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’∞ (2π ) 2πi ”t

and we thus get for the propagator
N +1 m (xn ’xn ’1 )2
’V (xn ,tn )
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
dt L(xt ,xt ,t)
i ¯ ¯ ™¯ ¯
≡ Dxt e , (A.9)

xt =x
Appendix A. Path integrals 505

where L in the continuum limit is seen to be Lagrange™s function
mx2 ’ V (xt , t) = xt · pt ’ H(xt , pt , t)
™ ™t ™
L(xt , xt , t) = (A.10)
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
¯ ¯ ™¯ ¯
S[xt ] = dt L(xt , xt , t) (A.11)

is the classical action expression for the path, xt , in question.3
The classical path is determined by stationarity of the action

=0 (A.12)
δxt xt =xc l

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)
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)
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 ,
xt f =xf
dt L(xt ,xt ,t)
¯ ¯ ™¯ ¯
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
dt L(xt ,xt ,t)
¯ ¯ ™¯ ¯

Dxt xt xt e
GH (t, t ) = dxf dxi ψ0 (xf ) ψ0 (xi ) (A.17)
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„ ]/
where the Euclidean action
SE [x„ ] = d„ LE (x„ , x„ )
™ (A.19)

is speci¬ed in terms of the Euclidean Lagrange function
mx2 + V (x„ ) ,
LE (x„ , x„ ) =
™ ™„ (A.20)
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
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
’ dt1 f (t2 )f (t1 ) sin(ω(t ’ t2 )) sin(ω(t1 ’ t ))
m2 ω 2 t t2


for a forced harmonic oscillator
1 1
mx2 ’ mω 2 x2 + f (t) xt
L(xt , xt , t) =
™ ™t (A.23)
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
1 1 ™2
= mx2 ’ V (x, t) m± R± ’ mω± R±
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
LI = ’x »± R± . (A.25)

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)
(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 ,


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

m± ω ± ω±
ρT (R± , R± ) = exp ’ ’ 2R± R±
2 2
(R± + R± ) cosh
ω kT
2 sinh kT±

m± ω ±
— . (A.29)
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
(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

»2 ω±
DK (t ’ t ) = cos(ω± (t ’ t ))
2m± ω± 2kT

1 ˆ ˆ
»2 {R± (t), R± (t )}
= (A.31)

»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

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

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)
S1 = ’ dt yt m¨t + · xt + VR (xt + yt /2) ’ VR (xt ’ yt /2)
¯ ¯ x¯ ™¯ (A.37)
¯ ¯ ¯ ¯
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
dω ’iωt K ω
DK (t) = DK (ω) = · ω coth
e D (ω) , . (A.38)
2π 2kT

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)

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