the square root of $-1$ took centuries, the same might be true of spinors" [115]. Textbooks on quantum field theory usually derive the Dirac equation by following P.A.M Dirac's original derivation and then show that the equation describes spin-$\frac{1}{2}$ particles [35][49][60][98][116][117][118]. We now reproduce the original derivation published in 1928 : we quote P.A.M Dirac but we use modern notations (including $\hat{p}_\mu \leftrightarrow i\hbar\frac{\partial}{\partial x^\mu}$). P.A.M Dirac explained : "The general interpretation of non-relativity quantum mechanics [...] is made possible by the wave equation [...] being linear in $\frac{\partial}{\partial t}$, so that the wave equation at any time determines the wave function at any later time. The wave equation of the relativity theory must also be linear [...] if the general interpretation is to be possible. [...] The symmetry between $\hat{p}_0$ and $\hat{p}_1, \hat{p}_2, \hat{p}_3$ required by relativity shows that, since the Hamiltonian we want is linear in $\hat{p}_0$, it must also be linear in $\hat{p}_1, \hat{p}_2, \hat{p}_3$. Our wave equation is therefore of the form

$$ (\hat{p}_0 + \alpha_1 \hat{p}_1 + \alpha_2 \hat{p}_2 + \alpha_3 \hat{p}_3 + \alpha_4 mc) \Psi = 0 \qquad (5.70) $$

[...] This agrees with

$$ (-\hat{p}_0^2 + \hat{p}_1^2 + \hat{p}_2^2 + \hat{p}_3^2 + m^2 c^2) \Psi = 0 \qquad (5.71) $$

if $\alpha_i \alpha_j + \alpha_j \alpha_i = \delta_{ij}$ **I**" [119]. With our notations $\delta_{ij}$ is the Kronecker symbol, **I** the identity matrix and $i, j = 1, 2, 3, 4$. Nowadays, the Dirac equation is rewritten in covariant form $(i\hbar\gamma^\mu\partial_\mu - mc)\Psi = 0$ where $\gamma^0 = \alpha^4$, $\gamma^i = \alpha^4\alpha^i$ and $i = 1, 2, 3$ such that $\gamma^\mu$ is a set of 4 anticommuting matrices : $[\gamma_\mu; \gamma_\nu]_+ = 2\eta_{\mu\nu}\mathbf{I}$. The solutions of the equations are spinors. Dirac's derivation appears unsatisfactory, as pointed out by S. Weinberg : "Dirac's analysis [...] would seem to rule out the existence of any particle of zero spin. [...] Today we know of a large number of spin zero particles." [120]. Indeed, Dirac's analysis considers that any relativistic quantum wave equation should be first order in the time derivative, and then first order in spatial derivatives too : it is clear that this reasoning can not be correct because spin-0 particles (like pions, kaons, and many atoms) are described by the Klein-Gordon equation which is second order in the time derivative. Similarly, massive spin-1 particles are described by Proca equations which are also second order in the time derivative (see equation (5.30)). Our aim is now to justify that spin-$\frac{1}{2}$ can in principle exist, and to show that the associated particles are described by spinors. This reasoning will allow us to to construct the Dirac equation. This approach is much longer than Dirac's approach but it clarifies the meaning of the Dirac equation. Furthermore, a link with general relativity will naturally appear.

The Lorentz group is well known but it is less known that the Lorentz group can be built from a more fundamental group : the reflection group. In any plan, the product of two reflections with respect to two strait lines containing the origin is a rotation, the rotation angle being twice the angle between the two strait lines. This statement is illustrated below :

<figure><img src="image_3.png" /><figcaption>Figure 1 : Illustration of a rotation ($P_0 \to P_2$) induced by two reflections ($P_0 \to P_1 \to P_2$) in a plane</figcaption></figure>

Starting from the position $P_0$, the flag reaches the position $P_2$ after two reflections with respects to the strait lines $\Pi_{01}$ and $\Pi_{12}$, the rotation angle being twice the angle between the two strait lines. To simplify the discussion, we will momentarily consider that 4-vectors are defined on a 4D Euclidean space : $\eta_{\mu\nu} \to I_{\mu\nu} = \text{diag}(1, 1, 1, 1)$ and $v_\alpha = I_{\alpha\beta}v^\beta = v^\alpha$. A 4-vector $v^\alpha$ becomes $-v^\alpha$ when a reflection with respect to an hyperplan orthogonal to it is applied. More generally, we choose a 4-vector $v^\alpha$ and write it as the sum of two 4-vectors : 1 parallel to $w^\alpha$, say $Cw^\alpha$ with $C$ a constant, and one orthogonal to $w^\alpha$, say $w_\perp^\alpha$, such that $w^\alpha w_{\alpha\perp} = 0$. Under a reflection with respect to an hyperplan orthogonal to $w^\alpha$, the vector $v^\alpha = Cw^\alpha + w_\perp^\alpha$ becomes $v'^\alpha = -Cw^\alpha + w_\perp^\alpha$. Evidently, the product of two reflexions oriented by the same vector is an identity operation ($v'^\alpha = v^\alpha$). Reflection matrices are called Househoulder matrices. Householder matrices $H'_{\mu\nu}$ are matrices of the form $H'_{\mu\nu}(w^\alpha) = I_{\mu\nu} - 2w_\mu w_\nu$ where $w^\alpha$ must be a unit vector. Nevertheless, it will be more convenient for us to work with their opposite $H_{\mu\nu}$ according to $H_{\mu\nu}(w^\alpha) = -H'_{\mu\nu}(w^\alpha) = -I_{\mu\nu} + 2w_\mu w_\nu$. Obvisouly,

$$ v'_{\mu} = H_{\mu\nu} v^{\nu} = (-I_{\mu\nu} + 2w_{\mu}w_{\nu})v^{\nu} = (-I_{\mu\nu} + 2w_{\mu}w_{\nu})(Cw^{\nu} + w_{\perp}^{\nu}) = Cw_{\mu} - w_{\mu\perp} \qquad (5.72) $$

Two projection matrices $P_{\parallel\mu\nu}(w^\alpha)$ and $P_{\perp\mu\nu}(w^\alpha)$ can be constructed from $H_{\mu\nu}(w^\alpha)$ according to

$$ H_{\mu\nu} = -I_{\mu\nu} + 2w_{\mu}w_{\nu} ; \quad P_{\parallel\mu\nu} = \frac{1}{2}(I_{\mu\nu} + H_{\mu\nu}) = w_{\mu}w_{\nu} ; \quad P_{\perp\mu\nu} = \frac{1}{2}(I_{\mu\nu} - H_{\mu\nu}) = I_{\mu\nu} - w_{\mu}w_{\nu} \qquad (5.73) $$
such that $P_{\parallel\mu\nu} + P_{\perp\mu\nu} = I_{\mu\nu}$ and

$$H_{\nu}^{\mu} H_{\sigma}^{\nu} = I_{\sigma}^{\mu} ; \quad H_{\sigma}^{\mu} P_{\parallel\mu\nu} = P_{\parallel\sigma\nu} ; \quad H_{\sigma}^{\mu} P_{\perp\mu\nu} = -P_{\perp\sigma\nu} \qquad (5.74)$$

$$P_{\parallel\mu\nu}v^{\nu} = (w_{\mu}w_{\nu})(Cw^{\nu} + w_{\perp}^{\nu}) = Cw_{\mu} ; \quad P_{\perp\mu\nu}v^{\nu} = (I_{\mu\nu} - w_{\mu}w_{\nu})(Cw^{\nu} + w_{\perp}^{\nu}) = w_{\perp\mu} \qquad (5.75)$$

We can now make an observation which we did not find in literature. Einstein's equations being

$$G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R = \frac{8\pi G}{c^4}T_{\mu\nu} = \frac{8\pi G}{c^4}\rho u_{\mu}u_{\nu} \qquad (5.76)$$

taking the trace gives $-2R = \frac{8\pi G}{c^2}\rho = \frac{8\pi G}{c^4}T$ where $T = g^{\mu\nu}T_{\mu\nu}$, and $R_{\mu\nu} = \frac{8\pi G}{c^4}(T_{\mu\nu} - \frac{1}{2}g_{\mu\nu}T)$. Then:

$$\frac{R_{\mu\nu}}{R} = g_{\mu\nu} - 2u_{\mu}u_{\nu} = -H_{\mu\nu}(u^{\alpha}) ; \quad \frac{T_{\mu\nu}}{T} = u_{\mu}u_{\nu} = P_{\parallel\mu\nu}(u^{\alpha}) ; \quad \frac{R_{\mu\nu}}{R} + \frac{T_{\mu\nu}}{T} = g_{\mu\nu} - u_{\mu}u_{\nu} = P_{\perp\mu\nu}(u^{\alpha}) \qquad (5.77)$$

When Einstein's equations are obeyed $\frac{R_{\mu\nu}}{R}$ is a Householder matrix (where the identity matrix $I_{\mu\nu}$ is naturally replaced by $g_{\mu\nu}$ because the space is not Euclidean).

We now illustrate the link between reflections and rotations, in a 4D Euclidean space. We first introduce two unit vectors $x^{\alpha}$ and $y^{\alpha}$. All orientations being equivalent, we choose $x^{\alpha}$ and $y^{\alpha}$ orthogonal to the first and fourth axes: $x^{\alpha} = (0, x_1, x_2, 0)$, $y^{\alpha} = (0, y_1, y_2, 0)$. The matrices $H_{\mu\nu}(x^{\alpha})$ and $H_{\mu\nu}(y^{\alpha})$ have similar forms with $x$ replaced by $y$ and their product is (the computation is given in [Annex 9]):

$$H_{\mu\nu}(x^{\alpha}) = \begin{bmatrix} -1 & 0 & 0 & 0 \\ 0 & -1+2x_1^2 & 2x_1x_2 & 0 \\ 0 & 2x_1x_2 & -1+2x_2^2 & 0 \\ 0 & 0 & 0 & -1 \end{bmatrix} ; \quad H_{\mu\beta}(x^{\alpha})H_{\nu}^{\beta}(y^{\alpha}) = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & \cos 2\theta & -\sin 2\theta & 0 \\ 0 & \sin 2\theta & \cos 2\theta & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \qquad (5.78)$$

where $\cos\theta = x_1y_1 + x_2y_2$ and $\sin\theta = \sqrt{1 - \cos^2\theta} = (x_2y_1 - x_1y_2)$. Equation (5.78) shows that the product of 2 reflection matrices is a rotation matrix, the rotation occurring in the plan spanned by the vectors orienting the reflections, while the rotation angle is twice the angle between the vectors: it was already illustrated on [Figure 1]. Acting on a vector $v^{\alpha} = (v_0, v_1, v_2, v_3)$, the rotation produces the vector $v'^{\alpha}$:

$$v'^{\alpha} = \begin{bmatrix} v_0' \\ v_1' \\ v_2' \\ v_3' \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & \cos 2\theta & -\sin 2\theta & 0 \\ 0 & \sin 2\theta & \cos 2\theta & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} v_0 \\ v_1 \\ v_2 \\ v_3 \end{bmatrix} = \begin{bmatrix} v_0 \\ v_1 \cos 2\theta - v_2 \sin 2\theta \\ v_2 \cos 2\theta + v_1 \sin 2\theta \\ v_3 \end{bmatrix} \qquad (5.79)$$

The transformation can be summarized by:

$$v_0' = v_0 ; \quad v_1' = \cos 2\theta \ v_1 - \sin 2\theta \ v_2 ; \quad v_2' = \cos 2\theta \ v_2 + \sin 2\theta \ v_1 ; \quad v_3' = v_3 \qquad (5.80)$$

To introduce the spinor representation of the above transformation, we must introduce a set of traceless matrices obeying anticommutation relations. In 4 and 5 dimensions, the minimal size of the matrices is $4 \times 4$. To construct the Dirac equation, it will be useful to work in 5 Euclidean dimensions. We choose our matrices $\alpha^M = (\alpha^\mu, \alpha^4) = (\alpha^0, \alpha^j, \alpha^4)$ where $j = 1, 2, 3$ in the following form:

$$\alpha^0 = \begin{pmatrix} 0_2 & I_2 \\ -I_2 & 0_2 \end{pmatrix} ; \quad \alpha^j = \begin{pmatrix} -i\sigma_j & 0_2 \\ 0_2 & i\sigma_j \end{pmatrix} ; \quad \alpha^4 = \begin{pmatrix} 0_2 & -iI_2 \\ -iI_2 & 0_2 \end{pmatrix} \qquad (5.81)$$

where

$$0_2 = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} ; \quad I_2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} ; \quad \sigma_1 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} ; \quad \sigma_2 = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} ; \quad \sigma_3 = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \qquad (5.82)$$

The above matrices $\alpha^M$ obey the anticommutation relations $[\alpha_M; \alpha_N]_+ = -2\mathbf{I}\delta_{MN}$ where $\mathbf{I}$ is the 4D identity matrix and $\delta_{MN}$ is a Kronecker symbol. The spinor representation of a 5-vector $v^M$ is given by $\alpha^M v_M = \alpha^0 v_0 + \alpha^1 v_1 + \alpha^2 v_2 + \alpha^3 v_3 + \alpha^4 v_4$. Any coordinate of the 5-vector can be extracted from its spinor representation according to the inversion formula $v^\beta = \frac{1}{4} \mathrm{Tr}(\gamma^\beta \alpha^M v_M)$ [121]: we can then use the spinor representation to compute any transformation of a 5-vector, and use the inversion formula to get each component of the transformed 5-vector. Using $\alpha_M \alpha_N + \alpha_N \alpha_M = -2\mathbf{I}\delta_{MN}$ with $M = 1$ and $N = 2$ gives

$$\alpha_1 \alpha_2 + \alpha_2 \alpha_1 = 0 \Rightarrow (\alpha_1)^2 \alpha_2 + \alpha_1 \alpha_2 \alpha_1 \Rightarrow -\alpha_2 + \alpha_1 \alpha_2 \alpha_1 = 0 \qquad (5.83)$$

Equation (5.83) holds for any value of $M$ and $N$ provided that $M \neq N$: the equation shows that it is trivial to represent a reflection oriented by any vector with spinors. If we consider a reflection oriented by a vector aligned with the direction $x_1$, its representation is indeed:

$$\alpha^M v_M' = \alpha^1(\alpha^M v_M)\alpha^1 \rightarrow \alpha^M v_M' = \alpha^0 v_0 - \alpha^1 v_1 + \alpha^2 v_2 + \alpha^3 v_3 + \alpha^4 v_4 \qquad (5.84)$$
The transformation is equivalent to the change $v_1 \to -v_1$. If an arbitrary direction is considered, $\alpha^1$ should be replaced by $\alpha^N w_N$ in the left hand side of equation (5.84), where $w_N$ is a unit vector orienting the reflection. The transformation would take the form: $\alpha^M v'_M = \alpha^N w_N (\alpha^M v_M) \gamma^P w_P$. The matrix $\alpha^N w_N$ is then the spinor representation of $H'_N(w^P)$: writing $\mathbf{H}(w^P) = \alpha^N w_N$ it is straightforward to check that

$$ \mathbf{H}(w^P)\alpha^M(Cw_M + w_{M\perp})\mathbf{H}(w^P) = \alpha^M(-Cw_M + w_{M\perp}) \quad (5.85) $$

Two successives reflections producing a rotation, we can easily represent a rotation. Defining again two unit vectors $x^M$ and $y^N$, the matrices representing a reflection associated to each vector will be $\mathbf{H}(x^M) = \alpha^N x_N$ and $\mathbf{H}(y^N) = \alpha^N y_N$ while the product of the two matrices will represent a rotation in the plan spanned by the two vectors. To illustrate this point, we assume that $x^M = (0, x_1, x_2, 0, 0)$ and $y^M = (0, y_1, y_2, 0, 0)$. The product $\mathbf{H}(x^M)\mathbf{H}(y^M)$ is given by $\mathbf{H}(x^M)\mathbf{H}(y^M) = (\alpha^1 x_1 + \alpha^2 x_2)(\alpha^1 y_1 + \alpha^2 y_2) = \mathbf{I} \cos\theta - \sin\theta \alpha^1 \alpha^2$ where $\alpha^1 \alpha^2 = \begin{pmatrix} -\mathbf{i}\sigma_3 & 0 \\ 0 & -\mathbf{i}\sigma_3 \end{pmatrix}$. Similarly, $\mathbf{H}(y^M)\mathbf{H}(x^M) = \mathbf{I} \cos\theta + \sin\theta \alpha^1 \alpha^2$. Then $\alpha^M v'_M = (\mathbf{H}(y^N)\mathbf{H}(x^N))(\alpha^M v_M)(\mathbf{H}(x^N)\mathbf{H}(y^N))$ gives:

$$ \gamma^\alpha v'_\alpha = \begin{pmatrix} e^{-i\theta} & 0 & 0 & 0 \\ 0 & e^{i\theta} & 0 & 0 \\ 0 & 0 & e^{-i\theta} & 0 \\ 0 & 0 & 0 & e^{i\theta} \end{pmatrix} \begin{pmatrix} -\mathbf{i}v_3 & -\mathbf{i}v_1 - v_2 & v_0 - \mathbf{i}v_4 & 0 \\ -\mathbf{i}v_1 + v_2 & \mathbf{i}v_3 & 0 & v_0 - \mathbf{i}v_4 \\ -v_0 - \mathbf{i}v_4 & 0 & \mathbf{i}v_3 & \mathbf{i}v_1 + 2 \\ 0 & -v_0 - \mathbf{i}v_4 & \mathbf{i}v_1 - v_2 & -\mathbf{i}v_3 \end{pmatrix} \begin{pmatrix} e^{i\theta} & 0 & 0 & 0 \\ 0 & e^{-i\theta} & 0 & 0 \\ 0 & 0 & e^{i\theta} & 0 \\ 0 & 0 & 0 & e^{-i\theta} \end{pmatrix} \quad (5.86) $$

$$ \begin{pmatrix} -\mathbf{i}v_3 & (-\mathbf{i}v_1 - v_2)e^{-2i\theta} & v_0 - \mathbf{i}v_4 & 0 \\ (-\mathbf{i}v_1 + v_2)e^{2i\theta} & \mathbf{i}v_3 & 0 & v_0 - \mathbf{i}v_4 \\ -v_0 - \mathbf{i}v_4 & 0 & \mathbf{i}v_3 & (\mathbf{i}v_1 + v_2)e^{-2i\theta} \\ 0 & -v_0 - \mathbf{i}v_4 & (\mathbf{i}v_1 - v_2)e^{2i\theta} & -\mathbf{i}v_3 \end{pmatrix} \quad (5.87) $$

We recognize the transformation of equation (5.79):

$$ v'_0 = v_0; \quad v'_1 = \cos 2\theta v_1 - \sin 2\theta v_2; \quad v'_2 = \cos 2\theta v_2 + \sin 2\theta v_1; \quad v'_3 = v_3 \quad (5.88) $$

If we are only concerned by rotations in the space $(x_1, x_2, x_3)$ the generators of rotations are conveniently chosen as $\mathbf{S}_k = \frac{i\hbar}{4}(\alpha_i \alpha_j - \alpha_j \alpha_i) = \frac{i\hbar}{2}\alpha_i \alpha_j$ where $i, j, k = 1, 2, 3$. A rotation of angle $\theta_3$ in the plane $(x_1, x_2)$ is written $\alpha^M v'_M = e^{-i\frac{S_3}{\hbar}\theta_3}\alpha^M v_M e^{i\frac{S_3}{\hbar}\theta_3}$: this expression reproduces equations (5.86) and (5.87) with $\theta$ replaced by $\frac{1}{2}\theta_3$. An important topological difference between the tensorial and the spinorial representations is that two elements of the spinorial representations correspond to the same tensorial element. In our exemple, the transformation of equation (5.79) can be represented by $\alpha^M v'_M = e^{-i\frac{S_3}{\hbar}\theta_3}\alpha^M v_M e^{i\frac{S_3}{\hbar}\theta_3}$ or $\gamma^M v'_M = (-e^{-i\frac{S_3}{\hbar}\theta_3})\alpha^M v_M (-e^{i\frac{S_3}{\hbar}\theta_3})$. The tensorial representation is 'single valued' while the spinorial representation is 'double valued'. The rotation matrices, or spin matrices $\mathbf{S}_i$, obey respectively the commutations and anti-commutations rules given in equation (5.89). Their squared norm is $\frac{3}{4}\hbar^2$ times the 4D identity matrix $\mathbf{I}$.

$$ \sum_{i=1}^{3} (\mathbf{S}_i)^2 = \frac{3}{4} \hbar^2 \mathbf{I} \quad ; \quad [\mathbf{S}_i, \mathbf{S}_j]_- = \mathbf{i} \hbar \epsilon_{ijk} \mathbf{S}_k \quad ; \quad [\mathbf{S}_i, \mathbf{S}_j]_+ = \frac{\hbar^2}{2} \mathbf{I} \delta_{ij} \quad (5.89) $$

To account for the signature of the metric in Minkowski space, we simply redefine $\alpha^0 \to -\mathbf{i}\alpha^0$ such that $(\alpha^0)^2 = \mathbf{I}$, while $\alpha^1, \alpha^2, \alpha^3, \alpha^4$ remain unchanged. With this change $\mathbf{H}(w^\alpha) = \gamma^\beta w_\beta$ is now identified with $H_{\mu\nu}(x^\alpha)$ instead of $H'_{\mu\nu}(x^\alpha)$ because now $\mathbf{H}(w^\alpha)\gamma^\beta(Cw_\beta + w_{\beta\perp})\mathbf{H}(w^\alpha) = \gamma^\beta(Cw_\beta - w_{\beta\perp})$: in Euclidian space we had $\mathbf{H}^2 = -\mathbf{I}$ where $\mathbf{I}$ is the 4D identity matrix but we have now $\mathbf{H}^2 = \mathbf{I}$. We can define the spinorial representations of the projectors $P_\parallel$ and $P_\perp$ according to $P_\parallel = \frac{1}{2}(\mathbf{I} + \mathbf{H})$ and $P_\perp = \frac{1}{2}(\mathbf{I} - \mathbf{H})$. The spinorial representation of equation (5.74) takes the following form:

$$ \mathbf{H}^2(w^\alpha) = \mathbf{I} w^\alpha w_\alpha = \mathbf{I} \quad ; \quad \mathbf{H}(w^\alpha)\mathbf{P}_\parallel(w^\alpha) = \mathbf{P}_\parallel(w^\alpha) \quad ; \quad \mathbf{H}\mathbf{P}_\perp(w^\alpha) = -\mathbf{P}_\perp(w^\alpha) \quad (5.90) $$

We end here our introduction of spinors.

Before introducing spin-$\frac{1}{2}$ particles, we consider again a spin-1 particle travelling along the $x_3$ direction (in the massless case it is a photon, in the massive case it is a Proca particle). We label helicity eigenstates $|\Lambda\rangle$ according to:

$$ |-1\rangle = \epsilon_{Li} \psi(x^\alpha) = \begin{pmatrix} 1 \\ -\mathbf{i} \\ 0 \end{pmatrix} \phi(x^\alpha); \quad |0\rangle = \epsilon_{3i} \psi(x^\alpha) = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} \psi(x^\alpha); \quad |+1\rangle = \epsilon_{Ri} \phi(x^\alpha) = \begin{pmatrix} 1 \\ \mathbf{i} \\ 0 \end{pmatrix} \phi(x^\alpha) \quad (5.91) $$

where $\psi(x^\alpha) = N e^{-i(\omega t - k_3 x^3)}$ is a normalized plane wave while $\epsilon_{Li}, \epsilon_{3i}, \epsilon_{Ri}$ with $i = 1, 2, 3$ form a complete basis of the 3D space (any vector is a linear combination of the 3 basis vectors). The spin matrices $\hat{\mathbf{S}}_1, \hat{\mathbf{S}}_2, \hat{\mathbf{S}}_3$
acting on the above states were defined in equation (5.50). Then $\hat{\mathbf{S}}_3|\Lambda\rangle = \Lambda|\Lambda\rangle$. We define the ladder operators $\hat{\mathbf{S}}_{\pm} = \hat{\mathbf{S}}_1 \pm i\hat{\mathbf{S}}_2$ such that :

$$ \begin{aligned} \hat{\mathbf{S}}_+ | -1 \rangle &= a_1 \hbar |0 \rangle ; \quad \hat{\mathbf{S}}_+^2 | -1 \rangle = a_2 \hbar^2 | +1 \rangle ; \quad \hat{\mathbf{S}}_+^3 | -1 \rangle = 0 \quad \text{since } \hat{\mathbf{S}}_+^3 = 0 \\ \hat{\mathbf{S}}_- | +1 \rangle &= a_{-1} \hbar |0 \rangle ; \quad \hat{\mathbf{S}}_-^2 | +1 \rangle = a_{-2} \hbar^2 | -1 \rangle ; \quad \hat{\mathbf{S}}_-^3 | +1 \rangle = 0 \quad \text{since } \hat{\mathbf{S}}_-^3 = 0 \end{aligned} \quad (5.92) $$

$a_{-2}$, $a_{-1}$, $a_1$ and $a_{-2}$ are constants. It is well known from the solutions of the Schrodinger equation in quantum mechanics (i.e. from the algebraic properties of the Laplace operator) that the eigenvalues $m\hbar$ of the orbital angular momentum operator along a given direction, say the direction $x_3$ associated to the component $\hat{L}_3 = i\hbar(x_2\frac{\partial}{\partial x_1} - x_1\frac{\partial}{\partial x_2})$, must be integers or half integers unit of $\hbar$ (i.e. $m$ must be an integer of half integer) [122]: the eigenvalues of the squared orbital angular momentum operator $|\vec{L}|^2$ are then of the form $l(l+1)\hbar^2$ where $m = -l, -l+1, ..., l-1, l$. The two constraints on the eigenvalues directly follow from the commutation relation $[\hat{L}_i; \hat{L}_j]_- = i\hbar\epsilon_{ijk}\hat{L}_k$: the explanation can be found in [Annex 10]. In equation (5.49), we introduced the spin operators $\hat{\mathbf{S}}_i$ of spin-1 particles : like orbital angular momentum operators, the spin operators $\hat{\mathbf{S}}_i$ obey the relation $[\hat{\mathbf{S}}_i; \hat{\mathbf{S}}_j]_- = i\hbar\epsilon_{ijk}\hat{\mathbf{S}}_k$. Consequently, the eigenvalues of the orbital angular momentum operators and the eigenvalues of the spin operator must obey the same algebraic constraints. In the case of the spin operator we write the eigenvalues of $|\vec{\mathbf{S}}|^2$ in the form $s(s+1)\hbar^2$. There is nevertheless an important difference. We represented a spin-1 particle travelling along the $x_3$ direction by combining a vector with a scalar function. By example the state with helicity $\Lambda = 1$ was written $| + 1 \rangle = \epsilon_{Ri}\psi(x^\alpha)$ in equation (5.91). In this state, the orbital angular momentum operator acts on the function $\psi(x^\alpha)$ while the spin operator acts on the vector $\epsilon_{Ri}$. The eigenvalues of $\hat{\mathbf{S}}_3$ are $m_s\hbar$ where $m = -1, 0, 1$, the 3 eigenvalues being respectively associated to the eigenvectors $\epsilon_{Li}, \epsilon_{3i}, \epsilon_{Ri}$. The 3 eigenvectors form a complete basis. In this complete basis, each eigenvector of $\hat{\mathbf{S}}_3$ is also an eigenvector of the squared spin operator $|\vec{\mathbf{S}}|^2 = \hat{\mathbf{S}}_1^2 + \hat{\mathbf{S}}_2^2 + \hat{\mathbf{S}}_3^2$ (see [Annex 10]). $|\vec{\mathbf{S}}|^2$ is proportional to the identity matrix and the proportionality constant must be of the form $s(s+1)$ where $s$ is the maximum value of $m_s$: then $s(s+1) = 2$. It can indeed be checked by direct computation that $\hat{\mathbf{S}}_1^2 + \hat{\mathbf{S}}_2^2 + \hat{\mathbf{S}}_3^2 = 2I_3$ where $I_3$ is the 3D identity matrix. When a given wave equation is solved to determine a wave function, the eigenvalues $m$ of the orbital angular momentum operators must be restricted to integer values, otherwise the wave function would not be single valued : consequently $l$ is an integer too. There is nevertheless no obvious reason the reject the possibility of particles with half integer eigenvalues of the spin operators. Indeed, in 1947, E. Wigner and V. Bargmann used group theory to classify the possible wave equations and they identified half integer eigenvalues with double-valued representations. We quote them with a change in notations: "Let $\Psi_a$ and $\Psi_b$ be the wave functions of the same state in two Lorentz frames a and b respectively. Then $\Psi_a = U(L)\Psi_b$ where $U(L)$ is a linear unitary operator which depends on the Lorentz transformation $L$ leading from a to b. By a proper normalization, $U$ is determined by $L$ up to a factor $\pm 1$. [...] . Moreover, the operators $U(L)$ form a single or double-valued representation of the inhomogeneous Lorentz group. [...] The representation $U(L)$ is single or double valued according to whether $m$ is integral of half integral" [101]. We then look for a spin-$\frac{1}{2}$ representation of the Lorentz group, which will allow us to describe particles with helicity $\Lambda = \pm\frac{\hbar}{2}$. Precisely, we look for a set of matrices such that $[\hat{\mathbf{S}}_i, \hat{\mathbf{S}}_j]_- = i\hbar\epsilon_{ijk}\hat{\mathbf{S}}_k$ and $\hat{\mathbf{S}}_1^2 + \hat{\mathbf{S}}_2^2 + \hat{\mathbf{S}}_3^2 = s(s+1)\hbar^2\mathbf{I} = \frac{1}{2}(\frac{1}{2}+1)\hbar^2\mathbf{I} = 3/4\hbar^2\mathbf{I}$ where $\mathbf{I}$ is the identity matrix. We recognize two of the three equalities given in equation (5.89). Conventionally assuming that the motion is aligned with the $x_3$ direction, $\hat{\mathbf{S}}_3$ will act on helicity eigenstates with two possible eigenvalues :

$$ \hat{\mathbf{S}}_3 | \pm \frac{1}{2} \rangle = \pm \frac{1}{2} \hbar | \pm \frac{1}{2} \rangle ; \quad \hat{\mathbf{S}}_+ | - \frac{1}{2} \rangle = a_+ \hbar | + \frac{1}{2} \rangle ; \quad \hat{\mathbf{S}}_- | + \frac{1}{2} \rangle = a_- \hbar | - \frac{1}{2} \rangle \quad (5.93) $$

where $a_+$ and $a_-$ are constants, $\hat{\mathbf{S}}_{\pm} = \hat{\mathbf{S}}_1 \pm i\hat{\mathbf{S}}_2$ and the conditions $0 = \hat{\mathbf{S}}_+^2 - \frac{1}{2} = \hat{\mathbf{S}}_-^2 + \frac{1}{2}$ must hold to restrict the possible values of $m_s$ to $\pm\frac{1}{2}$. If the helicity eigenstates constitute a complete basis, it follows that

$$ 0 = \hat{\mathbf{S}}_{\pm}^2 = \hat{\mathbf{S}}_1^2 - \hat{\mathbf{S}}_2^2 \pm i(\hat{\mathbf{S}}_1\hat{\mathbf{S}}_2 + \hat{\mathbf{S}}_2\hat{\mathbf{S}}_1) \quad (5.94) $$

Since $\hat{\mathbf{S}}_3^2$ acting on any state gives the eigenvalue $\frac{\hbar^2}{4}$, $\hat{\mathbf{S}}_3^2$ must be proportional to the identity : $\hat{\mathbf{S}}_3^2 = \frac{\hbar^2}{4}\mathbf{I}$. By symmetry, we deduce $\hat{\mathbf{S}}_1^2 = \hat{\mathbf{S}}_2^2 = \frac{\hbar^2}{4}\mathbf{I}$ such that, according to equation (5.94) $[\hat{\mathbf{S}}_1, \hat{\mathbf{S}}_2]_+ = 0$. By symmetry, we deduce that $[\hat{\mathbf{S}}_i, \hat{\mathbf{S}}_j]_+ = \frac{\hbar^2}{2}\mathbf{I}\delta_{ij}$ where $i, j = 1, 2, 3$. We recognize the last equality in equation (5.89) : there is then a deep link between spin-$\frac{1}{2}$ representations and anticommutative algebras. We can now identify the spin-$\frac{1}{2}$ representation with the double valued spinorial representation. We now look for an equation of motion for a given spinor field $\Psi(x^\alpha)$. By analogy with the cases of spin-0 and spin-1 particles, we would like an equation of the form $i\hbar\partial_\tau\Psi = mc^2\Psi = \hat{H}\Psi$ where $\hat{H}$ will be the Hamiltonian. The matrices $\hat{\mathbf{S}}_i$ acts on the spinor representation of a 5-vector $v_M$ given by $\alpha^M v_M$. Each column of the matrix $(\alpha^M v_M)$ will be identified a polarization vector which depends on $p_\alpha$. Furthermore, in rectangular coordinates, each polarization vector will be associated with a plane wave of the form $\psi = e^{-i\frac{S(x^\mu, \tau)}{\hbar}}$ where $S(x^\mu, \tau) = p_\mu x^\mu - mc^2\tau$. Since $p_\alpha$ must transform like a 4-vector under
a Lorentz transformation, it must transform like $v_\alpha$: we directly identify $(\alpha^M v_M)$ with $N(\alpha^M p_M)$ where $N$ is a normalization constant and $P_M = (p_\mu, mc)$. The spin matrices can then act on a matrix of the form $N(\alpha^M p_M)\psi$. Since $p^M p_M = p^\mu p_\mu - m^2 c^2 = 0$, we know that $(\alpha^M p_M)^2 = (\alpha^\mu p_\mu)^2 - m^2 c^2 = 0$. This observation if sufficient to produce the desired equation since $(\alpha^\mu p_\mu + \alpha_4 mc)N(\alpha^\nu p_\nu + \alpha_4 mc)\psi = 0$ implies:

$$
\begin{align}
\alpha^\mu p_\mu \mathbf{N}(\alpha^\nu p_\nu + \alpha_4 mc)\psi &= -\alpha_4 mc \mathbf{N}(\alpha^\nu p_\nu + \alpha_4 mc)\psi \nonumber \\
\alpha_4[\alpha^\mu p_\mu \mathbf{N}(\alpha^\nu p_\nu + \alpha_4 mc)]\psi &\alpha_4 = \alpha_4[-\alpha_4 mc \mathbf{N}(\alpha^\nu p_\nu + \alpha_4 mc)]\psi \alpha_4 \nonumber \\
\alpha_4 \alpha^\mu p_\mu \mathbf{N}(\alpha^\nu \alpha_4 p_\nu - mc)\psi &= mc \mathbf{N}(\alpha^\nu \alpha_4 p_\nu - mc)\psi \tag{5.95} \\
\alpha_4 \alpha^\mu p_\mu \mathbf{N}(-\alpha_4 \alpha^\nu p_\nu - mc)\psi &= mc(-\alpha_4 \alpha^\nu p_\nu - mc)\psi \nonumber \\
\alpha_4 \alpha^\mu p_\mu \mathbf{N}(\alpha_4 \alpha^\nu p_\nu + mc)\psi &= mc \mathbf{N}(\alpha_4 \alpha^\nu p_\nu + mc)\psi \nonumber \\
\gamma^\mu p_\mu \mathbf{N}(\gamma^\nu p_\nu + mc)\psi &= mc \mathbf{N}(\gamma^\nu p_\nu + mc)\psi \nonumber
\end{align}
$$

Where we used the notation $\alpha_4 \alpha^\mu = \gamma^\mu$. The gamma matrices are Dirac's matrices in Dirac's representation:

$$ \gamma^0 = \begin{pmatrix} I_2 & 0_2 \\ 0_2 & -I_2 \end{pmatrix} ; \quad \gamma^j = \begin{pmatrix} 0_2 & \sigma_j \\ -\sigma_j & 0_2 \end{pmatrix} ; \quad \sigma_1 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} ; \quad \sigma_2 = \begin{pmatrix} 0 & -\mathbf{i} \\ \mathbf{i} & 0 \end{pmatrix} ; \quad \sigma_3 = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \quad (5.96) $$

Calling $\Psi$ any column of the matrix $N(\gamma^\nu p_\nu + mc)\psi$ and using the fact that $p_\mu \psi = \mathbf{i}\hbar\partial_\mu \psi$, we get the Dirac equation

$$ \mathbf{i}\hbar\gamma^\mu\partial_\mu\Psi = mc\Psi \quad (5.97) $$

We can identify the Hamiltonian operator $\hat{H}$ with $\mathbf{i}\hbar c\gamma^\mu\partial_\mu$ and also write $\mathbf{i}\hbar\partial_\tau\Psi = \hat{H}\Psi = mc^2\Psi$. We previously introduced projection operators, defined with unit vectors. In a 5D Minkowski space $p^M$ is not a unit vector because it vanishes. It nevertheless deserves to be noticed that $N(\gamma^\nu p_\nu + mc)$ is proportional to

$$ \mathbf{P}_{\parallel}(\frac{p_{\alpha}}{m}) = \frac{1}{2}(\mathbf{I} + \mathbf{H}(\frac{p_{\alpha}}{m})) = \frac{1}{2}(\mathbf{I} + \gamma^{\alpha} \frac{p_{\alpha}}{m}) \quad (5.98) $$

which is the spinorial representation of $\frac{T_{\mu\nu}}{\hat{\Gamma}}$, as we have previously seen. This geometrical fact is exhibited in the expression of scattering cross sections. To illustrate this point, we reproduce lectures notes on particle physics [94]. The process considered in the lecture notes is called 'Electron-Muon scattering' and described by the following picture:

<figure><img src="image_4.png" /><figcaption>Figure 2: Illustration of Electron-Muon scattering</figcaption></figure>

On the picture above, the wave function of the incoming electron A has the form $u_A[p_A]\psi_A = u_A[p_A]e^{-i\frac{p_A\mu x^\mu}{\hbar}}$ where $u_A[p_A]$ is the polarization vector. Similarly, the wave function of the outgoing electron has the form $u_C[p_C]\psi_C = u_C[p_C]e^{-i\frac{p_C\mu x^\mu}{\hbar}}$. We recognize the eikonal approximation. We observe that each particle is represented by a mode of its wave equation (as previously discussed in [Section 5.2]). The scattering amplitude takes the form $|\overline{M}|^2 \propto L_{\text{electron}}^{\mu\nu} L_{\mu\nu}^{\text{muon}}$ where [94]:

$$ L_{\text{electron}}^{\mu\nu} = 4[p_A^\mu p_C^\nu + p_A^\nu p_C^\mu - \eta^{\mu\nu}(p_A^\sigma p_{C\sigma} - m^2c^2)] |\psi_A \psi_C|^2 \quad (5.99) $$

A similar expression holds for $L_{\mu\nu}^{\text{muon}}$. Fixing $p_A^\mu = p_C^\mu = p^\mu$ such that $\psi_A = \psi_C = \psi$ and using the equation $p_\mu \psi = \mathbf{i}\hbar\partial_\mu \psi$ in equation (5.99), we get:

$$ \frac{1}{8m^2|\psi|^2} L_{\text{electron}}^{\mu\nu} = \frac{\hbar^2}{2m^2} (\partial^\mu \psi \partial^\nu \psi^* + \partial^\nu \psi \partial^\mu \psi^*) - \eta^{\mu\nu} \left( \frac{\hbar^2}{2m} \eta^{\alpha\beta} (\partial_\alpha \psi \partial_\beta \psi^* + \partial_\beta \psi \partial_\alpha \psi^*) - c^2 |\psi|^2 \right) \quad (5.100) $$

We recognize the expression of the energy momentum of the scalar field given in equation (3.5) (the Lagrangian density $L_m$ appearing in equation (3.5) was defined in equation (3.1)).

To conclude this section, we provided two derivations of the geodesic equation from quantum physics. One of them introduced the proper time in quantum physics, enlightening the use of the 'Schwinger-DeWitt' technique in quantum gravity. Discussing the concepts of particles and polarization, we then suggested to extend general relativity to complex numbers. Finally, new results about the Dirac equation were established.
# 6 New results in canonical quantum gravity

R. Sorkin stated in 2020: "Loop Quantum Gravity is a species of canonical quantum gravity and I have strong reasons based on my own personal history to think that canonical quantum gravity is wrong. Canonical gravity will never be able to solve the difficulties which go by the name of the problem of time" [57]. Although the second quantization of gravity is not the subject of this paper, this section provides new results about the problem of time in canonical quantum gravity and loop quantum gravity.

We first introduce canonical quantum gravity and its link with loop quantum gravity. To quote existing literature, we temporarily adopt the (+, +, +, −) sign convention for the metric. Latin indices i, j will take the value 1, 2, 3 but not 0, x⁰ corresponding to the time coordinate. The essence of canonical quantization is to assume that the fields and their conjugate momenta obey commutation or anticommutation relations. The canonical formulation of classical gravity is known as ADM formulation (R. Arnowitt, S. Deser, C. W. Misner) and requires a foliation of space-time. The foliation of space-time is made according to: $ds^2 = (-\alpha^2 + \beta_i\beta^i)(dx^0)^2 + 2\beta_i dx^i dx^0 + \gamma_{ij}dx^i dx^j$. In a seminal paper published in 1967, B. DeWitt suggested to quantize this formulation of general relativity [123]. We now quote his paper. The metric and its inverse are used in the following form:

$$g_{\mu\nu} = \begin{bmatrix} \gamma_{ij} & \beta_j \\ \beta_i & -\alpha^2 + \beta_k\beta^k \end{bmatrix} \quad ; \quad g^{\mu\nu} = \begin{bmatrix} \gamma^{ij} & -\alpha^{-2}\beta^i\beta^j & \alpha^{-2}\beta^j \\ \alpha^{-2}\beta^i & -\alpha^{-2} & \alpha^{-2} \end{bmatrix} \qquad (6.1)$$

with $\gamma_{ik}\gamma^{jk} = \delta_i^j$ and $\beta^j = \gamma^{ij}\beta_i$. The gravitational Lagrangian density $\mathcal{L}$ introduced by B. DeWitt is:

$$\mathcal{L}\sqrt{g} = R\sqrt{g} = \alpha\gamma^{1/2}(K_{ij}K^{ij} - K^2 + R') + 2\partial_0(\gamma^{1/2}K) + 2\partial_i(\gamma^{1/2}K\beta^i - \gamma^{1/2}\gamma^{ij}\partial_i\alpha) \qquad (6.2)$$

where $g = -\det|g_{\mu\nu}| = \gamma\alpha^2$, $\gamma = \det|\gamma_{ij}|$, $K_{ij} = \frac{1}{2}(\nabla_j\beta_i + \nabla_i\beta_j - \gamma_{ij,0})$, $K^{ij} = \gamma^{ik}\gamma^{jl}K_{kl}$, $K = \gamma^{ij}K_{ij}$. Furthermore, $R'$ is the 3D Ricci scalar and $\nabla_j$ the 3D covariant derivative, both associated to the metric $\gamma_{ij}$. B. DeWitt indicates: "the contracted forms $R'$ and $K_{ij}K^{ij} - K^2$ will be refered to as intrinsic and extrinsic curvature, respectively. The last two terms [...] being total derivatives [...] may be dropped. The Lagrangian becomes $L = \int \alpha\gamma^{1/2}(K_{ij}K^{ij} - K^2 + R')d^3x$. [...] The momenta conjugate to $\alpha$, $\beta_i$ and $\gamma_{ij}$ will be denoted $\pi$, $\pi^i$ and $\pi^{ij}$ respectively". B. DeWitt uses functional derivatives on $L$ but we use partial derivatives on $\mathcal{L}$:

$$\pi = \frac{\partial \mathcal{L}}{\partial \alpha_{,0}} = 0 \quad ; \quad \pi^i = \frac{\partial \mathcal{L}}{\partial \beta_{i,0}} = 0 \quad ; \quad \pi^{ij} = \frac{\partial \mathcal{L}}{\partial \gamma_{ij,0}} = -\gamma^{1/2}(K^{ij} - \gamma^{ij}K) \qquad (6.3)$$

"the Hamiltonian [...] takes the form:

$$H = \int \pi\alpha_{,0} + \pi^i\beta_{i,0} + \pi^{ij}\gamma_{ij,0} d^3x - L = \int \pi\alpha_{,0} + \pi^i\beta_{i,0} + \alpha\mathcal{H}' + \beta_i X^i d^3x \qquad (6.4)$$

where

$$\mathcal{H}' = \frac{1}{2}\gamma^{-1/2}(\gamma_{ik}\gamma_{jl} + \gamma_{il}\gamma_{jk} - \gamma_{ij}\gamma_{kl})\pi^{ij}\pi^{kl} - \gamma^{1/2}R' = \gamma^{1/2}(K_{ij}K^{ij} - K^2 - R') \quad ; \quad X^i = -2\nabla_j\pi^{ij} \qquad (6.5)$$

[...] If desired, one can always assign definite values to $\alpha$ and $\beta_i$ which may be purely numerical [...]. Each choice corresponds to the imposition of certain conditions on the space-time coordinate. For example, one may choose $\alpha = 1, \beta_i = 0$. Another favorite choice is $K = \frac{1}{2}\gamma^{-1/2}\gamma_{ij}\pi^{ij} = 0$ ; $\partial_j(\gamma^{1/2}\gamma^{ij}) = 0$ which corresponds to the requirement that the volume of every hypersurface $x^0$ = constant be stationary under small time deformations, and that the spatial coordinates in each hypersurface be harmonic". B. DeWitt then explains that, in the quantization process, the momenta should be understood as conditions applied on a state vector $\Psi$ according to $\pi\Psi = \pi^i\Psi = \mathcal{H}'\Psi = X^i\Psi = 0$ (while, furthermore, $\Psi^\dagger K\Psi = \Psi^\dagger\nabla_j(\gamma^{1/2}\gamma^{ij})\Psi = 0$). The conditions $\mathcal{H}'\Psi = 0$ and $X^i\Psi = 0$ are respectively called Hamiltonian constraint and momentum constraints. A key problem of quantum gravity is to solve these constraints: the results of the previous sections will allow us to shed new light on the constraints. Nowadays, the constraint $\mathcal{H}'\Psi = 0$ is rewritten by replacing the momenta $\pi^{ij}$ by functional derivatives:

$$\mathcal{H}'\Psi = \left[ \frac{\kappa}{2}\gamma^{-1/2} \left( \gamma_{ik}\gamma_{jl} + \gamma_{il}\gamma_{jk} - \gamma_{ij}\gamma_{kl} \right) \left( -\mathbf{i}\frac{\delta}{\delta\gamma_{ij}} \right) \left( -\mathbf{i}\frac{\delta}{\delta\gamma_{kl}} \right) \right] - \kappa^{-1}\gamma^{1/2}R' \qquad (6.6)$$

Equation (6.6) is the famous Wheeler-DeWitt equation, where $\kappa$ is a constant [29] and $\gamma_{ik}\gamma_{jl} + \gamma_{il}\gamma_{jk} - \gamma_{ij}\gamma_{kl}$ is called the DeWitt metric. In 1986, A. Ashtekar suggested to replace the variables $\gamma_{ij}$ and $\pi^{ij}$ by other variables, now called Ashtekar's variables. It was the entryway to loop quantum gravity, as explained by C. Rovelli and F. Vidotto: "The canonical version of Loop Quantum Gravity was born from the discovery of 'loop' solutions of the Wheeler-DeWitt equation, namely the formal quantization of canonical general relativity, rewritten in the
Ashtekar variables" [1]. In 1999, B. DeWitt and A. Ashtekar nevertheless both agreed that a weakness of the Wheeler-DeWitt equation is its lack of manifest general covariance, arising from the ansatz used to define the metric in equation (6.1). Indeed, during a conference, they respectively declared : "it is a very bad equation, based on the relativity-violating procedure of cutting space-time into 'times slices'" and "the weakness of this is that one is taking space-time and dividing it into space and time" [29]. During the same conference, J. Stachel explained : "I share Bryce's unease with canonical quantization, which is based on singling out a global time function. Of course, in the end one hopes to show that the resulting quantum theory is independent of this choice; but I have the feeling it would be better never to paint oneself into this corner, out of which one must later so painfully try to paint oneself. But the local (i.e., the proper) time is a physically meaningful concept classically, and it must have some quantum-mechanical aspect or analog insofar as any sort of more-or-less conventional quantization of general relativity is contemplated." [29]. We can now use the results of the previous sections. Using the notations of the previous section, we relabel the metric of equation (6.1) with 5 coordinates :

$$g_{\mu\nu} \rightarrow \underline{g_{MN}}; \quad g^{\mu\nu} \rightarrow \underline{g^{MN}}; \quad \gamma_{ij} \rightarrow g_{\mu\nu}; \quad \alpha \rightarrow \Phi^{-1}; \quad \beta_i \rightarrow \frac{qA_\mu}{Mc}; \quad \nabla_i \rightarrow \nabla_\mu; \quad x^0 \rightarrow x^5 \qquad (6.7)$$

The variable $x^5$ now takes the place of the time coordinate $x^0$. In the previous section we justified that we could consider $\dot{x}^5 = M/m$ and $\phi = 1$ from the experimental point of view : in this context, $x^5$ is proportional to the proper time. It is expected that the metric will not depend on $x_5$ in vacuum but it does not need to be true where the matter energy momentum tensor does not vanish (the energy momentum tensor being the source in Einstein's field equations). Indeed, we previously observed that the Lagrangian of unstable particles possibly depends on the proper time. Using equation (6.7), we rewrite equation (6.1) according to :

$$\underline{g_{MN}} = \left[ \begin{array}{cc} \frac{qA_\mu}{Mc} & \frac{qA_\alpha}{Mc} \\ \frac{qA_\mu}{Mc} & -\frac{1}{\Phi^2} + \frac{qA_\alpha}{Mc} \frac{qA^\alpha}{Mc} \end{array} \right] ; \quad \underline{g^{MN}} = \left[ \begin{array}{cc} \Phi^2 \frac{qA^\mu}{Mc} & \Phi^2 \frac{qA^\nu}{Mc} \\ \Phi^2 \frac{qA^\mu}{Mc} & -\Phi^2 \end{array} \right] \qquad (6.8)$$

In the above metric, space and time are treated on equal footing, such that the 4D covariance is manifest. Equation (6.8) must now be compared with equation (4.9) : as a result, we identify equation (6.8) with the inverse metric of the Kaluza-Klein theory previously discussed, together with the $(+, -, -, -, -)$ sign convention. Consequently the 3D harmonic gauge condition $\partial_j(\gamma^{1/2}\gamma^{ij}) = 0$ introduced below equation (6.5) becomes the usual 4D harmonic gauge condition : $\partial_\mu(\sqrt{g}g^{\mu\nu}) = \sqrt{g}(\partial_\mu g^{\mu\nu} + g^{\mu\nu}\Gamma^\lambda_{\mu\lambda}) = -\sqrt{g}g^{\mu\lambda}\Gamma^\nu_{\mu\lambda} = 0$. To exhibit other features of mixing the Kaluza-Klein theory with the canonical quantization of gravity, we now simplify the discussion by imposing $\Phi = 1$, $g_{\mu\nu} = \eta_{\mu\nu}$ and the final condition $\partial_5 g_{MN} \rightarrow 0$ (we can not start with the assumption $\partial_5 g_{MN} = 0$ because partial derivatives with respect to $x_5$ appear in equation (6.3) when the change $x^0 \rightarrow x^5$ is made). In this context, $K_{ij}$ becomes $K_{\mu\nu} = \frac{q}{2Mc}(\partial_\mu A_\nu + \partial_\nu A_\mu)$ while the gauge condition $K = 0$ becomes the electromagnetic Lorenz gauge : $K = \eta^{\mu\nu}K_{\mu\nu} = \frac{q}{Mc}\partial_\alpha A^\alpha = 0$. In the Lagrangian density of equation (6.2), the term $K_{ij}K^{ij} - K^2$ becomes $K_{\mu\nu}K^{\mu\nu} - K^2$ and can be replaced by the Maxwell Lagrangian up to two boundary terms since :

$$\begin{align} \frac{1}{4} F_{\mu\nu} F^{\mu\nu} &= \frac{1}{4} (\partial_\mu A_\nu - \partial_\nu A_\mu) (\partial^\mu A^\nu - \partial^\nu A^\mu) = \frac{1}{4} (\partial_\mu A_\nu \partial^\mu A^\nu - \partial_\mu A_\nu \partial^\nu A^\mu - \partial_\nu A_\mu \partial^\mu A^\nu + \partial_\nu A_\mu \partial^\nu A^\mu) \nonumber \\ &= \frac{1}{4} (\partial_\mu A_\nu \partial^\mu A^\nu + \partial_\mu A_\nu \partial^\nu A^\mu + \partial_\nu A_\mu \partial^\mu A^\nu + \partial_\nu A_\mu \partial^\nu A^\mu) - \frac{1}{2} (\partial_\mu A_\nu \partial^\nu A^\mu + \partial_\nu A_\mu \partial^\mu A^\nu) \nonumber \\ &= \frac{M^2 c^2}{q^2} K_{\mu\nu} K^{\mu\nu} - \partial_\mu A_\nu \partial^\nu A^\mu = \frac{M^2 c^2}{q^2} K_{\mu\nu} K^{\mu\nu} - \partial_\mu (A_\nu \partial^\nu A^\mu) + A_\nu \partial^\nu \partial_\mu A^\mu \tag{6.9} \\ &= \frac{M^2 c^2}{q^2} K_{\mu\nu} K^{\mu\nu} - \partial_\mu (A_\nu \partial^\nu A^\mu) + \partial^\nu (A_\nu \partial_\mu A^\mu) - (\partial^\nu A_\nu) (\partial_\mu A^\mu) \nonumber \\ &= \frac{M^2 c^2}{q^2} (K_{\mu\nu} K^{\mu\nu} - K^2) - \partial_\mu (A_\nu \partial^\nu A^\mu) + \partial^\nu (A_\nu \partial_\mu A^\mu) \nonumber \end{align}$$

The momentum constraints $X^i = 0$ becomes $X^\nu = \partial_\mu(K^{\mu\nu} - \eta^{\mu\nu}K) = 0$ and implies Maxwell's equations :

$$0 = X^{\nu} = \frac{2q}{Mc} \left[ \frac{1}{2} \partial_{\mu} (\partial^{\mu} A^{\nu} + \partial^{\nu} A^{\mu}) - \eta^{\mu\nu} \partial_{\mu} \partial_{\alpha} A^{\alpha} \right] = \frac{q}{Mc} (\partial_{\mu} \partial^{\mu} A^{\nu} - \partial_{\mu} \partial^{\nu} A^{\mu}) = \frac{q}{Mc} \partial_{\mu} F^{\mu\nu} \qquad (6.10)$$

When the Lorentz gauge is assumed, the momentum constraints then take the simple form $\Box A^\mu = 0$. Finally, $\gamma_{ij}$ can be replaced by $g_{\mu\nu}$ in the Wheeler-DeWitt equation such that space and time are treated on equal footing.

To conclude, the above analysis allows us to identify the electromagnetic interaction of the Kaluza-Klein theory as an extrinsic curvature of the 4D space-time. In [Section 4] we mentioned the fact that the Kaluza-Klein mechanism is used in string theories : it is now clear that the geometry behind the Kaluza-Klein theory has similarities with the geometry assumed in canonical gravity, which is the basis of loop quantum gravity.
# 7 The theory of generally covariant quantum mechanics

In [Section 2] and [Section 4], we stated 3 postulates to modify general relativity: we generalized the concept of spacetime and replaced the usual matter Lagrangian density by the vanishing Lagrangian density of a quantum field. The 3 postulates are the foundations of generally covariant quantum mechanics, which is a theory of quantum physics in curved spacetime. Nevertheless, we will not use the 3 postulates as a basis anymore: to define generally covariant quantum mechanics, we will use the action given in equation (7.1) below. This action naturally follows from the 3 postulates. Before introducing the action of equation (7.1), it is interesting to illustrate that generally covariant quantum mechanics provides a bridge between general relativity and quantum mechanics: this can be done by showing how the theory solves some of the problems mentioned the book 'Conversations on quantum gravity' published by J. Armas in 2021. The first sentence of the book is "The holy grail of theoretical physics is to find the theory of everything that combines all the forces of nature, including gravity". The book contains a series of interviews of leading physicists about their views on the unification of general relativity and quantum mechanics. We quote P. Horava, H. Nicolai, and F. Dowker.

P. Horava, interviewed in 2011, said: "Our ancestors, who were working on this problem, have always been curious about which of the two paradigms of physics is more fundamental. Is it general relativity, described by some classical geometry, or is it quantum mechanics, described by the Schrödinger equation and a probabilistic interpretation of wave functions? Is it general relativity that needs to be modified in order to find a theory of quantum gravity? Is it quantum mechanics that needs to be modified? Or is it both?" [57]. Generally covariant quantum mechanics modifies quantum mechanics by incorporating the principle of general covariance in its formalism. Quantum mechanics describes matter with a complex field $\psi$ while electromagnetic and gravitational fields are usually described with real fields $A_\mu$ and $g_{\mu\nu}$. Since the fields can transform into each other, the theory suggests to treat them on equal footing by using complex fields $\psi, A_\mu, g_{\mu\nu}$.

H. Nicolai, interviewed in 2013, declared: "Einstein was led to his equations by way of two conceptually very simple principles, general covariance and the equivalence principle. Unfortunately, at this time I cannot see any such conceptual simplicity in the different approaches to quantum gravity" [57]. A fundamental idea of generally covariant quantum mechanics is to describe interactions (gravity and electromagnetism) and matter on equal footing. Matter is then described by fields and the proper time $\tau$ becomes a coordinate such that, (up to boundary terms) the free matter Lagrangian densities vanish, like the free Lagrangian densities of the gravitational and electromagnetic fields. It was deduced that the fields of unstable particles depend on $\tau$. Furthermore, we can write $c^2\tau^2 = (x^5)^2 - (x^6)^2$ to unify Einstein's gravity and Maxwell's electrodynamics. From the properties of Minkowski spacetime, one can show that the solutions of Maxwell's equations must be helicity eigenstates (see [Section 5.2]) and are consequently complex: then all fields, treated on equal footing, can be complex. There is nothing more. From the conceptual point of view, it is a very simple generalization of Einstein's gravity.

F. Dowker, interviewed in 2020, explained: "Today we are in a situation where we don't have a unified framework for fundamental physics and in particular there's a disjunction between our understanding of spacetime and our understanding of matter. [...] the starting points of quantum theory and general relativity are very different. [...] The problem is that quantum theory, as is usually understood, forces us into a position where time is treated as a parameter, while general relativity tells us that spacetime, including time, is a dynamical substance" [57]. Generally covariant quantum mechanics replaces $t$ by $\tau$ in the quantum domain but the methods of quantum field theory were not discussed (except the introduction of $\tau$ in the propagators of quantum field theory, quantum gravity, and string theory, see [Section 5.1]). The theory nevertheless provides a unified framework since it has been shown how each theory (general relativity or quantum mechanics) can be derived from the other, with the geodesic equation emerging in the eikonal approximation (see [Section 2] and [Section 5.1]).

To explore the physical content of generally covariant quantum mechanics, we now write its action explicitly. In its simplest form, the action of the theory combines the 5D Ricci scalar $\underline{R}$ and the Lagrangian density $\mathcal{L}_m$ of the spin-0 field given in equation (4.40). In the Lagrangian $\mathcal{L}_m$ we use $\phi$ in the form $\phi(x^\alpha, \underline{x}^5, x^6) = \psi(x^\alpha)e^{i\frac{Mc^5-c(\sqrt{M^2-m^2})x^6}{\hbar}} \approx \psi(x^\alpha)e^{i\frac{mc^2\tau}{\hbar}}$, as explained below equation (4.41). $|\phi|^2 = |\psi|^2$ is expressed in units of mass density. We will allow a slightly more complex matter Lagrangian density by adding a term $P(x^\alpha)$ to the matter Lagrangian $\mathcal{L}_m$ of equation (4.40): $P(x^\alpha)$ will be a pressure term. With $M^2 = \frac{c^2\mu_0q^2}{16\pi G}$, the action $S_T$ takes the form:

$$S_T = \int \left( \frac{\hbar^2}{2m^2} g^{MN} \left( \partial_M \phi \partial_N \phi^* + \partial_M \phi^* \partial_N \phi \right) + \left( \frac{M^2}{m^2} - 1 \right) c^2 |\phi|^2 + 2P - \frac{c^4 \underline{R}}{8\pi G} \right) \Phi \sqrt{\underline{g}} d^6 x \quad (7.1)$$

where

$$\underline{g}_{MN}(x^\alpha) = \begin{bmatrix} g_{\mu\nu} - \frac{\Phi^2 q A_\mu q A_\nu}{Mc^2} & \Phi^2 \frac{q A_\mu}{Mc} \\ \frac{\Phi^2 q A_\nu}{Mc} & -\Phi^2 \end{bmatrix} \quad ; \quad \underline{g}^{MN}(x^\alpha) = \begin{bmatrix} g^{\mu\nu} & \frac{q A^\nu}{Mc} \\ \frac{q A^\mu}{Mc} & -\frac{1}{\Phi^2} + \frac{q A_\alpha}{Mc} \frac{q A^\alpha}{Mc} \end{bmatrix} \quad (7.2)$$
which implies

$$ \underline{R} = R + \frac{8\pi G}{c^4} \left( \frac{\Phi^2}{2\mu_0} F^{\alpha\beta} F_{\alpha\beta} \right) - 2 \frac{\Box g \Phi}{\Phi} \quad ; \quad \sqrt{\det |g_{MN}|} = \sqrt{-\Phi^2 \det |g_{\mu\nu}|} = \Phi \sqrt{g} \qquad (7.3) $$

The gravitational and electromagnetic interactions are unified in $S_T$ but we could of course describe them as two independent interactions (by suppressing Postulate 3, see [Section 4]). M. Planck wrote in 1925: "As long as Natural Philosophy exists, its ultimate highest aim will always be the correlating of various physical observations into a unified system, and, where possible, into a single formula" [124]. The action $S_T$ is clearly not such a single formula (by example quarks, leptons and the nuclear forces are missing). As we will see in [Section 7.1], it is nevertheless possible to derive the following limiting cases from the single action $S_T$ given in equation (7.1):

- Limit n°1: Quantum electrodynamics of spin-0 particles,

- Limit n°2: Quantum physics in curved spacetime,

- Limit n°3: Einstein's gravity & Maxwell's electrodynamics,

- Limit n°4: Hydrodynamics of perfect fluids,

- Limit n°5: Newtonian mechanics,

- Limit n°6: Thermodynamics of perfect gases.

The aim of generally covariant quantum mechanics is to constitute a new step in the exploration of the secrets of Nature. As explained by F. Dyson, "science is not a collection of truths. It is a continuing exploration of mysteries" [125]. We will first establish the above limits and then discuss the interpretation of quantum mechanics suggested by generally covariant quantum mechanics.

## 7.1 Limiting cases of the theory

A. Einstein claimed in 1936: "Physics constitutes a logical system of thought which is in a state of evolution, whose basis cannot be distilled, as it were, from experience by an inductive method, but can only be arrived at by free invention. The justification (truth content) of the system rests in the verification of the derived propositions by sens experiences, whereby the relations of the latter to the former can only be comprehended intuitively. Evolution is proceeding in the direction of increasing simplicity of the logical basis. In order further to approach this goal, we must resign to the fact that the logical basis departs more and more from the facts of experience, and that the path of our thought from the fundamental basis to those derived propositions, which correlate with sense experiences, becomes continually harder and longer" [77]. We now examine how much physical informations can be derived from the single action $S_T$ of equation (7.1). In every limiting case discussed from [Section 7.1.1] to [Section 7.1.6], $S_{KK}$ is the mechanical action of the Kaluza-Klein theory such that the limit $\phi \to \mathcal{R}e^{-i\frac{S_{KK}}{\hbar}}$ means that we take the eikonal approximation. $S_{KK}$ was computed in equation (4.39).

### 7.1.1 Limit n°1: Quantum electrodynamics of spin-0 particles

This limit is obtained in the following approximation of $S_T$ given in equation (7.1):

$$ \Phi = 1 \ \& \ P = 0 \ \& \ g_{\mu\nu} = \eta_{\mu\nu} \qquad (7.4) $$

Then, the 4D Ricci scalar $R$ vanishes (since $g_{\mu\nu} = \eta_{\mu\nu}$). From equation (4.43), $S_T$ becomes:

$$ S_T = \int \frac{\eta^{\mu\nu}}{m^2} (\hbar\partial_{\mu} + i\mathbf{q}A_{\mu})\psi(\hbar\partial_{\nu} - i\mathbf{q}A_{\nu})\psi^* - c^2|\psi|^2 - \frac{1}{2\mu_0}F^{\alpha\beta}F_{\alpha\beta} \quad d^4x \, dx^5 \, dx^6 \qquad (7.5) $$

This is the action of a well known quantum field theory called scalar quantum electrodynamics or scalar QED [59][61][62]. The factor $\frac{1}{2}$ instead of the usual factor $\frac{1}{4}$ in front of $F^{\alpha\beta}F_{\alpha\beta}$ was explained below equation (4.17). The Lagrangian density of the above theory is invariant under $U(1)$ gauge transformations of the form: $\psi(x^\alpha) \to e^{i\frac{gB(x^\mu)}{\hbar}}\psi(x^\alpha)$; $A_\mu(x^\alpha) \to A_\mu(x^\alpha) - \partial_\mu B(x^\alpha)$. Naturally, the methods of quantum field theory (second quantization by example) must be introduced to produce quantum electrodynamics. Such methods produce divergences handled by the use of the renormalization group theory. In 1999, during a conference, A. Jaffe stated: "Yet in spite of these great successes, we do not know if the equations of relativistic quantum electrodynamics make mathematical sense. In fact the situation poses a dilemma. As a consequence of renormalization group theory, most physicists today believe that the equations of quantum electrodynamics in their simple form are inconsistent; in other words, we believe that the equations of electrodynamics have no solution at all!" [29]. During the same conference, F. Rohrlich explained: "QED is not a mathematically clean theory [...] Dirac has pointed out several times that he finds the QED perturbation expansion unsatisfactory: in the classical limit [...], QED should reduce to the classical electrodynamics of charged particles; but instead, the perturbation expansion diverges term by term!" [29]. We can nevertheless enlight a subject introduced by S. L. Glashow at the conference: "Why are
elementary particles forces described by gauge theories? Why is the gauge group what it is? There are many questions like this that simply cannot be answered within the context of [quantum field] theory. I call questions of this kind meta-questions" [29]. Generally covariant quantum mechanics provides an answer in the electromagnetic case (when Postulate 3 is taken into account) : the Lagrangian of the gauge field $A_\mu(x^\alpha)$ is invariant under a restricted form of the usual diffeomorphism invariance of general relativity (the restriction happens because the 5D metric only depends on the 4 spacetime coordinates, not the internal coordinate). Since the gauge field $A_\mu(x^\alpha)$ field appear in the phase of the complex field $\psi(x^\alpha)$, the U(1) gauge invariance follows.

### 7.1.2 Limit n°2 : Quantum physics in curved spacetime :

This limit is obtained when the following conditions are imposed in equation (7.1) :

$$ \Phi = 1 \ \& \ P = 0 \ \& \ q = 0 \qquad (7.6) $$

From $\Phi = 1$ and $q = 0$ we deduce and $qA_\mu = M = 0$ and $\underline{R} = R$. Furthermore, the matter Lagrangian density of $S_T$ reduces to the matter Lagrangian density of equation (2.12) with $A = \frac{\hbar^2}{m^2}$. Then $S_T$ becomes :

$$ S_T = \int \left( \frac{\hbar^2}{m^2} g^{\mu\nu} (\partial_{\mu} \psi \partial_{\nu} \psi^* + \partial_{\nu} \psi \partial_{\mu} \psi^*) - c^2 |\psi|^2 - \frac{c^4}{8\pi G} R \right) \sqrt{g} d^4 x \, dx^5 dx^6 \qquad (7.7) $$

This is the action of 'quantum field theory in curved spacetime'. This theory was explored by R. P Feynman in 1963 [113][126], but R. P Feynman used a real matter field instead of our complex field. He discussed the scattering of gravitons. In 1975, B. DeWitt published a long and famous paper on this theory. Like Feynman, he used a real matter field. He examined the introduction of the methods of quantum field theory in this theory, and stated : " Gary Gibbons has given the following completely covariant account of the situation that exists when there is a Killing vector $K^\mu$. [...] If $K^\mu$ is globally timelike one may introduce a coordinate $t$ upon which the metric does not depend and with respect to which $K^\mu$ takes the form $K^\mu = (1, 0, 0, 0)$. Furthermore, $K^\mu$ may be scaled so that $t$ gives directly the proper time measured by at least one clock [...] All this is just as in conventional particle physics. The only trouble with it is : it's wrong. It is not wrong in a technical mathematical sense. It simply provides a grossly inadequate foundation for the theory. Here are just some of the situations in which it fails : There may be no Killing vector at all, [...] There may be a global Killing vector, but it may not be everywhere timelike [...], Spacetime may be stationary only in limited regions" [127]. The fact that generally covariant quantum mechanics uses $\tau$ instead of $t$ evidently changes the foundation of the theory discussed by B. DeWitt.

Concerning the famous divergences of quantum field theory in curved spacetime, S. Weinberg suggested delving deeper into the quantum aspects, during his Nobel lecture in 1979 : "[...] suppose that there is a truly fundamental theory, characterized by an energy scale of order $10^{16}$ to $10^{19}$ GeV, at which strong, electroweak, and gravitational interactions are all united. It might be a conventional renormalizable quantum field theory, but at the moment, if we include gravity, we do not see how this is possible. [...] But if it is not renormalizable, what then determines the infinite set of coupling constants that are needed to absorb all the ultraviolet divergences of the theory? I think the answer must lie in the fact that the quantum field theory, which was born just fifty years ago from the marriage of quantum mechanics with [special] relativity, is a beautiful but not very robust child" [128]. Similarly, S. Hossenfelder suggested to look for alternatives to the quantization of gravity : "If gravity is not a fundamental interaction, but merely applicable in a classical limit, then quantizing gravitational degrees of freedom may not be the right way to go. Similarly, if quantization is only an emergent feature, or one approximately valid in the limits that we have tested, then the quantization of gravity again might not lead us to the right fundamental theory. Therefore, with quantum gravity we will mean any approach that is able to resolve the apparent tension between general relativity and quantum field theories" [129]. Interviewed by J. Arras in 2011, R. Penrose explained : "When people speak of quantum gravity, they usually think of applications of the rules of quantum mechanics to [...] Einstein's general theory of relativity [...]. what I'm saying is that you should be looking at the problem the other way around, that is, not how quantum theory affects the structures of spacetime but how the principles of general relativity will affect quantum mechanics" [57]. We now follow R. Penrose's suggestion by considering, one more time, superfluid Helium. We use two statements, (S₁) and (S₂). The statement (S₁) contains well established knowledge about superfluid helium while (S₂) recalls the validity domain of Einstein's equations.

- (S₁) : "Since a $^4$He atom contains two electrons, two protons and two neutrons, it is [...] a boson (it has total spin 0)" [40] where the $^4$He is the Helium atom. Furthermore, "the simplest example of a relativistic field theory deals with spin-0 particles described by the Klein-Gordon equation" [50]. In quantum physics, "A system of particles is describable by a complex wave function [...]. Macroscopic phase coherence (correlation of the quantum phase over macroscopic distances) gives rise to superfluidity" [130]. We denote the scalar field describing the atoms in the superfluid $\psi = \mathcal{R}e^{-i\frac{\phi}{\hbar}}$ where $\psi$ is the wave function. "The wave function corresponds to a condensate, or macroscopic number of particles. We can choose to normalize the wave function so that" the mass density of the condensate is $|\psi|^2 = \mathcal{R}^2 = \rho$.
— (S2) : At low energy, Einstein’s field equations are valid in the macroscopic domain, at least with a good approximation.

From equation (7.7), we compute the field equations by varying the action $S_T$ with respect to $\psi$ (or equivalently $\psi^*$) and $g^{\alpha\beta}$. The variations give respectively :

$$0 = -\Box_g \psi - \frac{m^2 c^2}{\hbar^2} \psi = -\partial_\mu \partial^\mu \psi - \Gamma^\mu_{\mu\nu} \partial^\nu \psi - \frac{m^2 c^2}{\hbar^2} \psi \quad (7.8)$$

$$\frac{c^4 (R_{\alpha\beta} - \frac{1}{2}g_{\alpha\beta}R)}{8\pi G} = \frac{\hbar^2}{2m^2} (\partial_\alpha \psi \partial_\beta \psi^* + \partial_\alpha \psi^* \partial_\beta \psi) - \frac{1}{2}g_{\alpha\beta} \left( \frac{\hbar^2}{m^2} g^{\mu\nu} (\partial_\mu \psi \partial_\nu \psi^* + \partial_\nu \psi \partial_\mu \psi^*) - c^2 |\psi|^2 \right) \quad (7.9)$$

With $\psi = \mathcal{R}e^{-i\frac{S}{\hbar}}$ and the computation given in [Annex 2], the right hand side of equation (7.9) reduces to $\rho \frac{\partial_\alpha S}{m} \frac{\partial_\beta S}{m} + \epsilon(\hbar^2) \approx \rho u_\alpha u_\beta$ where $\rho = \mathcal{R}^2$ because the eikonal approximation is valid for a macroscopic system : we have seen in [Section 2] that the velocity of the superfluid can be written $\vec{v}_s = \frac{\nabla S}{m}$ in the non relativistic limit, in agreement with the relativistic equality $\frac{\partial_\alpha S}{m} = u_\alpha$. We now consider that $\psi$ is an energy eigenstate : $\psi(t, \vec{r}) = e^{-i\frac{E_t}{\hbar}} \psi'(\vec{r})$ such that the mass density is independent of $t$ : $\rho = |\psi|^2 = |\psi'|^2 = \rho(\vec{r}) = \mathcal{R}^2(\vec{r})$. The fluid is then stationary. We use the linear approximation of Einstein’s equation in the rest frame of the superfluid. We write $g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}$ and we only keep $h_{00}$ to describe the Newtonian gravitational field. Einstein’s equation $R_{00} - \frac{1}{2}g_{00}R = \frac{8\pi G}{c^4} \rho c^2$ reduces to

$$-\frac{1}{2}\Box h_{00}(\vec{r}) = -\frac{1}{2}\frac{\partial h_{00}}{\partial t^2} + \frac{1}{2}\Delta h_{00}(\vec{r}) = \frac{8\pi G}{c^2}\rho(\vec{r}) \quad (7.10)$$

In the linear approximation, we write $g^{00} = 1 - h_{00}$ such that $g^{00}g_{00} = 1 - h_{00}^2 \approx 1$. The determinant of the metric is :

$$-g = \det |g_{\mu\nu}| = \det |\eta_{\mu\nu} + h_{00}(\vec{r})\delta_{\mu 0}\delta_{\nu 0}| = -(1 + h_{00}(\vec{r})) \quad (7.11)$$

such that $\Gamma^\mu_{\mu\nu} = \frac{\partial_\nu g}{2g}$ implies $\Gamma^\mu_{\mu 0} = 0$ and $\Gamma^\mu_{\mu i} = \frac{\partial_i h_{00}}{2(1+h_{00})}$ where $i = 1, 2, 3$. With $i, j = 1, 2, 3$, we can then rewrite equation (7.8) according to :

$$\begin{aligned} 0 &= -\hbar^2 \Box_g \psi - m^2 c^2 \psi \\ &= -\hbar^2 \partial_\mu (g^{\mu\nu} \partial_\nu \psi) - \hbar^2 \Gamma^\mu_{\mu\nu} \partial^\nu \psi - m^2 c^2 \psi \\ &= -\hbar^2 \left[ \partial_0 ((\eta^{00} - h^{00})\partial_0 \psi) + \partial_i ((\eta^{ij}\partial_j \psi)) \right] - \hbar^2 \frac{\Gamma^\mu_{\mu 0}}{\partial^0 \psi} - \hbar^2 \Gamma^\mu_{\mu i} \partial^i \psi - m^2 c^2 \psi \\ &= -\hbar^2 \Box \psi + h^{00} \hbar^2 \partial_0^2 \psi - \hbar^2 \Gamma^\mu_{\mu i} \partial^i \psi - m^2 c^2 \psi \end{aligned} \quad (7.12)$$

The term $h^{00}\hbar^2\partial_0^2\psi$ can be written $h^{00}(E/c)^2\psi$ since $\psi(t, \vec{r}) = \mathcal{R}e^{-i\frac{S}{\hbar}} = e^{-i\frac{E_t}{\hbar}}\psi'(\vec{r})$. We now write $E = E_N + mc^2$ where $E_N$ is the newtonian energy. The Newtonian approximation allows to replace $0 = -\hbar^2\Box\psi - m^2c^2\psi$ by $0 = 2m\hbar\frac{\partial}{\partial t}\psi + \hbar^2\Delta\psi + m^2c^2h^{00}\psi - \hbar^2\Gamma^\mu_{\mu i}\partial^i\psi$, as we have seen with equation (2.19). Neglecting the term $-\hbar^2\Gamma^\mu_{\mu i}\partial^i\psi$ and using the notation $V_G = \frac{c^2}{2}h^{00}$ we finally obtain :

$$i\hbar \frac{\partial}{\partial t} \psi + \frac{\hbar^2}{2m} \Delta \psi + mV_G \psi = 0 \quad ; \quad \Delta V_G = 4\pi G |\psi|^2 \quad (7.13)$$

The above equation is the Newton-Schrödinger equation : "This equation was proposed by Diosi and Penrose as an attempt to investigate quantum wave function collapse in the presence of a Newtonian gravitational potential. More recently, the same equation has been used, as in its original proposal to describe the evolution of boson or Bose-Einstein condensate stars" [132]. The validity of the Newton-Schrödinger equation is debated in literature. The interesting point is that this equation has been derived from statements (S1) and (S2), while the two statements were made in the context of quantum physics in curved spacetime.

### 7.1.3 Limit n°3 : Einstein’s gravity & Maxwell’s electrodynamics

This limit is obtained by imposing the following conditions in equation (7.1) :

$$\Phi \to 1 \ \& \ P = 0 \ \& \ \phi \to \mathcal{R}e^{-i\frac{S}{\hbar}\kappa} \ \& \ \mathcal{R}^2 = \rho \quad (7.14)$$

To derive Einstein’s gravity and Maxwell’s electrodynamics, we first re-express the matter Lagrangian density $\mathcal{L}_m$ of equation (7.1) :

$$\mathcal{L}_m = \frac{\hbar^2}{2m} g^{MN} (\partial_M \phi \partial_N \phi^* + \partial_N \phi \partial_M \phi^*) + \left(\frac{M^2}{m^2} - 1\right) c^2 |\phi|^2 \quad (7.15)$$
We use the mechanical Kaluza-Klein action to take the eikonal approximation of the Lagrangian density :

$$
\phi(x^\alpha, \underline{x}^5, x^6) = \mathcal{R}(x^\alpha) e^{-i \frac{S_{KK}(x^\alpha, \underline{x}^5, x^6)}{\hbar}} = \mathcal{R}(x^\alpha) e^{-i \frac{S(x^\alpha) + M x^5 - \sqrt{M^2 - m^2 x^6}}{\hbar}} = \psi(x^\alpha) e^{i \frac{M x^5 - \sqrt{M^2 - m^2 x^6}}{\hbar}} \quad (7.16)
$$

where $S(x^\alpha)$ is the usual 4D mechanical action. Writing "cc" to denotes the complex conjugate of the terms on the left, we get :

$$
\begin{align}
\mathcal{L}_m &= \frac{\hbar^2}{2m^2} g^{MN} (\partial_M \phi \partial_N \phi^* + \partial_N \phi \partial_M \phi^*) + (\frac{M^2}{m^2} - 1) c^2 |\phi|^2 \nonumber \\
\mathcal{L}_m &= \frac{\hbar^2}{2m^2} g^{MN} \left( \partial_M \mathcal{R} e^{-i \frac{S_{KK}}{\hbar}} - i \partial_M \frac{S_{KK}}{\hbar} \mathcal{R} e^{-i \frac{S_{KK}}{\hbar}} \right) \left( \partial_N \mathcal{R} e^{i \frac{S_{KK}}{\hbar}} + i \partial_N \frac{S_{KK}}{\hbar} \mathcal{R} e^{i \frac{S_{KK}}{\hbar}} \right) + \text{``cc''} \nonumber \\
&\quad + (\frac{M^2}{m^2} - 1) c^2 \mathcal{R}^2 \tag{7.17} \\
\mathcal{L}_m &= \frac{\hbar^2}{2m^2} g^{MN} \left[ \partial_M \mathcal{R} \partial_N \mathcal{R} + \mathcal{R}^2 \partial_N \frac{S_{KK}}{\hbar} \partial_M \frac{S_{KK}}{\hbar} + i (\mathcal{R} \partial_M \mathcal{R} \frac{\partial_N S_{KK}}{\hbar} - \mathcal{R} \partial_N \mathcal{R} \frac{\partial_M S_{KK}}{\hbar}) \right] + \text{``cc''} \nonumber \\
&\quad + (\frac{M^2}{m^2} - 1) c^2 \mathcal{R}^2 \nonumber \\
\mathcal{L}_m &= \frac{\hbar^2}{m^2} g^{MN} \partial_M \mathcal{R} \partial_N \mathcal{R} + \mathcal{R}^2 g^{MN} \frac{\partial_M S_{KK}}{m} \frac{\partial_N S_{KK}}{m} + (\frac{M^2}{m^2} - 1) c^2 \mathcal{R}^2 \nonumber
\end{align}
$$

In the eikonal approximation, we neglect the first term and take the limit $\frac{\partial_M S_{KK}}{m} \to u_M$ where $u_M$ is the velocity 5-vector computed in equation (4.23). Writing $(\frac{M^2}{m^2} - 1)c^2 = (u^6)^2$ and $\mathcal{R}^2(x^\alpha) = \rho(x^\alpha)$ we get :

$$
\mathcal{L}_m = \rho g^{MN} u_M u_N + \rho (u^6)^2 \quad (7.18)
$$

Accordingly, in the eikonal approximation, $S_T$ becomes :

$$
S_T = \int \left( \rho g^{MN} u_M u_N + \rho u^6 u^6 - \frac{c^4}{8\pi G} \underline{R} \right) \Phi \sqrt{g} d^6 x \quad (7.19)
$$

Using the definitions given in equations (7.2) and (7.3) we can rewrite $S_T$ more explicitly :

$$
\int \left[ \rho \left( g^{\mu\nu} u_\mu u_\nu - \Phi^2 \left(u^5 - \frac{q A_\mu u^\mu}{Mc}\right)^2 + u^6 u^6 \right) - \frac{c^4 R}{8\pi G} - \frac{\Phi^2}{2\mu_0} F_{\mu\nu} F^{\mu\nu} + \frac{c^4}{4\pi G} \frac{\Box_g \Phi}{\Phi} \right] \Phi \sqrt{g} d^6 x \quad (7.20)
$$

The action $S_T$ in equations (7.19) and (7.20) then reproduces the action of equations (4.11) and (4.18) respecti-
vely. Varying $S_T$ with respect to $g^{55}$ and $g^{5\nu}$ we get respectively, according to equations (4.27) and (4.30) :

$$
\frac{\Phi^2}{2} R + \frac{8\pi G}{c^4} \left( \frac{3\Phi^4}{4\mu_0} F^{\alpha\beta} F_{\alpha\beta} \right) = \frac{8\pi G}{c^4} \rho \frac{M^2 c^2}{m^2} \quad (7.21)
$$

$$
\Phi^2 \nabla_{\nu} F^{\nu\alpha} + 3\Phi F^{\nu\alpha} \partial_{\nu} \Phi = \mu_0 \frac{q}{m} \rho u^{\alpha} = \mu_0 j^{\alpha} \quad (7.22)
$$

With $\Phi = 1$ we obtain Maxwell's equations in curved spacetime :

$$
\nabla_{\nu} F^{\nu \alpha} = \mu_{0} j^{\alpha} \qquad (7.23)
$$

where $j^\alpha = \frac{q}{m} \rho u^\alpha$ is the electric current. Neglecting gravity, we recover the usual Maxwell's equation $\partial_\nu F^{\nu\alpha} = \mu_0 j^\alpha$. Varying now $S_T$ with respect to $g^{\mu\nu}$ we get, according to (4.33) :

$$
G_{\mu\nu} = \frac{8\pi G}{c^4} \left( \Phi^2 \left[ \frac{g^{\alpha\beta} F_{\nu\alpha} F_{\beta\mu} + \frac{1}{4} g_{\mu\nu} F^{\mu\alpha} F_{\nu\alpha}}{\mu_0} \right] + \left[ \frac{c^4}{8\pi G} \frac{\partial_\mu \partial_\nu \Phi - \Gamma^\sigma_{\mu\nu} \partial_\sigma \Phi - g_{\mu\nu} \Box_g \Phi}{\Phi} \right] + \left[ \rho u_\mu u_\nu \right] \right) \quad (7.24)
$$

Writing $\rho u_\mu u_\nu = T_{\mu\nu}$ in the limit $\Phi = 1$, we get :

$$
G_{\mu\nu} = \frac{8\pi G}{c^4} \left[ \frac{g^{\alpha\beta} F_{\nu\alpha} F_{\beta\mu} + \frac{1}{4} g_{\mu\nu} F^{\mu\alpha} F_{\nu\alpha}}{\mu_0} \right] + \frac{8\pi G}{c^4} T_{\mu\nu} \quad (7.25)
$$

Equations (7.23) and (7.25) are the Einstein-Maxwell equations. We can multiply equation (7.24) by $g^{\mu\alpha}g^{\nu\beta}$ and then take the divergence with $\nabla_\alpha$: the right hand side must vanish because $\nabla_\alpha G^{\alpha\beta} = 0$. The computations,
given in [Annex 6], produces :

$$0 = u^{\mu} \left[ \nabla_{\alpha}(\rho u^{\alpha}) + \rho \frac{u^{\alpha} \partial_{\alpha} \Phi}{\Phi} \right] + \rho \left[ \dot{u}^{\mu} + \Gamma^{\mu}_{\alpha\beta} u^{\alpha} u^{\beta} + \frac{q F^{\mu}_{\beta} u^{\beta}}{Mc} \left[ \Phi^{2}(\underline{u}^{5} - \frac{A_{\alpha} u^{\alpha}}{Mc}) \right] + \frac{\partial^{\mu} \Phi}{\Phi^{3}} \left[ \Phi^{2}(\underline{u}^{5} - \frac{A_{\alpha} u^{\alpha}}{Mc}) \right] \right] \quad (7.26)$$

In [Section 4], we justified that both term in the first [ ] independently vanish, see equations (4.19) to (4.21). We also derived that $[\Phi^2(\underline{u}^5 - \frac{A_\alpha u^\alpha}{Mc})] = \frac{M}{m}$ according to equation (4.22). We can then conclude :

$$0 = \rho \left[ \dot{u}^{\mu} + \Gamma^{\mu}_{\alpha\beta} u^{\alpha} u^{\beta} + \frac{q F^{\mu}_{\beta} u^{\beta}}{mc} + \frac{M^2}{m^2} \frac{\partial^{\mu} \Phi}{\Phi^3} \right] \quad (7.27)$$

It has been shown from equations (4.35) to (4.38) that the last term can be neglected in the low energy approximation. In this approximation, the above equation finally reduces to :

$$0 = \rho \left[ \dot{u}^{\mu} + \Gamma^{\mu}_{\alpha\beta} u^{\alpha} u^{\beta} + \frac{q F^{\mu}_{\beta} u^{\beta}}{mc} \right] \quad (7.28)$$

We recognize the geodesic equation combined with the Lorentz force. The action $S_T$ in equation (7.1) allowed us to derive Maxwell's equations in equation (7.23), Einstein's equations in equation (7.25), and the geodesic equation combined with Lorentz force in equation (7.28). Einstein's gravity and Maxwell electrodynamics are then unified in the action $S_T$ given in equation (7.1).

The above analysis is remarkable when quantum mechanics is taken into account. Discussing the ontology of quantum field theory, R. Röhlich explained in 1999 (we underline important words) : "Any interpretation of quantum mechanics (QM) must start with a formulation of the theory in its own right and not as 'quantized classical mechanics'. The notion of 'quantization' indicates that one has not been freed of the heuristic level on which the theory has been developed originally. [...] While QM is not the topic here, I cannot leave it without pointing out one important ontological difference due to its different cognitive level : the words 'particle' and 'field' have a meaning in QM entirely different from their meaning in classical physics. The so-called wave-particle conundrum is largely due to a metaphoric use of these words. This comes about because the classical dichotomy between fields and particles becomes confused in QM : the particle is described by a wave function which depends on space and time and is therefore a (linear!) field; and the linear electromagnetic field of classical mechanics takes on particle properties, the photons of QM. Both 'particle' and 'field' have a new and different meaning. One must therefore distinguish the classical meaning of these words from 'quantum particle' and 'quantum field'." [29]. In arbitrary spacetime, Maxwell's equations take the form $\nabla_{\nu} F^{\nu\alpha} = \mu_{0}j^{\alpha}$ where $\nabla_{\nu}$ involves the metric $g_{\alpha\beta}$ obeying equation (7.25) : accordingly, $g_{\alpha\beta}$ depends on $F^{\nu\alpha}$ and Maxwell's equations are not linear. In a theory unifying quantum mechanics and Einstein's field equations, it is clear that the quantum notion of photon can not be based on linear equations.

It now deserves to be recalled that the standard model of particle physics contains a massive scalar field : the BEH field or Brout-Englert-Higgs field [60][61][94]. In the standard model, the quarks, the leptons, and the gauge bosons associated to the weak interaction acquire their masses by the Higgs mechanism. In the Higgs mechanism, the scalar field fluctuates around a constant value, called the 'Vacuum Expectation Value' or VEV. This VEV correspond to the minimum of the scalar field potential. In 1989, S. Weinberg discussed the VEV and emphasized : "The real beginning of serious worry about the vacuum energy seems to date from the success of the idea of spontaneous symmetry breaking in the electroweak theory. In this theory, the scalar field potential takes the form (with $\mu^2 \ge 0, g \ge 0)$ : $V = V_0 - \mu^2\Phi^\dagger\Phi + g(\Phi^\dagger\Phi)^2$. At its minimum this takes the value $V_{min} = V_0 - \frac{\mu^4}{4q}$". S. Weinberg explained that this potential should contribute to the vacuum energy, and then to the cosmological constant. He concluded : "this example shows vividly how unnatural it is to get a reasonably small effective cosmological constant" [100]. To introduce generally covariant quantum mechanics, we focused on the massless photon and the massless graviton while matter was described by a scalar field to simplify the discussion (leptons and quarks are described by spinors) : the present theory nevertheless contains another massless scalar field, $\Phi(x^\alpha)$. We made the assumption that this scalar field should be massless (see Postulate 3 in [Section 4]) : this condition can of course be relaxed. If $\Phi(x^\alpha)$ is chosen massive, it is tempting to identify $\Phi$ or $\Phi^2$ with the scalar field of the standard model because they both fluctuate around a constant value and they are both associated to the mass terms of matter Lagrangians. E. Zeidler has already suggested to identify the scalar field of the Kaluza-Klein mechanism with the scalar field of the Higgs mechanism [132]. Furthermore, equation (7.24) shows that only the derivatives of $\Phi$ contribute to its energy momentum tensor, such that its VEV does not contribute to the vacuum energy. It is nevertheless required to include the weak interaction in the present theory to determine if $\Phi$ can be identified with the scalar field of the Higgs mechanism.
### 7.1.4 Limit n°4 - Hydrodynamics of perfect fluids

This limit is obtained when the following conditions are applied to equation (7.1) :

$$ \Phi = 1 \quad \& \quad q = 0 \quad \& \quad \phi \to \mathcal{R}e^{-i \frac{S_{KK}}{\hbar}} \quad \& \quad \mathcal{R}^2 = \rho + P \qquad (7.29) $$

From $\phi \to \mathcal{R}e^{-i \frac{S_{KK}}{\hbar}}$, we get, according to equation (7.17) :

$$ \mathcal{L}_m = \frac{\hbar^2}{m^2} g^{MN} \partial_M \mathcal{R} \partial_N \mathcal{R} + \mathcal{R}^2 g^{MN} \frac{\partial_M S_{KK}}{m} \frac{\partial_N S_{KK}}{m} + (\frac{M^2}{m^2} - 1) c^2 \mathcal{R}^2 \qquad (7.30) $$

From $\Phi = 1$ and $q = 0$ (such that $M = 0$), the above equations becomes :

$$ \mathcal{L}_m = \frac{\hbar^2}{m^2} g^{\mu\nu} \partial_\mu \mathcal{R} \partial_\nu \mathcal{R} + \mathcal{R}^2 g^{\mu\nu} \frac{\partial_\mu S_{KK}}{m} \frac{\partial_\nu S_{KK}}{m} - c^2 \mathcal{R}^2 \qquad (7.31) $$

In the eikonal approximation, we neglect the first term of equation (7.31) and we identify $\frac{\partial_\mu S_{KK}}{m}$ with $u_\mu$ where $u_\mu$ is the 4-velocity (if $q \neq 0$ we have $\frac{\partial_\mu S_{KK}}{m} = u_\mu + \frac{qA_\mu}{m}$). Using $\mathcal{R}^2 = \rho + P$, we rewrite $\mathcal{L}_m$ :

$$ \mathcal{L}_m = \left( \rho + \frac{P}{c^2} \right) \left( g^{\mu\nu} u_\mu u_\nu - c^2 \right) \qquad (7.32) $$

The action $S_T$ becomes :

$$ S_T = \int \left( \rho + \frac{P}{c^2} \right) (g^{\mu\nu} u_\mu u_\nu - c^2) + 2P - \frac{c^4}{8\pi G} R \sqrt{g} d^4 x \quad dx^5 dx^6 \qquad (7.33) $$

Varying this action with respect to $g^{\mu\nu}$ produces :

$$ \delta S_T = \int \left( \rho + \frac{P}{c^2} \right) u_\mu u_\nu - \frac{1}{2} g_{\mu\nu} \left( \rho + \frac{P}{c^2} \right) \left( g^{\alpha\beta} \overline{u_\alpha u_\beta} - c^2 \right) - P g_{\mu\nu} - c^4 \frac{R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R}{8\pi G} \sqrt{g} \delta g^{\mu\nu} d^4 x \quad dx^5 dx^6 \qquad (7.34) $$

The condition of stationary action, i.e. $\delta S_T = 0$, implies :

$$ G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R = \frac{8\pi G}{c^4} \left( \left( \rho + \frac{P}{c^2} \right) u_\mu u_\nu - g_{\mu\nu} P \right) \qquad (7.35) $$

The energy momentum tensor $(\rho + \frac{P}{c^2})u_\mu u_\nu - g_{\mu\nu}P$ is the usual tensor describing a perfect fluid [3]. It has the well known property that $g^{\mu\nu}((\rho + \frac{P}{c^2})u_\mu u_\nu - g_{\mu\nu}P) = \rho - 3P$. Equation (7.35) can be rewritten :

$$ G^{\alpha\beta} = R^{\alpha\beta} - \frac{1}{2} g^{\alpha\beta} R = \frac{8\pi G}{c^4} \left( \left( \rho + \frac{P}{c^2} \right) u^\alpha u^\beta - g^{\alpha\beta} P \right) \qquad (7.36) $$

Using the fact that $\nabla_\alpha G^{\alpha\beta} = \nabla_\alpha g^{\alpha\beta} = 0$, we deduce

$$ \begin{aligned} 0 &= \nabla_\alpha \left( \left( \rho + \frac{P}{c^2} \right) u^\alpha u^\beta - g^{\alpha\beta} P \right) \\ 0 &= u^\beta \nabla_\alpha \left( \left( \rho + \frac{P}{c^2} \right) u^\alpha \right) + \left( \rho + \frac{P}{c^2} \right) u^\alpha \nabla_\alpha u^\beta - \partial^\beta P \\ 0 &= u^\beta u^\alpha \partial_\alpha \rho + u^\beta \rho \nabla_\alpha u^\alpha + u^\beta u^\alpha \partial_\alpha \left( \frac{P}{c^2} \right) + u^\beta \left( \frac{P}{c^2} \right) \nabla_\alpha u^\alpha + \left( \rho + \frac{P}{c^2} \right) u^\alpha \nabla_\alpha u^\beta - \partial^\beta P \qquad (7.37) \\ 0 &= u^\beta \dot{\rho} + u^\beta \rho \nabla_\alpha u^\alpha + u^\beta \left( \frac{\dot{P}}{c^2} \right) + u^\beta \left( \frac{P}{c^2} \right) \nabla_\alpha u^\alpha + \left( \rho + \frac{P}{c^2} \right) u^\alpha \nabla_\alpha u^\beta - \partial^\beta P \\ 0 &= u^\beta \left( \dot{\rho} + \left( \rho + \frac{P}{c^2} \right) \nabla_\alpha u^\alpha \right) + u^\beta \left( \frac{\dot{P}}{c^2} \right) + \left( \rho + \frac{P}{c^2} \right) u^\alpha \nabla_\alpha u^\beta - \partial^\beta P \end{aligned} $$

Multiplying the above equation by $u_\beta$ and using the fact that $u_\beta u^\alpha \nabla_\alpha u^\beta = u^\alpha \partial_\alpha (u_\beta u^\beta) = u^\alpha \partial_\alpha (c^2) = 0$, we get :

$$ 0 = c^2 \dot{\rho} + (c^2 \rho + P) \nabla_\alpha u^\alpha + \dot{P} + \left( \rho + \frac{P}{c^2} \right) u_\beta u^\alpha \overline{\nabla_\alpha u^\beta} - \dot{P} \Leftrightarrow c^2 \left( \dot{\rho} + \left( \rho + \frac{P}{c^2} \right) \nabla_\alpha u^\alpha \right) = 0 \qquad (7.38) $$

Injecting this result in equation (7.37) and writing $u^\alpha \nabla_\alpha u^\beta = \dot{u}^\beta + \Gamma^\beta_{\alpha\sigma} u^\alpha u^\sigma$, we get :

$$ 0 = \left( \rho + \frac{P}{c^2} \right) \left( \dot{u}^\beta + \Gamma^\beta_{\alpha\sigma} u^\alpha u^\sigma \right) - \partial^\beta P + u^\beta \frac{\dot{P}}{c^2} \qquad (7.39) $$
In the non relativistic limit, we consider $\rho \gg \frac{P}{c^2}$ and $\partial^\beta P \gg u^\beta \frac{P}{c^2}$ such that the above equation becomes :

$$0 = \rho(\dot{u}^\beta + \Gamma_{\alpha\sigma}^\beta u^\alpha u^\sigma) - \partial^\beta P \qquad (7.40)$$

In 1916, K. Schwarzschild used Einstein's equations in vacuum to determine the metric outside a stationary spherical body, its mass being M. He derived the expression [133] :

$$ds^2 = \left(1 - \frac{2GM}{c^2R}\right)dt^2 - \frac{dR^2}{1 - \frac{2GM}{c^2R}} - R^2(d\theta^2 + \sin^2\theta d\phi^2) \qquad (7.41)$$

where $R = (r^3 + \alpha^3)^{\frac{1}{3}}$ is an auxiliary variable, $r$ is the radial coordinate and $\alpha$ an integration constant (K. Schwarzschild fixed $\alpha = \frac{2GM}{c^2}$). Nowadays, the Schwarzschild metric is derived by assuming that the radial part of the metric must have the form $-r^2(d\theta^2 + \sin^2\theta d\phi^2)$ [3][27][56][102] : the integration constant $\alpha$ does not appear, and the metric becomes :

$$ds^2 = \left(1 - \frac{2GM}{c^2r}\right)dt^2 - \frac{dr^2}{1 - \frac{2GM}{c^2r}} - r^2(d\theta^2 + \sin^2\theta d\phi^2) \qquad (7.42)$$

The subtlety is that $r$ and $R$ are not defined on the same domain, as discussed by I. Mol [134]. In the low energy limit, we write $g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}$ where $|h_{\mu\nu}| \ll 1$, and we neglect non linear terms, such that indices are upped (or lowered) with $\eta^{\mu\nu}$ (or $\eta_{\mu\nu}$). The equation of motion $0 = \rho(\dot{u}^\beta + \Gamma_{\alpha\sigma}^\beta u^\alpha u^\sigma) - \partial^\beta P$ reduces to $0 = \rho(\dot{u}^\beta + \Gamma_{00}^\beta u^0 u^0) - \partial^\beta P$ because $u^0 \gg |u^i|$ where $i = 1, 2, 3$. Furthermore, the metric being independent of the time coordinate :

$$\Gamma_{00}^{\beta} = \frac{\eta^{\alpha\beta}}{2}(h_{0\alpha,0} + h_{0\alpha,0} - h_{00,\alpha}) = -\frac{\eta^{\alpha\beta}}{2}h_{00,\alpha} = \partial^{\beta} \frac{GM}{c^2r} \qquad (7.43)$$

If we consider a fluid at the surface of the Earth, the spatial components of the equation of motion are :

$$0 = \rho(\dot{u}^i + \Gamma_{00}^i u^0 u^0) - \partial^i P \qquad (7.44)$$

where $i = 1, 2, 3$. With

$$u^0 \approx c \quad \& \quad \dot{u}^\beta = u^\alpha \partial_\alpha u^\beta = \frac{d}{d\tau} u^\beta \quad \& \quad -\partial^i h_{00} = \partial_i h_{00} \quad \& \quad -\partial^i P = +\partial_i P \qquad (7.45)$$

We can rewrite $0 = \rho(\dot{u}^i + \Gamma_{00}^i u^0 u^0) - \partial^i P$ in the form :

$$\begin{aligned} 0 &= \rho\left(\frac{d}{d\tau}\vec{u} + \vec{\nabla}\frac{h_{00}c^2}{2}\right) + \vec{\nabla}P \\ 0 &= \rho\left(\frac{d}{d\tau}\vec{u} + \frac{GM\vec{r}}{r^2}\right) + \vec{\nabla}P \end{aligned} \qquad (7.46)$$

In the non relativistic limit, we can replace $\frac{d}{d\tau}$ by $\frac{d}{dt}$ and $\vec{u} = \frac{\vec{v}}{\sqrt{1-|\vec{v}|^2/c^2}}$ by $\vec{v}$. In fluid mechanics, it is considered that $\frac{GM}{r^2}$ is constant because the distance between the different parts of the fluid and the center of the Earth does not vary significantly. It is then possible to define the gravity of Earth $g$ according to $g = \frac{GM}{r^2}$ : this definition uses the numerical values of the mass and the radius of the Earth instead of $M$ and $r$. Consequently, $g$ is an effective acceleration : in 2024 its Codata value is 9.80665 $m/s^2$. In hydrodynamics, it is also customary to neglect the curvature of the Earth such that it becomes possible to replace $\frac{GM}{r^2}\vec{r}$ by $V_G = gx_3$ : it is assumed that the reference frame is chosen such that the gravitational acceleration is aligned with the $x_3$ direction. Equation (7.48) can then be rewritten :

$$0 = \rho\left(\frac{d}{dt}\vec{v} + \vec{\nabla}V_G\right) + \vec{\nabla}P \qquad (7.47)$$

$$\rho \frac{d}{dt} \vec{v} = -\rho \vec{\nabla} V_G - \vec{\nabla} P \qquad (7.48)$$

$$\rho \frac{d}{dt} \vec{v} = -\rho \vec{g} - \vec{\nabla} P \qquad (7.49)$$

where we defined $\vec{g} = \vec{\nabla}V_G$. The above equation is the Euler equation in fluid mechanics : it describes an inviscid flow [17]. In equation (7.48) the left hand side $\frac{d}{dt}\vec{v}$ denotes $\frac{\partial}{\partial t}\vec{v} + (\vec{v} \cdot \vec{\nabla})\vec{v}$. Labelling $v_i$ and $v_j$ where $i, j = 1, 2, 3$ the components of $\vec{v}$, we observe that :

$$\begin{aligned} (\vec{v} \cdot \vec{\nabla}) v_j &= \sum_{i=1,2,3} v_i \frac{\partial}{\partial x^i} v_j = \sum_{i=1,2,3} \left( v_i \frac{\partial}{\partial x^j} v_i + v_i \left( \frac{\partial}{\partial x^i} v_j - \frac{\partial}{\partial x^j} v_i \right) \right) \\ &= \frac{1}{2} \frac{\partial}{\partial x^i} |\vec{v}|^2 + \sum_{i=1,2,3} v_i \left( \frac{\partial}{\partial x^i} v_j - \frac{\partial}{\partial x^j} v_i \right) \end{aligned} \qquad (7.49)$$
Since

$$ \frac{\partial}{\partial x^i} v_j - \frac{\partial}{\partial x^j} v_i = \frac{1}{m} \left( \frac{\partial}{\partial x^i} \frac{\partial S_{KK}}{\partial x^j} - \frac{\partial}{\partial x^j} \frac{\partial S_{KK}}{\partial x^i} \right) = 0 \qquad (7.50) $$

We deduce

$$ \frac{d}{dt} \vec{v} = \frac{\partial}{\partial t} \vec{v} + (\vec{v} \cdot \vec{\nabla}) \vec{v} = \frac{\partial}{\partial t} \vec{v} + \frac{1}{2} \vec{\nabla} |\vec{v}|^2 \qquad (7.51) $$

With equation (7.51) we rewrite equation (7.48) in the following form :

$$ \rho \frac{\partial}{\partial t} \vec{v} + \frac{1}{2} \rho \vec{\nabla} |\vec{v}|^2 = - \rho \vec{\nabla} V_g - \vec{\nabla} P \quad \Leftrightarrow \quad - \rho \frac{\partial}{\partial t} \vec{v} = \frac{1}{2} \rho \vec{\nabla} |\vec{v}|^2 + \rho \vec{\nabla} V_g + \vec{\nabla} P \qquad (7.52) $$

When the system is stationary (which implies $\frac{\partial}{\partial t} \vec{v} = 0$) and the density is constant ($\vec{\nabla} \rho = 0$) we get :

$$ 0 = \vec{\nabla} \left( \frac{1}{2} \rho |\vec{v}|^2 + \rho V_g + P \right) \Rightarrow \frac{1}{2} \rho |\vec{v}|^2 + \rho g x_3 + P = \text{constant} \qquad (7.53) $$

It is the Bernoulli equation, which describes a "steady flow of a nearly inviscid, nearly incompressible fluid with negligible heat transfer" [17].

### 7.1.5 Limit n°5 - Newtonian mechanics

This limit is obtained when :

$$ \Phi = 1 \ \& \ q = 0 \ \& \ \phi \to \mathcal{R} e^{i \frac{S_{KK}}{\hbar}} \ \& \ \mathcal{R}^2 = \rho + P \qquad (7.54) $$

Using the assumptions of equation (7.54), we derived equation $\rho \frac{d}{dt} \vec{v} = -\rho \vec{g} - \vec{\nabla} P$ in equation (7.48). Writing $\vec{F}_1 = -m\vec{g}$, $\vec{F}_2 = -m\frac{\vec{\nabla}P}{\rho}$ and $\frac{d}{dt}\vec{v} = \vec{a}$, we easily compute :

$$ \rho \frac{d}{dt} \vec{v} = - \rho \vec{g} - \vec{\nabla} P \quad \Leftrightarrow \quad m \vec{a} = - m \vec{g} - m \frac{\vec{\nabla} P}{\rho} = \sum_{i=1,2} \vec{F}_i \qquad (7.55) $$

We recognize Newton's second law : $m\vec{a} = \sum_i \vec{F}_i$

### 7.1.6 Limit n°6 - Thermodynamics of perfect gases

The kinetic theory of gases will be used to derive the usual laws of thermodynamics : the thermodynamic limit and the Newtonian limit correspond to the same approximation of generally covariant quantum mechanics. We have seen that the eikonal approximation allows to recover the classical limit from the quantum theory : in this approximation the modes of the fields behaves as pointlike particles. We first consider a gas composed of N identical particles in Minkowski spacetime. As we will see, a macroscopic description of the gas requires the introduction of new concepts, related to the statistical behavior of the particles. In the continuous approximation, the distribution of the positions and momenta of the particles is described by a function $f(\vec{x}, \vec{p})$ where $\vec{x}$ and $\vec{p}$ are functions of the time coordinate $t$. The function $f(\vec{x}, \vec{p})$ is called the distribution function in literature [135] [136]. The density of particles having momenta between $\vec{p}$ and $\vec{p} + d\vec{p}$ is :

$$ dn(\vec{x}) = f(\vec{x}, \vec{p}) d^3 p = f(\vec{x}, \vec{p}) dp^1 dp^2 dp^3 \qquad (7.56) $$

The density of particles is then defined as [135] :

$$ n(\vec{x}) = \int f(\vec{x}, \vec{p}) d^3 p = \int f(\vec{x}, \vec{p}) dp^1 dp^2 dp^3 \qquad (7.57) $$

Consequently, the mass density is $\rho(\vec{x}) = mn(\vec{x})$ where m is the mass of each particle. The number of particles occupying positions between $\vec{x}$ and $\vec{x} + d\vec{x}$ and momenta between $\vec{p}$ and $\vec{p} + d\vec{p}$ is given by :

$$ dN = f(\vec{x}, \vec{p}) d^3 x d^3 p = f(\vec{x}, \vec{p}) dx^1 dx^2 dx^3 dp^1 dp^2 dp^3 \qquad (7.58) $$

The number of particles is :

$$ N = \int f(\vec{x}, \vec{p}) d^3 x d^3 p = \int f(\vec{x}, \vec{p}) dx^1 dx^2 dx^3 dp^1 dp^2 dp^3 \qquad (7.59) $$

In flat space, $\vec{x}$ and $\vec{p}$ are conjugate variables with respect to the Fourier transform, as we have seen in [Section 2]. If the integration domain is infinite, the above equation is well defined only if $f$ is separable : $f(\vec{x}, \vec{p}) = g_1(\vec{x})g_2(\vec{p})$.
When the equilibrium is reached, the number $dN$ must be constant in time. The volume element $d^3xd^3p$ is in fact constant in time when the motion is determined by the Hamilton equations of motion with a standard Hamiltonian:

$$ \frac{dx^i}{dt} = \frac{\partial H}{\partial p^i} ; \quad \frac{dp^i}{dt} = -\frac{\partial H}{\partial x^i} \quad \text{where} \quad H(\vec{x}, \vec{p}) = \sqrt{|\vec{p}|^2 c^2 + m^2 c^4} + V(\vec{x}) \qquad (7.60) $$

Indeed, when $x^1 \to x^1 + \dot{x}^1 dt = x^1 + \frac{\partial H}{\partial p^1} dt = \tilde{x}^1$ and when $p^1 \to p^1 + \dot{p}^1 dt = p^1 - \frac{\partial H}{\partial x^1} dt = \tilde{p}^1$, the determinant of the transformation is

$$ J = \det \begin{bmatrix} \frac{\partial \tilde{x}^1}{\partial x^1} & \frac{\partial \tilde{x}^1}{\partial p^1} \\ \frac{\partial \tilde{p}^1}{\partial x^1} & \frac{\partial \tilde{p}^1}{\partial p^1} \end{bmatrix} = \det \begin{bmatrix} 1 + \frac{\partial^2 H}{\partial x^1 \partial p^1} dt & \frac{\partial^2 H}{\partial (p^1)^2} dt \\ -\frac{\partial^2 H}{\partial (x^1)^2} dt & 1 - \frac{\partial^2 H}{\partial p^1 \partial x^1} dt \end{bmatrix} = 1 - \left( \left( \frac{\partial^2 H}{\partial x^1 \partial p^1} \right)^2 - \frac{\partial^2 H}{\partial (x^1)^2} \frac{\partial^2 H}{\partial (p^1)^2} \right) dt^2 \quad (7.61) $$

which means

$$ \frac{dJ}{dt} = \lim_{dt \to 0} \frac{1 - \left( \left( \frac{\partial^2 H}{\partial x^1 \partial p^1} \right)^2 - \frac{\partial^2 H}{\partial (x^1)^2} \frac{\partial^2 H}{\partial (p^1)^2} \right) dt^2 - 1}{dt} = \lim_{dt \to 0} \left( \left( \frac{\partial^2 H}{\partial x^1 \partial p^1} \right)^2 - \frac{\partial^2 H}{\partial (x^1)^2} \frac{\partial^2 H}{\partial (p^1)^2} \right) dt = 0 \quad (7.62) $$

Then $d\tilde{x}^1 d\tilde{p}^1 = \sqrt{J} dx^1 dp^1 = dx^1 dp^1$. The same is true for any pair of conjugate coordinates $(x^i, p^i)$ where $i = 1, 2, 3$ such that $dx^1 dx^2 dx^3 dp^1 dp^2 dp^3$ is constant in time. The quantity $dN$ is then constant if and only if $\frac{d}{dt} f(\vec{x}, \vec{p}) = 0$. If the potential $V(\vec{x})$ is arbitrary, the energy $H(\vec{x}, \vec{p})$ (understood as the $0^{th}$ component of the 4-momentum) is the only constant of motion: the condition $\frac{d}{dt} f(\vec{x}, \vec{p}) = 0$ implies $f(\vec{x}, \vec{p}) = f_0(H(\vec{x}, \vec{p})) = f_0(\sqrt{|\vec{p}|^2 c^2 + m^2 c^4} + V(\vec{x}))$. Furthermore, we have seen below equation (7.59) that $f$ should be separable to be well defined on an infinite integration domain: $f(\vec{x}, \vec{p}) = g_1(\vec{x})g_2(\vec{p})$. We conclude that:

$$ f(\vec{x}, \vec{p}) = N_0 e^{-\beta H(\vec{x}, \vec{p})} = N_0 e^{-\beta \left( \sqrt{|\vec{p}|^2 c^2 + m^2 c^4} + V(\vec{x}) \right)} = N_0 e^{-\beta \sqrt{|\vec{p}|^2 c^2 + m^2 c^4}} e^{-\beta V(\vec{x})} = g_1(\vec{x}) g_2(\vec{p}) \quad (7.63) $$

where $N_0$ and $\beta$ are two constants. $f(\vec{x}, \vec{p})$ is integrable only if $\beta > 0$. We now consider the non relativistic limit when the interaction between the particles is negligible, i.e. $V(\vec{x}) = 0$, which means that the density $n(\vec{x})$ is approximatly constant. We assume that the particles are in a cubic box of volume $V = L^3$. With $V(\vec{x}) = 0$, we write $f(\vec{x}, \vec{p}) \approx N_0 e^{-\beta mc^2} e^{-\beta \frac{|\vec{p}|^2}{2m}} = N e^{-\beta \frac{|\vec{p}|^2}{2m}}$. Writing $\vec{p} = m\vec{v}$ and $N' = \sqrt{m^3 N}$, the corresponding distribution becomes, in spherical coordinates:

$$ \int f(\vec{x}, \vec{p}) d^3 x d^3 p = \sqrt{m^3 N} \int e^{-\beta \frac{m|\vec{v}|^2}{2}} d^3 v = N' \int_{-\infty}^{+\infty} e^{-\beta \frac{m|\vec{v}|^2}{2}} dv^1 dv^2 dv^3 = 4\pi N' \int_0^{+\infty} e^{-\beta \frac{m v^2}{2}} v^2 dv \quad (7.64) $$

where we assumed that the spherical symmetry is approximatly valid. Writing $\beta = k_B T$ where $k_B$ and $T$ will be defined later, we must fix $4\pi N' = N \left( \frac{m}{2\pi k_B T} \right)^{3/2}$ to fulfill equation (7.59):

$$ \int f(\vec{x}, \vec{p}) d^3 x d^3 p = 4\pi N' \int_0^{+\infty} e^{-\beta \frac{m v^2}{2}} v^2 dv = N \int_0^{+\infty} \left( \frac{m}{2\pi k_B T} \right)^{3/2} e^{-\beta \frac{m v^2}{2 k_B T}} v^2 dv = N \quad (7.65) $$

The function $\left( \frac{m}{2\pi k_B T} \right)^{3/2} e^{-\frac{m v^2}{2 k_B T}} v^2$ is well known in literature: it is the Maxwell molecular-velocity distribution function [135]. As explained in literature, "a variety of experimental verifications of the Maxwell velocity distribution have been made" [135]. By example, in 1946, O. Stern declared in his Nobel lecture: "by measuring the intensity (the number of cesium atoms arriving per second) at the end of the beam perpendicular to it as a function of the distance from the straight line, we get directly the distribution of velocities [...] the agreement with Maxwell's law is very good" [137]. From Maxwell's law, we can compute the internal energy $U$ of the system. Since we neglected the potential by assuming $V(\vec{x}) = 0$, the total energy of the particles, called $U$, is indeed obtained by adding the kinetic energy of each particle. The average kinetic energy per particles is $\overline{E_K} = \frac{U}{N}$

$$ \frac{U}{N} = \overline{E_K} = \frac{1}{2}mv^2 = \left( \frac{m}{2\pi k_B T} \right)^{3/2} \int_0^{+\infty} \frac{1}{2}mv^2 e^{-\frac{mv^2}{2k_B T}} v^2 dv = \frac{3}{2}k_B T \Leftrightarrow U = \frac{3}{2}Nk_B T \quad (7.66) $$

We use the relation $U = \frac{3}{2}Nk_B T = N\overline{E_K}$ to define the temperature $T$ and Boltzman's constant $k_B$ which determines the temperature scale. The temperature is then proportional to the average kinetic energy of the particles.

The concept of temperature $T$ is associated with the concept of heat $Q$: they are both fundamental concepts to derive the second law of thermodynamics. "Heat is defined as energy transferred by virtue of a temperature difference. It flows from regions of higher temperature to regions of lower temperature. [...] The basic modes of
heat transfer are conduction, radiation, and convection" [138]. The fact that heat flows from regions of higher temperature to regions of lower temperature can be deduced from our interpretation of temperature, based on the kinetic theory of gases. By example: "The conduction process takes place at the molecular level and involves the transfer of energy from the more energetic molecules to those with a lower energy level. This can be easily visualized within gases, where we note that the average kinetic energy of molecules in the higher-temperature regions is greater than that of those in the lower-temperature regions. The more energetic molecules, being in constant and random motion, periodically collide with molecules of a lower energy level and exchange energy and momentum. In this manner there is a continuous transport of energy from the high-temperature regions to those of lower temperature. In liquids [...] the molecular energy exchange process is qualitatively similar to that in gases" [138]. A description of the radiation process and the convection process can also be found in [138]. The above analysis explains why the net transfer of heat spontaneously occurs from higher-temperature regions to lower-temperature regions: it is a macroscopic consequence of the collision between microscopic particles. The second law of thermodynamics, according to R. Clausius' statement is: "Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time" [139]. The second law of thermodynamics then follows directly from our microscopic interpretation of temperature.

We define a thermal equilibrium by the absence of macroscopic (net) heat exchange between two systems which exchange heat at the microscopic level: the two systems then have equal temperatures. We conclude that "all systems which are in thermal equilibrium with a given system are also in thermal equilibrium with each other. [...] it is the zeroth law of thermodynamics" [140]. We consider again our cubic box of volume $V = L^3$ filled with particles. The collisions between the particles and the walls of the box are elastic collisions. We examine the motion of a particle along the $x_1$ direction. The time between two collisions with the walls is $L/v_1$ and the time between two collisions with the same wall is $\Delta t = 2L/v_1$ where $v_1$ is the velocity of the particle along the $x_1$ direction. When the particle collides with a wall, its change of momentum is $\Delta p_1 = 2mv_1$. This happens periodically, with period $2L/v_1$, for each particle inside the box. With $N$ particles, the force exerted on a wall is then $F = \frac{\Delta p_1}{\Delta t} = Nmv_1^2/L$ where $v_1^2$ is the average squared velocity along the $x_1$ direction. The analysis is independent of the velocity of each particle along the $x_2$ and $x_3$ directions. Since the symmetry of the problem implies that $v_1^2 = v_2^2 = v_3^2 = \frac{1}{3}v_1^2$, the pressure inside the box is $P = F/L^2 = \frac{N}{3V}mv_1^2 = \frac{2}{3V}NE_K = Nk_B T/V$. We recognize the perfect gas law:

$$PV = Nk_B T \qquad (7.67)$$

From the above analysis, we will consider 3 ways to vary the total energy of a system: the heat transfert $Q$, the work $W$, and matter exchanges. We already interpreted heat as a transfer of kinetic energy from the hotter to the cooler region. If matter is exchanged, each particle exchanged is associated to a transfert of kinetic energy (i.e. the kinetic energy of the exchanged particle). The work $W$ is interpreted according to the usual mechanical definition: $W = \int \vec{F} \cdot d\vec{x}$ where $\vec{F}$ is the force. The work made on a particle changes its kinetic energy $E_K$ since:

$$\frac{dW}{dt} = \frac{d}{dt} \int \vec{F} \cdot d\vec{x} = \frac{d}{dt} \int \vec{F} \cdot \vec{v} dt = \vec{F} \cdot \vec{v} = \frac{dm\vec{v}}{dt} \cdot \vec{v} = \frac{d}{dt} \left( \frac{1}{2} m \vec{v} \cdot \vec{v} \right) = \frac{d}{dt} \left( \frac{1}{2} m v^2 \right) = \frac{dE_K}{dt} \qquad (7.68)$$

Consequently, heat transfert, work, and matter exchange modify the amount of kinetic energy of a system, the internal energy $U$ being the sum of the kinetic energies of each particle. We then write the infinitesimal variation of the internal energy as:

$$dU = \delta Q + \delta W + \mu dN \qquad (7.69)$$

where $\delta Q$ is a small amount of heat, $\delta W$ a small amount of work, and $\mu dN$ the energy associated to a change of the number of particles by a small amount $dN$ when the other state variables (by example the temperature $T$ and the pressure $P$) are kept constant. $\mu$ is the chemical potential. The above expression was required to obtain $dU = \delta Q + \delta W$ when the number of particles is conserved ($dN = 0$), such that only heat and work contribute. Equation (7.69) is the first law of thermodynamics [140]: it has been deduced from the kinetic theory of gases (we assumed that matter is made of pointlike particles, allowing to interpret each term $\delta Q, \delta W$ and $\mu dN$ which modify the internal energy of the thermodynamical system). We used the symbol $\delta$ instead of $d$ to denote the small amount of heat $\delta Q$ and the small amount of work $\delta W$: this notation emphasizes the fact that "the work performed in a process, and also the transferred heat, not only depends on the initial and final state of the system, but also on the way of performing the process. This, however, means that work and heat are not suited to describing a macroscopic state in a unique way. They are not state quantities! Mathematically, this means that work and heat are not exact (i.e., total) differentials" [140]. On the contrary, $dU$ and $dN$ are exact differential because they only depend on the initial and final states, not on the process: this is obvious because $U$ is the sum of the kinetic energy of each particle inside the system, while $N$ is the number of particles. To illustrate the fact that heat and work depend on the way of performing the process, we compare an isothermal compression and an adiabatic compression, applied to a closed system ($dN = 0$). We consider a gas in a box of volume $V_i$ surrounded by a thermostat at temperature $T$, equal to the temperature of the gas. We consider two ways to reduce the volume of the box:
- Process n° 1 : We decrease the initial volume of the box (with mechanical work) from $V_i$ to $V_f$, sufficiently slowly to maintain the temperature of the gas equal to the temperature of the thermostat. This is an isothermal compression. When the volume $V_f$ is reached, the gas is in equilibrium (the gas remains in equilibrium during the whole process).

- Process n° 2 : We rapidly decrease the volume of the gas from $V_i$ to $V_f$, the temperature of the gas becomes greater than the temperature of the pressostat. This is an adiabatic compression. When the volume $V_f$ is obtained the gas is hotter than the thermostat, and some time is needed to reach equilibrium (the gas must exchange heat with the thermostat to reach equilibrium).

The first process is called reversible and requires less work than the second process which is called irreversible. "It is a daily experience that a process in an isolated system proceeds by itself until an equilibrium is reached. Since such processes do not reverse themselves, they are called irreversible. Examples of such processes are nearly all processes of daily life, in particular the expansion of a gas from a smaller into a larger volume or all processes which produce friction heat. It is characteristic for irreversible processes that they proceed over nonequilibrium states. On the other hand, processes which proceed only over equilibrium states are called reversible. A reversible process is an idealization which is, strictly speaking, non existent. [...] The importance of reversible changes of state is the following : for every small step of the process the system is in an equilibrium state with definite values of the state quantitites, so that the total changes of the state variables can be obtained by integrating over the infinitesimal reversible steps. For irreversible processes this is not possible. During an irreversible process it is in general not possible to attribute values to the state quantities" [140]. Reversible processes are theoretically important to define state functions and state variables. We now know that the work is not a state variable because it depends on the way of performing the process. When the equilibrium is reached in a cubic box filled with a gas, the force exerted on a face orthogonal to the $x_1$ direction is $F_1 = PL^2$. If the face slightly moves outward but almost remains in a state of equilibrium, the work done is $F_1dx_1 = PL^2dx_1 = PdV$. The work absorbed by the system is $\delta W = -PdV < 0$ : our sign convention is consistent with the fact that increasing the volume decreases the internal energy of the system. The volume $V$ is of course a state variable, like $N$ or $U$ : "the change of total energy is independent of the way the procedure takes place [...] The internal energy therefore possesses an exact differential. [...] the work has the form $\delta W_{rev} = -PdV$ only for reversible processes [140]". Furthermore, "The first law holds independently of whether a change of state is reversible or irreversible" [140] :

$$dU = \delta Q_{rev} + \delta W_{rev} + \mu dN = \delta Q_{irr} + \delta W_{irr} + \mu dN \quad (7.70)$$

"For reversible process one requires the least work or produces the most work respectively, while for irreversible processes a part of the work is always converted into heat" [140] : $\delta W_{irr} > \delta W_{rev}$. Consequently, $\delta Q_{irr} < \delta Q_{rev}$. From the mathematical point of view, the fact that $\delta Q_{rev}$ is not an exact differential can be easily shown. If $df(x_1, x_2, x_3)$ is an exact differential, then :

$$df = \frac{\partial f}{\partial x_1} dx_1 + \frac{\partial f}{\partial x_2} dx_2 + \frac{\partial f}{\partial x_3} dx_3 \quad \& \quad \frac{\partial}{\partial x_i} \left( \frac{\partial f}{\partial x_j} \right)_{x_k} = \frac{\partial}{\partial x_j} \left( \frac{\partial f}{\partial x_i} \right)_{x_k} \quad (7.71)$$

where $i, j, k = 1, 2, 3$. The notation $(...)(x_k)$ means that $x_k$ is kept constant when the partial derivative is computed (this widespread notation is useful to explicitly indicate the state variables used to define the state function $f$). Writing $\mu = \mu(N, T, V)$, $U = \frac{3}{2}Nk_B T$, $P = \frac{Nk_B T}{V}$, we compute $\delta Q_{rev}(N, T, V)$ :

$$\begin{aligned} \delta Q_{rev} &= dU + PdV - \mu dN \\ \delta Q_{rev} &= \frac{3}{2}Nk_BdT + \frac{3}{2}k_BTdN + \frac{Nk_BT}{V}dV - \mu dN \quad (7.72) \\ \delta Q_{rev} &= \frac{3}{2}Nk_BdT + \frac{Nk_BT}{V}dV + (-\mu + \frac{3}{2}k_BT)dN \end{aligned}$$

If we assume that the above equation has the form :

$$\delta Q_{rev} = \left(\frac{\partial Q}{\partial T}\right)_{N,V} dT + \left(\frac{\partial Q}{\partial V}\right)_{N,T} dV + \left(\frac{\partial Q}{\partial N}\right)_{T,V} dN \quad (7.73)$$

Then

$$\frac{\partial}{\partial V} \left( \frac{\partial Q}{\partial T} \right)_N = \frac{\partial}{\partial V} \left( \frac{3}{2} N k_B \right) = 0 \quad \neq \quad \frac{\partial}{\partial T} \left( \frac{\partial Q}{\partial V} \right)_N = \frac{\partial}{\partial T} \frac{N k_B T}{V} = \frac{N k_B}{V} \quad (7.74)$$

The condition expressed in the right hand side of equation (7.71) is not fulfilled in equation (7.74) : $\delta Q_{rev}$ is not an exact differential. Up to this point, we consider a reversible process, such that :

$$dU = \delta Q_{rev} - PdV + \mu dN \quad (7.75)$$
W. Greiner explains that the "task of thermodynamics is to define appropriate physical quantities (the state quantities), which characterize macroscopic properties of matter, the so-called macrostate, in a way which is as unambiguous as possible, and to relate these quantities by means of universally valid equations (the equations of state and the laws of thermodynamics)" [140]. The variables $\mu$, $P$ and $T$ are intensive variables : when the equilibrium is reached, intensive variables do not depend on the amount of matter of the system. On the contrary, $V$ and $N$ are extensive variables : they are proportional to the amount of matter, like $U$. We now introduce a state variable, $S$, such that $dU(S, N, V)$ takes the form :

$$ 
\begin{aligned}
dU &= \left(\frac{\partial U}{\partial S}\right)_{N,V} dS + \left(\frac{\partial U}{\partial V}\right)_{N,S} dV + \left(\frac{\partial U}{\partial N}\right)_{V,S} dN \\
dU &= \left(\frac{\partial U}{\partial S}\right)_{N,V} dS - P dV + \mu dN
\end{aligned}
\qquad (7.76) $$

Since $U(N, T, V) = \frac{3}{2}Nk_B T$, $S$ can not be identified with the temperature $T$ because $(\frac{\partial U}{\partial V})_{T,N} = 0$ while $(\frac{\partial U}{\partial V})_{S,N} = -P$ according to equation (7.76). The state variable $S$ will be called 'entropy'. Our aim is to identify (7.75) with equation (7.76). In equation (7.75) all quantities are expressed as functions of $T, N, V$ : the strategy is then to identify $\delta Q_{rev}(T, N, V)$ with $f(T, N, V)dS$ where $dS$ is an exact differential while $f(T, N, V)$ is a function which should be determined. According to the usual terminology, $f(T, N, V)$ is an 'integrating factor'. Equation (7.72) becomes :

$$ 
\begin{aligned}
\delta Q_{rev} &= dU + P dV - \mu dN = \frac{3}{2} N k_B dT + \left( P + \left( \frac{\partial U}{\partial V} \right)_{T,N} \right) dV + \left( \frac{3}{2} k_B T - \mu \right) dN \\
\frac{\delta Q_{rev}}{f} &= dS = \left( \frac{1}{f} \frac{3}{2} N k_B \right) dT + \left( \frac{1}{f} \frac{N k_B T}{V} \right) dV + \left( \frac{1}{f} \left( \frac{3}{2} k_B T - \mu \right) \right) dN
\end{aligned}
\qquad (7.77) $$

From the expression $\delta Q/f = dS$, we can put restrictions on $f(N, V, T)$. This has been done by M. Born by example [141]. We nevertheless follow a new approach. We first consider a box which exchanges an amount of heat $\delta Q$ with its surrounding. It is assumed that the exchange is reversible. The volume of the box is $V$, the number of particles inside the box is $N$ and the temperature is $T$. The heat exchange is represented on the figure below :

<figure><img src="image_5.png" /><figcaption>Figure 2 : Illustration of a box emitting heat</figcaption></figure>

The box can be conceptually divided into $\alpha$ virtual boxes, as illustrated with $\alpha = 7, 5$ on the figure below :

<figure><img src="image_6.png" /><figcaption>Figure 3 : The previous box is divided into 7,5 boxes</figcaption></figure>

The physical box has been conceptually split in 7 identical boxes of smaller size and a half box. Naturally, we have

$$ \delta Q_1 + \delta Q_2 + \delta Q_3 + \delta Q_4 + \delta Q_5 + \delta Q_6 + \delta Q_7 + \delta Q_{7,5} = \delta Q \qquad (7.78) $$

and

$$ \delta Q_1 = \delta Q_2 = \delta Q_3 = \delta Q_4 = \delta Q_5 = \delta Q_6 = \delta Q_7 = 2\delta Q_{7,5} = \delta Q/7.5 \qquad (7.79) $$

The symmetry of the problem implies that $\delta Q = 7, 5\delta Q_1$ where $\delta Q_1 = f(\frac{N}{7,5}, \frac{V}{7,5}, T)dS_1$ such that $\delta Q = 7, 5\delta Q_1 = 7, 5f(\frac{N}{7,5}, \frac{V}{7,5}, T)dS_1$. When the number $\alpha$ is arbitrary instead of being 7.5, we can write the following equation for the physical box :

$$ \alpha \delta Q_1 = \alpha f(\frac{N}{\alpha}, \frac{V}{\alpha}, T) dS_1 \qquad (7.80) $$

Deriving this equation with respect to $\alpha$, we get :

$$ \delta Q_1 = f(\frac{N}{\alpha}, \frac{V}{\alpha}, T) dS_1 + \alpha \frac{\partial f(\frac{N}{\alpha}, \frac{V}{\alpha}, T)}{\partial \alpha} dS_1 \qquad (7.81) $$
Since $\delta Q_{rev} = f(N, V, T)dS$, we must have $\delta Q_1 = f(\frac{N}{\alpha}, \frac{V}{\alpha}, T)dS_1$: the comparison with equation (7.81) immediately implies $\frac{\partial f(\frac{N}{\alpha}, \frac{V}{\alpha}, T)}{\partial \alpha} = 0$, which imposes the condition $f(\frac{N}{\alpha}, \frac{V}{\alpha}, T) = f(\frac{N}{V}, T)$. According to the perfect gas law, $\frac{N}{V} = \frac{P}{k_B T}$ such that $f$ could be expressed as a function of $T$ and $P$. Since $f$ depends on intensive variables, the function itself is intensive. $\delta Q = f dS$ being an extensive quantity, we conclude that the entropy is an extensive state variable.

We now consider a box filled with a mixture containing $N_1$ particles of a gas, and $N_2$ particles of another gas. The volume of the box is $V$ and the temperature of the mixture is $T$. Since the entropy is extensive, the entropy of the mixture is $dS = dS_1 + dS_2$ where $dS_1$ and $dS_2$ are the entropies of each gas. When heat is exchanged, without work (i.e. $PdV = 0$) and without exchange of matter (i.e. $dN = 0$), using $U = \frac{3}{2}Nk_B T$ we can write that the variation of the internal energy $dU$ of the mixture is equal to the heat exchanged $\delta Q$ by each gas:

$$dU = dU_1 + dU_2 = \frac{3}{2}(N_1 + N_2)k_B dT = f(\frac{N_1}{V}, T)dS_1 + f(\frac{N_2}{V}, T)dS_2 \quad (7.82)$$

And we can also write that the variation of internal energy is equal to the heat exchanged by the mixture, where $dS = dS_1 + dS_2$

$$dU = f(\frac{N_1 + N_2}{V}, T)dS = f(\frac{N_1 + N_2}{V}, T)(dS_1 + dS_2) \quad (7.83)$$

The comparison of equations (7.82) and (7.83) implies $f(\frac{N_1}{V}, T)dS_1 = f(\frac{N_1+N_2}{V}, T)dS_1$ and $f(\frac{N_2}{V}, T)dS_2 = f(\frac{N_1+N_2}{V}, T)dS_2$: this is possible only if $f(\frac{N_1}{V}, T) = f(T)$. Injecting $f = f(T)$ in equation (7.77) we get

$$dS = (\frac{1}{f(T)}\frac{3}{2}Nk_B)dT + (\frac{1}{f(T)}P)dV + \left(\frac{1}{f(T)}(\frac{3}{2}k_B T - \mu)\right) dN \quad (7.84)$$

The above equation should be combined with the condition $\frac{\partial}{\partial N}\frac{\partial S}{\partial T} = \frac{\partial}{\partial T}\frac{\partial S}{\partial V}$, required to define an exact differential. Consequently:

$$\frac{\partial}{\partial V} \left( \frac{1}{f(T)} \frac{3}{2} N k_B \right) = 0 = \frac{\partial}{\partial T} \left( \frac{1}{f(T)} \frac{N k_B T}{V} \right) \quad (7.85)$$

We then deduce that $f(T) = c_1 T$ where the constant $c_1$ would only rescale the entropy in the equation $\delta Q_{rev} = f(N, V, T)dS$: we then take $c_1 = 1$ such that $\delta Q_{rev} = T dS$. The condition that $dS$ is an exact differential now determines the expression of $\mu(T, V, N)$. Writing $S = S(T, V, N)$, equation (7.72) can be recast:

$$\begin{aligned} dS &= \frac{\delta Q_{rev}}{T} = \frac{1}{T} \left( \frac{3}{2} N k_B dT + \frac{N k_B T}{V} dV - \left( \mu - \frac{3}{2} k_B T \right) dN \right) \\ &= \frac{3 N k_B}{2 T} dT + \frac{N k_B}{V} dV - \left( \frac{\mu}{T} - \frac{3}{2} k_B \right) dN \\ &= \frac{\partial S}{\partial T} dT + \frac{\partial S}{\partial V} dV + \frac{\partial S}{\partial N} dN \end{aligned} \quad (7.86)$$

from which we deduce:

$$\begin{aligned} \frac{\partial}{\partial V} \frac{\partial S}{\partial N} &= \frac{\partial}{\partial N} \frac{\partial S}{\partial V} &\Leftrightarrow & -\frac{1}{T} \frac{\partial \mu}{\partial V} &= \frac{k_B}{V} &\Rightarrow & \mu = -k_B T \ln(V) + f_1(T, N) \\ \frac{\partial}{\partial T} \frac{\partial S}{\partial N} &= \frac{\partial}{\partial N} \frac{\partial S}{\partial T} &\Leftrightarrow & -\frac{1}{T} \frac{\partial \mu}{\partial T} + \frac{\mu}{T^2} &= \frac{3k_B}{2T} &\Rightarrow & \mu = -k_B T[\ln(T^{\frac{3}{2}}) + f_2(V, N)] \end{aligned} \quad (7.87)$$

Consequently $\mu = -k_B T[\ln(VT^{\frac{3}{2}}) + f_2(V, N)] + f_1(T, N)$. Since $\mu$ is an intensive variable it can only depend on $T$ and the ratio $V/N$. Introducing the constants $K$ and $\mu'$ such that $\ln(k_B^2 K) = \mu'$, we get [140]:

$$\mu(T, V, N) = -k_B T[\ln(\frac{V}{N}T^{\frac{3}{2}}) + \mu'] = \Leftrightarrow \mu(T, V, N) = -k_B T \ln(\frac{V}{N}(k_B T)^{\frac{3}{2}} K) \quad (7.88)$$

Our aim is now to derive Euler's equation [140]. Since $U, S, V, N$ are extensive quantities, it follows that:

$$U(\lambda S, \lambda V, \lambda N) = \lambda U(S, V, N) \Leftrightarrow U((1+\epsilon)S, (1+\epsilon)V, (1+\epsilon)N) = U + \epsilon U \quad (7.89)$$

Writing $\lambda = 1 + \epsilon$ where $\epsilon \ll 1$ we get:

$$U((1+\epsilon)S, (1+\epsilon)V, (1+\epsilon)N) = U(S, V, N) + \frac{\partial U}{\partial S}\epsilon S + \frac{\partial U}{\partial V}\epsilon V + \frac{\partial U}{\partial N}\epsilon N = U + \epsilon(TS - PV + \mu N) \quad (7.90)$$

Combining equations (7.89) and (7.90), we now deduce that Euler's equation is valid:

$$U = TS - PV + \mu N \Leftrightarrow U + PV = TS + \mu N \quad (7.91)$$
With $U + PV = \frac{3}{2}Nk_B T + Nk_B T = \frac{5}{2}Nk_B T$ we compute $S = \frac{5}{2}Nk_B - \frac{\mu N}{T}$ or

$$S(T, V, N) = Nk_B \left[ \ln\left(\frac{V}{N}(k_B T)^{\frac{3}{2}} K\right) + \frac{5}{2} \right] \qquad (7.92)$$

If the constant $K$ in equation (7.92) is chosen as $K = (\frac{2\pi m}{h^2})^{3/2}$ the entropy of a monoatomic gas becomes :

$$ \begin{aligned} S(T, V, N) &= Nk_B \left[ \ln \left( \frac{V}{N} (k_B T)^{3/2} \left( \frac{2\pi m}{h^2} \right)^{3/2} \right) + \frac{5}{2} \right] \\ S(U, V, N) &= Nk_B \left[ \ln \left( \frac{V}{N} \left( \frac{U}{N} \right)^{3/2} \left( \frac{4\pi m}{3h^2} \right)^{3/2} \right) + \frac{5}{2} \right] \end{aligned} \qquad (7.93) $$

In standard textbooks, this entropy is called the Sackur-Tetrode entropy [140][136]. It can be inverted to express $U = U(S, N, V)$. The Sackur-Tetrode entropy can be derived from quantum considerations. We have seen that a particle is a concept valid in the eikonal approximation of the field equations (the eikonal approximation is used to derive the classical limit of quantum mechanics). In the non relativistic limit, we consider the Schrödinger equation :

$$i\hbar \frac{\partial}{\partial t} \psi(x_1, x_2, x_3, t) = \frac{\hbar^2}{2m} \left( \frac{\partial^2}{\partial x_1^2} + \frac{\partial^2}{\partial x_2^2} + \frac{\partial^2}{\partial x_3^2} \right) \psi(x_1, x_2, x_3, t) \qquad (7.94)$$

Assuming that the field $\psi$ is defined in a cubic box of volume $V = L^3$ and obeys boundary conditions, each component $p_i$ of the momentum of each particle obeys the relation $p_i = mv_i = \frac{2\pi n_i h}{L} = \frac{n_i h}{L}$ where $i = 1, 2, 3$. Then we write $\vec{p} = \vec{n} \frac{h}{L}$ and $v = \frac{p}{m} = \frac{|\vec{p}|}{m} = \frac{|\vec{n}| h}{mL} = \frac{nh}{mL}$ where $\vec{n} = (n_1, n_2, n_3)$. Accordingly, when $L$ is sufficiently large to consider that $n$ is a continuous variable, the right hand side of equation (7.65) can be rewritten :

$$N = N \left( \frac{m}{2\pi k_B T} \right)^{3/2} \int_0^{+\infty} e^{-\frac{m}{2k_B T} \left( \frac{n h}{m L} \right)^2} \frac{n^2 h^3}{m^3 L^3} dn = \frac{N}{V} \left( \frac{1}{k_B T} \right)^{3/2} \left( \frac{h^2}{2\pi m} \right)^{3/2} \int_0^{+\infty} e^{-\frac{m}{2k_B T} \left( \frac{n h}{m L} \right)^2} n^2 dn \qquad (7.95)$$

The above equation is used to define the partition function per particle $\bar{Z}(T, V, N)$:

$$\bar{Z}(T, V, N) = \frac{1}{N} \int_{0}^{+\infty} e^{-\frac{m}{2k_{B}T} \left(\frac{n h}{m L}\right)^2} n^2 dn = \frac{V}{N} (k_{B}T)^{3/2} \left(\frac{2\pi m}{h^2}\right)^{3/2} \qquad (7.96)$$

From $\ln \bar{Z}$ we can compute the internal energy $U$, the pressure $P$ and the chemical potential $\mu$. Precisely :

$$U(T, V, N) = Nk_B T^2 \frac{\partial}{\partial T} \ln \bar{Z} = \frac{3}{2} N k_B T \qquad (7.97)$$

$$P(T, V, N) = Nk_B T \frac{\partial}{\partial V} \ln \bar{Z} = \frac{Nk_B T}{V} \qquad (7.98)$$

$$\mu(T, V, N) = -k_B T \ln \bar{Z} = -k_B T \ln \left( \frac{V}{N} (k_B T)^{3/2} \left( \frac{2\pi m}{h^2} \right)^{3/2} \right) \qquad (7.99)$$

Using $U = \frac{3}{2}Nk_B T$ and $P = \frac{Nk_B T}{V}$, it can now be shown by direct computation that :

$$ \begin{aligned} k_B T d(N \ln \bar{Z}) &= k_B T N \frac{\partial \ln \bar{Z}}{\partial T} dT + k_B T N \frac{\partial \ln \bar{Z}}{\partial V} dV + k_B T N \frac{\partial \ln \bar{Z}}{\partial N} dN + k_B T \ln \bar{Z} dN \\ k_B T d(N \ln \bar{Z}) &= \frac{3}{2} N k_B T dT + \frac{N k_B T}{V} dV - k_B T dN + k_B T \ln \bar{Z} dN \\ k_B T d(N \ln \bar{Z}) &= d\left(\frac{3}{2} N k_B T\right) - \frac{3}{2} k_B T dN + P dV - k_B T dN - \mu dN \end{aligned} \qquad (7.100) $$

$$k_B T d(N \ln \bar{Z}) = dU + P dV - \frac{5}{2} k_B T dN - \mu dN$$

$$k_B T d(N \ln \bar{Z} + \frac{5}{2} N) = dU + P dV - \mu dN$$

Defining the Sackur-Tetrode entropy $S$ according to

$$S = Nk_B \ln \bar{Z} + \frac{5}{2} N k_B = N k_B \left[ \ln \left( \frac{V}{N} (k_B T)^{3/2} \left( \frac{2\pi m}{h^2} \right)^{3/2} \right) + \frac{5}{2} \right] \qquad (7.101)$$

we finally get :

$$dU = T k_B d(N \ln \bar{Z} + \frac{5}{2} N) - P dV + \mu dN = \Leftrightarrow dU = T dS - P dV + \mu dN \qquad (7.102)$$
The usual properties of the entropy can be illustrated with two simple examples. We first consider a system A at temperature $T_A$ which spontaneously exchanges an infinitesimal amount of heat $\delta Q$ with a system B at temperature $T_B$ where $T_A > T_B$. We have concluded below equation (7.66) that the heat transfert will occur from A to B because $T_A > T_B$. Then $\delta Q = \delta Q_B = -\delta Q_A > 0$. Without work and without matter exchanged, we have $dU = \delta Q$. The total change of entropy is the sum of the changes of the entropies of the two systems :

$$dS = dS_B + dS_A = \frac{\delta Q_B}{T_B} + \frac{\delta Q_A}{T_A} = \delta Q(\frac{1}{T_B} - \frac{1}{T_A}) > 0 \rightarrow dS > 0 \quad (7.103)$$

The above example can be found in [135]. Now, we consider two gases A and B at temperature T in a box of volume 2V. There are $N_A$ particles of the gas A and $N_B$ particle of the gas B. The two gases are initially separated by a partition : each of them occupy half of the box, i.e. a volume $V_i = V$. When the partition is removed, the two gases mix in the total volume $V_f = 2V$. The mixing changes to total entropy according to :

$$\Delta S = \Delta S_A + \Delta S_B = N_A k_B \ln(\frac{V_f}{V_i}) + N_B k_B \ln(\frac{V_f}{V_i}) = (N_A + N_B) k_B \ln 2 > 0 \rightarrow \Delta S > 0 \quad (7.104)$$

This second example can be found in [136]. The previous considerations allowed to justify the three law of thermodynamics :

- Zeroth law : "all systems which are in thermal equilibrium with a given system are also in thermal equilibrium with each other"

- First law : $dU = TdS - PdV + \mu dN$

- Second law : "Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time"

In the first law, S has been identified with the Sackur-Tetrode entropy.

## 7.2 Interpretation of quantum mechanics

L. Smolin wrote in 2000 : "There are many different points of view about what quantum theory really asserts about reality and its relationship to the observer. [...] The founders of quantum theory, such as Einstein, Bohr, Heisenberg and Schrödinger, could not agree on these questions [...]. There is now no more agreement about what quantum theory means than when Einstein and Bohr first debated the question in the 1920s" [142]. In academic training, the standard interpretation of quantum mechanics is the Copenhagen interpretation : particles exhibit wave properties, with a probabilistic interpretation. In 1927, this interpretation was defended at the Solvay Conference by N. Bohr, M. Born, P. A. M. Dirac, W. Heisenberg and W. Pauli, against the deterministic but different points of view defended by E. Schrödinger, A. Einstein, H. A. Lorentz or L. De Broglie [143]. As explained by C. Cohen-Tannoudji, B. Diu and F. Laloë in their textbook : "After much trial and error which will not be described here, we arrived at the notion of wave-particle duality [...] : the corpuscular and wave aspects of light are inseparable; light behaves both as a wave and as a flow of particles, the wave allowing to compute the probability of a particle to appear" [83]. More explicitly, during his lectures, R. P Feynman explained to the students : "the absolute square of the amplitude $\psi(t, x_1, x_2, x_3)$ gives the probability density of finding the electron [...]. The wave function $\psi(t, x_1, x_2, x_3)$ for an electron in an atom does not, then, describe a smeared-out electron with a smooth charge density. The electron is either here, or there, or somewhere else, but wherever it is, it is a point charge. [...] The quantum behavior of atomic objects (electrons, protons, neutrons, photons, and so on) is the same for all, they are all 'particle waves,' or whatever you want to call them. So what we learn about the properties of electrons [...] will apply also to all 'particles,' including photons of light" [75]. Nevertheless, it should be remembered that it is nowadays clear that a photon is not a point particle, as we have seen in [Section 2]. R. P. Feynman explained that the probability of finding the point particle in a small region $d^3x$ is $|\psi(t, x_1, x_2, x_3)|^2 d^3x$ when the wave function is normalized such that $\int |\psi(t, x_1, x_2, x_3)|^2 d^3x = 1$. This explanation can be found in any textbook [21][34] [36][76][80][122]. Measuring the position of an electron produces the so called 'collapse of the wave function' [80][122] because it is meaningless to define the 'probability density of finding the electron' when the electron is observed in a certain position. Nevertheless, as emphasized by S. Weinberg : "in the Copenhagen interpretation a measurement changes the state of a system in a way that cannot itself be described by quantum mechanics [...]. This interpretation of quantum mechanics entails a departure from the dynamical assumptions of quantum mechanics during measurement. This is clearly unsatisfactory. If quantum mechanics applies to everything, then it must apply to a physicist's measurement apparatus, and to physicists themselves. On the other hand, if quantum mechanics does not apply to everything, then we need to know where to draw the boundary of its area of validity. Does it apply only to systems that are not too large? Does it apply if a measurement is made by some automatic apparatus, and no human reads the result?" [80]. We will now use generally covariant quantum mechanics to reconsider the Copenhagen interpretation.
### 7.2.1 The wave-particle duality and the complementarity principle

In 1905, A. Einstein published a famous paper on Planck's law of blackbody radiation. He called N the number of moles, T the temperature and $\beta = h/k_B$ where $k_B = R/N$ is the Boltzmann constant while R is the perfect gas constant. He wrote: "From existing observations of the blackbody radiation, it is clear that the law originally postulated by W. Wien [...] is not exactly valid. It is, however, well confirmed experimentally for large values of $\nu/T$. We shall base our analysis on this formula, keeping in mind that our results are only valid within certain limits". A. Einstein concluded: "Monochromatic radiation of low density (within the range of validity of Wien's radiation formula) behaves thermodynamically as though it consisted of a number of independent energy quanta of magnitude $R\beta\nu/N$" where $R\beta\nu/N = h\nu$ [144]. The energy quanta, discovered by A. Einstein in 1905, were called 'photons' up to 1926 [49]. In 1909, he published another famous paper in which, as told by L. Rosenfeld in 1973: "He found the surprising result that the fluctuation in the radiation field, which is governed by Planck's law, was the sum of the fluctuations that would arise from a classical radiation field and those that would arise from an assembly of independent photons" [145]. The two papers, written in 1905 and 1909, allow us to understand something important about the wave-particle duality. Planck's law of blackbody radiation, discovered in 1900, is:

$$ \frac{U}{V} = u = \int u_{\nu} d\nu = \int \frac{8\pi}{c^3} \frac{h\nu}{e^{\frac{h\nu}{k_B T}} - 1} \nu^2 d\nu \qquad (7.105) $$

where u is the energy density, U the energy inside the cavity, V the volume of the cavity and T the temperature. The two limiting cases are associated to suggestions published by W. Wien, J. W. S. Rayleigh and J. Jeans:

$$ \text{Wien : } u_{\nu} \rightarrow \frac{8\pi}{c^3} h\nu^3 e^{-\frac{h\nu}{k_B T}} \text{ when } \frac{h\nu}{k_B T} \gg 1 \quad ; \quad \text{Rayleigh-Jeans : } u_{\nu} \rightarrow \frac{8\pi}{c^3} k_B T \nu^2 \text{ when } \frac{h\nu}{k_B T} \ll 1 \qquad (7.106) $$

In 1909, A. Einstein considered a cavity of volume V with perfectly reflecting walls and assumed that a 'monochromatic radiation', with frequency in the interval $[\nu, \nu + d\nu]$, where $d\nu \ll \nu$, filled the cavity. Then:

$$ \frac{U}{V} = u = u_{\nu} d\nu = \frac{8\pi}{c^3} \frac{h\nu}{e^{\frac{h\nu}{k_B T}} - 1} \nu^2 d\nu \qquad (7.107) $$

A. Einstein computed, in a small region of the cavity, the mean square of the fluctuation of the energy, say $\sigma^2$:

$$ \sigma^2 = h\nu u_{\nu} V d\nu + \frac{c^3}{8\pi\nu^2} u_{\nu}^2 V d\nu = \frac{8\pi}{c^3} \left( h\nu \frac{h\nu}{e^{\frac{h\nu}{k_B T}} - 1} + \left( \frac{h\nu}{e^{\frac{h\nu}{k_B T}} - 1} \right)^2 \right) V \nu^2 d\nu \qquad (7.108) $$

If the radiation is made of classical particles, the expected result is $\sigma^2 = h\nu u_{\nu} V d\nu$ while, if the radiation is made of classical waves, the expected result is $\sigma^2 = \frac{c^3}{8\pi\nu^2} u_{\nu}^2 V d\nu$ [146][147][148] (details on the computations for physical waves have been published by H. A. Lorentz, P. Erhenfest, and P. Jordan [149][150]). The mean square of the fluctuation is the sum of the particles and the waves contributions: in 1909, A. Einstein explained that the law discovered by Planck in 1900 implied to generalize Maxwell's electrodynamics. In 1962, L. Rosenfeld gave a historical account of the discoveries made by M. Planck and A. Einstein between 1900 and 1909: "One was here led to describe the radiation by [...] the concept of particle. [...] This is also, like the monochromatic wave, an ideal conception. [...] the two are mutually exclusive. Yet, they are intimately related by the equations $E = \hbar\omega$, $\vec{p} = \hbar\vec{k}$. [...] These are at first sight paradoxical equations [...]. It is this logical problem which confers a very peculiar character on quantum mechanics. [...] the new logical instrument which was created by Bohr is called complementarity. Complementarity denotes the logical relation [...] between concepts which are mutually exclusive, and which therefore cannot be considered at the same time because that would lead to logical mistakes, but which are nevertheless both necessary in order to give a complete description of the situation" [151]. Two years later, he continued: "Complementarity is no system, no doctrine with ready-made precepts. [...] no formal definition of it can even be found in Bohr's writings [...]. The French are shocked by this breach of the Cartesian rules; they blame Bohr for indulging in 'clair-obscur' and schrouding himself in 'les brumes du Nord'. Germans in the thoroughness have been at work distinguishing several forms of complementarity and studying, in hundered of pages, their relations to Kant. Pragmatics Americans have dissected complementarity with the scalpel of symbolic logic and undertaken to define this gentle art of the correct use of words without any words at all" [152]. A. Einstein died in 1955. In 1951, in a letter to his friend M. Besso, he declared: "All these fifty years of conscious brooding have brought me no nearer to the answer to the question 'What are light quanta?'" [153]. In 1952 he continued: "we are as far from a truly rational theory (of the duality of light quanta and particles) as we were 50 years ago! [...] A truly rational theory would allow us to deduce the elementary particles (electron, etc.) and we would not be forced to state them a priori" [154]. The historical context being recalled, we now use generally covariant quantum mechanics to discuss equation (7.108). According to Planck's radiation law, in the
limit $\frac{h\nu}{k_B T} \ll 1$ the term associated to particles can be neglected in equation (7.108), while in the limit $\frac{h\nu}{k_B T} \gg 1$ the term associated to waves can be neglected. In the general case, the two concepts are not mutually exclusive, as assumed by the Copenhagen interpretation. This fact has been experimentally observed, an example was given by T. H. Yoon and M. Cho in 2021: "[T]he quantum object propagates through the double-path interferometer partly as particle-like [...] and partly as wave-like. [...] the wave-like and particle-like behaviors of the quanton (signal photon) are tunable quantities. [...] we anticipate that the interpretation based on the double-path interferometry experiments with ENBS (entangled nonlinear bi-photon source) will have fundamental implications for better understanding the principle of complementarity and the wave-particle duality relation quantitatively, leading to demystifying Feynman's mystery for the double-slit experiment explanation based on the quantum mechanics" [155]. As mentioned above, in 1905 A. Einstein had already introduced the energy quanta of magnitude $h\nu$ or the 'photons' as a limiting case associated to large frequencies (or short wavelengths). The photon is then associated to the eikonal approximation of Maxwell's equation, which allows to replace wave optics by geometric optics : in this approximation, photons follow the shortest path in agreement with the principle of least action. We have seen that massless particles are helicity eigenstates, which requires a complex electromagnetic field. We then generalize the usual electromagnetic Lagrangian density $-\frac{1}{4\mu_0}F^{\mu\nu}F_{\mu\nu}$ involving the real vector field $A_\mu$ by using $-\frac{1}{4\mu_0}F^{\mu\nu*}F_{\mu\nu}$ where $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$ and $A_\mu(x^\alpha) = A_\mu(x^\alpha)e^{-i\frac{S(x^\alpha)}{\hbar}} = \epsilon_\mu(x^\alpha)\mathcal{R}(x^\alpha)e^{-i\frac{S(x^\alpha)}{\hbar}}$. Here $\epsilon^\mu(x^\alpha)$ is one of the polarization vectors introduced in [Section 5.2] such that $\epsilon_\mu\epsilon^{\mu*} = -1$, a result which can be easily checked for all polarization vectors in equation (5.46) : to simplify the discussion we also assume that $\epsilon_\mu$ is real ($\epsilon_1^\mu = \frac{1}{\sqrt{2}}(\epsilon_R^\mu + \epsilon_L^\mu)$ is real by example). Then $A_\mu A^\mu = -\mathcal{R}^2$, while we write $\mathcal{R}^2 = \rho$.

$$ 
\begin{aligned}
-\frac{1}{4\mu_0} \mathcal{F}_{\mu\nu} \mathcal{F}^{\mu\nu*} &= -\left(\partial_\mu A_\nu - \frac{i\partial_\mu S}{\hbar} A_\nu - \partial_\nu A_\mu + \frac{i\partial_\nu S}{\hbar} A_\mu\right) \left(\partial^\mu A^\nu + \frac{i\partial^\mu S}{\hbar} A^\nu - \partial^\nu A^\mu - \frac{i\partial^\nu S}{\hbar} A^\mu\right) \\
&= -\frac{1}{4\mu_0} \left( F^{\mu\nu} F_{\mu\nu} + \frac{2}{\hbar^2} \partial_\nu S \partial^\nu S A^\mu A_\mu - \frac{2}{\hbar^2} (\partial_\nu S A^\nu)^2 \right) \\
&= -\frac{1}{4\mu_0} F^{\mu\nu} F_{\mu\nu} + \frac{\rho}{2\mu_0\hbar^2} \eta^{\mu\nu} \partial_\mu S \partial_\nu S + \frac{\rho}{2\mu_0\hbar^2} (\partial_\nu S \epsilon^\nu)^2
\end{aligned}
\qquad (7.109) $$

In the limit of long wavelengths, all terms involving partial derivatives of $S$ can be neglected, such that only the classical term $-\frac{1}{4\mu_0}F^{\mu\nu}F_{\mu\nu}$ remains : it is the Rayleigh-Jeans regime. In the limit of large frequencies $\partial_\mu S \to p_\mu$. As explained below equation (5.42), $\partial_\nu S\epsilon^\nu = p_\nu\epsilon^\nu$ vanishes. Furthermore each term of $\frac{1}{4\mu_0}F^{\mu\nu}F_{\mu\nu}$ is negligible when compared to each term of $\frac{\rho}{2\mu_0\hbar^2}\eta^{\mu\nu}\partial_\mu S\partial_\nu S$ in this limit : it is the Wien regime. $\frac{\rho}{2\mu_0\hbar^2}\eta^{\mu\nu}\partial_\mu S\partial_\nu S$ is the Lagrangian density of a massless particle. It deserves to be noticed that a spherically symmetric solution of the equation $\Box A^\mu = (\frac{1}{c^2}\frac{\partial^2}{\partial t^2} - \frac{\partial^2}{\partial r^2} - \frac{2}{r}\frac{\partial}{\partial r})A^\mu$ takes the form $A^\mu = \epsilon^\mu \frac{e^{-i\omega t + ik_r r}}{r} = \epsilon^\mu \mathcal{R}(r)e^{-i\frac{S(t,r)}{\hbar}}$ where $\omega = ck_r$. Using now the Lagrangian density $\mathcal{L}_{EM} = -\frac{1}{4\mu_0}F^{\mu\nu}F_{\mu\nu} + \frac{\rho}{2\mu_0\hbar^2}g^{\mu\nu}\partial_\mu S\partial_\nu S$ to define an electromagnetic action $S_{EM}$ in curved spacetime according to $S_{EM} = \int \mathcal{L}_{EM}\sqrt{g}d^4x$, we can easily compute the associated energy momentum tensor $T_{EM}^{\mu\nu}$:

$$ T_{EM}^{\mu\nu} = \frac{g^{\alpha\beta} F_{\nu\alpha} F_{\beta\mu} + \frac{1}{4} g_{\mu\nu} F^{\mu\alpha} F_{\mu\alpha}}{\mu_0} + \frac{1}{2\hbar^2\mu_0} \rho \partial^\alpha S \partial^\beta S - \frac{1}{2} g_{\mu\nu} \frac{\rho}{2\mu_0\hbar^2} g^{\mu\nu} \partial_\mu S \partial_\nu S \qquad (7.110) $$

The conversation law $\nabla_\alpha T_{EM}^{\alpha\beta} = 0$ implies :

$$ 0 = F_{\mu}^{\beta}(-\nabla_{\alpha}F^{\alpha\mu}) + \frac{1}{2\hbar^2\mu_0}\partial^{\beta}S\nabla_{\alpha}(\rho\partial^{\alpha}S) + \frac{1}{2\hbar^2\mu_0}\rho(\partial^{\alpha}S\partial_{\alpha}\partial^{\beta}S + \Gamma_{\sigma\alpha}^{\beta}\partial^{\sigma}S\partial^{\alpha}S) \qquad (7.111) $$

where $\nabla_\alpha F^{\alpha\mu}$ vanishes in vacuum according to Maxwell's equations (4.31). In the eikonal approximation $0 = \nabla_\alpha(\rho\partial^\alpha S)$ is the continuity equation while

$$ 0 = \partial^{\alpha} S \partial_{\alpha} \partial^{\beta} S + \Gamma_{\sigma\alpha}^{\beta} \partial^{\sigma} S \partial^{\alpha} S = p^{\alpha} \partial_{\alpha} p^{\beta} + \Gamma_{\sigma\alpha}^{\beta} p^{\sigma} p^{\alpha} = \hbar^2 (k^{\alpha} \partial_{\alpha} k^{\beta} + \Gamma_{\sigma\alpha}^{\beta} k^{\sigma} k^{\alpha}) \qquad (7.112) $$

is the equation of null geodesics [27][156], i.e. 'the shortest path', for the massless photon : we recover the laws of geometric optics. There is another simple way to derive the equations of null geodesics. In the eikonal approximation $g^{\mu\nu}\partial_\mu S\partial_\nu S = 0$ such that $0 = \partial^\beta(\frac{\rho}{2\mu_0\hbar^2}g^{\mu\nu}\partial_\mu S\partial_\nu S) = \frac{\rho}{2\mu_0\hbar^2}\partial^\beta(g^{\mu\nu}\partial_\mu S\partial_\nu S) + \frac{\partial^\beta\rho}{2\mu_0\hbar^2}g^{\mu\nu}\partial_\mu S\partial_\nu S$ while we already derived $\partial^\alpha S\partial_\alpha \partial^\beta S + \Gamma_{\sigma\alpha}^{\beta}\partial^\sigma S\partial^\alpha S = 0$ from the condition $\partial^\beta(g^{\mu\nu}\partial_\mu S\partial_\nu S) = 0$ in equation (5.3). We have then shown that the complex Maxwell's equations produces classical waves associated to 'photons' which behaves like particles in the geometric optics limit. On this subject G. F. R. Ellis, R. Maartens, and M. A. H. MacCallum explain in the book 'Relativistic cosmology' : "The basis of astronomical observations is the geometric optics limit of Maxwell's equations, supplemented by the quantum mechanical concept of a photon. The photon viewpoint enters into detector design (in the case of very distant galaxies, individual photons are detected)" [156]. Evidently, the geodesic path is only defined where $\rho \neq 0$ : the probability of finding a particle in a region where
$\rho = 0$ vanishes ($\rho$ being identified with $|\psi|^2$ for massive particles).

### 7.2.2 Multiparticle states

We now quote different authors discussing multiparticle systems and highlight apparent contradictions between assumptions widely used in quantum physics. In the first lines of his textbook 'Introduction to quantum mechanics' published in 1995, D. J. Griffith gave the following description of quantum mechanics: "There is no general consensus as to what its fundamental principles are, how it should be taught, or what it really means. Every competent physicist can do quantum mechanics, but the stories we tell ourselves about what we are doing are as various as the tales of Scheherazade, and almost as implausible. Richard Feynman [...] remarked 'I think I can safely say that nobody understands quantum mechanics' [122]". To describe multiparticle states of identical particles, the textbook stipulates: "Quantum mechanics neatly accommodates the existence of particles that are indistinguishable in principle: we simply construct a wave function that is noncommittal as to which particle is in which state. There are actually two ways to do it". Up to a normalisation constant, the expression given by D. J. Griffith is:

$$ \psi_{\pm}(\vec{r}_1, \vec{r}_2) = \psi_a(\vec{r}_1)\psi_b(\vec{r}_2) \pm \psi_b(\vec{r}_1)\psi_a(\vec{r}_2) \quad (7.113) $$

D. J Griffith continues: "Thus the theory admits two kinds of identical particles: bosons, for which we use the plus sign, and fermions, for which we use the minus sign" [122]. A generalization of equation (7.113) to states with an arbitrary number of particles can be easily found in literature (see [80][157] by example). Fixing $a = b$ in equation (7.113), we get $\psi_-(\vec{r}_1, \vec{r}_2) = 0$: this justifies the Pauli exclusion principle stating that no two identical fermions can be in the same state [158]. The Schrödinger equation describing two non-interacting identical particles in 1 dimension has the well known form [122][158][159][160][161]:

$$ i\hbar \frac{\partial \psi(t, r_1, r_2)}{\partial t} = \hat{H} \psi(t, r_1, r_2) = (E_1 + E_2) \psi(t, r_1, r_2) = \frac{\hbar^2}{2m} \frac{\partial^2 \psi(t, r_1, r_2)}{\partial r_1^2} + \frac{\hbar^2}{2m} \frac{\partial \psi(t, r_1, r_2)}{\partial r_2^2} \quad (7.114) $$

where $r_1, r_2$ are the coordinates of the two particles (in 3 dimensions, we would replace $r_1$ by $\vec{r}_1$ and $r_2$ by $\vec{r}_2$). E. Schrödinger published his equation in 1926 and examined the case of a diatomic molecule, writing $\psi$ as a function of 6 coordinates (3 coordinates for each atom of the molecule). He then observed: "the direct interpretation of this wave function of six variables in three dimensional space meets, at any rate initially, with difficulties of an abstract nature" [20]. The same year H. A. Lorentz indicated to him that a multiparticle wave function of the form $\psi = \psi(t, \vec{r}_1, \vec{r}_2, ..., \vec{r}_n)$ implies that it is impossible to "interpret the waves and vibrations physically" [20]. Also in 1926, A. Einstein wrote to P. Ehrenfest: "Schrödinger is, in the beginning, very captivating. But the waves in n-dimensional coordinate space are indigestible [...]" [163]. Similarly, to justify that Schrödinger's theory has a 'symbolical character', N. Bohr emphasized in 1927 that "there can be no question of an immediate connection with our ordinary conceptions because the 'geometrical' problem represented by the wave equation is associated with the so-called coordinate space, the number of dimensions of which is equal to the number of degrees of freedom of the system, and, hence, in general greater than the number of dimensions of ordinary space. [...] it would scarcely seem justifiable [...] to demand a visualization by means of ordinary spacetime pictures" [164]. In 1958, W. Heisenberg similarly recalled: "Born took up this idea of the probability waves and gave a clear definition of the mathematical quantity in the formalism, which was to be interpreted as the probability wave. It was [...] a wave in the many-dimensional configuration space, and therefore a rather abstract mathematical quantity. [...] One may call the waves in configuration space 'objective' when one wants to say that these waves do not depend on any observer; but one can scarcely call them 'real' unless one is willing to change the meaning of the word" [165]. In agreement with the Copenhagen interpretation, R. P. also explained to his students in the 1960s: "it must be emphasized that the wave function [...] is not like a real wave in space; one cannot picture any kind of reality to this wave as one does for a sound wave. Although one may be tempted to think in terms of 'particle waves' when dealing with one particle, it is not a good idea, for if there are, say, two particles, the amplitude to find one at $\vec{r}_1$ and the other at $\vec{r}_2$ is not a simple wave in three-dimensional space, but depends on the six space variables $\vec{r}_1$ and $\vec{r}_2$" [75].

It should now be observed that equation (7.113) significantly differs from the description of superfluid helium reproduced from J. F. Annett in [Section 2]. To describe the superfluid it was indeed possible to introduce a macroscopic wave function associated to many particles, without introducing the coordinates of each particle. The superfluid, with density $\rho(x^\alpha) = m|\psi_0(x^\alpha)|^2$ where $m$ is the mass of the Helium atom and $|\psi_0(x^\alpha)|^2$ the density of atoms, is visible with naked eyes. The macroscopic wave function is then defined in the usual spacetime. To describe the phenomenon of superconductivity in which electrons form pairs called 'Cooper pairs', J. F. Annett also states: "Ginzburg and Landau postulate that the order parameter $\psi$ should be a complex number, thinking of it as a macroscopic wave function for the superconductor in analogy with superfluid $^4$He [...] . In fact, we can even identify $|\psi|^2$ as the density of 'Cooper pairs' present in the sample" [40]. Similarly, to introduce Bose-Einstein condensates, he writes: "$n(\vec{r})$ is the density of atoms at point $\vec{r}$ in the trap. [...] As usual, the chemical potential $\mu$ is determined from the constraint of the constant total number of atoms in the trap, $N$, [where] $N = \int n(\vec{r})d^3r$.
[...] At zero temperature all the particles are in the condensate and $n(\vec{r}) = N|\psi_0(\vec{r})|^2$ where $\psi_0(\vec{r})$ is the ground state of the wave function" [40]. Remarkably, although equations (7.113) and (7.114) can be found in the chapter 'Atomic Physics' of the textbook 'Modern Physics' written by P.A. Tipler and R. A. Llewellyn [40], in the chapter 'The Schödinger Equation' the two authors discuss the scattering of beams of particles in 1 dimension and they indicate : "it is usual to normalize such wave functions in terms of the density of particles in the beam. Thus, $\int_a^b |\psi(x)|^2 dx = \int_a^b \rho dx = \int_a^b dN = N$ where dN is the number of particles in the interval dx and N is the number of particles in the interval $(b-a)$" [40]. To go further with this example, we quote the section entitled 'Relativistic scattering' of lectures notes on particle physics. The authors describe the scattering process in the rest frame of the target particles (a block of matter) : "we choose a finite volume V and normalize all the wave functions such that $\int_V |\psi|^2 d^3x = 1$. [...] For the classical wave function the density is $\rho = |\psi|^2$ so that the normalization gives one particle per volume V. This normalization is not Lorentz invariant. [...] for the plane wave solutions of the Klein-Gordon equation [...] it is now customary to define the box normalization such that there are 2E particles in the volume V. In other words $\rho = \frac{2E}{V}$. We refer to this as 2E particles per unit volume. [...] The number of beam particles that pass through unit area per unit time is given by $|\vec{v}_A|n_A$. The number of target particles per unit volume is $n_B$. For relativistic plane waves the density of particles n is proportional to $\rho = \frac{2E}{V}$ such that

$$ \text{flux} = |\vec{v}_A|n_A n_B \propto \frac{2|\vec{p}_A|c m_B c^2}{V} \qquad (7.115) $$

(Remember that in relativity $\vec{v} = \vec{p}/E$. For the Klein-Gordon waves we had indeed that the current density was $\vec{j} = \rho\vec{p}/E$)" [94]. Similarly, in the preface of his book 'Quantum field theory I', E. Zeidler emphasizes : "It is my intention to build a bridge between mathematicians and physicists". Later in the book, he discusses the Schrödinger equation and one can read : "the function $\psi$ describes a flow of identical particles on the real line which has current density vector $\vec{j}(x,t) = \rho(x,t)\vec{v}(x,t)$ with the particle density $\rho(x,t) = |\psi(x,t)|^2$ and the velocity vector $\vec{v}(x,t)$ at the point $x$ at time $t$. [...]" [40]. Superfluidity, Bose-Einstein condensates, superconductivity, and scattering of particles are 4 experimentally validated examples in which the wave function of a multiparticle system takes the form $\psi = \psi(x^\alpha)$ and not the form $\psi = \psi(x_1^\alpha, x_2^\alpha, \dots, x_n^\alpha)$ of equations (7.113) and (7.114). Furthermore, the examples of scattering validate our identification of the mass current $j_m^\alpha$ with $\rho \frac{\partial^\alpha S}{\partial \rho^\alpha}$ in the eikonal approximation (we normalized $\rho$ as a mass density). In [Section 2] we have seen that the same identification is used with superfluid Helium. We previously quoted H. A. Lorentz, N. Bohr, W. Heisenberg and R. P. Feynman to illustrate that it would be meaningless to interpret physically $|\psi|^2$ if $\psi$ is of the form $\psi = \psi(x_1^\alpha, x_2^\alpha, \dots, x_n^\alpha)$ because $\psi$ is not defined on the physical spacetime in this context. Nevertheless there are now good experimental reasons to believe that a wave function describing matter particles is in fact associated with the physical spacetime. By example, S. Cohen et al. claimed in 2013 : "The electron spatial distribution measured by our microscope is a direct macroscopic image of the projection of the microscopic squared modulus of the electron wave [...] Wave function microscopy aims at observing the spatial structure of the electronic wave function" [167]. Clearly, if $|\psi|^2$ can be measured in the physical space, then $|\psi|^2 = |\psi(x^\alpha)|^2$ where $x^\alpha = (ct, x^1, x^2, x^3)$ are the coordinates of the physical spacetime. Similarly, a paper published in 2021 indicates : "how to measure the absolute square of the wave function in momentum space $|\hat{\psi}(\vec{p})|^2$ in real experiments? The state-of-the art method is to use a COLd Target Recoil Ion Momen tum Spectroscopy (COLTRIMS) reaction microscope. The additional knowledge about the initial time [...] and position [...] of the electron in the spectrometer allows fo the reconstruction of the three-dimensional momentum distribution $|\hat{\psi}(\vec{p})|^2$ with a resolution of typically 1/100 atomic units" [168]. This illustrates again that the wave function must be defined in the physical space. Furthermore the form $\psi = \psi(x_1^\alpha, x_2^\alpha, \dots, x_n^\alpha)$ introduced in equation (7.113) has serious theoretical drawbacks. When a photon is studied, C. Cohen-Tannoudji, B. Diu, F. Laloë explain : "The information on a photon at the time t is given by the wave $\vec{E}(\vec{r}, t)$, solution of Maxwell's equations [...] $\vec{E}(\vec{r}, t)$ is interpreted as the probability amplitude for that a photon manifests its presence, at time t, at point $\vec{r}$ : this means that the corresponding probability is proportional to $|\vec{E}(\vec{r}, t)|^2$" [83]. Obviously all fields should be treated on equal footing because their particles transform into each other : we previously discussed the process $\pi_0 \rightarrow \gamma\gamma$. In this process, the pion is described by the Klein-Gordon equation (or the Schrödinger equation in the non relativistic limit) while the photons are described by Maxwell's equations. Although a plus sign must be used in equation (7.113) to describe bosons, photons are bosons but the coordinates $x_1, \dots, x_n$ of each photon do not appear in Maxwell's equations while the coordinates of each matter particles appear in the two-particle Schödinger equation (7.114). Furthermore, one must know how to compute the electric charge density $\rho_e(x^\mu)$ from the field $\psi(x^\mu)$ where $|\psi(x^\mu)|^2$ is interpreted as a probability density : by example, L. Landau and E. Lifshitz use the Schrödinger equation and indicate (we replace their electric charge $e$ by $q$ and their charge density $\rho$ by $\rho_e$) : "it is also easy to verify that the current and the charge density $\rho_e = q|\psi|^2$ statisfy [...] the continuity equation" [45]. Using this prescription, the Maxwell-Gauss law simply takes the form :

$$ \partial_{\alpha} F^{\alpha 0}(x^{\mu}) = \frac{\vec{\nabla} \cdot \vec{E}(x^{\mu})}{c} = \mu_0 j^0 \Leftrightarrow \vec{\nabla} \cdot \vec{E}(x^{\mu}) = \frac{j^0}{\epsilon_0 c} = \frac{\rho_e(x^{\mu})}{\epsilon_0} = \frac{q |\psi(x^{\mu})|^2}{\epsilon_0} \Rightarrow \vec{\nabla} \cdot \vec{E}(x^{\mu}) = \frac{q |\psi(x^{\mu})|^2}{\epsilon_0} \qquad (7.116) $$

In the above equation $\rho_e = j^0/c$ is the electric charge density. Maxwell's equations depend on the spacetime
coordinates $x^\mu$ but they do not depend on the coordinates of each matter particle or each photon : consequently there is no obvious way to rewrite equation (7.116) when $\psi(x^\mu)$ is replaced by $\psi(x_1^\mu, x_2^\mu)$ to describe the two particle state introduced in equation (7.113). Clearly we can not make the change $r_1 = r_2 = r$ in equation (7.113) since $\psi_-(r, r) = 0$. In the Copenhagen interpretation of quantum mechanics, the use of equations (7.113) and (7.114) is important because equation (7.113) justifies the Pauli exclusion principle. Nevertheless, we have identified 4 important problems with the notation $\psi(\vec{r}_1, \vec{r}_2)$ of equation (7.113) :

- **Problem 1** : Superfluidity, Bose-Einstein condensates, superconductivity, and scattering of particles are 4 experimentally validated examples in which the field describing a multiparticle system is written $\psi(x^\alpha)$ and not $\psi(x_1^\alpha, x_2^\alpha, \dots, x_n^\alpha)$;

- **Problem 2** : As explained by H. A. Lorentz, N. Bohr, W. Heisenberg, R. P. Feynman and many others, the wave function $\psi_+(\vec{r}_1, \vec{r}_2)$ is not defined in the physical spacetime, it is only a symbolical object : the fact that authors claim that it is possible to measure $|\psi|^2$ and $|\dot{\psi}|^2$ must then be explained;

- **Problem 3** : It must be explained how the two-particle wave function $\psi_+(\vec{r}_1, \vec{r}_2)$ can be consistently transformed into the electric charge density $\rho_e(\vec{r})$ appearing in the Maxwell-Gauss law of equation (7.116);

- **Problem 4** : The fact that electromagnetic fields and matter fields are not treated on equal footing must be explained. The explanation must be consistent with the fact that a wave function $\psi(x_1^\alpha, x_2^\alpha)$, obeying the Klein-Gordon equation (or the Schrödinger equation in the non relativistic limit), can be used to describe two $\pi_0$ mesons in a box, such that, if they both decay into photons, the box will only contain a field $A^\mu(x^\alpha)$ which will depend on the usual spacetime variables (and not on the coordinate of each photon). Said differently, at the beginning, the box contains two $\pi_0$ described by $\psi(x_1^\mu, x_2^\mu)$ and, when the $\pi_0$ have decayed, the box contains 4 photons described by $A^\mu(x^\alpha)$.

There is possibly a fifth problem. W. Pauli published his exclusion principle in 1925. Much later, in 1946, he declared during his Nobel lecture : "[...] an antisymmetrical function containing two particles in the same state is identically zero. [...] It is this possibility which actually holds for electrons. This situation appeared to me as disappointing in an important respect. Already in my original paper I stressed the circumstance that I was unable to give a logical reason for the exclusion principle or to deduce it from more general assumptions. I had always the feeling and I still have it today, that this is a deficiency"[169]. In his textbook on quantum mechanics, S. Weinberg wrote in 2013 : "today there is no interpretation of quantum mechanics that does not have serious flaws, and [...] we ought to take seriously the possibility of finding some more satisfactory other theory, to which quantum mechanics is merely a good approximation" [80]. The basic principle of generally covariant quantum mechanics is that all fields should be treated on equal footing. This clearly suggests to use the notation $\psi(x^\alpha)$ when many particles are involved : it is already done with superfluidity, superconductivity, Bose-Einstein condensates or particle scattering (and certainly other examples). But then a new way to express the exclusion principle is needed. In the non relativistic approximation, we consider two electrons, in a 1 dimensional box of length L. The interaction between the particles is neglected. As we have recalled in [Section 5.1], in particle physics, particles are identified with the modes of their field equations (this is why we used the notation $A^\mu = \epsilon^\mu e^{i\frac{-p\alpha x^\alpha}{\hbar}}$ to describe a photon, solution of Maxwell's equations). We write the wave function of the two electrons $\psi(t, x) = \psi_{n_1}(t, x) + \psi_{n_2}(t, x)$ with $\psi_{n_i}(t, x) = \frac{1}{\sqrt{L}}e^{i\frac{-E_{n_i}t+p_{n_i}x}{\hbar}}$ where $p_{n_i} = 2\pi n_i \hbar/L$, $E_{n_i} = \frac{p_{n_i}^2}{2m}$ and $i = 1$ or 2. With the normalization $\int_L |\psi_{n_i}|^2 dx = m$ where m is the mass, the electric charge in the box is :

$$ -\frac{q}{m} \int_L |\psi_{n_1} + \psi_{n_2}|^2 dx = -q(2 + \delta_{n_1 n_2}) \qquad (7.117) $$

where $\delta_{n_1 n_2}$ is a Kronecker delta. Since the electric charge in the box must be $-2q$, we must have $n_1 \neq n_2$ : the two particles should not be in the same state. This is an exclusion principle. In this example we considered the electric charge, but we could possibly consider the energy, the spin, or the mass (particles are classified by eigenvalues of mass and spin in particles physics [62]). Nevertheless, it must now be explained why bosons, which obey Bose-Einstein statistics, do not obey this exclusion principle associated to Fermi-Dirac statistics. Our exclusion principle followed from the assumption that the density of a multiparticle system would be given by $\rho = |\sum_n \psi_n|^2$ : we could totally avoid the modified principle by assuming $\rho = \sum_i |\psi_i|^2$ since this expression contains no interference between two eigenstates. In optical sciences, the expression $\rho = |\sum_i \psi_i|^2$ applies to completely coherent waves while the expression $\rho = \sum_n |\psi_n|^2$ applies to completely incoherent waves [170] [171]. It should then be recalled that "Coherence means the coordinated motion of several waves. When two or more waves are coherent, they will maintain a fixed and predictable phase relationship with each other. [...] As the emission acts occur without any coordination in the source, the resulting wave trains will not have any correlation in their phase values. [...] Consequently, the real light waves are incoherent"[170]. The dependence of photon statistics on the experimental set up is a well known fact : "The setup enables [...] to measure photon count distribution for both coherent and pseudothermal light sources yielding Poisson and Bose-Einstein distributions, respectively" [172] (see also [173]). It then seems possible to adopt a simple picture in which electronic waves are coordinated in a single atom (the exclusion principle is valid in the atom), while light waves produced by many independent atoms are incoherent (no exclusion principle in usual radiations), matter waves and light waves being
nevertheless described in the same manner (i.e. without the introduction of the coordinates of each particle in the Schrödinger equation). If no other suggestion is made to justify the exclusion principle, the only alternative is to resolve the four problems listed above and to give a logical reason for W. Pauli's exclusion principle : W. Pauli could not.

### 7.2.3 Operators and probabilities in quantum mechanics

We are now interested by the famous collapse of the wave function, and the associated probabilistic interpretation. To recall the Copenhagen interpretation, we will quote two well known textbooks (respectively written by W. Greiner and D. J. Griffith) and S. Weinberg's lecture notes. We are interested by a free particle in a box. We slightly modify the original notations used in the references but the text is kept unchanged. The reader interested by the missing details is invited to examine the original textbooks and lecture notes. We start with W. Greiner's textbook. "The question of how to interpret a wave describing a particle [...] was subject to discussion in the first years of quantum mechanics. [...] Max Born paved the way for the statistical interpretation of the wave describing a particle. He created the term 'guiding field'. The square of the amplitude of the wave function $\psi$ is the intensity. It ought to determine the probability of finding a particle at a certain place. [...] It is normalized to one, i.e. the amplitude of $\psi$ is chosen such that $\int_{-\infty}^{\infty} \psi\psi^* d^3x = 1$. This means that the particle must be somewhere in space. An example of wave functions that cannot be normalized [...] is the wave function $\psi(\vec{r}, t) = N e^{i(\vec{k}\cdot\vec{r} - \omega t)}$ where $N$ is a real constant. This plane wave describes the motion of a free particle with momentum $\vec{p} = \hbar\vec{k}$ and undefined locality. But we can define $\psi(\vec{r}, t) = N e^{i(\vec{k}\cdot\vec{r} - \omega t)}$ if we define all functions within a large, finite volume considered to be a cube with an edge length $L$. Very often a periodicity with period $L$ is selected as boundary condition; we require that $\psi(x_1, x_2, x_3) = \psi(x_1 + L, x_2, x_3) = \psi(x_1, x_2 + L, x_3) = \psi(x_1, x_2, x_3 + L)$. [...] hence it follows that $N = \frac{1}{\sqrt{L^3}} = \frac{1}{\sqrt{V}}$. Thus, we get the normalized wave function $\psi_{\vec{k}}(\vec{r}, t) = \frac{1}{\sqrt{V}} e^{i(\vec{k}\cdot\vec{r} - \omega_{\vec{k}}t)} = \psi_{\vec{k}}(\vec{r})e^{-i\omega_{\vec{k}}t}$; $\psi_{\vec{k}}(\vec{r}) = \frac{1}{\sqrt{V}} e^{i\vec{k}\cdot\vec{r}}$. The boundary conditions of our problem restrict the possible values of the vector $\vec{k} : \vec{k} = \frac{2\pi\vec{n}}{L}, \vec{k} = (k_1, k_2, k_3), \vec{n} = (n_1, n_2, n_3)$ where $n_1, n_2$ and $n_3$ take on all integer values. Hence, the momentum $\vec{p} = \hbar\vec{k} = \frac{2\pi\hbar}{L}\vec{n}$ is quantized. The same is true for the energy $E = \hbar\omega_{\vec{k}}$, and so for the frequency of the wave :

$$ \omega_{\vec{k}} = \frac{E}{\hbar} = \frac{|\vec{p}|^2}{2m\hbar} = \left(\frac{2\pi}{L}\right)^2 \frac{\hbar}{2m} (n_1^2 + n_2^2 + n_3^2) \quad (7.118) $$

[...] these normalized wave functions $\psi_k(\vec{r}, t)$ [...] constitute an orthonormal function system so that

$$ \int \psi_{\vec{k}}^{*} \psi_{\vec{k}'} dV = \delta_{n_1 n_1'} \delta_{n_2 n_2'} \delta_{n_3 n_3'} = \delta_{\vec{k} \vec{k}'} \quad (7.119) $$

[...] In addition, the $\psi_{\vec{k}}$ are a complete system. Then the following completeness relation is valid

$$ \int_V \psi\psi^* dV = \int_V |\psi|^2 dV = \sum_{k_1, k_2, k_3} |a_{\vec{k}}|^2 = \sum_{\vec{k}} |a_{\vec{k}}|^2 \quad (7.120) $$

where the $a_{\vec{k}}$ are the expansion coefficients of the arbitrary wave function $\psi$ in terms of a complete set of the $\psi_{\vec{k}}$,

$$ \psi = \sum_{\vec{k}} a_{\vec{k}} \psi_{\vec{k}} \quad (7.121) $$

[...] With the help of the normalization integral, we obtain

$$ 1 = \int_{-\infty}^{\infty} \psi\psi^* dV = \sum_{\vec{k}\vec{k}'} a_{\vec{k}}^* a_{\vec{k}'}^* \int_V \psi_{\vec{k}} \psi_{\vec{k}'}^* dV = \sum_{\vec{k}\vec{k}'} a_{\vec{k}} a_{\vec{k}'}^* \delta_{\vec{k}\vec{k}'} = \sum_{\vec{k}} a_{\vec{k}} a_{\vec{k}}^* = \sum_{\vec{k}} |a_{\vec{k}}|^2 \quad (7.122) $$

Now we interpret the quantity $|a_{\vec{k}}|^2$ as the probability of finding a particle with momentum $\vec{p} = \hbar\vec{k}$ in the state $\psi$. This interpretation is very reasonable in view of equation [(7.121)] and the fact that the $\psi_{\vec{k}}(\vec{r}, t)$ [...] are wave functions with definite momentum $\vec{p} = \hbar\vec{k}$ [...]. One of the fundamentals principles of quantum mechanics is the principle of linear superposition of states or, for short, the superposition principle. It states that a quantum-mechanical system which can take on the discrete states $\psi_n$ is also able to occupy the state $\psi = \sum_n a_n \psi_n$ [...]. Thus the wave equation for $\psi$ must be a linear differential equation. The superposition principle can only be satisfied as follows : if the $\psi_n$ are solution of the linear fundamental equation, a linear combination will also be because of the linearity of the equation" [76]. The above analysis is the standard interpretation of quantum mechanics, here applied to the case of a free particle in a box. The process of measurement described in S. Weinberg's textbook is : "If we measure an observable represented by an Hermitian operator $\hat{A}$, and the system is initially in
a normalized superposition $\sum_n c_n \psi_n$ of orthonormal eigenvectors of $\hat{A}$ with eigenvalues $a_n$, then the state will collapse during the measurement to a state in which the observable has a definite one of the values $a_n$, and the probability of finding the value $a_n$ is given by what is known as the Born rule, as $|c_n|^2$ [76]. In his textbook, D. J. Griffith completes the description: "But what if I made a second measurement, immediately after the first? On this question every one is in agreement: a repeated measurement (on the same particle) must return the same value. [...] We say that the wave function collapses up measurement. [...] There are, then, two entirely distinct kind of physical processes: "ordinary" ones, in which the wave function evolves in a leisurely fashion under the Schrödinger equation, and "measurements", in which $\psi$ suddenly and discontinuously collapses" [122]. The above quotes were used to recall the Copenhagen interpretation: we now examine the consistency of this interpretation. Since a free particle in a box is the simplest example in textbooks, we consider a 1 dimensional box. We write $\psi = \frac{1}{\sqrt{L}} \sum_n a_n e^{i(-\omega_k t + k_n x)}$ where $\omega_k = \frac{\hbar^2}{2m\hbar} = \frac{\hbar k^2}{2m} = \frac{\hbar}{2m} (\frac{2\pi}{L})^2 n^2$. The energy is again quantized, each energy level being associated to two possible values of the momentum: $p = \hbar k$ or $p = -\hbar k$. D. J. Griffith explains how the quantum operators allows us to measure an observable: "If you measure an observable $Q$ on a particle in the state $\psi$, you are certain to get one of the eigenvalues $\hat{Q}$. The probability of getting the particular eigenvalue $\lambda$ is equal to the absolute square of the $\lambda$ component of $\psi$, when expressed in the orthonormal basis of eigenvectors." [122]. We consider two physicists, performing measurements on a quantum state $\psi$. We imagine that the first physicist has measured the energy of the quantum system and obtained the eigenvalue $E = \frac{n_a^2}{2m} (\frac{\hbar}{L})^2$: there are then two possible states associated to this energy: $\psi_{n_a} = \frac{1}{\sqrt{L}} e^{i(-\omega_k t + k_{n_a} x)}$ and $\psi_{-n_a} = \frac{1}{\sqrt{L}} e^{i(\omega_k t - k_{n_a} x)}$. Then $\psi = a_{n_a} \psi_{n_a} + a_{-n_a} \psi_{-n_a}$ where $|a_{n_a}|^2 + |a_{-n_a}|^2 = 1$ is a possible state because it is an eigenstate of the energy operator, i.e. the Hamiltonian operator:

$$ \begin{aligned} \hat{H}\psi &= -\frac{\hbar^2}{2m} \frac{\partial}{\partial x^2} \psi = -\frac{\hbar^2}{2m} \frac{\partial}{\partial x^2} (a_{n_a} \psi_{n_a} + a_{-n_a} \psi_{-n_a}) = a_{n_a} \frac{p^2}{2m} \psi_{n_a} + a_{-n_a} \frac{p^2}{2m} \psi_{-n_a} \\ &= \frac{p^2}{2m} (a_{n_a} \psi_{n_a} + a_{-n_a} \psi_{-n_a}) = \frac{p^2}{2m} \psi \end{aligned} \quad (7.123) $$

The first physicist must now indicate to the second physicist that the quantum system is in the state $\psi = a_{n_a} \psi_{n_a} + a_{-n_a} \psi_{-n_a}$ with energy $E = \frac{n_a^2}{2m} (\frac{\hbar}{L})^2$, while "a repeated measurement (on the same particle) must return the same value". The second physicist, who is moving with velocity $v_0$, then answers that the states $\psi_{n_a}$ has energy $\frac{1}{2m} (p_a - mv_0)^2 = \frac{p^2}{2m} - p_a v_0 + \frac{1}{2}mv_0^2$ while the state $\psi_{-n_a}$ has energy $\frac{1}{2m} (-p_a - mv_0)^2 = \frac{p_a^2}{2m} + p_a v_0 + \frac{1}{2}mv_0^2$: the state measured by the first physicist is not an eigenstate of his Hamiltonian, then he can't measure the system in this state. The example shows that the combination of superposition, collapse of the wave function and covariance is not trivial. Generally covariant quantum mechanics provides a simple solution: each mode has been interpreted as a particle (in a similar manner, each mode is identified with a particle when cross sections are computed in particle physics). The momentum of the particle(s) is not given by an operator but by an integral over a momentum density. Along the direction $i$, the momentum $p_i$ is:

$$ p_i = \frac{i\hbar}{2m} \int \psi^* \frac{\partial}{\partial x^i} \psi - \psi \frac{\partial}{\partial x^i} \psi^* d^3 x \quad \text{if} \quad \int |\psi|^2 d^3 x = m \quad (7.124) $$

We similarly determined the momentum of the electromagnetic field from the Poynting vector in equation (5.53): all fields are then treated on equal footing.

We now discuss the Hydrogen atom and the Born rule. The Hydrogen atom is introduced with many details by D. J. Griffith [122], we only reproduce the main results. In spherical coordinates $r, \theta, \varphi$, the wave function $\psi(r, \theta, \varphi)$ is labelled by three quantum numbers $n, l, m$. We write $\psi = \psi_{nlm}(r, \theta, \varphi) = \mathcal{R}_{nl}(r) Y_l^m(\theta, \varphi)$. Here, $\mathcal{R}_{nl}(r) = \frac{1}{r} \frac{1}{(\frac{r}{an})^{l+1}} e^{-(\frac{r}{an})v(\frac{r}{an})}$ where $a$ is the Bohr radius and $v(\frac{r}{an})$ a polynomial. $Y_l^m(\theta, \varphi) = P_l^m(\theta) e^{im\varphi}$ are spherical harmonics and $P_l^m$ are the associated Legendre functions. Then, $\psi(t, r, \theta, \varphi) = \mathcal{R}_{nl}(r) P_l^m(\theta) e^{i(-\frac{E_N t + L_3 \varphi}{\hbar})}$ where $E_N$ is the non relativistic energy and $L_3 = m\hbar$ is a quantized orbital angular momentum. If we use the notation $\psi(t, r, \theta, \varphi) = \mathcal{R} e^{-i\frac{S}{\hbar}}$ we get

$$ \mathcal{R} = \mathcal{R}(r, \theta) = \mathcal{R}_{nl}(r) P_l^m(\theta) \quad \& \quad S = S(t, \varphi) = E_N t - L_3 \varphi \quad (7.125) $$

In the phase $S/\hbar$ of $\psi$, the function $S$ corresponds to the action of an electron according to the Bohr model of the Hydrogen atom: the electron follows circular orbits with quantized angular momentum. The Hydrogen atom is then an example of system in which the eikonal approximation is not valid: the variations of the phase and the relative variations of the amplitude have comparable magnitudes and then the system does not behaves classically, explaining why the Bohr model is more precise than the Newtonian model. The wave function $\psi$ obey the eigenstate equation:

$$ E_N \psi = -|E_N| \psi = \hat{H} \psi = -\frac{\hbar^2}{2m} \Delta \psi - \frac{q^2}{4\pi \epsilon_0 r} \psi \quad (7.126) $$
# 8 Comparison of the theory with experimental data

C. Rovelli wrote in 1998: *The arbiters in science are experiments, and not a single experimental result supports, not even indirectly, any of the current theories that go beyond the Standard Model and general relativity"* [175]. In 2020, the situation had not changed, according to R. Loll: *"In high-energy theoretical physics the main question is to understand quantum gravity. [...] It's a hard problem. We don't really have many candidate solutions, right? It's not that everyone has their own complete quantum gravity solution in a drawer and we just have to agree which is the right one. All approaches are highly incomplete, although not all have the same degree of incompleteness. And none of them has produced any suggestion for phenomenology, so what is our reality check? [...] It would therefore be rather naive to expect that solving quantum gravity will be given by a one-liner, with everything falling into place. In addition, the absence of guidance from experiment clearly hampers us in finding the correct answer"* [57]. This section compares generally covariant quantum mechanics with experimental data.

## 8.1 The physical meaning of the wave function $\psi$

In agreement with the Copenhagen interpretation of quantum mechanics, R. P. Feynman explained: *"the wave function [...] is not like a real wave in space; one cannot picture any kind of reality to this wave"* [75]. Indeed, standard quantum mechanics asserts that the wave function is defined in configuration space such that $\psi$ is not a physical field, as we have recalled in [Section 7.2]. This usual interpretation seems incompatible with experimental data since the same section explained that some authors published papers involving experimental measurements of $|\psi|^2$ or $|\hat{\psi}|^2$. To give a new example, we quote a paper entitled 'The quantum future of microscopy: wave function engineering of electrons, ions, and nuclei' published in 2020: *"The described approach for low-dose imaging applications is based on real-space interference effects and requires no modification of the phase profile of the free-electron wave function. Other quantum phenomena in electron microscopy involve the active engineering of an electron's wave function employing phase plates or laser-based modulators. [...] By scanning through the phase shift between the two pulses, the wave function can be entirely reconstructed."* [176]. An obvious example showing that the wave function is defined in the physical space is given by superfluid Helium: the density (or mass density) of Helium atoms, directly proportional to $|\psi|^2 = \rho$, is visible with naked eyes. Generally covariant quantum mechanics explicitly agrees with experimental data by considering that $\psi$ is a physical field, like the electromagnetic field $A_\mu$ or the gravitational field $g_{\mu\nu}$: this is theoretically expected because the fields can transform into each other (we previously mentioned the decay process $\pi^0 \to \gamma\gamma$).

According to quantum field theory, on which the standard model of particle physics is based, neutral particles are described by real fields while charged particles are described by complex fields. This fact is illustrated by the following quotes reproduced from various textbooks:

- "real fields [...] describes neutral particles. Charged particles are always described by complex fields" [98].

- "Having discussed the quantum theory of one real free Klein-Gordon field, we may adapt the results to describe the charged particle [...] in terms of a complex wave function" [177].

- "Such a particle is found in nature, the neutral $\pi^0$. [...] the $\pi^0$ will be represented [...] by a real solution $\psi = \psi^*$ of the Klein-Gordon equation" [116].

- "the complex Klein-Gordon field theory describes spin zero particles carrying electric charge while the real Klein-Gordon field theory describes charge neutral spin zero particles" [178].

- "Hitherto we have inspected Klein-Gordon equation both for a real, i.e. uncharged, and for a complex, i.e. charged, scalar field. [...] The $\pi^0$, being a neutral particle, is characterized by a real wave function whereas $\pi^+$ and $\pi^-$ have the same mass and opposite charges, i.e., we can define $\psi_{\pi+} = \psi$ and $\psi_{\pi-} = \psi^*$" [74].

On the contrary, generally covariant quantum mechanics suggests that all particles (including neutral particles) are described by complex fields: the decomposition of each field in amplitude and phase has been used to justify the wave-particle duality. Furthermore, if Postulate 3 is included, the electric charge is interpreted as a momentum along an internal direction of a generalized spacetime, described by the Kaluza-Klein theory introduced in [Section 4]. This difference between quantum field theory and generally covariant quantum mechanics deserves an experimental examination. We can provide three reasons to justify that all wave functions describing matter should be complex physical fields:

- **Non relativistic limit**: When the spin is neglected, all matter particles, including neutral particles, are described by the Schrödinger equation in the non relativistic limit: all experimental results are then compared with the complex solutions of this equation in the non relativistic limit. In 1994, M. V. Berry published a famous paper explaining: *"A quantal system in an eigenstate, slowly transported round a circuit by varying parameters in his Hamiltonian, will acquire a geometrical phase factor $e^{i\gamma}$ in addition to the familiar dynamical phase factor"* [179]. This quantum phase evidently requires a complex field, the quantum system being a charged particle or not. Nowadays, this subject is an active area of research, as explained in 2020: *"The geometric phase (GP) is an important concept in adiabatic theory. It represents
the phase acquired when a quantum state is adiabatically transported around a closed circuit in the para-
meter space, while remaining in the same Hilbert space. The concept of the GP has given rise to various
observable effects, and has now become a central topic across various research fields ranging from chemis-
try to condensed matter physics" [180]. Starting with a real relativistic field (obeying the Klein-Gordon
equation) describing a neutral particle, there is no reason to obtain a complex field (obeying the Schrö-
dinger equation) in the non relativistic limit : nevertheless, the complex nature of the wave function has
observable consequences.

**Interferometry** : The neutron is a neutral spin-$-\frac{1}{2}$ particle described by spinors. A well known experimental result is the $4\pi$ periodicity of its wave function [181]. We quote the book 'Neutron interferometry' published by H. Rauch and A. Werner in 2015 : "neutron interferometry also provides experimental access to the phases of neutron wave functions as they are perturbed and modified by gravity, inertia, motion, magnetic and electric fields, topology, and material media. [...] We begin here with a general discussion of the quantum phase of a matter wave evolving in space and time according to the Feynman-Dirac path integral along the trajectories defined by classical mechanics. This is identical to the Wentzel-Kramers-Brillouin (WKB) approximation and usually called the eikonal approximation in classical optics. [...] The phase accumulated on either path is a line integral over the Lagrangian $L$ in space-time" [182]. The authors then describe an interferometer and indicate : "This kind of interferometer has been used to measure the real and imaginary parts of the forward scattering amplitude for atoms and molecules" [182]. Since neutrons, atoms and molecules are neutral, interferometry clearly shows that neutral particles are described by complex wave functions with amplitude and phase. A simple example is, again, superfluid Helium : the electrically neutral superfluid is described by a complex wave function, as we have seen in [Section 2].

**Optical theorem** : When matter is concerned, Landau and Lifschitz indicate : "we obtain the following general relation between the total elastic scattering cross section and the imaginary part of the amplitude of scattering through an angle zero : $\Im[m[f(\vec{n}, \vec{n}')]] = \frac{k\sigma}{4\pi}$. This is called the optical theorem for scattering" [182]. When light is concerned, J. D. Jackson explains "A fundamental relation, called the optical theorem, connects the total cross section of a scatterer to the imaginary part of the forward scattering amplitude. The theorem follows from very general considerations of the conservation of energy and power flow, and has its counterpart in the quantum-mechanical scattering of particles" [37]. The measurable cross section $\sigma$ is then determined by the splitting of a complex field describing matter particles or photons into real and imaginary parts. Furthermore, the optical theorem is also a standard result of quantum field theory, involved when decay rates are concerned. Indeed, M. Maggiore explains : "The fact that the eigenvalues of the Hamiltonian become complex when we have an unstable particle must happen also in relativistic quantum theory". M. Maggiore then separates the mass of an unstable particle into real and imaginary parts according to $m = m_R - i\frac{\Gamma}{2c^2}$, where $m_R$ is the renormalized mass and $\gamma$ the decay rate of the process. He concludes : "The optical theorem states that this is a general result, and the imaginary part of m is always equal to $-\frac{\Gamma}{2}c^2$, where $\Gamma$ is the total decay rate (if there are many possible decay channels they all contribute to the imaginary part)" [61]. It should be noted that it is not possible to describe a neutral particle (like a $\pi^0$) with a real solution of the Klein-Gordon equation if the mass term is complex, as in the case of unstable particles (the $\pi^0$ being unstable).

Since generally covariant quantum mechanics describes all particles by physical complex fields, it exhibits
differences with standard quantum mechanics and quantum field theory. The difference with quantum field theory
is not trivial since quantum field theory uses $\psi$ and $\psi^*$ to define a particle and an antiparticle with opposite electric
charge : the second quantization of a complex scalar field then involves 4 creation and annihilation operators
(2 for particles and 2 for antiparticles) [178]. When the differences are considered, generally covariant quantum
mechanics agrees with the above experimental data (especially the non relativistic limit, the interferometry, the
optical theorem with complex mass) : in this context, it seems delicate to describe a $\pi^0$ with a real field, and
then to describe the charged pions $\pi^+$ and $\pi^-$ according to $\psi_{\pi^+} = \psi$ and $\psi_{\pi^-} = \psi^*$, as suggested in [74].

## 8.2 The wave particle duality and the quantum to classical transition

The previously mentioned textbook entitled 'Neutron interferometry' stipulates : "The more recent achievements of atom and molecular interferometry show that there is apparently no natural upper mass limit to which interferometry can be developed, but the experimental requirements become increasingly more stringent" [182]. A paper entitled 'Optics and interferometry with atoms and molecules', the authors indicate : "In this section we will outline the theoretical and design principles underlying the construction of an atom/molecule interferometer. [...] The difference between the phases accumulated along each path can be expressed in terms of the classical actions along these paths : $\phi = \frac{S_1-S_2}{\hbar}$. [...] The detected intensity, $I$, from the portion of the two interfering beams passing through the third grating is $I = |\psi|^2 = A_1^2 + A_2^2 + 2A_1A_2\cos\phi$ where $A_1$ and $A_2$ are the amplitudes of the two interfering beams and $\phi$ is their phase difference" [183]. We previously emphasized in [Section 7.2.2]
that E. Schrödinger examined the case of a diatomic molecule in 1926 by writing $\psi$ as a function of 6 variables in configuration space (3 variables for each atom of the molecule). Molecule interferometry shows that the wave function describing a diatomic molecule is a function of 3 variables, since the phase difference is a measurable quantity determined by 3 variables in the physical space according to $\varphi = \frac{S_1-S_2}{\hbar}$. Interferometry is an obvious example exhibiting the wavelike behavior of matter. It is often used to illustrate the complementarity principle discussed in [Section 7.2.1]. We quote two textbooks to describe this principle, according to the Copenhagen interpretation of quantum mechanics:

- "in any experimental situation in which a physical entity (matter or radiation) exhibits its wave properties, it is impossible to attribute corpuscular characteristics to it. The entity does behave like a particle if its wave packet is made compact enough [...] but when this is done, it loses the ability to display any wave properties [...] In other words, the particle and wave aspects of a physical entity are complementary and cannot be exhibited at the same time. This is the complementarity principle of Bohr" [184].

- "The wave and particle models are complementary; if a measurement proves the wave character of radiation or matter, then it is impossible to prove the particle character in the same measurement, and conversely" [159].

In [Section 7.2.1], we discussed the square of fluctuations of the blackbody radiation and observed that they contain a term associated to waves and a term associated to particles (see equation (7.108)) : the particle and wave aspects of light are simultaneously exhibited by the spectrum of the blackbody radiation. In [Section 2], we wrote $\psi = \mathcal{R}e^{-i\frac{S}{\hbar}}$ such that the Klein-Gordon equation implied (see equation (2.21)):

$$0 = -\hbar^2 \Box_g \psi - m^2 c^2 \psi = \left[ g^{\mu\nu} \partial_{\mu} S \partial_{\nu} S - m^2 c^2 - \hbar^2 \frac{\Box_g \mathcal{R}}{\mathcal{R}} \right] \psi + \frac{i\hbar}{\rho} [\nabla_{\mu}(\rho \partial^{\mu} S)] \psi \quad (8.1)$$

where each term between [ ] must vanish. The term $\hbar^2 \Box_g \mathcal{R}$ modifies the classical Hamilton-Jacobi equation $g^{\mu\nu} \partial_{\mu} S \partial_{\nu} S - m^2 c^2 = 0$ and then the classical laws are only valid if $\hbar^2 \Box_g \mathcal{R}$ can be neglected: this is the eikonal approximation. In this approximation, the complex field $\psi$ exhibits a particle-like behavior, with $S$ identified with the mechanical action along the classical path. When applied to the electromagnetic field, the same eikonal approximation produces photons. The concept of photon can also be derived from the blackbody radiation (this has been done by A. Einstein, as we have seen in [Section 7.2.1]). To exhibit the wave-like behavior of matter, one can reduce the values of the terms involved in the sum $g^{\mu\nu} \partial_{\mu} S \partial_{\nu} S - m^2 c^2$ such that they becomes comparable to the terms involved in $\hbar^2 \Box_g \mathcal{R}$. This means that one must use light particles (small mass) or slow particles (low velocity) since $\partial_{\mu} S \approx p_{\mu} = m u_{\mu}$ but there is 'no natural upper mass limit to which interferometry can be developed' as already deduced from previous experiments [182]. A simple way to decrease the velocity of the particles is to cool them, since the temperature is proportional to the average kinetic energy, as we have seen in [Section 7.1.6]. This analysis is confirmed by double slit experiments with heavy molecules: in 2013, interferences with molecules containing more than 800 atoms have been reported [185] and, in 2019, interferences with molecules containing more than 2000 atoms have been reported [186]. Equation (8.1) suggests that the non classical term $\hbar^2 \Box_g \mathcal{R}$ explores the entire experimental setup without decreasing, even if the amplitude $\mathcal{R}$ decreases when the distance from the center of gravity increase. This explains why the motion of a quantum object is different when one or two slits are open in a double slit experiment: the number of open slits modifies $\mathcal{R}$ which modifies $S$ according to equation (8.1) such that the mass current $j_m^\alpha = \rho \frac{\partial^\alpha S}{\partial m^\alpha} = \mathcal{R}^2 \frac{\partial^\alpha S}{\partial m^\alpha}$ is also modified. Furthermore, this interpretation contradicts the complementarity principle since the particle and wave aspects of matter are simultaneously exhibited. This is precisely the conclusion reached by L. Mairhofer and O. Passon in 2022, when they analyzed the experiment producing interferences with molecules containing more than 2000 atoms [186]. They wrote: "Here, two descriptions clash, the wave concept and the particle picture. While wave-particle duality claims that they are mutually exclusive, the description of the diffraction of complex molecules requires that we apply them both at the same time" [186]: interestingly, equation (8.1) contains the wave term $\hbar^2 \Box_g \mathcal{R}$ associated to the particle term $g^{\mu\nu} \partial_{\mu} S \partial_{\nu} S - m^2 c^2$, explaining why experiments exhibits simultaneously the particle and wave aspects of matter. Generally covariant quantum mechanics then describes the transition from quantum to classical phenomena as a continuous process. The opposite process, i.e. the transition from classical to quantum phenomena, is described by W. Schöllkopf in the case of a Bose-Einstein Condensate (BEC): "The phenomenon of BEC is intimately connected with wave-like behavior of particles in the quantum domain. At thermal energies, gaseous atoms move with fairly high velocity, and de Broglie wave-lengths are minute with respect to all other relevant length scales of the gas. As a result, quantum effects are usually negligible at these conditions. At sufficiently low temperatures, however, where the motion of atoms in a gas is greatly slowed down, de Broglie wavelengths become correspondingly large. In a simplified picture of BEC formation upon cooling of a gas, one can consider the atoms' de Broglie wavelengths becoming about as large as the average distance between the atoms. At this point, the quantum wave nature begins to play a decisive role as matter waves of individual atoms are no longer independent, but begin to overlap. Below a density-dependent critical temperature, this leads to the formation of a single macroscopic quantum state described by a wave-function of the part of the gas that BE-condensed" [188].
Generally covariant quantum mechanics disagrees with the Copenhagen interpretation when the complementary principle is considered: the above examples seem to show that the theory is consistent with experimental data (blackbody radiation and double slit experiments) while the complementarity principle is not. Interferometry is another experimental example showing that the wave function of a multiparticle state is not defined in configuration space.

## 8.3 Atom interferometers and gravity

In 2020, G. M. Tino published a paper entitled 'Testing gravity with cold atom interferometry: results and prospects'. We reproduce the abstract: "Atom interferometers have been developed in the last three decades as new powerful tools to investigate gravity. They were used for measuring the gravity acceleration, the gravity gradient, and the gravity-field curvature, for the determination of the gravitational constant, for the investigation of gravity at microscopic distances, to test the equivalence principle of general relativity and the theories of modified gravity, to probe the interplay between gravitational and quantum physics and to test quantum gravity models, to search for dark matter and dark energy, and they were proposed as new detectors for the observation of gravitational waves. Here I describe past and ongoing experiments with an outlook on what I think are the main prospects in this field and the potential to search for new physics" [189]. We have seen in [Section 8.2] that cold atoms are indeed needed to produce interferences if the atoms are heavy. Here we will mention 3 examples which combine quantum interferences and gravity.

- The experimental observation of a gravitational Aharonov-Bohm effect has been reported in 2020: "As Aharonov and Bohm argued in 1959, a particle in a spatial superposition is sensitive to the potential energy difference between its wave packets even if the field vanishes along their trajectories. A matterwave interferometer can therefore measure a phase shift due to the potential even if the interferometer arms are not deflected. This phase shift $\phi_{AB}$ is given by the action difference $\Delta S$ between arms according to the expression $\phi_{AB} = \Delta S/\hbar$" [190]. The idea of the authors was then to use a mass to produce two different gravitational potential along the interferometer arms: this is how they observed the gravitational Aharonov-Bohm effect.

- The experimental observation of gravitational redshift with an atom interferometer has been reported in 2010: "The gravitational redshift has been measured using clocks on a tower, an aircraft and a rocket, currently reaching an accuracy of $7 \times 10^{-5}$. Here we show that laboratory experiments based on quantum interference of atoms, enable a much more precise measurement, yielding an accuracy of $7 \times 10^{-9}$. Our result supports the view that gravity is a manifestation of space-time curvature, an underlying principle of general relativity that has come under scrutiny in connection with the search for a theory of quantum gravity. [...] The phase accumulation $\Delta\phi$ of each matter wave is thus given by a free evolution term $\Delta\phi_{free}$ and a light-atom interaction term $\Delta\phi_{light}$. The term $\Delta\phi_{free}$ is given by the elegant expression $\Delta\phi_{free} = \frac{1}{\hbar} \int L d\tau$" [191] where $L$ is the mechanical Lagrangian such that $\frac{1}{\hbar} \int L d\tau = \frac{S}{\hbar}$.

- The experimental observation of the Sagnac effect in superfluid liquids has been reported in 2008: "In this letter, we wish to consider some problems of interpretation of Sagnac-type experiments with beams of cold atoms and with superfluids. [...] The rotating frame of reference, [...] is treated as an accelerated frame from the standpoint of general relativity. In such frames, characterized by a space-time metric $ds^2 = g_{00}(dx^0)^2 + 2g_{0i}dx^0dx^i + g_{ij}dx^i dx^j$, clocks can be synchronized [...]. We first consider particles - electrons, neutrons, or atoms - represented by localized wave packets with a slowly varying overall phase $\phi$. These wave packets can be treated in the semiclassical approach: the phase is related to the classical action $\phi = S/\hbar$" [192].

Atom interferometers are a promising tool to test the effects of curved spacetimes on the phase of quantum wave functions, the phase being identified with the mechanical action of the particle divided by $\hbar$. The description of quantum wave functions in curved spacetimes is precisely the framework provided by generally covariant quantum mechanics while the experimental identification of the phase with the mechanical action according to $\phi = S/\hbar$ is simply the eikonal approximation.

## 8.4 The Schrödinger-Newton equation & the Einstein-Maxwell equations

In [Section 7.1.3], we derived the Schrödinger-Newton equation, by considering a self gravitating superfluid in the non relativistic limit:

$$i\hbar \frac{\partial}{\partial t}\psi + \frac{\hbar^2}{2m}\Delta\psi + mV_G\psi = 0 \quad ; \quad \Delta V_G = 4\pi G|\psi|^2 \qquad (8.2)$$