In this first real blog post, I want to show and explain a poster I have made. Last April the Emergence at All Scales consortium held its first annual meeting, here I did not only give an introductory presentation for my project, I also showcased a poster with the name ``Entropy in Noncommutative Geometry’’. Since this consortium consists of scientists with different backgrounds, the target audience consists of physicists, who have worked with notions such as entropy and field theory before, but have probably never heard of noncommutative geometry. While the poster was definitely made with this in mind, it didn’t stop me from using it at another conference: the Workshop on Interfaces of Spectral Theory, Operator Algebras and Noncommutative Geometry, in honour of Matthias Lesch.

Allow me to explain the poster, as if you were there at one of these events. Firstly, notice that the poster is split into two columns, each with its own function: basic theory on the left and on the right the example of two-point noncommutative space, together with some explanation on how to use the theory on the left.

Noncommutative geometry: spectral triples The first section immediately shows that it is targeted at an audience with a background in physics, who don’t often work with objects such as spectral triples and need motivation to have this initial starting data.

Second quantisation: Fock spaces They are on the other hand more well-versed in the use of Fock spaces, which takes the Hilbert space as space describing one particle, and then lifts it to a system with an arbitrary amount of particles. Since we want to describe fermionic like particles in this case, we take the anticommutative product: the wedge. To see this exterior algebra as a Hilbert space itself, we must put an inner product on it and that we can use the Dirac operator to do this. The reason for this is that the Dirac operator also allows for negative energy solutions, but this is in a sense unphysical and this choice is in a sense related to the ‘‘Dirac sea’’. By modifying what in noncommutative geometry is called the complex structure, denoted by $I$, we start working with $|D|$ instead of $D$, which has as eigenvalues the absolute values of the eigenvalues of $D$.

Entropy Seemingly unrelated, we can define the entropy for (specific, namely type-I,) states on any Hilbert space, using the formula $-\textrm{Tr}(\rho \log \rho)$ as often seen in physics. While you might first want to apply this to the original Hilbert space $\mathcal{H} $ from the spectral triple, this is not correct. Remembering that the entropy is a notion for a system with a large amount of particles, we want to apply this for the Hilbert space $\mathcal{F} = \bigwedge \mathcal{H}$.

KMS-states From (statistical) physics we also know that the entropy is a notion used for a system near equilibrium, i.e. we must look at states to which the system wants to converge. In this mathematical framework, these turn out to be the KMS-states, which are defined by the formal expression $\phi(a \sigma_t(b))|_{t = i \beta} = \phi(ba)$ and multiple things should be said about the objects in this expression. Firstly, the parameter $\beta$ is like the $\beta$ in physics, which is one over the temperature. This means that if $\beta$ approaches $0$, the entropy should maximise. This dependence on temperature gives rise to the local equilibriums we are describing here, which minimise something similar to the entropy: the free energy. Important to note is that we fix the value of $\beta$ for each solution and that the states depend on $\beta$ as only there the equation has to hold. Secondly, we consider time evolution operator $\sigma_t$ and have in mind that it should describe the dynamics in our system. In physics, we often think of a Hamiltonian $H$ as describing the dynamics and in this case they turn out to be related: if a Hamiltonian $H$ exists, it gives a time evolution via $\sigma_t(X) = e^{itH} X e^{-itH}$, you might recognize this as the evolution of operators in the Heisenberg picture of quantum mechanics.

Example: two-point space Next is the part describing the example, where we can actually find expressions for the KMS-states and compute the entropy. The starting point is writing down a spectral triple that describes the space in the figure, which is a result from the literature. There are also some results that also hold in more generality: the uniqueness of KMS-states and how it uses the lift of operators on Hilbert spaces to the corresponding Fock spaces. Here we don’t write out all these operators, which is clearly possible as operators in $\mathcal{F}$ are four by four matrices in this case.

Entropy for two-point space For the expression of the entropy you can either believe I did the calculations, or easily verify it via the spectral action principle, which can be found in the first source.

Entropic force and emergence When we look at the plot of the function from the entropy, we see that for $\beta \to 0$ we get maximal entropy, as we expected already, but also that $r \to \infty$ has the same effect. Since we have a principal of maximal entropy, we expect $r \to \infty$ to be the limit for this system. This is enforced by the entropic force, for which we again borrow a formula from physics. The last thing to mention is that we see the emergence of two points from a single one in this behaviour.