Quantum mechanics is an incredibly successful theory and yet the statistical nature of its predictions is hard to accept and has been the subject of numerous debates. The notion of inherent randomness, something that happens without any cause, goes against our rational understanding of reality. To add to the puzzle, randomness that appears in non-relativistic quantum theory tacitly respects relativity, for example, it makes instantaneous signaling impossible. Here, we argue that this is because the special theory of relativity can itself account for such a random behavior. We show that the full mathematical structure of the Lorentz transformation, the one which includes the superluminal part, implies the emergence of non-deterministic dynamics, together with complex probability amplitudes and multiple trajectories. This indicates that the connections between the two seemingly different theories are deeper and more subtle than previously thought.

The full mathematical structure of the Lorentz transformations contains both subluminal and superluminal terms. The superluminal part is usually discarded, on the premise that it makes no physical sense, and, as a consequence, a familiar classical picture of a particle moving along a well defined path is obtained. Here we show that if we retain the superluminal terms, and take the resulting mathematics of the Lorentz transformation seriously, then the notion of a particle moving along a single path must be abandoned and replaced by a propagation along many paths, exactly like in quantum theory.

The generalised Lorentz transformation can be derived in few simple steps [1]. Consider a classical 1 + 1 dimensional case (the 1 + 3 case will be discussed later) with an inertial frame (t', x') moving with the velocity V relative to the frame (t, x). We seek the most general form of the coordinate transformation between these frames that is consistent with the Galilean principle of relativity. It has to be a linear transformation, so that no point in spacetime is singled out, and its coefficient must depend only on the relative velocity V.

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We argue that ruling out a superluminal family of observers from special relativity, regardless whether such observers exist or not, is not necessary; it leads to a classical description of a particle moving along a well-defined single trajectory. In contrast, if one keeps both subluminal and superluminal solutions then non-deterministic behavior and non-classical motion of particles arise as a natural consequence.

The superluminal solutions appear quite naturally in general relativity. For instance a Schwarzschild solution to Einstein equations written in Schwarzschild coordinates has a peculiar property that time and radial coordinates change their metric signs at the event horizon. This is normally dismissed by arbitrarily stating that the Schwarzschild solutions only make sense above the event horizon, although written in (freely falling) Kruskal coordinates, they are smooth at the horizon. In order to resolve this puzzle we point out that Schwarzschild coordinates correspond to stationary observers placed at fixed distances from the horizon. Such observers can be subluminal only above the horizon, and under the horizon they would require superluminal motions. The sign flip in the metric therefore signifies the transition from a subluminal to a superluminal family of stationary observers residing under the event horizon in a fixed distance from the singularity.

We believe that our approach is more than a mathematical exercise and, if taken seriously, it may offer new valuable insights into deep connections between quantum theory and special relativity.

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New Journal of Physics Volume 22, March 2020