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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.