The (internal) geometry of the Standard Model is that of its Lie group, namely SU(3) x SU(2) x U(1). In perturbation theory about free fields, the fields take value in the algebra, beyond perturbation theory they explore the full group manifold. This is the way the dynamics is described in lattice formulations of the Standard Model.

This group manifold exists at every point in spacetime and it is known that spacetime transformations commute with the transformations that are symmetries of this manifold.

The statements about minimal length closed lines are, simply, wrong. The correct statements are much simpler, in fact, so there's no point in wasting time and effort doing wrong things.

The gauge group of the Standard Model is an input for describing the dynamics of the known particles, not an output. There are uncountably many Lie groups-and the corresponding field theories, based on them-in mathematics; Nature turns out to be described by the Lie groupSU(3) x SU(2) x U(1) (along with the direct product with the Lorentz group); moreover the known particles realize a very specific representation of that group. That's a discovery, not an invention.

Stam Nicolis > This group manifold exists at every point in spacetime and it is known that spacetime transformations commute with the transformations that are symmetries of this manifold.

This is a primitive standard approach where the group is endowed with the property of a geometric object, while geometric objects that have group symmetries are more interesting.

Stam Nicolis > The statements about minimal length closed lines are, simply, wrong. The correct statements are much simpler, in fact, so there's no point in wasting time and effort doing wrong things.

And I like this idea and it does not seem wrong. Quite the opposite, in contrast to the standard approach involving the symmetry breaking mechanism, the idea of the correspondence between the mass of an elementary particle and its length, measured in 8-dimensional space with a neutral metric, is very promising and may allow solving the problem of the hierarchy of the masses of elementary particles.

All these are just words-what matters are the results. The excitations described by the Standard Model are point-like objects, so their length is zero. Their masses are Lorentz invariant quantities, whose values aren't affected by the gauge group, since the coupling constants are independent quantities.

The only meaning that can be assigned to the term ``very promising'' is if it does, indeed, lead to the prediction of any mass hierarchy, of which that of the Standard Model is but one example. And the answer is quite simply that it can't, because the way the mass hierarchy is described in a field theory in general and in the Standard Model in particular doesn't have much to do with the geometry of the gauge group. And the reason is very simple: In the unbroken phase the gauge bosons are massless, while in the broken phase they become massive in a way that isn't sensitive to the geometry of the gauge group. The fermions become massive through the Yukawa couplings that, likewise, aren't sensitive to this geometry.

All this is known.

Stam Nicolis > The excitations described by the Standard Model are point-like objects, so their length is zero. Their masses are Lorentz invariant quantities, whose values aren't affected by the gauge group, since the coupling constants are independent quantities.

This is all about the standard model, and I suggest that you rise above the standard model in order to then descend to it, but see something more. So boldly expand the geometry of space-time to a doublet of Minkowski spaces, measure the lengths of minimal closed curves twisted around the product of the Clifford torus and the time torus, and you will be happy.

I will try to "liven up" the discussion even more. Earlier I mentioned the compactified isotropic cone of Minkowski space wrapped around the product S3xS1. Here it was meant that an arbitrary pseudo-Euclidean plane is mapped onto a torus, in which the defining circles of the torus correspond to isotropic straight lines of the plane. With such a winding of the Minkowski space, its isotropic cone is mapped onto the product of the projective plane and the circle RP2xS1, and the symmetry group of this product coincides with the group SU(2)xU(1).

So what? There are many ways to describe SU(2)xU(1) and SU(3) x SU(2) x U(1), that don't have anything to do with the fact that this is the gauge group of a quantum field theory, whose excitations transform under certain very particular representations.

The mass hierarchy of the particles of a quantum field theory is not determined by the symmetry group, unless the theory is integrable-and the Standard Model isn't integrable.

Stam Nicolis,

I understand that you are not interested in the geometric foundations of the Standard Model in my interpretation, so there is no need to repeat the same thing several times. Whoever does not want to go beyond the standard model, let him remain in place, and we will move on.

Well it would be useful to be able to ``move on'', by providing an example of the claims, namely a mass hierarchy in any gauge theory, that's determined in the way claimed. Such a claim, however, can't work in four spacetime dimensions for a non-integrable field theory, in general and for the Standard Model, in particular. One way to see this is by noticing that the SU(2) part of the Lorentz group refers to the rotation subgroup and doesn't have anything to do with the SU(2) gauge group of the Standard Model (or the SU(2) subgroup of any other gauge group). The action of the Lorentz group commutes with the action of the gauge group-that's been known since the work of Coleman and Mandula (their result refers to a non-trivial scattering matrix, that's why it doesn't constrain integrable theories, that don't display non-trivial scattering). That's why it can't lead to any constraints on the mass hierarchy-which people did study in the 1960s and was the motivation for Coleman and Mandula, in fact.

So any extension of the Standard Model is similarly constrained. The mathematical mistake that the Lorentz group may not commute with the gauge group, in four spacetime dimensions, can't lead to any physical consequences, however. So it might be a good idea to abandon it.

So, what can we say about the masses of fermions within the framework of the geometric interpretation of their closed curves on the product of the Clifford torus and the time torus? First, it should be noted that closed curves lying on a compactified isotropic cone (generating the gauge symmetry group) have zero length, and therefore can be associated with gauge bosons. Second, since for stable particles the time torus can be replaced by a circle, we will first look for minimal closed curves that entwine the product of the Clifford torus and the circle.

In addition, by studying the topology of nodes on the product of spheres S3 × S3 × S1, we can simplify this product to a 3-dimensional torus S1 × S1 × S1 without prejudice to understanding the topological properties of the node, and for clarity consider a closed ribbon lying on a 2-dimensional classical torus S1 × S1. Then, due to the fact that the node on the torus corresponds to the fundamental group of its complement, isomorphic to the corresponding braid group, it is quite fair to compare the elements of this group with the fundamental components of the elementary particle, as is done in the topological Bilson-Thompson model. Finally, note that developing this approach to the case of unstable particles of subsequent generations of fermions, it will be necessary to turn to the 4-torus.

Dear Igor Bayak

The question is, what is matter? what is particle? where is mass is coming and originate from? where the spherical shape is coming from? where the movement is originated from? when we done finding all these answers, then, we realize nothing in the universe is working mechanically (geometrically )

It is incorrect to count universe as a dumb, accidental entity, while this organization is holding over billions of galaxies and put them on right order.

Mr. Minkowski, or Einstein theory, is set for one solar system universe, where it is not even working with just our small solar system. We must look at universe much different from past century scientists.

https://www.academia.edu/43351041/How_the_Quantum_Universe_Works

regards

Javad Fardaei ,

You are right, geometry is just a foundation, and all the diversity of the universe in a variety of forms of moving matter. However, I do not raise these questions here. However, check out my essay

Research Proposal MATHEMATICAL NOTES ON THE NATURE OF THINGS

Let's go back to the problem of the fermion hierarchy. So, we have a model in which fermions can be represented by the minimal closed curves (torus knots) of the product of the Clifford torus and the circle. However, this is a static model that does not take into account the time component, and therefore, to pass to the dynamic model of fermions, it is necessary to consider fermions as torus knots in the product of the Clifford torus and the time torus. Whence the hierarchy of fermions in the form of three generations is a simple consequence of the hierarchy of torus knots on the torus of time in the form of a triad: the defining circle of the torus, the circle of Villarso and the trefoil knot. Thus, the stability of fermions depends on how the corresponding nodes are tied to the torus of time, the first generation of fermions moves along the defining circle of the torus of time, the second along the circle of Villarso, and the third along the trefoil node.

Developing the mathematical formalism of the geometric foundations of the standard model of elementary particles, it should be noted that the geometry of a compactified 8-dimensional space with a neutral metric naturally leads to the concept of a complex bispinor. Indeed, if we take pairs of coordinates from the Minkowski space and its dual space, which form a pseudo-Euclidean plane, wind this pseudo-Euclidean plane onto a torus, and then map the torus onto a sphere with punctured poles, then, as shown in

Preprint Chaotic dynamics of an electron

an arbitrary non-isotropic straight line of the pseudo-Euclidean plane corresponds to the certain circle sphere passing through the reference point of the sphere, and a certain complex number corresponds to the rotating circle of the sphere. Thus, four complex numbers can be interpreted as the coordinates of a rotating circle in a compactified 8-dimensional space with a neutral metric.

I proposed a hypothesis that fundamental particles are described by finite groups closely related to the symmetries of regular polyhedra. See article `Platonic Solids and Elementary Particles`

https://vixra.org/abs/1802.0246

Lev Verkhovsky ,

Thanks, that's a good observation. At the same time, let me note that a more rigorous presentation of the topic "group foundations of elementary particle physics" can be found in

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