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To first order in the loop expansion there is no wave function renormalization in the cp4 model. Nevertheless, the function ZeMCP) is nontrivial to this order even though it contains no logarithms. Using the effective action (9-115) it may be shown to be equal to
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ZeMCP)
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A2 cp2 1 + 6(4n)2 (2m2 + Ac(2)
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(9-130)
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The massless limit may be obtained as in (9-129). Calculations have been pursued to higher orders. Let us quote here the results up to second order with the corresponding diagrams. We use the shorter notations
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(4n)2
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(9-131)
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and we obtain: Diagram
~ [(1 + X)2 In (1 + x) - (x +
3;2)]
(X2 -
~X [
- [(1 8
+ x) In (1 + x) x
12 1 + x
In(l
x + x)+~- ] 1+x
x - - l n2 (1
[1 +
+ x) -
+ x)
(X2 {
-i In (1 + x) + -X1+ x
1+ x
x In (1
+ x) + 2XJ
2x ~x2
[ In(l
+ x) + 4A 3
+ x)
[In(l
+ x) + 4(A -
4 +-
x - - - + B} 3 (1 + X)2
(9-132)
Here B is a constant depending on the normalization conventions and A is given by
In u
1 - u + u2
3 ""
[1 +
~~- -
3p)2
+ 3p)2
(9-133)
The effective action discussed here is similar to the Euler-Heisenberg lagrangian of electromagnetism (Sec. 4-3-4). We leave it to the reader to speII out the details of this relationship. The S matrix has an expansi"on in powers of h similar to the one of Green's functions. Show that the two first terms for its normal kernel are given by (9-134)
QUANTUM FIELD THEORY
where 'Pclass is the solution of
+ m2 )'Pclass + V'('Pclass) =
(9-135)
with Feynman boundary conditions; 'Pclass is also given by the integral equation
'Pclass(x) = 'Pas(x) -
d yG F(X
y)V'['Pclass(Y)]
(9-136)
with 'Pas given by Eq. (9-85).
The effective action may be given a more physical definition. In particular, v"ff(CP) may be interpreted as the ground-state energy density under the constraint that the mean value of the field is equal to cP uniformly. It enables us to explore possible instabilities of the system (Chap. 11).
9-3 CONSTRAINED SYSTEMS
A number of dynamical systems may be described by observables submitted to constraints at fixed time. As a familiar instance we may quote electrodynamics. The quantization of such systems is not a straightforward matter and we shall soon encounter even more severe difficulties in the case of nonabelian gauge theories. Pa~h integrals provide an ideal framework to handle such systems because they maintain a close relationship with the classical case where a simple treatment may be given. The quantization method requires the elimination of as many pairs of canonically conjugate variables as there are constraints. The latter have to satisfy suitable compatibility conditions to be clarified below. This technical development can be skipped in a first reading and reconsidered when turning to the concrete application to gauge fields (Chap. 12), where use will be made of the result (9-159).
9-3-1 General Discussion
Let a classical system with n(n> 1) degrees of freedom be submitted first to a unique constraint
f(p, q) = 0
(9-137)
Call C the (2n - 1)-dimensional manifold in phase space characterized by (9-137). Our considerations wiII apply locally, i.e., in a neighborhood of a point belonging to C. Furthermore, we should not distinguish between two functions f and F which both vanish on C, that is, such that F(p, q) = rx(p, q)f(p, q). Let d be the ring of (differentiable) functions which vanish on C. We use the notation F ~ 0 to mean FE d. We may incorporate the constraint in the action using a time-dependent Lagrange multiplier A.(t). Equations of motion follow from the stationarity conditions of
FUNCTIONAL METHODS
dt [pq - h(p, q) - A(t)f(p, q)]
(9-138)
These include Eq. (9-137) obtained by varying A, together with
. oh of Pi= - - - A Oqi Oqi
(9-139)
Of course, on C the variables (Pi, qi) are too numerous. A natural compatibility condition is that the evolution (9-139) leaves the manifold C invariant. This reads in terms of Poisson brackets (9-140) More generally it follows that any F '" 0 will have a Poisson bracket with h belonging to .91: (9-141a) F '" 0 = {h, F} '" 0 .91 is stable under the Poisson bracket since it has a unique generator
F '" 0 = {J, F} '" 0
(9-141b)
and Eqs. (9-141a, b) imply that this ring of functions (but not necessarily an individual member) is stable under the evolution (9-139). Let F be an arbitrary fixed element in d. We define an equivalence relation E on C as follows. Consider the flow generated by F in phase space. In infinitesimal form it is described by the equations
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