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)

2π

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)

©

2π

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