Can the SR Concept of Reference Frames be Applied to Quantum Mechanics?

[LarryS]

SR "Reference Frames" in QM ?

The SR concept of “Reference Frame” cannot be transferred to the micro-world of QM because, due to the HUP, space and time are blurry in that world. Comments? (Thanks in advance).

 

[jtbell]

due to the HUP, space and time are blurry in that world.

Why do you think that is true?

 

[LarryS]

Why do you think that is true?

The degree (standard deviation) of indeterminacy of Time is linked, via the HUP, to the degree of indeterminacy of Energy .

Likewise for Position and Momentum.

In QM, in practice, one never finds Time and Space precisely defined.

Therefore, I believe that a set of perfectly defined Time and Space coordinates for the Quantum World is what physicists refer to as mathematical fiction.

 

[haushofer]

I never fully understood the meaning of this "uncertainty in energy and time" in QM. Griffiths calls dt the time that the energy changes a standard deviation or something like that. I know it is motivated by the fact that energy and time are the zero components of the 4-vectors momentum and position, but a completely satisfactory explanation seems to be rare.

I think you should be careful about phrasing

"In QM, in practice, one never finds Time and Space precisely defined".

What does this mean?

 

[ytuab]

To begin with, can HUP (uncertainty principle) coexist with the relativistic theory?

HUP is caused by the two properties of the Schrödinger wave function.
The wave function means the probability amplitude of the particles at the position (x,y,z),
and the derivative of the function means the momentum of them.

But in the relativistic QFT, if the wave function of Dirac (or KG) equation means the probability amplitude, when the particles are unequally distributed in space, this equation doesn't keep Lorentz invariant.
So, to keep Lorentz invariant strictly, we need to forget the ideas of the probability amplitude at a position and HUP, I think.
(Of course, we can use the Schrödinger wave function with the relativistic functions as the non-relativistic approximation.)

This is due to "the world of the wave function" of QM, I think.
When we use the relativistic theory in QM, the scope of the application is more restricted than the classical-mechanical particles to get the whole equation Lorentz invariant.

So we treat the relativistic particles such as electrons and photons by integrating the wave function in all space. For example, N relativistic particles with the energy hv are exinsting in all space.

I feel "they have reached their limit".
(It's an interesting question. so I want to listen to more other opinions.)

 

[Fredrik]

Spacetime is perfectly well-defined in relativistic QM. It's just Minkowski space, i.e. the same mathematical structure that's used in classical SR. The position of a particle on the other hand, that's another story. But it's the same story as in non-relativistic QM. The HUP doesn't change just because you replace non-relativistic spacetime with relativistic spacetime.

 

[ytuab]

The HUP doesn't change just because you replace non-relativistic spacetime with relativistic spacetime.

Can you explain this part in more detail?
Can you combine the idea of HUP with the relativistic QFT to form Lorentz invariant functions (not non-relativistic approximation)?
I would be glad if you show one concrete example of those wave functions here.
If the function is not Lorentz invariant, the form of Dirac (or KG) equation could possibly change in each reference frame.

As far as I know, the wave functions of the relativistic field equations (Dirac or KG) do not mean the probability amplitude.

In page 110 (the Story of Spin)
-------------------------------
The Dirac equation is also the relativistic field equation for the electron and it cannot be considered to be an equation of probability amplitude in x,y,z space. They insisted that a concept like "the probability of a particle to be at x in space" is meaningless for relativistic particles- be they electrons, photons ...
------------------------------------

So to incorporate the idea of HUP, we have to combine the non-rerativistic Schrödinger wave function to the relativistic fields. This could not keep Lorentz invariant.

How about the electron's movement ?
A nucleus with a large charge will cause an electron to have a high velocity. A higher electron velocity means an increased electron relativistic mass. These relativistic effects are experimentally showed.
So, the electrons are actually moving fast obeying the probability density of the Schrödinger equation (or other equations which show the probability density) ?
If so, why don't they radiate energy?

If the electrons are not actually moving as the electron clouds, why does the relativistic mass change occur?
In the relativistic theory, the particle's movement in one direction means our (observer's) movement in the opposite direction. So if we (observers) are actually moving in one direction and the relativistic effects of the observed particles are seen,
this means that these effects are caused by the particles' actual movement in the opposite direction?

 

[Fredrik]

Check out the derivation of the modern version of the HUP. There's no reference at all to spacetime structure. If you want to specifically consider the uncertainty principle for position and momentum, then yes, things do change when we go from non-relativistic QM to relativistic QM, because there's no position operator in the relativistic theory. (Apparently it's possible to construct one in a lot of cases, but not all. I don't think you can construct one for photons for example).

 

[atyy]

Reference frame is fine in non-relativistic and relativistic QM. In non-relativistic QM, particles localized in one frame are localized in another frame. In relativistic QFT, localization is defined by some "Newton-Wigner" state, in which a particle localized in one frame is not localized in another frame.

Teller, http://books.google.com/books?id=4f3S6DJ8OZIC&dq=teller+quantum+field&source=gbs_navlinks_s ,p87

 

[jimmy neutron]

Check out the derivation of the modern version of the HUP. There's no reference at all to spacetime structure. If you want to specifically consider the uncertainty principle for position and momentum, then yes, things do change when we go from non-relativistic QM to relativistic QM, because there's no position operator in the relativistic theory. (Apparently it's possible to construct one in a lot of cases, but not all. I don't think you can construct one for photons for example).

In NR QM the HUP seems intimately related to the commutator [xi,pj] = ihbar delta(i,j).
Presumably this does not generalise in RQM.

 

[Demystifier]

There is a way to treat time and space in QM on an equal footing:
http://xxx.lanl.gov/abs/0811.1905 [Int. J. Quantum Inf. 7 (2009) 595]