Majorana spinor pair creation in accelerated frames

130 Pages·2010·0.96 MB·English

by  Pietro Longhi

Alma Mater Studiorum (cid:4) Universita` di Bologna
FACOLTA` DI SCIENZE MATEMATICHE, FISICHE E NATURALI
Corso di Laurea Specialistica in Fisica
MAJORANA SPINOR
PAIR CREATION
IN ACCELERATED FRAMES
Tesi di Laurea in Teoria dei Campi
Relatore: Presentata da:
PIETRO LONGHI
Chiar.mo Prof.
ROBERTO SOLDATI
Sessione I
Anno Accademico 2009-2010

Ai miei genitori Nadia e Franco,
con gratitudine.

Contents
Abstract 1
Introduction 3
Acknowledgements 7
1 Quantum Field Theory in non-Minkowskian geometries 9
1.1 Generalized spin-0 field theory 9
1.2 Spin-0 fields in Rindler geometry 12
a. The Rindler frame 12
b. The scalar field 14
1.3 Defining spinors: the generalized theory of spin-1 fields 16
2
1.4 The Majorana field in Minkowski spacetime 21
1.5 Majorana spinors in Rindler geometry 27
a. Solving the Majorana-Rindler equation 28
b. The basis of helicity eigenstates 38
c. Study of the hermiticity of the Hamiltonian 42
2 The Unruh effect 45
2.1 The Bogolyubov transform 46
2.2 General theory of the Unruh effect 48
I
CONTENTS
2.3 Unruh effect for the spin-0 field 53
2.4 Unruh effect for the Majorana field 57
a. Finding the spinor algebraic RS-to-MS transformation 58
b. Consistency with the general theory of spinors in curved
spacetimes 62
c. Helicity-eigenstate normal modes 63
d. Canonical normal modes 65
e. Choosing the proper representation for helicity eigenstates 66
f. Choosing the proper representation for canonical modes 71
g. Digression: normalization of Rindler modes in RS 75
h. Comparing the helicity-eigenstate scheme with the canon-
ical modes one: advantages of each scheme 76
i. The thermal spectrum for Majorana fermions 78
2.5 A different derivation of the Unruh effect: helicity structure 80
2.6 Criticisms and discussions on the Unruh effect 86
3 Dark Matter 89
3.1 The idea: a connection with Dark Matter 89
3.2 Dark matter models 91
3.3 Heuristic evaluation of the energy density 94
3.4 Majorana-Unruh fermions in strong gravitational fields 98
4 Conclusions 103
Appendix 107
A.1 Orthonormality and completeness of Rindler modes 107
a. A study of the scalar Rindler modes 107
b. Proof: orthonormality of Majorana Rindler modes in MS 111
c. Proof: completeness of Majorana Rindler modes in MS 113
d. Proof: completeness of Unruh modes in MS 114
A.2 Alternative derivation of the Unruh effect 115
Table of constants 118
Bibliography i
II
Abstract
Nel primo capitolo presenteremo gli strumenti necessari alla riformu-
lazionedellateoriadeicampiinmanierageneralmentecovariante, studieremo
poi le teorie di campo scalare e di Majorana dal punto di vista di un osserva-
tore uniformemente accelerato. Eseguiremo uno studio esplicito e dettagliato
di entrambe le teorie, dal punto di vista classico dapprima, quantistico poi.
L’obiettivo del capitolo ´e quello di acquisire tutti gli strumenti necessari ad
un’analisi approfondita dell’effetto Unruh.
Il secondo capitolo ´e dedicato allo studio dell’effetto Unruh per i campi
scalare e di Majorana. Dopo aver speso qualche cenno sulla teoria delle
trasformazioni di Bogolyubov, tratteremo in maniera del tutto generale la
teoria dell’effetto Unruh: mostreremo che un oggetto del tutto naturale in
relativita´,comeunatrasformazionegeneraledicoordinate,pu´oindurreeffetti
drammaticisulloschemadiquantizzazionecomeportareall’inequivalenzatra
spazi di Fock. Procederemo analizzando questo inscindibile legame tra op-
eratori di seconda quantizzazione e sistemi di coordinate nei casi di campo
scalareediMajorana. Nelcasodiquest’ultimoseguiremoduepossibilistrade
equivalenti, di cui una ci permettera´ di formulare la teoria quantistica in
maniera particolarmente agevole, mentre l’altra avra´ il pregio di preservare
il significato fisico della trattazione. Il taglio della trattazione ´e prettamente
tecnico e particolare attenzione ´e posta nello studio dei modi di Unruh ot-
tenuti, si dimostra in particolare che: sono analitici su tutto lo spaziotempo
1
Abstract
di Minkowski, si riducono ai modi di Rindler opportunamente trasformati nei
rispettivi settori, sono un set ortonormale e completo e sono dunque adatti
per costruirvi una teoria quantistica. Lo studio dell’effetto Unruh si conclude
con un’analisi della struttura dei coefficienti di Bogolyubov per lo spinore di
Majorana, in cui stabiliremo le relazioni tra stati fisicamente osservabili dagli
osservatori inerziale e non. Il capitolo chiude con una rassegna sulle recenti
critiche e dispute su problemi di natura matematica legati a certe derivazioni
dell’effetto Unruh.
Nel terzo capitolo sfrutteremo infine i risultati ottenuti per il campo di
Majorana studiando la possibilita´ di generare materia oscura tramite il mec-
canismo di Unruh. Dapprima introdurremo alcune ipotesi necessarie per
giustificare la possibile presenza di un ipotetico campo di Majorana. Pre-
senteremo poi in sintesi alcuni tra i modelli piu´ recenti di distribuzioni di
materia oscura e i rispettivi candidati particellari. Svolgeremo dunque un
derivazione euristica della densit´a di tali fermioni generati tramite meccan-
ismo di Unruh, ipotizzando l’accelerazione cosmica come causa scatenante di
tale effetto. In una seconda parte studieremo lo stesso meccanismo in pre-
senzadiaccelerazionegravitazionaledabuconero, esplicitandol’analogiacon
l’effetto Hawking. Il capitolo conclude con una rassegna sugli attuali risultati
circa la distribuzione di materia oscura ai livelli galattico e di grande scala.
2
Introduction
Physics is just the refinement of everyday thinking
Albert Einstein
The unification of quantum field theories with the theory of general rela-
tivity is being, since the seventies, among the greatest efforts in fundamental
theoretical Physics, probably the most ambitious one. Einstein’s elegantly
simple idea is that of a geometrical universe, wherein spacetime and matter
are both main actors, shaping each other according to the laws of general
relativity. Quantum field theory is instead a conceptually complex theory,
which successfully describes the behavior and the properties of the matter
forming the universe at its most fundamental level. While in general rel-
ativity one completely ignores the fundamental structure of matter, in the
quantum theory of fields it is spacetime that is neglected, being treated as a
rigid stage which ignores the effects of the events taking place on it. Both
theories have been confirmed experimentally to the highest orders of preci-
sion, however all the data in our possess regard contexts wherein the effects
of one theory or the other become negligible. Nonetheless there are situations
in which both theories are important: it could be extreme phenomena like
those happening in presence of a black hole or like the origin of universe, but
it could be much more common situations as the presence of dark matter.
3
Introduction
Of the various attempts to unify these theories, we will deal with that
known as quantum field theory in curved spacetime. This semi-classical ap-
proach consists in generalizing the quantum theory of fields through a gener-
ally covariant formulation which makes is possible to incorporate the equiv-
alence principle within the theory. It is not expected to be an exact theory
of nature, but it should provide a good approximate description of those cir-
cumstances in which the effects of quantum gravity do not play a dominant
role. The most striking application of the theory is Hawking’s prediction that
black holes behave as black bodies, emitting a thermal spectrum of radiation
with temperature T κ 2π. There was however a very disturbing aspect of
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Hawking’s calculation: it appeared to show a divergent density of ultrahigh
energy particles in proximity of the horizon of the black hole. In order to gain
insight on this issue, Unruh made an operational choice of particle: a particle
is a state of the field which can induce a transition in a certain detector appa-
ratus. What Unruh found out was surprising: whenever in flat spacetime a
certain field is in the ordinary Minkowskian vacuum, an accelerated observer
perceives a thermal spectrum of particles of temperature T a 2π. These
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apparently paradoxical phenomena have their roots in a fundamental fact
that actually lies at the heart of quantum field theory in curved spacetimes:
thenotionofparticleisnotfundamentalinQFT,thequantumtheoryoffields
is, indeed, a theory of fields not particles. In order to better understand the
meaning of this last claim it is necessary to delve technically in the theory
of QFT in curved spacetimes. There are actually three main approaches
to the Unruh effect: (i) analysis of the response of accelerated detectors in
Minkowski spacetime (ii) Unruh’s original derivation which is based on QFT,
withoutreferencetothedetailsofdetectors(iii)thealgebraic approach, based
largely on the Bisognano-Wichmann theorem which essentially says that the
ordinary Minkowski vacuum, when restricted to observables localized in the
right Rindler wedge, satisfies the Kubo-Martin-Schwinger condition. In this
work we will deal exclusively with Unruh’s original derivation.
The first chapter begins with an overview of the tools that are necessary
to reformulate QFT in a generally covariant way. After reviewing the gen-
eralized Klein Gordon theory we begin to study the case of a scalar field in
Rindler spacetime. We solve the classical theory by finding normal modes
that are suitable for quantization, we then proceed for the quantum theory.
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