Obviously, it is necessary to start modeling the standard model by modeling its global symmetries. And since the basic symmetry group of the standard model is SU(3)xSU(2)xU(1), we will first model this group. This problem has a solution if, as the basic element of our constructor, we use generators of linear vector fields tangent to the product $S^{1}\times S^{3}$, in which the radius of the circle is equal to the radius of a three-dimensional sphere. However, we also need to simulate symmetry breaking, so we propose to change the geometry of the product so that the radii of the circle and sphere differ. Then the group of global symmetries of the standard model must be constructed from vector fields tangent to the product $S^{1}\times S^{3}\times s^{3}$, where R(S^3)>R(s^3).
I suggest that a rare participant in the discussion delve into the topic
Preprint Mathematical Notes on the Nature of Things (fragment)
S(U(3)×SU(2)×U(1) is the group of local, not global, symmetries of the Standard Model. How to model its dynamics is known, using a lattice rrgularization.
Stam Nicolis ,
Globality in this case means that we do not bind these symmetries to a point in space-time, so we do not need to talk about locality and dynamics yet. As always, you take the discussion aside.
Wrong, the gauge symmetries refer to spacetime and locality is one of the fundamental properties of the Standard Model. It would be a good idea to study quantum field theory, before mixing up global and local symmetries and making trivially wrong statements.
The global symmetry of the Standard Model is invariance under Lorentz transformations, rotations and translations of the spacetime coordinates.
Calculations can be done in perturbation theory about free fields and, beyond perturbation theory, using a lattice regularization and numerical methods.
The symmetries-along with the requirement that the equations of motion contain only up to two derivatives, with respect to spacetime, completely define the dynamics. Once upon a time this wasn't understood, it's been many years since it is.
I repeat once again - globality in this case means only that this group operates everywhere and always, that is, at any point in Minkowski space. Localization of the Lagrangian according to this group is the second question, so first I'm talking about modeling the group, and then we'll talk about modeling dynamics. And don't lecture me.
Which representations of the Lorentz group and which representations of the gauge group are needed for the Standard Model, is, also, known. How to calculate anything, also. There’s nothing to rediscover, it’s in textbooks, which it would be a good idea to read.
The question is not about the representations of the Lorentz group and the group of internal symmetries, but about the origin of these groups.
The symmetries (groups and representations) along with the requirement that the Lagrangian not contain terms with more than two derivatives, completely determines the Lagrangian, therefore the classical dynamics-and the rules of perturbation theory determine part of the quantum dynamics. The dynamics beyond perturbation theory can be and is studied on the lattice. That's it, there's nothing more to add.
The symmetries define the Lagrangian, but where exactly do these symmetries come from, that's the question.
While S1 is the group manifold of U(1) and S3 is the group manifold of SU(2), the group manifold of SU(3) is harder to describe. The way to do this is known, however, since the invention of lattice gauge theory: The gauge fields are group elements, Uμ(n)=exp(iθa(n)Ta), where the Ta are the Lie algebra generators, for U(1) (the identity), for SU(2) (the Pauli matrices) and for SU(3) (the Gell-Mann matrices), associated to the links of the lattice from node to node n+μ (in direction μ). The matter fields are associated to the nodes of the lattice.
The symmetries come from deductions from experiment. It's possible to write down an infinite number of mathematically consistent quantum field theories, the Standard Model is the particular quantum field theory that describes the matter content of our Universe, as deduced by experiment. And there's a constant exchange between theory and experiment, of course, since individual experiments can only provide partial information.
There is ``the" Standard Model (there's just one); there isn't ``a'' Standard Model.
Thought experiments are what theorists do. These lead to infinitely many quantum field theories and how to organize them is, also known: By the spacetime symmery group (which for relativistic theories is the Poincaré group) and the internal symmetry group-which is where work is done. That's how extension of the Standard Model are studied, where the internal symmetry group of the Standard Model is embedded in a larger group. How this can be done is known, what isn't known is how to organize the calculations in a particularly efficient way. That's where effort is focused.
Stam Nicolis : That's how extension of the Standard Model are studied, where the internal symmetry group of the Standard Model is embedded in a larger group.
Again, you're taking the question aside. It is not a question of including a group of internal symmetries in some large group, but of getting it from the first principles (geometry plus dynamics).
It's not possible to get a particular group from first principles, which, for the internal symmetry group can be stated as the cancellation of gauge (not global!) anomalies. There are infinitely many anomaly-free groups and/or anomaly-free representations. The Standard Model describes one particular choice, but there's nothing special about it, from the theoretical side; infinitely many other choices would be possible. It's an experimental fact that the Standard Model is described by the particular gauge group, with matter in a particular representation of this group.
Once you know part of the theory, it is possible to deduce-part of-the rest: So, once anomaly cancellation was understood, this implied that the discovery of the tau lepton implied the existence of the tau neutrino-though not its mass or its mixing angles. And it was possible to deduce the existence of a third family of quarks and their charges; though not their masses.
Similarly, it was possible to put some bounds on the mass of the Higgs boson, to know that it must be possible to discovery it-or the particles that realized the Brout-Englert-Higgs mechanism-at the LHC, but not its exact mass or self-coupling. Its discovery does imply that additional particles must exist; but their properties are not specified and models are required for concrete predictions to be made.
There is nothing impossible here. From the first principles, one can obtain not only the group of internal symmetries, but also the Lorentz group. It is enough to take a close look at the dynamic flows of the seven-dimensional sphere.
Of course it's not. There's nothing special about the seven-dimensional sphere, beyond the fact that it admits a Hopf fibration. And that doesn't have anything to do with the mathematical consistency of quantum field theories, relativistic or not. There are infinitely many manifolds in any dimension, each with its isometry group. Any one is as good as any other, mathematically.
The Lorentz group is the symmetry group of four-dimensional Minkowski spacetime. There's nothing special about it from the point of view of mathematics (it's the solution to Einstein's equaions with zero cosmological constant, so what? Einstein's equations admit solutions with positive as well as with negative cosmological constant) and there are infinitely many Lie algebras that can describe the charges of particles, created by fields on such a spacetime, or any other.
The symmetries come first and they determine the dynamics. And which symmetry groups are relevant for describing natural phenomena is the result of experiment. Since our universe is definitely not seven-dimensional, focusing on the seven sphere doesn't have anything to do with the Standard Model.
The Hopf bundle has nothing to do with it. I use another bundle, which is obtained as a result of mapping the hypersphere of an 8-dimensional space with a neutral metric onto the sphere of an 8-dimensional Euclidean space. You obviously haven't read the fragment
Preprint Mathematical Notes on the Nature of Things (fragment)
Once more, nothing singles out eight-dimensional space from any other space. It's possible to solve similar exercises in any dimension. They still don't have anything to do with the Standard Model.
There's nothing mathematically exclusive about our Universe, it's just one possibility among infinitely many, equally mathematically consistent, possibilities. Only expriment allows finding which possibility is realized.
Newtonian mechanics isn't mathematically inconsistent and there's no a priori reason why electrodynamics must be invariant under Lorentz transformations. It is possible to include terms that violate Lorentz invariance, in a mathematically consistent way. It's experiment that implies that these terms are absent.
An 8-dimensional space with a neutral metric is solely because its hypersphere of unit radius is homeomorphic to $R^4\times S^3$, and the Lie algebra of linear vector fields tangent to it is isomorphic to $sl(4,C)$, that is, to the Lie algebra of the Dirac matrix algebra.
So what? It's possible to obtain the same structures in infinitely many different ways. And, once more, the Dirac matrix algebra doesn't mean anything by itself. And it can be defined in dimensions other than four.
I'm simply stressing that it's wrong to claim that our Universe is uniquely singled out by certain mathematical structures, because these structures aren't unique. It's just not true that only our Universe can be described in a mathematically consistent way and any other can't.
Infinitely many Universes can be described in a mathematically consistent way; which one among them is ours can't be deduced without conducting experiments, which serve, precisely, this purpose, of eliminating what's, simply, possible. What theory does is set up the possibilities in a way that experiments can eliminate them.
So there is a need for sharp predictions from theory-and for sharp experiments, that can distinguish between different, mathematically consistent, possibilities.
The reason Newtonian mechanics was found insufficient for describing natural phenomena wasn't its mathematical inconsistency; it was because its symmetries-described by the Galilean group-weren't consistent with Maxwell's equations, which are invariant under the Lorentz group. And it was experiment-among them the experiment of Michelson and Morley-that hinted that the additional terms and/or deformations that would make Maxwell's equations invariant under Galilean transformations, were, in fact, absent. Since that time it has been possible to sharpen the experiments and now it's routine to test for violations of Lorentz invariance.
And it is now understood that Newtonian mechanics is a particular limit of relativistic mechanics. Just as it's understood that relativistic mechanics is a particular limit of general relativity and that classical physics is a particular limit of quantum physics and so on.
What isn't known is what is the quantum theory, whose classical limit is described by general relativity and its generalizations. Such a quantum theory would be relevant for describing the pre-inflationary universe and the near horizon geometry of black holes. Now it's very difficult to perform experiments in these régimes.
Leave any other universe to those who live there, and we will have to study the one we have. For example, we have Euclidean space and Hilbert space, and personally I am very interested in the first principles from which they are formed.
They’re defined axiomatically. And the consistency of the axioms doesn’t have anything to do with our Universe. Whatever the value of Newton's constant and of the cosmological constant, doesn't affect what Euclidean space or Hilbert space is. Nor does the matter content. Beings in other Universes will come upon the same notions, just the names will change.
The Universe we have isn't distinguished from any other possible Universe by specific mathematical structures as such, because the notions of Euclidean space and of Hilbert space don't single out our Universe from infinitely many others.
And what if the axioms themselves are a consequence. Have you considered this option?
They're not and this can be tested, because it's possible to construct more than one consistent axiomatic systems. It is possible, once more, to describe consistently, universes that are different from ours. Ours isn't special, from the mathematical point of view.
That the spacetime geometry of our Universe is described by de Sitter spacetime doesn't mean that Einstein's equations don't admit other solutions. Nothing would have changed for mathematics, had it turned out that the cosmological constant were zero, or negative, instead of positive. And nothing did change-for mathematics-when the cosmological constant was measured.
Here's another question - where do Einstein's equations come from? And they follow (as I think) from the condition of minimality (stability) of flows on a seven-dimensional sphere.
A brief formula for the theory of everything:
Classical physics = the principle of least action = the principle of least path on the seven-dimensional hypersphere of space with a neutral metric.
Quantum physics = the principle of the minimum phase = the principle of the most probable path wound on a seven-dimensional sphere of space with a Euclidean metric.
Einstein’s equations don’t have anything to do with the seven-sphere in particular. They’re the only equations that contain at most two derivatives and are invariant under general coordinate transformations=diffeomorphisms.
Once more, the seven-sphere doesn’t have anything particular that distinguishes it from any other shape.
Classical physics doesn’t single out the seven-sphere and quantum physics doesn’t either.
When gravitational effects can be neglected, spacetime is flat, with isometty group the Poincaré group. In this spacetime the properties of matter are described by a relativistic quantum field theory-but this isn’t sufficient to fix the internal symmetry group, beyond the requirement that it be anomaly-free. The Standard Model is one solution.
When gravitational effects can’t be neglected, it’s not completely understood what happens, because it’s not known how to describe the effects of quantum matter on the spacetime geometry.
Nor is it known how to describe quantum fluctuations of spacetime geometries, because the group of diffeomorphisms is a noncompact group.
The claim that the Standard Model can be described by gauge fields taking values in SO(8)-which is the translation of the statement that the group manifold is the seven-sphere, is, simply, wrong-it suffices to write down the action and compute the consequences. And, once more, there’s nothing singling out SO(8) from SO(n).
It is possible to embed the Standard Model group in SO(8)-and obtain a so-called ``Grand Unified Theory'' that way. But it will have all sorts of problems, that are known.
Once the Standard Model gauge group is known, it's straightforward to embed it in any bigger group. But the Standard Model gauge group isn't the only group of rank 5. Amd there's no mathematical reason that it must be of rank 5 at all.
You are inattentive, we do not put a group of internal symmetries into a large group, so forget about SO(8). With us, the dynamics of flows determines the geometry of the product of spheres, which in turn defines a group of internal symmetries.
And, once more, (a) there's nothing unique about that and (b) the group manifold of SU(3) isn't a sphere. So the claim that a product of spheres is something inevitable is wrong.
The statement that the dynamics of flow determines the geometry as that of a product of spheres represents a choice.There are plenty of flows that don't have anything to do with the geometry of products of spheres-and the Standard Model is one of these. Not all compact manifolds are spheres. SU(3) is one such example.
You start to fantasize. Have I mentioned group manifold somewhere? As for the generation of SU(3) by the algebra of linear vector fields, see chapter 3 at the link
Research Proposal MATHEMATICAL NOTES ON THE NATURE OF THINGS
Once more: The linear vector fields that describe the flow on spheres are the generators of the Lie algebras SO(n), for any n. One can, also, consider products, SO(n)xSO(m)x... and so on.
They don't have anything to do with the Standard Model.
The linear vector fields on SU(3), defined by linearizing exp(iθλ), where λ are the generators of SU(3), do not describe the flow on a sphere, because the group manifold of SU(3) isn't a sphere. The action of SU(3) doesn't leave any sphere invariant. Because SU(3) isn't isomorphic to SO(n) for any value of n.
SU(2) isn't isomorphic to SO(3), the isometry group of the 2-sphere, nor to SO(4), the isometry group of the 3-sphere. Nor is U(1) isomorphic to SO(2).
The relation is more complicated. So it's not true that the gauge group of the electroweak sector of the Standard Model can be written as SO(3) x SO(2). That's wrong. And the leptons don't belong to representations of SO(3) x SO(2), either, nor does the Higgs.
The group manifold of U(1) is the circle S^1 and the group manifold of SU(2) is the 3-sphere, S^3, so the group manifold of SU(2)×U(1) is S^3 × S^1. However they have unit radius each. It is not true that the BEH mechanism affects the radii of the gauge group manifolds.
You're fantasizing again. Where did I mention the group SO(3) x SO(2)? I said that SU(2) is generated by linear vector fields tangent to the spheres S^3, and these are not rotations from the group SO(3) or SO(4), but paired rotations in R^4 tangent to the torus S^1\times S^1 lying on the sphere S^3.
However, in the Standard Model, the U(1) factor is not the gauge group of electromagnetism, but that of weak hypercharge. SU(2) x U(1) has four generators, two are linear combinations of the Z and the photon. So when SU(2)×U(1) breaks to U(1), the initial and final U(1)s are not identical.
Math is different from physics. It's not possible to deduce what the gauge group of the Standard Model is by mathematical criteria: It's only possible to deduce the consequences, upon assuming a gauge group and an anomaly-free representation (which, for the Standard Model, requires taking into account quarks AND leptons) using mathematics.
I repeat. Describing a flow using a group manifold and generating a group by a flow (using tangent vector fields) are two different things.
I agree, mathematics is not physics. However, try to understand the structure of the electron without mathematics.
The properties of the electron-that it belongs to a particular representation of SU(2)_L x U(1)_Y and of the Poincaré groups-are inputs, not outputs.
Because, for the umpteenth time, there are infinitely many mathematically consistent quantum field theories and the Standard Model is but one of them-that one that is picked out by performing experiments.
There's no mathematical reason why weak interactions have to break parity and it would have been possible to cancel the chiral anomalies using representations of SU(2), not SU(2)_L; it's experiment that showed that the electrons emitted during beta decay are only left handed. There's no mathematical reason why the gauge group of the strong interactions must be SU(3).
Indeed there's no mathematical reason why the known properties of matter are described by the gauge group of the Standard Model and not by any other (and that includes electromagnetism and the weak interactions):
There's no mathematical reason why there are three non-gravitational interactions, with the properties they have, and not less or not more.
A Lie group is determined by the Lie algebra through exponentiation. There's no difference between describing a group-which is a manifold-by a flow and describing the group by its algebra, which defines the ``linear vector fields'' on the group manifold. Writing a differential equation and writing its solutions are two equivalent ways of representing the same functions and which one uses is, just, a matter of taste.
Once upon a time, maybe, (some) people thought these were distinct subjects. They aren't.
The theory of Lie groups is a subject in mathematics. One can study flows on the group manifolds of Lie groups. That by itself doesn't imply anything about the Universe we live in. There's no mathematical reason why spacetime must be described by the group of diffeomorphisms-which, in turn, doesn't imply that only the metric describes the spacetime geometry, or that additional fields are required-nor why matter-that can be created by an expanding Universe-must be described by particles that carry charges under the representation of a particular Lie group, rather than of any other.
Indeed there's no mathematical reason why the Universe must be expanding at all, nor that the expansion must be accelerating. Einstein's equations have solutions in any case-it's experiment that determined that the Universe is expanding and that the expansion is accelerating.
The electroweak gauge group, which decreases to the hypercharge group, is modeled using the Lie algebra of vector Killing fields of a 6-dimensional space with metric (4,2), where hyperbolic rotations changing the radii of the spheres of the product S^3xS^1 are responsible for the Higgs mechanism. As for the impossibility of a mathematical description of an electron, I disagree. You haven't tried knitting knots on the product of spheres yet.
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