’∞ (2π ) 2πi ”t

and we thus get for the propagator

N +1 m (xn ’xn ’1 )2

N

’V (xn ,tn )

i

1 dxn ”t 2”t

K(x, t; x , t ) = lim e n =1

’d/2 ’d/2

N ’∞ m m

n=1 2πi ”t

2πi ”t

xt =x

t

dt L(xt ,xt ,t)

i ¯ ¯ ™¯ ¯

≡ Dxt e , (A.9)

t

¯

xt =x

Appendix A. Path integrals 505

where L in the continuum limit is seen to be Lagrange™s function

1

mx2 ’ V (xt , t) = xt · pt ’ H(xt , pt , t)

™ ™t ™

L(xt , xt , t) = (A.10)

2

related to Hamilton™s function through a Legendre transformation. The integration

measure has here been obtained for the case where we take the piecewise linear

approximation for a path.2

Instead of formulating quantum dynamics in terms of operator calculus we have

thus exhibited it in a way revealing the underlying superposition principle, viz. ac-

cording to Feynman™s principle: each possible alternative path contributes a pure

phase factor to the propagator, exp{iS/ }, where

t

¯ ¯ ™¯ ¯

S[xt ] = dt L(xt , xt , t) (A.11)

t

is the classical action expression for the path, xt , in question.3

The classical path is determined by stationarity of the action

δS

=0 (A.12)

δxt xt =xc l

t

the principle of least action,4 or explicitly through the Euler-Lagrange equations

d ‚L ‚L

’ =0 (A.13)

™

dt ‚x ‚x

the classical equation of motion.

Formulating quantum mechanics of a single particle as the zero-dimensional limit

of quantum ¬eld theory amounts to focussing on the correlation functions of, for

example, the position operator in the Heisenberg picture, say the time-ordered cor-

relation function

GH (t, t ) ≡ T (ˆ H (t) xH (t ))

ˆ (A.14)

x

where the bracket refers to averaging with respect to some state of the particle, pure

or mixed, say for the ground state

GH (t, t ) = ψ0 |T (ˆ H (t) xH (t ))|ψ0 .

ˆ (A.15)

x

2 Other measures can be used, such as expanding the paths on a complete set of functions, so

that the sum over all paths becomes the integral over all the expansion coe¬cients.

3 In classical mechanics only the classical paths between two space-time points in question are

of physical relevance; however, stating the quantum law of motion involves all paths. The way in

which the various alternative paths contribute to the expression for the propagator was conceived

by Dirac [152], who realized that the conditional amplitude for an in¬nitesimal time step is related

to Lagrange™s function, L, according to

i ”t L(x,(x’x )/”t)

x, t + ”t|x , t ∝e

however, with L expressed in terms of the coordinates at times t and t + ”t. This gem of Dirac™s

was turned into brilliance by Feynman.

4 Or principle of stationary action, but typically the extremum is a minimum.

506 Appendix A. Path integrals

Noting, by inserting complete sets of eigenstates for the Heisenberg operators,

xH (t) |x, t = x |x, t , we have, for ti < t, t < tf ,

ˆ

xt f =xf

tf

dt L(xt ,xt ,t)

¯ ¯ ™¯ ¯

i

xf , tf |T (ˆ H (t) xH (t ))|xi , ti = Dxt xt xt e

ˆ , (A.16)

x ti

¯

xt i =xi

where on the right-hand side the order of the real position variables xt and xt is

immaterial since the path integral automatically gives the time-ordered correlation

function due to its built-in time-slicing de¬ning procedure (recall Eq. (A.9)). We

therefore have

xt f =xf

tf

dt L(xt ,xt ,t)

¯ ¯ ™¯ ¯

i

—

Dxt xt xt e

GH (t, t ) = dxf dxi ψ0 (xf ) ψ0 (xi ) (A.17)

ti

¯

xt i =xi

and equivalently for any number of time ordered Heisenberg operators, thereby rep-

resenting any time-ordered correlation function on path integral form.

Exercise A.1. Derive for a particle in a potential the path integral expression for

the imaginary-time propagator (consider the one-dimensional case for simplicity for

a start)

x( /kT )=x

G(x, x , /kT ) ≡ G(x, ’i /kT ; x , 0) = x|e’H/kT |x Dx„ e’SE [x„ ]/

ˆ

=

x(0)=x

(A.18)

where the Euclidean action

/kT

SE [x„ ] = d„ LE (x„ , x„ )

™ (A.19)

0

is speci¬ed in terms of the Euclidean Lagrange function

1

mx2 + V (x„ ) ,

LE (x„ , x„ ) =

™ ™„ (A.20)

2

where the potential energy is added to the kinetic energy.

Interpreting „ as a length, we note that the Euclidean Lagrange function LE

equals the potential energy of a string of length L ≡ /kT and tension m, placed in

the external potential V , and we have established that the imaginary-time propagator

is speci¬ed in terms of the classical partition function for the string.

In general, only for the case of a quadratic Lagrange function, i.e. for homoge-

neous external ¬elds, can the path integral for the propagator be performed, or rather

simply circumvented by shifting the variable of integration to that of the deviation

from the classical path, x(t) = xcl (t) + δxt , and recalling that the action is stationary

for the classical path, leading to

i

Scl (x,t;x ,t )

K(x, t; x , t ) = A(t, t ) e , (A.21)

Appendix A. Path integrals 507

i.e. speci¬ed in terms of the action for the classical path and a prefactor, the con-

tribution from the Gaussian ¬‚uctuations around the classical path, which can be

determined from the initial condition for the propagator Eq. (1.15).

Exercise A.2. Obtain the expression for the propagator, T ≡ t ’ t ,

mω imω

(x2 + x 2 ) cos ωT ’ 2xx

K(x, t, x , t ) = exp

2πi sin ωT 2 sin ωT

t t

2x 2x

dtf (t) sin(ω(t ’ t )) + dtf (t) sin(ω(t ’ t))

¯¯ ¯ ¯¯ ¯

+

mω mω

t t

t t

2

’ dt1 f (t2 )f (t1 ) sin(ω(t ’ t2 )) sin(ω(t1 ’ t ))

dt2

m2 ω 2 t t2

(A.22)

for a forced harmonic oscillator

1 1

mx2 ’ mω 2 x2 + f (t) xt

L(xt , xt , t) =

™ ™t (A.23)

t

2 2

by evaluating the classical action.

Consider a particle coupled weakly to N other degrees of freedom, i.e., linearly to

a set of N harmonic oscillators collectively labeled R = (R1 , R2 , ..., RN ). The total

Lagrange function, L = LS + LI + LE , is then

N

1 1 ™2

= mx2 ’ V (x, t) m± R± ’ mω± R±

22

LS ™ , LE = , (A.24)

2 2 ±=1

where the particle in addition is coupled to an applied external potential, V (x, t),

and the linear interaction with the environment oscillators is speci¬ed by

N

LI = ’x »± R± . (A.25)

±=1

At some past moment in time, t , the density matrix is assumed separable,

ρ(x, R, x , R , t ) = ρS (x, x ) ρE (R, R ), i.e. prior to that initial time the particle

did not interact with the environment of oscillators, the system and the environment

are uncorrelated. The equation, Eq. (3.13), for the density matrix speci¬es, by trac-

ing out the oscillator degrees of freedom, the density matrix for the particle at time

t in terms of its density matrix at the initial time according to

ρ(xf , xf , t) = dxi dxi J(xf , xf , t; xi , xi , t ) ρ(xi , xi ) (A.26)

508 Appendix A. Path integrals

and the propagator of the particle density matrix is

(1) (2)

xt =xf xt =xf

(2) (1)

i

(1) (2) (1) (2)

Dxt Dxt e F [xt , xt ] (A.27)

(S(xt )’S(xt ))

J(xf , xf , t; xi , xi , t ) = ¯ ¯

¯ ¯ ¯ ¯

(1) (2)

xt =xi xt =xi

where S is the action for the particle in the absence of the environment, and the

so-called in¬‚uence functional F is

R(t)=Rf Q(t)=Rf

(1) (2)

F [xt , xt ] DR(t) DQ(t)

= dRf dRi dRi ρE (Ri , Ri )

R(t )=Ri Q(t )=Ri

i (2) (1)

— SI [xt , R(t)] + SE [R(t)] ’ SI [xt , Q(t)] ’ SE [Q(t)]

exp ,

(A.28)

where SE is the action for the isolated environment oscillators, and SI is the action

due to the interaction, analogous to the external force term in Eq. (A.23) upon the

substitution f ’ » x for each of the couplings to the oscillators.

Assuming that the initial state of the oscillators is the thermal equilibrium state,

ρE (R, R ) = ± ρT (R± , R± ), the equilibrium density matrix is immediately obtained

from Eq. (1.21) and the result of Exercise A.2, in the absence of the force, as it is

obtained upon the substitution t ’ t ’ ’i /kT in Eq. (A.22), here T denotes the

temperature (or equivalently in view of Exercise A.1, the imaginary time variable

being interpreted as variable on the appendix part of the contour depicted in Figure

4.4)

m± ω ± ω±

ρT (R± , R± ) = exp ’ ’ 2R± R±

2 2

(R± + R± ) cosh

ω kT

2 sinh kT±

1/2

m± ω ±

— . (A.29)

ω

2π sinh kT±

The path integrals with respect to the oscillators are immediately obtained using the

result of Exercise A.2, and the remaining three ordinary integrals in Eq. (A.28) are

Gaussian and can be performed giving for the in¬‚uence functional

t t2

i

(1) (2) (2) (1) (2) (1)

F [xt , xt ] = dt1 [xt2 ’ xt2 ] D(t2 ’ t1 ) [xt1 + xt1 ]

exp dt2

¯ ¯

t t

t t2

1 (2) (1) (2) (1)

’ dt1 [xt2 ’ xt2 ] DK (t2 ’ t1 ) [xt1 ’ xt1 ]

dt2 , (A.30)

t t

Appendix A. Path integrals 509

where

»2 ω±

DK (t ’ t ) = cos(ω± (t ’ t ))

±

coth

2m± ω± 2kT

±

1 ˆ ˆ

»2 {R± (t), R± (t )}

= (A.31)

±

±

and

»2 1 ˆ ˆ

D(t ’ t ) = sin(ω± (t ’ t )) =

±

»2 [R± (t), R± (t )] (A.32)

±

2m± ω±

± ±

speci¬es the non-Markovian dynamics of the oscillator through a systematic dissi-

pative or friction term and the kinetic Green™s function, Eq. (A.31), describing the

¬‚uctuation e¬ects of the environment, the two physically distinctly di¬erent terms

being related by the ¬‚uctuation“dissipation relation.5 The in¬‚uence functional is

also immediately obtained by observing that the expression in Eq. (A.28) can be

put on contour form by letting the time variable reside on the contour depicted in

Figure 4.4, noting the force is vanishing on the appendix part of the contour. We

then obtain the exponent of the form as in Eq. (9.38), and by combining the retarded

and advanced terms the form in Eq. (A.30). This observation accounts for the iden-

ti¬cation in terms of the operator expressions for the thermal equilibrium oscillator

Green™s functions in Eq. (A.31) and Eq. (A.32).6

Introducing a continuum of oscillators and the coupling in such a way that the

spectral weight function of the oscillators

»2

δ(ω ’ ω± ) = i DR (ω) ’ DA (ω)

±

J(ω) = π = D(ω) (A.33)

2m± ω±

±

becomes the linear or Ohmic spectrum

= · ω θ(ωc ’ ω)

J(ω) (A.34)

™

the friction term becomes local as D(t) = ’· δ(t) in the limit of a large cut-o¬

’1

frequency, i.e. for times much larger ωc . We then obtain for the propagator of the

density matrix for the particle

i

Dx(t) Dy(t) exp

J(xf , xf , t; xi , xi , t ) = (S1 + S2 ) , (A.35)

5 Instead of brute force, the result follows straightforwardly from the expresssion for the generating

functional for a harmonic oscillator, Eq. (4.108), and handling the linear coupling according to

Eq. (9.41) and Eq. (9.27).

6 Essential for the structure in the expression in Eq. (A.30) is only that the coupling to the

environment oscillators is linear. The non-equilibrium, i.e. driven, spin-boson problem, representing

for example a monitored qubit coupled to a decohering dissipative environment, is discussed in

reference [153].

510 Appendix A. Path integrals

(2) (1) (2) (1)

where xt = (xt + xt )/2 and yt = xt ’ xt , and

t t

dt1 yt2 DK (t2 ’ t1 ) yt1

S2 = i dt2 (A.36)

t t

and (up to a boundary term which vanishes for initial and ¬nal states satisfying

y(t) = 0 = y(t ), which will be assumed in the following)

t

S1 = ’ dt yt m¨t + · xt + VR (xt + yt /2) ’ VR (xt ’ yt /2)

¯ ¯ x¯ ™¯ (A.37)

¯ ¯ ¯ ¯

t

where the Ohmic spectrum guarantees a friction force proportional to the velocity.

For the chosen type of coupling, the potential is the result of the interaction renor-

malized by a harmonic contribution, VR (x) = V (x) ’ ωc ·x2 /π. We have arrived at

the Feynman“Vernon path integral theory of dissipative quantum dynamics for the

case of an Ohmic environment [154, 155, 156, 157].

If the external potential is at most harmonic, the path integral with respect to

yt is Gaussian and can be performed giving an expression for the path probability

analogous to Eq. (12.9). We therefore obtain that the dissipative dynamics of the

quantum oscillator is a Gaussian stochastic process described by the Langevin equa-

tion, Eq. (12.1), however the noise is not just the classical thermal one of Eq. (12.2),

but includes the quantum noise due to the environment as the stochastic force is

described by the correlation function

ωc

dω ’iωt K ω

DK (t) = DK (ω) = · ω coth

e D (ω) , . (A.38)

2π 2kT

’ωc

The time scale of the correlations in the environment, tc , the measure of the non-

Markovian character of the dynamics, is set by the temperature according to

∞ ∞

’ t2

2 K

dt DK (t) = 1 . (A.39)