Alma Mater Studiorum Universit ` a di Bologna FACOLT ` A 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: Chiar.mo Prof. ROBERTO SOLDATI Presentata da: PIETRO LONGHI Sessione I Anno Accademico 2009-2010
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Alma Mater Studiorum Universita di Bologna
FACOLTA DI SCIENZE MATEMATICHE, FISICHE E NATURALI
Corso di Laurea Specialistica in Fisica
MAJORANA SPINORPAIR CREATION
IN ACCELERATED FRAMES
Tesi di Laurea in Teoria dei Campi
Relatore:
Chiar.mo Prof.ROBERTO SOLDATI
Presentata da:PIETRO LONGHI
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-12
fields 16
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-
lazione della teoria dei campi in maniera generalmente covariante, 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, come una trasformazione generale di coordinate, puo indurre effetti
drammatici sullo schema di quantizzazione come portare all’inequivalenza tra
spazi di Fock. Procederemo analizzando questo inscindibile legame tra op-
eratori di seconda quantizzazione e sistemi di coordinate nei casi di campo
scalare e di Majorana. Nel caso di quest’ultimo seguiremo due possibili strade
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 densita 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-
senza di accelerazione gravitazionale da buco nero, esplicitando l’analogia con
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 κ2π. There was however a very disturbing aspect of
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 a2π. These
apparently paradoxical phenomena have their roots in a fundamental fact
that actually lies at the heart of quantum field theory in curved spacetimes:
the notion of particle is not fundamental in QFT, the quantum theory of fields
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,
without reference to the details of detectors (iii) the algebraic 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.
4
Introduction
Thereafter we turn to the case of spinor fields. After explaining how the
connection with the Lorentz group can be achieved in a curved spacetime,
we begin to study the Majorana field in the frame of a Rindler observer. We
first give a schematic treatment of the generalized theory of spinor fields,
explaining how it is possible to preserve the connection with the Lorentz
transformation properties of these fields. Then we begin to study the par-
ticular case of a Majorana field in Rindler spacetime. We found that, in the
literature regarding the Unruh effect, scalar fields are overwhelmingly much
more studied than spinor ones, so we decided to put particular emphasis on
the development of such theory by analyzing it in detail. In particular, we
develop two quantization schemes: one allowing for a cleaner quantization
while the other providing a physical meaning to one-particle states in terms
of physical observables. The aim of this chapter is to acquire all the necessary
tools that we will need for a detailed study of the Unruh effect.
The second chapter is devoted to the study of the Unruh effect for the
scalar and the Majorana fields. We begin with a review of the theory of
Bogolyubov transformation for both bosonic and fermionic systems. Then
we outline the general theory of the Unruh effect: the aim is to show, on
the most general grounds, how a coordinate transformation, which is just a
natural operation in relativity, can have dramatic effects on the quantization
of fields, such as bringing to inequivalences between Fock spaces. The Unruh
effect is then studied first for the scalar field and then for the Majorana one.
The content of this chapter is to a large extent technical and its aim is to
prove that the Unruh modes that we find for the Majorana field have all the
required properties, such as: they are analytical over the whole Minkowski
spacetime, they reduce to Rindler modes within the corresponding sectors of
Rindler spacetime, they are complete and orthonormal. We eventually derive
the spectrum of Majorana particles which turns out to be an exactly thermal
fermionic distribution. We end our study of the Unruh effects with another
analysis of the structure of Bogolyubov coefficients for the Majorana spinor,
wherein we determine the relation between physically observable states for
the inertial and Rindler observers. The chapter ends with a review of the
recent critics and discussions regarding some mathematical issues involved in
certain derivations of the Unruh effect.
In the third chapter we take advantage of the results obtained for the Ma-
5
Introduction
jorana field by studying the possibility to generate dark matter by means of
the Unruh mechanism. We first make some necessary assumptions concerning
the hypothetical existence of a Majorana field, explaining why the Majorana
field could be plausible dark matter candidate. We then briefly review the
most recent models of galactic and extra-galactic distributions of DM and
the corresponding particle candidates. Thereafter we proceed evaluating the
energy density that the Unruh mechanism would produce, we assume the cos-
mic acceleration as the source of the effect. The derivation is heuristic since
the aim is to obtain an order of magnitude for the energy density, in order to
make a comparison with the observed values of dark matter density. Finally
we repeat the evaluation taking into account the strongest non-inertial field
that occurs in our universe, namely the gravitational field of a black hole.
We first exploit the analogy with the Hawking effect and then proceed to
evaluate the Majorana energy density produced by such gravitational field
by means of the Unruh mechanism.
6
Acknowledgements
Acknowledgements
This work was originally conceived as the investigation of an idea sug-
gested by Prof. Roberto Soldati, supervisor of this thesis. First and foremost
I would like to thank him for the guidance and the support he has provided
during the course of this work. Being both new to this fascinating field of
physics, I had the pleasure to discuss with him on many issues that came up
along the path, learning from him an enormous amount. I am also grateful
to Prof. Soldati for he gave me my first exposure to quantum field theory, in
a superb course which pushed me to undertake the way of theoretical physics
(at the time I was a student in experimental high energy physics). A thank
goes out also to Prof. Fabio Ortolani for valuable discussions on issues of
mathematical nature that arised within this work. I had the fortune to meet
many valuable professors and teachers during my studies, I cannot exempt
myself from spending a few words for two of them. Of Prof. Giovanni Carlo
Bonsignori I cannot forget the endless passion for science and for teaching
in all their forms, together with his intuitive way of understanding physics.
Of Stefano Valli, a dear teacher of mathematics of mine, I treasure the rigor-
ous and clear approach to mathematics and physics; his willingness to make
every student understand the subjects certainly finds satisfaction in his in-
credible teaching skills. Last, but not least, I am grateful to Luca Zambelli
and Aurelio Patelli for several enlightening discussions and for their interest
in this work.
The typesetting of this work has been carried out using the freeware program
Kile based on the LATEX standard. The figures were realized using the asymptote
language. I am grateful to all those people who made it possible for these projects
to begin and to remain free.
7
CHAPTER 1
Quantum Field Theory in non-Minkowskian geometries
I believe that ideas such as absolute certitude, absolute exactness, final
truth, etc., are figments of the imagination which should not be
admissible in any field of science... This loosening of thinking seems to
me to be the greatest blessing which modern science has given us. For
the belief in a single truth and in being the possessor thereof is the root
cause of all evil in the world.
Max Born
First the general treatment of Quantum Field Theory in curved space-
times is presented in this chapter for both the Klein-Gordon and the Dirac
fields. We will then turn to the study of Rindler spacetime (we will refer to
it as RS) and its causal structure. Finally we will present a detailed study of
the real scalar and Majorana spinor fields in RS. We will use natural units
i.e. ~ c 1
1.1 Generalized spin-0 field theory
This section is devoted to the study of the Klein-Gordon field as perceived
from a non-inertial point of view: the generalized theory of spin-0 fields
9
Chapter 1. Quantum Field Theory in non-Minkowskian geometries
is presented in order to give all the necessary tools to deal with general
coordinate transformations. In the next section we will focus on the case of
constant proper acceleration, i.e. the Rindler case.
We start with the generalized Lagrangian density for a real scalar field
where gpxq is the metric determinant. The coupling between the scalar field
and the gravitational field is given by the term ξRφ2, where ξ is a numerical
factor and R(x) is the Ricci scalar, the only possible local, scalar coupling of
this sort with the correct dimensions. The resulting action is
S »Lpxq d4x . (1.2)
By varying S with respect to φ and setting δS 0 we arrive at the generalized
Klein Gordon equationlm2 ξRpxqφpxq 0 (1.3)
where the D’Alembert operator reads
lφ gµν∇µ∇νφ pgq12Bµpgq12gµνBνφ (1.4)
There are two particular cases, corresponding to two values of ξ that are
interesting: the minimally coupled case (ξ 0) and the conformally coupled
case (ξ p14qrpn 2qpn 1qs) where the positive integer n is the number
of spacetime dimensions. In the latter case, if m 0 the scalar field equation
turns out to be conformally invariant.
Since we will be dealing with flat spacetimes, let us drop the gravitational
coupling term; then it is easy to check that a conserved current of the La-
grangian is
Jµ iφÐÑB µφ ∇ J pgq12Bµrpgq12Jµs 0 (1.5)
and more generally, also the vector current
Jµ12pxq iφ1pxqÐÑB µ
φ2pxq (1.6)
10
1.1. Generalized spin-0 field theory
satisfies the continuity equation, as long as φ1, φ2 satisfy (1.3). It is then
possible to define an invariant scalar product for the field as
pφ1, φ2q i
»Σ
φ1pxqÐÑBλφ2pxq dΣλ (1.7)
where Σ is a three dimensional spacelike hypersurface and
dΣλ 1
3!ελρστ dxρ ^ dxσ ^ dxτ
agpxq (1.8)
ε0123 1 εµνκλ gµτ ετνκλ (1.9)
is the invariant future-oriented hypersurface element. As for the Minkowskian
case, the solutions of (1.3) can be expanded in normal modes
φpxq ¸i
aiuipxq a:iu
i pxq
(1.10)
with the tuiu being a complete and orthonormal set of mode solutions.
Covariant quantization is achieved by imposing the commutation relations
rai, aj:s δij , etc. (1.11)
It is then straightforward to construct a vacuum state, a Fock space and
proceed as usual for the Minkowskian case. Although from the purely math-
ematical point of view we haven’t encountered any difficulties, these show
up as soon as we try to give a physical interpretation of what we just ob-
tained. Indeed it was clearly pointed out by Fulling in [22] that, while in
the Minkowskian case we could readily make a distinction between positive-
frequency and negative-frequency normal modes, this is no longer obvious in
a nonflat metric.
In MS (Minkowski spacetime), the vector BBt is a Killing vector of the space,
orthogonal to the spacelike hypersurfaces t constant and the well-known
modes
rp2πq32ωks12 exp tiωkt ik xu (1.12)
are eigenfunctions of this Killing vector. These modes are closely associated
with the natural coordinates pt, x, y, xq. In turn these coordinates are asso-
ciated with the Poincare group, which leaves the line element unchanged.
In curved spacetime the Poincare group is no longer a symmetry group. As
11
Chapter 1. Quantum Field Theory in non-Minkowskian geometries
a consequence there will be no Killing vectors that can be used to define
positive-frequency modes, at least in general. Further detail is given in [57]
This could be expected, as the very first and most important consequence
of the principle of General Covariance is just that coordinate systems are
physically irrelevant.
Summarizing, we have seen that, at least mathematically, a curved space-
time scalar quantum field theory is viable, although it is unclear what physi-
cal meaning could be attributed to it. We will now leave this interesting and
crucial point and return to it later on. Actually there are some very special
classes of spacetimes in which ’natural coordinates’ analogous to pt, x, y, zqfor MS may exist, together with a timelike Killing vector. This is indeed the
case for Rindler spacetime.
1.2 Spin-0 fields in Rindler geometry
We will now examine the behavior of a scalar field from the point of
view of an accelerated observer. This case, in which the metric is flat, is
nonetheless interesting, since it enjoys some important features due to non-
inertial quantum effects.
a. The Rindler frame
Consider an object moving with constant proper acceleration along the
xaxis through Minkowski spacetime. A typical example of this situation
could be a spaceship with an infinite energy supply and a propulsion engine
that exerts a constant force, another could be an electron inside an infinitely-
wide parallel plane condensator. Let us define the laboratory frame as the
usual inertial reference frame with the coordinates pt, x, y, xq, and the proper
frame as the accelerated system of reference that moves together with the
observer. To describe quantum fields as seen by an accelerated observer,
we need to use the proper coordinates pη, ξ, y, zq, which are also known as
12
1.2. Spin-0 fields in Rindler geometry
ξ=0+ ,
η=+∞ξ
=0 −,η= −∞
ξ=0 +,η= −∞
ξ=0− ,
η=+∞ξ=
const.
η = const.
x
ct
R
L
Figure 1.1: The Rindler wedge: the dashed lines are the event horizons where ξ 0 and
η 8, the hyperbolae are the trajectories of particles at rest with respect to the Rindler
observer (ξ constant), while the stright lines are equal-time spacelike hypersurfaces
with respect to the Rindler time variable (η constant). Notice that η increases towards
Minkowski’s past in the left wedge. The origin is a singularity of coordinate system.
Rindler coordinates. They are defined as
pt, x, y, zq Ñ pη, ξ, y, zq (1.13)
t ξ sinh aη x ξ cosh aη (1.14)
where a ¡ 0, these can be inverted to yield
ξ ?x2 t2 η a1arctanhptxq (1.15)
from the inversion formulae it is evident that this system of coordinates covers
only the region t2 x2, 8 y, z 8, which is called ”Rindler wedge”.
The rest of Minkowski’s space can be covered by changing the signs in the
right-hand side of equations (1.14). In the new frame of reference the sign of
time is reversed in the region x 0, i.e. for ξ 0 η increases as t decreases
and vice versa.
Finally, the metric tensor in Rindler space is gµν diagpa2ξ2,1,1,1q;A remarkable property of RS is that its causal structure is deeply different
from the MS one. Indeed the hypersurfaces tpt, x, y, zq;x tu play the role
13
Chapter 1. Quantum Field Theory in non-Minkowskian geometries
of event horizons, dividing MS into causally disconnected parts. This feature
of RS and its effects on the fields will be discussed in more detail at the end
of this chapter.
x
c t
P
Q
x
c t
P
Q
Figure 1.2: On the left: depicted is the world line of a particle moving through MS
as seen by a Minkowskian observer, the two points P,Q are causally connected from this
point of view. On the right: the particle lives in RS and propagates through an accelerated
background, from the point of view of a Rindler observer there is not a world line connecting
P to Q, any signal emitted from P will fall towards the future light-cone branch, which
plays the role of an event horizon. The dashed lines are world lines of particles at rest in
P in MS and RS respectively.
b. The scalar field
Rindler spacetime is flat, which means that equation 1.3 reduces to the
usual Klein Gordon equation, being Rpxq 0 at every point. The field, as
seen by the Rindler observer, will satisfy the KG equation which reads1
a2ξ2 B
2
Bη2 1
ξ BBξ
B2
Bξ2 B2
By2 B2
Bz2m2
φpη, ξ, y, zq 0
(1.16)
To find a functional form of φ satisfying (1.16) it is convenient to make use
of a partial Fourier transform
φpη, ξ, y, zq »R3
dk0 dkKp2πq32 φpk0,kK; ξq eik0ηikKxK (1.17)
xK py, zq kK pky, kzq (1.18)
14
1.2. Spin-0 fields in Rindler geometry
so that equation (1.16) becomesB2
Bξ2 1ξ BBξ β2 k0
2
a2x2
φpk0,kK; ξq 0 (1.19)
β a
kK2 m2 (1.20)
If we perform a Wick rotation by substituting ik0 E, we are left with the
well-known Bessel equation B2
Bξ2 1
ξ BBξ
β2 k0
2
a2x2
φpk0,kK; ξq 0 (1.21)
the solutions of the above differential equation are known to be
φpk0,kK; ξq C1pkKqIiνpβξq C2pkKqKiνpβξq (1.22)
iν E
a ik0
a(1.23)
the functions Iiνpβξq diverge exponentially for large positive ξ and must
consequently be rejected. Finally the full solution to (1.16) reads
φpη, ξ,xKq » 0
8» 8
0
dk0?
2π
»d2kK2π
fpkKqKik0apβξqeik0ηikKxK
p2πq32» 8
0
dk0
»d2kK
rfpkKq exp pikK xK ik0ηq fpkKq exp pik0η ikK xKqs
where we took into account the reality condition for the scalar field that
brings
φpk0,kK; ξq φpk0,kK; ξq Ñ fpkKq fpkKq (1.24)
being Kik0apβξq Kik0apβξq. This solution holds within the right part of
the Rindler wedge; as discussed earlier it suffices to make the appropriate
changes of sign in the coordinates and repeat the above procedure to obtain
solutions of (1.16) in the remaining parts of MS. The normal modes
Then the full solution to the Covariant Majorana equation (1.99) in RS
reads:
ψpxq pRqψpxq pLqψpxq (1.150)
whereas
pR,Lqψpxq ¸
a,k0,kK
pR,Lqfa,k0,kKpR,LqUa,k0,kKpη, ξ,xKq
((1.151)
¸
a,k0,kK
pR,Lqga,k0,kKpR,LqVa,k0,kKpη, ξ,xKq
((1.152)
It is straightforward to check that the following identities hold:
pR,LqUa,k0,kKpη, ξ,xKq pR,LqUa,k0,kKpη, ξ,xKq
pR,LqVa,k0,kKpη, ξ,xKq
pR,LqVa,k0,kKpη, ξ,xKq
(1.153)
36
1.5. Majorana spinors in Rindler geometry
then, by virtue of the purely imaginary form of the Majorana gammas and by
conjugating the Majorana equation, we see that the classical Majorana field
must be real also in RS. It follows immediately that fa,k0,kK rfa,k0,kKsand so for ga,k0,kK . Then the form (1.151) of the field is then manifestly self-
conjugated (complex-conjugation). Note that, unlikely to the Minkowskian
case, positive- and negative-frequency solutions are not well-separated here,
indeed we have an integral over k0 which ranges over R. In order to achieve
field quantization we will need to separate positive frequency modes from the
negative frequency ones, which can be obtained by splitting the domain of
integration, as follows
pR,Lqψpxq ¸a,kK
» 8
0
» 0
8
dk0 pR,Lqfa,k0,kK
pR,LqUa,k0,kKpη, ξ,xKq
¸a,kK
» 8
0
dk0pR,Lqfa,k0,kK
pR,LqUa,k0,kKpη, ξ,xKq pc.c.q
(1.154)
the same holding true for the V modes. Notice that expression (1.154) ex-
hibits clearly PT invariance.
By virtue of eq.s (1.149) and (1.147) we can infer that the standard or-
thonormality relations hold true for the canonical modespAqUa,k0,kK ,
pA1qUa1,k10,kK1 δA,A1 δpk0 k10q δpkK kK
1q (1.155)
where A,A1 R,L. Moreover, independence of the positive-frequency modes
from the negative frequency ones occurspAqUa,k0,kK ,
pA1qUa1,k10,kK
1
δpk0 k10q δpkK kK
1q δA,A1 (1.156)
0
since we are restricting the field expansions to positive values of the variable
k0.
Quantization is achieved introducing the usual creation-annihilation op-
erators satisfying
δaa1 δA,A1δpkK kK1q δpk0 k0
1q tpAqfa,k0,kK ,pA1qf :
a1,k01,kK1u (1.157)
tpAqga,k0,kK ,pA1qg:
a1,k01,kK1u
all the other anticommutators vanishing.
37
Chapter 1. Quantum Field Theory in non-Minkowskian geometries
b. The basis of helicity eigenstates
We have seen how it is possible to solve the Majorana equation and cast
its solution in the form of an expansion on normal modes, in such a way that
the procedure of quantization is straightforward. Actually the expansion we
buildt doesn’t help us understand what physical meaning to give each normal
mode, in order to recover some more information let us stress that our solu-
tions are built on the distinction between the eigenspinors of the matrix 9γ0 9γ1,
whose eigenvalues are doubly degenerate. This means that the eigenspinors
Θpqa , with a 1, 2 are not univocally defined, indeed any combination
Υ AΘpq1 BΘ
pq2 (1.158)
will still satisfy 9γ0 9γ1 Υ Υ.
We can then search for any observable commuting with 9γ0 9γ1, so to pre-
serve our construction of stationary solutions. If we try, by analogy with the
Minkowskian case, to check r 9γ0 9γ1, hs we find out that spinors with helicity
12
along the acceleration axis are the only suitable ones. This could have
been naively expected, since an acceleration along the Ox-axis means that
none of the stationary solutions can have zero-momentum in that direction,
which does not allow for stationary solutions with spin along the Oy or Oz
axes.
Indeed, it is straightforward to check that
Σ1M i
0 σ3
σ3 0
Σ1M , 9γ
09γ1 0 (1.159)
the matrix Σ1M has two doubly degenerate eigenvalues λ 1, we can then
look for combinations of stationary normal modes that satisfy
Σ1M
A pR,LqU1,k0,kKpη, ξ,xKq B pR,LqU2,k01,kK
1pη, ξ,xKq
A pR,LqU1,k0,kKpη, ξ,xKq B pR,LqU2,k01,kK
1pη, ξ,xKq
(1.160)
since we look for stationary solutions, we must immediately impose the con-
dition k0 k01. Before we try to solve the abovementioned problem, let us
point out some formulae that will turn out to be useful, actually one can
verify by direct inspection that
Σ1M Θ
1 iΘ2 Σ1
M Θ1 pkKq i Θ
2 pkKq (1.161)
38
1.5. Majorana spinors in Rindler geometry
and, since pΣ1Mq2 1, (1.161) implies also that
Σ1M Θ
2 iΘ1 Σ1
M Θ2 pkKq i Θ
1 pkKq (1.162)
if, for simplicity, we restrict ourselves to the right-Rindler wedge, it follows
that
Σ1M Ut1
2u,k0,kKpxq i p2πq32 eik
0ηikKxK vt21u,k0,kK
pξq (1.163)
then we can recast equation (1.160) asΣ1M
Σ1M
ψÒpxqψÓpxq
ψÒpxqψÓpxq
(1.164)
together withψÒpxqψÓpxq
A B
C D
U1,k0,kKpxqU2,k0,kK
1pxq
U
v1,k0,kKpxq eikKxK
v2,k0,kK1pxq eikK
1xK
p2πq32eik
0η
(1.165)
where A,B,C,D are complex-valued 4 4 linear operators. The above con-
ditions explicitly read:
i
Av2,k0,kK
1 eikK1xK B v1,k0,kK eikKxK
C v2,k0,kK1 eikK
1xK D v1,k0,kK eikKxK
Av1,k0,kK eikKxK B v2,k0,kK1 eikK
1xK
C v1,k0,kK eikKxK D v2,k0,kK1 eikK
1xK
we see that it is sufficient that we take A,B,C,D P C and we readily get
the conditions
A iB C iD (1.166)
β β1 kK 0 kK1 (1.167)
i.e. only for particles travelling in the direction of the acceleration we get a
helicity eigenstate:
ψÒÓ ψÒÓk0,kKΣ1M ψÒÓk0,0 ψÒÓk0,0 (1.168)
39
Chapter 1. Quantum Field Theory in non-Minkowskian geometries
the corresponding with Ò while the with Ó. Hence by requiring ψÒÓ to
be normalized, and making the convenient choice kK1 kK, we finally get,
up to a phase factor
U 1?2
1 i1 i
U :U 1 (1.169)
Then we obtain the new normal modes
ψÒ,k0,kKpxq α θpξq eik
0ηikKxK
iβΥÒ
K 12 ik0
a
pβξq uÒpkKqK 12 ik0
a
pβξq
(1.170)
ψÓ,k0,kKpxq α θpξq eik
0ηikKxK
iβΥÓ
K 12 ik0
a
pβξq uÓpkKqK 12 ik0
a
pβξq
(1.171)
together with
α cpk0,kKq2
π32 12π2
bcoshpπk0aq
aβ
ΥÒ 1
2
i
i
1
1
ΥÓ 1
2
ii1
1
uÒpkKq 12
m ky ikz
m ky ikz
im iky kz
im iky kz
uÓpkKq 12
m ky ikz
m ky ikz
im iky kz
im iky kz
In the same way one obtains the second set of eigenspinors of Σ1
M :ψÒ,k0,kK
pxqψÓ,k0,kK
pxq
1?
2
i 1
i 1
V1,k0,kKpxqV2,k0,kKpxq
(1.172)
ψÒ,k0,kKpxq
ψÓ,k0,kKpxq
1?
2V
V1,k0,kKpxqV2,k0,kKpxq
ψÒ,k0,kK
pxq α θpξq eik0ηikKxK (1.173)
iβΥÒ
K 12 ik0
a
pβξq uÒpkKqK 12 ik0
a
pβξq
ψÓ,k0,kKpxq α θpξq eik
0ηikKxK
iβΥÓ
K 12 ik0
a
pβξq uÓpkKqK 12 ik0
a
pβξq
40
1.5. Majorana spinors in Rindler geometry
together with
ΥÒ 1
2
1
1
ii
ΥÓ 1
2
1
1
i
i
uÒpkKq 12
im iky kz
im iky kz
m ky ikz
m ky ikz
uÓpkKq 12
im iky kz
im iky kz
m ky ikz
m ky ikz
Notice that, had we used U in place of V , we would have obtained a phase
factor i for ψÒ,k0,kKand a i for ψÓ,k0,kK
. We chose to use V just for a
matter of convenience.
These new normal modes are mutually orthogonal and normalized. Let
us compactify our notation according to
ι pr, k0,kK, σq ι P O (1.174)
where r Ò, Ó, σ ,.
The same reasoning can be applied in order to recover left-Rindler wedge
helicity eigenstates, the procedure is the same. For the sake of brevity let
us just summarize the results since we will need them later on. It turns out
that these modes are the left-Rindler wedge counterparts of those in (1.170)
and (1.171):
LψÒ,k0,kKpxq α θpξq eik
0ηikKxK (1.175)
iβΥÒ
K 12 ik0
a
pβξq uÒpkKqK 12 ik0
a
pβξq
LψÓ,k0,kKpxq α θpξq eik
0ηikKxK (1.176)
iβΥÓ
K 12 ik0
a
pβξq uÓpkKqK 12 ik0
a
pβξq
Now that the explicit form of ψιpxq is completely clear, once again we
stress the fact that one has
Σ1M uÒpkKq uÒpkKq (1.177)
Σ1M uÓpkKq uÓpkKq
i.e. ψιpxq has a definite helicity iff kK 0.
41
Chapter 1. Quantum Field Theory in non-Minkowskian geometries
This could be expected since we have seen that the only component of
the relativistic spin operator that commutes with 9γ0 9γ1, i.e. with the operator
that defines the stationary normal modes, is Σ1M , hence helicity could be well-
defined only for particles moving along the x direction.
The modes ψipxq satisfy Majorana’s equation by their definition, it readily
follows that
HMψιpxq iBηψιpxq k0ψιpxq (1.178)
by virtue of their status of stationary solutions, actually the last identity
could be proven by direct inspection quite easily (the proof involves the use
of recursion relations for the McDonald functions), however we will omit
these passages since they are not interesting.
Hence these normal modes correspond to particles of opposite helicity
(when they move along the direction of the acceleration) and positve energy
k0 in the Rindler frame.
We could write the field expansion by substituting (1.165) and (1.172)
into (1.154) however this would lead to a complication since we’d need to
introduce other constant spinors together with the ΥÒÓ and the uÒÓ pkKq. A
simpler way to write down the field expansion is using equation (1.151), in
which the energy index runs continuously over R, that brings
pRqψpxq ¸
pÒÓq,µ,kK
aÒÓµkK ψ
ÒÓµkKpxq aÒÓµkK ψ
ÒÓµkKpxq c.c.
(1.179)
c. Study of the hermiticity of the Hamiltonian
Let us now check that the Majorana-Rindler Hamiltonian is actually Her-
mitean: rewriting equation (1.99) as
iBη aξ mloomoonHm
i aξ 9γ09γ1
Bξ 1
2ξ
looooooooooomooooooooooon
Hξ
i aξ 9γ09γ2 By 9γ0
9γ3 Bzloooooooooooooomoooooooooooooon
HK
(1.180)
42
1.5. Majorana spinors in Rindler geometry
Hermiticity must be checked w.r.t. the invariant scalar product defined by
pψ1pxq , ψ2pxqq »
Σ
dΣα ψ1pxq γαpxqψ2pxq (1.181)
if we choose Σ tx P RS|η 0udΣα dΣ0
?g ε0βγδ3!
dxβdxγdxδ
aξ dξdydz
ñ pψ1pxq , ψ2pxqq »
Σ
dξdxK ψ:1pxqψ2pxq (1.182)
hermiticity of Hm and of HK can be easily checked by standard procedures,
let us focus on the two terms within Hξ separately: without loss of generality
we can work with right-Rindler-wedge modes only and getpRqψ1 ,i aξ 9γ0
9γ1 Bξ pRqψ2
» 8
0
dξ
»R2
dxK ψ:1
i aξ 9γ0
9γ1 Bξψ2
»R2
dxKi aξ ψ:1 9γ0
9γ1 ψ2
ξ8ξ0
» 8
0
dξ
»R2
dxKi aξ 9γ0
9γ1 Bξ ψ1
:ψ2
» 8
0
dξ
»R2
dxKi a 9γ0
9γ1 ψ1
:ψ2 (1.183)
while the second piece readspRqψ1 ,
i aξ 9γ0
9γ1 1
2ξ
pRqψ2
i aξ 9γ0
9γ1 1
2ξ
pRqψ1 ,
pRqψ2
the same happens in left Rindler wedge. Eventually we come to the hermitic-
ity conditionpRqψ1 , HξpRqψ2
pRqψ1Hξ,pRqψ2
»R2
dxK iaξ ψ:1 9γ
09γ1ψ2
ξ0
ξ ψ:1 9γ09γ1ψ2
ξ0
In conclusion it turns out that the Hamiltonian is hermitean only if we restrict
to fields satisfying
ξψpxq ξÑ0Ñ 0 (1.184)
This is Physically relevant, as it tells us that, in order for the evolution oper-
ator to be unitary, the event horizons must play the role of mirrors. Equiva-
lently, any field that does not satisfy condition (1.184) enjoys an absorption-
creation contribution to the evolution operator, as if it were manipulated
43
Chapter 1. Quantum Field Theory in non-Minkowskian geometries
from a hidden source. Hence a field crossing the light-cone event horizon
would be unphysical. This means that we should work in a restricted Hilbert
space, satisfying the mirror condition.
Eventually we can safely say that, since H is hermitean, its eigenstates
form a complete and orthonormal basis of the one-particle Hilbert space,
henceforth they are suitable for quantization. This ultimately confirms the
validity of our quantization scheme.
44
CHAPTER 2
The Unruh effect
Any intelligent fool can make things bigger, more complex, and more
violent. It takes a touch of genius -and a lot of courage- to move in the
opposite direction.
Albert Einstein
This chapter is devoted to the presentation of the Unruh effect, which
arises as a natural consequence for the inequivalence between the Minkowski-
Fock and the Rindler-Fock quantization schemes. We will see how a Rindler
observer perceives a thermal bath of particles as he moves through the quan-
tum state that coincides with the vacuum in the Minkowski-Fock represen-
tation. At this point, before we begin our treatment of the Unruh effect, we
stress that the Physical interpretation of these results is not completely clear,
thus they should be regarded as a working mathematical scheme, based on
some assumptions which may be too strong; nowadays the debate over the
validity of these predictions is still open between theoretical Physicists as no
experiment has yet shed light on this rather complicated argument.
45
Chapter 2. The Unruh effect
2.1 The Bogolyubov transform
To begin our study of the Unruh effect we will first introduce a well-
known mathematical framework known as the Bogolyubov transform. As we
will show in the next sections, the role of Bogolyubov transforms is absolutely
central in QFT in curved spacetime. This is due to the fact that they de-
scribe the relations among the normal-mode solutions of the field equations
in different frames, which in turn yield a connection between the different
quantization schemes. In other words, Bogolyubov transforms encode how
canonical quantization is affected by the frame of reference.
Since we will be dealing with canonical quantization, we are mainly inter-
ested in Bogolyubov transformations applied to continuous sets of harmonic
oscillators. Let us begin with the simplest case of just one bosonic har-
monic oscillator, let a, a: be the annihilation-creation operators satisfying
ra, a:s 1, then one may define a hermitean operator N a:a which eigen-
states form a complete-orthonormal discrete set of the one-particle Hilbert
space (for a rigorous and exhaustive treatment see [6], chap. V)
t|nyun N |ny n |ny n 0, 1, 2 a|0y 0 |ny pa:qn?
n!|0y
a:|ny ?n 1|n 1y a|ny ?
n |n 1y¸|nyxn| 1 xm|ny δm,n
then one can introduce another set of operators, b, b: satisfying the same
algebra and use them to build another complete-orthonormal basis of the
one-particle Hilbert space
t|nyun N |ny n |ny n 0, 1, 2 b|0y 0 |ny pa:qn?
n!|0y
b:|ny ?n 1|n 1y b|ny ?
n |n 1y¸|nyxn| 1 xm|ny δm,n
since both bases are complete, it must be possible to express each state of
the first in terms the second basis’ states, and vice versa
|ly ¸n
cln |ny (2.1)
46
2.1. The Bogolyubov transform
in order to find the expansion coefficients, it is necessary to introduce a
general transformation between the two couples of operators
b α a β a: b: α a: β a (2.2)
requiring that b, b: actually satisfy the usual algebra one finds the fundamen-
tal relation
1 rb, b:s rα a β a: , α a: β as |α|2 |β|2 (2.3)
that is characteristic of the bosonic Bogolyubov transform. Henceforth the
α, β coefficient are not wholly independent and the whole Bogolyubov trans-
formation may be described by three parameters θ1, θ2, γ such that α eiθ1 cosh γ, β eiθ2 sinh γ. It is important to notice that if β 0 then the
two ground states are inequivalent, indeed
b|0y pα a β a:q |0y β|1y βÑ0Ñ 0 (2.4)
hence |0y |0y. Indeed
0 b|0y pα a β a:q
¸j
c0j |jy
by virtue of the orthonormality, it follows immediately that
c0p2kq β
α
k p2k 1q!!p2kq!!
12
c00 (2.5)
c01 0 c0p2k1q β
α
k p2kq!!p2k 1q!!
12
c01 0
the coefficient c00 can be evaluated by requiring that x0|0y 1. Indeed
convergence of the series is guaranteed by
|c00|2 x0|0y 8k0
β
α
k p2k 1q!!p2kq!!
12
¤ 8
k0
βαk 1
1 |β||α|
since |β| |α| by virtue of 2.3.
All the other coefficients follow by application of (2.5). Moreover also the
cjk with j ¥ 1 are found by recursive action of b: on |0y.
47
Chapter 2. The Unruh effect
It is interesting to analyze the Bogolyubov transform when involving
fermionic harmonic oscillators. To cut a long story short, the Hilbert space
only consists of the states |0y, |1y and all one has to do is replace eq. (2.3)
with
1 tb, b:u tα a β a: , α a: β au |α|2 |β|2 (2.6)
however when one tries to repeat the above machinery to find the connection
between |0y, |1y and |0y, |1y it turns out that they are independent.
One can as well generalize to multiple sets of harmonic oscillators, defining
a Bogolyubov transform that mixes operators with different frequencies, this
is just what occurs in QFT in curved spacetime. We will investigate this
interesting case further on, while beginning to dig into the Unruh effect.
2.2 General theory of the Unruh effect
We saw in Chapter 1 that it is generally possible, both for scalars and
spinors, to solve the field equations and to quantize the field once we find
a suitable (i.e. complete and orthonormal) basis; what is still missing is a
physical interpretation of the theory: the major weakness of QFT in curved
spacetime with respect to standard QFT is that the latter gives a notion of
particle which is consistent with what is observed, while the former doesn’t
generally give the notion of particle! What makes a particle a particle in
the framework of standard QFT is the clear distinction between positive and
negative-frequency normal modes. This distinction is made according to the
action of the Killing vector Bt; the existence of such a vector is not trivial in
general, indeed most classes of spacetimes do not admit the existence of such a
Killing vector, this clearly precludes the possibility of distinguishing positive-
frequency modes from the negative-frequency ones, in such a situation the
concept of particle remains obscure.
However there are also situations with a high degree of symmetry which
admit a Killing vector playing the role of Bt. In such cases it is possible
to use a quantization scheme analogous to the standard one: associating
annihilation operators with the positive-frequency modes in the field decom-
position and the creation operator with the negative-frequency ones. Rindler
spacetime is obviously one of such cases.
48
2.2. General theory of the Unruh effect
The next question to ask is whether this quantization scheme is equivalent
to the common one, i.e. to that of standard QFT, or more appropriately if
there is any kind of relation between the two. Indeed, what can one expect
as equivalence between quantization schemes? The form of normal modes
depends on the coordinate system, more precisely on the metric. To clarify
the concept of equivalence between two different quantization schemes, let
us introduce the related Bogolyubov transformation.
For the sake of simplicity, let us introduce a free autoconjugated field
φpxq, over a certain Riemaniann manifold D, its Lagrangian density Lrφ, gswill be a functional of the metric and of the field. As usual it will be possible
to derive the field equations and to express the solutions in form of normal
modes decomposition. If we also assume that a time-like Killing vector is
admitted, we can divide the normal modes with respect to the sign of their
frequency and come to the standard form for the quantized field
φpxq ¸ι
aι uιpxq aι
: uιpxq
(2.7)
where the index ι is a label for a set of eigenvalues belonging to a complete set
of commuting observables which we shall call O, while uιpxq are positive fre-
quency modes and uιpxq are the negative-frequency counterparts. The sum
is understood to be extended over the whole set of complete and orthonormal
modes. The construction of the Fock space proceeds from the definition of
the vacuum state |0y
aι|0y 0 @ι P O (2.8)
and by recurrent action of the creation operator on |0y, paying attention
to the operator algebra, depending on the field’s spin. Next we turn to a
different coordinate frame, in which the metric takes a different form from
the previous case, and solve again the equations of motion, generally they
will differ from the ones above, due to the fact that gµν is different in the
lagrangean and so will the differential operators; hence the corresponding
normal modes will be different from the first set. Let us express the field in
the new coordinate system as
ψpx1q ¸κ
bκ vκpx1q bκ
: vκpx1q, x1 T x (2.9)
49
Chapter 2. The Unruh effect
Again the set of all the vκpx1q will be complete and orthonormal, they are di-
vided into positive-frequency (vκpx1q) and negative frequency (vκpx1q) modes,
and the construction of a Fock space proceeds in the usual way. The new
vacuum state will be called |0q. Let us assume that the new normal modes
tvκpx1qu are complete and orthonormal, then it must be possible to express
it in relation to the former set of modes, through a set of coefficients:
vκpx1q ¸ι
rακιuιpxq βκιuιpxqs
vκpx1q
¸ι
rακιuι pxq βκιuιpxqs (2.10)
The field lagrangean is covariant, and so are the field equations, hence the
following must hold true
φpxq ψpT xq @x P D (2.11)
ψpx1q ¸κ
#bκ
¸ι
rακιuιpxq βκιuιpxqs
b:κ¸ι
rακιuι pxq βκιuιpxqs+
¸ι
¸κ
ακιaκ βκιa
:κ
vκpx1q
ακιa
:κ βκιaκ
vκpx1q
(by comparison with (2.7) we obtain the Bogolyubov transformation between
creation/annihilation operators belonging to the two different Fock represen-
tations.
bκ ¸ι
ακιaι βκιa
:ι
b:κ
¸ι
ακιa
:ι βκιaι
(2.12)
We can now give a clear definition for the equivalence of two Fock represen-
tations, precisely the two representations will be said equivalent iff
βικ 0 @ι, κ (2.13)
As it is, the above condition looks just as a mathematical condition, which
tells us that if two representations are equivalent their creation/annihilation
50
2.2. General theory of the Unruh effect
operator do not mix in the Bogolyubov transformation which relates them,
and vice versa. To catch a glimpse of how profound this condition actually is,
let us suppose that the representations were inequivalent, then it is instructive
to inspect the state |0y by the point of view of the second observer, to this
end we shall evaluate the number of particles seen by the two observers, it is
straightforward from (2.12) that
x0|N1|0y x0|¸ι
aι:aι
|0y 0 (2.14)
x0|N2|0y x0|¸κ
bκ:bκ
|0y ¸κ
¸ι
|βκι|2
(2.15)
hence, while for the first observer the |0y state contains no particles (as by
definition), the second perceives the presence of particles belonging to any
state, depending on the nature of the Bogolyubov transformation; precisely
the number of particles in state κ is given by°ι |βκι|2.
Let us go back a little: where does it all come from? The key feature of
this inequivalence is the nature of the Bogolyubov transformation, which is
given by eq. (2.10), indeed it is easy to check that
βκι puι , vκq (2.16)
in turn this only depends on the analytical form of the two sets of normal
modes, which in turn depend on the coordinate transformation. Finally, we
obtained that a coordinate transformation, which is just a natural operation
in general relativity, can have dramatic effects on the quantum treatment of
fields, leading to discrepant particle interpretations.
The Unruh effect is just a particular case of such a situation, precisely it
predicts that a thermal bath of particles is detected by an observer moving
with constant acceleration through Minkowski’s vacuum. In particular the
spectrum of particles perceived is Planckian for a scalar field and has a similar
form for higher spin fields.
The unfamiliar reader would (reasonably) be skeptical about any theory
in disagreement with the field algebra of the Minkowski-Fock representation.
To cite Fulling in [22], ’...if any other proposed theory disagrees with this
one, so much the worse for that theory...’. Indeed the Unruh effect has been
both sustained and pitched by theoretical Physicists during the years, as it
51
Chapter 2. The Unruh effect
presents some peculiarities, one for all, Rindler coordinates do not cover the
whole Minkowski spacetime, not to mention the fact that its causal structure
is deeply inequivalent to the MS one. A complete understanding of this
problem doesn’t seem to be achieved yet, nor from the mathematical point
of view (what effects does incompleteness of space have on quantization?),
nor from the Physical one. On the other hand, experimental data do not tell
us much more.
The Unruh effect is particularly interesting, since in presence of a gravi-
tational (or cosmological!) field every point of space time has a gravitational
acceleration associated with it, hence the study of the Unruh effect is con-
nected with the local behavior of quantum fields in presence of gravity. Let
us give a well-known example of this fact: let us consider the Schwarzschild
metric
ds2
1 2GM
r
dt2
1 2GM
r
dr2 r2 dΩ2 (2.17)
if we perform the coordinate change ξ r1 2GMrs12 the metric turns
into
ds2 ξ2 dt2
4GM
r1 ξ2s22
dξ2
2GM
1 ξ2
2
dΩ2 (2.18)
in such a way that, in the vicinity of the Schwarzschild radius one has ξ 0,
bringing
ds2 ξ2 dt2 p4GMq2 dξ2 p2GMq2 dΩ2 (2.19)
id est, upon a rescaling of the radial coordinate we get a Rindler-like metric
with corresponding acceleration along the radial direction.
Nonetheless the main reason of interest in the Unruh effect is that it
provides the simplest case for understanding how to properly quantize a field
in a non-Minkowskian background: Rindler spacetime enjoys many features
(such as event horizons and a singularity of the coordinate system) in common
with more complicated metrics, and these features are the fundamental ones
that give rise to many of the issues encountered in curved-spacetime QFT.
In the following we give a treatment of this interesting effect both for
scalar and for Majorana fields. At the end of the chapter we will instead
discuss some notable opinions about it.
52
2.3. Unruh effect for the spin-0 field
2.3 Unruh effect for the spin-0 field
In order to study the Unruh effect for a scalar field, we need two inequiva-
lent quantization schemes to be compared. We derived the so-called Rindler-
Fulling quantized field in the previous chapter, so we proceed with calculat-
ing its Bogolyubov coefficients with respect to the Minkowsian quantization
scheme. To begin we evaluate these for modes within the right-Rindler wedge,
a generalization to other regions of spacetime are straightforward.
The well-known normal modes expansion for a scalar field in an inertial
frame reads
φpxq »
dkckhkpxq c:kh
kpxq
(2.20)
where ck, ck: are the standard second quantization annihilation-creation op-
erators, which obey the canonical commutation relationsck, ck1
: δ pk k1q (2.21)
rck, ck1s ck:, ck1:
0 (2.22)
while hkpxq are the Minkowski modes and they can be cast in the form
hkpxq r2ωkp2πq3s12 exp tiωkt ik xu (2.23)
ωk pk2 m2q12. (2.24)
In the Rindler-Fulling scheme, we have instead
φpxq »
dkK
» 8
0
dk0ak0,kKuk0,kKpxq a:k0,kK
uk0,kKpxq
(2.25)
where the normal modes read
uk0,kKpxq d
sinhπ k0
a
8π4 a
Kipk0aqpβ ξq eik0ηikKxK (2.26)
as we discussed previously. We start by evaluating the α coefficients, let us
introduce the shorthand notation k pk0,kKq.
αkk1 phk, uk0,kKq i
» 8
0
dξ
»dxK
?g hkgηηÐÑB ηuk0,kK
η0
53
Chapter 2. The Unruh effect
in order to evaluate this amplitude, let us employ a trick due to Takagi,
which we already used for the normalization of Rindler-Fulling modes (see
appendix). Since the scalar product does not depend on the hypersurface Σ
that we choose, as long as it is spacelike, we push it up close to the horizon
H (see figure 1 in the appendix). To make it more explicit, we introduce
the null coordinates defined as
u aη log pξλq v aη log pξλq λ P R (2.27)
then the condition of pushing the hypersurface towards the H horizon con-
sists in taking the limit uÑ 8 inside the integral; it is easy to see that in
these coordinates the hypersurface oriented element reads
dΣv dv dxK (2.28)
so that integration over the transverse coordinates keeps unchanged and we
are left with:
αkk1 i C
»dv lim
uÑ8
eiλ2peveuqpω1k11qÐÑBv (2.29)
Kipk0aqβ λ e
vu2
ei
k0
auv
2
with
C δpkK k1Kqc
sinhpπk0aq8 π3 aω1
(2.30)
in the limit uÑ 8 we can take advantage of an expansion for small values
of the argument of Bessel functions (see formula (16))
Kipk0aqβ λ e
vu2
α
ei
k0
avu
2 Reik0
avu
2
α αpk0,kKq iπ
2 sinhπk0
a
Γ1 ik
0
a
β λ
2
i k0
a
, R α
α
the amplitude then reads
C α
»dv lim
uÑ8
eiλ2peveuqpω1k11q
i k0 aR
eik0
av
eik0
au R ei
k0
aviλ
2pω1 k11qev ei
λ2pω1k11qpeveuq
C α
»dv
eiλ2
evpω1k11qi k0
av R
k0
a λ
2pω1 k11q ev
eik0
au ei
λ2pω1k11qevv λ
2pω1 k11q
54
2.3. Unruh effect for the spin-0 field
where we dropped the terms eu in taking the limit; besides this, let us note
that the amplitude we are evaluating is between normal modes, in particular
the Mikowskian ones (the plane waves), do have an infinite norm as they
actually lie out of the proper one-particle Hilbert space. Under this light, we
might expect to get an infinite value of this amplitude, indeed we’ll see that
this is the case, reasonably. Actually when one deals with Physical particle
states, one works with wave packets, i.e. one smears the field operator with
a test function fpkq. Indeed what we are evaluating is precisely an improper
Bogolyubov coefficient, which is to be intended in a distributional sense. In
this spirit, and taking into account that we are still working in the limit
u Ñ 8, we may drop the last line by virtue of the Riemann Lebesgue
lemma, as intended for integration over k0 when smearing over a Rindler
wave packet
C αR
»dv ei
λ2
evpω1k11qi k0
av
k0
a λ
2pω1 k11q ev
(2.31)
by changing variable according to z ev, we get
C αR
» 8
0
dz eiλ2z pω1k11q zi
k0
a
k0
a z λ
2pω1 k11q
(2.32)
and, by employing another substitution y z λ2pω1 k11q, where ω1 ¡ k11 by
definition, we finally come to
C αR
» 8
0
dy
k0
a y
yi
k0
a1 eiy
λ
2pω1 k11q
i k0
a
the last integral can be easily evaluated by contour integration, with a rota-
tion of π2, and it reads
i k0
aeπk0
2a Γpik0aq eπk0
2a Γp1 ik0aq
2 eπk0
2a Γp1 ik0aqfinally, we found that the value of αk,k1 reads
αk,k1 i δpkK kK1qω1 k11ω1 k11
ik0
2a
eπk0
2a
8 π aω1 sinh
πk0
a
12
(2.33)
55
Chapter 2. The Unruh effect
The β coefficient can be obtained with a similar procedure, by substituting
the normal mode uk0,kK by its complex-conjugate, as in equation (2.16), this
corresponds to switching
k1 Ñ k1 ωk1 Ñ ωk1 (2.34)
finally one finds
βk,k1 i δpkK kK1qω1 k11ω1 k11
ik0
2a
eπk0
2a
8 π aω1 sinh
πk0
a
12
(2.35)
it is now easy to compute the number of Rindler-Fulling particles seen by
an accelerated observer moving through the state |0y, by virtue of formula
(2.15)
Npk0,kKq »
dk1 |βk,k1 |2 (2.36)
1
4 π a
1
e2πk0
a 1
»dk11
1apk11q2 β2
just as we expected, this quantity is logarithmically divergent; in order to
recover a mathematical sense, let us go back a little and employ the so-called
proper states for the Miknowski-normal modes:
Hk1pxq »
dk1 gpk1qhk1pxq
where g is the Fourier transform of some wave-packet within the one-particle
Hilbert space.
Then the number of detected particles is given by suitably modifying eqs
(2.7) and (2.15) as follows
φP pxq ¸k1
ak1Hk1pxq a:k1H
k1pxq
(2.37)
x0|N2,P |0y ¸k
¸k1
|Bk,k1 |2
(2.38)
(2.39)
where
Bk,k1 pHk1 , ukq phk1 , ukq gpk1q βk,k1 g
pk1q (2.40)
56
2.4. Unruh effect for the Majorana field
in full analogy with the reasoning of the previous section, whence the number
of detected particles reads
x0|N2,P |0y »
dk
»dk1 |βk,k1 |2 |gpk1q|2 (2.41)
1
4π a
» 8
0
dk0 1
e2πk0
a 1
»dkK
»dk11
|gpk11,kKq|2bkK
2 pk11q2 m2
hence, we obtained the spectrum of particles, up to a multiplicative factor,
which is given by the last two integrals, the convergence being ensured by our
assumptions on g. The spectrum of particles is Planckian, with temperature
T a
2π ~a
2πckB(2.42)
this represents a canonical ensemble with temperature T, called Davies-
Unruh temperature.
2.4 Unruh effect for the Majorana field
As we pointed out, in order to study the Unruh effect for the Majorana
field one needs to compare the Rindler quantization scheme with the usual
Minkowskian one. More precisely it is necessary to evaluate the coefficients
of the Bogolyubov transformation that occurs between the two sets of normal
modes, the modes on which the quantization schemes are built on. Once these
coefficients are found one can use them to determine how the Rindler cre-
ation/annihilation operators are related to the original Minkowskian ones.
Eventually this machinery allows one to evaluate the spectrum of Rindler
quanta that are present in Minkowski’s vacuum state, which is just the oc-
currence of the Unruh effect.
Let us begin; first of all we shall compactify our notation for the Majorana
field: we’ll drop the indices kK as they do not play an important role in our
discussion, so the Rindler modes will read
ψÒÓ,µpxq (2.43)
where it is understood that µ k0a and that it carries with itself the
quantum numbers kK. Unless otherwise specified, we understand x as the
generic Rindler-space coordinate pη, ξ,xKq.
57
Chapter 2. The Unruh effect
a. Finding the spinor algebraic RS-to-MS transfor-
mation
The first important difference between the scalar and the spinor cases is
the following: when we compared the Rindler modes to the Minkowskian
modes for the scalar field all we had to do was to make a change of vari-
able, instead in the case of a spinor field one needs to take into account the
algebraic transformation under which the spinor field undergoes when a gen-
eral coordinate transformation is performed. Our first step will then be to
find such transformation operator: let us consider the particular coordinate
transformation between Minkowski and Rindler observers
t ξ sinh aη x ξ cosh aη (2.44)
our aim is to find the spinor algebraic transformation that accounts for this
coordinate transformation. One could naively argue that the case of an
observer with constant proper acceleration is nothing but that of a time-
varying Lorentz boost with velocity β apη η0q, where η is the Rindler
time. This naive point of view is intuitive and actually appropriate, as we
will see.
By differentiating (2.44) one obtains
dt dξ sinh aη aξ dη cosh aη (2.45)
dx dξ cosh aη aξ dη sinh aη
id est the variation of Minkowskian coordinates subject to variation of the
Rindler ones, from these formulæ one can extrapolate the local linearized
version of the coordinate transformation, which in the neighbourhood of a
certain pη0, ξ0,xK0q reads: t
x
xK
Lµν
τ
ξ
xK
cosh aη0 sinh aη0
sinh aη0 cosh aη0
I2
τ
ξ
xK
(2.46)
58
2.4. Unruh effect for the Majorana field
where we introduced the Rindler proper time τ , that is the line element of an
observer at rest w.r.t the Rindler frame, indeed the metric turns Minkowskian
upon the substitution ξ0η Ñ τ :
ds2 paξq2 dη2 dξ2 dxK2
Ñ dτ 2 dξ2 dxK2 (2.47)
actually Lµν can be expressed in term of its generator:
Lµν exp paη0 J1q (2.48)
J1
0 1 0 0
1 0 0 0
0 0 0 0
0 0 0 0
(2.49)
that is just the boost generator along the x-axis for flat-spacetime QFT, in-
deed the whole structure of this transformation is that of a boost with the
identification β Ø aη0, hence this is actually a boost of time-varying velocity
which increases linearly with time, the action of a constant proper accelera-
tion is manifest. Moreover, the fact that this transformation holds locally is
a consequence of the fact that one can find a Lorentz boost transformation
only between MS and an instantaneous rest frame of a Rindler observer i.e.
an inertial frame which at time η η0 has the same velocity as the Rindler
observer but that can be related to MS by a simple Lorentz boost by virtue
of its inertiality.
Since the metric is given by (2.47), the gamma matrices are just the
Minkowskian gamma matrices since the Vierbeins read V µα δµα and they
transform as contravariant vectors under a coordinate transformation
pxµq1 Lµν xν Ñ pγµpxqq1 Lµνpxq γνpxq (2.50)
Lµνpxq 9γν
by direct inspection one has
gµν Lµα L
νβ gαβ (2.51)
hence confirming that L is a Lorentz-like transformation in the sense that
it leaves the metric invariant (one must keep in mind that L is just the
59
Chapter 2. The Unruh effect
local, linearized version of (2.44)); henceforth the new set of gammas will be
equivalent to the former in the sense that
tpγµq1, pγνq1u 2gµν tγµ, γνu (2.52)
this means that one can find a certain real matrix S satisfying
pγµq1 S1γµS (2.53)
the matrix S can be rendered unique, up to a sign, by imposing the normal-
ization constraint
detS 1 (2.54)
indeed, if we assume that there exist another matrix T such that S1γαS T1γαT then it must also hold that γα ST1 ST1 γα which entails ST1 λI i.e. S λT , this proves that the normalization condition fixes the arbi-
trary multiplicative factor up to a sign.
From (2.50) and (2.53) it follows that
Lµν γν S1 γµ S (2.55)
since we have required S to be normalized we can, without loss of gener-
ality, set S exp
12εαβ J αβ
and expand (2.55) up to first order in the
transformation parameters
γµ εµνγν
I εαβ
2J αβ
γµ
I εαβ
2J αβ
(2.56)
εµνγν 1
2εαβ
γµ,J αβ
(2.57)
now, let us recall that εαβ εβα since L leaves the metric unchanged,
henceforth if we antisymmetrize the l.h.s. of the last equation according to
1
2εαβ pgµαγβ gµβγαq 1
2εαβ
γµ,J αβ
(2.58)
we can finally get rid of the transformation parameters ε without the indeter-
minacy of a possible additive symmetric factor. The solution to this equation
is well known and reads
J αβ 1
4
γα, γβ
(2.59)
60
2.4. Unruh effect for the Majorana field
just as one could naively expect, it turns out that the generators of the spin
algebraic transformation for this particular Lorentz-like transformation are
actually the Lorentz group generators.
Hence we have found the general form of a Lorentz-like spinor algebraic
transformation, whereas in our case
ε01 ε10 aη0 all other components vanishing (2.60)
that leads to a Minkowski-to-Rindler algebraic spinor transformation which
infinitesimal form reads
S I 1
29γ09γ1 aη0 I i
2εµνΣ
µν (2.61)
Σµν i
4rγµ, γνs
hence the finite transformation reads
ψ1px1q SpLqψpxq exp
aη0
29γ09γ1ψpxq (2.62)
finally, if we require a transformation that follows the Rindler observer through-
out its whole motion, we can just rewrite (2.62) according to
SpLq expaη
29γ09γ1
(2.63)
Notice that one could obtain the above transformation rule also by means
of eq (1.50), indeed recalling that the only non-vanishing Fock-Ivanenko co-
efficient is Γη, that was obtained in (1.105) it is straightforward that
ψ1RSpxq NpXqψMSpXqNpXpxqq exp
!» x
dx1µ Γµpx1q) exp
!» x
dη1a
29γ09γ1)
exp! aη
29γ09γ1) S1pLq (2.64)
in full accordance with our previous derivation.
The fact that the spinor transformation rule resembles a time-varying
boost along the acceleration axis is clearly intuitive in terms of classical me-
chanics. Nonetheless it also tells us that Rindler-Fulling modes are actually
eigenstates of the generator of boosts along the acceleration axis, indeed in
literature they are sometimes referred to as boost modes.
61
Chapter 2. The Unruh effect
b. Consistency with the general theory of spinors in
curved spacetimes
In order to enforce our derivation of the spinor representation of the RS-
to-MS coordinate transformation, let us show that it is consistent with the
well-known generally covariant form of the (Dirac) Majorana equation. In
what follows we shall call ψMpXq, ψRpxq respectively the Minkowskian and
Rindler spinors linked by
X L x ψMpXq SpLqψRpxq (2.65)
X P MS x P RS
then it is convenient to start from the flat Majorana equation:
0 i 9γ0Bt i 9γ1Bx i~9γK ~BK m
ψMpXq
i 9γ0Bt i 9γ1Bx i~9γK ~BK m
eaη2
9γ09γ1
ψRpxqinverting (2.44) one gets
ξ ?x2 t2 aη arctanh
t
x(2.66)
Bt t?x2 t2
Bξ 1
ax
1
1 tx
2 Bη
sinhpaηq Bξ 1
aξcoshpaηq Bη (2.67)
Bx x?x2 t2
Bξ 1
a
1
1 tx
2
t
x2
Bη
coshpaηq Bξ 1
aξsinhpaηq Bη (2.68)
substituting into our wave equation bringsi 9γ0
sinhpaηq Bξ 1
aξcoshpaηq Bη
i 9γ1
coshpaηq Bξ 1
aξsinhpaηq Bη
i~9γK ~BK m
eaη2
9γ09γ1
ψRpxq
(2.69)
62
2.4. Unruh effect for the Majorana field
#
eaη2
9γ09γ1
i 9γ0
sinhpaηq Bξ 1
aξcoshpaηq Bη 9γ0 9γ1
2ξcoshpaηq
i 9γ1
coshpaηq Bξ 1
aξsinhpaηq Bη 9γ0 9γ1
2ξsinhpaηq
eaη2
9γ09γ1 i~9γK ~BK m
+ψRpxq
#
eaη2
9γ09γ1
i9γ0
aξ
coshpaηq 9γ0
9γ1 sinhpaηq Bη a
29γ09γ1
i 9γ1coshpaηq 9γ0
9γ1 sinhpaηq Bξ
eaη2
9γ09γ1 i~9γK ~BK m
+ψRpxq
eaη2
9γ09γ1 pi∇ mq ψRpxq (2.70)
which completes the proof.
c. Helicity-eigenstate normal modes
So far we have obtained the explicit algebraic spinor operator that carries
out the transformation of spinors from Rindler coordinates to Minkowskian
ones, and confirmed its validity. What we still lack is a suitable expression for
the McDonald functions in Minkowski coordinates, however literature offers a
variety of integral representations and we may employ the following suitable
one (see [59] §6.22)
Kνpzq 1
2e
iπν2
» 8
8dθ eiz sinh θνθ <pzq ¡ 0 (2.71)
indeed it is straightforward that
Kνpβξq eaην 1
2e
iπν2
» 8
8dθ eiβξ sinh θνpθaηq
1
2e
iπν2
» 8
8dθ eiβξpcosh aη sinh θ sinh aη cosh θqνθ
1
2e
iπν2
» 8
8dθ eipωtkxxqνθ
63
Chapter 2. The Unruh effect
where we set ω β cosh θ and kx β sinh θ and used relations (2.44) to
emphasize that these functions can be regarded as a superposition of positive-
frequency or negative-frequency 2-dimensional plane waves, since the funda-
mental relation β2 k2x ω2 of flat-spacetime QFT is satisfied. Actually, in
what follows, we will use the representation
Kνpβξq eaην 1
2e
iπν2
» 8
8dθ eikxxiωtνθ (2.72)
If we recall that, by construction:
9γ09γ1 ΥÒÓ
ΥÒÓ 9γ0
9γ1 uÒÓ uÒÓ (2.73)
SpLqΥÒÓ e
aη2 ΥÒÓ
SpLquÒÓ eaη2 uÒÓ (2.74)
eventually the Rindler modes in Minkowskian coordinates read, within the
right Rindler wedge
RpMqψ
ÒÓ,µpxq α θpξq eiµaηikKxK
σÒÓ iβ SpLqΥÒÓ
Kiµ12pβξq SpLquÒÓKiµ12pβξq
α θp?x2 t2q eikKxK
σÒÓ iβ eaηpiµ 1
2qΥÒÓKiµ12pβξq eaηpiµ 1
2quÒÓKiµ12pβξq
α
2θp?x2 t2q eikKxK
σÒÓ iβΥÒÓ
eiπ2 piµ 1
2q» 8
8dθ eiωtikxxpiµ 1
2qθ
uÒÓ eiπ2 piµ 1
2q» 8
8dθ eiωtikxxpiµ 1
2qθ R
pMqψÒÓ,µpXq
σÒ , σÓ
together with their counterparts
RpMqψ
ÒÓ,µpxq α θpξq eiµaηikKxK
σÒÓ iβ SpLqΥÒÓ
Kiµ12pβξq SpLquÒÓKiµ12pβξq
(2.75)
64
2.4. Unruh effect for the Majorana field
α θp?x2 t2q eikKxK
σÒÓ iβ eaηpiµ 1
2qΥÒÓKiµ12pβξq eaηpiµ 1
2quÒÓKiµ12pβξq
α
2θp?x2 t2q eikKxK
σÒÓ iβΥÒÓ
eiπ2 piµ 1
2q» 8
8dθ eiωtikxxpiµ 1
2qθ
uÒÓ eiπ2 piµ 1
2q» 8
8dθ eiωtikxxpiµ 1
2qθ R
pMqψÒÓ,µpXq
σÒ , σÓ
where X P MS; clearly one can choose among all the four combinations of
integral representations for each set of modes, all being equivalent within the
right Rindler wedge.
d. Canonical normal modes
For reasons that will become clear later, we want to find the explicit
expression of modes (1.149) for a Minkowskian observer.
By making use of (2.72) and noticing that
9γ09γ1 Θ
a Θa 9γ0
9γ1 Θa pkKq Θ
a pkKq (2.76)
SpLqΘa e
aη2 Θ
a SpLq Θa pkKq e
aη2 Θ
a pkKq (2.77)
if, just for convenience, we call canonical modes those given in (1.149), then
we can write down their Minkowskian version within the right Rindler wedge
as:
RpMqUa,µ,kKpxq
cpk0,kKqp2πq 3
2
θpξq eiµaηikKxK
β SpLqΘ
aKiµ12pβξq SpLqΘaKiµ12pβξq
cpk0,kKq
p2πq 32
θp?x2 t2q eikKxK
β eaηpiµ 1
2qΘaKiµ12pβξq eaηpiµ 1
2qΘaKiµ12pβξq
(2.78)
65
Chapter 2. The Unruh effect
cpk0,kKq2
52π
32
θp?x2 t2q eikKxK
βΘ
a eiπ2 piµ 1
2q» 8
8dθ eiωtikxxpiµ 1
2qθ
Θa ei
π2 piµ 1
2q» 8
8dθ eiωtikxxpiµ 1
2qθ R
pMqUa,µ,kKpXq(2.79)
together with their counterparts
RpMqVa,µ,kKpxq
cpk0,kKqp2πq 3
2
θpξq eiµaηikKxK
β SpLqΘ
aKiµ12pβξq SpLqΘaKiµ12pβξq
(2.80)
cpk0,kKqp2πq 3
2
θp?x2 t2q eikKxK
β eaηpiµ 1
2qΘaKiµ12pβξq eaηpiµ 1
2qΘaKiµ12pβξq
cpk0,kKq
252π
32
θp?x2 t2q eikKxK
βΘ
a eiπ2 piµ 1
2q» 8
8dθ eiωtikxxpiµ 1
2qθ
Θa ei
π2 piµ 1
2q» 8
8dθ eiωtikxxpiµ 1
2qθ R
pMqVa,µ,kKpXq(2.81)
where X P MS; again one can choose among all the four combinations of
integral representations for each set of modes, all being equivalent within the
right Rindler wedge.
e. Choosing the proper representation for helicity eigen-
states
We now come to a crucial point: we need to compare two quantization
schemes, but in order to build a Fock space related to Minkowski space-
time one needs a complete-orthonormal basis that covers the whole MS. Un-
66
2.4. Unruh effect for the Majorana field
ruh found that it is possile to extend the above expressions of the Rindler-
Majorana modes, by combining different integral representations: in the fol-
lowing we present Unruh’s original procedure.
We will first discuss the procedure for the Majorana-Rindler helicity-
eigenstate modes, thereafter we will repeat the whole procedure for the canon-
ical modes, finally the benefits of each of the two representations will be clear.
Recall that the two representation are completely equivalent and linked to
each other by a unitary transformation, as we have already shown.
First we must drop the Heaviside θ-terms, we shall then make a choice
for the integral representations to use: we’ll define
pMqψÒÓ,µpX|aq def Aa
2eikKxK (2.82)
σÒÓ iβΥÒÓ
eiπ2 piµ 1
2q» 8
8dθ eipωtkxxqpiµ 1
2qθ
uÒÓ eiπ2 piµ 1
2q» 8
8dθ eipωtkxxqpiµ 1
2qθ
pMqψÒÓ,µpX|`q def A`
2eikKxK (2.83)
σÒÓ iβΥÒÓ
eiπ2 piµ 1
2q» 8
8dθ eipωtkxxqpiµ 1
2qθ
uÒÓ eiπ2 piµ 1
2q» 8
8dθ eipωtkxxqpiµ 1
2qθ
where we introduced the two different normalization coefficientsAa, A` which
are due since normalization in MS is generally different from the one in RS;
actually it is not difficult (see appendix) to obtain thatpMqψ
ÒÓ,µpX|aq,pMq ψ
òó,µ1pX|aq
MS
|Aa|2eπµ8aπ4β δÒÓ,òóδpk0 k01qδpkK kK
1qpMqψ
ÒÓ,µpX|`q,pMq ψ
òó,µ1pX|`q
MS
|A`|2eπµ8aπ4β δÒÓ,òóδpk0 k01qδpkK kK
1qpMqψ
ÒÓ,µpX|`q,pMq ψ
òó,µ1pX|aq
MS
0 @pµ,kK, ÒÓq; pµ1,kK1,òóq(2.84)
so that we may suitably choose, up to a phase factor
Aa eπµ2
π2?
8aβA` e
πµ2
π2?
8aβ(2.85)
67
Chapter 2. The Unruh effect
Actually the same can be done for the pMqψÒÓ,µpXq, namely we define two
different representations as:
pMqψÒÓ,µpX|aq def Ba
2eikKxK (2.86)
σÒÓ iβΥÒÓ
eiπ2 piµ 1
2q» 8
8dθ eipωtkxxqpiµ 1
2qθ
uÒÓ eiπ2 piµ 1
2q» 8
8dθ eipωtkxxqpiµ 1
2qθ
pMqψÒÓ,µpX|`q def B`
2eikKxK (2.87)
σÒÓ iβΥÒÓ
eiπ2 piµ 1
2q» 8
8dθ eipωtkxxqpiµ 1
2qθ
uÒÓ eiπ2 piµ 1
2q» 8
8dθ eipωtkxxqpiµ 1
2qθ
where we introduced the two different normalization coefficients Ba, B`; ac-
tually it can be obtained in the way as before thatpMqψ
ÒÓ,µpX|aq,pMq ψ
òó,µ1pX|aq
MS
|Ba|2eπµ8aπ4β δÒÓ,òóδpk0 k01qδpkK kK
1qpMqψ
ÒÓ,µpX|`q,pMq ψ
òó,µ1pX|`q
MS
|B`|2eπµ8aπ4β δÒÓ,òóδpk0 k01qδpkK kK
1qpMqψ
ÒÓ,µpX|`q,pMq ψ
òó,µ1pX|aq
MS
0 @pµ,kK, ÒÓq; pµ1,kK1,òóq(2.88)
so that we may suitably choose, up to a phase factor
Ba Aa B` A` (2.89)
Actually, one can check (see appendix) that these two sets of modes enjoy
completeness separately w.r.t. MS, i.e.¸ÒÓ,µ,kK
pMqψ
ÒÓ,µpX|`q b pMqψ
ÒÓ :,µpX 1|`q (2.90)
pMqψÒÓ,µpX|aq b pMqψ
ÒÓ :,µpX 1|aq
X0X01
δpXX1q¸ÒÓ,µ,kK
pMqψ
ÒÓ,µpX|`q b pMqψ
ÒÓ :,µpX 1|`q (2.91)
pMqψÒÓ,µpX|aq b pMqψ
ÒÓ :,µpX 1|aq
X0X01
δpXX1q
68
2.4. Unruh effect for the Majorana field
By virtue of (2.72) one can easily infer that the following identity holds
true within the left Rindler wedge
Kνpβξq eaην 1
2e
iπν2
» 8
8dθ eipωtkxxqνθ
1
2e
iπν2
» 8
8dθ eiωtikxxνθ (2.92)
as we will see shortly, this clarifies the reason of our choice on integral rep-
resentations (2.86) and (2.87). From this crucial observation follows the
celebrated Unruh trick: let’s consider the combinations
RÒÓ,µpXq e
πµ2 pMqψ
ÒÓ,µpX|aq e
πµ2 pMqψ
ÒÓ,µpX|`q?
2 coshπµ(2.93)
LÒÓ,µpXq eπµ2 pMqψ
ÒÓ,µpX|aq e
πµ2 pMqψ
ÒÓ,µpX|`q?
2 coshπµ(2.94)
these modes enjoy some crucial features:
• They all satisfy the MS version of the Majorana equation, by construc-
tion.
• They are analytical over the whole MS, by construction.
• They are orthonormal in MS, by virtue of equations (2.84).
• As shown in A.1 d. they form a complete set in MS, by virtue of
equation (2.90).
• They enjoy:$'''&'''%RÒÓ,µpXq R
pMqψÒÓ,µpxq in the right Rindler wedge
RÒÓ,µpXq 0 in the left Rindler wedge
LÒÓ,µpXq LpMqψ
ÒÓ,µpxq in the left Rindler wedge
LÒÓ,µpXq 0 in the right Rindler wedge
the last point needs to be verified, let us do so.
69
Chapter 2. The Unruh effect
If we confine ourselves to the right part of the wedge
RÒÓ,µpXq e
πµ2 Aa e
πµ2 A`a
2 coshpπµq eikKxK (2.95)
σÒÓ iβ eaηpiµ 1
2qΥÒÓKiµ12pβξq eaηpiµ 1
2quÒÓKiµ12pβξq
1
α 1
2π2
dcoshpπµq
aβR
pMqψÒÓ,µpXq (2.96)
SpLq RpRq ψ
ÒÓ,µpxq x L1 X
LÒÓ,µpXq eπµ
2 Aa eπµ2 A`a
2 coshpπµq eikKxK
σÒÓ iβ eaηpiµ 1
2qΥÒÓKiµ12pβξq eaηpiµ 1
2quÒÓKiµ12pβξq
0 (2.97)
upon recalling that µ k0a. While, turning to the left Rindler wedge
RÒÓ,µpXq 1
4π2aaβ coshpπµq eikKxK (2.98)
#σÒÓ iβΥÒÓ
eπµi
π2 piµ 1
2q» 8
8dθ eiωtikxxpiµ 1
2qθ
eπµiπ2 piµ 1
2q» 8
8dθ eiωtikxxpiµ 1
2qθ
uÒÓ
eπµi
π2 piµ 1
2q» 8
8dθ eiωtikxxpiµ 1
2qθ
eπµiπ2 piµ 1
2q» 8
8dθ eiωtikxxpiµ 1
2qθ+
it is now sufficient to use the following formulæ
eπµiπ2 piµ 1
2q eiπ2 piµ 1
2q piq (2.99)
eπµiπ2 piµ 1
2q eiπ2 piµ 1
2q piq (2.100)
together with equation (2.92) to achieve that
RÒÓ,µpXq 0 (2.101)
by a similar, straightforward reasoning one can see that within the left part
of the wedge also the following holds true
LÒÓ,µpXq LpMqψ
ÒÓ,µpxq (2.102)
70
2.4. Unruh effect for the Majorana field
which completes the proof of the properties of Unruh modes.
One can define as well the combinations
RÒÓ,µpXq e
πµ2 pMqψ
ÒÓ,µpX|aq e
πµ2 pMqψ
ÒÓ,µpX|`q?
2 coshπµ(2.103)
LÒÓ,µpXq eπµ2 pMqψ
ÒÓ,µpX|aq e
πµ2 pMqψ
ÒÓ,µpX|`q?
2 coshπµ(2.104)
and verify quite easily that, also for these modes, all the above properties are
satisfied.
Summarizing, we found two sets of normal modes:
• The!pMqψ
ÒÓ,µpX|aq , pMqψ
ÒÓ,µpX|`q , pMqψ
ÒÓ,µpX|aq , pMqψ
ÒÓ,µpX|`q
), which
are orthonormal and complete (orthonormality and completeness hold
for the set of ψ separately from the set of ψ) and analytical on MS.
Another important feature of these modes is that they can be regarded
as the superposition of purely positive-frequency plane waves or purely
negative frequency ones with respect to Minskowskian time. Henceforth
a Fock space built one them must be equivalent to the usual standard
flat QFT Fock space.
• The!LÒÓ,µpXq , RÒÓ
,µpXq , LÒÓ,µpXq , RÒÓ,µpXq
), these are the so-called
Unruh modes; they are complete, orthonormal (again the set enjoys
orthonormality and completneness separately from the set) and an-
alytical on MS and reduce to Rindler modes within the corresponding
sectors of the wedge.
f. Choosing the proper representation for canonical
modes
Within this subsection we use the so-called canonical modes and repeat
the machinery of the previous subsection, quite quickly one might definepMqU1,µ,kKpX|aqpMqU2,µ,kKpX|aq
U :
pMqψ
Ò,µpX|aq
pMqψÓ,µpX|aq
(2.105)
pMqU1,µ,kKpX|`qpMqU2,µ,kKpX|`q
U :
pMqψ
Ò,µpX|`q
pMqψÓ,µpX|`q
(2.106)
71
Chapter 2. The Unruh effect
where U is given by (1.165), in this manner we just get
pMqUa,µ,kKpX|aq def Aa?8
eikKxK (2.107)
βΘ
a eiπ2 piµ 1
2q» 8
8dθ eipωtkxxqpiµ 1
2qθ
Θa ei
π2 piµ 1
2q» 8
8dθ eipωtkxxqpiµ 1
2qθ
pMqUa,µ,kKpX|`q def A`?8
eikKxK (2.108)
βΘ
a eiπ2 piµ 1
2q» 8
8dθ eipωtkxxqpiµ 1
2qθ
Θa ei
π2 piµ 1
2q» 8
8dθ eipωtkxxqpiµ 1
2qθ
notice that the normalization factor differs by a multiplicative factor?
2
from eqs. (2.82), (2.83), this accounts for the fact that the tΘa , Θ
a u aren’t
normalized while the tΥÒÓ , u
ÒÓ u are, indeed:
U
Θ1
Θ2
i
?2
ΥÒ
ΥÓ
(2.109)
V
Θ1
Θ2
i
?2
ΥÒ
ΥÓ
(2.110)
U
Θ1 pkKq
Θ2 pkKq
?2
uÒpkKquÓpkKq
(2.111)
V
Θ1 pkKq
Θ2 pkKq
?2
uÒpkKquÓpkKq
(2.112)
By virtue of (2.84) it is obvious thatpMqUa,µ,kKpX|aq,pMq Ua1,µ1,kK1pX|aq
MS
δa,a1 δpk0 k01q δpkK kK
1qpMqUa,µ,kKpX|`q,pMq Ua,µ,kKpX|`q
MS
δa,a1 δpk0 k01q δpkK kK
1qpMqUa,µ,kKpX|`q,pMq Ua,µ,kKpX|aq
MS
0 @pµ,kK, ÒÓq; pµ1,kK1,òóq(2.113)
72
2.4. Unruh effect for the Majorana field
Actually the same can be done for the pMqVa,µ,kKpXq, namely
pMqV1,µ,kKpX|aqpMqV2,µ,kKpX|aq
V :
pMqψ
Ò,µpX|aq
pMqψÓ,µpX|aq
(2.114)
pMqV1,µ,kKpX|`qpMqV2,µ,kKpX|`q
V :
pMqψ
Ò,µpX|`q
pMqψÓ,µpX|`q
(2.115)
where V is given by (1.172), in this manner we just get
pMqVa,µ,kKpX|aq def Aa?8
eikKxK (2.116)
βΘ
a eiπ2 piµ 1
2q» 8
8dθ eipωtkxxqpiµ 1
2qθ
Θa ei
π2 piµ 1
2q» 8
8dθ eipωtkxxqpiµ 1
2qθ
pMqVa,µ,kKpX|`q def A`?8
eikKxK (2.117)
βΘ
a eiπ2 piµ 1
2q» 8
8dθ eipωtkxxqpiµ 1
2qθ
Θa ei
π2 piµ 1
2q» 8
8dθ eipωtkxxqpiµ 1
2qθ
By virtue of (2.88) it is obvious that
pMqVa,µ,kKpX|aq,pMq Va1,µ1,kK1pX|aq
MS
δa,a1 δpk0 k01q δpkK kK
1qpMqVa,µ,kKpX|`q,pMq Va,µ,kKpX|`q
MS
δa,a1 δpk0 k01q δpkK kK
1qpMqVa,µ,kKpX|`q,pMq Va,µ,kKpX|aq
MS
0 @pµ,kK, ÒÓq; pµ1,kK1,òóq(2.118)
Also for canonical modes, it can be shown (see appendix) that they enjoy
73
Chapter 2. The Unruh effect
completeness w.r.t. MS separately, in the sense that¸a,µ,kK
pMqUa,µ,kKpX|`q b pMqU :a,µ,kKpX 1|`q (2.119)
pMqUa,µ,kKpX|aq b pMqU :a,µ,kKpX 1|aqX0X01
δpXX1q¸a,µ,kK
pMqVa,µ,kKpX|`q b pMqV:a,µ,kKpX 1|`q (2.120)
pMqVa,µ,kKpX|aq b pMqV:a,µ,kKpX 1|aqX0X01
δpXX1qOnce again, if we notice that the following identity holds true within the
left Rindler wedge
Kνpβξq eaην 1
2e
iπν2
» 8
8dθ eiωtikxxνθ (2.121)
From this crucial observation follows the celebrated Unruh trick: let’s con-
sider the combinations
URa,µ,kKpXq eπµ2 pMqUa,µ,kKpX|aq e
πµ2 pMqUa,µ,kKpX|`q?
2 coshπµ(2.122)
ULa,µ,kKpXq eπµ2 pMqUa,µ,kKpX|aq e
πµ2 pMqUa,µ,kKpX|`q?
2 coshπµ(2.123)
As for helicity-eigenstate Unruh modes, also the canonical Unruh modes
enjoy some crucial features:
• They all satisfy the MS version of the Majorana equation, by construc-
tion.
• They are analytical over the whole MS, by construction.
• They enjoy orthonormality in MS, by virtue of equations (2.113).
• They enjoy completeness in MS, by virtue of equations (2.119).
• They enjoy:$'''&'''%URa,µ,kKpXq R
pMqUa,µ,kKpxq in the right Rindler wedge
URa,µ,kKpXq 0 in the left Rindler wedge
ULa,µ,kKpXq LpMqUa,µ,kKpxq in the left Rindler wedge
ULa,µ,kKpXq 0 in the right Rindler wedge
74
2.4. Unruh effect for the Majorana field
proof of the last point follows readily from the results on helicity-eigenstate
Unruh modes, we shall not repeat the procedure here.
One can define as well the combinations
VRa,µ,kKpXq eπµ2 pMqVa,µ,kKpX|aq e
πµ2 pMqVa,µ,kKpX|`q?
2 coshπµ(2.124)
VLa,µ,kKpXq eπµ2 pMqVa,µ,kKpX|aq e
πµ2 pMqVa,µ,kKpX|`q?
2 coshπµ(2.125)
and verify quite easily that, also for these modes, all the above properties are
satisfied.
Summarizing, we found again two sets of normal modes: