### Is there Minkowski space-time?

The issue of the non-universality and interaction-dependence of Lorentz transformations was discussed here and here . It was shown that in a rigorous quantum relativistic setting, the Lorentz transformations (for the energy-momentum of particles as well as for the position-time of events) depend on the interaction acting in the physical system. The boost transformations of particle observables are sensitive to the presence of other particles around and on interactions with them.

This means that one cannot rigorously introduce the idea of Minkowski space-time. Indeed, this idea works only in the case when all physical systems obey exactly the same boost transformation laws - the Lorentz transformations. Only then one can say that these are not simply transformations of individual particle observables, but global transformations of space-time coordinates (similar to universal and interaction-independent spatial shifts and rotations). This is the story told to us by Einstein's special relativity, and this is an old story that should be changed.

- Q.: Are you saying that particle observables, such as momentum and energy, do not transform according to covariant tensor laws?
- A.: Yes.
- Q.: Then how do they transform when the velocity of the observer changes?
- A.: The exact transformation laws are very close to the usual Lorentz formulas, but they include small corrections that depend on the state of the system of particles and on the interaction acting there. There is no universal formula.
- Q.: Can you say something more definitive?
- A.: Yes, I can. One can perform a rigorous calculation of the slowing down of the decay of fast moving particles and show that Einstein's time dilation formula holds only approximately.
- Q.: But this formula was confirmed in particle decay experiment to a great precision.
- A.: It is true. However, the corrections to the decay laws of moving particles predicted by my approach are several orders of magnitude smaller than the resolution of modern instruments.
- Q.: I find it difficult to believe that the Minkowski space-time picture that served us well for almost 100 years should be abandoned. A huge part of theoretical physics is built upon this formalism.
- A.: Let me give you one more argument. Please compare two kinds of inertial transformations: time translations and boosts. It doesn't look surprising that the result of a time translation is interaction-dependent. The time evolution of an interacting system is quite different from the time evolution of a non-interacting system. Why boosts should be different? In the instant form of Dirac's dynamics, the boost operator (just like the Hamiltonian) is interaction-dependent. Therefore we should expect that boost transformations of observables in the interacting system should be different from the transformations in the non-interacting system. There is a full analogy with the time evolution.
- Q: If we abandon the Minkowski space-time and covariant tensor transformation laws, then there is nothing left from the theory of relativity.
- A: I do not agree. The theory becomes more complicated than Einstein's special relativity with universal space-time. This is true. However, the most important ingredient of any relativistic theory is not affected: the invariance of physical laws with respect to the Poincare group of inertial transformations is strictly maintained. Together with the laws of quantum mechanics, this is sufficient for performing all kinds of quantum relativistic calculations.

## 3 Comments:

I remember seeing your paper appear on the arxiv because I did a similar (in spirit) toy calculation: Consider a charged classical particle orbiting in a constant magnetic field without radiation. Treat the particle/period as a clock. Now give the particle some translational velocity as well and calculate the period. In other words, Lorentz symmetry is not exact as it is for plane EM waves. You'll find that the period of the particle dilatates as expected only when the translational speed is much greater than the orbital speed. (unless i made a mistake :)

Jefimenko wrote a paper in the AJP doing essentially the same thing, i.e., considering time dilatation as a dynamical process in real clocks.

It's very interesting to me that this inconsistency can exist in classical dynamics and E&M. For my GR course I tried computing the same periods including gravity with the hopes that it would rescue the exact Lorentz invariance. No such luck. Weird, eh?

s. mchugh:

Do you have your calculations of the particle orbiting in the magnetic field published? I am very interested. Why don't you put it on www.arxiv.org/physics, for example?

Jefimenko wrote a paper in the AJP...I think you are talking about the paper [1]. There is a similar (in spirit) approach in [2]. I like their way of thinking, but I wouldn't necessarily trust the obtained numerical results, because, to my taste, they use too many assumptions about the validity of relativistic mechanics and classical electromagnetism. I wouldn't bet my money on the exactness of the empirical Lorentz force formula.

What I like about the studies of the dynamical character of time dilation in particle decays (see this paper and references therein, especially Khalfin and Shirokov) is that they are free of any additional assumptions. They use only laws of quantum mechanics and the Poincare group properties of inertial transformations, which is the exact mathematical description of the principle of relativity.

[1] O.D. Jefimenko, "Direct calculation of time dilation",

Am. J. Phys.,64, (1996) 812.[2] D.K. de Vries and W.M. de Muynk, "A dynamical approach to time dilation and length contraction"

Found. Phys. Lett.,9, (1996) 133.I think you are talking about the paper [1]. There is a similar (in spirit) approach in [2]. I like their way of thinking, but I wouldn't necessarily trust the obtained numerical resultsyes, that's the paper. i agree that getting numerical result to compare with experiment, particle decay rates are the best way to go.

no, i've not published anything about it since it seemed pretty textbook.

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