The Common Logic of Quantum Universe—Part I: The Case of Non-relativistic Quantum Mechanics

Abstract

One of the most challenging and fascinating issue in mathematical and theoretical physics concerns the possibility of identifying the logic underlying the so-called quantum universe, i.e., Quantum Mechanics and Quantum Gravity. Besides the sheer difficulty of the problem, inherent in the actual formulation of Quantum Mechanics—and especially of Quantum Gravity—to be used for such a task, a crucial aspect lies in the identification of the appropriate axiomatic logical proposition calculus to be associated to such theories. In this paper the issue of the validity of the conventional principle of non-contradiction (PNC) is called into question and is investigated in the context of non-relativistic Quantum Mechanics. In the same framework a modified form of the principle, denoted as 3-way PNC is shown to apply, which relates the axioms of quantum logic with the physical requirements placed by the Heisenberg Indeterminacy Principle.

Appendices

Appendix 1: The Logic of Classical Mechanics (Newtonian Cosmology)

Let us recall here briefly some elementary notions of Classical Mechanics (CM), also known as the so-called Newtonian cosmology. We consider for definiteness, as a possible realization of CM, a classical N-body system $S_{N}$. We shall assume for this purpose that $S_{N}$ is formed by N point particles of mass m which are described by the canonical state ${\mathbf{x}}\equiv \left( \mathbf{r},\mathbf{p}\right) \in \Gamma ^{N}$, with $\mathbf{ x}\equiv ({\mathbf{x}}_{1},\ldots,\mathbf{x_{N}}),$ ${\mathbf{r}}\equiv ({\mathbf{r}} _{1},\ldots,{\mathbf{r}}_{N})$ and ${\mathbf{p}}\equiv ({\mathbf{p}}_{1},\ldots,\mathbf{p }_{N})$ being respectively the N-body state, configuration and canonical momentum vectors. Here $\Gamma ^{N}$ is the N-body phase space $\Gamma ^{N}=\Omega ^{N}\times V^{N}$, with $\Omega ^{N}=\prod \limits _{i=1,N}\Omega$ and $V^{N}=\prod \limits _{i=1,N}V$ being respectively the N-body configuration and momentum spaces, and where $\Omega$ and V are the corresponding 1-body configuration and momentum spaces. Thus, in particular, the 1-body configuration space $\Omega \equiv E^{3}$ is identified with the whole Euclidean space on ${\mathbb{R}} ^{3}$, while the classical universe (or space-time) coincides with the so-called Galilean product space-time $\Omega \times I$, with $I\equiv E$ denoting the 1-dimensional (Euclidean) time axis spanned by the absolute time t. In CM, by assumption, the dynamics of $S_{N}$ is governed by the classical Hamilton–Jacobi equation

$$\begin{aligned} \frac{\partial S({\mathbf{r}},t)}{\partial t}+H(\mathbf{r},\nabla S({\mathbf{r}} ,t),t)=0, \end{aligned}$$

(108)

where $H({\mathbf{r}},\nabla S({\mathbf{r}},t),t)=\frac{1}{2m}\left( \nabla S\right) ^{2}+U({\mathbf{r}},t)$ and $U({\mathbf{r}},t)$ are respectively the classical Hamiltonian and potential acting on the classical N-body system $S_{N}$. As a consequence, by setting ${\mathbf{p}}=\nabla S({\mathbf{r}},t)$ it follows that the canonical state $x(t)=\left( {\mathbf{r}}(t),\mathbf{p} (t)\equiv \nabla S({\mathbf{r}}(t),t)\right)$ is uniquely determined by the corresponding Hamilton equations, namely

$$\begin{aligned} \left\{ \begin{array}{l} \frac{d{\mathbf{r}}}{dt}=\frac{1}{m}{\mathbf{p}}, \\ \frac{d{\mathbf{p}}}{dt}=-\nabla U(\mathbf{r},t), \end{array} \right. \end{aligned}$$

(109)

together with the initial value problem

$$\begin{aligned} {\mathbf{x}}(t_{0})={\mathbf{x}}_{0}=\left\{ {\mathbf{r}}_{0},\mathbf{p} _{0}\right\} , \end{aligned}$$

(110)

with ${\mathbf{x}}_{0}\in \Gamma ^{N}$ being suitably-prescribed. Then, by assumption the solution of the initial-value problem (109)–(110) exists (Newton’s law) and is unique (Newton’s deterministic principle). Denoting it by ${\mathbf{x}}(t)= {\chi }({\mathbf{x}}_{0},t_{0},t)$ and ${\mathbf{x}}(t_{0})= {\chi }({\mathbf{x}}(t),t,t_{0})$ its inverse map, it then follows that the phase-space mapping (i.e., a one-parameter bijection in the phase space $\Gamma ^{N}$)

$$\begin{aligned} {\mathbf{x}}(t_{0})\rightarrow {\mathbf{x}}(t)={\chi }({\mathbf{x}} _{0},t_{0},t) \end{aligned}$$

(111)

determines a classical dynamical system, to be identified as the $N-$body classical dynamical system $S_{N}.$

Then, a classical observable is realized by an arbitrary real vector phase-function $f({\mathbf{x}},t)\in {\mathbb{R}} ^{k}$ with $k\ge 1$, while the classical expectation value of a classical observable $f({\mathbf{x}},t)\in {\mathbb{R}} ^{k}$ (with $k\ge 1$) is defined by the image of the map corresponding to a prescribed arbitrary extended state $\left( {\mathbf{x}},t\right)$, i.e.,

$$\begin{aligned} \left( {\mathbf{x}},t\right) \in \Gamma ^{N}\times I\rightarrow f({\mathbf{x}} ,t). \end{aligned}$$

(112)

The logic of Classical Mechanics is the logic of classical expectation values of classical observables. For definiteness, let us consider the case of scalar classical observables $\alpha$, in which $f({\mathbf{x}},t)\in {\mathbb{R}}$. If $a\in {\mathbb{R}}$ we say that the logical sentence "the observable $f({\mathbf{x}},t)$ assumes the scalar value a for the state ${\mathbf{x}}\in \Gamma ^{N}$" is true if the image of the map (112), namely $f({\mathbf{x}} ,t)$, coincides with a, namely if $a=f({\mathbf{x}},t)$, being $a\in {\mathbb{R}}$ a real constant. Therefore, this means that the set $S=\{{\mathbf{x}}\in \Gamma ^{N}|a=f({\mathbf{x}},t)\}$ is non-empty. This is the set of all true propositions, while the complementary set, namely ${\mathcal{C}}S\equiv \Gamma ^{N}-S$ is the set of false propositions. As a consequence:

• 1. The set S identifies the classical truth in CM, while its complementary set, i.e., $CS\equiv \Gamma ^{N}-S$ corresponds to classical falsehood (the set of false propositions).

• 2. The expectation value ${\mathbf{a}}=f({\mathbf{x}},t)$ induces a partition of the phase-space $\Gamma ^{N}$ into two subsets $F=\{S,\Gamma ^{N}-S\}$.

• 3. Axiom of independence for measurement of independent observables: given an arbitrary set of independent observables $f_{i}( {\mathbf{x}},t)$ ($i=1,2f$, with f denoting the degree of freedom of the system), their expectation values $a_{i}=f_{i}({\mathbf{x}},t)$ are independent too.

In conclusion we can state that the logic of Classical Mechanics obeys the axioms of Classical Logic, namely:

a. All logical sentences are either true or false (2-way Principle of non contradiction).

b. All logical propositions are decidable (Deterministic principle).

c. All logical propositions are associative with respect to the logical connectives $\wedge$ (“and”), $\vee$ “or”, and $\lnot$“not” so that

$$\begin{aligned} A\wedge \left( B\vee C\right)= & {} \left( A\wedge B\right) \vee \left( A\wedge C\right) , \end{aligned}$$

(113)

$$\begin{aligned} A\wedge \left( \lnot B\vee \lnot C\right)= & {} \left( A\wedge \lnot B\right) \vee \left( A\wedge \lnot C\right) . \nonumber \end{aligned}$$

(114)

The last property follows because propositions of this type can be combined by the logical connectives “and”, “or”, and “not”. The structure of propositional logic for CM is obviously that of classical logic, i.e., a Boolean algebra of subsets of the state space.

Appendix 2

Let us proceed here with the explicit evaluation of the standard deviations introduced in Sect. 3, $\sigma _{\Delta r_{i}}^{2}$ and $\sigma _{\pi _{\Delta r_{j}}}^{2}$, prescribed in terms of Eqs. (48) and (49) via the definition of the local scalar product (18). The quantum wave function is taken of the Madelung form (8), while the quantum PDF and phase-function $\rho ({\mathbf{r}} ,t)$ and $S^{(q)}({\mathbf{r}},t)$ take respectively the N-body Gaussian-like GLP-quantum PDF (27) and the second-degree polynomial (29). To evaluate explicitly the standard deviation (48) we notice that the following identities hold

$$\begin{aligned} \overline{\Delta r_{j}}= & {} \left\langle \psi \left| \Delta r_{j}\psi \right. \right\rangle _{L}=\int d^{3N}{\mathbf{r}}\rho (\mathbf{r},t)\Delta r_{j}=\int d^{3N}\Delta {\mathbf{r}}\rho _{G}(\Delta {\mathbf{r}})\Delta r_{j}=0, \end{aligned}$$

(115)

$$\begin{aligned} \sigma _{\Delta r_{j}}^{2}= & {} \left\langle \psi \left| \left( \Delta r_{j}- \overline{\Delta r_{j}}\right) ^{2}\psi \right. \right\rangle _{L}=\int d^{3N}{\mathbf{r}}\rho (\mathbf{r},t)\Delta r_{j}^{2}=\int d^{3N}\Delta {\mathbf{r}}\rho _{G}(\Delta {\mathbf{r}})\Delta r_{j}^{2}=\frac{r_{th}^{2}}{2}. \end{aligned}$$

(116)

Furthermore, one obtains

$$\begin{aligned} {\overline{\pi }}_{\Delta r_{j}}=\left\langle \psi \left| \pi _{\Delta r_{j}}\psi \right. \right\rangle _{L}=\int d^{3N}\Delta {\mathbf{r}}\rho _{G}(\Delta {\mathbf{r}})\left[ a(\mathbf{r},t)\Delta r_{j}\mathbf{+}b_{j}( \mathbf{r},t)\right] =b_{j}(\mathbf{r},t), \end{aligned}$$

(117)

and furthermore

$$\begin{aligned} \sigma _{\pi _{\Delta r_{j}}}^{2}=\left\langle \psi \left| \left( \pi _{\Delta r_{j}}-{\overline{\pi }}_{\Delta r_{j}}\right) ^{2}\psi \right. \right\rangle _{L}=\left\langle \left( \pi _{\Delta r_{j}}^{*}-\overline{ \pi }_{\Delta r_{j}}\right) \psi \left| \left( \pi _{\Delta r_{j}}- {\overline{\pi }}_{\Delta r_{j}}\right) \psi \right. \right\rangle _{L}, \end{aligned}$$

(118)

where

$$\begin{aligned} \left\langle \left( \pi _{\Delta r_{j}}^{*}-{\overline{\pi }}_{\Delta r_{j}}\right) \psi \left| \left( \pi _{\Delta r_{j}}-{\overline{\pi }} _{\Delta r_{j}}\right) \psi \right. \right\rangle _{L}=\frac{4\hbar ^{2}}{ r_{th}^{4}}\int d^{3N}\Delta {\mathbf{r}}\rho _{G}(\Delta {\mathbf{r}})\Delta r_{j}^{2}, \end{aligned}$$

(119)

so that

$$\begin{aligned} \sigma _{\pi _{\Delta r_{j}}}=\frac{2\hbar ^{2}}{r_{th}^{2}}. \end{aligned}$$

(120)

Appendix 3

Let us introduce for the quantum wave-function $\psi ({\mathbf{r}},t)$ the asymptotic representation

$$\begin{aligned} \psi ({\mathbf{r}}^{\prime },r_{i},\varepsilon r_{i},t)\equiv e^{\frac{i}{ \hbar }r_{(i)}\pi _{r_{i}}^{(o)}}{\widehat{\psi }}({\mathbf{r}}^{\prime },\varepsilon r_{i},t), \end{aligned}$$

(121)

with ${\widehat{\psi }}({\mathbf{r}}^{\prime },\varepsilon r_{i},t)$ being a suitably smooth complex function, ${\mathbf{r}}^{\prime }\equiv \left( r_{1},r_{i-1},r_{i+1},\ldots,r_{3N}\right)$, $\varepsilon$ being an infinitesimal real parameter and $\pi _{{\mathbf{r}}_{i}}^{(o)}$ a constant asymptotic real eigenvalue of the quantum momentum $\pi _{r_{i}}$ assumed independent of $({\mathbf{r}},t)$ and $\varepsilon$. Then, denoting $r_{i1}=\varepsilon r_{i}$, by construction the wave function $\psi (\mathbf{r }^{\prime },r_{i},r_{i1},t)$ satisfies the asymptotic eigenvalue equation:

$$\begin{aligned} \pi _{{\mathbf{r}}_{i}}\psi ({\mathbf{r}}^{\prime },r_{i},r_{i1},t)=\pi _{ {\mathbf{r}}_{i}}^{(o)}e^{\frac{i}{\hbar }r_{(i)}\pi _{{\mathbf{r}}_{i}}^{(o)}} {\widehat{\psi }}({\mathbf{r}}^{\prime },r_{i1},t)-\varepsilon i\hbar e^{\frac{i }{\hbar }r_{(i)}\pi _{{\mathbf{r}}_{i}}^{(o)}}\frac{\partial }{\partial r_{i1}} {\widehat{\psi }}({\mathbf{r}}^{\prime },r_{i1},t). \end{aligned}$$

(122)

The Born rule $\rho ({\mathbf{r}},t)\equiv$ $\left| \psi ({\mathbf{r}} ,t)\right| ^{2}$ then delivers

$$\begin{aligned} \rho ({\mathbf{r}},t)\equiv \rho ({\mathbf{r}}^{\prime },r_{i1},t)=\left| {\widehat{\psi }}({\mathbf{r}}^{\prime },r_{i1},t)\right| ^{2}, \end{aligned}$$

(123)

where for $\rho ({\mathbf{r}}^{\prime },r_{i1},t)$ we assume a Gaussian-like solution of the type (27). Then, denoting $\Delta {\mathbf{r}} \equiv \left\{ \Delta r_{1},\ldots,\Delta r_{3N}\right\}$ and $\Delta \mathbf{r }^{\prime }\equiv \left\{ \Delta r_{1},\Delta r_{i-1},\Delta r_{i+1}...,\Delta r_{3N}\right\}$ this yields

$$\begin{aligned} \rho ({\mathbf{r}}^{\prime },r_{i1},t)\equiv \rho _{G}^{\prime }(\Delta {\mathbf{r}}{^{\prime }})\rho _{Gi}(\varepsilon \Delta r_{i})\exp \left\{ -\int \limits _{t_{o}}^{t}dt^{\prime }\nabla _{{\mathbf{R}}}\cdot \nabla _{ {\mathbf{R}}}S(\Delta {\mathbf{r}}^{\prime },\varepsilon \Delta r_{i},t^{\prime })\right\} , \end{aligned}$$

(124)

where $\rho _{G}^{\prime }(\Delta {\mathbf{r}}{^{\prime }})$ and $\rho _{Gi}(\varepsilon \Delta r_{i})$ denote respectively the $\left( 3N-1\right)$- and 1-dimensional Gaussian distributions

$$\begin{aligned} \rho _{G}^{\prime }(\Delta {\mathbf{r}}{^{\prime }})= & {} \frac{1}{\pi ^{3(N-1)/2}r_{th}^{3(N-1)}}\exp \left\{ -\frac{\left( \Delta {\mathbf{r}} ^{\prime }\right) ^{2}}{r_{th}^{2}}\right\} , \\ \rho _{Gi}(\varepsilon \Delta {\mathbf{r}}_{i})= & {} \frac{1}{\pi ^{3/2}r_{th}^{3}}\exp \left\{ -\frac{\left( \varepsilon \Delta {\mathbf{r}} _{i}\right) ^{2}}{r_{th}^{2}}\right\} . \end{aligned}$$

(125)

Therefore, by construction the normalization

$$\begin{aligned} \int \limits _{-\infty }^{\infty }d\Delta r_{i}\frac{1}{\pi ^{1/2}r_{th}}\exp \left\{ -\frac{\left( \varepsilon \Delta {\mathbf{r}}_{i}\right) ^{2}}{ r_{th}^{2}}\right\} =\frac{1}{\varepsilon } \end{aligned}$$

(126)

holds, which implies that $\lim _{\varepsilon \rightarrow 0}\rho ({\mathbf{r}} ^{\prime },\varepsilon {\mathbf{r}}_{i},t)$ is not dynamically consistent in the limit $\varepsilon \rightarrow 0$. In terms of $\psi ({\mathbf{r}}^{\prime },r_{i},\varepsilon r_{i},t)$ and $\rho ({\mathbf{r}}^{\prime },\varepsilon r_{i},t)$, however, the standard deviation $\sigma _{\pi _{r_{i}}}^{2}\equiv \left\langle \left\langle \psi ({\mathbf{r}}^{\prime },r_{i},\varepsilon r_{i},t)\left| \left( D\pi _{r_{i}}\right) ^{2}\psi ({\mathbf{r}}^{\prime },r_{i},\varepsilon r_{i},t)\right. \right\rangle \right\rangle$ can be formally evaluated. In particular one finds that

$$\begin{aligned} \sigma _{\pi _{r_{i}}}^{2}\equiv \left\langle \left\langle (D\pi _{r_{i}})^{*}\psi ({\mathbf{r}}^{\prime },r_{i},\varepsilon r_{i},t)\left| \left( D\pi _{r_{i}}\right) \psi ({\mathbf{r}}^{\prime },r_{i},\varepsilon r_{i},t)\right. \right\rangle \right\rangle , \end{aligned}$$

(127)

where straightforward algebra yields

$$\begin{aligned} \left\{ \begin{array}{l} (D\pi _{r_{i}})^{*}\psi ^{*}({\mathbf{r}}^{\prime },r_{i},\varepsilon r_{i},t)=\varepsilon i\hbar e^{-\frac{i}{\hbar }r_{(i)}\pi _{r_{i}}^{(o)}} \frac{\partial }{\partial r_{i1}}{\widehat{\psi }}^{*}({\mathbf{r}}^{\prime },r_{i1},t), \\ (D\pi _{r_{i}})\psi ({\mathbf{r}}^{\prime },r_{i},\varepsilon r_{i},t)=-\varepsilon i\hbar e^{\frac{i}{\hbar }r_{(i)}\pi _{r_{i}}^{(o)}} \frac{\partial }{\partial r_{i1}}{\widehat{\psi }}({\mathbf{r}}^{\prime },r_{i1},t). \end{array} \right. \end{aligned}$$

(128)

Hence, noting that one expects $\hbar ^{2}\left\langle \left\langle \left| \frac{\partial }{\partial r_{i1}}{\widehat{\psi }}({\mathbf{r}} ^{\prime },r_{i1},t)\right| ^{2}\right\rangle \right\rangle \sim 1/O(\varepsilon )$ it follows

$$\begin{aligned} \sigma _{\pi _{r_{i}}}^{2}\equiv \varepsilon ^{2}\hbar ^{2}\left\langle \left\langle \left| \frac{\partial }{\partial r_{i1}}{\widehat{\psi }}( {\mathbf{r}}^{\prime },r_{i1},t)\right| ^{2}\right\rangle \right\rangle \sim O(\varepsilon ), \end{aligned}$$

(129)

and hence the asymptotic estimate

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\sigma _{\pi _{r_{i}}}^{2}=0. \end{aligned}$$

(130)

indeed holds.