1C/66/10
INTERNATIONAL ATOMIC ENERGY AGENCY
INTERNATIONAL CENTRE FOR THEORETICAL
PHYSICS
INCONSISTENCY OF
CANONICAL COMMUTATION RELATIONS
AMONG CURRENT DENSITIES
S. OKUBO
1966
PIAZZA OBERDAN
TRIESTE
IC/66/10
INTERNATIONAL ATOMIC ENERGY AGENCY
INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS
INCONSISTENCY OF CANONICAL COMMUTATION RELATIONS
AMONG CURRENT DENSITIES*
SUSUMU
OKUBO*
TRIESTE
February 1966
+ Submitted to Nuovo Cimento
:
On leave of absence from the University of Rochester, Rochester, N.Y., U.S.A.
ABSTRACT
Ordinary commutation relations among current densities
have been shown to lead to inconsistency without assuming the
conservation law of currents.
Also it is demonstrated that the
ordinary PS-PV meson theory and also the <r-model will give rise
to similar inconsistencies in general.
-1-
INCONSISTENCY OF CANONICAL COMMUTATION RELATIONS
AMONG CURRENT DENSITIES
SI.
INTRODUCTION AND SUMMARY OF RESULTS
One promising approach for broken symmetry theory of elementary
particles is that of the algebra of currents.
Among many interesting
results thus obtained, it may suffice to mention for instance calculation
of the axial vector renormalization constant in the |3-decay by ADLER
and WEISBERGER
and a recent successful explanation of anomalous
2
magnetic moments of the nucleon by FUBINI, ROSSETTI and FURLAN .
The method of the algebra of currents is based upon equal-time
commutation relations of spatially integrated currents, which can be
easily derived if we assume special models like that of the quark model
for instance.
The reason for using the spatially integrated form rather
than the unintegrated density itself is of course in order to avoid the
possible existence of inconsistencies such as have been originally found
3
by SCHWINGER . Indeed, the last mentioned author has shown that an
equal-time commutation relation
leads to an internal inconsistency.
between electric current densities
However, we note that his argument
does not apply for the important case of axial vector currents since there
is no conservation law in general for the latter.
The purpose of this note is two-fold.
First of all, we shall show
that one can dispense with the assumption of the conservation law for
currents in order to prove the existence of an inconsistency.
our
argument
vector ones.
is
Hence,
applicable for axial vector currents as well as for
Secondly, it will be shown in Section III that ordinary
canonical quantization procedure for the pseudoscalar meson theory with
pseudovector coupling (hereafter abbreviated as PS-PV theory) will lead
to a similar inconsistency.
Analogously, the so-called or-model which
4
has been introduced in connection with the partially conserved axial
-2-
vector current hypothesis (hereafter abbreviated as P.C.A.C.) will give
rise to an inconsistency in the sense that some bare quantities such as
bare masses or charge must be infinite except for some exceptional case.
These examples may indicate that there is a difficulty in the present
quantum field theory and it will give us some warnings on indiscriminate
uses of algebra of currents, especially for unintegrated commutation
relations.
§n
INCONSISTENCY OF COMMUTATION RELATIONS AMONG
CURRENT DENSITIES
As an illustration, let us consider the quark model.
Then, for
instance, vector current densities may be given by
T^h)
?tt)y^ >* fCx)
(la)
where \a(a = l, . . , 9) are some 3 x 3 matrices defined by GELL-MANN .
Then, assuming equal-time commutation relations among <p(x) and lp(x) ,
one can formally find the following:
(2b)
SCHWINGER
3
has shown that Eq. (2a) leads to an inconsistency if one
notes the conservation law
However, his argument does not apply for axial vector commutation
-3-
relation Eq. (2b) since
a^>(x) is not in general conserved.
But we
shall prove that this conservation hypothesis is really not necessary for
proof of inconsistency, so that our argument is also applicable for
Eq. (2b).
More generally, we shall show the following:
Let jM(x)
be a local Lorentz vector satisfying a condition
j
where the bracket in the left-hand side of the above equation indicates
the vacuum expectation value and where "JJ, (x) is defined by
(4)
so that ^(x) is again a Lorentz 4-vector.
Then, we claim that the
validity of Eq. (3) for fi k 4 must imply identical vanishing of j (x)
and hence that Eq. (3) is inconsistent for non-trivial current j^(x) .
In this proof we do not assume the conservation law 9 j (x) - 0 at all
and hence our proof is applicable for axial vector current case Eq. (2l>)
as well as Eq. (2a}, or, more generally, for the case that L (x) is an
arbitrary linear combination of vector and axial vector currents.
Now let us prove our assertion.
quantities:
-4-
To this end, define the following
Utilizing the ordinary argument in derivation of the LEHMANN-KALLEN
g
representation , one finds that one can write these Green functions
into the following form:
where P1$ P2, (\ and p2 are scalar functions of m defined by
(7b)
-y\
with condition p2 + m2 = 0 and where
AJ ±J
f x - y , m) is defined by
(8)
Now, if j^(x) is self-conjugate, i.e. if j^fx) satisfies the additional
relation
as v/ a) te)
and a»(aVx) defined in Eqs. (1) do, then of course we must
-5-
have
J^)
fM
£M
= &(•")
(9)
However, to prove Eq. (9), this assumption of being self-conjugate is
actually unnecessary in view of the following well-known argument ,
The micro-causality demands that we should have
Hence, G^ (x - y) = G^ (x-y) for space-like region (x-y) 2 > 0 ,
which proves the validity of Eq. (9) again .
Therefore, one finally finds the following spectral representation
for the vacuum expectation value of the commutator:
>e = { 2
where
Eq. (11) will be our starting point. Now, before going into our proof,
we note that the spectral weights p, (m) and p2 (m) are non-negative
and satisfy the following inequality:
-6-
The first upper bound can be easily proved if we multiply p pv to both
sides of Eq. (7a) and note p 2 = -m 2 ;
i
where we used Eq. (4). One may remark'that the upper bound m 2 p (m)
= p1 (m) can be attained if and only if we have the conservation law
djj^ (x) = 0 as we shall see shortly. Next, the lower inequality can be
proved by taking v a v k 4 t say p = v - 3 , for instance, in Eq. (7a) %
Then, one gets
'
Therefore, we obtain
However, the spatial momentum p3 can be chosen arbitrarily as long
as the condition p2 + m2 = 0 is maintained. Especially, one may choose
pg = 0 to obtain px (m) s£ 0.
Now, we are in a position to prove our original statement. Suppose
that we have the equal-time commutation relation Eq. (3) for a k 4 ,
Then, noting the well-known relations
Eq. (11) leads to
-7ft &QL-3) > J
Because of the inequalities Eq. (13), this implies that we must have
-7-
Hence, from Eqs. (7) and (9), one gets
However, then the argument of JOHNSON and FEDERBUSH
implies that jjj(x) and hence j^fx) must be identically zero.
immediately
There-
fore, we conclude that the equal-time commutation relations Eqs. (2)
for ju S 4 are inconsistent.
Note that this inconsistency can be proved
only for commutation relations between the fourth and non-fourth components of currents, and we cannot prove anything in this manner for
commutation relations between two fourth component currents or among
non-fourth components of currents.
Also, one may note that if we
relax the condition Eq. (3) so that we have either of the following two
equations instead of Eq. (3):
(15)
0^
then we can no longer prove any inconsistency.
may be consistent.
Thus, these equations
This result may have some relevance in application
of algebra of currents in broken symmetry problems.
explanation of these facts is well known.
The possible
It is essentially due to the
use of mathematically undefined products of two unbounded operators
as in Eqs. (1).
The ordinary procedure
for instance, to interpret Eq. (la) as
-8-
to avoid this difficulty is,
where e is a space-like vector with no fourth component and the limit
is supposed to be taken after final calculations of commutators.
Such
a procedure will in general induce additional non-zero derivative contributions in the right-hand side of Eq. (2a), thus avoiding the inconsistency.
If one integrates once as in Eq. (15), such derivative terms will give no
contribution at all.
Hence, Eq. (15) may be consistent.
However,,
these statements are rather over-simplified and more careful discussions
are really needed on this point.
Finally, in closing this section, we remark that one can prove a
9
theorem analogous to that of COLEMAN by our method. The theorem
stated is as follows.
Let us define
= (A
for some y, k 4 where jjj(x) is a self-conjugate conserved vector
current, i . e . .
Then we assert that it is impossible to have
= O
(18)
i. e., X(t) cannot annihilate the vacuum state for an arbitrary but fixed
time t , unless j^fx) is identically zero.
To show this statement, let us suppose that Eq. (X8) holds valid,
then of course, we have
Noting Eq. (11), this means that we must have
-9-
for arbitrary x 0 .
Therefore, we must have p% (m) = 0 .
Because
of our assumption of the conservation law 9^ jM(x) - 0 , we have now
Hence,
p (m) = 0 also.
Thus,
p, (m) = p9 (m) = 0 identically, and,
2
l
repeating the previous argument,
a
j (x) must vanish identically.
Our theorem indicates that for electric current j (x) there can
exist no exact symmetry group which contains the operator
d3x jy (x) (n % 4) as one of its infinitesimal generators.
This is
because if such an exact symmetry group exists at all, then the vacuum
state must belong to a singlet representation of this group and hence
Eq. (18) must be valid - an impossibility, as has been shown above.
§HI. INCONSISTENCY OF THE PS-PV MESON THEORY
As an application of the method explained in the previous section,
we shall show here that the PS-PV meson theory leads in general to an
inconsistent result.
The Lagrangean density of this theory is given by
(19)
^a
If one wishes, we may add other terms not involving derivatives of Ta
in the right-hand side of Eq. (19).
Now, the canonical quantization
can be achieved ordinarily by defining the canonical conjugate variable
7^(x) as follows:
—
—
—»
w
fiui-f\
A
'A
-v
/u/n
\
(20)
-10-
'
Then, the canonical commutation relations are given by
(21)
Of course, in addition to Eq. (21) we ordinarily assume equal-time anticommutation relation
between fi(x) and ^(x) .
ing, we need not assume it.
However, in the follow-
Note thati commutation relations Eq. (21)
give so-called self-consistent quantization in the sense that the Hamiltonian equations based upon them reproduce ©.xactly the original Lagrangean
equation.
Now, define an axial vector current Jj/ a ' (x) by
(22)
£
However, as has been explained in the end of the previous section, we
must be careful how to define J ^ (x) since it involves a product of two
operators.
As in the previous section, we have to interpret it as
(23)
where e is a space-like vector without the time-component and the
limit € —f 0 is to be understood to be performed in the last stage of
-11-
calculation.
Now on the basis of the quantization condition Eq. (21),
one finds
, 3?k;t)l=
Hence, letting
CTV«, Tfn: 6)1=6,
ft;-Jj
e—*0 , we may write it as
f\
(*%)
(25)
Of course, Eq. (25) is immediate if we start from Eq. (22).
Now let us consider the Lehmann-Kallen representation for.commutator between J^
(x) and
4>fi(y) . Then, taking account of the
charge-independence, one may write
where we took account of the micro-causality condition.
and taking x 0
=
Setting n = 4
y0 in Eq. (26), the first relation of Eq. (25) gives us
(27)
when we note Eq. (14).
Furthermore, let us take a derivative of both
sides of Eq. (26) with respect to y0 and set xQ = y0
„
Then for
H % 4 , one finds
Henpe in view of Eq. (27), we find
] >
-12-
& ^ * +;
(28)
On the other hand, Eqs. (20) and (22) give us
Therefore, when we note the second equation of Eq. (25), we can rewrite
Eq. (28) as
However, in the previous section, we have proved that this is impossible
unless JM(a) (x) vanishes identically, i. e., g0 = 0 . Therefore, one
concludes that the commutation relation Eq. (21) or (25) for canonical
f •
p
variables of two different fields leads to an inconsistent answer unless
we have no derivative coupling.
We find that our argument is also applicable for the scalar meson
"i
theory with vector coupling. In that case, we know that we can solve
the problem exactly . However, it has been pointed out also
that
the Lehmann-Kalle'n representation may not hold in such a case because
of high singularity in Green functions. Probably, the existence of the
Lehmann-Kjalle'n representation is a very stringent condition which may
not be enjoyed in some cases.
§IV INCONSISTENCY OF THE d-MODEL
!'
This model has been investigated in connection with the partiallyconserved axial vector current hypothesis (P. C. A. C.). The Lagrangean
density of this theory is given by
V
^(t)
-13-
(30)
where, for simplicity, we did not take account of the boson-boson interactions.
Note that the meson field
<^(x) (a = 1, 2, 3) and the <r-field
cr(x) have exactly the same bare mass /Jo and the same bare coupling
constant g0 .
Actually, all quantities involving a bilinear form
0 (x) Q. ij/(x) in Eq. (30) should be interpreted as a normal product
With this understanding, the equation of motion is given by
(32)
(33)
Now let us define an axial vector current j,^
(x) by
(34)
1 %&.) T
_
Then, utilizing equations of motion Eqs. (31), (32) and (33), one finds
the P.C.A.C.
(35)
Now, in our theory, the equal-time canonical commutation relations are
-14-
given by
< 36 >
and in addition we shall have
Our proof of inconsistency of the a-model is somewhat involved and
rather similar in spirit to the attempt of KALLEN
an inconsistency in the quantum electrodynamics.
who tried to show
To that end, we
shall use again the Lehmann-Kall&i representation Eq. (11);
Then/ again, we have the inequality Eq. (13), i . e . ,
-™* &(**)&
SSM ~£ O
(39)
I
Using the P . C. A. C. Eq. (35), one finds now
2
ft*
-15-
fM^u-lw)
(40)
where p(m) is given by
Because of the inequality Eq. (39), we must have
f(-")
-2
0
(42)
Similarly t one gets
First, let us set i/ = 4 with x 0
c
y0
in Eq. (40), then noting the defi-
nition Eq. (34) together with the commutation relations Eqs.{36) and (37),
one obtains
(44>
Similarly, taking a derivative of both sides of Eq, (43) with respect to
x 0 and setting then a^ = y0
ft")
, one finds
=
/JL^.^M5"
L
do ^°
J
Finally, evaluating the quantity
at. x0 = y0
, Eq. (43) gives
where we used Eqs. (32) and (37).
Furthermore, note that the equation of motion Eq. (33) leads to
-16-
(45)
S»
(47)
Actually, the right-hand side of Eq. (47) is a little ambiguous since we
may interpret it as either —Tr S^{0) or -—Tr s£(0)
where
°
*
(48)
However, these quantities — T r s( (0) or — T r s ' (0) a r e in general
infinite as we shall see shortly.
Then, of course <cr(x)> = oo . In
4m2
• /""*
2
that case Eq. (44) demands that either —o -Po or / dm pfm) must
»0
wO
be infinite. The first case indicates that one of ir^ and n0 is infinite
unless g0 = 0 $ which is a trivial case. Thus, the theory is not
consistent unless trivial in this case. On the other hand, if
dm p(m) = + oo then noting the positiveness condition Eq. (42) for p(m),
we see that Eq. (45) demands that
^•^n2 must be infinite also. Then
again, repeating the same argument, we conclude that one of mQ and
HQ must be divergent unless we have the trivial case g = 0 , again.
In the above argument, we have assumed that <<r^x)^ is
divergent. However, this is not necessarily true always. It could
happen that <cr(x)^ may be finite. To investigate this case in further
detail, and to show that this case also leads to an inconsistency, we shall
consider the Lehmann-Kalle*n representation
(49)
-17-
where the spectral weights ? (m) and ?2 (m) must .be positive:
^O
(50)
although we shall not use this constraint.
Now, evaluating the vacuum expectation value
at x 0 = y0
, the first
equation of Eq. (49) together with Eqs. (31) and
(36) gives us
where we have set <d,(x)^
= 0 because of the parity conservation.
e, behaves as
Note that near the light cone, A (x-yf in), for instance,
The behaviour of A*(x-y, m) function is of a similar character on the
light cone.
Hence, in order to have a finite value for Tr S^ (0) or
Tr s£(0) so as to make <cr(x)^
in Eq. (47) finite, it is necessary for
us to have
t tjH -~\(^f\ « Q
if we use the Lehmann-Kalleri representation Eq. (49).
(52)
Then, Eq. (51)
simply becomes
< <5-bC)>c =
— ~-°
(53)
Inserting this into the right-hand side of Eq. (44), one finds now
Hence, taking account of the positive definiteness condition Eq. (42), we
-18-
obtain
(54)
or because of Eq. (41), this is equivalent to
As has often been remarked in the previous sections, this relation
implies the exact conservation law
-
O
(56)
because of the Johnson-Federbush argument. Then, by virtue of the
P. C.A.C. Eq. (35), we must have either of the following alternatives:
(57)
csr
However, the first case ju0 = 0 is impossible because the well-known
argument * of LEHMANN gives us n = 0 automatically, where /u
13
is the physical pion mass. Also, the same argument
demands that
we have no interaction at all: i. e., g0 = 0 . Thus, the only possibility
is that we should have m0 = 0 . Therefore, any theory with m0 ^5 0
leads to an inconsistency. For m0 = 0 we cannot say, unfortunately,
whether the itheory is still inconsistent or not. However, in that case,
4
it is well known that the Lagrangean Eq. (30) is invariant under the
following infinitesimal transformations:
-
2 &.
(58)
-19-
for arbitrary real infinitesimal numbers €a(a - 1, 2, 3).
Then, the
physical masses ofCT(X)and <f>a fx) must be exactly the same, which
is not so good from the experimental viewpoint.
ACKNOWLEDGMENTS
A part of this work has been done when the author was in the University of Rochester.
He would like to express his gratitude to Profes-
sors R.E. Marshak, R. Hagen and G. Gttralnik.
He is also grateful to
Professors Abdus Salam and P. Budini and the IAEA for the hospitality
extended to him at the International Centre for Theoretical Physics,
Trieste.
-20-
REFERENCES
1
A, ADLER, Phys. Rev. Letters 14, 1051 (1965).
W.I. WEISBERGER, ibid 14, 1047 (1965).
2
S. FUBINI, G. FURLAN and C. ROSSETTI, preprint (1965).
3
J. SCHWINGER, Phys. Rev. Letters 3, 296 (1959).
K. JOHNSON, Nucl. Phys. 2jj, 431 (1961).
4
M. GELL-MANN and M. LEVY, Nuovo Cimento 16, 705 (1960).
5
M. GELL-MANN, Phys. Rev. 12_5, 1067 (1962).
6
H. LEHMANN, Nuovo Cimento Lt, 342(1954).
G. KALLEN, Helv. Phys. Acta 23,201 (1950).
7
Here we have followed the argument given by S. S. SCHWEBER:
"An Introduction to Relativistic Quantum Field Theory", Evanston,
111., Row-Peterson (1961).
8
P . FEDERBUSH and K. JOHNSON, Phys. Rev. 120, 1926 (1960).
9
S. COLEMAN, Phys. Letters 1JJ, 144 (1965).
10
J. SCHWINGER, see reference 3 and a recent preprint (1965).
11
S. OKUBO, Nuovo Cimento ^ 9 , 574(1961).
12
G. KALLEN, Kgl. Danske Videnskab. Selskab, M a t . - F y s . Medd.,
21., 12 (1953).
F o r other references and criticisms on this paper, see a paper by
K. JOHNSON, Ann. Phys. (N. Y.) ^ 0 , 536 (1960).
13
W. S. H E L L M A N a n d P . ROMAN, Phys. Letters L3, 336 (1964)
and also see Nuovo Cimento ^ 7 , 779 (1965).
-21-
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