Comment on 'Quantum principle of relativity'

No particles (or signals) can move faster than light. Yet we sometimes use the notion of superluminal particles in order to better understand the consequences of existence of the speed limit. For example Rindler [1, p 54] considers a superluminal signal going from a spacetime point A to B and causing some effect at that point (a glass breaking). Rindler pointed out that there is an another reference frame in which point B foreruns point A. In this frame it seems that the glass breaks spontaneously (without cause). Rindler concludes 'Since in macro-physics no such uncaused events are observed, nature must have a way to prevent superluminal signals'. Recently Dragan and Ekert [2] proposed to reconsider superluminal particles in the realm of quantum world in which intrinsic random events (which we cannot predict and which seem to have no causes) occurs. Although quantum mechanics is in full agreement with observations it is still true that in words of Dragan and Ekert 'The notion of inherent randomness, something that happens without any cause, goes against our rational understanding of reality' [2]. These authors present arguments that a non-deterministic dynamics is a natural consequence of the principle of relativity. Their result, if correct, would be very profound as connecting relativistic and quantum physics at a very deep level. Their main argument is based on the Galilean principle of relativity i.e. on the postulated equivalence of all inertial frames without excluding the possibility of existence of superluminal ones. It is highlighted that this principle allows two branches of coordinate transformations (instead of a single branch usually exploited in relativistic physics). Dragan and Ekert state that these two branches correspond to subluminal and superluminal families of observers. In this Comment we show that the proposed extension of the Lorentz transformation can be interpreted in a natural way without invoking superluminal phenomena. In our proposal the branch corresponding to superluminal observers can be interpreted as a relabelling of time and space coordinates for subluminal observers.

It is sufficient for presenting our argument to consider simple 1 + 1 dimensional case. For both time and space dimensions we will use coordinates ξ_j (j = 1, 2) with the units of length. Let us consider two inertial observers O and O' with O' moving relative to O with dimensionless velocity V (velocity is measured in the units of c) using coordinate systems ξ_j and ${\xi }_{j}^{\prime }$ respectively. Simple arguments based on the symmetry between O and O' (figures 1(a) and (b)) lead to a linear transformation between coordinate systems of the usual form

$$\begin{align}\hfill & {\xi }_{1}^{\prime }=\gamma (V)({\xi }_{1}-V{\xi }_{2}),\hfill \\ \hfill & {\xi }_{2}^{\prime }=\gamma (V)({\xi }_{2}-V{\xi }_{1})\hfill \end{align} \tag{ 1 }$$

with γ(V) being symmetric or antisymmetric function of V. A possible next step (in the common derivation of Lorentz transformation) is to consider O and O' with reversed space coordinates (see figure 1(c)). Then the transformation of equation (1) gives

$$\begin{align}\hfill & {\tilde{\xi }}_{1}^{\prime }=\gamma (-V)({\tilde{\xi }}_{1}-(-V){\tilde{\xi }}_{2}),\hfill \\ \hfill & {\tilde{\xi }}_{2}^{\prime }=\gamma (-V)({\tilde{\xi }}_{2}-(-V){\tilde{\xi }}_{1}).\hfill \end{align} \tag{ 2 }$$

After plugging

$$\begin{align}\hfill & {\tilde{\xi }}_{1}={\xi }_{1},\hfill \\ \hfill & {\tilde{\xi }}_{2}=-{\xi }_{2},\hfill \\ \hfill & {\tilde{\xi }}_{1}^{\prime }={\xi }_{1}^{\prime },\hfill \\ \hfill & {\tilde{\xi }}_{2}^{\prime }=-{\xi }_{2}^{\prime }\hfill \end{align} \tag{ 3 }$$

to equation (2) one obtains

$$\begin{align}\hfill & {\xi }_{1}^{\prime }=\gamma (-V)({\xi }_{1}-V{\xi }_{2}),\hfill \\ \hfill & {\xi }_{2}^{\prime }=\gamma (-V)({\xi }_{2}-V{\xi }_{1}),\hfill \end{align} \tag{ 4 }$$

and finally concludes that γ(V) must be a symmetric function i.e. γ(−V) = γ(V).

Zoom In Zoom Out Reset image size

Dragan and Ekert propose to ignore this conclusion and to proceed further without fixing the parity of γ(V). This is acceptable from the pure mathematical point of view. However, one has to be aware that as a consequence the standard (or natural) physical interpretation of mathematical expressions thus obtained can be misleading. Let us proceed along the line proposed by Dragan and Ekert. They postulate that γ(V) fulfils

$$\begin{equation}\frac{\gamma (V)\gamma (-V)-1}{{V}^{2}\gamma (V)\gamma (-V)}=K\end{equation} \tag{ 5 }$$

with some constant K. (Note, than in the symmetric case γ(−V) = γ(V), this postulate can be justified by considering three observers and mutual transformations between them (equations (3)–(7) in [2].)

Now we can analyze the two mentioned branches of transformation, which emerge after fixing the parity of γ(V).

First let us consider the symmetric case γ(−V) = γ(V). In this case equation (5) gives

$$\begin{equation}\gamma (V)=\pm \frac{1}{\sqrt{1-K{V}^{2}}}.\end{equation} \tag{ 6 }$$

Thus we obtain a family (parametrized by τ, K, V) of coordinate transformations of the form

$$\begin{equation}\left[\begin{matrix}\hfill {\xi }_{1}^{\prime }\hfill \\ \hfill {\xi }_{2}^{\prime }\hfill \end{matrix}\right]={\Lambda}(\tau ,K,V)\left[\begin{matrix}\hfill {\xi }_{1}\hfill \\ \hfill {\xi }_{2}\hfill \end{matrix}\right],\end{equation} \tag{ 7 }$$

where

$$\begin{equation}{\Lambda}(\tau ,K,V)=\tau \frac{1}{\sqrt{1-K{V}^{2}}}\left[\begin{matrix}\hfill 1\hfill & \hfill -V\hfill \\ \hfill -V\hfill & \hfill 1\hfill \end{matrix}\right]\end{equation} \tag{ 8 }$$

with τ = ±1 and KV² < 1. Let us now emphasize that starting from a concrete physical scenario (figure 1 and equation (1)) we obtain a continuum of possible coordinate transformations. This redundancy arises from the fact that observers can use arbitrary coordinate systems. It is often highlighted (usually in the context of curved spacetimes) that coordinates have no direct physical meaning. Of course this is also true in flat spacetimes. A connection between coordinates and measurable quantities (distances and time intervals) has to be established with the use of some apparatus (e.g. clocks, meter sticks or teodolits, light signals) ([2], p 41). This connection then can be represented by a metric tensor. However, among many (abstract) coordinate systems one can choose coordinates established by physical means (grids of meter sticks and synchronised clocks). We will label such 'measurement induced' coordinates by ct and x. The metric tensor expressed in these coordinates has a simple form

$$\begin{equation}g=\left[\begin{matrix}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -1\hfill \end{matrix}\right]\end{equation} \tag{ 9 }$$

and measurable space and time displacements are simply given by differences of coordinates. Note, that it is only after fixing both, the physical scenario (figure 1) and coordinate systems ξ₁ = ct, ξ₂ = x, ${\xi }_{1}^{\prime }=c{t}^{\prime }$, ${\xi }_{2}^{\prime }={x}^{\prime }$, that one can fix parameters τ = 1, K = 1 and interpret V as the velocity of O' relative to O restricted to |V| ⩽ 1. In this case

$$\begin{equation}\left[\begin{matrix}\hfill c{t}^{\prime }\hfill \\ \hfill {x}^{\prime }\hfill \end{matrix}\right]={\Lambda}(1,1,V)\left[\begin{matrix}\hfill ct\hfill \\ \hfill x\hfill \end{matrix}\right].\end{equation} \tag{ 10 }$$

One can as well use the transformation Λ(−1, 1, V) with ξ₁ = ct, ξ₂ = x, ${\xi }_{1}^{\prime }=-c{t}^{\prime }$, ${\xi }_{2}^{\prime }=-{x}^{\prime }$ with the same interpretation of V. On the other hand if one decides to use e.g. K = −1 there will be no restrictions to V. In this case |V| > 1 is of course not a sign of a superluminality. It simply means that there is no possibility to interpret V as a relative velocity between observers. For example in the limit of infinite V

$$\begin{equation}{\Lambda}(1,-1,\infty )=\left[\begin{matrix}\hfill 0\hfill & \hfill -1\hfill \\ \hfill -1\hfill & \hfill 0\hfill \end{matrix}\right],\end{equation} \tag{ 11 }$$

which can be interpreted as transformations between two observers at rest.

Let us now turn to the antisymmetric case. To make a clear distinction between the two branches of transformation (originated from the choice of the γ(V) parity) we introduce some new notation in the antisymmetric case. Namely the parameter V will be denoted by W and coordinates used by observer O' will be denoted by η_j . γ(−W) = −γ(W) leads to

$$\begin{equation}\gamma (W)=\pm \frac{W}{\vert W\vert }\frac{1}{\sqrt{K{W}^{2}-1}}.\end{equation} \tag{ 12 }$$

It follows that

$$\begin{equation}\left[\begin{matrix}\hfill {\eta }_{1}\hfill \\ \hfill {\eta }_{2}\hfill \end{matrix}\right]=L(\tau ,K,W)\left[\begin{matrix}\hfill {\xi }_{1}\hfill \\ \hfill {\xi }_{2}\hfill \end{matrix}\right]\end{equation} \tag{ 13 }$$

with

$$\begin{equation}L(\tau ,K,W)=\tau \frac{W}{\vert W\vert }\frac{1}{\sqrt{K{W}^{2}-1}}\left[\begin{matrix}\hfill 1\hfill & \hfill -W\hfill \\ \hfill -W\hfill & \hfill 1\hfill \end{matrix}\right].\end{equation} \tag{ 14 }$$

In this case it is not so obvious from the physical point of view, how to choose the value of K. For simplicity let us choose K = 1 and concentrate on physical interpretation of the transformation L(−1, 1, W) with |W| ⩾ 1. In figure 2 we present spacetime diagrams showing worldlines of some massive (with non-zero invariant mass) subluminal (from the point of view of observer O) particles and light rays. First notice that the light rays seem to propagate in an expected way. In both coordinate systems the light rays fulfil the same equation ξ₁ = ±ξ₂ and η₁ = ±η₂. The most important observation, however, is that looking at figure 2(b) it seems that particles are propagating with superluminal velocities. Is this observation correct?

Zoom In Zoom Out Reset image size

Before answering this question, let us notice that this observation is crucial for the argument of Dragan and Ekert. If we have a given particle propagating subluminally and superluminally with respect to two equally 'legal' observers, then considering superluminal particles seriously cannot be avoided. On the other hand the existence of superluminal particles cannot be compatible with local and deterministic dynamics, hence unexpected and exciting connection between the relativity postulate and the inherent randomness of quantum mechanics can be obtained.

However, we suggest that the above observation could be wrong. Observation, that particles in figure 2(b) propagate superluminally is based on hidden assumption that there exist observer O' (called by Dragan and Ekert a superluminal observer) whose measurements of the time interval Δt_O' and the distance Δx_O' are connected to coordinates η_j in a simple way

$$\begin{align}\hfill & c{\Delta}{t}_{{O}^{\prime }}={\Delta}{\eta }_{1},\hfill \\ \hfill & {\Delta}{x}_{{O}^{\prime }}={\Delta}{\eta }_{2}.\hfill \end{align} \tag{ 15 }$$

It seems, that this assumption is based on the analogy between equations (10) and (13) and not on any actual consistent construction of a 'measurement induced' coordinate system for O'.

Let us show that one can choose a more natural interpretation of η_j coordinate system. Figure 3(a) presents a space-time diagram with two light rays emitted from the origin of the coordinate system to the left and to the right (from the point of view of observer O). Figure 3(b) presents the same rays in the η_j coordinates diagram. What we mean by a natural interpretation is to assume that the horizontal axis in figure 3(b) represents the time coordinate, whereas the vertical one represents the space coordinate. It means that the transformation L(−1, 1, W) interchanges space and time coordinates. We have seen a similar effect before in the transformation Λ(1, 1, ∞). So we propose to replace equation (15) by

$$\begin{align}\hfill & c{\Delta}{t}_{{O}^{\prime }}={\Delta}{\eta }_{2},\hfill \\ \hfill & {\Delta}{x}_{{O}^{\prime }}={\Delta}{\eta }_{1}.\hfill \end{align} \tag{ 16 }$$

With this choice of coordinates particles in figure 2(b) propagate with a subluminal velocity. We can now prove that one can construct a measurement system consistent with our proposition in equation (16) such that these equations together with equation (13) lead to

$$\begin{equation}\left[\begin{matrix}\hfill c{\Delta}{t}_{{O}^{\prime }}\hfill \\ \hfill {\Delta}{x}_{{O}^{\prime }}\hfill \end{matrix}\right]=\left[\begin{matrix}\hfill 0\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 0\hfill \end{matrix}\right]L(-1,1,W)\left[\begin{matrix}\hfill c{\Delta}{t}_{O}\hfill \\ \hfill {\Delta}{x}_{O}\hfill \end{matrix}\right].\end{equation} \tag{ 17 }$$

It is easy to show that

$$\begin{equation}L(-1,1,W)=\left[\begin{matrix}\hfill 0\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 0\hfill \end{matrix}\right]{\Lambda}(1,1,{W}^{-1}).\end{equation} \tag{ 18 }$$

Thus the coordinate transformation L(−1, 1, W) with parameter W ⩾ 1 together with an interchanging space and the time coordinates is equivalent to the standard transformation Λ(1, 1, V) with velocity V = W⁻¹ ⩽ 1. It follows that standard textbook construction based on meter sticks and clocks is consistent with equation (16) (see also figure 4). Moreover we have

$$\begin{align}\hfill {L}^{-1}(-1,1,W)& ={\Lambda}(1,1,-{W}^{-1})\left[\begin{matrix}\hfill 0\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 0\hfill \end{matrix}\right]\hfill \\ \hfill & =L(-1,1,-W).\hfill \end{align} \tag{ 19 }$$

It should be so because in order to return to the original coordinate system we should first exchange space and time coordinates and then perform the Lorentz transformation with the opposite velocity. This leads to antisymmetric γ(W) as given in equation (12).

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Let us emphasize that the coordinate system (η₁, η₂) can only be called superluminal because equation of motion of the point η₂ = 0, as seen by observer O, is

$$\begin{equation}x=Wct,\end{equation} \tag{ 20 }$$

with W > 1. However, it is well known that 'Arbitrary large velocities are possible for moving points that carry no information (...)' ([1], p 56).

Let us point that there are two possible notions of superluminality. The first one (let us call it the coordinate superluminality) is based on the coordinate velocity. The second one (the geometric superluminality) considers a spacetime displacement as superluminal if the square of the spacetime interval is less than 0 ((Δs)² = (Δx)² + (Δy)² + (Δz)² − (Δt)² < 0). For standard subluminal coordinate systems these two notions are equivalent. However, in the case of superluminal coordinate systems they disagree. In figure 4 particle wordlines connecting two events—X and Y are presented. It is clear that

$$\begin{equation}\left\vert \frac{{\Delta}{\xi }_{2}}{{\Delta}{\xi }_{1}}\right\vert < 1\end{equation} \tag{ 21 }$$

whereas

$$\begin{equation}\left\vert \frac{{\Delta}{\eta }_{2}}{{\Delta}{\eta }_{1}}\right\vert > 1.\end{equation} \tag{ 22 }$$

Thus, according to the first notion of superluminality the particle is subluminal relative to observer O and superluminal relative to O'.

On the other hand let us consider a square of the spacetime interval between events X and Y

$$\begin{equation}{{\Delta}s}^{2}={({\Delta}{\xi }_{1})}^{2}-{({\Delta}{\xi }_{2})}^{2}.\end{equation} \tag{ 23 }$$

Although expressed here with the use of coordinates, this is a purely geometrical notion, i.e. has an absolute, independent on the coordinates meaning. It can be stated e.g. as a proper time measured by a clock moving from X to Y with a constant speed (see [1], p 93). All observers (independent of their motion and of coordinate systems which they use) have to agree about (so measured) proper time. From figure 4 or equation (23) one sees that Δs² > 0. So it must be that Δs² > 0 for observer O' too. Of course one can as well express Δs² with the use of (η₁, η₂) coordinates. Using equation (13) in equation (23) gives

$$\begin{equation}{{\Delta}s}^{2}=-{({\Delta}{\eta }_{1})}^{2}+{({\Delta}{\eta }_{2})}^{2}.\end{equation} \tag{ 24 }$$

Once again it can be seen that η₁ plays the role of a spatial coordinate, whereas η₂ plays the role of a time coordinate.

So (in the sense of the geometric superluminality) the particle wordline depicted in figure 4 must be considered as subluminal for both observers O and O'.

Note that what is needed in the argument of Dragan and Ekert is the geometric superluminality (a possibility of sending particles between spatially separated events Δs² < 0). Unfortunately, what is provided by introducing the second branch of transformations is only the existence of the coordinate superluminality. A causal partition of events by light cones is absolute, not only for users of the Lorentz transformations ${\Lambda}\left(1,1,V\right)$, but also for users of the newly introduced transformations $L\left(-1,1,W\right)$. It is not true that a particle at rest relative to observer O will be considered superluminal by observer O' (in the geometric sense of superluminality). Being superluminal is not relative.

We have proved that the second branch of the coordinate transformation (based on antisymmetricity of $\gamma \left(V\right)$) given by equation (14) can be interpreted in a consistent way without the notion of superluminal particles or superluminal observers. In particular we have shown that the branch corresponding to superluminal observers can be interpreted as a relabelling of time and space coordinates for subluminal observers. This interpretation is coherent with the standard causal structure induced by light cones and expressed by the sign of the spacetime interval squared. It must be remembered that coordinates alone have no physical meaning. Hence in our opinion the considered transformation does not have to correspond to real physical observers but is just renaming of coordinates.

We thank A Dragan for comments on the manuscript. This work has been supported by the Polish National Science Centre (NCN) under the Maestro Grant No. DEC-2019/34/A/ST2/00081.

All data that support the findings of this study are included within the article (and any supplementary files).