Fortschritte der Physik 7, 291 -328 (1959)
The T C P Theorem and its Applications1).
G. GRAWERT
Institut fur Theoretische Physik der Freien Universitat Berlin 2,
G . LUDERS
Nuz-Planck-lnstitut fur Physik und Astroph ysik, Miinchen
and
H. ROLLNIK
Institut f iir Theoretische P h ysik der Universitat Heidelberg z,
Introduction
I n quantum field theory we can define certain unitary or antiunitary symmetry
operators in the HILBERTspace of physical states. The subjects of discussion in
this article are the three discrete symmetry operations of space reflection P ,
time reflection T and charge conjugation or more precisely particle-antiparticle
conjugation C.
The physical meaning of these operators is the following: Space reflection P :
The state of particles with space coordinates ri with momenta pi and (for spins
different from zero) with spin directions ui is transformed into the state of the
same particles a t -ri with momenta -pi and unchanged spin directions ui.
- Time reflection T : The state of particles at q with momenta pi and spin
directions uiis transformedinto the state of the same particles a t ri too, but with
reversed momenta -pi and reversed spins --ai. Furthermore the direction of
time is reversed, that means, incoming particles are transformed into outgoing
particles and vice versa. - Charge conjugation: The particles a t ri,pi, cri are
transformed into the corresponding antiparticles without changing space coordinates, momenta and spins.
Precise definitions and discussions of the symmetry operators and products of
them are presented for free fields in chapter I.
The postulate of invariance of quantum field theory under one of these operations always implies certain restrictions in the Hamiltonian of the interaction.
However it is an important fact that the invariance under the product T C P
generally is guaranteed without any restrictions in the coupling constants.
The T C P theorem says (we give a n preliminary version a t this place): A local
quantum field theory, which is invariant under LORENTZ-rotations and which
l)
2,
8)
22
The chapters I andIIare by G. GRAWERT
andH. ROLLNIK;
chapter I11 is by G. LUDERS.
Present address: Inst. f. Theor. Physik der Univ. Gottingen.
On leave of absence from Freie UniversitBt Berlin.
Zeitschrift ,,Fortschritte der Physik"
292
G . GRAWERT,G . LUDERS,H. ROLLNIK
includes the usual causal commutativity resp. anticommutativity of the field
operators, is always invariant under the product operation TCP.
A detailed discussion of this theorem and its different versions is the subject of
chapter 11.
Physically the T C P theorem means for instance that the transition probabilities
are equal for the following two processes: 1. particles a t ri with pi, oireact to
give particles a t r; with pi, 0:. - 2. the corresponding antiparticles a t --r;
withpi, -a: react t o give antiparticles a t - riwithpi, - oi.
The general result, which is formulated in the theorem, was already discovercd
some time ago, when the invariance of quantum field theory under the thrce
operations separately was unquestioned.
Two examples first showed t h a t in a theory with LORENTZand P-invariance
the postulate of T-invariance on the one hand and t h a t of C-invariance on thc
other hand implied exactly the same restrictions in the interaction Hamiltonian.
BIEDENHARN
and ROSE [ I ] as well as TOLHOEKand DE GROOT [2] examined
the (at t h a t time) most general expression for the interaction between four
spinor fields
HInt =
2 gi q ( I ) oiv(2)y(3)oiv(4)
*
I
(iruns over the well known invariants S , V , T, A , P).
T-invariance as well as C-invariance implies t h a t all the five coupling constants gi
have the same phase factor. By analogous arguments it was shown (LUDERS,
OEHME, THIRRING
[3]; PAIS,JOST
[a]; LUDERS[ 5 ] )that, postulating T-invariance or resp. C-invariance, we are forbidden t o couple a neutral scalar (or pseudovector) boson field t o a Dirac field simultaneously with and without derivatives.
Because of these results LUDERS [5, 61 conjectured and proved, that every
P-invariant relativistic quantum field theory automatically is invariant under
the product CT. PAULI
[7] generalized this result by abandoning P-invariance
in the premises. Only postulating invariance under proper LORENTZ-transformations he proved the invariance under TCP. An analogous proof is contained
in an article of BELL
PAULI’Sproof only discussed the algebra of operators
and made use of a n inversion of the order of field operators, a n operation which
is difficult t o represent by a n operator in HILBERTspace. By adding a Hermitean
conjugation t o PAULI’Sproof, one shows the invariance of the theory under a n
antiunitary operator T C P in HILBERTspace (LUDERS[ 6 , 91). JOST
[IO] finally
gave a very satisfactory proof of the theorem for that version of quantum
field theory, which starts with the vacuum expectation values of operator products. (Implicitly contained but not explicitly stated was the theorem in some
articles of SCHWINGER
[II].)
Originally the T C P theorem had been of rather academic interest only. The
discovery of parity violation in weak interactions attached more importance t o
it. I n Chapter I11 some of the consequences of the theorem are discussed. Firstly
is shown that the T C P transformation can be used t o prove the occurrence of
particles and antiparticles in quantized relativistic field theories provided that
there exists some generalized charge. As a matter of fact, even if C invariance
or C P invariance are postulated the assumption of the existence of generalized
charge is indispensable for the proof of the occurrence of “dressed” particles
and antiparticles. Secondly the role is discussed which the TCP theorem plays
in connection with the empirical analysis of the various symmetry properties.
[a].
The TCP Theorem and its Applications
293
On the one hand, if parity is not conserved a t least one of the two other symmetry
properties ( T ,C ) has t o be violated too. On the other hand, the T C P theorem
can be used t o perform, under certain additional assumptions, indirect tests of
C invariance (by actually testing T P instead).
CHAPTER I
The Discrete Symmetry-Operations
Let us start with a quantum field theory, which is invariant under proper (inhomogeneous) LORENTZ-transformations !& and let us ask for additional symmetry
properties, which may hold in the theory.
To have a clear-cut answer one evidently must state all the physical assumptions
being fulfilled by the fields in question besides the transformation properties.
Some rather weak assumption, which allows t o derive a n additional symmetry,
is stated in the T C P theorem. But for a mathematical formulation of the operations C , P and T separately it is convenient t o make the most stringent assumption concerning our field theory: we consider only f r e e p a r t i c l e s i. e. linear
field equations.
At first sight one would formulate our task a8 follows: We have t o construct all
symmetry operations U , unitary or antiunitary [12], which transform the field
operators q r ( x ) ( T labels the independent components of the field) into pi(..’),
the space-time coordinates x’ of which being transformed by one of the four
discrete LORENTZ-transformations.
Identity E (x; = xp),Space reflection P ( x i = - x k , xi = x41).
Time reversal T ( x i = xk, xi = - x4),Inversion (x; = - 4.
But we shall show a t once that in such a way we are lead t o a much bigger class
of symmetry operations than is usually considered. Having in mind the application of the operations t o interacting fields we must restrict ourselves from
the beginning t o l o c a l s y m m e t r y - o p e r a t i o n s , i. e. pi(x’) must be connected with y r ( x )by a linear transformation of the form
Thus linear integral transformations are excluded. I n the “normal” case of a,
field with non-vanishing mass the additional supposition of locality exactly
leads to the symmetry group generated by charge conjugation, space reflection
and time-reversal. For vanishing mass one gets the socalled PAULI-PURSEY
group in addition. We shall restrict our calculations t o typical examples, the
complex scalar field and the Dirac spinor field.
1. Space- Time fixed: Charge Conjugation
a) S c a l a r - f i e l d
We consider a field p ( x ) which is invariant under proper LORENTZ-transformations. In generd tp (2) is a non Hermitean operator, which obeys the equations
(0- P 2 ) V ( X ) = 0
1)
22’
9
[pt(X1):p(X2)I=
Latin indices run over 1 through 3, Greek ones over 1 through 4.
- XP).
(2)
G. GRAWERT,
G. LUDERS,
H. ROLLNIK
294
The operator U , transforming (x)into y ' ( x ) , has t o commute with energymomentum. Consequently the vacuum is invariant unde; U (up t o a phase,
which is set equal t o one). Further U must act on the basis vectors of the one
part,icle HILBERT
space in the following way
@;(It)
= u @,(k)
=
2 afl0*@,
d=f
(E) ,
0=
f.
(3)
the two particles, which are described by the non Hermitean p (x)
Because of the unitary or antiunitary character of U the matrix aaa,, must be
unitary (cf. App. 1). From (3) we derive for the creation operators
0 distinguishes
Specifying son, in .some way, the operator U is defined uniquely for the basis
vectors only. It becomes a unitary operator in whole HILBERT
space, if we
assume U t o be linear; U becomes antiunitary, if it is antilinear. Passing from (3)
t o the coordinate representation this difference will be quite apparent. Using
we get
if the operator U is linear. (Here we have introduced the positive and negative
frequency part of q~(x)according t o (4)).If U is antilinear we have instead of (5):
Consequently antiunitary operations always contain a time reversal, which will
be discussed later on.
Coming back t o (5) we observe that y'( x) in general is connected with v (x)by a n
integral transformation. U only becomes a local transformation, if we require
1
-
a++= a_- = a1,
a + _= a:,
= az.
(3a)
Because of the unitary we furthermore have
This equation forbids the multiplication of all the one particle states by the same
phasefactor (for this would mean a,, = a-- = eia , a + _= a_+= 0 ) .
Summing up we can state : All local symmetry operations which do not change
space time are unitary and they are given either by
y'(4
=
r p ( 4 = Ep?(x)E+,]
(5a)
or alternatively by
v'(x) = r c p t ( x )= C p i ( X ) C+.
7 and q c are arbitrary phasefactors. The group (5a), which is connected continuously with the identity, defines by its infinitesimal generator a generalized
charge &I)
4?= i (v,Lp - vt v,w) dgP
/
U
1)
Here we use the abbreviation T , for
~ the derivatives of the field.
The TCP Theorem and its Applications
295
with the property
[ Q )Q, ($11 = Q,($1
The unitary operator E is equal t o eiaQ with 7 = eta . The operator C defined
by (5b) anticommutes with Q.
CQ = - Q C .
It is called charge conjugation [13].It transforms eigenstates of Q into eigenstates
*
with the opposite eigenvalue.
We add some supplementary remarks. C may be expressed by Q and the operators (j and N
6=
d3k (a: a-
+ a! a - ) )
N = j d3k (a: a+
+ a’
a-) ,
in the following way
This is possible since C is connected CQntinuously with the identity by the (in
general nonlocal) transformations (3‘). The observables Q and Q obviously don’t
commute, but N commutes with Q and Q. As far as one considers bare particles
one can easiIy prepare the eigenstates of Q as well as those of Q. In nature there
are only eigenstates of Q in the normal case, i. e, among the “old” particles.
For the KO-mesons exclusively one can prepare physical eigenstates of Q.
On the whole one must distinguish three possible cases :
a ) p(x ) describes a dublett of neutral particles (e.g. the KO-mesons)Q as well as Q
have physical significance. Producing KO-mesons by collision of particles with
strangeness zero, eigenstates of Q are created. I n decaying these states are transformed into the eigenstates Kl and K , of Q.
b) The field Q , ( x )is charged (e. g. n*-mesons). The physical particle states are
characterized by momentum, angular momentum and charge. The charge is
defined by the electrical interaction with an “unit charge”. I n nature we always
find both signs of the charge, therefore the symmetry operations with UQUt = & Q
are allowed physically. The lower sign refers t o the charge conjugation.
c) Q, (2)is a Hermitean operator (e. g. the no-rnesons).The transformations (5a, b)
degenerate to
-
y ’ ( 4 = =k f J ( 4
(5c)
Correspondingly the operators Q and Qvanish identically.
The extension to vector fields is trivial. I n the special case of real vector fields there
are just the transformations
2 96
G. GRAWERT,G. LUDERS,H. ROLLNIK
b) S p i n o r f i e l d
Having discussed the scalar case extensively, we can start a t once with the
local Ansatz for the transformed spinorl) 2,
y'(4
=A
y(4
+ By+(x)
(6)
By direct calculation one easily proves that y ' ( x ) as well as y ( x ) obey the Dirac
equation if the matrices A and B are given by
*
A=All,
B=AZey,T3),
(6a)
where 1, are arbitrary complex numbers.
I n the reverse (6a) is a necessary condition in order t o have y 7 ( x )obey the
Dirac equation if the particle mass m does not vanish. I n the case of vanishing
m the most general matrices meeting the mentioned requirement are
A
=
A, 1 + 1;y5
)
B
= (A2
+ 1; y s ) c 7: .
(6b)
Because of the form of B we introduce the abbreviation
y " 4 = c y I ( 4 = cy: y t ( 4 ,
and we can rewrite transformation (6) in the form of
y'(4
+
+ + 1;
(11 A i 7 5 ) y ( 4
(7)
y " ( 4*
I n the case of non-vanishing mass one has t o put A: ==0 ( i = 1, 2).
=
0 2
(8)
y5)
Furthermore (8) must not alter the commutation relations
W(%)}
-
hJ((.l)) = i S ( X 1 - x2)
{Y(%),Y(%)} = 0
(9)
From this requirement one obtains by elementary, but somewhat lengthy
calculations the following conditions for the coefficients of (7)
2
2 (I&
i=l
9
+ IAJ)z = 1 ,
A1 A, = A; A: , Re (A, A';
+ A, A';)
=0
.
(10)
I n the case of Spinor fields with non-vanishing mass these conditions imply the
existence of the following two symmetry groups
y ( z ) = q y ( x )E E y ( x )E' and y ' ( a ) = qcyc(z)
= C y ( s ) C' . (11)
They exactly correspond t o the groups (5a, b) in the case of scalar fields. Indeed
if m
0 , the equations (10) are synonymous t o (3b).
For the neutrino field the most general transformation (8) obeying conditions (10)
is given by (PAULI-PURSEY
group [ l a ] )
+
y(4
y'(x)
= a eiay,
with
]aI2
+ b e"au.y5yC(4,
+ lbI2 = 1 ,
(11')
a real.
l ) According to t h e usual convention y' shall be a column-vector if multiplied by a matrix
from t h e left. One has per definitionem Atp = y~ AT', Bij = ij BT' etc.
The index "T" denotes the transposed matrix.
2, We propose t o give another way of introducing the operator C in t h e spinor case. 0
course t h e method of part a) can be used a t this place as well, and vice versa the way o
reasoning of this part can be used for scalar fields too.
The matrix c is defined by: cy,, c-l = - y r , - C T = c, c t = c-l
.
The TCP Theorem and its Applicetions
297
(11’) may be constructed from the following transformations
y ’ ( z ) = 7 e“Y5 y ( x )= E y ( x ) E’
(a: real)
(ila)
yf (2) = qc eLa’Y6y c(x)= C y (2) Ct
(af: real)
(11b)
( p : real)
(11 c)
y’(x) = c o s p y ( x )
+ sinpy5yc(x).
The operator C from (11)and (11b) again has the property of charge conjugation
CQ=-QC.
For the charge Q the equations hold
E = e i a Q , [ Q , p ( x ) ] = y ( x ) , Q=i/Wyrydoi’.
U
We again add some supplementary remarks. I n the case of fields without mass
the charge conjugation C can be obtained continuously from the identity. This
is possible because of the new transformation group (11c).
Up t o now the transformations ( l l a ) t o ( i l c ) are allowed only for the “bare”
neutrino field. It depends on the physical nature of the neutrino whether they
may be applied on physical states too. Again there are three possibilities :
a ) There is no conservation of fermions. All operations ( i i a ) to ( i i c ) lead t o
physically realizable states.
b) The physical neutrinos have a fermion charge defined by Q. I n nature you
may find only eigenstates of Q . Only the “gauge transformation” ( i i a ) and the
charge conjugation C are allowed.
c) Majorana field: y = ye. The gauge transformations ( I l a ) only make sense.
The generator of this transformations
;r W r Y s Y d a p
may be considered as a sort of charge (“pseudocharge”). With respect to this
new charge there may exist fields with and without conservation law. I n the
frame of the two-component theory it is given by the difference of the number of
right-hand and left-hand neutrinos.
The equation of motion and the commutation relations remain invariant under
the transformations (11).But because of the existence of inequivalent representations of the commutation relations ( Q ) , this fact does not ensure (11) t o generate
unitary transformations in the HILBERTspace of free particles. By direct
calculation however we can pass from (11) t o the corresponding transformations
in the creation operators and thus directly prove that (11) defines unitary transformations in HILBERT
space.
2. Xpace reflection
With the methods of the preceding section one easily finds the most general
transformation involving space reflection :
a ) scalar field
V’P, t) =q p
V(-Y,
t ) = P y ( r , 4 P+,
(12)
G. GRAWERT,
G. LUDERS,H. ROLLNIK
298
b) neutral vector field
c) spinor field
or
y / ( r ,t ) = qp‘y4yC(--r, t ) = P’y ( r ,t ) P / + .
(14’)
All transformations are unitary. The second possibilities (12’) resp. (14’) evidently are obtained from (12) resp. (14) by a charge conjugation.
It is rather academic t o ask whether P or P’gives the “true” space reflection,
under which the interactions should be invariant according t o ‘(a-priori”reasons.
Let us consider a system of charged particles, e. g. the positronium, and
let us fix by an external measurement the space-time cordinates of the particles
only. I n such a situation the charges of the particles come into play only by
their mutual interaction, and the operations P and P’ cannot be distinguished.
But in many experiments the charges will be fixed (‘absolutely” by comparing
them with an (arbitrary, but fixed) e x t e r n a l unit charge. (E. g. in the experiments with mu-mesons this comparison will be carried through by absorbtion of
negative muons in matter). I n such cases P and P‘ are distinguishable.
But postulating a n a-priori invariance principle regarding to the space reflection
of thew h ole world - in case this is a meaningful postulate a t all in the framework
of quantum-theory - we don’t know whether we should require the invariance
under P or under P’.
3. Time reversal and Inversion. Products of Symmetry Operations
There are no unitary symmetry transformations involving time reversal. Arguments concerning this fact were given already in the first section. The transformations (3’)7 which still contain unitary and antiunitary operations, only
lead t o a time reversal if U is a n antilinear operator.
A direct proof of this statement can be given for scalar fields (and more generally
for each boson field) by using the linear Ansatz
v?’(r,t)=Ag?(fr,-tt)+B9+(~:,--).
If y’ is connected with 9 by a n unitary transformation, y’ has t o obey the same
commutation relation as y . But this requirement leads t o a contradiction ;
(IA(2+
J B / 2 ) d ( f r 1f r 2 , t z -
tl)
=-(IAl2+
py)A(rl -r2,4-tz)=
= d (r, - r Z ,t, - t 2 ) .
In fermion case this argument does not work. The transformation
Y’(T9
1) = r Y 1 Y z Y 3 Y ( r , -
0.
The TCP Theorem and its Applications
299
(Racah time reversal) [I51 does not alter the commutation relations (9). But
this transformation leads t o an inequivalent representation of these relations,
and maps the whole one-particle HILBERT
space into the zero vector.
The physical basis of all these arguments is given by the fact t h a t a n unitary
time reversal must lead t o negative energies. Because of the commutation
relations between T and the translations in time direction
TeiPr+iPOt= e i P r - i P , t T ,
where POdenotes the energy, one could conclude for unitary T
T P , = - POT.
The antiunitary transformations with time reversal can be written as product of
T d r , t ) T' = q T q ( r , - t ) ;
T y(r,t ) Ti
= qT
C t y5
YP,-4
(15)
and the charge conjugation (5b) or (11). Therefore besides (15) the transformation
T ' v ( ~ , tT"
)
= r(p*v'(r,
-t);
T ' y ( r ,t ) T''
= r y C t Y 5 Y C ( r,
t ) (15')
is possible too. (15) will be called the WIGNERtime reversal [16].
Concerning the last case, the inversion of space and time, we refer t o table 1,
where all the discrete symmetry operations are given.
I n the second table we list the squares of the symmetry operators, which will
give us a hint for a practical choice of the phase factors in table 1.
In the boson case we can choose all phases in such a way that the squares of the
operators are proportional t o the unit operator. But in the case of fermions
independently of the choices of the qi:
pa
pit2 =
-y,
T 2 y Ti'
= -p
)
12yIf2= - Y .
(16)
T a b l e 1. The discrete symmetry operations for scalar, spinor and neutral vector fields
U
E
C
P
p'
T
T
J
OrJ'
G . GRAWERT,G . LUDERS,H. ROLLNIK
300
T a b l e 3. Phasefactors for T C P
U
7
T a b l e 4. Transformations of the creation operators (up t o a phasefactor) for scalar fields (a)
and Diracspinors (b)
P , T , (P' C, usw.)
C , P T C , (0,.
. .,)
PC, T C , (P', . . .)
a t (k)
a+ (k)
a+(-4
P
T
PT
C
CY
CT
CPT
Tab1 e 5. Transformation rules for the observables
P
T
c
PT
PC
CT
PTC
Q
Q
-&
Q
-&
-&
-&
- pk
-
Pk
Pk
PIC
- Pk
- Pk
Pk
PO
PO
PO
PO
P@
Po
Po
Mi
-Mj
-
Mi
Mj
xi
-Mj
-
Mj
301
The TCP Theorem and its Applications
Analogously the following relations hold for all choices of the phases in table 1
C2Y Ct2 = y ,
(PT’)y(PT’)’ = -(T‘P)y(T’P)’,
( P 0 ) y(PO)t
= - ( 0 P ) y(OP)+
*
(16”)
The last two relations can be confirmed by direct calculation.
To illustrate the TCP theorem we calculate i n what way TCP transforms the
spinor y. We easily obtain
where the phase q depends on the order of the three operators and is given in
table 3.
I n the following two tables we list the transformation of creation operators of
the particles by symmetry operators. We use the decomposition (4) for the
scalar field and the corresponding FOURIERtransformation
+
of the Diracspinor. Here u($(k) denotes the solutions of (iy,k@ m ) u = 0,
and a:,, (k)the creation operator of a particle of type 0 (particle-antiparticle),
of the momentum k and the spin component r in direction of the z-axis ( r = &).
Disregarding phasefactors one obtains: table 4 a and 4b.
Finally we list the transformation rules of the observables charge Q, energymomentum P, and angular momentum Mi = i Jkl l ) with respect t o our
operations. Prom these rules for the operators we can derive the transformation
properties of particle states, stated in the introduction. Without question the
gauge transformations E commute with Q, P and M . I n table 5 the rules for
T, PT etc. are essentially based on its antilinear character. E. g. T must
commute with all spatial translations and it is because of the antiunitarity of T
that we must conclude in accordance with table 5 .
4 . InvariarLce of the interactions : Determination of the phases
For each of the symmetry operations : charge conjugation, space reflection and
time reversal we have found a set of unitary resp. antiunitary operators in
HILBERT
space (table 1). But evidently each set can be written as a product of a
special operator of this set - fixing arbitrarily the phases qi2) - and the set of
gauge transformations. The arbitrariness of the choice of phase can be reduced
by postulating that the squares of the symmetry operators are equal t o the
identity. (This postulate is possible, but there is no stringent reason for it.)
It restricts the phases of P, T’ and 0‘ t o f 1, f i. One obtains further restrictions if t h e products of certain operators shall be equal. I n order t o let all
operators commute or anticommute, one must choose the phases equal t o f 1
for boson fields, equal t o f 1, f i for fermion fields. Each combination is
Here ( j k l )is a cyclic permutation of (123) and J k l an infinitesima.1rotation.
Even after having fixed the the operators U contain arbitrary phasefactors. As usual
we fix them by setting U
X = xvacuum.
~
~
~
~
~
~
I)
2,
G. GRAWERT,G. LUDERS,H. ROLLNIK
302
allowed. I n t h a t way we get ( a t most double valued)') representations of the
abstract Abelian group generated by the elements
1, P,T and C
with the multiplication table
P2 = T2 = C2 = 1 , p' = P C
J
=PT =
=C
P , T' = T C = C T ,
(18)
TP, 0 = PTC.
Only double valued representations l ) with the properties (16') and ( 16'') can
occur. However the representation of the symmetry group (18) that is generated
in the case of some distinct field, can be determined by means of interactions
only.
At this place we make explicit use of the presumption, t h a t the quantum field
theory can be derived from a LAGRANGian, which is composed additively of a
free-field-LAGRANGian and an interaction term, and furthermore t h a t we can
introduce an interaction (Tomonaga-Schwinger-)picturein the usual way. The
operators G , P and T and their products operate on the state vectors of this
picture, and the following discussions of this paragraph always refer t o it.
If there exists a n interaction invariant under the operation U between two
fields, we can fix the relative phases. Indeed the interaction described by the
term L , in the LAGRANGian, is invariant under u, if the transformed operator
U L , U t can be obtained from L , by a gauge transformation E; that means there
is a choice of the phases of U in such a way that
U L , U'
= L,
(19)
is fulfilled. This equation determines the phases of U for the fields containedin L,.
Let us consider in detail the YUKAWA-inteI%CtiOnbetween a spin 0 - field A (x)
and two spinor fields yl(x), y z ( x ) with non-vanishing mass. The most general
expression for L, invariant under proper LORENTZ-transformations and certain
gauge transformations is given by
Lw = g$1 yeA
+ g'
y2 A
y l y5
+f
wlyP
YZ A'P
+ f'
y1 Y P y5
YZ
PI'
$- h*c'
(20)
We ask for the restrictions of (20) following from invariance under C , P or T
We abbreviate the LAGRANGian (20) by the symbol (9, 9'; f , f ' ) . By straightforward calculations one can prove the following transformation properties
of L,:
C(g,g';f,f')C'
=
(qcg', - q c g ' * ; q c r , - q c f ' * ) ,
P(g,9'; f f ' ) Pt = ( q p g , - q d ;q P f ,-- q p f ' ) ,
T ( g ,9'; f , 1') T' = (TTS*,?ITS'*; ??Tf,* r l T f ' * ) .
7
(21a )
(21b
(21 c)
Bere we have introduced
Ti
qT(1)qi(2) qi(0)
(i = C , P ,T)
means: If the representation of the LORENTZ-groupis double-valued, we are allowed
to represent the identity of the discrete symmetry group by
1 and - 1 too.
1) That
+
303
The TCP Theorem and its Applications
are the phases of the boson field.) I n ( 2 1 ~ the
) antifor convenience. (qi(0)
unitarity has been taken care of.
By combining (21a- c) in any order we obtain
@ ( g , 9’; f , f’)
0’ = a h , 9’; f > f’).
(21’)
Here 0 stands for the product of C, P and T in a certain order. a is a phasefactor which depends on this order. By the choice qc = q p qT and ‘ ;7 = 1
we arrive at ci = 1 in each case. This is a special case of the TGP-theorem:
The interaction (20) is invariant under T’CP without any restriction on the
coupling constants.
On the contrary invariance under one of the symmetry operations only holds
for special values of the coupling constants. Prom (21) and (19) we conclude the
following necessary conditions :
C-invariance
Re(g’ g*) = Re(g’ f * ) = Re(f’ g*) = Re(/’ f * ) =
(22 a)
= Im(g* f ) = Im(g’* f’) = 0 .
I. e. the phases of g and f (resp. of g’ and f’) are equal as well as those of g and g‘
etc.
g* g’ = f* f’ = g* f’ = f* g’ = 0
P-invariance
T-invariance
.
(22b)
Im(g* 9’) = Im(f* f ’ ) = Im(g* f’) = I m ( f * 9’) =
Im(g’* f ’ ) = 0
.
= Im(g* f ) =
(22c)
1.e. the phase of all coupling constants are equal.
Evidently P-invariance holds if
g’
= f’ = 0
and q $ ( l ) q p ( 2 ) q p ( 0 )= 1
or
g =f =0
and
qC(1)qp(2)q p ( 0 )= - 1
.
Consequently a P-invariant interaction fixes the phase q p (0) of A ( x) relative t o
yl y2 but it cannot fix the absolute phase of the boson field. Only if yl = y2,
in which case A ( x ) must be a neutral field, the phase is determined uniquely by
the form of the (P-invariant)interaction. E. g. by this way of reasoning we may
recognize the pseudoscalar nature of the neutral pions. [I71 For charged mesons
on the contrary the q p ( 0 )cannot be determined absolutely. It is pure convention
if we put equal the phases of the neutron and the proton. Only after having
accepted this convention the phases of charged pions can be determined by experiment [I?‘]. From our arguments we immediately see t h a t there are neutral
particles, the phase q p of which is not determined by any interaction. I n the case
of KO-mesons e. g. the two fermion fields in strong YUKAWA
interactions must
be different because of conservation of strangeness.
The case y1 fy2 exhibits special features for charge conjugation and time
reversal too. The Lagrangian L , is Hermitean only if g is real and g’, f , f’ are
imaginary. Hence from (22a) and (22c) we conclude
charge conjugation:
time reversal:
f f‘
= g f = g’
f
g g’ = g f = g f’ = 0
=0
(23a)
.
(23b)
304
G. GRAWERT,G.
r.UDERS,
H. ROLLNIK
Both the conditions imply that a neutral scalar field (9‘ = f’ = 0 ) must not be
coupled with and without derivatives simultaneously [3, 4, 51. Again the underlying reason is the T C P theorem : a P-invariant interaction must be transformed
by C in the same manner as by 1’.Another conclusion from ( 2 2 ) and (23) is :
I n the case of interactions without derivatives P-and T-invariance automatically are coupled and C-invariance always holds.
If the conditions (22a, c) are satisfied the phases must be chosen according t o
8’ =
q z ~ ( l ) q c ( 2 ) q c ( o=
) --9 = - 9*
8’”
f
--
f*
=
f‘
-f*
g
9’
f
f’
$(l) q T ( 2 ) qT(o)= g*
- = - = - = -gf*
f*
f’*
Again the phases of the boson field are fixed if y1 = y,.
I n exactly the same way we can get invariance conditions for a n interaction of
a vector field A , with two spinor fields. We denote again the couplings without
derivatives as g-couplings, the couplings with derivatives as f -couplings; the
invariance conditions are synonymous with (22a) through (22 c). A dserencc
arises for y, = yz. I n this case the Hermitean character of L , puts a different
restriction: f’ must be real g, g‘ and f must be imaginary. Hence (23a, b) have
t o be replaced by
c: g g ’ = g ’ f ‘ = g ’ f = 0
1’:f f’
= g’ f ’ = g f ’ =
0.
Now one concludes from C- as well as from T-invariance, that a pseudovectorfield (g = f = 0) must not be coupled with and without derivatives at the same
time. As a further conclusion we remark : the electromagnetic interaction
including the PAULIterm (only g and f # 0 ) is C- and T-invariant.
The phases qc, q p are intimately connected with charge- und space-parity of one
particle states. The charge parity only can be defined in the case of “neutral”
states, i. e. eigenstates of the charge Q with eigenvalue zero. Let x be such a
state:
(P,s) = C ( P , 8 ) x ( P , 4 .
(24)
I n this equation p denotes the four-vector of energy-momentum with
p , p” = - m2,and s distinguishes the different spin states. Now ~ ( ps), can be
constructed from a special statex (p,,, so) with the help of proper LORENTZ-transformations and spin operators, each of which commutes with the charge conjugation C. Hence the parity c in (24) cannot depend on p and s. It only depends
on the nature of the particle.
The connection of c with the phase qc can be derived from the transformation law
cx
C a J ( k , s ) C ’ = q c a l ( k , s ) , C a l ( k , s ) C += q c a J ( k , s ) .
Accordingly the parity c is equal t o qc times the charge parity of the vacuum,
which we conventionally have put equal t o unity. Even without this convention
the ratio cJcZ of the charge parities of two particles A,, A , is equal t o
v c (1)/vc( 2 )
and can be determined by experiment, if A, and A , interact by a C-invariant force. E. g. from the existence of a process
N
+ A,
+N
+
A28
(254
The TCP Theorem and its Applications
305
where N is a n additional "genuinely" neutral particle, the equality of the charge
parities of the two fields can be inferred. I n practice the decaying processes are
important,
A1+nA,,
(25b)
from the existence of which it follows: c1 = c:. n being even (e. g. no-+ 2 y )
one has c1 =
1 ; if in addition the decays with odd n do not occur the parity
of A, is determined too: c, = - 1. There are (up t o now) only two genuinely
neutral bosons ( y , 7co) among the elementary particles- therefore the concept
of charge parity is important especially for composed particles (e. g. the positronium [a, 18, 191.
Space parity can be defined for particle states with non-vanishing mass. Here
we consider the coordinate system in which the particle is a t rest, and we define
+
P X ( 0 ,s) = P X ( 0 , S ) .
Again the parity p does not depend on the spin s. We can proceed quite analogously t o the case of charge parity.
P a : ( k , s ) P += q > a : ( - k , s ) ;
P a ' ( k , s ) P += & q P a t ( - I c , s ) .
in the second equation the upper sign holds for bosons, the lower one for fermions
[one has to use y4 U + (- k ) = & u* (- k ) ] . I n the case of bosons the
phases must be real, if we impose the condition P2 = 1 mentioned above,
therefore the parities of particles and antiparticles are equal. I n the fermion
case this remark holds only for q p = & i. But a particle-antiparticle state (both
particles a t rest)
Pa:(O,sfal(O,s')Xvae= - [qpj2,:(0,s)at(0,s')Xvac.
always has a negative parity. This is the reason for the frequent statement [I91
t h a t the parities of particles and antiparticles in the fermion case are opposite.
To get a n experimental determination of the parity one can use the reactions
(25a, b) again. In (25a) the particle N may be arbitrary in this case, but one
must take account of the relative motion of the particles. The processes (25a, b)
are restricted by various conversation laws (of charge, of baryonic number etc.).
Indeed these laws determine, which of the relative parities can be measured by
experiments.
A parity for time reversal cannot be defined. Because of the antiunitarity of T
the concept of a n eigenstate of T makes no sense (of App. 1).
5 . Physical meaning of time reversal
I n order t o discuss the physical interpretation of time reversal we investigate
the transformation properties of the S-matrix under T resp. J. We infer from
commutativity of T and the interaction Hamiltonian and from the antilinearity
of T :
T t S T = TtP[ei/Htod'"]T= $ [ e - i / R w d ' z ] = 8-1,
(Here P denotes the time-ordered product,
we have
(TXA
P the inverse order.) Therefore
8 TXF) = (TtsTXF, X A ) =
(XF9
SXA)
-
(26)
G.GRAWERT,
G. LUDERS,H. ROLLNIK
306
Here we have used the definition of the anti-Hermitean conjugate operator,
(cf. App. I.) We apply (26) t o free particle states lkl, s,; k,, s,), which are
characterized by momenta ki and spin directions si. Because of
-L
Tpl,s,;k,>s,) = 1-kl, -81;
-82)
( u p t o a phase) eq. (26) means: the probabilities for the Processes ( a ) and ( b )
of figure 1 are equal. I. e. if the transition rate from the state A t o the state P
Fig. 1 Processes connected by time reversal T.
is given by W , the transition probability from F‘ t o A‘ is given by the same
number, the state A‘ (resp. 3’’)
differing from A (resp. F ) by inversion of all
momenta and angular momenta :
W ( k , : sl,k,, s2+ k ; s;, k ; ) 8;) = W ( - k ; , - s;, - k;, - s;+
)
-k1, -s1, -k2, - s2j.
(27 a )
/”
d y-5,
-sz
-s;
Having invariance under inversion (i. e. the
product PT) the processes of figure l a and
figure 2 proceed with equal probabilities (observe
Jlk,,
- 51
’
81; k ? , S 2 ) = 1161, -81;
k,,
-32)).
If only particles with vanishing spin are involved, the J-invariance leads to the principle
of detailed balance :
W (k,, k , -+ k;, kb) = W ( k i , kl-+ kl, k,) . (27b)
I n the case of non-vanishing spins this equality
only holds after averaging over the spin directions: semi-detailed balance [ZO].It is easy t o formulate this principle in the
most general manenr. The initial and the final state are characterized by
a set of quantum numbers, part of which remains unchanged under inversion while the rest is multiplied by (- 1). The equation (27b) is valid
after having averaged over the “minus”-quantities [ZI]. (Unfortunately the
notion of detailed balance is used with different meanings in the literature.
Some authors e. g. denote eq. (27a) as this principle.)
With figures 1 and 2 we also can give a simple interpretation of the TGPtheorem: According t o this theorem the process 1a involving particles and the
process 2 involving the corresponding antiparticles have equal probabilities.
Fig. 2 The space-reflected process of
flg. 1b.
The TCP Theorem and its Applications
307
6. Discrete Xymmetry operations in general field theory
The discussions of § 4 make use of the existence of a n interaction Lagrangian
and of a n interaction picture for the time dependence. I n the general relativistic quantum theory of interacting fields, we only assume that the field
operators have the asymptotic behaviour of free fields a t f infinity.
More precisely: We have two sets of field operators pin(x), yi,(x), .. . and
Tout( x),?pout(x),... which obey the field equations and commutation relations
of free (scalar, spinor, .. .) fields, and which are looked upon as describing the
physical situation in the far future or the far past respectively. Furthermore
we have operators p ( x ) , y ( x ) . . . which interpolate for finit,e times between
the incoming and outgoing fields, that is we have for the Fourier transformed
operators a ( k , t )
Lim ( @ ] a ( k , t ) I Y=
) (di aout( k ) Y ) etc.
t-5-
I
l i n
for arbitrary (normalizable) state vectors di, Y.
The S-operator is defined by
flt p in (x)X
= pout (2)
etc.
Now it is possible t o introduce the symmetry operations for incoming and outgoing fields separately. All the formulae for the unitary symmetry operations
(which don’t include time reversal) in 9 1-3 can be used a t once, if only the
field operators as well as the symmetry operators themselves are augmented
by “in” or “out” subscripts. E. g. we define a n operator Cin for scalar fields by
Cin pin
(XI
-
Ci+n = qc pi+n(x)
(Cp. formula 5b.)
For time reversal one requires (in the case of scalar fields)
Tin pin (r,t )
Tout pout
(
~
= TTvout (r,- t ) 2
t ) Tkt
3
= qr Fin (r,- t ) 8
because of the discussed physical meaning of this operation and its antiunitarity. (Analogous equations follow for products of time reversal with any
one of the unitary symmetry operators.) Similarly one may define symmetry
operators for a n arbitrary space like surface IT (in certain cases)
C,p(x) CJ = r e p t (x)
for
x
E
IT,
using the interpolating operators ~ ( x y) (, x ) ,. . . But in the spirit of modern
field theory we shall not use these generalisations and restrict our considerations
t o the asymptotic operators Ci,, Gout etc.
The in- and out- symmetry operators are connected via the X-matrix in a
similar way as the field operators. (Up t o phase factors) we have
X‘Cj,X
= Gout,
Xi Pins= Pout,
St T i n s t = Tout
etc.
Invariance under charge conjugation resp. space reflection now means that
the phase factors qc or 'yip of the various fields can be chosen in such a way t h a t
resp.
Ci,S = SCi,
Pi, S = Spin
23 Zeitschrift ,,Fortschritte der Physik“
or
or
Ci, = Gout ,
Pin= Pout.
G. GRAWERT,G. LUDERS,H. ROLLNIK
308
Invariance with respect t o time reversal means
Ti,&’’
= XTi,
or
1’.
111 =
- T out .
(Remark again the antilinearity of T in the commutation property with the
S-matrix.)
The general theory of “interpolating” fields without explicit use of Lagrangians
can be cast into a version, which takes the vacuum expectation values of
products of field operators as basic quantities, as WIGHTMAN
[27] has demonstrated.
Invariance of the theory with respect t o the symmetry operations then means
certain properties of the vacuum expectation values. We list these properties
for the case of scalar fields only.
C-invariance ;
P-invariance
T-invariance
CHAPTER I1
T C P Theorem
Suppose we are discussing some quantum field theory with Lagrangian explicitly
given. Then by direct calculation we can show, t h a t the invariance of the
Lagrangian under proper Lorentz-transformations always implies its invariance
with respect t o TCP. An example of such a calculation can be found in
paragraph I, 4.
To give a general proof of the T C P theorem we take up a slightly modificd way
of reasoning. We don’t take the explicit construction of the operators T , C and P
as a starting-point. But we put forward the following question: Given any field
theory, which is invariant with respect t o the proper LORENTZ-group, is it
possible t o find some further symmetry-operation, which leaves invariant the
theory! After having proved the existence of such an operation, we shall show,
that it is physically identical (up t o a phase factor) with the product T C P .
1. T C P Theorem in the algebra of field operators
We make the following assumptions :
1) The field equations are of local type.
2) The Lagrangian is invariant under the proper LORENTZ-group.
The T C P Theorem and its Applications
309
3) The usual connection between spin and statistics holds, .i e. the field operators
at space-like points of a field with integral spin commute, those of a field with
half -integral spin anticommute l ) .
4) Boson fields commute with each of the other fields, kinematically independent Fermi fields anticommute 2 ) .
5) Every product of field operators is completely symmetrized with respect t o
Boson fields and antisymmetrized with respect to Fermi fields 3).
The first postulate, stated in a more detailed way, means, that all the field
quantities are spinors or tensores of finite rang, and that the interaction is local
and contains derivatives only up t o some finite order. Without loss of generality
we can assume, that the “elementary” field operators behave as irreducible
representations of the proper LORENTZ-group.These irreducible representations
- as is well known - can be characterized by two integral numbers, and thus
under a proper LORENTZ-transformation /1 any field u, (x)is transformed:
D ( A )U , ( X) D ( A )
=
C
drr,( A )u,, (klX) ,
r’
where the matrix d, r, belongs t o one of the representations D (n,rn)4 ) .
The representations and thus the field quantities can be classified into four
classes *) :
class I:
n,rn even
I1:
n, rn odd
I11:
n, even, rn odd
n, odd, rn even.
IV :
It is clear, t h a t every product of field operators just belongs t o a definite class4).
I n the following we denote by u A a field quantity belonging t o the class A .
l) A derivation of this postulate starting with causality and definiteness of energy (as
the main premises) has been given some time ago for free fields by W. PAULI
[22].Recently
proofs for interacting fields have been given by G. LUDERSand B. ZUMINO[23] and
[Za].
independently by N. BURGOYNE
2, A discussion and a first derivation of this postulate is given by G . LUDERS[25].Discussions of the relevancy of this presumption for the TCP theorem can be found at the
[2GJ
same place and in a paper b y KINOSHITA
3, In formulae t h a t means: Instead of the “simple” products u1 . ?c2 ... UN we take the
quantities
where P runs over all permutations of 1, 2, . . ., N . p (P)
is t h e number of transpositions of
Fermi fields in the permuted ordering il, i,, . . ., ix compared t o the original one 1,2, . . ., A7.
For instance t,he Dirac current i e (7y p tp has t o be replaced by
I n quantities, which contain each kinematically independent field only linearly, of course
the simple and the symmetrized products coincide beoause of postulate 4.
”) Compare Appendix 2.
LJ*
310
G. GRAWERT,
G. LUDERS, H. ROLLNIK
We introduce the following “transformation of the classes’’ l) :
i
(x)--t UI (- x)
(x) - 2611 (- x)
UIII (x)
i UIII (- x)
UIV (x)+ - i U 1 v ( - - 2 ) .
UI
UII
-+
--f
Now we are in a position t o state PAULI’s theorem [7]: If we transform the
elementary field operators according t o (28), and if we invert the ordering of field
operators in every product, then every symmetrized product is transformed
according t o (28) likewise.
The transformation (28) plus the inversion of operators in products means a
mapping of the algebra of field operators onto itself. It has been named strong
reflection.
The theorem clearly implies that every LORENTZ-COVariant field equation is
invariant under strong reflection. For a covariant equation is an equation
between quantities of one class only, and consequently under strong reflection
the whole equation is just multiplied by a factor. The Lagrangian, belonging
t o class I , ( m = n = O), is invariant with respect to strong reflection. Hence we
have the theorem : Any quantum field theory, which obeys the postulates 1 t o 5,
is invariant under strong reflection [7].
For physical interpretation we remark, that the energy-momentum tensor is a
quantity of class I, the electric current a quantity belonging t o class I12).
Thus strong reflection has the physical meaning of space-time reflection plus
change of the sign of electric charges.
To see what happens we treat two examples : The product
belongs to class I1 and under (28) and inversion of the ordering of field operators it is
transformed into
thus f(r)-+ -f(- 5) as it should be according t,o PAULI’S
theorem. The Dirac-Spinor with
four components is Composed by a two-component spinor of class 111 and a two-component
spinor of class IV.
1) This transformation already has been introduced by W. PAULI
in his discussion
connection between spin and statistics [eel.
2) Compare Appendix 2.
6f
the
The TCP Theorem and its Applications
311
Transformation (28) means’)
w(4
-+
y(z) -+ y(-
i 1’5 w(--z),
2) i
y6 ’
The quantity
(part of the energy-momentum tensor) is transformed into
thus
fp’
(x)+ fl.” (- z) as it should be, f”” being a quantity of class I.
2. Proof of PAULI’S
theorem
n(x)may denote any symmetrized product containing
nA factors of class A.
Transformation of the elementary field operators according to (28) yields
n ( x ) -3- (- l)%in111 (- i)”m IT(-2).
The inversion of the ordering of operators yields another factor
(-
1) ’ l a (a111 i“IV) ( ~ I I I+ ~
I V
-1)
(the exponent being the number of transpositions of Fermi fields which occurs
by inverting
Use of the multiplication table of classes (compare appendix 2) and explicit evaluation of the product of factors - 1 and i leads t o the
following table
n(x)).
n ( z )belongs Strong reflection
maps n ( z ) into
. n ~n~~nm . n ~ ~t o class
I
I1
I11
IV
IV
I11
I1
I
g
=
even,
-
n(-4
n(- 2)
in(-2)
-in(-z)
in(-2)
- n(- z)
- i n ( - z)
n(-4
u = odd.
l) Because of the form of y ( x ) as it is written down, we have t o use VAN DER WAERDEN’S
representation of the y-matrixes : The infinitesimal LoREmz-transformations yp y y with
p =/= v are
- (2)
with two-rowed matrices A, B. We take ys Hermitean: y6 =
312
G. GRAWERT,
G. LUDERS,H. ROLLNIK
Comparison of the last two columns with (28) shows t h a t everything is transformed as was stated in the theorem.
3. T C P Theorem in Hilbert space
The version of the TCP theorem due t o PAULI,which was discussed in section 1
of this chapter, establishes an invariance within the algebra of operators.
It contains no reference t o a HILBERTspace, in which these operators act.
The strong reflection is a mapping of the algebra onto itself, which cannot be
rcpresented.by a n unitary or a n antiunitary operator in HILBERTspace. This is
evident from the fact, that strong reflection prescribes an inversion of the
ordering of operators in products [ 9 ] .
Furthermore we can examine for instance the application of strong reflection
t o a free Dirac spinor field. Decomposing the Dirac spinor into positive and
negative frequency parts, t h a t is into production and annihilation operators,
wc get
Compare this formula with (4‘) in paragraph I. 3 ! If we try the Ansatz
iy, y(- x) = U y ( x ) U-’
with unitary operator U in HILBERTspace - (unitary, since y’(x) and y(x
fulfill the same field equations and commutation relations) - we have t o
conclude
U a X - (k) U - l = a - 7 ,- (k).
But this rclation is clearly impossible for a n unitary operator.
I n order t o substitute the strong reflection by an operator in HILBERTspace,
we have t o add a second mapping of the algebra of operators, which again
inverts the ordering of operators. However this mapping must not change the
physical observables. The Hermitean conjugation answers these demands. At
this moment we have t o introduce in addition t o the five postulates of 11, 1
a sixth one in order t o guarantee the invariance of quantum field theory :
6) The Lagrangian is self-adjoint.
The Hermitean conjugation of course means complex conjugation of all c-numbers. Hence the symmetry operator has to be a n antilinear operator.
Restricting the statement t o scalar, Dirac spinor and vector fields we can present
t h e thcorem [9]:
Every quantum field theory of interacting fields of spins 0, 1/2 and 1, which
obeys the postulates 1 to 6 is invariant with respect t o the following antiunitary
operation in Hilbert space
This transformation we get by taking the “complex conjugate” of (28).
The proof of this theorem is complete, if we demonstrate, t h a t 0 indeed is an
antiunitary operator in Hilbert space. To do this we only have t o compare (29)
313
The XCP Theorem and its Applications
with (Table 1 of I. 3) : 0 is up t o a phase factor identical with the product of the
three operators in Hilbert space T , C and P. C and P are unitary, T is antiunitary, hence 0 is antiunitary.
This comparison also gives the physical interpretation of the operator 0 and of
the physically relevant invariance formula for the S-matrix elements
( 0X A ,
0 X B ) = (XB7
i30)
XA) '
4. T C P Theorem in terms of vacuum expectation values
WIGHTMAN
[27] has demonstrated, that any quantum field theory is completely
defined by the set of vacuum expectation values of field operator products.
A proof of the TCP theorein in terms of these vacuum expectation values is due
[lo].
t o JOST
Postulates :
1) Invariance of the theory with respect t o the proper LORENTZ-group.
2 ) Definiteness of the energy; existence of a vacuum.
3) Weak causality :
+
For all ( n 1)-tuplesofspace-timepoints(s,, . . ., X , ~ + I ) ,for whichCli(g-sXi.tl)
is always a space -like vector, if only l i 2 0 and 2 l i = 1, we have the relation
cqJ1(x1)412(52) * - 9 - 1 - 1 ( G L + l ) ? O
I
= (-l)'<Yn+l
(%+1) *
*
(31)
.Y2(xz)Y1(~1))0
(where (T is the number of transpositions of Fermi fields in the inverted product
compared t o the original one).
This weak causality is certainly guaranteed, if we impose the usual causality
requirements for operators formulated in postulates 3 and 4 of 11.1, but it is
indeed much weaker than these requirements.
JOST'S
theorem :
I n any quantum fields theory, which meets the postulates 1 to 3, the vacuum
expectation values are invariant with respect t o 0 ,i. e. we have
+
for every ( n 1)-tuple of space-time points.
If for instance the n 1 fields yi are scalar fields1), equation (32) reads
(Ml(4
M2(%)
..
+
M/ L
tl
(%-tl))0 = (Y! (-
XI) Y I (-
x2)
*.
.Mnt+l(--
%L+l))?J
*
(32')
The proof of this theorem is based on results concerning the analytic properties
of vacuum expectation values, which have been derived by BARGMAN-HALL
and WIGHTMAN.We shall state these results without giving the proofs or
detailed discussions and we shall restrict ourselves t o scalar fields [27, 281.
l)
scalar with reepect to the proper LORENTZ-group.
314
G. GRAWERT,G. LDDERS,H. ROLLNIK
The vacuum expectation value of a product of n + I scalar field operators
yi(xi) is because of invariance under translations a function of the n variables
ti = xi - xi+lonly.
(33)
= F ( t I , 52, . . - > 6,).
This function F (El,. . . , 5,) is the boundary value of a function F (C1,. . . , C,)
of complex four-vectors, which is defined and analytic in a region R,,consisting
of all points (Cl,. . ., 5,) with I m 5, (i = I, . ., n) element of the forward light
cone. We have
(411(~1)4)2(~2).
*
*~)n+l(~n+l))o
.
With respect to the proper LORENTZ-group we have
F ( A 51, . , A 5,) = F(51,. . -,5,) *
. . ., c,) over R, can be continued analytically in a n unam1
.
(rl,
The function F
biguous way into the region R,I, t h a t arises if we apply all complex LORENTZtransformations of determinant
1 t o all the points of R,.
The continued function fulfills
+
P ( A 5; > . . . , (1G ) = P ( G >. . . , 5,)
+
for all complex LORENTZ-transformations il of determinant
1.
Though R, contains no real points, the region R,'will contain points (el, ... , en)
with real four-vectors pi.
If we start with a function over the region of real points contained in RL , then
we have - if any - exactly one analytic continuation into the whole of .R{L.
Hence the vacuum expectation values a t the real points of RL fix in a n unambiguous way these values everywhere.
The region of real points contained in RI, has been determined by JOST
[lo].
It consists of all real (pl, . . ., en) for which CAiei is space-like if Ai 2 0 and
L':ili = 1. (This is just the requirement for ( n 1) -tuples written down in thc
postulate of weak causality).
Having these results a t our disposal the proof of JOST'S
theorem is rather
elementary. We again restrict ourselves t o scalar fields and consider the function
F(Cl>
. . ., 5%)defined in (33) as well as the following function
+
Q(51,
. . . 5, n ) = (pi (Xl)Q)Zt(%)
The postulate of weak causality
(G =
. .p ~ - r l ( ~ , L + l ) ) l l ~
*
0) means
p(51,.. . > 5,) = G*(51> . . > t n )
(35)
for all real points of 36.One of the complex LORENTZ-transformations, that
leave invariant the expectation values, is the space-time inversion ti-+ - Ei.
(Note t h a t with the help of complex LORENTZ-transformationsthe inversion can
be obtained continuously from the identity.) Hence we can state
F(51,
f
. . , L ) = G*
(6-
51,
'
'
., - En)
(36)
for all real points of Ri.Now we continue t o all points of RL and get
E"(51, . . ., 5,)
= G*(-
C;, . . ., - 5 9 .
(37)
The T C P Theorem and its Applications
.
(c1,
~
315
If
. . ., 5,) is a n element of R,, so is (-(?, . ., -5;). Consequently we can
go t o the limit I m Ti 3 0 in equation (37) as described in (34).We conclude, that
(36) holds everywhere.
However (36) is exactly the same equation as (32). Thus we have proved
JOST’S
theorem.
( I t is interesting t o know, that the following reversed theorem holds too : The
postulates 1 and 2 together with TCP-invariance imply weak causality. That is
TCP-invariance and weak causality are equivalent requirements1). Proof :
Starting now from equation (36) t o be fulfilled everywhere, we a t once conclude,
that (35)holds for all real points of R,, because of invariance of the WIGHTMAN
functions within Rk under the complex LORENTZ-group.)
CHAPTER I11
Applications of the Theorem ’)
1. Particles and Antiparticles
Empirically one observes that some particles occur in nature in one type only
whereas others occur in two types. So all neutral x mesons are identical but the
charged ones exist with either charge. I n the latter case one speaks of particles
and antiparticles though, logically, there exists no property of being a particle
or being a n antiparticle but only the relation that the two types of a particle
are antiparticles t o each other. We shall show t h a t in order t o justify the
existence of such particle twins one needs, besides the T C P theorem, the concept of generalized charge.
By generalized charge we mean a n additive quantum number3) carried by the
individual particles which is strictly conserved. All generalized charges of the
vacuum are equal t o zero by definition. I n nature there appear to exist three
such generalized charges :
1. electric charge,
2. leptonic charge,
3. baryonic charge.
The electric charge pla,ys a particular role in that it is not only a n additive
quantum number but also measures the strength of interaction of a particle
with the electromagnetic field. As long as one restricts oneself t o strong and
electromagnetic interactions there is a further generalized charge called strangeness. Empirically all particle-antiparticle pairs are indeed characterized by
one or several non-vanishing generalized charges :
1. by the electric charge: e;’ e - ; p+,p-; K+, K-; x+,x-; p+, p-;
2. by the leptonic charge: v, V; e+, e-; p+, p-;
3. by the baryonic charge: pf, p-; n, 5;AO,
no.
Indeed this is the accurate version of JOST’S
theorem, he proved in his original paper [ l o ] .
Modified version of a series of lectures delivered at the Seminar on High Energy Physics
in Oberwolfach, Black Forest, Fall 1958.
3) We remind the reader t h a t parity for instance is a multiplicative quantum number.
I)
2,
316
G. GRAWERT,
G. LUDERS,H. ROLLNIK
Presumably there exist antibaryons t o all baryons ; they would be characterized
by ( a t least) their baryonic charge. Within the range of validity of strong and
electromagnetic interactions there is also a particle-antiparticle pair characterized solely by its non-vanishing strangeness, i. e. the pair KO,
I n Ch. I1 i t was shown that, in a rather wide class of relativistic field theories,
there exists an anti-unitary transformation 0 (the TC P transformation) which
anticommutes with all operators Q of generalized charges
KO.
{ 0 , Q=}0 .
It commutes with the momentum operators1)
[0,Pa] = 0 .
(1)
Now, a one-particle state @ with restmass p is defined by the eigenvalue problem
+
( P U P , p2) Q,
=
0;
(3)
for a one-particle state certain additional conditions are to be satisfied. Since Q
defines a conserved quantity
[Q?
P
P1
(4)
=0
the state @ can be chosen as an cigenstate t o Q with eigenvalue (charge) q
(5)
Q@ = 4 0 .
I n the spirit of super selection rules 1291 one would even say that onlyeigenstates t o Q have a physical meaning. From Eq. (2) i t follows that @@ is a oneparticle state with the same mass
(POP"+ pu")@@
=
0;
(6)
cccause of Eq. ( 1 ) its charge is opposite. So, if @ (or the corresponding particle)
iarries no generalized charge of any sort one cannot argue t h a t Q, and O O are
ntrinsically different states. They might just represent states of the same
particle (i. e. belong t o the same linear manifold transformed into itself under
the proper inhomogeneous LORENTZgrOUp). For non-vanishing charge, however, the two states certainly are intrinsically different; then there are two types
of particles with equal mass which are antiparticles t o each other. One easily
shows that they also have the same spin. The whole argument, strictly speaking,
refers t o stable particles only; unstable particles shall be discussed later2).
An instructive example for the role of generalized charge in the particle-antiparticle concept is presented by the neutral K mesons. Within the range of
validity of strong and electromagnetic interactions there is a particle-antiparticle
pair characterized by its non-vanishing strangeness. When the weak interactions are "switched on" this generalized charge breaks down and the mass
degeneracy is split into two in general distinct masses; that the actual K mesons
then become unstable is irrelevant for the present general argument.
Let us add a few remarks of a more formal character. As shown in Ch. I, Sect. 1 a,
the operator&is the generator of gauge transformations of the first kind, a t least
l) Because of the anti-linearity of 0 it is important to use hermitian operators P, (i. e. the
0-component instead of the 4-component) in this equation.
z, We might mention that the requirement of non-vanishing charge cannot be circumvented
by postulating C or C P invariance.
The TCP Theorem and its Applications
31 7
for bare fields. This observation throws some light on the use of non-hermitian
fields in quantum theory. Since hermitian fields can be regarded as the basic elements, non-hermitian fields are reasonable concepts only if real and imaginary
part are interwoven by gauge invariance. If one starts with non-hermitian bare
fields one will, in general, not find mass dubletts for physical states (defining
physical non-hermitian fields) if there is no gauge invariance (or corresponding
generalized charge). I n the presence of such gauge invariance, however, the
TCP theorem immediately leads t o mass dubletts and justifies the use of
physical non-hermitian fields.
The previous remarks do not apply t o unstable particles. Though t o our knowledge the formal role of unstable particles in quantized field theory has not
been clarified completely so far it is rather evident
that i t should be possible t o extract their properties
,b.
,
from the S matrix (which connects incoming and outgoing states of stable particles only). We shall base the
following discussion on the conjecture1) that, in a
process between stable decay products of an unstable
Fig. 3 Transition from state a t o state b by
virtue of an intermediate unstable
particle.
Fig. 4 Particle with spin “up” in a magnetic fleld generated by
the current in a loop.
particle (Fig. 3), this unstable particle makes itself felt though a resonance
due t o its “propagator” in the form
r
That restmass p and reciprocal mean life
occur in just this combination is
inferred from the observation t h a t the “wave function” (whatever the exact
meaning of this word might be) of the unstable particle has t o obey the time
dependence exp [- i ( p - i r / 2 )t ] . One might further conjecture that the numerator of the resonance term can be factorized in a way that exhibits the decay
matrix but this further conjecture shall not be discussed here.
If initial and final state carry some non-vanishing charge one certainly will
attribute the same charge t o the intermediate unstable particle2). Now, equation (11, 30) or, in the present notation
The same final results were obtained without this conjecture by LUDERSand ZUMINO
[30]. These authors use a formulation which can be proved in quantum mechanics but is,
perhaps, less applicable in quantum field theory. Cf. also the pioneer work by LEE, OEHME,
and YANG[31]. It is the merit of the latter authors to have recognized the practical applicability of the TCP theorem.
2, This assumption of non-vanishing charge is essentially equivalent to the postulate in [31]
that particle and antiparticle do not decay into common channels.
l)
318
G. GRAWERT,
G. LUDRRS,H. ROLLNIK
permits t o go over t o processes between the corresponding antiparticles. The
unstable particle which now links initial and final state carries the opposite
charge and should be regarded as antiparticle t o the one considered originally. Mass and lifetime of both particles are the same as follows immediately from a combination of Eqs. (8) and ( 7 ) l ) . We mention already a t this
occasion and come back t o it in Sect. 2 that, because of the reversal of the
order of events in time, one cannot conclude from the TCP theorem t h a t
also branching ratios into corresponding channels are the same for particle
and antiparticle. They are the same, however, if C holds; they are also the
same after summation over spin directions if only CP holds as actually appears
t o be the case in weak interactions.
Let us finally show that, as a consequence of the TCP theorem, (stable) particles
and antiparticles have opposite magnetic dipole moments 2). “Opposite” here
of course means with respect t o the direction of the spin. Instead of giving
a formal argument we wish t o discuss a Gedankenexperiment (Fig. 4). To
measure the magnetic moment of a particle one has t o bring it into a n external
magnetic field; this magnetic field may be generated by a loop of a wire conducting a current of electrons. Let us assume the spin of the particle (using for
a moment this unsatisfactory terminology) to be <‘up’’. Let the spatial reflection contained in TCP be done with respect t o the center of the loop. Then,
after application of the TCP transformation, one obtains an antiparticle a t
the same position but with spin “down”. What happens to the magnetic field
under this transformation? Originally there is a current of negative electrons in
the wire with the direction of motion opposite t o the direction of the current. Under T the direction of the electrons is reversed; C transforms the negative electrons
into positiee ones so that the electric current (and the magnetic field) is the same
after application of TC. P finally does not affect the current in the loop. Therefore, the magnetic field stays unchanged under TCP.As a consequence of the
TCP theorem the energy of the particle in the magnetic field with spin u p and
of the antiparticle with spin down are the same. Because of the occurrenco
of a term p H in the expression for the energy, the magnetic moment has
t o be the same with respect to the external field in both cases. This means
it is numerically equal b u t opposite in orientation with respect t o the spin,
for particle and antiparticle 3 ) .
2 . Experimental Tests of Xymmetry Operations
Before the end of 1966 the only experimental problem in connection with
symmetry operations was the determination of intrinsic parities and, for
strictly neutral particles, also of charge parities; as mentioned in Appendix 1 no
I ) As pointed out to the author by G. WANDERS
it actually suffices to postulate that there
is Some procedure by which one can extract properties of the unstable particle from the
S matrix.
2, Similar results hold for higher static moments and also for the distribution of charge and
moment densities in space.
3, Replacing the loop by a condenser one can show that electric dipole moments (if they
exist) are also opposite for particle and antiparticle. This observation fills a gap in
LANDAU’S proof [32] that C P invariance insures vanishing electric dipole moments. It is,
however, easier to argue directly in a Gedankenexperiment with T invariance and to apply
the TCP theorem afterwards.
The TCP Theorem and its Applications
319
quantum number is connected with the operation of time reversal. The classical paper by LEE and YANG[33] opened a new field of experimental problems,
the empirical decision of invariance or non-invariance under particular symmetry operations. The TCP theorem enters into this type of problems in two
ways [ 3 1 ] .
1. If one of the three symmetries T , C , or P is violated (as P manifestly
is in weak interactions) then a t least one of the two others is also violated1).
Empirically it appears that also C is violated but there is so far no evidence
for T violation.
2. Invariance under C or CP can be tested without comparing a process
between some particles with a process between the corresponding antiparticles. One simply tests T P or T instead (which, however, as we shall
see requires additional assumptions).
Evidently a general theorem like the TCP theorem is needed only if one wants
t o test symmetry properties in a way which is independent of detailed assumptions on the interaction mechanisms. Otherwise one could calculate observable
quantities in terms of the coupling constants in t)he assumed interactions; then
one would, from the experimental results, draw conclusions on these coupling
constants and compare the conclusions with the restrictions following from
invariance under the various symmetry operations. I n the following we shall
only be concerned with model independent tests of symmetry operations. This
problem shall, however, be discussed under a wider aspect t,han that of the
application of the TCP theorem.
Because of the smallness of all cross sections the only experiments in the field
of weak interactions which practically can be performed, are observations in
connection with decays. Our first aim shall be t o express such experiments in
a mathematical language. For this purpose we introduce the decay matrix
( a 1 M l k ) which connects states Ic of the unstable particle (strictly speaking
spin states for momentum zero) with states a of the decay products (with total
momentum zero and total energy equal t o the rest energy of the unstable
particle). We do not enter here into the problem of actual calculation of this
matrixz). I n a particular experiment3) one creates a n initial situation of the
unstable particles (e. g. alignment of the spins) which can be described by a
density matrix ( k @init I k’) . Further one performs particular observations on
the decay products by means of a special set up of detectors. Also this observation on the decay products can be characterized by some density matrix4)
(a’ 1 @anal1 a ) . The (non-normalized) transitions probability from the initial into
the final situation is then given by
I
w(init -+ final) =
There are also lecture notes by R. GATTOfrom the fall of 1956 in which this role of the
TCP theorem is pointed out.
2) Cf. Eq. (16) and the remarks in connection with Eq. (28).
We only discuss idealized experiments without localization in space.
4) Only in very rare cases the final observation selects pure states (even if one disregards the
fact that in actual experiments the momenta of the observed particles fall into some finite
volume in momentum space). Normally one does not observe all decay products simultaneously or one does a t least not measure all spins.
320
G . GRAWERT,G. LUDERS,H. ROLLNIK
Before discussing experimental tests of the various symmetry operations, which
are all based on Eq. (9), we shall list the behaviour of momenta, spins, and
charges of particles under the symmetry operations (Table 6). This table is a
condensed version of Table 5. Reversal of charges (q -+ - q ) means that
particles are t o be replaced by antiparticles. It is also noted whether the order of
events in time stays unchanged or is reversed.
-y
S
time order
j reversed
1 unchanged
unchanged
reversed
The experimental test of parity
conservation or parity violation
is now quite straight forward.
The logical procedure goes as
follows : One tentatively assumes
parity conservation, draws conclusions on observable quantities from this assumption and
tests the conclusions experimentally. If they are not satisfied,
parity is violated. For parity
conservation one has
Fig 5a Alignment of the spin of an unstable particle, and momentum of the observed decay prcduct. b. Reflected
situation.
( a p q k ) = ( P a [ M I P k ) .(10)
(This relation is almost evident;
it can be checked using the explicite first order formula Eq. (16) or other perturbation theoretical expressions.) Let us define reflected density matrices by
( k I @Pinit 1 k’)
( P k I p i n i t / Pk‘) >
(a’ I e p a n n i j a )
( Pa’
I @final 1 P a )
(11)
*
They obviously characterize reflected situations in which all momenta are reversed but all spins remain unchanged (cf. Table 6). By a change of the variables
of integration (with Jacobian equal t o unity) it follows from Eq. (9) t h a t
w (init -+ final) = J ( p aI N I ~
k
)
(qeinitl
P V ) ( P ~ ’ ~ M I P ( ~ ’~ : *~ ’ pl a )~ a k~d l c~ ’ d~ a a, a ’
(12)
which after application of Eqs. (10) and (11)simplifies t o
w (init -+ final) = w (Pinit -+ P final).
If this relation is not satisfied parity is necessarily violated.
(13)
The TCP Theorem and its Applications
32 1
We discuss a simple example (i. e. the WU experiment). Let the spin sinlt
of
the unstable particles be aligned with respect t o some particular direction
(“up”). Let us observe one kind of decay products with momentum pAnal.
This is shown schematically in Fig. 5a. Fig. 5 b shows the reflected situations
P init and P final :the initial spin is still ‘(up))but the momentum of the secondary
is reversed. (Because of the unquestioned invariance under rotations only the
angle between the direction of alignment of the primary spin and the momentum
of the secondary, is relevant.) If parity is Conserved, i. e. if Eq. (13) holds, t,he
probability for both processes is the same and the angular distribution is symmetric with respect t o the plane perpendicular t o the direction of alignment.
I n a concise way which, however, requires additional interpretation one says
that parity is violated if one “observes” the scalar product s i n i t . p n n a l . Indeed,
if parity were conserved one would observe positive and negative values of
this quantity with equal frequency.-We mention in paranthesis that the same
formal expression s p is also characteristic for the longitudinal polarization of
decay products from a n unpolarized source.
All parity experiments can be formalized in this way: From the spins characteristic for the preparation of the initial situation and from the spins and momenta
characterisbic for the observations on the decay products one forms rotationally
invariant expressions. These quantities then require physical interpretation as
in the above case. If the quantities contain a n odd number of momenta i t means
that they change sign under parity (cf. Table 6) and t h a t their being (‘observed”
indicates parity violation in the process in question1).
Invariance under charge conjugation can in principle be tested in a n analogous
way. Eq. (10) is t o be replaced by
-
and Eq. (13) by
w (init -+ final) = w (C init -+ C final).
(15)
Therefore one has t o compare the decay of some particle with the decay of the
corresponding antiparticle. The usual tests of charge conjugation invariance
are, however, done on one type of particles only; in this case one applies the
TCP theorem.
Performing the TCP transformation (or the operation of time reversal) on a
decay process one meets with the difficulty that the operation reverses the
order of events in time. So, contrary to the result of an application of P or C,
a decay process is not mapped into another decay process. It is, therefore, not
surprising that useful conclusions on decay processes can only be drawn from
the invariance under symmetry operations which reverse the time order, if
some additional assumptions are made.
The consequences of TCP invariance for decay processes shall be analysed
under the following assumptions [30] : The particles in question are stable under
strong interactions (where the terminus “strong” stands for both strong and
electromagnetic interactions in the usual sense)z, and it suffices t o treat the
l) This rule appears t o be simpler than a formulation in which one has t o distinguish
between scalars and pseudoscalars.
2, More generally one might speak of defining (instead of strong) and decay (instead of
weak) interactions.
322
G.GRAWERT,G . LUDERS, H. ROLLNIK
weak interactions, responsible for the decay, to first order. The strong interactions then define a one-particle state !Pk(labelled by momentum and spin k
of the particle) which will decay after switching on weak interactions. Strong
interactions also lead t o scattering states !Paout of the decay products which
are characterized by outgoing particles with quantum numbers a. Working
in an interaction representation i n which the field operators obey field equations
including strong interactions and in which the change of the state vector with
time is governed by the weak interaction Hamiltonian H ( t ) one calculates the
decay matrix ( a I M I k ) from1),2)
+ m
2nd4(k-P23a)(a[JfIk) = J ( F o o u t , H ( t )
!Pk)dt*
(16)
--m
I n terms of the decay matrix the reciprocal mean life I
‘ is given by 3,
p r = 252Jl(nlM,k)/”*(k--p,)da
(17)
with integration over all final states a. The letter ,u stands for the rest mass of
the unstable particle. Similarly, branching ratios are obtained by integration
in Eq. (17) over selected final states only. Under the assumption of TCP invariance O F k is a stable antiparticle state
@!Pk =
(18)
!Pea
with momentum unchanged and spin reversed (cf. Table 6). I n the same way
one has
Olu,ou, = y e a i n .
(19)
The replacement of “out” by “in” is a consequence of the reversal of time
order; the state !Peain is characterized by incoming antiparticles with unchanged momenta and reversed spins. The “in” states are related to the “out”
states through the S matrix of strong interactions
Y e a i n = S!Pe,nout
= j d b (OblXlOa) Yebout.
(20)
For the weak decay interaction H ( t ) the postulate of TCP invariance means
OH(t)@-’
=fI(-t).
(21)
I n this way one finally finds4)
(,\Hilt) = j d b ( @ b I M \ @ k ) * (Ob\XIOa} = / d b j @ b \ M l O k ) * ( a I s \ b )
(22)
(cf. Eq. (8)).
Strictly speaking relativistic normalization of momentum eigenstates is assumed but this
technical detail is unimportant for the general argument.
2, The decay matrix in Eq. (16) refers t o arbitrary total momentum whereas the one used in
Eqs. (9) t o (15) was specialized t o total momentum zero.
The differential d a is a product of relativistically invariant momentum differentials.
4, I n quite the same way i t follows from T invariance t h a t (alMlk) =
db(TblM(Tk)*
l)
/
1
( T b [ S I T,)= d b (Tb I M [ Tk)* ( a I
S1 b ) with well-known consequences (cf. [ 3 4 ] ) . In
the literatur i t is sometimes stated t h a t this formula expresses the final state interaction
between the decay products though it actually is of a wider validity.
The 1'CP Theorem and its Applications
323
T,ct us make two applications of Eq. ( 2 2 ) : 1. The equality of lifetimes of
particles and antiparticles as a consequence of T C P invariance (which was
proved in Sect. 1) can be checked with this first order formula. Inserting
Eq. (22)into Eq. (17) one obtains
,d=
2 n / ( O b l M / O k ) *(OcIM]Ok)(a]S]b)(aIX]c)*dadbdc(23)
which, after application of the unitarity condition
J (alSlb) (aISlc)* d a = ( c j 4
(24)
p r = 2 x 1 ]( Obl M IOk) I2S4(k--pb) d b .
(25)
reduces to
Here the right hand side now obviously is equal to the product of the mass
and the reciprocal lifetime of the antiparticle (the masses are t o this order not
modified by weak interactions and, therefore, equal for particle and antiparticle). - 2. If strong interactions are negligeable for the particular decay
process (practically no scattering between the decay products)
(apjb) = (a/b)
(26)
one obtains from Eq. (22)
( ~ 1 M l k=
) (OaIM/Ok)*.
(27)
If Eq. ( 2 7 ) holds in the presence of T C P invariance we shall say that the
particular decay process is "essentially of first order". Our derivation was given
for a first order process in the strict sense; Eq. (27) holds, however, under more
gcneral premises. Let us assume t h a t the decay matrix can be calculated in
perturbation theory. Then it can be represented by a sum of terms in which
thc numerators are products of matrix elements of the interaction Hamiltonian
H (both strong and weak) and the denominator consists of products of energy
differences. From T C P invariance it follows for the matrix elements
(ClHld) = (OcjHIOd)".
(28)
The energy denominators are real apart from the famous i s . This i s only
matters for vanishing energy denominators, i. e. if a n intermediate state could
also be final state. If the contributions from such intermediate states are negligeable, Eq. (27)is true. I n the presence of final state interactions there are always
intermediate states which also occur as final states and so Eq. (27)is not valid;
indeed, under additional assumptions it is t o be replaced by Eq. (22). It has
occasionally been discussed whether, for instance, beta decay occurs not directly
but through some intermediate states; such intermediate states, however,
cannot be possible final states since, otherwise, they would empirically occur
as (probably more frequent) additional decay channels.
Now we come back t o the problem of testing C invariance without going over
to antiparticles. Under the assumption of C invariance one has
( O a l M I O k )= ( C O a l M I C O k ) = ( T P a I M I T P k ) .
24 Zeitschrtft ,.FortschrItte der Physik"
(29)
324
G . GRAWERT,G. LUDERS,H. ROLLNIK
This relation can either be insertedl) into Eq. (22) or into Eq. (27) (when thc
process is essentially of first order). I n the latter case one obtains
( a l M J k )= ( T P a J M J T P E ) " .
(30)
We discuss the consequences of Eq. (30). Introducing T P transformed density
matrices by
(kIeTPinitIk')
(a'
I
eTPfina1
a)
(TPk' [ ~ i n iJt T P k ) ,
(TPa
I @final I T P a ' )
(31)
we obtain in an analogous way as in connection with P invariance the following
relat,ion
w (init --t final) = w ( T P init --t T P final).
(32)
The T P reversed situations are obtained by the replacement p -+ p , s --t - s
(cf. Table 6). If one wants to test invariance under charge conjugation without
comparing particle processes with antiparticle processes one has t o ,,observe"
quantities, invariant under rotations, which contain a n odd number of spins.
An example is given by the quantity s . p which was already discussed in coiinection with parity violation experiments. Conclusions on the violation of
charge conjugation can in this way, however, only be drawn in the case of
negligeable final state interactions (e. g. only for terms proportional t o (aZ)O
in beta decay phenomena).
Model independent tests of time reversal invariance (which have nothing to d o
with T C P ) proceed in a n analogous manner. If a process is essentially of first
order one has
( a ] M l k )= ( T a l M ] T k ) *
(33)
and obtains from this equation
w (init -+ final) = w ( T init -+ T final)
(34)
where the time reversed situations are characterized by reversed momenta and
spins. To test time reversal invariance one has t o observe rotationally invariant
quantities which contain a n odd number of factors (like (pl x p2). s).
Experimental tests of TCP invariance could in principle be done applying
Eq. (22) or (27). For the validity of these equations additional assumptions
are t o be satisfied. For a direct test of TCP invariance one would have t o conipare reactions between some particles with reactions involving the corresponding antiparticles. Cross sections of reactions produced by weak interactions
are, however, much too small t o permit such a n experiment.
Appendix 1
Antiunitary operators in Hilbert space.
An operator T mapping any linear manifold D of the Hilbert space H onto
another linear manifold W cH is called antilinear, if the following relation
Essentially the equation obtained in this way was applied by R. GATTO[35] to the
analysis of A0 decay. Because of the strong final state interactions this process certainly is
not essentially of first order.
l)
325
The TCP Theorem and its Applications
holds for any vectors f , g of I) and complex numbers a, b
T(af
+ b g ) = a* Tf + b*
T g.
(A 1)
Multiplication of a n antilinear operator by a linear operator again yields an
antilinear operator. The product of two antilinear operator is linear. Because
of equation (A 1) the notion of a n eigenspace of a n antilinear operator wiIl not
make sense. For, p being a n eigenvector with eigenvalue a, the vector eia Q: is
eigenvector with eigenvalue e-ZLa a.
If D is dense in H , we can define in an unambiguous way the Antihermitean
conjugated operator T' by
( T f ,9 ) = (T' g ,
f)
(A 2)
Proof : The quantity ( T f ,g) is a linear funtional with respect to f . Hence we can
apply RIESZ' theorem: corresponding t o every g there exists another vector g'
in such a way that ( T f ,g) = ( g ' , f ) holds. T' is the operator which maps g into g' .
We could have called T' the transposed operator with respect t o T,which in
contrast t o the case of linear operators now can be defined independently of
vector basis in Hilbert space. The same symbol for Antihermitean conjugation
as well as for Hermitean conjugation points t o the important fact, that the
following relations hold for linear L and antilinear T :
(Tl TJ'
=
T i Ti , (TL)' = L' T ' , (Tl LT2)'
=
TJL'T:.
By a n antiunitary operator W we understand an opcrator, which maps H onto H
and which obeys
(A 3)
( W g ,W f ) = (f99)
for any vectors f , g in Hilbert space. Straightforward arguments show the
antilinearity of W and the existence of a n inverse W-l , which is just identical
with W t .
Suppose { q ~ , , } is a complete system of orthogonal and normalized vectors.
Expanding W qP with respect t o this system we have
Remark that (A 5 ) is just the same restriction, which holds for the matrix
representation of a n unitary operator. Now for the reverse, it is easy to show:
An operator defined by (A 4),mapping H onto H and obeying (A 5 ) is unitary,
if it is linear; it is antiunitary, if it is antilinear. The product of two antiunitary
operators is unitary.
Hence any antiunitary operator W can be constructed starting from a fixed
special antiunitary operator W , by multiplying W,, with an unitary operator U .
As operator W,, we can take the complex conjugation in some coordinate system,
e. g. in r-representation :
K ( r / f )=
(W*
which has the simple properties
K2=l,
24.
K=K'.
G. GRAWERT,
G. LUDEKS,H. ROLLNIK
32 6
Appendix 2
For the readers' convenience we think it worthwile to compile some well-known
results concerning the finite-dimensional irreducible representations of the proper
homogeneous LORENTZ-group. The proper LORENTZ-group is the group of
transformations of four - vectors
Xi' =
3 5, a,"
V
1, /I
+
with Det a; = sign a: =
1. Generators of this group are the six infinitesimal transformations J,,, = - J,, (infinitesimal rotations in the pv-plane),
which obey the commutation relations
[J,Y,
JXAI-
= -
gpx
JYI
- % A J,',
+
S P A Jvx
+
g v x J,A.
If we introduce
we can calculate
[ M i , Nk]- = 0
(anyj, k ) .
Consequently the operators Mi as well as the N i behave as angular momentum
operators and we can use the well-known results concerning irreducible representations of the three dimensional rotation group.
The finite-dimensional irreducible representations of the proper LORENTZ
group can be characterized by two integers n, m, the eigenvalue of C Nl being
, that of
2 M j being
. The representation D ( n ,m ) (some
(t y )
authors prefer the notation D -. - for this representation)has the dimension
(n 1) ( m 1).
The formula for reducing the direct product of irreducible representations again
into irrcducible representations reads
+
D(%,ml)
+
x W n , , m,)
+ n2,m, + 4
@D(nl+n2--2, mi+mz)
0... O D ( % ,+ n2,jml - m z / )
0... @D(ni+n,-2, Iml-mzI)
OD(in,--n,', ml+mz)
O ... @D(lnl-nn,I, Iml-mEI).
= D(nl
A classification of representations into four classes has been introduced in
chapter 11, 1. It is trivial, that all the representations in the right-hand side
of the reduction formula belong exactly t o one class. We have the following
multiplication table of classes.
The TCP Theorem and its Applications
327
+
M i iVi being the physical angular momentum operator, the field operators
belonging t o classes I and I1 represent fields with integral spin, those of classes
111 and I V represent fields with half-integer spin, Class I contains tensors of
even rang, class I1 tensors of odd rang. For example we note
D ( 0 , 0) : scalars
D (0, 1) and D (1,0): two-component-spinors
D (1, I ) : vectors
D (2, 0 ) and D (0, 2 ) : antisymmetric tensor (of 2n'Erang)
D ( 2 , 2 ) : symmetric tensors with trace 0.
The basis vectors in spaces underlying representations D (n,m) on one hand
and D( m , n) on the other can be chosen in such a way, that the representation
matrices for D ( n , m)are just the conjugate-complex of that for D ( m , n ) . Consequently if u is a quantity of class 111, the operator t i t has to be of class IV.
One of the authors (H. R.) wishes to thank the ,,Gorresgesellschaft zur Pflege der Wisuenschaften" for a grant in aid.
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