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7/28/2019 lec11.ps
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Lecture XI 32
Lecture XI: Statistical Mechanics and Semi-Classics
Connection of Path Integral to Classical Statistical Mechanics
Consider flexible string held under constant tension and confined to gutter potential
x
u
V(u)
Potential energy stored in spring due to line tension:
from segment x to x + dx, dVT = T
extension [(dx2 + du2)1/2 dx] T dx (xu)2/2
VT[xu]
dVT =1
2
L0
dx T (xu(x))2
External (gutter) potential: Vext[u] L0
dx V[u(x)]
According to Boltzmann principle, equilibrium partition function
Z= tr eF = Du(x) exp L0
dx
T
2(xu)
2 + V(u)
cf. quantum mechanical transmission amplitude Mapping:
Z=b.c.
Dq(t) exp
i
t0
dt
mq2
2 V(q)
Wick rotation t
i
imaginary (Euclidean) time path integralt
0
idt (tq)2
0
d(q)2,
t0
idtV(q) 0
dV(q)
Z=b.c.
Dqexp
1
0
dm
2(q)
2 + V(q)
N.B. change of relative sign!
(a) Classical partition function of one-dimensional systemcoincides with quantum mechanical amplitude
Z= dqq|eiHt/|qt=i
where time is imaginary, and plays the role of temperature
Lecture Notes October 2005
7/28/2019 lec11.ps
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Lecture XI 33
More generally, path integral for d-dimensional quantum system
corresponds to classical statistical mechanics of d + 1-dimensional system
(b) Quantum partition function
Z= tr(eH) =
dqq|eH|q
i.e. Zcan be interpreted through dynamical transition amplitude q|eiHt/|qevaluated at imaginary time t = i.
(c) In semi-classical limit ( 0), PI dominated by stationary configurations of actionS[p, q] = dt(pq H(p, q))
S = S[p + p,q+ q] S[p, q]=
dt [pq+ pq ppH qqH] + O(p2, q2,pq)
=
dt [p (q pH) + q( p qH)] + O(p2, q2,pq)
i.e. Hamiltons classical e.o.m.: q = pH, p = qH with b.c. q(0) = qI, q(t) = qF(Similarly, with Lagrangian formulation : S = 0
(dtq
q) L(q, q) = 0)
qqI
qF h
1/2
t
q
q(t)
Contributions to PI from fluctuations around classical paths?
Usually, exact evaluation of PI impossible resort to approximation schemes...
Saddle-point and Stationary Phase analysis
Consider integral over single variable
I =
dz ef(z)
Integral dominated by minima of f(z); suppose unique i.e. f(z0) = 0
Lecture Notes October 2005
7/28/2019 lec11.ps
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Lecture XI 34
Taylor expand around minimum: f(z) = f(z0) + (z z0) 0 f(z0) +12 (z z0)2f(z0) +
I ef(z0)
dz e(zz0)2f(z0)/2 =
2
f(z0)ef(z0)
Example : (s + 1) =
0
dzzsez =
0
dz ef(z), f(z) = z s ln z
f(z) = 1 s/z i.e. z0 = s, f(z0) = s/z20 = 1/si.e. (s + 1) 2se
(ss ln s)
Stirlings formula
If minima not on contour of integration deform contour through saddle-pointe.g. (s + 1), s complex
What if exponent complex? Fast phase fluctuations cancellationi.e. expand around region of slowest (i.e. stationary) phase and use identity
dz eiaz2/2 =
2
aei/4
Can we apply same approach to analyse the FPI?Yes: but we we must develop new technology;
basic tool of QFT the Gaussian functional integral!
Lecture Notes October 2005