n(x,t)=aa ( )=5::1 ( ) oCP x, t uuoCP x, t in VS .NET

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Similarly, we define the Poisson bracket of two functionals Ll and L2 of the fields cP and n at time t as follows: {Ll L2} = -{L2 Lt} ' ,
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fd3X[~ bcp(x,2t) - bcp(x, t) bn(x, t) bL ~~J bn(x, t)
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These functional derivatives are to be taken with a grain of salt. For instance, if L is expressed as a space integral over a density involving gradients of the fields, a suitable integration by parts is always understood. In particular,
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{n(x, t), cp(y, t)} {n(x, t), n(y, t)}
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(1-133)
and, for instance,
It is easy to see that the field equations (1-44) take the form:
8o<p(x, t) = {H, <p(x, t)} = -<- - H un(x, t)
(1-134)
8on(x, t)
{H, n(x, t)}
= - -<- -
u<p(x, t)
28 QUANTUM FIELD THEORY
in complete analogy with nonrelativistic dynamics, when we have identified Hamilton's function H as the integral of the energy density:
d 3 x E)00(x, t)
(1-135)
expressed in terms of <p(x, t), V<p(x, t), and n(x, t). Equation (1-134) expresses the fact that H generates time translations of the system. The reader should find it easy to generalize it to space translations and infinitesimal Lorentz transformations.
1-2-3 Internal Symmetries
Systems such as those of particle physics are endowed with internal symmetries which playa prominent part in analyzing their spectrum and interactions. As far as we know most ofthese symmetries are, however, only approximate, being broken by forces weaker than the ones under consideration. It is nevertheless useful to pretend at first that these are exact invariances and to study the pattern of violations as a second approximation. Let us briefly derive Noether's theorem in this classical context. Let 4h ... CPN stand for N interacting fields. Some of them can be real or complex and carry Lorentz indices. For each value of x we consider them as a vector in an N-dimensional space where we are given a representation of a group G. The latter can leave certain subspaces invariant (such will necessarily be the case if we have fields having inequivalent transformation properties under the Lorentz group). If G is a compact Lie group we shall write T'(s = 1, ... , r), the corresponding generators with T' antihermitian, and
(1-136)
Recall that the structure constants C S ,S2S3 can be chosen totally antisymmetric for a compact group. For brevity we do not distinguish the group from its linear realization on cp. Let us consider space-time-dependent variations of the fields
bcp(x) = bCl.s(x)T'cp(x) bcpt(x) = -bCl.s(x)cpt(x)T'
(1-137)
under which we have Sf(cp) -4 Sf(cp + bcp) = il!, where il! is a function of cp, OIlCP (and their complex conjugates), bCl.., and 0llbCl.s. A variation around the stationary solution will yield
0= M
f d x bCl..(x)
ail! ail!} obCl..(x) - all o[ollbCl..(x)]
(1-138)
We define the corresponding currents
ail! j:(x) = o[ollbCl.s(x)]
(1-139)
Only the part of the lagrangian containing field derivatives will contribute to
CLASSICAL THEORY
jf. Noether's theorem follows from (1-138) and gives the divergence of these currents as
.Jl _ 02(x) oJlJs(x) - Obr:J. s
(1-140)
where we have noted that 02(x)/Obr:J.s(X) can be identified with the same derivative for constant variations br:J.s. A current jf will be conserved if the corresponding term on the right-hand side of (1-140) vanishes, which means that the original lagrangian is invariant under the one-dimensional subgroup of G generated by 7,. Stated differently, to each such subgroup is associated a conserved charge
Qs =
d3 x
j~(x, t)
dd;s =
d3 x
o~f(x, t) = 0
(1-141)
Let us return to the general case without assuming the conservation of the currents. This does not prevent us from defining the charges Qs(t) at time t. In what follows, t will be fixed and we shall not write it explicitly. If g; stands for a functional of the fields and conjugate momenta at time t, we want to compute the Poisson bracket (1-142) The dependence of 2 on Oobr:J.s(x) arises -only from the dependence of 2 on OOcPa. To simplify, let us assume that the fields are real (in which case 7, is real and antisymmetric). If this were not the case, we would consider separately the real and imaginary parts. Then it follows that
.0 02 Js (x) = O[Oobr:J.s(X)]
QS(X) =
We have, therefore,
(1-143)
d 3 x n(x)T'cP(x)
(1-144) In particular,
{Qs, cP(x)} = T'cP(x)
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