We note the semi-group property of the evolution operator

U (t, t ) U (t , t ) = U (t, t ) (3.10)

and the unitarity of the evolution operator, U † (t, t ) = U ’1 (t, t ), as a state vec-

tor has the scalar product with itself of modulus one enforced by the probability

interpretation of the state vector. As a consequence, U † (t, t ) = U (t , t).

Exercise 3.1. Show that

U † (t, t ) ≡ [U (t, t )]† = T e dt H(t)

t ¯¯

˜i , (3.11)

t

˜

where the anti-time-ordering symbol, T , orders the time sequence oppositely as com-

pared with the time-ordering symbol, T , as the adjoint inverts the order of a sequence

of operators. Use this (or the unitarity of the evolution operator, I = U † (t, t ) U (t, t ))

to verify

‚U † (t, t ) ‚U (t, t )

= U † (t, t ) H(t) ,

’i ’i = U (t, t ) H(t ) . (3.12)

‚t ‚t

The dynamics of a mixture is described by the time dependence of the statistical

operator which, according to Eq. (3.6), is

ρ(t) = U (t, t ) ρ(t ) U † (t, t ) (3.13)

and the statistical operator satis¬es the von Neumann equation

dρ(t)

i = [H(t), ρ(t)] . (3.14)

dt

A diagonal element of the statistical operator, ψ|ρ(t)|ψ , gives the probability for

the occurrence of the arbitrary state |ψ at time t (explaining the use of the word

density matrix).

An important set of mixtures in practice for an isolated system (i.e. the Hamilto-

nian is time independent) is the stationary states in which all physical properties are

time independent. The statistical operator is thus for stationary states a function of

the Hamiltonian of the system, ρ = ρ(H).

For an isolated system, the evolution operator takes the simple form

U (t, t ) = e’

i

(t’t )H

. (3.15)

The generator of time displacements is the only operator in the Heisenberg picture

which, in general, is time independent, and the quantity it represents we call the

energy. For an isolated system, the Hamiltonian thus represents the energy.

3.1.2 The Heisenberg picture

Instead of having the dynamics described by an equation of motion for a state vector

or realisticly by a statistical operator, the Schr¨dinger picture discussed above, it

o

3.1. Quantum dynamics 57

is convenient to transfer the dynamics to the physical quantities, resembling in this

feature the dynamics of classical physics. In this so-called Heisenberg picture, the

state of the system

≡ U † (t, tr ) |ψ(t) = |ψ(tr )

|ψH (3.16)

is time independent, according to Eq. (3.3) and Eq. (3.12), whereas the operators

representing physical quantities are time dependent

AH (t) ≡ U † (t, tr ) A U (t, tr ) . (3.17)

At the arbitrary reference time, tr , the two pictures coincide, the evolution operator

satisfying U (t, t) = 1.

We note that if {|a }a is the set of eigenstates of the operator A then the Heisen-

berg operator has the same spectrum but di¬erent eigenstates

|a, t ≡ U † (t, tr ) |a .

AH (t) |a, t = a |a, t , (3.18)

The operator representing a physical quantity in the Heisenberg picture satis¬es

the equation of motion7

‚AH (t)

= [AH (t), HH (t)] ,

i (3.19)

‚t

where

HH (t) ≡ U † (t, tr ) H(t) U (t, tr ) . (3.20)

Introducing the ¬eld in the Heisenberg picture

ψH (x, t) = U † (t, tr ) ψ(x) U (t, tr ) (3.21)

we obtain its equation of motion

‚ψH (x, t)

= [ψH (x, t), H(t)] .

i (3.22)

‚t

Often context allows us to leave out the subscript, writing ψ(x, t) = ψH (x, t).

In the Heisenberg picture, only the equal time anti-commutator or commutator,

for fermions or bosons, respectively, of the ¬elds is in general a simple quantity, of

course the c-number function:

[ψ(x, t), ψ † (x , t)]s = δ(x ’ x ) . (3.23)

At unequal times, the anti-commutator or commutator of the ¬elds are, owing to

interactions, complicated operators whose unravelling will be done in terms of the

correlation functions of the ¬elds, the Green™s functions we introduce in the next

section.

7 Ifthe Schr¨dinger operator is time dependent, such as can be the case for the current operator

o

in the presence of a time-dependent vector potential representing a classical ¬eld, of course an

additional term appears.

58 3. Quantum dynamics and Green™s functions

Exercise 3.2. Show that the probability density for a particle to be at position x

at time t

n(x, t) = Tr(ρ(t) ψ † (x) ψ(x)) (3.24)

can be rewritten in terms of the ¬elds in the Heisenberg picture

n(x, t) = Tr(ρ ψ † (x, t) ψ(x, t)) , (3.25)

where ρ is an arbitrary statistical operator at the reference time where the two

pictures coincide.

For an isolated system, where the Hamiltonian is time independent, the quantum

¬eld (or any other) operator in the Heisenberg picture is related to the operator in

the Schr¨dinger picture in accordance with Eq. (3.17), which in that case becomes

o

(the coincidence with the Schr¨dinger picture is chosen to be at time t = 0)

o

ψ(x) e’

i i

Ht Ht

ψ(x, t) = e . (3.26)

Exercise 3.3. Show that the time evolution of a free ¬eld in the Heisenberg picture

speci¬ed by the free or kinetic energy Hamiltonian in Eq. (2.21), is given by

ap (t) = ap e’

i

pt

, (3.27)

where p = p2 /2m is the kinetic energy of the free particle with momentum p, and

the coincidence with the Schr¨dinger picture is chosen to be at time t = 0.

o

Show the commutation relations for the free ¬elds is

1

†

e p·(x’x )’ p (t’t ) .

i i

[ψ0 (x, t), ψ0 (x , t )]s = (3.28)

Vp

For the case of an isolated system of particles interacting through an instanta-

neous two-particle interaction, Eq. (2.59), the Hamiltonian transformed according to

Eq. (3.17) can be expressed in terms of the ¬elds in the Heisenberg picture

2

1 ‚

†

HH (t) = dx ψH (x, t) ψH (x, t)

2m i ‚x

1 † †

dx dx ψH (x, t) ψH (x , t) V (2) (x, x ) ψH (x , t) ψH (x, t) (3.29)

+

2

and according to Eq. (3.19), H(t) = H, i.e. the Hamiltonian in the Heisenberg picture

is the Hamiltonian, representing the energy of the system.

Our interest shall be the case of non-equilibrium situations where a system is

coupled to external classical ¬elds, for example the coupling of current and density

of charged particles to electromagnetic ¬elds as represented by the Hamiltonian

2

1 ‚

†

’ eA(x, t)

HA,φ (t) = dx ψH (x, t) + eφ(x, t) ψH (x, t) , (3.30)

2m i ‚x

3.1. Quantum dynamics 59

where the quantum ¬elds are in the Heisenberg picture.

Considering the case of two-particle interaction, and using the operator identities

[A, BC] = [A, B]C ’ B[C, A] = {A, B}C ’ B{C, A} (3.31)

for bose or fermi ¬elds, respectively, and their commutation relations, the equation

of motion for the ¬eld in the Heisenberg picture becomes

‚ψ(x, t)

dx V (2) (x, x ) ψ † (x , t)ψ(x , t) ψ(x, t) ,

i = h(t) ψ(x, t) + (3.32)

‚t

where h = h(’i∇x , x, t) is the free single-particle Hamiltonian, which can be time-

dependent due to external classical ¬elds. For example, for the case of a charged

particle coupled to an electromagnetic ¬eld

2

1 ‚

h(’i ∇x , x, t) = ’ eA(x, t) + e•(x, t) . (3.33)

2m i ‚x

The dynamics of a system, speci¬ed by the time dependence of the quantum ¬eld in

the Heisenberg picture, is thus described in terms of higher-order expressions in the

¬eld operators.

Exercise 3.4. Multiply Eq. (3.32) from the left by ψ † (x, t), and obtain also the

adjoint of this construction. Obtain the continuity or charge conservation equation

in the multi-particle space

‚ n(x, t)

+ ∇x · j(x, t) = 0 , (3.34)

‚t

where

n(x, t) = ψ † (x, t) ψ(x, t) (3.35)

and

ψ † (x, t) ∇x ψ(x, t) ’ (∇x ψ † (x, t)) ψ(x, t)

j(x, t) = (3.36)

2mi

are the probability current and density operators on the multi-particle space in the

Heisenberg picture.

Exercise 3.5. Show that the commutation relation for the displacement ¬eld oper-

ator in the Heisenberg picture at equal times is

‚uβ (x , t)

= i δ±β δ(x ’ x )

u± x, t), ni M (3.37)

‚t ’

re¬‚ecting the canonical commutation relations of non-relativistic quantum mechanics

for the position and momentum operators of the ions in a lattice.

Exercise 3.6. Show that the phonon ¬eld in the Heisenberg picture satis¬es the

equal-time commutation relation (neglecting the ultraviolet or Debye cut-o¬, ωD ’

∞)

c2

‚φ(x , t)

= ’i x δ(x ’ x ) .

φ(x, t), (3.38)

‚t 2

’

60 3. Quantum dynamics and Green™s functions

Exercise 3.7. Show that, for an isolated system of identical particles interacting

through an instantaneous two-body interaction, V (x ’ x ), the ¬eld operator in the

Heisenberg picture, say in the momentum representation, satis¬es the equation of

motion (recall Exercise 2.10 on page 44)

dap (t) dq

dp V (’q) a† +q (t) ap (t) ap+q (t) .

i = p ap (t) + (3.39)

p

(2π )3

dt

Show that the Hamiltonian in the Heisenberg picture can be expressed in the

form

1 dap (t)

† †

H(t) = p ap (t) ap (t) + i ap (t) . (3.40)

2p dt

Exercise 3.8. Obtain the equation of motion for the electron and phonon ¬elds in

the Heisenberg picture for the case of longitudinal electron“phonon interaction.

Any property of a physical system is expressed in terms of a correlation function

of ¬eld operators taken with respect to the state in question. In Section 3.3, we turn

to introduce these, the Green™s functions. But ¬rst, we will take a short historical

detour.

3.2 Second quantization

Quantum ¬eld theory, as presented in the previous chapters, is simply the quantum

mechanics of an arbitrary number of particles. For the non-relativistic case the

practical task was to lift the N -particle description to the multi-particle state space.

Quantum ¬elds are often referred to as second quantization, which in view of our

general introduction of quantum ¬eld theory for many-body systems is of course a

most unfortunate choice of language. The misnomer has its origin in the following

analogy.

Consider the Schr¨dinger equation for a single particle, say in a potential

o

2

‚ψ(x, t) 1 ‚

i = + V (x, t) ψ(x, t) . (3.41)

‚t 2m i ‚x

Next, interpret the equation as a classical ¬eld equation ` la Maxwell™s equations. A

a

di¬erence is, of course, that the ¬eld is complex, and in the case of the electromagnetic

¬eld there are additional ¬eld components. The Schr¨dinger equation, Eq. (3.41),

o

can be derived from the variational principle

δ dt dx L = 0 , (3.42)

where the Lagrange density is

2

‚ψ(x, t)

—

∇x ψ — (x, t) · ∇x ψ(x, t) ’ ψ — (x, t) V (x, t) ψ(x, t).

L = ψ (x, t) i ’

‚t 2m

(3.43)

3.2. Second quantization 61

The conjugate ¬eld variable is then

‚L

= i ψ — (x, t)

Π(x, t) = (3.44)

‚ ‚ψ(x,t)

‚t

in analogy with the canonical momenta in classical mechanics

‚L

p= (3.45)

™

‚x

and the variables of the ¬eld, x, is in the analogy equivalent to the labeling, i, of the

mechanical degrees of freedom, and Π(x) corresponds to pi .

Analogous to Hamilton™s function

pi xi ’ L

H= ™ (3.46)

i

enters the Hamilton function for the classical Schr¨dinger ¬eld

o

‚ψ(x, t)

H= ’L

dx dt Π(x, t) . (3.47)

‚t

In analogy with the canonical commutation relations

[pi , xj ] = δij , [xi , xj ] = 0 , [pi , pj ] = 0 (3.48)

i

the quantum ¬eld theory of the corresponding species is then obtained from the

classical Schr¨dinger ¬eld by imposing the quantization relations for the quantum

o

¬elds (not distinguishing them in notation from their classical counterparts)

δ(x ’ x )

[Π(x, t), ψ(x , t)] = (3.49)

i

and

[ψ(x, t), ψ(x , t)] = 0 , [Π(x, t), Π(x , t)] = 0 . (3.50)

Since according to Eq. (3.45), Π(x, t) = i ψ † (x, t), these are the commutation rela-

tions for a bose ¬eld, Eq. (1.101) and Eq. (1.102). The Hamiltonian, Eq. (3.47), is seen

to be identical to the Hamiltonian operator on the multi-particle space, Eq. (2.19).

In this presentation the bose particles emerge as quanta of the ¬eld in analogy to

the quanta of light in the analogous second quantization of the electromagnetic ¬eld

(recall Section 1.4.2). The quantum ¬eld theory of fermions is similarly constructed

as quanta of a ¬eld, but this time anti-commutation relations are assumed for the

¬eld.8

8A practicing quantum ¬eld theorist need thus not carry much baggage, short-cutting the road

by second quantization.

62 3. Quantum dynamics and Green™s functions

3.3 Green™s functions

An exact solution of a quantum ¬eld theory amounts, according to Eq. (3.32), to

knowing all the correlation functions of the ¬eld variables; needless to say a mis-

sion impossible in general. We shall refer to these correlation functions generally as

Green™s functions.9 We shall also use the word propagator interchangeably for the

various types of Green™s functions.

To get an intuitive feeling for the simplest kind of Green™s function, the single-

particle propagator, consider adding at time t1 a particle in state p1 to the arbitrary

state |Ψ(t1 ) , i.e. we obtain the state a† 1 |Ψ(t1 ) , which at time t has evolved to

p

the state

(t) = e’iH(t’t1 ) a† 1 |Ψ(t1 ) = e’iHt a† 1 (t1 ) |ΨH ,

|Ψp1 (3.51)

p p

t1

where in the last equality we have introduced the creation operator and state vector

in the Heisenberg picture (choosing the time of coincidence with the Schr¨dinger

o

picture at time tr = 0). Similarly, we could consider the state where a particle at

time t1 is added in state p1 . The amplitude for the event that the ¬rst constructed

state is revealed in the other state at the arbitrary (and irrelevant) moment in time

t is then

(t) = ΨH |ap1 (t1 ) a† 1 (t1 ) |ΨH

Ψp1 t1 (t)|Ψp1 (3.52)

p

t1

and the single-particle Green™s function is a measure of the persistence, in time span

|t1 ’ t1 |, of the single-particle character of the excitation consisting of adding a

particle to the system (or determining the persistence of a hole state when removing

a particle upon the interchange a ” a† ).

3.3.1 Physical properties and Green™s functions

Physical quantities for a many-body system such as the average (probability) density

of the particles or the average particle (probability) current density are speci¬ed in

terms of one-particle Green™s functions. For a system in an arbitrary state described

by the statistical operator ρ, the average density at a space-time point for a particle

species of interest is (recall the result of Exercise 3.2 on page 58, which amounted to

employing the cyclical property of the trace)

†

n(x, t) = Tr(ρ ψH (x, σz , t) ψH (x, σz , t)) , (3.53)

σz

where the quantum ¬eld describing the particles ψH (x, σz , t) is in the Heisenberg

picture with respect to the arbitrary Hamiltonian H(t), and Tr denotes the trace in

the multi-particle state space of the physical system in question. The reference time

where the Schr¨dinger and Heisenberg pictures coincide is chosen as the moment