The Heisenberg Spectral Triple and Associated Zeta Functions

Date

2024-01-03

Authors

Steed, Brendan

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Abstract

The construction of Butler, Emerson, and Schultz [2] produced a certain spectral triple, which they called the Heisenberg cycle, by way of the quantum mechanical annihilation and creation operators, d/dx ± x, along with their relationships to the harmonic oscillator, -d^2/dx^2 + x^2; Where all of these operators are de fined (initially) to act on smooth functions over R. In particular, their Heisenberg cycle was over a crossed-product generated by the natural translation action on the (commutative) C*-algebra of uniformly continuous, bounded, functions on R. In this thesis, we generalize the Heisenberg cycle of Butler, Emerson, and Schultz to allow for the construction of a spectral triple over a crossed-product generated by the natural translation action on the C*-algebra of uniformly continuous, bounded, functions on a Euclidean space, V , of arbitrary finite dimension n. For such a generalization, the annihilation and creation operators are replaced using the exterior derivative and codifferential, exterior and interior multiplication by a certain differential 1-form, and the relationship these four operators have to the n-dimensional harmonic oscillator acting on differential forms. Similarly to [2], we will show that our generalized Heisenberg cycle provides a new way of producing spectral triples over crossed-products of the form C(M) ⋊_α Γ, where Γ is a discrete subgroup of V and α : V x M →M is a smooth V -action on a compact manifold M. In Chapter 1, we introduce the problem and briefly discuss some historical background behind Alain Connes program of noncommutative geometry, as well as touch on some elementary constructions in multi-linear algebra. Chapter 2 is where we de ne the classes of differential forms which appear most frequently in this thesis. Therein, we also rigorously de fine the operators mentioned in the paragraph above, and use them to produce the so-called Dirac-Heisenberg which will be associated to our generalization of the Heisenberg cycle. For the first half of Chapter 3, we discuss some basic C*-algebra theory and introduce the crossed-product native to the Heisenberg cycle. In the latter half of that chapter, we verify that our Heisenberg cycle satis es the conditions of a spectral triple, compute an integral formula for the resulting ζ-functions, and show how one uses the Heisenberg cycle to produce spectral triples over crossed-products generated by smooth actions of V on compact manifolds.

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Keywords

Noncommutative geometry, K-theory, K-homology

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