For example, is it possible to consider the gauge non-compact group SU(3)xSL(2,C)xU(1) instead of the Higgs scalar field, which breaks the gauge symmetry of the standard model? Have you come across such studies?
Igor Bayak, As for models of the kind you are imagining, in general the conditions for them to have the right symmetry structure, would forbid non-compact symmetries, in particular when they are supposed to obey unitarity
I have found recently that in curved space the photon acquires mass.
https://www.sciencedirect.com/science/article/abs/pii/S0030402621015102
I understand that observables in quantum mechanics must be real quantities and therefore unitary operators correspond to them, but on the other hand, nothing prohibits us from also having complex observables. After all, no one prohibits multivalued functions either.
In the case of a photon, this is like a transition from the U(1) group to the group of conformal transformations.
Complex numbers are fine, unboundedness is much more of a problem. Ordinary theories don’t allow it. We do have exceptions of course - think of the Lorentz group, or Poincar’e. This is one of the reasons why gravity is so hard.
Dear Igor Bayak
If we hypothesize that during the evolution of the universe there was a period that there was no matter, only vacuum space (flat Higgs field), the only “available” quantum fields are the universal electric field, its corresponding magnetic field (together the electromagnetic field) and the flat Higgs field. According to Einstein (1920) and Eric Verlinde (2011) gravity is an emergent force field. That means there must be matter first.
The consequence is that during that period of time the universal electric field and its corresponding magnetic field represent all the dynamics in the universe. This is in line with the 2 universal conservation laws: the law of conservation of energy (electric field) and the law of conservation of momentum. Momentum represents a local quantity of energy and its direction (vectors).
In other words, all the dynamics in the universe is created by the universal electric field and its corresponding magnetic field.
So the question arise if the decrease of one or more scalars of the flat Higgs field means a supply of energy from the scalar to the universal electric field. Or that the decrease of one or more scalars facilitate the electric field to concentrate more energy to the amount of the concentration of energy that forced the scalar(s) to decrease.
If we choose the first possibility we have to explain a synchronous process to supply energy from the universal electric field to the lattice of the Higgs field to keep all the energy in the universe conserved. That is not realistic.
Recently Sabina Hossenfelder mentioned a publication of Einstein that describes a more or less similar idea (the creation of matter out of vacuum space).
A. Einstein (1923); “Spielen Gravitationsfelder im Aufbau der materiellen Elementarteilchen eine wesentliche Rolle?” Das Relativitätsprinzip. Fortschritte der Mathematischen Wissenschaften in Monographien. Vieweg+Teubner Verlag, Wiesbaden. https://doi.org/10.1007/978-3-663-19510-8_10
I never heard of it so I was really surprised.
With kind regards, Sydney
Gerard t Hooft If boundlessness in this case means arbitrariness in choosing the coefficients by which the generators of the Lie algebra of a non-compact group are multiplied, then the coupling constant of the unitary group also evolves.
Yet I have not seen any convincing arguments against the gauge group SL(2,C). The Lie algebra of this group over the field of real numbers is the sum su(2)+i*su(2) and thus, in comparison with su(2), we have a doubling of the generators of the Lie algebra of a non-compact group, which can be interpreted as the replenishment of charge generators with mass generators.
It can also be added that the Lie algebra of a special linear group SL(2,C) has a geometric interpretation as the algebra of linear vector fields tangent to the Clifford torus S^1 x S^1, where some generators are responsible for paired Euclidean rotations of the torus (rotations of the defining circles of the torus), and others for paired pseudo-Euclidean rotations of the torus (hyperbolic rotations of the circles of the torus, i.e. compression and stretching of them diameters). In this case, the first (charge) generators generate a compact group, and together with the second (mass) generators they form a non-compact group.
Igor Bayak , The (by now) conventional view as to why we need the Higgs mechanism, is that it is indispensable for having infinite integrals cancel each other out so that a particle model ends up being renormalizable. If the infinities do not cancel you can’t do the calculations needed to predict the outcome of experimental measurements. The Higgs self interactions generate precisely the effects needed to distinguish particles with mass from massless particles. As they are in the real world. This is why some people say that “the Higgs particle is responsible for mass”. If you have an 'alternative theory’ you need a lot of explaining to do of why it can do something similar. But such theories do exist: QCD for example.
the central point with the higgs mechanim is, that gauge symmetry is not broken.
Gerard t Hooft , I understand that you managed to eliminate infinities using the Higgs mechanism. Then you will be interested to see how you can eliminate infinities using the Riemann zeta function
Preprint Chaotic dynamics of an electron
Another question is how to convey this metaphysical essay to the physics community.
everyone, who wants to get a thorough understanding of the Higgs mechanism, should consult the following brilliant book:
P.Nair Quantum Field Theory-A modern perspective
Gerard t Hooft , If you're interested, let me touch on the motivation behind my approach. In classical physics, calculating the energy of an electron leads to infinity, which is eliminated by switching to a spherical electron model. In turn, in quantum mechanics, where an electron is a point particle, the divergence in the energy value is eliminated only by a mathematical trick - renormalization. At the same time, there are divergences when calculating the sum in the Riemann zeta function, which are eliminated when specifying the function on a compact (spherical-toroidal) manifold.
Gerard t Hooft you refer to the Riemann zeta function. if that means , that you use hawking zeta function renormalization method, you would again use a renormalization method. compactifying the base space of a theory is the simplest way of getting rid of divergencies.
The SL(2,C) group (that describes Lorentz transformations) is non-compact, whereas the groups that describe the internal symmetries of matter (color-the SU(3) group, the electroweak charges-the SU(2)xU(1) group (the U(1) factor is NOT electromagnetism, it's weak hypercharge) are compact groups. It is the property that SL(2,C) is a non-compact group that leads, inevitably, to the emergence of spacetime singularities at the classical level and the failure of the approach to quantization, that works for compact groups, for gravity, upon introducing gauge fields that transform under SL(2,C), since some of them will have kinetic terms of the ``wrong sign'', but aren't gauge artifacts.
That, indeed, is the problem with formulating a quantum theory of gravity: That of finding what is the quantum theory, whose classical limit is a gauge theory with this (SL(2,C) a non-compact) gauge group. Classical gravity is, indeed, a gauge theory, with, to be precise, as gauge group the group of diffeomorphisms of pacetime, which can be cast as the theory where Lorentz transformations are a local symmetry, not a global symmetry.
The masses of the quarks and thus of the hadrons do receive a contribution from the Brout-Englert-Higgs mechanism, since they do have electroweak interactions; but the major contribution to the masses of these particles comes from QCD, that describes the strong interactions. Even in the absence of electroweak interactions it is possible to show-by numerical simulations-that quarks and hadrons are massive particles, in the presence of the strong interactions, even if the quarks are taken to be massless in the classical action (it's not possible to produce such mass terms in perturbation theory and a mathematical proof is, still, not available).
However this way of describing the mass of particles doesn't work for the electroweak theory: If one starts with massless leptons in the classical action, they won't acquire a mass by electromagnetic or weak interactions, even when going beyond perturbation theory; nor will the W and Z bosons. Why this is so is understood, even though a mathematical proof, that is fully rigorous, is, also, still, absent.
Hence the need for the Bout-Englert-Higgs mechanism, that does provide a consistent description in the framework of perturbation theory about free fields.
There have been attempts to replace the Brout-Englert-Higgs mechanism, that involves scalar fields, by a mechanism of condensation of fermionic fields, that produces scalars through condensation, similar to what occurs in superconductivity, where the electrons form Cooper pairs that behave as scalars; this mechanism is called ``technicolor''. Unfortunately it, inevitably, implies effects that are in contradiction with experiment and making these smaller than the experimental bounds is not at all trivial.
The discovery, 12 years ago, now, of the Higgs boson, shows that at the level of the Standard Model, such scalar fields can exist by themselves and, at the energies probed by the LHC, don't need to be described as bound states of other particles.
Stam Nicolis there are also problems with the bound state of massless spinors to form a boson. S.Weinberg gave a thorough discussion on this. topic.
Stam Nicolis, Let's keep things simple. Let us have a group U(1) of gauge transformations that acts on the plane (x,y). If we add a scalar field to it in the form of local deformation (compression or stretching), we obtain conformal transformations of the plane. On the other hand, if we have a group of gauge transformations C(1), then we immediately have the same group of conformal transformations of the plane, since to preserve the local algebra of complex numbers it is necessary to satisfy the Cauchy-Riemann conditions.
Article Applications of the local algebras of vector fields to the m...
Massless particles of any spin have ``infrared'' issues, by definition. How to deal with these is known: Introduce a regulator (the lattice will do) and then study very carefully the combination of taking the lattice spacing to zero and the lattice size to infinity. It is this step that cannot, yet, be implemented, in full generality, in a mathematically rigorous way beyond perturbation theory about free fields (except for field theories in two spacetime dimensions); but it can be realized by numerical simulations, that can provide hints for the extrapolation to the scaling limit.
Regarding the claim about a scalar field, charged under U(1) gauge transformations, in two spatial dimensions, the first remark is that the classical action is invariant under spatial conformal transformations, only if the scalar field is massless. Adding a mass term-which is consistent with the spacetime symmetries of the classical action-explicitly breaks conformal invariance. Not to mention the fact that these transformations don't commute with three-dimensional Lorentz transformations, which is the group one is, usually, interested in in 2+1-dimensional field theories.
If the interest is in field theories in Euclidian signature, then the statement would be that the conformal symmetry of the classical action isn't protected from quantum corrections: First of all, there can't be electromagnetic waves in two dimensions and, second, there can't be propagating particles, charged under the gauge field-that can only define a constant electric field-in two dimensions. Only massive, electrically neutral particles (the simplest of which would be dipoles) can propagate. So the reason these particles are massive isn't an alternative to the Brout-Englert-Higgs mechanism, simply because such a mechanism doesn't make sense in two spacetime dimensions.
The Cauchy-Riemann conditions describe the propagation of free fields, not interacting fields. They're equivalent to the fields satisfying the Laplace equation in Euclidian signature, which is the wave equation in Lorentzian signature.
Stam Nicolis
it can be simulated on lattices of size 3 times 3 times 3 points, what is ridiculous.
No, it's been possible to carry out simulations on lattices much much larger in three, four and higher dimensions, for many years now. It's useful not to live in the past.
the lack of a continuum limes in more than 2 dimensions is another obstacle.
The inertial mass of the electron is deeply connected to the Zitterbewegung motion of its charge:
Article Electron Structure, Ultra-Dense Hydrogen and Low Energy Nucl...
Preprint Charge Clusters, Low Energy Nuclear Reactions and Electron Structure
Lee Hively ,
Could you comment on my theoretical attempts to reduce the energy of thermonuclear fusion?
https://www.researchgate.net/publication/354866262_On_the_induction_of_a_low-energy_nuclear_reaction_in_a_supersonic_plasma_vortex
Giorgio Vassallo ,
in my opinion, the fact that a photon has no mass, while an electron does, is due to the fact that they have different gauge symmetry groups. For a photon this is a compact group, but for an electron it is not. In geometric language, this can be explained by the difference between the symmetry group of the circle and the Clifford torus. For a photon, the symmetries of a circle are sufficient, and for an electron, the symmetries of a Clifford torus embedded in a 3-dimensional sphere are sufficient, and non-compact deformations (compression, stretching) of the diameters of the torus are responsible for the mass.
Igor, I looked at your paper and have several comments: (1) Your English is excellent; (2) I would have liked to see your derivations, perhaps in an appendix; (3) Ball lightning is a spherical tokamak IMHO, which has been funded by both the USA and UK on the basis of ideas originated by my former supervisor, Y.-K. M. (Martin) Peng at Oak Ridge National Laboratory, Oak Ridge, Tennessee, USA and Princeton University, Princeton, New Jersey, USA. That plasma configuration is more stable than conventional tokamaks and is still being pursued experimentally; (4) You never specified the frequency (omega2) in your diagram. Its value determines whether you heat ions, electrons, or get some other plasma-wave mode from the many possibilities; (5) You don't say whether the external conditions are a vacuum or air. From the context, I assume that this test would be done in air, which leads to many difficulties, though vacuum has its own set of problems; (6) Regardless of the heating mode, a rotating vortex that's heated by whatever means is simply a variant of a spherical tokamak, which is promising, but will melt the chamber wall before you get to fusion-scale temperatures; (7) Moreover, you need three, simultaneous conditions for a fusion reactor: a high enough density (n) [10^(19)-10^(20)/m^3], for a long enough time (tau) (typically many seconds); at a high enough temperature (T) (typically 10 keV for ions). Experimentalists have made substantial progress on raising the value of n*T*tau by many orders of magnitude, since the 1950s; Your idea is novel, but uses conventional physics to produce fusion power. Revolutionary physics is needed for success IMHO.
Lee Hively ,
Thank you so much for your comment, but let me disagree about the need for a revolutionary approach. If you look closely at my other articles, you will notice that I am friendly with revolutionary physics. However, in this case, if we take into account the approximation of three nuclei, classical physics will be sufficient. However, perhaps the modified Schrödinger equation [1] will come in handy. As for vacuum and frequency, I agree - there may be options. Please look at my latest diagram
https://www.lenr-forum.com/forum/thread/6850-i-ask-members-to-join-my-effort-to-create-a-fundamentally-new-lenr-device/?postID=206809#post206809
1.
Preprint Chaotic dynamics of an electron
the central point with the Higgs mechanism is, that gauge symmetry is n o t broken and that this is of upmost importance. see the brilliant book about QFT of P.Nair.
Gerard t Hooft
nobody knows the mechanism, by which masses are produced in QCD. the only calculations done, were numerical, done on a lattice by martin lüscher. and he was not able to continue his results to the continuum, until today. unfortunately, we do not live on a lattice. and numerical calculations give no hint , which mechanism produces masses in QCD.
Dear Stam Nicolis ,
you told us (3): "...taking the lattice spacing to zero and the lattice size to infinity."
Why do you think this is necessary? What, if at PLANCK-level you can see this:
Preprint One Unique Possible Universe
It seems as if you actually can have a basic 3D-grid and nevertheless have a continuum at VEV-level and above...
...the special point seems to be, that there exists a smallest wave's period time, associated with PLANCK-time. This makes the related wave equation non-linear.
Dear Andreas Schwarz The reason it's necessary to perform calculations using many different lattice sizes and spacings and extrapolate to the limiting case is because nothing should depend on the choice of lattice size or spacing. If something non-trivial does turn out to occur at a particular scale, that scale will not depend on the lattice properties, either. How this occurs is known, when studying QCD, for instance. Lattice calculations provide much more than hints, it would be useful to study them. Cf. for instance M. Lüscher's lecture notes on QCD: https://luscher.web.cern.ch/luscher/lectures/LesHouches97.pdf and Rajan Gupta's lectures, Article Introduction to Lattice QCD
Some more recent lecture notes: https://indico.iihe.ac.be/event/946/contributions/2173/attachments/1748/2021/talk.pdf
The lattices in question-that are relevant for particle physics-are four-dimensional, not three-dimensional and what they allow the calculation of are the correlation functions that define the equilibrium properties of fields with a bath of quantum fluctuations, when gravitational fluctuations don't play a role, therefore the Planck scale doesn't play any role, either. The only scale that might be expected to emerge would be the Grand Unification scale.
The reason it doesn't make sense to study gravity using a lattice discretization (i.e. trying to discretize the Einstein-Hilbert action) is because the gauge group of gravity, that of the diffeomorphisms, is non-compact, so the measure over the group isn't well-defined. That's the problem of quantum gravity: To construct the theory, whose classical limit is invariant under diffeomorphisms. This doesn't have anything to do with the Brout-Englert-Higgs mechanism, however.
Dear Stam Nicolis ,
thanks a lot. The links are very useful lecture.
Generally, if you want to describe a medium which has a smallest wave period, naturally, i.e., not introduced by some explicit value which is experimentally measured: How can this be done?
My "Ansatz" was the 3D-grid, simulating a DIRAC, and analyzing the result: I got the smallest wave's period, naturally, in time-step units and related propagation in space-step units per time-step units. Principally, this could be scaled arbitrarily small, but I associated both with PLANCK-time and speed-of-light in vacuum.
How could such approach be done without discretization? Especially the smallest wavelength?
A medium that has a cutoff on the wavelength of a propagating wave has a dispersion relation that realizes this. The problem is that such a cutoff isn’t consistent with Lorentz invariance; indeed the lattice discretization breaks Lorentz (rather rotation invariance, in euclidian signature) to the hypercubic subgroup.
A way to achieve this in the continuum would be by either a dispersive medium or by boundaries, both of which, also, break translation invariance (since the medium defines a particular rest frame).
Dear Stam Nicolis ,
"The problem is that such a cutoff isn’t consistent with Lorentz invariance"
If this cutoff is very, very far away from the actually usefull zone, let's say by a factor of ≥5∙10¹⁶ (PLANCK-level from VEV-level), it should be approximately consistent to LORENTZ invariance, at least for all things we need to calculate, correct? Or is the error much larger than I expect?
It is, in the sense that Lorentz violation at high energies doesn’t remain limited to high energies. And that has been studied.
Dear Stam Nicolis ,
in the case of the wave equation, with a very small damping term added, makes it non-linear. This added term gets more and more neglectible as the wavelength increases.
I would expect that for wavelengths 10¹⁶ times longer than the cut-off, and for not too much wave periods, this term actually does not distort notibly.
For a PLANCK-oscillator, its redshift at distance of VEV-level is just about 10⁻¹⁷ - this is very less, but, if it is reflected to the oscillator's center, it is enough to generate a "beat" frequency period, exactly one PLANCK-time long, a constructive resonance, keeping the oscillator running for a while, as long as not disturbed...
For very much wave periods, i.e., for very long travel distance, there is a more and more significant effect of redshift z...
Thus, it seems that at VEV-level the distortions can be neglected, hence approximately LORENTZ-invariant, while at larger scales the redshift makes it more and more variant...
Do you agree?
The problem with such arguments is that there must be prefaced with a reason about the term that breaks global Lorentz invariance. And such a reason is known: It's called gravity, where global Lorentz invariance becomes a local symmetry. The reason, indeed, why it is a good approximation not to take into account spacetime curvature effects at the LHC is because the energy of particles at the LHC is about 10^4 GeV, the Planck scale is about 10^19 GeV so the RHS of Einstein's equations differs from 0 by 10^(-15), which quantifies why flat spacetime is an excellent approximation when studying phdnomena using the LHC.
Dear Stam Nicolis ,
"It's ... gravity, where global Lorentz invariance becomes a local symmetry."
"...flat spacetime is an excellent approximation when studying phenomena using the LHC."
Thanks a lot! Since my idea of the non-linear medium is the basic mechanism of gravitation, the gravitation-related invariance get into action not before VEV-level (LHC-level), the smallest particle-relevant oscillation with the highest energy.
Therefore, this model of a non-linear medium is actually suitable and "an excellent approximation".
Dear Stam Nicolis ,
... additionally, usually, if accelerating an oscillator, LORENTZ is out of order, nevertheless.
Thus, if the center of an oscillator is PLANCKian, that does not cause a problem, also.
These issues, however, don't have anything to do with the Brout-Englert-Higgs mdchanism, nor with any other ways by which particles can acquire mass. Furthermore, the Planck units are just the result of dimensional analysis-there's no evidence that something particular occurs at the Planck energy.
Andreas Schwarz, Stam Nicolis
You can read about the relativistic mass as a characteristic of an inertial manifold on page 8 (equation 1.1.7)
Research Proposal MATHEMATICAL NOTES ON THE NATURE OF THINGS
Once more: That (inertial) mass is one of the invariants that label relativistic objects-particles or not-is known for more than a century. That the symmetries of the weak interactions imply that the W and Z bosons and the leptons must be massless is known for more than sixty years. That the Brout-Englert-Higgs mechanism can describe how these particles acquire their mass, thanks to their interactions with a scalar field, is known for as much as long. That the quarks, even though their weak interactions can't be neglected, when studying the weak interactions of the leptons, get most of their mass from the strong interactions is known for more than fifty years (they get a small part of their mass from the interaction with the Higgs through their weak interactions). So it would be a good idea to study the technical properties of the Standard Model, that are now well-established, both from the theoretical and the experimental side. There are many lecture notes on the subject and there's no excuse not studying them.
Before the discovery of the Higgs boson there was considerable activity in trying to replace it by bound states of new fermions,whose scalar condensates could, in principle, play the role of the Higgs boson (these are the so-called ``technicolor'' theories).However it turned out that such constructions inevitably predict processes that haven't been observed. Since its discovery, therefore, people have been forced to deal with the fact that a spin-0 particle can exist at the same energies as the others, without being, inevitably, composite. That's a very brief summary of the state of the art.
Stam Nicolis, I'm more interested in where the mass comes from, not where the mass in the standard model comes from.
Igor Bayak The Brout-Englert-Higgs mechanism is about where the mass of the one-particle excitations of gauge fields and of matter fields comes from, when the symmetries forbid the appearance of mass terms. It's not limited to the Standard Model.
If you're not interested in that, there's no point in the discussion.
Alternatives to this mechanism describe the scalar field as a composite object, that's the only difference.
Stam Nicolis,
However, I am interested in where the standard model of elementary particles comes from.
The Standard Model of particle physics comes from its symmetries, global Lorentz invariance and gauge invariance under the Lie group SU(3) x SU(2)LxU(1)Y. The symmetries imply the dynamics. For more, on how these ``internal'' symmetries were discovered, read here: https://cds.cern.ch/record/2217096/files/9789814733519_0002.pdf
and here: https://arxiv.org/pdf/hep-ph/9210207
In fact one shouldn't confuse history-how the Standard Model was discovered-with logic. Logically one starts with the symmetries and they determine the dynamics and then one can set up experiments that can test this. The Standard Model is the starting point for understanding the subatomic world, up to a certain energy scale (or down to a certain length scale) not the end point!
To understand where the Standard Model comes from means understanding what ``beyond the Standard Model'' means, what ``grand unification'' might mean. For the moment the only hints for physics beyond the Standard Model are the fact that neutrinos are massive and the existence of dark matter. However these discoveries don't imply enough constraints for the theory of which the Standard Model is a part. Experiments aren't yet focused enough-there are many ways to describe the extensions theoretically.
Dear Stam Nicolis ,
"For the moment the only hints for physics beyond the Standard Model are the fact that neutrinos are massive and the existence of dark matter."
Very interesting: please offer a document which shows this. And please not those, which just suppose that it should exist... Are there finally any experimental results?
There have been experimental results on this for decades now. The evidence for the existence of dark matter is inferred from the motion of ordinary matter. This motion implies either the existence of a new form of matter-called dark matter, since it can't carry electric, magnetic or color charge, like known matter; whether it can carry weak charge hasn't been conclusively ruled out yet-or a modification of general relativity that would imply violations of the equivalence principle. It is, apparently, still possible to parametrize these in such a way as to keep them below experimental precision, but, at this point, this is just an equivalent way of describing the same thing.
Dear Stam Nicolis ,
galaxy rotation curves I knew and wanted to be excluded from the answer.
What I wanted to know whether there actually are "proves" for "matter", behaving as "Dark Matter" - and as I assumed, there are none, just the possibilities, "if there were such matter, then ...".
To my opinion, the latter point of your statement fits:
"... a modification of general relativity that would imply violations of the equivalence principle"
The problem is due to that the intrinsic mechanism of gravitation is not yet fully understood (no "Grand Unification" yet) and thus, GR actually will get a correction to cope with that, meanwhile keeping the equivalence principle.
We'll see...
In addition to physical phenomena that go beyond the standard concepts (dark matter, etc.), there is (for some people) dissatisfaction with the standard concept itself. Therefore, the highest achievement would be the derivation of standard representations (not only quantum, but also classical) from the first principles. At least, this is exactly the purpose of the monograph that I recommend to you.
Preprint Mathematical Notes on the Nature of Things (fragment)
Andreas Schwarz If the particle content of dark matter were known, there wouldn't be any issue. This does not mean that the rotation curves of galaxies aren't appropriate for inferring the existence of dark matter-just as they're useful for inferring the fact that galaxies are, also, made up of ordinary matter.
Grand Unification does not refer to gravity, but to the internal symmetries-that are, still, not understood within a unified framework.
Classical gravity is understood and the fact that it describes spacetime singularities can be deduced from the property that it's a gauge theory, whose gauge group is noncompact. This doesn't affect the properties of matter, nor does it affect the properties of the spacetime geometry in a way that would be relevant for describing what can be described in terms of dark matter. It is known that the most general formulation of gravity is supergravity. What's not known, regardng gravity, is how scalar-tensor theories can be described as decoupling limits of supergravity.
Igor Bayak ``First principles'' doesn't mean anything by itself. The Standard Model is, almost, determined by its symmetries and everything follows from that, quantum and classical, along with the non-relativistic limit.
What isn't fixed by the symmetries is the number of families and the representations of the internal symmetry group. The representations of the internal symmetry group are constrained only by the condition of anomaly cancellation in the electroweak sector (that is realized within each family).
Stam Nicolis
Symmetries can follow from first principles. For example, an inertial manifold (a set of limit cycles of a dynamical system) in the form of a torus covering a sphere without polar caps has symmetries of the SU(2) group. As for particle families, they can also follow from first principles. For example, particle families could be interpreted as a bunch of inertial manifolds.
Not all inertial manifolds have SU(2) symmetry, the gauge group of the Standard Model isn't SU(2) and the number of inertial manifolds is as arbitrary as the number of families. So no first principles here.
There''s an infinite number of Lie groups and an infinite number of representations within each. Trying to deduce the Standard Model from this is pointless.
Stam Nicolis
I gave you a link, but you ignore it. But in vain. If you had gone through the text, you would have noticed that the inertial manifold is the source of the Newtonian gravitational potential.
The Newtonian gravitational potental is (a) the non-relativistic approximation to the spacetime description of general relativity, whose gauge group is that of diffeomorphisms of four-manifolds in Lorentzian signature and (b) completely irrelevant for describing non-gravitational interactions, among other reasons because of the equivalence principle.
Minkowski space also follows from a dynamical system in a metaphysical extension of Euclidean space. See equation 1.1.7 and 1.1.8 on page 8. However, I am sure that a pseudo-Riemannian manifold can also be obtained by considering a dynamical system filled with inertial manifolds (particles).
Preprint Mathematical Notes on the Nature of Things (fragment)
apologies in advance if the below is a repea;
The Higgs mechanism is the standard method used in the Standard Model of particle physics to give mass to the W and Z bosons, as well as to provide the framework for the masses of fermions. It involves the spontaneous breaking of gauge symmetry via the Higgs field, a scalar field that acquires a nonzero vacuum expectation value.
However, there have been alternative approaches proposed to address the origin of mass and gauge symmetry breaking that do not rely on the Higgs mechanism or that propose different frameworks for symmetry breaking.
1. Technicolor
Overview
Technicolor is an alternative model where a new strong interaction, similar to the strong force in Quantum Chromodynamics (QCD), is responsible for the symmetry breaking. Instead of the Higgs field, massless fermions called technifermions form bound states (technimesons) that act similarly to the Higgs boson. These bound states acquire mass dynamically due to the strong force, breaking the electroweak symmetry.
Challenges
Technicolor models typically struggle to explain the large mass of the top quark and often predict additional particles that have not been observed. They also face difficulties in producing realistic mass spectra for quarks and leptons.
2.Extra Dimensions
Overview
Theories with extra spatial dimensions, such as models involving large extra dimensions or warped extra dimensions (e.g., the Randall-Sundrum model), suggest that the mechanism for electroweak symmetry breaking could originate from higher-dimensional interactions. In some of these models, the Higgs boson could be a manifestation of a component of a higher-dimensional gauge field or a brane fluctuation.
Challenges
These models often predict deviations from the Standard Model that have not been observed experimentally, such as new particles or modifications to gravitational forces at small scales.
3. Composite Higgs Models
Overview
In composite Higgs models, the Higgs boson is not an elementary particle but rather a bound state of more fundamental particles. This idea is somewhat similar to technicolor but typically involves a new confining interaction at a lower scale, which can naturally produce a lighter Higgs boson.
Challenges
These models must carefully explain the observed properties of the Higgs boson, such as its relatively low mass, without introducing fine-tuning.
4. Gauge Symmetry Breaking via Non-Compact Groups
Overview
The idea of using non-compact groups, such as \( SU(3) \times SL(2, \mathbb{C}) \times U(1) \), instead of the standard compact gauge groups, is a more speculative approach. Non-compact groups can lead to different symmetry-breaking mechanisms, where the usual scalar field Higgs mechanism might be replaced by other means, such as gravitational effects or dynamical symmetry breaking in a non-compact space.
Challenges
The use of non-compact groups presents mathematical and phenomenological challenges, including ensuring the theory remains renormalizable and physically consistent. Furthermore, non-compact groups can lead to negative-norm states (ghosts) that are problematic for unitarity.
5. Dynamical Symmetry Breaking
Overview
Some models propose that the breaking of gauge symmetries happens dynamically through the vacuum structure of the theory, without needing a fundamental scalar field. Examples include models where gauge interactions become strong at low energies, leading to the formation of condensates that break the symmetry.
Challenges
Dynamical symmetry breaking models must explain how to generate the correct mass spectrum for particles while avoiding unwanted massless particles or other inconsistencies.
6. Conformal Symmetry and Scale Invariance
Overview
In some approaches, the idea is that there is an underlying conformal symmetry or scale invariance that is spontaneously broken, giving rise to masses for particles. In such models, the Higgs boson could be an emergent degree of freedom associated with the breaking of scale invariance.
Challenges
These models must account for the observed scale of electroweak symmetry breaking and ensure that they remain consistent with precision electroweak measurements.
Research on Non-Compact Groups and Gauge Theories
There has been some theoretical exploration of using non-compact groups like \(SL(2, \mathbb{C})\) as part of gauge symmetries. However, these approaches are not widely adopted due to the difficulties in constructing consistent and phenomenologically viable models. They often face issues with unitarity, renormalizability, and consistency with known experimental data.
Harri Shore
Thanks for the thorough analysis of alternative approaches to the Higgs mechanism. However, in my opinion, first of all it is necessary to grasp the alternative of a material point in the form of an inertial manifold. See equation 1.1.4 and 1.1.6 on pages 7,8.
Research Proposal MATHEMATICAL NOTES ON THE NATURE OF THINGS
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