# Quantum mechanics vs. postulates of special relativity

1. Apr 2, 2009

### Heirot

Imagine a thought experiment: A is conducting a classical mechanics experiment (e.g. bouncing a ball on the table) and B is moving relative to A at speed so great that the table of A is contracted to the size of an atom. According to the postulates of special relativity, the physics of A and B must be the same. But A is using classical mechanics while B must use quantum mechanics. So, the sheer existence of quantum mechanics must necessary invalidate the postulates of special relativity. Any opinions?

2. Apr 2, 2009

### jambaugh

There is nothing wrong within QM with the location of the ball being localized to a size smaller than an atom. All QM says is that then the momentum of the ball must be uncertain to a degree inversely proportional to that atomic length. Given that (as seen by B) the momentum of the ball must be HUGE for this amount of relativistic contraction then even if the uncertainty of the momentum is a very small percentage of its total momentum, that uncertainty can still be huge and thus consistent with Heisenberg's uncertainty principle.

The key point here is that the uncertainty principle is dictated by the magnitude of the commutator of the momentum and position operators. This commutator is (in standard QM) the identity operator (times hbar) which has norm 1 (times hbar). This is true in all inertial frames via SR since the identity is a scalar w.r.t. Lorentz transformations. So if the uncertainty principle is obeyed in one frame it is in all frames.

3. Apr 2, 2009

### Heirot

I was actually implying that there something wrong (incomplete) with the postulates of relativity. The atomic scale in the problem is there just to point out that A i B use "different" physics - classical and quantum, while describing the same phenomena. So the laws of physics cannot be the of the same form in all inertial frames.

4. Apr 2, 2009

### jambaugh

But there is no such contradiction. You are using different physics by assumption. You are not finding a reason to use different physics from the experiment. Quantum mechanics applies to all scales. It is just that measurements above the atomic scale typically don't show much difference between classical and quantum predictions.

Similarly Special Relativity applies at all scales of relative velocity. It is just that at typical velocities we don't see much difference between relativistic and non-relativistic predictions.

So it is not that QM "suddenly kicks in" when you make the ball appear atomic size. It was there all along and quantum issues for the B observer map directly to quantum issues for the A observer. You can't assume facts as seen by the A observer which violate QM and then show that QM is violated by the B observer. You've already assumed a contradiction.

Rather you are via your boosted B observer simply magnifying what we otherwise would consider insignificant quantities (the uncertainty in the momentum and or position of the ball) as seen by the A observer. They are not absent, we rather do not include them when we aren't trying to localize the ball's (center of mass) position and its momentum to the umpteenth decimal place.

5. Apr 2, 2009

### Heirot

So, is it fair to say that, when dealing with classical physics, we should stick to the Galileian transformations because Lorentz transformations would (in principle) require QM?

6. Apr 2, 2009

### robphy

I wouldn't say that.
We don't get the electromagnetic waves we see with the Galilean transformations.

In principle, as far as we can tell,
everything requires Quantum Physics (i.e. something like Quantum Mechanics) and everything requires General Relativity (or something akin to it).
It's just that in certain situations, some "limiting approximations" may be appropriate to use [since the full machinery might be too difficult or impossible to apply].

Depending on one's approach to a unified theory of Quantum Physics and Gravitation,
one could argue that
Quantum is not quite right, or General Relativity is not quite right,
or [more likely] both are not quite right.
Each side has its own set of problems... conceptual, computational, etc..., as well as foundational... that make it difficult for them play nicely together in all situations.

7. Apr 2, 2009

### Staff: Mentor

Also, since the standard model is relativistic it is hard to see how it would pose any conundrum here.

8. Apr 2, 2009

### alxm

Special relativity and QM have been merged for some time. You get the Dirac equation or Klein-Gordon equation instead of the Schrödinger equation.

9. Apr 7, 2009