Compare · A Theory of Everything

Spectral Pati-Salam Unification vs Connes-Chamseddine Spectral Action

← Back to Spectral Pati-Salam Unification

Non-Commutative Geometry· within family

	Spectral Pati-Salam Unification Frontier	Connes-Chamseddine Spectral Action Frontier
Proposed	2013	1996
Key figures	Ali Chamseddine, Alain Connes, Walter van Suijlekom	Alain Connes, Ali Chamseddine
In one sentence	Chamseddine, Connes, and van Suijlekom showed in 2013 that replacing the Standard Model internal algebra with the next simplest algebra compatible with the spectral triple axioms yields a Pati-Salam model with gauge group SU(4) x SU(2)_L x SU(2)_R, unifying quarks and leptons without invoking a larger simple group like SU(5) or SO(10).	Connes showed in 1996 that a noncommutative algebra encoding spacetime plus the Standard Model's internal symmetries, plus a single Dirac operator acting on it, automatically produces the bosonic Lagrangian of the Standard Model and general relativity when the spectral action is evaluated.
Predictions	The framework predicts an intermediate stage where the Pati-Salam symmetry breaks down to the Standard Model. The spectral geometry does not fix the energy at which this happens, that scale is a free input, but once it is chosen, the masses of the extra gauge bosons (the heavy SU(4) and SU(2)_R carriers) are set, so finding or ruling out those particles at a given energy tests the choice Leptoquark gauge bosons with specific quantum numbers under SU(4) couple quarks and leptons in the Pati-Salam manner and are accessible at colliders if the breaking scale is near the TeV range A right-handed neutrino gauge field under SU(2)_R is present, producing an additional source of Majorana mass distinct from the see-saw mechanism of the 2007 variant Protons should be far more stable than the simplest competing grand-unified theory, minimal SU(5), predicts. The Pati-Salam structure avoids the leading process that lets protons decay in SU(5), so the two make sharply different forecasts. Large underground detectors that watch tanks of water for a single proton decay can test this: SU(5)'s shorter lifetime is already in tension with the data, while Pati-Salam's longer one is not yet reachable	The Standard Model gauge groups SU(3) x SU(2) x U(1) and the Higgs doublet are not independent choices but are fixed by the geometry of the finite noncommutative space, once that space is specified by the spectral triple axioms Gravitational and gauge couplings unify at the Planck scale as a consequence of the spectral geometry, providing a top-down constraint on the relationship between Newton's constant and the gauge coupling strengths The Higgs scalar is a fluctuation of the internal geometry rather than a fundamental particle in the traditional sense, and its potential is geometrically determined up to the scale of the internal space A see-saw mechanism for neutrino masses emerges naturally in the extended spectral triple that includes right-handed neutrinos in the Hilbert space, as shown in the 2007 Chamseddine-Connes-Marcolli revision
Where it breaks	The scale at which Pati-Salam breaks to the Standard Model is a free parameter in the NCG framework; there is no derivation from the spectral geometry of the breaking scale itself, only of the gauge structure and particle content above it No direct experimental evidence for Pati-Salam breaking has been found; leptoquarks predicted by the model have not been observed at the LHC through Run 3 within the minimal mass range the model targets The variant requires additional [[scalar-field\|scalar fields]] to break the gauge symmetry stepwise from SU(4) x SU(2)_L x SU(2)_R to the Standard Model; the scalar sector introduces new parameters that are not tightly constrained by the spectral geometry The finite algebra classification identifies the Pati-Salam algebra as the next simplest option, but there is no derivation of why nature would choose the Standard Model algebra over the Pati-Salam one; the framework does not explain the selection	The finite-dimensional internal algebra must be selected from outside the spectral geometry framework; the axiomatic constraints narrow the choices but do not uniquely select the Standard Model algebra The 2007 Higgs mass prediction of approximately 170 GeV was falsified by the LHC in 2012; subsequent revisions introduced a scalar singlet to shift the prediction to 125 GeV, prompting criticism that the model was adjusted post-observation The framework is semiclassical: the spectral action is treated as a classical functional from which quantum field theory is derived perturbatively; genuine quantum gravity or a non-perturbative completion within the NCG framework has not been demonstrated The finite spectral triple for the Standard Model requires three fermion generations to be postulated; the framework does not explain why there are exactly three
Key unresolved problem	The symmetry-breaking scale problem: the theory produces the unified Pati-Salam force structure but cannot say at what energy it splits down into ordinary forces, leaving its main testable signature, where the new particles would show up, unpinned.	The finite algebra problem: the framework's core rules narrow down the basic mathematical building block, the internal algebra, but cannot single out the one that gives our Standard Model, so the theory's most important input is still picked by hand.
Reader vote	No votes yet	No votes yet

Spectral Pati-Salam Unification

2013 · Frontier

Connes-Chamseddine Spectral Action

1996 · Frontier

Proposed

2013

1996

Key figures

Ali Chamseddine, Alain Connes, Walter van Suijlekom

Alain Connes, Ali Chamseddine

In one sentence

Chamseddine, Connes, and van Suijlekom showed in 2013 that replacing the Standard Model internal algebra with the next simplest algebra compatible with the spectral triple axioms yields a Pati-Salam model with gauge group SU(4) x SU(2)_L x SU(2)_R, unifying quarks and leptons without invoking a larger simple group like SU(5) or SO(10).

Connes showed in 1996 that a noncommutative algebra encoding spacetime plus the Standard Model's internal symmetries, plus a single Dirac operator acting on it, automatically produces the bosonic Lagrangian of the Standard Model and general relativity when the spectral action is evaluated.

Predictions

The framework predicts an intermediate stage where the Pati-Salam symmetry breaks down to the Standard Model. The spectral geometry does not fix the energy at which this happens, that scale is a free input, but once it is chosen, the masses of the extra gauge bosons (the heavy SU(4) and SU(2)_R carriers) are set, so finding or ruling out those particles at a given energy tests the choice
Leptoquark gauge bosons with specific quantum numbers under SU(4) couple quarks and leptons in the Pati-Salam manner and are accessible at colliders if the breaking scale is near the TeV range
A right-handed neutrino gauge field under SU(2)_R is present, producing an additional source of Majorana mass distinct from the see-saw mechanism of the 2007 variant
Protons should be far more stable than the simplest competing grand-unified theory, minimal SU(5), predicts. The Pati-Salam structure avoids the leading process that lets protons decay in SU(5), so the two make sharply different forecasts. Large underground detectors that watch tanks of water for a single proton decay can test this: SU(5)'s shorter lifetime is already in tension with the data, while Pati-Salam's longer one is not yet reachable

The Standard Model gauge groups SU(3) x SU(2) x U(1) and the Higgs doublet are not independent choices but are fixed by the geometry of the finite noncommutative space, once that space is specified by the spectral triple axioms
Gravitational and gauge couplings unify at the Planck scale as a consequence of the spectral geometry, providing a top-down constraint on the relationship between Newton's constant and the gauge coupling strengths
The Higgs scalar is a fluctuation of the internal geometry rather than a fundamental particle in the traditional sense, and its potential is geometrically determined up to the scale of the internal space
A see-saw mechanism for neutrino masses emerges naturally in the extended spectral triple that includes right-handed neutrinos in the Hilbert space, as shown in the 2007 Chamseddine-Connes-Marcolli revision

Where it breaks

The scale at which Pati-Salam breaks to the Standard Model is a free parameter in the NCG framework; there is no derivation from the spectral geometry of the breaking scale itself, only of the gauge structure and particle content above it
No direct experimental evidence for Pati-Salam breaking has been found; leptoquarks predicted by the model have not been observed at the LHC through Run 3 within the minimal mass range the model targets
The variant requires additional [[scalar-field|scalar fields]] to break the gauge symmetry stepwise from SU(4) x SU(2)_L x SU(2)_R to the Standard Model; the scalar sector introduces new parameters that are not tightly constrained by the spectral geometry
The finite algebra classification identifies the Pati-Salam algebra as the next simplest option, but there is no derivation of why nature would choose the Standard Model algebra over the Pati-Salam one; the framework does not explain the selection

The finite-dimensional internal algebra must be selected from outside the spectral geometry framework; the axiomatic constraints narrow the choices but do not uniquely select the Standard Model algebra
The 2007 Higgs mass prediction of approximately 170 GeV was falsified by the LHC in 2012; subsequent revisions introduced a scalar singlet to shift the prediction to 125 GeV, prompting criticism that the model was adjusted post-observation
The framework is semiclassical: the spectral action is treated as a classical functional from which quantum field theory is derived perturbatively; genuine quantum gravity or a non-perturbative completion within the NCG framework has not been demonstrated
The finite spectral triple for the Standard Model requires three fermion generations to be postulated; the framework does not explain why there are exactly three

Key unresolved problem

The symmetry-breaking scale problem: the theory produces the unified Pati-Salam force structure but cannot say at what energy it splits down into ordinary forces, leaving its main testable signature, where the new particles would show up, unpinned.

The finite algebra problem: the framework's core rules narrow down the basic mathematical building block, the internal algebra, but cannot single out the one that gives our Standard Model, so the theory's most important input is still picked by hand.

Reader vote

No votes yet