns (P.) hatte bereits gezeigt, dass allgemein bei ganzE i n s t e i n-B o s e-Statistik, bei halbzahligem Spin aus (II) allein zahligem Spin aus (I) allein F e r m i-D i r a c-Statistik folgt. Ferner hatte der andere von uns (B.) gezeigt, dass fiir eine gewisse Klasse von Teilchen, (die alle bisher in der Natur beobachteten umfasst, und die dadurch‘charakterisiert ist, dass sie hochstens durch einen Undor von gegebener Stufe beschrieben wird), aus dem durch ein spezielles Transformationsgesetz spezialisierten Postulat (III) fiir ganzen Spin E.-B.-Statistik, fiir halbzahligen Spin F.-D.-Statistik gefolgert werden kann. In der vorliegenden Note wird in den typischen Fallen von Spin 0 und Spin Q durch Beispiele gezeigt, dass im allgemeinen Fall mehrerer Undoren von gleicher Stufe aus (III) nicht mehr eindeutig auf die Statistik der Teilchen geschlossen werden kann, wahrend (II) bzw. (I) fiir ganzen bzw. halben Spin hiezu stets hinreichend bleiben. In den speziellen Fallen des Skalarfeldes, des Vektorfeldes und des D i r a c-Elektrons, wo nur ein einziger Undor gegebener Stufe in die Theorie eingeht, folgt dagegen das Transformationsgesetz der Ladungskonjugierung eindeutig, so dass hier (III) zur Festlegung der Statistik ausreicht.
In a recknt paper one of the authors 1) has a general principle, frpm which the statistical behaviour
3 1. Introduction.
indicated
Physica VII
177 12
178
W. PAUL1
ANll
F. J. BELINFANTE
(that is, the sign occurrin, u in the commutation relations of the components) of all particles and quanta empirically known until can be deduced in a simple way. This principle was the following. the field of the particles and quanta in question be described by of undors “) yk ,....R,,,which a,re functions of I the co-ordinates
field now Let a set
and which satisfy certain first order partial differential equations (field equations), in which some constants of the dimension of a charge may occur. Under an infinitesimal L o r e n t z transformation or spatial rotation of the frame of reference, so that the components of a given vector 5 are transformed according to *) sy = & &JP”, these undors are transformed
(1)
according to “)
m = 4 8w~” r, rr Y,
(2)
where
Here the y:) are matrices operating on the index kj of Yk,..,.R,, and satisfying the relations $ yq G $7 Yt” + $J Yll” = 2gp,. (4) Operating on’ the conjugate linear operator
complex Yt....i,,
of an undor by the
g = fi gn,
(5)
j=l
where g(i) is a matrix fying the relations
operating
on the index hi of Y~I,,..~,, and satis-
$’ Qi, = .gcny(j)* >
(6)
pe find again an undor ‘P = ix*. *) Greek indices run from 0 to 3: summation signs are omitted. the definition (4) of the matrices ylL, which differs a little from previous papers of one tif the authors I) *) 8) ‘). ’
(7) Attention is drawn the definition given
to in
STATISTICAL
BEHAVIOUR
OF ELEMENTARY
179
PARTICLES
.
We can normalize
E in such a way that ‘pe = y,
(8)
by putting jp g(i)* = 1. Instead of by (5)-(7), from Y* by
we might have constructed \r(f.’ =
cgi,diy*,
where G is a scalar operator - for instance a product number of factor yt), which are defined by t) yf’ = iy’f’ yy yy’ y$
(9) an undor ‘IF) . (10) of an even *)
(11) In a similar way we can form an undor again from Y* by means of *) YQ = QY*
or Y(Q) = GQY*, with Q = yi’)E,
(14
so that yQQ = - y.
(13) Now, it can be shown “) ‘) that, if the undors Y describing the fields of experimentally known particles or quanta (such as electrons, nuclons §), photons, mesons), are transformed according to (7), we obtain a set of undors Y2, which satisfy again a set of field equations, in which, however, all (real) constants of the dimension of a charge, for instance e, have been changed into their opposite: eL = - e* (14) This is true only “) for transformation *) If we require reflection through
that uf and yg shall transform the origin, this reflection must
T1 . ...k.. = f
Yip
according to (7) and not to in exactly the same way under be given by
a spatial
Yy Q....k,.
An under, for which the reflection takes this form, we shall call an ordirrary undor. If we require that y and 'I'Q shall transform in exactly the same way, however, must not be an ordinary under, but a quantity (,,quasi-under”), for which reflection defined by . . . . yo(d Yk 1.. .k *. %, . . ..k. = f iyp An odd number of factors yv) changes an ordinary undor into quasi-under into an ordinary one. t) Attention is drawn to the fact that there is a difference of defined in this paper I) and the ys defined in some papers of one of 5) The particle that is a proton in its charged state and a neutron have called a nuclorr.
a quasi-undor,
and
\r is
a
a sign between the ys the authors “). in its neutral state, we
‘1’80
W.
(12). Further,
PAUL1
AND
F. J. BELINFANTE
of the field equations ‘and a similar formulae (such as for instance the total energy or the total electric charge) exist ody on account of the commutation relations holding between the undors. By the transformation invariance
this invariance
of all Physically significant
Y + YL together therefore,
with e + eL,
(15)
where YL is given by YL = Y”,
(16)
we pass from one description of the fields of particles and quanta (by the undors Y) to another description (by Yg), which is conqbletely equivalent to the original one and was called - according to a terminology of K r a m e r s “) - the charge-conjugated description of the physical world. On the other hand it is plain that, postulating that the transformation (15) specified by (16) shall leave all physically significant quantities unaltered *), we obtain some information about the commutation relations holding between some undor components. This information is sufficient, indeed, to determine the sign occurring in the commutation relations, that is, to make a definite choice between commutativity and anticommutativity ,of the field components Y and Y*, though the c-number occurring in the usual commutation relations (for instance some a-function) remains entirely undeterm.ined in this way. It was shown by the one of us 1) that for a certain collection of hypothetical particles t). described by one undor of an arbitrarily given rank it follows from this postulate that particles with an integer spin must satisfy E i n s t e i n-B o s e statistics and those with a half-odd spin must satisfy F e r m i-D i r a c statistics. On the other hand it was shown by the other of us ‘), in a very general way, that a similar result for arbitrary particles described by a field satisfying a set of linear homogeneous differential equations could be obtained from the following two postulates : of the theory. Since the trans*) One may call this property the .,charge-invariance” formation (15) was specified by (16), we shall call it spccijied charge-invariance in the following. We admit that the name ,,charge-invariance ” is a little misleading since this property may exist for neutral as well as for charged particles. t) All particles experimentally known until now are contained in this collection.
STATISTICAL
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181
(I) for these particles there is no infinite number of stat& of negative energy ;
(II) observables in points connected by a space-like vector aye purely comvnuta tive. These two postulates can be understood from a physical point of view. The first postulate is necessary since otherwise particles would drop into states of lower and lower energy creating an infinite number of quanta or pairs of particles. According to the second postulate simultaneous measurements in points connected by a space-like vector must be always possible; as it should be, since the field in the one point cannot influence the field in the other point, if the connecting vector is space-like. Invariance under the transformation (15) with the specificatiofi (16) is a specialized case of invariance under some unspecified transformation (15) of the undors together with inversion of the signs of the constants of the dimension of a charge (14). Invariance of all physical quantities under another transformation than (16), but still of the type of (14) - (IS), would also have meant that there is a description of the physical world, in which every “elementary charge” has the opposite sign. The existence of such a generalized charge-conjugated description may perhaps be infered from some speculations on a possible symmetry between positive and negative charges, which - though not existing on the earth - may still be a fundamental property of nature. (Anyhow, we know that it exists for all known particles l)). Therefore, without specialization by (16) *), we shall postulate (III) invariance of all physically significant formulae and quantities under some transformation (1 S), where Yp represents some set of undors +(l), $t2) . . . and YL a similar set of undors +(uL, +c2jL, . . . . transforming in the same way as ‘I!. We do not postulate, however, that after the transformation from Y? to YL - which we shall call “unspecified charge-conjugation” - the components of the undor #(“jr. are linear combinations of the components of only +(“)* again. In the following we shall discuss the bearing of the postulates (I), (II) and (III) on the question of the statistical behaviour of hypothetical particles with a spin 0 or B/2. For a spin 2 tZ similar considerations - though more complicate - are possible. *) One might of the theory.
call
the property
postulated
by (III)
the
,,unspecified
clrarge-illvariallce”
182
W.
PAUL1
AND
F. J. BELINFANTE
$ 2. Particles OYquanta with a spin 0. In this section we shall discuss some possibilities of a quantum-theory of particles, which are (at least partly) described by a (complex) function s of the co-ordinates Y, which is a scalar at least with respect to spatial rotations and L o r e n t z transformations and which satisfies - if all external forces are neglected - a K 1 e i n-G o r d o n equation
(0 - 2)s = 0,
(cl = VT,;
v, = a/w;
K = mcpi).
Here we shall discuss the case of particles or quanta vanishing mass (x $r 0) *). Putting in this case
(17)
with a non-
qb = V”S,
we find from (17) :
(18)
its = V’cp,.
(19) The equations (18)-( 19) together form the first order field equations. They can be derived from a L a g r a n g i a n Li =I J-fff
L dx dy dz dt,
(20)
where the Lag r an g i an function L depending on the field components s, (py,s*, etc. and (linearly) on their (first order) derivatives is given by L = I< {rpp*“(Xcp”-
V&s) -
s*(xs -
so that, if K is a real constant, the total is real. For the total energy we find “) M== xKfff
relations,
= c(x, x’) ;
[s(x) ; s(d)]
= [s”(x) ; s*(i)]
with
a wwislting
They
can be derived
vp,
energy
= 0;
order
equations
(24) are given
by
(l8--19a)
= 0.
from L = R { p*“(p”
and the total
(24
(23)
mpss the first
p” = v,s,
c-number).
we shall asszlme that
[s(x) ; s*(d)] particles
(21)
L a g r a n g i a n fi (20)
{s*s +VfO$‘pV} dx dy dz (-
As regards the commutation they are of the form
*) For scalar
V/f”)},
is given kf=
- V,s)
(Zln)
+ s*V,(p”]
in this case by ll’,~o~:~V
dn dy dz (-
c-number).
The further discussion of the possible commutation relations tion in this c.ase is analogous to that for scalar particles witn
and laws of charge-conjugaa non-vanishing mass.
!2&)
STATISTICAL
BEHAVIOUR
OF ELEMENTARY
where the bracket symbol [A-; B] denotes either [A ; B]- or the anti-commutator [A ; B] + : [A;B]*
183
PARTICLES
the commutator
=ABfBA,
(25)
and where c(x, x’) is a c-number function of X, and xi. The commutation relations for the other field components are derived from (23)-(24) by means of the equations (18). From (23) - (24) we conclude that c is a scalar function, which, on account of (17)) must satisfy the equations (0 - 2) c(x, x’) = (0 ’ - 2) c(x, x’) = 0,
(26)
( Cl ’ operates on x’ like 0 on X) . Further we remark that the postulate (II) of 5 1 is satisfied, if we assume for all pairs of field components ql, q2 : [qi;, whe,reTdenotes
x,,) ; q2G, xc,)] = 0 for x0 = xi but : #2,
(27)
xi, x2, x3. Since (27) should hold for s with s* as well
as for s with ‘po, * both c and Voc must vanish for x0 = x& z# ~7 so that from the scalar character of c it follows that c must be equal to 8) ‘) c(x, x’) = ic . 9(x - x’). (28) Here C is a constant and
S(x) = SIG x0)=
(&-)“~~~~1‘&.’
sin;oxo dk, dk, dk,
(29)
(k” = + 2/k: + k; + k: + x2) is a solution of (26) satisfying
the conditions
sgo,=0;(g0)x*=0 =s$j. sb=sb*; All other solutions
of (26) are excluded by the postulate
(II). We
remark that 9 depends on 1;I and x0 only and is an odd function of x0, so that B*(x) = - %I(- x). (31) The commutation
relations now must read [s(x); s*(x’)]-
= iaD(x
- x’),
(32)
184
‘W.
since F erm i-Dira for x = x’ (compare
PAUL1
AND
F. J. BELINFANTE
c statistics are excluded (23) with (28), (30)), from
by the fact that,
ss* + s*s = 0
(33)
it would follow that the field vanishes entirely ‘) : s = 0. Further
iC must be purely’imaginary _
[s(x); s*(d)]*
(34) on account of (3!) and
= [s(d);
s*(x)].
(35)
The total energy (22) is automaiically positive, in.accordance with postulate (I). If e is the charge *) of the particles under consideration, we can introduce their interaction with the M ax we 11 i a n field by changing in (17) - (2 1) V, into a = 9” + (e/W% > (36) where 5?&denotes the electromagnetic potential four-vector. From s, cp,, etc.,.we must now derive quantities sL, cp,“,etc., satisfying the same field equations except for a change of the sign of e - that is, equations, in which D, has been changed into Df everywhere. The only possibility of such a charge-conjugation is (apart from an arbitrary factor) given by complex conjugation : sL = s*, (therefore,
cpk = cp,*; PL = s, (byte = (PJ.
(37)
For E i n s t e i n-B o s e statistics the total L a g r an g i an Li (20)- (21) is automatically invariant under this transformation (37), so that also the postulate (III) is satisfied. It must be emphasized that the fact that the law of charge-conjugation is uniquely determined by the L a g r a n g i a n (20) - (2 1) is a consequence of the fact that, besides s* and cpg, no other quantities transforming in the same way as s and (pv and describing particles with a charge (- e) are considered as components of the field. If such field components would have occurred, there might also have been some ambiguity in the commutation relations. Still any anti*) If the charge of the field under consideration would vanish, the necessity of taking the conjugate complex of the field components for the purpose of charge-conjugation would follow only, if some interaction exists between the neutral particles under consideration and some other particles, which possess an electric charge. Of course, for neutral particles the possibility exists that they are described by rral tensors or, more generally, hy neutrcttors 2) 1).
STATISTICAL
BEHAVIOUR
OF ELEMENTARY
PARTICLES
185 .
commutativity of the. form of (23) - (24) can be excluded in this case, 0% account of the postztlate (II) 7), so that all laws of chargeconjugation, which do not yield charge-invariance in the case of E i n s t e i n-B o s e statistics, must be excluded on the strenghth of this postulate (II). The proof that only the postulate (II) - and no combination of (I) and (III) - necessitates E i n s t e i n-B o s e statistics for fields consisting of more than one scalar, is given by the following example of a possible theory of particles with vanishing spin but obeying Fe r m i-D i r a c statistics, in which the total energy is strictly positive and in which charge-invariance exists. (The postulate (II) will not be satisfied, of course.) This example was given by one of the authors in a previous paper a). Let us introduce the operator z/x* ‘- A defined by
fG x0)= fff da
. f G
x0)
=
J/T
a$
x0) ei’.3dk,
n (x
x0)
eiz.
;’
dk, dk,,
k” dk, dk, dk, ,
(38)
(k” = + 2/X*+ k: + k$ + %I, and the function
53>,(x) defined by
2,j3///eJ.fT 6,(;xo)= (‘-A-
dk, dk, dk,,
(39)
so that ‘;or(z) = + S,(- x). We remark that Si f isb satisfies the equations (9, f w*(x) = (5% f q (- 4, (40) V”(Si f &) = f ii/x* - A. (9, f is). Now, we first consider the field given by (17) - (19), but we shall split up the scalars and the four-vector cp,, into two parts according to s = s(4) + CJ-); vYs(f) = qJl;-t’, (41) V”s(k) = & idx*
- A. s(f).
On account of (42), the commutation take the following form (compare (40)) :
relations
(42) for s(+) and s(-)
[s’~t’(x) ; w*(x’)J
= CT*) . (S, f CD) (x - x’),
[S(f)(X) ; P’*(d)]
= 0,
[s; s] =
[s*; s*] = 0, etc.
(43)
186
W.
PAUL1
AND
I;.
J. BELINFANTE
Here C(+) and C(-) are real constants. - The equations ( 17) - ( 19) and (42) remain valid under the foliowing law of charge-conjugation : +P z S(T)* (44) Now, instead of regarding (20) - (2 1) as the total L a g r a n g i a n of our system - in which, after substitution of (41), the integrals over time of the cross products of terms denoted by (+) and by (-) would vanish - we shall interchange s(-) and &-I* and regard L = K {($+)*v (xcpi+’ - vg(+)) + q+)v (qp*
-
-
v”&l*)
St+)* (&f) -
-
&.+J+)“) +
SC-1(xs(-l*
-
v&‘*“)}
as the L a g r a n g i a n function. From this L a g r a n g i a n field equations (18) - (19) can be derived for s(+) and for s(-); two equations (42) must then be added as additional conditions. the total energy we find the following strictly positive expression H-z
,K/Jy{s(+)*
(45) the the For 8) :
St+) + ,+-) s(-)* + + i (cpt+‘* cp!,+’+ cpt-’ &;-‘*)} v=o
dx dy dz.
(46)
Charge-invariance (III) and positive energy (I) are now ensured by (44) and (46) for F e r M i-D i r a c as well as for E i n s t e i nR o s e statistics. Therefore, we may regard the bracket symbols in (43) (compare (25)) as anti-commutators; at least, if in (43) we take C(+)/C(-) > 0, in order to avoid relations of the type of (33). Thus we see that the fiostulates (I) and (III) certainly do not exclude F e Y m i-D i r a c statistics in the case of spin = 0, in this more general case that more than one scalar and its conjugate complex are regarded as field components; the postulate (II), however, always suffices for this purpose in the case of integer spin ‘). $3. Particles OYquanta with a spin 812. In this section we shall discuss some possibilities of a quantum-theory of particles, which are (at least partly) described by a four-component function JCRof the co-ordinates, which transforms like an undor of the first rank “) at least with respect to spatial rotations and L o r e n t z transformations and which satisfies - if all external forces are neglected - a K 1 e i n-G o r d o n equation (0
-
x2) + = 0,
(0 .= VT”).
(47)
STATISTICAL
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OF
ELEMENTARY
187
PARTICLES
.
We shall put V,, = y”V,,
so that V$ = Cl ,
where yv are the matrices determined by (2)-(4). easily shown that the four-component quantity vo, + = -
(48) Then it can be
xy,
(49)
transforms in exactly the same way as 4 did. With the definition (491, the equation (47) takes, on account of (48), the form of v,,x
= - XqJ.
(50)
The equations (49)- (50) together form the first order field equations. They take a particularly simple form, if “) X=&t.
(5.1)
Admitting that x may be negative, we can choose the + sign in (Sl), so that in this particular case (51) the first order equations (49)-(50) pass into the D i r a c equation (V,, + 4 JI = 0. Now we introduce
(54
the matrix y” = y5 defined by (11) and put
ap=yoyp=-y"y~,(p=0,1,2,3,5),
(53)
so tha.t I) a"=
1;
ys = - *a, a2a3.
Then it can be proved that a hermitian 8ap = ($ap)+,
matrix8
(p=O,
(54 exists such that “) *)
1,2,&S),
(55)
where the cross t denotes the adjoint (hermitian conjugate) of a matrix **). Th is matrix 8 shall be normalized in such a way that e+%# represents the electric charge density of the field. Further we put 9) p = i9y” (= Ba1a2a3a5 = P+). (56) *) This
l *)
matrix
8 was called
A in a previous
paper
of the one of the authors
Since 0 is a matrix of one column and four rows, the ndjoint $t is a matrix row and four columns. Its elements are conjugate complex to those of 4. 0) This matrix p was called p in a previous paper of the one of the authors5) n a paper of the other *) I.).
‘). with
OrlC
but @p
188
W.
Now the equation g i a n (20) with or, in covariant
PAUL1
AND
F. J. BELINFANTE
(52) can be derived
from a scalar
Lag
r a n-
L = - K++P (Vo, + x)$5 undor-notation 2),
(57)
L = iK+*’ (V, ’ + xS;) +k,
(58)
’ IJP zz iJr+p (s I#+)
(59)
where we have put *) and
Vlk = (Y’)lk. V” = (?,Jlk,
(48~)
so that V,,‘V~k=8~.Vvv”=6~,
0.
(48b)
The total energy is now given by H = (iK/c) / fJ J/+% al)/% . dx dy dz (-
c-number).
(60)
In analogy .to the preceding section, we shall nssume that commutation relations are of the form of [+k(X)
;
v’(x’,l
=
%+,
the (61)
%‘)a
(62)
Since ck(x, x’) must be a mixed undor of the second rank (compare (61)) and must satisfy the equations (26) and (27), on account of (47) and the postulate (II), the undor c:(x, x’) must be of the form of t-$(x, d) = c(vk’ + 6?,8;) sb(x - x’),
(63)
where C and a are constants and where S(X) is given by (29). In particular case (52) the constant a is determined by (V,, + x)c = 0.
our
(64)
Inserting (0 - x2)% = 0. we find (compare
(264
(48))
C(Vo, + x) (V,, + a)% = 0 = qv,
+ x) (V,, - x)9,
or (a + x) (V,, + X)?Q = 0, *) Attention variant undors of the authofs
is drawn to the fact defined by (59) differs on undor calculus 2).
that the relation by a factor i from
between covariant and contrathat defined in the paper of one
STATISTICAL
BEHAVIOUR
OF ELEMENTARY
PARTICLES
189 .
so that and c=C(V,~-X)~.
a=--x
(65) Making use of (3 l), of (59) and ‘(48a) and of (compare (53), (55), (56)) pyp = -
= anti-hermitian
i8af‘
= -
(pyp)t = - ypt@, (66)
we find (compare (35)) that C must be real, again. The equation (52) is charge-invariant only under the transformation “) “) “) (J” = cp I ;E+*. - (67) Charge-invariance of all physical quantities derivable from the L ag r a n g i a n (20), (57) is obtained for (67) only in the case of artticommtitativity of the-components of +* and + J), so that postulate (III) requires F e r m i-D i r a c statistics. This is also required in order to make the total energy (60) strictly positive after quantization of the field, so that the postulates (I) and (III) are equivalent in this case. Again everything is complicated, if we admit that the field may consist of more undors than (c,and +” (or $*) only. For imtance the original equations (49) - (50) without the restriction (51) may then describe the field. In this case it is convenient to introduce the undors x*,1/2
(68)
= 4Jkxx,
which satisfy the first order field equations PO, f 4 y f) = 0,
(69)
so that we may regard this field as describing a mixture of D i r a c particles (52) of positive and negative mass (& J&/C). The equations (69) are invariant under either of the following transformations (where YS = j;yr*, YQ = QY*, compare (6)-(7) and (12)): either
Yf*,
= ac*,Y&,
with
1a(*) I2 = 1,
(704
Or
Yt*)
= a(,,Y&,
with (a: ar)2 = 1,
Wb)
so that
Y&
= Y (*)
or perhaps
YfiL:I) = - Y(*).
(704
In the case of (70b) the fields of the particles with positive and negative mass are “interchanged” under charge-conjugation. The field equations (69) can be derived from a L a g r a n g i a n (20) with L = - I((+) ‘r:+, P(Vo, + x) y(+) - K(-,y:-)
P(Vo, - 4 Y(-,> (71)
190
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PAUL1
AND
F. J. BELINFANTE
so that the total energy is equal to IT=
W4ff.f
W(+, ‘r:+, 8 aY,+,Pt + + K,-, YT-, B aY?,-,/at} dx dy dz (-
c-number).
In analogy to (61), (62), (65) this field can be quantized rq*&);
c'r,*,,w
~$&~,I = C(,)W d ‘r,,,(x')l = 0,
7 4)
w
(72)
according to
- 4,
C(*) = real,
(73)
further [Y;Y]
=[YyY”]
= 0.
(74) For .2: = x’ we find from (73), on account of (48ci), (30) and (59), (56) : --t
[rc*&4; y+*&)l = cc*) W) G *
In the usual representations of matrix remains unity as long other representations of undors these particular representations
(75)
undors the matrix 8 is unity “). This as only unitary transformations to are admitted “). If we make use of only, we can simplify (75) to
P&,(4 ; WC*&)>*1 = cc*) GL
Pa)
so that in the case of F e r m i-D i r a c statistics the C(,, now must be positive constants, whereas from (75) it follows that Y[+) vanishes, if C(+) = 0, and that Y+, vanishes, if C,-, = 0 (compare (33) - (34)). In this connection it is of importance to remark that the relations (73)- (74) are canonical commutation relations (from which for instance ii?@ = [F; Hj- follows) only, if C(+) K(+, = C,-, K,-,
-
AC,
(76) commutation relatians arc excluded in the case of F e r m i-D i I a c statistics, if K(+, : K,-, < 0. On the other hand E i n s t e i n-B o s e statistics are excluded entirely by postulate (I), since no choice of K(,, and C(*) would make the energy (72) strictly
so that canonical
positive in this case. If we postulate (III) charge-invariance of all quantities derivable from the L a g r an g i a n given by (71), the law (70b) of chargeconjugation yields F e r m i-D i r a c statistics only, if la(+) I2 = 1 : [a(.-) 12= K1+, : K,-,, but E i n s t e i n-B o s e statistics,
(77)
if
I a(+) I2 = 1 : la(-) I2 = -
K(+) : K+
,
STATISTICAL
BEHAVIOUR
OF ELEMENTARY
PARTICLES
191 .
whereas for any other choice of K(,, and a(,) charge-invariance is possible neither for E.-B. nor for F.-D. statistics. Therefore, it is not possible to exclude E i n s t e i n-B o s e statistics by the postulates (II) and (III) a 1one in our case. If, however, we combine the postulates (III) and (I), we can exclude the law (70b) of charge-conjugation, if the relation (77) does not exist between the constants K(,, in the L a g r a n g i a h and the constants a(*:)’ occurring in this law of charge-conjugation. For instance K(,, : Kc-, 5 0 would not be consistent with (77). On the other hand we find for any Kc+, and K(-, F e r m iD i r a c statistics from the postulate of charge-invariance, if we define charge-conjugation in the usual way by (70n) and not by (70b). In this case (70n) the postulates (I) and (III) are equivalent, again. $4. Conclusion. In the preceding sections we have seen that, as soon as two scalars (38), (40) or two undors of a given rank (68)- (71) are introduced, where one scalar (21) or one undor (5 1) - (57) would suffice, there are several possibilities of building up a quantized field theory of free elementary particles. Generally neither the L a g r a ng i a n, nor the commutation relations, nor the law of charge-conjugation (which was left unspecific d by (15) without b(16)) is then a priori unambiguously determined by the first order field equations. Still the postulates of positive energy (I) and of commutativity of observables in points connected by a space-like vector (II) are sufficient in these cases to decide between commutativity and anticommutativity ‘), though the c-numbers’occurring’in the commutation relations are no longer uniquely determined. The postulate of unspecijied charge-invariance of all physically significant quantities (III) will in these cases in general not be sufficient for this purpose (compare the discussion of (44)- (45) in 5 2 and of (70b) and (77) in $3), though in the last case considered in 5 3 it was quite sufficient for this purpose to postulate specified charge-invariance (that is, invariance under (15) z&t/z the specification (16)). In the case of integer spin (field of tensors) the postulates (I) and (III) are both entirely superfluous, since they are fulfilled automatically, if (II) is satisfied. On the other hand, there exist cases with integer spin and Fe r m i-D i r a c statistics, where (I) and (III) are fulfilled but not (II).
192
STATISTICAL
BEHAVIOUR
OF ELEMENTARY
PARTICLES
The first order field equations of the particles actually knowrt zcntil now have the property of determining the law of charge-conjugation unambiguously, whereas there is only one way of building up for them the (first order) L a g r a n g i a n without introducing superfluous quantities. Then, the commutation relations are entirely determined by either the postulates (I) and (II), or the postulates (III) and (II), 1‘f we assume that they are of the usual type (compare for instance (23)- (24) or (61)-(62)). In this case the postulate (III) determines the sign to be used in the bracket symbols [A; B], whereas (II) determines the c-numbers, to which these bracket symbols must be equal. On the other hand, if .these c-numbers are known (from (II)), the sign in the bracket symbols is again determined automatically in the case of integer spin ‘), whereas in the case of half odd spin it can now be deduced from (III) as well as from (I). Received
Dec.
23rd,
1939.
REFERENCKS I) F. J. B e 1 i n f a n t e, Physica 8, 870, 1939. 2) F. J. B e 1 in f a n t e, Physica 6, 849, 1939. 3) F. J. B e 1 i n f a n t e, Physica 0, 887, 1939. 4) F. J. Belinfante, ,Tpeory of Heavy Quanta” (thesis L.cicic:u 19>9), 111, 5 4. (M. Nijhoff, The Hague, 1939). 5) W. Pa u 1 i, Ann. Inst. H. PoincarC U, 109, 1936. 6) H. A. I< r a m e r s, Proc. rop. Acad. Amsterdam 10: 814, 1937. 7) W. P a o 1 i , ,.Bericht iiber die allgemeinen Eigenschnften der Elcmentartei!chen”, I, 5 3. (8mc Conseil d. Phys. Solvay, 1939). In the press. 8) W. P a u 1 i, Ann. Inst. H. Poincar6 C, 137, 193b.
VII,
no 3
:
Maart
1’940
ON THE STATISTICAL BEHAVIOUR OF KNOWN AND UNKNOWN ELEMENTARY PARTICLES by W. PAUL1 I’hysikalischcs Institut Instituut voor
and F. J. BELINFANTE
dcr EidgenGssischen Technischen Hochschule, Theoretische Natuurkunde der Rijks-Universiteit,
Zurich (Schwciz) Leiden
Zusammenfassung Es wird untersucht, in wie, weit Spekulationen iiber die Statistik willktirlicher, hypothetischer Teilchen in einer relativistisch invarianten Theorie moglich sind, werin alle oder ein Teil der folgenden drei Postulate vorausgesetzt werden: (I) die Energie ist positiv; (II) Observable an verschiedenen Raum-Zeitpunkten mit raumartiger Verbindungslinie sind kommutativ; (III) es gibt zwei Hquivalente Beschreibungen der Natur, in welchen die Elementarladungen entgegengesetzte Vorzeichen haben und in welchen einander entsprechende Feldgrossen sich bei L o r e n t zTransformationen in gleicher Weise transformieren. Der eine von uns (P.) hatte bereits gezeigt, dass allgemein bei ganzE i n s t e i n-B o s e-Statistik, bei halbzahligem Spin aus (II) allein zahligem Spin aus (I) allein F e r m i-D i r a c-Statistik folgt. Ferner hatte der andere von uns (B.) gezeigt, dass fiir eine gewisse Klasse von Teilchen, (die alle bisher in der Natur beobachteten umfasst, und die dadurch‘charakterisiert ist, dass sie hochstens durch einen Undor von gegebener Stufe beschrieben wird), aus dem durch ein spezielles Transformationsgesetz spezialisierten Postulat (III) fiir ganzen Spin E.-B.-Statistik, fiir halbzahligen Spin F.-D.-Statistik gefolgert werden kann. In der vorliegenden Note wird in den typischen Fallen von Spin 0 und Spin Q durch Beispiele gezeigt, dass im allgemeinen Fall mehrerer Undoren von gleicher Stufe aus (III) nicht mehr eindeutig auf die Statistik der Teilchen geschlossen werden kann, wahrend (II) bzw. (I) fiir ganzen bzw. halben Spin hiezu stets hinreichend bleiben. In den speziellen Fallen des Skalarfeldes, des Vektorfeldes und des D i r a c-Elektrons, wo nur ein einziger Undor gegebener Stufe in die Theorie eingeht, folgt dagegen das Transformationsgesetz der Ladungskonjugierung eindeutig, so dass hier (III) zur Festlegung der Statistik ausreicht.
In a recknt paper one of the authors 1) has a general principle, frpm which the statistical behaviour
3 1. Introduction.
indicated
Physica VII
177 12
178
W. PAUL1
ANll
F. J. BELINFANTE
(that is, the sign occurrin, u in the commutation relations of the components) of all particles and quanta empirically known until can be deduced in a simple way. This principle was the following. the field of the particles and quanta in question be described by of undors “) yk ,....R,,,which a,re functions of I the co-ordinates
field now Let a set
and which satisfy certain first order partial differential equations (field equations), in which some constants of the dimension of a charge may occur. Under an infinitesimal L o r e n t z transformation or spatial rotation of the frame of reference, so that the components of a given vector 5 are transformed according to *) sy = & &JP”, these undors are transformed
(1)
according to “)
m = 4 8w~” r, rr Y,
(2)
where
Here the y:) are matrices operating on the index kj of Yk,..,.R,, and satisfying the relations $ yq G $7 Yt” + $J Yll” = 2gp,. (4) Operating on’ the conjugate linear operator
complex Yt....i,,
of an undor by the
g = fi gn,
(5)
j=l
where g(i) is a matrix fying the relations
operating
on the index hi of Y~I,,..~,, and satis-
$’ Qi, = .gcny(j)* >
(6)
pe find again an undor ‘P = ix*. *) Greek indices run from 0 to 3: summation signs are omitted. the definition (4) of the matrices ylL, which differs a little from previous papers of one tif the authors I) *) 8) ‘). ’
(7) Attention is drawn the definition given
to in
STATISTICAL
BEHAVIOUR
OF ELEMENTARY
179
PARTICLES
.
We can normalize
E in such a way that ‘pe = y,
(8)
by putting jp g(i)* = 1. Instead of by (5)-(7), from Y* by
we might have constructed \r(f.’ =
cgi,diy*,
where G is a scalar operator - for instance a product number of factor yt), which are defined by t) yf’ = iy’f’ yy yy’ y$
(9) an undor ‘IF) . (10) of an even *)
(11) In a similar way we can form an undor again from Y* by means of *) YQ = QY*
or Y(Q) = GQY*, with Q = yi’)E,
(14
so that yQQ = - y.
(13) Now, it can be shown “) ‘) that, if the undors Y describing the fields of experimentally known particles or quanta (such as electrons, nuclons §), photons, mesons), are transformed according to (7), we obtain a set of undors Y2, which satisfy again a set of field equations, in which, however, all (real) constants of the dimension of a charge, for instance e, have been changed into their opposite: eL = - e* (14) This is true only “) for transformation *) If we require reflection through
that uf and yg shall transform the origin, this reflection must
T1 . ...k.. = f
Yip
according to (7) and not to in exactly the same way under be given by
a spatial
Yy Q....k,.
An under, for which the reflection takes this form, we shall call an ordirrary undor. If we require that y and 'I'Q shall transform in exactly the same way, however, must not be an ordinary under, but a quantity (,,quasi-under”), for which reflection defined by . . . . yo(d Yk 1.. .k *. %, . . ..k. = f iyp An odd number of factors yv) changes an ordinary undor into quasi-under into an ordinary one. t) Attention is drawn to the fact that there is a difference of defined in this paper I) and the ys defined in some papers of one of 5) The particle that is a proton in its charged state and a neutron have called a nuclorr.
a quasi-undor,
and
\r is
a
a sign between the ys the authors “). in its neutral state, we
‘1’80
W.
(12). Further,
PAUL1
AND
F. J. BELINFANTE
of the field equations ‘and a similar formulae (such as for instance the total energy or the total electric charge) exist ody on account of the commutation relations holding between the undors. By the transformation invariance
this invariance
of all Physically significant
Y + YL together therefore,
with e + eL,
(15)
where YL is given by YL = Y”,
(16)
we pass from one description of the fields of particles and quanta (by the undors Y) to another description (by Yg), which is conqbletely equivalent to the original one and was called - according to a terminology of K r a m e r s “) - the charge-conjugated description of the physical world. On the other hand it is plain that, postulating that the transformation (15) specified by (16) shall leave all physically significant quantities unaltered *), we obtain some information about the commutation relations holding between some undor components. This information is sufficient, indeed, to determine the sign occurring in the commutation relations, that is, to make a definite choice between commutativity and anticommutativity ,of the field components Y and Y*, though the c-number occurring in the usual commutation relations (for instance some a-function) remains entirely undeterm.ined in this way. It was shown by the one of us 1) that for a certain collection of hypothetical particles t). described by one undor of an arbitrarily given rank it follows from this postulate that particles with an integer spin must satisfy E i n s t e i n-B o s e statistics and those with a half-odd spin must satisfy F e r m i-D i r a c statistics. On the other hand it was shown by the other of us ‘), in a very general way, that a similar result for arbitrary particles described by a field satisfying a set of linear homogeneous differential equations could be obtained from the following two postulates : of the theory. Since the trans*) One may call this property the .,charge-invariance” formation (15) was specified by (16), we shall call it spccijied charge-invariance in the following. We admit that the name ,,charge-invariance ” is a little misleading since this property may exist for neutral as well as for charged particles. t) All particles experimentally known until now are contained in this collection.
STATISTICAL
BEHAVIOUR
OF ELEMENTARY
PARTICLES
181
(I) for these particles there is no infinite number of stat& of negative energy ;
(II) observables in points connected by a space-like vector aye purely comvnuta tive. These two postulates can be understood from a physical point of view. The first postulate is necessary since otherwise particles would drop into states of lower and lower energy creating an infinite number of quanta or pairs of particles. According to the second postulate simultaneous measurements in points connected by a space-like vector must be always possible; as it should be, since the field in the one point cannot influence the field in the other point, if the connecting vector is space-like. Invariance under the transformation (15) with the specificatiofi (16) is a specialized case of invariance under some unspecified transformation (15) of the undors together with inversion of the signs of the constants of the dimension of a charge (14). Invariance of all physical quantities under another transformation than (16), but still of the type of (14) - (IS), would also have meant that there is a description of the physical world, in which every “elementary charge” has the opposite sign. The existence of such a generalized charge-conjugated description may perhaps be infered from some speculations on a possible symmetry between positive and negative charges, which - though not existing on the earth - may still be a fundamental property of nature. (Anyhow, we know that it exists for all known particles l)). Therefore, without specialization by (16) *), we shall postulate (III) invariance of all physically significant formulae and quantities under some transformation (1 S), where Yp represents some set of undors +(l), $t2) . . . and YL a similar set of undors +(uL, +c2jL, . . . . transforming in the same way as ‘I!. We do not postulate, however, that after the transformation from Y? to YL - which we shall call “unspecified charge-conjugation” - the components of the undor #(“jr. are linear combinations of the components of only +(“)* again. In the following we shall discuss the bearing of the postulates (I), (II) and (III) on the question of the statistical behaviour of hypothetical particles with a spin 0 or B/2. For a spin 2 tZ similar considerations - though more complicate - are possible. *) One might of the theory.
call
the property
postulated
by (III)
the
,,unspecified
clrarge-illvariallce”
182
W.
PAUL1
AND
F. J. BELINFANTE
$ 2. Particles OYquanta with a spin 0. In this section we shall discuss some possibilities of a quantum-theory of particles, which are (at least partly) described by a (complex) function s of the co-ordinates Y, which is a scalar at least with respect to spatial rotations and L o r e n t z transformations and which satisfies - if all external forces are neglected - a K 1 e i n-G o r d o n equation
(0 - 2)s = 0,
(cl = VT,;
v, = a/w;
K = mcpi).
Here we shall discuss the case of particles or quanta vanishing mass (x $r 0) *). Putting in this case
(17)
with a non-
qb = V”S,
we find from (17) :
(18)
its = V’cp,.
(19) The equations (18)-( 19) together form the first order field equations. They can be derived from a L a g r a n g i a n Li =I J-fff
L dx dy dz dt,
(20)
where the Lag r an g i an function L depending on the field components s, (py,s*, etc. and (linearly) on their (first order) derivatives is given by L = I< {rpp*“(Xcp”-
V&s) -
s*(xs -
so that, if K is a real constant, the total is real. For the total energy we find “) M== xKfff
relations,
= c(x, x’) ;
[s(x) ; s(d)]
= [s”(x) ; s*(i)]
with
a wwislting
They
can be derived
vp,
energy
= 0;
order
equations
(24) are given
by
(l8--19a)
= 0.
from L = R { p*“(p”
and the total
(24
(23)
mpss the first
p” = v,s,
c-number).
we shall asszlme that
[s(x) ; s*(d)] particles
(21)
L a g r a n g i a n fi (20)
{s*s +VfO$‘pV} dx dy dz (-
As regards the commutation they are of the form
*) For scalar
V/f”)},
is given kf=
- V,s)
(Zln)
+ s*V,(p”]
in this case by ll’,~o~:~V
dn dy dz (-
c-number).
The further discussion of the possible commutation relations tion in this c.ase is analogous to that for scalar particles witn
and laws of charge-conjugaa non-vanishing mass.
!2&)
STATISTICAL
BEHAVIOUR
OF ELEMENTARY
where the bracket symbol [A-; B] denotes either [A ; B]- or the anti-commutator [A ; B] + : [A;B]*
183
PARTICLES
the commutator
=ABfBA,
(25)
and where c(x, x’) is a c-number function of X, and xi. The commutation relations for the other field components are derived from (23)-(24) by means of the equations (18). From (23) - (24) we conclude that c is a scalar function, which, on account of (17)) must satisfy the equations (0 - 2) c(x, x’) = (0 ’ - 2) c(x, x’) = 0,
(26)
( Cl ’ operates on x’ like 0 on X) . Further we remark that the postulate (II) of 5 1 is satisfied, if we assume for all pairs of field components ql, q2 : [qi;, whe,reTdenotes
x,,) ; q2G, xc,)] = 0 for x0 = xi but : #2,
(27)
xi, x2, x3. Since (27) should hold for s with s* as well
as for s with ‘po, * both c and Voc must vanish for x0 = x& z# ~7 so that from the scalar character of c it follows that c must be equal to 8) ‘) c(x, x’) = ic . 9(x - x’). (28) Here C is a constant and
S(x) = SIG x0)=
(&-)“~~~~1‘&.’
sin;oxo dk, dk, dk,
(29)
(k” = + 2/k: + k; + k: + x2) is a solution of (26) satisfying
the conditions
sgo,=0;(g0)x*=0 =s$j. sb=sb*; All other solutions
of (26) are excluded by the postulate
(II). We
remark that 9 depends on 1;I and x0 only and is an odd function of x0, so that B*(x) = - %I(- x). (31) The commutation
relations now must read [s(x); s*(x’)]-
= iaD(x
- x’),
(32)
184
‘W.
since F erm i-Dira for x = x’ (compare
PAUL1
AND
F. J. BELINFANTE
c statistics are excluded (23) with (28), (30)), from
by the fact that,
ss* + s*s = 0
(33)
it would follow that the field vanishes entirely ‘) : s = 0. Further
iC must be purely’imaginary _
[s(x); s*(d)]*
(34) on account of (3!) and
= [s(d);
s*(x)].
(35)
The total energy (22) is automaiically positive, in.accordance with postulate (I). If e is the charge *) of the particles under consideration, we can introduce their interaction with the M ax we 11 i a n field by changing in (17) - (2 1) V, into a = 9” + (e/W% > (36) where 5?&denotes the electromagnetic potential four-vector. From s, cp,, etc.,.we must now derive quantities sL, cp,“,etc., satisfying the same field equations except for a change of the sign of e - that is, equations, in which D, has been changed into Df everywhere. The only possibility of such a charge-conjugation is (apart from an arbitrary factor) given by complex conjugation : sL = s*, (therefore,
cpk = cp,*; PL = s, (byte = (PJ.
(37)
For E i n s t e i n-B o s e statistics the total L a g r an g i an Li (20)- (21) is automatically invariant under this transformation (37), so that also the postulate (III) is satisfied. It must be emphasized that the fact that the law of charge-conjugation is uniquely determined by the L a g r a n g i a n (20) - (2 1) is a consequence of the fact that, besides s* and cpg, no other quantities transforming in the same way as s and (pv and describing particles with a charge (- e) are considered as components of the field. If such field components would have occurred, there might also have been some ambiguity in the commutation relations. Still any anti*) If the charge of the field under consideration would vanish, the necessity of taking the conjugate complex of the field components for the purpose of charge-conjugation would follow only, if some interaction exists between the neutral particles under consideration and some other particles, which possess an electric charge. Of course, for neutral particles the possibility exists that they are described by rral tensors or, more generally, hy neutrcttors 2) 1).
STATISTICAL
BEHAVIOUR
OF ELEMENTARY
PARTICLES
185 .
commutativity of the. form of (23) - (24) can be excluded in this case, 0% account of the postztlate (II) 7), so that all laws of chargeconjugation, which do not yield charge-invariance in the case of E i n s t e i n-B o s e statistics, must be excluded on the strenghth of this postulate (II). The proof that only the postulate (II) - and no combination of (I) and (III) - necessitates E i n s t e i n-B o s e statistics for fields consisting of more than one scalar, is given by the following example of a possible theory of particles with vanishing spin but obeying Fe r m i-D i r a c statistics, in which the total energy is strictly positive and in which charge-invariance exists. (The postulate (II) will not be satisfied, of course.) This example was given by one of the authors in a previous paper a). Let us introduce the operator z/x* ‘- A defined by
fG x0)= fff da
. f G
x0)
=
J/T
a$
x0) ei’.3dk,
n (x
x0)
eiz.
;’
dk, dk,,
k” dk, dk, dk, ,
(38)
(k” = + 2/X*+ k: + k$ + %I, and the function
53>,(x) defined by
2,j3///eJ.fT 6,(;xo)= (‘-A-
dk, dk, dk,,
(39)
so that ‘;or(z) = + S,(- x). We remark that Si f isb satisfies the equations (9, f w*(x) = (5% f q (- 4, (40) V”(Si f &) = f ii/x* - A. (9, f is). Now, we first consider the field given by (17) - (19), but we shall split up the scalars and the four-vector cp,, into two parts according to s = s(4) + CJ-); vYs(f) = qJl;-t’, (41) V”s(k) = & idx*
- A. s(f).
On account of (42), the commutation take the following form (compare (40)) :
relations
(42) for s(+) and s(-)
[s’~t’(x) ; w*(x’)J
= CT*) . (S, f CD) (x - x’),
[S(f)(X) ; P’*(d)]
= 0,
[s; s] =
[s*; s*] = 0, etc.
(43)
186
W.
PAUL1
AND
I;.
J. BELINFANTE
Here C(+) and C(-) are real constants. - The equations ( 17) - ( 19) and (42) remain valid under the foliowing law of charge-conjugation : +P z S(T)* (44) Now, instead of regarding (20) - (2 1) as the total L a g r a n g i a n of our system - in which, after substitution of (41), the integrals over time of the cross products of terms denoted by (+) and by (-) would vanish - we shall interchange s(-) and &-I* and regard L = K {($+)*v (xcpi+’ - vg(+)) + q+)v (qp*
-
-
v”&l*)
St+)* (&f) -
-
&.+J+)“) +
SC-1(xs(-l*
-
v&‘*“)}
as the L a g r a n g i a n function. From this L a g r a n g i a n field equations (18) - (19) can be derived for s(+) and for s(-); two equations (42) must then be added as additional conditions. the total energy we find the following strictly positive expression H-z
,K/Jy{s(+)*
(45) the the For 8) :
St+) + ,+-) s(-)* + + i (cpt+‘* cp!,+’+ cpt-’ &;-‘*)} v=o
dx dy dz.
(46)
Charge-invariance (III) and positive energy (I) are now ensured by (44) and (46) for F e r M i-D i r a c as well as for E i n s t e i nR o s e statistics. Therefore, we may regard the bracket symbols in (43) (compare (25)) as anti-commutators; at least, if in (43) we take C(+)/C(-) > 0, in order to avoid relations of the type of (33). Thus we see that the fiostulates (I) and (III) certainly do not exclude F e Y m i-D i r a c statistics in the case of spin = 0, in this more general case that more than one scalar and its conjugate complex are regarded as field components; the postulate (II), however, always suffices for this purpose in the case of integer spin ‘). $3. Particles OYquanta with a spin 812. In this section we shall discuss some possibilities of a quantum-theory of particles, which are (at least partly) described by a four-component function JCRof the co-ordinates, which transforms like an undor of the first rank “) at least with respect to spatial rotations and L o r e n t z transformations and which satisfies - if all external forces are neglected - a K 1 e i n-G o r d o n equation (0
-
x2) + = 0,
(0 .= VT”).
(47)
STATISTICAL
BEHAVIOUR
OF
ELEMENTARY
187
PARTICLES
.
We shall put V,, = y”V,,
so that V$ = Cl ,
where yv are the matrices determined by (2)-(4). easily shown that the four-component quantity vo, + = -
(48) Then it can be
xy,
(49)
transforms in exactly the same way as 4 did. With the definition (491, the equation (47) takes, on account of (48), the form of v,,x
= - XqJ.
(50)
The equations (49)- (50) together form the first order field equations. They take a particularly simple form, if “) X=&t.
(5.1)
Admitting that x may be negative, we can choose the + sign in (Sl), so that in this particular case (51) the first order equations (49)-(50) pass into the D i r a c equation (V,, + 4 JI = 0. Now we introduce
(54
the matrix y” = y5 defined by (11) and put
ap=yoyp=-y"y~,(p=0,1,2,3,5),
(53)
so tha.t I) a"=
1;
ys = - *a, a2a3.
Then it can be proved that a hermitian 8ap = ($ap)+,
matrix8
(p=O,
(54 exists such that “) *)
1,2,&S),
(55)
where the cross t denotes the adjoint (hermitian conjugate) of a matrix **). Th is matrix 8 shall be normalized in such a way that e+%# represents the electric charge density of the field. Further we put 9) p = i9y” (= Ba1a2a3a5 = P+). (56) *) This
l *)
matrix
8 was called
A in a previous
paper
of the one of the authors
Since 0 is a matrix of one column and four rows, the ndjoint $t is a matrix row and four columns. Its elements are conjugate complex to those of 4. 0) This matrix p was called p in a previous paper of the one of the authors5) n a paper of the other *) I.).
‘). with
OrlC
but @p
188
W.
Now the equation g i a n (20) with or, in covariant
PAUL1
AND
F. J. BELINFANTE
(52) can be derived
from a scalar
Lag
r a n-
L = - K++P (Vo, + x)$5 undor-notation 2),
(57)
L = iK+*’ (V, ’ + xS;) +k,
(58)
’ IJP zz iJr+p (s I#+)
(59)
where we have put *) and
Vlk = (Y’)lk. V” = (?,Jlk,
(48~)
so that V,,‘V~k=8~.Vvv”=6~,
0.
(48b)
The total energy is now given by H = (iK/c) / fJ J/+% al)/% . dx dy dz (-
c-number).
(60)
In analogy .to the preceding section, we shall nssume that commutation relations are of the form of [+k(X)
;
v’(x’,l
=
%+,
the (61)
%‘)a
(62)
Since ck(x, x’) must be a mixed undor of the second rank (compare (61)) and must satisfy the equations (26) and (27), on account of (47) and the postulate (II), the undor c:(x, x’) must be of the form of t-$(x, d) = c(vk’ + 6?,8;) sb(x - x’),
(63)
where C and a are constants and where S(X) is given by (29). In particular case (52) the constant a is determined by (V,, + x)c = 0.
our
(64)
Inserting (0 - x2)% = 0. we find (compare
(264
(48))
C(Vo, + x) (V,, + a)% = 0 = qv,
+ x) (V,, - x)9,
or (a + x) (V,, + X)?Q = 0, *) Attention variant undors of the authofs
is drawn to the fact defined by (59) differs on undor calculus 2).
that the relation by a factor i from
between covariant and contrathat defined in the paper of one
STATISTICAL
BEHAVIOUR
OF ELEMENTARY
PARTICLES
189 .
so that and c=C(V,~-X)~.
a=--x
(65) Making use of (3 l), of (59) and ‘(48a) and of (compare (53), (55), (56)) pyp = -
= anti-hermitian
i8af‘
= -
(pyp)t = - ypt@, (66)
we find (compare (35)) that C must be real, again. The equation (52) is charge-invariant only under the transformation “) “) “) (J” = cp I ;E+*. - (67) Charge-invariance of all physical quantities derivable from the L ag r a n g i a n (20), (57) is obtained for (67) only in the case of artticommtitativity of the-components of +* and + J), so that postulate (III) requires F e r m i-D i r a c statistics. This is also required in order to make the total energy (60) strictly positive after quantization of the field, so that the postulates (I) and (III) are equivalent in this case. Again everything is complicated, if we admit that the field may consist of more undors than (c,and +” (or $*) only. For imtance the original equations (49) - (50) without the restriction (51) may then describe the field. In this case it is convenient to introduce the undors x*,1/2
(68)
= 4Jkxx,
which satisfy the first order field equations PO, f 4 y f) = 0,
(69)
so that we may regard this field as describing a mixture of D i r a c particles (52) of positive and negative mass (& J&/C). The equations (69) are invariant under either of the following transformations (where YS = j;yr*, YQ = QY*, compare (6)-(7) and (12)): either
Yf*,
= ac*,Y&,
with
1a(*) I2 = 1,
(704
Or
Yt*)
= a(,,Y&,
with (a: ar)2 = 1,
Wb)
so that
Y&
= Y (*)
or perhaps
YfiL:I) = - Y(*).
(704
In the case of (70b) the fields of the particles with positive and negative mass are “interchanged” under charge-conjugation. The field equations (69) can be derived from a L a g r a n g i a n (20) with L = - I((+) ‘r:+, P(Vo, + x) y(+) - K(-,y:-)
P(Vo, - 4 Y(-,> (71)
190
W.
PAUL1
AND
F. J. BELINFANTE
so that the total energy is equal to IT=
W4ff.f
W(+, ‘r:+, 8 aY,+,Pt + + K,-, YT-, B aY?,-,/at} dx dy dz (-
c-number).
In analogy to (61), (62), (65) this field can be quantized rq*&);
c'r,*,,w
~$&~,I = C(,)W d ‘r,,,(x')l = 0,
7 4)
w
(72)
according to
- 4,
C(*) = real,
(73)
further [Y;Y]
=[YyY”]
= 0.
(74) For .2: = x’ we find from (73), on account of (48ci), (30) and (59), (56) : --t
[rc*&4; y+*&)l = cc*) W) G *
In the usual representations of matrix remains unity as long other representations of undors these particular representations
(75)
undors the matrix 8 is unity “). This as only unitary transformations to are admitted “). If we make use of only, we can simplify (75) to
P&,(4 ; WC*&)>*1 = cc*) GL
Pa)
so that in the case of F e r m i-D i r a c statistics the C(,, now must be positive constants, whereas from (75) it follows that Y[+) vanishes, if C(+) = 0, and that Y+, vanishes, if C,-, = 0 (compare (33) - (34)). In this connection it is of importance to remark that the relations (73)- (74) are canonical commutation relations (from which for instance ii?@ = [F; Hj- follows) only, if C(+) K(+, = C,-, K,-,
-
AC,
(76) commutation relatians arc excluded in the case of F e r m i-D i I a c statistics, if K(+, : K,-, < 0. On the other hand E i n s t e i n-B o s e statistics are excluded entirely by postulate (I), since no choice of K(,, and C(*) would make the energy (72) strictly
so that canonical
positive in this case. If we postulate (III) charge-invariance of all quantities derivable from the L a g r an g i a n given by (71), the law (70b) of chargeconjugation yields F e r m i-D i r a c statistics only, if la(+) I2 = 1 : [a(.-) 12= K1+, : K,-,, but E i n s t e i n-B o s e statistics,
(77)
if
I a(+) I2 = 1 : la(-) I2 = -
K(+) : K+
,
STATISTICAL
BEHAVIOUR
OF ELEMENTARY
PARTICLES
191 .
whereas for any other choice of K(,, and a(,) charge-invariance is possible neither for E.-B. nor for F.-D. statistics. Therefore, it is not possible to exclude E i n s t e i n-B o s e statistics by the postulates (II) and (III) a 1one in our case. If, however, we combine the postulates (III) and (I), we can exclude the law (70b) of charge-conjugation, if the relation (77) does not exist between the constants K(,, in the L a g r a n g i a h and the constants a(*:)’ occurring in this law of charge-conjugation. For instance K(,, : Kc-, 5 0 would not be consistent with (77). On the other hand we find for any Kc+, and K(-, F e r m iD i r a c statistics from the postulate of charge-invariance, if we define charge-conjugation in the usual way by (70n) and not by (70b). In this case (70n) the postulates (I) and (III) are equivalent, again. $4. Conclusion. In the preceding sections we have seen that, as soon as two scalars (38), (40) or two undors of a given rank (68)- (71) are introduced, where one scalar (21) or one undor (5 1) - (57) would suffice, there are several possibilities of building up a quantized field theory of free elementary particles. Generally neither the L a g r a ng i a n, nor the commutation relations, nor the law of charge-conjugation (which was left unspecific d by (15) without b(16)) is then a priori unambiguously determined by the first order field equations. Still the postulates of positive energy (I) and of commutativity of observables in points connected by a space-like vector (II) are sufficient in these cases to decide between commutativity and anticommutativity ‘), though the c-numbers’occurring’in the commutation relations are no longer uniquely determined. The postulate of unspecijied charge-invariance of all physically significant quantities (III) will in these cases in general not be sufficient for this purpose (compare the discussion of (44)- (45) in 5 2 and of (70b) and (77) in $3), though in the last case considered in 5 3 it was quite sufficient for this purpose to postulate specified charge-invariance (that is, invariance under (15) z&t/z the specification (16)). In the case of integer spin (field of tensors) the postulates (I) and (III) are both entirely superfluous, since they are fulfilled automatically, if (II) is satisfied. On the other hand, there exist cases with integer spin and Fe r m i-D i r a c statistics, where (I) and (III) are fulfilled but not (II).
192
STATISTICAL
BEHAVIOUR
OF ELEMENTARY
PARTICLES
The first order field equations of the particles actually knowrt zcntil now have the property of determining the law of charge-conjugation unambiguously, whereas there is only one way of building up for them the (first order) L a g r a n g i a n without introducing superfluous quantities. Then, the commutation relations are entirely determined by either the postulates (I) and (II), or the postulates (III) and (II), 1‘f we assume that they are of the usual type (compare for instance (23)- (24) or (61)-(62)). In this case the postulate (III) determines the sign to be used in the bracket symbols [A; B], whereas (II) determines the c-numbers, to which these bracket symbols must be equal. On the other hand, if .these c-numbers are known (from (II)), the sign in the bracket symbols is again determined automatically in the case of integer spin ‘), whereas in the case of half odd spin it can now be deduced from (III) as well as from (I). Received
Dec.
23rd,
1939.
REFERENCKS I) F. J. B e 1 i n f a n t e, Physica 8, 870, 1939. 2) F. J. B e 1 in f a n t e, Physica 6, 849, 1939. 3) F. J. B e 1 i n f a n t e, Physica 0, 887, 1939. 4) F. J. Belinfante, ,Tpeory of Heavy Quanta” (thesis L.cicic:u 19>9), 111, 5 4. (M. Nijhoff, The Hague, 1939). 5) W. Pa u 1 i, Ann. Inst. H. PoincarC U, 109, 1936. 6) H. A. I< r a m e r s, Proc. rop. Acad. Amsterdam 10: 814, 1937. 7) W. P a o 1 i , ,.Bericht iiber die allgemeinen Eigenschnften der Elcmentartei!chen”, I, 5 3. (8mc Conseil d. Phys. Solvay, 1939). In the press. 8) W. P a u 1 i, Ann. Inst. H. Poincar6 C, 137, 193b.