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.
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 , while the analysis
of  has been done within the QMUPL model . In spite of the fact that the two models should
give the same predictions in an appropriate limit , the formula derived in  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: email@example.com
†Electronic address: firstname.lastname@example.org
‡Electronic address: email@example.com
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:
dx[M(x) − ?M(x)?t]dWt(x) −
dx[M(x) − ?M(x)?t]2dt
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) =
g (x − y)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
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
As shown e.g. in , 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
g(y − x)ξt(x)d3x, (5)
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) =
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:
2m∇ψ∗· ∇ψ + V ψ∗ψ − ?√γm
HRcontains the kinetic term for the electromagnetic field:
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
The electromagnetic potential A(x,t) takes the form:
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:
2m∇ψ∗· ∇ψ + V ψ∗ψ + HR,(12)
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:
h∗(k − q − p)
h(j − q − p)
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:
|h(k − q − p)|21
= Θq+p,q+p, (87)
where we have used the box normalization to continuum normalization replacement
d3(k − j) (88)
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  is negligible, and the Golden Rule formula used as the basis for the calculations of ,
 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  then be suppressed?
IX.SPONTANEOUS EMISSION FROM THE VACUUM
The Feynman rules suggest that also the following process is possible:
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
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:
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)
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(j,t) ≡˜ N
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
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)
FL(x − y) ≡
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).
 Q. Fu, Phys. Rev. A56, 1806 (1997).
 S.L. Adler and F.M. Ramazanoˇ glu, J. Phys. A 40, 13395 (2007).
 A. Bassi and D. D¨ urr, J. Phys. A 42, 485302 (2009).
 S.L. Adler, J. Phys. A 40, 2935 (2007).
 S.L. Adler and A. Bassi, Science 325, 275 (2009).
 G.C. Ghirardi, P. Pearle and A. Rimini, Phys. Rev. A 42, 78 (1990).
 L. Di´ si, Phys. Rev. A 40, 1165 (1989); 42, 5086 (1990).
 S.L. Adler and A. Bassi, J. Phys. A 40, 15083 (2007). See also references therein.
 G.C. Ghirardi, A. Rimini and T. Weber, Phys. Rev. D 34, 470 (1986).