# 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.

## Description

## Keywords

Noncommutative geometry, K-theory, K-homology