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In Veltman's Diagrammatica, the full Lagrangian of the standard model is spelt out. This has around a hundred terms. This is way too many for even the most dedicated physicist (or physically inclined mathematician) to work on, except by piece by piece.

In the Connes-Lott-Barrett-Chamseddine model, based on non-commutative geometry, the standard model is derived from a spectral action with a simple geometric input, spacetime is multiplied with a 'fat' non-commutative point:

$\mathbb{C} \oplus \mathbb{H}_L \oplus \mathbb{H}_R \oplus M_3(\mathbb{C})$

It is zero-dimensional, classically but has KO-theory dimension 6. This reproduces the full standard model including the Higgs and neutrino mixing. It turns out that the bimodule over the dual of this fat point, which is the sum of all irreducible irreps of odd dimension gives one full generation of the fermions. It's worth noting that the spectral action is a generalisation of the Einstein-Hilbert action.

The theory is not limited to just the standard model. In a later paper with Chamseddine, Connes shows how NCG can model a grand unification theory like the Pati-Salam SU(5) GUT.

Although non-commutative geometry is a geometry without a geometry. I mean by this that they work with a non-commutative algebra to be thought of as the algebra of functions on a non-commutative geometry which has not bern constructed yet. I'm not so sure that this will be the case in the near future (or perhaps the far near future given the density of maths required to understand and work with NCG). One of the standard examples of a non-commutative space, according to Connes, is the irrational torus where classical tools do not give any information but non-commutative tools can. However, diffeology, which is a generalisation of classical differential geometry, does and it gives roughly similar results to that of Connes.

I'd also add that Mathilde Marcolli has elaborated an explanation of the fractional quantum Hall effect and which has taken its point of departure from Bellisard, van Elst & Schulz-Baldes The Non-Commutative Geometry of the Quantum Hall Effect (1994) and which she calls the earliest work on physics in NCG. They show that the magnetic field turns the Brillioun zone into a non-commutative torus.

In Veltman's Diagrammatica, the full Lagrangian of the standard model is spelt out. This has around a hundred terms. This is way too many for even the most dedicated physicist (or physically inclined mathematician) to work on, except by piece by piece.

In the Connes-Lott-Barrett-Chamseddine model, based on non-commutative geometry, the standard model is derived from a spectral action with a simple geometric input, spacetime is multiplied with a 'fat' non-commutative point:

$\mathbb{C} \oplus \mathbb{H}_L \oplus \mathbb{H}_R \oplus M_3(\mathbb{C})$

It is zero-dimensional, classically but has KO-theory dimension 6. This reproduces the full standard model including the Higgs and neutrino mixing. It turns out that the bimodule over the dual of this fat point, which is the sum of all irreducible irreps of odd dimension gives one full generation of the fermions. It's worth noting that the spectral action is a generalisation of the Einstein-Hilbert action.

The theory is not limited to just the standard model. In a later paper with Chamseddine, Connes shows how NCG can model a grand unification theory like the Pati-Salam SU(5) GUT.

Although non-commutative geometry is a geometry without a geometry. I mean by this that they work with a non-commutative algebra to be thought of as the algebra of functions on a non-commutative geometry which has not bern constructed yet. I'm not so sure that this will be the case in the near future (or perhaps the far near future given the density of maths required to understand and work with NCG). One of the standard examples of a non-commutative space, according to Connes, is the irrational torus where classical tools do not give any information but non-commutative tools can. However, diffeology, which is a generalisation of classical differential geometry, does and it gives roughly similar results to that of Connes.

I'd also add that Mathilde Marcolli has elaborated an explanation of the fractional quantum Hall effect and which has taken its point of departure from Bellisard, van Elst & Schulz-Baldes The Non-Commutative Geometry of the Quantum Hall Effect (1994) and which she calls the earliest work on physics in NCG. They show that the magnetic field turns the Brillioun zone into a non-commutative torus.

She has also published a recent book with Connes, titled Non-commutative Geometry, Quantum Fields and Motives. Here as a reviewer at the AMS says in 2007:

So the cat is out of the bag. What greater mathematical objective can there be to realise the RH [Riemann Hypothesis] for number fields and the RH for function fields as two sides of the same coin, the discriminators being as it were, algebraic geometry and (or versus) non-commutative geometry? This is manifestly one aspect of the rationale for what the entire sweeping programme is about, with a complementary aspect of quantum physics in its post-Feynman form.

What a wild, wild ride!

An understatement.

I imagine the latter is the reason behind another book that Marcolli has published, this time simply titled Feynman Motives.

In Veltman's Diagrammatica, the full Lagrangian of the standard model is spelt out. This has around a hundred terms. This is way too many for even the most dedicated physicist (or physically inclined mathematician) to work on, except by piece by piece.

In the Connes-Lott-Barrett-Chamseddine model, based on non-commutative geometry, the standard model is derived from a spectral action with a simple geometric input, spacetime is multiplied with a 'fat' non-commutative point:

$\mathbb{C} \oplus \mathbb{H}_L \oplus \mathbb{H}_R \oplus M_3(\mathbb{C})$

It is zero-dimensional, classically but has KO-theory dimension 6. This reproduces the full standard model including the Higgs and neutrino mixing. It turns out that the bimodule over the dual of this fat point, which is the sum of all irreducible irreps of odd dimension gives one full generation of the fermions. It's worth noting that the spectral action is a generalisation of the Einstein-Hilbert action.

The theory is not limited to just the standard model. In a later paper with Chamseddine, Connes shows how NCG can model a grand unification theory like the Pati-Salam SU(5) GUT.

Although non-commutative geometry is a geometry without a geometry. I mean by this that they work with a non-commutative algebra to be thought of as the algebra of functions on a non-commutative geometry which has not bern constructed yet. I'm not so sure that this will be the case in the near future (or perhaps the far near future given the density of maths required to understand and work with NCG). One of the standard examples of a non-commutative space, according to Connes, is the irrational torus where classical tools do not give any information but non-commutative tools can. However, diffeology, which is a generalisation of classical differential geometry, does and it gives roughly similar results to that of Connes.

In Veltman's Diagrammatica, the full Lagrangian of the standard model is spelt out. This has around a hundred terms. This is way too many for even the most dedicated physicist (or physically inclined mathematician) to work on, except by piece by piece.

In the Connes-Lott-Barrett-Chamseddine model, based on non-commutative geometry, the standard model is derived from a spectral action with a simple geometric input, spacetime is multiplied with a 'fat' non-commutative point:

$\mathbb{C} \oplus \mathbb{H}_L \oplus \mathbb{H}_R \oplus M_3(\mathbb{C})$

It is zero-dimensional, classically but has KO-theory dimension 6. This reproduces the full standard model including the Higgs and neutrino mixing. It turns out that the bimodule over the dual of this fat point, which is the sum of all irreducible irreps of odd dimension gives one full generation of the fermions. It's worth noting that the spectral action is a generalisation of the Einstein-Hilbert action.

The theory is not limited to just the standard model. In a later paper with Chamseddine, Connes shows how NCG can model a grand unification theory like the Pati-Salam SU(5) GUT.

Although non-commutative geometry is a geometry without a geometry. I mean by this that they work with a non-commutative algebra to be thought of as the algebra of functions on a non-commutative geometry which has not bern constructed yet. I'm not so sure that this will be the case in the near future (or perhaps the far near future given the density of maths required to understand and work with NCG). One of the standard examples of a non-commutative space, according to Connes, is the irrational torus where classical tools do not give any information but non-commutative tools can. However, diffeology, which is a generalisation of classical differential geometry, does and it gives roughly similar results to that of Connes.

I'd also add that Mathilde Marcolli has elaborated an explanation of the fractional quantum Hall effect and which has taken its point of departure from Bellisard, van Elst & Schulz-Baldes The Non-Commutative Geometry of the Quantum Hall Effect (1994) and which she calls the earliest work on physics in NCG. They show that the magnetic field turns the Brillioun zone into a non-commutative torus.

In Veltman's Diagrammatica, the full Lagrangian of the standard model is spelt out. This has around a hundred terms. This is way too many for even the most dedicated physicist (or physically inclined mathematician) to work on, except by piece by piece.

In the Connes-Lott-Barrett-Chamseddine model, based on non-commutative geometry, the standard model is derived from a spectral action with a simple geometric input, spacetime is multiplied with a 'fat' non-commutative point:

$\mathbb{C} \oplus \mathbb{H}_L \oplus \mathbb{H}_R \oplus M_3(\mathbb{C})$

It is zero-dimensional, classically but has KO-theory dimension 6. This reproduces the full standard model including the Higgs and neutrino mixing. It turns out that the bimodule over the dual of this fat point, which is the sum of all irreducible irreps of odd dimension gives one full generation of the fermions. It's worth noting that the spectral action is a generalisation of the Einstein-Hilbert action.

Although non-commutative geometry is a geometry without a geometry. I'm not so sure that this will be the case in the near future. One of the standard examples of a non-commutative space, according to Connes, is the irrational torus where classical tools do not give any information but non-commutative tools can. However, diffeology, which is a generalisation of classical differential geometry, does and it gives roughly similar results to that of Connes.

The theory is not limited to just the standard model. In a later paper with Chamseddine, Connes shows how NCG can model a grand unification theory like the Pati-Salam SU(5) GUT.

In Veltman's Diagrammatica, the full Lagrangian of the standard model is spelt out. This has around a hundred terms. This is way too many for even the most dedicated physicist (or physically inclined mathematician) to work on, except by piece by piece.

In the Connes-Lott-Barrett-Chamseddine model, based on non-commutative geometry, the standard model is derived from a spectral action with a simple geometric input, spacetime is multiplied with a 'fat' non-commutative point:

$\mathbb{C} \oplus \mathbb{H}_L \oplus \mathbb{H}_R \oplus M_3(\mathbb{C})$

It is zero-dimensional, classically but has KO-theory dimension 6. This reproduces the full standard model including the Higgs and neutrino mixing. It turns out that the bimodule over the dual of this fat point, which is the sum of all irreducible irreps of odd dimension gives one full generation of the fermions. It's worth noting that the spectral action is a generalisation of the Einstein-Hilbert action.

The theory is not limited to just the standard model. In a later paper with Chamseddine, Connes shows how NCG can model a grand unification theory like the Pati-Salam SU(5) GUT.

Although non-commutative geometry is a geometry without a geometry. I mean by this that they work with a non-commutative algebra to be thought of as the algebra of functions on a non-commutative geometry which has not bern constructed yet. I'm not so sure that this will be the case in the near future (or perhaps the far near future given the density of maths required to understand and work with NCG). One of the standard examples of a non-commutative space, according to Connes, is the irrational torus where classical tools do not give any information but non-commutative tools can. However, diffeology, which is a generalisation of classical differential geometry, does and it gives roughly similar results to that of Connes.