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Lecture X 29

Lecture X: Feynman Path Integral

Although the second quantisation provides a convenient formulation of many-body systems,it admits solution only for systems that are effectively free. In our choice of applications,we were careful to consider only those systems for which interaction effects could be consid-ered as small, e.g. large spin in quantum magnetism of weak interaction in the dilute Bosegas. Yet interactions can have a profound effect leading to transitions to new phases withelementary excitations very different from the bare particles. To address such phenom-ena, it is necessary to switch to a new formulation of quantum mechanics. However, todo so, it will be necessary to leave behind many-body theories and return to single-particlesystems.

Motivation:

Alternative formulation of QM (cf. canonical quantisation) Close to classical construction i.e. semi-classics easily retrieved Effective formulation of non-perturbative approaches Prototype of higher-dimensional field theories

Time-dependent Schrodinger equation

it| = H|Formal solution: |(t) = eiHt/|(0) =

n

eiEnt/|nn|(0) Time-evolution operator

|(t) = U(t, t)|(t), U(t, t) = e i H(tt)(t t) N.B. Causal Real-space representation:

(q, t) q|(t) = q|U(t, t) dq|qq|

|(t) = dq U(q, t; q, t)(q, t),where U(q, t; q, t) = q|e i H(tt)|q(t t) propagator or Green function

it H

U(t t) = i(t t) N.B. t(t t) = (t t)

Physically: U(q, t; q, t) describes probability amplitude for particle to propagatefrom q at time t to q at time t

Construction of Path Integral

Feynmans idea: separate time evolution into N discrete time steps t = t/N

eiHt/ = [eiHt/]N

Lecture Notes October 2005

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Lecture X 30

Then separate the operator content so that momentum operators stand to the left

and position operators to the right:

eiHt/ = eiTt/eiVt/ + O(t2)

qF|[eiHt/]N|qI qF|eiTt/eiVt/ . . . eiTt/eiVt/|qI

Inserting at resol. of id. =

dqn

dpn|qnqn|pnpn|, and using q|p = 12

eiqp/,

eiVt/|qnqn|pnpn|eiTt/ = |qneiV(qn)t/qn|pneiT(pn)t/pn|,and

pn+1

|qn

qn

|pn

=

1

2

eiqn(pnpn+1)/

qF|eiHt/|qI =N1

n=1qN=qF,q0=qI

dqn

Nn=1

dpn2

exp

i

t

N1n=0

V(qn) + T(pn+1) pn+1 qn+1 qn

t

qI

qF

t n

p

N1 2

Phase

Space

t

i.e. at each time step, integration over the classical phase space coords. (qn, pn)

Contributions from trajectories where (qn+1 qn)pn+1 > are negligible motivates continuum limit

qF|eiHt/|qI =

q(t)=qF,q(0)=qI

D(q, p) N1n=1

qN=qF,q0=qI

dqn

Nn=1

dpn2

exp

i

t0

dt t

N1n=0

(

H(q, p|t=tn) V(qn) + T(pn+1)

pq|t=tn pn+1

qn+1 qnt

)

Hamiltonian formulation of Feynman Path Integral:Propagator expressed as functional integral

qF|eiHt/|qI =q(t)=qF,q(0)=qI

D(q, p)exp i

Action t0

dtLagrangian

(pq H(p, q))

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