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{Qs,n(x)} = -n(x)T S = TSn(x)
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(1-145)
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{br:J.sQs, cP(x)}
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bcP(x)
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{br:J.sQs, n(x)} = bn(x)
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where bcP(x) and bn(x) are the variations (1-137) of the fields under infinitesimal transformations. In other words, the charges QS generate these infinitesimal transformations through Poisson brackets in the same way that the hamiltonian
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30 QUANTUM FIELD THEORY
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generates time translations. Similarly, we have
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{QSI, QS2} = -
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d 3x n(x)[PI, P2] <fJ(X) = - CS,S2S3 QS3
(1-146)
as should obviously be the case. The invariance of the lagrangian was not required to derive the above relations. Indeed, if H stands for the hamiltonian we easily derive that
dQS off at (t) = {H(t), QS(t)} = f d x oba (x, t)
(1-147)
We can generalize Eq. (1-146) to the equal-time Poisson brackets of the time components of the currents with the following result:
Ub'(x, t),jfJ(y, t)} = -CS,S2 S3 b 3 (x - y)j'o'(x, t)
(1-148)
In their quantized version these relations will give rise to the important developments of current algebra (see Chap. 11).
The reader will find it instructive to study similarly the equal-time Poisson brackets of the energy momentum tensor: {0 00 (x, t), 0 00 (y, t)} {0 00 (x, t), 0 0k (y, t)}
= = =
[0 0k (x, t) [0
kl (X,
y) 3(x - y) t) -lI000(y, t)] ofb
+ 0 0k (y, t)] ot,P(x -
(1-149)
{0 0k (x, t), 0 01 (y, t)}
[0 0k (y, t) of
+ 0 01 (x, t) on b3 (x =
from which we recover the Lie algebra of the Poincare group:
{P", r}
{J"', pa}
p"g,a _ p"g"a
(1-150)
To close this section let us study a simple lagrangian involving two real fields <PI and <P2, which we shall describe using the complex (independent) quantities
<PI + i<P2
<PI - i<P2
<P*(x) .... e'e. <p*(x)
(1-151)
Let us assume the dynamics invariant under internal rotations in the (1,2) space or, equivalently, under (1-152)
The constant e appearing here will be identified with the elementary coupling constant to the electromagnetic field, i.e., the electric charge. For instance, Ii' assumes the form: (1-153) with Van arbitrary smooth function, for instance, a polynomial. Noether's theorem yields a conserved current (1-154)
CLASSICAL THEORY 31
We can, of course, directly verify this conservation using the equations of motion:
(D+m )</>=-o</>*
(1-155)
The charge (1-156) is therefore time independent. We now wish to couple this system to the electromagnetic field. Since we are given a conserved current we might think of using it on the right-hand side of Maxwell's equations. But some care must be exercised since the coupling of </> to the vector potential A" might modify the structure of the current itself. We seek therefore a total lagrangian given as a sum of three pieces-S!'em = _!F2, S!'scalar given by (1.153), and an interaction part S!'int(</>'</>*, A"): (1-157) Let us assume that the current 1" given by Noether's theorem applied to the full lagrangian (1-157) is indeed the electromagnetic current. In other words, we have
J" = j"
oS!'
O[O"IX(X)]
= 0 F"' = - ~ " oA,(x)
oS!'
(1-158)
It is easy to convince oneself that the following interaction:
S!'int = - ieA"</>* o"</>
+ e2A2</>*</>
(1-159)
fulfills these conditions in such a way that (1-160) The full lagrangian is therefore (1-161) leading to the coupled equations (1-162) We observe that (1-161) follows from a principle of minimal coupling to the electromagnetic field, according to which any derivative 0" acting on a charged field (with charge e) has to be replaced by the covariant derivative 0" + ieAw As a consequence, the lagrangian is not only invariant under the transformations (1-152) with constant IX but under more general x-dependent (i.e., local) gauge transformations:
</>(x) -> e-iea(x) </>(x) </>*(x) -> eiea(x) </>*(x)
(1-163)
What was initially a trick to derive Noether's theorem has now become a deeper property of electromagnetic couplings which generalizes the gauge invariance already discussed for the free electromagnetic field. This idea suitably extended to noncom mutative groups yields very interesting model field theories (see Chap. 12).
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