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