While proving that no matter can reach the speed of light (a fact which I call the kinetic energy barrier), Einstein uses the fact that he can calculate the velocity and position of an electron. However, if quantum effects apply, then it seems to create a problem in Einstein's assumptions themselves. How is the proof of the kinetic energy barrier true even though quantum effects exist in nature?

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    $\begingroup$ Related: physics.stackexchange.com/q/6202/2451 and links therein. $\endgroup$ – Qmechanic Dec 28 '11 at 16:03
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    $\begingroup$ Related - How can a photon have no mass and still travel at the speed of light? $\endgroup$ – voix Dec 28 '11 at 16:20
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    $\begingroup$ Hi nikhil, and welcome to Physics Stack Exchange! I edited your question for a couple of reasons: (1) It was really two questions in one, and we generally prefer that separate questions be asked in their own separate posts; (2) the first part of your question was already covered by those linked by Qmechanic and voix. Rather than close this as a duplicate, I figured I'd just make this post about the quantum thing. If after reading the linked questions you're still confused about how photons can travel at the speed of light, you're welcome to post a new question about it. $\endgroup$ – David Z Dec 28 '11 at 22:19
  • $\begingroup$ I got stuck st this point: the is no limit to the kinetic energy an object can have, rather it's kinetic energy goes to infinity as its velocity goes to c. $\endgroup$ – Stephen McAteer Dec 29 '11 at 1:17
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    $\begingroup$ @Ron: yes, in this case that's exactly what I meant to do. As it was, the question was partially a duplicate of the other one, but it had another part that wasn't a duplicate. So rather than closing it outright, I just edited out the part that was a duplicate. (Also I know that "speed of light barrier" is the common term but I opted to preserve nikhil's wording of "kinetic energy barrier" because I didn't think it made the question any less understandable. But that's a fine edit to make.) $\endgroup$ – David Z Dec 29 '11 at 20:05

It is well established by now that the speed of light barrier does not apply to quantum particles, and this property makes the construction of relativistic quantum field theories and other relativistic quantum systems tightly constrained. The argument that you can't transmit signals faster than light is fine, but particles are not necessarily signals, because if you make a particle over here, and measure a particle over there, you might not be measuring the particle you created, but another identical particle you created from the vacuum.

So relativistic field theory, with its faster-than-light particles, requires that there are no unique particles, that all particles have identical copies. Further, the faster than light motion can be back-in-time in different frames, and back-in-time motion means that every particle must have a back-in-time partner, called its antiparticle.

The quantum field restores locality. So even though quantum particles can propagate faster than light, information can't propagate faster than light. The quantum fields are the quantities which tell you what information you can gain locally by experiments. The particles in a Hamiltonian formulation are nonlocally related to the quantum fields (but the two are related more simply in a particle path integral formulation).

The proof that quantum particles cannot be restricted to less than light speed is simple: the restriction that the energy is positive means that the frequency is positive, while the restriction to forward propagation inside the light cone means that the propagator vanishes outside the future lightcone, so in particular, into the past. and there is no function which vanishes in a half-plane whose Fourier transform also vanishes in a big unbounded region like this. This is covered in this answer: the causality and the anti-particles

  • $\begingroup$ Assuming a reliable setup, using an entangled trigger i can be sure that Schrödinger's cat is dead before the light of its corpse reaches me. $\endgroup$ – Cees Timmerman Oct 17 '13 at 17:15
  • $\begingroup$ However, I think when people have in mind "violation of the speed of light barrier" they specifically mean transmission of information and causal influences (or something equivalent to it) into the "elsewhere" region of an event. $\endgroup$ – The_Sympathizer Oct 21 '18 at 5:27
  • $\begingroup$ QFT, as you say, does not permit this, even if some unobservable processes may be interpreted as "going faster than light". $\endgroup$ – The_Sympathizer Oct 21 '18 at 5:27

Light speed being invariant is necessary for there to be "quantum effects" (if by that, you mean probabilistic phenomena).

Since light speed is invariant, fixed at a rate of 1 lp per 1 tp, and no measurements are possible below the Planck scale, therefore any speed slower than c must be described as a probability at that scale.

Even if you plot the position of a wavefront as smoothly continuous regardless of scale, when you superimpose the smooth path of that wavefront on graph paper taking each square to be 1 lp by 1 tp, since the slope of the path is the velocity, at speeds below c there will increasingly be occurrences where two or more consecutive steps forward in time may fall between two consecutive steps forward in distance.

Alice, without knowing the initial state of the system, upon observing the particle at a certain point, even if she knew its velocity, would never be able to know for certain where the particle would be at the next Planck time, because there would always be a probability less than 100% that particle would move 1 lp relative to her frame of reference.

Considering that, it seems that the invariance of c is indeed what forces speeds slower than c to be expressed as probabilities at the scale below which observation cannot theoretically distinguish differences. Perhaps this seems like a circular argument, but, consider that if the maximum possible speed (c) could change, then the Planck length would change too.

Regarding the other answer that says quantum particles can go faster than light speed, I'm not sure how useful that answer is, even if it's true, because a "quantum particle" that moves faster than light is not in the same category of things as everything else we think of as a "particle." Furthermore in light of the work of Joy Christian and John Bush, et al., there does seem to be some debate on whether the interpretation of QM accurately represents what is actually going on behind the scenes. Although there's no doubt as to QM's predictive accuracy, the cosmological picture it paints is by no means scientifically proven the same way that the speed of light is.

If we limit ourselves to only a discussion about what is directly observable and what actually constitutes "information," I think we can avoid some semantic confusion. When we say "speed barrier" we are referring to a velocity that an observable or real object can travel at, and we should leave out of the discussion any imaginary or virtual things that cancel out of the equation by the time any effect can be observed.

Correct me if I'm wrong, please.

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    $\begingroup$ The first sentence is obviously false, because we have a perfectly consistent non-relativistic theory of quantum mechanics. $\endgroup$ – WillO Nov 16 '15 at 4:52
  • $\begingroup$ I'm curious why you think quantum mechanics being non-relativistic means that the invariance of light speed is not crucial for quantum mechanics being probabilistic in nature? It seems to me that the limitations of our ability to observe quantum things stem in part from the fact we cannot observe things with photons without shooting them at the speed of light. If photons could be bounced off of things rather slowly perhaps it would allow us to make more precise, and less probabilistic observations. Of course, I am just a layman, reasoning about these things from what limited reading I've done. $\endgroup$ – CommaToast Nov 17 '15 at 6:03
  • $\begingroup$ This does not provide an answer to the question. To critique or request clarification from an author, leave a comment below their post. - From Review $\endgroup$ – Jon Custer Nov 17 '15 at 21:10
  • $\begingroup$ @JonCuster: While this answer seems to me to be wrong and meandering, it is an attempt to answer the question. CommaToast, you seem to assume that quantum mechanics presupposes Planck length and Planck time are limits of measurement. This is wrong, it is a heuristic thing we say because those are the scale where quantum gravity becomes definitely non-neglegible, but neither time nor space are discrete in standard quanutm mechanics, as you can see by the position operator having continuous spectrum and the time evolution being a smooth one-parameter group. $\endgroup$ – ACuriousMind Nov 17 '15 at 22:11
  • $\begingroup$ @ACuriousMind - indeed, I perhaps selected the incorrect option - which would you (and the community) prefer for this? $\endgroup$ – Jon Custer Nov 17 '15 at 22:57

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