Are there any accepted relativistic interpretations of quantum mechanics?

[pines-demon]

From the few interpretations of quantum mechanics that I know, let's say:

• Copenhagen
• Many Worlds (MWI)
• Bohmian mechanics
• Superdeterminism
• Retrocausality/transactional
• Objective-collapse
• Qbism*
• Consistent histories*

I believe the only one that accepts a relativistic version in Copenhagen interpretation and only because it is completely agnostic to many points. Bohmian mechanics has always been criticized for not being field-like or and unable to be relativistic. Per Measurements in QFT I understand that MWI does not have an accepted relativistic version. I would argue that Superdeterminism and objective-collapse are more like ingredients of interpretation and barely count for relativistic concerns. And retrocausality just throws causality in the drain. And the others with a * I do not know even know what they mean or how they solve anything.

Anyway is there an interpretation in quantum mechanics with an accepted realtivistic formulation?

 

[Morbert]

Consistent histories is readily applicable to any quantum theory, where a "quantum theory" is a theory whose C*-algebra of observables is non commutative. Under consistent histories, complementary "classical" boolean lattices are constructed from this algebra, and hence frameworks for constructing reliable logical propositions and inferences about quantum systems. In a sense, the name "consistent histories" buries the lede. It should be called something like complementary lattices interpretation.

 

[pines-demon]

Consistent histories is readily applicable to any quantum theory, where a "quantum theory" is a theory whose C*-algebra of observables is non commutative. Under consistent histories, complementary "classical" boolean lattices are constructed from this algebra, and hence frameworks for constructing reliable logical propositions and inferences about quantum systems. In a sense, the name "consistent histories" buries the lede. It should be called something like complementary lattices interpretation.

Could you recommend any introduction to this interpretation?

 

[Morbert]

Could you recommend any introduction to this interpretation?

"Classical Equations for Quantum Systems" by James Hartle and Murray Gell-Mann is a good paper.

"Understanding Quantum Mechanics" by Roland Omnes and "Consistent Quantum Theory" by Robert Griffith are good books.

[edit] - The paper probably isn't good as an introduction. Better to start with the books and move to the paper after.

 

[Demystifier]

Bohmian mechanics has always been criticized for not being field-like or and unable to be relativistic.

For a response to such critiques see my https://arxiv.org/abs/2205.05986

 

[Fra]

• Qbism*

...
And the others with a * I do not know even know what they mean or how they solve anything.

Anyway is there an interpretation in quantum mechanics with an accepted realtivistic formulation?

IMO, in an observer centered view like qbism, I would take "relativity" to be a statement about the relations between different observers/agents; which is indirectly a statement about the "environment" or embedding of the quantum system, from where observations are made.

Ie. the "relativisitic formulation" is not so much a statement about the quantum system, as it is about the relations within the equivalence class of "descriptions of the quantum system", from the perspective of a set of "inertial observers".

Weird paradoxal things still happen when you try to make this "observer structure" part of the quantum system (just like you try to make "macroscopic measuremen devices" part of a quantum system. But this I think is an limitation with the theory, and I dont' see how a pure interpretation can solve it.

/Fredrik

 

[RUTA]

From the few interpretations of quantum mechanics that I know, let's say:

• Copenhagen
• Many Worlds (MWI)
• Bohmian mechanics
• Superdeterminism
• Retrocausality/transactional
• Objective-collapse
• Qbism*
• Consistent histories*

I believe the only one that accepts a relativistic version in Copenhagen interpretation and only because it is completely agnostic to many points. Bohmian mechanics has always been criticized for not being field-like or and unable to be relativistic. Per Measurements in QFT I understand that MWI does not have an accepted relativistic version. I would argue that Superdeterminism and objective-collapse are more like ingredients of interpretation and barely count for relativistic concerns. And retrocausality just throws causality in the drain. And the others with a * I do not know even know what they mean or how they solve anything.

Anyway is there an interpretation in quantum mechanics with an accepted realtivistic formulation?

The spacetime distribution of quantum events per the Hilbert space (kinematics) of QM is perfectly compatible with the Minkowski spacetime (M4 kinematics) of SR, as long as one does not add non-Lorentz-invariant dynamics in an attempt to account constructively for spacelike correlated quantum events in time-ordered fashion. This holds even though the time evolution of the state vector in a fixed-dimensional Hilbert space (dynamics) is governed by Schrödinger's equation which is not Lorentz invariant, because the kinematics is only concerned with the distribution of quantum outcomes in M4. Schrödinger's equation is simply the low energy approximation of the Lorentz-invariant Klein-Gordon equation of quantum field theory, which accounts for distributions in M4 requiring Hilbert spaces of many different dimensions (Fock space).

 

[gentzen]

Schrödinger's equation is simply the low energy approximation of the Lorentz-invariant Klein-Gordon equation of quantum field theory

This is simply nonsense.

 

[RUTA]

This is simply nonsense.

Zee, A. (2003). Quantum Field Theory in a Nutshell. Princeton University Press, Princeton, p. 173

 

[gentzen]

Zee, A. (2003). Quantum Field Theory in a Nutshell. Princeton University Press, Princeton, p. 173

I guess you are talking about a section that starts with "For clarity consider first the relativistic Klein-Gordon
equation $$(\partial^2 + m^2)\Phi = 0 \qquad (2)$$ for a free scalar field. A mode with energy E=..." and ends with "Dropping the term $(\partial^2/\partial t^2)\varphi$ as small compared to $−2im(\partial/\partial t)\varphi$, we find Schrödinger’s equation, as we had better: ...(3)"
What you get here is a Schrödinger equation for a single particle. But didn't you start with an equation describing an entire field, i.e. potentially multiple particles? That is no surprise in the book, because the derivation focused on a single "mode with energy E=...". So the book is fine from my perspective.

But your claim is nonsense, because the Klein-Gordon equation describes bosons, while the nonrelativistic Schrödinger equation is used for both bosons and fermions. A derivation which works fine for the case of a single particle doesn't necessarily generalize for the case of multiple particles!

 

[Demystifier]

because the Klein-Gordon equation describes bosons,

That's not necessarily true. A function $\psi(x)$ that satisfies Dirac equation, satisfies also the Klein-Gordon equation.

 

[gentzen]

That's not necessarily true. A function $\psi(x)$ that satisfies Dirac equation, satisfies also the Klein-Gordon equation.

You mean I should have written "because the Lorentz-invariant Klein-Gordon equation of quantum field theory describes bosons" instead?
Also, your statement still seems to talk about the single particle case, so it doesn't yet invalidate my statement that "A derivation which works fine for the case of a single particle doesn't necessarily generalize for the case of multiple particles!". More generally, I hope that despite of your criticism of details of my post, you still agree that the statement about Schrödinger's equation which I called nonsense is at least very misleading.

 

[Demystifier]

Also, your statement still seems to talk about the single particle case

There are also Dirac and Klein-Gordon equations for many-particle states. See e.g. my https://arxiv.org/abs/1205.1992 Sec. 8.3.3.

 

[RUTA]

I guess you are talking about a section that starts with "For clarity consider first the relativistic Klein-Gordon
equation $$(\partial^2 + m^2)\Phi = 0 \qquad (2)$$ for a free scalar field. A mode with energy E=..." and ends with "Dropping the term $(\partial^2/\partial t^2)\varphi$ as small compared to $−2im(\partial/\partial t)\varphi$, we find Schrödinger’s equation, as we had better: ...(3)"
What you get here is a Schrödinger equation for a single particle. But didn't you start with an equation describing an entire field, i.e. potentially multiple particles? That is no surprise in the book, because the derivation focused on a single "mode with energy E=...". So the book is fine from my perspective.

But your claim is nonsense, because the Klein-Gordon equation describes bosons, while the nonrelativistic Schrödinger equation is used for both bosons and fermions. A derivation which works fine for the case of a single particle doesn't necessarily generalize for the case of multiple particles!

You've Zee's point. He has the general prescription for taking "the nonrelativistic limit of a quantum field theory" at the bottom of p. 172 so that you obtain the SE from the KG equation ("as we had better"). A QM Hilbert space can represent more than one particle, yet it is surely the case that the SE governs the time evolution of the state vector in Hilbert space. Zee is simply showing you how to get the relevant nonrelativistic diffusion equation from the relevant relativisitic wave equation, since this it is the Hamiltonian for this equation that dictates the evolution in a fixed-dimensional Hilbert space, i.e., if you start with 2 particles you always have 2 particles. That's QM, it's the nonrelativistic limit of QFT where you can start with 2 particles and end up with many more than 2 (Hilbert space --> Fock space). Consider the alternative hypothesis, i.e., QM isn't the nonrelativistic limit of QFT, but some other theory is. Then you'd expect so see some deviation between QM predictions and experimental data where both QM and this hypothetical alternative nonrelativistic limit of QFT conflict but are expected to hold. I don't know of any such deviations between experimental data and QM.

Likewise, people who think QM and SR are "incompatible" (e.g., "On the Incompatibility of Special Relativity and Quantum Mechanics" https://arxiv.org/pdf/1704.02587.pdf) can't possible be talking about the theories themselves. Both theories work beautifully in their realms of applicability. Quantum entanglement is usually blamed for this supposed incompatibility between the theories, but physicists do not believe the experimental evidence in accord with QM refutes SR. So, it's not the physics that is problematic, it's the physicists and philosophers who have preconceived notions of reality that don't allow them to understand the empirical evidence in support of the theories. The problem stems from trying to account dynamically/constructively for the QM correlations between spacelike related measurement outcomes that violate Bell inequalities. Doing so necessarily violates locality, statistical independence, intersubjective agreement, or the uniqueness of experimental outcomes. QM does not contain such a causal mechanism, so it doesn't necessitate the violation of any of those things.

I could say a lot more, but I don't have time. My colleagues and I have a book coming out this summer with Oxford UP titled, "Einstein's Entanglement: Bell Inequalities, Relativity, and the Qubit" where we go into great detail for the general reader on how to understand entanglement in principle fashion so as to avoid these undesirable consequences of "constructive efforts." I'll leave it there.

 

[gentzen]

I could say a lot more, but I don't have time. My colleagues and I have a book coming out this summer with Oxford UP titled, "Einstein's Entanglement: Bell Inequalities, Relativity, and the Qubit" where we go into great detail for the general reader on how to understand entanglement in principle fashion so as to avoid these undesirable consequences of "constructive efforts." I'll leave it there.

Fine, you don't need to say much. Just clearly replace the statement

Schrödinger's equation is simply the low energy approximation of the Lorentz-invariant Klein-Gordon equation of quantum field theory

by something which is less disparaging towards nonrelativistic QM. As I said, the Klein-Gordon equation only applies to bosons, and this is not a limitation of nonrelativistic QM. It is true that nonrelativistic QM has trouble creating or destroying particles, and hence has a hard time with photons, because they are massless.

Please also see my point. I have invested huge efforts to understand nonrelativistic QM good enough to be able to somehow use it in my daily job. I have some knowledge of QFT, but far less than of QM. It is fine for me to read your references to Zee, or Demystifier's paper. But the way this discussion here went feels very bad to me.

 

[Morbert]

As an aside: An interesting paper (that I unfortunately cannot find a preprint of )

They present the Schroedinger equation in QFT as $$\sum_{k=1}^n \frac{\partial}{\partial x_k^0}\psi(A|x_1,\dots,x_n) = H\psi(A|x_1,\dots,x_n)$$where the Hilbert space is a linear space of wave functionals and $$\psi(A|x_1,\dots,x_n) =N \int D\left[\phi\right]\phi(x_1)\dots\phi(x_n)e^{-S\left[\phi\right]}$$

 

[PeterDonis]

the way this discussion here went feels very bad to me.

When you use the word "nonsense" to describe someone else's post, you should expect to feel bad. It would have been better to dial back your response to something like asking for a reference. Which you got anyway.

 

[A. Neumaier]

It is true that nonrelativistic QM has trouble creating or destroying particles, and hence has a hard time with photons, because they are massless.

Standard QM can handle a finite number of free relativistic particles (including photons) easily in the momentum picture!

The free Hamiltonian for a single particle of mass $m\ge 0$ and spin $s$ is $H_0=\sqrt{p^2+m^2}$ acting on the Hilbert space $L^2(R^3,C^{2s+1})$ - except in the massless spin $s\ge 1$ case, where the Hilbert space is the subspace of helicity $\pm s$ states. Thus for photons ($m=0,s=1$), $\psi$ must satisfy $p\cdot\psi(p)=0$.

In a finite-dimensional truncated Fock space one can also handle several free particles and their interaction, incuding their creation and annihilation unless the interaction allows the creation of too many particles. Thus the production of soft photons (Bremsstrahlung) cannot be modeled, but many other creation and annihilation processes can!

Indeed, quantum optics routinely and successfully uses truncated Fock spaces for the approximate description of processes involving photons.

 

[gentzen]

When you use the word "nonsense" to describe someone else's post, you should expect to feel bad. It would have been better to dial back your response to something like asking for a reference. Which you got anyway.

I didn't describe his post as nonsense. I objected to a specific statement. Then I got a reference as reply, which explains where that statement came from, but basically just confirms that the statement as made is objectionable. Then I explained in detail why the statement is still objectionable, despite the reference. So far so good, nothing that would feel bad to me.

But then the wiggling and objections started, picking on details and giving the false impressions as if those details would invalidate my points. I don't want PF to become the original source for the statement that "Schrödinger's equation is simply the low energy approximation of the Lorentz-invariant Klein-Gordon equation of quantum field theory", backed up by ... I don't want this, because this statement is disparaging towards Schrödinger's equation, and misleading.

Edit: And the word "nonsense" is my reaction to the word "simple". I find it unbelievably condescending to use words like "of course", "simple", "..." in the context of stuff related to QFT.

 

[PeterDonis]

I didn't describe his post as nonsense. I objected to a specific statement.

You're quibbling. You used the word "nonsense".

the word "nonsense" is my reaction to the word "simple". I find it unbelievably condescending to use words like "of course", "simple", "..." in the context of stuff related to QFT.

And that's fine if that's your reaction. But if you're going to go the route of expressing that reaction in a public post, don't complain later that the way the discussion went "feels bad" to you. You used the word "nonsense", so own it. If you don't like the way the discussion went as a result, then next time you might want to choose a more moderate response, as I said.

 

[PeterDonis]

But then the wiggling and objections started, picking on details and giving the false impressions as if those details would invalidate my points. I don't want PF to become the original source for the statement that "Schrödinger's equation is simply the low energy approximation of the Lorentz-invariant Klein-Gordon equation of quantum field theory", backed up by ... I don't want this, because this statement is disparaging towards Schrödinger's equation, and misleading.

It should be evident to you from the discussion that others posting in this thread do not share your viewpoint here. I personally can't even make sense of your claim that the statement is "disparaging towards Schrödinger's equation"--how can you "disparage" a physics equation? Are you afraid its feelings will get hurt?

As far as the statement in Zee's book, I read that book years ago and saw no issue with it. So my general reaction to your objections is similar to that of other posters in the thread.

 

[bhobba]

To throw the cat among the pigeons, does QFT require any interpretation over and above the principles of QM?

Weinberg is famous for his view (he calls it a folk theorem) that any theory looks like a QFT at large enough distances and the EFT view of QFT
https://philarchive.org/archive/CARTQF-5

If that is true, what is there to interpret?

That then raises the issue of how natural an interpretation is when applied to the QFT subset of QM.

Also, there is the issue, likely requiring a separate thread, of what are the principles of QM? Most I have seen (Ballentine is an exception) include the Schrödinger equation, so those are ruled out.

Thanks
Bill

 

[physika]

From the few interpretations of quantum mechanics that I know, let's say:

• Copenhagen
• Many Worlds (MWI)
• Bohmian mechanics
• Superdeterminism
• Retrocausality/transactional
• Objective-collapse
• Qbism*
• Consistent histories*

Objective Collapse is not an interpretation, it is a theory by is own.


......

 

[PeterDonis]

Objective Collapse is not an interpretation, it is a theory by is own.......

Some versions are, such as the GRW stochastic collapse model, since they explicitly change the math. But I'm not sure all interpretations that call themselves "objective collapse" do that.

 

[gentzen]

Some versions are, such as the GRW stochastic collapse model, since they explicitly change the math. But I'm not sure all interpretations that call themselves "objective collapse" do that.

The only such "objective-collapse" interpretation I am aware of (which doesn't change the math) is the infamous "consciousness causes collapse" interpretation (which should not be called von Neumann–Wigner interpretation despite wikipedia, because von Neumann and Wigner described something else). I would argue that this should not deter us from using "objective-collapse theory" just like wikipedia and SEP, referring specifically to those collapse interpretations that do change the math.

So back to @physika's point of whether using the word "interpretation" for those theories is OK. Given that wikipedia and SEP take care not to call them interpretations, it feels like a good idea in general to follow that practice. But this practice also includes the Penrose interpretation. It seems to be a question of the effort and disruption it would cause, to insist to never refer to such theories using the word "interpretation".

 

[physika]

Some versions are, such as the GRW stochastic collapse model, since they explicitly change the math. But I'm not sure all interpretations that call themselves "objective collapse" do that.

(Ghirardi, Rimini & Weber)
As Ghirardi says

"Collapse Theories qualify themselves as rival theories of quantum mechanics and one can easily identify some of their physical implications which, in principle, would allow crucial tests discriminating between the two"

https://plato.stanford.edu/Entries/qm-collapse/

 

[PeterDonis]

As Ghirardi says

Of course he's going to say that, since it is true for his "objective collapse" interpretation.

 

[physika]

"tests discriminating between the two"

https://plato.stanford.edu/Entries/qm-collapse/

Tests will tell.

 

[gentzen]

To throw the cat among the pigeons, does QFT require any interpretation over and above the principles of QM?
[...]
That then raises the issue of how natural an interpretation is when applied to the QFT subset of QM.

Peter Woit posted an interesting link on February 14, 2024, which might be relevant:

While trying to figure out how best to pass from QM to QFT, I’ve kept coming across various aspects of this that I’ve always found confusing, never seen a good explanation of. Today I ran across a wonderful article by Thanu Padmanabhan, who I knew about just because of his very good introductory book on QFT, Quantum Field Theory: The why, what and how. The article is called “Obtaining the Non-relativistic Quantum Mechanics from Quantum Field Theory: Issues, Folklores and Facts” and subtitled “What happens to the anti-particles when you take the non-relativistic limit of QFT?” It contains a lot of very clear discussion of issues that come up when you try and think about the QM/QFT relationship, a sort of thing I haven’t seen anywhere else.

Especially the following quote from https://arxiv.org/abs/1712.06605 ("Obtaining the Non-relativistic Quantum Mechanics from Quantum Field Theory: Issues, Folklores and Facts" by T. Padmanabhan)

The dichotomy of description between NRQM and QFT does not originate just from the square root in the Hamiltonian or from the demand of Lorentz invariance, as it is sometimes claimed. The real difficulty has its origin in the necessary existence of antiparticles to ensure a particular notion of relativistic causality. Because of these conceptual issues, it turns out that one cannot, in fact, obtain some of the popular descriptions of NRQM by any sensible limiting procedure applied to QFT. To obtain NRQM from QFT in a seamless manner, it is necessary to work with NRQM expressed in a language closer to that of QFT. This fact has several implications, especially for the operational notion of space coordinates in quantum theory.

hints at similar issues as the following quote from https://arxiv.org/abs/1205.1992 ("Relativistic Quantum Mechanics and Quantum Field Theory" by H. Nikolic), a link posted by Demystifier earlier in this thread:

In our quest towards relativistic Bohmian mechanics, as a byproduct we realize that even non-Bohmian relativistic QM and QFT should be first made “more relativistic” than they are in the usual formulation, i.e., that time and space should be treated more symmetrically. First, the usual single-time wave function should be generalized to the many-time wave function, such that each particle has its own spacetime coordinate. Second, $\psi^2$ should be reinterpreted as a probability density in spacetime, rather than that in space.

 

[A. Neumaier]

Peter Woit posted an interesting link on February 14, 2024, which might be relevant:

Especially the following quote from https://arxiv.org/abs/1712.06605 ("Obtaining the Non-relativistic Quantum Mechanics from Quantum Field Theory: Issues, Folklores and Facts" by T. Padmanabhan)

You forgot to mention the main result (substantiated in Section 6) of the paper:

the transition from QFT to NRQM is not possible if your aim is to reproduce many of the conventional descriptions of NRQM. Towards the end of the paper, I will describe how this can be achieved using one specific formulation of NRQM.

This means that - in contrast to what you suggest - when properly done, there are no issues at all with the nonrelativistic limit of QFT!