where the state is speci¬ed, i.e. when the arbitrary statistical operator, ρ, repre-

senting the state of the system is speci¬ed. Here σz describes an internal degree of

9 Thus

using the notion in a broader sense than in mathematics, where it denotes the fundamental

solution of a linear partial di¬erential equation as discussed in Appendix C.

3.3. Green™s functions 63

freedom of the identical particles in question. For example, in the case of electrons

this is the spin degree of freedom, and the density is the sum of the density of elec-

trons with spin up and down, respectively.10 The average density is expressed in

terms of the diagonal element of the so-called G-lesser Green™s function

n(x, t) = ±i G< (x, σz , t, x, σz , t) , (3.54)

σz

where11

†

= “i Tr(ρ ψH (x , σz , t ) ψH (x, σz , t))

G< (x, σz , t, x σz , t )

†

≡ “i ψH (x , σz , t ) ψH (x, σz , t) , (3.55)

where upper (lower) sign corresponds to bosons (fermions), respectively, and we have

introduced the notation that the bracket means trace of the operators in question

weighted with respect to the state of the system, all quantities in the Heisenberg

picture,

. . . ≡ Tr(ρ . . .) . (3.56)

For the case of a pure state, ρ = |Ψ Ψ|, we see that G< (x, t, x , t ) is the amplitude

for the transition at time t to the state ψH (x , σz , t ) |Ψ , where a particle with

spin σz is removed at position x from state |Ψ , given the system at time t is

in the state ψH (x, σz , t) |Ψ where a particle with spin σz is removed at position

x (assuming t < t , otherwise we are dealing with the complex conjugate of the

opposite transition). Equivalently, it is the amplitude to remain in the state |Ψ after

removing at time t a particle with spin σz at position x and restoring at time t a

particle with spin σz at position x . For the case of a mixture, an additional statistical

averaging over the distribution of initial states takes place. Average quantities, such

as the probability density, can thus be expressed in terms of the one-particle Green™s

function.

The average electric current density for an assembly of identical fermions having

charge e in an electric ¬eld represented by the vector potential A is, according to

Eq. (2.47),

e ‚ ‚

’ G< (x, σz , t, x , σz , t)

j(x, t) =

2m ‚x ‚x

σz

x =x

e2

G< (x, σz , t, x, σz , t) ,

+ i A(x, t) (3.57)

m σz

the particles assumed to have an internal degree of freedom, say spin as is the case

for electrons.

10 One can, of course, also encounter situations where interest is in the density of electrons of a

†

given spin, in which case one studies n(x, σz , t) = Tr(ρ ψH (x, σz , t) ψH (x, σz , t)) .

11 The annoying presence of the imaginary unit is for later convenience with respect to the Feynman

diagram rules. However, one is entitled to the choice of favorite for de¬ning Green™s functions, and

the corresponding adjustment of the list of Feynman rules.

64 3. Quantum dynamics and Green™s functions

From the equation of motion for the ¬eld operator, Eq. (3.32), the equation

of motion for the Green™s function G-lesser becomes, for the case of two-particle

interaction (assuming spin independent interaction so the spin degree of freedom is

suppressed or using inclusive notation),

‚

’ h(t) G< (x, t, x , t ) = dx V (2) (x, x ) G(2) (x, t, x , t, x , t, x , t ) ,

i

‚t

(3.58)

where

G(2) (x, t, x , t, x , t, x , t ) = ±i ψ(x, t) ψ(x , t) ψ † (x , t) ψ † (x , t ) (3.59)

is a so-called two-particle Green™s function since it involves the propagation of two

particles. The dynamics of a system, speci¬ed by the time dependence of the one-

particle Green™s function, is thus described in terms of higher-order correlation func-

tions in the ¬eld operators. The equation of motion for the one-particle Green™s

function thus leads to an in¬nite hierarchy of equations for correlation functions con-

taining ever increasing numbers of ¬eld operators, describing the correlations set up

in the system by the interactions.12 Since there is no closed set of equations for re-

duced quantities such as Green™s functions, approximations are, in practice, needed

in order to obtain information about the system. On some occasions the system pro-

vides a small parameter that allows controlled approximations; a case to be studied

later is that of electron“phonon interaction in metals. In less controllable situations

one in despair appeals to the tendency of higher-order correlations to average out

for a many-particle system, when it comes to such average properties as densities

and currents, so that the hierarchy of correlations can be broken o¬ self-consistently

at low order. We shall discuss such situations in Section 10.6 and in Chapter 12 in

the context of applying the e¬ective action approach to such di¬ering situations as

a trapped Bose“Einstein condensate and classical statistical dynamics, respectively.

3.3.2 Stable of one-particle Green™s functions

The correlation function G-lesser appeared in the previous section most directly

as related to average properties such as densities and currents. However, we shall

encounter various types of quantum ¬eld correlation functions, i.e. various kinds of

Green™s functions that appear for reasons of their own. For de¬niteness we collect

them all here, though they are not needed until later. The rest of this chapter can

thus be skipped on a ¬rst reading if one shares the view that things should not be

called upon before needed.

We shall also encounter the so-called G-greater Green™s function

† †

G> (x, t, x , t ) = ’i ψH (x, t) ψH (x , t ) = ’iTr(ρ ψH (x, t) ψH (x , t )) , (3.60)

the amplitude for the process of an added particle at position x at time t given a

particle is added at position x at time t , the one-particle propagator in the presence

of interaction with all the other particles.

12 Analogous to the BBGKY-hierachy in classical kinetics or for any description of a system in

terms of a reduced, i.e. partially traced out, quantity.

3.3. Green™s functions 65

We shall later, for reasons of calculation in perturbation theory, encounter the

time-ordered Green™s function

†

G(x, t, x , t ) = ’i T (ψH (x, t) ψH (x , t )) (3.61)

and we note (valid for both bosons and fermions, recalling the minus sign convention

when two fermi ¬elds are interchanged)

G< (x, t, x , t ) t >t

G(x, t, x , t ) = (3.62)

G> (x, t, x , t ) t>t .

In perturbation theory, the time-ordered Green™s function appears because of the

crucial role of time-ordering in the evolution operator, Eq. (3.7). Quantum dynamics

is ruled by operators, non-commuting objects. However, as shown in Chapter 5, the

necessity of the time-ordered Green™s function is only in one version of perturbation

theory, and then an additional analytic continuation needs to be invoked. Or, if

one is interested only in ground state properties, then perturbation theory can be

formulated in closed form involving only the time-ordered Green™s function. The

general real-time perturbation theory valid for non-equilibrium situations will be

formulated in Chapter 5 in terms of essentially two Green™s functions, and in a way

which displays physical information of systems most transparently.

Finally, in this set-up we shall also later encounter the anti-time-ordered Green™s

function

†

˜ ˜

G(x, t, x , t ) = ’i T (ψH (x, t) ψH (x , t )) , (3.63)

˜

where T anti-time orders, i.e. orders oppositely to that of T . We note that the

time-ordered and anti-time-ordered Green™s functions can be expressed in terms of

G-greater and G-lesser, for example

˜

G(x, t, x , t ) = θ(t ’ t ) G< (x, t, x , t ) + θ(t ’ t) G> (x, t, x , t ) , (3.64)

where θ is the step or Heaviside function.

Recalling Eq. (3.58), we note for the free Green™s functions the relations

G’1 (x, t) G< (x, t, x , t ) = 0 G’1 (x, t) G> (x, t, x , t ) = 0

, (3.65)

0 0

0 0

and for the time-ordered

G’1 (x, t) G0 (x, t, x , t ) = δ(x ’ x ) δ(t ’ t ) (3.66)

0

and anti-time-ordered

G’1 (x, t) G0 (x, t, x , t ) = ’ δ(x ’ x ) δ(t ’ t ) ,

˜ (3.67)

0

where

‚

G’1 (x, t) = ’h ∇x , x, t

i , (3.68)

0

‚t i

66 3. Quantum dynamics and Green™s functions

which for the case of a charged particle coupled to an electromagnetic ¬eld is

2

‚ 1 ‚

G’1 (x, t) ’ ’ eA(x, t) ’ e•(x, t)

= i . (3.69)

0

‚t 2m i ‚x

Introducing

G’1 (x, t, x , t ) = G’1 (x, t) δ(x ’ x ) δ(t ’ t ) (3.70)

0 0

we obtain a quantity on equal footing with the Green™s function, the inverse free

Green™s function (here in the position representation) as

(G’1 — G0 ) (x, t, x , t ) =

ˆ ˆ δ(x ’ x ) δ(t ’ t ) , (3.71)

0

where — signi¬es matrix multiplication in the spatial and time variables, i.e. internal

integrations over space and for the latter internal integration from minus to plus

in¬nity of times.

Exercise 3.9. The equation of motion for the free phonon ¬eld is (recall Section

2.4.3)

2 φ(x, t) = 0 . (3.72)

Show that the time-ordered free phonon Green™s function

D0 (x, t, x , t ) = ’i T (φ(x, t) φ(x , t )) (3.73)

therefore satis¬es the equation of motion

2 D0 (x, t, x , t ) = ’ x ) δ(t ’ t ) .

x δ(x (3.74)

i

Exercise 3.10. From the equation of motion for the ¬eld operator, show that the

equation of motion for the time-ordered Green™s function is

‚

’ h0 (t) G(x, t, x , t ) δ(x ’ x )δ(t ’ t )

i =

‚t

’ i T ([ψ(x, t), Hi (t)] ψ † (x , t )) , (3.75)

where Hi (t) is the interaction part of the Hamiltonian in the Heisenberg picture.

Other combinations of ¬eld correlations will be of importance in Chapter 5 when

the real-time perturbation theory of general non-equilibrium states are considered,

viz. the retarded Green™s function

’iθ(t ’ t ) [ψ(x, t) , ψ † (x , t )]“

GR (x, t, x , t ) =

θ(t ’ t ) G> (x, t, x , t ) ’ G< (x, t, x , t )

= (3.76)

3.3. Green™s functions 67

and advanced Green™s functions

iθ(t ’ t) [ψ(x, t) , ψ † (x , t )]“

GA (x, t, x , t ) =

’θ(t ’ t) G> (x, t, x , t ) ’ G< (x, t, x , t )

= (3.77)

and the Keldysh or kinetic Green™s function

= ’i [ψ(x, t) , ψ † (x , t )]±

GK (x, t, x , t )

= G> (x, t, x , t ) + G< (x, t, x , t ) , (3.78)

where upper and lower signs, as usual, are for bose and fermi ¬elds, respectively.

Introducing the notation s = ’s, the two kinds of statistics can be combined leaving

¯

the Green™s functions in the forms

’iθ(t ’ t ) [ψ(x, t) , ψ † (x , t )]s

GR (x, t, x , t ) = (3.79)

¯

and

GA (x, t, x , t ) = iθ(t ’ t) [ψ(x, t) , ψ † (x , t )]s (3.80)

¯

and

’i [ψ(x, t) , ψ † (x , t )]s = GS (x, t, x , t )

GK (x, t, x , t ) = (3.81)

where the superscript on the last Green™s function also could remind us of it being

symmetric with respect to the quantum statistics.

Exercise 3.11. Show that the density, up to a state independent constant, can be

expressed in terms of the kinetic Green™s function according to

n(x, t) = ± i GK (x, σz , t, x, σz , t) . (3.82)

σz

Exercise 3.12. Show that the current density can be expressed in terms of the

kinetic Green™s function according to (in the absence of a vector potential)

e ‚ ‚

’ GK (x, t, x , t)

j(x, t) = . (3.83)

2m ‚x ‚x

x =x

The presence of a vector potential just adds the diamagnetic term (recall Eq. (3.57))

in accordance with gauge invariance, ’i ∇ ’ ’i ∇ ’ eA.

We note the relationship

GR (x, t, x , t ) ’ GA (x, t, x , t ) = G> (x, t, x , t ) ’ G< (x, t, x , t ) (3.84)

68 3. Quantum dynamics and Green™s functions

irrespective of the quantum statistics of the particles. The above combination is of

such importance that we introduce the additional notation for the spectral weight

function

= i(GR (x, t, x , t ) ’ GA (x, t, x , t )) = [ψ(x, t) , ψ † (x , t )]“

A(x, t, x , t )

= i G> (x, t, x , t ) ’ G< (x, t, x , t ) . (3.85)

We note as a consequence of the equal time anti-commutation or commutation rela-

tions of the ¬eld operators, that the spectral function at equal times satis¬es

δ(x ’ x )

A(x, t, x , t) = (3.86)

irrespective of the state of the system.

Exercise 3.13. Introduce the mixed or Wigner coordinates13

x+x

r= x’x

R= , (3.87)

2

and

t+t

, t =t’t .

T= (3.88)

2

Show that the spectral weight function expressed in these variables satis¬es the

sum-rule

∞

dE

A(E, p, R, T ) = 1 . (3.89)

2π

’∞

Exercise 3.14. Verify the relations, valid for both bosons and fermions,

—

GA (x, t, x , t ) = GR (x , t , x, t) (3.90)

and

—

GK (x, t, x , t ) = ’ GK (x , t , x, t) (3.91)

and

—

A(x, t, x , t ) = (A(x , t , x, t)) (3.92)

and

—

G< (x, t, x , t ) = ’ G< (x , t , x, t) (3.93)

and

—

G> (x, t, x , t ) = ’ G> (x , t , x, t) . (3.94)

Note the relations are valid for arbitrary states.

13 There will be more about Wigner coordinates in Section 7.2.

3.3. Green™s functions 69

For the case of a hermitian bose ¬eld, such as the phonon ¬eld, additional useful

relations exist

DR (x, t, x , t ) = DA (x , t , x, t) (3.95)

and

DK (x, t, x , t ) = DK (x , t , x, t) (3.96)

and

D> (x, t, x , t ) = D< (x , t , x, t) . (3.97)

We thus have that DR(A) (x, t, x , t ) are real functions, whereas DK (x, t, x , t ) is

purely imaginary.

Above, the Green™s function are displayed in terms of the ¬elds in the position

representation. Equally, we can introduce the Green™s function displayed in the

momentum representation, related to the above by Fourier transformation, or for

that matter in any representation speci¬ed by a complete set of states, say the energy

representation speci¬ed in terms of the eigenstates of the Hamiltonian.

Correlation functions of the quantum ¬elds can be obtained by di¬erentiation of

a generating functional. For example, to generate time-ordered Green™s functions we

introduce

∞

+ ψ † (x,t) · — (x,t))

Z[·, · — ] = T ei dx dt (ψ(x,t) ·(x,t)

(3.98)

’∞

generating for example the time-ordered Green™s function for bosons, Eq. (3.61), by

di¬erentiating twice with respect to the complex c-number source function ·,14 to

give

δ 2 Z[·, · — ]

= ’i T (ψ(x, t) ψ † (x , t )) .

G(x, t; x , t ) = i — (3.99)

δ· (x , t )δ·(x, t)

·=0=· —

The generating functional is a device we shall consider in detail in Chapter 9.

The Green™s functions introduced in this section are the correlation functions for

the case of an arbitrary state. Before we embark on the construction of the general

non-equilibrium perturbation theory and its diagrammatic representation starting