Quantum Mechanics and Special Relativity (violations)

[cgnicholls]

Would anybody please be able to explain to me in what ways the differing interpretations of Quantum Mechanics might violate Special Relativity?
For instance, in the Copenhagen Interpretation non-locality is proposed to explain quantum entanglement effects. What exactly does non-locality mean in this instance and how would it affect special relativity?

Any comments would be greatly appreciated.

Thank you,
Chris

 

[DrChinese]

Would anybody please be able to explain to me in what ways the differing interpretations of Quantum Mechanics might violate Special Relativity?
For instance, in the Copenhagen Interpretation non-locality is proposed to explain quantum entanglement effects. What exactly does non-locality mean in this instance and how would it affect special relativity?

Any comments would be greatly appreciated.

Thank you,
Chris

It is generally considered that QM and SR are not directly in conflict. The reason is that no signal or cause can propagate faster than c. The discovery of QM did not require any modification to either SR or General Relativity (GR).

However, there are possibilities that QM implies non-locality. The usual citation is that the wave function collapses faster than c (thought to be instantaneous in the view of most, and certainly greater than 10,000c per a recent experiment).

I would say that the topography of space-time - when considered in the 10+ dimensions that are thought to exist (not all of these are space-like) - is an open question in physics. It is possible that there is a dimension in which all space-like points are essentially local. However, this is speculation.

There are a number of threads on this forum, many recent, which discuss this subject in more detail.

 

[Naty1]

As a complement to post #2, you can also keep in mind that relativity proposes "nothing" can exceed c. quantum mechanics does not directly violate that limit: while some non locality and entanglement come awfully close, so far it has not been shown that any information can travel faster than c.

On the other hand there are some faster than light phenomena. One well known one is the expansion of space/time via the big bang; another is the "scissor blade intersection"...and the movement of a searchlight beam, a perfectly rigid rod, etc...but none of these has been shown to violate relativity...none can be used to transmit information...

Faster Than Light, a book by Nick Herbert (1989) discusses these phenomena and others.

 

[ajw1]

As a complement to post #2, you can also keep in mind that relativity proposes "nothing" can exceed c. quantum mechanics does not directly violate that limit: while some non locality and entanglement come awfully close, so far it has not been shown that any information can travel faster than c.

Is it not fair to say that when there are no hidden variables, information must have been transferred at a speed exceeding c (in 4D space-time), because of the correlated behaviour of entangled particles?

It seems to me irrelevant that this process cannot be used to tranfer usefull information.

 

[malawi_glenn]

Can someone tell us why the need of Relativistic Quantum Field Theory if there is no conflict? ;-)

 

[Civilized]

Can someone tell us why the need of Relativistic Quantum Field Theory if there is no conflict? ;-)

The conflict arises when we compare the theory to experiment, since the universe is relativistic, in other words non-relativistic QM is mathematically consistent but is in conflict with the real world. Then we find that relativistic QM is inconsistent because it treats the number of particles as fixed, and so to build a consistent relativistic quantum theory we introduce quantum fields.

 

[DrChinese]

Is it not fair to say that when there are no hidden variables, information must have been transferred at a speed exceeding c (in 4D space-time), because of the correlated behaviour of entangled particles?

It seems to me irrelevant that this process cannot be used to tranfer usefull information.

True, but where is the error in SR then? All of the theory of SR remains intact, and that is in turn embedded in GR. So perhaps they describe different areas, i.e. have a different scope.

 

[meopemuk]

The conflict arises when we compare the theory to experiment, since the universe is relativistic, in other words non-relativistic QM is mathematically consistent but is in conflict with the real world. Then we find that relativistic QM is inconsistent because it treats the number of particles as fixed, and so to build a consistent relativistic quantum theory we introduce quantum fields.

Exactly. If we limit ourselves with interactions that do not change the number of particles, then relativistic QM can be consistently formulated (see, e.g., B. Bakamjian and L. H. Thomas, "Relativistic particle dynamics. II", Phys. Rev. 92 (1953), 1300, and further works in this spirit). If we allow particle-number-changing interactions (that are obviously present in nature), then we need to use relativistic QFT.

 

[Fredrik]

Can someone tell us why the need of Relativistic Quantum Field Theory if there is no conflict? ;-)

This reply to DrChinese seemed very strange to me at first, but I realized after a while that you must be using a strange and old-fashioned definition of the term "quantum mechanics". You're probably defining QM as the theory of "wave equations". That definition seems so...1920.

You were probably taught this stuff the same way I was: 1. Let's try to replace the Schrödinger equation with a relativistic equation. 2. Oops, we don't get a conserved current. QM and SR must not be compatible. 3. Let's promote this useless wavefunction to an operator instead and call this procedure "second quantization". 4. Hey, this seems to be a useful idea. We seem to have solved the problem. Let's call this "quantum field theory".

The thing is, there never was a conflict between QM and SR. The above just means that step 1 was the wrong way to get the word "relativistic" into the theory. I would say that QM is the idea that states are represented by the one-dimensional subspaces of a complex separable Hilbert space, and that their time evolution is determined by the requirement that the symmetry group of spacetime must have a unitary representation on the Hilbert space. When you think of QM this way, step 1 above seems pretty strange. The obvious way to go from "non-relativistic" to "relativistic" is to just replace the Gallilei group with the Poincaré group, and there's no need to ever use the term "second quantization".

I don't think of QFT as the next step after QM. To me, QFTs are just specific theories of matter and interactions in the framework of QM.

 

[ajw1]

The thing is, there never was a conflict between QM and SR.

I'm not so sure about that. Someone like G. 't Hoofd (nobel price winner) still seems to see problems (http://www.narcis.info/publication/RecordID/oaidspacelibraryuunl187422671/Language/nl/repository_id/uudare/;jsessionid=8erp6u5xr3uz" ).
Does QFT clarify the instantaneous non-local behaviour of entangled particles?

 

[meopemuk]

You were probably taught this stuff the same way I was: 1. Let's try to replace the Schrödinger equation with a relativistic equation. 2. Oops, we don't get a conserved current. QM and SR must not be compatible. 3. Let's promote this useless wavefunction to an operator instead and call this procedure "second quantization". 4. Hey, this seems to be a useful idea. We seem to have solved the problem. Let's call this "quantum field theory".

The thing is, there never was a conflict between QM and SR. The above just means that step 1 was the wrong way to get the word "relativistic" into the theory. I would say that QM is the idea that states are represented by the one-dimensional subspaces of a complex separable Hilbert space, and that their time evolution is determined by the requirement that the symmetry group of spacetime must have a unitary representation on the Hilbert space. When you think of QM this way, step 1 above seems pretty strange. The obvious way to go from "non-relativistic" to "relativistic" is to just replace the Gallilei group with the Poincaré group, and there's no need to ever use the term "second quantization".

I agree completely. The worst thing is that most QFT textbooks still present the theory in this outdated and confusing fashion. They begin from "classical fields", "Dirac equation", "Dirac sea", "second quantization" and other irrelevant stuff. As far as I know, the only exception is Weinberg's "The quantum theory of fields".

I don't think of QFT as the next step after QM. To me, QFTs are just specific theories of matter and interactions in the framework of QM.

Exactly. QFT is just a specific example of quantum mechanics in which the number of particles is allowed to change.

 

[Fredrik]

I'm not so sure about that. Someone like G. 't Hoofd (nobel price winner) still seems to see problems (http://www.narcis.info/publication/RecordID/oaidspacelibraryuunl187422671/Language/nl/repository_id/uudare/;jsessionid=8erp6u5xr3uz" ).
Does QFT clarify the instantaneous non-local behaviour of entangled particles?

I expressed that part a bit poorly. I meant that the apparent conflict between SR and QM in "step 2" above isn't a problem, since "step 1" isn't the right way to go.

The article you referenced doesn't address the issue of whether QM and SR are compatible. It's about the possibility of having a deterministic theory that's more fundamental than QM.

QFT doesn't clarify anything about entanglement. That isn't really relevant here though, since entanglement doesn't violate SR.

 

[Fredrik]

As far as I know, the only exception is Weinberg's "The quantum theory of fields".

That book is what taught me to think this way. Chapter 2 is one of my favorite chapters in any book because of that.

Ballentine's QM textbook is the only one I know that treats non-relativistic QM this way. He downplays the role of "quantization" of classical theories, and defines the important observables by identifying them with the generators of a representation of the Galilei group.

 

[Peeter]

Pauli's wave mechanics book introduces the Schrödinger equation using relativistic context right from the beginning. His little quantum text starts off, not with the Bohr model or black bodies, ... but with a lighting fast two page treatment of SR.

In a nutshell, paraphrasing slightly, Pauli's argument goes like so:

- For one of the components of the electric or magnetic field in a vacuum (say \psi) we have:

<br /> \frac{1}{c^2}\frac{\partial^2 \psi}{\partial t^2} - \nabla^2 \psi = 0<br />

- Fourier transforming this equation, and absorbing some of the factors into a weighting function we have

<br /> \psi = \int A(k) e^{i(\omega t - k \cdot x)} d^3 k\quad\quad\quad(1)<br />

- application of the wave equation operator to (1) shows that this is still a solution to the wave equation provided

<br /> \omega^2 - c^2 k^2 = 0\quad\quad\quad(2)<br />

- From the photoelectric effect, we have the relation between frequency and energy

<br /> E = \hbar \omega<br />

so the null vector constraint on the angular frequency and wave number vector above (2) is in fact an \hbar scaled energy momentum invariant Lorentz length.

- This is where the debroglie hypothesis comes in, where the relativisitic energy momentum invariant length expressed in terms of frequency becomes

<br /> \omega^2 - c^2 k^2 = m^2 c^4 /\hbar^2<br />

- if one assumes that (1) also expresses the solution of a wave equation where the Lorentz length is non-null, then the associated wave equation for this solution, if conformity with SR is desired, must be:

<br /> \frac{1}{c^2}\frac{\partial^2 \psi}{\partial t^2} - \nabla^2 \psi = - \frac{m^2 c^2}{\hbar^2} \psi<br />

(ie: KG eqn)

This is actually close to the starting point to Pauli's treatment, and I'm filling in many of the gaps as I percieve them (Pauli says "see optics" and some of the above that I interpolate that to mean).

- The next step in the treatment, which would be more cumbersome to type up here (especially from memory and without paper to make notes on) is a small momentum approximation for \omega

<br /> \omega \sim m c^2 \left(1 + \frac{\hbar^2 k^2}{2m} \right)<br />

- substitution and some algebra writing

<br /> \psi = \psi&#039; e^{i m c^2 t}<br />

the KG equation expressed in terms of \psi&#039; takes the form of Schrödinger's equation (albeit with an extra second order time derivative scaled by \hbar^2 that Pauli neglects ... my assumption is that this neglect is due to the smallness of the second power in \hbar).

Schrödinger's equation at its root (at least as Pauli presents it) appears to be deeply relativistic. This isn't obvious when most introductory texts don't mention relativity at all, and the spatial and time derivatives aren't even of the same order.

I quite liked his approach, although it was too consise for my liking. According to the wikipedia page on KG it sounds like this also the historical approach that Schrödinger used.

After reading Pauli's intro, it's kind of ironic that Sdredniki's QFT book motivates the KG equation as a relativisitic correction of Schrödinger's equation, when it sounds like (according the the wikipedia KG page) Schrödinger actually started with this relativistic matter wave equation.

 

[meopemuk]

Hi Peeter,

Schroedinger-Pauli handwavings about "wave equations" looked cool 80 years ago, but today they are completely obsolete. The modern way to formulate relativistic QM and QFT is through unitary representations of the Poincare group. Note that Weinberg in his "The quantum theory of fields" mentions Dirac's equation only in the middle of the book as some kind of technical side effect.

 

[Dale]

Is it not fair to say that when there are no hidden variables, information must have been transferred at a speed exceeding c (in 4D space-time), because of the correlated behaviour of entangled particles?

It seems to me irrelevant that this process cannot be used to tranfer usefull information.

Two quick points. First, the smallest unit of information is a bit. If it cannot transmit a bit then it is not information that is transmitted by definition. Second, even if information were transmitted FTL that is not incompatible with SR by itself, only incompatible with SR and causality together.

 

[malawi_glenn]

Yes, when I mean Quantum Mechanics, I mean old fashioned non-relativistic Quantum Mechanics ala Scrodinger equation. To me, QM and SR COMBINES into QFT.

 

[ajw1]

The article you referenced doesn't address the issue of whether QM and SR are compatible. It's about the possibility of having a deterministic theory that's more fundamental than QM.

In this article the author doesn't mention what he considers to be problems between QM and SR. My only point here is that he clearly uses this as an argument for his theory.

Two quick points. First, the smallest unit of information is a bit. If it cannot transmit a bit then it is not information that is transmitted by definition. Second, even if information were transmitted FTL that is not incompatible with SR by itself, only incompatible with SR and causality together.

In my opinion a bit is transferred between entangled particles (again if one accepts hidden variables is a no-go). The only problem is that we cannot have the system transfer a bit we have chosen.
So there has been a FTL transmission that has affected a particle's state at a distance. Of course one can ignore this issue by arguing that causality is not broken.

 

[Peeter]

Schroedinger-Pauli handwavings about "wave equations" looked cool 80 years ago, but today they are completely obsolete. The modern way to formulate relativistic QM and QFT is through unitary representations of the Poincare group. Note that Weinberg in his "The quantum theory of fields" mentions Dirac's equation only in the middle of the book as some kind of technical side effect.

My point has nothing to do with relativistic QM nor QFT, and to be honest most of what you wrote is meaningless to me.

Pauli's book is an introductory text on non-relativistic quantum mechanics. With the original poster was looking for "ways the differing interpretations of Quantum Mechanics might violate Special Relativity", it seemed reasonable to me to point out that at least one approach that arrives at the familiar non-relativistic form of the Schrödinger's equation relies completely on special relativistic arguments. My attempt to make that point was likely too verbose, but I found it personally helpful to walk through the sequence of steps in my own words.

Now, you can call that handwaving if you like, but compared to what can be found in other introductory quantum texts, I personally rank it as a step up in plausibility. There is a logical sequence, and when some of the missing steps are filled in, none of the steps are suprising.

One can compare this to the energy conservation argument in French and Taylor's introductory quantum mechanics. It's a well constructed argument but unnatural argument. For reference that argument is essentially the same as found in wikipedia's "heuristic derivation" ... but the wiki article has the associated discussion omitted. Alternately, look at how Schrödinger's equation is "derived" in Liboff's text where he says "operatorize the Hamiltonian" and pulls \mathbf{p} \sim -i \hbar \nabla out of a magic hat. There's no shortage of handwaving to be found in introductory QM texts. Bohm's Quantum Theory is one that I've seen that doesn't do much of it, but the cost is 200 pages and nine chapters to get to the starting point of most other QM books (an excellent book but not for the impatient).

 

[Fredrik]

Yes, when I mean Quantum Mechanics, I mean old fashioned non-relativistic Quantum Mechanics ala Scrodinger equation. To me, QM and SR COMBINES into QFT.

When I wrote my previous posts in this thread, it had completely slipped my mind that there's a term I'm comfortable with for the theory you call "quantum mechanics". I see that Peeter used it. It's "wave mechanics".

I wouldn't say that WM and SR "combines into QFT", since it's actually the incompatibility of the two (i.e. the failed attempts to make WM relativistic) that led to "second quantization" and QFTs, but I realize of course that you're aware of this, and that this is in fact what you had in mind. So I don't disagree with your thoughts, but I had to point out the fact that what you said doesn't accurately represent what you were thinking.

 

[malawi_glenn]

hehe well I guess that quite much in this thread is quite "ill defined" and missunderstood ;-)

For instance, OP's first post might reflect that what he means with QM is "good old fashioned" non-rel QM. I don't know really, we should perhaps wait till he comes back :-)