PRELIMINARY REPORT ON PROGRESS IN SPECTRAL UNIFICATION MODELS

Let us indulge ourselves & imagine we can wear Seven-League Boots in order to engage through a Grand Tour of Physics with a spirit strong enough to address seven myth or mysteries of the standard models of particle physics & cosmology. To do so we will borrow the magic boots of several Giants who tailored these models or extracted the essential marrow from both.

Public advertisement: several published peer-reviewed articles and books on Solid Theoretical Research into Natural Geometric Structures have been copycatted in the making of the following mashup. Hyperlinks to professional scientific content are provided to pay due tribute to the real authors. Any specific claim, hypothesis or conjecture not duly borrowed from published work by physicists, mathematical physicists or mathematicians are only mine and can be recognized as they are placed into the symbols [...]. Hopefully they have been reduced to a minimum and quarantined essentially in the discussion section. If some expert or any educated reader thinks that the following presentation is a knowing-just-enough-to-be-dangerous-naive way to look at the spectral standard model and its possible extensions I urge her or him to let me know how and why she or he thinks so and leave a comment to this post or emailing me at laboussoleestmonpays@gmail.com.


The elimination of the logical inconsistencies ... requires a radical reconstruction of the theory, and in particular, the rejection of a Riemannian geometry dealing, as we have seen here, with quantities which are unobservable in principle, and perhaps also the rejection of our ordinary concepts of space and time, replacing them by some much deeper and non-evident concepts. "Wer's nicht glaubt, bezahlt einen Thaler."


Perhaps we should not try to solve the hierarchy problem now, 
but rather wait until we have understood how to incorporate
gravity (i.e. completed the unification). 


What will be the thematic melodies of 21st century theoretical physics?
... The one area where big fundamental revolutions may perhaps take place, I believe, is astrophysics. Such puzzles like dark matter and dark energy might in time be replaced by beautiful new concepts, very much like the Fitzgerald contraction hypotheses being replaced by Einstein's special relativity a century ago.


“You can't always get what you want … but if you try … you might find … you get what you need”


The Brave Little Model Tailor addressing seven myths
or mysteries of particle physics & cosmology at one blow

ENS & Université--  "Orsay Saclay, Paris (alma mater)
 Laboratoire des Champs Magnétiques Intenses de Grenoble (former PhD laboratory)
& Lycée [Georges Brassens*, Marianne] Académie de Créteil (current position)

(1st draft : 1 Apr 2017, 2nd version 15 Nov 2019, 3rd version March-April-July 2023)


CONTENTS:

PROLEGOMENA

  I   INTRODUCTION

 II   SPECTRAL GEOMETRY FOR A FINER SPACETIME MODEL

III   YANG-MILLS+NONCOMMUTATIVE GEOMETRY  = YANG-MILLS-HIGGS

IV   QUANTA OF SPACETIME GEOMETRY => MIMETIC GRAVITY?

V   KEYSTONES OF THE SPECTRAL STANDARD MODEL CONSTRUCTION

VI  PEBBLES FOR A PATHWAY TO SPECTRAL UNIFICATION OF FUNDAMENTAL INTERACTIONS
 
VII ADDRESSING SEVEN MYTHS OR MYSTERIES OF THE STANDARD MODELS OF PARTICLE PHYSICS & COSMOLOGY + SOME CURRENT FLAVOR ANOMALIES

VIII ON THE WAY WITH YOUNG SPACETIME QUANTUM GEOMETRY EXPLORERS

DISCUSSION: A WARNING

AFTERWORD

SOME VIDEOS


PROLEGOMENA :

I attempt to outline, without any technical details but providing relevant bibliography, the possible physical consequences behind the spectral noncommutative geometrisation of the Standard Model (SM)  & the spectacular progress achieved since the discovery of the Higgs Boson on spectral models of gravity coupled to matter. I did not try, actually I am not expert, to offer an unbiased presentation. My aim is not to be critical, but rather to transmit a part of the excitement and enthusiasm that we enjoy, when we have the clear feeling of possibly being on the right track or at least having both a working compass & a good marine chronometer. I want to share this excitement and enthusiasm with everybody, even the most skeptical and pessimistic. I aim at convincing you that as John Iliopoulos recently advocated:
  • non-commutative geometry has come to stay! It is part of gauge theories... 
  • This approach may provide the way we are all looking for to unify gauge theories and Gravity.
  • It may give some predictions for the Standard Model parameters. (In the traditional framework of renormalisable perturbation theory such a prediction is impossible)
  • It may offer a new insight for the mysteries of dark matter and dark energy.
I am aware and I anticipate the criticism you may have. I am aware of the fact that when no new results beyond the spectral standard model are obtained, its validity is not examined, and when unsettled questions remain unaddressed, I argue that we do not yet have a complete spectral picture of gravity coupled to quantum matter (the issues of quantum gravity & second quantisation have just started to be addressed) and the formulation of the spectral action principle requires an extension to accommodate consistently the quanta of geometry recently uncovered. You are advised to keep that in mind in evaluating the significance of these ideas, but also to agree that, on the face of it, they are sufficiently successful to command some attention. 



INTRODUCTION

Among the major scientific breakthroughs of the last decade, the discovery of the Higgs boson at the LHC by ATLAS & CMS, the taking and analysis of the highest-resolution photograph ever of the early universe by the Planck mission and the direct detection of gravitational waves by LIGO & VIRGO are spectacular successes crowning decades of effort by a large community of scientists exploring the infinitely large and the infinitely small. 

Without exaggeration they have established experimentally that quantum physics, which is normally assumed to be relevant to explain matter-radiation interactions at the atomic and subatomic scale is also useful to probe spacetime geometry dynamics and works on the scale of the entire Universe with Einstein theory, both being valid in a fantastic range from 10^-27 cm to 10^+28 cm.  These discoveries reinforce three fundamental concepts in physics: the generation of the mass of elementary particles, the dynamics of space-time geometry and the coupling of gravity with matter. At the same time, they open up new fields of exploration for deepening the understanding of these concepts at the quantum level. Moreover, the cosmological observations (and astrophysical ones at the galactic scale) show that the current description of gravity coupled to matter is incomplete. Among the open questions are: the nature of dark matter; the absence of antimatter in the cosmos; the origin of the acceleration of the expansion of the universe today and in the first moments just after the Big Bang. 

Last but not least, the current absence of clear signals of new physics beyond the SM on all frontiers of particle physics leaves also a lot of crucial questions unanswered. What is the range of applicability of the SM? What stabilizes the Higgs-boson mass? What lessons does electroweak symmetry breaking hold for unified theories of the strong, weak, and electromagnetic interactions? What will be the next symmetry recognized in Nature? What is dark matter made of? All these questions could be related to particle physics in the primordial universe and require a coherent quantum theory treatment including gravity effects. Supersymmetry and extra-dimensions at the TeV scales have been examples of popular theoretical constructions offering tentative answers to several of these questions. But there has been no hint of an experimental signal up until now to confirm the relevance of any of them.

As the LHC is preparing for Run 3 in 2021 and discussions about the future of high energy collider physics are underway, astrophysicists are polishing, adjusting and calibrating their next earth or space telescopes, antenna, interferometers and spectrometers, expecting to map the geometry of the Universe, the polarisation of the cosmic microwave background, probing further compact objects collisions or instabilities with multimessengers and looking further away for the cosmic dark ages.

 To assure a bountiful harvest of experimental results requires some care about sowing, building the proper next generation of accelerator and choosing the most relevant triggers in particle detectors to have the best chance to land on a new continent while navigating the roaring 10 TeV scale. It might be the right time then to sift the wheat from the chaff and put into questions some theoretical prejudices against:

I hope the following roadmap made of a short list of article titles & a few slogans extracted from a rich and dynamic bibliography on spectral NCG will provide food for thought and raise some interest in the community of physicists looking for new enlightenment on the black box of the SM & the dark sector of the concordance model.

This text aims also at showing one should not give up on a 100 year-old  dream as physicists are not lost in mathematics but know that human nature needs time to distill physical models from mathematical grape harvests.
II   SPECTRAL GEOMETRY FOR A FINER SPACETIME MODEL



 V   KEYSTONES OF THE SPECTRAL STANDARD MODEL CONSTRUCTION


VI  PEBBLES FOR A PATHWAY TO SPECTRAL UNIFICATION OF FUNDAMENTAL INTERACTIONS

    (i) Gravitation und electrizität & Perturbations of the metric and Weyl’s eichinvarianz
   (ii) About time, gravitational theory, quantum theory and thermodynamics
  (iii) Spectral action scale invariance & Quantum gravity boundary terms from spectral action of NC spaces
  (iv) Lessons from Quantum Field Theory - Hopf Algebras and Spacetime Geometries
   (v) A Lorentzian version of the NCG of the SM of particle physics             
  (vi) NCG as a Framework for Unification of all Fundamental Interactions Part I.
 (vii) Noncommutative Geometry, Quantum Fields and Motives        
(viii) Beyond the Spectral SM: Emergence of Pati-Salam Unification 
  (ix) Quanta of Geometry & Axiomatisation of [spacetime model spectral aufbau for] physics
   (x) Spectral action with zeta function regularization & Wick rotation and fermion foubling in NCG
  (xi) Second quantization and the spectral action & Principle of locality in NCG
 (xii) Resolving cosmological singularities  & Asymptotically free mimetic gravity         

         
VII ADDRESSING SEVEN MYTHS OR MYSTERIES OF THE STANDARD MODELS OF PARTICLE PHYSICS & COSMOLOGY + SOME CURRENT FLAVOR ANOMALIES

    (i) Why the SM in 4D spacetime: bottom-up & top-down approaches merging in spectral NCG
   (ii) Four remarks about the naturalness problem in the framework of spectral unification models
  (iii) The Higgs mass & stability of the electroweak vacuum
  (iv) Leptogenesis & Neutrino phenomenology in nonsupersymmetric grand unification theories
   (v) Cosmology with Mimetic Gravity
 (vi) Mimetic gravity, astrophysical evidence for dark matter and physical time in quantum cosmology
 (vii) Gauge coupling unification in the spectral Pati-Salam Models
(viii) B-decay anomalies and scalar leptoquarks in unified Pati-Salam models from NCG


VIII ON THE WAY WITH YOUNG SPACETIME QUANTUM GEOMETRY EXPLORERS



DISCUSSION: A WARNING

At the time being, building a genuine Spectral Unification Model (SUM) able to make specific experimental predictions beyond the SM (in a way similar to studies testing vanilla leptogenesis scenario in minimal non-supersymmetric SO(10) GUTs for instance) remains a challenging task not necessarily appealing for phenomenologists as the prospects for direct experimental signatures testable at the current & foreseeable low/high energy or precision frontiers seems a priori daunting.

On the other hand SUMs are pretty falsifiable as any upcoming experimental evidence for supersymmetric particles at the TeV scale or a proton disintegration signal would probably kill them all. I emphasize that any SUM would be required to be computed from the mathematically quite rigid noncommutative geometric framework based on few axioms [which number has decreased since its inception] as the experimental data set validating the SM has increased (I take as granted that neutrino mixing belongs to the current SM-2019 in the spectral noncommutative geometric perspective).

I imagine the situation for SUM building may become as more favorable as conventional dark matter particles search will stay unsuccessful.

On a more cheerful tone the fact that one can envision on a pretty sound basis, in lieu of the string landscape, a specific bridge from the Higgs scale to a Pati-Salam gauge couplings unification scale (spanning roughly fourteen orders of magnitudes in energy from the 125 GeV Higgs mass to the  O(1016) GeV natural scale of spectral action) is a remarquable progress that should provide incentives to address for instance the hierarchy problem in a broader perspective namely explicitely connecting it with the issue of cosmological inflation. It is indeed worth reminding that the spectral action includes naturally a dilaton field which guarantees the scale invariance of the SM interactions, and provides a mechanism to generate mass hierarchies. Moreover considerations of scale invariance require including new terms in the spectral action (see end of paragraph 11 in "Quanta of Space-Time...") that makes it possible to avoid cosmological and black hole singularities in GR and build asymptotically free mimetic gravity.



ACKNOWLEDGEMENTS
I dedicate this text to Roland Omnès & François Le Diberder, the first quantum physics professor & teaching assistant I had. I thank Alain Connes for answering a couple of naïve questions once at a séminaire d'algèbres d'opérateurs. I would like to extend the warmest thanks also to Thomas Schücker and Thierry Masson for very kind and helpful email exchanges in the past.  



AFTERWORD

I cannot resist to table here three beautiful French anagrams imagined by Etienne Klein & Jacques Perry-Salkow (respectively physicist-philosopher of science & pianist-poet). Generally speaking anagrams embody vividly how non-commutativity is omnipresent in everyday life in the written language. With hindsight possibly gained by the former reading I think the three following ones can chant and summarize vividly the conceptual insights offered by spectral unification models on three salient  experimental or observational features of contemporary particle physics and astronomy.

The first one
Le boson scalaire de Higgs = l'horloge des anges ici-bas.
(The Higgs boson scalar "=" the angels' clock down here)
was explained by Etienne Klein in the following way:
... certaines [anagrammes] sont particulièrement troublantes comme « le boson scalaire de Higgs »… Elle a un sens profond : dans l’univers primordial, avant d’interagir avec l’évanescent boson de Higgs, les particules élémentaires n’avaient pas de masse ; dès qu’elles en ont acquis une, elles se sont trouvées ici-bas affublées d’un temps propre, c’est-à-dire d’une sorte d’horloge portative. 
... some [anagrams] are particularly troubling like "the Higgs scalar boson" ... It has a deep meaning: in the primordial universe, before interacting with the evanescent Higgs boson, elementary particles had not mass; as soon as they have acquired one, they have found themselves down here decked with a proper time, that is, a kind of portable clock.

The second one reads


The last one

is an evocative recap to this modest report on an ongoing Odyssey.


SOME VIDEOS

Alain Connes: Quanta of Geometry (mathematical lecture) 
(at Erwin Schrödinger International Institute for Mathematics and Physics March 2015)



Ali Chamseddine: Quanta of Geometry (physics lecture)
(at Future Prospects for Fundamental Particle Physics and Cosmology Workshop: May, 2015)



Walter van Suijlekom: Semigroup of inner perturbations in Non Commutative Geometry
(at Hausdorff Trimester Program Non-commutative Geometry and its Applications. 23.09.2014)



(Organized by Europe GNCG Seminar, February 2, 2021)






Personal notes

Fig. 1: The place of the Spectral Unification Model in the Grand Tour of Physics. 

The picture above is a personal tentative update of the popular 1980s Glashow's cosmic Uroboros that epitomizes the possible connexion between the physics of the two infinities since the raise of grand unification theories in the 70s. The Higgs boson scalar discovery at LHC & the lack of any dark matter or supersymmetric particle detection up to now fits very well with the spectral standard model which derives the Lagrangian & the spectrum of fundamental particles in the Standard Model of microscopic physics from a noncommutative geometric model of spacetime. The latter offers also a new perspective on the cosmological issues of dark matter, dark energy & inflation thanks to an unexpected connexion with the theory of mimetic gravity. The keystone of this vision is the Chamseddine-Connes-Mukhanov equation written at the "top" of what I nickname the Grand Tour of Physics. To embody the scientific aim to build a spectral unification model I have picked the popular French fairy tale character Hop-o'-My-Thumb wearing seven-league boots to make a giant leap from the infinitely small to the infinitely big in lieu of the more traditional Glashow's snake that fascinated me when I saw it for the 1st time. 

In this framework the recently discovered Higgs boson mass value provides a scale for the fine structure of spacetime that is more subtle (neither continuous nor discrete and 0D) than some ordinary continuous more than 4D spaces proposed in the past. It consists in two sheets of 4D Euclidean spacetime separated by a distance of 10⁻²⁰ cm but "connected" thanks to the Higgs field. For such a noncommutative space to appear as a solution of the C²M equation requires an even finer structure at ultra-high energy scale (a third sheet with a 10⁻²⁹ cm distance separation) associated with a hypothetical ultra-heavy Higgs boson brother. This "Big broson" triggers the seesaw mechanism accounting on the one side for the very weak mass of left-handed neutrinos and on the other side for a baryogenesis through leptogenesis scenario implying three super-heavy right-handed neutrinos. Last but not least, the C²M equation implies a volume quantization of spacetime geometry leading to mimetic dark matter and dark energy phenomenology at astrophysical and cosmological scales.


Fig 2: This is the last pages of the Abdus Salam's transcription of the 1st 1988 DIRAC MEMORIAL LECTURES  where the author of this post saw for the very 1st time the Glashow's snake or rather Cosmic Uroboros (Cambridge University Press 1990) 


Notes :
v5 (July 2023): remove one epigraph by Ken Wilson to replace it by Pierre Ramond one. 
v6 (February 2024) : video of Lisa Glaser was added.











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