Geometry of Time-Spaces : Non-Commutative Algebraic by Olav arnfinn audal

By Olav arnfinn audal

This can be a monograph approximately non-commutative algebraic geometry, and its program to physics. the most mathematical inputs are the non-commutative deformation conception, moduli thought of representations of associative algebras, a brand new non-commutative idea of section areas, and its canonical Dirac derivation. The publication begins with a brand new definition of time, relative to which the set of mathematical velocities shape a compact set, implying specified and normal relativity. With this version in brain, a common Quantum thought is constructed and proven to slot with the classical thought. specifically the "toy"-model used as instance, comprises, as a part of the constitution, the classical gauge teams u(1), su(2) and su(3), and consequently additionally the idea of spin and quarks, and so on.

Readership: Researchers in geometry and quantum physics.

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Additional info for Geometry of Time-Spaces : Non-Commutative Algebraic Geometry, Applied to Quantum Theory

Example text

Put, A(σ) := P h∞ (A)/(δ n tp − Γp ) where σ := (δ n tp − Γp ) is the two-sided δ-ideal generated by the defining equations of σ. Obviously δ induces a derivation δσ ∈ Derk (A(σ), A(σ)), also called the Dirac derivation, and usually just denoted δ. Notice that if σi , i = 1, 2, are two different order n dynamical systems, then we may well have, A(σ1 ) A(σ2 ) P h(n−1) (A)/(σ∗ ), as k-algebras, see the Introduction. 2 Quantum Fields and Dynamics For any integer n ≥ 1 consider the schemes Simpn (A(σ)) and Spec(C(n)), and the corresponding (almost uni-) versal family, ρ˜ : A(σ)) → EndSpec(C(n)) (V˜ ) Mn (C(n)).

The difference is that whereas for finite n, the set Simpn (A) has a nice, finite dimensional scheme structure, this is, in general, no longer true for the set, HrA nor for the set of fields, F(A; R), as the physicists call it, unless we put some extra conditions on the fields, so called decorations. If R is Artinian of length n, then the corresponding F(A; R) does exist and has a nice structure, both as commutative and as non-commutative scheme. e. on the set of surjective homomorphisms k[x1 , x2 , x3 ] → R = k 2 .

Given two such points, (qi , pi ), i = 1, 2, an easy calculation proves, dimk Ext1P hA (k(q1 , p1 ), k(q2 , p2 )) = 1, for q1 = q2 dimk Ext1P hA (k(q1 , p1 ), k(q2 , p2 )) = 3, for q1 = q2 , , p1 = p2 dimk Ext1P hA (k(q1 , p1 ), k(q2 , p2 )) = 6, for (q1 , p1 ) = (q2 , p2 ) Put xj (qi , pi ) := qi,j , dxj ((qi , pi ) := pi,j , αj = q1,j − q2,j , βj = p1,j − p2,j . See that for any element α ∈ Homk (k((q1 , p1 )), k((q2 , p2 ))) we have, xj α = q1,j α, αxj = q2,j α, dxj α = p1,j α, αdxj = p2,j α, with the obvious identification.

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