# Quantum-Wave Equation and Heisenberg Inequalities of Covariant Quantum Gravity

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## Abstract

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## 1. Introduction

- Issue #1: the canonical g-quantization. The issue concerns the quantization, here referred to as g-quantization, of the classical canonical state $x=\left(\right)open="\{"\; close="\}">g,\pi $ with $\pi \equiv \left(\right)open="\{"\; close="\}">{\pi}_{\mu \nu}$ being the classical reduced-dimensional canonical momentum conjugate to the continuum field tensor $g\equiv \left(\right)open="\{"\; close="\}">{g}_{\mu \nu}\left(r\right)$. This is realized by means of a correspondence principle between the classical state $x=\left(\right)open="\{"\; close="\}">g,\pi $ and the corresponding quantum variables ${x}^{\left(q\right)}=\left(\right)open="\{"\; close="\}">{g}^{\left(q\right)}\equiv g,{\pi}^{\left(q\right)}$, with ${\pi}^{\left(q\right)}\equiv \left(\right)open="\{"\; close="\}">{\pi}_{\mu \nu}^{\left(q\right)}$ being the corresponding quantum operator. In such a context, the question arises of the unique prescription of the quantum-wave function and the corresponding quantum wave-equation associated with g-quantization. These should be understood respectively as quantum wave-function and quantum wave-equation of the universe and, therefore, to hold for arbitrary realizations of the background space-time. According to [8], the four-scalar (i.e., obtained by saturation of four-tensors) quantum state $\psi $ should dynamically evolve with respect to an invariant proper-time parameter s defining the canonical Hamiltonian flow. Hence, besides $g\equiv \left(\right)open="\{"\; close="\}">{g}_{\mu \nu}\left(r\right)$ $\psi $ is parametrized in terms of the prescribed metric tensor $\widehat{g}\left(r\right)$ of the background space-time, as well as the four-position ${r}^{\mu}$ and the proper time s, whose physical meaning in the context of QG remains nevertheless to be specified.
- Issue #2: the quantum Hamilton variational principle. This is about the search for a variational principle for the same quantum wave equation, which may provide an additional “a posteriori” justification of its physical validity. The form of the same equation, in fact, should be consistent with the existence of real symmetric functionals, which are bilinear with respect to the quantum wave function $\psi $, while satisfying the principle of manifest-covariance for the Hamiltonian functional. In agreement with the variational setting developed for the classical derivation of the Einstein equations, also in the quantum regime, we seek the implementation of a synchronous variational principle characterized by having integral differential four-volume and/or line elements, which are left invariant during synchronous variations (see the details in [6]).
- Issue #3: the g-quantization Heisenberg inequalities. This concerns the problem of quantum measurement and, more precisely, the possible validity of a suitable Heisenberg principle, which may provide inequalities appropriate for the treatment of g-quantization and relate the standard deviations of the quantum observables ${x}^{\left(q\right)}$.

## 2. The Classical Hamiltonian Structure of SF-GR

#### 2.1. Reduced-Dimensional Hamiltonian Representation

#### 2.2. Hamilton–Jacobi Equation

## 3. Hamilton–Jacobi $\mathit{g}$-Quantization

- First, the canonical quantization rule, herein referred to as the g-quantization rule, prescribing the mapping between the classical and quantum Hamiltonian structures:$$\left(\right)open="("\; close=")">x=\left(\right)open="\{"\; close="\}">g,\pi \Rightarrow \left(\right)open="("\; close=")">{x}^{\left(q\right)}=\left(\right)open="\{"\; close="\}">{g}^{\left(q\right)},{\pi}^{\left(q\right)}$$$$\left(\right)$$
- Second, the quantum-wave equation advancing in proper-time the same quantum state. This is provided by the CQG-wave equation:$$i\hslash \frac{\partial}{\partial s}\psi \left(s\right)=\left(\right)open="["\; close="]">{H}_{R}^{\left(q\right)},\psi \left(s\right)$$

- It is manifestly covariant, i.e., it retains its form under the action of arbitrary local point transformations (1), which preserve by construction the differential manifold of the space-time $\left(\right)$.
- It is an evolution equation, which is parametrized in terms of the proper-time s, i.e., the Riemann distance, which is associated with the background space-time for subluminal trajectories.
- It advances in proper-time the four-scalar wave-function $\psi \left(s\right)$, the associated configuration-space quantum PDF being prescribed by Equation (25).
- The same wave equation holds in principle for arbitrary initial conditions, as well as for arbitrary external sources, as is appropriate for the treatment of problems of QG and quantum cosmology.

## 4. The Quantum Hamilton Variational Principle

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## 5. The Variational Quantum Hydrodynamic Equations

## 6. Generalized Heisenberg Inequalities for $\mathit{g}$-Quantization

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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Cremaschini, C.; Tessarotto, M.
Quantum-Wave Equation and Heisenberg Inequalities of Covariant Quantum Gravity. *Entropy* **2017**, *19*, 339.
https://doi.org/10.3390/e19070339

**AMA Style**

Cremaschini C, Tessarotto M.
Quantum-Wave Equation and Heisenberg Inequalities of Covariant Quantum Gravity. *Entropy*. 2017; 19(7):339.
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**Chicago/Turabian Style**

Cremaschini, Claudio, and Massimo Tessarotto.
2017. "Quantum-Wave Equation and Heisenberg Inequalities of Covariant Quantum Gravity" *Entropy* 19, no. 7: 339.
https://doi.org/10.3390/e19070339