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A composite model of the standard model particles was recently derived using the Dirac Lagrangian on a spacetime where time does not advance along the worldlines of fundamental dust particles, called an 'internal spacetime'. The aim of internal spacetime geometry is to model certain quantum phenomena using (classical) degenerate spacetime metrics. For example, on an internal spacetime, tangent spaces have variable dimensions, and spin wavefunction collapse is modeled by the projection from one tangent space to another. In this article, we show that the combinatorial structure of the internal Dirac Lagrangian yields precisely the standard model trivalent vertices, together with two new additional (longitudinal) Z vertices that generate the four-valent boson vertices. In particular, we are able to derive electroweak parity violation for both leptons and quarks. We also obtain new restrictions on the possible spin states that can occur in certain interactions. Finally, we determine the trivalent vertices of the new massive spin- 2 boson predicted by the model. [ABSTRACT FROM AUTHOR]

'Internal spacetime' is a modification of general relativity that was recently introduced as an approximate spacetime geometric model of quantum nonlocality. In an internal spacetime, time is stationary along the worldlines of fundamental (dust) particles. Consequently, the dimensions of tangent spaces at different points of spacetime vary, and spin wavefunction collapse is modeled by the projection from one tangent space to another. In this paper, we develop spinors on an internal spacetime, and construct a new Dirac-like Lagrangian ℒ = ψ ̄ (i ∂ / − ω ̂) ψ whose equations of motion describe their couplings and interactions. Furthermore, we show that hidden within ℒ is the entire standard model: ℒ contains precisely three generations of quarks and leptons, the electroweak gauge bosons, the Higgs boson, and one new massive spin- 2 boson; gluons are considered in a companion paper. Specifically, we are able to derive the correct spin, electric charge, and color charge of each standard model particle, as well as predict the existence of a new boson. [ABSTRACT FROM AUTHOR]

Let A be a nondegenerate dimer (or ghor) algebra on a torus, and let Z be its center. Using cyclic contractions, we show the following are equivalent: A is noetherian; Z is noetherian; A is a noncommutative crepant resolution; each arrow of A is contained in a perfect matching whose complement supports a simple module; and the vertex corner rings e i A e i are pairwise isomorphic. [ABSTRACT FROM AUTHOR]

To incorporate quantum nonlocality into general relativity, we propose that the preparation and measurement of a quantum system are simultaneous events. To make progress in realizing this proposal, we introduce a spacetime geometry that is endowed with particles which have no distinct points in their worldlines; we call these particles 'pointons'. This new geometry recently arose in nonnoetherian algebraic geometry. We show that on such a spacetime, metrics are degenerate and tangent spaces have variable dimension. This variability then implies that pointons are spin- 1 2 fermions that satisfy the Born rule, where a projective measurement of spin corresponds to an actual projection of tangent spaces of different dimensions. Furthermore, the 4-velocities of pointons are necessarily replaced by their Hodge duals, and this transfer from vector to pseudo-tensor introduces a free choice of orientation that we identify with electric charge. Finally, a simple composite model of electrons and photons results from the metric degeneracy, and from this we obtain a new ontological model of photon polarization. [ABSTRACT FROM AUTHOR]

We introduce a theory of geometry for nonnoetherian commutative algebras with finite Krull dimension. In particular, we establish new notions of normalization and height: depiction (a special noetherian overring) and geometric codimension. The resulting geometries are algebraic varieties with positive-dimensional points, and are thus inherently nonlocal. These notions also give rise to new equivalent characterizations of noetherianity that are primarily geometric. We then consider an application to quiver algebras whose simple modules of maximal dimension are one dimensional at each vertex. We show that the vertex corner rings of are all isomorphic if and only if is noetherian, if and only if the center of is noetherian, if and only if is a finitely generated -module. Furthermore, we show that is depicted by a commutative algebra generated by the cycles in its quiver. We conclude with an example of a quiver algebra where projective dimension and geometric codimension, rather than height, coincide. [ABSTRACT FROM AUTHOR]

Let ψ : A → A ′ be a cyclic contraction of dimer algebras, with A non-cancellative and A ′ cancellative. A ′ is then prime, noetherian, and a finitely generated module over its center. In contrast, A is often not prime, nonnoetherian, and an infinitely generated module over its center. We present certain Morita equivalences that relate the representation theory of A with that of A ′ . We then characterize the Azumaya locus of A in terms of the Azumaya locus of A ′ , and give an explicit classification of the simple A -modules parameterized by the Azumaya locus. Furthermore, we show that if the smooth and Azumaya loci of A ′ coincide, then the smooth and Azumaya loci of A coincide. This provides the first known class of algebras that are nonnoetherian and infinitely generated modules over their centers, with the property that their smooth and Azumaya loci coincide. [ABSTRACT FROM AUTHOR]

We introduce new mathematical aspects of the Bell states using matrix factorizations, non-noetherian singularities, and noncommutative blowups. A matrix factorization of a polynomial p consists of two matrices ϕ1, ϕ2 such that ϕ1ϕ2 = ϕ2ϕ1 = p id. Using this notion, we show how the Bell states emerge from the separable product of two mixtures, by defining pure states over complex matrices rather than just the complex numbers. We then show in an idealized algebraic setting that pure states are supported on non-noetherian singularities. Moreover, we find that the collapse of a Bell state is intimately related to the representation theory of the noncommutative blowup along its singular support. This presents an exchange in geometry: the nonlocal commutative spacetime of the entangled state emerges from an underlying local noncommutative spacetime. [ABSTRACT FROM AUTHOR]

Abstract: A superpotential algebra is square if its quiver admits an embedding into a two-torus such that the image of its underlying graph is a square grid, possibly with diagonal edges in the unit squares; examples are provided by dimer models in physics. Such an embedding reveals much of the algebras representation theory through a device we introduce called an impression. Let A be a square superpotential algebra, Z its center, and the maximal ideal at the origin of Spec Z. Using an impression, we [•] give a classification of all simple A-modules up to isomorphism, and give algebraic and homological characterizations of the simple A-modules of maximal k-dimension; [•] show that Z is a 3-dimensional normal toric domain and is Gorenstein, by determining transcendence bases and Z-regular sequences; and [•] show that is a noncommutative crepant resolution of , and thus a local Calabi–Yau algebra. A particular class of square superpotential algebras, the algebras, is considered in detail. We show that the Azumaya and smooth loci of the centers coincide, and propose that each ramified maximal ideal sitting over the singular locus is the exceptional locus of a blowup shrunk to zero size. [Copyright &y& Elsevier]

Let Z be the center of a nonnoetherian dimer algebra on a torus. Although Z itself is also nonnoetherian, we show that it has Krull dimension 3, and is locally noetherian on an open dense set of Max Z. Furthermore, we show that the reduced center Z / nil Z is depicted by a Gorenstein singularity, and contains precisely one closed point of positive geometric dimension. [ABSTRACT FROM AUTHOR]