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

We note that, owing to quantum e¬ects, the noise is not white but blue
2
kT 1
D (t) = ’ 2· ωc |t|
K
, 1. (A.40)
2 πkT |t|
sinh
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
symmetries

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[φ]
δS = d1 F1 [φ] = 0. (B.2)
δφ1
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
3
‚ (x)
δS = ’ dx jμ (x) (B.4)
‚xμ
μ=0

in order to vanish for the global case considered above. If the ¬eld φ(t, x) satis¬es
the classical equation of motion
δS[•]
=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
3
‚jμ (x)
=0 (B.6)
‚xμ
μ=0


511
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
(B.10)
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
δ
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
δS[φ]
F1 [φ] = 0 (B.13)
δφ1
which leaves also the measure Dφ invariant, in which case Eq. (B.12) becomes the
Ward identity
δ
J1 F1 Z[J] = 0 (B.14)
iδJ
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,
ˆ
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)
‚t
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
o
view of propagation is not mandatory in non-relativistic quantum mechanics where the quantum
numbers describing the particle species are conserved.


513
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
o
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
ˆ
’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
propagator
’iθ(t ’ t )G(x, t; x , t )
GR (x, t; x , t ) =

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

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)
‚t
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
ˆ
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
ˆ
’ 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 )
x
1 2
ˆ p ’ eA(ˆ )
ˆ
H= (C.18)
x
2m
the imaginary-time propagator satis¬es the equation
2
‚ 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
ˆ2
ip
GR (p, t; p , t ) = ’iθ(t ’ t ) p|e’ |p
(t’t )
(C.20)
2m
0
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
ˆˆ
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’
i
p (t’t )
(C.22)
0

determined by the energy of the state in question

p2
= (C.23)
p
2m
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’
ˆ
i
|x
GR (x, t; x , t ) = H0 (t’t )
0


d/2
m 2
i m (x’x )
’iθ(t ’ t )
= e . (C.24)
t ’t
2
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
‚t
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 )
(D.1)
E
’∞

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)
E


Analogously we obtain that the advanced Green™s operator is the solution of the same
equation
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
∞ +
i0
1 (’)
dE e’
i
ˆ R(A)
ˆ R(A) Et
G (t) = GE (D.5)
2π ’∞ + i0
(’)




517
518 Appendix D. Analytic properties of Green™s functions


and the hermitian property, Eq. (C.14), leads to the relationship

GA = [GR — ]† .
ˆ ˆ (D.6)
E E

We introduce the Green™s operator
§R
ˆ
⎨ GE for mE > 0
ˆ
GE ≡ (D.7)
© ˆA
GE for mE < 0

for which we have the spectral representation
| » »|
1
ˆ
GE = = (D.8)
E’ »
ˆ
E’H »

in terms of the eigenstates, | , of the Hamiltonian
»

ˆ
H| |
= . (D.9)
» » »

The analytical properties of the retarded and advanced Green™s operators leads,
by an application of Cauchy™s theorem, to the spectral representations
∞ ˆ
dE AE
ˆ R(A) =
GE (D.10)
’∞ 2π E ’ E (’) i0
+


where we have introduced the spectral operator, the discontinuity of the Green™s
operator across the real axis
ˆ ˆE ˆE ˆ ˆ
≡ i(GR ’ GA ) = i(GE+i0 ’ GE’i0 )
AE

ˆ
= 2π δ(E ’ H) = 2π | » | δ(E ’ ») . (D.11)
»
»

Equivalently, we have the relationship between real and imaginary parts of, say,
position representation matrix elements

m GR (x, x , E )
dE
e G (x, x , E) = P
R
(D.12)
E ’E
’∞ π

and

e GR (x, x , E )
dE
m G (x, x , E) = ’P
R
. (D.13)
E ’E
’∞ π

The Kramers“Kronig relations due to the retarded propagator is analytic in the upper
half-plane.
The perturbation expansion of the propagator in a static potential is seen to be
equivalent to the operator expansion for the Green™s operator
1 1 1
ˆ
GE = = =
ˆ ˆ ˆ ˆ ˆ ˆ
(E ’ H0 )(1 ’ (E ’ H0 )’1 V )
E’H E ’ H0 + V
Appendix D. Analytic properties of Green™s functions 519


1 1
=
ˆ ˆ ˆ
1 ’ (E ’ H0 )’1 V E ’ H0

1
1 + (E ’ H0 )’1 V + (E ’ H0 )’1 V (E ’ H0 )’1 V + ...
ˆ ˆ ˆ ˆ ˆ ˆ
=
ˆ
E ’ H0

ˆ ˆ ˆˆ ˆ ˆˆ ˆˆ
= G0 (E) + G0 (E)V G0 (E) + G0 (E)V G0 (E)V G0 (E) + ... , (D.14)

where
1
ˆ
G0 (E) = (D.15)
ˆ
E ’ H0
is the free Green™s operator.
The momentum representation of the retarded (advanced) propagator or Green™s
function in the energy variable can be expressed as the matrix element
R(A)
ˆ |p
GR(A) (p, p , E) = p| GE (D.16)

of the retarded (advanced) Green™s operator
1
i0)’1
ˆ R(A) = ˆ
≡ (E ’ H +
GE (D.17)
(’)
ˆ (’) i0
E’H +


the analytical continuation from the various half-planes of the Green™s operator.
Other representations are obtained similarly, for example,

ˆ R(A)
x| GE |x .
GR(A) (x, x , E) = (D.18)

The hermitian property Eq. (D.6) gives the relationship

[GR (x, x , E)]— = GA (x , x, E — ) (D.19)

and similarly in other representations.
ˆ
Employing the resolution of the identity in terms of the eigenstates of H

ˆ | »|
I= (D.20)
»
»

we get the spectral representation in, for example, the position representation

ψ» (x)ψ» (x )
R(A)
G (x, x , E) = . (D.21)
E ’ » (’) i0
+
»

The Green™s functions thus have singularities at the energy eigenvalues (the energy
spectrum), constituting a branch cut for the continuum part of the spectrum, and
simple poles for the discrete part, the latter corresponding to states which are nor-
malizable (possible bound states of the system).
520 Appendix D. Analytic properties of Green™s functions


Along a branch cut the spectral function measures the discontinuity in the Green™s
operator
ˆ ˆ
≡ x|i(GE+i0 ’ GE’i0 )|x
A(x, x , E)

= i GR (x, x , E) ’ GA (x, x , E)

= ’2 mGR (x, x , E)


ψ» (x)ψ» (x ) δ(E ’
= 2π ») . (D.22)
»

ˆ
From the expression (P (x) = |x x|)
ˆ ˆ
A(x, x, E) = 2π T r(P (x)δ(E ’ H)) = 2π | x| |2 δ(E ’ ») (D.23)
»
»

we note that the diagonal elements of the spectral function, A(x, x, E), is the local
density of states per unit volume: the unnormalized probability per unit energy
for the event to ¬nd the particle at position x with energy E (or vice versa, the
probability density for the particle in energy state E to be found at position x).
Employing the resolution of the identity we have

δ(E ’ ≡ 2πN (E) ,
dx A(x, x, E) = 2π ») (D.24)
»

where N (E) is seen to be the number of energy levels per unit energy, and Eq. (D.24)
is thus the statement that the relative probability of ¬nding the particle somewhere
in space with energy E is proportional to the number of states available at that
energy.
We also note the completeness relation
dE
A(x, x , E) = δ(x ’ x ) (D.25)

σ

where the integration (and summation over discrete part) is over the energy spectrum.
The position and momentum representation matrix elements of any operator are
related by Fourier transformation. For the spectral operator we have (assuming the
system enclosed in a box of volume V )

x|p A(p, p , E) p |x
A(x, x , E) =
pp


1 p·x’ i p ·x
i
= e A(p, p , E) , (D.26)
V
pp

and inversely we have

= N ’1 dx dx e’ p·x+ i p ·x
i
ˆ
p|AE |p
A(p, p , E) = A(x, x , E) , (D.27)
Appendix D. Analytic properties of Green™s functions 521


where the normalization depends on whether the particle is con¬ned or not, N =
V, (2π )d .
ˆ
For the diagonal momentum components of the spectral function we have (P (p) =
|p p|)

ˆ ˆ
A(p, p, E) = 2π T r(P (p) δ(E ’ H)) = 2π | p| |2 δ(E ’ ») (D.28)
»
»

describing the unnormalized probability for a particle with momentum p to have
energy E (or vice versa). Analogously to the position representation we obtain

A(p, p, E) = 2π N (E) . (D.29)
p

We have the momentum normalization condition
§
⎨ δ(p ’ p )
dE
A(p, p , E) = (D.30)
©

δp,p
σ

depending on whether the particle is con¬ned or not.
Let us ¬nally discuss the analytical properties of the free propagator. Fourier
transforming the free retarded propagator, Eq. (C.22), we get (in three spatial di-
mensions for the pre-exponential factor to be correct), mE > 0,


’m e pE |x’x |
i
R
G0 (x, x , E) = , pE = 2mE (D.31)
2π 2 |x ’ x |

the solution of the spatial representation of the operator equation, Eq. (D.3),
2
E’ G0 (x, x , E) = δ(x ’ x ) , (D.32)
x
2m

which is analytic in the upper half plane.

The square root function, E, has a half line branch cut, which according to
the spectral representation, Eq. (D.21), must be chosen along the positive real axis,
the energy spectrum of a free particle, as we choose the lowest energy eigenvalue to
have the value zero. In order for the Green™s function to remain bounded for in¬nite
separation of its spatial arguments, |x ’ x | ’ ∞, we must make the following choice
of argument function
§√
⎨E for eE > 0

E≡ . (D.33)
©
|E| for
i eE < 0
rendering the free spectral function of the form

m sin( 1 pE |x ’ x |)
A0 (x, x , E) = θ(E) (D.34)
|x ’ x |
π2
522 Appendix D. Analytic properties of Green™s functions


and we can read o¬ the free particle density of states, the number of energy levels
per unit energy per unit volume,1
§ m
⎪ d=1
⎪ 2π 2 2 E



1 m
N0 (E) ≡ d=2 ,
A0 (x, x; E) = θ(E) (D.35)
2π 2

2π ⎪


© m√2mE
d=3
2π 2 3

where for completeness we have also listed the one- and two-dimensional cases.
The spectral function for a free particle in the momentum representation follows,
for example, from Eq. (D.28)

A0 (p, E) ≡ A0 (p, p, E) = 2π δ(E ’ p) , (D.36)

and describes the result that a free particle with momentum p with certainty has
energy E = p , or vice versa.




1 Thisresult is of course directly obtained by simple counting of the momentum states in a given
energy range, because for a free particle constrained to the volume Ld , there is one momentum
state per momentum volume (2π /L)d . However, the above argument makes no reference to a
¬nite volume.
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Index

Abrikosov ¬‚ux lattice, 461 Bose“Einstein condensation, 51, 301, 351,
Abrikosov“Shubnikov phase, 461 353
absorption vertex, 129 Bose“Einstein distribution, 73, 98
action, 306, 321, 325 bosons, 6
free, 317 Brownian motion, 194, 450
adiabatic switching, 83
canonical ensemble, 49, 70
adjoint operator, 8
grand, 50, 70
analytical continuation, 142
canonical formulation, 281
analytical properties of the free propaga-
central limit theorem, 329
tor, 521
charge imbalance, 250
annihilation operator
charge-density wave, 231
bose, 23
chemical potential, 50
fermion, 16
classical electrodynamics, 28
anti-time-ordering, 56
classical equation of motion, 310, 511
antisymmetric subspace, 9
classical ¬eld, 297
appendix contour, 91
closed contour, 85
Aristotelian dynamics, 449
closed time path, 84
auxiliary ¬eld, 453
coherence length, 231, 368
average ¬eld, 297
coherent backscattering, 384, 387
BCS-energy gap, 220 collision integral
BCS-pairing, 219 electron“phonon, 208
BCS-state, 39 electron-electron, 214
BCS-theory, 217, 219 electron-impurity, 188
Bogoliubov equations, 364 collision rate, 435
Bogoliubov“Valatin transformation, 221 commutator, 3
Boltzmann conductivity, 377 condensate density, 356
Boltzmann equation, 188 condensate wave function, 353, 363
Boltzmann factor, 323 condensation energy, 221
Boltzmann propagator, 193, 412 conductance, 159
Boltzmannian paths, 401 conductance ¬‚uctuations, 442
Boltzmannian motion, 192 conductance tensor, 159
path of, 192 conductivity, 191
bose ¬eld, 24 minimum metallic, 377
Bose function, 98 conductivity diagram, 161
Bose gas, 351 conductivity diagrams, 377
Bose“Einstein condensate, 51, 77 conductivity tensor, 158


531
532 Index


continuity equation, 13, 14, 191, 195 Drude theory, 190
contour ordered Green™s function dual space, 9
inverse free, 105 dyadic notation, 467
contour ordering, 85, 88 dynamic melting, 493
contour variable, 87 Dyson equation, 116
Cooperon, 379, 381, 390 equilibrium, 138
Cooperon equation, 403 left-right subtracted, 179
creation operator matrix, 135
bose, 23 non-equilibrium, 303
fermion, 14 Dyson equations
critical phenomena, 258, 290 non-equilibrium, 116
critical velocity, 493 Dyson™s formula, 84
current correlation function, 169 Dyson“Beliaev equation, 357
current density, 13, 40, 63, 67 Dyson“Schwinger equation, 279
current response, 155
e¬ective action, 296, 299, 323
current response function, 155
two-particle irreducible, 343
current vertex, 157
Eilenberger equations, 232
cyclotron frequency, 215
Einstein relation, 244
elastic medium, 26
d™Alembertian, 27
electric ¬eld ¬‚uctuations, 172
Debye cut-o¬, 59
electron“electron interaction
Debye model, 26
di¬usion enhanced, 436
deformation potential, 46, 410
electron“hole excitations, 112
delta functional, 314
electron“phonon interaction, 45, 46, 200,
density, 67
410
probability, 13
electron“photon Hamiltonian, 48
density matrix, 22, 48, 56
electron“photon interaction, 48
density operator, 38
emission vertex, 129
current, 40
energy, 56
density response, 153
energy gap, 220, 222
density response function, 153
energy relaxation rate, 431, 435, 436
density“density response, 434
Euclidean action, 197, 506
diamagnetic current, 40
evolution operator, 3, 55
dielectric function, 434
exact impurity eigenstates, 430
di¬usion approximation, 194
exclusion principle, 16, 18, 21
di¬usion constant, 194, 198
di¬usion equation, 196
Fermi energy, 22
di¬usion propagator, 194, 196
Fermi ¬eld, 21
Di¬uson, 197, 381, 382, 401, 431
Fermi function, 98
diluteness parameter, 351
Fermi gas, 22
dirty superconductor, 242
Fermi momentum, 37
displacement ¬eld, 26, 27, 410
Fermi sea, 22, 111
dissipation, 169
Fermi surface, 22
distribution function, 187
Fermi wavelength, 231, 232, 377
distribution functions for superconductors,
Fermi™s Golden Rule, 168
245
Fermi“Dirac distribution, 73
Index 533


fermion“boson interaction, 45, 107, 125 full, 118
fermions, 6 grand canonical, 72
Feynman rules, 113 Greater, 64
Feynman-Vernon theory, 510 imaginary time, 140

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