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The mass levels and the broken symmetry in terms of inequivalent representations
ANNALS
OF PHYSICS:
The Mass
26, 336-363 (1964)
Levels and the Inequivalent HIROOMI
Department
of Physics,
Institute
for
in Terms
of
UMEZAWA* University
YASUSHI Dublin
Broken Symmetry Representations
of Tokyo,
Tokyo,
Japan
TAKAHASHI
Advanced
Studies,
Dublin,
Ireland
AND SUSUMU Department
of Physics,
KAMEFUCHI Imperial
College,
t London,
England
It is pointed out that inequivalent representations inherent in the theory of quantized fields in an infinite volume play an essential role in deriving mass levels and broken symmetries. A set of continuous parameters is introduced to specify these representations. By the requirement that the Hamiltonian shall tend to that of asymptotic fields as the volume becomes infinite, suitable representations are singled out. This requirement further determines possible values of the mass, which exhibit, in general, broken symmetries. It is also shown that with our method Nambu’s “Superconductor theory” of elementary particles can be formulated in a consistent manner. I. INTRODUCTION
It is an unpleasant feature of the present theory of particle physics that some of the symmetries established in strong phenomena are violated in the mass spectrum and weak phenomena. An interesting possibility has been suggested by Nambu (1) to derive broken symmetries from a fully symmetric interaction. He introduced a massequation with an analogy to the Bardeen method of superconductors, which has been interpreted, in our previous paper (2), in terms of inequivalent representations. The aim of this paper is to apply the method of inequivalent representation to relativistic field theory. * Present address: Istituto di Fisica Teorica, Universita di Napoli, Napoli, Italy. t Present address : Department of Physics, Tokyo University of Education, Tokyo, Japan. 336
MASS
LEVELS
AND
BROKEN
SYMMETRY
337
It is now well-known that, when dealing with quantized fields, one is forced to consider various representations which are not mutually equivalent (3). Suppose that a field is confined in a box with a finite volume V. There exist then a countable infinite number of momentum states, each of which can be occupied by any number of particles in the case of Bose fields or by at most one particle in the case of Fermi fields. Consequently, the state vector space becomes nonseparable. This has been discussed by several authors in connection with the van Hove model (3). This kind of problem does not arise, however, when one introduces a maximum (or a cutoff) momentum for each particle. Throughout this paper we assume that a cutoff function exists in the Hamiltonian, although it is not written explicitly. Even with such Hamiltonians we still meet the problem of inequivalent representations when we are concerned with a system of infinite volume, since there appears a continuous distribution of momentum states. It is possible, however, to avoid the use of momentum states by employing an orthonormal set of wave functions which vanish with a reasonable speed at the spatial infinity (i.e., the wave packets). Since such a set of wave functions is an enumerable one, we are led in this way to a separable space of state vectors. This vector space is a special one chosen by the suitable boundary condition. However, we have some reason to believe that the representation so chosen may not always be a reasonable one: this is due to a delicacy of the behavior of the field at the spatial infinity. To see this we may recall the problem of vacuum polarization in quantum electrodynamics. Here, the calculations show that the total charge is decreased by the renormalization factor 2:” which is due to the effect of the vacuum polarization and therefore that the conservation of the total charge is violated. Schwinger (4) suggested, however, that the conservation of the charge may be recovered when one takes into account the charge which is distributed over the whole space with a constant but infinitesimally small density. It is evident that the charge distribution of this kind can not be obtained by means of the wave packets. This may not be a difficulty in quantum electrodynamics in the wave packet representation, where the renormalized charge is conserved throughout reactions. However, as the theorems in Section II show, an accumulation of physical quantities distributed infinitesimally over the whole space is not always vanishing but depends on the choice of representations. Thus, a question arises as to how one should choose the reasonable representations from among the mutually inequivalent ones. The consideration in the present paper strongly suggests that this question is intimately connected with problems of the broken symmetry and the mass spectrum. To study our question above, we begin with a case of a finite volume V and introduce a set of real continuous parameters { 0, p) : each set of values of these parameters specifies the representation. Referring to each representation we calculate the limit (V 3 00) of operators. This operation (2) will be called in the following the V-limit and denoted by V-lim. Although, for a finite V, these
338
UMEZAWA,
TAKAHASHI,
AND
KAMEFUCHI
representations are connected with each other by unitary transformations, many of them may turn out to be inequivalent to each other in the limit V -+ a,. Supposing that the parameters { 0, (p) represent a continuous set of inequivalent representations, we shall show that the V-limit of operators also depends on the parameters (8, ~1. It should be noted that the V-limit is not quite the same as the weak limit, because the basis of the representation in which the limit is carried out may depend on the volume V. To perform the V-limit of operators consistently, we adopt the convention that all calculations are carried out exclusively in a fixed representation and the operators belonging to different representations are not considered simultaneously. We shall base our arguments on this convention, expecting that such a procedure may be justified in future by rigorous mathematics. When considering a special representation, we denote the annihilation operator of the Fermi particle and antiparticle by CY~”and ,&’ respectively, and the vacuum by @o, i.e., (YgT*o= pk&l
= 0,
where k denotes the momentum of the particle state. We may introduce the operators
(1.1)
or the antiparticle
and r the spin
(1.2) with
c-numbers
p1 and pz satisfying I Pl
the relation I2 +
I P2 I2 =
1.
O-
(1.3)
Although these operators obey the same commutation relations as those of a! and 8, &’ and fikT cannot be called the annihilation operators. This is because, as will be shown in Section II, in the representation under consideration there is no state 5, satisfying the relations &?&
= /3kkrS0= 0.
(1.4)
On the other hand, the representation of a one-particle wave function can be changed by unitary transformations in the same Hilbert space. For example, the annihilation operator can be defined in the spherical wave representation as well as in the plane wave representation. As will be discussed in detail in Sections II and III, most of the inequivalent representations are not physically acceptable. The values of the parameters for the acceptable representation are obtained by the condition that the V-limit of the Hamiltonian shall be diagonal when it, is written in terms of the creation
MASS
and annihilation
LEVELS
AND
BROKEN
339
SYMMETRY
operators :
V-lim [x] = g: -
z
3
,?L(a;l’cuk” + /3LtPk’) + c-number
(1.5’)
with Ek = dk2
(1.6)
+ AP.
This condition leads us to a set of equations, which determines the mass .!I/ together with the parameters (19, ‘p} . If the field equations for a system in a finite box have an invariant property with respect to a certain unitary transformation which thus leads to a conservation law, it can happen that the representations which are related to each other by this unitary transformation turn out to be mutually inequivalent in the limit V + 00. In such a case the V-limit of the Hamiltonian loses the invariant property and we observe the broken invariance. To recover the invariance we need to take account of the effects of matrix elements of the order of (l/V), effects of which are hidden in the measurement of masses. The situation will be illustrated in Section IV by the Nambu-Heisenberg model. We shall see in Section IV that the requirement (1.5) allows more than one inequivalent representation, which correspond to different masses of the field under consideration. There seems to be no difficulty in evaluating transition matrices, as far as the transitions between particles belonging to the same representation are concerned (cf. Section V). It is not clear, however, at the present stage how to deal with transitions between particles belonging to the different representations. Section V will be devoted to a discussion of an important relation between the V-limit and the asymptotic condition. II.
INEQUIVALENT
REPRESENTATIONS QUANTITIES
AND
THE
J’-LIMIT
OF
FIELD
We shall consider a system of a fermion field confined in a box with a finite volume V and denote the annihilation operators for particles and antiparticles by akr and bk7, respectively: [akr, a?]+ = [bk’, bit]+ = &zL . Here the subscript k refers states: 1’ = 1 for the state of positive helicity) and r (i.e., the state of negative by ~0 :
to the momentum and the superscript of the spin parallel to the momentum = 2 for that of the spin antiparallel to helicity). The vacuum of the fermion akTQO = bkrt2,, = 0.
( 2.1) r to the helicity (i.e., the state the momentum field is denoted (2.2)
340
UMEZAWA,
TAKAHASHI,
AND
JLAMEFUCHI
Let us now consider a general set of unitary transformations condition:
P’, Gl = 0,
G satisfying the
[L, Gl = 0,
(2.3)
where P and L are the total momentum and total angular momentum operators of the system. It is obvious that the set of states G!& covers all the states with zero momentum and zero angular momentum. Since the vacuum should be invariant under the spatial translation, we may regard this set as candidates for the physical vacuum state. It is clear that they are not necessarily the ground states of the free Hamiltonian for the fermion system. This, however, is not a difficulty, as the procedure of taking the V-limit changes the structure of the Hamiltonian (cf. Section III). Rather, we are looking for the representation in which the requirement for the Hamiltonian (1.5) is satisfied. The unitary transformation satisfying (2.3) has in general the following form: G = exp[iF], where the hermitian s
(2.4)
operator F consists of terms such as
d32a. . . a&>ra . -a a+(z), (1)
s d3xa. . . a$(x>ra ... a+(x)a .-- a$(x)r’a .-- a+(x) with p, r’ spinor matrices. As will be seen in Section V, it is sufficient to consider the case in which F is of bilinear form in # and 4. We further simplify the situation by restricting [G, N] = 0
(2.6)
N = g [af+uk’ - G+bk’].
(2.7)
with
In other words, we consider the case in which the fermion number of the vacuum states is zero. To construct G, we shall introduce the following hermitian operators : T::;
= $$&J;+ b:\ + KI, a;), (2.8)
MASS
LEVELS
AND
BROKEN
Any combination of aLtb?k , lLak’, altakr and b$b?, can be written combination of TCi’(i = 1, 2, 3, 4). It can easily be seen that iTi? , Ti12T)]= iT i,“r’,
as a linear (2.9)
etc.,
(2.10,
[T:::: , T/i::] = 0, [Tj:? , T!yJ] = 0 for The relation for rotation. written as
341
SYMMETRY
(k, r) # (I, s).
(2.11)
(2.9) shows that Tf+) , T:,“,’ , and T:,3,’ behave as a set of generators Then the general form of the unitary transformation G can be (2.12)
kr
Here 2.13) with Sc3) = g exp[it@ Ti?],
2.14)
Sc4) = n exp[iOi:‘,TT:P,‘].
2.15’)
k,r
we can easily check the following
relations:
akr = Gak ‘G-’ =f? i6(kvr)[~~~ ekr.akr
+ eiqp(kxr’sin /jk’.bi_l],
(2.16)
@Lk = GbLkG-’ =e i”(kvr)[~~~ ekr.bTk - eip(k,r) sin ekr.a;+],
(2.17)
where
(2)
ok.7
. (1)
-
dk,r
=
t
e reG(kA k
2.18)
1
and qk, r) = ei:: - ei,?,
(2.19)
~(k, r> = eiy? + eil”) .
(2.20)
The parameters 6 and Y (and therefore OC3) and eC4’)induce an independent change of each basic state vector. Hence it is enough for us to consider only the case in which they are zero: e’4’ e(3) 0. (3.21) k.r
=
k,r
=
342
UMEZAWA,
We can further
TAKAHASHI,
assume without
AND
KAMEFUCHI
loss of generality
0 s ek’ 6 7r/2,
that
0 s (o(k, r) < 27r.
and that, when 9kr = 7r/2, the parameter Thus, we have
(2.22)
V( Ic, r) is zero.
(2.23)
(2.24)
(2.25) The state
(2.26) k,r
is the vacuum
in the representation
under consideration:
ffk%, = pkk’@O= 0. The operator w(k,
(2.27)
r) is defined by w(k,
r) = cos ek’ - eicockqr)sin &‘.d+b???.
(2.28)
It should be noted that creation operators of different helicities are never combined in TJ$ in (2.8). This is due to the second condition in (2.3). This condition also requires that ok’ and (o( k, r) should not depend on the direction of the momentum k. In the next section we shall further restrict the values of parameters (ok’, pk’} by (1.5) and then see that the resulting theory becomes invariant under the orthochronous Lorentz transformation. Recall the relation 9(x) Here Ukr and equation:
vkr
= (l/dv)
are the positive
g
[ukrak’e’b
+ vkrb;+e-“k”].
and negative frequency
solutions
(2.29) of the Dirac
(2.30) (-i$$,
+
m)vk’
=
0,
where m is the bare mass of the fermion (cf. Section III) (k, iC&) with Wk = dkz + m2. They are normalized as
and 5 the four vector
MASS
LEVELS
AND
BROKEN
343
SYMMETRY
(2.31)
(?A;*, UkS) = a,, ) (v;*, VkS) = a,, ) and related to each other by the charge conjugation’: v;, = C,&B* Substituting
(2.25) into (2.29) +(x)
= (l/dV)
(2.32)
.
we obtain
F [z4kr( 19,(o)cykreikx + vkl’(0, cP)PL+eC”k”]
( 2.33)
with uk’( 8, q) = ukr cos &’ + v~ke-iP(k’r’ sin ok’, vkr(
8, p)
=
vkT cos
ekr
-
u?-keirp(k”)
sin
(2.34)
ekr .
It is evident that Ukr( 8, (0) and vk*( 8, (p) coincide with ukr and uk’ respectively, when &’ = 0. Regarding a0 as our vacuum state, we shall construct basic vectors as follows: rt a0 ) @;+$I ) /3;+cx;+$l , . . . . %J ) LYk The general form of these vectors
is
@[kr, Is, ‘. . : pt, qu, . . - ] = &,+P,“+ . . . a;+&+ . . * a+) . We shall denote these states by $( 8, (p), c&( 8, ‘p), . . . , with a suitable subscript to distinguish various states, where @,( 0, (p) is the vacuum defined by (2.26). When we consider the limit V ---f 00, the transformation (2.26) loses its sense and the basic vectors $( 0, lo), aI( 0, CO), . . . for each set (0, (01 constitute the basis of the respective inequivalent representation. This may be seen by the relation : ;;
(a(e,
We shall illustrate
+a@,
(2.35)
4))
= 0
for
(0, P} z {e’, &I.
(2.35)
as follows: d”k log x(k)]
= o,
(2.36)
where
cp(k, r>- cp’(k, r> x(k) = [Coft(ek’ - e;‘)cos2 2
>
1 It should of (2.32).
be remarked
that
we have
the same
helicity
(2.37)
superscript
(T) in the both
sides
344
UMEZAWA,
TAKAHASHI,
AND
KAMEFUCHI
Taking into account (2.22)) we see that the equality in (2.37) holds only when (0, (o) = {O’, 9’1. The relation (2.35) shows that G& does not possess any well defined limit for V -+ ~0 and therefore that the vacuum c&( 0, (o) of each inequivalent representation cannot be defined by (2.26). Nevertheless, we shall proceed further by adopting the following assumption, expecting that our procedure could be formulated by profound mathematics for a nonseparable vector space: The assumption is that for a system of infinite volume a set of inequivalent representations is classified by the parameters { 0, ~1. When we fix the values of the parameters (8, (01, we get a set of creation and annihilation operators (akr, Pit and their hermitian conjugate). When we say that we are given a field operator $, this means that t/ is given as an operator defined with reference to a special representation. In other words, # is given by (2.33) in terms of (CYST,Pkr). It is only because of the normalization of the operators akr and Pkr that V appears in (2.33): Since k is a continuous variable, we could have normalized them in such a way that [ah’, al+]+ = 6,&k
- 1).
However, the specific feat#ures of our theory will be explicitly manifested when we describe the theory through the intermediary of the V-limit convention. To be more precise, we define the limit of operators, say Q, with respect to a representation specified by ( 8, (o} as follows:
(we, 4, V-h-n [&Me, (0)) = pp(e,
dw(4
d>,
(2.38)
and the whole calculation is carried out in the { 0, (PI-representation. The relation (2.38) defines the limit of the operator Q as an operator only in the Hilbert space, whose basic vectors are related to the set %( 8, (o), %( 0, P), . . . by a unitary transformation. The limit of Q can be defined without specifying the one-particle representation (plane wave, spherical wave, etc.), although we shall use in this paper the plane wave representation. When considering a representation specified by ( 8, (p} , we are given the annihilation operators akr and @hr. We can then define the operators &* and &’ by (1.2). These operators can not be called the annihilation operators, because in the Hilbert space under consideration (i.e., in the { 0, (p}-representation) there is no state & which satisfies the relations in ( 1.4). What the condition (1.5) implies is that the V-limit of the Hamiltonian should be diagonal when it is written in terms of the creation and annihilation operators. Let us now introduce the normal products : iL(xdrLp(22) where
the order of operators
. *. :
CY~‘,@kr, c$, and ,f$ are so arranged
(2.39) that all the
MASS
LEVELS
AND
BROKEN
SYMMETRY
345
annihilation operators appear on the side right to the creation operators. Considering (2.33), we see that the c-number coefficients in (2.39) depend on the parameters (0, (01. Then we have following theorems which play a central role in our theory. THEOREM 1. The V-limit qf all the normal products vanishes : V-lim [ : ~J~(x~)#~(x~) THEOREM
.. . : ] = 0.
( 2.40)
2. We have V-lim
ils
d3z : ~J~(x)$p(x)
...
1
:
= 0
for
n > 2,
(2.41)
where n denotes the ,numbev of Jield operators in the bracket. THEOREM 3.
1
d3x : $a(~)~~(~)
.. . :
= 0.
Taking into account the fact that each operator # contributes to the product by the volume factor l/z/v, we can illustrate the proof for (2.40) by
A similar argument also gives us Theorems 2 and 3. Let us denote the vacuum expectation value of an operator (Q>o = (@o(@, (o), Since the vacuum
@o(
8, q) is invariant
Q*o(e, cp>1.
under the spatial translation,
do not depend on x but on the parameters Since we have the relations
from (2.41) and (2.42)
& by
follow the theorems:
(0, p}.
the quantities
346
UMEZAWA,
THEOREM
[S
AND
KAMEFUCHI
4.’ V-lim
V-lim
TAKAHASHI,
d3&x!(XM3(d
d3x &(x,lj#dx>
t [J
1
= /2x
: iL!W~a(x)
1
= cdf+P)?
(2.45)
:
(2.47)
+ an injinite c-number,
= Ca,(4#) /- d3x: $a(x>~~(x>: + Cde,#) /- d3x: &(x>~a: (248) -
Ca,(e, +) j- d3x : &(X)&T(X)
: - C,,(e, cp> /- d3x : $a(x)J/v(x)
:
+ an injinite c-number. In these relations the c-numbers are proportional to the volume V as is the zero-point energy. These c-numbers are thus amalgamated into the vacuum energy. Quite general1 y, we can show that the V-limit of s
d3dhM,h)
.. .
for any power of field operators is a linear combination of various and a c-number. Since %+,is invariant under space rotation, it is clear that Cr = ($W)O = 0
for
r = y, y-is, L-h, -4,
bilinear terms
(2.49)
which can also be checked by direct calculations. To calculate ($I’$) for other I’s, we make use of (2.33) together with (2.34). Since (fikrmk8), (Bkrr~kS), (tori%), and (Greta) with r = 74, y5, 1 or YIYS are invariant under the space rotation, they would not be zero unless r # s. It will be proved in the next section that (2.50) 2 This conductor.
theorem has been stated in ref. B which dealt with the B.C.S.-model A theorem of similar kind was also stated in ref. 6.
for the super-
MASS
LEVELS
AND
where Ok and (ok are the functions
347
SYMMETRY
of Ic ( = 1k 1) only and er is +l -1
7 e = Taking
BROKEN
for for
1’ = 1, ~=2.
(2.51)
(2.50) into account we obtain for I’ = 1, y4 , y4y5 , and y5 the relation:
Here A,(k)
= cos2 6b(iik’ruk’) + $5 sin
+ sin” &(
2ok
cos
fik7rl)kr)
(ok{ (tik%kr)
+
- $$if’ sin ii?& sin (ok{ (ckmk’) B,(k)
= cos’ ek( fikrr&‘) - 4/i sin
+
2ek
= cos2 ek(fik’r&r) -
We obtain,
co8
(ok{ ( fikrr2)kr)
+
-
sin2
( iikrrukr) -
6keSiPck”)(
(2.54)
]
( tikrrUkr)),
nkrrukr)
(2.55)
cos ek sin &ei”k’r) ( (iikrr?kkr) from
(&‘rUk’)},
+ sin” &(?&‘I&‘)
+ @ Er sin 26, sin (ok{ (?ik7mk7) e,(k)
(2.53)
( ckrhkr)l
-
(ekrr&)‘)].
(2.52), (2.56)
for r under consideration. Taking into account the relations in the Appendix, we can show that C, = V-lim [i(h
J/)01= &
C, = V-lim [($$)01 = - &
j”
Ca = V-lim [(&A
75
y7)01 = 0.
and (-4.20)
d”k sin 288 sin (Pk,
(2.57)
1
2 cos 28~ - -!! sin 28k cos (Pk , Ok
C, = V-lim [($-yJ +),,I = &a
(A.17)
/ d”k = 26(x)
Wk
Ixzcl ,
(2.58) (2.59) (2.60)
It is to be remarked here that (2.49) leads us to (2.60), which is necessary for the theory to be invariant under the orthochronous Lorentz transformation.
348
UMEZAWA,
TAKAHASHI,
AND
KAMEFUCHI
It is not surprising, however, that C, # 0, although the corresponding spatial parts vanish. Indeed, the value of C, does not depend on the choice of the parameters (8, ~1, and therefore it is equal to the value of the zero point charge density: C, = Tr[r.S+(O)].
(2.61)
It has been known for some time that, in order to secure the Lorentz invariance of the theory, we have to construct the vector term in such a way as
J, = 4hhhk - s/‘u,$‘l, where #’ is the charge conjugate
(2.62)
of #. It is easy to see that
C;’ = V-lim (J&
= 0.
(2.63)
We are then left with C, and C, which are not zero. III.
THE
V-LIMIT
OF
HAMILTONIAN
AND
THE
MASS
EQUATION
Let us now show that only in special representations, i.e., only for special sets of the parameters { 0, (p}, is the condition (1.5) satisfied for a given X. We shall consider a fermion field # whose Hamiltonian is given by 3-2
=
X0
+
Xint
7
(3.1)
+ m)9.
(3.2)
where x0 =
s
d3&ya
It is to be understood here that the Hamiltonian operator is defined with reference to the Hilbert space with the parameters { 8, (o} . This means that # in (3.2) is defined by (2.33) in terms of akr and Pk . The bare mass m is so defined that each term in the interaction Hamiltonian contains more than two field operators #. Thus, the interaction Hamiltonian consists of terms like
where $ is a boson field, X denotes the coupling constant, and r and I” are spinor matrices. Let us consider the second term in (3.3) as an example. As was shown by (2.48), its V-limit consists of terms bilinear in # and a c-number. Due to the Lorentz invariance of the theory, these bilinear terms should be of the form:
Here S and g depend on the set of parameters
{ 0, q}.
MASS
In this
way,
LEVELS
AND
BROKEN
SYMMETRY
349
we see that V-lim [I
d”,TiC] = [ d3ccE0 + c-number,
(35)
where x, = x0 + 6X”. Our next task is to fix the parameters Hamiltonian :
(3.6)
(0, cpJ so that ??o has the form of free
x0 = c &(4+d k,r
(3.7)
+ P;+Pk’) + wo )
where Ek = dk2
+ M”.
(3.8)
The mass ill and the zero point energy Wo are also determined when the parameters are fixed. Relations (2.33) and (2.34) together with those in Appendix lead to x” = c ‘Ok cos 20/((Y;+C$
+ p;+,&‘)
k,r
wk sin
2&‘[eiqa;+$?k
+
e?&ak’]
+
2
g
sin ‘ok’,
Wk
(3.9)
-F
i
s
&gNtx
= p
cr sin 28k’ sin &&+akr
+ &‘fik’)
-ie’
( cos2 ok’ + sin’ ek’ezi’)C$+fi~t,
(3.10)
+ ie’ (cos’ ekr + sin* e,‘e-““‘)@‘@~I
s
&b d3x = z
k (m
COS
- CL
2ok’ -
k
Sin
m eiq sin 2&* + ’
k,r
Wk
2okr
+ 5 cr sin
COS p)(LYitd
(cos2 ok7 - sin”
+
p,
fl;+@k’)
ekre2”)
Wk
+ sin 28k’ + ’
28k"sin
1
(cos2 ekr _ sina ekre-*i9
aL+p?k
1-j
(3.11) @:k
akr
Ok
-
~~(wc0s2ekr--
k sin
28k’
cos
p).
Here cp is an abbreviation of ‘p( I%,r). The Hamiltonian zo defined by (3.6) is of the form (3.7), parameters (13,(e) satisfy the following relation:
if and only if the
350
UMEZAWA,
TAKAHASHI,
wk sin 2&’ + igc’ (~0s’ &*e+
AND
KAMEFUCHI
+ sin” ekreiV)
k sin 2ekr + - (~0s’ ekre+
1
- sin’ &‘ei’)
= 0.
Wk
(3.12)
When (3.12) holds, we obtain Ek
=
Wk COS 2okr
-
gEr Sin
2okr’Sin
Cp( k, T)
+ f ’
(m cos 2ekr - k sin
Wk
wo = c k.r
&Ok sin”
ok*
+
-
Erg
f i
sin
2okr
(m
cos
(3.13)
28; cos cp(k, r)),
sin &k, r) 2ekr
-
sin
k
2ok’
coS
p(k,
r) ) -
Wk
= -2cEk.
1
(3.14)
k
The imaginary
part of (3.12) gives
US
g cos cp(k, r) = fe’ i
sin cp(k, r).
(3.15)
1
cos p(k, r) = *:f; dg”
+
(3.16)
f2(k2/~k2)
’
1
sin cp(k, r) = fg -&”
+
I.
f2(k2/Wk2)
(3.17)
’ *
The real part of (3.12) gives [Wk
$
f(m/Uk)l
Sin +
2ek’ [@‘Sin
(3.18) (p( k,
r)
+
f( ii/WI)
coS
(P( k, r)]
coS
%k’
=
0,
or [Ok
+
.f(m/Cdk)]
Sin
2ok’
f
dg2
+
j2(k2/uk2)
COS %kr
=
0.
(3.19)
It is clear that &’ satisfying (3.19) is independent of the helicity state. Then, considering (3.16) and (3.17) we can prove (2.50). Requiring that Ek should be positive, we find, after some straightforward calculation, EI, = dm, COS 2ekr
=
(3.20)
(l/Ek)[Ws
sin 2&* = (I/&)
+
l/g”
f(m/mk)l,
+ f2(~2/ws2)
(3.21) (3.22)
MASS
LEVELS
AND
BROKEN
351
SYMMETRY
(3.23) 1
sin cp(& r) = -g dg2
+
r
j2(k2/Wk2)
(3.24)
”
M2 = (m + f)” + g2.
(3.25)
In deriving these results we have taken into account the relation (2.22) which means that sin 28k’ is positive. As was stated before, f and g are functions of 13and (o, depending on the structure of interaction Hamiltonians. Let us write these functions as j( 8, 9) and g(6, cp). Furthermore, (3.21), (3.22), (3.23), and (3.24) give us gk and (ok as functions of j and g: &( j, g) and (ok(j, g) . In this way, we obtain a set of equations f = fwt
91, df,
a,
(3.26)
9 = au,
91, df,
9)).
(3.27)
Solving these equations we can determine the values of j and g, and then the values of the parameters (0, ~1. The mass value M is now given by (3.25). We have seen that the relation ( 1.5) holds in the representations in which we have (3.21), (3.22)) (3.23)) and (3.24). In these representations, (2.33) and (2.34) give us l.3.28) with Ukr(
j,
9)
=
uk’u(
2)kr(j,
9)
=
2/krU(j,
[kk
@k
j,
g)
+
9)
-
u:kl”(
j,
u:kli*(
(3.29)
g), j,
(3.30)
g),
where u(
j,
g>
=
+
Wk
+
f
(3.31)
E)]“’
l/2
‘(”
We can further
‘)
=
-
&
[Ek
+
Wk
:
1
j(?‘fZ/wk)]
[f (k/d
- ierg].
(3.32)
prove that
(“L*(f, g>‘uk’(f, 9)) = &* (d*(f,
g> ‘uk*(f,
9))
@*(.f,
g> -v8k(f,
9))
M*(f,
9)
9)
.Gf,
= = 1
=
‘%s
(3.33) (3.34)
352
UMEZAWA,
TAKAHASHI,
AND
KAMEFUCHI
and that F u.Ldf, shXf,
F &&
s> = kk
[-i&-h
s>&df, 9) = -kk
+ m + f - igr&3,
(3.35)
[~K,Y~+ m -I- f - in&~,
(3.36)
where K, is the four vector (k, iEk) . These relations lead to &a
d3kuk’(f, g)ak’(f,
r
g)eiKps#
(3.37)
= i (Yrar - m - f + Q%)A+(z:M), d3kvkr(f7
g)okk(fT
g>
iK@,, e
(3.38)
= 5 hap
- m - f + igrs)h(z:M).
Here A+(x:M) and h(z:M) are the positive and negative frequency part, respectively, of A(x:M), the invariant delta function with the mass M. Introducing the time dependent operator #(x) by
%~~akr+ e)kr(f,g)e-iK@*pkr’]
l/4x) =
(3.39)
we obtain
bhGGN+
1 = i (-ha,- - m - f -I- igsyL)A(x - x’:M),
(~M(~)k(~‘>l)~ = MYrap - m - f + imhb
- x’:M.
(3.40) (3.41)
Let us note the relation
(Trap - m - f + igy6)y4 = ei”Y5(y,d, - M)y4e+‘rs
(3.42)
with a constant 4 being defined by sin24 Equation
= -5,
cos2cp = & (m + f).
(3.43)
(3.41) then gives us (!w(x),
ivx’)]>o = %s(y,a, - M)y,S’y,A,(x
- x’:M)
(3.44)
with
S = exp [i+ys].
(3.45)
MASS
LEVELS
AND
BROKEN
353
SYMMETRY
Thanks to the relation (3.44) we may formulate the perturbation calculation for transition matrices by means of the Feynman technique as follows: Each internal line corresponds to the propagator $&!!iF(x
-
x’:nf)
=
>d(r,,a,,
-
hf)A,(.c
-
(3.46)
a’:hf)
and each vertex I? is replaced by (3.47)
r’ = y&qrs. The transition matrices will be considered in Section VI. We finally note that, in the representation under consideration, from (2.57), (2.58), (3.21), (3.22), (3.23), and (3.24) the relations
we obtain
(3.48)
which yield (m + f)C, IV.
THE
NAMBU-HEISENBERG
Let us now apply our theory the Hamiltonian is
When m = 0, the Hamiltonian
(3.50)
= gc, . MODEL
to the Nambu-Heisenberg
(4.1) is invariant i6-0 *-+e 4
model (1, 6) where
under the transformation (4.2)
with 4 being a real c-number, and this is the case which Heisenberg and Kambu considered. Here we begin with the general case in which m is not necessarily zero. We first note that the Fierz transformation gives us the relation
- bhn6~)(h.&Yh~)l + G&4+,
(4.3)
where C, is defined by (2.59). The last term on the right hand side of (4.3) appears due to the change of order of field operators. Denoting the charge conjugate of #Jby $‘, we have the relation:
h,+ = %rs7,$ - $‘dl The relation
(4.3) now reads as
+
G%4
*
(4.4)
354
UMEZAWA,
av”
TAKAHASHI,
- @-Y51c/)2 = -MJPJ,
AND
KAMEFUCHI
- bI%aXw%~6~)1 +
%Cv” ,
(4.3’)
where J,, is defined by (2.62). We shall disregard the last c-number term, which can be amalgamated into the vacuum energy. The theorem (2.48) now says that, in the V-limit, the interaction Hamiltonian ; 1 d34 W”
- hiH21
(4.5)
becomes d3L&iGl = x
s
d32(C&
s
+ iC&sJ/J,
(4.6)
where we again dropped a c-number which will be amalgamated into the vacuum energy. In deriving (4.6), we have taken into account the relations (2.57)) (2.58), (2.60), and (2.63). Comparing (4.6) with (3.4), we get f = XC,) Thus, by substituting relations :
g = xc,.
(4.7)
(4.7) into (3.25), (3.48), and (3.49) we get the following
X, = c &((Y;+(Y~’ + @?/3k’) + c-number,
(4.8)
k,r Ek
=
dm,
M2
=
(m
(4.9) +
XC,)”
+
h2Cp2,
(4.10) (4.11) (4.12)
By Eqs. (4.11) and (4.12) the mass M can be determined. bilities to be discussed:
There are two possi-
(;;)3 c, = 0, M = m- -M/g, a% m= 0, ’ + & s -= Ek ‘* Note that the second case appears only when m = 0 and X < 0. The first case with m = 0 and M # 0 is a special case of (4.14). We remark further that (4.13) with m = 0 is the equation obtained by Nambu (1))
MASS
LEVELS
AND
BROKEN
35.5
SYMMETRY
We shall begin with the first case with m # 0. The equation (4.13) gives the massM in the form of power series d% s z+
M=m-&n
(4.15)
‘..I
This is the massgiven by the chain approximation in the ordinary perturbation calculation. Let us now consider the second case. The solution of (4.14) gives us a mass value which is missed by the ordinary perturbation calculation. Since vz = 0, (3.21) shows that the parameter ok can be determined by M as cos 2ek = k/E/s .
(4.16)
The quantities C, and C, are restricted only by the relation M2 = A’( c,2 + C,‘).
(4.17)
The mass1Malone cannot determine Pk , because tanp(lc, I’) = (C,/C,)e
(4.18)
as is shown by (3.23) and (3.24). It is remarkable that if (4.1) is invariant under the transformation (4.2) (i.e., nl = 0) we can choose a Hilbert space in which the theory loses the invariance (M # 0). This result is mainly due to the theorems in Section II. Another way of saying this is that the transformation (4.2) cannot be represented by any unitary transformation in the Hilbert space concerned. Our result suggests that the broken symmetries observed in the mass spectrum might be explained by a Lagrangian of fully symmetric character, without explicitly introducing any interaction with broken symmetry. We might further expect that the transition matrices would lose their symmetric character only through the mass spectrum. This is true in the case of the Nambu-Heisenberg model, provided that the transition matrices are calculated by the perturbation theory. As was mentioned in Section III, each vertex in the Feynman diagram may be replaced by (3.47) if one usesat the same time the propagator (3.46). This replacement, however, does not give rise to any change in the transition matrices at all, since the interaction Hamiltonian in (4.1) is invariant under the transformation (4.2). Thus, in this case the only agent for violating the invariance under the transformation (4.2) is the appearance of nonzero M in the propagator. V. THE
V-LIMIT
AND
THE
ASYMPTOTIC
CONDITION
When talking about physical particles, we describe them in terms of the asymptotic fields #in or #Out. To do this we must first express the Heisenberg field $ in terms of tiin or It”“” and then consider all the possible inequivalent
356
UMEZAWA,
TAKAHASHI,
AND
KAMEFUCHI
representations for the latter fields. In this sense our argument in the previous sections may be regarded as an approximation in which 9 is simply replaced by tiin. By substituting the expression of 9 in terms of, say, #in, the Hamiltonian X[#] can be rewritten in the following form: X[#]
=
%3[~‘“]
where X&i”] is the free Hamiltonian (1.5) to @n, then the representation
+
Xint
(5.1)
[$‘“I,
of the field +P. If we apply the requirement for @” is fixed by the condition
V-lim X[#] = X&Ji”],
(5.2)
V-limXi,t
(5.3)
or what is the same thing, [$‘“I = 0.
In the case of the Lee model a direct calculation shows that the above condition can in fact be satisfied (7). In this connection it should be noted that the transformation G defined by (2.23) with Gin instead of $ covers all the possible representations for P: it is not necessary to include, in F, terms containing more than two P’s (cf. (2.5) ) . This is because the field equation for gin is linear. The operator tPn can, then, be written in terms of the parameters (f, g) , following the recipe given in Section III, as follows,
p(x) = --& g [ukr(f,g)alP” + uk’(f, sM~+e-i”“l,
(5.4)
where uk’(f, g) and ukr(f, g) are defined by (3.29) and (3.30) and the four vector K, is K, = (k, i&J,
&
= dk2
+ (m + f)” + g2.
(5.5)
The in-field @” satisfies the equation (r&
+ m + f + igra)iwx)
= 0.
(5.6)
The set of parameters (f, g) which specifies the in-field representation will be determined so that the V-limit of the Hamiltonian has the form of (5.2)) where ?&[I/@] = g El,(CX;+f& + &+&‘) To rewrite
(5.7)
(5.7) we note the relations
(h’&, + M)%‘(M) (+,K, with
+ c-number.
+ M)u~‘(M)
= 0,
(5.8)
= 0
(5.9)
MASS
LEVELS
AND
BROKEN
SYMMETRY
357
w+r(M) = s%r(f, 9)
(5.10)
hC(M)
(5.11)
= S~kr(.f, g), fif = d(?n
where S is defined by (3.45). prove that (u;*(A!f)
__ + f)” + g”,
By considering
.Uk8(M))
(~l*(&f)&~(M))
(3.33)
and (3.34))
= (&*(M)v,“(A!f)) = (u;*(M)dk(AZ))
we can further
= 6,, )
(5.13)
= 0.
(5.14)
Making use of the relations (5.13), (5.14), and
we can prove that (5.7) is written as
zi[$P] = 1 d3x(ip) (ya + M) (sip) (5.16) = d32p( ya + m + f + ig~J~‘“. We shall write the field equation for the spinor field # as (5.17)
(Yr4.l + m># = 71,
where m denotes the bare massand, therefore, 71consists of terms containing at least two operators. The solution of (5.17) can be given by $(x)
= #‘“(x)
+ /’ d4r’S(x - .r’,f, g)q(r’,, --m
(5.18)
where
and S(.r
-
t’,f,
g)
=
(Q,
-
111 -
j
+
igy,)A(.r
-
a’:N)
(5.20)
(cf. (3.40)). In (5.18) we suppose that the adiabatic factor exp [-c 1t 11. (lim e ---) 0) is present in the integral”. Solving (5.18), we obtain4 $ written in 3 To respect should 4 It limiting bound so that side of
avoid the use of the adiabatic factor, we may consider the weak limit (in time) with to the wave packet.sin the in-field representation.In this casethe limiting process be performed in such a way that l/(sV) + 0. can happen that the asymptotic limit of the $‘s is not irreducible. This is due to the process c -+ 0. In such a case we meet bound states. In other words, when there are states, we should introduce the asymptotic fields of the composite particles, Bi*, all the asymptotic fields form an irreducible operator ring. Then, in the right hand (5.211, we have terms which contain Ri”. The coefficients of these terms can be cal-
358
UMEZAWA,
terms of normal products
TAKAHASHI,
AND
KAMEFUCHI
of gin as l)(x)
= Z”“(f,
g)lp
+ . .. )
(5.21)
where the expansion coefficients depend on the parameters (f, g) and Z(f, g) is the renormalization constant. The operators J, are in this way defined in the in-field representation. Up to here we have considered the system in the finite volume. For example, $, 9, and A(x:M) in (5.18) are of the forms: #(xl
= F IL&, Wb,
4(x> = 7 rj(k, Wb, A(x:M)
= -+
F
1 dkoei”%(k,,k,,
+ M2)t(ko).
Substituting (5.21) into # in EM], we have many terms containing more than three operators. Let us now consider the limit V --) co. Then, many of the representations specified by (f, g) turn out to be mutually inequivalent. Among terms in the Hamiltonian we shall consider, as an example, a term of the form of
Now, Theorem 5 in Section II implies that this is decomposed into several bilinear terms and c-number terms in the limit V + 03. In this way we see that the V-limit of X[#] contains only the bilinear terms of the in-field operators and c-number terms. Collecting all the bilinear terms, we obtain s A$%&
+ m + w,
9) + +Xf,
g)dP,
(5.23)
where the Lorentz invariance of the theory is taken into account. The c-numbers F(f, g) and G(f, g) depend on the parameters (f, g). Since (5.23) should coincide with (5.16), we obtain the equations f = w,
s>,
g = G(f, s),
(5.24)
which fix the values of parameters (f, g) and thus the value of mass M (cf. (5.12)). The relations in (5.24) in the case of the Nambu-Heisenberg model lead to (4.11) and (4.12), when we make the approximation of taking only the culated, when the coefficients of terms containing @‘* only are known. The can be calculated by solving (5.18) by means of the successive substitution. cussion was given in ref. 8.
latter coefficients A detailed dis-
MASS
LEVELS
AND
BROKEN
359
SYMMETRY
first term in (5.21) into account and of taking the renormalization constant 2 to be one. We can also handle the problem in a different way when we start with the expansion (5.21)) leaving the coefficients undetermined.6 Substituting (5.21) into the field equation (5.17) and collecting the terms with the same powers of #in, we obtain the equations (Y,& + m + W,
g) + G(f, .. . , .. .
g)dP
= 0, ( 5.25)
Here dots stand for equations which determine the coefficients of the normal products of higher power in #in in (5.21). As an example we shall consider the Nambu-Heisenberg Model. The field equation is (Y&4 + ml+ Substituting
= -N($4>
.ti -
(5.26)
(ih6~h~l.
(5.21) into (5.26) we obtain
( -yrd, + r?z)p
= - XZ[( p%p)p
-
(py@p)y6p]
+ *. . .
(5.27)
We shall disregard the terms denoted by dots and approximate 2 by unity. Rewriting the equation in terms of normal products by means of the relations i$in~6$in = i:@ny6@n: + C, , etc.,
(5.28)
and collecting the terms linear in tiin, we get (-I$, which
+ m + xc, + ixc,ys)p”
= 0,
(5.29)
leads to f = xc,,
g = xc,.
(5.30)
These relations together with (3.48) and (3.49) give us the equations (4.11) and (4.12). This line of argument can be formulated by means of the Feynman diagram, where each internal line represents the propagator defined by (3.41) and the external line $@. The interaction Hamiltonian contains the counter terms - (f@P -I- ig~‘“/k’“> . It is the essential point of the theory that the propagator and the in-field operators depend on the parameters (f, g). Applying the Dyson method, we can obtain the mass shift term (i.e., bilinear term) and the renormalization constants Z(f, g). The mass shift term has the form Wf,
g)+P
+ dG(f,
(5.31)
g)PWin
s A similar approach has also been proposed by Ezawa and Kikkawa cation).
(private
communi-
360
UMEZAWA,
TAKAHASHI,
AND
KAMEFUCHI
with Wf,
s> = w,
s> - f,
Since f and g should be so chosen that
sG(.f, g> = Wf, s> - g.
(5.32)
6F = 8G = 0, we come to the relations in
(5.24). The relation (5.2) states that the V-limit of the Hamiltonian is just the free Hamiltonian. Yet we cannot drop Xint in (5.1) from the theory, when we want to calculate the reaction cross sections. This is due to the following reason. To deal with reactions we need to calculate the transition matrix T whose V-limit is zero : V-lim
T = 0,
(5.33)
or lim (n:in 1 T 1m:in)
= 0.
V-W
The cross section u~+~ is finite, however,
since
u~+~(Y 1(n:in 1 T 1m:in)
l*Vz,
(5.34)
where I( >0) depends on the number of particles participating in the reaction. In other words, to calculate the cross section we should consider, not the V-limit of the Xi,,[+‘“] itself, but the V-limit of Xint[$in], multiplied by a suitable power of the volume V. Summarizing, the mass values can be calculated by the V-limit of the Hamiltonian, while the evaluation of cross sections needs the V-limit of interaction terms multiplied by suitable powers of V. Let us close this section by noting that, when we deal with fields with various degrees of freedom such as isospin, the parameters f and g are not necessarily c-numbers but matrices in the space of isospin and some other degrees of freedom. APPENDIX.
We shall derive the relations
CALCULATION
(2.57),
(2.58),
OF
(2.59))
Cr
and (2.60). Our notation
is: ukr: wave function of particle with momentum k and positive (r = 1) or negative (T = 2) helicity, 4: wave function of antiparticle with momentum k and positive (T = 1) or negative (r = 2) helicity, and
MASS
LEVELS
AND
BROKEN
361
SYMMETRY
&=[uifJi 1. 0
(A.3)
0
Taking the direction of k as the third axis, we have the relations: (ir,k, ( -ir,k,
+ nL)ukT = 0,
iz;3ulcr= crUkr,
+ nL)vkr = 0,
&VkT =
u;,,
= C,&:s
1 { -1
l.A.5)
)
where C is the charge conjugation matrix, w = dk2 + m2 and er =
(A.41
- Ervkr,)
for for
I%Z the four vector (k, iw) with r=l, r=2.
In deriving (A.4), we have taken into account the fact that the wave function vkr of the antiparticle with energy momentum (Ic,) is a solution of the Dirac equation whose energy momentum is ( --k,). For & and & , we have the relations 23 ul,k = - ETUTk )
23 v; = ETVLk )
(A.7)
since the helicity state changes when the direction of the momentum is inverted. Let us now make use of the representations of Dirac matrices in which 74
=
p3
.
(A.81
In such representations we have c = pz 22.
(A.91
Define the zero spinors Uk3= Uk4= Vk3= Vk4= 0,
(A.lO)
so that the superscript r runs from 1 to 4. Then u;,, , v;,, and z&,* are given by ul,o =
&a =
1
(--ir,k,
+ m> -1
ar
d2w(w + m) 1 ~4(hb
dWw
+
m)
1+ m> 2
(A.ll)
, a?
1
VLR,== which give
+ m> [
d24w
Y4 P222zi
(A.12)
,
1
7
l2r
(A.13)
362
UMEZAWA,
a;,, =
TAKAHASHI,
1
AND
1 + Y4 ___ (--Y/Au 2
2/2W(W + m)
KAMEFUCHI
+ m>
1 r(l
(A.14)
,
(A.15)
It is then easy to show that
?,i~k7rU,l$r =
r mw/
for
r = 1,
0
for
~=Ys,
i 1
for
r=y4,
L(~/u)E~
To evaluate im we need to introduce tion. Let us consider two cases: (i> (ii)
for a further
Y =
ZP2,
Y&i =
y=
--I;1
Ys
=
I (A.17)
I r = ~~7~.1 specialization
-/a,
74
=
P3,
-/32,
Y4
=
P3
of the representa(A.18)
*
(A.19)
The results are, in the case (i) , -k/o
tikklrxk =
--ie' 0 I -(m/w)cr
r=l, for r=iy6, for r = y4, for r = y4y6, for
(A.20)
and, in the case (ii),
(A.21)
In the discussionsin the text use is made of the representation (i) . However, all the formulas of case (ii) can be obtained from those of the case (i) by simply replacing qp(k, T) by co(k, r) + (r/2). ACKNOWLEDGMENT
The authors are indebted of us (Y.T.) acknowledges sity of Tokyo. RECEIVED:
to Dr. H. Ezawa and Dr. B. Misra for the hospitality extended to him during
June 25,1963
helpful discussions. One his visit to the Univer-
MASS
LEVELS
AND
BROKEN
SYMMETRY
363
Note added in proof: Concerning the matter of V-limit, the following remark will prove useful to clarify the situation. What our theorems l-5 really imply is the fact that the interaction between separate particles disappears at the limit V + 00 if the wave function is so normalized that the total probability of finding a particle in an in&de volume is unity. The proof of the theorems employed in our paper does not make sense, however, since the plane wave cannot be normalized in an infinite volume. Consequently, they serve only for a heuristic purpose. In order to overcome this difheulty, we reformulated the theory in terms of wave packets and found that our theory can be presented without use of the theorems 1-5 to find a set of inequivalent representations in which the Hamiltonian is bilinear and diagonal. All the calculations presented in the paper still hold true in the lowest approximation in our new formulation. This point will be discussed in a separate paper. REFEREXCES 1. Y. NAMBU AND G. JONA-LASINIO, Phys. Rev. 122, 345 (1961); ibid. 134, 246 (1961); Y. XAMBU AND D. LURIE, Phys. Rev. 126,1429 (1962). 2. Y. TAKAHASHI AND H. UMEZAWA, to be published. 8. L. VAN HOVE, Physica 18, 145 (1952); A. S. WIGHTMAN AND S. S. SCHWEBER, Phys. Rev. 98, 312 (1955); R. HAAG, Kgl. Danske Videnskab Selskab, Mat. Fys. Medd. 29, No. 12 (1955); H. EZAWA, to be published. 4. J. SCHWINGER, Phys. Rev. 76, 790 (1949). 6. R. HAAG, Nuovo Cimento (10) %, 287 (1962). 6’. W. HEISENBERG, Z. Naturjorsch. 14a, 441 (1959). Earlier papers are quoted there. 7. H. EZAWA, K. KIKKAWA, AND H. UMEZAWA, Nuovo Cimento (10) 23, 751 (1962). 8. H. EZAWA, K. KIKKAWA, AND H. IJMEZAWA, Nuovo Cimento (10) 26, 1141 (1962); H. EZAWA, T. MUTA, AND H. UMEZAWA, Progr. Theoret Phys. (Kyoto), in press.
OF PHYSICS:
The Mass
26, 336-363 (1964)
Levels and the Inequivalent HIROOMI
Department
of Physics,
Institute
for
in Terms
of
UMEZAWA* University
YASUSHI Dublin
Broken Symmetry Representations
of Tokyo,
Tokyo,
Japan
TAKAHASHI
Advanced
Studies,
Dublin,
Ireland
AND SUSUMU Department
of Physics,
KAMEFUCHI Imperial
College,
t London,
England
It is pointed out that inequivalent representations inherent in the theory of quantized fields in an infinite volume play an essential role in deriving mass levels and broken symmetries. A set of continuous parameters is introduced to specify these representations. By the requirement that the Hamiltonian shall tend to that of asymptotic fields as the volume becomes infinite, suitable representations are singled out. This requirement further determines possible values of the mass, which exhibit, in general, broken symmetries. It is also shown that with our method Nambu’s “Superconductor theory” of elementary particles can be formulated in a consistent manner. I. INTRODUCTION
It is an unpleasant feature of the present theory of particle physics that some of the symmetries established in strong phenomena are violated in the mass spectrum and weak phenomena. An interesting possibility has been suggested by Nambu (1) to derive broken symmetries from a fully symmetric interaction. He introduced a massequation with an analogy to the Bardeen method of superconductors, which has been interpreted, in our previous paper (2), in terms of inequivalent representations. The aim of this paper is to apply the method of inequivalent representation to relativistic field theory. * Present address: Istituto di Fisica Teorica, Universita di Napoli, Napoli, Italy. t Present address : Department of Physics, Tokyo University of Education, Tokyo, Japan. 336
MASS
LEVELS
AND
BROKEN
SYMMETRY
337
It is now well-known that, when dealing with quantized fields, one is forced to consider various representations which are not mutually equivalent (3). Suppose that a field is confined in a box with a finite volume V. There exist then a countable infinite number of momentum states, each of which can be occupied by any number of particles in the case of Bose fields or by at most one particle in the case of Fermi fields. Consequently, the state vector space becomes nonseparable. This has been discussed by several authors in connection with the van Hove model (3). This kind of problem does not arise, however, when one introduces a maximum (or a cutoff) momentum for each particle. Throughout this paper we assume that a cutoff function exists in the Hamiltonian, although it is not written explicitly. Even with such Hamiltonians we still meet the problem of inequivalent representations when we are concerned with a system of infinite volume, since there appears a continuous distribution of momentum states. It is possible, however, to avoid the use of momentum states by employing an orthonormal set of wave functions which vanish with a reasonable speed at the spatial infinity (i.e., the wave packets). Since such a set of wave functions is an enumerable one, we are led in this way to a separable space of state vectors. This vector space is a special one chosen by the suitable boundary condition. However, we have some reason to believe that the representation so chosen may not always be a reasonable one: this is due to a delicacy of the behavior of the field at the spatial infinity. To see this we may recall the problem of vacuum polarization in quantum electrodynamics. Here, the calculations show that the total charge is decreased by the renormalization factor 2:” which is due to the effect of the vacuum polarization and therefore that the conservation of the total charge is violated. Schwinger (4) suggested, however, that the conservation of the charge may be recovered when one takes into account the charge which is distributed over the whole space with a constant but infinitesimally small density. It is evident that the charge distribution of this kind can not be obtained by means of the wave packets. This may not be a difficulty in quantum electrodynamics in the wave packet representation, where the renormalized charge is conserved throughout reactions. However, as the theorems in Section II show, an accumulation of physical quantities distributed infinitesimally over the whole space is not always vanishing but depends on the choice of representations. Thus, a question arises as to how one should choose the reasonable representations from among the mutually inequivalent ones. The consideration in the present paper strongly suggests that this question is intimately connected with problems of the broken symmetry and the mass spectrum. To study our question above, we begin with a case of a finite volume V and introduce a set of real continuous parameters { 0, p) : each set of values of these parameters specifies the representation. Referring to each representation we calculate the limit (V 3 00) of operators. This operation (2) will be called in the following the V-limit and denoted by V-lim. Although, for a finite V, these
338
UMEZAWA,
TAKAHASHI,
AND
KAMEFUCHI
representations are connected with each other by unitary transformations, many of them may turn out to be inequivalent to each other in the limit V -+ a,. Supposing that the parameters { 0, (p) represent a continuous set of inequivalent representations, we shall show that the V-limit of operators also depends on the parameters (8, ~1. It should be noted that the V-limit is not quite the same as the weak limit, because the basis of the representation in which the limit is carried out may depend on the volume V. To perform the V-limit of operators consistently, we adopt the convention that all calculations are carried out exclusively in a fixed representation and the operators belonging to different representations are not considered simultaneously. We shall base our arguments on this convention, expecting that such a procedure may be justified in future by rigorous mathematics. When considering a special representation, we denote the annihilation operator of the Fermi particle and antiparticle by CY~”and ,&’ respectively, and the vacuum by @o, i.e., (YgT*o= pk&l
= 0,
where k denotes the momentum of the particle state. We may introduce the operators
(1.1)
or the antiparticle
and r the spin
(1.2) with
c-numbers
p1 and pz satisfying I Pl
the relation I2 +
I P2 I2 =
1.
O-
(1.3)
Although these operators obey the same commutation relations as those of a! and 8, &’ and fikT cannot be called the annihilation operators. This is because, as will be shown in Section II, in the representation under consideration there is no state 5, satisfying the relations &?&
= /3kkrS0= 0.
(1.4)
On the other hand, the representation of a one-particle wave function can be changed by unitary transformations in the same Hilbert space. For example, the annihilation operator can be defined in the spherical wave representation as well as in the plane wave representation. As will be discussed in detail in Sections II and III, most of the inequivalent representations are not physically acceptable. The values of the parameters for the acceptable representation are obtained by the condition that the V-limit of the Hamiltonian shall be diagonal when it, is written in terms of the creation
MASS
and annihilation
LEVELS
AND
BROKEN
339
SYMMETRY
operators :
V-lim [x] = g: -
z
3
,?L(a;l’cuk” + /3LtPk’) + c-number
(1.5’)
with Ek = dk2
(1.6)
+ AP.
This condition leads us to a set of equations, which determines the mass .!I/ together with the parameters (19, ‘p} . If the field equations for a system in a finite box have an invariant property with respect to a certain unitary transformation which thus leads to a conservation law, it can happen that the representations which are related to each other by this unitary transformation turn out to be mutually inequivalent in the limit V + 00. In such a case the V-limit of the Hamiltonian loses the invariant property and we observe the broken invariance. To recover the invariance we need to take account of the effects of matrix elements of the order of (l/V), effects of which are hidden in the measurement of masses. The situation will be illustrated in Section IV by the Nambu-Heisenberg model. We shall see in Section IV that the requirement (1.5) allows more than one inequivalent representation, which correspond to different masses of the field under consideration. There seems to be no difficulty in evaluating transition matrices, as far as the transitions between particles belonging to the same representation are concerned (cf. Section V). It is not clear, however, at the present stage how to deal with transitions between particles belonging to the different representations. Section V will be devoted to a discussion of an important relation between the V-limit and the asymptotic condition. II.
INEQUIVALENT
REPRESENTATIONS QUANTITIES
AND
THE
J’-LIMIT
OF
FIELD
We shall consider a system of a fermion field confined in a box with a finite volume V and denote the annihilation operators for particles and antiparticles by akr and bk7, respectively: [akr, a?]+ = [bk’, bit]+ = &zL . Here the subscript k refers states: 1’ = 1 for the state of positive helicity) and r (i.e., the state of negative by ~0 :
to the momentum and the superscript of the spin parallel to the momentum = 2 for that of the spin antiparallel to helicity). The vacuum of the fermion akTQO = bkrt2,, = 0.
( 2.1) r to the helicity (i.e., the state the momentum field is denoted (2.2)
340
UMEZAWA,
TAKAHASHI,
AND
JLAMEFUCHI
Let us now consider a general set of unitary transformations condition:
P’, Gl = 0,
G satisfying the
[L, Gl = 0,
(2.3)
where P and L are the total momentum and total angular momentum operators of the system. It is obvious that the set of states G!& covers all the states with zero momentum and zero angular momentum. Since the vacuum should be invariant under the spatial translation, we may regard this set as candidates for the physical vacuum state. It is clear that they are not necessarily the ground states of the free Hamiltonian for the fermion system. This, however, is not a difficulty, as the procedure of taking the V-limit changes the structure of the Hamiltonian (cf. Section III). Rather, we are looking for the representation in which the requirement for the Hamiltonian (1.5) is satisfied. The unitary transformation satisfying (2.3) has in general the following form: G = exp[iF], where the hermitian s
(2.4)
operator F consists of terms such as
d32a. . . a&>ra . -a a+(z), (1)
s d3xa. . . a$(x>ra ... a+(x)a .-- a$(x)r’a .-- a+(x) with p, r’ spinor matrices. As will be seen in Section V, it is sufficient to consider the case in which F is of bilinear form in # and 4. We further simplify the situation by restricting [G, N] = 0
(2.6)
N = g [af+uk’ - G+bk’].
(2.7)
with
In other words, we consider the case in which the fermion number of the vacuum states is zero. To construct G, we shall introduce the following hermitian operators : T::;
= $$&J;+ b:\ + KI, a;), (2.8)
MASS
LEVELS
AND
BROKEN
Any combination of aLtb?k , lLak’, altakr and b$b?, can be written combination of TCi’(i = 1, 2, 3, 4). It can easily be seen that iTi? , Ti12T)]= iT i,“r’,
as a linear (2.9)
etc.,
(2.10,
[T:::: , T/i::] = 0, [Tj:? , T!yJ] = 0 for The relation for rotation. written as
341
SYMMETRY
(k, r) # (I, s).
(2.11)
(2.9) shows that Tf+) , T:,“,’ , and T:,3,’ behave as a set of generators Then the general form of the unitary transformation G can be (2.12)
kr
Here 2.13) with Sc3) = g exp[it@ Ti?],
2.14)
Sc4) = n exp[iOi:‘,TT:P,‘].
2.15’)
k,r
we can easily check the following
relations:
akr = Gak ‘G-’ =f? i6(kvr)[~~~ ekr.akr
+ eiqp(kxr’sin /jk’.bi_l],
(2.16)
@Lk = GbLkG-’ =e i”(kvr)[~~~ ekr.bTk - eip(k,r) sin ekr.a;+],
(2.17)
where
(2)
ok.7
. (1)
-
dk,r
=
t
e reG(kA k
2.18)
1
and qk, r) = ei:: - ei,?,
(2.19)
~(k, r> = eiy? + eil”) .
(2.20)
The parameters 6 and Y (and therefore OC3) and eC4’)induce an independent change of each basic state vector. Hence it is enough for us to consider only the case in which they are zero: e’4’ e(3) 0. (3.21) k.r
=
k,r
=
342
UMEZAWA,
We can further
TAKAHASHI,
assume without
AND
KAMEFUCHI
loss of generality
0 s ek’ 6 7r/2,
that
0 s (o(k, r) < 27r.
and that, when 9kr = 7r/2, the parameter Thus, we have
(2.22)
V( Ic, r) is zero.
(2.23)
(2.24)
(2.25) The state
(2.26) k,r
is the vacuum
in the representation
under consideration:
ffk%, = pkk’@O= 0. The operator w(k,
(2.27)
r) is defined by w(k,
r) = cos ek’ - eicockqr)sin &‘.d+b???.
(2.28)
It should be noted that creation operators of different helicities are never combined in TJ$ in (2.8). This is due to the second condition in (2.3). This condition also requires that ok’ and (o( k, r) should not depend on the direction of the momentum k. In the next section we shall further restrict the values of parameters (ok’, pk’} by (1.5) and then see that the resulting theory becomes invariant under the orthochronous Lorentz transformation. Recall the relation 9(x) Here Ukr and equation:
vkr
= (l/dv)
are the positive
g
[ukrak’e’b
+ vkrb;+e-“k”].
and negative frequency
solutions
(2.29) of the Dirac
(2.30) (-i$$,
+
m)vk’
=
0,
where m is the bare mass of the fermion (cf. Section III) (k, iC&) with Wk = dkz + m2. They are normalized as
and 5 the four vector
MASS
LEVELS
AND
BROKEN
343
SYMMETRY
(2.31)
(?A;*, UkS) = a,, ) (v;*, VkS) = a,, ) and related to each other by the charge conjugation’: v;, = C,&B* Substituting
(2.25) into (2.29) +(x)
= (l/dV)
(2.32)
.
we obtain
F [z4kr( 19,(o)cykreikx + vkl’(0, cP)PL+eC”k”]
( 2.33)
with uk’( 8, q) = ukr cos &’ + v~ke-iP(k’r’ sin ok’, vkr(
8, p)
=
vkT cos
ekr
-
u?-keirp(k”)
sin
(2.34)
ekr .
It is evident that Ukr( 8, (0) and vk*( 8, (p) coincide with ukr and uk’ respectively, when &’ = 0. Regarding a0 as our vacuum state, we shall construct basic vectors as follows: rt a0 ) @;+$I ) /3;+cx;+$l , . . . . %J ) LYk The general form of these vectors
is
@[kr, Is, ‘. . : pt, qu, . . - ] = &,+P,“+ . . . a;+&+ . . * a+) . We shall denote these states by $( 8, (p), c&( 8, ‘p), . . . , with a suitable subscript to distinguish various states, where @,( 0, (p) is the vacuum defined by (2.26). When we consider the limit V ---f 00, the transformation (2.26) loses its sense and the basic vectors $( 0, lo), aI( 0, CO), . . . for each set (0, (01 constitute the basis of the respective inequivalent representation. This may be seen by the relation : ;;
(a(e,
We shall illustrate
+a@,
(2.35)
4))
= 0
for
(0, P} z {e’, &I.
(2.35)
as follows: d”k log x(k)]
= o,
(2.36)
where
cp(k, r>- cp’(k, r> x(k) = [Coft(ek’ - e;‘)cos2 2
>
1 It should of (2.32).
be remarked
that
we have
the same
helicity
(2.37)
superscript
(T) in the both
sides
344
UMEZAWA,
TAKAHASHI,
AND
KAMEFUCHI
Taking into account (2.22)) we see that the equality in (2.37) holds only when (0, (o) = {O’, 9’1. The relation (2.35) shows that G& does not possess any well defined limit for V -+ ~0 and therefore that the vacuum c&( 0, (o) of each inequivalent representation cannot be defined by (2.26). Nevertheless, we shall proceed further by adopting the following assumption, expecting that our procedure could be formulated by profound mathematics for a nonseparable vector space: The assumption is that for a system of infinite volume a set of inequivalent representations is classified by the parameters { 0, ~1. When we fix the values of the parameters (8, (01, we get a set of creation and annihilation operators (akr, Pit and their hermitian conjugate). When we say that we are given a field operator $, this means that t/ is given as an operator defined with reference to a special representation. In other words, # is given by (2.33) in terms of (CYST,Pkr). It is only because of the normalization of the operators akr and Pkr that V appears in (2.33): Since k is a continuous variable, we could have normalized them in such a way that [ah’, al+]+ = 6,&k
- 1).
However, the specific feat#ures of our theory will be explicitly manifested when we describe the theory through the intermediary of the V-limit convention. To be more precise, we define the limit of operators, say Q, with respect to a representation specified by ( 8, (o} as follows:
(we, 4, V-h-n [&Me, (0)) = pp(e,
dw(4
d>,
(2.38)
and the whole calculation is carried out in the { 0, (PI-representation. The relation (2.38) defines the limit of the operator Q as an operator only in the Hilbert space, whose basic vectors are related to the set %( 8, (o), %( 0, P), . . . by a unitary transformation. The limit of Q can be defined without specifying the one-particle representation (plane wave, spherical wave, etc.), although we shall use in this paper the plane wave representation. When considering a representation specified by ( 8, (p} , we are given the annihilation operators akr and @hr. We can then define the operators &* and &’ by (1.2). These operators can not be called the annihilation operators, because in the Hilbert space under consideration (i.e., in the { 0, (p}-representation) there is no state & which satisfies the relations in ( 1.4). What the condition (1.5) implies is that the V-limit of the Hamiltonian should be diagonal when it is written in terms of the creation and annihilation operators. Let us now introduce the normal products : iL(xdrLp(22) where
the order of operators
. *. :
CY~‘,@kr, c$, and ,f$ are so arranged
(2.39) that all the
MASS
LEVELS
AND
BROKEN
SYMMETRY
345
annihilation operators appear on the side right to the creation operators. Considering (2.33), we see that the c-number coefficients in (2.39) depend on the parameters (0, (01. Then we have following theorems which play a central role in our theory. THEOREM 1. The V-limit qf all the normal products vanishes : V-lim [ : ~J~(x~)#~(x~) THEOREM
.. . : ] = 0.
( 2.40)
2. We have V-lim
ils
d3z : ~J~(x)$p(x)
...
1
:
= 0
for
n > 2,
(2.41)
where n denotes the ,numbev of Jield operators in the bracket. THEOREM 3.
1
d3x : $a(~)~~(~)
.. . :
= 0.
Taking into account the fact that each operator # contributes to the product by the volume factor l/z/v, we can illustrate the proof for (2.40) by
A similar argument also gives us Theorems 2 and 3. Let us denote the vacuum expectation value of an operator (Q>o = (@o(@, (o), Since the vacuum
@o(
8, q) is invariant
Q*o(e, cp>1.
under the spatial translation,
do not depend on x but on the parameters Since we have the relations
from (2.41) and (2.42)
& by
follow the theorems:
(0, p}.
the quantities
346
UMEZAWA,
THEOREM
[S
AND
KAMEFUCHI
4.’ V-lim
V-lim
TAKAHASHI,
d3&x!(XM3(d
d3x &(x,lj#dx>
t [J
1
= /2x
: iL!W~a(x)
1
= cdf+P)?
(2.45)
:
(2.47)
+ an injinite c-number,
= Ca,(4#) /- d3x: $a(x>~~(x>: + Cde,#) /- d3x: &(x>~a
Ca,(e, +) j- d3x : &(X)&T(X)
: - C,,(e, cp> /- d3x : $a(x)J/v(x)
:
+ an injinite c-number. In these relations the c-numbers are proportional to the volume V as is the zero-point energy. These c-numbers are thus amalgamated into the vacuum energy. Quite general1 y, we can show that the V-limit of s
d3dhM,h)
.. .
for any power of field operators is a linear combination of various and a c-number. Since %+,is invariant under space rotation, it is clear that Cr = ($W)O = 0
for
r = y, y-is, L-h, -4,
bilinear terms
(2.49)
which can also be checked by direct calculations. To calculate ($I’$) for other I’s, we make use of (2.33) together with (2.34). Since (fikrmk8), (Bkrr~kS), (tori%), and (Greta) with r = 74, y5, 1 or YIYS are invariant under the space rotation, they would not be zero unless r # s. It will be proved in the next section that (2.50) 2 This conductor.
theorem has been stated in ref. B which dealt with the B.C.S.-model A theorem of similar kind was also stated in ref. 6.
for the super-
MASS
LEVELS
AND
where Ok and (ok are the functions
347
SYMMETRY
of Ic ( = 1k 1) only and er is +l -1
7 e = Taking
BROKEN
for for
1’ = 1, ~=2.
(2.51)
(2.50) into account we obtain for I’ = 1, y4 , y4y5 , and y5 the relation:
Here A,(k)
= cos2 6b(iik’ruk’) + $5 sin
+ sin” &(
2ok
cos
fik7rl)kr)
(ok{ (tik%kr)
+
- $$if’ sin ii?& sin (ok{ (ckmk’) B,(k)
= cos’ ek( fikrr&‘) - 4/i sin
+
2ek
= cos2 ek(fik’r&r) -
We obtain,
co8
(ok{ ( fikrr2)kr)
+
-
sin2
( iikrrukr) -
6keSiPck”)(
(2.54)
]
( tikrrUkr)),
nkrrukr)
(2.55)
cos ek sin &ei”k’r) ( (iikrr?kkr) from
(&‘rUk’)},
+ sin” &(?&‘I&‘)
+ @ Er sin 26, sin (ok{ (?ik7mk7) e,(k)
(2.53)
( ckrhkr)l
-
(ekrr&)‘)].
(2.52), (2.56)
for r under consideration. Taking into account the relations in the Appendix, we can show that C, = V-lim [i(h
J/)01= &
C, = V-lim [($$)01 = - &
j”
Ca = V-lim [(&A
75
y7)01 = 0.
and (-4.20)
d”k sin 288 sin (Pk,
(2.57)
1
2 cos 28~ - -!! sin 28k cos (Pk , Ok
C, = V-lim [($-yJ +),,I = &a
(A.17)
/ d”k = 26(x)
Wk
Ixzcl ,
(2.58) (2.59) (2.60)
It is to be remarked here that (2.49) leads us to (2.60), which is necessary for the theory to be invariant under the orthochronous Lorentz transformation.
348
UMEZAWA,
TAKAHASHI,
AND
KAMEFUCHI
It is not surprising, however, that C, # 0, although the corresponding spatial parts vanish. Indeed, the value of C, does not depend on the choice of the parameters (8, ~1, and therefore it is equal to the value of the zero point charge density: C, = Tr[r.S+(O)].
(2.61)
It has been known for some time that, in order to secure the Lorentz invariance of the theory, we have to construct the vector term in such a way as
J, = 4hhhk - s/‘u,$‘l, where #’ is the charge conjugate
(2.62)
of #. It is easy to see that
C;’ = V-lim (J&
= 0.
(2.63)
We are then left with C, and C, which are not zero. III.
THE
V-LIMIT
OF
HAMILTONIAN
AND
THE
MASS
EQUATION
Let us now show that only in special representations, i.e., only for special sets of the parameters { 0, (p}, is the condition (1.5) satisfied for a given X. We shall consider a fermion field # whose Hamiltonian is given by 3-2
=
X0
+
Xint
7
(3.1)
+ m)9.
(3.2)
where x0 =
s
d3&ya
It is to be understood here that the Hamiltonian operator is defined with reference to the Hilbert space with the parameters { 8, (o} . This means that # in (3.2) is defined by (2.33) in terms of akr and Pk . The bare mass m is so defined that each term in the interaction Hamiltonian contains more than two field operators #. Thus, the interaction Hamiltonian consists of terms like
where $ is a boson field, X denotes the coupling constant, and r and I” are spinor matrices. Let us consider the second term in (3.3) as an example. As was shown by (2.48), its V-limit consists of terms bilinear in # and a c-number. Due to the Lorentz invariance of the theory, these bilinear terms should be of the form:
Here S and g depend on the set of parameters
{ 0, q}.
MASS
In this
way,
LEVELS
AND
BROKEN
SYMMETRY
349
we see that V-lim [I
d”,TiC] = [ d3ccE0 + c-number,
(35)
where x, = x0 + 6X”. Our next task is to fix the parameters Hamiltonian :
(3.6)
(0, cpJ so that ??o has the form of free
x0 = c &(4+d k,r
(3.7)
+ P;+Pk’) + wo )
where Ek = dk2
+ M”.
(3.8)
The mass ill and the zero point energy Wo are also determined when the parameters are fixed. Relations (2.33) and (2.34) together with those in Appendix lead to x” = c ‘Ok cos 20/((Y;+C$
+ p;+,&‘)
k,r
wk sin
2&‘[eiqa;+$?k
+
e?&ak’]
+
2
g
sin ‘ok’,
Wk
(3.9)
-F
i
s
&gNtx
= p
cr sin 28k’ sin &&+akr
+ &‘fik’)
-ie’
( cos2 ok’ + sin’ ek’ezi’)C$+fi~t,
(3.10)
+ ie’ (cos’ ekr + sin* e,‘e-““‘)@‘@~I
s
&b d3x = z
k (m
COS
- CL
2ok’ -
k
Sin
m eiq sin 2&* + ’
k,r
Wk
2okr
+ 5 cr sin
COS p)(LYitd
(cos2 ok7 - sin”
+
p,
fl;+@k’)
ekre2”)
Wk
+ sin 28k’ + ’
28k"sin
1
(cos2 ekr _ sina ekre-*i9
aL+p?k
1-j
(3.11) @:k
akr
Ok
-
~~(wc0s2ekr--
k sin
28k’
cos
p).
Here cp is an abbreviation of ‘p( I%,r). The Hamiltonian zo defined by (3.6) is of the form (3.7), parameters (13,(e) satisfy the following relation:
if and only if the
350
UMEZAWA,
TAKAHASHI,
wk sin 2&’ + igc’ (~0s’ &*e+
AND
KAMEFUCHI
+ sin” ekreiV)
k sin 2ekr + - (~0s’ ekre+
1
- sin’ &‘ei’)
= 0.
Wk
(3.12)
When (3.12) holds, we obtain Ek
=
Wk COS 2okr
-
gEr Sin
2okr’Sin
Cp( k, T)
+ f ’
(m cos 2ekr - k sin
Wk
wo = c k.r
&Ok sin”
ok*
+
-
Erg
f i
sin
2okr
(m
cos
(3.13)
28; cos cp(k, r)),
sin &k, r) 2ekr
-
sin
k
2ok’
coS
p(k,
r) ) -
Wk
= -2cEk.
1
(3.14)
k
The imaginary
part of (3.12) gives
US
g cos cp(k, r) = fe’ i
sin cp(k, r).
(3.15)
1
cos p(k, r) = *:f; dg”
+
(3.16)
f2(k2/~k2)
’
1
sin cp(k, r) = fg -&”
+
I.
f2(k2/Wk2)
(3.17)
’ *
The real part of (3.12) gives [Wk
$
f(m/Uk)l
Sin +
2ek’ [@‘Sin
(3.18) (p( k,
r)
+
f( ii/WI)
coS
(P( k, r)]
coS
%k’
=
0,
or [Ok
+
.f(m/Cdk)]
Sin
2ok’
f
dg2
+
j2(k2/uk2)
COS %kr
=
0.
(3.19)
It is clear that &’ satisfying (3.19) is independent of the helicity state. Then, considering (3.16) and (3.17) we can prove (2.50). Requiring that Ek should be positive, we find, after some straightforward calculation, EI, = dm, COS 2ekr
=
(3.20)
(l/Ek)[Ws
sin 2&* = (I/&)
+
l/g”
f(m/mk)l,
+ f2(~2/ws2)
(3.21) (3.22)
MASS
LEVELS
AND
BROKEN
351
SYMMETRY
(3.23) 1
sin cp(& r) = -g dg2
+
r
j2(k2/Wk2)
(3.24)
”
M2 = (m + f)” + g2.
(3.25)
In deriving these results we have taken into account the relation (2.22) which means that sin 28k’ is positive. As was stated before, f and g are functions of 13and (o, depending on the structure of interaction Hamiltonians. Let us write these functions as j( 8, 9) and g(6, cp). Furthermore, (3.21), (3.22), (3.23), and (3.24) give us gk and (ok as functions of j and g: &( j, g) and (ok(j, g) . In this way, we obtain a set of equations f = fwt
91, df,
a,
(3.26)
9 = au,
91, df,
9)).
(3.27)
Solving these equations we can determine the values of j and g, and then the values of the parameters (0, ~1. The mass value M is now given by (3.25). We have seen that the relation ( 1.5) holds in the representations in which we have (3.21), (3.22)) (3.23)) and (3.24). In these representations, (2.33) and (2.34) give us l.3.28) with Ukr(
j,
9)
=
uk’u(
2)kr(j,
9)
=
2/krU(j,
[kk
@k
j,
g)
+
9)
-
u:kl”(
j,
u:kli*(
(3.29)
g), j,
(3.30)
g),
where u(
j,
g>
=
+
Wk
+
f
(3.31)
E)]“’
l/2
‘(”
We can further
‘)
=
-
&
[Ek
+
Wk
:
1
j(?‘fZ/wk)]
[f (k/d
- ierg].
(3.32)
prove that
(“L*(f, g>‘uk’(f, 9)) = &* (d*(f,
g> ‘uk*(f,
9))
@*(.f,
g> -v8k(f,
9))
M*(f,
9)
9)
.Gf,
= = 1
=
‘%s
(3.33) (3.34)
352
UMEZAWA,
TAKAHASHI,
AND
KAMEFUCHI
and that F u.Ldf, shXf,
F &&
s> = kk
[-i&-h
s>&df, 9) = -kk
+ m + f - igr&3,
(3.35)
[~K,Y~+ m -I- f - in&~,
(3.36)
where K, is the four vector (k, iEk) . These relations lead to &a
d3kuk’(f, g)ak’(f,
r
g)eiKps#
(3.37)
= i (Yrar - m - f + Q%)A+(z:M), d3kvkr(f7
g)okk(fT
g>
iK@,, e
(3.38)
= 5 hap
- m - f + igrs)h(z:M).
Here A+(x:M) and h(z:M) are the positive and negative frequency part, respectively, of A(x:M), the invariant delta function with the mass M. Introducing the time dependent operator #(x) by
%~~akr+ e)kr(f,g)e-iK@*pkr’]
l/4x) =
(3.39)
we obtain
bhGGN+
1 = i (-ha,- - m - f -I- igsyL)A(x - x’:M),
(~M(~)k(~‘>l)~ = MYrap - m - f + imhb
- x’:M.
(3.40) (3.41)
Let us note the relation
(Trap - m - f + igy6)y4 = ei”Y5(y,d, - M)y4e+‘rs
(3.42)
with a constant 4 being defined by sin24 Equation
= -5,
cos2cp = & (m + f).
(3.43)
(3.41) then gives us (!w(x),
ivx’)]>o = %s(y,a, - M)y,S’y,A,(x
- x’:M)
(3.44)
with
S = exp [i+ys].
(3.45)
MASS
LEVELS
AND
BROKEN
353
SYMMETRY
Thanks to the relation (3.44) we may formulate the perturbation calculation for transition matrices by means of the Feynman technique as follows: Each internal line corresponds to the propagator $&!!iF(x
-
x’:nf)
=
>d(r,,a,,
-
hf)A,(.c
-
(3.46)
a’:hf)
and each vertex I? is replaced by (3.47)
r’ = y&qrs. The transition matrices will be considered in Section VI. We finally note that, in the representation under consideration, from (2.57), (2.58), (3.21), (3.22), (3.23), and (3.24) the relations
we obtain
(3.48)
which yield (m + f)C, IV.
THE
NAMBU-HEISENBERG
Let us now apply our theory the Hamiltonian is
When m = 0, the Hamiltonian
(3.50)
= gc, . MODEL
to the Nambu-Heisenberg
(4.1) is invariant i6-0 *-+e 4
model (1, 6) where
under the transformation (4.2)
with 4 being a real c-number, and this is the case which Heisenberg and Kambu considered. Here we begin with the general case in which m is not necessarily zero. We first note that the Fierz transformation gives us the relation
- bhn6~)(h.&Yh~)l + G&4+,
(4.3)
where C, is defined by (2.59). The last term on the right hand side of (4.3) appears due to the change of order of field operators. Denoting the charge conjugate of #Jby $‘, we have the relation:
h,+ = %rs7,$ - $‘dl The relation
(4.3) now reads as
+
G%4
*
(4.4)
354
UMEZAWA,
av”
TAKAHASHI,
- @-Y51c/)2 = -MJPJ,
AND
KAMEFUCHI
- bI%aXw%~6~)1 +
%Cv” ,
(4.3’)
where J,, is defined by (2.62). We shall disregard the last c-number term, which can be amalgamated into the vacuum energy. The theorem (2.48) now says that, in the V-limit, the interaction Hamiltonian ; 1 d34 W”
- hiH21
(4.5)
becomes d3L&iGl = x
s
d32(C&
s
+ iC&sJ/J,
(4.6)
where we again dropped a c-number which will be amalgamated into the vacuum energy. In deriving (4.6), we have taken into account the relations (2.57)) (2.58), (2.60), and (2.63). Comparing (4.6) with (3.4), we get f = XC,) Thus, by substituting relations :
g = xc,.
(4.7)
(4.7) into (3.25), (3.48), and (3.49) we get the following
X, = c &((Y;+(Y~’ + @?/3k’) + c-number,
(4.8)
k,r Ek
=
dm,
M2
=
(m
(4.9) +
XC,)”
+
h2Cp2,
(4.10) (4.11) (4.12)
By Eqs. (4.11) and (4.12) the mass M can be determined. bilities to be discussed:
There are two possi-
(;;)3 c, = 0, M = m- -M/g, a% m= 0, ’ + & s -= Ek ‘* Note that the second case appears only when m = 0 and X < 0. The first case with m = 0 and M # 0 is a special case of (4.14). We remark further that (4.13) with m = 0 is the equation obtained by Nambu (1))
MASS
LEVELS
AND
BROKEN
35.5
SYMMETRY
We shall begin with the first case with m # 0. The equation (4.13) gives the massM in the form of power series d% s z+
M=m-&n
(4.15)
‘..I
This is the massgiven by the chain approximation in the ordinary perturbation calculation. Let us now consider the second case. The solution of (4.14) gives us a mass value which is missed by the ordinary perturbation calculation. Since vz = 0, (3.21) shows that the parameter ok can be determined by M as cos 2ek = k/E/s .
(4.16)
The quantities C, and C, are restricted only by the relation M2 = A’( c,2 + C,‘).
(4.17)
The mass1Malone cannot determine Pk , because tanp(lc, I’) = (C,/C,)e
(4.18)
as is shown by (3.23) and (3.24). It is remarkable that if (4.1) is invariant under the transformation (4.2) (i.e., nl = 0) we can choose a Hilbert space in which the theory loses the invariance (M # 0). This result is mainly due to the theorems in Section II. Another way of saying this is that the transformation (4.2) cannot be represented by any unitary transformation in the Hilbert space concerned. Our result suggests that the broken symmetries observed in the mass spectrum might be explained by a Lagrangian of fully symmetric character, without explicitly introducing any interaction with broken symmetry. We might further expect that the transition matrices would lose their symmetric character only through the mass spectrum. This is true in the case of the Nambu-Heisenberg model, provided that the transition matrices are calculated by the perturbation theory. As was mentioned in Section III, each vertex in the Feynman diagram may be replaced by (3.47) if one usesat the same time the propagator (3.46). This replacement, however, does not give rise to any change in the transition matrices at all, since the interaction Hamiltonian in (4.1) is invariant under the transformation (4.2). Thus, in this case the only agent for violating the invariance under the transformation (4.2) is the appearance of nonzero M in the propagator. V. THE
V-LIMIT
AND
THE
ASYMPTOTIC
CONDITION
When talking about physical particles, we describe them in terms of the asymptotic fields #in or #Out. To do this we must first express the Heisenberg field $ in terms of tiin or It”“” and then consider all the possible inequivalent
356
UMEZAWA,
TAKAHASHI,
AND
KAMEFUCHI
representations for the latter fields. In this sense our argument in the previous sections may be regarded as an approximation in which 9 is simply replaced by tiin. By substituting the expression of 9 in terms of, say, #in, the Hamiltonian X[#] can be rewritten in the following form: X[#]
=
%3[~‘“]
where X&i”] is the free Hamiltonian (1.5) to @n, then the representation
+
Xint
(5.1)
[$‘“I,
of the field +P. If we apply the requirement for @” is fixed by the condition
V-lim X[#] = X&Ji”],
(5.2)
V-limXi,t
(5.3)
or what is the same thing, [$‘“I = 0.
In the case of the Lee model a direct calculation shows that the above condition can in fact be satisfied (7). In this connection it should be noted that the transformation G defined by (2.23) with Gin instead of $ covers all the possible representations for P: it is not necessary to include, in F, terms containing more than two P’s (cf. (2.5) ) . This is because the field equation for gin is linear. The operator tPn can, then, be written in terms of the parameters (f, g) , following the recipe given in Section III, as follows,
p(x) = --& g [ukr(f,g)alP” + uk’(f, sM~+e-i”“l,
(5.4)
where uk’(f, g) and ukr(f, g) are defined by (3.29) and (3.30) and the four vector K, is K, = (k, i&J,
&
= dk2
+ (m + f)” + g2.
(5.5)
The in-field @” satisfies the equation (r&
+ m + f + igra)iwx)
= 0.
(5.6)
The set of parameters (f, g) which specifies the in-field representation will be determined so that the V-limit of the Hamiltonian has the form of (5.2)) where ?&[I/@] = g El,(CX;+f& + &+&‘) To rewrite
(5.7)
(5.7) we note the relations
(h’&, + M)%‘(M) (+,K, with
+ c-number.
+ M)u~‘(M)
= 0,
(5.8)
= 0
(5.9)
MASS
LEVELS
AND
BROKEN
SYMMETRY
357
w+r(M) = s%r(f, 9)
(5.10)
hC(M)
(5.11)
= S~kr(.f, g), fif = d(?n
where S is defined by (3.45). prove that (u;*(A!f)
__ + f)” + g”,
By considering
.Uk8(M))
(~l*(&f)&~(M))
(3.33)
and (3.34))
= (&*(M)v,“(A!f)) = (u;*(M)dk(AZ))
we can further
= 6,, )
(5.13)
= 0.
(5.14)
Making use of the relations (5.13), (5.14), and
we can prove that (5.7) is written as
zi[$P] = 1 d3x(ip) (ya + M) (sip) (5.16) = d32p( ya + m + f + ig~J~‘“. We shall write the field equation for the spinor field # as (5.17)
(Yr4.l + m># = 71,
where m denotes the bare massand, therefore, 71consists of terms containing at least two operators. The solution of (5.17) can be given by $(x)
= #‘“(x)
+ /’ d4r’S(x - .r’,f, g)q(r’,, --m
(5.18)
where
and S(.r
-
t’,f,
g)
=
(Q,
-
111 -
j
+
igy,)A(.r
-
a’:N)
(5.20)
(cf. (3.40)). In (5.18) we suppose that the adiabatic factor exp [-c 1t 11. (lim e ---) 0) is present in the integral”. Solving (5.18), we obtain4 $ written in 3 To respect should 4 It limiting bound so that side of
avoid the use of the adiabatic factor, we may consider the weak limit (in time) with to the wave packet.sin the in-field representation.In this casethe limiting process be performed in such a way that l/(sV) + 0. can happen that the asymptotic limit of the $‘s is not irreducible. This is due to the process c -+ 0. In such a case we meet bound states. In other words, when there are states, we should introduce the asymptotic fields of the composite particles, Bi*, all the asymptotic fields form an irreducible operator ring. Then, in the right hand (5.211, we have terms which contain Ri”. The coefficients of these terms can be cal-
358
UMEZAWA,
terms of normal products
TAKAHASHI,
AND
KAMEFUCHI
of gin as l)(x)
= Z”“(f,
g)lp
+ . .. )
(5.21)
where the expansion coefficients depend on the parameters (f, g) and Z(f, g) is the renormalization constant. The operators J, are in this way defined in the in-field representation. Up to here we have considered the system in the finite volume. For example, $, 9, and A(x:M) in (5.18) are of the forms: #(xl
= F IL&, Wb,
4(x> = 7 rj(k, Wb, A(x:M)
= -+
F
1 dkoei”%(k,,k,,
+ M2)t(ko).
Substituting (5.21) into # in EM], we have many terms containing more than three operators. Let us now consider the limit V --) co. Then, many of the representations specified by (f, g) turn out to be mutually inequivalent. Among terms in the Hamiltonian we shall consider, as an example, a term of the form of
Now, Theorem 5 in Section II implies that this is decomposed into several bilinear terms and c-number terms in the limit V + 03. In this way we see that the V-limit of X[#] contains only the bilinear terms of the in-field operators and c-number terms. Collecting all the bilinear terms, we obtain s A$%&
+ m + w,
9) + +Xf,
g)dP,
(5.23)
where the Lorentz invariance of the theory is taken into account. The c-numbers F(f, g) and G(f, g) depend on the parameters (f, g). Since (5.23) should coincide with (5.16), we obtain the equations f = w,
s>,
g = G(f, s),
(5.24)
which fix the values of parameters (f, g) and thus the value of mass M (cf. (5.12)). The relations in (5.24) in the case of the Nambu-Heisenberg model lead to (4.11) and (4.12), when we make the approximation of taking only the culated, when the coefficients of terms containing @‘* only are known. The can be calculated by solving (5.18) by means of the successive substitution. cussion was given in ref. 8.
latter coefficients A detailed dis-
MASS
LEVELS
AND
BROKEN
359
SYMMETRY
first term in (5.21) into account and of taking the renormalization constant 2 to be one. We can also handle the problem in a different way when we start with the expansion (5.21)) leaving the coefficients undetermined.6 Substituting (5.21) into the field equation (5.17) and collecting the terms with the same powers of #in, we obtain the equations (Y,& + m + W,
g) + G(f, .. . , .. .
g)dP
= 0, ( 5.25)
Here dots stand for equations which determine the coefficients of the normal products of higher power in #in in (5.21). As an example we shall consider the Nambu-Heisenberg Model. The field equation is (Y&4 + ml+ Substituting
= -N($4>
.ti -
(5.26)
(ih6~h~l.
(5.21) into (5.26) we obtain
( -yrd, + r?z)p
= - XZ[( p%p)p
-
(py@p)y6p]
+ *. . .
(5.27)
We shall disregard the terms denoted by dots and approximate 2 by unity. Rewriting the equation in terms of normal products by means of the relations i$in~6$in = i:@ny6@n: + C, , etc.,
(5.28)
and collecting the terms linear in tiin, we get (-I$, which
+ m + xc, + ixc,ys)p”
= 0,
(5.29)
leads to f = xc,,
g = xc,.
(5.30)
These relations together with (3.48) and (3.49) give us the equations (4.11) and (4.12). This line of argument can be formulated by means of the Feynman diagram, where each internal line represents the propagator defined by (3.41) and the external line $@. The interaction Hamiltonian contains the counter terms - (f@P -I- ig~‘“/k’“> . It is the essential point of the theory that the propagator and the in-field operators depend on the parameters (f, g). Applying the Dyson method, we can obtain the mass shift term (i.e., bilinear term) and the renormalization constants Z(f, g). The mass shift term has the form Wf,
g)+P
+ dG(f,
(5.31)
g)PWin
s A similar approach has also been proposed by Ezawa and Kikkawa cation).
(private
communi-
360
UMEZAWA,
TAKAHASHI,
AND
KAMEFUCHI
with Wf,
s> = w,
s> - f,
Since f and g should be so chosen that
sG(.f, g> = Wf, s> - g.
(5.32)
6F = 8G = 0, we come to the relations in
(5.24). The relation (5.2) states that the V-limit of the Hamiltonian is just the free Hamiltonian. Yet we cannot drop Xint in (5.1) from the theory, when we want to calculate the reaction cross sections. This is due to the following reason. To deal with reactions we need to calculate the transition matrix T whose V-limit is zero : V-lim
T = 0,
(5.33)
or lim (n:in 1 T 1m:in)
= 0.
V-W
The cross section u~+~ is finite, however,
since
u~+~(Y 1(n:in 1 T 1m:in)
l*Vz,
(5.34)
where I( >0) depends on the number of particles participating in the reaction. In other words, to calculate the cross section we should consider, not the V-limit of the Xi,,[+‘“] itself, but the V-limit of Xint[$in], multiplied by a suitable power of the volume V. Summarizing, the mass values can be calculated by the V-limit of the Hamiltonian, while the evaluation of cross sections needs the V-limit of interaction terms multiplied by suitable powers of V. Let us close this section by noting that, when we deal with fields with various degrees of freedom such as isospin, the parameters f and g are not necessarily c-numbers but matrices in the space of isospin and some other degrees of freedom. APPENDIX.
We shall derive the relations
CALCULATION
(2.57),
(2.58),
OF
(2.59))
Cr
and (2.60). Our notation
is: ukr: wave function of particle with momentum k and positive (r = 1) or negative (T = 2) helicity, 4: wave function of antiparticle with momentum k and positive (T = 1) or negative (r = 2) helicity, and
MASS
LEVELS
AND
BROKEN
361
SYMMETRY
&=[uifJi 1. 0
(A.3)
0
Taking the direction of k as the third axis, we have the relations: (ir,k, ( -ir,k,
+ nL)ukT = 0,
iz;3ulcr= crUkr,
+ nL)vkr = 0,
&VkT =
u;,,
= C,&:s
1 { -1
l.A.5)
)
where C is the charge conjugation matrix, w = dk2 + m2 and er =
(A.41
- Ervkr,)
for for
I%Z the four vector (k, iw) with r=l, r=2.
In deriving (A.4), we have taken into account the fact that the wave function vkr of the antiparticle with energy momentum (Ic,) is a solution of the Dirac equation whose energy momentum is ( --k,). For & and & , we have the relations 23 ul,k = - ETUTk )
23 v; = ETVLk )
(A.7)
since the helicity state changes when the direction of the momentum is inverted. Let us now make use of the representations of Dirac matrices in which 74
=
p3
.
(A.81
In such representations we have c = pz 22.
(A.91
Define the zero spinors Uk3= Uk4= Vk3= Vk4= 0,
(A.lO)
so that the superscript r runs from 1 to 4. Then u;,, , v;,, and z&,* are given by ul,o =
&a =
1
(--ir,k,
+ m> -1
ar
d2w(w + m) 1 ~4(hb
dWw
+
m)
1+ m> 2
(A.ll)
, a?
1
VLR,== which give
+ m> [
d24w
Y4 P222zi
(A.12)
,
1
7
l2r
(A.13)
362
UMEZAWA,
a;,, =
TAKAHASHI,
1
AND
1 + Y4 ___ (--Y/Au 2
2/2W(W + m)
KAMEFUCHI
+ m>
1 r(l
(A.14)
,
(A.15)
It is then easy to show that
?,i~k7rU,l$r =
r mw/
for
r = 1,
0
for
~=Ys,
i 1
for
r=y4,
L(~/u)E~
To evaluate im we need to introduce tion. Let us consider two cases: (i> (ii)
for a further
Y =
ZP2,
Y&i =
y=
--I;1
Ys
=
I (A.17)
I r = ~~7~.1 specialization
-/a,
74
=
P3,
-/32,
Y4
=
P3
of the representa(A.18)
*
(A.19)
The results are, in the case (i) , -k/o
tikklrxk =
--ie' 0 I -(m/w)cr
r=l, for r=iy6, for r = y4, for r = y4y6, for
(A.20)
and, in the case (ii),
(A.21)
In the discussionsin the text use is made of the representation (i) . However, all the formulas of case (ii) can be obtained from those of the case (i) by simply replacing qp(k, T) by co(k, r) + (r/2). ACKNOWLEDGMENT
The authors are indebted of us (Y.T.) acknowledges sity of Tokyo. RECEIVED:
to Dr. H. Ezawa and Dr. B. Misra for the hospitality extended to him during
June 25,1963
helpful discussions. One his visit to the Univer-
MASS
LEVELS
AND
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SYMMETRY
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Note added in proof: Concerning the matter of V-limit, the following remark will prove useful to clarify the situation. What our theorems l-5 really imply is the fact that the interaction between separate particles disappears at the limit V + 00 if the wave function is so normalized that the total probability of finding a particle in an in&de volume is unity. The proof of the theorems employed in our paper does not make sense, however, since the plane wave cannot be normalized in an infinite volume. Consequently, they serve only for a heuristic purpose. In order to overcome this difheulty, we reformulated the theory in terms of wave packets and found that our theory can be presented without use of the theorems 1-5 to find a set of inequivalent representations in which the Hamiltonian is bilinear and diagonal. All the calculations presented in the paper still hold true in the lowest approximation in our new formulation. This point will be discussed in a separate paper. REFEREXCES 1. Y. NAMBU AND G. JONA-LASINIO, Phys. Rev. 122, 345 (1961); ibid. 134, 246 (1961); Y. XAMBU AND D. LURIE, Phys. Rev. 126,1429 (1962). 2. Y. TAKAHASHI AND H. UMEZAWA, to be published. 8. L. VAN HOVE, Physica 18, 145 (1952); A. S. WIGHTMAN AND S. S. SCHWEBER, Phys. Rev. 98, 312 (1955); R. HAAG, Kgl. Danske Videnskab Selskab, Mat. Fys. Medd. 29, No. 12 (1955); H. EZAWA, to be published. 4. J. SCHWINGER, Phys. Rev. 76, 790 (1949). 6. R. HAAG, Nuovo Cimento (10) %, 287 (1962). 6’. W. HEISENBERG, Z. Naturjorsch. 14a, 441 (1959). Earlier papers are quoted there. 7. H. EZAWA, K. KIKKAWA, AND H. UMEZAWA, Nuovo Cimento (10) 23, 751 (1962). 8. H. EZAWA, K. KIKKAWA, AND H. IJMEZAWA, Nuovo Cimento (10) 26, 1141 (1962); H. EZAWA, T. MUTA, AND H. UMEZAWA, Progr. Theoret Phys. (Kyoto), in press.