In the following analysis, the Riemann Hypothesis is presented as a natural Gedankenexperiment in Quantum Mechanics that reveals the super-deterministic nature of the Universal Wave Function. This forces us to reconsider claims that the theory of algorithmic probability, where probabilities are of a deterministic and frequentist nature, lack careful epistemic and physical justification.
If anything at all in our universe is correct, it has to be the Riemann Hypothesis, if for no other reasons, so for purely esthetical reasons.-Atle Selberg
We generalise the double-slit to countably many-slits. In this generalisation, we suppose that all \(N \geq 1\) experiments are observed simultaneously where for the Nth experiment a choice is made by the Universal Wave Function as to which slit the particle will go through. Assuming that the experimenter is free to choose between different measurement settings, for the Nth experiment we have \(N\) slits that are equiprobable.
Within this thought experiment, the Riemann Zeta function proves to be essential in the global analysis of expected Quantum Measurements.
In generalising the double-slit experiment to \(N\) slits we may model the slit chosen by the photon using the integer \(X \sim U([1,N])\) with Shannon Entropy \(H(X) = \ln N\). This may be encoded using a Quantum State represented by a density matrix \(\rho \in \mathbb{C}^{N \times N}\) with Von Neumann Entropy:
\[\begin{equation} S \circ \rho = \ln N \tag{1} \end{equation}\]
In order to effectively model Quantum Measurements using Unitary Operators we may use the Dirac Delta function:
\[\begin{equation} P(\xi) = \delta(\xi - \ln N) \tag{2} \end{equation}\]
where \(\xi\) represents an entropy-valued observable.
Moreover, we’ll note that (2) allows a unique decoding of Quantum States via Quantum Measurement using the fact that the Von Neumann entropy is invariant to Unitary transformations:
\[\begin{equation} \rho' = U \circ \rho \implies S \circ \rho' = S \circ \rho \tag{3} \end{equation}\]
a result generally known as the Law of Conservation of Information.
Besides being an effective indicator function, it is worth noting that (2) may be viewed as a generalised distribution using the Shannon Source Coding theorem. If the experiment is repeated \(n\) times, as \(n \to \infty\) we may expect that \(\forall \epsilon > 0, \lvert S \circ \rho - \ln N \rvert < \epsilon\).
The event of interest may be represented using the random variable \(X\) sampled from the Cartesian Product:
\[\begin{equation} X \sim \lim_{N \to \infty} U\big(\prod_{i=1}^N [1,n] \big) \tag{4} \end{equation}\]
where we are implicitly using the Axiom of Countable Choice.
Its associated frequency distribution is given by:
\[\begin{equation} F_{H(X)} = \sum_{n=1}^{\infty} P(H(X_n) = \ln n) \tag{5} \end{equation}\]
where \(\forall n \in \mathbb{N}, X_n \sim U([1,n]) \implies H(X_n) = \ln n\) and assuming that the experiments are independent we may perform measurements in a sequential manner so we have:
\[\begin{equation} F_{H(X)} = \sum_{n=1}^{\infty} \delta(\xi - \ln n) \tag{6} \end{equation}\]
where \(\xi\) is used to denote entropy as well as an arrow of time.
Calculating the expectation with the Laplace Transform, we obtain the Riemann Zeta function:
\[\begin{equation} \mathcal{L} \{F_{H(X)}\}(s) = \mathbb{E}\big[\sum_{\xi} e^{-s\xi} \big] = \zeta(s) \tag{7} \end{equation}\]
Crucially, this may be formulated as a global analysis of the superposition of Quantum States:
\[\begin{equation} \mathbb{E}[\Psi] = \sum_{n=1}^{\infty} \int_{\mathbb{R}_{+}} e^{-s\xi} \cdot \delta(\xi - \ln n) d\xi \tag{8} \end{equation}\]
where \(\Psi_n = e^{-s \cdot \xi}\) represents the nth Quantum Measurement, \(\ln n\) represents the Von Neumann entropy associated with the nth measurement and \(\lvert \Psi_n \rvert^2\) represents the probability that we observe a photon at the slit \(X_n \sim U([1,n])\).
We may note that the zeros of the Riemann Zeta function control the expected behaviour of the Universal Wave function. Thus, the location of the zeros may be determined by considering that the instant a measurement is made its corresponding uncertainty(and hence entropy) vanishes.
For the nth experiment, we’ll note that:
\[\begin{equation} \xi = \ln n \iff - \ln \lvert \Psi_n \rvert^2 = \ln n \tag{9} \end{equation}\]
Hence, the Maximum Entropy criterion determines Quantum Measurements. Thus, we have:
\[\begin{equation} \mathbb{E}\Big[\frac{d\Psi}{d\xi}\Big] = -s \cdot \zeta(s) = 0 \tag{10} \end{equation}\]
and using (9) and the Method of Quantum Amplitudes, we may deduce that the entropy terms vanish if and only if:
\[\begin{equation} \forall n \in \mathbb{N}, \lvert \Psi_n \rvert^2 = \lvert e^{-s \cdot \ln n} \rvert^2 = \frac{1}{n} \tag{11} \end{equation}\]
From (10) and (11), it follows that:
\[\begin{equation} \forall s \in \mathbb{C}, \zeta(s) = 0 \iff \textbf{Re}(s) = \frac{1}{2} \tag{12} \end{equation}\]
Thus, we may conclude that all the non-trivial zeros of the Riemann Zeta function must lie on the critical line:
\[\begin{equation} \zeta(\frac{1}{2} + it) = \sum_{n=1}^{\infty} \frac{1}{\sqrt{n}} \cdot e^{-i \cdot \ln n \cdot t} \tag{13} \end{equation}\]
As we have demonstrated, \(\zeta\) has countably many zeroes along the critical line where each zero controls the outcome of countably many experiments simultaneously. On the other hand, if all slits were equiprobable as posited by Everettian Quantum Mechanics then \(\zeta\) would have uncountably many zeroes as there are uncountably many ways the entropy terms may vanish. As the Riemann Zeta function is analytic on \(\mathbb{C} \setminus \{1\}\), it would be identical to zero so we have a contradiction.
Thus, from this analysis we may conclude that the assumption of statistical independence is incorrect(though it may be useful) and that Quantum Measurements are super-deterministic. Although such a view may be counter-intuitive, it is consistent with advanced by one of the founding fathers of Probability Theory:
We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes.—Pierre Simon Laplace
an epistemological framework advanced in 1773 in his famous philosophical essay on probability theory [8].
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Laplace, Pierre Simon, A Philosophical Essay on Probabilities. Dover Publications. 1951.
For attribution, please cite this work as
Rocke (2022, March 8). Kepler Lounge: The Von Neumann Entropy and the Riemann Hypothesis. Retrieved from keplerlounge.com
BibTeX citation
@misc{rocke2022the,
author = {Rocke, Aidan},
title = {Kepler Lounge: The Von Neumann Entropy and the Riemann Hypothesis},
url = {keplerlounge.com},
year = {2022}
}