Special relativity in a discrete quantum universe

The hypothesis of a discrete fabric of the universe, the “Planck scale,” is always on stage since it solves mathematical and conceptual problems in the infinitely small. However, it clashes with special relativity, which is designed for the continuum. Here, we show how the clash can be overcome within a discrete quantum theory where the evolution of fields is described by a quantum cellular automaton. The reconciliation is achieved by defining the change of observer as a change of representation of the dynamics, without any reference to space-time. We use the relativity principle, i.e., the invariance of dynamics under change of inertial observer, to identify a change of inertial frame with a symmetry of the dynamics. We consider the full group of such symmetries, and recover the usual Lorentz group in the relativistic regime of low energies, while at the Planck scale the covariance is nonlinearly distorted.

Figure 1

Top: body-centered-cubic lattice. Each pair of nearest neighbor is connected by a blue link. Each point of the lattice has eight nearest neighbors, e.g., the red points are the nearest neighbors of the yellow one. Bottom: Brillouin zone $\mathsf{B}$ of the bcc lattice. The zone is a rhombic dodecahedron in which the opposite faces are identified.

Figure 2

The green surface represents the dispersion relation in Eq. (8) where we fixed $k_z=0$ (${\omega }_{\mathbf{k}}=arccos{\lambda }^{\pm }\left(\mathbf{k}\right)$ have the same plot). The red surface is the usual relativistic dispersion relation ${\omega }_{\mathbf{k}}^2=k_x^2+k_y^2$. Notice that the two surfaces get closer approaching the origin for $\mathbf{k}\to 0$.

Figure 3

Top left figure: surfaces $\lambda \left(\mathrm{k}\right)=0$ in Eq. (3) (yellow) and $cos\left(2k_y\right)=0$ (red planes) inside the Brillouin zone (transparent). Top middle figure: ${\mathsf{B}}_0$ zone (red X shaped). Top right figure: ${\mathsf{B}}_0$ (red) and ${\mathsf{B}}_1$ (blue). Bottom left to right: ${\mathsf{B}}_1,{\mathsf{B}}_2,{\mathsf{B}}_3$. Bottom right: region ${\mathsf{B}}_1$ represented in a properly translated Brillouin zone. In this paper, the Lorentz transformations are those that leave the dispersion relations of the Weyl QW invariant, and act on the Weyl spinor independently of the wave vector. In such way they are nonlinear in $(\omega ,\mathrm{k})$ and linear over the Weyl spinor. Therefore, the Lorentz group acts as a group of diffeomorphisms over the Brillouin zone $\mathsf{B}$. The four domains ${\mathsf{B}}_i\subset \mathsf{B}$ are Lorentz invariant (up to a null-measure set, see Fig. 6 in the Appendix). More precisely, a point $(\omega ,\mathrm{k})$ with $\mathrm{k}\in {\mathsf{B}}_i$ and ${sin}^2\omega -{\left|\mathrm{n}\left(\mathrm{k}\right)\right|}^2=0$ is mapped to a point $({\omega }^{\prime },{\mathrm{k}}^{\prime })$ with ${sin}^2{\omega }^{\prime }-{\left|\mathrm{n}\left({\mathrm{k}}^{\prime }\right)\right|}^2=0$ and ${\mathrm{k}}^{\prime }\in {\mathsf{B}}_i$. Moreover, the map $\mathrm{n}$ maps each ${\mathsf{B}}_i$ into the same set (up to null-measure set; see Fig. 6). Since the kinematics of a wave vector $\mathrm{k}$ depends only on the vector $\mathrm{n}\left(\mathrm{k}\right)$, we can conclude that the ${\mathsf{B}}_i$ regions are kinematically equivalent and they can be interpreted as four different massless Weyl fermions. Because of the identification of the boundary points in the Brillouin zone, all the ${\mathsf{B}}_i$ regions have the same X shape as ${\mathsf{B}}_0$. This is evident in the bottom right figure, in which we see that the region ${\mathsf{B}}_1$ (in red), when represented in a properly translated Brillouin zone (in blue), has the same X shape as the region ${\mathsf{B}}_0$. Considering the identification of the boundary points of the Brillouin zone in Fig. 1, one realizes that the opposite arms of the X are glued together, resulting in a solid double-torus (genus two). This result is rigorously proved in the Appendix where we show that the ${\mathsf{B}}_i$ regions are diffeomorphic to a solid ball pierced by two arches of ellipses (Fig. 6 in the Appendix).

Figure 4

The distortion effects of the Lorentz group for the discrete Planck-scale theory represented by the quantum walk in Eq. (2). Left figure: the orbit of the wave vectors $\mathbf{k}=(k_x,0,0)$, with $k_x\in \{0.05,0.2,0.5,1,1.7\}$ under the rotation around the $z$ axis. Right figure: the orbit of wave vectors with $\left|\mathbf{k}\right|=0.01$ for various directions in the $(k_x,k_y)$ plane under the boosts with $\beta$ parallel to $\mathbf{k}$ and $\left|\beta \right|\in [0,tanh4]$.

Figure 5

The green surface represents the orbit of the wave vector $\mathbf{k}=(0.3,0,0)$ under the full rotation group $\text{SO}\left(3\right)$.

Figure 6

Left figure: region ${\mathsf{Q}}_a$. Right figure: $\mathsf{H}$ zone in red inside the unit ball. In the left figure, the tubes around the arches ${\mathrm{e}}_{+}{\left(\mathsf{T}\right)}_1$ and ${\mathrm{e}}_{-}{\left(\mathsf{T}\right)}_2$ emphasize the piercing of the ball by the one-dimensional holes along the elliptic arches ${\mathrm{e}}_{+}{\left(\mathsf{T}\right)}_1$ and ${\mathrm{e}}_{-}{\left(\mathsf{T}\right)}_2$. The region ${\mathsf{Q}}_a$ is clearly homeomorphic to a solid torus of genus two. Because of this nontrivial topological feature, the set $\left\{(\omega ,\mathrm{m})\text{such}\text{that}\right|\omega |\le \frac{\pi }{2},\mathrm{m}\in \mathrm{n}\left({\mathsf{B}}_i\right),{sin}^2\omega -|\mathrm{m}{|}^2=0\}$ cannot be diffeomorphic to any Lorentz-invariant region of ${\mathbb{M}}^4$. However, it is possible to remove from the region ${\mathsf{Q}}_a$ a null-measure set such that the resulting topology is trivial. This can be done by removing the set $\mathsf{H}$ (red zones in the right figure), resulting in a star-shaped open set in ${\mathbb{R}}^3$.