I'd just like to add a point, as I have read this paper, and read it a while ago.
This is the prime example I'd say of modeling using Heisenberg. Versus modeling using Schrodinger.
We are modeling a space of coefficients \(\{a_j\}_{j=0}^\infty \in \mathbb{C}\) as we apply flow matrices \(H^t A\) where the object \(A = (a_0,a_1,...,a_j,...)\); and \(H^t\) is a matrix semi group, \(H^t A = (b_0(t),b_1(t),...,b_j(t),...)\).
I am not trying to say any of Daniel's or Gottfried's deliberations are wrong. I'm trying to say, that we can do the same thing with integrals. Schrodinger's approach is the same thing as Heisenberg--it's just a matter of language.
To begin we project this space:
$$
\begin{align}
\sigma(A) &= \sum_{j=0}^\infty a_j z^j\\
\sigma(H^tA) &= \sum_{j=0}^\infty b_j(t)z^j\\
\end{align}
$$
The above matrix solution works perfectly. And what both of you guys are talking about is perfect. The difference I would say, is that much of this can be represented with Fourier transforms/integral transforms. And that ultimately, we are saying the same thing.
This is absolutely seen most obviously, going back about a hundred years to the great war. Ramanujan had written this odd equation which is:
$$
\Gamma(t)H^{-t} = \int_0^\infty \left(\sum_{n=0}^\infty H^n \frac{(-x)^n}{n!}\right)x^{t-1}\,dx\\
$$
This holds for square matrices/infinite matrices/general linear operators \(H\). Not to mention it holds for holomorphic functions \(H(t) : \mathbb{C}_{\Re(t) > 0} \to \mathbb{C}\) which are appropriately bounded (Carlson would go on to win a Fields' medal because he rigorously proved this result with no exception).
This relates perfectly to fractional calculus, because: the Riemann-Liouville Differintegral/ the Exponential Differintegral is written as:
$$
\frac{d^{-z}}{dx^{-z}}\Big{|}_{x=0} \vartheta(x) = \frac{1}{\Gamma(z)}\int_0^\infty \vartheta(-x)x^{z-1}\,dx\\
$$
Where, it owes its name to:
$$
\frac{d^{-z}}{dx^{-z}} e^x = e^x\\
$$
So if we apply this differintegral; then:
$$
\vartheta(x) = \sum_{n=0}^\infty H^n\frac{x^n}{n!} = e^{Hx}\\
$$
Now when we apply the differintegral:
$$
\frac{d^{-z}}{dx^{-z}} e^{Hx} = H^{-z}e^{Hx}\\
$$
The majority of this appears in Ramanujan's notebooks; and is additionally used in a lot of mathematics--but primarily number theory. I just want to play devil's advocate and rejustify what you guys are already seeing. Please remember that \(H\) is an operator on a hilbert space, or a linear operator on a vector space. Which are \(n\times n\) or \(\infty \times \infty\) scenarios.
So \(H\) acts on something, let's say the sequence \(A = (a_0,a_1,...,a_j,...)\). If \(H\) is well enough behaved, then \(H^{-z}\) is discoverable through this integral transform. And what's more, it's exactly what you guys are talking about. It's just integrals instead of infinite matrices. Heisenberg vs Schrodinger.
We can write:
$$
H^{-z} \textbf{v} = H^{-z} (v_0,v_1,...,v_j,...) = (u_0(-z),u_1(-z),...,u_j(-z),...)\\
$$
Or we can write:
$$
\int_0^\infty e^{-Hx}x^{z-1}\,dx \textbf{v}\\
$$
Where both produce the same result!
Either way, love you guys. Thanks for reminding me of this paper, Daniel
Regards, James.
EDIT:
As you guys might not get what I mean by "Heisenberg vs. Schrodinger" I'll give a little history lesson.
The idea of infinite square matrices was invented by Heisenberg. Where these infinite square matrices acted on an infinite vector space (this is all countable). Heisenberg talked about how there were eigen values to these well developed matrices. The matrices were measurables of momentum/position; discrete measurements you could apply to a vector. Hence, Heisenberg's inequality \(PQ - QP < \hbar/2\). This is heisenberg's construction of quantum physics.
Schrodinger is all about waves. It's all about functions which look like waves. And these waves oscillate, and have the same frequencies as the infinite vector space. So we take \(\mathbf{v} = (v_0,v_1,...,v_j,...)\), and translate it into a wave \(\nu(x)\). Now, in Schrodinger's language, we can just take integrals of \(\nu\) which do the same thing as Heisenberg, just a different language.
There were feuds about this for a good while. People couldn't believe both mathematicians/physicists were saying the same thing. Until Von Neumann coined the term "Hilbert Space". Which then showed that both constructions were equivalent if we viewed "infinite matrices like wave functions". This was based primarily off of Hilbert's study of waves, and integral operators on waves.
I'm just trying to stream line everything, boys.
, so just remember it's correct, I may have just fucked up some variable changes, lol. Too busy to double check every single number, but the main idea is absolutely true!!!