Why shall Quantum Mechanics accept Relativity?

[dpa]

Hi all,
Quantum Mechanics has everything weird and unacceptable to classical theory and its foundations with phenomena like:
Uncertainity/Entanglement/Decoherence/Tunneling and let's say almost all Quantum Mechanical Phenomena are non classical.:shy:

My silly question is why do we expect QM to follow relativity? I don't know, but is it solely because we are chasing a fantasy where all forces unite and unification of QM and GR is the only way that is apparent to us?

 

[Nabeshin]

Well for 1) we know relativity works. This has been confirmed by a bunch of experiments. These experiments give us a very strong inkling that any theories of nature need to be lorentz invariant, which nonrelativistic QM obviously isn't, thus the push for a relativistic version, from a theoretical standpoint. For 2), more practically, relativistic quantum mechanics is ridiculously well tested and precise, perhaps the most well tested theory we have, and it works in every single case.

 

[DaveC426913]

Quite simply, to be acceptable, QM must be consistent with all the observations we've so far seen in the universe, even those observations that are consistent with relativity.

 

[dpa]

///
more practically, relativistic
quantum mechanics is ridiculously
well tested and precise, perhaps
the most well tested theory we
have, and it works in every single
case.
///
any examples... Such theories include ST, QFT right?
//

 

[Pengwuino]

any examples... Such theories include ST, QFT right?

I believe they were talking specifically about QED.

 

[Nabeshin]

I believe they were talking specifically about QED.

You believe correctly.

 

[dpa]

then my question was about why has QM to be compatible with GR

 

[Fredrik]

then my question was about why has QM to be compatible with GR

Because both are theories about matter. The matter that QM makes predictions about is clearly influenced by gravity (in the real world, but not in QM). The matter that GR makes predictions about, and describes as behaving in a completely classical way, is clearly not behaving in a completely classical way. So the fact that each of these theories is so successful proves that the other one is wrong.

That doesn't imply that a unified theory exists, but it explains why it would be desirable to have one.

 

[RUTA]

Because both are theories about matter. The matter that QM makes predictions about is clearly influenced by gravity (in the real world, but not in QM). The matter that GR makes predictions about, and describes as behaving in a completely classical way, is clearly not behaving in a completely classical way. So the fact that each of these theories is so successful proves that the other one is wrong.

Well, both could be wrong or both could be right, too (non-contradiction is an assumption).

 

[RUTA]

My silly question is why do we expect QM to follow relativity? I don't know, but is it solely because we are chasing a fantasy where all forces unite and unification of QM and GR is the only way that is apparent to us?

QM is not as incompatible with SR as you might think. Kaiser writes [1], p. 706,

"For had we begun with Newtonian spacetime, we would have the Galilean group instead of the [restricted Poincaré group]. Since Galilean boosts commute with spatial translations (time being absolute), the brackets between the corresponding generators vanish, hence no canonical commutation relations (CCR)! In the [c→∞ limit of the Poincaré algebra], the CCR are a remnant of relativistic invariance where, due to the nonabsolute nature of simultaneity, spatial translations do not commute with pure Lorentz transformations." (Italics in original).

Bohr and Ulfbeck also realized that the “Galilean transformation in the weakly relativistic regime” ([2], Sect. D of part N, p. 28) is needed to construct a position operator for QM, and this transformation “includes the departure from simultaneity, which is part of relativistic invariance.” Specifically, they note that the commutator between a “weakly relativistic” boost and a spatial translation results in “a time displacement,” which is crucial to the relativity of simultaneity. Thus they write [2], p. 24,

“For ourselves, an important point that had for long been an obstacle, was the realization that the position of a particle, which is a basic element of nonrelativistic quantum mechanics, requires the link between space and time of relativistic invariance.”

So, the essence of non-relativistic quantum mechanics—its canonical commutation relations—is entailed by the relativity of simultaneity. Harvey Brown called this "the footprint of relativity" (private conversation). The corresponding transformations leaving the Schrodinger dynamics invariant are not Lorentz, but they do contain relativity of simultaneity, so they're not Galilean either [3].

1. Kaiser, G.: J. Math. Phys. 22, 705–714 (1981)
2. Bohr, A., Ulfbeck, O.: Rev. Mod. Phys. 67, 1–35 (1995)
3. Stuckey, W.M., et al: Found. Phys. 38, No. 4, 348 – 383 (2008), see section 2. quant-ph/0510090.

 

[Fredrik]

Well, both could be wrong or both could be right, too (non-contradiction is an assumption).

Would you care to elaborate? This doesn't make sense to me. Would you also say that the statements "x=0" and "x≠0" can both be correct, since non-contradiction is an assumption?

QM is not as incompatible with SR as you might think. Kaiser writes [1], p. 706,

"For had we begun with Newtonian spacetime, we would have the Galilean group instead of the [restricted Poincaré group]. Since Galilean boosts commute with spatial translations (time being absolute), the brackets between the corresponding generators vanish, hence no canonical commutation relations (CCR)! In the [c→∞ limit of the Poincaré algebra], the CCR are a remnant of relativistic invariance where, due to the nonabsolute nature of simultaneity, spatial translations do not commute with pure Lorentz transformations." (Italics in original).

This is very interesting. I didn't know this.

 

[bohm2]

My silly question is why do we expect QM to follow relativity? I don't know, but is it solely because we are chasing a fantasy where all forces unite and unification of QM and GR is the only way that is apparent to us?

Up till now, the search for unification has been quite successful. Not only in physics but also in other sciences. The unification of biology with chemistry is one fairly recent example. Henrik Zinkernagel writes:

The quest for unification is a major drive behind the search for quantum gravity. For instance, Kiefer writes in the introduction to his recent book on quantum gravity concerning the main motivations for this theory:

The first motivation is unification. The history of science shows that a reductionist viewpoint has been very fruitful in physics (Weinberg 1993). The standard model of particle physics is a quantum field theory that has united in a certain sense all non-gravitational interactions. [...] The uni-versal coupling of gravity to all forms of energy would make it plausible that gravity has to be implemented in a quantum framework too.

As discussed below, it is not always the case that unification coincides with reductionism. In any case, it is true that the idea of unification between the different natural phenomena, and the theories that describe them, has been a guiding principle in physics at least since the days of Galileo. Indeed, the “success” history of physics can, at least partly, be portrayed as the history of unification.

The Philosophy behind Quantum Gravity
http://philsci-archive.pitt.edu/4050/1/PhilBehindQGFinalWref.pdf

I think there's a difference between unification and reduction though. Sometimes unification may not be possible because the reducing or more "fundamental" science has had to undergo radical revision for unification to proceed. Unification of chemistry with physics is an example. It didn't occur until classical physics was replaced by QM.

 

[StevieTNZ]

Have all attempts at unifying gravity and QM kept all the interpretational problems that we associate with QM?

 

[strangerep]

Kaiser writes [1], p. 706,

"For had we begun with Newtonian spacetime, we would have the Galilean group instead of the [restricted Poincaré group]. Since Galilean boosts commute with spatial translations (time being absolute), the brackets between the corresponding generators vanish, hence no canonical commutation relations (CCR)! In the [c→∞ limit of the Poincaré algebra], the CCR are a remnant of relativistic invariance where, due to the nonabsolute nature of simultaneity, spatial translations do not commute with pure Lorentz transformations." (Italics in original).

1. Kaiser, G.: J. Math. Phys. 22, 705–714 (1981)

A thought-provoking point of view. Thanks for mentioning it.

For those who can't access that JMP article, the argument can also be found in Kaiser's book, http://arxiv.org/abs/0910.0352 . See pp93-95 et seq.

In the standard QM treatment, one merely starts with the Galilean algebra, notes that it must be represented projectively, and observes that this implies room for an extra central element (in contrast to Poincare which admits no such extra central element). But Kaiser's viewpoint circumvents such manipulations: we now start with Poincare and perform a group contraction corresponding to the nonrelativistic limit $\frac{v}{c} \to 0$.

However, that limit doesn't make sense for photons, since $v/c = 1$, -- which is also food for thought...

 

[lugita15]

This raises an interesting issue. Presumably there are countless possible symmetry groups which reduce to the Galilean group in some limit or other. Yet if we form a quantum theory out of the Hilbert space representations of any such group, in general this theory will not yield the canonical commutation relation in the limit as this group reduces to the Galilei group. But the commutation relation is a consequence of the representation theory of the Galilei group itself. So we have a situation where, in the limit as group G reduces to the Galilei group, the representation theory of the former need not reduce to the representation theory of the latter. However, in the case where G is the Poincare group, this does hold - a relativistic quantum theory correctly reduces in the Galilean limit.

Do I understand all this correctly? If so, would this have allowed us to deduce special relativity from the structure of nonrelativistic quantum mechanics if Einstein hadn't come along?

 

[strangerep]

Do I understand all this correctly?

I don't think so...

Presumably there are countless possible symmetry groups which reduce to the Galilean group in some limit or other. Yet if we form a quantum theory out of the Hilbert space representations of any such group, in general this theory will not yield the canonical commutation relation in the limit as this group reduces to the Galilei group.

Such group "reduction" is actually called an "Inonu-Wigner contraction". It proceeds by a continuous deformation of the structure constants of the original group. If we start with (rigged) Hilbert space that carries a (projective) representation of the original group, then I don't see why we wouldn't end up a Hilbert space carrying the CCRs in the Galilei limit. (It might be a different Hilbert space of course.)

But the commutation relation is a consequence of the representation theory of the Galilei group itself.

That's not enough. You also need the free Newtonian dynamics to identify the physical mass parameter. See Ballentine section 3.4, especially "case (i)" on pp80-82.

 

[lugita15]

Such group "reduction" is actually called an "Inonu-Wigner contraction". It proceeds by a continuous deformation of the structure constants of the original group. If we start with (rigged) Hilbert space that carries a (projective) representation of the original group, then I don't see why we wouldn't end up a Hilbert space carrying the CCRs in the Galilei limit. (It might be a different Hilbert space of course.)

Then let me ask you this. Under what circumstances does a symmetry group, which has the Galilei group as an Inonu-Wigner contraction, have boosts and spatial translations commuting, and under what circumstances does such a group have them not commuting? Because that is the issue that is claimed to lead to the canonical commutation relations.

 

[strangerep]

Under what circumstances does a symmetry group, which has the Galilei group as an Inonu-Wigner contraction, have boosts and spatial translations commuting, and under what circumstances does such a group have them not commuting?

I don't know how to answer that in any useful way.

Suppose we pick a point on a 2D plane -- call the point (x,y). Your question is a bit like asking "under what circumstances are lines which lead to (x,y) straight or curved?"

 

[lugita15]

I had assumed that it was rare for a group whose Inonu-Wigner contraction is the Galilei group to have boosts and translation not commuting, and that the Poincare group (and those that contain it) are one of the few (or perhaps the only ones) that satisfy this condition. Am I wrong in that impression? If having them not commute is a fairly common property of groups which reduce to the Galilei group, then Kaiser's result wouldn't be that significant to me.

 

[strangerep]

I had assumed that it was rare for a group whose Inonu-Wigner contraction is the Galilei group to have boosts and translation not commuting, and that the Poincare group (and those that contain it) are one of the few (or perhaps the only ones) that satisfy this condition. Am I wrong in that impression?

Take any Lie algebra, modify its structure constants a little, and you (probably) have a different Lie algebra (modulo those modifications which are equivalent to a change of basis in the original Lie algebra).

So there are infinitely many.

If having them not commute is a fairly common property of groups which reduce to the Galilei group, then Kaiser's result wouldn't be that significant to me.

What matters is which groups are already well-proven experimentally to be of deep importance in physics. Poincare fits that description -- it's already well established from over a century of special relativity.

Another reason why I find Kaiser's result to be thought provoking is that it banishes the old puzzle about why there's no straightforward extension of the position-momentum CCRs
to an analogous time-energy relation.

 

[lugita15]

Take any Lie algebra, modify its structure constants a little, and you (probably) have a different Lie algebra (modulo those modifications which are equivalent to a change of basis in the original Lie algebra).

So there are infinitely many.

OK, what about "fundamentally different" in some sense or another?

 

[strangerep]

OK, what about "fundamentally different" in some sense or another?

The important phrase in my previous post was "well-proven experimentally to be of deep importance in physics.". Even if you found a "fundamentally different" group, you still must show that it is physically important in its own right.

But such open-ended questions are too speculative for my tastes, and are not something that interests me any further.

 

[faen]

Hi all,
Quantum Mechanics has everything weird and unacceptable to classical theory and its foundations with phenomena like:
Uncertainity/Entanglement/Decoherence/Tunneling and let's say almost all Quantum Mechanical Phenomena are non classical.:shy:

My silly question is why do we expect QM to follow relativity? I don't know, but is it solely because we are chasing a fantasy where all forces unite and unification of QM and GR is the only way that is apparent to us?

Even if quantum mechanics is non classical, it is consistent with classical mechanics. We try to figure out how it can be consistent with GR as well.

 

[RUTA]

Would you care to elaborate? This doesn't make sense to me. Would you also say that the statements "x=0" and "x≠0" can both be correct, since non-contradiction is an assumption?

Yes, this is a violation of the principle of non-contradiction. And, yes, it doesn't "make sense" to me either because the principles of logic are just a reflection of what "makes sense." How am I applying it to physics? When we had Maxwell's eqns and Newtonian mechanics, both of which worked in their respective realms, we had a contradiction in that Maxwell's eqns are Lorentz inv while Newtonian mech is Galilean inv. Thus, assuming non-contradiction, we "knew" that at least one of these theories wasn't "correct." And, of course, special relativity came along to show us it was Newtonian mech that was only a v << c approximation to the "correct" theory. Now we have QM and GR, both of which work in their respective realms, but they contradict one another, e.g., spacetime is flat, spacetime is locally Lorentz inv, etc. So, we "assume" that at least one of these theories is "wrong."

 

[andresB]

My two cents.

maybe not general relativity, but at least some theory of gravitation, (presumably ,but not necesarily, one that have in its postulate the princile of equivalence) is necesary for the consistance of quantum mechanics.

maybe I am wrong, but the Einstiein clock in the box paradox (and the bohr solution) seems to imply that