Lorentz invariance of the current operators implies that they satisfy the
well-known commutation relations with the representation operators of the
Lorentz group. It is shown that if the standard construction of the current
operators in quantum field theory is used then the commutation relations are
broken by the Schwinger terms.Comment: 7 pages, LaTe

Lev, Felix M.

English

arXiv

Lorentz invariance of the current operators implies that they satisfy the well-known commutation relations with the representation operators of the Lorentz group. It is shown that if the standard construction of the current operators in quantum field theory is used then the commutation relations are broken by the Schwinger terms

Lev, F M

CERN Document Server

THE PROBLEM OF CONSTRUCTING THE
CURRENT OPERATORS IN QUANTUM FIELD
THEORY
Felix M. Lev
Laboratory of Nuclear Problems, Joint Institute for Nuclear Research,
Dubna, Moscow region 141980 Russia (E-mail: lev@nusun.jinr.dubna.su)
Abstract
Lorentz invariance of the current operators implies that they sat-
isfy the well-known commutation relations with the representation
operators of the Lorentz group. It is shown that if the standard con-
struction of the current operators in quantum eld theory is used then
the commutation relations are broken by the Schwinger terms.
PACS numbers: 03.70+k, 11.10, 11.30-j, 11.40.-q
In any relativistic quantum theory the system under consideration is de-
scribed by some unitary representation of the Poincare group. The electro-
magnetic or weak current operator J^(x) for this system (where  = 0; 1; 2; 3
and x is a point in Minkowski space) should satisfy the following necessary
conditions.
Let U^ (a) = exp({P^a) be the representation operator corresponding to
the displacement of the origin in spacetime translation of Minkowski space by
the four-vector a. Here P^ = (P^ 0; P^) is the operator of the four-momentum,
P^ 0 is the Hamiltonian, and P^ is the operator of ordinary momentum. Let
also U^ (l) be the representation operator corresponding to l 2 SL(2; C). Then
U^ (a)−1J^(x)U^(a) = J^(x− a) (1)
U^(l)−1J^(x)U^(l) = L(l) J^
(L(l)−1x) (2)
where L(l) is the element of the Lorentz group corresponding to l and a sum
over repeated indices ;  = 0; 1; 2; 3 is assumed.
1
As follows from Eq. (1)
J^(x) = exp({P^x)J^(0)exp(−{P^x) (3)
Therefore, if the operators P^ are known, the problem of constructing J^(x)
can be reduced to that of constructing the operator J^(0) with the correct
properties.
Let M^ (M^ = −M^) be the representation generators of the Lorentz
group. Then, as follows from Eq. (2), Lorentz invariance of the current
operator implies
[M^ ; J^(0)] = −{(gJ^(0)− gJ^(0)) (4)
where g is the metric tensor in Minkowski space.
When the S-matrix is calculated in the framework of perturbation theory,
Eq. (3) is used explicitly while Eq. (4) is not. It is shown instead (see, for
example, Refs. [1, 2]) that Lorentz invariance can be ensured by using the
appropriate denition of the T-product (taking into account the Schwinger
terms [3]). However when we consider electromagnetic or weak properties
of strongly interacting systems, perturbation theory does not apply and,
for consistency, Eqs. (1) and (2) must be satised. One usually believes
that, although the actual construction of the operator J^(x) in QCD is very
dicult technical problem, in principle QCD makes it possible to construct
this operator in such a way that Eqs. (1) and (2) will be satised. The
purpose of the present paper is to pay attention to the fact that the standard
procedure of constructing the current operators in quantum eld theory leads
to the operators which do not satisfy Eq. (2).
In the standard formulation of quantum eld theory the operators P^; M^
are given by
P^ =
Z
T^ d(x); M^ =
Z
M^d(x) (5)
where T^  and M^

 are the energy-momentum and angular momentum ten-
sors and d(x) = (x −  )d4x is the volume element of the space-like
hypersurface dened by the time-like vector  (2 = 1) and the evolution
parameter  . The initial commutation relations for the eld operators are
given on the hypersurface (x).
For simplicity we will consider the well-known case of QED where the
interaction Lagrangian is given by Lint(x) = −eJ^(x)A^(x), e is the electron
2
charge and A^(x) is the operator of the photon eld. Then it is easy to show
(see, for example, Refs. [4, 5]) that
P^  = P  − 
Z
Lint(x)(x−  )d
4x;
M^ = M −
Z
Lint(x)(x
 − x)(x−  )d4x (6)
where we use P  and M to denote the four-momentum operator and the
generators of the Lorentz group in the case when all interactions are turned
o.
It is well-known (see, for example, Refs. [4, 6]) that the operator J^(x)
in QED is given by
J^(x) =
1
2
[ ^ (x); γ ^(x)] (7)
where  ^(x) is the Heisenberg operator of the electron-positron eld. If x = 0
this operator coincides with the free operator  (0) in the interaction repre-
sentation and therefore, as follows from Eq. (7), J^(0) = J(0) where J(x)
is the current operator in the free theory.
The most often considered case is  = 0,  = (1; 0; 0; 0). Then (x −
 )d4x = d3x and the integration in Eq. (6) is taken over the hyperplane
x0 = 0. Therefore, as follows from Eq. (6), P^ = P, and, as follows from
Eq. (3), the operator J^(0;x)  J^(x) is free, i.e. J^(x) = J(x). It is also
obvious that A^(x) = A(x).
Since J(0) satises Eq. (4) if M^ = M , it follows from the second
expression in Eq. (6) that J^(0) will satisfy Eq. (4) ifZ
xiA(x)[J
(x); J(0)]d3x = 0 (i = 1; 2; 3) (8)
for all .
It is well-known that if the standard equal-time commutation relations
are used naively then the commutator in Eq. (8) vanishes for all ;  and
therefore this equation is satised. However the famous Schwinger result [3]
is
[J i(x); J0(0)] = C
@
@xi
(x) (9)
where C is some (innite) constant. Therefore Eq. (8) is not satised and
the current operator J^(x) does not satisfy Lorentz invariance.
3
Let us now consider the following question. While the arguments given
in Ref. [3] prove that the commutator in Eq. (9) cannot vanish, one might
doubt whether the singularity of the commutator is indeed given by the right
hand side of this expression. However it is easy to show that only this form of
the commutator is compatible with the continuity equation @J^(x)=@x = 0.
Indeed, as follows from this equation and Eq. (3), [J^(0); P^] = 0. Since
J^(0) = J(0) and J(0) satises the condition [J(0); P] = 0, it follows
from Eq. (6) that the continuity equation is satised only ifZ
A(x)[J
(x); J0(0)]d3x = 0 (10)
In turn this relation is satised only if [J0(x); J0(0)] = 0 and Eq. (9) is valid,
since div(A(x)) = 0.
As pointed out by Dirac [7], any physical system can be described in
dierent forms of relativistic dynamics. By denition, the description in the
point form implies that the operators U^ (l) are the same as for noninteracting
particles, i.e. U^ (l) = U(l) and M^ = M , and thus interaction terms can be
present only in the four-momentum operators P^ (i.e. in the general case P^  6=
P  for all ). The description in the instant form implies that the operators of
ordinary momentum and angular momentum do not depend on interactions,
i.e. P^ = P, M^ = M (M^ = (M^23; M^31; M^12)), and therefore interaction
terms may be present only in P^ 0 and the generators of the Lorentz boosts
N^ = (M^01; M^02; M^03). In the front form with the marked z axis we introduce
the + and - components of the four-vectors as x+ = (x0 + xz)=
p
2, x− =
(x0 − xz)=
p
2. Then we require that the operators P^+; P^ j; M^12; M^+−; M^+j
(j = 1; 2) are the same as the corresponding free operators, and therefore
interaction terms may be present only in the operators M^−j and P^−.
In quantum eld theory the form of dynamics depends on the choice of
the hypersurface (x). In particular, it is clear from the above consideration
that the choice  = 0,  = (1; 0; 0; 0) leads to the instant form [7]. The front
form can be formally obtained from Eq. (6) as follows. Consider the vector
 with the components
0 =
1
(1− v2)1=2
; j = 0; 3 = −
v
(1− v2)1=2
(j = 1; 2) (11)
Then taking the limit v ! 1 in Eq. (6) we get
P^  = P  − !
Z
Lint(x)(x
+)d4x;
4
M^ = M −
Z
Lint(x)(x
! − x!)(x+)d4x (12)
where the vector ! has the components !− = 1, !+ = !j = 0. It is obvious
that the generators (12) are given in the front form and that’s why Dirac [7]
related this form to the choice of the light front x+ = 0.
By analogy with the above consideration it is easy to show that the stan-
dard light front quantization also leads to the current operator which does
not satisfy Eq. (2). Let us also note that if the theory should be invariant
under the space reflection or time reversal, the corresponding representation
operators in the front form U^P and U^T are necessarily interaction dependent
(this is clear, for example, from the relations U^PP+U^
−1
P = U^TP
+U^−1T = P^
−)
and the operator J^(0) should satisfy the conditions
U^P (J^
0(0); J^(0))U^−1P = U^T (J^
0(0); J^(0))U^−1T = (J^
0(0);−J^(0)) (13)
Therefore it is not clear whether these conditions are compatible with the re-
lation J^(0) = J(0). However this diculty is a consequence of the diculty
with Eq. (2) since, as noted by Coester [8], the interaction dependence of the
operators U^P and U^T in the front form does not mean that there are discrete
dynamical symmetries in addition to the rotations about transverse axes.
Indeed, the discrete transformation P2 such that P2 x := fx0; x1;−x2; x3g
leaves the light front x+ = 0 invariant. The full space reflection P is the
product of P2 and a rotation about the 2-axis by . Thus it is not an
independent dynamical transformation in addition to the rotations about
transverse axes. Similarly the transformation TP leaves x+ = 0 invariant
and T = (TP )P2R2().
The fact that the choice J^(0) = J(0) is incompatible with Lorentz
invariance was pointed out by several authors investigating relativistic ef-
fects in nuclear physics (see, for example, Refs. [9, 10]). In our opinion this
problem is in fact algebraic. Indeed, in quantum eld theory the standard
quantization procedure implies that J^(0) = J(0) while some of the opera-
tors M^ contain interaction terms. Since there is no reason to believe that
they commute with J(0), it is reasonable to conclude that Eq. (4) is not
satised and therefore Eq. (2) is not satised too.
At the same time, if J^(0) = J(0) then Eq. (4) is obviously satised in
the point form. In Ref. [7] the point form was related to the hypersurface
t2 − x2 > 0; t > 0, but as argued by Sokolov [11] the point form should
5
be related to the hyperplane orthogonal to the four-velocity of the system
under consideration. In view of the above discussion this problem deserves
investigation.
Let us summarize our discussion. Lorentz invariance of the operator
J^(x) implies that the commutation relations (2) are satised. However if
the standard construction of J^(x) in quantum eld theory is used then the
inevitable presence of the Schwinger terms leads to the conclusion that these
relations are not satised.
Acknowledgments
The author is grateful to F.Coester, S.B.Gerasimov and O.Yu.Shevchenko
for valuable discussions. This work was supported by grant No. 93-02-3754
from the Russian Foundation for Fundamental Research.
References
[1] J.D.Bjorken, Phys. Rev. 147, 1467 (1966).
[2] C.Itzykson and J.-B.Zuber. Quantum Field Theory (McGraw-Hill Book
Company, New York 1982).
[3] J.Schwinger, Phys. Rev. Lett. 3, 296 (1959).
[4] A.I.Akhiezer and V.B.Berestetsky. Quantum Electrodynamics (Nauka,
Moscow 1969).
[5] N.N.Bogolubov and D.V.Shirkov. Introduction to the Theory of Quan-
tized Fields (Nauka, Moscow 1976).
[6] J.D.Bjorken and S.D.Drell. Relativistic Quantum Fields (McGraw-Hill
Book Company, New York 1976).
[7] P.A.M.Dirac, Rev. Mod. Phys. 21, 392 (1949).
[8] F.Coester. Private Communication of May 23, 1995.
[9] I.L.Grach and L.A.Kondratyuk, Yad. Fiz. 39, 316 (1984).
6
[10] P.L.Chung, F.Coester, B.D.Keister and W.N.Polyzou, Phys. Rev. C37,
2000 (1988).
[11] S.N.Sokolov, in Problems in High Energy Physics and Quantum Field
Theory (IHEP, Protvino, 1984) p. 85.
7


The problem of constructing the current operators in quantum field theory

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The Problem of Constructing the Current Operators in Quantum Field Theory

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