QUANTUM FIELD THEORY in .NET framework

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QUANTUM FIELD THEORY
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The ground state energy is given by the formula
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. = hm [ - -1 In -Z(g)]
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(9-199)
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with Z(g) expressed through (9-198). The expansion of Z(g)/Z(O) in powers of g reads Z(g) Z(O) = Zk =
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1 ~!1)k Z(O) !!2<p exp [fP dt - 0
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(9-200)
cjJZ
+ <pz + kIn (fP dt <p4)] 2 0
For large k we evaluate Zk using the steepest-descent method. We look for a saddle point <Pc(t) such that <Pc(O) = <Pc([3) and minimizing the effective action I= that is, <Pc - <Pc
dt <P ; <P - k In
(fP dt <p4 )
(9-201)
+ ~p--'--- = 0
dt' <p~(t')
4k<pg
(9-202)
Rescaling <Pc according to P
4k ]
if;(t)
(9-203)
o dt' if;4(t')
we obtain the equation (9-204) expressing the fact that if; is of order unity while <Pc grows like k 1/Z . As compared to the usual equation of motion, Eq. (9-204) exhibits two changes of sign beyond the rescaling which has eliminated the magnitude of the coupling. First, the potential, and hence the force, has reversed its sign due to the rotation to euclidean time (another way to put it is to replace t by it so that the acceleration is reversed). Second, the effective coupling is negative as shown by the relative sign between the harmonic (if;) and anharmonic (if;3) contributions. The effective potential is depicted in Fig. 9-6. The equation is translation ally invariant in time and admits the symmetry if; --+ - if;. By relabeling t we may use a symmetric interval in time [ - [3/2, [3/2] instead of [0, [3]. In the limit [3 --+ 00 the interval is infinite and the solution which minimizes the action is found to be if;r(t) = cosh (t - r)
(9-205)
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Figure 9-6 Effective potential in euclidean space V(q;) = - q;2/2 + q;4/4. The limiting motion is shown in dotted lines.
up to a global sign and with an arbitrary origin in time T. Clearly this expression fulfils the periodic boundary conditions which are here supplemented by the condition that the action be finite. We "thus have an infinity of degenerate saddle points satisfying
dt-=2
+ 00
4>2 2 dt-=2 3
(9-206)
(9-207)
In this limit the relation between
({Jc
;p reads
({Jc=~4({J
f3k-
(9-208)
Due to the degeneracy of the saddle points which follow from the symmetries of the problem, the integration over quadratic deviations from the minimum requires some care. It is proper to call it a quantization problem since we look for the quantum frequencies of oscillations around a classical extremum of the action. The question will arise whenever a classical saddle point will be the starting point of an approximation to a path integral and degeneracies arise due to a continuous invariance. One or several modes will be of zero frequency. Here they correspond to time translations and should first be isolated to yield a factor f3 after integration over T. It is, of course, crucial to keep f3 finite, albeit
QUANTUM FIELD THEORY
large, even though it may be taken infinite for those quantities which admit a finite limit. The problem bears a close relationship with the quantization of constrained systems studied in Sec. 9-3. Thus we want to treat separately the collective mode arising from translational motion. To do this we introduce in the path integral a factor unity in the form
dTb(8tp-T)
(9-209)
-P12
where 8tp is defined implicitly by the condition
dt ljJ(t - 8tp)cp(t) = 0
(9-210)
and the function ljJ is the normalized derivative of (jJ corresponding, say, to T = 0:
ljJ(t)
(f dt cp2 )1/2 "'
<p(t)
(9-211)
The reason for this choice will soon appear clear. If cp is translated by an amount T through cp(t) -4 cp,(t) = cp(t - T), this leaves the limiting action as well as the path integral measure invariant but 8tp is changed into 8tp + T. Thus the integral over T can be performed explicitly and yields, as expected, a factor [3. Then
[3 (_l)k e- I ,
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