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dt t D (t) = , tc = ,
2·kT 2kT 2·kT
’∞ ’∞

We note that, owing to quantum e¬ects, the noise is not white but blue
kT 1
D (t) = ’ 2· ωc |t|
, 1. (A.40)
2 πkT |t|
We note that the damping term S2 in Eq. (A.36) limits the excursions of y(t).
In the high temperature limit, kT ωc , quantum excursions yt of the particle are
suppressed, and the integration with respect to y(t) is Gaussian and the remain-
ing integrand in the path integral in Eq. (A.35) is the probability distribution for
a given realization of a classical path, Eq. (12.9). The corresponding Markovian
stochastic process in the Wigner coordinate, xt , is described by the Langevin equa-
tion, Eq. (12.1), and we recover the theory of classical stochastic dynamics discussed
in Section 12.1. At high enough temperatures, all quantum interference e¬ects of
the particle are suppressed by the thermal ¬‚uctuations, and the classical dissipative
dynamics of the particle emerges. We note how potentials, alien to classical dy-
namics but essential in quantum dynamics, disappear as the classical limit emerges,
delivering only the e¬ect of the corresponding classical force, ’V (xt ).
Appendix B

Path integrals and

A virtue of the path integral formulation is that symmetries of the action easily lead
to exact relations between various Green™s functions, the Ward identities.
An in¬nitesimal symmetry transformation
φ1 ’ φ1 + F1 [φ] (B.1)
is one that leaves the action invariant, i.e.
δS = d1 F1 [φ] = 0. (B.2)
If the in¬nitesimal symmetry transformation is not global, i.e. is not a constant
in¬nitesimal, but an in¬nitesimal function of space and time, (t, x), the variation of
the action under the transformation
φ(t, x) ’ φ(t, x) + (t, x) (B.3)
will in general not vanish, but takes the form, x = (t, x) = xμ
‚ (x)
δS = ’ dx jμ (x) (B.4)

in order to vanish for the global case considered above. If the ¬eld φ(t, x) satis¬es
the classical equation of motion
=0 (B.5)
δ•(t, x)
the action is stationary with respect to arbitrary variations, and assuming (t, x) to
vanish for large arguments, a partial integration leads to the continuity equation
‚jμ (x)
=0 (B.6)

512 Appendix B. Path integrals and symmetries

and the existence of the conserved quantity, the constant of motion

Q= dx j0 (t, x) . (B.7)

A symmetry of the action thus implies a conservation law, Noether™s theorem.
Returning to the global transformation, Eq. (B.1), the measure in the path inte-
gral representation of the generating functional

Dφ ei[φ]+iφ J
Z[J] = (B.8)

changes with the Jacobian according to
δF1 [φ] δF1 [φ]
Dφ ’ Dφ Det δ12 ’ = Dφ 1 ’ + O( 2 ) (B.9)
δφ2 δφ1
and since the generating functional is invariant with respect to the transformation
Eq. (B.1), we obtain

δF1 [φ] δS[φ]
Dφ 1 ’ + O( 2 )
ei[φ]+iφ J
Z[J] = 1+ i + iJ F1 [φ]
δφ1 δφ1
and thereby

δS[φ] δF1 [φ]
Dφ ei[φ]+iφ J + J1 F1 [φ] + i =0 (B.11)
δφ1 δφ1

or equivalently
⎛⎛ ⎞ ⎞
δ δ
δS δF1 iδJ
⎝⎝ + J1 ⎠ F1 ⎠ Z[J] = 0 .
+i (B.12)
δφ1 iδJ δφ1

In the event that the transformation, Eq. (B.1), is a translation, i.e. just a ¬eld
independent constant, F1 [φ] = f1 , Eq. (B.12) simply becomes the Dyson“Schwinger
equation, Eq. (9.32), (recall also Eq. (10.42)).
The real advantage of the path integral formulation presents itself if the transfor-
mation, F1 [φ], is a symmetry of the action
F1 [φ] = 0 (B.13)
which leaves also the measure Dφ invariant, in which case Eq. (B.12) becomes the
Ward identity
J1 F1 Z[J] = 0 (B.14)
relating various Green™s functions, for example the vertex function and the one-
particle Green™s functions.
Appendix C

Retarded and advanced
Green™s functions

In this appendix we shall consider the properties of the retarded and advanced Green™s
functions for the case of a single particle. When it comes to calculations Green™s
functions are convenient, and even more so when many-body systems and their in-
teractions are considered as studied in the main text.
The retarded Green™s function or propagator for a single particle is de¬ned as
(the choice of phase factor is for convenience of perturbation expansions)
’iG(x, t; x , t ) for t ≥ t
GR (x, t; x , t ) ≡ (C.1)
0 for t < t ,
where the propagator for a single particle already was considered in Appendix A,
G(x, t; x , t ) = x, t|x , t = x|U (t, t )|x . The retarded propagator for a particle
whose dynamics is speci¬ed by the Hamiltonian H, satis¬es the equation

’H δ(x ’ x ) δ(t ’ t )
GR (x, t; x , t ) =
i (C.2)
which in conjunction with the condition

GR (x, t; x, t ) = 0 for t<t (C.3)

speci¬es the retarded propagator. The source term on the right-hand side of Eq. (C.2)
represents the discontinuity in the retarded propagator at time t = t , and is recog-
nized by integrating the left-hand side of Eq. (C.2) over an in¬nitesimal time interval
around t , and using the initial condition1

’i δ(x ’ x ) .
GR (x, t + 0; x , t ) = (C.4)
1 The retarded propagator also has the following interpretation: prior to time t the particle
is absent, and at time t = t the particle is created at point x , and is subsequently propagated
according to the Schr¨dinger equation. In contrast to the relativistic quantum theory, this point of
view of propagation is not mandatory in non-relativistic quantum mechanics where the quantum
numbers describing the particle species are conserved.

514 Appendix C. Retarded and advanced Green™s functions

Or one recalls that the derivative of the step function is the delta function. The re-
tarded Green™s function is thus the fundamental solution of the Schr¨dinger equation
and rightfully the mathematical function introduced by Green. The inverse opera-
tor to a di¬erential equation is expressed as an integral operator with the Green™s
function as the kernel. In the context of many-body theory we have used the label
Green™s in the less speci¬c sense, just referring to correlation functions.
The retarded Green™s function propagates the wave function forwards in time, as
we have for t > t for the wave function at time t

ψ(x, t) = i dx GR (x, t; x , t ) ψ(x , t ) (C.5)

in terms of the wave function at the earlier time t , and has the physical meaning of a
probability amplitude for propagating between the two space-time points in question.
According to Eq. (C.1), the retarded propagator is given by
’iθ(t ’ t ) x|U (t, t )|x .
GR (x, t; x , t ) = (C.6)
By direct di¬erentiation with respect to time it also immediately follows that the
retarded propagator satis¬es Eq. (C.2).
We note, according to Appendix A, the path integral expression for the retarded
’iθ(t ’ t )G(x, t; x , t )
GR (x, t; x , t ) =

xt =x
dt L(xt ,xt )
i ¯
’i θ(t ’ t ) Dxt e ¯ ™¯
= . (C.7)

xt =x

We shall also need the advanced propagator
0 for t > t
GA (x, t; x , t ) ≡ (C.8)
iG(x, t; x , t ) for t ¤ t ,
which propagates the wave function backwards in time, as we have for t < t for the
wave function at time t

ψ(x, t) = ’i dx GA (x, t; x , t ) ψ(x , t ) (C.9)

in terms of the wave function at the later time t .
The retarded and advanced propagators are related according to
GA (x, t; x , t ) = [GR (x , t ; x, t)]— . (C.10)
The advanced propagator is also a solution of Eq. (C.2), but zero in the opposite
time region as compared to the retarded propagator.
We note that, in the position representation, we have
ˆ = i[GR (x, t; x , t ) ’ GA (x, t; x , t )]
G(x, t; x , t ) = x|U (t, t )|x

≡ A(x, t; x , t ) , (C.11)
Appendix C. Retarded and advanced Green™s functions 515

where we now have introduced the notation A for the Green™s function G, and also
refer to it as the spectral function.
Introducing the retarded and advanced Green™s operators
ˆ ˆ ˆ ˆ
GR (t, t ) ≡ ’iθ(t ’ t ) U (t, t ) , GA (t, t ) ≡ iθ(t ’ t) U (t, t ) (C.12)

we have for the evolution operator
ˆ ˆ ˆ ˆ
U (t, t ) = i(GR (t, t ) ’ GA (t, t )) ≡ G(t, t ) ≡ A(t, t ) (C.13)

and the unitarity of the evolution operator is re¬‚ected in the hermitian relationship
of the Green™s operators
GA (t, t ) = [GR (t , t)]† .
ˆ ˆ (C.14)
The retarded and advanced Green™s operators are characterized as solutions to
the same di¬erential equation
‚ ˆˆ ˆ
’ H GR(A) (t, t ) = δ(t ’ t ) I
i (C.15)
but are zero for di¬erent time relationship.
The various representations of the Green™s operators are obtained by taking ma-
trix elements. For example, in the momentum representation we have for the retarded
propagator the matrix representation
GR (p, t; p , t ) = ’iθ(t ’ t ) p, t|p , t = p|GR (t, t )|p . (C.16)

Exercise C.1. De¬ning in general the imaginary-time propagator
’ H („ ’„ )
G(x, „ ; x , „ ) ≡ θ(„ ’ „ ) x|e |x (C.17)

show that for the Hamiltonian for a particle in a magnetic ¬eld described by the
vector potential A(ˆ )
1 2
ˆ p ’ eA(ˆ )
H= (C.18)
the imaginary-time propagator satis¬es the equation
‚ 1
∇x ’ eA(x) G(x, „ ; x , „ ) ≡ δ(x ’ x ) δ(„ ’ „ )
+ (C.19)
‚„ 2m i

and write down the path integral representation of the solution.

The free particle propagator in the momentum representation
GR (p, t; p , t ) = ’iθ(t ’ t ) p|e’ |p
(t’t )
516 Appendix C. Retarded and advanced Green™s functions

is given by

δ(p ’ p )
= GR (p, t ’ t )
GR (p, t; p , t ) = GR (p, t, t ) p|p , (C.21)
0 0 0 δp,p

where the Kronecker or delta function (depending on whether the particle is con¬ned
to a box or not) re¬‚ects the spatial translation invariance of free propagation. The
compatibility of the energy and momentum of a free particle, [H0 , p] = 0, is re¬‚ected
in the de¬nite temporal oscillations of the propagator

GR (p, t, t ) = ’iθ(t ’ t ) e’
p (t’t )

determined by the energy of the state in question

= (C.23)
the dispersion relation for a free non-relativistic particle of mass m.
Fourier transforming, i.e. inserting a complete set of momentum states, we obtain
for the free particle propagator in the spatial representation

’iθ(t ’ t ) x|e’
GR (x, t; x , t ) = H0 (t’t )

m 2
i m (x’x )
’iθ(t ’ t )
= e . (C.24)
t ’t
2π i(t ’ t )

Exercise C.2. Show that the free retarded propagator in the momentum represen-
tation satis¬es the equation

’ δ(p ’ p ) δ(t ’ t ) .
GR (p, t; p , t ) =
i (C.25)
p 0
Appendix D

Analytic properties of
Green™s functions

In the following we shall in particular consider the analytical properties of the Green™s
functions for a single particle. However, by introducing the Green™s operators, results
are taken over to the general case of a many-body system.
For an isolated system, where the Hamiltonian is time independent, we can for any
complex number E with a positive imaginary part, transform the retarded Green™s
operator, Eq. (C.12), according to

1 i
ˆ ˆ
d(t ’ t ) e GR (t ’ t ) .
GR = E(t’t )

The Fourier transform is obtained as the analytic continuation from the upper half
plane, mE > 0. According to Eq. (C.15) we have, for mE > 0, the equation

ˆˆ ˆ
E ’ H GR = I . (D.2)

Analogously we obtain that the advanced Green™s operator is the solution of the same
ˆˆ ˆ
E ’ H GE = I (D.3)

for values of the energy variable E in the lower half plane, mE < 0, and by analytical
continuation to the real axis

1 i
ˆE ˆ
GA ≡ Et
GA (t) .
dt e (D.4)

We note the Fourier inversion formulas
∞ +
1 (’)
dE e’
ˆ R(A)
ˆ R(A) Et
G (t) = GE (D.5)
2π ’∞ + i0


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