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Human resources and mobility activity
Marie Curie actions
Intra-European Fellowships
Part B Section 1
QGNC
QGNC-partB1
Contents
B1.1 Scientific quality of the project... 3
Scientific/technogical quality, inter-multi disciplinary aspects... 3
Research methodology... 5
Originality, innovation, relationship to the state of the art... 6
Timeliness and relevance of the project... 6
BI.2 Quality of the researcher... 7
Curriculum Vitae of the applicant... 7
Description of major accomplishments... 11
2
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B1.1 Scientific quality of the project
Scientific/technogical quality, inter-multi disciplinary aspects
There is a strong theoretical evidence that the usual continuum model of spacetime is
not correct at very short distances (i.e. the Planck scale lp ' 1, 6.10−33 cm). It is rather
straightforward to show that if the principles of quantum theory hold right down to very small
length scales where gravitational forces come into play then the familiar picture of spacetime
as a manifold must be abandoned. But since we have no experimental information and there is
no firmly established quantum theory of gravity, there is no certainty on what spacetime looks
like at the Planck scale. Understanding the intimate geometry of spacetime is thus a key open
problem for theoretical physics. As well this is a challenge to mathematics: in the same way
that General Relativity would not have been possible without Riemann’s transition from flat
to curve spaces, it is most likely that Planck scale physics asks for a formulation of geometry
more flexible than usual differential geometry.
Several theories propose a (incomplete) treatment of quantum gravity, but even the most
popular of them (string theory, loop quantum gravity) has not been experimentally tested
for the energy scale corresponding to the Planck length, roughly 1019 Gev, is far beyond our
experimental possibilities. However it has been recently stressed that quantum gravity theories,
by giving a non continuum structure to spacetime, may lead to some observational effects1 : for
instance the ”fuzziness” of spacetime may introduce some additional sources of noise in gravity
interferometers which could be sought experimentally with LIGO/VIRGO-type interferometers.
As well a suitable deformation of the concept of spacetime symmetry at Planck scale yields in
some pictures a modification of the usual relativistic energy-momentum dispersion relation
(E 2 = p2 + m2 ) by adding a correction term depending on the continuum-breaking scale,
E 2 = p2 + m2 + f (p, m; Lp ).
(1)
This leads to a wavelength dependence of the speed of massless particles3 which, even for small
f , could lead to observably-large effects. In particular the associated relative time delays in
the arrival times of (nearly-)simultaneously emitted photons could be detectable in contexts
involving high-energy radiation coming from cosmological distances, as in the case of gamma
ray bursts. Certain gamma-ray observatories soon to be operational (GLAST satellite in 2005)
will have excellent sensitivity toward this type of effect.
The magnitude and precise form of these effects depends on the specific structure of the
quantum spacetime picture. For example in ref.[2] the deformation of the dispersion relation
and its observational effects have been analyzed for a string-theory motivated picture, namely
the canonical noncommutativity of the coordinates
[xµ , xν ] = iθµ,ν
(2)
where θµ,ν are parameters. Similarly a ”Lie-algebra” noncommutativity has been considered
β
xβ .
[xµ , xν ] = iθµ,ν
(3)
In both cases the noncommutativity of spacetime is described in a quantum-group approach.
Namely one posits a noncommutative algebraic structure by a deformation of the classical
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algebra just as quantum groups are deformation of classical Lie algebras. However the physical
status of such spacetimes is not clear. Naive questions like ”what are the points of a quantum
spacetime ?” or ”what happens to the notion of distance ?” have no clear answer.
These questions, among others, are addressed by the conceptual approach to Noncommutative Geometry (NCG) developed by Connes6 (1982’s Fields medalist). NCG provides a formulation of standard geometric and topological concepts (like spin and differential structures)
within a purely algebraic framework. Riemannian geometry is encompassed as a particular
case, commutative, of a more general theory. NCG sheds a new light on classical oppositions,
like discrete versus continuous, by giving access to spaces obtained as the product of a continuous manifold by a discrete part7 . From a physicist point of view this corresponds to a
description of the Standard Model of elementary particles where the exterior degrees of freedom of general relativity and the internal gauge degrees of freedom of isospin are described in a
unified manner5 . For free one obtains the Higgs fields as the scalar part of the (noncommutative
equivalent of the) Levi-Civita connection.
To summarize, the present situation is the following: one the one hand more and more
phenomenology is done on quantum spaces of type (2) or (3). But most often the noncommutativity of the coordinates is an ad-hoc postulate or a consequence of candidate-theories to
quantum gravity. On the other hand NCG proposes a discrete structure for spacetime which is
based solely on the well known physics of the Standard Model. But due to the mathematical
difficulty of Connes’ approach, the phenomenology of the underlying noncommutative space
has been little studied. The knowledge of the structure of spacetime at the Planck scale would
certainly benefit from a combination of these two approaches. Participating in this cooperation
is the primary aim of this project.
Besides the Planck scale structure of spacetime, NCG has interesting implications for physics
in the way it implements symmetries. In agreement with the commutative case where
Diff(M ) ' Aut(C ∞ (M ))
with M a Riemannian manifold, the role of the diffeomorphism group is played in NCG by the
automorphism group of a suitable algebra A. For A noncommutative, Aut(A) splits in two:
inner automorphisms In(A) that can be implemented by unitaries† and outer automorphisms
.
obtained as the quotient Out(A) = Aut(A)/In(A). This distinction, which remains hidden in
the commutative case (In(A) reduces to the identity) is crucial in NCG because the physical
interpretation of In(A) radically differs from the one of Out(A). The former act as gauge
transformations and induces fluctuation of the metric (see ”major accomplishment section”
for more explanation). The later provides the geometry with an intrinsical dynamic‡ that has
been used by Rovelli8 to address the issue of time in quantum gravity (the so called thermal
time hypothesis). In a recent work12 we have based on this hypothesis an adaptation of Unruh
effect§ for an observer with finite lifetime. A secondary aim of the project is to take advantage
of Rome’s exceptional scientific environment (especially specialists of local quantum physics
like S. Doplicher or R. Longo) to test the validity of the thermal time hypothesis.
†
α ∈ In(A) ⇔ α(a) = uau∗ ∀a , where u = u∗ = u−1 ∈ A
the beautiful idea that ”Von Neumann algebras naturally evolve with time”, see ref.[7] for instance
§
i.e. the thermalisation of the vacuum state of a quantum field theory on Minkovsky spacetime for an eternal
uniformly accelerated observer
‡
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Research methodology
1. On the one side (from quantum space to NCG) we aim at finding some characteristic
physical effects associated with the NCG description of the Standard Model, also hoping to
identify experimental contexts in which these predictions could be tested. During my PhD I
have studied the metric aspect of NCG and the space underlying the Standard Model. The
lastest can be thought as a two sheet model, two copy of a classical Euclidean manifold whose
distance from one another fluctuates with the Higgs field11 . More precisely – this was shown
in my thesis – the Higgs field is the component of the metric in a discrete internal dimension.
From a phenomenological point of view our strategy will be to describe the propagation of
some signal on the two sheet spacetime of the Standard Model, then to deduce a pertubation
of the dispersion relation in the spirit of eq.(1). Each point on a sheet is in fact a couple (x, p)
where x is a point of the manifold and p either the pure state¶ of C or the pure state of the
algebra of quaternions H. Unitaries of C yields the U (1) gauge group of electromagnetism,
unitaries of H the SU (2) gauge group of weak interaction, and the distance between the two
sheets evolves with the Higgs field. Thus the propagation on this space should involve some
specific particle-physics process with interaction with the Higgs boson. The important point in
determining this process will be to understand the exact physical status of the pure states, as
well as the transition from one sheet to the other (what does it mean for a physical system to
go from a pure state of electromagnetism to a pure state of the weak interaction ?).
2. On the other side (from NCG to quantum space) we will re-examine within the rigorous
mathematical framework of NCG some of the physical analysis developed for quantum spaces.
In particular the notion of distance is not available in quantum space whereas NCG provides
a definition which i) coincides with Riemannian geodesic distance in the commutative case,
ii) still makes sense in a quantum context where the classical definition (distance = length
of the shortest path) has no meaning (since the notion of path between points is ill-defined in
quantum mechanics). Our strategy will be to export NCG definition of the distance to quantum
spaces, and then interpret it in the ”spacetime fuzzyness” framework of ref.[1] that leads to a
measurable noise in laser interferometers. Specifically we intend to use the NCG distance as
the quantitative tool to ”measure” the fuzzyness of quantum spacetime.
3. As explained above I intend to take advantage of Rome’s scientific environment to
test the relevance of the thermal time hypothesis in various contexts. Our work in ref.[12] is
based on Hislop-Longo’s computation9 of the modular groupk for a diamond’s shape region
of Minkovsky space. In this example it turns out that the modular group coincides with the
proper-time translation of a uniformly accelerated observer with finite lifetime. However there
is no general result on that matter. Hence the first way of testing the robustness of the thermal
time hypothesis is to study for what other regions of Minkovsky spacetime the modular group
may have a geometrical action. Some results exist on this point. Being in Rome, where Longo
is, would allow me to get more familiar with them. Second I intend to test our result about
the diamond in the context of black hole physics. It is well known that Unruh effect coincides
with Hawking radiation for an observer moving on the horizon of an eternal black hole, with
¶
we recall that a state of an algebra A over a field K is a linear positive application from A to K with norm
one. Pure states are those that do not decompose as convex combination of two other states.
k
the modular group of region D of Minkovsky space is a particular one-parameter group of outer automorphisms of the algebra of local observables on D. See Tomita’s book for a full treatment of the theory.
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acceleration given by the surface gravity. The analogy is no longer possible with Kerr black
holes because the Killing field that generates the horizon in Kerr metric has spacelike orbit at
infinity13 and cannot be identified to any physical proper time. Our result [12] could be relevant
in this context, but first has to be adapted to curved spacetime. S. Doplicher (who also works
in Rome) pointed me out the points deserving a careful analysis while going from flat to curve
space (essentially the properties of the vacuum and the KMS condition). Being in Rome would
give me the great opportunity to work with this respected and long experimented researcher.
Originality, innovation, relationship to the state of the art
The search of Planck-scale quantum-spacetime effect within Connes framework is an innovating task that may provide a second way (the first one being the prediction of the Higgs
mass4 ) to test the relevance of this approach for physics. Note that there are not so many
experimental signature from theory ”beyond the standard model”. Any improvement in this
matter would be highly relevant. Exporting NCG distance to quantum spaces is also a relevant
task since in quantum spaces a metric tensor is (sometimes) available but its interpretation
as a length element is not clear10 . The relevance of the thermal time hypothesis in black hole
physics is developed in section above.
Timeliness and relevance of the project
Although the number of papers on noncommutative spacetime increased very much within
the last years, interactions between the various approaches remains weak. Sooner or later the
different communities will have to talk: on the one side phenomenology will be at some point
confronted to the questions addressed by Connes (e.g. what is a spinor on a quantum space ?).
On the other side if NCG really says something about the geometry of physical spacetime,
then it should be confronted to experiment one day or another. The sooner, the better. Also
some of the experiments allowing to test the ideas of quantum gravity phenomenology are now
launched (e.g. Ligo/Virgo) but have not yet delivered their first results. Thus the subject of
the proposal is really challenging: new ideas can be developed with a reasonable hope to be
confirmed or falsified within a close future.
References
1
2
3
4
5
6
7
8
9
10
11
12
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G. Amelino Camelia, Nature 398 (1999) 216-218; Physical Review D62 (2000) 024015/1-18.
G. Amelino-Camelia, L. Doplicher, S. Nam, Y.S. Seo, Phys.Rev. D67 (2003) 085008.
G. Amelino Camelia, Nature 408 (2000) 661-664.
L. Carminati, B. Iochum, T. Schücker, Eur. Phys. J. C8 (1999) 697-709.
A. H. Chamseddine, A. Connes, Commun. Math. Phys. 186 (1996) 737-750.
A. Connes, Noncommutative geometry, Academic Press (1994).
A. Connes, Commun. Math. Phy. 182 (1996) 155-176.
A. Connes, C. Rovelli, Class. Quantum Grav 11 2899-2917 (1994).
P. D. Hislop, R. Longo, Comm. Math. Phys. 84 71-85 (1982).
J. Madore, Introduction to noncommutative differential geometry, Camb.U.P. (1995).
P. Martinetti, R. Wulkenhaar, J.Math.Phys. 43 (2002) 182-204.
P. Martinetti, C. Rovelli, Class. Quant. Grav 20 (2003) 4919-4932.
R M. Wald, “Qft in Curved Spacetime”, University of Chicago Press (Chicago, 1994).
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BI.2 Quality of the researcher
During my PhD I worked on NCG, especially the metric aspect of the theory. According to
my PhD’s advisor report ”Noncommutative geometry is a mathematically demanding subject
from which P. Martinetti has learnt a good amount of mathematical rigour. The involved ingredients for solving the addressed questions are a large part of functional analysis, in particular
C ∗ -algebras and related topics on operators, differential geometry, with synthesis in noncommutative geometry, gauge theory and particle physics. He managed to use all these theories in
a short time and proved that he was able to produce good and concluding results and to be
interested and efficient in different areas”.
After my Ph.D I have enlarged my research horizon by studying quantum effect in strong
gravitational field, such as Unruh effect and Hawking black hole radiation, in collaboration
with C. Rovelli (see his assessment). I have also worked on renormalization within the Hopfalgebraic framework of Connes and Kreimer. This is related to noncommutative geometry
but not directly to this project although some surprises may happen since Hopf algebra is an
important tool in the description of the symmetry of quantum spacetimes.
My main center of interest is quantum gravity and related topics (I consider that NCG,
especially the metric aspect, is one related topic. This is precisely the object of the present
proposal). However due to my background I have found more easily some postdoctoral positions
in mathematics laboratory (like IST in Lisbon, within the EU Network ”Geometric Analysis”).
I learnt there quantities of passionating things (especially on the amazing link between NCG,
statistical physics and number theory, see the work of Bost-Connes recently revisited by ConnesMarcolli). The only problem is that my cv looks more and more as the one of a mathematician,
so I find more easily some postdocs in math, then my cv is more and more mathematical etc
etc. That is why having the possibility to work for two years in Rome would be a very precious
opportunity to re-oriente myself towards ”real” physics. It would also assess of my willing to
open my research area beyond the scope of mathematical-physics and to confront myself to
various communities of researchers.
I believe that the best way of showing my initiative and independent thinking is to openly
confess my doubt about writing myself a text describing my independent thinking. To say it
in a rough way, although one is more and more asked to sell oneself, I do not think science is
about selling. Nevertheless let me underline my tenacity to work on difficult, and relatively out
of fashion, subjects, like the computation of NCG distance. I have a good knowledge of Connes
theory and a strong desire to confront it with the ideas of quantum gravity phenomenology.
Curriculum Vitae of the applicant
Education
2004: various summer schools on topics non directly connected to my research (”Biology for
Physicist and Mathematician”, Marseille august 2004; various schools of cosmology).
1998-01: Ph.D. in mathematical physics, CPT Marseille and université de Provence.
Subject: Distances en géométrie non commutative. Supervisor: B. Iochum
1997-98: postgraduate diploma, DEA mathematical physics, CPT Marseille and université de
Toulon et du Var.
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1993-97: undergaduate studies of physics - université Lyon 1 and Ecole Normale Supérieure de
Lyon.
1993-95: undergaduate studies of philosophy - université Lyon 3.
1992-93: classe préparatoire in litterature and philosophy - lycée du Parc, Lyon.
1991-92: baccalauréat maths and physics.
Work experience
2004-05: postdoc visiting at CPT and teaching assistant at Ecole Supérieure d’architecture,
Marseille Luminy.
2003-04: postdoc at IST, Lisboa. Financial support eu-network geometric analysis.
2003: visiting Perimeter Institute for theoretical physics, Waterloo, Canada.
2002-03: postdoc at université Mohammed 1, Oujda Morocco. Financial support by the Agence
Universitaire de la Francophonie.
2001-02: teaching assistant at université de Provence in mechanics and fortran programing.
1998-01: tutorial at university in mathematics, special relativity, quantum mechanics and electromagnetism; with T. Schücker (CPT Marseille).
Publications, thesis and proceedings
• Carnot-Caratheodory metric from gauge fluctuation in noncommutative geometry, work
in progress.
• What kind of noncommutative geometry for quantum gravity ?, proceeding of Lusofona
workshop on noncommutative geometry and quantum gravity, to be published in Mod.
Phys. Lett. A, gr-qc/0501022.
• An algebraic Birkhoff decomposition for the continuous renormalization group, with F.
Girelli and T. Krajewski, J. Math. Phys. 45 (2004) 4679-4697, hep-th/0401157.
• A brief remark on Unruh effect and causality, gr-qc/0401116.
• Diamonds’s Temperature: Unruh effect for bounded trajectories and thermal time hypothesis, with C. Rovelli, Class. Quant. Grav 20 (2003) 4919-4932, gr-qc/0212074.
• A brief introduction to the noncommutative geometry description of particle physics standard model, notes from lectures given at Oujda university, Morocco, (2003), math-ph/0306046.
• Discrete Kaluza-Klein from scalar fluctuations in noncommutative geometry with R. Wulkenhaar, J.Math.Phys. 43 (2002) 182-204, 2002, hep-th/0104108.
• Distance in finite spaces from non commutative geometry, with B. Iochum and T. Krajewski, J. Geom. Phy. 31 (2001) 100-125, hep-th/9912217.
• Wave-Function renormalization and the Hopf algebra of Connes and Kreimer with Florian
Girelli and Thomas Krajewski, Mod. Phys. Lett. A16 (2001) 299-303, hep-th/0104108.
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• Le modèle standard en géométrie non commutative, proceeding of ”Rencontres avec les
jeunes chercheurs”, Aussois 2000, http://lyopcs1.in2p3.fr/aussois2000.
Talks
conferences
(future) Carnot-Carathéodory metric from gauge fluctuation in Noncommutative Geometry, international conference on high energy and mathematical physics, Marrakech, april 2005;
meeting ”Sur la géométrie de Finsler”, CIRM Marseille, november 2005.
- On the time interpretation of Von Neumann algebra automorphisms, workshop noncommutative manifold, Sissa Trieste, october 2004, organised by G. Landi et L. Dabrowski.
- What kind of noncommutative geometry for quantum gravity ? first workshop on noncommutative geometry ans quantum gravity, universidade Lusofona, Lisbon july 2004,
organised by A. Mikovic, N. Costa Dia, J. Prata.
- Noncommutative geometry for quantum gravity ? 40th winter school on theoretical physics,
Ladek Zdrój, Poland february 2004, organised by G. A. Camelia (Roma La Sapienza) and
J. K. Glikman(Wroclaw).
- A physical introduction to the Dirac operator, workshop geometry in Lisbon, IST, january
2004, organised by P. Almeida.
- La distance en géométrie non commutative et le champ de Higgs, GDR 2001, Marseille,
local organizer: S. Lazzarini (CPT Marseille).
- Modèle standard en géométrie non commutative, Rencontres avec les jeunes chercheurs,
Aussois december 2000, organised by (among others) D. Davesne (IPN Lyon).
invited talks
(future) Distance de Carnot-Carathéodory et fluctuation de la métrique en géométrie non commutative, Université de Caen, march 2005, invited by L. Vainerman; Institut Desargues,
Université Lyon 1, april 2005, invited by par A. Frabetti;.
- Effet Unruh pour trajectoire bornée et l’hypothèse du temps thermodynamique, LAPTH,
Annecy, janvier 2005, invited par L. Gallot.
- Une décomposition de Birkhoff algébrique pour le groupe de renormalisation continu, university Mohammed 1, Oujda february 2004, invited by E. H. Tahri and T. Ouali.
- Distances in noncommutative geometry, universidade Lusofona, Lisbon december 2003,
invited by A. Mikovic.
- Unruh effect for bounded trajectories and the thermal time hypothesis, Perimeter Institute,
Waterloo october 2003, invited by F. Girelli.
- Distance in noncommutative geometry, Sissa Trieste, march 2002, invited by L. Dabrowski.
- Neutrinos massifs et modèle standard en géométrie non commutative, Institut des sciences
nucléaires, Grenoble march 2001, invited by D. Santos.
working groups
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- Hopf algebra and renormalization, seminar of the centre for analysis, dynamical systems
and geometry, IST Lisbon, june 2004.
- Introduction to noncommutative geometry and its application to physics, IST, Lisbon april
2003.
- many interventions in CPT, Marseille: ncg working group, relativity lunch, rendez-vous
des thésards.
course -Introduction à la géométrie non commutative et ses applications à la physique, university
Mohammed 1, Oujda november - december 2002.
Abroad experience
2003-04: postdoc at IST, Lisboa, Portugal.
oct 2003: invited guest in Perimeter institute for theoretical physics, Waterloo, Canada.
2002-2003: postdoc at université Mohammed 1, Oujda Morocco.
1997: training in experimental cosmology (project MITO on cosmic strings), university la
Sapienza, Roma, Italy.
1996: observational training in Mount Stromlo observatory (MACHos project), Canberra, Australia.
Contacts
Lisboa: Paulo Almeida: [email protected]
Aleksandar Mikovic: [email protected]
Marseille: Carlo Rovelli, Thomas Schücker (for teaching), Bruno Iochum: [email protected]
Thomas Krajewski: [email protected]
Oujda: E.H Tahri: [email protected]
Others
-Computers: practice of Mathematica and Mapple, notions in C++, fortan-90 and 77,
creation of webpages.
-Languages: french, english, italian (read, written, spoken), good notions of portuguese,
notions of german.
-from 1998 to 2002: worked as an actor in a theatrical company (l’Egregore, Marseille).
Funded projects
In 2002 I was awarded a six months postdoctoral fellowship from the Agence Universitaire
de la Francophonie. From april 2003 to july 2004 I was postdoc at IST Lisboa within the
European Network Geometric Analysis.
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Description of major accomplishments
Discrete Kaluza Klein from scalar fluctuation in noncommutative geometry
Abstract: We compute the metric associated to noncommutative spaces described by a
tensor product of spectral triples. Well-known results of the two-sheets model (distance on a
sheet, distance between the sheets) are extended to any product of two spectral triples. The
distance between different points on different fibres is investigated. When one of the triples
describes a manifold, one finds a Pythagorean theorem as soon as the direct sum of the internal
states (viewed as projections) commutes with the internal Dirac operator. Scalar fluctuations
yield a discrete Kaluza-Klein model in which the extra component of the metric is given by the
internal part of the geometry. In the standard model, this extra component comes from the
Higgs field.
The aim of this work was to compute the NCG distance in the standard model. During a
seminar, R. Wulkenhaar had shown that the distance in a geometry obtained by the product
of the continuum by an internal C2 space was the geodesic distance in a discrete Kaluza-Klein
model. I used this seminal work to obtain a similar result for the geometry of the standard
model. The task was much more involved since instead of C2 one is dealing with the internal
algebra C+H+M3 (C). The solution was to restrict to von Neumann algebras and consider only
normal states, in order to have projectors associated to states. In this way the computation
became tractable and I managed to isolate sufficient (but non necessary) conditions to ensure
the product of geometry is orthogonal in the sense of Pythagora’s theorem.
During this work I gain good skills in differential geometry, C ∗ -algebras and Von Neumann
algebras.
Diamonds temperature: Unruh effect for bounded trajectories and the thermal
time hypothesis
Abstract: We study the Unruh effect for an observer with a finite lifetime, using the
thermal time hypothesis. The thermal time hypothesis maintains that: (i) time is the physical
quantity determined by the flow defined by a state over an observable algebra, and (ii) when
this flow is proportional to a geometric flow in spacetime, temperature is the ratio between
flow parameter and proper time. An eternal accelerated Unruh observer has access to the local
algebra associated to a Rindler wedge. The flow defined by the Minkowski vacuum of a field
theory over this algebra is proportional to a flow in spacetime and the associated temperature
is the Unruh temperature. An observer with a finite lifetime has access to the local observable
algebra associated to a finite spacetime region called a “diamond”. The flow defined by the
Minkowski vacuum of a (four dimensional, conformally invariant) quantum field theory over
this algebra is also proportional to a flow in spacetime. The associated temperature generalizes
the Unruh temperature to finite lifetime observers.
Furthermore, this temperature does not vanish even in the limit in which the acceleration
is zero. The temperature associated to an inertial observer with lifetime T , which we denote
as “diamond’s temperature”, is TD = 2~/πkb T . This temperature is related to the fact that a
finite lifetime observer does not have access to all the degrees of freedom of the quantum field
theory.
The starting point of this paper was the reading of Haag’s book on local quantum physics.
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There I found that the modular group does not have a geometrical action only for the Rindler
wedge, but also for diamonds in Minkovski space. From that point I started to adapt the
algebraic approach to Unruh effect (based on the modular group of the Rindler wedge) to the
case of diamonds. I obtained a temperature of the vacuum that depends on both the observer’s
lifetime and proper-time. Carlo Rovelli played an important role by replacing the result in
its physical context. Especially he emphasized the fact that the temperature I found was not
ex-nihilo but can be viewed as a correction to classical Unruh effect. He suggested that I
should sign the paper alone since, according to him, he helped me with fruitful discussions but
I made most of the work. Together we decided that a double signature was more likely to draw
attention on a paper on a relatively isolated subject.
Carnot-Carathéodory metric from gauge fluctuation in noncommutative geometry
This is a work in progress that should be put on the arXiv in a near future. However a
well detailed draft is available on request. In my thesis I have computed the NCG distance
in the Standard Model only taking into account the scalar part of the NCG equivalent of the
Levi-Civita connection. When the gauge part of the connection is non zero, a formula for the
distance is announced in ref.[7]. According to this statement the NCG distance should equal
the Carnot-Carathéodory (CC) distance defined by the connection. Given two point x, y, the
CC distance between x and y is the length of shortest path between x and y whose tangent
vector is everywhere horizontal (i.e. is constant with respect to the covariant derivative defined
by the connection). It is not difficult to show that the NCG metric and the CC metric have the
same connected component. As easy is to show that they are equal when the connection is flat.
However, very surprisingly and after lots of doubts, I discovered that the two distances are not
always equal, depending on the holonomy of the connection. I sent this result to Connes who
explained me that its statement (the equality of the two metrics) was indeed rather qualitative
and that the question should be considered as open. He suggested that I push further the
analysis of the condition of the holonomy. That is why the paper is not yet published, although
it already contains some nontrivial results.
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Human resources and mobility activity
Marie Curie actions
Intra-European Fellowships
Part B Section 1
QGNC