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# Generally Covariant Quantum Mechanics

Jeremy Kerneis

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# Generally Covariant Quantum Mechanics

## A geometric unification of general relativity and quantum mechanics

**Author**: Jérémy Kerneïs jeremy.kerneis@proton.me

**Date**: Thursday, 8th May 2025 (Last update: Friday, 30th May 2025)

**DOI**: KERNEIS, J. (2025). Generally Covariant Quantum Mechanics. Zenodo.

https://doi.org/10.5281/zenodo.16934481

## Abstract

In a book written by C. Rovelli and F. Vidotto in 2013, one can read: "our successful theories are based on contradictory hypotheses. A good student following a general relativity class in the morning and a quantum field theory class in the afternoon must think her teachers are chumps, or haven't been talking to one another for decades. They teach two totally different worlds. [...] Copernicus, Einstein, and, to a lesser extent, Newton, have understood something new about the world without new data - just comparing apparently contradictory successful theories. We are in the same privileged situation. We lack their stature, but we are not excused from trying hard" [1]. Nowadays, the two leading candidates to unify general relativity and quantum mechanics are string theory and loop quantum gravity. The main problem of all existing approaches is the lack of experimental data. The theory introduced here, generally covariant quantum mechanics, is based on ideas developed by R. Penrose. He wrote in 2014: "the case for 'gravitizing' quantum theory is at least as strong as that for quantizing gravity. Accordingly, the principles of general relativity must influence, and actually change, the very formalism of quantum mechanics" [2]. Generally covariant quantum mechanics provides interesting links with string theory and loop quantum gravity and starts with two remarkable successes:

- quantum electrodynamics, quantum physics in curved spacetime, Einstein's gravity & Maxwell's electrodynamics, hydrodynamics of perfect fluids, Newtonian mechanics, thermodynamics of perfect gases can all be derived from a single action (see [Section 7]);

- the foundations of the theory can be compared to existing experimental data (see [Section 8]).

In the present paper, the word *classical* refers to the macroscopic domain (general relativity being then a classical theory) while the word *quantum* refers to the microscopic domain. Long computations are left in the annexes. Our sign conventions are those of Landau and Lifschitz [3]. Finally, we use the Einstein summation convention (with greek or capital letters) and the usual notations for partial derivatives :

$$u_{\alpha}u^{\alpha} = \sum_{\alpha} u_{\alpha}u^{\alpha} \quad ; \quad g_{\mu\nu,\alpha} = \partial_{\alpha}g_{\mu\nu} = \frac{\partial g_{\mu\nu}}{\partial x^{\alpha}}$$

## Contents

<table><tr><td>1</td><td>New results about the principle of least action</td><td>2</td></tr><tr><td>2</td><td>Derivation of quantum mechanics from general relativity</td><td>4</td></tr><tr><td>3</td><td>Derivation of general relativity from quantum physics</td><td>11</td></tr><tr><td>4</td><td>New results in general relativity (Kaluza-Klein theory)</td><td>13</td></tr><tr><td>5</td><td>New results in quantum physics</td><td>20</td></tr><tr><td>6</td><td>New results in canonical quantum gravity</td><td>37</td></tr><tr><td>7</td><td>The theory of generally covariant quantum mechanics</td><td>39</td></tr><tr><td>8</td><td>Comparison of the theory with experimental data</td><td>63</td></tr><tr><td>9</td><td>Conclusion</td><td>71</td></tr><tr><td>10</td><td>References</td><td>74</td></tr><tr><td>11</td><td>Annexes : Detailed computations</td><td>79</td></tr></table>
# 1 New results about the principle of least action

About 2000 years ago, Hero of Alexandria stated that light follows the shortest path [4][5][6]. His discovery was largely forgotten until the intellectual and philosophical movement that took place in Europe between the 17th and 19th centuries, a period called 'the Age of Enlightenment'. According to C. Lanczos, "it is the only period of cosmic thinking in the entire history of Europe since the time of the Greeks" [7]. Indeed, P. De Fermat explained in 1662: "nature always acts by the easiest means [...] or in any case by the shortest time, in order to soften its work and to complete its operation sooner" [8]. In the 18th century, the principle of least action (or shortest path, or shortest time) became the basis of the Lagrangian formulation of classical mechanics which encompasses Newton's laws. In 1902, H. Poincaré analyzed: "The very statement of the principle of least action has something shocking to the mind. To go from one point to another one, a material molecule [...] will follow the geodesic line, i.e., the shortest path. It seems that this molecule knows the point where one wants it to go, that it anticipates the time needed to reach it along such or such path, and then chooses the most convenient path. In a sense, the statement presents this molecule as a free animated being. It is clear that it would be better to replace it by a less shocking statement where, as philosophers would say, the final causes would not appear to replace the efficient ones" [9]. In 1900, D. Hilbert put forth 23 problems that helped set the research agenda for mathematics in the 20th century. He stated: "the problem of the shortest line plays a chief and historically important part in the foundations of geometry, the theory of curved lines and surfaces, in mechanics and variational calculus". His 23rd problem was to develop the mathematical methods associated to the principle of least action [10]. The present section introduces a new formulation of this principle.

We use the letter $\tau$ to parametrize a path. In general relativity, $\tau$ is the proper time. The variables $x^\alpha(\tau)$ and $\dot{x}^\alpha(\tau) = \frac{dx^\alpha}{d\tau}$ are the position and the velocity. It is assumed that they are independent variables. The Lagrangian is $L(x^\alpha, \dot{x}^\alpha, \tau)$. The action $S$ is the integral of the Lagrangian with respect to $\tau$. Explicitly, $S = \int L(x^\alpha, \dot{x}^\alpha, \tau)d\tau$. As R. P. Feynman explained to his students between 1961 and 1964, in classical physics "the miracle is that the true path is the one for which that integral is least" [11]. Precisely, the laws of motion are derived from the condition that the action is stationary on any interval $[\tau_1, \tau_2]$. On any interval $[\tau_1, \tau_2]$, we then consider an arbitrary infinitesimal variation of the path which changes the variables according to: $x^\alpha \to x^\alpha + \delta x^\alpha$ and $\dot{x}^\alpha \to \frac{d}{d\tau}(x^\alpha + \delta x^\alpha) = \dot{x}^\alpha + \delta \dot{x}^\alpha$. The boundary conditions $\delta x^\alpha(\tau_1) = \delta x^\alpha(\tau_2) = 0$ are imposed such that the initial and final positions remain unchanged. When the path is varied, $S$ becomes $S + \delta S$. By definition, $S$ is least, or more precisely stationary, if $\delta S = 0$ to the first order. With $\delta \dot{x}^\alpha \frac{\partial L}{\partial \dot{x}^\alpha} = \frac{d}{d\tau}(\delta x^\alpha \frac{\partial L}{\partial \dot{x}^\alpha}) - \delta x^\alpha \frac{d}{d\tau} \frac{\partial L}{\partial \dot{x}^\alpha}$, we get:

$$ \delta S = \int_{\tau_1}^{\tau_2} \delta L d\tau = \int_{\tau_1}^{\tau_2} \delta x^\alpha \frac{\partial L}{\partial x^\alpha} + \delta \dot{x}^\alpha \frac{\partial L}{\partial \dot{x}^\alpha} d\tau = \int_{\tau_1}^{\tau_2} \delta x^\alpha \left( \frac{\partial L}{\partial x^\alpha} - \frac{d}{d\tau} \frac{\partial L}{\partial \dot{x}^\alpha} \right) d\tau + \int_{\tau_1}^{\tau_2} \frac{d}{d\tau} \left( \delta x^\alpha \frac{\partial L}{\partial \dot{x}^\alpha} \right) d\tau \quad (1.1) $$

Since $\delta x^\alpha(\tau_1) = \delta x^\alpha(\tau_2) = 0$, the last term vanishes. Since $\delta x(\tau)$ is arbitrary, the condition $\delta S = 0$ implies the laws of motion, i.e. the famous Euler-Lagrange equations [12][13][14][15]:

$$ \frac{\partial L}{\partial x^{\alpha}} - \frac{d}{d\tau} \frac{\partial L}{\partial \dot{x}^{\alpha}} = 0 \qquad (1.2) $$

In his lecture on classical mechanics (1842-1843), C. G. J. Jacobi emphasized: "This is the 'principe de la moindre action' wrongly called the principle of least action. [...] In almost all textbooks, even in the best, those of Poisson, Lagrange and Laplace, the principle has been presented so that, in my view, it is impossible to understand. [...] The minimum of the above integral is further to be understood in the sense that when the initial and final positions are given, the integral, among all possible paths from one position to another, would be a minimum for the one actually described. [...] It is difficult to find a metaphysical reason for the principle of least action if it is expressed in this correct form, as is necessary" [16]. Now we provide a new form of the equations of motion given in equation (1.2). We introduce an auxiliary action $S_0 = \zeta(S)$ where $\zeta$ is an arbitrary monotonic function: consequently, $S$ is stationary if and only if $S_0$ is stationary. We define the Lagrangian $L_0(x^\alpha, \dot{x}^\alpha, \tau)$ according to $S_0 = \int L_0 d\tau \Leftrightarrow L_0 = \frac{dS_0}{d\tau} = \frac{d\zeta(S)}{d\tau} = \frac{dS}{d\tau} \frac{d\zeta(S)}{dS} = L \frac{d\zeta(S)}{dS}$. The action $S_0$ is stationary if:

$$ 0 = \frac{d}{d\tau} \left( \frac{\partial L_0}{\partial \dot{x}^{\alpha}} \right) - \left( \frac{\partial L_0}{\partial x^{\alpha}} \right) = \frac{d}{d\tau} \left( \frac{\partial L}{\partial \dot{x}^{\alpha}} \frac{d\zeta}{dS} + L \frac{\partial S}{\partial \dot{x}^{\alpha}} \frac{d^2\zeta}{dS^2} \right) - \left( \frac{\partial L}{\partial x^{\alpha}} \frac{d\zeta}{dS} + L \frac{\partial S}{\partial x^{\alpha}} \frac{d^2\zeta}{dS^2} \right) \quad (1.3) $$

$$ 0 = \left( \left( \frac{d}{d\tau} \frac{\partial L}{\partial \dot{x}^{\alpha}} \right) \frac{d\zeta}{dS} + \frac{\partial L}{\partial \dot{x}^{\alpha}} L \frac{d^2\zeta}{dS^2} + \frac{d}{d\tau} \left( L \frac{\partial S}{\partial \dot{x}^{\alpha}} \right) \frac{d^2\zeta}{dS^2} + L^2 \frac{\partial S}{\partial \dot{x}^{\alpha}} \frac{d^3\zeta}{dS^3} \right) - \left( \frac{\partial L}{\partial x^{\alpha}} \frac{d\zeta}{dS} + L \frac{\partial S}{\partial x^{\alpha}} \frac{d^2\zeta}{dS^2} \right) \quad (1.4) $$

We factorize each term with the corresponding derivative of $\zeta(S)$:

$$ 0 = \frac{d^3 \zeta}{dS^3} \left[ L^2 \frac{\partial S}{\partial \dot{x}^\alpha} \right] + \frac{d^2 \zeta}{dS^2} \left[ L \left( \frac{\partial L}{\partial \dot{x}^\alpha} - \frac{\partial S}{\partial x^\alpha} \right) + \frac{d}{d\tau} \left( L \frac{\partial S}{\partial \dot{x}^\alpha} \right) \right] + \frac{d\zeta}{dS} \left[ \frac{d}{d\tau} \frac{\partial L}{\partial \dot{x}^\alpha} - \frac{\partial L}{\partial x^\alpha} \right] \quad (1.5) $$
Since $\zeta$ is arbitrary, each term between [ ] must vanish : we examine them from left to right. $\frac{\partial S}{\partial \dot{x}^\alpha} = 0$ implies that $S$ can explicitly depend on $\tau$ and $x^\alpha$ but not on $\dot{x}^\alpha$: indeed, if $S$ depends on $\dot{x}^\alpha$ then $L = \frac{dS}{d\tau}$ depends on $\ddot{x}^\alpha$. With the generalized momentum $p_\alpha$ defined by $p_\alpha = \frac{\partial L}{\partial \dot{x}^\alpha}$, the second term between [ ] proves the well known equality $p_\alpha = \frac{\partial L}{\partial \dot{x}^\alpha} = \frac{\partial S}{\partial x^\alpha}$ [12][13][14][15]. $S$ is then similar to the velocity potential in fluid mechanics [17]. Finally, we recognize the Euler-Lagrange equations applied to $L$ in the last [ ] of equation (1.5). From $\frac{\partial S}{\partial \dot{x}^\alpha} = 0$ we deduce $L = \frac{dS}{d\tau} = \frac{\partial S}{\partial \tau} + \dot{x}^\mu \frac{\partial S}{\partial x^\mu} \Leftrightarrow \frac{\partial S}{\partial \tau} = L - \dot{x}^\mu \frac{\partial S}{\partial \dot{x}^\mu}$. With $\frac{\partial L}{\partial \dot{x}^\alpha} = \frac{\partial S}{\partial x^\alpha}$ we get $\frac{\partial S}{\partial \tau} = L - \dot{x}^\mu \frac{\partial L}{\partial \dot{x}^\mu}$. The Hamiltonian $H$ being the Legendre transform of the Lagrangian according to $H = \frac{\partial L}{\partial \dot{x}^\mu} \dot{x}^\mu - L$ [12][13][14][15], we have derived the famous Hamilton Jacobi equation $\frac{\partial S}{\partial \tau} + H = 0$ [12][13][14][15]. Injecting $\frac{\partial L}{\partial \dot{x}^\alpha} = \frac{\partial S}{\partial x^\alpha}$ and $L = \frac{dS}{d\tau}$ in the Euler-Lagrange equations, we obtain a new expression of the equations of motion in classical physics :

$$0 = \frac{d}{d\tau} \left( \frac{\partial L}{\partial \dot{x}^{\alpha}} \right) - \frac{\partial}{\partial x^{\alpha}} \left( L \right) \Leftrightarrow 0 = \frac{d}{d\tau} \left( \frac{\partial S}{\partial x^{\alpha}} \right) - \frac{\partial}{\partial x^{\alpha}} \left( \frac{dS}{d\tau} \right) = \left[ \frac{d}{d\tau}; \frac{\partial}{\partial x^{\alpha}} \right]_{-} S \quad (1.6)$$

where $[A; B]_{-} = AB - BA$ defines a commutator. To enlighten the meaning of this new expression, we momentarily stop treating the position $x^\alpha$ and the velocity $\dot{x}^\alpha$ as independent variables. For this purpose we introduce the underlined operator $\underline{\partial}_\alpha$ which does not obey the convention to treat $x^\alpha$ and $\dot{x}^\alpha$ as independent variables and we keep the notation $\underline{\partial}_\alpha$ for the operator which obey the convention (i.e. the operator which does not act on $\dot{x}^\alpha$). As in fluid mechanics, the material derivative [17] acting on $\dot{x}^\alpha$ is $\partial_\tau \dot{x}^\alpha + \dot{x}^\beta \partial_\beta \dot{x}^\alpha = \frac{d}{d\tau} \dot{x}^\alpha = \ddot{x}^\alpha$. Naturally, $\underline{\partial}_\beta S(\tau, x^\alpha) = \partial_\beta S(\tau, x^\alpha)$ for any function $S$ which depends on $x^\alpha$ and $\tau$ but not $\dot{x}^\alpha$. In equations (1.7) and (1.8), we compute two expressions of $\frac{d^2}{d\tau^2} S(\tau, x^\alpha)$ and then, in equation (1.9), we subtract (1.8) from (1.7) :

$$\frac{d^2 S}{d\tau^2} = \frac{d}{d\tau} (\partial_{\tau} S + \dot{x}^{\alpha} \partial_{\alpha} S) \quad \Rightarrow \quad \frac{d^2 S}{d\tau^2} = \frac{d}{d\tau} \partial_{\tau} S + \dot{x}^{\alpha} \frac{d}{d\tau} \partial_{\alpha} S + \ddot{x}^{\alpha} \partial_{\alpha} S \quad (1.7)$$

$$\begin{aligned} \frac{d^2 S}{d\tau^2} = \frac{\partial_{\tau} \frac{dS}{d\tau} + \dot{x}^{\beta} \underline{\partial}_{\beta} \frac{dS}{d\tau}}{\partial_{\tau} \frac{dS}{d\tau} + \dot{x}^{\beta} \underline{\partial}_{\beta} \frac{dS}{d\tau}} = \frac{\partial_{\tau} (\partial_{\tau} S + \dot{x}^{\alpha} \partial_{\alpha} S) + \dot{x}^{\beta} \underline{\partial}_{\beta} (\partial_{\tau} S + \dot{x}^{\alpha} \partial_{\alpha} S)}{\partial_{\tau} \frac{dS}{d\tau} + \dot{x}^{\beta} \underline{\partial}_{\beta} \frac{dS}{d\tau}} = \frac{\partial_{\tau} (\partial_{\tau} S + \dot{x}^{\alpha} \partial_{\alpha} S)}{\partial_{\tau} \frac{dS}{d\tau} + \dot{x}^{\beta} \underline{\partial}_{\beta} \frac{dS}{d\tau}} \\ + \left[ \underline{\partial}_{\tau} \dot{x}^{\alpha} \right] \partial_{\alpha} S + \dot{x}^{\beta} \underline{\partial}_{\beta} \dot{x}^{\alpha} \right] \partial_{\alpha} S \Rightarrow \frac{d^2 S}{d\tau^2} = \frac{\partial_{\tau} \frac{dS}{d\tau} + \dot{x}^{\alpha} \partial_{\alpha} \frac{dS}{d\tau}}{\partial_{\tau} \frac{dS}{d\tau} + \dot{x}^{\alpha} \partial_{\alpha} \frac{dS}{d\tau}} + \ddot{x}^{\alpha} \partial_{\alpha} S \end{aligned} \quad (1.8)$$

The underlined operators have disappeared and we will not need them anymore. The subtraction gives :

$$0 = \dot{x}^{\alpha} \left[ \frac{\partial}{\partial x^{\alpha}}; \frac{d}{d\tau} \right]_{-} S + \left[ \frac{\partial}{\partial \tau}; \frac{d}{d\tau} \right]_{-} S \quad (1.9)$$

To make a comparison later, we now compute the total derivative of any function $L(\tau, x^\alpha, \dot{x}^\alpha)$ :

$$\begin{aligned} \frac{dL}{d\tau} = \frac{\partial L}{\partial \tau} + \dot{x}^{\alpha} \frac{\partial L}{\partial x^{\alpha}} + \ddot{x}^{\alpha} \frac{\partial L}{\partial \dot{x}^{\alpha}} & \Rightarrow \frac{dL}{d\tau} = \frac{\partial L}{\partial \tau} + \dot{x}^{\alpha} \frac{\partial L}{\partial x^{\alpha}} + \frac{d}{d\tau} (\dot{x}^{\alpha} \frac{\partial L}{\partial \dot{x}^{\alpha}}) - \dot{x}^{\alpha} \frac{d}{d\tau} \frac{\partial L}{\partial \dot{x}^{\alpha}} \\ & \Rightarrow 0 = \dot{x}^{\alpha} \left[ \frac{\partial L}{\partial x^{\alpha}} - \frac{d}{d\tau} \frac{\partial L}{\partial \dot{x}^{\alpha}} \right] + \left[ \frac{\partial L}{\partial \tau} - \frac{d}{d\tau} (L - \dot{x}^{\alpha} \frac{\partial L}{\partial \dot{x}^{\alpha}}) \right] \end{aligned} \quad (1.10)$$

The comparison of equations (1.9) and (1.10) will be given below equation (1.11). Here we begin a new derivation of the Euler-Lagrange equations. $x^\alpha$ and $\dot{x}^\alpha$ being independent variables, $S(x^\sigma, \tau)$ obeys equation (1.11) :

$$\frac{\partial}{\partial x^{\alpha}} \frac{dS}{d\tau} = \frac{\partial}{\partial x^{\alpha}} \frac{\partial S}{\partial \tau} + \frac{\partial}{\partial x^{\alpha}} \left( \dot{x}^{\beta} \frac{\partial S}{\partial x^{\beta}} \right) = \frac{\partial}{\partial \tau} \frac{\partial S}{\partial x^{\alpha}} + \dot{x}^{\beta} \frac{\partial}{\partial x^{\beta}} \frac{\partial S}{\partial x^{\alpha}} = \frac{d}{d\tau} \frac{\partial S}{\partial x^{\alpha}} \Rightarrow \left[ \frac{\partial}{\partial x^{\alpha}}; \frac{d}{d\tau} \right]_{-} S = 0 \quad (1.11)$$

Writing $S(\tau, x^\sigma) = \int L(\tau, x^\sigma, \dot{x}^\sigma)d\tau \Leftrightarrow \frac{dS}{d\tau} = L$, it follows that $\frac{\partial L}{\partial \dot{x}^\alpha} = \frac{\partial}{\partial \dot{x}^\alpha} \frac{dS}{d\tau} = \frac{\partial (\partial_\tau S + \dot{x}^\mu \partial_\mu S)}{\partial \dot{x}^\alpha} = \frac{\partial \dot{x}^\mu}{\partial \dot{x}^\alpha} \frac{\partial S}{\partial x^\mu} = \delta^\mu_\alpha \frac{\partial S}{\partial x^\mu} = \frac{\partial S}{\partial x^\alpha}$. Injecting $\frac{dS}{d\tau} = L$ and $\frac{\partial S}{\partial x^\alpha} = \frac{\partial L}{\partial \dot{x}^\alpha}$ in equation (1.11), we obtain $\frac{\partial}{\partial x^\alpha} \frac{dS}{d\tau} = \frac{d}{d\tau} \frac{\partial S}{\partial x^\alpha} \Leftrightarrow \frac{\partial L}{\partial x^\alpha} = \frac{d}{d\tau} \frac{\partial L}{\partial \dot{x}^\alpha}$. We have derived the Euler-Lagrange equations. Defining $p_\alpha = \frac{\partial L}{\partial \dot{x}^\alpha}$, the previous results also implies $p_\alpha = \frac{\partial S}{\partial x^\alpha}$. Furthermore, $L = \frac{dS}{d\tau} = \frac{\partial S}{\partial \tau} + \dot{x}^\mu \frac{\partial S}{\partial x^\mu} = \frac{\partial S}{\partial \tau} + \dot{x}^\mu \frac{\partial L}{\partial \dot{x}^\mu}$ or $\frac{\partial S}{\partial \tau} = L - \dot{x}^\mu \frac{\partial L}{\partial \dot{x}^\mu}$. Then $\left[ \frac{\partial}{\partial \tau}; \frac{d}{d\tau} \right]_{-} S = \frac{\partial L}{\partial \tau} - \frac{d}{d\tau} (L - \dot{x}^\alpha \frac{\partial L}{\partial \dot{x}^\alpha})$ : the last commutator of equation (1.9) is equal to the last term between [ ] in equation (1.10). Defining $H = \dot{x}^\alpha \frac{\partial L}{\partial \dot{x}^\alpha} - L$, the previous result also implies $\frac{\partial S}{\partial \tau} = -H$. To summarize, we made the assumption that $x^\sigma(\tau)$ and $\dot{x}^\sigma(\tau)$ are independent variables when partial derivatives are applied and, from an arbitrary function $L(x^\sigma, \dot{x}^\sigma, \tau)$, we respectively defined $S(\tau, x^\alpha) = \int L(x^\sigma, \dot{x}^\sigma, \tau)d\tau$, $p_\alpha = \frac{\partial L}{\partial \dot{x}^\alpha}$, $H = \dot{x}^\alpha \frac{\partial L}{\partial \dot{x}^\alpha} - L$. Then we successively derived :

$$\frac{\partial L}{\partial x^{\alpha}} = \frac{d}{d\tau} \frac{\partial L}{\partial \dot{x}^{\alpha}} \quad ; \quad \frac{\partial S}{\partial x^{\alpha}} = p_{\alpha} \quad ; \quad \frac{\partial S}{\partial \tau} + H = 0 \quad (1.12)$$

To conclude, we provided a new expression of the equations of motion in classical physics, with a commutator of operators acting on the action. This expression illuminated another way to derive the Euler-Lagrange equations without introducing variations or boundary conditions : this formulation then offers a new perspective on the "the final causes", "the miracle" and the lack of "metaphysical reason" discussed by H. Poincaré, R. P. Feynman and C. G. J. Jacobi. The well known equalities $p_\alpha = \frac{\partial S}{\partial x^\alpha}$ and $\frac{\partial S}{\partial \tau} + H = 0$ will be used in the following sections.
## 2 Derivation of quantum mechanics from general relativity

In 2011, C. DeWitt-Morette wrote: "The structures of quantum physics and gravitation physics do not constitute a single logical system. Digging deeper into their respective structures may reveal a common root." [18]. The aim of the present section is to show that quantum physics and gravitation physics have a common geometrical root. Quantum mechanics will be derived from an amended version of general relativity, by introducing 2 postulates: the most attractive characteristic of the 2 postulates is that they simplify the physical content of general relativity. The first postulate sheds new light on an old conflict between general relativity and quantum mechanics, notably described by C. Kiefer and P. Peters: "It is common knowledge that the physicists' view of time evolved in 1905 when Einstein [...] introduced the notion of proper time. But special relativistic time is still an absolute concept [...]. In contrast, spacetime is dynamical in general relativity, and an arbitrary spacetime does not possess any timelike Killing vector. In that sense, it seriously departs from what is needed in quantum mechanics" [19]. The first postulate uses the fact that the proper time is not dynamical in general relativity while the second postulate is intimately linked with the origins of the Schrödinger equation. Indeed, it was already known that geometrical optics corresponds to the eikonal approximation of wave optics in 1926, when E. Schrödinger introduced his famous equation and suggested to interpret classical mechanics as the eikonal approximation of a 'wave mechanics': "The true mechanical process is realised or represented in a fitting way by the wave processes in configuration space, and not by the motion of image points in this space. The study of the motion of image points, which is the object of classical mechanics, is only an approximate treatment, and has, as such, just as much justification as geometrical or 'ray' optics has, compared with the true optical process" [20]. Similarly, in a textbook published in 1961, A. Messiah called $\psi$ the wave function and explained how classical mechanics can be derived from quantum mechanics: "The second way to formulate the classical approximation consists in linking the system to a statistical mixture of classical systems [...] The conditions of validity of this classical approximation are exactly those of geometrical optics. [...] In the classical approximation, $\psi$ describes a fluid of non-interacting classical particles of mass $m$ [...] : the density and current density of this fluid at each point of space are at all times respectively equal to the probability density and the probability current density of the quantum particle at that point" [21]. A similar analysis has been published by P.A.M Dirac [22]. In the eikonal approximation, our second postulate allows to identify the quantum Lagrangian density with its classical counterpart.

Our first postulate will be stated and its consequences will be immediately examined while the discussion of its meaning will be postponed to the end of this section. Before introducing the postulate we quote a book dedicated to the recent reform of the international system of units. In the book, C. Bordé wrote: "A generalized framework for fundamental metrology: 5D geometry combining spacetime and proper time. [...] we must give coherence to the new system of units [...] we must inscribe it in a unified physical framework for fundamental metrology that contains a proper description of spacetime, proper time and mass" [23]. The suggestion made by C. Bordé has been applied by him to the Ramsey-Bordé interferometer, which is an atom interferometer [24]. Other authors also used atom interferometers to introduce the interplay between proper time and quantum mechanics [25] [26]. The link with the above quote from C. Kiefer and P. Peters is immediate since spacetime is dynamical while the proper time is not.

**Postulate 1:** Matter Lagrangian densities of general relativity are defined on a space of 5-vectors such that their action vanishes. The proper time $\tau$ is the fifth coordinate (conventionally, the proper time remains invariant and Lagrangian densities depend on $x^\alpha$ only).

Applying Einstein summation convention to capital letters ($M$ and $N$ below), Postulate 1 is used to rewrite the relativistic length $c^2d\tau^2 = ds^2 = c^2dt^2 - (dx^1)^2 - (dx^2)^2 - (dx^3)^2$ in the form:

$$0 = (dx^0)^2 - (dx^1)^2 - (dx^2)^2 - (dx^3)^2 - ds^2 = \eta_{\mu\nu}dx^\mu dx^\nu - (dx^4)^2 = \eta_{MN}dx^M dx^N \quad (2.1)$$

where $dx^4 = ds = cd\tau$ and $dx^0 = cdt$ with $c$ the speed of light, 5D vectors being respectively:

$$dx^M = (dx^\mu, dx^4) = (dx^0, d\vec{x}, ds); \quad dx_M = \eta_{MN} dx^M = (dx_\mu, dx_4) = (dx^0, -d\vec{x}, -ds) \quad (2.2)$$

$\eta_{MN} = \text{diag}(1, -1, -1, -1, -1)$ generalises the Minkowski metric for 5 variables, the fifth one being the proper time, with a spacelike signature. The momenta of massless particles (gravitons, photons) and the momenta of massive particles are now both null-vectors on a 5-vectors space: the symmetry between massive and massless particles is increased while the physical content of general relativity is simplified because the mass is not anymore an external parameter, it is a dynamical parameter associated to the $x_4$ direction: the link with the Higgs boson is discussed in [Section 7.1.3] (this discussion requires the results given in [Section 4]). In [Section 4], we will relax the requirement that $s$ should be invariant but, in agreement with Postulate 1, we temporarily consider that only the 4-dimensional part of the metric can be affected by coordinate transformations: $\eta_{\mu\nu} \rightarrow \eta_{\alpha\beta} \frac{\partial x'^\alpha}{\partial x^\mu} \frac{\partial x'^\beta}{\partial x^\nu} =$
$g_{\mu\nu}$. Consequently, $g_{44} = \eta_{44}$ and $g_{\mu4} = 0$. The reason to momentarily forbid coordinate transformations involving the proper time is to leave general relativity unaffected by the introduction of the 5-vectors space. We now recall well established results of general relativity but we treat the proper time $\tau = s/c = x^4/c$ as a neutral coordinate (and $\dot{x}^4 = c$ as a constant). According to Postulate 1, the 5D Lagrangian of a particle, say $L_m$, vanishes. We write $L_m = L_m(x^\mu, \dot{x}^\mu)$ in the form [27] [28]:

$$0 = L_m = \frac{m}{2}(g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu} - (\dot{x}^4)^2) = \frac{m}{2}(g^{\mu\nu}\dot{x}_{\mu}\dot{x}_{\nu} - (\dot{x}_4)^2) \quad ; \quad \dot{x}_{\mu} = g_{\mu\nu}\frac{dx^{\nu}}{d\tau} \quad ; \quad g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu} = c^2 \qquad (2.3)$$

With $p_\alpha = \frac{\partial L_m}{\partial \dot{x}^\alpha} = m\dot{x}_\alpha$, the generally covariant Hamiltonian is $H_g = \frac{\partial L_m}{\partial \dot{x}^\alpha} - L_m = \frac{m}{2}\dot{x}_\alpha\dot{x}^\alpha + \frac{mc^2}{2} = \frac{g^{\mu\nu}p_\mu p_\nu}{2m} + \frac{mc^2}{2} = H_g(p_\mu, x^\mu)$. Then $H_g(p_\mu, x^\mu) = mc^2$ is a constant of the motion. The mechanical action becomes [12] [15]:

$$S = \int L_m d\tau = \int L_m - \frac{\partial L_m}{\partial \dot{x}^\alpha} \dot{x}^\alpha d\tau + \int \frac{\partial L_m}{\partial \dot{x}^\alpha} \dot{x}^\alpha d\tau = \int -mc^2 d\tau + \int p_\alpha dx^\alpha = \int p_M dx^M \qquad (2.4)$$

With $\frac{\partial S}{\partial x^\alpha} = p_\alpha$ and $H_g(p_\alpha, x^\alpha) = H_g(\frac{\partial S}{\partial x^\alpha}, x^\alpha)$, the Hamilton Jacobi equation for a particle in a gravitational field takes the well known form $\frac{\partial S}{\partial \tau} + H_g(\frac{\partial S}{\partial x^\alpha}, x^\alpha) = 0 \Leftrightarrow -mc^2 + \left(\frac{g^{\mu\nu}\partial_\mu S\partial_\nu S}{2m} + \frac{mc^2}{2}\right) = 0 \Leftrightarrow g^{\mu\nu}\partial_\mu S\partial_\nu S - m^2c^2 = 0$ [3][28]. We identify the Lagrangian $L_m$ with the Hamilton Jacobi equation since, using $m\dot{x}_\alpha = \partial_\alpha S$:

$$0 = L_m = \frac{m}{2}(g^{\mu\nu}\frac{\partial_\mu S}{m}\frac{\partial_\nu S}{m} - c^2) \qquad (2.5)$$

We now consider that space is occupied by an enormous amount of particles such that we use the continuous approximation. We change our notations according to the convention used in general relativity: $\dot{x}^\mu = \frac{\partial^\mu S(x^\alpha)}{m} \rightarrow u^\mu(x^\alpha)$, $\dot{x}^4 = c \rightarrow u^4 = c$, $\frac{m}{2} \rightarrow \rho(x^\alpha)$ and the vanishing Lagrangian $L_m$ is replaced by the vanishing Lagrangian density $\mathcal{L}_m$. $u^\mu$ is the 4-velocity of the particle located at the position $x^\alpha$, in the continuous approximation. $\rho$ is the mass density (with units of mass per volume). For later use, we note that the change $L_m = 0 \rightarrow \mathcal{L}_m = 0$ allows us to rewrite equation (2.5) in the local form:

$$0 = \mathcal{L}_m = \rho(g^{\mu\nu}\frac{\partial_\mu S}{m}\frac{\partial_\nu S}{m} - c^2) \Leftrightarrow 0 = \mathcal{L}_m = \rho(g^{\mu\nu}u_\mu u_\nu - c^2) = \rho g^{\mu\nu}u_\mu u_\nu - \rho u_4^2 \qquad (2.6)$$

The proper time remaining unaffected by coordinate transformations, the 5D Einstein Hilbert action $S_{EH5}$ (including the gravitational Lagrangian density $R$ and the matter Lagrangian density $\mathcal{L}_m$) takes the form:

$$S_{EH5} = \int \left( \mathcal{L}_m - \frac{c^4}{8\pi G} R \right) \sqrt{g} d^4 x ds = \int \left( \rho(g^{\mu\nu}u_\mu u_\nu - u_4^2) - \frac{c^4}{8\pi G} R \right) \sqrt{g} d^4 x ds \qquad (2.7)$$

where $R = g^{\mu\nu}R_{\mu\nu}$ is the usual Ricci scalar (the 5D Ricci scalar is simply the 4D Ricci scalar since Postulate 1 imposes the conditions $g_{44} = \eta_{44}$ and $g_{\mu4} = 0$), $G$ is Newton's gravitational constant, and $g = -\det|g_{\mu\nu}|$. General relativistic boundary terms are excluded from the discussion (see Hawking-Gibbons boundary term in [27], [28]). We now consider an infinitesimal variation of the metric, together with the conditions $\delta g^{44} = \delta g^{\mu4} = 0$ since no transformation of coordinates involving the proper time is allowed. The principle of stationary action gives [27]:

$$\begin{align}
\delta S_{EH5} &= 0 = \int \left( \rho u_\mu u_\nu - \frac{1}{2} g_{\mu\nu} \mathcal{L}_m - \frac{c^4}{8\pi G} (R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R) \right) \sqrt{g} \delta g^{\mu\nu} \, d^4 x ds \tag{2.8} \\
&\Rightarrow 0 = \rho u_\mu u_\nu - \frac{1}{2} g_{\mu\nu} \mathcal{L}_m - \frac{c^4}{8\pi G} \left( R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R \right) \Leftrightarrow \frac{8\pi G}{c^4} \rho u_\mu u_\nu = \frac{8\pi G}{c^4} T_{\mu\nu} = G_{\mu\nu}
\end{align}$$

Einstein's equations, with $G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R$ the Einstein tensor and $T_{\mu\nu} = \rho u_\mu u_\nu$ the energy momentum tensor, thus hold. It deserves to be noticed that the condition $\mathcal{L}_m = 0$, following from Postulate 1, was in fact needed to derive Einstein's equations from the Einstein Hilbert action. Using the property $\nabla_\alpha G^{\alpha\beta} = \partial_\alpha G^{\alpha\beta} + \Gamma^\beta_{\alpha\sigma} G^{\alpha\sigma} + \Gamma^\alpha_{\alpha\sigma} G^{\beta\sigma} = 0$ [27], where $\Gamma^\beta_{\alpha\sigma}$ is a Christoffel symbol, we deduce $\nabla_\alpha T^{\alpha\beta} = 0$:

$$\begin{align}
0 &= \nabla_{\alpha} T^{\alpha\beta} = \partial_{\alpha} (\rho u^{\alpha} u^{\beta}) + \Gamma^{\beta}_{\alpha\sigma} \rho u^{\alpha} u^{\sigma} + \Gamma^{\alpha}_{\alpha\sigma} \rho u^{\sigma} u^{\beta} \nonumber \\
0 &= u^{\beta} [\partial_{\alpha} (\rho u^{\alpha}) + \Gamma^{\alpha}_{\alpha\sigma} \rho u^{\sigma}] + \rho [u^{\alpha} \partial_{\alpha} u^{\beta} + \Gamma^{\beta}_{\alpha\sigma} u^{\sigma} u^{\alpha}] \tag{2.9}
\end{align}$$

The term $[\partial_\alpha(\rho u^\alpha) + \Gamma^\alpha_{\alpha\sigma}\rho u^\sigma] = \nabla_\alpha(\rho u^\alpha) = \nabla_\alpha j^\alpha_m$ is the covariant form of the continuity equation, with $j^\alpha_m$ the mass current. The other term of equation (2.9) is the geodesic equation since $u^\alpha \partial_\alpha u^\beta = \dot{u}^\beta (\leftrightarrow \dot{x}^\alpha \partial_\alpha u^\beta = \frac{du^\beta}{d\tau})$. It is well known that the continuity equation and the geodesic equation independently vanish. A simple proof is obtained by contracting equation (2.9) with $u_\beta$ since after a bit of computation, the contraction gives:

$$0 = [\partial_{\alpha}(\rho u^{\alpha}) + \Gamma^{\alpha}_{\alpha\sigma}\rho u^{\sigma}] + \rho [u^{\alpha}\partial_{\alpha}(u_{\beta}u^{\beta})] \rightarrow 0 = [\partial_{\alpha}(\rho u^{\alpha}) + \Gamma^{\alpha}_{\alpha\sigma}\rho u^{\sigma}] + \rho \frac{d}{ds}[g_{\alpha\beta}u^{\alpha}u^{\beta}] \qquad (2.10)$$
We can now state the second postulate, and use it to include quantum mechanics.

**Postulate 2 :** In general relativity, matter Lagrangian densities are constructed from complex fields obeying wave equations, at least in the low energy approximation (conventionnaly, the energy momentum tensor is kept real)

In the present section, we will only consider spin-0 fields (other fields are introduced in [Section 5], the geometrical meaning of the spin being discussed in the same section). The introduction of complex fields in matter Lagrangian densities evidently suggests to begin with a complex version of general relativity. At this stage, general relativity is deliberately left insensitive to the introduction of complex fields, thanks to the reality condition conventionally imposed on the energy momentum tensor (the metric remains real because the energy momentum tensor is the source of the gravitational field in Einstein's equations). In [Section 5.2], it will be recalled that the metric $g_{\mu\nu}$ obeys a wave equation in the linearized (or low energy) approximation when the harmonic gauge is imposed [27] while the electromagnetic 4-potential $A_\mu$ obeys a wave equation when the Lorenz gauge is imposed. The assumption that matter Lagrangian densities should be constructed from fields obeying wave equations (at least in the low energy approximation) then simplifies again the physical content of general relativity : only fields obeying wave equations in the low energy limit will appear in the Einstein Hilbert action. Our two postulates can be understood as symmetry postulates between matter and interactions, and then between matter and spacetime. An article written by D. J. Gross in 1999 can be used to clarify the new idea contained in our two postulates : "There have been two major revolutions completed in this century : relativity (special and general) and quantum mechanics. These were associated with two of the three dimensional parameters of physics : h, Planck's quantum of action, and c, the velocity of light. [...] the last dimensional parameter we need in order to establish a set of fundamental dimensional units of nature is Newton's gravitational constant" [29]. The constants c and G are directly provided by Postulate 1 which contains general relativity but h is not provided. In his scientific autobiography, M. Planck wrote : "The velocity of light is to the theory of relativity as the elementary quantum of action is to the quantum theory : it is its absolute core." [30]. The new approach discussed in the present section will enable us to derive the absolute core of quantum mechanics, i.e. the constant h.

The simplest Lagrangian density $\mathcal{L}_m(\phi, \partial_\mu\phi, \phi^*, \partial_\mu\phi^*)$ consistent with Postulate 1 and Postulate 2 is :

$$S_m = \int \mathcal{L}_m \sqrt{g} d^4 xds = \int \left( \frac{A}{2} g^{\mu\nu} (\partial_\mu \phi \partial_\nu \phi^* + \partial_\mu \phi^* \partial_\nu \phi) - A \partial_4 \phi \partial_4 \phi^* \right) \sqrt{g} d^4 xds \quad (2.11)$$

where A is a constant. In quantum field theory, $\mathcal{L}_m$ would be the Lagrangian density of a massive scalar field or spin-0 field. According to Postulate 1, $\mathcal{L}_m$ should not depend explicitly on $s = x^4$. Then, $\phi(x^\alpha, x^4)$ takes the form $\phi(x^\alpha, x^4) = \psi(x^\alpha)e^{-ik_4 x^4}$ such that $-A\partial_4\phi\partial_4\phi^* = -Ak_4^2|\psi(x^\alpha)|^2$ depends on $x^\alpha$ only. $k_4$ is independent of $x^\alpha$ otherwise the term $-Ak_4^2$ could be interpreted as a dynamical metric term $g_{44}$ while we imposed the condition $g_{44} = \eta_{44}$. Now, we conventionally fix $A = c^2/(k_4)^2$ such that the term $-A\partial_4\phi\partial_4\phi^* = -c^2|\phi|^2 = -c^2|\psi|^2$ in equation (2.11) is directly identified with the term $-\rho u_4^2 = -\rho c^2$ in equation (2.6) : $\rho \leftrightarrow |\psi|^2$. We rewrite $S_m$ :

$$S_m = \int \mathcal{L}_m \sqrt{g} d^4 xds = \int \left( \frac{c^2}{2k_4^2} g^{\mu\nu} (\partial_\mu \psi \partial_\nu \psi^* + \partial_\mu \psi^* \partial_\nu \psi) - c^2 |\psi|^2 \right) \sqrt{g} d^4 xds \quad (2.12)$$

As usual, the equations of motion are obtained by imposing $\delta S_m = 0$ which implies $\delta(\mathcal{L}_m \sqrt{g}) = 0$ :

$$0 = \delta\psi \frac{\partial(\mathcal{L}_m \sqrt{g})}{\partial\psi} + \delta\partial_\mu\psi \frac{\partial(\mathcal{L}_m \sqrt{g})}{\partial\partial_\mu\psi} + \text{"cc"} = \delta\psi \left( \frac{\partial(\mathcal{L}_m \sqrt{g})}{\partial\psi} - \partial_\mu \frac{\partial(\mathcal{L}_m \sqrt{g})}{\partial\partial_\mu\psi} \right) + \partial_\mu (\delta\psi \frac{\partial(\mathcal{L}_m \sqrt{g})}{\partial\partial_\mu\psi}) + \text{"cc"} \quad (2.13)$$

where "cc" denotes the complex conjugate of the terms on the left, and where $\partial_\mu(\delta\psi \frac{\partial(\mathcal{L}_m \sqrt{g})}{\partial\partial_\mu\psi})$ can be ignored since it is a boundary term when the 4D volume integral of the Lagrangian density is performed in the action $S_m$. Then we obtain :

$$0 = \delta\psi \left( \frac{\partial(\mathcal{L}_m \sqrt{g})}{\partial\psi} - \partial_\mu \frac{\partial(\mathcal{L}_m \sqrt{g})}{\partial\partial_\mu\psi} \right) + \text{"cc"} = \delta\psi \left[ -\Box_g \psi^* - k_4^2 \psi^* \right] \sqrt{g} + \delta\psi^* \left[ -\Box_g \psi - k_4^2 \psi \right] \sqrt{g} \quad (2.14)$$

where $\Box_g$ is the generally covariant Laplacian : $\Box_g \psi = \partial_\mu \partial^\mu \psi + \Gamma^\mu_\mu \partial^\nu \psi$. The variations $\delta\psi$ and $\delta\psi^*$ being arbitrary, $\psi$ obey the 4D Helmholtz equation $\Box_g \psi + k_4^2 \psi = 0$, while $\phi$ obey the 5D wave equation $\Box_g \phi - \partial_4^2 \phi = 0$.

We can now derive the relation $p_\mu = \hbar k_\mu$ where $k_\mu = (\omega/c, \vec{k})$ is the 4-wave vector, $\vec{k}$ the wave vector, and $\hbar$ a universal constant identified with the reduced Planck constant : $2\pi\hbar = h$ where h is the Planck constant introduced by M. Planck in 1900. All derivations of the relation $p_\mu = \hbar k_\mu$ that we have found in literature
assume that $\hbar$ exists: our task is to deduce that this constant must exist. When $\mu = 0$, $p_\mu = \hbar k_\mu$ becomes the Planck-Einstein relation: $E/c = \hbar\omega/c \Leftrightarrow E = \hbar\omega = h\nu$, where $\nu$ is the frequency. According to his friend C. Lanczos, a few years before his death in 1955, A. Einstein made the following remark: "If anybody tells you that he understands what $E = h\nu$ means, tell him that he is a liar" [31]. Analogously, N. Bohr was inclined to consider that the quantum of action $h$ could not be explained in atomic physics, like life in biology. By example, in 1932 he claimed that "the existence of life must be considered as an elementary fact that can not be explained, but must be taken as a starting point in biology, in a similar way as the quantum of action [...] taken together with the existence of the elementary particles, form the foundations of atomic physics" [32] and, in 1937 he stated that "the existence of life itself would have to be regarded in biology [...] as no more subject to analysis than the existence of the quantum of action in atomic physics" [33]. We will nevertheless analyse the existence of the quantum of action $h = 2\pi\hbar$ by deriving the relation $p_\mu = \hbar k_\mu$. Our derivation is made in rectangular coordinates, gravity being neglected. The solutions $\psi(x^\alpha)$ of the Helmholtz equation in rectangular coordinates, when $g_{\mu\nu} = \eta_{\mu\nu}$ and $\Gamma^\mu_{\mu\nu} = 0$, such that $\eta^{\mu\nu}\partial_\mu\partial_\nu\psi + k_4^2\psi = 0$, are plane waves of the form $\psi(x^\alpha) = \mathcal{R}e^{-i\mathbf{k}_\mu x^\mu}$ where $\mathcal{R}$ is constant and $k_\mu k^\mu = k_4^2$. Consequently, neglecting gravity we have:

$$ \mathcal{L}_m = \frac{c^2}{2k_4^2} \eta^{\mu\nu} (\partial_\mu \psi \partial_\nu \psi^* + \partial_\mu \psi^* \partial_\nu \psi) - c^2 |\psi|^2 = \left(\frac{1}{k_4^2} \eta^{\mu\nu} k_\mu k_\nu - 1\right) c^2 |\psi|^2 = 0 \quad (2.15) $$

Plane waves are idealized solutions, or approximations, widely used in physics. By example, plane waves are used to compute the scattering cross sections of spin-0 particles [34][35][36] : in this context, they are called 'momentum eigenfunctions' [36]. Nevertheless, planes waves do not exist, even when photons are considered: "Even in the most monochromatic light source [...] one deals with a finite (although perhaps small) spread of frequencies" [37]. An important fact is that "$[t]he phase of a plane wave is an invariant quantity, the same in all coordinate frames. This is because the elapsed phase of a wave is proportional to the number of wave crests that have passed the observer. Since this is merely a counting operation, it must be independent of coordinate frame" [37]. The phase of $\psi(x^\alpha) = \mathcal{R}e^{-i\mathbf{k}_\mu x^\mu}$ being $-k_\mu x^\mu$, we deduce that the 4-wave vector $k^\mu$ must transform like $x^\mu$, and hence like $p^\mu$, under Lorentz transformations. According to Postulate 2, in the plane wave approximation, we must identify the energy momentum tensor of particles with the energy momentum tensor of plane waves. Precisely, we must identify $T_{\mu\nu} = \rho\dot{x}_\mu\dot{x}_\nu$ with $T_{\mu\nu} = \frac{c^2}{2(k_4)^2}(\partial_\mu\psi\partial_\nu\psi^* + \partial_\mu\psi^*\partial_\nu\psi) - \frac{1}{2}g_{\mu\nu}\mathcal{L}_m = \mathcal{R}^2c^2k_\mu k_\nu/(k_4)^2$ where equation (2.15) has been used. Since $\mathcal{R}^2 = |\psi|^2 = \rho$, we deduce $\dot{x}_\mu = ck_\mu/|k_4| = ck_\mu/k^4$ such that $p_\mu = \frac{mc}{|k_4|}k_\mu$. When $\mu = 0$ we get $p_0 = \frac{mc}{|k_4|}k_0 \Rightarrow \frac{p_0}{k_0} = \frac{mc}{|k_4|}$ such that, when $\mu = 0, 1, 2, 3$ we get:

$$ \frac{p_0}{k_0} = \frac{p_1}{k_1} = \frac{p_2}{k_2} = \frac{p_3}{k_3} = \frac{mc}{|k_4|} \quad (2.16) $$

The above equation can be deduced in a different way. The $i^{th}$ component $v_i$ of the group velocity of a wave is defined according to $\frac{v_i}{c} = \frac{\partial\omega/c}{\partial k^i} = \frac{ck_i}{\omega} = \frac{k_i}{k_0}$ where $\omega = c\sqrt{|\vec{k}|^2 + (k_4)^2}$ because $k_\mu k^\mu = k_4^2$. In flat spacetime the components of the 4-momentum of a massive particle are proportional to the mass because $p_\mu = \frac{\sqrt{(E/c)^2 - |\vec{p}|^2}}{\sqrt{(E/c)^2 - |\vec{p}|^2}} p_\mu = \frac{1}{\sqrt{1^2 - (v/c)^2}} \sqrt{(E/c)^2 - |\vec{p}|^2} \frac{p_\mu}{E} = \gamma mv_\mu = m\dot{x}_\mu$ where $|\vec{p}|$ is the norm of the 3D momentum, $\gamma = \frac{1}{\sqrt{1 - (v/c)^2}}$, $v = \frac{|\vec{p}|}{m}$, and $\frac{v_\mu}{c} = (1, \frac{\vec{v}}{c}) = (1, \frac{d\vec{x}}{d\tau})$. Since the plane wave approximation corresponds to a definite value of $k_\mu$, the group velocity $\frac{v^\mu}{c} = (1, \frac{\vec{v}}{c}) = \frac{k^\mu}{k_0} = (1, \frac{\vec{k}}{k_0})$ must be identified with the particle velocity $\frac{v^\mu}{c} = (1, \frac{\vec{v}}{c})$ where $\vec{v} = \frac{d\vec{x}}{d\tau}$: indeed, "[i]f an energy density is associated with the magnitude of a wave (or its absolute square) it is clear that in this approximation the transport of energy occurs with the group velocity, since that is the rate at which the pulse travels along" [37]. Considering a plane wave propagating along a given direction, we have then the equality of three 3D vectors: $\vec{p}c/E = \vec{v}/c = \vec{k}c/\omega$. We now rewrite the equality $p_\mu = \gamma mv_\mu = \frac{E}{c^2}v_\mu$ in the form $p_\mu = \frac{E}{c^2}v_\mu = \frac{E}{k_0c}k_\mu = \frac{p_0}{k_0}k_\mu$. If $\mu = 1$ we obtain $p_1 = \frac{p_0}{k_0}k_1 \Rightarrow \frac{p_0}{k_0} = \frac{p_1}{k_1}$ such that, when $\mu = 0, 1, 2, 3$ we get:

$$ \frac{p_0}{k_0} = \frac{p_1}{k_1} = \frac{p_2}{k_2} = \frac{p_3}{k_3} \quad (2.17) $$

Equation (2.17) can be written in form $p_\mu = \hbar k_\mu$ where $\mu = 0, 1, 2, 3$ which implies $m^2c^2 = p^\mu p_\mu = \hbar^2 k^\mu k_\mu = \hbar^2 k_4^2$ such that equation (2.16) follows. Equation (2.16) can be written as $p_M = \hbar k_M$ where $M$ varies from 0 to 4 while $\hbar$ is covariantly constant because $p_\mu$ and $k_\mu$ transform similarly under Lorentz transformations, as explained below equation (2.15). Now we consider a plane wave $\psi$, associated to a particle having mass $m$. We also assume that the particle is a bound state of two particles having masses $m'$ and $m''$ respectively. We neglect the interaction energy such that $m = m' + m''$. In the plane wave approximation, $\psi$ is then the sum of two planes waves and we must have $mc = (m' + m'')c = \hbar(k_4' + k_4'') = \hbar k_4$ and $p_\mu = \gamma mv_\mu = \gamma(m' + m'')v_\mu = \hbar(k_\mu' + k_\mu'') = \hbar k_\mu$. As a result, $\hbar$ is independent of the mass. We conclude that $\hbar$ is a universal constant (identified with the reduced Planck constant).
With $p^4 = mc = \hbar k^4$ and $x^4 = c\tau$ we get $\phi(x^\alpha, x^4) = \phi(x^\alpha, c\tau) = \psi(x^\alpha)e^{i\frac{mc^2\tau}{\hbar}}$ while the Helmholtz equation $\Box_g\psi + (k_4)^2\psi = 0$ takes the form $\Box_g\psi + (\frac{mc}{\hbar})^2\psi = 0$: we recognize the Klein-Gordon equation in curved spacetime. Nowadays, this equation is used to describe various particles, called spinless bosons (like $\pi$ mesons, $K$ mesons or many atoms: the helium atom will be discussed below). In rectangular coordinates, the generally covariant Laplacian is replaced by the usual Laplacian and the general solution of the Klein-Gordon equation takes the form of an inverse Fourier transform (which is a superposition of plane waves). With $p_\mu = \hbar k_\mu$, we use the following definitions of the Fourier transform (on the left) and the inverse Fourier transform (on the right):

$$ \hat{\psi}(p_\alpha) = \frac{1}{(2\pi)^2} \int \psi(x^\alpha) e^{i\frac{p_\alpha x^\alpha}{\hbar}} d^4x \quad ; \quad \psi(x^\alpha) = \frac{1}{(2\pi\hbar^2)^2} \int \hat{\psi}(p_\alpha) e^{-i\frac{p_\alpha x^\alpha}{\hbar}} d^4p \qquad (2.18) $$

Usual Heisenberg inequalities (also called Fourier inequalities) are included in the present theory through equation (2.18) since "[a] non zero function and its Fourier transform cannot be sharply localized. When translated in the language of quantum mechanics, [it means] that the value of a pair of canonically conjugate observables such as position and momentum cannot both be precisely determined in any quantum state" [38]. With our 4D definition of the Fourier transform, Heisenberg inequalities hold for 4 pairs of canonically conjugate observables: $(x_1; p_1)$, $(x_2; p_2)$, $(x_3; p_3)$, $(ct; E/c)$. Each inequality is of the form $\Delta x_1 \Delta p_1 \ge \hbar/2$ where $\Delta x_1$ and $\Delta p_1$ are the standard deviations while the saturation of the inequality is obtained if $\psi$ has a gaussian shape. N. Bohr gave the following comment about these inequalities: "The limitation in the classical concepts expressed through $[\Delta x_1 \Delta p_1 \ge \hbar/2]$, is, besides, closely connected with the limited validity of classical mechanics, which in the wave theory of matter corresponds to the geometrical optics in which the propagation of waves is depicted through 'rays'" [39]. Injecting the ansatz $\psi(x^\alpha) = \psi(t, \vec{x}) = e^{-i\frac{mc^2t}{\hbar}}\psi_S(t, \vec{x})$ in the Klein-Gordon equation $0 = \Box\psi + (\frac{mc}{\hbar})^2\psi$ we deduce:

$$ 0 = \left( -\left(\frac{mc}{\hbar}\right)^2 \psi_S - 2i \frac{m}{\hbar} \frac{\partial}{\partial t} \psi_S + \frac{1}{c^2} \frac{\partial}{\partial t^2} \psi_S - \Delta \psi_S + \left(\frac{mc}{\hbar}\right)^2 \psi_S \right) e^{-i\frac{mc^2t}{\hbar}} \qquad (2.19) $$

where $\Delta$ is the 3D Laplacian. Multiplying the above equation by $\frac{\hbar^2}{2m}$, and neglecting the term with the factor $\frac{1}{c^2}$ to consider the non relativistic limit, we derive the Schrödinger equation for a free particle:

$$ 0 = -i\hbar \frac{\partial}{\partial t} \psi_S - \frac{\hbar^2}{2m} \Delta \psi_S \quad \Rightarrow \quad i\hbar \frac{\partial}{\partial t} \psi_S = -\frac{\hbar^2}{2m} \Delta \psi_S \qquad (2.20) $$

The most striking effect induced by assuming that the matter Lagrangian density $L_m$ should be built from (complex) fields obeying wave equations is quantization: we illustrate this effect with superfluid helium, in the non relativistic limit (the Schrödinger equation is used). In this context, He I and He II designate the two phases of the helium. In a book introducing the properties of superfluid helium, J. F. Annett uses a field $\psi_0$ linked to our field $\psi$ by the relation $\psi_0 = \frac{\sqrt{n_0}}{\rho}\psi$ where $n_0$ is the density of helium atoms. Writing $\vec{r} = (x_1, x_2, x_3)$, he states: "In this chapter we shall focus on the properties of $^4$He. [...] let us imagine that we can characterize the state of helium by a local wave function $\psi_0(\vec{r})$ at every point in space. [...] We can choose to normalize the wave function so that the density of particles in the condensate is $n_0 = |\psi_0(\vec{r})|^2$. Therefore we can write the complex wave function as $\psi_0(\vec{r}) = \sqrt{n_0(\vec{r})}e^{i\theta(\vec{r})}$. Using the Schrödinger equation, J. F. Annett defines the conserved current $j_0 = \frac{\hbar}{2mi}(\psi_0^*\vec{\nabla}\psi_0 - \psi_0\vec{\nabla}\psi_0^*) = n_0\frac{\hbar}{m}\vec{\nabla}\theta$ where $\vec{\nabla}$ is a 3D gradient and he continues: "Since the condensate density is $n_0$ and the [...] current of particles equals a density times a velocity, we see that this can be interpreted as showing that the condensate flows with particle velocity $v_s$ where $v_s = \frac{\hbar}{m}\vec{\nabla}\theta$. This equation is the fundamental defining relation for the superfluid velocity, $v_s$. The existence of the macroscopic wave function in He II leads to a quantization of the superflow. [...] Now consider a flow along a closed tube. [...] Consider any path which extends around the whole tube length [...]. We can define the flow circulation by the integral $\kappa = \frac{\hbar}{m} \oint \vec{\nabla}\theta.d\vec{r} = \frac{\hbar}{m}\Delta\theta$. Here $\Delta\theta$ is the change in the phase angle theta after going around the tube. But if the macroscopic wave function, $\psi_0(\vec{r}) = \sqrt{n_0}e^{i\theta(\vec{r})}$ is to be uniquely defined we must have $\psi(\vec{r}) = \psi(\vec{r})e^{i\Delta\theta}$ which implies $\Delta\theta = 2\pi n$ where n is an integer. The circulation of the flows is therefore quantized. [...] This flow quantization has been confirmed in a number of experiments" [40]. Similarly, to derive the Hydrogen's spectrum with his equation, E. Schrödinger stated in 1926: "The limitation of n to integral values is necessary so that the surface harmonic may be single-valued" [41]. From the mathematical point of view, integers numbers appear in the solutions of the Schrödinger equation because looking for the solutions of the Laplacian is equivalent to looking for the solutions of a Sturm-Liouville operator with specific boundary conditions [42]. In quantum physics, the integer numbers associated to boundary conditions correspond to 'first quantization'. The other kind of quantization, called 'second quantization' will be discussed in [Section 5.2].

The plane wave solution $\psi = \mathcal{R}e^{-ik_\mu x^\mu} = \mathcal{R}e^{-i\frac{p_\mu x^\mu}{\hbar}} = \mathcal{R}e^{i\theta(x^\mu)}$ and the above (non relativistic) fundamental defining relation $v_s = \frac{\hbar}{m}\vec{\nabla}\theta \Leftrightarrow mv_s = \hbar\vec{\nabla}\theta$ suggests a deep link between the phase $\theta(x^\mu)$ of the field $\psi$ and $S/\hbar$
where $S$ is the mechanical action, since $\partial_\alpha S = p_\alpha$. Interestingly, it is the phase of a signal ($\theta$ for us), not its amplitude ($\mathcal{R}$ for us), which usually carries the essentiel information about the signal (an illustration is given in [Annex 1]). To examine the link between the phase and $S/\hbar$, we inject the ansatz $\psi = \mathcal{R}(x^\alpha)e^{-i\frac{S(x^\alpha)}{\hbar}}$ in the curved spacetime version of the Klein Gordon equation. The computation given in [Annex 1] produces:

$$0 = -\hbar^2 \Box_g \psi - m^2 c^2 \psi = \left[ g^{\mu\nu} \partial_\mu S \partial_\nu S - m^2 c^2 - \hbar^2 \frac{\Box_g \mathcal{R}}{\mathcal{R}} \right] \psi + \frac{i\hbar}{\rho} [\nabla_\mu (\rho \partial^\mu S)] \psi \quad (2.21)$$

where $\rho = \mathcal{R}^2$. Each term between [ ] must vanish. The term $\nabla_\mu (\rho \partial^\mu S) = \frac{i\hbar}{2} \nabla_\mu (\psi^* \partial^\mu \psi - \psi \partial^\mu \psi^*) = 0$ is a Wronskian equation, or continuity equation. Furthermore, identifying $S$ with the action, this term corresponds to the continuity equation $\nabla_\mu (\rho u^\mu) = \nabla_\mu j_m^\mu = 0$ of equation (2.10) since $\frac{\partial^\mu S}{m} = u^\mu$. The first term between [ ] shows that $S$ cannot be exactly identified with the mechanical action which would obey the simpler relation $g^{\mu\nu} \partial_\mu S \partial_\nu S - m^2 c^2 = 0$. Nevertheless, if the variations of the amplitude $\mathcal{R}$ are negligible when compared to the variations of the phase $S/\hbar$, the term $\hbar^2 \frac{\Box_g \mathcal{R}}{\mathcal{R}}$ can be neglected and $S$ can be identified with the mechanical action. Since the term $S/\hbar$ determines the frequency of the wave while the mechanical action determines the classical path through the principle of least action, the analogy between matter and light is immediate. Indeed, in optical sciences, the eikonal approximation corresponds to the short wavelength domain (or high frequency domain with the wavelength $\lambda$ given by $\lambda = c/\nu$). "The branch of optic which is characterized by the [...] limiting case $\lambda \to 0$ is known as geometrical optic, since in this approximation optical laws may be formulated in the language of geometry. The energy may then be regarded as being transported along certain curves (light rays)" [43]. Since light and matter obey wave equations, the eikonal approximation allows to recover the classical path in both cases. As stated by G. T'Hooft: "The rules of quantum mechanics work, but only if all natural phenomena in the world of the small are subjected to the same rules. This includes viruses, bacteria, even people. However, the bigger and heavier an object is, the harder it becomes to observe the quantum mechanical deviations from the ordinary, 'classical' laws of movement" [44]. Bigger and heavier objects are indeed associated to larger energies, and then larger frequencies or shorter wavelengths. Writing $\psi(x^\alpha) = \mathcal{R}(x^\alpha)e^{-i\frac{S(x^\alpha)}{\hbar}}$ to derive the classical motion from the Schrödinger equation is made in the books of Landau and Lifschitz [45] or P. A. M. Dirac [22], by example. To understand the non relativistic version of equation (2.21), P. A. M. Dirac indicates to interpret $\rho$ as the density of a fluid, the motion of the fluid being determined by $S$: this is in precisely what has been done here, in a generally covariant context. Keeping the ansatz $\psi(x^\alpha) = \mathcal{R}(x^\alpha)e^{-i\frac{S(x^\alpha)}{\hbar}}$ and using the relation $mc = \hbar k^4$, equation (2.12) takes the form:

$$S_m = \int \mathcal{L}_m \sqrt{g} d^4 x ds = \int \left( \frac{\hbar^2}{2m^2} g^{\mu\nu} (\partial_\mu \psi \partial_\nu \psi^* + \partial_\mu \psi^* \partial_\nu \psi) - c^2 |\psi|^2 \right) \sqrt{g} d^4 x ds \\ = \int \rho \left( g^{\mu\nu} \frac{\partial_\mu S}{m} \frac{\partial_\nu S}{m} - c^2 - \frac{\hbar^2}{m^2} \frac{\Box_g \mathcal{R}}{\mathcal{R}} \right) \sqrt{g} d^4 x ds \quad (2.22)$$

The computation, including a neglected boundary term, is given in [Annex 1]. Comparing equations (2.21) and (2.22), we check that, up to the neglected boundary term, $\mathcal{L}_m$ vanishes when the Klein-Gordon equation is obeyed. Clearly, the quantum Lagrangian density of equation (2.22) reproduces the classical Lagrangian density of equation (2.6) in the the eikonal approximation, i.e. when $-\frac{\hbar^2}{m^2} \frac{\Box_g \mathcal{R}}{\mathcal{R}}$ can be neglected.

A well established fact now deserves to be mentioned. There exists a deep geometrical link between:

- (a) the separability of the mechanical action according to $S(x^0, x^1, x^2, x^3) = S_0(x^0) + S_1(x^1) + S_2(x^2) + S_3(x^3)$, in the Hamilton-Jacobi equation for a particle in curved space $g^{\mu\nu} \frac{\partial_\mu S}{m} \frac{\partial_\nu S}{m} - c^2 = 0$;

- (b) the separability of $\psi$ according to $\psi(x^0, x^1, x^2, x^3) = \psi_0(x^0)\psi_1(x^1)\psi_2(x^2)\psi_3(x^3)$, in the Helmholtz equation $\Box_g \psi + (k_4)^2 \psi = 0$ or $\Box_g \psi + (\frac{mc}{\hbar})^2 \psi = 0$;

- (c) Einstein's equations $R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R = \frac{8\pi G}{c^4}T_{\mu\nu}$;

To illustrate this purpose, we quote C. P. Boyer, E. G. Kalnins, and W. Miller: "It follows easily from Stackel's original paper that any orthogonal separable system for [the Hamilton-Jacobi equation] also separates [the Helmholtz equation]. [...] Eisenhart's great contribution was to show that the Robertson condition amounted to the requirement $R_{\mu\nu} = 0$ $\mu \neq \nu$ where $R_{\mu\nu}$ is the Ricci tensor of the Riemann manifold. It follows immediately from this that for all Einstein spaces [...] the Helmholtz and Hamilton Jacobi equations separate in exactly the same orthogonal coordinate systems." [46]. Said simply, there exists a deep geometrical link between the equations of classical mechanics (a), quantum mechanics (b), and general relativity (c): it could certainly not be different.

We finally discuss the meaning of our Postulate 1, used to write the relation $ds^2 = dx_\mu dx^\mu$ in the form $0 = dx_\mu dx^\mu - ds^2$. We quote C. Bordé again: "Mass and proper time are entangled concepts which correspond to conjugate variables in classical mechanics and to non-commuting operators in quantum mechanics" [24]. Using $t$ as differentiation parameter, the usual action of a particle in Minkowski space is $S = \int mc ds dt =$
$\int mc^2 \sqrt{1 - v^2/c^2} dt$ while the corresponding Hamiltonian is $H = c\sqrt{p^2 + (mc)^2}$. Then:

$$ \frac{\partial H}{\partial mc} = c \frac{mc}{\sqrt{p^2 + (mc)^2}} = c \sqrt{\frac{(mc)^2}{p^2 + (mc)^2}} = c \sqrt{\frac{(mc)^2 + p^2 - p^2}{p^2 + (mc)^2}} = c \sqrt{1 - \frac{p^2 c^2}{E^2}} = c \sqrt{1 - \frac{v^2}{c^2}} = \frac{ds}{dt} \quad (2.23) $$

The above computation suggests to identify the mass as the conjugate momentum of $s = c\tau$ in Minkowski space. The Hamilton equation for the momentum $mc$ would then be $\frac{dmc}{dt} = -\frac{\partial H}{\partial s}$ and, since $H$ is independent of $s = c\tau$, the mass could be identified with a constant of motion, associated to the invariance of the Hamiltonian under variations of the proper time. Furthermore, if we replace the 4D Fourier transform applied to $\psi(x^\alpha)$ in equation (2.18) by a 5D Fourier transform applied to $\phi(x^\alpha, x^4) = \phi(x^\alpha, c\tau)$, a fifth Heisenberg inequality associated to the pair $(\tau; mc\dot{s})$ or $(\tau; mc^2)$ emerges. Thought experiments have been used to suggest that this fifth pair should be physical [47]. In high energy physics, the Heisenberg inequality associated to particle decays is interpreted in correlation with the pair $(ct; E/c)$ and not the pair $(\tau; mc\dot{s})$ because the possible existence of the fifth pair is a consequence of Postulate 1. The usual interpretation is now reconsidered. We quote C. E. Ibanez: "The Breit-Wigner distribution is used to model resonances (i.e. unstable particles) in high energy physics. [...] Any particle with a finite lifetime [$\Delta\tau$] has a mass distribution of non-zero width, explained by the Heisenberg uncertainty principle [$\Delta\tau = \hbar/\Gamma$]" [48]. From the operational point of view, D. J. Griffith explains (he writes $\Gamma = \Delta mc^2$): "You obviously cannot determine the lifetime of such a particle by measuring the length of its track. [...] Instead, you make a histogram of mass measurements, and invoke the uncertainty principle [...] so we get $\Delta\tau = \hbar/\Delta mc^2$. Thus the spread in mass is a measure of the particle's lifetime" [49]. To go further, we now use a thought experiment. We consider the decay of a $\pi_0$ meson into 2 photons, a process usually written: $\pi_0 \rightarrow \gamma\gamma$. It is known that the 2 photons are not structureless quanta (plane waves) of the electromagnetic field. According to quantum field theory, "[...] to work with strictly normalizable states in the continuum limit, the momentum eigenstates have to be smeared out by constructing waves packets" [50]. The same conclusion holds for particles obeying the Klein-Gordon equation, like the $\pi_0$, since: "[s]trictly speaking the theory should be formulated using [...] spatially localized wave packets" [50]. Furthermore, the spatiotemporal and spectral profiles of a single photon can be measured [51], modified [52], or controlled before the production of the photon by judiciously adapting the source [53]: the shape of a single photon's wave packet can then be appropriately chosen for a given use [54]. This engineering makes extensive use of Fourier transform theory, so physicists must take into account the Heisenberg inequality which relates the spatiotemporal and spectral distributions of an isolated photon [55]. When only the time-energy (or time-frequency) pair is concerned, the concept of 'time-bandwidth product' is introduced and this quantity admits a minimum value, determined by the saturation of the Heisenberg inequality (the saturation corresponds to a gaussian shape). We then consider that an isolated $\pi_0$ is a wave packet, such that it has a shape, like the photons produced in the process $\pi_0 \rightarrow \gamma\gamma$. We can easily imagine that in the future a physicist will be able to observe and modify the shape of $\pi_0$ mesons, by interactions in each spacetime region where each $\pi_0$ is located. As today with a single photon, the physicist of the future will be confronted with 4 Heisenberg inequalities associated to each $\pi_0$, one for each pair $(ct; E/c)$, $(x_1; p_1)$, $(x_2; p_2)$, $(x_3; p_3)$. The Heisenberg inequality associated to the mass and lifetime of the $\pi_0$ mesons will appear to the physicist as different from the others: while the four usual inequality refer to local measurements on a single particle, the last one refers to measurements made on many $\pi_0$ at different times and places. Precisely, at each time and place, the energy and the momentum of each $\pi_0$ should be measured to compute each mass, and then an histogram should be made to observe $\Delta mc^2$. Furthermore, at each time and place, the physicist should measure the time required to observe the decay of each $\pi_0$ and calculate the corresponding proper time for each $\pi_0$: then a second histogram should be made to observe $\Delta\tau$ and verify the relation $\Delta\tau\Delta mc^2 = \hbar > \hbar/2$, in agreement with the Breit-Wigner distribution. Computations given in literature conclude from the Breit-Wigner distribution that an unstable particle will decay according to $|\psi|^2 \sim e^{-\frac{\Gamma\tau}{\hbar}}$. The possible alternative now considered would be to describe the decay with the 5D field $\phi$ such that $|\phi|^2 \sim e^{-\frac{\Gamma\tau}{\hbar}}$. In the first case $|\psi|^2 = \rho \sim e^{-\frac{\Gamma\tau}{\hbar}}$: the Lagrangian density $\mathcal{L}_m$ depends explicitly on $t$. In the second case $|\phi|^2 = \rho \sim e^{-\frac{\Gamma\tau}{\hbar}}$: $\mathcal{L}_m$ depends explicitly on $\tau = s/c$. According to Noether's theorem, the energy should not be conserved if the Lagrangian $L_m$ depends on $t$, while the mass should not be conserved if $L_m$ depends on $\tau$: it is experimentally the mass which is not conserved in a decay process (the process $\pi_0 \rightarrow \gamma\gamma$ is an obvious example). This analysis suggests that the proper time is, in some way, a coordinate. More details will be provided in [Section 5.1].

To conclude this section, we derived quantum mechanics (including $p_\mu = \hbar k_\mu$) from an amended version of general relativity, with 2 postulates which simplify the physical content of general relativity. The first postulate replaced the usual relation $ds^2 = dx_\mu dx^\mu$ by $0 = dx_\mu dx^\mu - ds^2$. From the mathematical point of view, it was innocuous: usual general relativity was left unaffected. The second postulate was used to replace the classical matter Lagrangian by the quantum matter Lagrangian: it was then possible to identify the Klein Gordon Lagrangian density with the classical matter Lagrangian density, in the eikonal approximation. In the end of the section, the interpretation of the proper time as a coordinate was discussed.
### 3 Derivation of general relativity from quantum physics

In a famous book C. W. Misner, K. S. Thorne and J. A. Wheeler provide 6 routes to derive general relativity [56]: the aim of this section is to provide a 7th route, based on quantum physics. During an interview, A. Ashtekar declared in 2011: "In my opinion, the biggest puzzle is how to reconcile two existent and completely different views of the world. On one hand we have general relativity and on the other we have quantum field theory. These theories are conceptually very different from each other. In fact, in their foundations, they start with almost opposite assumptions. Moreover, their mathematics and conceptual frameworks are very distinct, and hence they give rise to two different views of the world, which seem hard to put together" [57]. D. Dieks gives the following comment about the conceptual framework of quantum field theory: "The question of how to handle localized systems is vital for the interpretation of relativistic quantum field theory. Without an answer to it the theory is incomplete at best and inconsistent at worst. This is because the fields in quantum field theory do not attach values of physical magnitudes to spacetime points. Rather, they are fields of operators, with a standard interpretation in terms of macroscopic measurement results. [...] What we would like to do is to provide an interpretation in which not only operators, but also properties are assigned to spacetime regions [...]. Obviously, the above programme is ambitious and addresses complicated and controversial issues" [58]. In this section, an interpretation of quantum field theory in which 'properties are assigned to spacetime regions' will emerge along with general relativity. Spin-0 fields are the simplest fields: therefore, standard textbooks on quantum field theory usually introduce the second quantization of spin-0 fields (or Klein-Gordon fields) before introducing the second quantization of fields with non-zero spin [50][59][60][61]. The starting point is then the following Klein-Gordon action in Minkowski space:

$$S_m = \int \mathcal{L}_m d^4 y = \int \frac{\hbar^2}{2m^2} \eta^{\mu\nu} (\partial_\mu \psi \partial_\nu \psi^* + \partial_\mu \psi^* \partial_\nu \psi) - c^2 |\psi|^2 d^4 y \quad (3.1)$$

where the Lagrangian density has been symmetrized by using $\eta^{\mu\nu}\partial_\mu\psi\partial_\nu\psi^* = \frac{1}{2}\eta^{\mu\nu}(\partial_\mu\psi\partial_\nu\psi^* + \partial_\mu\psi^*\partial_\nu\psi)$, while the coordinates have been labeled $y^\mu = (y^0, y^1, y^2, y^3)$. The equations of the motion are obtained by extremizing the action. Precisely the action must remain invariant (to the first order of approximation) if the fields $\psi$ and $\psi^*$ are varied by adding an arbitrary infinitesimal field to each of them [62]: $\psi \to \psi + \delta\psi$, $\psi^* \to \psi^* + \delta\psi^*$, $\partial_\mu\psi \to \partial_\mu\psi + \partial_\mu\delta\psi$ and $\partial_\mu\psi^* \to \partial_\mu\psi^* + \partial_\mu\delta\psi^*$

$$ \begin{aligned} 0 = \delta S_m &= \int \frac{\partial \mathcal{L}_m}{\partial \psi} \delta \psi + \frac{\partial \mathcal{L}_m}{\partial \partial_\mu \psi} \partial_\mu \delta \psi + \text{``cc''} d^4 y \\ &= \int \left( \frac{\partial \mathcal{L}_m}{\partial \psi} - \partial_\mu \frac{\partial \mathcal{L}_m}{\partial \partial_\mu \psi} \right) \delta \psi + \partial_\mu \left[ \frac{\partial \mathcal{L}_m}{\partial \partial_\mu \psi} \delta \psi \right] + \text{``cc''} d^4 y \end{aligned} \quad (3.2) $$

where "cc" denotes the complex conjugate of the terms on the left. The last term between [ ] appears inside the expression of a total derivative and does not contributes to $\delta S_m$ under appropriate boundary conditions. The term between ( ) must then vanish, which produces the equations of motion. The equations of motion are the Klein-Gordon equation for $\psi$ and $\psi^*$: $-\hbar^2\Box\psi - m^2c^2\psi = 0$. Instead of introducing second quantization, the strategy is now to express the action in curvilinear coordinates: an arbitrary change of coordinates is obtained by writing $y^\mu = y^\mu(x^\alpha)$ where $x^\alpha$ is any new coordinate system. Then $\partial_\mu$ becomes $\frac{\partial x^\alpha}{\partial y^\mu}\partial_\mu$. We write $g_{\alpha\beta} = \eta_{\mu\nu}\frac{\partial y^\mu}{\partial x^\alpha}\frac{\partial y^\nu}{\partial x^\beta}$ such that $\eta^{\mu\nu}\partial_\mu\psi\partial_\nu\psi^*$ becomes $g^{\alpha\beta}\partial_\alpha\psi\partial_\beta\psi^*$ while $d^4y$ becomes $\sqrt{g}d^4x$ where $g = -\det|g_{\alpha\beta}|$. Expressed in the new coordinate system, the action takes the form:

$$S_m = \int \mathcal{L}_m \sqrt{g} d^4 x = \int \left( \frac{\hbar^2}{2m^2} g^{\alpha\beta} (\partial_\alpha \psi \partial_\beta \psi^* + \partial_\alpha \psi^* \partial_\beta \psi) - c^2 |\psi|^2 \right) \sqrt{g} d^4 x \quad (3.3)$$

The process introduced the tensor field $g^{\alpha\beta}(x^\gamma)$. We require again that the action remains invariant if the fields are varied by adding an arbitrary infinitesimal field to each of them: $\psi \to \psi + \delta\psi$, $\psi^* \to \psi^* + \delta\psi^*$, $\partial_\mu\psi \to \partial_\mu\psi + \partial_\mu\delta\psi$, $\partial_\mu\psi^* \to \partial_\mu\psi^* + \partial_\mu\delta\psi^*$ and $g^{\alpha\beta} \to g^{\alpha\beta} + \delta g^{\alpha\beta}$. Evidently $\sqrt{g}$ is varied when $g^{\alpha\beta}$ is varied. Equation (3.2) is modified according to:

$$\delta S_m = \int \left( \left( \frac{\partial \mathcal{L}_m}{\partial \psi} - \nabla_\mu \frac{\partial \mathcal{L}_m}{\partial \partial_\mu \psi} \right) \delta \psi + \nabla_\mu \left[ \frac{\partial \mathcal{L}_m}{\partial \partial_\mu \psi} \delta \psi \right] + \text{``cc''} + T_{\alpha\beta} \delta g^{\alpha\beta} \right) \sqrt{g} d^4 x \quad (3.4)$$

where "cc" still denotes the complex conjugate of the terms on the left, $\nabla_\mu$ is the covariant derivative induced by the factor $\sqrt{g}$ when the integration by parts is made, and $T_{\alpha\beta}$ is given by:

$$T_{\alpha\beta} = \frac{\partial \mathcal{L}_m}{\partial g^{\alpha\beta}} - \frac{1}{2} g_{\alpha\beta} \mathcal{L}_m = \frac{\hbar^2}{2m^2} (\partial_\alpha \psi \partial_\beta \psi^* + \partial_\alpha \psi^* \partial_\beta \psi) - \frac{1}{2} g_{\alpha\beta} \mathcal{L}_m \quad (3.5)$$

The equations of motion for $\psi$ and $\psi^*$ are now $-\hbar^2\Box\psi - m^2c^2\psi = 0$ where $\Box_g$ is the covariant Laplacian. The variations $\delta\psi$ and $\delta g^{\alpha\beta}$ being arbitrary, the term $T_{\alpha\beta}\delta g^{\alpha\beta}$ produces $\delta S_m \neq 0$ in equation (3.4). To maintain
the condition $\delta S_m = 0$, we add a term $\mathcal{L}_g$ to the Lagrangian density $\mathcal{L}_m$, in order to cancel the term $T_{\alpha\beta}\delta g^{\alpha\beta}$. The equations of motion for $\psi$ and $\psi^*$ remain unaffected if $\mathcal{L}_g$ is independent of $\psi$ and $\psi^*$: then $\mathcal{L}_g$ should be interpreted as the Lagrangian density of the free field $g^{\alpha\beta}$. If $\mathcal{L}_m$ becomes $\mathcal{L}_m + \mathcal{L}_g$ in equation (3.3), the variation $\delta S_m$ in equation (3.4) becomes:

$$ \delta S_m = \int \left( \left( \frac{\partial \mathcal{L}_m}{\partial \psi} - \nabla_\mu \frac{\partial \mathcal{L}_m}{\partial \partial_\mu \psi} \right) \delta \psi + \nabla_\mu \left[ \frac{\partial \mathcal{L}_m}{\partial \partial_\mu \psi} \delta \psi \right] + \text{``cc''} + \left( T_{\alpha\beta} + \frac{\partial \mathcal{L}_g}{\partial g^{\alpha\beta}} - \frac{1}{2} g_{\alpha\beta} \mathcal{L}_g \right) \delta g^{\alpha\beta} \right) \sqrt{g} d^4 x \quad (3.6) $$

The condition $\delta S_m = 0$ now implies the following equations of motion for the field $g^{\alpha\beta}$:

$$ T_{\alpha\beta} + \frac{\partial \mathcal{L}_g}{\partial g^{\alpha\beta}} - \frac{1}{2} g_{\alpha\beta} \mathcal{L}_g = 0 \qquad (3.7) $$

The tensor $\frac{\partial \mathcal{L}_g}{\partial g^{\alpha\beta}} - \frac{1}{2} g_{\alpha\beta} \mathcal{L}_g$ should be built from $g_{\mu\nu}$ and its first two derivatives, by analogy with the field equations of the Klein-Gordon field or the electromagnetic field (otherwise the boundary conditions required to determine the solutions of the field equations would differ from other fields). Furthermore, when the equations of the motion for $\psi$ and $\psi^*$ are obeyed, it is shown in [Annex 2] that $\nabla_\beta T^{\alpha\beta} = 0$. Since $T_{\alpha\beta} = T_{\beta\alpha}$ and $\nabla_\alpha T^{\beta\alpha} = 0$, the tensor $\frac{\partial \mathcal{L}_g}{\partial g^{\alpha\beta}} - \frac{1}{2} g_{\alpha\beta} \mathcal{L}_g$ should be also symmetric and divergenceless. It is a theorem due to D. Lovelock that the unique tensor satisfying such constraints is given by [56][63]:

$$ \frac{\partial \mathcal{L}_g}{\partial g^{\alpha\beta}} - \frac{1}{2} g_{\alpha\beta} \mathcal{L}_g = \frac{1}{\kappa} \left( R_{\alpha\beta} - \frac{1}{2} g_{\alpha\beta} R \right) + \Lambda g_{\alpha\beta} = \frac{1}{\kappa} \left( \frac{\partial R}{\partial g^{\alpha\beta}} - \frac{1}{2} g_{\alpha\beta} R \right) + \Lambda g_{\alpha\beta} \qquad (3.8) $$

where $\kappa$ and $\Lambda$ are two constants which should be determined experimentally, $R$ is the Ricci scalar and $R_{\alpha\beta}$ the Ricci tensor. A boundary term has been omitted to write $R_{\alpha\beta} = \frac{\partial R}{\partial g^{\alpha\beta}}$ [27][28]. The constant $\Lambda$ is the cosmological constant and will not be discussed: we consider $\Lambda = 0$. The constant $\kappa$ is discussed below. To get the classical limit, we take the eikonal approximation: we write $\psi = \mathcal{R}(x^\gamma)e^{-i\frac{S(x^\gamma)}{\hbar}}$ where $S$ is identified with the mechanical action. Writing $\mathcal{R}^2 = \rho$, it is shown in [Annex 2] that $T^{\alpha\beta} = \rho \frac{\partial^\alpha S}{\partial m} \frac{\partial^\beta S}{\partial m} + \epsilon(\hbar^2)$ where the epsilon term is by definition neglected in the eikonal approximation. Then $\frac{\partial^\beta S(x^\gamma)}{\partial m}$ is identified with the 4-velocity $u^\beta(x^\gamma)$. As we have seen in [Section 2], the same considerations are used to describe the motion of superfluid Helium: normalizing $\psi_0 = \sqrt{n_0} e^{i\theta}$ such that $n_0$ is identified with the density of atoms in the superfluid, the velocity of the superfluid is $\vec{v}_s = \frac{\hbar}{m} \vec{\nabla}\theta$. The tensor $T^{\alpha\beta} \approx \rho \frac{\partial^\alpha S}{\partial m} \frac{\partial^\beta S}{\partial m} \approx \rho u^\alpha u^\beta$ is now identified with the energy momentum tensor. In the classical limit, we take $T^{\alpha\beta} = \rho u^\alpha u^\beta$ and the condition $\nabla_\alpha T^{\alpha\beta} = 0$ becomes:

$$ \nabla_\alpha T^{\alpha\beta} = \nabla_\alpha (\rho u^\alpha u^\beta) = 0 \quad \Rightarrow \quad [u^\beta \nabla_\alpha (\rho u^\alpha)] + [u^\alpha \partial_\alpha u^\beta + \Gamma_{\alpha\sigma}^\beta u^\alpha u^\sigma] = 0 \qquad (3.9) $$

Contracting the equation on the right hand side with $u_\beta$ shows that $\nabla_\alpha (\rho u^\alpha) = 0$ and $u^\alpha \partial_\alpha u^\beta + \Gamma_{\alpha\sigma}^\beta u^\alpha u^\sigma = 0$ which correspond to the continuity equation and the geodesic equation: the two equations determines the classical motion of the particles associated with the fields $\psi$ and $\psi^*$ in the background field $g_{\alpha\beta}$. Since the energy momentum tensor $T_{\alpha\beta}$ is the source in the equations of the motion of $g_{\alpha\beta}$, we can finally identify $g_{\alpha\beta}$ with the gravitational field. Consistency with the Newtonian limit imposes $\kappa = -\frac{8\pi G}{c^4}$ where $G$ is Newton's constant. Einstein's equations $R_{\alpha\beta} - \frac{1}{2}g_{\alpha\beta}R = \frac{8\pi G}{c^4}T_{\alpha\beta}$ follows. Furthermore $\mathcal{L}_g = -\frac{R}{8\pi G} = -\frac{g^{\alpha\beta}R_{\alpha\beta}}{8\pi G}$. When there is no source, $R = 0$ and the action of the gravitational field vanishes, like the action of the Klein-Gordon field in equation (2.22). The above analysis addressed the conceptual question introduced by D. Dieks (see the quote above equation (3.1)): an interpretation of quantum physics in which quantum fields have physical magnitudes associated to each spacetime point has been provided, the classical limit being obtained in the eikonal approximation. Superfluid Helium is an example in which such physical magnitudes are experimentally observed: the superfluid, with density $n_0 = |\psi_0|^2$, is indeed visible with naked eyes.

According to literature, quantum mechanics and general relativity are two drastically different theories. By example, U. Bannier stated in 1988: "One of the most fascinating challenges of contemporary physics is the unification of Einstein's general relativity theory with quantum theory. Many attempts are made, but the goal seems to be still far off. It even is not clear how much of the conceptual and technical structures of both theories will survive an unification, because they seem to be fundamentally different" [64]. Nevertheless, in [Section 2] we derived quantum mechanics from the action of general relativity (we derived the Klein-Gordon equation which reproduces the Schrödinger equation in the non relativistic limit): the constants $c$ and $G$ where assumed and the constant $\hbar$ was derived. In the present section we derived general relativity from the action of the Klein-Gordon field: the constants $c$ and $\hbar$ were assumed and the constant $G$ was derived.
# 4 New results in general relativity (Kaluza-Klein theory)

The aim of this section is to unify the Lagrangian densities of general relativity and quantum electrodynamics (applied to a spin-0 matter field). According to S. Weinberg, "*Einstein devoted the last 30 years of his life to an unsuccessful search for a 'unified field theory', which would unite general relativity - his own theory of spacetime and gravitation - with Maxwell's theory of electromagnetism*" [65]. During his quest, A. Einstein published papers on the five dimensional Kaluza-Klein theory, at least between 1927 [66] and 1943 [67]. With W. Pauli he writes in 1943: "*When one tries to find a unified theory of gravitational and electromagnetic fields, he cannot help the feeling that there is some truth in Kaluza's five dimensional theory*" [67]. The Kaluza-Klein theory was introduced by T. Kaluza who sent his results to A. Einstein in 1919, and it was first published in 1921. To our best knowledge, the Kaluza-Klein theory contains 3 severe drawbacks: the meaning of the fifth dimension is not explained, the fact that the metric should not depend on the fifth dimension is not justified, and worse, the theory contains a scalar field which modifies the electromagnetic Lorentz's force without justifying that such a modification is not experimentally observed. Literature then admits that the original theory fails to unify electricity and gravity. Nevertheless, the Kaluza-Klein theory has become a basic tool of string theory. In this context, J. Polshinski explains: "*A second idea is that spacetime has more than four dimensions, with the additional ones so highly curved as to be undetectable at current energies [...] This opens the possibility of unifying the gauge interactions and gravity (the Kaluza-Klein mechanism) [...] Each of these ideas - grand unification, extra dimensions, and supersymmetry - has attractive features and is consistent with the various tests of the Standard Model. It is plausible that these will be found as elements of a more complete theory of fundamental physics. It is clear, however, that something is still missing*" [68]. In this section we will reconsider the original Kaluza-Klein theory. Nevertheless, our own interest will be limited to the connexions between the Kaluza-Klein theory and the results exposed in the previous sections. As tensorial computations are very long, they will often be left in the annexes.

## 4.1 Classical unification of Einstein's gravity and Maxwell's electrodynamics

Our first postulate was the mathematically innocuous change: $ds^2 = dx_\mu dx^\mu \to 0 = dx_\mu dx^\mu - ds^2$. We observed at the end of [Section 2] that this change nevertheless suggested to go one step further than the special theory of relativity, by including the proper time in the list of the coordinates. We now suggest to go a second step further by assuming $c^2 d\tau^2 = ds^2 = (dx_5)^2 - (dx_6)^2$ such that $0 = dx_\mu dx^\mu - (dx_5)^2 + (dx_6)^2$. We then begin with a new postulate, which constitutes our first contribution to the Kaluza Klein theory. In this postulate $\mu_0$ is the vacuum magnetic permability, $q$ the unit electric charge, and $m$ the mass of a particle with charge $q$.

**Postulate 3: Postulate 1 is amended. In a flat space labelled with coordinates $(y^\alpha, y^4)$, the proper time has an internal structure such that $c^2 = \dot{s}^2 = (\dot{y}^4)^2 = (\dot{y}^5)^2 - (\dot{y}^6)^2$ where $(\dot{y}^5)^2 = \frac{c^4 \mu_0 g^2}{16\pi G m^2}$: $y^6$ is invariant, electromagnetism corresponds to coordinates transformations $y^5 = y^5(x^\mu, x^5)$, 4 dimensional transformations remain unaffected, $y^\alpha = y^\alpha(x^\mu)$. Associated fields are massless.**

According to Postulate 3, the proper time has a hyperbolic internal structure: for a given particle, the momentum $m\dot{y}^5 = \sqrt{\frac{c^4 \mu_0 g^2}{16\pi G}}$ will allow us to fix the ratio between the magnitudes of the gravitational and electromagnetic interactions. In the present paper, this hyperbolic structure will not be discussed: indeed, an equation is solved, the definition domain is an input data. Our definition domain, or our spacetime, has been specified by postulates 1 and 3, and it was simply chosen to reproduce known physics. It will be convenient to introduce the mass scale $M$ according to $M^2 = \frac{c^2 \mu_0 g^2}{16\pi G}$ such that $(\dot{y}^5)^2 = c^2 \frac{M^2}{m^2}$ and $(\dot{y}^6)^2 = c^2(\frac{M^2}{m^2} - 1)$. Then $M \approx 0.0427 m_P$ where $m_P = \sqrt{\frac{\hbar c}{G}}$ is the Planck mass. To clarify and illustrate the meaning of Postulate 3, we start from the vanishing classical Lagrangian density $\mathcal{L}_m$ in flat space. Flat space being labelled by coordinates $(y^0, y^1, y^2, y^3, y^5, y^6) = (y^M, y^6)$ we apply a coordinate transformation to express the Lagrangian density in another coordinate system $(x^M, x^6)$, in agreement with Postulate 3 ($y^6 = x^6$ since the last coordinate is not included in the transformation). To distinguish 5 dimensional quantities from usual 4 dimensional quantities, 5 dimensional quantities will be underlined (quantities associated to the sixth coordinate will not be underlined). $\eta_{MN}$ is the diagonal matrix $diag(1, -1, -1, -1, -1)$:

$$ 
\begin{aligned}
0 = \mathcal{L}_m &= \rho \left( (\dot{y}^0)^2 - (\dot{y}^1)^2 - (\dot{y}^2)^2 - (\dot{y}^3)^2 - \underline{(\dot{y}^5)^2} + (\dot{y}^6)^2 \right) \\
\mathcal{L}_m &= \rho \left( \eta_{MN} \frac{\dot{y}^M}{\dot{y}^N} + (\dot{y}^6)^2 \right) = \rho \left( \eta_{\alpha\beta} \dot{y}^\alpha \dot{y}^\beta - \underline{(\dot{y}^5)^2} + (\dot{y}^6)^2 \right) \\
\mathcal{L}_m &= \rho \left( \eta_{\alpha\beta} \frac{\partial y^\alpha}{\partial x^\mu} \frac{\partial y^\beta}{\partial x^\nu} \dot{x}^\mu \dot{x}^\nu - \underline{\left( \dot{x}^5 \frac{\partial y^5}{\partial x^5} + \dot{x}^\mu \frac{\partial y^5}{\partial x^\mu} \right)} + (\dot{x}^6)^2 \right)
\end{aligned}
\quad (4.1) $$
We now conventionally relabel $\eta_{\alpha\beta} \frac{\partial y^\alpha}{\partial x^\mu} \frac{\partial y^\beta}{\partial x^\nu} = g_{\mu\nu}$, $\frac{\partial y^5}{\partial x^5} = \Phi$ and $\Phi^{-1} \frac{\partial y^5}{\partial x^\mu} = -q \frac{A^\mu}{Mc}$ such that $\mathcal{L}_m$ becomes:

$$0 = \mathcal{L}_m = \rho \left( g_{\mu\nu} \dot{x}^{\mu} \dot{x}^{\nu} - \Phi^2 \left( \dot{x}^5 - \frac{q A_{\mu}}{Mc} \dot{x}^{\mu} \right)^2 + (\dot{x}^6)^2 \right) \quad (4.2)$$

$$0 = \mathcal{L}_m = \rho \left( \left( g_{\mu\nu} - \Phi^2 \frac{q A_{\mu}}{Mc} \frac{q A_{\nu}}{Mc} \right) \dot{x}^{\mu} \dot{x}^{\nu} + \left( \Phi^2 \frac{q A_{\mu}}{Mc} \dot{x}^{\mu} \dot{x}^{5} + \Phi^2 \frac{q A_{\nu}}{Mc} \dot{x}^{\nu} \dot{x}^{5} \right) - \Phi^2 \dot{x}^{5} \dot{x}^{5} \right) + \rho \dot{x}^{6} \dot{x}^{6} \quad (4.3)$$

Apart from the additional term $\rho \dot{x}^6 \dot{x}^6$ (and some signs associated to our sign conventions), equation (4.2) is the starting point of the Kaluza-Klein theory. We immediately see that the change $\dot{x}^5 \to -\dot{x}^5$ is equivalent to the change $q \to -q$: in the theory presently discussed, charge conjugation is then simply a space inversion applied to the fifth coordinate. This change is also equivalent to the change $\dot{x}^\mu \to -\dot{x}^\mu$, corresponding to combined parity and time reversal operations. The unused term $\rho \dot{x}^6 \dot{x}^6$ is not esthetical: the present paper is only concerned by gravitational and electromagnetic interactions, while 2 other interactions exist (the strong and the weak interactions). Obviously, a relevant discussion of the term $\rho \dot{x}^6 \dot{x}^6$ would require the inclusion of the other interactions in the present theory. Curvature will be determined by Einstein's equations, as usual. In equation (4.2), $g_{\mu\nu} = g_{\mu\nu}(x^\alpha)$, $A^\mu = A^\mu(x^\alpha)$ and $\Phi = \Phi(x^\alpha)$: indeed, according to Postulate 3, the 5 dimensional metric is independent of $x^5$ otherwise the fields would not be massless. The present interpretation allows to solve 2 of the 3 drawbacks of the original Kaluza-Klein theory: the fifth coordinate is interpreted as a part of the proper time and the fields $g_{\mu\nu}$, $A^\mu$, $\Phi$ only depend on $x^\alpha$ because the energy-momentum tensor, which is the source of the gravitational field, only depend on $x^\alpha$ (the energy-momentum tensor is derived from the Lagrangian density $\mathcal{L}_m$ which only depends on $x^\alpha$ according to Postulate 1, at least when stable matter is considered). To simplify the discussion, we momentarily introduce the particle Lagrangian $L_m$ according to

$$0 = L_m = \frac{m}{2} \left( g_{\mu\nu} \dot{x}^{\mu} \dot{x}^{\nu} - \Phi^2 \left( \dot{x}^5 - \frac{q A_{\mu}}{Mc} \dot{x}^{\mu} \right)^2 + (\dot{x}^6)^2 \right) \quad (4.4)$$

Making the change $A_\mu \to 0$, $\Phi \to 1$, $(\dot{x}^5)^2 - (\dot{x}^6)^2 \to (\dot{x}^4)^2$, we recognize equation (2.3). With $g_{\mu\nu} \dot{x}^\mu \dot{x}^\nu = 1$ while $(\dot{x}^6)^2 = (\dot{y}^6)^2 = (\frac{M^2}{m^2} - 1)c^2$ is constant, we know that $\Phi^2(\dot{x}^5 - \frac{q A_\mu}{Mc} \dot{x}^\mu)^2 = \frac{M^2}{m^2}c^2$. The Euler-Lagrange equations associated to $\dot{x}^5$ and $x^\alpha$ are given in equations (4.5) and (4.6):

$$0 = \frac{d}{d\tau} \frac{\partial L_m}{\partial \dot{x}^5} - \frac{\partial L_m}{\partial \dot{x}^5} = - \frac{d}{d\tau} \left[ \Phi^2 \left( \dot{x}^5 - q \frac{A_\mu}{Mc} \dot{x}^\mu \right) \right] \quad (4.5)$$

We deduce that $[\Phi^2(\dot{x}^5 - \frac{q A_\mu}{Mc} \dot{x}^\mu)]$ is a constant of motion. Since $\Phi^2(\dot{x}^5 - \frac{q A_\mu}{Mc} \dot{x}^\mu)^2 = \frac{M^2}{m^2}c^2$, $\Phi$ and $\dot{x}^5 - \frac{q A_\mu}{Mc} \dot{x}^\mu$ are two constants of the motion. When the spacetime flat, $\Phi = 1$ and $\dot{x}^5 = \frac{M}{m}c$: we will then consider that $\Phi = 1$ and $\dot{x}^5 - \frac{q A_\mu}{Mc} \dot{x}^\mu = \frac{M}{m}c$ along the path of our particle, since $\frac{q A_\mu}{Mc} \dot{x}^\mu$ is an infinitesimal quantity from the experimental point of view. The fact that $\dot{x}^5$ is an enormous quantity for usual particles has been an argument against the original Kaluza-Klein theory since it appeared very surprising to include a velocity much greater than the speed of light in general relativity. In the present theory, it is a consequence of the hyperbolic internal structure given to the proper time and so a topological consequence of the space used to apply general relativity. Using the notation $F_{\mu\alpha} = \partial_\mu A_\alpha - \partial_\alpha A_\mu = A_{\alpha,\mu} - A_{\mu,\alpha}$, the other Euler-Lagrange equations, together with an additional term $\frac{g^{\alpha\beta}}{m}$, are:

$$\begin{align}
0 &= \frac{g^{\alpha\beta}}{m} \left( \frac{d}{d\tau} \frac{\partial L_m}{\partial \dot{x}^{\alpha}} - \frac{\partial L_m}{\partial x^{\alpha}} \right) \nonumber \\
0 &= g^{\alpha\beta} \frac{d}{d\tau} \left( g_{\alpha\nu} \dot{x}^{\nu} - \left[ \Phi^2 \left( \dot{x}^5 - \frac{q A_{\mu} \dot{x}^{\mu}}{Mc} \right) \right] \left( -\frac{q A_{\alpha}}{Mc} \right) \right) \nonumber \\
&\quad - g^{\alpha\beta} \left( \frac{1}{2} g_{\mu\nu,\alpha} \dot{x}^{\mu} \dot{x}^{\nu} - \left[ \Phi^2 \left( \dot{x}^5 - \frac{q A_{\mu}}{Mc} \dot{x}^{\mu} \right) \right] \left( -\frac{q A_{\alpha,\mu}}{Mc} \dot{x}^{\mu} \right) - \frac{\partial_{\alpha} \Phi}{\Phi^3} \left[ \Phi^2 \left( \dot{x}^5 - \frac{q A_{\mu}}{Mc} \dot{x}^{\mu} \right) \right] \right)^2 \tag{4.6} \\
0 &= g^{\alpha\beta} \left( g_{\alpha\nu} \ddot{x}^{\nu} + \frac{1}{2} (g_{\alpha\nu,\mu} + g_{\mu\alpha,\nu}) \dot{x}^{\mu} \dot{x}^{\nu} + \frac{q A_{\alpha,\mu}}{m} \dot{x}^{\mu} - \frac{1}{2} g_{\mu\nu,\alpha} \dot{x}^{\mu} \dot{x}^{\nu} - \frac{q A_{\mu,\alpha}}{m} \dot{x}^{\mu} + c^2 \frac{M^2}{m^2} \frac{\partial_{\alpha} \Phi}{\Phi^3} \right) \nonumber \\
0 &= \ddot{x}^{\beta} + \Gamma^{\beta}_{\mu\nu} \dot{x}^{\mu} \dot{x}^{\nu} + \frac{q}{m} F^{\beta}_{\mu} \dot{x}^{\mu} + c^2 \frac{M^2}{m^2} \frac{\partial^{\beta} \Phi}{\Phi^3} \nonumber \\
&\Rightarrow \ddot{x}^{\beta} + \Gamma^{\beta}_{\mu\nu} \dot{x}^{\mu} \dot{x}^{\nu} = \frac{q}{m} F^{\beta}_{\mu} \dot{x}^{\mu} - c^2 \frac{M^2}{m^2} \frac{\partial^{\beta} \Phi}{\Phi^3} \nonumber
\end{align}$$

We recognize the gravitational geodesic equation together with the electromagnetic Lorentz force [3][37] and a new term, proportional to $c^2 \frac{M^2}{m^2}$ where $c^2 \frac{M^2}{m^2}$ is an enormous quantity for usual particles. We will use the field equations to justify why this new term has not been experimentally identified yet. Since:

$$\dot{x}_{\beta} (\ddot{x}^{\beta} + \Gamma^{\beta}_{\mu\nu} \dot{x}^{\mu} \dot{x}^{\nu}) = \dot{x}_{\beta} \ddot{x}^{\beta} + \frac{1}{2} (g_{\beta\mu,\nu} + \underline{g_{\nu\beta,\mu} - g_{\mu\nu,\beta}}) \dot{x}^{\mu} \dot{x}^{\nu} \dot{x}^{\beta} = \frac{1}{2} \frac{d}{d\tau} (g_{\mu\beta} \dot{x}^{\mu} \dot{x}^{\beta}) = \frac{1}{2} \frac{d}{d\tau} (1) = 0 \quad (4.7)$$
contracting the last line of equation (4.6) with $\dot{x}_{\beta}$ gives :

$$ \frac{\dot{x}_{\beta}(\ddot{x}^{\beta} + \Gamma^{\beta}_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu})}{m(F_{\beta\mu}\dot{x}^{\mu}\dot{x}^{\nu})} = \frac{q}{M^2} \frac{\dot{x}_{\beta}\partial^{\beta}\Phi}{\Phi^3} = 0 \quad (4.8) $$

where we used $F_{\mu\beta} = -F_{\beta\mu}$. The equality $\dot{x}_{\beta}\partial^{\beta}\Phi = \frac{d\Phi}{d\tau} = 0$ shows again that $\Phi$ is constant of the motion.

We now examine the field equations of the theory. As $g_{66}$ is equal to 1, we only consider the 5-dimensional part of the metric to compute the Ricci scalar. The 5D metric $g_{MN}$ and its inverse $g^{MN}$, are :

$$ g_{MN} = \begin{bmatrix} g_{\mu\nu} - \frac{\Phi^2 \frac{qA_{\mu}}{Mc} \frac{qA_{\nu}}{Mc}}{\Phi^2 \frac{qA_{\nu}}{Mc}} & \Phi^2 \frac{qA_{\mu}}{Mc} \\ \frac{qA_{\mu}}{Mc} & -\Phi^2 \end{bmatrix} ; \quad g^{MN} = \begin{bmatrix} g^{\mu\nu} & \frac{qA^{\nu}}{Mc} \\ \frac{qA^{\mu}}{Mc} & -\frac{1}{\Phi^2} + \frac{qA_{\alpha}}{Mc} \frac{qA^{\alpha}}{Mc} \end{bmatrix} \quad (4.9) $$

Then $g^{\mu\nu} = g^{\nu\mu}$. 4D indices are upped (or lowered) with $g^{\mu\nu}$ (or $g_{\mu\nu}$). Up to a sign, the square root of the determinant of the 5D metric is $\Phi\sqrt{g}$. We now work with the continuous approximation : $\dot{x}^{\mu} \to u^{\mu}(x^{\alpha})$, $\dot{x}^{5} \to u^{5}(x^{\alpha})$, $\dot{x}^{6} \to u^{6}$, $\frac{mc}{2} \to \rho(x^{\alpha})$. In 5D notation, we use Postulate 3 to generalize the Lagrangian density $\mathcal{L}_m = \rho g^{\mu\nu}u_{\mu}u_{\nu} - \rho u_4^2 = 0$ of equation (2.6) according to :

$$ 0 = \mathcal{L}_m = \rho g^{MN} u_M u_N + \rho u^6 u^6 = \rho g^{MN} u_M u_N + \rho \left( \frac{M^2}{m^2} - 1 \right) c^2 \quad (4.10) $$

The Einstein Hilbert action $S_{EH6}$ corresponding to the present theory is :

$$ S_{EH6} = \int \left( \rho g^{MN} u_M u_N + \rho u^6 u^6 - \frac{c^4}{8\pi G} \underline{R} \right) \Phi \sqrt{g} d^6 x \quad (4.11) $$

where $\underline{R} = g^{MN}R_{MN}$ is a 5D Ricci scalar. All results given from equations (4.12) to (4.38) follow directly from this action. Since we imposed the conditions $g^{\mu6} = 0$ and $g^{66} = 1$ with Postulate 3, we fix $\delta g^{\mu6} = 0$ and $\delta g^{66} = 0$ when the action $S_{EH6}$ is varied. The variation gives (boundary terms are still omitted from the discussion) :

$$ 0 = \delta S_{EH6} = \int \left( \rho \left( u_M u_N - \frac{1}{2} g_{MN} \underline{\mathcal{L}}_m \right) - \frac{c^4}{8\pi G} \left( R_{MN} - \frac{1}{2} g_{MN} \underline{R} \right) \right) \Phi \sqrt{g} \delta g^{MN} d^6 x \quad (4.12) $$

from which we deduce the 5D Einstein's equations :

$$ \underline{G}_{MN} = \underline{R}_{MN} - \frac{1}{2} g_{MN} \underline{R} = \frac{8\pi G}{c^4} \rho \underline{u}_M \underline{u}_N = \frac{8\pi G}{c^4} \underline{T}_{MN} \quad (4.13) $$

The expression of the 5D Ricci scalar $\underline{R}$ and the Einstein tensor corresponding to the metric given in equation (4.2), are computed in [Annex 3]. The final results are :

$$ \underline{R} = R + \frac{8\pi G}{c^4} \left( \frac{\Phi^2}{2\mu_0} F^{\alpha\beta} F_{\alpha\beta} \right) - 2 \frac{\Box g}{\Phi} \quad (4.14) $$

$$ \underline{G}_{55} = \frac{\Phi^2}{2} R + \frac{8\pi G}{c^4} \left( \frac{3\Phi^4}{4\mu_0} F^{\alpha\beta} F_{\alpha\beta} \right) \quad (4.15) $$

$$ \underline{G}_{\mu 5} = -\frac{\Phi^2}{2} \frac{q}{Mc} \nabla_{\nu} F^{\mu}_{\nu} - \frac{3\Phi}{2} \frac{q}{Mc} F_{\nu\mu} \partial^{\nu} \Phi - \frac{qA_{\mu}}{Mc} \underline{G}_{55} \quad (4.16) $$

$$ \underline{G}_{\mu\nu} = R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R - \frac{8\pi G}{c^4} \frac{\Phi^2}{\mu_0} \left( g^{\alpha\beta} F_{\nu\alpha} F_{\beta\mu} + \frac{1}{4}g_{\mu\nu}F^{\mu\alpha}F_{\nu\alpha} \right) \\ + \left( -\frac{\partial_{\mu}\partial_{\nu}\Phi - \Gamma^{\sigma}_{\mu\nu}\partial_{\sigma}\Phi}{\Phi} + g_{\mu\nu}\frac{\Box g}{\Phi} \right) - \frac{qA_{\mu}}{Mc}\underline{G}_{5\mu} - \frac{qA_{\nu}}{Mc}\underline{G}_{5\nu} - \frac{qA_{\mu}}{Mc}\frac{qA_{\nu}}{Mc}\underline{G}_{55} \quad (4.17) $$

In the expression of the 5D Ricci scalar $\underline{R}$ given in equation (4.14), the electromagnetic Lagrangian $\frac{1}{2\mu_0}F^{\alpha\beta}F_{\alpha\beta}$ is twice the usual Lagrangian of classical electrodynamics. In classical electrodynamics, written in Minkowski spacetime, $A^{\mu}$ takes the form $A^{\mu} = (\frac{V}{c}, \vec{A})$ where $V$ is the scalar potential, and $\vec{A}$ the vector potential. In this context, it is assumed that when $A^{\mu} = (\frac{V}{c}, \vec{A})$ is varied, $A_{\mu} = (\frac{V}{c}, -\vec{A})$ is similarly varied. Such assumption is manifestly invalid in the present theory based on general relativity, since $A^{\mu} = g^{\mu5}$ while $A_{\mu} = g_{\mu\nu}A^{\nu} = \frac{g_{\mu5}}{\Phi^2}$. The variations of $A^{\mu}$ and $A_{\mu}$ being separated, the electromagnetic Lagrangian of the theory must be equal to twice the usual Lagrangian. The expressions of the Einstein tensors given from equations (4.15) to (4.17) are discussed below. The Einstein Hilbert action $S_{EH6}$ given in equation (4.11) can be written :

$$ S_{EH6} = \int \left[ \rho \left( g^{\mu\nu} u_{\mu} u_{\nu} - \Phi^2 \left( u^5 - \frac{qA_{\mu} u^{\mu}}{Mc} \right)^2 + u^6 u^6 \right) - \frac{c^4 R}{8\pi G} - \frac{\Phi^2}{2\mu_0} F_{\mu\nu} F^{\mu\nu} + \frac{c^4}{4\pi G} \frac{\Box g}{\Phi} \right] \Phi \sqrt{g} d^6 x \quad (4.18) $$
Making the change $A_μ \to 0$, $\Phi \to 1$, $(u^5)^2 - (u^6)^2 \to (u^4)^2$, we recognize the action of equation (2.7). We did not see the results presented now in the literature we consulted. Injecting the geometric property $\nabla_M G^{MN} = 0$ [27] in equation (4.13) implies $\nabla_M T^{MN} = 0$. This last equation can be split in two parts :

$$0 = \nabla_M (\rho u^M u^N) = u^N \left[ \partial_M (\rho u^M) + \Gamma_{QM}^Q \rho u^M \right] + \rho \left[ \underline{u}^M \partial_M u^N + \Gamma_{QM}^N \rho u^M u^Q \right] \quad (4.19)$$

Repeating the analysis given in equation (4.7) with $g_{MN}u^M u^N = c^2(1 - \frac{M^2}{m^2})$, we can contract equation (4.19) with $u_N$ such that the second term between [ ] cancels. Christoffel symbols are computed in [Annex 3] where it is shown that $\Gamma_{\alpha 5}^{\alpha} + \Gamma_{55}^{5} = 0$. When the contraction with $u_N$ is done, the remaining vanishing term is then :

$$0 = \underline{u_N} u^N \left[ \partial_\mu (\rho u^\mu) + \partial_5 (\rho u^5) + \left( \Gamma_{\alpha\beta}^\alpha + \Gamma_{5\beta}^5 \right) \rho u^\beta + \left( \Gamma_{\alpha 5}^\alpha + \Gamma_{5\alpha}^5 \right) \rho u^5 \right] \\ 0 = c^2 \left(1 - \frac{M^2}{m^2}\right) \left[ \partial_\alpha (\rho u^\alpha) + \left( \Gamma_{\nu\alpha}^\nu + \frac{\partial_\alpha \Phi}{\Phi} \right) \rho u^\alpha \right] = c^2 \left(1 - \frac{M^2}{m^2}\right) \left[ \nabla_\alpha (\rho u^\alpha) + \rho \frac{u^\alpha \partial_\alpha \Phi}{\Phi} \right] \quad (4.20)$$

When $N = \mu$ in equation (4.19), the expression of $\nabla_M (\rho u^M u^\mu)$ computed in [Annex 4], can be expanded :

$$0 = \nabla_M (\rho \underline{u}^M u^\mu) = u^\mu \left[ \nabla_\alpha (\rho u^\alpha) + \rho \frac{u^\alpha \partial_\alpha \Phi}{\Phi} \right] \\ + \rho \left[ \dot{u}^\mu + \Gamma^\mu_{\alpha\beta} u^\alpha u^\beta + \frac{q F^\mu_\beta}{Mc} [\Phi^2(\underline{u}^5 - \frac{A_\alpha u^\alpha}{Mc})] + \frac{\partial^\mu \Phi}{\Phi^3} [\Phi^2(\underline{u}^5 - \frac{A_\alpha u^\alpha}{Mc})]^2 \right] \quad (4.21)$$

The first term between the big [ ] vanishes according to equation (4.20). With $[\Phi^2(\underline{u}^5 - \frac{A_\alpha u^\alpha}{Mc})] = \frac{M^2}{m^2}c^2$, the second term between the big [ ] is the local version of equation (4.6), in the continuous approximation. Repeating the computation of equation (4.8), we can contract the above equation with $u_\mu$ to deduce that $u^\mu \partial_\mu \Phi = 0 \Leftrightarrow \frac{d\Phi}{d\tau} = 0$. Then we conclude from equation (4.20) that $\nabla_\alpha (\rho u^\alpha) = \nabla_\alpha j_m^\alpha = 0$: the usual 4D continuity equation holds. When $N = 5$ in equation (4.19), the explicit expression of $\nabla_M \rho u^M u^5$ computed in [Annex 4] is :

$$0 = \nabla_M \rho \underline{u}^M u^5 = \underline{u}^5 \left[ \nabla_\alpha (\rho u^\alpha) + \rho \frac{u^\alpha \partial_\alpha \Phi}{\Phi} \right] + \frac{\rho}{\Phi^2} u^\beta \partial_\beta [\Phi^2(\underline{u}^5 - \frac{q A_\alpha u^\alpha}{Mc})] \quad (4.22)$$

We recover the fact that the quantity $\Phi^2(\underline{u}^5 - \frac{q A_\alpha u^\alpha}{Mc})$ is a constant and we already justified its value : $\frac{M}{m}c$. To write the explicit form of the field equations given by equation (4.13), we now need to compute the components of the energy momentum tensor to equalize them with those of the Einstein tensor. The results are given from equation (4.27) to equation (4.33). The velocities $\underline{u}_5$ and $\underline{u}_\mu$ are computed according to $\underline{u}_M = g_{MN} \underline{u}^N$ :

$$\underline{u}_5 = g_{\mu 5} \underline{u}^\mu + g_{55} \underline{u}^5 = \Phi^2 \frac{q A_\mu}{Mc} \underline{u}^\mu - \Phi^2 \underline{u}^5 = -\Phi^2 \left(\underline{u}^5 - \frac{q A_\mu}{Mc} \underline{u}^\mu\right) = -\frac{M}{m} c \\ \underline{u}_\mu = g_{\mu \nu} \underline{u}^\nu + g_{55} \underline{u}^5 = \left(g_{\mu \nu} - \Phi^2 \frac{q A_\mu}{Mc} \frac{q A_\nu}{Mc}\right) \underline{u}^\nu + \Phi^2 \frac{q A_\mu}{Mc} \underline{u}^5 \quad (4.23) \\ = \underline{u}_\mu + \frac{q A_\mu}{Mc} \left[\Phi^2 \left(\underline{u}^5 - \frac{q A_\nu}{Mc} \underline{u}^\nu\right)\right] = \underline{u}_\mu + \frac{q A_\mu}{m}$$

The components $\underline{T}_{MN}$ of the energy momentum tensor are then :

$$\underline{T}_{55} = \rho \underline{u}_5 \underline{u}_5 = \rho c^2 \frac{M^2}{m^2} \quad (4.24)$$

$$\underline{T}_{\mu 5} = \rho \underline{u}_5 \underline{u}_\mu = -\rho c \frac{M}{m} \underline{u}_\mu - \rho c \frac{M}{m} \frac{q A_\mu}{m} \quad (4.25)$$

$$\underline{T}_{\mu \nu} = \rho \underline{u}_\mu \underline{u}_\nu = \rho \underline{u}_\mu \underline{u}_\nu + \rho \left( \underline{u}_\mu \frac{q A_\nu}{m} + \underline{u}_\nu \frac{q A_\mu}{m} + \frac{q A_\mu}{m} \frac{q A_\nu}{m} \right) \quad (4.26)$$

We now use Einstein's equations (4.13). We first equalize $\underline{G}_{55}$ given by equation (4.15) with $\frac{8\pi G}{c^4} \underline{T}_{55}$ given by equation (4.24) :

$$\frac{\Phi^2}{2} R + \frac{8\pi G}{c^4} \left( \frac{3\Phi^4}{4\mu_0} F^{\alpha\beta} F_{\alpha\beta} \right) = \frac{8\pi G}{c^4} \rho \frac{M^2 c^2}{m^2} \quad (4.27)$$

From this, we get the useful result $-\frac{q A_\mu}{Mc} \underline{G}_{55} = -\frac{8\pi G}{c^2} \rho \frac{q A_\mu}{mc} \frac{M}{m}$. With this result, we can equalize $\underline{G}_{5\mu}$ given by equation (4.16) with $\frac{8\pi G}{c^4} \underline{T}_{5\mu}$ where $\underline{T}_{5\mu}$ is given by equation (4.25) :

$$-\frac{\Phi^2}{2} \frac{q}{Mc} \nabla_\nu F_\mu^\nu - \frac{3\Phi}{2} \frac{q}{Mc} F_{\nu\mu} \partial^\nu \Phi - \frac{8\pi G}{c^2} \rho \frac{q A_\mu}{mc} \frac{M}{m} = -\frac{8\pi G}{c^4} \rho c \frac{M}{m} \underline{u}_\mu - \frac{8\pi G}{c^4} \rho c \frac{M q A_\mu}{m^2} \quad (4.28)$$
$$ -\Phi^2 \nabla_{\nu} F_{\mu}^{\nu} - 3\Phi F_{\nu\mu} \partial^{\nu} \Phi = -\frac{2Mc}{q} \frac{8\pi G}{c^4} \rho c \frac{M}{m} u_{\mu} = \frac{16\pi G M^2}{\mu_0 c^2 q^2} \left(\mu_0 \frac{q}{m} \rho u_{\mu}\right) = -\mu_0 j_{\mu} \quad (4.29) $$

where we previously defined M such that $\frac{16\pi G M^2}{\mu_0 c^2 q^2} = 1$. Then $j_{\mu} = \frac{q}{m} \rho u_{\mu}$ is identified with the electric current. The above equation can be rewritten:

$$ \Phi^2 \nabla_{\nu} F^{\nu\alpha} + 3\Phi F^{\nu\alpha} \partial_{\nu} \Phi = \mu_0 j^{\alpha} \quad (4.30) $$

Taking the divergence of each side of this equation gives $0 = \nabla_{\alpha} j^{\alpha} + \frac{j^{\alpha} \partial_{\alpha} \Phi}{\Phi}$ (the computation is given in [Annex 5]): the electric current is then covariantly conserved, in agreement with the continuity equation. When the interaction energy is small, we can work in the linearized approximation: the metric is then written $g_{MN} = \eta_{MN} + h_{MN}$ where $|h_{MN}| \ll 1$ and only the linear terms are conserved in the equations. If this is done with the above equation, $\nabla_{\nu} \to \partial_{\nu}$ and $\Phi$ must be written $\Phi(x^{\alpha}) = 1 + h_{55}(x^{\alpha})$: we now rewrite the previous equation with this notation and neglect the remaining non linear terms:

$$ \partial_{\nu} F^{\nu\alpha} + 2h_{55} \partial_{\nu} F^{\nu\alpha} + (h_{55})^2 \partial_{\nu} F^{\nu\alpha} + 3(1 + h_{55})^2 F^{\nu\alpha} \partial_{\nu} h_{55} = \mu_0 j^{\alpha} \quad \to \quad \partial_{\nu} F^{\nu\alpha} = \mu_0 j^{\alpha} \quad (4.31) $$

We recognize the Lorentz covariant representation of Maxwell's equations [3][37], interpreted as the linear approximation, or low energy limit, of the present theory. From equations (4.27) and (4.28), we deduce:

$$ \begin{aligned} & -\frac{qA_{\mu}}{Mc}G_{5\nu} - \frac{qA_{\nu}}{Mc}G_{5\mu} - \frac{qA_{\mu}}{Mc}\frac{qA_{\nu}}{Mc}G_{55} \\ &= -\frac{8\pi G}{c^4} \rho \left( \frac{qA_{\mu}}{Mc} \left( -\frac{M}{m} cu_{\nu} - \frac{Mc}{m} \frac{qA_{\nu}}{m} \right) + \frac{qA_{\nu}}{Mc} \left( -\frac{M}{m} cu_{\mu} - \frac{Mc}{m} \frac{qA_{\mu}}{m} \right) + \frac{qA_{\mu}}{Mc} \frac{qA_{\nu}}{Mc} \frac{M^2 c^2}{m^2} \right) \\ &= \frac{8\pi G}{c^4} \rho \left( \frac{qA_{\mu}}{m} u_{\nu} + \frac{qA_{\nu}}{m} u_{\mu} + \frac{qA_{\mu}}{m} \frac{qA_{\nu}}{m} \right) \end{aligned} \quad (4.32) $$

This equation is used to simplify the equality $G_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}$ where $G_{\mu\nu}$ is given by equation (4.17) while $T_{\mu\nu}$ is given by equation (4.26). With the 4D Einstein tensor given by $G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R$, we get:

$$ \begin{aligned} G_{\mu\nu} - \frac{8\pi G}{c^4} \frac{\Phi^2}{\mu_0} \left( g^{\alpha\beta} F_{\nu\alpha} F_{\beta\mu} + \frac{1}{4} g_{\mu\nu} F^{\mu\alpha} F_{\mu\alpha} \right) + \left( -\frac{\partial_{\mu} \partial_{\nu} \Phi - \Gamma_{\mu\nu}^{\sigma} \partial_{\sigma} \Phi}{\Phi} + g_{\mu\nu} \frac{\Box g}{\Phi} \right) &= \frac{8\pi G}{c^4} \rho u_{\mu} u_{\nu} \\ G_{\mu\nu} &= \frac{8\pi G}{c^4} \left( \Phi^2 \left[ \frac{g^{\alpha\beta} F_{\nu\alpha} F_{\beta\mu} + \frac{1}{4} g_{\mu\nu} F^{\mu\alpha} F_{\mu\alpha}}{\mu_0} \right] + \left[ \frac{c^4}{8\pi G} \frac{\partial_{\mu} \partial_{\nu} \Phi - \Gamma_{\mu\nu}^{\sigma} \partial_{\sigma} \Phi - g_{\mu\nu} \Box g}{\Phi} \right] + \left[ \rho u_{\mu} u_{\nu} \right] \right) \end{aligned} \quad (4.33) $$

In equation (4.33), the first term between [ ] is the usual electromagnetic stress energy tensor [3][37] while the second term between [ ] is interpreted as the energy momentum tensor of the field $\Phi$. The last term is the usual source: the matter energy momentum tensor. In usual general relativity, when the pressure of matter can be neglected, the gravitational Einstein-Maxwell-Euler equations take the form [69]:

$$ G_{\mu\nu} = \frac{8\pi G}{c^4} \left( \left[ \frac{g^{\alpha\beta} F_{\nu\alpha} F_{\beta\mu} + \frac{1}{4} g_{\mu\nu} F^{\mu\alpha} F_{\mu\alpha}}{\mu_0} \right] + \left[ \rho u_{\mu} u_{\nu} \right] \right) \quad (4.34) $$

where $F_{\mu\nu}$ must be computed from Maxwell's equations. When $\rho = 0$, equation (4.34) reduces to the Einstein-Maxwell equations [70][71][72]). Obviously, equation (4.33) reproduces equation (4.34) in the limit $\Phi \to 1$: the 5D equation (4.33) generalizes the Einstein-Maxwell equations (4.34), with the electromagnetic energy momentum tensor interpreted as a part of the 5D gravitational field. Since $\nabla_{\alpha} G^{\alpha\beta} = 0$, we know that we can multiply the right-hand side of equation (4.33) by $g^{\mu\alpha} g^{\nu\beta}$ and then take the divergence with $\nabla_{\alpha}$ to get a vanishing result. The computation is given in [Annex 6]. It gives back the right hand side of equation (4.21), i.e. the continuity equation and the geodesic equation combined with the Lorentz force together with the extra term $\frac{\partial^{\mu} \Phi}{\Phi^3} [\Phi^2(\frac{u^5}{M^2} - \frac{A_{\alpha} u^{\alpha}}{M^2})]^2 = \frac{M^2 c^2}{m^2} \frac{\partial^{\mu} \Phi}{\Phi^3}$. The last manifest drawback of the present theory is then precisely this extra term, which appears with the Lorentz force in equation (4.21): the quantity $\frac{M^2 c^2}{m^2}$ being enormous for usual particles, we must explain why this term has no obvious experimental effect. We already identified that Maxwell's equations are obtained in the linear, or low energy, approximation of the present theory. Our task is then to examine the effect of this additional term in the low energy approximation. We will neglect gravity since we are only interested by the electromagnetic part of the theory. Electrons, with their electric charge $-q$, have the biggest charge to mass ratio $|q/m|$: we then examine the effect of $\Phi$ on the electromagnetic force between two electrons. In this purpose, electrons will be modeled by simple spherical bodies (the spin or an internal structure are neglected). We initially consider the electric field produced by a first electron, in its rest frame. Outside the electron, where $\rho = 0$, the electric field $\vec{E} = -\vec{\nabla}V - \frac{\partial \vec{A}}{\partial t}$ is computed from equation (4.31):
$\partial_{\nu} F^{\nu 0} = \frac{1}{c} \vec{\nabla} \cdot \vec{E} = 0$. We call $r = \sqrt{x_1^2 + x_2^2 + x_3^2}$ the distance from the center of the electron. Writing $\frac{1}{\epsilon_0} = \mu_0 c^2$, the usual stationary solution, which is independent of the time coordinate $t$, is well known: $\vec{E} = -\frac{\mu_0 c^2 q}{4\pi r^2} \frac{\vec{r}}{r} = -\frac{q}{4\pi\epsilon_0 r^2} \frac{\vec{r}}{r}$. The corresponding electromagnetic potential is $A^0 = \frac{V}{c} = -\frac{\mu_0 c^2 q}{4\pi c r} = -\frac{q}{4\pi\epsilon_0 c r}$. In the rest frame of the electron, the electron produces no current such that there is no magnetic field. Contracting equation (4.33) with $g^{\mu\nu}$ when $\rho = 0$ gives $-2R = -3 \frac{\Box_g \Phi}{\Phi}$ (the trace of the electromagnetic energy momentum tensor vanishes [3][37]). Neglecting gravity ($\Box_g \to \Box$) and injecting $R = \frac{3}{2} \frac{\Box_g \Phi}{\Phi}$ in equation (4.27) when $\rho = 0$ gives:

$$ \frac{3}{4} \Phi \Box \Phi = - \frac{8\pi G}{c^4} \left( \frac{3\Phi^2}{4\mu_0} F^{\alpha\beta} F_{\alpha\beta} \right) \Leftrightarrow \Phi \Box \Phi = \frac{16\pi G}{c^6 \mu_0} \Phi^2 |\vec{E}|^2 = \frac{16\pi G}{c^4} \Phi^2 \epsilon_0 |\vec{E}|^2 = \Phi^2 \frac{G q^2}{c^4 \pi \epsilon_0} \frac{1}{r^4} \quad (4.35) $$

In the linear approximation, we reproduce the analysis given in equation (4.31) and suppress the time derivatives to get a stationary solution: $\Box \to -\Delta$ where $\Delta$ is the 3D Laplacian. With $\Phi = 1 + \underline{h}_{55}$, equation (4.35) becomes:

$$ (1 + \underline{h}_{55}) \Delta(1 + \underline{h}_{55}) = \frac{8\pi G}{c^4} (1 + \underline{h}_{55})^2 \epsilon_0 |\vec{E}|^2 = (1 + \underline{h}_{55})^2 \frac{G q^2}{c^4 \pi \epsilon_0} \frac{1}{r^4} \quad (4.36) $$

Neglecting non linear terms ($|\underline{h}_{55}| \ll 1$), the above equation written in spherical coordinates produces:

$$ \Delta h_{55} = \frac{\partial^2 h_{55}}{\partial r^2} + 2 \frac{\partial h_{55}}{\partial r} = \frac{8\pi G}{c^6 \mu_0} |\vec{E}|^2 = \frac{G \mu_0 q^2}{\pi c^2} \frac{1}{r^4} \Rightarrow h_{55} = \frac{G}{c^4} \frac{q^2}{2\pi \epsilon_0} \frac{1}{r^2} = \frac{G \mu_0 q^2}{c^2 2\pi} \frac{1}{r^2} \quad (4.37) $$

With this result, we can now consider the electromagnetic force produced by our electron on a second electron. Since the above solution does not depend on time, when $\alpha = 0$ the expression $\ddot{x}^\alpha = -\frac{q}{m} F^\alpha_\beta \dot{x}^\beta - c^2 \frac{M^2}{m^2} \frac{\partial^\alpha \Phi}{\partial \Phi^3}$ reduces $\ddot{x}^0 = -\frac{q}{m} F^0_\beta \dot{x}^\beta - c^2 \frac{M^2}{m^2} \frac{\partial^0 \Phi}{\partial \Phi^3}$: in this case, $\Phi$ has no effect on the $0^{th}$ component of the electromagnetic force. We are then only interested by the cases $\alpha = 1, 2, 3$. In the low energy limit, we approximate $\dot{x}^0 = \frac{d\dot{t}}{d\tau} \approx c$. The electron does not produce a magnetic field in its rest frame, the electromagnetic force per unit mass is then:

$$ \begin{aligned} \dot{x}^i &= \frac{-q}{m} F^i_0 \dot{x}^0 - c^2 \frac{M^2}{m^2} \frac{\partial^i h_{55}}{(1 + \underline{h}_{55})^3} = \frac{-q}{m} \mathbf{E}^i - c^2 \frac{M^2}{m^2} \partial^i h_{55} = \frac{\mu_0 c^2 q^2}{4\pi m r^2} \frac{x^i}{r} - \frac{M^2}{m^2} \frac{G \mu_0 q^2}{\pi r^3} \frac{x^i}{r} \\ &= \frac{\mu_0 c^2 q^2}{4\pi m r^2} \frac{x^i}{r} \left( 1 - \frac{4GM}{c^2 r} \frac{M}{m} \right) = \frac{\mu_0 c^2 q^2}{4\pi m r^2} \frac{x^i}{r} \left( 1 - \frac{1}{mc^2} \frac{\mu_0 c^2 q^2}{4\pi r} \right) = \frac{-q}{m} \mathbf{E}^i \left( 1 - \frac{qV}{mc^2} \right) \end{aligned} \quad (4.38) $$

In the linear, or low energy approximation, the term $\frac{qV}{mc^2}$ should be neglected (the low energy approximation precisely corresponds to the approximation $\frac{\mu_0 c^2 q^2}{4\pi r m c^2} = \frac{qV}{mc^2} \ll 1$). This term being neglected, the electromagnetic force reduces to the Lorentz force: $m\ddot{x}^i = -q\mathbf{E}^i$. When the linear approximation holds, if the above result is true in the rest frame of the first electron, it must be true in all frames. The present theory then reduces to general relativity combined with classical electrodynamics, in the low energy limit: precisely, Maxwell's equations and the Lorentz force follow from the low energy approximation of the present purely gravitational theory.

## 4.2 From 5D general relativity to quantum electrodynamics

We now examine the effect of Postulate 3 on the matter Lagrangian of a spin-0 field. We generalize equation (2.4) and write the action $S_{KK}(x^\alpha, \underline{x}^5, x^6)$ associated to the Lagrangian $L_m$ of equation (4.4) as $S_{KK} = \int \frac{\partial L_m}{\partial \dot{x}^M} dx^M + \int \frac{\partial L_m}{\partial \dot{x}^6} dx^6$ such that $\frac{\partial S_{KK}}{\partial \dot{x}^M} = \frac{\partial L_m}{\partial \dot{x}^M}$ and $\frac{\partial S_{KK}}{\partial x^6} = \frac{\partial L_m}{\partial x^6}$. Then:

$$ S_{KK}(x^\alpha, \underline{x}^5, x^6) = m \left( \int g_{\mu\nu} \dot{x}^\nu + \frac{qA_\mu}{mc} dx^\mu - \int c \frac{M}{m} dx^5 + \int c \sqrt{\frac{M^2}{m^2} - 1} dx^6 \right) \quad (4.39) $$

We previously used Postulate 3 to generalize equation (2.6) according to equation (4.10): with the constant $A$ previously fixed to be $A = \frac{c^2}{(k_4)^2} = \frac{\hbar^2}{m^2}$, we similarly generalize equation (2.12) to $\phi = \phi(x^\alpha, \underline{x}^5, x^6)$:

$$ S_m = \int \mathcal{L}_m d^6 x = \int \frac{\hbar^2}{2m^2} g^{MN} \left( \frac{\partial_M \phi}{\partial_N \phi} \phi^* + \frac{\partial_M \phi^*}{\partial_N \phi} \phi \right) + \left( \frac{M^2}{m^2} - 1 \right) c^2 |\phi|^2 \Phi \sqrt{g} d^6 x \quad (4.40) $$

To simplify the computations, we momentarily neglect the convention that the energy momentum tensor should be real. We use the fact that $\frac{1}{2} g^{MN} (\partial_M \phi \partial_N \phi^* + \frac{\partial_M \phi^*}{\partial_N \phi}) = \frac{g^{MN}}{g^{NM}} \partial_M \phi \partial_N \phi^*$ if $g^{MN} = g^{NM}$ to rewrite $S_m$:

$$ S_m = \int \frac{\hbar^2}{m^2} \left( g^{\mu\nu} \partial_\mu \phi \partial_\nu \phi^* + \frac{qA^\mu}{Mc} \partial_\mu \phi \partial_\nu \phi^* + \frac{qA^\nu}{Mc} \partial_\nu \phi^* \partial_\mu \phi + \left(\frac{-1}{\Phi^2} + \frac{qA^\alpha}{Mc} \frac{qA_\alpha}{Mc}\right) \partial_\nu \phi \partial_\mu \phi^* \right) \\ + \left(\frac{M^2}{m^2} - 1\right) c^2 |\phi|^2 \Phi \sqrt{g} d^6 x \quad (4.41) $$
where equation (4.9) was used. We now write $\phi(x^\alpha, x^5, x^6) = \psi(x^\alpha)e^{i\frac{Mcx^5-c(\sqrt{M^2-m^2})x^6}{\hbar}} \approx \psi(x^\alpha)e^{i\frac{mc^2\tau}{\hbar}}$ because in flat space $\frac{d}{d\tau}(Mcx^5 - c(\sqrt{M^2-m^2})x^6) = Mc\dot{x}^5 - c(\sqrt{M^2-m^2})\dot{x}^6 = mc^2 = \frac{d}{d\tau}(mc^2\tau)$. In the eikonal approximation, we could write $\phi(x^\alpha, x^5, x^6) = \psi(x^\alpha)e^{i\frac{Mcx^5-c(\sqrt{M^2-m^2})x^6}{\hbar}} \approx \mathcal{R}(x^\alpha)e^{-i\frac{S_{KK}}{\hbar}}$ where $\psi$ is complex and $|\phi|^2 = |\psi|^2 = \mathcal{R}^2\rho(x^\alpha)$. Using $\hbar\partial_5\phi = iMc\phi$, we express $S_m$ as a function of $\psi$ instead of $\phi$:

$$S_m = \frac{1}{m^2} \int g^{\mu\nu}(\hbar\partial_{\mu} + iqA_{\mu})\psi(\hbar\partial_{\nu} - iqA_{\nu})\psi^* + (M^2c^2(\frac{-1}{\Phi^2} + 1) - m^2c^2)|\psi|^2 \Phi \sqrt{g}d^6x \quad (4.42)$$

Neglecting gravity ($g^{\mu\nu} \to \eta^{\mu\nu}$) and taking the approximation $\Phi \to 1$, $S_m$ reduces to:

$$S_m = \frac{1}{m^2} \int \eta^{\mu\nu}(\hbar\partial_{\mu} + iqA_{\mu})\psi(\hbar\partial_{\nu} - iqA_{\nu})\psi^* - m^2c^2|\psi|^2 d^6x \quad (4.43)$$

The above expression of $\mathcal{L}_m$ is the Lagrangian density of a spin-0 field interacting with the electromagnetic field, a quantum field theory called "scalar quantum electrodynamics" [59][61] or 'scalar QED' [62] in literature. Varying the action $S_m$ given by equation (4.40) in the usual way gives:

$$
\begin{align}
0 &= -\hbar^2 \left( \frac{\partial_M}{\partial \mu} \frac{\partial^M}{\partial \mu} + \Gamma^N_{NM} \frac{\partial^M}{\partial \mu} \right) \phi + (M^2 - m^2) c^2 \phi + \text{"cc"} \nonumber \\
0 &= -\hbar^2 \frac{\partial_M}{\partial \mu} (g^{MN} \frac{\partial_N}{\partial \mu}) \phi - \hbar^2 \Gamma^N_{NM} \frac{\partial^M}{\partial \mu} \phi + (M^2 - m^2) c^2 \phi + \text{"cc"} \nonumber \\
0 &= -\hbar^2 \partial_\mu \partial^\mu \phi - \hbar^2 \partial_\mu \left( \frac{q A^\mu}{Mc} \partial_5 \phi \right) - \hbar^2 \partial_5 \left( \frac{q A^\mu}{Mc} \partial_\mu \phi \right) - \hbar^2 \partial_5 \left( \left( -\frac{1}{\Phi^2} + \frac{q A_\alpha}{Mc} \frac{q A^\alpha}{Mc} \right) \partial_5 \phi \right) \nonumber \\
&\quad - \hbar^2 \left( \Gamma^\nu_{\nu\mu} \frac{\partial^\mu}{\partial \nu} + \Gamma^5_{5\mu} \frac{\partial^\mu}{\partial \nu} \right) \phi - \hbar^2 \left( \Gamma^\nu_{\nu5} + \Gamma^5_{55} \right) \frac{\partial^5}{\partial \nu^5} \phi + (M^2 - m^2) c^2 \phi + \text{"cc"} \tag{4.44} \\
0 &= -\hbar^2 \partial_\mu \partial^\mu \phi - i\hbar \partial_\mu (q A^\mu \phi) - i\hbar q A^\mu \partial_\mu \phi - \left( \frac{M^2 c^2}{\Phi^2} - q A^\alpha q A_\alpha \right) \phi \nonumber \\
&\quad - \hbar^2 \Gamma^\nu_{\nu\mu} \frac{\partial^\mu}{\partial \nu} \frac{\partial^\mu}{\partial \nu} \phi + (M^2 - m^2) c^2 \phi + \text{"cc"} \nonumber
\end{align}
$$

where "cc" denotes the complex conjugate of the term on the left. Neglecting again gravity ($g^{\mu\nu} = \eta^{\mu\nu}$ such that $\Gamma^\nu_{\nu\mu} = 0$) and taking the approximation $\Phi = 1$ (such that $\partial_\mu\Phi = 0$), we get:

$$0 = \left[ -\eta^{\mu\nu}(\hbar\partial_{\mu} + iqA_{\mu})(\hbar\partial_{\nu} + iqA_{\nu})\psi - m^2c^2\psi \right] e^{i\frac{Mcx^5+c(\sqrt{M^2-m^2})x^6}{\hbar}} \quad (4.45)$$

The expression between $[]$ must vanish, which gives the equation obeyed by $\psi(x^\alpha)$: we recognize the motion of a Klein-Gordon field interacting with the electromagnetic field, according to quantum field theory [35] [73] [74]. Our sign convention being $(---)$ while $\partial_\mu = \frac{\partial}{\partial x^\mu}$, we rewrite this equation in the form:

$$(\hbar\partial_0 + iqA_0)(\hbar\partial_0 + iqA_0)\psi = \left(\sum_{i=1}^{3}(\hbar\partial_i + iqA_i)^2 - m^2c^2\right)\psi \quad (4.46)$$

Assuming that the Lagrangian density used in the action of equation (4.41), (and especially $\rho$ and $A_0 = V/c$), should not depend on $t$, we get the non relativistic limit by writing $\psi = e^{-i\frac{mc^2t}{\hbar}}\psi_S$. With $\psi = e^{-i\frac{mc^2t}{\hbar}}\psi_S$, the left hand side of equation (4.46) becomes $e^{-i\frac{mc^2t}{\hbar}}(\hbar\partial_0 + i(qA_0 - mc))(\hbar\partial_0 + iq(A_0 - mc))\psi_S$ and only the terms proportional to $m$ remains (since $i\hbar\partial_0\psi_S \ll mc\psi_S$ and $qA_0\psi_S = \frac{V}{c}\psi_S \ll mc\psi_S$) such that we get:

$$e^{-i\frac{mc^2t}{\hbar}}(-mc(2i\hbar\partial_0 - 2qA_0 + mc))\psi_S = e^{-i\frac{mc^2t}{\hbar}}\left(\sum_{i=1}^{3}(\hbar\partial_i + iqA_i)^2 - m^2c^2\right)\psi_S \quad (4.47)$$

$$\Rightarrow \quad i\hbar\partial_t\psi_S = \frac{(-i\hbar\vec{\nabla} - q\vec{A})^2}{2m}\psi_S + V\psi_S \quad (4.48)$$

where $A_i = -A^i$. $\psi_S$ obeys the Schrödinger equation for a particle in an electromagnetic field [21][75][76].

We now conclude this section. In agreement with Postulate 3 we extended general relativity to a 5D domain, assigning an internal structure to the proper time. It was then possible to derive classical electrodynamics combined with 4D general relativity, in the low energy limit (non linear terms appear in Maxwell's equations when the low energy approximation is not valid anymore). Postulate 3 was then combined with Postulate 2 to generalize the matter Lagrangian density discussed in [Section 2] : it became possible to derive 'scalar quantum electrodynamics' or 'scalar QED' from the Lagrangian density of a spin-0 field combined with 5D gravity. The Schrödinger equation for a charged particle in an electromagnetic field was obtained in the non relativistic limit.
# 5 New results in quantum physics

A. Einstein wrote in 1936: "it has been pointed out that the introduction of a space-time continuum may be considered as contrary to nature in view of the molecular structure of everything which happens on a small scale. It is maintained that perhaps the success of the Heisenberg method points to a purealy algebraical method of description of nature, that is, to the elimination of continuous functions from physics. Then, however, we must also give up, on principle, the spacetime continuum. [...] At the present time, however, such a program looks like an attemp to breathe in empty space. In view of this situation, it seems to be entirely justifiable seriously to consider the question as to whether the basis of field physics cannot by any means be put in harmony with quantum phenomena. Is this not the only basis which [...] can be adapted to the requirement of the general theory of relativity? The belief, prevailing among the physicists of today, that such an attemps would be hopeless, may have its root in the unwarranted assumption that such a theory must lead, in first approximation, to the equations of classical mechanics for the motion of corpuscules [...]. As a matter of fact, up to now we have never succeeded in a field-theoritical description of corpuscules free of singularities [...]. One thing, however, is certain : if a field theory results in a representation free of singularities, then the behavior of theses corpuscules in time is determined solely by the differential equations of the field" [77]. To begin this section, we briefly discuss the link between the classical motion of particles and the differential equations of their field. We consider a Klein-Gordon field, describing spin-0 particles, in curved spacetime. The field equation is $-\hbar^2\Box_g\psi - m^2c^2\psi = 0$. In [Section2], we wrote the solution in the form $\psi = \mathcal{R}(x^\alpha)e^{-i\frac{S(x^\alpha)}{\hbar}}$ with $\mathcal{R}^2 = |\psi|^2 = \rho$. With the computation given in [Annex 1], it was observed that the Klein-Gordon equation becomes :

$$0 = -\hbar^2\Box_g\psi - m^2c^2\psi = \left[ g^{\mu\nu}\partial_{\mu}S\partial_{\nu}S - m^2c^2 - \hbar^2\frac{\Box_g\mathcal{R}}{\mathcal{R}} \right]\psi + \frac{i\hbar}{\rho}[\nabla_{\mu}(\rho\partial^{\mu}S)]\psi \quad (5.1)$$

In the eikonal approximation the variations of the amplitude $\mathcal{R}$ can be neglected when compared to the variations of the phase $S/\hbar$: in this approximation we can then identify $S$ with the mechanical action such that $\partial_\mu S/m$ becomes the 4-velocity $u_\mu$. Since each term between $[]$ must vanish in (5.1) we conclude that $g^{\mu\nu}\partial_\mu S\partial_\nu S - m^2c^2 = 0$ in the eikonal approximation: we recognize the Hamilton Jacobi equation. We will act on the Hamilton Jacobi equation with the partial derivative $\partial^\beta = g^{\alpha\beta}\partial_\alpha$. We will use the two computational tricks given in equation (5.2), the first one being the well known fact that the covariant derivative commutes with the metric :

$$\begin{gathered} 0 = \nabla_{\mu}g^{\alpha\beta} = \partial_{\mu}g^{\alpha\beta} + \Gamma_{\mu\sigma}^{\alpha}g^{\sigma\beta} + \Gamma_{\mu\sigma}^{\beta}g^{\alpha\sigma} \Leftrightarrow \partial_{\mu}g^{\alpha\beta} = -\Gamma_{\mu\sigma}^{\alpha}g^{\sigma\beta} - \Gamma_{\mu\sigma}^{\beta}g^{\alpha\sigma} \\ (\partial_{\alpha}g^{\mu\nu})\partial_{\mu}S\partial_{\nu}S = (\partial_{\alpha}g^{\mu\nu})g_{\mu\sigma}\partial^{\sigma}S\partial_{\nu}S = (\partial_{\alpha}\underbrace{\overline{g^{\mu\nu}g_{\mu\sigma}}}_{\text{g}} - g^{\mu\nu}g_{\mu\sigma,\alpha})\partial^{\sigma}S\partial_{\nu}S = -g_{\mu\sigma,\alpha}\partial^{\sigma}S\partial^{\mu}S \end{gathered} \quad (5.2)$$

Applying $\partial^\beta = g^{\alpha\beta}\partial_\alpha$ on the left hand side of the Hamilton Jacobi equation gives :

$$\begin{align} 0 &= g^{\alpha\beta}\partial_{\alpha}(g^{\mu\nu}\partial_{\mu}S\partial_{\nu}S) - \underbrace{g^{\alpha\beta}\partial_{\alpha}(m^2c^2)}_{\text{g}} \nonumber \\ 0 &= 2g^{\alpha\beta}g^{\mu\nu}\partial_{\mu}\partial_{\alpha}S\partial_{\nu}S + g^{\alpha\beta}[(\partial_{\alpha}g^{\mu\nu})\partial_{\alpha}S\partial_{\nu}S] \nonumber \\ 0 &= 2g^{\mu\nu}\partial_{\mu}(g^{\alpha\beta}\partial_{\alpha}S)\partial_{\nu}S - 2g^{\mu\nu}[\partial_{\mu}g^{\alpha\beta}]\partial_{\alpha}S\partial_{\nu}S - g^{\alpha\beta}g_{\mu\sigma,\alpha}\partial^{\sigma}S\partial^{\mu}S \nonumber \\ 0 &= 2\partial^{\mu}S\partial_{\mu}\partial^{\beta}S + 2g^{\mu\nu}(\Gamma_{\mu\sigma}^{\alpha}g^{\sigma\beta} + \Gamma_{\mu\sigma}^{\beta}g^{\alpha\sigma})\partial_{\alpha}S\partial_{\nu}S - g^{\sigma\beta}g_{\mu\alpha,\sigma}\partial^{\alpha}S\partial^{\mu}S \nonumber \\ 0 &= 2(\partial^{\mu}S\partial_{\mu}\partial^{\beta}S + g^{\mu\nu}\Gamma_{\mu\sigma}^{\beta}g^{\alpha\sigma}\partial_{\alpha}S\partial_{\nu}S) + 2g^{\mu\nu}\Gamma_{\mu\sigma}^{\alpha}g^{\sigma\beta}\partial_{\alpha}S\partial_{\nu}S - g^{\sigma\beta}g_{\mu\alpha,\sigma}\partial^{\alpha}S\partial^{\mu}S \tag{5.3} \\ 0 &= 2(\partial^{\mu}S\partial_{\mu}\partial^{\beta}S + \Gamma_{\mu\sigma}^{\beta}\partial^{\sigma}S\partial^{\mu}S) + g^{\sigma\beta}(2\Gamma_{\mu\sigma\alpha} - g_{\mu\alpha,\sigma})\partial^{\alpha}S\partial^{\mu}S \nonumber \\ 0 &= 2m^2(u^{\mu}\partial_{\mu}u^{\beta} + \Gamma_{\mu\sigma}^{\beta}u^{\mu}u^{\sigma}) + g^{\sigma\beta}(g_{\alpha\mu,\sigma} + g_{\sigma\alpha,\mu} - g_{\mu\sigma,\alpha} - g_{\mu\alpha,\sigma})\partial^{\alpha}S\partial^{\mu}S \nonumber \\ 0 &= 2m^2(u^{\mu}\partial_{\mu}u^{\beta} + \Gamma_{\mu\sigma}^{\beta}u^{\mu}u^{\sigma}) + g^{\sigma\beta}(g_{\sigma\alpha,\mu} - g_{\mu\sigma,\alpha})\partial^{\alpha}S\partial^{\mu}S \nonumber \end{align}$$

As expected by A. Einstein in 1936, the geodesic equation (which describes the motion of corpuscules in curved spacetime) can be derived solely from the differential equation of the Klein-Gordon field: the derivation simply considers a fluid with density $\rho = \mathcal{R}^2$ and uses the eikonal approximation. The aim of the present section is to explore others facets of quantum physics by using the geometric approach of the previous sections. The discussion will include general covariance, polarization, complex general relativity and spinors.

## 5.1 Generally covariant formulation of quantum mechanics

In 1972, F. Dyson stated: "The most glaring incompatibility of concepts in contemporary physics is that between Einstein's principle of general coordinate invariance and all the modern schemes for a quantum-mechanical description of nature. [...] In practice all serious quantum-mechanical theories are based either on the Poincaré group P or the Galilei group G. This means that a class of preferred inertial coordinate-systems is postulated a priori, in flat contradiction to Einstein's principle. The contradiction is particularly uncomfortable, because
Einstein's principle of general coordinate invariance has such an attractive quality of absoluteness" [78]. Our aim is now to introduce a generally covariant formulation of quantum mechanics: we will use the Klein-Gordon equation, understood as the relativistic extension of the Schrödinger equation. It is firstly useful to recall well known results of non relativistic classical and quantum mechanics. In the Hamiltonian formulation of non relativistic classical mechanics, calling H the non relativistic Hamiltonian, the equations of motion are:

$$ \frac{dx^i}{dt} = \frac{\partial H}{\partial p^i} \quad ; \quad \frac{dp^i}{dt} = -\frac{\partial H}{\partial x^i} \quad \text{where } i = 1, 2, 3 \qquad (5.4) $$

As explained in literature, "we can develop a structure of classical mechanics, paralleling the Hamiltonian formulation, expressed solely in terms of Poisson brackets. [...] Historically this Poisson bracket formulation, [...], was especially useful for carrying out the original transition from classical to quantum mechanics" [13]. Indeed, the Poisson brackets can be used to compute to total derivative of any function $O(x^1, x^2, x^3, p^1, p^2, p^3)$ where we simplify the discussion by assuming that $O$ does not explicitly depend on the time coordinate $t$:

$$ \frac{dO}{dt} = \sum_{i=1,2,3} \left( \frac{dx^i}{dt} \frac{\partial O}{\partial x^i} + \frac{dp^i}{dt} \frac{\partial O}{\partial p^i} \right) = \sum_{i=1,2,3} \left( \frac{\partial O}{\partial x^i} \frac{\partial H}{\partial p^i} - \frac{\partial O}{\partial p^i} \frac{\partial H}{\partial x^i} \right) = [O; H]_{PB} \qquad (5.5) $$

The last equality defines the Poisson bracket $[O; H]_{PB}$ between $O$ and $H$. The transition from classical mechanics to quantum mechanics is obtained by Dirac's quantization rule (in the Heisenberg picture) [36][79]. With $\hat{H}$ the Hamiltonian operator,

$$ \frac{dO}{dt} = [O; H]_{PB} \quad \rightarrow \quad \frac{d\hat{O}}{dt} = \frac{1}{i\hbar}[\hat{O}; \hat{H}]_{-} \qquad (5.6) $$

The second equality in equation (5.6) is the Heisenberg equation of motion [21] [36][78][80]. When an obversible $\hat{O}$ does not depend explicitly on time, the time derivative of its average value $\langle \hat{O} \rangle$ is given by [34][75][81][82]:

$$ \frac{d}{dt} \langle \hat{O} \rangle = \langle \psi | [\hat{O}; \hat{H}]_- | \psi \rangle \qquad (5.7) $$

where [34][75][81][82]

$$ \langle \hat{O} \rangle = \langle \psi | \hat{O} | \psi \rangle = \int \psi^* \hat{O} \psi \, d^3 x \quad ; \quad \langle \psi | [\hat{O}; \hat{H}]_- | \psi \rangle = \int \psi^* [\hat{O}; \hat{H}]_- \psi \, d^3 x. \qquad (5.8) $$

It is now enlighting to deduce the famous Ehrenfest theorem, published by P. Ehrenfest in 1927. This theorem is used to justify the classical limit of quantum mechanics in many textbooks [34] [79][80][81] [83][84]. In Newtonian mechanics the time independent 1D Hamiltonian of a particle is $H(p, x) = \frac{p^2}{2m} + V(x)$. Then

$$ \begin{aligned} \frac{dx}{dt} &= [x; H]_{PB} = \left( \frac{\partial x}{\partial x} \frac{\partial H}{\partial p} - \frac{\partial p}{\partial x} \frac{\partial H}{\partial x} \right) = \frac{p}{m} & \rightarrow \frac{dx}{dt} &= \frac{p}{m} \\ \frac{dp}{dt} &= [p; H]_{PB} = \left( \frac{\partial p}{\partial x} \frac{\partial H}{\partial p} - \frac{\partial p}{\partial x} \frac{\partial H}{\partial x} \right) = -\frac{\partial V}{\partial x} & \rightarrow \frac{dp}{dt} &= -\frac{\partial V}{\partial x} \end{aligned} \qquad (5.9) $$

We now examine the quantum version of the same system. The quantum operator associated to the momentum is $\hat{p} = -i\hbar \frac{\partial}{\partial x}$ while the quantum operator associated to the position is $\hat{x} = x$ [75][76][80]. Using the 1D Hamiltonian operator $\hat{H} = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} + V(x)$, the 1D Schrödinger equation becomes:

$$ i\hbar \frac{\partial}{\partial t} \psi(x, t) = E_N \psi(x, t) = \hat{H} \psi(x, t) = \left( -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} + V(x) \right) \psi(x, t) \qquad (5.10) $$

where $E_N$ is the Newtonian (or non relativistic) energy. The Ehrenfest theorem is obtained by computing equation (5.7) when $O = \hat{x}$ and $O = \hat{p}$ [34][80][83]:

$$ \begin{aligned} \frac{d}{dt} \langle \hat{x} \rangle &= \frac{1}{i\hbar} \langle \psi | [\hat{x}, \hat{H}]_- | \psi \rangle = \langle \psi | \frac{\hat{p}}{m} | \psi \rangle = \langle \frac{\hat{p}}{m} | & \rightarrow \frac{d}{dt} \langle \hat{x} \rangle = \langle \frac{\hat{p}}{m} | \\ \frac{d}{dt} \langle \hat{p} \rangle &= \frac{1}{i\hbar} \langle \psi | [\hat{p}, \hat{H}]_- | \psi \rangle = -\langle \psi | \frac{\partial V}{\partial x} | \psi \rangle = \langle -\frac{\partial V}{\partial x} | & \rightarrow \frac{d}{dt} \langle \hat{p} \rangle = \langle -\frac{\partial V}{\partial x} | \end{aligned} \qquad (5.11) $$

Equation (5.11) is the Ehrenfest theorem which justifies the classical laws of equation (5.9). Nevertheless, it contains an important problem. The Schrödinger equation (5.10) is solved by $\psi(x, t) = e^{-i\frac{E_N t}{\hbar}} \psi'(x)$ where:

$$ E_N \psi'(x) = \hat{H} \psi'(x) = \left( -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} + V(x) \right) \psi'(x) \qquad (5.12) $$
Consequently $|\psi(x,t)|^2 = |\psi'(x)|^2$ depends on the single variable $x$, not on $t$. Then $\langle \hat{x} \rangle = \int \psi^* x \psi \, dx = \int x |\psi'(x)|^2 dx$ is a constant because the single variable $x$ is integrated out. It follows that $\frac{d}{dt} \langle \hat{x} \rangle = 0 \neq \langle \frac{\hat{p}}{m} \rangle$ which contradicts the Erhenfest theorem. An explicit derivations of the Erhenfest can be found in [34] : when the author uses the hermiticity of the Hamiltonian, an implicit integration by parts containing an omitted term is performed. This term should not be omitted and it will explicitly appear in our generally covariant formulation of quantum mechanics. Furthermore, starting from the Heisenberg equation of motion in the right hand side of equation (5.6), multiplying each side from the left by $\langle \psi |$ and from the right by $| \psi \rangle$ produces :

$$ \langle \psi | \frac{d}{dt} \hat{O} | \psi \rangle = \langle \frac{d}{dt} \hat{O} \rangle = \langle \psi | [\hat{O}; \hat{H}]_- | \psi \rangle \quad (5.13) $$

The above equation is not similar to equation (5.7), from which the Ehrenfest theorem has been derived. To extend the above analysis to general relativity, we use the generally covariant Lagrangian $L_m(x^\mu, \dot{x}^\mu) = \frac{1}{2}(mg_{\mu\nu}\dot{x}^\mu\dot{x}^\nu - mc^2)$ previously introduced in equation (2.3). The associated Euler-Lagrange equations are :

$$ \frac{d}{d\tau} \frac{\partial L_m}{\partial \dot{x}^\alpha} = \frac{\partial L_m}{\partial x^\alpha} \Leftrightarrow \frac{d}{d\tau} m \dot{x}^\alpha = \frac{1}{2} m g_{\mu\nu,\alpha} \dot{x}^\mu \dot{x}^\nu \Leftrightarrow \dot{p}_\alpha = \frac{1}{2m} g_{\mu\nu,\alpha} p^\mu p^\nu \quad (5.14) $$

The associated generally covariant Hamiltonian $H_g$ is :

$$ H_g(x^\alpha, p_\alpha) = \dot{x}^\alpha \frac{\partial L}{\partial \dot{x}^\alpha} - L = \dot{x}^\alpha p_\alpha - \frac{1}{2}(mg_{\mu\nu}\dot{x}^\mu\dot{x}^\nu - mc^2) = \frac{1}{2m}g^{\mu\nu}p_\mu p_\nu + \frac{1}{2}mc^2 \quad (5.15) $$

Since $g^{\mu\nu}p_\mu p_\nu = m^2c^2$, $H_g = mc^2$. The associated classical equations of motion are :

$$ \dot{x}^{\alpha} = \frac{\partial H_g}{\partial p_{\alpha}} = \frac{1}{m} g^{\mu\alpha} p_{\mu} = \frac{p^{\alpha}}{m} \quad ; \quad \dot{p}_{\alpha} = -\frac{\partial H_g}{\partial x^{\alpha}} = -\frac{1}{2m} \partial_{\alpha} g^{\mu\nu} p_{\mu} p_{\nu} = \frac{1}{2m} g_{\sigma\mu,\alpha} p^{\sigma} p^{\mu} \quad (5.16) $$

where we used $\partial_\alpha g^{\mu\nu} p_\mu p_\nu = \partial_\alpha g^{\mu\nu} (g_{\mu\sigma} p^\sigma) p_\nu = [\partial_\alpha (g^{\mu\nu} g_{\mu\sigma}) - g^{\mu\nu} g_{\sigma\mu,\alpha}] p^\sigma p_\nu = -g_{\sigma\mu,\alpha} p^\sigma p^\mu$ to compute the right hand side : this result will be used in equation (5.18) too. Generalizing the Poisson brackets to general relativity, the total derivative of any function $O(x^\lambda, p_\lambda)$ becomes :

$$ \frac{dO}{d\tau} = \left( \frac{dx^\alpha}{d\tau} \frac{\partial O}{\partial x^\alpha} + \frac{dp_\alpha}{d\tau} \frac{\partial O}{\partial p_\alpha} \right) = \left( \frac{\partial O}{\partial x^\alpha} \frac{\partial H_g}{\partial p_\alpha} - \frac{\partial O}{\partial p_\alpha} \frac{\partial H_g}{\partial x^\alpha} \right) = [O, H_g]_{PB} \quad (5.17) $$

The geodesic equation $\ddot{x}^\lambda + \Gamma^\lambda_{\mu\nu}\dot{x}^\mu\dot{x}^\nu = 0 \Leftrightarrow \dot{p}^\lambda + \frac{\Gamma^\lambda}{m} p^\mu p^\nu = 0$ can be obtained from the Poisson brackets, by fixing $O = p^\lambda = g^{\lambda\sigma} p_\sigma$ :

$$ \begin{aligned} \dot{p}^{\lambda} &= \frac{d(g^{\lambda\sigma} p_{\sigma})}{d\tau} = [g^{\lambda\sigma} p_{\sigma}, H_g]_{PB} = \left[ \frac{g^{\lambda\sigma}}{\partial x^{\alpha}} p_{\sigma} \frac{\partial H_g}{\partial p_{\alpha}} - g^{\lambda\sigma} \frac{\partial p_{\sigma}}{\partial x^{\alpha}} \frac{\partial H_g}{\partial x^{\alpha}} \right] = [\partial_{\alpha} g^{\lambda\sigma} p_{\sigma} \frac{p^{\alpha}}{m} + g^{\lambda\alpha} \frac{g_{\mu\nu,\alpha}}{2m} p^{\mu} p^{\nu}] \\ &= \frac{1}{m} [-g^{\lambda\sigma} g_{\mu\sigma,\alpha} p^{\mu} p^{\alpha} + g^{\lambda\alpha} \frac{g_{\mu\nu,\alpha}}{2} p^{\mu} p^{\nu}] = -\frac{g^{\lambda\sigma}}{2m} [(g_{\mu\sigma,\alpha} + g_{\alpha\sigma,\mu}) p^{\mu} p^{\alpha} - g_{\mu\nu,\sigma} p^{\mu} p^{\nu}] = -\frac{1}{m} \Gamma^{\lambda}_{\mu\nu} p^{\mu} p^{\nu} \end{aligned} \quad (5.18) $$

For later comparison, we rewrite the above equation in the form :

$$ [p^{\lambda}, H_g]_{PB} = \frac{dp^{\lambda}}{d\tau} = -\frac{1}{m} \Gamma^{\lambda}_{\mu\nu} p^{\mu} p^{\nu} \quad (5.19) $$

Equation (5.19) is generally covariant but classical : it is possible to derive its quantum version from the Klein-Gordon equation written in arbitrary space. Using again the notations of [Section 2], we know that if $\psi(x^\alpha)$ solves the generally covariant Klein-Gordon equation $\hbar^2 \Box_g \psi + m^2 c^2 \psi = 0$, then $\phi(x^\alpha, \tau) = \psi e^{i \frac{mc^2\tau}{\hbar}}$ solves the equation $\hbar^2 \Box_g \phi - \frac{\hbar^2}{c^2} \frac{\partial^2}{\partial\tau^2} \phi = 0$ because $\frac{\hbar^2}{c^2} \frac{\partial^2}{\partial\tau^2} \phi = -m^2 c^2 \phi$. Furthermore $-i\hbar \frac{\partial}{\partial\tau} \phi = mc^2 \phi$ : the function $\phi$ is an eigenstate of the operator $-i\hbar\partial_s$ with eigenvalue $mc^2$. This is interesting because particles are classified by eigenvalues of mass and spin the Standard Model of particles physics [62] (mass and spin are the Casimir invariants of the Poincaré group). It can be noticed that the state $\phi(x^\alpha, \tau) = \phi_1 + \phi_2$ such that $\hbar^2 \Box_g \phi_1 + m^2 c^2 \phi_1 = \hbar^2 \Box_g \phi_2 + m^2 c^2 \phi_2 = 0$ is an eigenstate of $-i\hbar \frac{\partial}{\partial\tau} \phi$ since $-i\hbar \frac{\partial}{\partial\tau} (\phi_1 + \phi_2) = mc^2 \phi_1 + mc^2 \phi_2 = mc^2 \phi$. Such property is not valid in the non relativistic limit, when the Schrödinger equation is used : $\psi_S = \psi_{S1} + \psi_{S2}$ is not an eigenstate of the Schrödinger Hamiltonian in general since $i\hbar \frac{\partial}{\partial t} (\psi_{S1} + \psi_{S2}) = E_1 \psi_{S1} + E_2 \psi_{S2}$. In [Section 2], we suggested that an unstable particle would decay according to $|\phi|^2 \sim e^{-\frac{\Gamma\tau}{\hbar}}$, a result which is obtained by writing $\phi(x^\alpha, \tau) = e^{i \frac{mc^2\tau}{\hbar}} \psi(x^\alpha)$ such that $|\phi|^2 = |\psi(x^\alpha)|^2 e^{-\frac{\Gamma\tau}{\hbar}}$. This is equivalent to the introduction of the complex mass $m = m_0 + i \frac{\Gamma}{2c^2}$. Using this complex mass in Minkowski space, the 5D wave equation $-\Box\phi + \frac{1}{c^2} \frac{\partial^2}{\partial\tau^2} \phi$ will produces the 4D Klein-Gordon equation :

$$ -\Box\psi - \frac{m^2 c^2}{\hbar^2} \psi = -\Box\psi - \frac{(m_0 c^2 + i \frac{\Gamma}{2})^2}{\hbar^2 c^2} \psi = 0 \quad (5.20) $$
This is the Klein-Gordon equation for an unstable particle [85]. When $m_0c^2 \gg \Gamma/2$, the propagator associated to the above equation takes the form :

$$ \frac{1}{p_{\mu}p^{\mu} - m^2c^2} = \frac{1}{p_{\mu}p^{\mu} - (m_0 + i\frac{\Gamma}{2c^2})^2c^2} \approx \frac{1}{p_{\mu}p^{\mu} - m_0^2c^2 + im_0\Gamma} \quad (5.21) $$

This expression is (the complex conjugate of) the propagator of an unstable scalar particle [59][61][62]. Furthermore, the eigenvalue $mc^2$ is the value of the generally covariant Hamiltonian $H_g = \frac{1}{2m}g^{\mu\nu}p_{\mu}p_{\nu} + \frac{1}{2}mc^2 = mc^2$ introduced above : we can then write $-i\hbar\partial_{\tau}\phi = H_g\phi = \frac{1}{2m}(-\hbar^2\Box_g + m^2c^2)\phi$. Defining the generally covariant Hamiltonian operator $\hat{H}_g = \frac{-\hbar^2\Box_g}{2m} + \frac{1}{2}mc^2$, we get a generally covariant Schrödinger equation for $\phi$ :

$$ -i\hbar \frac{\partial}{\partial \tau} \phi = mc^2 \phi \Leftrightarrow -i\hbar \frac{\partial}{\partial \tau} \phi = \hat{H}_g \phi \quad (5.22) $$

A quote from K. Kiefer and P. Peter now deserves to be given. They wrote in 2021 : "The problem of time in quantum gravity has been with us for more than fifty years, having been originally discussed by Peter Bergmann and his group and by Paul Dirac in the late 1950s. [...] In short, time in quantum theory is absolute, whereas it is dynamical in general relativity. This 'incompatibility' a priori renders complicated the intertwining of both theories into a working quantum theory of gravity, which is indeed yet to be formulated" [19]. With equation (5.22), the proper time is explicitly introduced in quantum physics while the time coordinate $x^0 = ct$ and the space coordinates $x^1, x^2, x^3$ appear on equal footing : this is a very simple way to solve the 'problem of time'. For any functions $O = O(x^\alpha)$ and $\psi = \psi(x^\alpha)$, it is shown in [Annex 7] that :

$$ \int \psi^*[O\Box_g\psi - \Box_g(O\psi)]\sqrt{g}d^4x = \int (\psi\partial^\nu\psi^* - \psi^*\partial^\nu\psi)\partial_\nu O\sqrt{g}d^4x - \int \partial_\mu(\partial^\mu O|\psi|^2\sqrt{g})d^4x \quad (5.23) $$

The last term is a boundary term which vanishes under appropriate boundary conditions. Evidently $\psi$ can be replaced by $\phi = \psi e^{i\frac{mc^2\tau}{\hbar}}$ in the above equation. We previously observed below equation (2.21) that the term $-\frac{i\hbar}{2}(\psi\partial^\nu\psi^* - \psi^*\partial^\nu\psi)$ can be identified with the covariantly conserved mass current $j_m^\mu$ such that $\nabla_\mu j_m^\mu = 0$ if $\psi$ solves the Klein-Gordon equation : $\nabla_\mu j_m^\mu = 0$ corresponds to a 4D Wronskian equation. Using the polar decomposition $\psi = Re^{-i\frac{S}{\hbar}}$ and the definition $\hat{H}_g = \frac{-\hbar^2\Box_g}{2m} + mc^2$, equation (5.23) can be rewritten :

$$ \begin{aligned} & \frac{1}{i\hbar} \int \psi^*[\hat{H}_g\psi - \hat{H}_g(O\psi)]\sqrt{g}d^4x = \frac{-i\hbar}{2m} \int (\psi\partial^\nu\psi^* - \psi^*\partial^\nu\psi)\partial_\nu O\sqrt{g}d^4x \\ & \Leftrightarrow \int (\psi^* \frac{1}{i\hbar} [O: \hat{H}_g]_- \psi) \sqrt{g}d^4x = \int \psi^* (\frac{\partial^\nu S}{m} \partial_\nu O) \psi \sqrt{g}d^4x \end{aligned} \quad (5.24) $$

In [Section 1], we introduced the material derivative in classical physics (by analogy with fluid mechanics) and we established the relation $\partial^\mu S = p^\mu = m\dot{x}^\mu$, if $S$ is the mechanical action : this relation can be used on any function $O(x^\alpha)$ to write

$$ \frac{d}{d\tau}O = \dot{x}^\nu \partial_\nu O = \frac{\partial^\nu S}{m} \partial_\nu O \quad (5.25) $$

such that, by example, $\dot{p}^\lambda = \frac{d}{d\tau}\partial^\lambda S = \frac{\partial^\nu S}{m}\partial_\nu\partial^\lambda S$. Furthermore, in [Section 2], we established the fact that the phase of a quantum field can be identified with the mechanical action divided by $\hbar$ in the eikonal approximation and we concluded that the classical limit emerges from the eikonal approximation. We now introduce the notation $\langle \hat{O} \rangle = \int \psi^* \hat{O} \psi \sqrt{g}d^4x$ where $\hat{O}$ is any operator (and $\hat{O} = O$ if $\hat{O}$ is a scalar function). Since $S$ can be identified with the action in the eikonal approximation, in the classical limit we can use equation (5.25) to write equation (5.24) in the form :

$$ \langle \frac{1}{i\hbar} [O; \hat{H}_g]_- \rangle = \langle \frac{dO}{d\tau} \rangle = \langle \frac{\partial^\nu S}{m} \partial_\nu O \rangle \quad (5.26) $$

We know from equation (5.3) that $\partial^\beta(\partial_\mu S\partial^\mu S) = 2(\partial^\mu S\partial_\mu\partial^\beta S + \Gamma_{\mu\sigma}^\beta\partial^\sigma S\partial^\mu S)$ : if we now fix $O(x^\alpha) = \partial^\lambda S = g^{\lambda\sigma}\partial_\sigma S$ we easily deduce

$$ \frac{\partial^\nu S}{m} \partial_\nu O = \frac{\partial^\nu S}{m} \partial_\nu \partial^\lambda S = -\frac{\Gamma_{\nu\beta}^\lambda}{m} \partial^\nu S \partial^\beta S + \frac{\partial^\lambda(\partial^\nu S \partial_\nu S)}{2m} \quad (5.27) $$

In the eikonal approximation $\partial^\nu S = p^\nu$ such that $\frac{\partial^\nu S}{m}\partial_\nu O = \dot{x}^\nu\partial_\nu O = \frac{dO}{d\tau}$ and, since we fixed $O = \partial^\lambda S \Leftrightarrow O = p^\lambda$, equation (5.27) becomes :

$$ \frac{dO}{d\tau} = \frac{\partial^\nu S}{m} \partial_\nu O = \frac{dp^\lambda}{d\tau} = -\frac{1}{m} \Gamma_{\nu\beta}^\lambda p^\nu p^\beta + \frac{\partial^\lambda(p^\nu p_\nu)}{2m} \quad (5.28) $$
where the last term vanishes because $p^\nu p_\nu = m^2c^2$ is a constant. Combining equation (5.26) with equation (5.28) and $O = p^\lambda$ we get :

$$ < \frac{1}{i\hbar} [p^\lambda; \hat{H}_g]_- > = < \frac{dp^\lambda}{d\tau} > = < -\frac{1}{m} \Gamma_{\nu\beta}^\lambda p^\nu p^\beta > \quad (5.29) $$

In the above equation (which replaces the Erhenfest theorem), the equality $< \frac{1}{i\hbar} [p^\lambda; \hat{H}_g]_- > = < \frac{dp^\lambda}{d\tau} >$ corresponds to the Heisenberg equation of motion in quantum mechanics while the equality $< \frac{dp^\lambda}{d\tau} > = - < \frac{1}{m} \Gamma_{\nu\beta}^\lambda p^\nu p^\beta >$ corresponds to the geodesic equation in general relativity. The generally covariant version of the Heisenberg equation of motion $< \frac{1}{i\hbar} [p^\lambda; \hat{H}_g]_- > = < \frac{dp^\lambda}{d\tau} >$ is valid in the eikonal approximation when appropriate boundary conditions are fulfilled (see the neglected boundary term in equation (5.23)). The single equation (5.29) identifies the motion of quantum mechanics with the motion of general relativity, in the eikonal approximation : the classical counterpart of equation (5.29) is of course equation (5.19). In [Section 2] we observed that the quantum Lagrangian density $\mathcal{L}_m = -\frac{\hbar^2}{2m^2} g^{\mu\nu} (\partial_\mu \psi \partial_\nu \psi^* + \partial_\mu \psi^* \partial_\nu \psi) - c^2 |\psi|^2$ reduces to the classical Lagrangian density $\mathcal{L}_m = \rho (g^{\mu\nu} \frac{\partial_{\mu}S}{mc} \frac{\partial_{\nu}S}{mc} - 1)$ in the eikonal approximation (see the analysis below equation (2.22)). Then, in [Section 3], we concluded that the quantum motion reduces to the geodesic motion in the eikonal approximation, see equation (3.9). Here we obtain again the same result, from purely quantum considerations (we did not use Einstein's equations or the Euler-Lagrange equations). In equation (5.29), the formulation of the motion is background-independent because $g_{\mu\nu}$ is not fixed. Interestingly, L. Smolin wrote in a famous book : "The key question for a quantum theory of gravity is then the following : Can we extend to quantum theory the principle that space has no fixed geometry ? That is, can we make quantum theory background-independent, at least with regard to the geometry of space ? If we can do this, we will automatically merge gravity and quantum theory, because gravity is already understood to be an aspect of dynamical spacetime geometry" [86]. Our analysis to derive the equality $< \frac{1}{i\hbar} [p^\lambda; \hat{H}_g]_- > = < \frac{dp^\lambda}{d\tau} > = - < \frac{1}{m} \Gamma_{\nu\beta}^\lambda p^\nu p^\beta >$ can be extended according to Postulate 3 to include the electromagnetic interaction but this extension will not be discussed. We nevertheless discuss the case of massive spin-1 particles (by example, the $\rho$ meson is an unstable spin-1 particle). The following Proca Lagrangian density $\mathcal{L}_{Proca}$ [87] and its equation of motion describe spin-1 particles [62] :

$$ \mathcal{L}_{Proca} = \frac{\hbar^2}{2m^2} \mathcal{F}_{\mu\nu}^* \mathcal{F}^{\mu\nu} - c^2 \mathcal{A}_{\mu}^* \mathcal{A}^{\mu} \quad ; \quad \nabla_{\mu} \mathcal{F}^{\mu\nu} = - \frac{m^2 c^2}{\hbar^2} \mathcal{A}^{\nu} \quad (5.30) $$

where $\mathcal{F}_{\mu\nu} = \partial_\mu A_\nu(x^\alpha) - \partial_\nu A_\mu(x^\alpha)$. We know from the comment below equation (4.30) that we can take the divergence of the equations of motion to conclude that $A^\nu$ obey the Lorenz gauge $\nabla_\nu A^\nu = 0$ : this is obvious in flat space-time since $0 = \partial_\mu \partial_\nu \mathcal{F}^{\mu\nu} = -\frac{m^2 c^2}{\hbar^2} \partial_\nu A^\nu = 0$. The condition $\nabla_\nu A^\nu = 0$ implies $\nabla_\mu \mathcal{F}^{\mu\nu} = \Box_g A^\mu - R^\mu_\nu A^\nu$ [27] where $R^\mu_\nu$ is the Ricci tensor, such that the equations of motion become :

$$ \frac{m^2 c^2}{\hbar^2} A^\mu = - \Box_g A^\mu + R^\mu_\nu A^\nu \Leftrightarrow mc^2 A^\mu = \frac{1}{2} mc^2 A^\mu - \frac{\hbar^2}{2m} \Box_g A^\mu + \frac{\hbar^2}{2m} R^\mu_\nu A^\nu = \hat{H}_g A^\mu + \frac{\hbar^2}{2m} R^\mu_\nu A^\nu \quad (5.31) $$

Introducing $\phi^\mu(x^\alpha, \tau) = A^\mu(x^\alpha) e^{i \frac{m c^2 \tau}{\hbar}}$ we can replace $A^\mu(x^\alpha)$ by $\phi^\mu(x^\alpha, \tau)$ in the Proca Lagrangian while the generally covariant Schrodinger equation for the massive spin-1 field $\phi^\mu$ takes the form :

$$ -i\hbar \frac{\partial}{\partial \tau} \phi^\mu = mc^2 \phi^\mu \Leftrightarrow -i\hbar \frac{\partial}{\partial \tau} \phi^\mu = \hat{H}_g \phi^\mu - \frac{\hbar^2}{2m} R^\mu_\nu \phi^\nu \quad (5.32) $$

Furthermore, the usual relation $|\vec{r}|^2|\vec{p}|^2 = (\vec{r} \cdot \vec{p})^2 + |\vec{r} \times \vec{p}|^2$ between the 3-vectors $\vec{r}$ and $\vec{p}$ extends as follow between two 4-vectors in Minkowski space :

$$ k_\mu k^\mu A^\nu A_\nu = (k_\mu A^\mu)^2 + |\vec{B}|^2 - |\vec{\mathcal{E}}|^2 \quad \text{where} \quad \vec{\mathcal{E}} = -\vec{k} A_0 + k_0 \vec{A} \quad \text{and} \quad \vec{B} = \vec{k} \times \vec{A} \quad (5.33) $$

In rectangular coordinates, the solutions of the Proca equation are plane waves and the condition $\partial_\mu A^\mu = 0$ implies $k_\mu A^\mu = 0$. We can then check that the Lagrangian density of massive spin-1 particles vanishes, like the Klein-Gordon Lagrangian density (and the gravitational Lagrangian density in vacuum) :

$$ \mathcal{L}_{Proca} = \frac{\hbar^2}{2m^2c^2} \mathcal{F}_{\mu\nu}^* \mathcal{F}^{\mu\nu} - \mathcal{A}_{\mu}^* \mathcal{A}^{\mu} = \frac{\hbar^2}{m^2c^2} (|\vec{B}|^2 - |\vec{\mathcal{E}}|^2) - \mathcal{A}_{\nu}^* \mathcal{A}^{\nu} = \frac{\hbar^2}{m^2c^2} k_{\mu} k^{\mu} \mathcal{A}_{\nu}^* \mathcal{A}^{\nu} - \mathcal{A}_{\nu}^* \mathcal{A}^{\nu} = 0 \quad (5.34) $$

We introduced the proper time, interpreted as a physical coordinate, in quantum physics. The proper time has already been introduced in quantum field theory, quantum gravity, or string theory without interpreting it as a coordinate : this subject deserves to be discussed. In rectangular coordinates, $\phi(x^\alpha, \tau)$ takes the form $Ne^{i \frac{p_\mu x^\mu - m c^2 \tau}{\hbar}}$ where $N$ is a normalization constant. Without interactions, we have $p_\mu x^\mu - mc^2 \tau = (p_\mu \dot{x}^\mu - mc^2) \tau = (\frac{p^\mu p_\mu - m^2 c^2}{m}) \tau$ such that $\phi(x^\alpha, \tau) = Ne^{i \frac{(p^\mu p_\mu - m^2 c^2) \tau}{m \hbar}}$. We now give a quote from E. Witten in the context of string theories (we
adapt his sign convention) : in momentum space, "the Feynman propagator for a scalar field of mass m and momentum p<sub>μ</sub> is"

$$
\frac{-\mathbf{i}}{p^{\mu}p_{\mu} - m^2c^2 + \mathbf{i}\epsilon} \quad (\equiv \frac{1}{m\hbar} \frac{-\mathbf{i}m\hbar}{p^{\mu}p_{\mu} - m^2c^2 + \mathbf{i}\epsilon}) \qquad (5.35)
$$

where we added the obvious equality in the right hand side. "ε is a small positive quantity that should be taken to
zero at the end of a calculation [...] The [...] propagator can similarly be written as an integral over proper time
τ" [88]. With our notations and s = cτ the equation given by E. Witten is analogous to :

$$
\frac{-i m \hbar}{p^{\mu} p_{\mu} - m^2 c^2 + i \epsilon} = \int_{0}^{\infty} e^{i \frac{(p^{\mu} p_{\mu} - m^2 c^2 + i \epsilon) \tau}{m \hbar}} d \tau \quad (5.36)
$$

In the context of string theories and quantum field theory, τ is usually called the Schwinger proper time. An
illustration of its use in quantum field theory is given by M. D. Schwartz with the following interpretation : a
"propagator is the amplitude for a particle to propagate from x to [x'] in proper time [τ], integrated over [τ]. [...]
The Schwinger proper time method is not only useful for calculating loops using quantum mechanics, it also gives
a new perspective on the semi-classical and non-relativistic limits of quantum field theory" [62]. Literature on
string theories also indicates : "String theories should reduce to ordinary four-dimensional field theories at low
energies. Yet the formulation of the two are so different that such a connection, if it exists, is not immediately
obvious. With the Schwinger proper-time representation, [...], it has been shown that field theories can indeed be
written in a string-like manner". When equation (5.36) is used in the context of quantum gravity, it is called the
'Schwinger DeWitt technique' [89]. This technique allows to define a propagator K(x<sup>α</sup>, x'<sup>α</sup>, τ) between the point
x<sup>α</sup> and x'<sup>α</sup>. This propagator obeys an equation of the form -iħ ∂<sub>τ</sub>/∂τ K(x<sup>α</sup>, x'<sup>α</sup>, τ) = Ω<sub>g</sub> K(x<sup>α</sup>, x'<sup>α</sup>, τ) where Ω<sub>g</sub> is a
generally covariant differential operator. When Ī<sub>g</sub> = Ī<sub>g</sub>, we easily recognize the generally covariant Schrödinger
equation of a spin-0 field given in equation (5.22). The introduction of the proper time has been made in quantum
field theory, string theory and quantum gravity but without interpreting it as a physical coordinate : it was only
introduced to produce a convenient "parametric representation of a propagator" [89]. According to Postulate 1,
the present theory suggests a much more profound meaning.

5.2 Particles, spin and complex general relativity

We are now interested by the polarizations, or spins, of electromagnetic and gravitational fields. This subject
will naturally lead us to discuss the descriptions of particles occuring in quantum physics, and to motivate the
extension of general relativity to complex numbers. The results introduced here will be used to discuss the Dirac
equation in the next subsection. According to L. Susskind : "Naively, the spin can be pictured as a little arrow
that points in some direction but that naive picture is too classical to accurately represent the real situation" [90].
It is well known that the "total angular momentum of the electromagnetic wave can often be separated into two
parts, the spin angular momentum (wave polarization) and the orbital angular momentum" [91]. The spin of
an arbitrary field can be deduced from the transformation properties of the field under the Lorentz group [61].
"Written in terms of three-vectors, the spin of the electromagnetic field reads : " [50]

$$
\vec{S} = \int \epsilon_0 \vec{E} \times \vec{A} \, d^3 x \qquad (5.37)
$$

where $\vec{S}$ is the electromagnetic spin [61], $\vec{E} = -\vec{\nabla}V - \partial_t \vec{A}$ is the electric field, and $\vec{A}$ is the electromagnetic vector potential. Another way to derive the expression of the electromagnetic spin is to start from the Poynting vector $\vec{P} = \epsilon_0 \vec{E} \times \vec{B}$ where $\vec{B} = \vec{\nabla} \times \vec{A}$ is the magnetic field. $\vec{P}$ being a momentum density, we easily define an angular momentum density $\vec{J} = \vec{r} \times \vec{P}$ where $\vec{r}$ is the position vector. A total angular momentum $\vec{J}$ can then be computed [92][93] :

$$
\vec{J} = \int \vec{\mathcal{J}} d^3 x = \int \epsilon_0 \vec{r} \times \vec{E} \times \vec{B} d^3 x = \int \epsilon_0 \sum_{i=1}^{3} \mathbf{E}_i (\vec{r} \times \vec{\nabla}) A_i d^3 x + \int \epsilon_0 \vec{E} \times \vec{A} d^3 x = \vec{L} + \vec{S} \quad (5.38)
$$

where an integration by parts has been performed, $\vec{L}$ being the orbital angular momentum. In quantum field
theory, the second quantization of the electromagnetic field, which is a spin-1 field, is made by introducing
creation and annihilation operators. The helicity is then introduced because "the spin $\vec{S}$ and orbital angular
momentum $\vec{L}$ cannot be fixed seperately for vectors fields. However the helicity, the projection $\Lambda = \frac{\vec{S} \cdot \vec{k}}{|\vec{k}|}$ of the spin
in the direction of motion, is a well-defined quantity" [50]. Only two polarizations contributes to the helicity
when the second quantization of the electromagnetic field is performed. The creation and annihilation operators
are respectively $\hat{a}_{\lambda}^{\dagger}(\vec{k})$ and $\hat{a}_{\lambda}(\vec{k})$ where the values $\lambda = 1$ and $\lambda = 2$ correspond to each of the two polarizations. The helicity operator takes the form [50]:

$$ \hat{\Lambda} = -\mathbf{i} \int [\hat{a}_1^\dagger(\vec{k}) \hat{a}_2(k) - \hat{a}_2^\dagger(\vec{k}) \hat{a}_1(k)] d^3k = \int [\hat{a}_R^\dagger(\vec{k}) \hat{a}_R(k) - \hat{a}_L^\dagger(\vec{k}) \hat{a}_L(k)] d^3k \quad (5.39) $$

where $\hat{a}_R(\vec{k}) = \frac{1}{\sqrt{2}}[\hat{a}_1(\vec{k}) + \mathbf{i}\hat{a}_2(\vec{k})]$ and $\hat{a}_L(\vec{k}) = \frac{1}{\sqrt{2}}[\hat{a}_1(\vec{k}) - \mathbf{i}\hat{a}_2(\vec{k})]$. "Thus the spin projection in the direction of motion (which here is chosen to be parallel to the $[x_3]$ axis) is carried by quanta with positive helicity which contribute +1 unit and quanta with negative helicity which contribute -1 unit (the units of course are $\hbar$)" [50]. Our aim is now to discuss the spin of light in a classical context. In standard textbooks introducing quantum field theory, authors often make use of the Coulomb gauge, and not only the Lorenz gauge, to show that the photon has only 2 polarizations [60][62][94]. We first provide a proof which uses only the Lorenz gauge : the comparison between electromagnetic waves and gravitational waves will then be easier. In Minkowski spacetime, the electromagnetic Lagrangian density $\mathcal{L}_{EM}$ and the four equations of motion (Maxwell's equations) are respectively:

$$ \mathcal{L}_{EM} = -\frac{1}{4\mu_0} F_{\mu\nu} F^{\mu\nu} \quad ; \quad \partial_{\mu} F^{\mu\nu} = 0 \qquad (5.40) $$

This Lagrangian density and the equations of motion are invariant under the gauge transformation $A_{\mu}(x^{\alpha}) \rightarrow A_{\mu}(x^{\alpha}) + \partial_{\mu}B(x^{\alpha})$ where $B(x^{\alpha})$ is an arbitrary function : there is a redundancy in the description of the field. We now impose the Lorenz gauge $\partial_{\mu}A^{\mu} = 0$ which gives us a 5th equation implying that Maxwell's equations take the form of wave equations : $\partial_{\mu}F^{\mu\nu} = 0 \rightarrow \Box A^{\nu} = 0$. When the Lorenz gauge is imposed, the redundancy is not suppressed. Indeed, the 5 equations remains invariant under the transformation $A_{\mu}(x^{\alpha}) \rightarrow A_{\mu}(x^{\alpha}) + \partial_{\mu}B(x^{\alpha})$ if $\Box B = 0$ : this is called a restricted gauge transformation by J. D. Jackson [37], because the function $B(x^{\alpha})$ is not arbitrary anymore. Up to now, the letter $\lambda$ will be used to label polarizations and Einstein's summation convention will not be applied to this letter. We express $A_{\mu}$ according to equation (5.41), defined on a 3D domain because $k_0, k_1, k_2, k_3$ are not independent (since $\Box A^{\nu} = 0$ implies $k_{\alpha}k^{\alpha} = 0$):

$$ A_{\mu}(x^{\alpha}) = \frac{1}{(\sqrt{2\pi})^3} \int \frac{1}{\sqrt{k_0}} \sum_{\lambda=0}^{3} a_{\lambda}(\vec{k}) \epsilon_{\lambda\mu}(\vec{k}) e^{-i k_{\alpha} x^{\alpha}} d^3 k \quad (5.41) $$

The vectors $\epsilon_{0\mu}, \epsilon_{1\mu}, \epsilon_{2\mu}, \epsilon_{3\mu}$ are called polarization vectors. To examine their properties, we consider a single plane wave travelling along the $x_3$ direction with 4-wave vector $k^{\mu} = (k_3, 0, 0, k_3)$ and $k_{\mu} = (k_3, 0, 0, -k_3)$:

$$ A_{\mu} = N \sum_{\lambda=0}^{3} a_{\lambda}(\vec{k}) \epsilon_{\lambda\mu}(\vec{k}) e^{-i(\omega t - k_3 x^3)} \quad (5.42) $$

where $N$ is a normalization constant. We take the polarization vectors in the form $\epsilon_{0\mu} = (1, 0, 0, 0)$, $\epsilon_{1\mu} = (0, 1, 0, 0)$, $\epsilon_{2\mu} = (0, 0, 1, 0)$, $\epsilon_{3\mu} = (0, 0, 0, 1)$. The Lorenz gauge $\partial^{\mu}A_{\mu} = 0$ implies $\sum_{\lambda} k^{\mu}a_{\lambda}\epsilon_{\lambda\mu} = 0$ such that the constraint $a_0 = -a_3$ follows. From this constraint only 3 independent polarizations remain : we only need 3 polarization vectors. To use only 3 polarizations, we redefine $(\epsilon_{0\mu} - \epsilon_{3\mu}) \rightarrow \epsilon_{0\mu} = (1, 0, 0, -1)$ such that now $k^{\mu}\epsilon_{0\mu} = k^{\mu}\epsilon_{1\mu} = k^{\mu}\epsilon_{2\mu} = 0$ which forces $\partial^{\mu}A_{\mu} = 0$. It should be noticed that $\epsilon_{0\mu}$ is now proportional to $k_{\mu}$. If the wave travels along an arbitrary direction with wave vector $\vec{k}$, the 3 polarization vectors takes the form:

$$ \epsilon_{1\mu} = \frac{1}{|\vec{k}|}(0, -\vec{k}'); \quad \epsilon_{2\mu} = \frac{1}{|\vec{k}|}(0, -\vec{k}'); \quad \epsilon_{0\mu} = \frac{1}{|\vec{k}|}(|\vec{k}|, -\vec{k}) = (1, -\frac{\vec{k}}{|\vec{k}|}) = \frac{k_{\mu}}{|\vec{k}|} \quad (5.43) $$

where the conditions $\vec{k}.\vec{k}' = \vec{k}.\vec{k}'' = \vec{k}'.\vec{k}'' = 0$ and $|\vec{k}| = |\vec{k}'| = |\vec{k}''| = |\vec{k}''|$ must be fulfilled (we will not need to fix the explicit expressions of $\vec{k}'$ and $\vec{k}''$ for a given $\vec{k}$). With the polarization vectors defined according to equation (5.43), we can write $A_{\mu}(x^{\alpha}) + \partial_{\mu}B(x^{\alpha})$ in the form:

$$ A_{\mu}(x^{\alpha}) + \partial_{\mu}B(x^{\alpha}) = \frac{1}{(\sqrt{2\pi})^3} \int \frac{1}{\sqrt{k_0}} \sum_{\lambda=0}^{2} a_{\lambda}(\vec{k}) \epsilon_{\lambda\mu}(\vec{k}) e^{-i k_{\alpha} x^{\alpha}} d^3 k - \frac{1}{(\sqrt{2\pi})^3} \int \frac{1}{\sqrt{k_0}} b(\vec{k}) \mathbf{i} k_{\mu} e^{-i k_{\alpha} x^{\alpha}} d^3 k \quad (5.44) $$

The conditions $0 = \partial^{\mu}A_{\mu} = \Box A_{\mu} = \Box B$ are then fullfilled and we can always choose $b(\vec{k}) = -\mathbf{i}\frac{a_0(\vec{k})}{|\vec{k}|}$ such that $a_0(\vec{k})\epsilon_{0\mu} - b(\vec{k})\mathbf{i}k_{\mu} = 0$. Then, the gauge invariance always allows us to suppress the 0th polarization to write $A_{\mu}$:

$$ A_{\mu}(x^{\alpha}) = \frac{1}{(\sqrt{2\pi})^3} \int \frac{1}{\sqrt{k_0}} \sum_{\lambda=1}^{2} a_{\lambda}(\vec{k}) \epsilon_{\lambda\mu}(\vec{k}) e^{-i k_{\alpha} x^{\alpha}} d^3 k \quad (5.45) $$
We recover the well known result that electromagnetic waves have only two physically independent polarizations, the polarization vectors being orthogonal to the direction of propagation. If $\vec{k}$ is arbitrary, the nonvanishing components of $\epsilon_{1\mu}$ and $\epsilon_{2\mu}$ are restricted to the 3D space $(x_1, x_2, x_3)$. In optical sciences, one defines two other polarization vectors according to $\epsilon_{R\mu} = \frac{1}{\sqrt{2}}(\epsilon_{1\mu} + i\epsilon_{2\mu})$ and $\epsilon_{L\mu} = \frac{1}{\sqrt{2}}(\epsilon_{1\mu} - i\epsilon_{2\mu})$ [56]. In this context, the polarization vectors are called Jones' vectors [95]. Considering a plane wave travelling along the $x_3$ direction, the Jones' vectors are:

$$ \epsilon_1^\mu = - \begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \end{bmatrix} ; \quad \epsilon_2^\mu = - \begin{bmatrix} 0 \\ 0 \\ 1 \\ 0 \end{bmatrix} ; \quad \epsilon_3^\mu = - \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix} \quad \epsilon_R^\mu = - \frac{1}{\sqrt{2}} \begin{bmatrix} 0 \\ 1 \\ \mathbf{i} \\ 0 \end{bmatrix} ; \quad \epsilon_L^\mu = - \frac{1}{\sqrt{2}} \begin{bmatrix} 0 \\ 1 \\ -\mathbf{i} \\ 0 \end{bmatrix} \qquad (5.46) $$

The vanishing $0^{th}$ components of the above vectors are often omitted in literature (in this case we will write the polarization vectors $\epsilon_1^i, \epsilon_2^i, \epsilon_3^i, \epsilon_R^i$ and $\epsilon_L^i$ with $i=1,2,3$). We immediately deduce from equations (5.45) and (5.46) that $A^\mu = (0, A^1, A^2, 0)$. Furthermore $\vec{E} = -\vec{\nabla}\vec{V} - \partial_t \vec{A} = \mathbf{i}\omega\vec{A}$ and $\vec{B} = \mathbf{i}\vec{k} \times \vec{A}$. Then, since $\vec{k} \cdot \vec{A} = 0$, we deduce $c^2|\vec{B}|^2 = c^2|\vec{k}|^2|\vec{A}|^2 = \omega^2|\vec{A}|^2 = |\vec{E}|^2$. Consequently:

$$ \mathcal{L}_{EM} = -\frac{1}{4\mu_0} F_{\mu\nu} F^{\mu\nu} = \frac{1}{2\mu_0 c^2} (|\vec{E}|^2 - c^2|\vec{B}|^2) = 0 \qquad (5.47) $$

The Lagrangian density of the free electromagnetic field vanishes in Minkowski spacetime (like the gravitational Lagrangian density $R$). The polarization vectors $\epsilon_1^\mu$ and $\epsilon_2^\mu$ correspond to horizontal and vertical polarizations respectively. $\epsilon_R^\mu$ and $\epsilon_L^\mu$ correspond to right handed and left handed polarizations, respectively [95]. As explained by M. Schwartz: "these are the states of circularly polarized light in the $[x_3]$ direction, which are helicity eigenstates. In general, the polarizations of massless particles with spin $\ge 0$ can always be taken to be helicity eigenstates" [62]. In the same context, M. Maggiore points out that "massless particles have only one degree of freedom, and are characterized by the value $\Lambda$ of their helicity. [...] From the point of view of the representations of the Poincare group, a massless particle with helicity $\Lambda$ and a massless particle with helicity $-\Lambda$ are logically two different species of particles" [61]. Although the real basis provided by $\epsilon_1^\mu$ and $\epsilon_2^\mu$ is mathematically equivalent to the complex basis provided by $\epsilon_R^\mu$ and $\epsilon_L^\mu$, helicity eigenstates (circularly polarized photons) are complex. Still considering a plane wave travelling along the $x_3$ direction, we then generalize the definition of the spin according to [93]:

$$ \vec{S} = \int \epsilon_0 \vec{E}^* \times \vec{A} \, d^3 x = \int -\mathbf{i} \frac{\epsilon_0}{\omega} \vec{E}^* \times \vec{E} \, d^3 x \qquad (5.48) $$

Complex fields are useful in optical sciences, as illustrated by the following quote reproduced from the book "Principles of optics" by M. Born and E. Wolf: "discussing monochromatic waves fields we have found it useful to regard each real function as the real part of an associated complex wave function. In the present chapter we shall be concerned with polychromatic fields. It will again be useful to employ a complex representation" [43]. Each component $S_i$, $i=1,2,3$, of the spin $\vec{S} = (S_1, S_2, S_3)$, can be written in the form:

$$ S_i = \int \frac{\epsilon_0}{\hbar\omega} \begin{bmatrix} \mathbf{E}_1^* & \mathbf{E}_2^* & \mathbf{E}_3^* \end{bmatrix} \hat{\mathbf{S}}_i \begin{bmatrix} \mathbf{E}_1 \\ \mathbf{E}_2 \\ \mathbf{E}_3 \end{bmatrix} \, d^3 x \qquad (5.49) $$

where the three matrices $\hat{\mathbf{S}}_1, \hat{\mathbf{S}}_2$ and $\hat{\mathbf{S}}_3$ are [96]:

$$ \hat{\mathbf{S}}_1 = \mathbf{i}\hbar \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end{bmatrix} ; \quad \hat{\mathbf{S}}_2 = \mathbf{i}\hbar \begin{bmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ -1 & 0 & 0 \end{bmatrix} ; \quad \hat{\mathbf{S}}_3 = \mathbf{i}\hbar \begin{bmatrix} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \qquad (5.50) $$

such that $[\hat{\mathbf{S}}_i, \hat{\mathbf{S}}_j] = \mathbf{i}\hbar\epsilon_{ijk}\hat{\mathbf{S}}_k$ where $i, j, k = 1, 2, 3$ and $\epsilon_{ijk}$ is the Levy-Civita tensor. Then,

$$ \hat{\mathbf{S}}_3 \epsilon_R^i = -\frac{1}{\sqrt{2}} \mathbf{i} \hbar \begin{bmatrix} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} \mathbf{i} \\ 0 \end{bmatrix} = -\hbar \frac{1}{\sqrt{2}} \begin{bmatrix} 1 \\ \mathbf{i} \\ 0 \end{bmatrix} = \hbar \epsilon_R^i \quad \rightarrow \quad \hat{\mathbf{S}}_3 \epsilon_R^i = \hbar \epsilon_R^i \qquad (5.51) $$

Similarly, $\hat{\mathbf{S}}_3 \epsilon_L^i = -\hbar \epsilon_L^i$ and $\hat{\mathbf{S}}_3 \epsilon_3^i = 0$: the eigenvectors of $\hat{\mathbf{S}}_3$ are $\epsilon_R^i, \epsilon_L^i$ and $\epsilon_3^i$. The electric field of a right handed photon travelling along the $x_3$ direction $\vec{E} = \mathbf{i}\omega\vec{A} = \mathbf{i}\omega\epsilon_R^i N e^{-i(\omega t - k_3 x^3)}$ contains the vector part $\epsilon_R^i$ and the plane wave part $\mathbf{i}\omega N e^{-i(\omega t - k_3 x^3)}$: since $\vec{E} \times \vec{E} = 0$, the vector part must be complex to get a nonvanishing spin in equation (5.48). From $A^\mu = (0, A^1, A^2, 0)$ and equation (5.48), we immediately see that $\vec{S} = (0, 0, S_3)$ such
that $\Lambda = \frac{\vec{s} \cdot \vec{p}}{|\vec{p}|} = \frac{\vec{s} \cdot \vec{k}}{|\vec{k}|} = S_3$. Considering a plane wave propagating along the $x_3$ direction in a box of volume $V$, we fix $N = \sqrt{\frac{\hbar}{V\epsilon_0\omega}}$ and rewrite equation (5.42) with only two polarizations:

$$A^{\mu} = \sqrt{\frac{\hbar}{V\epsilon_0\omega}} (a_1\epsilon_1^{\mu} + a_2\epsilon_2^{\mu}) e^{i(\omega t - k_3 x^3)} = \sqrt{\frac{\hbar}{V\epsilon_0\omega}} (a_R\epsilon_R^{\mu} + a_L\epsilon_L^{\mu}) e^{i(\omega t - k_3 x^3)} \quad (5.52)$$

where $a_1, a_2, a_R$ and $a_L$ are the expansion coefficients. The coefficients $a_1$ and $a_2$ become operators when the second quantization is introduced. In equation (5.52), we normalized $A^\mu$ such that the energy and momentum in the box are:

$$ 
\begin{aligned}
E &= \int_V \frac{\epsilon_0}{2} |\vec{E}|^2 + \frac{1}{2\mu_0} |\vec{B}|^2 \, d^3x = \int_V \epsilon_0 |\vec{E}|^2 \, d^3x = \hbar\omega (|a_R|^2 + |a_L|^2) \frac{1}{V} \int_V d^3x = \hbar\omega (|a_R|^2 + |a_L|^2) \\
\vec{p} &= \int_V \vec{P} \, d^3x = \int \epsilon_0 \vec{E}^* \times \vec{B} \, d^3x = \hbar\vec{k} (|a_R|^2 + |a_L|^2) \frac{1}{V} \int_V d^3x = \hbar\vec{k} (|a_R|^2 + |a_L|^2)
\end{aligned}
\quad (5.53) $$

We now inject equation (5.52) in equations (5.48) and (5.49) to compute the helicity of a single plane wave travelling along the $x_3$ direction inside a box of volume $V$:

$$ \Lambda = S_3 = i\hbar(a_1^*a_2 - a_2^*a_1) \frac{1}{V} \int_V d^3x = i\hbar(a_1^*a_2 - a_2^*a_1) = \hbar (|a_R|^2 - |a_L|^2) \quad (5.54) $$

More details about the concepts of spin, helicity and chirality of electromagnetic waves can be found in [97]. Interpreting now a photon as an helicity eigenstate of a single mode (solution of Maxwell's equations), a spin up photon corresponds to the combination $|a_R\rangle = 1, a_L = 0$ in equation (5.52) while a spin down photon corresponds to the combination $a_R = 0, |a_L\rangle = 1$. In their introductory course to particle physics, F. Halzen and A. Martin give a very similar interpretation of the photon although they do not state that it is a helicity eigenstate: "We see that the wavefunction $A^\mu$ for a free photon satisfies the equation $\Box A^\mu = 0$ which has solutions $A^\mu = \epsilon^\mu(\vec{k})e^{ik_\nu x^\nu}$. The 4-vector $\epsilon^\mu$ is called the polarization vector of the photon" [35]. The above discussion of the spin of electromagnetic waves allows us to suggest a definition of the concept of particle and to enlighten the relation between quantum and classical physics. First, it should be recalled that textbooks on quantum physics usually indicate that the spin of particles is a purely quantum effect, without classical analog. By example, Landau and Lifshitz state: "an elementary particle must be assigned a certain 'intrinsic' angular momentum [...] This property of elementary particles is peculiar to quantum theory (it disappears in the limit $\hbar \to 0$), and therefore has in principle no classical interpretation. The intrinsic angular momentum of a particle is called its spin" [45]. Indeed, it is commonly considered that Planck's constant is a purely quantum constant. In particular, Planck failed to introduce his constant in classical physics, as he acknowledged in his autobiography: "My futile attempts to fit the elementary quantum of action somehow into the classical theory continued for a number of years, and they cost me a great deal of effort. Many of my colleagues saw in this something bordering on a tragedy" [30]. In this context, it deserves to be noticed that our analysis to derive $E = \hbar\omega$ and $\vec{p} = \hbar\vec{k}$ in equation (5.53) was based on classical electrodynamics: $\hbar$ was used to normalize $A_\mu$ in equation (5.52) where $A_\mu$ is the classical potential solving Maxwell's equations in vacuum. To explicitly show a difference between the above classical electromagnetic field (with $\hbar$ used to normalize $A^\mu$) and the second quantized electromagnetic field of quantum field theory, we compare their Hamiltonians when the potential $A^\mu$ is expanded in the form of orthogonal plane waves, travelling along the $x_3$ direction, and vanishing at the walls $x_3 = 0$ and $x_3 = L$ of cubic box. We call $V = L^3$ the volume of the box.

$$ A^{\mu} = \sum_{n} \sqrt{\frac{\hbar}{V\epsilon_0\omega_n}} (a_{1n}\epsilon_{1}^{\mu} + a_{2n}\epsilon_{2}^{\mu}) e^{-i(\omega_n t - k_{3n}x^3)} \quad (5.55) $$

where $E_n/\hbar = w_n = ck_n = cn/L = p_n c/\hbar$. The classical energy $E$, or the classical Hamiltonian $H$, is then:

$$ E = H = \int_V \frac{\epsilon_0}{2} |\vec{E}|^2 + \frac{1}{2\mu_0} |\vec{B}|^2 \, d^3x = \sum_{n,\lambda} \hbar\omega_n a_{\lambda n}^* a_{\lambda n} \quad (5.56) $$

where $\lambda = 1, 2$. In quantum field theory, the process of second quantization replaces $a_{\lambda n}^*$ and $a_{\lambda n}$ by the quantum operators $\hat{a}_{\lambda n}^\dagger$ and $\hat{a}_{\lambda n}$ obeying the commutation relations $[\hat{a}_{\lambda n}^\dagger; \hat{a}_{\lambda'n'}]_- = \delta_{\lambda\lambda'}\delta_{nn'}$ such that the quantum Hamiltonian is [98][99]:

$$ \hat{H} = \frac{1}{2} \sum_{n, \lambda} \hbar \omega_n (\hat{a}_{\lambda n}^{\dagger} \hat{a}_{\lambda n} + \hat{a}_{\lambda n} \hat{a}_{\lambda n}^{\dagger}) = \sum_{n, \lambda} \hbar \omega_n (\hat{a}_{\lambda n}^{\dagger} \hat{a}_{\lambda n} + \frac{1}{2}) = \sum_{n, \lambda} \hbar \omega_n \hat{a}_{\lambda n}^{\dagger} \hat{a}_{\lambda n} + \sum_{n, \lambda} \frac{1}{2} \hbar \omega_n \quad (5.57) $$
where $\hat{a}_{\lambda n}^{\dagger} \hat{a}_{\lambda n} = \hat{n}$ is the particle number operator. The above equation is the last equation in which no Einstein summation convention was applied to the letter $\lambda$. Comparing equations (5.56) and (5.57), we see that the introduction of creation and annihilation operators produces the divergent vacuum energy $\sum_{n,\lambda} \frac{1}{2} \hbar \omega_n$ [99]. The vacuum energy is associated to the famous "cosmological constant problem" when general relativity is taken into account [100]. According to M. Schwartz, "first quantization refers to the discrete modes, for example, of a particle in a box. Second quantization refers to the integer numbers of excitations of each of these modes" [62]. In our example, the first quantization then corresponds to the discrete values of $E_n/\hbar = w_n = ck_{n3} = cn/L = p_n c/\hbar$ imposed by the boundary conditions while the second quantization corresponds to $\hat{a}_{\lambda n}^{\dagger} \hat{a}_{\lambda n} = \hat{n}$ where operators have been introduced. Our definition of a photon as a helicity eigenstate of the form $A^{\mu} = \epsilon^{\mu}(\vec{k}) e^{i k_{\nu} x^{\nu}}$ solving Maxwell's equations (which describes spin-1 particles) makes no use of second quantization. Since a spin-1 field transforms according to the (unitary) spin-1 representation of the Lorentz group, our photon is defined according to Wigner's famous representation theory: "The wave functions [...] describing the possible states of a quantum mechanical system form a linear vector space V [...]. A classification of all unitary representations of the Lorentz group [...] amounts, therefore, to a classification of all possible relativistic wave equations" [101]. Explicitly, if $\Lambda_{\mu}^{\nu}$ is a Lorentz transformation corresponding to a rotation of angle $\theta$ in the plane $(x_1; x_2)$, i.e :

$$ \Lambda_{\mu}^{\nu} = e^{i\mathbf{M}_{12}\theta} = \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & \cos\theta & \sin\theta & 0 \\ 0 & -\sin\theta & \cos\theta & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}; \quad \mathbf{M}_{12} = \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & -\mathbf{i} & 0 \\ 0 & \mathbf{i} & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix} \qquad (5.58) $$

then, the photon field given in equation (5.52) with $a_R = 1$ and $a_L = 0$ transforms as :

$$ A'^{\nu} = \Lambda_{\mu}^{\nu} A^{\mu} = \sqrt{\frac{\hbar}{V\epsilon_{0}\omega}} e^{i(\omega t - k_{3}x^{3})} \Lambda_{\mu}^{\nu} \epsilon_{R}^{\mu} = \sqrt{\frac{\hbar}{V\epsilon_{0}\omega}} e^{i(\omega t - k_{3}x^{3})} e^{i\theta} \epsilon_{R}^{\nu} \Leftrightarrow A'^{\nu} = e^{i\theta} A^{\nu} \qquad (5.59) $$

Since "any plane wave $\psi$ which is transformed by a rotation of any angle $\theta$ about the direction of propagation into $\psi' = e^{i\frac{\Lambda}{\hbar}\theta}\psi$ is said to have helicity $\Lambda$" [102], it is clear that the right handed photon has helicity $\Lambda = \frac{\vec{S} \cdot \vec{p}}{|\vec{p}|} = 1$ and is then a spin-1 particle. It will be interesting to compare this representation of the photon with the representation of the graviton given below.

Now we want to recall that gravitational waves have only 2 polarizations, like electromagnetic waves: the discussion will allow us to motivate the extension of general relativity to complex numbers. We follow C. W. Misner, K. S. Thorne and J.A. Wheeler with slightly different notations [56]. In this context, the metric is written $g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}$ where $h_{\mu\nu}$ is a small perturbation, such that only first order terms are kept. "Linearized theory, one recalls, is a weak-field approximation to general relativity. [...] linearized theory describes gravity by a symmetric, second-rank tensor field $\bar{h}_{\mu\nu}$. In the standard gauge, this field satisfies the gauge condition $\partial_{\alpha}\bar{h}^{\mu\alpha} = 0$. It is shown in [Annex 8] that the usual gravitational harmonic gauge $g^{\alpha\beta}\Gamma^{\mu}_{\alpha\beta} = 0$ produces $\partial_{\alpha}\bar{h}^{\mu\alpha} = 0$ in the linear approximation. "throughout linearized theory, indices of $\bar{h}_{\mu\nu}$ are raised and lowered with the Minkowski metric $\eta_{\alpha\beta}$. In this gauge the propagation equations for vacuum gravitational fields are the familiar wave equations $\Box\bar{h}_{\mu\nu} = 0$. [...] the metric perturbation $h_{\mu\nu}$ is related to the gravitational field $\bar{h}_{\mu\nu}$ by

$$ h_{\mu\nu} = \bar{h}_{\mu\nu} - \frac{1}{2}\eta_{\mu\nu}\bar{h} \quad ; \quad \bar{h}_{\mu\nu} = h_{\mu\nu} - \frac{1}{2}\eta_{\mu\nu}h \quad ; \quad h = h_{\alpha}^{\alpha} = -\bar{h} = -\bar{h}_{\alpha}^{\alpha} \qquad (5.60) $$

Actually, the coordinates of linearized theory are not fully fixed by the conditions $\partial_{\alpha}\bar{h}^{\mu\alpha} = 0$. There remains an ambiguity embodied in further 'gauge changes' $\xi_{\mu}$ which satisfy a restrictive condition $\Box\xi_{\mu} = 0$ in order to preserve $\partial_{\alpha}\bar{h}^{\mu\alpha} = 0$. [...] then

$$ \bar{h}_{\mu\nu} \rightarrow \bar{h}_{\mu\nu} + \partial_{\mu}\xi_{\nu} + \partial_{\nu}\xi_{\mu} - \eta_{\mu\nu}\partial_{\alpha}\xi^{\alpha} \qquad (5.61) $$

is the gauge change. The simplest of all solutions to the linearized equations $\Box\bar{h}_{\mu\nu} = 0$ is the monochromatic, plane-wave solution $\bar{h}_{\mu\nu} = \mathcal{R}_e[\epsilon_{\mu\nu}e^{i\frac{p_{\alpha}x^{\alpha}}{\hbar}}]$. Here $\mathcal{R}_e[\ ]$ means that one must take the real part of the expression in brackets while $A_{\mu\nu}$ (amplitude) and $p_{\mu}$ satisfying $p_{\alpha}p^{\alpha} = 0$, $\epsilon_{\mu\alpha}p^{\alpha} = 0$ (consequences of $\Box\bar{h}_{\mu\nu} = 0$ and $\partial_{\alpha}\bar{h}^{\mu\alpha} = 0$ respectively). At first sight the amplitude $\epsilon_{\mu\nu}$ of this plane wave appears to have six independent component (ten, less the four orthogonality constraints $\epsilon_{\mu\alpha}p^{\alpha} = 0$). But this cannot be right! [...] the gravitational field in general relativity has only two dynamic degrees of freedom, not six. Where has the analysis gone astray? One went astray by forgetting the arbitraryness tied up in the gauge freedom. The plane-wave vector $\xi_{\mu} = iC_{\mu}e^{i\frac{p_{\alpha}x^{\alpha}}{\hbar}}$ with four arbitrary constant $C_{\mu}$ generates a gauge transformation that can change arbitrarily four of the six independent component of $\epsilon_{\mu\nu}$. One gets rid of this arbitrariness by choosing a specific gauge" [56]. $\epsilon_{\mu\nu}$ is the polarization tensor [27]. In the transverse traceless gauge, only the spatial component of $h_{\mu\nu}$ are non zero ($h_{\mu0} = 0$) and
$h = 0$ such that $\bar{h}_{\mu\nu} = h_{\mu\nu}$ [56] : the non vanishing components of the polarization tensor are then restricted to the 3D space $(x_1, x_2, x_3)$, like in the electromagnetic case. Finally, "consider a plane, monochromatic wave propagating in the $x_3$ direction. In the transverse traceless gauge [...], the gauge constraints reveal that the only nonvanishing components of $h_{\mu\nu}$ are" [56] :

$$h_{11} = -h_{22} ; \quad h_{12} = h_{21} \qquad (5.62)$$

Then Misner et al. introduce the gravitational analogs of $\epsilon_{R\mu}$ and $\epsilon_{L\mu}$ [56], we call them $\epsilon_{R\mu\nu}$ and $\epsilon_{L\mu\nu}$ :

$$\epsilon_{R\mu\nu} = \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 1 & \mathbf{i} & 0 \\ 0 & \mathbf{i} & -1 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix} ; \quad \epsilon_{L\mu\nu} = \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 1 & -\mathbf{i} & 0 \\ 0 & -\mathbf{i} & -1 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix} \qquad (5.63)$$

$\epsilon_{R\mu\nu}$ and $\epsilon_{L\mu\nu}$ correspond to circularly polarized gravitational waves [103]. If $\Lambda^\mu_\alpha$ is a Lorentz transformation corresponding to a rotation of angle $\theta$ in the plane $(x_1, x_2)$, i.e, if $\Lambda^\mu_\alpha$ is given by equation (5.58), under the corresponding rotation, the polarization tensor $\epsilon_{R\mu\nu}$ becomes :

$$\epsilon'_{R\alpha\beta} = \Lambda^\mu_\alpha \Lambda^\nu_\beta \epsilon_{R\mu\nu} = e^{i2\theta} \epsilon_{R\alpha\beta} \qquad (5.64)$$

The polarization tensor $\epsilon_{R\mu\nu}$, which is the gravitational analog of the polarization vector $\epsilon_{R\mu}$, then transforms like a field with helicity $\Lambda = 2$ [102] (it should be remembered that the complete gravitational field $\bar{h}_{\mu\nu}$ contains a tensorial part and a plane wave part). To describe circularly polarized gravitational waves, the corresponding tensor is then complex, like in the electromagnetic case.

Our aim is now to motivate the extension of general relativity to complex numbers. In 1950, A. Einstein suggested that the introduction of hermitian metrics could be the most natural way to generalize general relativity [104]. The extension of general relativity to complex numbers (with a symmetric complex metric, with a hermitian metric or with complex spacetime) has been suggested in many theoretical contexts since 1950. Such extensions appear especially in theories which aim to unify general relativity and quantum physics : this includes canonical quantum gravity, loop quantum gravity, twistor theory and string theory. A few authors can be cited. In 1983 J. B. Hartle and S. W. Hawking considered a hermitian Wheeler-DeWitt operator in the context of canonical quantum gravity [105] and, in 1990, J. Halliwell and J. B Hartle considered complex metrics, complex Einstein's equations and complex diffeomorphisms in the same context [106]. In 1986, R. Penrose and W. Rindler considered a complexified Minkowski space in the context of twistor theory [107]. In 1988, A. Ashtekar, T. Jacobson and L. Smolin considered complex Einstein's equations in the context of canonical quantum gravity and loop quantum gravity [108]. In 2005, J. Polshinski considered complex manifolds (and especially Kähler manifolds) in string theory [109]. Finally, in 2017, M. Atiyah et al. introduced again the concept of complexified Minkowski space to combine the ideas of string theory and twistor theory [110]. These examples illustrate that complex general relativity is a notorious subject in theoretical physics. In [Section 2] we derived quantum mechanics from general relativity by introducing complex matter fields, but it was required to impose a ad hoc reality condition on the energy momentum tensor of the complex matter fields, precisely because the gravitational field is real in general relativity. A rather artificial asymmetry between matter fields and the gravitational field was introduced to describe a real gravitational field interacting with complex matter fields. It is obvious that this asymmetry is not elegant but it is not obvious that it is coherent : it is indeed well known that gravity couples to all forms of matter and energy [27], while the quantum theory describes unstable particles with complex masses [85] [111] and complex energies [112]. Clearly, unstable particles constitute the norm, while the few stable particles (electrons, protons, photons...) are the exception : the unification of general relativity with quantum theory intrinsically suggests to extend general relativity to complex numbers. The major problem here remains the absence of experimental data. Nevertheless, another path, connected to experimental data, suggests the extension of general relativity to complex numbers. To describe circularly polarized electromagnetic and gravitational waves, it is required to introduce complex numbers since helicity eigenstates are complex vectors. We will now use the complex polarization of gravitational waves to derive the definition of a complex electromagnetic entity (used in optical sciences). Using the Lorenz gauge, we previously concluded that the polarization vectors of an electromagnetic wave have non vanishing components in the space $(x_1, x_2, x_3)$ only. Similarly, using the transverse traceless gauge in the gravitational case, we have $h_{\mu 0} = 0$ such that only the spatial components $h_{ij}$, where $i, j = 1, 2, 3$, are non vanishing. Using the transverse traceless gauge in the linear approximation, we have :

$$\bar{h} = h ; \quad \bar{h}_{\mu\nu} = h_{\mu\nu} ; \quad g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu} ; \quad G_{\mu\nu} = -\frac{1}{2} \Box h_{\mu\nu} \qquad (5.65)$$

In the linear approximation, we now use this gauge to consider the spatial part $(i, j = 1, 2, 3)$ of the Einstein-
Maxwell equations previously given in equation (4.34), with $\rho = 0$:

$$G_{ij} = -\frac{1}{2}\Box h_{ij} = 8\pi G \left( \frac{\eta^{\alpha\beta} F_{\alpha i} F_{j\beta} + \frac{1}{4}\eta_{ij} F^{\mu\alpha} F_{\mu\alpha}}{\mu_0} \right) = \frac{8\pi G}{\mu_0} \left( \frac{1}{2}(|\vec{E}|^2 + |\vec{B}|^2)\delta_{ij} - (\mathbf{E}_i \mathbf{E}_j + \mathbf{B}_i \mathbf{B}_j) \right) \quad (5.66)$$

where $\vec{E}$ is the electric field and $\vec{B}$ is the magnetic field. The term on the right-hand side is the negative of the Maxwell stress tensor [37]. We will now neglect the magnetic field because "the microscopic forces exerted by the electric field of the wave on matter are much larger than the forces produced by the magnetic field, and so the temporal evolution of the electric field is chosen as representative of the property called polarization" [91] (some details can be found in [43]). Neglecting the magnetic field, we rewrite equation (5.66) in the explicit form:

$$-\frac{\mu_0}{16\pi G} \Box h_{ij} = \frac{1}{2} |\vec{E}|^2 \delta_{ij} - (\mathbf{E}_i \mathbf{E}_j) \Leftrightarrow \frac{\mu_0}{16\pi G} \Box h_{ij} = -\frac{1}{2} |\vec{E}|^2 \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} + \begin{bmatrix} \mathbf{E}_1 \mathbf{E}_1 & \mathbf{E}_1 \mathbf{E}_2 & \mathbf{E}_1 \mathbf{E}_3 \\ \mathbf{E}_2 \mathbf{E}_1 & \mathbf{E}_2 \mathbf{E}_2 & \mathbf{E}_2 \mathbf{E}_3 \\ \mathbf{E}_3 \mathbf{E}_1 & \mathbf{E}_3 \mathbf{E}_2 & \mathbf{E}_3 \mathbf{E}_3 \end{bmatrix} \quad (5.67)$$

If general relativity is extended to complex numbers, equation (5.67) would be a particular case of a more general equation involving complex fields. By example, with $g_{ij} = \eta_{ij} + h_{ij}$ hermitian, equation (5.67) could become:

$$\frac{\mu_0}{16\pi G} \Box h_{ij} = -\frac{1}{2} |\vec{E}|^2 \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} + \begin{bmatrix} \mathbf{E}_1 \mathbf{E}_1^* & \mathbf{E}_1 \mathbf{E}_2^* & \mathbf{E}_1 \mathbf{E}_3^* \\ \mathbf{E}_2 \mathbf{E}_1^* & \mathbf{E}_2 \mathbf{E}_2^* & \mathbf{E}_2 \mathbf{E}_3^* \\ \mathbf{E}_3 \mathbf{E}_1^* & \mathbf{E}_3 \mathbf{E}_2^* & \mathbf{E}_3 \mathbf{E}_3^* \end{bmatrix} \quad (5.68)$$

As explained by R. P. Feynman: "To represent plane wave gravitons, we shall use the fields $h_{\mu\nu} = \epsilon_{\mu\nu}e^{ik_{\alpha}x^{\alpha}}$ [113] where $\epsilon_{\mu\nu}$ is the polarization tensor of the gravitational field [27]. Since $\epsilon_{\mu\nu}$ contains the information associated to the polarization of the gravitational field, it is clear from equation (5.64) that we cannot describe gravitational waves with helicity $\Lambda = 2$ with a real polarization tensor because helicity eigenstates are complex vectors: gravitational waves with helicity $\Lambda = 2$ requires a complex metric. Furthermore, since both sides of equation (5.68) must be of the same nature, the right hand side should contain the information associated to the polarization of the electromagnetic field. Then, if we generalize general relativity according to the hermitian equation (5.68), the right hand side should allow us to define a polarization tensor, or polarization matrix, for the electromagnetic field. Remarkably, it is indeed the case, without any reference to gravity, according to optical sciences: using the symbols $\langle\rangle$ to denote time averaging over the measurement time, "the 3D coherency matrix or 3D polarization matrix, which contains all the second-order measurable information about the state of polarization (including intensity) of an electromagnetic wave, is defined as the following 3 × 3 Hermitian matrix:

$$R = \begin{bmatrix} < \mathbf{E}_1 \mathbf{E}_1^* > & < \mathbf{E}_1 \mathbf{E}_2^* > & < \mathbf{E}_1 \mathbf{E}_3^* > \\ < \mathbf{E}_2 \mathbf{E}_1^* > & < \mathbf{E}_2 \mathbf{E}_2^* > & < \mathbf{E}_2 \mathbf{E}_3^* > \\ < \mathbf{E}_3 \mathbf{E}_1^* > & < \mathbf{E}_3 \mathbf{E}_2^* > & < \mathbf{E}_3 \mathbf{E}_3^* > \end{bmatrix} \quad (5.69)$$

As Fano pointed out in a seminal paper on the description of states in quantum mechanics by means of density matrices (Fano 1957), an appropriate basis for coherency matrices is a set of Hermitian trace-orthogonal operators, as is that constituted by the 3 × 3 identity matrix together with the eight Gell-Mann matrices" [91]. With equation (5.69), equation (5.68) can be rewritten: $\langle \frac{\mu_0}{16\pi G} \Box h_{ij} \rangle = -\frac{1}{2} \text{Tr}[R] + R$ where R is the 3D polarization matrix of the electromagnetic field and $\text{Tr}[R]$ its trace. In 1946, A. Einstein and E. G. Straus discussed the generalization of general relativity to complex numbers with a hermitian metric [114]: they hoped to identify the symmetric part of the metric with the gravitational field and the antisymmetric (imaginary) part with the electromagnetic field. The above analysis associates the complex numbers appearing in the metric with the circular polarization (or spin) of gravitational waves but not with the electromagnetic field. In [Section 4], we reconsidered the 5D Kaluza-Klein theory to identify a 5D gravitational field with the usual gravitational and electromagnetic fields. The linearized approximation of the 5D Kaluza-Klein theory is discussed in [Annex 8]. We reproduce here the conclusion given in [Annex 8]: "Counting simultaneously the 2 possible polarizations of electromagnetic waves and the 2 possible polarizations of gravitational waves can be done in the linearized approximation of the 5D Kaluza-Klein theory. The gravitational part is not affected while the electromagnetic part is automatically produced: the 5D gauge invariance produces the 4D gravitational gauge invariance and the electromagnetic gauge invariance while the 5D harmonic gauge produces the 4D gravitational harmonic gauge and the electromagnetic Lorenz gauge".

## 5.3 New results about the Dirac equation

M. Atiyah stated: "No one fully understands spinors. Their algebra is formally understood but their general significance is mysterious. In some sense they describe the 'square root' of geometry and, just as understanding