Random variables seemingly arise in two contexts. In a microscopic system, the collapse of wave function produces truly random outcomes, since Bell inequality disallows any possibility of hidden variables. For brevity, let us call quantum systems to be first kind random.
But what is the nature of macroscopic randomness? In fact, classical mechanics describes the evolution of a classical system in the phase space, and it is possible to see it as completely deterministic. At this point, chaos theory proposes to explain the mystery by demonstrating that deterministic systems may produce apparently random results too. Let us call classical systems to be second-kind random.
What is the relation between randomness of both kinds? Is one of them more primitive, and another an emerging phenomenon generated by it? Or are both equally fundamental? How does mathematics characterize their similarity and differences?
On meditating on this, we would realize the probability theory doesn’t forbid a completely static interpretation. After all, the event space is just a static collection of outcomes, and the expectation is just an integration. The present theory of probability is undoubtedly rigorous, but the very core concept lying within, the nature of probability, lies only in our heart. We have defined everything except the most crucial one: what randomness is. Are they something we see everyday but can’t say? Or is it something we will never know?
Consider the first linear system:
first:
\(\mathbf{A} \vphantom{ \mathbf{A}}_{1}\)\(\mathbf{x} \vphantom{ \mathbf{x}}_{1}\)\(\;=\;\)\(\left[\vphantom{\;\begin{matrix} 1 ,\\ 0 ,\\ 0 ,\\ 0 ,\\ \end{matrix}\; \;\begin{matrix} 0 ,\\ 0 ,\\ 1 ,\\ 0 ,\\ \end{matrix}\; \;\begin{matrix} 0 ,\\ 1 ,\\ 0 ,\\ 0 ,\\ \end{matrix}\; \;\begin{matrix} 0\\ 0\\ 0\\ 1\\ \end{matrix}\;}\right.\)\(\;\begin{matrix} 1 ,\\ 0 ,\\ 0 ,\\ 0 ,\\ \end{matrix}\;\)\(\;\begin{matrix} 0 ,\\ 0 ,\\ 1 ,\\ 0 ,\\ \end{matrix}\;\)\(\;\begin{matrix} 0 ,\\ 1 ,\\ 0 ,\\ 0 ,\\ \end{matrix}\;\)\(\;\begin{matrix} 0\\ 0\\ 0\\ 1\\ \end{matrix}\;\)\(\left.\vphantom{\;\begin{matrix} 1 ,\\ 0 ,\\ 0 ,\\ 0 ,\\ \end{matrix}\; \;\begin{matrix} 0 ,\\ 0 ,\\ 1 ,\\ 0 ,\\ \end{matrix}\; \;\begin{matrix} 0 ,\\ 1 ,\\ 0 ,\\ 0 ,\\ \end{matrix}\; \;\begin{matrix} 0\\ 0\\ 0\\ 1\\ \end{matrix}\;}\right]\)\(\,\cdot\,\)\(\dfrac{1}{\surd\! 2}\)\(\left[\vphantom{\;\begin{matrix} 1\\ 1\\ 0\\ 0\\ \end{matrix}\;}\right.\)\(\;\begin{matrix} 1\\ 1\\ 0\\ 0\\ \end{matrix}\;\)\(\left.\vphantom{\;\begin{matrix} 1\\ 1\\ 0\\ 0\\ \end{matrix}\;}\right]\)
\(\;=\;\)\(\mathbf{y} \vphantom{ \mathbf{y}}_{1}\)
\(\;=\;\)\(\left\langle\vphantom{1 ,\, 0 ,\, 1 ,\, 0}\right.\)\(1\)\(,\,\)\(0\)\(,\,\)\(1\)\(,\,\)\(0\)\(\left.\vphantom{1 ,\, 0 ,\, 1 ,\, 0}\right\rangle\)
And the second linear system:
second:
\(\mathbf{A} \vphantom{ \mathbf{A}}_{2}\)\(\mathbf{x} \vphantom{ \mathbf{x}}_{2}\)\(\;=\;\)\(\left[\vphantom{\;\begin{matrix} 0 ,\\ 0 ,\\ 1 ,\\ 0 ,\\ \end{matrix}\; \;\begin{matrix} 0 ,\\ 1 ,\\ 0 ,\\ 0 ,\\ \end{matrix}\; \;\begin{matrix} 1 ,\\ 0 ,\\ 0 ,\\ 0 ,\\ \end{matrix}\; \;\begin{matrix} 0\\ 0\\ 0\\ 1\\ \end{matrix}\;}\right.\)\(\;\begin{matrix} 0 ,\\ 0 ,\\ 1 ,\\ 0 ,\\ \end{matrix}\;\)\(\;\begin{matrix} 0 ,\\ 1 ,\\ 0 ,\\ 0 ,\\ \end{matrix}\;\)\(\;\begin{matrix} 1 ,\\ 0 ,\\ 0 ,\\ 0 ,\\ \end{matrix}\;\)\(\;\begin{matrix} 0\\ 0\\ 0\\ 1\\ \end{matrix}\;\)\(\left.\vphantom{\;\begin{matrix} 0 ,\\ 0 ,\\ 1 ,\\ 0 ,\\ \end{matrix}\; \;\begin{matrix} 0 ,\\ 1 ,\\ 0 ,\\ 0 ,\\ \end{matrix}\; \;\begin{matrix} 1 ,\\ 0 ,\\ 0 ,\\ 0 ,\\ \end{matrix}\; \;\begin{matrix} 0\\ 0\\ 0\\ 1\\ \end{matrix}\;}\right]\)\(\,\cdot\,\)\(\dfrac{1}{\surd\! 2}\)\(\left[\vphantom{\;\begin{matrix} 1\\ 1\\ 0\\ 0\\ \end{matrix}\;}\right.\)\(\;\begin{matrix} 1\\ 1\\ 0\\ 0\\ \end{matrix}\;\)\(\left.\vphantom{\;\begin{matrix} 1\\ 1\\ 0\\ 0\\ \end{matrix}\;}\right]\)
\(\;=\;\)\(\mathbf{y} \vphantom{ \mathbf{y}}_{2}\)
\(\;=\;\)\(\left\langle\vphantom{0 ,\, 1 ,\, 1 ,\, 0}\right.\)\(0\)\(,\,\)\(1\)\(,\,\)\(1\)\(,\,\)\(0\)\(\left.\vphantom{0 ,\, 1 ,\, 1 ,\, 0}\right\rangle\)
It is conceivable to find some physical systems represented by first and second. It might be a lottery machine. Suppose the components of state vectors \(\mathbf{x} \vphantom{ \mathbf{x}}_{1}\)\(,\,\)\(\mathbf{x} \vphantom{ \mathbf{x}}_{2}\) stands respectively for slots of balls drawn out and ruled out. For simplicity, consider only two balls. In the beginning, both of them lie in the drawn-out slot represented by the initial state vector \(\left(\vphantom{1 \,/\, \surd\! 2}\right.\)\(1\)\(\,/\,\)\(\surd\!\)\(2\)\(\left.\vphantom{1 \,/\, \surd\! 2}\right)\)\(\left\langle\vphantom{1 ,\, 1}\right.\)\(1\)\(,\,\)\(1\)\(\left.\vphantom{1 ,\, 1}\right\rangle\). The host rolls the box fiercely, until one of the ball is stuck at the drawn-out slot, and another at the ruled-out slot, resulting final state vectors \(\mathbf{y} \vphantom{ \mathbf{y}}_{1}\)\(,\,\)\(\mathbf{y} \vphantom{ \mathbf{y}}_{2}\). Suppose it is so designed that only one of the balls would enter the ruled-out slot. Which transition matrix of \(\mathbf{A} \vphantom{ \mathbf{A}}_{1}\)\(,\,\)\(\mathbf{A} \vphantom{ \mathbf{A}}_{2}\) is applicable, is decided by the the angle by which the box rotates, the slightest perturbation of dust, the rough surface it lands on, and so on. They are so complicated that we believe that both events occur with probability \(1\)\(\,/\,\)\(2\) .
They might also signify a Stern-Gerlach experiment on a beam of electrons. Suppose the components of state vectors \(\mathbf{x} \vphantom{ \mathbf{x}}_{1}\)\(,\,\)\(\mathbf{x} \vphantom{ \mathbf{x}}_{2}\) stands respectively for two initial states of the sample and of the apparatus. Let that the first and second components mean spin up and spin down. Matrices \(\mathbf{A} \vphantom{ \mathbf{A}}_{1}\)\(,\,\)\(\mathbf{A} \vphantom{ \mathbf{A}}_{2}\) are the unitary evolution within a time period in the Schrödinger picture, resulting final state vectors \(\mathbf{y} \vphantom{ \mathbf{y}}_{1}\)\(,\,\)\(\mathbf{y} \vphantom{ \mathbf{y}}_{2}\). Initially the electrons are in a mixed state \(\left(\vphantom{1 \,/\, \surd\! 2}\right.\)\(1\)\(\,/\,\)\(\surd\!\)\(2\)\(\left.\vphantom{1 \,/\, \surd\! 2}\right)\)\(\left\langle\vphantom{1 ,\, 1}\right.\)\(1\)\(,\,\)\(1\)\(\left.\vphantom{1 ,\, 1}\right\rangle\), and the apparatus either imposes \(\mathbf{A} \vphantom{ \mathbf{A}}_{1}\)\(,\,\)\(\mathbf{A} \vphantom{ \mathbf{A}}_{2}\) to it, keeping one component in and projecting another away.
From the thought experiments above, I maintain that there is actually nothing random about these allegedly random objects, and the perceived randomness comes from the environment. For a classical system, the measured target is deterministic, and it is the external force exerted, which looks to be random. For a quantum system, the measured target’s wave function similarly evolves in a deterministic manner in the Schrödinger picture, and the measurement filters out alternative components, leaving them correlated with the apparatus, in a way determined by the current state of the environment. The only difference between them is that each lottery ball has a clear position, and each electron doesn’t have a position without being seen. But thus seeing either collection as an ensemble, they don’t exhibit much difference.
I suspect that it isn’t meaningful to talk about true randomness, because mathematically, we can’t adequately elucidate it, and perhaps never will. To me, in accordance with the Bayesian view, probability is a number assigned to the system when we don’t have finer information, just as in the cases above, we deem it evident that the environment is truly random. Even for quantum physics, as far as I see, this doesn’t contradict Copenhagen or other interpretations.
❧ May 15, 2019; August 23, 2021