commutation relations (3.91) for the ¬elds and the equal-time, canonical commuta-

tors, [rni (t), plj (t)] = i δnl δij , for the charged particles. After a bit of algebra, the

Heisenberg equations are found to be

‚A (r, t)

E (r, t) = ’ , (4.33)

‚t

‚B (r, t)

∇ — E (r, t) = ’ , (4.34)

‚t

1 ‚E (r, t)

= µ0 j⊥ (r, t) ,

∇ — B (r, t) ’ 2 (4.35)

c ‚t

pn (t) ’ qn A (rn (t) , t)

drn (t)

vn (t) ≡ = , (4.36)

dt Mn

dpn (t)

= qn E (rn (t) , t) + qn vn (t) — B (rn (t) , t) ’ qn ∇¦ (rn (t)) , (4.37)

dt

where vn (t) is the velocity operator for the nth particle, j⊥ (r, t) is the transverse

part of the current density operator

j (r, t) = δ (r’rn (t)) qn vn (t) , (4.38)

n

and the Coulomb potential operator is

1 ql

¦ (rn (t)) = . (4.39)

|rn (t) ’ rl (t)|

4π 0

l=n

½½ Interaction of light with matter

This potential is obtained from a solution of Poisson™s equation ∇2 ¦ = ’ρ/ 0 , where

the charge density operator is

ρ (r, t) = δ (r’rn (t)) qn , (4.40)

n

by omitting the self-interaction terms encountered when r ’ rn (t). Functions f (rn )

of the position operators rn , such as those in eqns (4.35)“(4.40), are de¬ned by

f (rn ) ψ (r1 , . . . , rN ) = f (rn ) ψ (r1 , . . . , rN ) , (4.41)

where ψ (r1 , . . . , rN ) is any N -body wave function for the charged particles.

The ¬rst equation, eqn (4.33), is simply the relation between the transverse part of

the electric ¬eld operator and the vector potential. Faraday™s law, eqn (4.34), is then

redundant, since it is the curl of eqn (4.33). The matter equations (4.36) and (4.37)

are the quantum versions of the classical force laws of Coulomb and Lorentz.

The only one of the Heisenberg equations that requires further explanation is eqn

(4.35) (Amp`re™s law). The Heisenberg equation of motion for E can be put into the

e

form

pni ’ qn Ai (rn (t) , t)

1 ‚Ej (r, t)

∆⊥ (r ’ rn (t)) qn

(∇ — B (r, t))j ’ = µ0 ,

ji

c2 ‚t Mn

n

(4.42)

but the signi¬cance of the right-hand side is not immediately obvious. Further insight

can be achieved by using the de¬nition (4.36) of the velocity operator to get

pn (t) ’ qn A (rn (t) , t)

= vn (t) . (4.43)

Mn

Substituting this into eqn (4.42) yields

1 ‚Ej (r, t)

∆⊥ (r ’ rn ) qn vni (t)

(∇ — B (r, t))j ’ = µ0 ji

c2 ‚t n

d3 r ∆⊥ (r ’ r ) ji (r , t) ,

= µ0 (4.44)

ji

where ji (r , t), de¬ned by eqn (4.38), can be interpreted as the current density oper-

ator. The transverse delta function ∆⊥ projects out the transverse part of any vector

ji

¬eld, so the Heisenberg equation for E (r, t) is given by eqn (4.35).

Parity and time reversal—

4.4

The quantum Maxwell equations, (4.34) and (4.35), and the classical Maxwell equa-

tions, (B.2) and (B.3), have the same form; consequently, the ¬eld operators and the

classical ¬elds behave in the same way under the discrete transformations:

r ’ ’r (spatial inversion or parity transformation) ,

(4.45)

t ’ ’t (time reversal) .

Parity and time reversal— ½½

Thus the transformation laws for the classical ¬elds”see Appendix B.3.3”also apply

to the ¬eld operators; in particular,

E (r, t) ’ EP (r, t) = ’E (’r, t) under r ’ ’r , (4.46)

E (r, t) ’ ET (r, t) = E (r, ’t) under t ’ ’ t . (4.47)

In classical electrodynamics this is the end of the story, since the entire physical

content of the theory is contained in the values of the ¬elds. The situation for quantum

electrodynamics is more complicated, because the physical content is shared between

the operators and the state vectors. We must therefore ¬nd the transformation rules for

the states that correspond to the transformations (4.46) and (4.47) for the operators.

This e¬ort requires a more careful look at the idea of symmetries in quantum theory.

According to the general rules of quantum theory, all physical predictions can be

2

expressed in terms of probabilities given by | ¦ |Ψ | , where |Ψ and |¦ are normalized

state vectors. For this reason, a mapping of state vectors to state vectors,

|˜ ’ |˜ , (4.48)

is called a symmetry transformation if

2 2

| ¦ |Ψ | = | ¦ |Ψ | , (4.49)

for any pair of vectors |Ψ and |¦ . In other words, symmetry transformations leave

all physical predictions unchanged. The consequences of this de¬nition are contained

in a fundamental theorem due to Wigner.

Theorem 4.1 (Wigner) Every symmetry transformation can be expressed in one of

two forms:

(a) |Ψ ’ |Ψ = U |Ψ , where U is a unitary operator;

(b) |Ψ ’ |Ψ = Λ |Ψ , where Λ is an antilinear and antiunitary operator.

The unfamiliar terms in alternative (b) are de¬ned as follows. A transformation Λ

is antilinear if

Λ {± |Ψ + β |¦ } = ±— Λ |Ψ + β — Λ |¦ , (4.50)

and antiunitary if

—

¦ |Ψ = Ψ |¦ = ¦ |Ψ , where |Ψ = Λ |Ψ and |¦ = Λ |¦ . (4.51)

Rather than present the proof of Wigner™s theorem”which can be found in Wigner

(1959, cf. Appendices in Chaps 20 and 26) or Bargmann (1964)”we will attempt to

gain some understanding of its meaning. To this end consider another transformation

given by

|Ψ ’ |Ψ = exp (iθΨ ) |Ψ , (4.52)

where θΨ is a real phase that can be chosen independently for each |Ψ . For any value of

θΨ it is clear that |Ψ ’ |Ψ is also a symmetry transformation. Furthermore, |Ψ

and |Ψ di¬er only by an overall phase, so they represent the same physical state.

½¾¼ Interaction of light with matter

Thus the symmetry transformations de¬ned by eqns (4.48) and (4.52) are physically

equivalent, and the meaning of Wigner™s theorem is that every symmetry transforma-

tion is physically equivalent to one or the other of the two alternatives (a) and (b).

This very strong result allows us to ¬nd the correct transformation for each case by a

simple process of trial and error. If the wrong alternative is chosen, something will go

seriously wrong.

Since unitary transformations are a familiar tool, we begin the trial and error

process by assuming that the parity transformation (4.46) is realized by a unitary

operator UP :

†

EP (r, t) = UP E (r, t) UP = ’E (’r, t) . (4.53)

In the interaction picture, E (r, t) has the plane-wave expansion

ωk

aks eks ei(k·r’ωk t) + HC ,

E (r, t) = i (4.54)

2 0V

ks

and the corresponding classical ¬eld has an expansion of the same form, with aks re-

placed by the classical amplitude ±ks . In Appendix B.3.3, it is shown that the parity

transformation law for the classical amplitude is ±P = ’±’k,’s . Since UP is linear,

ks

† †

P

UP E (r, t) UP can be expressed in terms of aks = UP aks UP . Comparing the quantum

and classical expressions then implies that the unitary transformation of the annihi-

lation operator must have the same form as the classical transformation:

†

aks ’ aP = UP aks UP = ’a’k,’s . (4.55)

ks

The existence of an operator UP satisfying eqn (4.55) is guaranteed by another well

known result of quantum theory discussed in Appendix C.4: two sets of canonically

conjugate operators acting in the same Hilbert space are necessarily related by a

unitary transformation. Direct calculation from eqn (4.55) yields

a P , aP † = ’a’k,’s , ’a† ,’s = δkk δss ,

ks k s ’k

(4.56)

= [’a’k,’ss , ’a’k ,’s ] = 0 .

aP , a P s

ks k

Since the operators aP satisfy the canonical commutation relations, UP exists. For

ks

more explicit properties of UP , see Exercise 4.4.

The assumption that spatial inversion is accomplished by a unitary transformation

worked out very nicely, so we will try the same approach for time reversal, i.e. we

assume that there is a unitary operator UT such that

†

ET (r, t) = UT E (r, t) UT = E (r, ’t) . (4.57)

The classical transformation rule for the plane-wave amplitudes is ±T = ’±— , so

ks ’k,s

the argument used for the parity transformation implies that the annihilation operators

satisfy

aks ’ aT = aT = UT aks UT = ’a†

†

’k,s . (4.58)

ks ks

All that remains is to check the internal consistency of this rule by using it to evaluate

the canonical commutators. The result

½¾½

Stationary density operators

a T , aT † = ’a† , ’a’k ,s = ’δkk δss (4.59)

ks k s ’k,s

is a nasty surprise. The extra minus sign on the right side shows that the transformed

operators are not canonically conjugate. Thus the time-reversed operators aT and aT †

ks ks

†

cannot be related to the original operators aks and aks by a unitary transformation,

and UT does not exist.

According to Wigner™s theorem, the only possibility left is that the time-reversed

operators are de¬ned by an antiunitary transformation,

ET (r, t) = ΛT E (r, t) Λ’1 = E (r, ’t) . (4.60)

T

Here some caution is required because of the unfamiliar properties of antilinear trans-

formations. The de¬nition (4.50) implies that ΛT ± |Ψ = ±— ΛT |Ψ for any |Ψ , so

applying ΛT to the expansion (4.54) for E (r, t) gives us

T

ωk

a†

ΛT E (r, t) Λ’1 ’aT e— e’i(k·r’ωk t) eks ei(k·r’ωk t) ,

= i +

ks ks ks

T

2 0V

ks

(4.61)

where

T

a† = ΛT a† Λ’1 .

aT = ΛT aks Λ’1 , (4.62)

ks ks ks T

T

Setting t ’ ’t in eqn (4.54) and changing the summation variable by k ’ ’k yields

ωk

a’ks e’ks e’i(k·r’ωk t) ’ a† e— ei(k·r’ωk t) .

E (r, ’t) = i (4.63)

’ks ’ks

2 0V

ks

After substituting these expansions into eqn (4.60) and using the properties e’k,’s =

eks and e— = ek,’s derived in Appendix B.3.3, one ¬nds

k,s

T

a† = ’a†

= ’a’k,s ,

aT ’k,s . (4.64)

ks ks

T

This transformation rule gives us a† = aT † and

ks ks

aT , aT †s = ’a’k,s , ’a† ,s = δkk δss ; (4.65)

ks k ’k

consequently, the antiunitary transformation yields creation and annihilation opera-

tors that satisfy the canonical commutation relations. The magic ingredient in this

approach is the extra complex conjugation operation applied by the antilinear trans-

formation ΛT to the c-number coe¬cients in eqn (4.61). This is just what is needed

to ensure that aT is proportional to a’k,s rather than to a† , as in eqn (4.58).

ks ’k,s

4.5 Stationary density operators

The expectation value of a single observable is given by

X (t) = Tr [ρX (t)] = Tr [ρ (t) X] , (4.66)

which explicitly shows that the time dependence comes entirely from the observable

in the Heisenberg picture and entirely from the density operator in the Schr¨dinger

o

½¾¾ Interaction of light with matter

picture. The time dependence simpli¬es for the important class of stationary density

operators, which are de¬ned by requiring the Schr¨dinger-picture ρ (t) to be a constant

o

of the motion. According to eqn (3.75) this means that ρ (t) is independent of time,

so the Schr¨dinger- and Heisenberg-picture density operators are identical. Stationary

o

density operators have the useful property

ρ, U † (t) = 0 = [ρ, U (t)] , (4.67)

which is equivalent to

[ρ, H] = 0 . (4.68)

Using these properties in conjunction with the cyclic invariance of the trace shows

that the expectation value of a single observable is independent of time, i.e.

X (t) = Tr [ρX (t)] = Tr (ρX) = X . (4.69)

Correlations between observables at di¬erent times are described by averages of

the form

X (t + „ ) Y (t) = Tr [ρX (t + „ ) Y (t)] . (4.70)

For a stationary density operator, the correlation only depends on the di¬erence in the

time arguments. This is established by combining U (’t) = U † (t) with eqns (3.83),

(4.67), and cyclic invariance to get

X (t + „ ) Y (t) = X („ ) Y (0) . (4.71)

4.6 Positive- and negative-frequency parts for interacting ¬elds

When charged particles are present, the Hamiltonian is given by eqn (4.28), so the

free-¬eld solution (3.95) is no longer valid. The operator aks (t)”evolving from the

annihilation operator, aks (0) = aks ”will in general depend on the (Schr¨dinger-o

†

picture) creation operators ak s as well as the annihilation operators ak s . The unitary

evolution of the operators in the Heisenberg picture does ensure that the general

decomposition

F (r, t) = F (+) (r, t) + F (’) (r, t) (4.72)

will remain valid provided that the initial operator F (+) (r, 0) (F (’) (r, 0)) is a sum

over annihilation (creation) operators, but the commutation relations (3.102) are only

valid for equal times. Furthermore, F (+) (r, ω) will not generally vanish for all negative

values of ω. Despite this failing, an operator F (+) (r, t) that evolves from an initial

operator of the form

F (+) (r, 0) = Fks aks eik·r (4.73)

ks

is still called the positive-frequency part of F (r, t).

½¾¿

Multi-time correlation functions

4.7 Multi-time correlation functions

One of the advantages of the Heisenberg picture is that it provides a convenient way

to study the correlation between quantum ¬elds at di¬erent times. This comes about

because the state is represented by a time-independent density operator ρ, while the

¬eld operators evolve in time according to the Heisenberg equations.

Since the electric ¬eld is a vector, it is natural to de¬ne the ¬rst-order ¬eld

correlation function by the tensor

(1) (’) (+)

Gij (x1 ; x2 ) = Ei (x1 ) Ej (x2 ) , (4.74)

where X = Tr [ρX] and x1 = (r1 , t1 ), etc. The ¬rst-order correlation functions are

directly related to interference and photon-counting experiments. In Section 9.1.2-B

we will see that the counting rate for a broadband detector located at r is proportional

(1) (1)

to Gij (r, t; r, t). For unequal times, t1 = t2 , the correlation function Gij (x1 ; x2 ) rep-

resents measurements by a detector placed at the output of a Michelson interferometer

with delay time „ = |t1 ’ t2 | between its two arms. In Section 9.1.2-C we will show

that the spectral density for the ¬eld state ρ is determined by the Fourier transform

(1)

of Gij (r, t; r, 0). The two-slit interference pattern discussed in Section 10.1 is directly

(1)

given by Gij (r, t; r, 0).

We will see in Section 9.2.4 that the second-order correlation function, de¬ned

by

(2) (’) (’) (+) (+)

Gijkl (x1 , x2 ; x3 , x4 ) = Ei (x1 ) Ej (x2 ) Ek (x3 ) El (x4 ) , (4.75)

is associated with coincidence counting. Higher-order correlation functions are de¬ned

(’) (’)

similarly. Other possible expectation values, e.g. Ei (x1 ) Ej (x2 ) , are not related

to photon detection, so they are normally not considered.

In many applications, the physical situation de¬nes some preferred polarization

directions”represented by unit vectors v1 , v2 , . . .”and the tensor correlation func-

tions are replaced by scalar functions

(’) (+)

G(1) (x1 ; x2 ) = E1 (x1 ) E2 (x2 ) , (4.76)

(’) (’) (+) (+)

G(2) (x1 , x2 ; x3 , x4 ) = E1 (x1 ) E2 (x2 ) E3 (x3 ) E4 (x4 ) , (4.77)

—

(+)

where Ep = vp · E(+) is the projection of the vector operator onto the direction vp .

For example, observing a ¬rst-order interference pattern through a polarization ¬lter

is described by

G(1) (x; x) = e · E(’) (x) e— · E(+) (x) , (4.78)

where e is the polarization transmitted by the ¬lter.

If the density operator is stationary, then an extension of the argument leading to

eqn (4.71) shows that the correlation function is unchanged by a uniform translation,

(1)

tp ’ tp + „, tp ’ tp + „ , of all the time arguments. In particular Gij (r, t; r , t ) =

½¾ Interaction of light with matter

(1)

Gij (r, t ’ t ; r , 0), so the ¬rst-order function only depends on the di¬erence, t ’ t , of

the time arguments.

The correlation functions satisfy useful inequalities that are based on the fact that

Tr ρF † F 0, (4.79)

where F is an arbitrary observable and ρ is a density operator. This is readily proved

by evaluating the trace in the basis in which ρ is diagonal and using Ψ F † F Ψ 0.

Choosing F = E (+) (x) in eqn (4.79) gives

G(1) (x; x) 0, (4.80)

(+) (+)

(x1 ) · · · En

and the operator F = E1 (xn ) gives the general positivity condition

G(n) (x1 , . . . , xn ; x1 , . . . , xn ) 0. (4.81)

A di¬erent sort of inequality follows from the choice

n

(+)

F= ξa Ea (xa ) , (4.82)

a=1