Contextual Unification of Classical and Quantum Physics

Abstract

Following an article by John von Neumann on infinite tensor products, we develop the idea that the usual formalism of quantum mechanics, associated with unitary equivalence of representations, stops working when countable infinities of particles (or degrees of freedom) are encountered. This is because the dimension of the corresponding Hilbert space becomes uncountably infinite, leading to the loss of unitary equivalence, and to sectorisation. By interpreting physically this mathematical fact, we show that it provides a natural way to describe the “Heisenberg cut”, as well as a unified mathematical model including both quantum and classical physics, appearing as required incommensurable facets in the description of nature.

Appendices

Appendix 1: Proof of the Sectorisation Limit Lemmata

1.1 Proof of the Vector Sectorisation Limit Lemma

Let $\mathcal{C}$ and $\mathcal{C}'$ be two distinct sectors of $\mathcal{H} = \otimes _{\alpha \in I}{\mathcal{H}}_\alpha$.

Let $\mathcal{C} \ni \vert \Phi \rangle = \otimes _{\alpha \in I} \vert \phi _\alpha \rangle$ and $\mathcal{C}'\ni \vert \Psi \rangle = \otimes _{\alpha \in I} \vert \psi _\alpha \rangle$.

By Breakdown theorem a), one has $\langle \Psi \vert \Phi \rangle = \prod _{\alpha \in I} \langle \psi _\alpha \vert \phi _\alpha \rangle =0$

By the definition of the convergence of infinite products, the fact that this infinite complex product converges to value 0 means that $\forall \varepsilon >0, \exists \hbox { a finite } J_\varepsilon \subset I$ such that $\forall I_N =\{ \alpha _1,...,\alpha _N\}\subset I$, all $\alpha _i$’s different and $J_\varepsilon \subset I_N$, and then $\vert \prod _{\alpha \in I_N} \langle \psi _\alpha \vert \phi _\alpha \rangle \vert < \varepsilon .$ Thereby $\vert \Phi _N\rangle = \otimes _{\alpha \in I_N} \vert \phi _\alpha \rangle$ and $\vert \Psi _N\rangle = \otimes _{\alpha \in I_N} \vert \psi _\alpha \rangle$. $\square$

1.2 Proof of the Operator Sectorisation Limit Lemma

As $\vert \Psi \rangle \in \mathcal{C} \ni \vert \Phi \rangle$, $\#\{\alpha \in I, \vert \psi _\alpha \rangle \ne \vert \phi _\alpha \rangle \}< \infty$. As $\hat{A}$ is defined over ${\mathcal{H}}$, and ${\mathcal{H}}$ being the direct sum of all sectors, including $\mathcal{C}$, by the principle of the excluded middle, $\forall \vert \Psi \rangle \in \mathcal{C}$, either ${\hat{A}} \vert \Psi \rangle \in \mathcal{C}$ or ${\hat{A}} \vert \Psi \rangle \notin \mathcal{C}$.

• ${\hat{A}} \vert \Psi \rangle \in \mathcal{C} \Leftrightarrow {\hat{A}}$ has non-zero matrix elements only for $(\vert \psi _\alpha \rangle ,\vert \phi _\alpha \rangle ), \alpha \in I_N:=\{\alpha _1,...,\alpha _N \}$ finite , i.e. $p(\hat{A})< \infty$. In this case, ${\hat{A}}$ is of the form ${\hat{A}} = {\hat{A}}_N \otimes \otimes _{\alpha \in I{\setminus } I_N }\hat{I}_\alpha$, where ${\hat{A}}_N$ can be viewed as an $N\times N$ complex matrix that allows computing the inner product $\langle {\Phi }\vert {\hat{A}} \vert \Psi \rangle$ to a finite complex value.

• If ${\hat{A}}$ acts on an infinite number of $\alpha _i$’s i.e. $p(\hat{A})$ infinite, one still has to prove that ${\hat{A}} \vert \Psi \rangle$ relies on convergent sequences. For this let us examine $\hat{A} \vert \Psi \rangle$. Since $\vert \Psi \rangle \in {\mathcal{H}}$

  $$\begin{aligned} \vert \Psi \rangle = \sum _{k \in K} { \Psi _k} \otimes _{\alpha \in I} \vert \psi _\alpha ^k\rangle \end{aligned}$$

  (A1)

  where the $\{ \vert \psi _\alpha ^k\rangle \}_{\alpha \in I}, k\in K$ are $\# K$ convergent sequences. Now ${\hat{A}}$ is not necessarily the tensor product of operators in each ${\mathcal{H}}_{\alpha }$, but at least shall decompose on the sum of such products,

  $$\begin{aligned} {\hat{A}} = \sum _{p} {a}_p \otimes _{\alpha \in I} {\hat{A}}^p_\alpha \hbox { and } {\hat{A}}\vert \Psi \rangle = \sum _{k,p} { \Psi _k} {a}_p\otimes _{\alpha \in I} {\hat{A}}^p_\alpha \vert \psi _\alpha ^k\rangle . \end{aligned}$$

  (A2)

  So

  $$\begin{aligned} \langle \Psi \vert {\hat{A}}^\dagger {\hat{A}}\vert \Psi \rangle = \sum _{k,l} \Psi _k^*\Psi _l \sum _{p,q}{a}_p^*{a}_q{\prod _{\alpha \in I}}\langle \psi _\alpha ^k\vert {\hat{A}}_\alpha ^{p \dagger }{\hat{A}}_\alpha ^q \vert \psi _\alpha ^l\rangle \end{aligned}$$

  (A3)

  and $\langle \Psi \vert \Psi \rangle = \sum _{k,l} \Psi _k^*\Psi _l$

Now, as ${\hat{A}}$ is bounded, $\exists c\in \mathbb {R}, \langle \Psi \vert {\hat{A}}^\dagger {\hat{A}} \vert \Psi \rangle \le c \langle \Psi \vert \Psi \rangle$. As $\langle \Psi \vert \Psi \rangle$ is bounded so is $\langle \Psi \vert {\hat{A}}^\dagger {\hat{A}} \vert \Psi \rangle$ which entails that ${\hat{A}} \vert \Psi \rangle$ relies on convergent sequences $\{{\hat{A}}_\alpha ^q \vert \psi _\alpha ^l\rangle \}_{\alpha \in I}, \forall q,l$. Therefore ${\hat{A}} \vert \Psi \rangle \in \mathcal{H} {\setminus} \mathcal{C}$ so the Vector sectorisation limit lemma applies to $\langle \Phi \vert \hat{A} \vert \Psi \rangle .$. $\square$

Appendix 2: About Ontology and Infinities

The CSM ontology relies on the general idea of contextual objectivity, as introduced in [6]. In this framework the systems within contexts are objectively real, modalities are actual (repeatable) events, and projectors in Hilbert spaces are mathematical tools used to calculate non-classical probabilities. For CSM there is only one factual reality, the role of agents is the same as in classical probabilities, and there are no hidden variables—but the usual state vector $\vert \psi \rangle$ must be completed by specifying the measurement/observation context in order to define a modality. The so-called quantum non-locality is better understood as predictive incompleteness, associated with the fundamental randomness that is a necessary consequence of contextual quantisation and is fully compatible with relativistic causality [13]. All these features fit well in the mathematical framework presented in this article, and since there is no free lunch, the price to pay is embedding infinities in the formalism.

A very interesting discussion on the role of infinities in physics and mathematics can be found in a conference given in 1925 by Hilbert [35]. On the one hand Hilbert writes that infinities (as defined by Cantor) are a required ingredient for the completeness on mathematics: “No one shall drive us out of the paradise which Cantor has created for us”. On the other hand he writes that “The infinite is nowhere to be found in reality, no matter what experiences, observations, and knowledge are appealed to”. Our point of view is somehow different: we consider that, with respect to natural sciences, mathematics are basically a language used to describe (or to model) physical reality: they are neither reality itself, nor are we constrained to any sort of ontological identity between physics and mathematics.

Admitting that, there is no reason for not using the full power of the mathematical language, including infinities. In practice this is done already in differential or integral calculus; but here these ideas are pushed further, towards Cantor-type incommensurability, with not only numbers but full algebraic structures appearing at the infinite limit. It is true that such concepts are not easy to grasp, and may lead to currently intractable calculations; but this should not be a reason for the physicists not to enter Cantor’s paradise.