The Common Logic of Quantum Universe—Part II: The Case of Quantum Gravity

Abstract

The logical structure of quantum gravity (QG) is addressed in the framework of the so-called manifestly covariant approach. This permits to display its close analogy with the logics of quantum mechanics (QM). More precisely, in QG the conventional 2-way principle of non-contradiction (2-way PNC) holding in Classical Mechanics is shown to be replaced by a 3-way principle (3-way PNC). The third state of logical truth corresponds to quantum indeterminacy/undecidability, i.e., the occurrence of quantum observables with infinite standard deviation. The same principle coincides, incidentally, with the earlier one shown to hold in Part I, in analogous circumstances, for QM. However, this conclusion is found to apply only provided a well-defined manifestly-covariant theory of the gravitational field is adopted both at the classical and quantum levels. Such a choice is crucial. In fact it makes possible the canonical quantization of the underlying unconstrained Hamiltonian structure of general relativity, according to an approach recently developed by Cremaschini and Tessarotto (2015–2021). Remarkably, in the semiclassical limit of the theory, Classical Logic is proved to be correctly restored, together with the validity of the conventional 2-way principle.

Appendix 1: Proper-Time Parametrization of Spacetime

We start recalling the notion of proper time in the context of GR and its physical interpretation. Its geometric definition follows by introducing a mapping between a subset of the real axis $I\subseteq {\mathbb {R}}$ and the same background spacetime $\left\{ {\mathbf {Q}}^{4},\widehat{g} (r)\right\}$ of the form $r=r(s)$, being r an arbitrary $4-$position (“point”) of $\left\{ {\mathbf {Q}}^{4},\widehat{g}(r)\right\}$ and $s\in I\subseteq {\mathbb {R}}$ being a suitable $4-$scalar, to be denote here as “proper time”. Therefore, the geometric meaning of s depends on its precise prescription. This is obtained by means of two definitions:

• (A) Proper time s

  First, identifying s with the arc length $ds^{2}=\widehat{g}_{\mu \nu }dr^{\mu }dr^{\nu }$ measured along an appropriate finite-length geodetic defined with respect to the background space-time $\left\{ {\mathbf {Q}}^{4}, \widehat{g}(r)\right\}$, i.e.,

  $$\begin{aligned} C_{(r_{o},r_{1})}=\left\{ \left. r\right| r=r(s^{\prime }),r_{o}=r(s_{o}),r_{1}=r(s_{1}),s^{\prime }\in \left[ s_{o},s_{1}\right] \ ,r_{o}\in \varvec{\Sigma }_{0}^{3},r_{1}\in \varvec{\Sigma } _{1}^{3}\right\} , \end{aligned}$$

  (132)

  $\Sigma _{0}^{3}$ and $\Sigma _{1}^{3}$ being two suitable subsets of ${\mathbf {Q}}^{4}$ (see below). Denoting $\frac{dr^{\mu }}{ds}=t^{\mu }(r)$ the tangent to the curve at r and $\nabla _{\mu }$ the covariant derivative evaluated with respect to the background spacetime$\left\{ {\mathbf {Q}}^{4},\widehat{g}(r)\right\}$, then by construction along the same curve if follows identically that $\nabla _{\mu }t^{\mu }(r)=0$.

• (B) Family of geodetics $\left\{ C_{(r_{o},r_{1})}\right\}$

The family $\left\{ C_{(r_{o},r_{1})}\right\}$ is defined in such a way that:

• (1) For fixed proper times $s_{o}$ and $s_{1}$ (with $s_{o}<$ $s_{1}$), each curve $C_{(r_{o},r_{1})}\in \left\{ C_{(r_{o},r_{1})}\right\}$ has the extrema $r_{o}=r(s_{o}),r_{1}=r(s_{1})$ crossed by the same curve respectively at proper times $s_{o}$ and $s_{1}$. In addition, by assumption: a) all curves $C_{(r_{o},r_{1})}$ belong to the same connected subset ${\mathbf {Q}}_{1}^{4}$ of spacetime $\left\{ {\mathbf {Q}}^{4},\widehat{g} (r)\right\}$ which has everywhere the same signature; b) the lower and upper extrema $r_{o}=r(s_{o})$ and $r_{1}=r(s_{1})$ belong to two smooth hypersurfaces $\Sigma _{0}^{3}$ (“lower” boundary) and $\Sigma _{1}^{3}$ (“upper” boundary) of the subset of ${\mathbf {Q}}_{1}^{4}$. Hence, ${\mathbf {Q}} _{1}^{4}$ is the subset of ${\mathbf {Q}}^{4}$ having lower and upper boundaries $\Sigma _{0}^{3}$ and $\Sigma _{1}^{3}$ where the extrema of all curves $C_{(r_{o},r_{1})}$ lie.

• (2) For each point r of the said subset of ${\mathbf {Q}}_{1}^{4}$ there is a unique curve $C_{(r_{o},r_{1})}\in$ $\left\{ C_{(r_{o},r_{1})}\right\}$ which belongs to it.

• (3) Two geodetics of the family never cross each other.

• (4) All geodetic curves $C_{(r_{o},r_{1})}$ are prescribed in such away to be mono-energetic, i.e., so that, in the frame in which at $r_{o}=r(s_{o})$ the spacetime is locally flat, the corresponding tangent $4-$vectors $t^{\mu }(r_{o})\equiv \left. \frac{dr^{\mu }}{ds}\right| _{r_{o}}$ have all the same zero component of the $4-$velocity.

The proper-time parametrization ($s-$parametrization) of spacetime is then realized by parametrizing the background metric tensor $\widehat{g} (r)=\left\{ \widehat{g}_{\mu \nu }(r)\right\}$ and the canonical tensor fields of the Hamiltonian formulation $x_{R}=\left\{ g_{\mu \nu },\pi ^{\mu \nu }\right\}$, and in particular the variational tensor field $g=\left\{ g_{\mu \nu }\right\}$, in terms of the proper time s. Thus, at the classical level, the $s-$parametrization of $\widehat{g}(r)$ is obtained by means of the representation

$$\begin{aligned} \widehat{g}(r)\rightarrow \widehat{g}(r(s))\equiv \left\{ \widehat{g}_{\mu \nu }(r(s))\right\} . \end{aligned}$$

(133)

Instead, the corresponding parametrization of the variational tensor field $g=\left\{ g_{\mu \nu }\right\}$ and its conjugate canonical momentum $\pi =\left\{ \pi ^{\mu \nu }\right\}$ is obtained letting

$$\begin{aligned} g\rightarrow & {} g(s)\equiv g(r(s),s), \end{aligned}$$

(134)

$$\begin{aligned} \pi\rightarrow & {} \pi (s)\equiv \pi (r(s),s). \end{aligned}$$

(135)

Finally, regarding the physical interpretation, this amounts to assume that the Hamiltonian structure of GR (to be suitably identified) corresponds to disturbances (or signals) of the background metric tensor which propagate in the space-time and occur with finite, i.e., sub-luminal, speed of propagation measured by the proper-time s. Such an interpretation becomes, nevertheless, obvious in the quantum formulation where such “signals” are interpreted in terms of massive gravitons (see Refs. [11]).