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arXiv:1011.3941v1 [quant-ph] 17 Nov 2010

On spontaneous photon emission in collapse models

Stephen L. Adler∗

Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, USA

Angelo Bassi†and Sandro Donadi‡

Department of Physics, University of Trieste,

Strada Costiera 11, 34151 Trieste, Italy and

Istituto Nazionale di Fisica Nucleare, Trieste Section, Via Valerio 2, 34127 Trieste, Italy

We reanalyze the problem of spontaneous photon emission in collapse models. We show

that the extra term found by Bassi and D¨ urr is present for non-white (colored) noise, but its

coefficient is proportional to the zero frequency Fourier component of the noise. This leads

one to suspect that the extra term is an artifact. When the calculation is repeated with the

final electron in a wave packet and with the noise confined to a bounded region, the extra

term vanishes in the limit of continuum state normalization. The result obtained by Fu and

by Adler and Ramazanoˇ glu from application of the Golden Rule is then recovered.

I.INTRODUCTION

In a previous series of articles [1–3], the problem of the spontaneous emission of radiation from

charged particles, as predicted by collapse models, was analyzed in detail. The interest in this

kind of problem arises from the fact that it currently sets the strongest upper bound on these

models [4, 5]. The analysis of [1, 2] has been done by using the CSL model [6], while the analysis

of [3] has been done within the QMUPL model [7]. In spite of the fact that the two models should

give the same predictions in an appropriate limit [3], the formula derived in [3] turns out to be

twice bigger than that of [1, 2], in the case of radiation emitted from a free particle. While this

factor of 2 difference is unimportant in the free particle case and for a white noise, as we will show

here it gives rise to awkward terms in the case of a colored noise, leading one to suspect that the

extra term is not physical.

We wish to come back on this issue, in order to clarify some mathematical details regarding the

derivation of the radiation formula.

∗Electronic address: adler@ias.edu

†Electronic address: bassi@ts.infn.it

‡Electronic address: donadi@ts.infn.it

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II. THE CSL MODEL FOR CHARGED PARTICLES

In the CSL model, the standard Schr¨ odinger equation is modified by adding terms which cause

the collapse of the wave function:

d|ψt? =

?

−i

?Hdt +

√γ

m0

?

dx[M(x) − ?M(x)?t]dWt(x) −

γ

2m2

0

?

dx[M(x) − ?M(x)?t]2dt

?

|ψt?;

(1)

here H is the standard quantum Hamiltonian of the system and the other two terms induce the

collapse. The mass m0is a reference mass, which is taken equal to that of a nucleon. The parameter

γ is a positive coupling constant which sets the strength of the collapse process, while M(x) is a

smeared mass density operator:

M (x) =

?

j

mjNj(x),Nj(x) =

?

g (x − y)ψ†

j(y)ψj(y) d3y,(2)

ψ†

j(y,s), ψj(y,s) being, respectively, the creation and annihilation operators of a particle of type

j at the space point y. The smearing function g(x) is taken equal to

g(x) =

1

?√2πrC

?3e−x2/2r2

C, (3)

where rC is the second new phenomenological constant of the model. Wt(x) is an ensemble of

independent Wiener processes, one for each point in space, which are responsible for the random

character of the evolution; the quantum average ?M(x)?t = ?ψt|M(x)|ψt? is responsible for its

nonlinear character.

As shown e.g. in [8], the averaged density matrix evolution associated to Eq. (1) can also

be derived from a standard Schr¨ odinger equation with a random Hamiltonian. Such an equation

does not lead to the state vector reduction, because it is linear; nevertheless, since they both

reproduce the same noise averaged density matrix evolution, and since physical quantities like the

photon emission rate can be computed from the noise averaged density matrix, the non-collapsing

equation can equally well be employed to compute such quantities. The advantage of this second

approach is that, being based on a linear (stochastic) Schr¨ odinger equation, it is much easier from

the computational point of view. In our case, the stochastic Hamiltonian is:

HTOT= H − ?√γ

?

j

mj

m0

?

N(y,t)ψ†

j(y)ψj(y)d3y(4)

where:

N(y,t) =

?

g(y − x)ξt(x)d3x, (5)

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and ξt(x) = dWt(x)/dt is a white noise field, with correlation function E[ξt(x)ξs(y)] = δ(t −

s)δ(x − y). As such, N(x,t) is a Gaussian noises field, with zero mean and correlation function:

E[N(x,t)N(y,s)] = δ(t − s)F(x − y),F(x) =

1

(√4πrC)3e−x2/4r2

C.(6)

The purpose of this article is to reconsider the analysis of the emission of radiation from a free

charged particle, previously discussed in the literature. Accordingly, in the following we will be

interested only in one type of particle, so from now on we will drop the sum over j.

The Hamiltonian HTOT can be written in terms of an Hamiltonian density HTOT. For the

systems we are interested in studying, we can identify three terms in HTOT:

HTOT = HP+ HR+ HINT.(7)

HPcontains all terms involving the matter field, namely its kinetic term, possibly the interaction

with an external potential V , and the interaction with the collapsing-noise:

HP =

?2

2m∇ψ∗· ∇ψ + V ψ∗ψ − ?√γm

m0Nψ∗ψ. (8)

HRcontains the kinetic term for the electromagnetic field:

HR=1

2

?

ε0E2

⊥+B2

µ0

?

,(9)

where E⊥ is the transverse part of the electric component and B is the magnetic component.

Finally HINT contains the standard interaction between the quantized electromagnetic field and

the non-relativistic Schr¨ odinger field:

HINT = i?e

mψ∗A · ∇ψ +e2

2mA2ψ∗ψ.(10)

The electromagnetic potential A(x,t) takes the form:

A(x,t) =

?

p,λ

αp

?

ǫp,λapei(p·x−ωpt)+ ǫ∗

p,λa†

pe−i(p·x−ωpt)?

,(11)

where αp=??/2ε0ωpL3and ωp= pc. We are quantizing fields in a cubical box of size L.

To analyze the problem of the emission rate, we will use a perturbative approach. We identify

the unperturbed Hamiltonian as that of the matter field (interaction with the noises excluded) plus

the kinetic term of the electromagnetic field:

H0 =

?2

2m∇ψ∗· ∇ψ + V ψ∗ψ + HR,(12)

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and we assume that its eigenstates and eigenvalues are known. In particular, we assume that the

matter part H0is diagonalizable. The perturbed term then is:

H1 = i?e

mψ∗A · ∇ψ +e2

2mA2ψ∗ψ − ?√γm

m0Nψ∗ψ.(13)

Such a division of HTOTin H0+ H1is justified by the fact that the effects of spontaneous col-

lapses driven by the noise field are very small at microscopic scales. This is also true for the

electromagnetic effects we are interested in computing.

III.FEYNMAN RULES

• ?

The Feynman diagrams for our model can be derived in a standard way, by means of the Dyson

series and Wick theorem. They are:

1. External lines (the symbol • denotes the generic space-time vertex (x,t)):

• = uk(x)e−i

p,λ

• = αpǫp,λei(p·x−ωpt)

k

?

?Ekt

• ?

k= u∗

k(x)e

i

?Ekt

?

p,λ= αpǫ∗

p,λe−i(p·x−ωpt)

? • = N (x,t)

The functions uk(x) are the eigenstates of −?2

2m∇2+ V , and Ekis the associated eigenvalue:

?

−?2

2m∇2+ V

?

uk(x) = Ekuk(x).

Since the noises field N is treated classically, there is no distinction between incoming and outgoing

lines.

2. Internal lines. The propagators for the matter field and for the photons, are:

•

1

?

2• = F12,

•

1

?

2• = Plm

12,

with 1 ≡ (x1,t1), 2 ≡ (x2,t2) and:

F12 ≡ F(x1,t1;x2,t2) = θ (t2− t1)

?

k

uk(x2)u∗

k(x1)e−i

?Ek(t2−t1)

(14)

Plm

12

≡ Plm(x1,t1;x2,t2) = θ(t1− t2)

?

?

k,λ

α2

kǫl

k,λǫ∗m

k,λei[k·(x1−x2)−ωk(t1−t2)]

+ θ (t2− t1)

k,λ

α2

kǫm

k,λǫ∗l

k,λei[k·(x2−x1)−ωk(t2−t1)].(15)

3. Vertices. There are three types of vertices, corresponding to the three terms in the interaction

Hamiltonian H1:

????

= i?e

m∇,

??

??

=

e2

m,

????

= −?√γm

m0.

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In the first vertex, the derivative acts always on the incoming external line. In the second vertex,

e2/m appears in place of e2/2m (as one would naively expect by inspecting at Eq. (13)) in order

to take properly into account the multiplicity of the diagrams. The same rule applies also to the

standard scalar QED (without the noise term).

4. One has to integrate over space and time in all vertices

1

(i?)n

n

?

j=1

?tf

ti

dtj

?

L3dxj

Note that there is no factorial term 1/n! coming from the Dyson’s series, because this is canceled

by the multiplicity of the diagram1. Only diagrams containing double photon propagators, like:

??????

????

do not follow this rule. In such a case, one has to multiply by a factor 1/2 for each such loop.

IV.PHOTON EMISSION PROBABILITY AT FIRST PERTURBATIVE ORDER

At first order in√γ and e, there are two different contributions to the process of photon emission,

coming from the interaction of the free particle with the noise field:

i

?

2

?

1

?

p,λ

???

f

?

2

?

1

?

???

According to the rules previously outlined, the contribution of the first diagram is:

−1

?2αp

?

i?e

m

??

−?√γm

m0

??

L3dx1

k

?tf

ti

dt1

?t1

ti

dt2eiωpt1e−i

?Eit2e

i

?Eft1e−i

?Ek(t1−t2)

×

?

?

L3dx2ui(x2)e−ip·x1ǫ∗

p,λ· [∇uk(x1)]u∗

k(x2)u∗

f(x1)N(x2,t2),(16)

while the contribution of the second diagram is:

−1

?2αp

?

i?e

m

??

−?√γm

m0

??

L3dx1

k

?tf

ti

dt1

?t1

ti

dt2eiωpt2e−i

?Eit2e

i

?Eft1e−i

?Ek(t1−t2)

×

??

L3dx2uk(x1)u∗

k(x2)e−ip·x2ǫ∗

p,λ· [∇ui(x2)]u∗

f(x1)N(x1,t1).(17)

1More precisely, a diagram containing n vertices has a factor 1/n! in front, coming from thew Dyson’s expansion.

However, there are n! such identical diagrams, differing only in the way the vertices are numbered.

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Summing these two contributions, the transition amplitude Tfibecomes:

Tfi = −1

?2αp

?

i?e

m

??

−?√γm

m0

??

k

?tf

ti

dt1

?t1

ti

dt2e

i

?(Ef−Ek)t1e

i

?(Ek−Ei)t2

×

?

?f|eiωpt1e−ip·ˆ xǫ∗

p,λ· ∇|k??k|N(ˆ x,t2)|i? + ?f|N(ˆ x,t1)|k??k|eiωpt2e−ip·ˆ xǫ∗

p,λ· ∇|i?

?

,(18)

where we have introduced the position operator ˆ x. It is convenient to rewrite the above expression

in a more compact form. Since the correlation function (6) of the noise is a product of its time and

space components, as far as the average values are concerned we can replace N(x,t) with ξtN(x),

where ξtis a white noise in time, while N(x) is a Gaussian noise in space, with zero mean and

correlator F(x − y). We also introduce the following two operators:

?

Rp≡ αp

i?e

m

?

e−ip·xǫp,λ· ∇,

N ≡ −?√γm

m0

N(ˆ x).(19)

The first operator refers to the radiative contribution (hence the symbol R), the second one to

the interaction with the noise (hence the symbol N). Defining moreover the matrix elements

Rp

can write Eq. (18) in the following way:

ki≡ ?k|Rp|i? and Nki≡ ?k|N|i?, considering photons with linear polarization (ǫ∗

p,λ= ǫp,λ), we

Tfi = −1

?2

?

k

?tf

ti

dt1

?t1

ti

dt2e

i

?(Ef−Ek)t1e

i

?t(Ek−Ei)t2

×

?

eiωpt1ξt2Rp

fkNki+ eiωpt2ξt1NfkRp

ki

?

.(20)

This is the final expression of the first-order transition amplitude for a charged particle to emit a

photon, as a consequence of the interaction with the noise field. The particle might be free, as we

will consider in the next section, or interacting with an external potential.

V.EMISSION RATE FOR A FREE PARTICLE

In the case of a free charged particle, the initial and final states and the generic eigenstate of

HP are:

ui(x) =

1

√L3,uf(x) =eiq·x

√L3,uk(x) =eik·x

√L3,(21)

and we have chosen the reference frame where the particle is initially at rest. The corresponding

eigenvalues are Ek= ?2k2/2m, and similarly for Eiand Ef. The matrix elements for the radiative

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part can now be easily computed:

Rp

fk= ?f|Rp|k? =

1

L3

?

?

L3dxe−iq·x

−?e

m

?

αp

?

i?e

m

?

e−ip·xǫp,λ· ∇

?

eik·x

= αp

?

(ǫp,λ· q)δk,q+p, (22)

Nki = −?

Rp

Nfk = −?

√

λm

m0

1

L3

?

dxN(x)e−ik·x

(23)

ki= ?k|Rp|i? = 0,

√

(24)

λm

m0

1

L3

?

dxN(x)ei(k−q)·x

(25)

As we see, the contribution given by the second Feynman diagram is null. Therefore, in squaring

Eq. (20), taking the average with respect to the noise, we obtain the relatively simple expression:

E|Tfi|2=

1

?4

?

k

?

j

Rp∗

fjRp

fkE[N∗

jiNki] (26)

×

?t

0

dt1

?t1

0

dt2

?t

0

dt3

?t3

0

dt4eiat1eibt2eict3eidt4δ (t2− t4),

where we have set ti= 0 and tf= t, and moreover we have defined the constants:

a ≡1

?(Ef+ ?ωp− Ek),b ≡1

?(Ek− Ei),c ≡ −1

?(Ef+ ?ωp− Ej),d ≡ −1

?(Ej− Ei),

(27)

We focus the attention on the temporal part. We have:

T =

?t

?t

?t

1

ca

0

dt1

?t1

?t

?t

0

dt2

?t

?t

?t

0

dt3

?t3

?t

0

dt4eiat1eibt2eict3eidt4δ (t2− t4)

=

0

dt1

0

dt2

0

dt3

0

dt4eiat1eibt2eict3eidt4δ (t2− t4)θ(t1− t2)θ(t3− t4)

=

0

dt2

t2

dt1

t2

dt3eiat1ei(b+d)t2eict3

=

?

ei(c+a)t1 − eigt

ig

+ eiatei(g+c)t− 1

i(g + c)

+ eictei(g+a)t− 1

i(g + a)

+1 − ei(g+c+a)t

i(g + c + a)

?

,(28)

where we have defined:

g ≡ b + d =

1

?(Ek− Ej).(29)

Because of the relation a + b + c + d = a + c + g = 0, Eq. (28) simplifies to:

T =

1

ac

?e−igt− 1

ig

+eiat− 1

ia

+eict− 1

ic

− t

?

,(30)

We are now ready to replace the matrix elements Rp∗

for the free particle. The indices k,j become vector indices k,j labeling the wave number, and the

fjand Rp

fkwith the explicit expressions (22)

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constraints given by the deltas in the R terms (see Eq. (22)) suppress the two sums in Eq. (26).

This moreover implies:

g = 0 (31)

a = −c =

1

?(Ef+ ?ωp− Eq+p) =

?

pc −?p2

2m−?q · p

m

?

,(32)

Then expression (30) for T simplifies to:

T =

2

a3[at − sin(at)],(33)

We now focus on the remaining part of Eq. (26):

1

?4

?

k

?

j

Rp∗

fjRp

fkE[N∗

jiNki] =

1

?4α2

p

??e

m

?2

(ǫp,λ· q)2E[N∗

(p+q)iN(p+q)i],(34)

where we have taken into account the constraints coming from the Kronecker delta in Eq. (22).

The stochastic average gives:

E[N∗

(p+q)iN(p+q)i] = ?2γ

?m

m0

?21

L6

?

L3dx1

?

L3dx2ei(p+q)·(x1−x2)F (x1− x2). (35)

We make the change of variable: x = x1− x2and y = x1+ x2(the Jacobian is 1/8) and we use

the rule:

?+L

2

−L

2

dx1

?+L

2

−L

2

dx2f (x1,x2) =1

2

?+L

0

dx

?+(L−x)

−(L−x)

dy [f (x,y) + f (−x,y)],(36)

thus obtaining:

E[N∗

(p+q)iN(p+q)i] =

?m

= ?2γ

m0

?2

1

8L6

3?

i=1

?+L

0

dxi

?+(L−xi)

−(L−xi)

dyi

1

√4πrc

?

ei(p+q)ixi+ e−i(p+q)ixi?

e−x2

i/4r2

C. (37)

The integral over yigives:

1

2L

?+(L−xi)

−(L−xi)

dyi = 1 −xi

L.

(38)

The second term vanishes in the large L limit, so we can ignore it. We are left with:

E[N∗

(p+q)iN(p+q)i] = ?2γ

?m

m0

?21

L3

?+L

−L

dxei(p+q)·xF (x).(39)

In the large L limit, the integral gives the Fourier transform of the correlation function F. Taking

into account the form (6) of F, and collecting all pieces, we have:

E|Tfi|2= Λ(ǫp,λ· q)2e−(q+p)2r2

C

at − sin(at)

a3

,(40)

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where:

Λ = 21

?4α2

p

??e

m

?2

?2γ

?m

m0

?21

L3=

1

L6

γ?e2

ε0cm2

0p

(41)

collects all constant terms.

The emission rate Γ(p) can be computed from the transition probability, by differentiating over

time, and by summing over the momentum q of the outgoing particle and the polarization λ of

the emitted photon, according to the formula:

dΓ

d3p

=

?L

2π

?6?

dq

?

λ

∂

∂tE|Tfi|2. (42)

Let us choose the axes so that p = (0,0,p); in this way a, as given by Eq. (32), becomes a function

only of p and qz. The sum over polarizations then gives?

cancel with each other, so we can take safely the limit L → +∞. The sum over q then becomes a

triple integral. The two integrals over qxand qycan be easily computed, being Gaussian, and we

λ(ǫp,λ· q)2= q2

x+ q2

y. All factors L

obtain:

dΓ

d3p

= 2Λ

?√π

rC

??√π

2r3

C

??

dqze−(qz+p)2r2

C

1 − cos(at)

a2

.(43)

The above integral can be rewritten in the following way:

?

dqze−(qz+p)2r2

C1 − cos(at)

a2

=

m

?p

?

dz e−z2β2 1 − cos[(D − z)t]

(D − z)2

,(44)

where we have defined the following new quantities: z = ?p(qz+ p)/m, D = pc + ?p2/2m and

β = mrc/?p.Since β ≃ 10−13s and D ≃ pc ≃ 1019s−1for a non relativistic electron and

for radiation in the KeV region, the Gaussian term in Eq. (43) is vanishing small in the region

where 1 − cos[(D − z)t]/(D − z)2is most appreciably different from zero. Around the origin,

where the Gaussian is not negligible, the denominator varies slowly, and one can approximate

1/(D − z)2∼ 1/D2, and bring it out of the integral. What remains, apart the Gaussian term, is

1 − cos[(D − z)t]. The second term oscillates very rapidly and gives a negligible contribution to

the integral. Thus only the first term survives. When integrating moreover over all directions in

which the photon can be emitted, the emission rate becomes:

dΓ

dp

=

λ?e2

2π2ε0c3m2

0r2

cp,

(45)

with λ ≡ γ/8π3/2r3

expression, we have neglected the oscillating term, which averages to zero over typical experimental

cequal to the collapse rate first introduced in the GRW model [9]. In the above

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times. The above result is expressed in SI units. The transformation to CGS units simply requires

the replacement ε0→ 1/4π, in which case we obtain twice the results of [2].

The mathematical reason for such a difference lies in the type of approximations used to obtain

the final formula. Going back to Eq. (30), in [2] the following approximation was made:

1

ac

?e−igt− 1

ig

+eiat− 1

ia

+eict− 1

ic

− t

?

≃ −t

ac.

(46)

While this is legitimates in general, it gives problems in the free particle case. Here, as we have seen,

g = 0, meaning that the oscillating term depending on g becomes linear in t. This contributions

sums with the other linear term, giving the factor of 2 difference. We also note that the remaining

two oscillating terms are mathematically important, though physically negligible.Since for a

free particle a = −c, they reduce to a cosine, which makes sure that the integral in Eq. (43) is

convergent. Without it, the pole at the denominator would produce a divergence.

VI.EMISSION RATE IN THE NON-WHITE NOISE CASE

The factor of 2 difference, while not being significant in the white-noise case, has serious con-

sequences in the colored-noise case. To see this, we now generalize Eq. (45) to the case where the

collapsing noise has a correlation function which is not white in time:

E[N(x,t)N(y,s)] = f(t − s)F(x − y).(47)

We can start from Eq. (26) for the average transition probability:

E|Tfi|2=

1

?4

?

k

?

j

?

L3dz Rp

fkNki(z)Rp∗

fjN∗

ji(z) (48)

×

?t

0

dt1

?t1

0

dt2

?t

0

dt3

?t3

0

dt4eiat1eibt2eict3eidt4f (t2− t4),

where now f replaces the Dirac delta. The coefficients a,b,c and d are the same as in (27). The

only effect of the non-white noise is to modify the time dependent part of the transition probability,

which we consider separately:

T =

?t

0

dt1

?t1

0

dt2

?t

0

dt3

?t3

0

dt4eiat1eibt2eict3eidt4f(t2− t4). (49)

This can be rewritten as follows:

T =

?t

0

dt2

?t

t2

dt1

?t

0

dt4

?t

t4

dt3eiat1eibt2eict3eidt4f (t2− t4)

= −1

ac

?t

0

dt2

?t

0

dt4

?eiat− eiat2??eict− eict4?eibt2eidt4f (t2− t4).(50)

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There are four terms, which all have the following structure:

I ≡

?t

0

du

?t

0

dveiαueiβvF(u − v) =

?t

0

du

?t

0

dve

i

2[(α+β)(u+v)+(α−β)(u−v)]f (u − v).(51)

We perform the change of variable x = u − v and y = u + v. In these new variables, the integral

changes as follows:

?t

0

du

?t

0

dvf(u,v) =

1

2

?t

0

dx

?2t−x

x

dy [f(−x,y) + f(x,y)].(52)

In our case, the integrating variables separate, and the integral over y can be easily performed,

giving:

?2t−x

x

dy e

i

2(α+β)y= 4e

i

2(α+β)t

(α + β)

sin1

2(α + β)(t − x). (53)

Then, taking into account that f(u − v) = f(x) = f(−x), the double integral I reduces to:

?t

I = 4e

i

2(α+β)t

(α + β)

0

dxf(x)sin1

2(α + β)(t − x)cos1

2(α − β)x.(54)

Going back to Eq. (50), taking into account the relation a + b + c + d = 0, we can write:

T = −4

ac

?

e−i

2gt

g

?t

?t

?t

?t

0

dxf(x)sin

?1

?1

?1

2g(t − x)

?

?

?

cos

?1

?1

?1

?

2(b − d)x

?

−e

i

2at

a

0

dxf(x)sin

2a(t − x) cos

2ax

?

−e

i

2ct

c

0

dxf(x)sin

2c(t − x)cos

2cx

?

+1

2

0

dxf(x)(t − x)cos[(a + b)x].(55)

Of course, in the white noise case f(x) = δ(x), the above equation reduces to Eq. (30).2

In

computing the matrix elements Rp

the Kronecker delta of Eq. (22), which implies a = −c and g = 0. Accordingly, the expression for

T further simplifies to:

ijand Nijfor the free particle, a further constraint comes from

T =

2

a2

4

acos

??t

0

?1

dxf(x)(t − x)[cos(bx) + cos[(a + b)x]]

??t

−

2at

0

dxf(x)sin

?1

2a(t − x)

?

cos

?1

2ax

??

.(56)

2Note that

case.

?t

0dxδ(x)g(x) =

1

2g(0), for a general function g(x), must be used in the reduction to the white noise

Page 12

12

The next step, in computing the emission rate, is to compute the time derivative. Differentiating

with respect to the upper limit of the integrals, produces terms proportional to f(t), which vanish

in the large time limit, as we assume that the correlation function has a finite correlation time.

The remaining terms coming from the second line produce oscillating terms, which average to zero.

Thus, the only significant term, in the large time limit, is:

∂T

∂t

− − − →

t?∞

1

a2

?˜f(b) +˜f(a + b)

?

, (57)

where we have defined the Fourier transform of the correlation function:

˜f(ω) ≡ 2

?+∞

0

dtf(t)cos(ωt) =

?+∞

−∞

dtf(t)eiωt. (58)

Finally, in computing the integral over the final momentum of the particle (see Eq. (43)), we have

have approximated the Gaussian term by a Dirac delta, meaning that we are imposing q ≃ −p.

This implies:

a =

?

?(q + p)2

2m

pc −?p2

2m−?p · q

m

?

−→

?

pc +?p2

2m

?

≃ pc(59)

b =

−→ 0. (60)

Thus we have:

dΓ

dp

????

NON-WHITE

=

1

2

?˜f(0) +˜f(pc)

?

×dΓ

dp

????

WHITE

.(61)

The second term is the expected one: the probability of emitting a photon with momentum p is

proportional to the weight of the Fourier component of the noise corresponding to the frequency

ωp= pc. The first term instead is independent of the photon’s momentum, and looks suspicious.

Precisely this term, in the white-noise limit, is responsible for the factor of 2 difference, as one can

easily check. In the remaining sections we analyze the origin of such an unexpected term.

VII. COMPUTATION USING A GENERIC FINAL STATE FOR THE CHARGED

PARTICLE

At the end of Section V we have discussed that the factor of 2 difference arises because g as

defined in Eq. (29) becomes 0, the reason being that the deltas in the R terms (see Eq. (22)) force

Ekto be equal to Ej. One could then expect that by considering a generic final state for the

outgoing particle—in place of the more artificial plane wave—such a constraint is removed. We

Page 13

13

discuss such a possibility in this Section. So, instead of a final state for the particle with definite

momentum, let us now take a normalized wave packet:

ui(x) =

1

√L3,uf(x) =

?

∆

h(∆)ei(q+∆)·x

√L3

,uk(x) =eik·x

√L3, (62)

where h(∆) normalizes the wave function:

1 =

?

L3dx|ψf|2=

1

L3

?

∆′

?

∆

h∗?∆′?h(∆)

?

L3dxe−i(q+∆′)·xei(q+∆)·x=

?

∆

|h(∆)|2.(63)

The matrix elements (22)–(25) now become:

Rp

fk= ?f |Rp|k? =

?

?

∆

h∗(∆)?f∆|Rp|k? = αp

?

−e?

m

??

∆

h∗(∆)[ǫp· (q + ∆)]δk,q+∆+p. (64)

Nki = −?

Rp

Nfk = −?

√

λm

m0

1

L3

dxN(x)e−ik·x

(65)

ki= ?k|Rp|i? = 0,

√

(66)

λm

m0

1

L3

?

ki= 0, the formula of E|Tfi|2is still given by Eq. (26), where the temporal

part takes the same form as in (30):

∆

h(∆)

?

dxN(x)ei(k−q−∆)·x

(67)

Since also in this case Rp

E|Tfi|2=

1

?4

?

k

?

j

Rp∗

fjRp

fkE[N∗

jiNki] T,(68)

with T given in Eq. (30). The two Kronecker deltas coming from Eq. (64) set: k = q + ∆ + p and

j = q + ∆′+ p. Accordingly, the coefficients a, c and g, defined in (27) and (29), become:

a =(Ef+ ?ωp− Eq+∆+p)

?

,c = −

?Ef+ ?ωp− Eq+∆′+p

?

?

,g =Eq+∆+p− Eq+∆′+p

?

. (69)

Moreover, we have:

E[N∗

jiNki] = ?2γ

?m

m0

?21

L6

?

L3dx1

?

L3dx2e−i(p+q)·(x1−x2)e−i(∆·x1−∆′·x2)F(x1− x2).(70)

The two exponents can be rewritten as: −i(p + q)·(x1−x2)−(i/2)(∆−∆′)·(x1−x2)−(i/2)(∆+

∆′) · (x1+ x2). We now make the change of variables: x = x1− x2, y = x1+ x2, as we did after

Eq. (35). The integral over y produces L3δ∆,∆′ plus extra terms which vanish in the large L limit.

Thus, as in the previous section, a, c and g take the values:

a = −c =

(Ef+ ?ωp− Eq+∆+p)

?

,g = 0.(71)

Page 14

14

The integral over x gives the Fourier transform of the correlator F; accordingly, the transition

probability reduces to:

E|Tfi|2= Λ

?

∆

|h(∆)|2[ǫp· (q + ∆)]2e−(q+∆+p)2r2

c

?at − sin(at)

a3

?

,(72)

with Λ defined as in (41). As we see, the structure is minimally modified from that of Eq. (40),

and the answer of [3] for the reduction rate is still obtained.

VIII. COMPUTATION WITH A NOISE CONFINED IN SPACE

The calculation of the previous Section shows that the reason why g = 0 also for an outgoing

wave packet, is because a δ∆,∆′ appears, which arises from the integral over space with respect to

the variable y = x1+ x2. This suggests that the problem can be avoided by considering a noise

which is confined to a finite region of space. We analyze this case here.

Let us suppose that the correlation function of the noise is:

E[N(x,t)N(y,s)] = f(t − s)F(x − y)e−(x+y)2/ℓ2,(73)

where ℓ is an appropriate cut off. We start from Eq. (68)

E|Tfi|2=

1

?4

?

k

?

j

Rp∗

fjRp

fkE[N∗

jiNki]T, (74)

where the temporal part T is given by Eq. (55). The two Kronecker deltas coming from Eq. (64)

set: k = q + ∆ + p and j = q + ∆′+ p. In this case, Eq. (70) is replaced by:

E[N∗

jiNki] = ?2γ

?m

m0

?21

L6

?

L3dx1

?

L3dx2e−i(p+q)·(x1−x2)e−i(∆·x1−∆′·x2)F(x1− x2)e−(x+y)2/ℓ2.

(75)

As before, we perform the change of variables: x = x1− x2, y = x1+ x2. In integrating over the

new variables, we use the rule:

?+L

2

−L

2

dx1

?+L

2

−L

2

dx2f (x1,x2) =1

2

?L

0

dx

?+(L−x)

−(L−x)

dy [f (x,y) + f (−x,y)].(76)

In our case:

f(x1,x2) = ei(q+∆′+p)·x1e−i(q+∆+p)·x2F(x1− x2)e−(x1+x2)2/l2

= ei

(77)

?

q+∆′+∆

2

+p

?

·xe

i

2(∆′−∆)·yF(x)e−y2/l2=

3?

i=1

fi(xi,yi),(78)

fi(xi,yi) = e

i

2(j+k)ixie

i

2(∆′−∆)iyi

1

√4πrC

e−x2

i/4r2

Ce−y2

i/l2, (79)

Page 15

15

where we have used the Kronecker delta constraints to replace q + p +1

2(∆ + ∆′) by1

2(j + k).

Thus we arrive at the following expression:

E[N∗

jiNki] = ?2γ

?m

m0

?2

1

8L6

3?

i=1

?L

e−x2

0

dxi

?+(L−xi)

−(L−xi)

dyi2cos

?1

2(j + k)ixi

?

× e

i

2(∆′−∆)iyi

1

√4πrC

i/4r2

Ce−y2

i/l2.

(80)

Since e−x2

i/4r2

C has a cutoff at |xi| ∼ rC≪ L, xinever approaches L. So we can write (in the large

L limit):

E[N∗

jiNki] = ?2γ

?m

?1

m0

?21

?+L

L3

3?

i=1

?L

?

0

dxi2cos

?1

2(j + k)ixi

?

1

√4πrC

e−x2

i/4r2

C

×

3?

j=1

2L

−L

dyj

e

i

2(∆′−∆)jyje−y2

j/l2.(81)

To summarize, substituting Eq. (81) into Eq. (74) and noting the Kronecker delta in Rp

effect of having considered a final wave packet in place of a plane wave, and of having confined the

fk, the

noise in space, is that the double Kronecker delta δk,q+pδj,q+pis replaced by:

Kjk =

?

∆

?

∆′

h∗(∆)h(∆′)δk,q+∆+pδj,q+∆′+p

3?

j=1

?1

2L

?+L

−L

dyj

?

e

i

2(∆′−∆)·ye−y2/l2.(82)

One can easily check that when ℓ = ∞, the triple integral reduces to δ∆,∆′. Kjkthen becomes:

?

δ∆,0, i.e. when the final state is a plane wave. Thus both a final wave-packet state and a noise

∆|h(∆)|2δk,q+∆+pδj,q+∆+p, which implies j = k. The same happens when ℓ < ∞, but h(∆) =

confined in space are necessary in order to avoid the factor of 2 term.

Assuming that ℓ ≪ L (thus extending the integration limits to infinity), and using the Kronecker

deltas, we can rewrite Eq. (82) as follows:

Kjk = h∗(k − q − p)h(j − q − p)

1

8L3

?

dye

i

2(k−j)·ye−y2/l2.(83)

The diagonal elements of Kjk, which multiply the factor of 2 term, give a contribution proportional

to:

?

k

Kkk =

?

k

|h(k − q − p)|21

8L3

?

dye−y2/l2

∝

?ℓ

L

?3

≪ 1.(84)

Accordingly, the undesired term becomes negligible. Let us consider the remaining terms. The

overall transition probability can be written as:

E|Tfi|2=

?

k

?

j

ΘkjKkj,(85)

Page 16

16

where Θkjcollects all terms which are not crucial for the subsequent evaluation. As in normal

experimental situations, we assume that the noise is confined in a region which is much bigger

than the typical width of the wave packet: ℓ ≫ ∆−1. We can then write:

E|Tfi|2≃ Θq+p,q+p

k

?

h∗(k − q − p)

?

k−j

h(j − q − p)

1

8L3

?

dye

i

2(k−j)·ye−y2/l2.(86)

In the above expression, we have used the fact that h is peaked where its argument is 0, and we

have shifted the second sum from j to k − j. The integral, being proportional to e−(k−j)2/16ℓ2is

appreciably different from zero only for |k − j|ℓ ? 1, i.e. |k − j| ? ℓ−1≪ ∆. This means that we

can take: h(j − q − p) = h((k − p − q) − (k − j)) ≃ h(k − p − q). So we have:

E|Tfi|2≃ Θq+p,q+p

k

?

|h(k − q − p)|21

8L3

?

dy

?

k−j

e

i

2(k−j)·ye−y2/l2

= Θq+p,q+p, (87)

where we have used the box normalization to continuum normalization replacement

1

L3

?

k−j

→

1

(2π)3

?

d3(k − j) (88)

to write

1

8L3

?

k−j

e

i

2(k−j)·y= δ(y).(89)

So this term is not suppressed for ℓ ≪ L, provided that we take: L ≫ rCand L ≫ ℓ ≫ ∆−1. All

these conditions are consistent with typical experimental situations. In this regime, the extra term

found in [3] is negligible, and the Golden Rule formula used as the basis for the calculations of [1],

[2] gives the entire answer.

In terms of continuum state normalization, the point g = 0 that gave the extra contribution from

the first line of Eq. (55) is a set of measure zero, and so this point does not give a contribution

to ∂T/∂t in the large t limit. Since the dependence on l has dropped out, the final answer is

independent of the assumed spatial cutoff on the noise. So once the box size L → ∞, we can

then take the limit l → ∞ and eliminate the noise spatial cutoff l. We pose the question: If the

calculation is repeated without a spatial cutoff on the noise, but with the initial and final electron

wave functions taking the interaction with the noise into account (in analogy with the distorted

wave Born approximation), will the extra term found in [3] then be suppressed?

IX.SPONTANEOUS EMISSION FROM THE VACUUM

The Feynman rules suggest that also the following process is possible:

?

1

??Æ?

2

?

Page 17

17

which corresponds to a photon emitted from the vacuum. The analytical expression for such a

process contains two internal particle’s propagators F12F21∝ θ(t2− t1)θ(t1− t2) (see Eq. (14)),

giving a zero contribution. Thus at the non-relativistic level there is no spontaneous photon

emission from the vacuum. However, such a process is expected not to vanish at the relativistic

level.

Acknowledgements

A.B. acknowledges partial financial support for the grant PRIN 2008 of MIUR, Italy. He also

wishes to acknowledge the hospitality of the Institute for Advanced Study in Princeton, where part

of this work has been done. S.L.A. acknowledges the hospitality of the Abdus Salam International

Centre for Theoretical Physics, where this work was completed. He also acknowledges support of

the Department of Energy under grant DE-FG02-90ER40642.

Appendix: Noise in the box

Let us consider a Gaussian noise, with zero mean and correlation function:

E[N(x,t),N(y,s)] = δ(t − s)F(x − y). (90)

Writing it in Fourier components:

N(x,t) =

1

(2π)3

?

dkeik·x˜ N(k,t), (91)

we easily find the following relation for the correlation function in momentum space:

E[˜ N(k,t),˜ N(k′,s)] = (2π)3δ(t − s)δ(k + k′)˜F(k),(92)

where:

˜F(k) ≡

?

dxe−ik·xF(x)(93)

is the Fourier transform of the spatial correlator. In placing the noise in a box of size L, we select

only the Fourier components with the correct boundary conditions. therefore we define:

NL(x,t) ≡

1

L3

+∞

?

j=−∞

ei2π

Lj·x˜ NL(j,t),

˜ NL(j,t) ≡˜ N

?2π

Lj,t

?

.(94)

Page 18

18

From Eq. (92) on can write the correlation function of˜ NL(j,t):

E[˜ NL(j,t),˜ NL(j′,s)] = L3δ(t − s)δj,−j′˜F(2π

Lj), (95)

from which one finds the following correlator for the noise in the box NL(x,t):

E[NL(x,t),NL(y,s)] = δ(t − s)FL(x − y′), (96)

with:

FL(x − y) ≡

1

L3

+∞

?

j=−∞

ei2π

Lj·(x−x′)˜F(2π

Lj).(97)

One can easily prove that in the limit L → ∞, the noise NL(x,t) as defined in Eq. (94), converges

to N(x,t), and the correlation function FL(x − y) as defined in Eq. (97), converges to F(x − y).

[1] Q. Fu, Phys. Rev. A56, 1806 (1997).

[2] S.L. Adler and F.M. Ramazanoˇ glu, J. Phys. A 40, 13395 (2007).

[3] A. Bassi and D. D¨ urr, J. Phys. A 42, 485302 (2009).

[4] S.L. Adler, J. Phys. A 40, 2935 (2007).

[5] S.L. Adler and A. Bassi, Science 325, 275 (2009).

[6] G.C. Ghirardi, P. Pearle and A. Rimini, Phys. Rev. A 42, 78 (1990).

[7] L. Di´ si, Phys. Rev. A 40, 1165 (1989); 42, 5086 (1990).

[8] S.L. Adler and A. Bassi, J. Phys. A 40, 15083 (2007). See also references therein.

[9] G.C. Ghirardi, A. Rimini and T. Weber, Phys. Rev. D 34, 470 (1986).