Spohn H. Dynamics of Charged Particles and their Radiation Field

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13.7 Dipole and single-photon approximation 173

energy levels of the harmonic oscillator are equidistant, several emitted photons

will interfere, which makes the emission spectrum distinct from, say, the hydrogen

atom; compare with section 17.4.

Even though the equations of motion (13.128) are linear, their solution is not

a back of the envelope computation and one often resorts to yet another approx-

imation, the rotating wave approximation. One starts from (13.126) with the har-

monic potential

mω

, which already includes the last summand in (13.126),

and rewrites the harmonic oscillator in terms of its creation and annihilation oper-

ator b, b

∗

. Then

H =

ω

∗

b − i



/2mω

(b − b

∗

) · eE

⊥ϕ

(0) + H

. (13.129)

In the coupling, one ignores the counter-rotating terms ba and b

∗

, which results

= (b, a) · h (b, a)

. (13.130)

Our notation emphasizes that the rotating wave Hamiltonian H

is quadratic in

(b, a) and should be regarded as the second quantization of the one-particle Hamil-

tonian h . The one-particle space is

K = C

⊕ (L

) ⊗ C

), the C

subspace

corresponding to b, b

∗

.Awave function in K is of the form (χ , ψ(k, λ)), χ the

one-particle amplitude for the oscillator and ψ(k,λ) the one-particle photon am-

plitude. h acting on a pair (χ , ψ) is deﬁned by



ψ(k,λ)





ω

χ −



λ=1,2



keϕ

∗



√

ω/mω

(k)ψ(k,λ)

−

eϕ

√

ω/mω

(k) · χ + ωψ(k,λ)



(13.131)

ϕ = ϕ(k), ω = ω(k). h will reappear as the Friedrichs–Lee Hamiltonian. For

e = 0, the eigenvalue

ω

is embedded in the continuous spectrum [0, ∞). The

coupling turns this eigenvalue into a resonance; compare with section 17.3.

Afurther popular variant is to set ω

= 0in(13.127) and to regard the Hamil-

tonian as describing a freely propagating charge. One ﬁnds that the mass of the

particle is increased due to the coupling with the ﬁeld. However, quantitatively

such a result cannot be trusted, since the dipole approximation is based on the as-

sumption that the electron remains close to the origin. There is no such mechanism

for a free particle.

(ii) Single-photon approximation

We restrict the Fock space to

C ⊕ h. Then the wave functions ψ are pairs

(ψ

(x), ψ

(x, k, λ)). ψ

(x) is the wave function for an electron and no photon

174 Quantizing the Abraham model

present, while ψ

(x, k,λ) is the wave function for the electron plus one photon

with momentum

k and helicity λ. The correspondingly restricted Pauli–Fierz

Hamiltonian is denoted by H

. From (13.39), setting N = 1,  = 1 = c, one infers

ψ)

(x) =

(x) +



λ=1,2



ke ˆϕ

√

2ω

ik·x

· pψ

(x, k,λ),

ψ)

(x, k,λ) =



+ ω



(x, k,λ)+ e ˆϕ

√

2ω

−ik·x

· pψ

(x),

(13.132)

where the A

contribution has been neglected. H

is a two-particle problem with a

translation-invariant interaction. The electron has kinetic energy

. The photon

can be either “dead” (ψ

) or “alive” (ψ

). The kinetic energy is zero in the dead

state and

ω in the alive state. Through the interaction a photon is either created or

annihilated, which corresponds to a transition between dead and alive. Because of

the 1/

√

ω-factor, this interaction has a long range and decays only as r

−3/2

in the

relative distance between the electron and the photon.

Notes and references

Section 13.1

The Hamiltonian form of the Abraham model in the Coulomb gauge is standard

and explained in Cohen-Tannoudji et al. (1989) and Sakurai (1986), for example.

Section 13.2

The name “Pauli–Fierz” is not accurate historically. The Hamiltonian (13.42) ap-

pears at the beginning of paragraph two of Pauli and Fierz (1938) as a matter of

fact, without citation. Pauli and Fierz study the generation of infrared photons in

Compton scattering. Cohen-Tannoudij et al. (1989) call (13.42) “of basic impor-

tance” and Milonni (1994) refers to (13.42) simply as “the Hamiltonian”. Thus

despite its fundamental nature the Hamiltonian (13.42) carries no speciﬁc name in

the literature. Lately, “nonrelativistic quantum electrodynamics” and “Pauli–Fierz”

have become common usage in some quarters. We stick to the latter convention,

which is certainly better than to be speechless.

The quantization of the electromagnetic ﬁeld as a system of harmonic oscilla-

tors was common knowledge right after the advent of quantum mechanics through

the work of Dirac (1927), Landau (1927), Jordan and Pauli (1928), Fermi (1930),

and Landau and Peierls (1930), and was immediately applied to atomic radiation

Notes and references 175

by many quantum theorists. The systematic derivation of the Hamiltonian (13.42)

is not so well documented and was presumably regarded as more or less obvious,

although the advantage of the Coulomb gauge was only slowly realized. The re-

view articles by Breit (1932) and by Fermi (1932) and the research monograph by

Heitler (1936, 1958) explain the quantization in its modern form, in essence. Since

“one cannot comb the hair on a sphere”, the polarization vectors e

(k) are necessar-

ily discontinuous in k, which causes poor decay in their Fourier transform. We refer

to Lieb and Loss (2004) for a formulation using only the transverse projection.

The size of atoms as based exclusively on the Coulomb Hamiltonian is a long-

standing open problem. We refer to Lieb (1990, 2001).

Textbooks on nonrelativistic quantum electrodynamics are listed in Notes and

References to section 3.2.

Section 13.3

Criteria for self-adjointness are given in Reed and Simon (1980, 1975). In our con-

text the Kato–Rellich theorem has been applied by Nelson (1964b) and Fr

ohlich

(1974), amongst others. Self-adjointness without restriction on the magnitude of

the charge is proved by Hiroshima (2000b, 2002). A review is Hiroshima (2001).

Section 13.5

A more detailed treatment of conservation laws is Huang (1998).

Section 13.6

Casimir (1948) discovered the attraction of two conducting plates through vacuum

ﬂuctuations. Casimir and Polder (1948) compute the attractive force between two

atoms, the retarded van der Waals force, and the force between an atom and a wall.

The forces are minute and direct experimental evidence had to wait for a while. We

refer to Sparnaay (1958) and Lamoreaux (1997). On the theoretical side a complete

coverage is Milloni (1994), Huang (1998), with the ﬁnite-temperature corrections

discussed by Schwinger et al. (1978), Bordag et al. (2000), and Feinberg et al.

(2000).

Section 13.7

Apparently the ﬁrst systematic study of the dipole approximation with a harmonic

external potential is Kramers (1948) and van Kampen (1951). Various aspects are

covered by Senitzky (1960), Schwabl and Thirring (1964), Ford, Kac and Mazur

176 Quantizing the Abraham model

(1965), Ullersma (1966), Ford, Lewis and O’Connell (1988a, 1988b), Grabert,

Schramm and Ingold (1988), Unruh and Zurek (1989). A mathematical study is the

series by Arai (1981, 1983a, 1983b, 1990, 1991). Since the dipole approximation

provides a reasonable description of radiation processes, one might regard the har-

monic potential as the lowest-order approximation and expand in the anharmonic-

ity. This program has been carried through in Maassen (1984), Spohn (1997),

Maassen, Gut

a and Botvich (1999), and Fidaleo and Liverani (1999). If the an-

harmonicity is small, in fact so small that the external potential remains convex

and grows as

mω

for large q, then the convergence of the time-dependent

Dyson series can be controlled uniformly in t.With such a strong estimate one can

show that qualitatively the properties of the damped harmonic oscillator persist

into the nonlinear regime.

The dipole approximation is not restricted to a single particle. For example one

may consider two harmonically bound charges with their center of charge at r

and

. Then the kinetic energies are approximated by ( p

− c

−1

))

/2m

, j =

1, 2. Denoting R =|r

−r

|, one is interested in the ground state energy, E(R),

as a function of the separation. Because of retardation E(R)

∼

−R

−7

for large R

and E(R)

∼

−R

−6

in an intermediate regime.

If φ

= 0, then the Hamiltonian (13.123) can be unitarily transformed to H



( p

/2m

eff

) + H

. m

eff

agrees with the effective mass of the Abraham model to

lowest order in |v|/c; compare with section 4.1.

The single-photon approximation was already used in disguise by Dirac (1927)

and Weisskopf and Wigner (1930). It is instructive to extend this approximation

by cutting Fock space at N photons (H

ubner and Spohn, unpublished manuscript;

Skibsted 1998). If one artiﬁcially adds to the space of single-photon wave functions

a one-dimensional subspace for a “dead” photon, then the theory has a structure

very similar to an (N + 1)-particle Schr

odinger equation. The photons interact

only indirectly through the atom. The cluster decomposition consists of n free

photons and N − n photons bound by the atom, n = 0, 1,...,N .

The statistical mechanics connection

Models from quantum mechanics can be converted into statistical mechanics

systems through the Wick rotation t

; −it.Within quantum ﬁeld theory this

technique has been very powerful, both in proving qualitative properties and as a

computational tool. Besides these more practical aspects, the statistical mechanics

formulation is an additional source of intuition which cannot so easily be extracted

from the Schr

odinger differential equation. The price to pay is that, in essence only

ground state properties can be handled. Truly time-dependent problems must be

treated in physical time. For charges interacting with the Maxwell ﬁeld the Wick

rotation is equally attractive. There is one additional bonus: since the ﬁeld Hamil-

tonian is quadratic and since the coupling to the ﬁeld is linear, as ﬁrst observed by

Feynman, the Gaussian integration over the Maxwell ﬁeld can be done explicitly.

This results in a fairly concise statistical-mechanical description for the particles.

In Euclidean language the possible paths of the charge and the ﬁelds become

ﬂuctuating quantities. To distinguish in notation we use t → q

for a random path

of the charge and t → A

(x) for a random history of the transverse vector ﬁeld.

E(·) refers to expectation with respect to the measure of integration, either spec-

iﬁed through the context or indicated by a subscript. Sometimes we also use the

statistical mechanics shorthand · for averages.

14.1 Functional integral representation

Forasingle particle, subject to the potential V (x), the imaginary time Schr

odinger

equation is, setting

 = 1 = m,

∂

ψ =−H

ψ, H

=−

 + V (14.1)

and its solution for t ≥ 0isconstructed through the Trotter product formula as

−tH

ψ)(x) = lim

n→∞

t/2n

−tV/n

)

ψ(x). (14.2)

177

178 The statistical mechanics connection

We recognize exp[

t]asthe transition probability for a Brownian motion, whose

paths will be denoted here by t → q

.Brownian motion is a Gaussian process and

therefore deﬁned through the mean and covariance. Explicitly,

E(q

) = 0 , E(q

sα

tβ

) = δ

αβ

min(s, t ). (14.3)

If the Brownian motion starts at x,weindicate the start point as a subscript

in the expectation and have

) = x, E

((q

− x)

) = δ

αβ

min(s, t ),

α, β = 1, 2, 3. In particular the transition probability is obtained as

({q

∈ d

y}) = (2πt)

−3/2

exp[−(y − x)

/2t]d

y = (e

t/2

)(x, y)d

y .

(14.4)

Writing out (14.2) in position space representation, one infers the Feynman–Kac

formula

−tH

ψ)(x) = E



exp



−



dsV(q

)



ψ(q

)



. (14.5)

The Brownian motion path has acquired a non-Gaussian weight, which is the ex-

ponential of the potential energy integrated along the path q

The statistical mechanics connection becomes more obvious upon discretizing

time in units of τ.Wesetφ

= q

nτ

,φ

∈ R

. Then, in approximation, (14.5) reads



...d

δ(φ

− x) exp



−

2τ

N −1



j=0

(φ

j+1

− φ

)



×exp



− τ



j=1

V (φ

)



ψ(φ

), (14.6)

N τ = t. The statistical mechanics model lives on a one-dimensional lattice and

has at each site a continuous “spin” with three components. The ﬁrst exponen-

tial is a quadratic nearest-neighbor interaction and, except for the normalization,

represents a discrete-time Gaussian random walk in

. The potential can be com-

bined with the Lebesgue measure as exp[−τ V (φ

)]d

and thus provides a non-

Gaussian single-site measure.

For (14.5) to make sense one needs some minimal conditions on V to en-

sure that the expectation is deﬁned. An obvious sufﬁcient condition is to have

V ≥ c

> −∞. Equation (14.5) indicates that very roughly there are three families

of potentials: (i) Binding, V increases at inﬁnity. Under the measure in (14.5) q

has in essence bounded ﬂuctuations. For t →∞the path measure for q

becomes

a stationary diffusion process. (ii) No binding, e.g. a repulsive potential decay-

ing to zero at inﬁnity or a bounded periodic potential. A typical path q

ﬂuctuates

and diffuses to inﬁnity as a Brownian motion with some effective diffusion coefﬁ-

cient. (iii) Local binding, like an attractive square-well potential. For the purpose

14.1 Functional integral representation 179

of discussion let us set the potential as λV with V attractive near the origin and

decaying to zero at inﬁnity. For large λ the potential dominates and q

is con-

ﬁned as a stationary diffusion process. As λ decreases, q

makes longer and longer

excursions until it unbinds at some critical λ

.Forλ<λ

the Brownian motion

dominates. Since Brownian motion is recurrent in dimension d = 1, 2, one has

= 0, whereas for d = 3 generically λ

> 0.

It is of use to translate the path properties of the particle to spectral properties

of the particle Hamiltonian H

=−

 + V .Wedenote by  the continuum edge

of H

and, if it exists, by ψ

the unique ground state of H

,i.e. H

= E

.In

case (i) the spectrum of H

is purely discrete, formally  =∞.Inthe second case

has a purely continuous spectrum and no eigenvalues. For a locally binding

potential which decays to zero at inﬁnity, case (iii), the continuum edge is  = 0.

For sufﬁcient attraction there are bound states with an energy below ,inparticu-

lar E

< 0. In dimension d = 1, 2anarbitrarily weak attraction results in a bound

state, whereas for d ≥ 3aminimal strength is required. As is well understood,

there is more complicated spectral behavior around with various borderline cases.

For our purposes the schematic classiﬁcation above will sufﬁce.

Our goal is to extend the Feynman–Kac formula (14.5) to e

−tH

with H the

Pauli–Fierz Hamiltonian. This will be done in two steps. Firstly we study a one-

particle Hamiltonian including an external vector potential, and secondly we write

−tH

in terms of a suitable Gaussian measure. Combining both elements yields

the desired generalization.

Let us assume then that the quantum particle is subject to a magnetic ﬁeld and

denote the corresponding vector potential by a(x),todistinguish from the ﬂuctu-

ating vector potential A

used later on. The imaginary time Schr

odinger equation

becomes

∂

ψ =−H

ψ, H

(−i∇−a)

+ V . (14.7)

Then, as before, we represent e

−tH

through the Trotter product formula. The vec-

tor potential yields a term proportional to ˙q,ascan be guessed from the corre-

sponding classical action. More precisely one obtains

−tH

ψ)(x) = E



exp



− i



·a(q

) −



ds∇·a(q

) −



dsV(q

)



ψ(q

)



(14.8)

The stochastic integral appearing in (14.8) is deﬁned as Ito integral, which means

that the discretization of a(x) is evaluated at the left end point,



· a(q

) = lim

n→∞



m=1

a(q

(m−1)/n

) · (q

m/n

− q

(m−1)/n

). (14.9)

180 The statistical mechanics connection

This limit exists almost surely with respect to Brownian motion. Through the

Ito convention one picks up in (14.8) the additional term containing ∇·a.It

disappears, if in (14.9) we were to use the, in our context perhaps more nat-

ural, Feynman–Stratonovich midpoint rule where a(q

(m−1)/n

) is replaced by

(a(q

m/n

) + a(q

(m−1)/n

)). Note that in the Coulomb gauge the stochastic inte-

gral does not depend on the particular choice of the rule for the discretization,

since ∇·a = 0.

On a purely formal level, following Feynman, the quantum propagator is written

as a sum over all paths from x



to x in the time span t “weighted” by the exponential

of the classical action,

−iH

)(x, x



) =





0≤s≤t

δ(q

− x



)δ(q

− x) exp





dsL(q

, ˙q

)



(14.10)

with the classical Lagrangian L(q, ˙q) =

˙q

− V (q) +˙q · a(q). Note that com-

pared to the right side of (14.8) the role of x and x



has been interchanged. Upon

Wick rotation t

; −it and time reversal q

; q

t−s

(14.10) becomes

−tH

)(x, x



) =





0≤s≤t

δ(q

− x)δ(q

− x



)

× exp



−





˙q

+ V (q

) + i ˙q

· a(q

)





. (14.11)

One recognizes the potential term and the stochastic integral −i



ds ˙q

· a(q

)

with the mid point rule. The exponential of the kinetic term combines with the

inﬁnite-product Lebesgue measure to Brownian motion, denoted by

in (14.8),

which starts at x according to the factor δ(q

− x).

We turn to the functional integral for the Maxwell ﬁeld, which we can think

of as an inﬁnite collection of harmonic oscillators. Let us ﬁrst recall the single

harmonic oscillator with Hamiltonian

H =



− ∂

+ ω

− ω



(14.12)

as a differential operator acting on L

(R, dx).Ithas the normalized eigenvectors

|n, n = 0, 1,...,i.e.

H|n=ε

|n,ε

= ωn . (14.13)

|0 is the ground state of H .Inthe position representation H ψ

= 0 with

(x)

√

ω/πe

−ωx

. Thus, alternatively we can use the linear span of the

|n’s as the Hilbert space of states. This corresponds to the Fock space

F over

14.1 Functional integral representation 181

the one-particle space C, which means ψ ∈ F is of the form ψ = (ψ

,ψ

,...),

ψ =



∞

n=0

|n.Afurther, as it will turn out natural, choice is the Hilbert space

= L

(R,ψ

(x)

dx) with weight given by the square of the ground state wave

function.

Of course, these Hilbert spaces are unitarily equivalent. Of interest is the uni-

tary map from

F to H

which is achieved through the Wick ordering of poly-

nomials. We regard x as a random variable on

R equipped with the normalized

Gaussian measure ψ

(x)

dx. Then, denoting expectation by ·, the Wick order

of x is deﬁned recursively through : x

: = 1, ∂

: x

: = n: x

n−1

:, and : x

:=0,

n = 1, 2,....Note that the Wick order depends both on the random variable and

on the underlying measure. Thus :1: = 1, : x: = x −x=x,:x

: = x

− 2xx −

x

+2x

= x

− (1/2ω), etc., in our case. Let P

denote the n-th Hermite

polynomial,

(x) =

[n/2]



j=0

(n − 2 j)! j!

(

−

)

n−2 j

, (14.14)

with [n] the integer part. Then the Wick-ordered mononomial of order n is given

: x

: = (2ω)

−n/2

(

√

2ωx). (14.15)

One has

: x

::x

:=: x

: , : x

:

= (2ω)

−n

n!δ

. (14.16)

By linearity Wick order extends to all ﬁnite polynomials. Let us also introduce

∗

√

2ω

(ωx − ∂

), a =

√

2ω

(ωx + ∂

) (14.17)

as creation and annihilation operators of the harmonic oscillator. Their Wick or-

der means that all annihilation operators are moved to the right, e.g. : aa

∗

: = a

∗

Then



√

2ω

∗

+ a)



: |0=(2ω)

−n/2

√

n!|n. (14.18)

Comparing with (14.16) the map U from

F to H

should be deﬁned through



√

2ω

∗

+ a)



: |0 → : x

: (14.19)

182 The statistical mechanics connection

and extended by linearity. By the very construction the closure of U as a linear

map

F → H

is then unitary. Note that e

−tH

is implemented as

U e

−tH

−1

: x

: =:(e

−ωt

: = e

−nωt

: x

: . (14.20)

Through the Feynman–Kac formula (14.5) we boost (14.12) to a Gaussian

stochastic process denoted by x

.Ittakes real values, is stationary in time, has

mean zero, and covariance

E(x

) =

2ω

−ω|t−s|

. (14.21)

We recognize x

as the stationary Ornstein–Uhlenbeck process governed by the

stochastic differential equation

=−ωx

dt + db

, (14.22)

where b

is standard one-dimensional Brownian motion. Note that

E( f (x

)) = E( f (x

)) =ψ

, f ψ

=1, f 



dxψ

(x)

f (x),

(14.23)

( f (x

)g(x

)) =ψ

, f e

−|t−s|H

gψ

=f, e

|t−s|L

g

, (14.24)

where we used the similarity transformation

−1

−tH

= e

, t ≥ 0 , (14.25)

with L the generator of the Ornstein–Uhlenbeck process x

L =−ωx∂

∂

. (14.26)

According to (14.23) the Ornstein–Uhlenbeck process x

has ψ

(x)

as stationary

measure. With probability one t → x

is continuous and we may choose C(R, R),

the space of all continuous functions over

R,aspath space. In fact, x

has in

essence bounded ﬂuctuations and increases at most logarithmically for large t.

The point of our exercise is that it carries over essentially verbatim to the

inﬁnite-dimensional setting, except for the ﬂat Hilbert space L

(R, dx). H

plays

the role of the harmonic oscillator. The boson Fock space over the transverse

vector ﬁelds L

⊥

, R

) plays the role of the Fock space over C. The Ornstein–

Uhlenbeck process x

is replaced by the inﬁnite-dimensional Ornstein–Uhlenbeck

process A

(x). Let us start with the latter. A

(x) is a Gaussian process with mean

zero and covariance



tα

(x) A



)



= (2π)

−3



ik·(x−x



)

⊥

αα



(k)

2ω

−ω|t−t



, (14.27)