## ABSTRACT

In this section we shall take up the problem of quantizing the electromagnetic field in free space. In the absence of charges and currents the classical electric and magnetic fields satisfy the free-space Maxwell equations given in Eqs. (2.1)-(2.4). The free fields are transverse, and for reasons which will become clear as we proceed, conveniently represented by the gauge invariant transverse vector potential AT (r, t). In terms of AT (r, t) the electric (ET (r, t)) and magnetic (B(r, t)) fields are given by

ET (r, t) = − ∂ ∂t AT (r, t), (15.1)

B(r, t) =∇×AT (r, t). (15.2) The transverse vector potential satisfies the homogeneous wave equation

AT (r, t) = 0 (15.3)

in free space; cf. the analysis in Sec. 10.2, resulting in Eq. (10.31). In order to quantize the electromagnetic field in a manner which leads up to the

monochromatic plane-wave photon concept, we imagine the field to be contained in a large cubic box of side L. The transverse vector potential can then be presented as a Fourier series, and hence specified in terms of an infinite but denumerable number of (vectorial) Fourier expansion coefficients. The procedure above is not essential for the quantization of the free field. Thus, no physically meaningful result will depend on L when L is much larger than all relevant wavelengths entering the given problem. In the limit L→∞ the Fourier series representation of the field goes into a Fourier integral representation. From a formal point of view it might seem more natural to base the free-field quantization on the continuous mode representation of the field in an infinite space domain. For the study of certain problems in quantum electrodynamics it is an advantage to make use of the continuous mode representation. For simplicity, not necessity, we impose the well-known periodic boundary conditions on the field [155, 211, 53]. With these, the plane-wave Fourier expansion of the transverse vector potential takes the form

AT (r, t) = L − 3

∑ q

( AT,q(t)e

iq·r + c.c. ) , (15.4)

where the wave vectors q are given by

q = 2π

L (n1, n2, n3), n1, n2, n3 = 0,±1,±2, · · · . (15.5)

The transversality of the vector potential implies that

q ·AT,q(t) = 0, (15.6)

of

and the vectorial Fourier amplitudeAT,q(t) therefore can be resolved after a pair of generally complex orthonormal base vectors, cf. the discussion in Sec. 2.5. For later convenience, the decomposition is written in the form

AT,q(t) =

( ~

2ǫ0ωq

αqs(t)εqs, (15.7)

where ωq = c|q|(= cq). In order for the expansion

AT (r, t) = ∑ q,s

( ~

2ǫ0L3ωq

) 1 2 ( αqs(t)εqse

iq·r + c.c. )

(15.8)

to be a solution to the wave equation in Eq. (15.3), the time dependent amplitude αqs(t) must satisfy the harmonic oscillator equation(

d2

dt2 + ω2q

) αqs(t) = 0, ∀q, s. (15.9)

It will prove convenient to utilize the solution

αqs(t) = αqs(0)e −iωqt (15.10)

to the oscillator equation in what follows. With this choice we reach the following monochromatic plane-wave expansion of the transverse vector potential:

AT (r, t) = ∑ q,s

( ~

2ǫ0V ωq

) 1 2 [ αqs(0)εqse

i(q·r−ωqt) + c.c. ] , (15.11)

where V = L3 is the field confinement volume. It can be shown [53] that the decomposition in Eq. (15.11) leads to a compact expression

for the total energy of the transverse electromagnetic field, viz.,

HT = ǫ0 2

∫ V

( E2T (r, t) + c

2B2(r, t) ) d3r

= ∑ q,s

~ωqα ∗ qs(t)αqs(t)

= ∑ q,s

~ωqα ∗ qs(0)αqs(0). (15.12)

It appears from this result that the energy can be expressed as a sum of energies for the individual (qs)-modes, ~ωq|αqs|2. On physical grounds, and from the analysis in Sec. 2.7 we know that the energy of the free field is time independent, in agreement with what appears explicitly from the last member of Eq. (15.12). When the transverse field interacts with charged matter, the time dependence of αqs(t) is not so simple as in Eq. (15.10). The last but one member of Eq. (15.12) still holds, but HT will depend on time, in general. For the purpose of field quantization it is useful to write the energy in the symmetrized form

HT = 1

∑ q,s

~ωq ( α∗qs(t)αqs(t) + αqs(t)α

) , (15.13)

or in an equivalent form with αqs(0) and α ∗ qs(0). In preparation of the canonical quantization

procedure, αqs(t) and its complex conjugate are replaced by a pair of real canonical variables Qqs(t) and Pqs(t):

αqs(t) = (2~ωq) − 1

2 (ωqQqs(t) + iPqs(t)) , (15.14)

α∗qs(t) = (2~ωq) − 1

2 (ωqQqs(t)− iPqs(t)) . (15.15) Expressed in terms of Qqs(t) and Pqs(t), the energy in the free field takes the form

HT = 1

∑ qs

( P 2qs(t) + ω

) . (15.16)

Formally, this will be recognized as the energy of a system of independent one-dimensional harmonic oscillators one for each (q, s)-mode. The quantities Qqs(t) and Pqs(t) take the role of scalar “coordinate” and “momentum” for the given mode. The expression for αqs(t) in Eq. (15.10) is the solution to the differential equation(

d

dt + iωq

) αqs(t) = 0, (15.17)

and if one inserts Eq. (15.14) for αqs(t) into Eq. (15.17), and divides the resulting equation into its real and imaginary parts one obtains the relations

d

dt Qqs(t) = Pqs(t), (15.18)

d

dt Pqs(t) = −ω2qQqs(t). (15.19)

With the help of these, one immediately obtains the “usual” set of Hamilton equations of motions for the canonical variables, viz.,

∂HT ∂Pqs

= d

dt Qqs(t), (15.20)

∂HT ∂Qqs

= − d dt Pqs(t). (15.21)

The field quantization is accomplished by (i) replacing the classical canonical variables by Hermitian operators (operator symbol:ˆ) that is

Qqs(t)⇒ Qˆqs(t) = Qˆ†qs(t), (15.22) Pqs(t)⇒ Pˆqs(t) = Pˆ †qs(t), (15.23)

where † stands for Hermitian conjugate, and (ii) assuming that these operators satisfy the following equal-time commutation relations:[

Qˆqs(t), Pˆq′s′(t) ] = i~δqq′δss′ (15.24)[

Qˆqs(t), Qˆq′s′(t) ] = [ Pˆqs(t), Pˆq′s′(t)

] = 0. (15.25)

The amplitude αqs(t) and its complex conjugate α ∗ qs(t) are elevated to the operator level

inserting the replacements in (15.22) and (15.23) into Eqs. (15.14) and (15.15). Hence,

αqs(t)⇒ aˆqs(t) = (2~ωq)− 1

( ωqQˆqs(t) + iPˆqs(t)

) , (15.26)

α∗qs(t)⇒ aˆ†qs(t) = (2~ωq)− 1

( ωqQˆqs(t)− iPˆqs(t)

) , (15.27)

of

The Hermiticity of Qˆqs(t) and Pˆqs(t) implies that the operators associated to αqs(t) and α∗qs(t) are each other’s Hermitian conjugate, as indicated in the notation above. The operator aˆqs(t) is manifest non-Hermitian. With the help of Eqs. (15.24) and (15.25), the reader may show that the equal-time commutation relations for aˆqs(t) and aˆ

† qs(t) are[

aˆqs(t), aˆ † q′s′(t)

] = δqq′δss′ , (15.28)

[aˆqs(t), aˆq′s′(t)] = [ aˆ†qs(t), aˆ

] = 0. (15.29)

For reasons to be given in Sec. 15.3, aˆqs(t) and aˆ † qs(t) are called the annihilation and creation

operator, respectively. The canonical quantization procedure implies that the classical transverse vector poten-

tial is replaced by a field operator, i.e.,

AT (r, t)⇒ AˆT (r, t) = ∑ q,s

( ~

2ǫ0V ωq

) 1 2 ( aˆqs(t)εqse

iq·r + h.c. )

= ∑ q,s

( ~

2ǫ0V ωq

) 1 2 [ aˆqs(0)εqse

i(q·r−ωqt) + h.c. ] , (15.30)

where h.c. means the Hermitian conjugate of the preceding term. The last member of Eq. (15.30) follows because the annihilation and creation operators for free fields have the time dependencies

aˆqs(t) = aˆqs(0)e −iωqt, (15.31)

aˆ†qs(t) = aˆ † qs(0)e

iωqt. (15.32)

From the expression for AˆT (r, t) explicit formulas for the transverse electric (EˆT (r, t)) and

magnetic (Bˆ(r, t)) field operators can immedeately be written down utilizing that

EˆT (r, t) = − ∂ ∂t AˆT (r, t), (15.33)

Bˆ(r, t) =∇× AˆT (r, t). (15.34)

The canonical quantization procedure implies that the classical Hamiltonian of the transverse field [Eq. (15.16)] turns into the Hamilton operator

HˆT = 1

∑ qs

( Pˆ 2qs(t) + ω

) . (15.35)

The inverse transformation to the one given by Eq. (15.26) and (15.27), namely,

Qˆqs(t) =

( ~

2ωq

) 1 2 ( aˆqs(t) + aˆ

) , (15.36)

Pˆqs(t) = i

( ~ωq 2

) 1 2 ( aˆ†qs(t)− aˆqs(t)

) , (15.37)

allows one to express HˆT in terms of the annihilation and creation operators. Respecting the order of these operators we find

HˆT = 1

∑ q,s

~ωq ( aˆ†qs(t)aˆqs(t) + aˆqs(t)aˆ

) . (15.38)

This expression also follows directly from Eq. (15.13), by the replacements in Eqs. (15.26) and (15.27), because the order of the commuting classical amplitudes αqs(t) and α

has been respected in this equation. Using the commutator relation in Eq. (15.28) for (qs) = (q′s′), viz., [

aˆqs(t), aˆ † qs(t)

] = 1, (15.39)

the Hamilton operator of the transverse electromagnetic field finally can be written in the very important form

HˆT = ∑ q,s

~ωq

( aˆ†qs(t)aˆqs(t) +

) . (15.40)

The terms in this equation which contain aˆ†qsaˆqs-factors relate to the photon concept in quantum optics, as we soon shall see. The terms containing 12 -factors relate to the so-called photon vacuum, a concept to be studied in Part VII. From many observations it has been concluded that the photon vacuum possesses physical properties. The quantized theory hence has demonstrated that the electromagnetic fields are more than “just” the photons.