andFlavour Oscillations

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Sede amministrativa del Dottorato di Ricerca

DIPARTIMENTO DI FISICA

XXVI CICLO DEL

DOTTORATO DI RICERCA IN FISICA

Electromagnetic Radiation Emission and Flavour Oscillations in Collapse Models

Settore scientifico-disciplinare FIS/02

DOTTORANDO: COORDINATORE DEL DOTTORATO DI RICERCA:

Sandro Donadi Prof. Paolo Camerini

Firma:

SUPERVISORE DI TESI:

Dr. Angelo Bassi Firma:

ANNO ACCADEMICO 2012/2013

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List of abbreviations and symbols

Name Value Description

GRW Ghirardi-Rimini-Weber collapse model.

CSL Continuous Spontaneous Localization collapse model.

QMUPL Quantum Mechanics with Universal Position

Localizations collapse model.

~ 6, 6× 10⁻³⁴ J s reduced Planck constant dened as ~ = _2π^h with h being the Planck constant.

e −1, 6 × 10⁻¹⁹ C charge of the electron.

c 3× 10⁸ m s⁻¹ speed of light.

ǫ₀ 8, 9× 10⁻¹² F m⁻¹ vacuum permittivity.

m0 9, 4× 10⁸ eV c⁻² mass of the nucleon.

rC 10⁻⁷ m typical size of the localizations.

λ localization rate in collapse models.

λ (GRW) 10⁻¹⁶ s⁻¹ value of λ proposed by Ghirardi, Rimini and Weber.

λ (ADLER) 10⁻⁸ s⁻¹ value of λ proposed by Adler.

γ λ8π^3/2r_c³ cm³ s⁻¹ localization constant for the CSL model.

λ (QMUPL) _2r^λ2

c cm⁻² s⁻¹ localization constant for the QMUPL model.

m g mass of the particle.

κ N m⁻¹ force constant of the harmonic oscillator.

ω0

pκ/m s⁻² natural frequency of the harmonic oscillator.

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List of publications directly related to this thesis

1. S. L. Adler, A. Bassi, S. Donadi. On spontaneous photon emission in collapse models;

Journ. Phys. A: Math. Theor. 46, 245-304 (2013).

Link to Arxiv: http://arxiv.org/abs/1011.3941.

The most important contents of this article are reported in CHAPTER 3.

2. A. Bassi, S. Donadi. Spontaneous photon emission from a non-relativistic free charged particle in collapse models: A case study; Phys. Lett. A 378, 761-765 (2014).

3. S. Donadi, A. Bassi, D.-A. Deckert. On the spontaneous emission of electromagnetic radiation in the CSL model; Annals of Physics 340, Issue 1, 70-86 (2014).

4. S. Donadi, A. Bassi, C. Curceanu, L. Ferialdi. The eect of spontaneous collapses on neutrino oscillations; Found. Phys. 43, 1066-1089 (2013).

5. S. Donadi, A. Bassi, C. Curceanu, A. Di Domenico, B. C. Hiesmayr. Are Collapse Models Testable via Flavor Oscillations?; Found. Phys. 43, 813-844 (2013).

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Link to Arxiv:http://arxiv.org/abs/1207.6000.

6. M. Bahrami, S. Donadi, L. Ferialdi, A. Bassi, C. Curceanu, A. Di Domenico, B. C.

Hiesmayr. Are collapse models testable with quantum oscillating systems? The case of neutrinos, kaons, chiral molecules; Sci. Rep.3, 1952 (2013).

The most important contents of this article are reported in CHAPTER 7 and 8.

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Chapter 1 Introduction and main results

In the last decades the interest of the scientic community in better understanding the limits of validity of quantum mechanics has increased. Indeed, many scientist (among them, e.g., the Nobel laureates A. Leggett and S. Weinberg [1, 2]) now believe quantum mechanics, as it is now, is a phenomenological theory and not a fundamental one.

Among the reasons why quantum mechanics should be modied, perhaps the most important is the so called measurement problem. The problem can be stated as follow: it is well known that, for microscopic systems, the quantum superposition principle holds. This has been veried in many experiments during the last century, e.g., diraction and interference experiments [3, 4, 5, 6, 7, 8]). However, when we move toward the macroscopic scale, the superposition principle seems to break down: we never see macro-objects in superposition of dierent position states in our daily life. In order to explain this behavior, the wave packet reduction postulate (or collapse postulate) has been introduced in quantum mechanics. This poses a problem: since measurement instruments and observers are made of the same atoms that are supposed to follow quantum mechanical rules, it is not clear why for these systems the Schrödinger equation does not hold and one has to use the wave packet reduction postulate. Moreover, even accepting the presence of two completely dierent dynamics the rst one, given by the Schrödinger equation which is linear and deterministic while the other one, described by the wave packet reduction postulate, is non linear and stochastic the theory does not explain clearly which systems should be considered as measurement instrument and which should not. To quote Bell [9]: What exactly qualies some physical systems to play the role of measurer. Was the wave function of the world waiting to jump for thousands of millions of years until a single-celled living creature appeared? Or did it have to wait a little

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longer for some better qualied system...with a PhD?. Therefore in quantum mechanics, the limit between micro systems, obeying a linear dynamics and macro systems, obeying a non linear collapse dynamics, is ambiguous.

In the last century dierent solutions for this problem were proposed. Some of them change the interpretation of the theory keeping the same physical predictions of quantum mechanics. This is the case, for example, of Bohmian Mechanics [10, 11, 12, 13]. A dierent way out of the problem is given by collapse models [14, 15, 16, 17]. The idea underlying these models is the following: each physical system interacts with a noise eld which induces the collapse of the wave function in space. These models are engineered in a such way that the eect of the noise is almost negligible for microscopic systems but, because of an amplication mechanism, this interaction becomes predominant for macro systems. As such, within a unique dynamics, collapse models provide an explanation of why micro systems have a quantum behavior while macro systems behave classically.

Collapse models make predictions dierent from quantum mechanics, hence they can be tested. Many dierent phenomena and experimental data have been studied so far and for all of them the predictions of quantum mechanics and of collapse models were compared. The dierence between these predictions is too small to be detected with the current technology and so there is not yet a decisive test of collapse models. However, these experiments set quite interesting bounds on collapse models' parameters. The state of the art of this research and the related bounds on the parameters of collapse models can be found in [18, 19].

In this thesis we focus two phenomena where the predictions of quantum mechanics and collapse models are dierent: (i) electromagnetic radiation emission from charged systems and (ii) avour oscillations. We analyzed both of them and obtained a quantitative prediction of the deviations from standard quantum behavior.

In the rst part of the thesis we study the electromagnetic radiation emission in collapse models. The interest in this phenomenon is due to the fact that so far it sets the strongest bound on the parameters of collapse models [18, 19]. Since in collapse models any system always interacts with a noise (that induces the collapse of the wave function), if the system is charged this interaction induces the emission of radiation. To give an intuitive semi- classical picture, one can think of a system being accelerated by the noise and, because of this acceleration, it emits radiation.

In the literature there are several computations of the radiation emission rate. In [20, 21] the calculation was carried out to the rst perturbative order using the Continuous

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Spontaneous Localization (CSL) model. In [22] the formula for emission rate was found doing exact computations using the simpler QMUPL (Quantum Mechanics with Universal Position Localizations) model. The results of these calculations were not the same, as they were supposed to be. In order to clarify this issue, in the rst part of chapter 3 we repeat the computations performed in [20, 21]. We study the radiation emission in the Continuous Spontaneous Localization (CSL) model for a free particle, taking a white noise. Treating both the electromagnetic and the noise interactions as perturbations, we nd that the formula for the emission rate is:

d dkΓ_k

white

= λ~e²

2π²ε0c³m²₀r²_Ck, (1.1) where ~, c and ε0 have the usual meaning, λ and rC are two parameters introduced in collapse models, m0 the mass of a nucleon, k is the photon wave vector. Eq. (1.1) diers from the result found in [20, 21] but it agrees with the result found in [22]. We show that the origin of the discrepancy is that in [20, 21] some relevant contributions to the emission rate were neglected [23]. The next step is to generalize the perturbative calculations using a colored noise. We present this computation in the second part of chapter 3 and we show that the emission rate formula is:

d

dkΓk = 1 2

d dkΓk

white

· [ ˜f (0) + ˜f (ωk)], (1.2) where

f (ω) :=˜ Z _+∞

−∞

f (s)e^iωsds, (1.3)

is the noise spectral density where f(s) is the noise eld time correlation. The second term inside the square bracket on the right side of Eq. (1.2) is the expected one: the probability of emitting a photon with momentum k is proportional to the spectral density of the noise correlation at the frequency ωk = kc. On the other hand, the rst term is related to the spectral density of the noise at zero energy. Due to the presence of this term, even a noise with very low energy can induce an emission of high energetic photons. This is unexpected as the typical picture is that the noise gives energy to the particle, and such energy is converted into that of the emitted photon. Understanding the origin of this term and how to avoid it is one of the main task of this thesis. In the third and fourth part of chapter 3 we study a prescription to avoid the presence of the unphysical term ˜f (0). We show that, if one takes wave packets instead of plane waves as nal states and if one connes the noise in space,

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then the unphysical term is not present anymore. However, this procedure seems an ad-hoc solution and it is not clear if it is valid also for systems dierent from the free particle.

Moreover, it forces us to change the model by imposing a noise connement.

A deeper insight into the problem is obtained working with the QMUPL model, where an exact treatment of the problem is possible. Indeed, in chapter 4 we show that, in the case of the free particle, the unphysical term is still present. However, for a harmonic oscillator, the rate is¹:

dΓ dk ≃ 1

2 d dkΓk

white

·h

e^−γtf (0) + ˜˜ f (ωk)i

, (1.4)

where γ = ^ω_2m⁰²^β and β = _6πǫ^e²₀_c3. We see that the unphysical term, for large times, is suppressed by the exponential damping factor e^−γt. However, it is important to note that treating electromagnetic interaction at the lowest order, which is equivalent of setting β = 0, implies γ = 0and so the unphysical term ˜f (0)is not suppressed anymore. The same problem arises if we set ω0 = 0, which is the free particle case. Therefore, from this analysis we proved that in order to get a physically meaningful result, rst the particle cannot be treated as completely free, and second the electromagnetic interaction cannot be treated at the lowest perturbative order. In chapter 5 we use the above results to compute the emission rate with the CSL model. Here an exact treatment of the problem is not possible. However, from the previous analysis with the QMUPL model, we see that the damping factor e^−γt responsible for the decay of the unphysical term ˜f (0), does not depend on the noise. This means that the noise can be treated perturbatively, but the higher orders terms of the electromagnetic interaction must be considered. This is exactly what we do in chapter 5: we nd the emission rate for a harmonic oscillator in the CSL model treating the electromagnetic interaction exactly and the noise interaction perturbatively. The emission rate, in the free particle limit, is the one given in Eq. (1.2) without the unphysical term ˜f (0).

The method used in chapter 5 gives a meaningful result, but it requires treating the electromagnetic interaction exactly when solving the Heisenberg equations. This can be done only for simple systems. For more complicated systems, one has to resort to perturbation theory. Therefore, we look for a way to include the decay behavior due to the electromagnetic interaction into the perturbative approach. In chapter 6 we show that this can be done by

1In this formula we used the symbol ≃ instead of = because this is not the exact result of our computation, that is given by the much more complicated formula in Eq. (4.24). However, in order to avoid useless complication, here we can refer to this simplied version of that equation, which catches all the important points.

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taking into account the possibility that, because of the electromagnetic self interaction, the propagator decays. Indeed, as discussed for example in [24], the presence of any external perturbation makes the eigenstates of the unperturbed Hamiltonian unstable so that they can decay. We computed the emission rate for a generic system taking into account this eect and nally we found a general formula where the unphysical term is not present.

To conclude, we briey discuss the experimental bounds on the collapse parameter coming from the emission of radiation. Following the analysis of [20], it is show that a mass proportional coupling between the noise and the particle is required for the model to be consistent with experimental data.

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PART I: RADIATION EMISSION IN COLLAPSE MODELS Main results

- We nd a formula for the radiation emission rate in collapse models.

- We show that, in order to get a correct emission rate, the electromagnetic interaction cannot be treated at the lowest perturbative order.

Main steps of our analysis and relative results

Step 1 We compute the emission rate from a free particle to the lowest perturbative order using the CSL model .

The rate is given by Eq. (1.1): it is in agreement with the result found in [22] and twice of the result found in [20, 21]. We claried this discrepancy showing that in [20, 21] some relevant

contributions were neglected.

Step 2 We extend the calculation to a colored noise, obtaining the result in Eq. (1.2).

A problem arises: the emission rate contains an unphysical term, proportional to ˜f (0), which implies emission of high energy photons also in presence of weak noises.

Step 3 The calculation of the emission rate is repeated by taking wave packets as nal states and conning the noise. The unphysical factor disappears.

Problem: this procedure seems an ad-hoc solution and requires to modify the model conning the noise.

Step 4 To better understand the origin of the unphysical factor we compute the emission rate from a free particle and a harmonic oscillator using the QMUPL model. Besides the dipole approximation,

the calculation is carried out treating the interactions with the electromagnetic eld and the noise exactly.

The formula for the rate of a harmonic oscillator has the structure given in Eq. (1.4).

The unphysical term is not present when:

1) The particle is not completely free (ω06= 0);

2) The electromagnetic interaction is not treated at the lowest order (β 6= 0).

Step 5 We check if the results found with the QMUPL model are true also for the CSL model.

In order to do that, we solve the Heisenberg equations, treating exactly the electromagnetic interaction and perturbatively the interaction with the noise.

We compute the emission rate for a harmonic oscillator and we show that the unphysical term is not present.

Step 6 We introduce the eect of higher order terms of the electromagnetic interaction in the

lowest order perturbative calculations by taking into account the decay of the the propagator.

The result we nd is a formula for the emission rate from a generic system and a generic collapse model which does not contains unphysical terms anymore.

Table 1.1: Summary of the main results and the most important steps of the rst part of the thesis

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In the second part of the thesis we study the phenomenon of avour oscillations in collapse models. Flavour oscillations arise whenever avour eigenstates of a particle are dierent from its mass eigenstates. Then avour eigenstates are supposed to be linear superposition of mass eigenstates. According to quantum mechanics, during the time evolution the mass eigenstates of a free particle change by acquiring dierent phase factors, depending on their mass. Therefore, a particle that is initially in some avour eigenstates (which are the ones that are measured in practice), after some time has a non zero probability to be in another

avour eigenstate. This probability shows an oscillatory behavior in the course of time.

Collapse models describe a dierent evolution for the mass eigenstates due to the constant interaction with the noise. As a consequence, the formula for the evolution of the oscillations changes. In fact, by treating the noise as perturbation, we show that in collapse models the oscillation probability is damped by an exponential factor. We performed this computation for two dierent types of particles: neutrinos in chapter 7 and mesons in chapter 8. The decay rate ξjk in the case of neutrinos is:

ξjk = γ

16π^3/2r_C³m²₀c⁴

m²_jc⁴

E_i^(j) −m²_kc⁴ E_i^(k)

!2

, (1.5)

where γ = λ8π^3/2r³_c and the label j(k) refers to the mass eigenstate with mass mj(mk) and energy E_i^(j)(E_i^(k)), with E_i^(j) = q

p²_ic²+ m²_jc⁴ and pi the momentum of the particle. Using available experimental data, we quantify the damping factors for these particles. The result for cosmogenic, solar and laboratory neutrinos are summarized in the following table:

cosmogenic solar laboratory

E(eV) 10¹⁹ 10⁶ 10¹⁰

t(s) 3.15× 10¹⁸ 5× 10² 2, 13× 10⁻² ξijt 2.31× 10⁻⁵⁵ 3.66× 10⁻⁴⁵ 1.56× 10⁻⁵⁷

For each type of neutrinos, in the rst line we report the typical order of magnitude of their energies, in the second line their typical times of ight and in the third line the damping factor obtained using Eq. (1.5). We see that, on the contrary to what was claimed in a previous work in the literature [25], the eect is very small.

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Regarding mesons, the decay rate ΛCSL in the non relativistic regime is:

Λ_CSL = γ (mj − m^k)²

16π^3/2r³_Cm²₀ (1.6)

Notice that this decay rate is equivalent to the one found for neutrinos in the non relativistic limit. In the following table we report, for dierent type of mesons, the decay rate ΛCSL

computed using Eq. (1.6) and the typical decay widths Γ due to the weak interactions:

K-mesons B-mesons Bs-meson D-mesons Λ_CSL(s⁻¹) 1.5 × 10⁻³⁸ 1.4× 10⁻³⁴ 1.7× 10⁻³¹ 3.2× 10⁻³⁷

Γ(s⁻¹) 1.2× 10¹⁰ 6.6× 10¹¹ 6.6× 10¹¹ 2.4× 10¹²

Compared to neutrinos, the collapse eect for mesons is stronger because they have larger masses, but still too small to be detectable. Moreover, it is also many orders of magnitude smaller than the decay due to weak interactions, so mesons decay before we can see any collapse eect in their oscillation. We also considered the case of a pair of entangled mesons and we showed that even in this case the eect is undetectable.

PART II: FLAVOUR OSCILLATIONS IN COLLAPSE MODELS Main results

- The eect of collapse models is to damp the avour oscillations.

- We compute the damping factor for neutrinos and mesons.

Neutrinos Neutrino oscillations are damped by an exponential factor with decay rate given in Eq. (1.5).

The eect is very small: the exponents in the damping factor are in a range between 10⁻⁴⁵ and 10⁻⁵⁷, depending on which type of neutrino is considered.

Mesons We compute the eects of collapse models on non relativistic mesons. The oscillations are damped by an exponential factor with decay rate given in Eq. (1.6).

This decay rate is many order of magnitudes smaller than the one due to the weak interactions.

The computation is repeated for a pair of entangled particles. We show that the noise acts

independently on each particle, so entanglement does not play any special role for this phenomenon.

Table 1.2: Summary of the main results of the second part of the thesis

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Chapter 2 Collapse models

In this chapter we introduce collapse models, explain their most important features and how they resolve the measurement problem. Collapse models modify the Schrödinger dynamics in a non linear way to include the collapse of the state vector. To avoid the possibility of having faster than light signaling (for example by manipulating entangled systems) the new dynamics must be also stochastic. Indeed, it has been proved that a non linear and deterministic time evolution allows faster than light signaling [26, 27]. The deviations of this new dynamics from the Schrödinger dynamics should be small for microscopic systems, in order to avoid contradictions with experiments. On the other hand, when a macroscopic system is considered, in order to solve the measurement problem the new dynamics should assure a well dened position to the center of mass of the system. Collapse models fulll all of these requirements [16, 17]. Therefore, within an unique dynamics, collapse models describe the typical quantum behavior of the microscopic systems and explain the collapse of the wave function for a macro system.

In the rst section of this chapter we introduce and explain the main properties of the

rst collapse model proposed in the literature: the Ghirardi-Rimini-Weber (GRW) model [14].

In the second section we introduce the dynamical equation of collapse models in the most general way, and we study its main features. Then, in the third section, we introduce the Continuous Spontaneous Localization (CSL) model [15] and the Quantum Mechanics with Universal Position Localization (QMUPL) model [28]. The CSL and the QMUPL models are the ones that we will use in this thesis.

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2.1 The GRW model

The GRW model was proposed in 1986 by Ghirardi, Rimini and Weber. In the GRW model each elementary constituent of matter is subject to random and spontaneous localizations.

The localizations amount to the collapse of the wave function in space. The important point is that these localizations processes are supposed to be part of the laws of nature, not something that happens only when an observer performs a measurement on the system.

2.1.1 Postulates of the GRW model

The postulates of the GRW model are the following:

1. Each particle is subject to spontaneous and random localizations, described by a Pois- sonian process with the mean rate λi (where "i" labels the i-th particle of the system).

2. The localization process for the i-th particle changes the state vector as follow:

|Ψi −→ Lⁱ_a|Ψi kLⁱa|Ψik, with

Lⁱ_a = πr_c²_−3/4 e⁻

(qi−a)²

2r2c ,

where qi is the position operator of the i-th particle, a is the point around which the particle is localized, and rc is the typical size of the localization.

3. The probability of having a localization around the point a is:

Pⁱ(a) =

Lⁱ_a|Ψi ².

4. When there is no localization, the system evolves according to the Schrödinger equation:

i~d

dt|Ψ (t)i = H |Ψ (t)i , where H is the standard quantum Hamiltonian.

The rst postulate describes the probability of having a localization at a certain time. The localization rates λi are chosen to be equal for all particles to λ = 10⁻¹⁶ s⁻¹ and this is a new

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parameter of the model. Such a choice implies that the probability of having a localization (and therefore a deviation from the behavior predicted by the Schrödinger equation) for a single particle is very small. However, because of an amplication mechanism that we will describe later, the rate of localization for a macroscopic object turns out to be much larger, making the localization more eective. The second postulate describes in a precise way what is meant by localization: the wave function of the system is coarse-grained with the spatial resolution rC. The parameter rC is the second new parameter introduced by the model and is chosen to be equal to 10⁻⁷ m. The coarse-graining is precisely expressed by multiplying the wave function with a gaussian centered at a with width rC. The third postulate sets the probability of having a localization around a given point a. This is the GRW model counterpart of the Born rule in quantum mechanics. It has been recently proven that this choice of probability is the only way to avoid the possibility of having faster than light signaling [27].

2.1.2 GRW master equation

Here we derive the equation for the density matrix in the GRW model. The dynamics of the GRW model evolves pure states into statistical mixtures. Indeed, when a localization happens, the state of the system |ψ (t)i evolves into one of the states _kL^Lⁱ^aⁱ_a^|Ψ(t)i_|Ψ(t)ik := |^Ψⁱ^a^(t)i

k|Ψⁱa(t)ik

with probability Pⁱ(a, t) = k|Ψⁱa(t)ik². This means that before a localization occurs the density matrix of the system is given by the pure state |Ψ (t)i hΨ (t)|, while after it is given by a statistical mixture of pure states |^Ψⁱ^a^(t)ih^Ψⁱ^a^(t)|

k|Ψⁱa(t)ik² , each one weighted with the probability Pⁱ(a, t). Therefore, a localization of the i-th particle of the system corresponds to the following change of the density matrix ρ(t):

ρ (t) = |Ψ (t)i hΨ (t)| localization−→ ρ (t) = Z

da|Ψⁱa(t)i hΨⁱa(t)|

k|Ψⁱa(t)ik² Pⁱ(a, t) =

= Z

da

Ψⁱ_a(t)

Ψⁱ_a(t) =

Z

daLⁱ_a|Ψ (t)i hΨ (t)| Lⁱa . (2.1) If we introduce the map:

Ti[ρ(t)] :=

Z

daLⁱ_aρ(t) Lⁱ_a (2.2)

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then the localization process can be described as:

ρ (t)−→ Ti[ρ (t)] . (2.3)

Now we have all what we need to derive the master equation for the GRW model. Since the localizations follow a Poissonian statistic, in the innitesimal time dt there is a probability λ_idt to have a collapse whose dynamics is given by Eq. (2.3) and a probability 1 − λidt that no collapse happens, so the system evolves accordingly to the Schrödinger equation. This means that the matrix density ρ(t) evolves as:

ρ (t + dt) = (1− λidt)

ρ (t)− i

~[H, ρ (t)] dt

+ λ_idtT_i[ρ (t)] , (2.4)

which is equivalent to:

d

dtρ (t) =−i

~[H, ρ (t)]− λi(ρ (t)− Ti[ρ (t)]) . (2.5) The rst term in the right hand side of above equation gives the standard Schrödinger evolution while the second term describes the collapse of the wave function. If we focus on the behavior of the matrix elements in the position basis ρt(x, y) = hx |ρ (t)| yi, Eq. (2.5) becomes:

dρt(x, y) dt =−i

~[H, ρt(x, y)]− λⁱh

1− e⁻^4r2¹^c^(x−y)

2i

ρt(x, y) . (2.6) This equation shows how, as a consequence of the collapse, the o diagonal elements are suppressed.

Let us introduce the master equation for a system composed by many particles. The derivation is similar to the one particle case and the nal result is:

d

dtρ (t) =−i

~[H, ρ (t)]− XN

i=1

λi(ρ (t)− Tⁱ[ρ (t)]) . (2.7)

2.1.3 The amplication mechanism

As already explained in the introduction, one of the major achievements of collapse models is to guarantee well dened positions for macroscopic objects. This is possible because of

amplication mechanism, which we describe now for the GRW model. Let us consider a

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macroscopic object composed of N microscopic constituents. Follwing [29], we focus on the study of the dynamics of the center of mass of the system. The position operator of the i-th particle qi can be written in term of the center of mass position operator Q and the relative coordinates rj (j = 1, 2, .., N − 1) as follows:

q_i = Q +X

j

cijr_j (2.8)

where cij are real coecients. Since we are interested in the time evolution of the center of mass, we want to derive the dynamics for the reduced density matrix ρ_CM = Tr_rel[ρ] (here

Tr_rel signies the trace over the relative coordinates) starting from Eq. (2.7). The non trivial part amounts to computing the partial trace of the term containing the maps Ti[ρ]

introduced in Eq. (2.2). In order to do that, it is convenient to rewrite this maps as follows¹:

Ti[ρ] = 1 πr²_c

3/2Z

da e⁻

(qi−a)²

2r2c ρ e⁻

(qi−a)²

2r2c

= r_c² π~²

3/2Z

dp e⁻^p2r2^2~2^ceⁱ^~^p^·qⁱρ e⁻^~ⁱ^p^·qⁱ

= r_c² π~²

3/2Z

dp e⁻^p2r2^2~2^ce

i

~p· Q+P

j

cijr_j

!

ρ e⁻

i

~p· Q+P

j

cijr_j

!

. (2.9)

The equivalence between the rst and the second line of Eq. (2.9) can be veried by computing Ti[ρ]between two generic position eigenstates. The partial trace of Eq. (2.9) can be computed using the fact that center of mass and relative coordinates commute and the cyclic property

1For sake of clarity we recall that a and p are vectors which components are real numbers while qi, Q and rj are vector operators.

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of the trace:

Tr_rel[Ti[ρ]] = r_c² π~²

3/2Z

dp e⁻^p2r2^2~2^c Tr_rel



e

i

~p· Q+P

j

c_ijr_j

!

ρ e⁻

i

~p· Q+P

j

c_ijr_j

!



= r_c² π~²

3/2Z

dp e⁻^p2r2^2~2^c Trrel

e

i

~

P

j

cijp·rj

eⁱ^~^p^·Qρ e⁻^~ⁱ^p^·Qe⁻

i

~

P

j

cijp·rj

= r_c² π~²

3/2Z

dp e⁻^p2r2^2~2^c e^~ⁱ^p^·Qρ e⁻^~ⁱ^p^·Q:= T_CM[ρ] (2.10) The above equation says that the localization of any particle (that is described by the maps Ti[ρ]) implies the localization of the center of mass of the object. Therefore, taking the partial trace with respect to the relative coordinates of the master equation given in Eq. (2.7), we get:

d

dtρ_CM(t) =−i

~[H_CM, ρ_CM(t)]− λ (ρCM(t)− TCM[ρ (t)]) . (2.11) with λ =

_N P

i=1

λi

. This means that, even if the localization rate of a single particle is very small (λi = 10⁻¹⁶ s⁻¹), the localization rate for the center of mass of a macroscopic object (where the number of particles is of order of Avogadro number 10⁻²³) is of the order 10⁷ s⁻¹. This implies that in the GRW model any macroscopic superposition is suppressed in a time scale of order 10⁻⁷ s.

Before concluding our discussion about the GRW model, we recall that in literature two dierent value for λ has been proposed. The rst one is λ = 10⁻¹⁶ s⁻¹ proposed by Ghirardi Rimini and Weber. This value is, somehow, the smallest possible choice: if one takes a smaller value, then the eect of the collapse becomes too weak and the localization of macroscopic systems becomes inecient. The second value for λ has been proposed by Adler [18]. It is based on the requirement that the process of latent image formation yields to a well localized position. This process involves a relatively small number of atoms, which implies that the amplication mechanism is not very eective. Therefore, in order for the latent image to be well localized, the strength of the noise has to be increased, according to Adler, by eight orders of magnitude. This implies λ = 10⁻⁸ s⁻¹.

The GRW model is the rst important and fully consistent collapse model. Compared to

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quantum mechanics, the model provides a clear and unambiguous description of the collapse of the wave function. However, the model as it is cannot be applied to systems containing identical particles. This is due to the fact that the collapse, as described in this model, does not preserve the symmetry of the wave function. This problem has been solved in the CSL model, which we will introduce soon.

2.2 The general structure of collapse equations

In the following we will introduce two other important collapse models: the CSL model and the QMUPL model. In these models, contrary to what happens in the GRW model where the collapse is described by discrete jumps, the collapse is described as a continuous process due to the interaction with an external noise. The dynamics of both models are described by stochastic dierential equations having the same mathematical structure. Recently it has been proven that, under quite general assumptions (e.g., no-faster-than-light signaling and the conservation of probability), any non linear modication of the Schrödinger equation described by a stochastic dierential equation must have the structure of collapse models [27].

Therefore, although collapse models are phenomenological (in particular the physical origin of the collapse is still unknown), they are the only possible consistent way to introduce non linear modications of quantum mechanics [27].

We rst summarize the general features of collapse equations. Their dynamical structure is:

d|φ^ti =

"

−i

~H dt + √ γ

XN i=1

(Ai− hAⁱi^t) dWi, t − γ 2

XN i=1

(Ai− hAⁱi^t)²dt

#

|φ^ti, (2.12)

where H is the standard Hamiltonian of the system, Ai are set of commuting self-adjoint operators, hAiit :=hφ (t) |Ai| φ (t)i, γ is a constant which sets the strength of the coupling with the noise eld and Wi, t are set of independent standard Wiener processes, one for each operator Ai. Notice that the dynamics is non linear, because of the presence of the terms hAⁱi^t, and stochastic because of the presence of the Wiener processes Wi, t. The rst term on the right hand side of the above equation is the standard Schrödinger evolution, and the other two terms describe the collapse of the state vector. In order to better understand how the collapse works and how it is related to the operators Ai, let us study Eq. (2.12) by neglecting the Schrödinger term. In such a case, it can be shown that any initial state evolves to one

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of the (common) eigenstates of the operators Ai [30]. In fact, for any given operator A that commutes with each of the operators Ai, its variance VA(t) :=hA²it− hAi²t is given by:

E[VA(t)] = VA(0)− 4γ XN

i=1

Z t 0

ds E

C_A,A² _i(s)

, (2.13)

where E denotes the noise average and CA,A_i(t) :=h(A − hAi^t)(Ai− hAⁱi^t)i^tis the correlation between the operator A and Ai. Taking into account that VA(t) must be positive for any time t and the second term in the right hand side of Eq. (2.13) is always negative, it must be that CA,Ai(t)→ 0 when t → ∞ for any i. Therefore, if we choose A as one of the operators Aj, this implies that its variance VAj(t) = CAj,Aj(t)goes to zero for large enough times. This is equivalent to saying that the state has evolved to one of the eigenstates of Aj. As we will see, in order to guarantee macroscopic objects to have well dened positions, the operators Ai are chosen to be functions of the position operator.

2.2.1 The master equation

Here we derive the master equation associated to Eq. (2.12). Using the It o product rule we have:

d|φtihφt| = (d|φti) hφt| + |φti (dhφt|) + (d|φti) (dhφt|) = (2.14)

=

"

−i

~H dt + √γ XN

i=1

(Ai− hAⁱi^t) dWi, t − γ 2

XN i=1

#

|φ^tihφ^t|

+ |φ^tihφ^t|

"

i

~H dt + √ γ

XN i=1

(Ai − hAⁱi^t) dWi, t − γ 2

XN i=1

#

+ γ XN

i=1

(Ai − hAiit)|φtihφt|(Ai− hAiit)dt.

A direct calculation shows that all the terms involving hAⁱi^t and hAⁱi²t multiplied with dt cancel each other. Moreover, the expectation value of the terms containing dWi, t is zero.

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Therefore the equation for ρ (t) = E [|φ^tihφ^t|] is:

dρ (t) dt = i

~ [ρ (t) , H]− γ 2

XN i=1

A²_iρ (t) + ρ (t) A²_i − 2Aiρ (t) Ai

. (2.15)

Rewriting the second term in a more compact way, we get:

dρ (t) dt = i

~[ρ (t) , H]− γ 2

XN i=1

[Ai[Ai, ρ (t)]] . (2.16)

One should note that Eq. (2.16) has the typical structure of Lindblad equation. Here, for simplicity we derived Eq. (2.16) for the case of a white noise. In the next chapters we will need also the master equation in the case of colored noise. Its derivation can be found in [30], here we report only the nal result which is valid to the rst order in γ:

dρ (t) dt =−i

~[H, ρ (t)]− γ XN i,j=1

Z t 0

dsDij(t, s) [Ai, [Aj(s− t) , ρ (t)]] , (2.17)

where Dij(t, s) is the correlation between the i-th noise at time t with the j-th noise at time s.

2.2.2 The imaginary noise trick

In this thesis we are interested in computing the eect of collapse models on the emission of electromagnetic radiation from charged particles and on avours oscillations. In both cases we have to compute appropriate expectation values, averaged over the noise. Therefore we can use a very useful mathematical trick, known as imaginary noise trick. Consider a generalization of the collapse equation Eq. (2.12) of the form:

d|ψ^ti =

"

−i

~Hdt +√γ XN

i=1

(ξAi− ξRhAⁱi^t)dWi, t−γ 2

XN i=1

(|ξ|²A²_i − 2ξξRAihAⁱi^t+ ξ_R²hAⁱi²t)dt

#

|ψ^ti, (2.18)

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where ξ is a generic complex number. When ξ = 1 Eq. (2.18) reduces exactly to the collapse equation in Eq. (2.12). On the contrary, if one takes ξ = i the equation reduces to:

d|ψti =

"

−i

~Hdt + i√ γ

XN i=1

AidWi, t− γ 2

XN i=1

A²_idt

#

|ψti. (2.19)

This equation is written in the It o form. The corresponding Stratonovich form, which is the one we are interested in, is given by the same equation without the third term on the right hand side term [31]. If we introduce the white noises as wi, t:= ^dW_dt^{i, t}, then the Stratonovich form of Eq. (2.19) can be written as:

i~d|ψti dt =

"

H− ~√ γ

XN i=1

Aiwi, t

#

|ψti, (2.20)

which is a Schrödinger equation with random potentials. The dynamics given in Eq. (2.20) is completely dierent from the one given by Eq. (2.12). In particular Eq. (2.12) describes an evolution that leads to the collapse of the wave function while in the dynamics given by Eq. (2.20) there is no collapse at all. However, one can show that the master equation associated to Eq. (2.18) is:

d

dtρ(t) = −i

~[H, ρ(t)] + γ 2|ξ|²

XN i=1

2 Aiρ(t) Ai− {A²i, ρ(t)}

. (2.21)

The important point is that the equation for ρ depends only from the modulus of ξ. This means that Eq. (2.12) (ξ = 1) and Eq. (2.20) (ξ = i), despite the fact that they describe a very dierent dynamics for the state vector, turn out to be completely equivalent at the statistical level. Therefore, as far as we are interested in making physical predictions, we will work with Eq. (2.20) which is much easier to handle compared to Eq. (2.12).

2.3 The CSL model

Here we introduce the CSL model. The equation that describes the evolution of state vectors in the CSL model has the form of Eq. (2.12) with a particular choice of the localization operators Ai. The operators are chosen in order to satisfy the following requirements:

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1. A macroscopic object must be well localized in space;

2. The model should be able to describe systems of identical particles.

Both conditions lead to the choice of a continuous set of operators N(x), one for each point of space x:

Ai −→ N (x) =X

j

X

s

Z

dyg (y− x) ψj^†(y, s) ψj(y, s) . (2.22) Here ψj^†(y, s)and ψj(y, s)are respectively the creation and annihilation operator of a particle of type j in the point y with spin s and

g (y− x) = 1

√2πr_c3e⁻

(y−x)2

2r2c (2.23)

is a gaussian smearing function that has the same role as the one introduced for the GRW model. Accordingly, the CSL dynamics reads:

d|φ^ti =

−i

~Hdt +√ γ

Z

dx (N (x)− hN(x)i) dW^t(x)−γ 2

Z

dx (N (x)− hN(x)i)²dt

|φ^ti.

(2.24) Since the dynamics in the above equation is written using the second quantization formalism, the second requirement mentioned above is automatically fullled. In order to understand the connection with the GRW model, it is useful to study the master equation in the position representation. For one particle this is given by:

dρt(x, y) dt =−i

~[H, ρt(x, y)]− λGRW

h1− e⁻^4r2¹^c^(x−y)²i

ρt(x, y) (2.25) where λGRW := _8π_3/2^γ _r₃

c is equivalent to the parameter λ introduced for the GRW model if one takes γ = 10⁻³⁰ cm³ s⁻¹. Therefore Eq. (2.25) is exactly the same like Eq. (2.6). This result is not true anymore in the case of many particles when they are identical. Let us also mention that also for the CSL model there is an amplication mechanism similar to the one of the GRW model [15, 16].

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2.4 The mass proportional CSL model

In the CSL model the noise eld inducing the collapse of the wave function acts in the same way on dierent types of particles. However, there are both theoretical and experimental reasons to believe that the eect of the collapse should be mass proportional [16, 20]. On the experimental side, for example, the data on the spontaneous radiation emission from Germanium falsify the original CSL model [20]. On the other hand, when a mass proportional coupling is considered, there is no disagreement between the theoretical prediction and the experimental data. We will discuss this issue in the conclusion of Part I. On the theoretical side, one of the main reasons to believe that the strength of the collapse eect should be mass proportional is the following. Consider two systems, with the same mass but composed of a dierent number of particles. According to the original CSL model, they are localized in dierent ways, depending on their number of particles. However, if we consider the total mass as a measure of macroscopicity, then we would expect both systems to be localized in the same way. This is exactly what happens using the mass proportional CSL model.

In addition, taking a mass proportional coupling suggests that the noise eld may have a connection with gravity. This possibility has been proposed by Penrose and Diosi [32, 33, 34]

and recently reconsidered in [35].

The mass proportional CSL dynamics is given by:

d|φti =

−i

~Hdt +

√γ m0

Z

dx (M (x)− hM(x)i) dWt(x)− γ 2m²₀

Z

dx (M (x)− hM(x)i)²dt

|φti, (2.26) where m0 is a reference mass, chosen to be equal to the mass of a nucleon, and

M (x) =X

j

mj

X

s

Z

dyg (y− x) ψ^†j(y, s) ψj(y, s) (2.27)

with mj the mass of the particle type j.

As explained before, in many computations it is convenient to use the imaginary noise trick introduced in section 2.6; in such a case the dynamics is given by a Schrödinger equation with the Hamiltonian:

HTOT= H − ~√

γX

j

mj

m0

X

s

Z

dy w(y, t)ψ^†_j(y, s)ψj(y, s). (2.28)

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Here w(x, t) is a Gaussian noise, with zero mean and the correlation function:

E[w(x, t)w(y, s)] = f (t− s)F (x − y), F (x) = 1 (√

4πrC)³e^−x²^/4r^C². (2.29) In the case of a white noise in time, the function f(t) is simply a Dirac delta δ(t). From now on, in order to simplify the writing, we will refer to the mass proportional CSL model simply as CSL model.

2.5 The QMUPL model

The QMUPL model was introduced for the rst time by Diosi in [28]. The model has a very simple coupling between the noise and the particles. The localization operators Ai are chosen to be position operators: Ai = qi with i = 1, 2, 3 labeling the three space directions.

Therefore, for a single particle, the dynamics of the model is:

d|φti =

−i

~H dt + √

λ (q− hqit) · dWt − λ

2(q− hqit)²dt

|φti, (2.30) where λ is the coupling constant with noise, analogous to the γ introduced for the CSL model. This model, compared to the CSL, has the disadvantage of not holding for system of identical particles. However, it has the great advantage of being much easier to handle compared to the CSL model. Indeed, as we will see in the next chapters, some problems that with the CSL model can be studied only perturbatively, in the QMUPL model can be solved exactly. Moreover, this model is not that much dierent from the CSL model as it might seem. The master equation of the QMUPL model in the position basis is:

dρt(x, y) dt =−i

~[H, ρt(x, y)]− λ

2 (x− y)²ρt(x, y) . (2.31) If we compare Eq. (2.31) with Eq. (2.25) we see that, for any system whose typical size is smaller than rC, the gaussian in Eq. (2.25) can be expanded in Taylor series and the relevant term is exactly the one in Eq. (2.31), if one sets λ = ^λ^GRW_2r2

c . Therefore, for a large number of systems, the predictions of the QMUPL model are practically equivalent to the CSL predictions. As in the case of the CSL model, for the calculations we will also use the imaginary noise trick. Then the dynamics given by Eq. (2.30) is replaced by a Schrödinger

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equation with Hamiltonian:

H_TOT= H − ~√

λ q· w(t). (2.32)

Our review on collapse models ends here. For the rest of the thesis we will present original work done by using these models.

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Part I

Radiation emission

30

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In this part of the thesis we present the problem of the electromagnetic radiation emission in collapse models. In collapse models no system is completely free because of the interaction with the noise. The direct eect of this interaction is the localization of the center of mass of the system. However, as an indirect eect, if the system is composed of charged particles the collapse noise induces the emission of radiation. As a consequence, collapse models predict the spontaneous radiation emission even for systems, e.g., a free particle or an atom in its ground state, that should not emit according to quantum mechanics.

The radiation phenomenon provides so far the strongest upper bound on the collapse parameter λ. Therefore, research on radiation emission in collapse models has been quite active. After some preliminary results by P. Pearle and collaborators using the GRW model [36, 37, 38], the rst theoretical calculation using the CSL model has been carried out by Q. Fu [20], to the rst meaningful perturbative order, for a free particle. The calculation has been conrmed by S.L. Adler and F.M. Ramazanoglu [21], and generalized to hydrogenic atoms and to non-white noises. More recently A.Bassi and D. Dürr have done a similar calculation using the QMUPL model [22]. Since the QMUPL model is simpler, from the mathematical point of view, than the CSL model, it allows an exact analytical treatment of the problem. Moreover, as discussed in chapter 2, the QMUPL model should reproduce the same results of the CSL model, as far as the system's size is smaller than rC = 10⁻⁷ m. However, the free particle's photon-emission rate in the case of white noise turns out to be twice larger than the CSL prediction. This discrepancy, which initially seemed of little importance, revealed many subtleties in the implementation of standard quantum eld perturbative methods in the context of stochastic models; in particular, for the case of colored noises, terms appear in the radiation emission formula, which look unphysical [23].

In the following chapters we study the radiation emission problem in detail. In chapter 3 we recompute the radiation emission from a free particle using the CSL model. We show that, mathematically speaking, the emission rate found in [20, 21] is wrong, while the correct result is the one found in [22]. Then we repeat the computation using a non white noise. Here a problem arise, because an unphysical factor is present in the emission rate. A possible way out of the problem is discussed in the last section of chapter 3, where we change the model by conning the noise and performing a more realistic calculation using wave packets instead of plane waves as nal states. A deeper insight into the problem is given in chapter 4, where we show that the unphysical terms disappear if higher order contributions are considered, without having to change the model. Accordingly in chapter 5, using the CSL model and

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treating the electromagnetic interaction exactly, we give a derivation of the emission rate formula for a harmonic oscillator which does not contain any unphysical term. In chapter 6 we generalize this result to a generic system, showing that the unphysical term disappears if the decay of the propagator is taken into account. To conclude, following the analysis done in [20], we compare the predicted rate with the available data on the spontaneous emission from Germanium, in order to derive a bound on the collapse parameter λ.

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Chapter 3 Perturbative calculation of the emission rate in CSL model

In this chapter we compute the emission rate from a free particle in the CSL model. We start by repeating the computation already done in [20, 21], where a white noise was considered.

Then we extend the calculation to a colored noise. In this case a problem arises: the formula for the emission rate, as we will see, contains an unphysical term. Because of this term, even a weak noise can induce the emission of photons with high energies. We show that a possible way to avoid the presence of this term is to carry out the perturbative calculation by taking wave packets as nal states and conning the noise.

3.1 The CSL model for charged particles

As explained in chapter 2, in order to compute physical predictions such as like the emission rate, one can use a Schrödinger dynamics given by the Hamiltonian in Eq. (2.28) which is:

H_TOT= H − ~√

γX

j

mj

m0

X

s

Z

dy w(y, t)ψ^†_j(y, s)ψj(y, s). (3.1) We are interested only in one type of particle, so from now on we will drop the sum over j. We will also neglect the spin degree of freedom since its eect is negligible compared to the other electromagnetic terms that we are considering.

Following the quantum eld theory approach, we write the Hamiltonian HTOTin terms of a Hamiltonian density HTOT. For the systems we are studying, we can identify three terms

33

andFlavour Oscillations

DIPARTIMENTO DI FISICA

XXVI CICLO DEL

DOTTORATO DI RICERCA IN FISICA

Electromagnetic Radiation Emission and Flavour Oscillations in Collapse Models

Sandro Donadi Prof. Paolo Camerini

Dr. Angelo Bassi Firma:

Contents

I Radiation emission 30

II Flavour oscillations 106

List of abbreviations and symbols

List of publications directly related to this thesis

Chapter 1

Introduction and main results

Chapter 2

Collapse models

2.1 The GRW model

2.1.1 Postulates of the GRW model

2.1.2 GRW master equation

2.1.3 The amplication mechanism

2.2 The general structure of collapse equations

2.2.1 The master equation

2.2.2 The imaginary noise trick

2.3 The CSL model

2.4 The mass proportional CSL model

2.5 The QMUPL model

Part I

Radiation emission

Chapter 3

Perturbative calculation of the emission rate in CSL model

3.1 The CSL model for charged particles

andFlavour Oscillations

DIPARTIMENTO DI FISICA

XXVI CICLO DEL

DOTTORATO DI RICERCA IN FISICA

Electromagnetic Radiation Emission and Flavour Oscillations in Collapse Models

Sandro Donadi Prof. Paolo Camerini

Dr. Angelo Bassi Firma:

Contents

I Radiation emission 30

II Flavour oscillations 106

List of abbreviations and symbols

List of publications directly related to this thesis

Chapter 1

Introduction and main results

Chapter 2

Collapse models

2.1 The GRW model

2.1.1 Postulates of the GRW model

2.1.2 GRW master equation

2.1.3 The amplication mechanism

2.2 The general structure of collapse equations

2.2.1 The master equation

2.2.2 The imaginary noise trick

2.3 The CSL model

2.4 The mass proportional CSL model

2.5 The QMUPL model

Part I

Radiation emission

Chapter 3

Perturbative calculation of the emission rate in CSL model

3.1 The CSL model for charged particles

2.1.3 The amplication mechanism