Using correlations to experimentally search for Quantum Gravity effects

(1)

Quantum Gravity effects

Dipartimento di Fisica dell’Università di Pisa Corso di Laurea Magistrale in Fisica

Candidate Luca Marchetti ID number 493884 Thesis Advisor Dr. Giancarlo Cella Academic Year 2017/2018

(2)

Using correlations to experimentally search for Quantum Gravity effects Master thesis. Università di Pisa

This thesis has been typeset by LA_{TEX and the Pisathesis class.}

(3)

(4)

(5)

Abstract

This master thesis project is a phenomenological work on a putative Quantum Gravity effect, the spacetime fuzziness. After a general discussion about Quantum Gravity phenomenology and a review of the theoretical evidences leading to such an effect, a theoretical model describing spacetime fuzziness and an experimental proposal devoted to its detection are discussed. The phenomenological model is based on stochastic fluctuations of the tetrad field. The propagation of a minimally coupled electromagnetic field is studied in such a stochastic spacetime, and a differential equation for the two point correlation of the electromagnetic field is obtained in a perturbative way. This equation is then reinterpreted in a way which makes is applicable also to complex interferometric apparatuses, such as the one proposed here. This is based on a particular kind of cavity, called sloshing cavity, which allows for a continuous and dynamical switching between two configurations, characterized by different spacetime correlation. This switching is possible if in the cavity an amplitude modulated light is injected. The whole apparatus is then composed by an interferometric setup where the two arms are two identical sloshing cavities. This setup is analysed in detail and input-output relations are provided both for the unperturbed and for the perturbed electromagnetic field, the last one being generated by spacetime vibrations. Lastly, the possible noises affecting this apparatus are briefly discussed and some proposals to reduce their contribution to a quantum spacetime noise measurement are qualitatively described.

(6)

(7)

Acknowledgments

I am profoundly tankful to Professor Giancarlo Cella for his patience, sympathy and kindness and also for his sincere commitment to this master thesis project. I am also thankful to the Comparative Quantum Gravity Group, whose discussions were of remarkable help to have a better comprehension of the Quantum Gravity problem and of the main proposals devoted to its solution. Lastly, I owe a debt of gratitude to the VIRGO Data Analysis Group, who helped me to understand better the physics behind an interferometric precision measurement.

(8)

(9)

Notations and Terminology

Indices Index notation follows the common use in the field of Quantum Gravity and General Relativity: Greek indices µ, ν, · · · = 0, 1, 2, 3 are 4d spacetime tangent indices. Latin in-dices i, j, · · · = 0, 1, 2, 3 are 4d Lorentz tan-gent indices. However, notice that in Ap-pendixA, Greek indices from the beginning of the alphabet α, β, · · · = 1, 2, 3 are 3d spa-tial indices. This notation will be used con-sistently also in Section 2.3. Anyway, since these are the only places in the thesis were 3d hypersurfaces are defined, in the rest of the thesis, α, β, . . . , will be used on the same footing as the indices µ, ν, . . . . Lastly, since in Chapters3and4, there will be no need to distinguish between flat and curved indices, we will use, throughout all those chapters, Greek indices µ, ν, . . . .

Coordinates Spacetime coordinates on a 4d manifold will be usually denoted as x, y, . . . , while coordinates on a 3d manifold (for ex-ample spatial coordinates) will be denoted as x, y, . . . . Thus the components of a spacetime coordinate x are

xµ= (t, x) .

Metric The metric tensor will be always de-noted as g_µν, while the Minkowski metric will be denoted as ηµν (except in Appendix A and in Section 2.3, where, being neces-sary the distinction between flat and curved coordinates, the Minkowski metric will be denoted η_ab). The signature of the metric tensor will be considered “mostly plus” in Chapter 2 and in Appendix A, in order to make the results more easily linkable to the Quantum Gravity literature, while will be considere “mostly minus” in Chapters3and 4, where a more “experimental” choice was more appropriate.

Fourier transform Fourier transform will be performed using the convention

f (x) =

Z _d4_p (2π)4e

−ip·x_{f (p) ,}

(0.1)

where the Fourier transform of f (p) will be denoted f (p) (or, equivalently, ˜f (p)). The same holds for frequency Fourier transform. Frequency variables will be denoted as ω or Ω.

Units Throughout all the thesis natural units , where both the speed of light and the Planck constant divided by 2π are set equal to 1: c = } = 1. The dimension of a quantity A will be denoted as [A], and in natural units, a number will be assigned to it, for example, [A] = 1 means that the A has the dimension of a mass (or, equivalently, of an energy). Sometimes, however } and c will be appear, essentially where it is important to stress the physical dimensions of a quantity.

Symbols

Ep is the Planck energy, Ep∼ p

1/G ∼ 2 · 1027 eV.

Lp is the Planck scale, Lp= Ep−1∼ 10−35m. tp is the Planck time, tp ∼ 5 · 10−44s.

rC is the Compton wavelength, rC ∼ m−1. rS is the Schwarzschild radius, rS ∼ Gm ∼

L2_pM .

κ is the DSR mass scale.

G is the Newton constant, G ∼ E_p−2.

Λ is the cosmological constant. `s is the string length.

α0 is essentially the inverse of the string ten-sion, T ≡ (2πα0)−1.

gs is the string coupling constant.

D indicates a functional measure of integra-tion.

{·, ·} indicates Poisson brackets.

(10)

P indicates the procedure of path-ordering.

ˆ· the hat over a quantity denotes the operator nature of that quantity, except for Appendix A, where there the treatment is only classi-cal and the ˆ· is used to denote a basis vector of the tangent space. Moreover, we want to mention, here, that, in order to make the notation less cumbersome, in Section2.2no hat will be placed upon operators.

h·i represents the ensamble average of a stochastic quantity.

E the subscript E is used to represent a vari-able defined in the Euclidean space.

∇ represents the covariant derivative with re-spect to the Levi-Civita connection.

g is one of the parameter of the model. L is the scale of the model, whose equivalent

energy is EL, but it is also the distance of the

arms of the experimental apparatus, which we however usually denote τ ≡ L/c to avoid confusion.

represents the wave or d’Alambert operator.

· represents the time average of a quantity.

C denotes quantities related to the carrier field.

S denotes quantities related to the sidebands field.

r is in general used for the reflectivity of an optical surface and it is taken to be real. t is in general used for the transmissivity of an

optical surface and it is taken to be real. ωs is the sloshing frequency.

(11)

4.3 The proposal . . . 90 4.3.1 Preliminaries . . . 91 4.3.2 Sloshing cavity . . . 94 4.3.3 Interferometric setup . . . 102 4.4 Sources of noise . . . 104 4.4.1 Laser noise . . . 104 4.4.2 Quantum noise . . . 105 4.4.3 Thermal noise . . . 111 5 Conclusions 115 A Canonical General Relativity 119 A.1 Gauge symmetries and constraints . . . 119

A.1.1 Primary and secondary constraints . . . 119

A.1.2 First and second class functions . . . 121

A.2 Canonical General Relativity . . . 122

A.2.1 The ADM action . . . 122

A.2.2 Legendre transform and Hamiltonian formulation . . . 124

A.2.3 Constraints analysis . . . 125

A.3 Ashtekar Canonical Gravity . . . 126

A.3.1 Tetrad formulation of gravity . . . 127

A.3.2 Canonical tetrad gravity . . . 129

(13)

B Gauge transformations 133

B.1 Preliminaries . . . 133

B.1.1 Flows and Lie derivatives . . . 134

B.1.2 Knight diffeomorphisms . . . 135

B.1.3 Interpreting the results . . . 136

B.2 Gauge transformations and perturbation theory . . . 137

B.2.1 Perturbations of spacetime and gauge choices . . . 137

B.2.2 Gauge transformations . . . 138

B.3 Gauge transformations of the metric tensor . . . 139

C Mathematica 141 C.1 Perturbative expansion. . . 141

C.2 Correlations . . . 144

C.2.1 Approximations and scalarization. . . 146

(14)

(15)

1

Introduction

This master thesis project should be framed in the context of Quantum Gravity phenomenology. Even though this research area is still in its early phase of development, it is, in our opinion, already enough mature to be addressed seriously. In this thesis work, we are going to study a supposed physical effect of Quantum Gravity, the so called “spacetime fuzziness”. The picture of a “spacetime foam” rather than a smooth spacetime is something that was introduced very long ago, since the pioneering work of Wheeler [81]:

“On an atomic scale the metric appears flat, as does the ocean to an aviator far above. The closer the approach, the greater the degree of irregularity. Finally, at distances of the order L∗ [. . . ] the character of the space undergoes an essential change [. . . ]. Multiple connectedness develops, as it does on the surface of an ocean where waves are breaking.”

In this work we will provide a phenomenological model quantitatively describing this spacetime foam, a phenomenon which is not only an intriguing suggestion, but which appears to connect on the quiet all the main approaches to Quantum Gravity. This done, we will propose an interferometric experimental setup designed to be a possible detector for such a fuzziness.

Before facing these problems, however, it is essential to understand what does “Quantum Gravity phenomenology” mean and to which extent one can say that what is doing is really Quantum Gravity phenomenology. This chapter will be essentially devoted to this purpose. In Section1.1, we will discuss some general aspects of Quantum Gravity phenomenology, starting from a characterization of the “Quantum Gravity problem”, and arriving to an identification of the main features needed for a Quantum Gravity experiment. Next, in Section 1.2, we will describe what are the physical effects that are expected to arise in a typical Quantum Gravity scenario. As we are going to see explicitly in the next chapter, usually these expectations are not very clearly quantified by the main Quantum Gravity models. Still, the “theoretical evidences” pointing to their presence are often so copious that they cannot be ignored. Lastly, Section1.3will be an example of a Quantum Gravity phenomenological model. We will explore how it is mathematically formulated starting from physical principles arising from some kind of “theoretical evidences”, and we will discuss how its phenomenology is carried out with the use of the so called “test theories”.

(16)

1.1 Quantum Gravity phenomenology

Our present comprehension of the Universe is based on two solid pillars: Quantum Mechanics and General Relativity. However, these two ways of looking at Nature seem to have very little in common. Quantum Mechanics has shown its power in the last years in the form of relativistic quantum field theory. The typical computations performed in the framework of “traditional” quantum field theory are cross-sections for transitions between different particle states, and usually, such computations are performed in a flat spacetime. On the other hand, General Relativity is a purely classical theory, which usually neglects any quantum properties of the components of the Universe.

However we should mention here that even though the previous discussion is true in most of the cases, there are formulations of quantum field theory in a curved, classical background. In this particular context, the background is determined from classical sources by the classical Einstein equations and the quantum fields propagate as test fields in such a spacetime. The other side of the coin consists in taking into account the effects of quantum fields on the spacetime geometry, using as source the expectation value of the stress-energy operator, properly renormalized. This theory is usually called semiclassical gravity, but still, “the scope and limitations of the theory are not so well understood” [43]. Anyway, it should be stressed that this is just the self-consistent superposition of Quantum Mechanics and General Relativity: it is not a description of the quantum nature of gravity.

At first sight the situation may seem odd: how can we trust two theories (at the same time) whose description of the Universe is so different? The point is that when we perform a quantum field theory experiment, usually the effects of gravity are so weak that we can neglect them. On the other hand, our belief in General Relativity is based on observations in which the gravitational interaction is very strong: typically such experiments involve “enormous objects” such as stars or planets. But those systems are composed by a vast number of elementary particles, so that their quantum properties are completely unimportant. It is therefore clear that the only reason why both the language of Quantum Mechanics and General Relativity are successful is that we are performing two widely separated classes of experiments.

Nonetheless, relativistic quantum field theory and General Relativity live in a shared ground: the spacetime. But here, again, their perception of it is completely different. In quantum field theory spacetime is a fixed background, a stage where the play takes place. On the contrary, in General Relativity spacetime is gravity and gravity is spacetime. It is therefore a dynamical quantity and certainly one of the main actors of the play.

Is the “Quantum Gravity problem” a true problem? Maybe (at least from the philo-sophical point of view) having two theories which perceive things in such a different way could be problematic. Anyway, in today experiments and at the sensitivity we are able to achieve, we have found nothing worthy to declare that Quantum Gravity is a true problem.

But let us consider a simple scattering process, a typical one performed in a particle-physics laboratory. Say that we want to collide two particles with energy of 1012eV each. The results we will obtain would not be surprising: relativistic quantum field theory makes a definite prediction for the scattering amplitude, and since the energy we are considering is small (with respect to the Planck energy E_p), we can neglect gravity. The prediction of quantum field theory shall turn out to be confirmed by our experiments and everything is going to be fine. However, even if we are not presently capable to build an identical setup with two “in” states with 1030 eV each, relativistic quantum field theory makes a definite prediction for such an experiment. Still, 1030 eV is a enormous amount of energy, and we can not neglect gravity as we did previously. We

(17)

have to resort to General Relativity, but, sadly, we are going to find uncontrollable divergences1 which will not produce anything useful.

Surely, if we would use this “trans-Planckian” argument to convince you that Quantum Gravity is a true scientific problem, you could be rightly skeptical, since our argument is missing a point we already stressed at the beginning of this paragraph: we are not presently able to perform such an experiment (and maybe we will never be able to do it). So, who cares? The early Universe, for example. As far as we know, during the first stages of evolution of the Universe there were many particles with energy comparable to Ep, and they played a central role in the subsequent growth of the Universe. Of course this is not a controlled nor a repeatable experiment, but we can test Quantum Gravity proposals against it. Different Quantum Gravity theories give different description of the Universe: surely a good one should provide predictions for observables that are compatible with those we measure in our Universe.

Quantum Gravity problem must be treated as a scientific problem We are therefore facing a true Quantum Gravity problem, but unfortunately, for (too) many years this problem was faced as if we could not obtain any clue from experiments. However, as clearly pointed out in [10],

“If there is to be a “science” of the quantum-gravity problem, this problem must be treated just like any other scientific problem, seeking desperately the guidance of experimental facts, and letting those facts take the lead in the development of new concepts. Clearly physicists must hope this works also for the quantum-gravity problem, or else abandon it to the appetites of philosophers.”

Obviously there is no guarantee of success, especially if we are correctly expecting that the novel physics will show at the Planck scale (L_p ≡ E−1

p ∼ 10−35 m), but, still, we must try: even the smallest manifestation of Quantum Gravity could lead us to great developments on the theory side. We already know it will be a very difficult task, but, if the onset of Quantum Gravity effects is smooth2, it is not an impossible one. Think for example at the Brownian motion: in that case, small, microscopic processes lead to measurable effects on macroscopic scales. Another comforting example is the search for proton decay. Within certain grandunified theories, the proton is predicted to decay, with a probability that goes like [m_proton/EGU T]4∼ 10−64. This is certainly an astonishing suppression factor, but we managed to obtain full sensitivity to this effect through a “smart trick”: the lifetime of a single proton according to these grandunified theories is of order of 1039 s, but monitoring 1033 protons, our sensitivity is dramatically increased.

Incidentally, the two example discussed above have something in common: the small, new effect was detected thanks to some kind of “amplifier”. In the first case it was the large number of microscopic processes cumulating in one macroscopic effect, while in the second one the amplifier was the enormous number of protons monitored. This gives us already an intuition about the kind of experiments we need to test Quantum Gravity effects: those endowed with a big amplifying factor. This is a topic which we shall discuss again later.

1_{This is due to the fact that General Relativity is not renormalizable and can only be understood as an Effective}

Field theory. This in turn means that beyond the leading order it predicts corrections proportional to (E/Ep)2, [21] where Ep∼ 1028eV is the Planck energy. Evidently, when E ∼ Epsuch a theory breaks down.

2

This means that Quantum Gravity can leave even a minute, impalpable trace at energies smaller than Ep. From now on, we will only consider Quantum Gravity effects with smooth onset, since otherwise there would be no hope to find evidences of Quantum Gravity, at least in the near future.

(18)

1.1.1 Where to look for Quantum Gravity effects?

Gravity is a two-faced-beast. This is encoded in the fact the gravitational field is, unlike all others, something that describes both gravitational interaction of particles and the structure of spacetime. However, as we have discussed above, usually these two sides of gravity are not faced on the same footing. For example, a particle physicist would try to take a background Riemannian manifold as spacetime, treating (perturbatively) “gravitons” as mediators of the gravitational interaction. On the other hand, a relativist would try to understand which kind of structure can substitute the Riemannian manifold of the General Relativity, therefore not necessarily looking at the perturbative description of gravity. It is then clear that the Quantum Gravity problem can assume different forms for physicist with different “inclinations”.

In general, we can try to distinguish two classes of regimes in which we could detect Quantum Gravity effects (with smooth onset): the first is where our theories are completely inapplicable, the other is the opposite one, where quantum effects of gravity show themselves as some small corrections on the well known (and tested) physics.

The unexplored: quantum black-hole regime

This is the wild and completely unknown jungle where our theories do not dare to venture. An example of this hypothetical situation is the one we discussed at the beginning: if we scatter two particles with an impact parameter of order of the Planck lenght with an exchange of energy between the two particle of the scale of the Planck energy, we are packing an amount of energy of the order of the Planck energy in a region of the order of the Planck length. We have obviously no experience of such a situation and both General Relativity and Quantum Mechanics are important:

. From one side we know that a particle with a rest energy M cannot be localized better than its Compton wavelenght3

rC ∼ M−1.

. On the other hand General Relativity assigns to any localized amount of rest energy M an horizon of the size of the Schwarzschild radius,

rS∼ GM ∼ L2pM .

When M ∼ L−1_p the Compton and the Schwarschild radii are of the same order of magnitude, and obviously General Relativity and Quantum Mechanics have to strongly compete.

Another situation similar in a certain sense to the one just described above can be imagined if we think about a macroscopic isolated black hole and we try to describe its future. Because of Hawking radiation it will undergo an evaporation, eventually reaching a Planck scale size (and Planck scale rest energy). And again we are left only with questions.

Clearly, in the “quantum black hole regime” we cannot resort to a description of only one face of gravity: we have to understand both how spacetime changes and how gravity interact to give a satisfactory answer. Theoretical research about this regime is as much ambitious as fascinating, but this is clearly experimentally inaccessible. Evidences for Quantum Gravity effects should be searched elsewhere.

3_{This is the Quantum Mechanical resolution limit: it is not possible to localize a particle with a precision}

better than the Compton wavelenght. In fact, from ∆x∆p& 1, for relativistic particles we get ∆x∆E & 1. If therefore the position uncertainty is smaller than the Compton length, ∆x ≤ E−1, we find E−1∆E& 1, and the uncertainty on the energy of the particle is bigger that its rest mass, making the notion of particle itself unclear.

(19)

Small corrections: the graviton-exchange regime and the quantum spacetime regime A key assumption of this thesis is that Quantum Gravity effects can be detected as small cor-rections to experimental results we are able to predict with our theories, at least at first approx-imation. Still, even agreeing on such a statement, as we stressed above, particle physicists and relativists can have a different concept of “small corrections regime”.

The graviton-exchange regime For a particle physicist the most natural environment to look for Quantum Gravity effects is the one involving the gravitational interaction aspects of Quantum Gravity. Typical examples are studies on long-range corrections to the Newton po-tential. Clearly, focusing on such scales, one stays far away from the dangerous jungle described above, but it is however natural to expect that the quantum nature of gravitational interaction should leave some remnants on the Newton law.

This possibility has been investigated with Effective Field Theory techniques applied to the Einstein-Hilbert linearized theory. As it is shown for example in [31],

∆VN ∼ G2_M

r3 ∼ L2_p r2VN,

where M is the mass of the source of the gravitational potential. It is clear that, at least for this example, the correction is fantastically small. In fact, our knowledge of the Newtonian regime of gravity extends to scales no shorter that 10−6 m, so that L2

p/r2 would be of the order of 10−(70−12)= 10−58.

The quantum spacetime regime Having given the particle physicist point of view it should be fair to turn to the relativist’s one. The nature of the Quantum Gravity problem strongly suggests us that the “true” description of the spacetime cannot be in terms of smooth, Rie-mannian manifolds. Obviously, we are far from having understood how this changes, but as we will discuss in the following, the arguments suggesting a “spacetime quantization” are solid and copious. Clearly, such a striking change in our notion of spacetime should be expected to happen in the “quantum black-hole regime”, but, as already stressed above, we have no access to that wild world. Still, the quantum features of the spacetime should leave a trace on the “Minkowski limit” of Quantum Gravity. On this well-studied context we have a lot of clear and high quality data, and we can hopefully find some unexpected new effect. Obviously, such effects would be very small, but our sensibility in this regime is incredibly high: maybe enough to probe the quantum spacetime structure.

The ideas developed in this thesis, involving laser interferometry, are related to this particular regime.

1.1.2 How deep must we dig?

It is clear that, in everyday life, gravity is not quantum. Still, it is an open question the order of magnitude at which quantum effects in gravity are not negligible. There are a lot of arguments pointing to the Planck scale as the typical scale of Quantum Gravity. Following [10] we would like to describe a couple of these arguments which are of substantial importance.

1. The first is in a certain sense, an (indirect) “experimental” argument. In fact, we know that the running of the couplings of electroweak and strong forces point toward an unification scale. The striking fact is that this unification of extrapolated couplings (based on the little information we have at scales below the TeV) happens at a scale which is not very far from

(20)

the Planck scale. Sure, it could be just an lucky accident, but it is still surprising to find two completely unrelated scales in such a close correspondence. For the sake of clarity, we are talking about an unification scale of non-gravitational couplings of the order of (1027 eV)−1 (actually (2 · 1026 eV)−1), while the Planck length is of the order of (1028 eV)−1. This could even suggest that there could be a full unification of all forces, but it seems that at a scale of (1027 eV)−1 gravity is not strong enough to compete with the other forces.

2. The second argument is about unitarity. Historically, unitarity has been a successful way to determine the scale at which the theory breaks down (think for example of the Fermi theory of weak interactions). In particular, in [40], using Effective Field Theories (EFT) techniques, it is shown that unitarity is violated for the EFT description of gravity roughly in an order of magnitude of the Planck scale.

It is fair, however, to stress that the arguments outlined above do not provide themselves in-controvertible evidences for the Planck length to be the Quantum Gravity scale. In fact, there are other important alternatives to these estimates. For example, we could think of a “running Newton constant”, subjected to some kind of renormalization group effect. This would imply, for example, that the length scale of spacetime quantization, L_QST, which we naively assumed to be likepG(∞), where G(∞) is the measured large distances Newton constant, should change like LQST∼

q

G(LQST).

So, to sum up, the situation is the following: there are compelling arguments to believe the Planck scale could be the scale of Quantum Gravity, but, at the same time, there are other valid arguments to believe that this is not the case. This is the reason why in the model we will describe in Chapter 3 we will not assume a specific value for the scale: L will be simply a free parameter (but still not reasonably far from the Planck length) to be determined through experimental observations.

1.1.3 A (very) good magnifying glass is needed

In the last two subsections we described the first two steps needed to understand the Quantum Gravity problem from a phenomenological point of view. We have seen what are the experimental situations to which we can have access and what (probably) is the typical scale for the effects we are considering. The last step is to identify what features are need by a Quantum Gravity experiment.

First of all, our experiment should be a quantum spacetime experiment rather than a Quan-tum Gravity one, since, as we have stressed earlier, there are little chances to open the doors of the quantum black-hole regime. Obviously this implies that we are looking for minute effects, whose magnitude is given by [Lp/λ]α, where Lp is the Planck lenght and λ is the wavelenght of the particles involved in the experiment. The consequences of such a dramatic suppression factor are twofold: on one side it is clear that, even if we are incorrigible optimists, for example supposing that α = 1 and using the shorter wavelenghts we are able to produce, ∼ 10−19 m, the factor [Lp/λ] ∼ 10−16, is still a very big one and the effects we are trying to see are really small4. This is of course discouraging but, on the other hand, such a disarming suppression factor greatly simplifies the problem of figuring out what could be a good experimental context, since of course such experiments should enjoy remarkable properties.

4

Actually, we could be even more optimistic. We could think of use for example astroparticles, whose wave-lenghts could even be of 10−27m, inducing a suppression of 10−8. Of course these would be “observations” rather than “experiments”, but still this possibility deserves a mention.

(21)

The most important is the one anticipated above: a good quantum spacetime experiment should be a very good amplifier : it should have a mechanism accounting to a large amplification of these tiny Quantum Gravity effects.

Interferometric experiments have such an amplifying factor. For example, the gravitational wave Advanced LIGO has a strain sensitivity of better than 10−23/√Hz around 100 Hz [1]. This is a huge amplifying factor which, monitoring the signal for a time T , by statistics, could give a sensitivity for amplitudes of order A ∼ 10−23(1 s /T )1/2. In a long run experiment, this sensitivity could be in principle enough to detect minute effects such as quantum spacetime ones. Another interesting example involves cosmology. In a scenario in which quantum spacetime effects are implemented as modified dispersion relations it is clear that if we consider photons (composing a signal whose source lasted only a very short time) propagating through very large (for example, cosmological) distances, eventually we will see a “time spreading” of the signal, and the cumulative effect could be detectable. Here, obviously, the amplifying factor is the enormous distance travelled by photons.

1.2 Candidate effects

From the considerations exposed in the previous section we see that we should look for small effects in the quantum spacetime regime, that such effects would be characterized by a scale which could be the Planck scale and that to observe them we need an experimental apparatus which should be a very good amplifier. Still, there is a missing piece in our discussion: what kind of effects are we expecting?

Each Quantum Gravity theory provides a different picture of the small scale structure (quan-tum black-hole regime) of the Universe. Still, after the “coarse graining” needed to describe the features of quantum spacetime we expect to find effects which are common to all those theories. But here we immediately find a problem. As pointed out in [64] by Amelino-Camelia, we are considering such complex theories that usually it is an incredibly hard (sometimes an impossible) task to give a definite prediction for a particular Quantum Gravity effect. The only thing we can do, therefore, is to find some “theoretical evidence” for a particular effect in a Quantum Gravity theory, and then do phenomenology using some test theories. Of course, the variety of theoretical quantum gravity scenarios produces a rather broad collection of hypothesis con-cerning possible quantum spacetime effects. In this section we are going to briefly list some of them.

1.2.1 Planck-scale modifications of classical spacetime and CPT symmetries

Symmetry tests are very reliable tests, since symmetries impose strong conditions on physical observables: for example, they could impose some of them to be exactly zero, a situation which lend itself to experimental tests (more about this in Section4.2). Moreover, it is easy to test spacetime symmetries very sensitively [10], and it is therefore natural to expect that introducing some “quantum” features of spacetime, its symmetries would be heavily affected. These are the reasons why, for Quantum Gravity phenomenology, a scenario involving some modifications of the classical symmetries is so appealing.

Classical spacetime symmetries For example, let us consider the Minkowski limit, where the spacetime is described by a classical, Minkowski space. Such a spacetime admits a clas-sical Lie-algebra of Poincaré symmetries, whose smoothness should be under severe scrutiny if this classical spacetime is replaced by a quantized or discretized version. Therefore, a very

(22)

active quantum spacetime phenomenology research area is the one which studies possible Quan-tum Gravity induced departures from Poincaré/Lorentz symmetries. There are essentially two possibilities which were explored in the past:

1. The first involves some symmetry-breaking mechanism of Poincaré/Lorentz symmetry. 2. The second possibility, maybe more stimulating from the theoretical point of view, is one

involving a “spacetime quantization” which actually deforms but does not break spacetime symmetries [7,8].

The study of such situations is certainly “encouraged” by the major Quantum Gravity proposals. For example, as we are going to see in the next chapter, Loop Quantum Gravity predicts some sort of intrinsic discretization of spacetime and, even thought the Minkowski limit of the theory is not still well understood, there are some arguments which suggest that such discretization could produce departures from classical Poincaré symmetries.

CPT symmetry Another typical feature of Quantum Gravity models is non-locality, at least in the form of some sort of limitation to the localizability of a spacetime event. But locality is a key ingredient for CPT invariance of a theory. It seems therefore reasonable to seek clues of Quantum Gravity in tests of CPT symmetry and indeed, a reach amount of literature can be found about this topic (collected, for example, in [10], section 2.2.2). Anyway, in this case the situation is even worse than in the case of classical spacetime symmetries. The reason is that the understanding of the fate of CPT symmetry in a Quantum Gravity context requires an handling of the candidate theories which at the moment we simply still do not have [10].

1.2.2 Decoherence and modifications of the de Broglie relation

We have already discussed about the importance of the assumption of a classical spacetime background in Quantum Mechanics tests. If such a background is no more provided some departures from Quantum Mechanics may arise. A first consequence can be the appearance of some kind of “decoherence”, i.e., some loss of quantum coherence, and it was at first motivated by test theories of quantum spacetime (see, for example [9]). A second consequence can be found in the modification of the usual de Broglie relation. In fact, such a relation, in ordinary Quantum Mechanics, is induced by the properties of the classical background spacetime that is assumed. In particular, it reflects the properties of differential calculus on that manifold [10]. In fact, in Quantum Mechanics the momentum observable is described in terms of derivative operator, which once it acted on a wave function with wavelength λ, gives the de Broglie relation p = }/λ. If the spacetime is no more classical, one must adopt new rules for the differential calculus [55,75] and therefore the wavelength-momentum relation should be modified.

1.2.3 Spacetime fuzziness

Wheeler’s intuition of “spacetime foam” [81], even though not very clearly operatively defined, is something that motivated many authors to look for types of spacetime characterized by the impossibility to define sharply the notion of distance between two events. In other words, these spacetime are typically endowed with a “fuzzy” metric. This context will be the one extensively studied on this thesis: we are going first to review in which models such an effect appears, and then we are going to construct a toy model implementing it and a detector to measure it. Therefore, at this point we are not going to go deeper: we will have plenty of time to discuss this scenario.

(23)

1.2.4 Planck-scale modifications of the Equivalence Principle

Let us conclude this short list by noticing that the Equivalence Principle, ensuring that (under appropriate conditions) two point particles would follow the same geodesic independently of their mass, should not be entirely trusted in a quantum spacetime context. Indeed, locality is fundamental for the present formulation of Equivalence Principle: it is clearly not applicable to extended bodies and presumably also not to “delocalized point particles”, i.e., point particles whose position is affected by irreducible uncertainties. In view of the model adopted to describe spacetime fuzziness in this thesis, it is interesting to cite the result found in [35], where it was obtained a violation of Equivalence Principle starting from a fluctuating metric.

1.3 The example of Doubly Special Relativity

Quantum Gravity phenomenology is without doubt still in its earliest stages of development. Because of this, it could not be perfectly clear what does it mean to construct a Quantum Gravity phenomenological model. And, as far as we know, there are actually very few of them which really deserve such a name. Among the most promising ones, Doubly Special Relativity (DSR) is, in our opinion, a very interesting proposal, both because of its intriguing working assumptions and because of the phenomenology it implies. It could be therefore of some help to quickly review the ideas behind its construction, since they can provide some guidance for the construction of the phenomenological model in Chapter3.

1.3.1 Motivations and definitions

DSR was introduced in [7,8], as an attempt to provide a different perspective on the fate of the Lorentz symmetry in Quantum Gravity. In fact, as we have already discussed above, there is a general expectation that Lorentz symmetry would be somehow violated in Quantum Gravity. This belief led to the conclusion that the usual energy-momentum dispersion relation would be consistently modified, assuming usually a form E2 _{= p}2 _{+ m}2 _{+ ηL}n

pp2En+ O(Ln+1p , En+2). Anyway, in all the relevant papers on these subjects (see [9] and references therein), it have been assumed that such modifications were induced by some breakdown of the Lorentz symmetry, assumption that implies the emergence of a preferred class of inertial observers. The striking analogy with this situation and the one faced at the advent of Special Relativity, which soon took the place of Galileo’s Relativity by the introduction of an observer-independent scale and of a “modified energy-momentum relation” was highlighted and taken seriously in [7,8]. Maybe, we are again facing a situation which does not demand the introduction of a preferred frame, but rather some new, modified transformation rules between inertial observers.

Definitions Even tough the motivations described above do not provide a well defined math-ematical formalism for the implementation of such a model, the physical scenario is anyway rather clear. In DSR one looks for a “rewriting” of the Relativity postulates similar to the one performed in the transition Galileo → Einstein. As in that case, in order to describe high-velocity particles, there appeared the need to introduce an invariant high-velocity scale c, it is at least plausible that, in order to describe high-energy particles, we should introduce a new invariant scale other than c, therefore substituting Special Relativity with some kind of “Doubly Special Relativity”. Therefore, we could roughly summarize the principles of DSR as follows:

(24)

DSR2 There exist two observer independent scales: a velocity scale c, identified with the speed of light, and a mass scale κ (or equivalently, a length scale λ = κ−1).

Since in DSR only the high-energy sector of the theory is modified, we could assume that in the limit κ → ∞, we should recover Special Relativity: hence, it is reasonable to assume that the present operative characterization of the velocity scale c would be preserved. Therefore, c is again the velocity of massless low energy particles. However, we see that, at least from the operative point of view, there is a difference between c and κ. From one side, we have just said that we can measure c by measuring the speed of light with energy E κ, while, on the other side, at the moment, there is no available operative definition of κ.

Besides these kind of problems, we know what such a definition of our theory actually means: from special relativity we learnt that scales appears as parameters of transformations between different reference frames. So, our DSR model should provide some transformation rules between inertial frames depending both on c and κ.

1.3.2 The κ-Poincaré framework for DSR

As we have already stressed, DSR does not bring with itself a well defined mathematical frame-work for its construction. Because of this reason there could be many different ways to implement a DSR scenario. One of the most interesting is without doubt the κ-Poincaré/κ-Minkosowki Hopf algebra framework. Let us discuss it briefly.

DSR as a limit of Quantum Gravity? If the physical principles on which DSR is based are correct, it is quite natural to expect that it can emerge as some limit of Quantum Gravity. Usually, we take for granted that Special Relativity is the “correct” limit of (Quantum) Gravity, in the case of a flat spacetime. From the “gravitational point of view”, flat Minkowski spacetime corresponds to configurations of the gravitational field in which the field itself is vanishing. However, the gravitational field is related to the energy content of the Universe; thus, when taking the flat spacetime limit (i.e., G → 0, Λ → 0, if in our theory we are considering also a non-vanishing cosmological constant) we should deal not only with a (vanishing) gravitational field, but with particles, too. And it is possible that the trace of particles’ back reaction on spacetime might be present as some global information. In other words, we are contemplating the possibility for DSR to arise as a limit of (Quantum) General Relativity coupled to point particles in the topological field theory limit.

This situation is not so implausible and the perfect stage on which it can be tested is 2 + 1 gravity, which does not possess local degrees of freedom and it is therefore a topological field theory. It is quite well established [63], that the excitations of 3d Quantum Gravity with a cosmological constants transform under representations of the quantum deformed de Sitter algebra SO_{q(3, 1), with z = log q given, in the small limit of Λ}}2/κ2 as z ≈√_{Λ}/κ, where κ is} the inverse of the 2 + 1 dimensional gravitational constant (which has the dimension of a mass). In the limit z → 0, this algebra becomes the standard SO(3, 1) algebra, and this is the reason for the notation SO_q(3, 1). Long story short, there is a standard way to obtain the Poincaré algebra from the de Sitter algebra. This two-step procedure goes as follows:

1. First, one identifies the momenta Pµ as appropriately rescaled generators M0,µ of the SO(3, 1) algebra.

2. Second, one takes the Inömü-Wigner contraction limit (Λ → 0) [15].

This limit is considered in [52], and the important result is that the structure one obtains is still a quantum algebra, called (three-dimensional) κ-Poincaré algebra. The reason for this name is

(25)

as above: it turns out that for κ → ∞, we obtain the standard Poincaré algebra. Moreover, there exists a basis, called bicrossproduct basis, where the algebra looks as follows:

[Ni, Nj] = −M ij, [M, Ni] = ijNj, [Ni, Pj] = δij κ 2 1 − e−2P0/κ₊P 2 κ2 − PiPj κ , [Ni, P0] = Pi [M, Pi] = ijPj, [P0, Pi] = 0 = [P2, P1] .

Therefore, in such a basis, the Lorentz sector is not modified, as we can see from the first line above (M is the rotations generator and Ni the boosts generators): the first DSR principle is satisfied. Moreover, it has been shown that κ is an observer-independent scale [49], so that also the second DSR principle is satisfied: the above algebra is exactly of the form needed in DSR.

In four dimensions things are not so simple, mainly because of two reasons. The first is that now the relevant algebra is only conjectured to be the quantum deformed de Sitter algebra SOq(3, 2), with z = log q behaving in the limit of small Λκ−2as z ≈ Λκ−2. Second, once the limit is properly taken, it turns out that we have to deal with a one-parameter family of contractions labeled by a real, positive parameter r: only for r = 1 a κ-Poincaré algebra is selected out (for 0 < r < 1 the result is a standard Poincaré algebra, for r > 1 the contraction does not exists) [50]. Therefore one should reasonably ask why Quantum Gravity should single out r = 1. Hopf algebras framework for DSR theories From the discussion above we have not only seen explicitly that a DSR scenario can be compatible with a Quantum Gravity theory, but we have also learnt that we can built a DSR model with the mathematical apparatus of quantum de-formed algebras (Hopf algebras). We can therefore take the κ-Poincaré algebra in bicrossproduct basis, now in four dimensions, as an efficient implementation of the DSR principles (forgetting all the aforementioned problems arising in four dimensions):

[M_i, Mj] = iijkMk, [Mi, Nj] = iijkNk [Ni, Nj] = −iijkMj, [M_i, Pj] = iijkPk, [Mi, P0] = 0 , [Ni, P0] = iPi, [Ni, Pj] = iδij _κ 2(1 − e −2P0/κ₊ 1 2κP 2_{− i}1 κPiPj.

Let us now take a closer look at this algebra. There are at least two points which deserve a mention:

. First, contrarily to the usual Lie algebra, we see that at the right hand side of the commutation relations we find smooth functions of the generators. Moreover, we see that, as above, the Lorentz sector of the algebra is not deformed. Therefore, the Lorentz symmetry is not broken, but rather non-linearly realized in its action on momenta [50]. Anyway, this does not mean that DSR is nothing but a “non-linear version” of Special Relativity. We will see why later. . The Casimir operator of this algebra is given by [50]:

4κ2sinh2P0 2κ− P

2_eP0/κ_{= M}2_, _(1.1)

which implies that the value |P| = κ corresponds to infinite energy P₀ = ∞ and the speed of massless particles v = ∂P0/∂|P| increases with momentum and diverges for |P| = κ.

However, the most striking property of the κ-Poincaré algebra is that it can be made an Hopf algebra, as we mentioned for the the three-dimensional case discussed above. The peculiarity

(26)

of an Hopf algebra is that it possesses a coproduct rule5, i.e., a map ∆ : A → A ⊗ A, where A is an (unital associative) algebra, satisfying some requirements6 _{that make it in a sense dual}

to the algebra multiplication rule · : A ⊗ A → A. This map essentially provides a rule on how the algebra acts on products (of functions, and more importantly for physical applications, on multiparticle states). In the bicrossproduct basis it is given by [50]

∆(Mi) = Mi⊗ 1 + 1 ⊗ Mi, ∆(Ni) = Ni⊗ 1 + e−P0/κ⊗ Ni+ 1 κijkPj⊗ Mk, ∆(P₀) = P₀⊗ 1 + 1 ⊗ P₀, ∆(P_i) = P_i⊗ 1 + e−P0/κ_{⊗ P} i.

This additional structure is enough to provide the entire phase space of DSR. This can be done essentially dually pairing the momentum coalgebra with the coordinate algebra [50]. In other words, the coordinates algebra is defined through the momentum coalgebra. The result of this (actually surprisingly simple) procedure is a κ-Minkowski non-commutative spacetime,

[X₀, Xi] ≡ X0Xi− XiX0 = − i

κXi, [Xi, Xj] = 0 ,

from which we see immediately that DSR, formulated in the κ-Poincaré Hopf algebras language is not Special Relativity in non-linear disguise. This is essentially because if the non-linearity disappears in the algebra sector of the κ-Poincaré because of an appropriate choice of the basis, it appears in the coproduct sector. By defining the coordinate algebra through the momentum coalgebra we see that this “compensation mechanism” makes the DSR phase space completely different from the Special Relativistic one [50]. We will dwell no more in other details on this fascinating mathematical framework, but a comprehensive discussion can be found in [54].

1.3.3 DSR phenomenology

If we really want to claim that we are doing DSR phenomenology, “a minimum requisite would be the availability of a theoretical framework whose compatibility with the DSR principles was fully established and characterized in terms of genuinely observable features” [9]. Such a condition is not presently satisfied even for the Hopf algebra scenario we briefly discussed above, which still represents one of the best developed attempt of finding DSR compatible theories [9]. So, we are left with two possibilities: the first is to postpone all reasoning about phenomenology to better times, the second is to reduce DSR to some “toy DSR test theories”. They are clearly of very small extent, but, contrarily to the “full theory” (either because its predictions seem impossible to be actually computed, such as in Quantum Gravity theories, or simply because the theory is still not unequivocally defined, such as the above DSR theory), their predictions for a particular effect are much more manageable. This second path is particularly significant for the spacetime fuzziness that we are going to introduce in the next chapter, so let us describe how these “toy test theories” are characterized.

5

Actually, it also possesses a counity : A → K where K is the field over which the algebra is defined, and an antipodal map S : A → A. The counit is in some sense dual to the unit map η : K → A, while the antipodal map is can be interpreted as a “generalized inverse”.

6_{Explicitly these requirements are that ∆(a · b) = ∆(a) · ∆(b), where a, b ∈ A and ∆(1) = 1 ⊗ 1, where 1 is the}

(27)

Wavelength dependence of the speed of light For example, let us consider only the relation between energy and momentum observables. From the form of the Casimir (1.1), we can write down a dispersion relation which, at the lowest non-trivial order in the limit κ → ∞, takes the form

E2 ' p2+ m2+ 1 κp

2_{E .} _(1.2)

This, of course, is not enough to determine an energy dependence of the speed of photons, but if one assumes also that v ≡ dE/ dp, one gets the following velocity formula [9]:

v ' 1 − m 2 2E2 + E κ , v ≡ dE d|p|. (1.3)

Therefore, our “toy DSR test theory” is characterized by the above dependence of the velocity on the energy (and also by the requirement that v ≡ dE/ dp). So, this “toy theory” forgets about all the assumptions of the DSR model: it is simply stated by the two properties in (1.3). On such a theory is finally possible to do phenomenology, and actually there is a very straight-forward way to investigate the possibility expressed by (1.3): whereas in Special Relativity the time of flight of two photons emitted simultaneously but with different energy is exactly the same, a little delay can be produced if equation (1.3) holds. The Universe provides us with a perfect observational situation for these kind of “time of flight tests”. In fact, gamma-ray bursts [11], reaches us from cosmological distances, corresponding to T ∼ 1017 s. Moreover, microbursts within a single bursts can have very short duration, of the order of 10−3 s (or even 10−4 s). Photons, therefore, are emitted roughly at the same time (up to an uncertainty of 10−3 s). On the other hand, photons in these bursts can be very energetic: even in the GeV-range. So, if we consider a ∆E ∼ 1 Gev between two photons, and assuming κ ∼ E_p, we see that we can find ∆t ∼ 10−2 s, which is greater than the uncertainty in the time of emission within a single microburst.

We want to stress something that may be obvious: we looked for an observation which is in the “theory jurisdiction”. We did neither asked it something about dynamics, nor even about energy-momentum conservation. Still evidences of effects predicted by (1.3) could be of great importance for development of the DSR theory.

Disclaimer

In this section about DSR we intended to provide a valuable example of how a phenomenologi-cal model, based on a physiphenomenologi-cal intuition can be constructed with the guidance of full Quantum Gravity theories (in this case this led us to the intuition of using Hopf algebras as the mathe-matical framework of the theory) and of how it is possible to do phenomenology on it, invoking the use of limited “test theories”. However, since we were moved by only these motivations, we deliberately missed many important points both in the description of the model and in its phenomenological aspects. Here, we want to stress only a few things that should be very clear (but that we suspect in our discussion they were not):

. We treated DSR only with the mathematical apparatus of Hopf algebras. This could lead to the conclusion that a DSR theory can be trated interchangeably with the concept of a κ-Poincaré algebra. This would be “inappropriate in the same sense that the physical proposal of Einstein’s special relativity cannot be strictly identified with the mathematics of Lorentz and Poincaré” [9]. In particular, this is even more inappropriate for DSR, since it has been shown that some aspects of the κ-Poincaré framework are not compatible with the principles on which DSR is based. For example, it has been argued that in the bicrossproduct basis

(28)

above, the sum of particles momenta is limited by the scale κ. This can be acceptable for microscopic particles, but it appears obviously unreasonable for macroscopic objects: because of this, that problem is usually called the soccer ball problem [50].

. There is nothing special in the bicrossproduct basis described above: we could in principle use the scale κ to redefine new energies and momenta as analytic functions of the old ones. An interesting point is that these arbitrariness in the choice of the momentum variables can be interpreted as a coordinate transformation on some “energy-momentum space”, which appear to be a constant curvature manifold [50]. For example, in another basis, called classical basis, the rules of composition of energy-momentum7 take the form (at the first order in λ ≡ κ−1) [50]

Ea+ Eb+ λpapb' Ec+ Ed+ λpcpd,

pa+ pb+ λ(Eapb+ Ebpa) ' pc+ pd+ λ(Ecpd+ Edpc) ,

for a process a + b ↔ c + d. These rules are often taken as energy-momentum conservation rules in purely kinematic test theories.

. The above example of the dependence of velocity on the energy is particularly debated. In fact, a more careful analysis studying the group velocity of a wave packet shows that in the definition of group velocity actually there is an ordering ambiguity due to the fact that the phase space of κ-Poincaré DSR is not trivial [50]. Another computation, where one derives the velocity starting already from the phase space of the κ-Poincaré DSR indicates that the speed of massless particles equals 1 [50]. Anyway, we want to stress again that this is absolutely irrelevant in our discussion, whose aim was to show to which extent one talks about a “test theory”.

For a more detailed review on DSR, a pedagogical introduction is [50], while in [9] one can find interesting references and many clarifying discussions.

7_{These rules can be defined starting from the coproduct rules, which, as we have mentioned above, essentially}

(29)

2

Motivations

In Section1.2we discussed some of the possible effects arising from a quantum spacetime struc-ture. From that discussion, we have mentioned that it is very difficult to state clearly if an effect is present or not in a given Quantum Gravity theory. Still, it is possible to look for “theoretical evidences” in favor of the emergence of a particular effect which can also inspire the construction of appropriate phenomenological models. These models, contrarily to the more complex Quantum Gravity theories, should give definite predictions for the effects under study (we will discuss in more detail this topic in Section 3.1). These two steps (the individuation of a particular effect from the aforementioned “theoretical evidences” and the construction of a phenomenological model) will be addressed respectively in this and in the next chapter.

We have already briefly discussed in Chapter1which is the physical effect we are looking for, introducing the so called spacetime fuzziness. This will be the object of our research throughout all this chapter. But why did we chose this specific effect? To answer this question, let us forget about spacetime fuzziness, and let us try to understand which is an effect common to most of Quantum Gravity theories. Since it should be shared by a large number of theories it has to be something coming from principles on which each theory should agree. Well, surely they should believe that (at least at low energies) General Relativity and Quantum Mechanics are true. Let us see what naturally happens if one tries to combine their fundamental principles.

Quantum Mechanics brings with it the Heisenberg uncertainty principle. On the other hand General Relativity tells us that spacetime is a dynamical entity. It affects particles and at the same time suffers from particles backreaction. Now, an uncertainty of quantum nature on the position of a particle implies an uncertainty on its momentum. But since energy interacts with gravity, this also implies an uncertainty in the geometry of spacetime, which, in turns, introduces an additional uncertainty in the position of the particle. We can therefore write the total position uncertainty as

δx = δgx + ∆x ,

where ∆x is the usual Heisenberg uncertainty and δgx is the one added by the gravitational interaction. We made geometry suffer from quantum fluctuations. This should not be surprising: if we want to quantize the gravitational interaction we should expect some kind of quantum fluctuations of the field.

We could now ask what is the scale of these fluctuations. Assume we want to resolve a region of radius L. Clearly we need a photon with wavelenght smaller than L, so that its energy would be grater that L−1. We are therefore plugging in the system an energy density ρ of L−4. Now

(30)

we could use Einstein equations, which roughly are ∂2g ∼ L2pρ & L2p/L4. We therefore are left with g_{& L}2_p/L2, so that the length we are measuring has an uncertainty of pgL2 _{& Lp}_.

From these kind of arguments we see that the notion of event in Quantum Gravity is not sharp anymore: we cannot localize with arbitrary precision an event in the spacetime, making it somewhat “fuzzy”. It seems therefore, that in such a fuzzy spacetime, a minimum lenght arise

“beyond which the very concepts of space and time lose their meaning [...]: there is no way of going beyond this border and its existence may be inferred through the relativistic corrections to the Newtonian theory” [34].

A minimum lenght scenario actually is provided by many Quantum Gravity theories and is justified by some sound theoretical arguments regarding thought experiments. In the rest of this section we are going to explain them.

2.1 Thought experiments

Historically, thought experiments were precious tools for physicists who could not, for example because of a lack of technologies, test the limits of a theory. Not only a thought experiment can explore ranges of parameter space of a theory which are inaccessible to real experiment, but it can also tie the theory to reality, carefully examining what are the implications of a physical measurement.

2.1.1 The Heisenberg microscope

e− γ λ y x

Figure 2.1. A schematic rep-resentation of the Heisenberg gamma ray microscope. The wavy lines represent light, the ellipse represents the lens and the thick, solid dot and line rep-resent respectively the electron and its motion.

Let us briefly review how the good old Heisenberg microscope works. Suppose that we want to determine the position in the x-direction of a particle using a light microscope. The uncertainty in this position measurement is determined by the wavelength of the light impinging on the electron1 and by the aperture of the lens: the wider the cone, the higher order of diffractions are able to interfere at the image plane of the objective, and hence the less “blurred” the image is. The resolution of the microscope and consequently the uncertainty on the particle position is given by the Rayleigh formula,

∆x ∼ λ sin  =

2π

ω sin , (2.1a) where λ is the light wavelength and ω its frequency. The im-portant point now it that, even if the momentum of the incident radiation were completely known, there still would be an upper limit on the accuracy with which this momentum changes. In fact, looking again at Figure2.1, we see that all we can say about the scattered light is that it is situated somewhere in the cone with half-aperture . This uncertainty on the direction of the photon implies that we can only say that the transverse momentum ac-quired by the photon is in the range [−p_γsin , p_γsin ]. Hence we have an uncertainty on the x-component of the electron given by ∆px∼ 2ω sin . (2.1b)

1_{The smaller the wavelength, the more sensitive the microscope: if the wavelength is too large, light no longer}

(31)

Combining these two results we arrive at

∆x∆px ∼ 1 , (2.2)

neglecting factors of order one. This is the usual Heinsenberg uncertainty relation, which we know being a fundamental property of the quantum nature of matter, not simply a feature of this particular measurement method.

Heisenberg microscope with Newtonian gravity Already in the Newtonian approxima-tion, following the treatment in [58], we can see how gravity changes the picture above. Let us define the region where the particle and the photon strongly interact by its radius R. To measure any effect arising from their interaction, the time elapsed since the moment the inter-action actually takes place must be grater than the time needed by the photon to travel the distance R, so τ _{& R. Next, we should realize that the photon carries an energy which makes it} interact gravitationally with the particle. The acceleration induced by this gravitational pull is at least a_{& Gω/R}2. If we now assume that the particle is non-relativistic, (hence being much slower than the photon), this pull lasts until the photon leaves the region of strong interaction. The particle acquires hence a velocity v_{& aR & Gω/R. Thus, in the time R the particle have} travelled a distance L_{& Gω. But again, the direction of the photon is unknown to within an} angle , meaning that we also are unable to tell the direction of the acceleration and the motion of the particle, too. Projecting on the x-axis, we have another source of uncertainty,

∆x_{& Gω sin ,} (2.3)

which combined with the one of the usual Heisenberg microscope, (2.1a), leads to ∆x_&

√

G = Lp. (2.4)

The conceptual meaning of this discussion is clear: if we want to reduce the uncertainty (2.1a) we need a photon with very high energy, which creates a non-negligible gravitational field. The effect of this field on the particle’s motion is independent on the nature of the particle, meaning that, contrarily to what we are used to with other forces, we cannot make the uncertainty deriving from it arbitrarily small by considering a very heavy particle. In fact, the above result is independent of the mass of the particle: it is a fundamental limitation on the possibility of localizing any kind of particle.

General relativistic Heisenberg microscope Let us now face the problem in a more so-phisticated manner, including also relativistic effects. Moreover, let us replace the photons with an arbitrary test particle, whose mass is given by µ (so that the photon case can be obtained by taking the limit µ → 0). The computation is pretty long, so it is useful first to list its main steps:

1. First of all we need the gravitational field generated by the test particle, affecting the motion of the measured one. To do this, we are going to assume that the motion of the test particle is uniform approximately during all its journey. Hence we will write down its gravitational field in a frame where the test particle is at rest, and then we will turn back to the laboratory frame.

2. The test particle should not turn into a black hole, or we are going to loose otherwise any hope to measure it. This fact will impose conditions on the gravitational field and on the velocity of the test particle.

Using correlations to experimentally search for Quantum Gravity effects

Quantum Gravity effects

Abstract

Acknowledgments

Notations and Terminology

Contents

1

Introduction

1.1

Quantum Gravity phenomenology

1.2

Candidate effects

1.3

The example of Doubly Special Relativity

2

Motivations

2.1

Thought experiments