Non-product geometries for particle physics and cosmology

143 Pages·2022·2.857 MB·English

by  Arkadiusz Bochniak

Jagiellonian University in Kraków
Faculty of Physics, Astronomy and Applied Computer Science,
Institute of Theoretical Physics, Department of Field Theory
Non-product geometries
for particle physics and cosmology
Arkadiusz Jakub Bochniak
PhD thesis
written under the supervision of prof. dr hab. Andrzej Sitarz
Kraków 2022
Wydział Fizyki, Astronomii i Informatyki Stosowanej
Uniwersytet Jagielloński
Oświadczenie
Ja niżej podpisany Arkadiusz Jakub Bochniak doktorant (nr indeksu: 1102259)
Wydziału Fizyki, Astronomii i Informatyki Stosowanej Uniwersytetu Jagiellońskiego
oświadczam, że przedłożona przeze mnie rozprawa doktorska pt. „Non-product
geometries for particle physics and cosmology” jest oryginalna i przedstawia
wyniki badań wykonanych przeze mnie osobiście, pod kierunkiem prof. dr. hab.
Andrzeja Sitarza. Pracę napisałem samodzielnie.
Oświadczam, że moja rozprawa doktorska została opracowana zgodnie z Ustawą
o prawie autorskim i prawach pokrewnych z dnia 4 lutego 1994 r. (Dziennik Ustaw
1994 nr 24 poz. 83 wraz z późniejszymi zmianami).
Jestem świadom, że niezgodność niniejszego oświadczenia z prawdą ujawniona
w dowolnym czasie, niezależnie od skutków prawnych wynikających z ww. ustawy,
może spowodować unieważnienie stopnia nabytego na podstawie tej rozprawy.
Kraków, dnia 9 lutego 2022,
...................................................
podpis doktoranta
Contents
1 Introduction 9
1.1 Noncommutative geometry . . . . . . . . . . . . . . . . . . . . . . 9
1.1.1 The idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.1.2 Fredholm modules and spectral triples . . . . . . . . . . . . 11
1.1.3 Grading and reality . . . . . . . . . . . . . . . . . . . . . . 13
1.1.4 The first-order condition and Hodge property . . . . . . . . 14
1.1.5 Reconstruction theorem and beyond . . . . . . . . . . . . . 16
1.1.6 Twisted spectral triples . . . . . . . . . . . . . . . . . . . . 19
1.2 The spectral action principle . . . . . . . . . . . . . . . . . . . . . 19
1.2.1 The general construction . . . . . . . . . . . . . . . . . . . 20
1.2.2 Spectral action in terms of Wodzicki residue . . . . . . . . . 22
1.3 Gauge theories and almost-commutative geometries . . . . . . . . . 23
1.4 Spectral Standard Model . . . . . . . . . . . . . . . . . . . . . . . 27
1.4.1 The fermion doubling problem . . . . . . . . . . . . . . . . 30
1.4.2 Lorentzian formulation . . . . . . . . . . . . . . . . . . . . 32
2 New directions 34
2.1 Non-product geometry for the Standard Model . . . . . . . . . . . 35
2.1.1 Formulation of the model and its basic predictions . . . . . 36
2.1.2 Bosonic spectral action . . . . . . . . . . . . . . . . . . . . 45
2.1.3 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
2.2 Non-product geometry for modified gravity theories . . . . . . . . . 76
2.2.1 Doubled geometries - dynamical stability of solutions for
FLRW models . . . . . . . . . . . . . . . . . . . . . . . . . 78
3
2.2.2 Analysis of the interaction term . . . . . . . . . . . . . . . 101
2.2.3 Towards generalized bimetric models . . . . . . . . . . . . . 113
2.2.4 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
3 Conclusions and outlook 124
4
Content of the thesis
The organization of the thesis is as follows. Section 1 is devoted to the in-
troduction to noncommutative geometry together with a brief review of the most
important results in the field, chosen by the author as relevant for the rest of the
thesis. In section 2 the performed research is motivated and its results are then
collected. Each of the subsections 2.1 and 2.2 begin with a brief introduction to
the problem. Then each of the articles (or preprints) is reproduced, preceded by
an appropriate commentary containing the summary of the obtained results. Each
subsection is closed with a brief summary and discussion of possible further research
directions in the subject. Finally, in section 3 closing remarks are contained.
This thesis consists of the following articles and preprints, to which the contri-
bution of each author was equal:
1. A. Bochniak and A. Sitarz, Spectral geometry for the standard model without
fermion doubling, Phys. Rev. D 101, 075038 (2020), DOI: 10.1103/Phys-
RevD.101.075038.
2. A. Bochniak, A. Sitarz and P. Zalecki, Spectral action and the electroweak
θ-terms for the Standard Model without fermion doubling, J. High Energ.
Phys. 12, 142 (2021), DOI: 10.1007/JHEP12(2021)142.
3. A. Bochniak and A. Sitarz, Stability of Friedmann-Lemaître-Robertson-Walker
solutions in doubled geometries, Phys. Rev. D 103, 044041 (2021), DOI:
10.1103/PhysRevD.103.044041.
4. A. Bochniak and A. Sitarz, Spectral interaction between universes, arXiv:
2201.03839.
5
5. A. Bochniak Towards modified bimetric theories within non-product spectral
geometry, arXiv: 2202.03765.
6
Acknowledgments
First of all I would like to express my sincere gratitude to my supervisor An-
drzej Sitarz for inviting me into the noncommutative world at the first year of my
Bachelor studies. I really appreciate the guidance, constructive criticism, scientific
inspiration, posing ambitious problems, as well as patience and help on each level
of my studies at Jagiellonian University.
I thank my family, especially my parents, for the support during my studies.
My appreciation also goes out to Sasha Kovalska for her loving support during the
whole of my PhD studies. I know that sometimes my scientific and bureaucratic
problems were unbearable. Thank you once again for your patience!
I offer my sincere thanks to my teachers, faculty staff, student colleagues,
friends and all the people who inspired and guided me, with whom I had the
opportunity to collaborate, interact or discuss several (not only scientific) prob-
lems at different stages of my career. It is not possible to provide a complete
list, but let me here just mention some of them: Henryk Arodź, Hari Bercovici,
Piotr Bizoń, Dawid Brzemiński, Karol Capała, Ewa Cygan, Andrzej Czarnecki,
Maciej Dołęga, Leszek Hadasz, Paweł Kasprzak, Masoud Khalkhali, Piotr Kor-
cyl, Wojciech Kucharz, Marcin Kulczycki, Marek Jarnicki, Zohar Nussinov, Piotr
Niemiec, Teresa Niemiec, Gerardo Ortiz, Piotr Pikul, Kevin M. Pilgrim, Joanna
Popielec, Michał Praszałowicz, Andrzej Rostworowski, Błażej Ruba, Alexander
Seidel, Piotr M. Sołtan, Jan Stochel, Vladimir Turaev, Thomas E. Williams, Jacek
Wosiek, Adam Wyrzykowski, Paweł Zalecki, George Zoupanos and Błażej Żmija.
Many thanks to Marta Górska who first showed me the beauty of physics. Special
thanks to Ewa Witkowska for solving seemingly unsolvable bureaucratic problems
during my PhD studies.
7
I would like to thank the Department of Mathematics of the Indiana University
Bloomington, USA, for the hospitality during the Fulbright Junior Research Award
scholarship funded by the Polish-US Fulbright Commission.
Part of this work was supported by the National Science Centre, Poland, Grant
No. 2018/31/N/ST2/00701. I would like to acknowledge also the Descartes pro-
gram for PhD students at the Jagiellonian University as well as the Faculty of
Physics, Astronomy and Applied Computer Science of the Jagiellonian University
and its staff.
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1 Introduction
1.1 Noncommutative geometry
Geometric structures existing behind physical theories play a crucial role in
understanding phenomena predicted by these models. The most recognizable the-
ory formulated in a purely geometric language is Einstein’s theory of gravitation
- the General Relativity [1]. The attractive idea that fundamental interactions
can be encoded by using objects of topological or geometric nature was further
extended far beyond the gravitational interactions. In particular, gauge theories
can be described in terms of connections and their curvatures defined on certain
fiber bundles [2].
1.1.1 The idea
Before making an attempt to geometrize physics one has to first answer the
question “What a geometry really is?”. The common understanding of geometric
objects as sets of points, equipped with additional structures that allow us to
decide, for example, how far are these points from each other, appears to not be
fully satisfactory. Fortunately, there is a different, but equivalent, description.
Instead of considering the set of points itself, one can concentrate on functions
defined on it. More precisely, to a locally compact Hausdorff space one can
M
associate the C -algebra C( ) of continuous complex-valued functions on .
∗
M M
The Gelfand-Naimark duality theorem [3] assures us that the topology of the space
can be reconstructed out of the data encoded in C( ). This is the first step
M M
to an alternative formulation of the notion of geometry, which can be further easily
generalized. Indeed, observe that the C -algebra C( ) is commutative, and it
∗
M
9
is natural to say that if we replace it with a noncommutative C -algebra A, it
∗
describes noncommutative topological space.
The aforementioned duality theorem allows us to store the information about
the topology of a space, but a given topological space can be equipped with further
structures. To attempt a reformulation of this part, one has to first make use of
the Gelfand-Naimark-Segal (GNS) construction [3, 4, 5] that allows us to treat
elements of the C -algebra A as operators acting on certain Hilbert space .
∗
H
The possibility of rewriting topological data in terms of algebraic objects can be
further extended, and certain dictionaries can be built. In particular, open (dense)
subsets corresponds to (essential) ideals and compactification of a topological space
can be translated into the unitalization of a C -algebra. For other constructions,
∗
we refer the reader to [6, 7].
The first crucial step towards introducing noncommutative analogs of differ-
ential and metric structures that one could consider on a given topological space
is to replace C -algebras A with their dense -subalgebras . Relaxing the C -
∗ ∗
∗ A
condition has many advantages as, in particular, it allows for the use of certain
differential operators. In classical differential geometry this means that instead of
the C - algebra C( ) we work with the -algebra C ( ) of smooth functions
∗ ∞
M ∗ M
on . The requirement of smoothness has plenty of implications, and this choice
M
is very natural in the context of differential geometry, where smooth vector fields
(sections of certain smooth vector bundles) play an important role.
The fundamental object in differential geometry is the space of one-forms
Ω1( ), from which the whole differential calculus can be built. This set has a
M
natural structure of C ( )-bimodule, and the exterior derivative d is a deriva-
∞
M
tion C ( ) Ω1( ), which is then extendable to Ω ( ) by requiring the
∞ •
M → M M
10