Axiomatization of quantum mechanics and physics in general ?

[bhobba]

What!? http://www.mth.kcl.ac.uk/~streater/piron.html

Amusing.

I have been refreshing my memory on this stuff and came across:
http://arxiv.org/pdf/0811.2516.pdf

Added Later:
Whoops - posted the wrong paper - now fixed

It seems I was remiss in assuming Pirons axioms led to the Hilbert space formalism - there are 5 - not three - and they do not rule out quaternion Hilbert spaces.

'Starting from the set L of all operational propositions of a physical entity and introducing five axioms on L he proved that L is isomorphic to the set of closed subspaces L(V ) of a generalized Hilbert space V whenever these five axioms are satisfied [6]'

[6] Piron, C. (1964), Axiomatique quantique

Which is of course the paper Patrick has posted in French.

One must go to the theorem of Soler to do that and evoke a sixth plane transitivity axiom.

But that is neither here nor there really - Piron ESSENTIALLY does it.

Its just that 'essentially' isn't quite the same as true in formal logic.

Thanks
Bill

 

[atyy]

By careful choice of the axioms you get exactly a Hilbert space - but some bits are not as 'natural' as one would like.

Soler's Theorem is a bit more of an advance in the natural department:
http://golem.ph.utexas.edu/category/2010/12/solers_theorem.html
http://arxiv.org/pdf/math/9504224v1.pdf
http://arxiv.org/pdf/quant-ph/0105107v1.pdf

But, while so tantalisingly close, still isn't quite there yet.

John Baez discussed it in some of his finds articles - if I remember correctly that is. You can almost hear him weep - if only - it would be just so beautiful if it was. I think its the natural reaction of those with a mathematical bent to this stuff (and of course I am one).

So the Piron-Soler sort of reasoning leads to infinite dimensional Hilbert spaces?

OTOH, the Hardy and Chirinell et al approaches lead to finite dimensional Hilbert spaces?

 

[bhobba]

So the Piron-Soler sort of reasoning leads to infinite dimensional Hilbert spaces?

Yes - and of course finite as well.

OTOH, the Hardy and Chirinell et al approaches lead to finite dimensional Hilbert spaces?

Yes - but for me that's not a worry - I simply generalise via Rigged Hilbert Spaces.

Thanks
Bill

 

[bhobba]

Well, we could just use Busch's theorem.

Ahhhh. But do the axioms of Piron map to a POVM. His theorem shows they map to projection operators, or equivalently subspaces (which is the same thing) but POVM's are another matter.

Thanks
Bill

 

[bhobba]

OK, googling suggests I was too hasty there. Here's a very interesting blog post, with interesting comments too: http://blog.computationalcomplexity.org/2014/01/fermats-last-theorem-and-large.html.

Interesting.

On the surface it doesn't seem to contradict what you said. Nor do I reasonably expect it to - Russell tried it with the Principa - I don't think anyone wants to repeat that tome.

I have to go and get some lunch will put on my thinking hat about it when I return.

Thanks
Bill

 

[atyy]

Ahhhh. But do the axioms of Piron map to a POVM. His theorem shows they map to projection operators, or equivalently subspaces (which is the same thing) but POVM's are another matter.

Hmmm, how about doing Piron + Gleason's in 3d, then defining 2d QM as resulting from projective measurements in 3d or higher? Basically this is adding an axiom that says 2d QM is defined by having a Naimark extension, ie. we need at least 3d, so Gleason's will apply.

Physically, I think this is saying we can move the Heisenberg cut outwards.

 

[microsansfil]

Can I suggest that both of you are right, and talking about different things?

Perhaps, however it is not what there are more interesting.

I did not know there was so much work on the topic quantum logic. Express the foundations of quantum mechanics in the language of the logic of mathematics.

Here "A New Approach to Quantum Logic".

The message of the book is of interest to a broad audience consisting of logicians, mathematicians, philosophers of science, researchers in Artifficial Intelligence and last but not least physicists. These communities, however, strongly differ in their scientific backgrounds. Normally, a physicist has no training in mathematical logic, and a logician is by no means expected to master the Hilbert space formalism of quantum mechanics.
This fact constitutes a major problem in any attempt to present the topic of quantum logic in a way accessible to the broad audience to which, in principle, it is of interest.

This seem open a new perspective to quantum mechanics or then be a deadlocked.

What about relativistic quantum mechanics logic ?

Patrick

 

[bhobba]

What about relativistic quantum mechanics logic ?

Axiomatic QFT (which is relativistic QM) is a whole new ball game.

It's mathematically way above my current level with tomes of VERY deep mathematics supporting it.

Its not based on Hilbert Spaces like standard QM, but draws heavily on Rigged Hilbert Spaces and distribution theory:
http://www.mathematik.uni-muenchen.de/~schotten/LNP-cft-pdf/08_978-3-540-68625-5_Ch08_23-08-08.pdf

The standard reference is Gelfand and Vilenkin - Generalized Functions. I have studied it and even with my math background its - how to put it - challenging - meaning very non trivial.

BTW its the correct formalism for QM as well - but in axiomatic QFT its unavoidable. And that's just to start with - QFT scales rather 'inspiring heights'.

Thanks
Bill

 

[bhobba]

Physically, I think this is saying we can move the Heisenberg cut outwards.

One could use Neumark's theorem to show in lower dimensions resolutions of the identity looks like POVM's and its very reasonable to assume probabilities etc are not altered, but reasonable, and formally provable are two different things.

Thanks
Bill

 

[bhobba]

This seem open a new perspective to quantum mechanics or then be a deadlocked.

Its well known - the reference by Varadarajan details it pretty well.

It is, as far as foundations is concerned, as I have said previously, our most penetrating formalism.

Pirons axioms, and even better with Solers theorem, ESSENTIALLY implies the QM formalism.

The issue is in that word - essentially - eg you need extra assumptions of a seemingly ad-hoc variety to rigorously make it work.

But above all its - HARD.

Thanks
Bill

 

[Fredrik]

What about relativistic quantum mechanics logic ?

The difference between non-relativistic QM and special relativistic QM is just a choice between the Galilean group and the Poincaré group. You postulate that there's a homomorphism from one of these groups into the group of automorphisms of the lattice. I you choose the former group, the result is non-relativistic QM. If you choose the latter, the result is special relativistic QM.

Axiomatic QFT (which is relativistic QM) is a whole new ball game.

I view relativistic quantum field theories as theories defined within the framework of special relativistic QM (as defined above). I believe that there are also non-relativistic QFTs, but I have never studied one.

The axioms of (axiomatic) QFT are supposed to define what a quantum field theory is, so yes, it's definitely a very different game.

 

[bhobba]

I view relativistic quantum field theories as theories defined within the framework of special relativistic QM (as defined above). I believe that there are also non-relativistic QFTs, but I have never studied one.

I have off and on been studying QFT over the years from various books such as Zee and others.

Mathematically - yea - I got it. But physically it didn't gel.

Then I came across thew following book recently released:
https://www.amazon.com/dp/019969933X/?tag=pfamazon01-20

The Kindle price was pretty good so I took a punt.

Really good at explaining what it means.

Its not a well known fact, bu5t still true, that QM can be reformulated as a the3ory of creation and anhilation operators:
http://math.bu.edu/people/mak/Styer Am J Phys 2002.pdf

See interpretation F.

What that book does is explain that view step by step then shows how it applies to QFT so you immediately know what the formalism means.

Thanks
Bill

 

[kith]

But does Hardy fail to mention the assumption of non-contextuality? Or is it in there, and just in a more natural or "reasonable" way, as he intends?

What does it even mean for an inherently probabilistic theory to be (non-)contextual?

 

[atyy]

What does it even mean for an inherently probabilistic theory to be (non-)contextual?

My understanding about non-contextuality in the context of Gleason's theorem comes from the comments by Peres in http://books.google.com/books?id=IjCNzbJYONIC&dq=peres+quantum&source=gbs_navlinks_s.

If understand Peres correctly, he says Gleason's theorem assumes that if Pu and Pv are orthogonal projectors, then <Pu + Pv> = <Pu> + <Pv>. However, there isn't a unique way to write P = Pu + Pv = Px + Py, so the assumption is that <Pu> + <Pv> = <Px> + <Py>. The assumption is non-trivial since measurements of Pu and Pv usually require different experimental setups from those that measure Px and Py.

Hardy doesn't even assume Hilbert spaces or anything, but he does derive the Born rule. Most people believe his derivation is correct and complete, so it's most likely that he has not left out an assumption such as non-contextuality. Rather, he has other axioms which do the work, and I'm wondering which of his axioms do that, and whether one can understand them as equivalent to non-contextuality in Gleason's.

 

[microsansfil]

The difference between non-relativistic QM and special relativistic QM is just a choice between the Galilean group and the Poincaré group. You postulate that there's a homomorphism from one of these groups into the group of automorphisms of the lattice. I you choose the former group, the result is non-relativistic QM. If you choose the latter, the result is special relativistic QM.

"A principle of modern mathematics holds in this lesson: when you are dealing with an entity S with a measure of structure, try to determine its group of automorphisms, the group of transformations of its components that preserve the structural relations. You can expect to gain a deep understanding of the constitution of S in this way. "Hermann Wey

http://en.wikipedia.org/wiki/Representation_theory_of_the_Poincaré_group

Does this mathematical logic view could be useful in the search for unification of General relativity and Quantum mechanics ?

Patrick

 

[Fredrik]

"A principle of modern mathematics holds in this lesson: when you are dealing with an entity S with a measure of structure, try to determine its group of automorphisms, the group of transformations of its components that preserve the structural relations. You can expect to gain a deep understanding of the constitution of S in this way. "Hermann Wey

http://en.wikipedia.org/wiki/Representation_theory_of_the_Poincaré_group

Does this mathematical logic view could be useful in the search for unification of General relativity and Quantum mechanics ?

I know almost nothing about that. I think that loop quantum gravity is an attempt to develop "general relativistic quantum mechanics" in a way that's similar to what I was talking about, but I don't really know.

 

[kith]

If understand Peres correctly, he says Gleason's theorem assumes that if Pu and Pv are orthogonal projectors, then <Pu + Pv> = <Pu> + <Pv>. However, there isn't a unique way to write P = Pu + Pv = Px + Py, so the assumption is that <Pu> + <Pv> = <Px> + <Py>. The assumption is non-trivial since measurements of Pu and Pv usually require different experimental setups from those that measure Px and Py.

So the assumption is called non-contextuality because for a single observable Pu+Pv, I can use all kinds of different experimental setups -where different setups correspond to different bases of the eigenspace of Pu+Pv- to measure it's expectation value?

I don't have access to Peres at the moment and an immediate follow-up question is how does this relate to Bohmian mechanics and it's contextuality? By what mathematical elements are observables even represented in Bohmian mechanics?

Hardy doesn't even assume Hilbert spaces or anything, but he does derive the Born rule. Most people believe his derivation is correct and complete, so it's most likely that he has not left out an assumption such as non-contextuality. Rather, he has other axioms which do the work, and I'm wondering which of his axioms do that, and whether one can understand them as equivalent to non-contextuality in Gleason's.

I haven't seen his derivation but since the first four axioms also apply to classical probability theory, it certainly has to do with the fifth. I would also be interested in seeing how the Hilbert space formalism and Hardy's formulation are related exactly.

 

[atyy]

So the assumption is called non-contextuality because for a single observable Pu+Pv, I can use all kinds of different experimental setups -where different setups correspond to different bases of the eigenspace of Pu+Pv- to measure it's expectation value?

I don't have access to Peres at the moment and an immediate follow-up question is how does this relate to Bohmian mechanics and it's contextuality? By what mathematical elements are observables even represented in Bohmian mechanics?

I don't understand this issue very well. As I understand it, the non-contextuality in Gleason's theorem seems to have nothing to do with hidden variables, since it is just about measures on states in Hilbert space.

However, by some corollary of Gleason's, there is apparently a link between the non-contextuality there and in hidden variable theories. It is discussed by in this link given by bhobba to an article by Bell http://fy.chalmers.se/~delsing/QI/Bell-RMP-66.pdf, and also in this proof by Busch of a Gleason-like theorem, but pertaining to POVMs instead of POMs http://arxiv.org/abs/quant-ph/9909073. Maybe bhobba or Fredrik can explain in more detail here?

I haven't seen his derivation but since the first four axioms also apply to classical probability theory, it certainly has to do with the fifth. I would also be interested in seeing how the Hilbert space formalism and Hardy's formulation are related exactly.

Here's Hardy's first derivation http://arxiv.org/abs/quant-ph/0101012. The fifth axiom just says that there's a continuous reversible transformation between pure states. Isn't that met by classical mechanics in phase space?

 

[kith]

Just a quick note:

The fifth axiom just says that there's a continuous reversible transformation between pure states. Isn't that met by classical mechanics in phase space?

Hardy's states are probability vectors (p₁,...,p_n). For pure states, all but one of the p_i are zero so the pure states correspond to the elements of the n-dimensional standard basis and there's no continuous transformation between them.

However, I am not sure if Hardy's formulation of QM works for infinite dimensional systems like a particle in space.

 

[atyy]

Just a quick note:

Hardy's states are probability vectors (p₁,...,p_n). For pure states, all but one of the p_i are zero so the pure states correspond to the elements of the n-dimensional standard basis and there's no continuous transformation between them.

However, I am not sure if Hardy's formulation of QM works for infinite dimensional systems like a particle in space.

Perhaps that is why Hardy's derivation doesn't work for continuous variables. For discrete variables, it is obvious that classically, a particle is either in one box or the other, whereas in the quantum case, it can be in a superposition of being in both boxes. As I understand it, the Chiribella et al derivation is also only of finite dimensional quantum mechanics.

On the other hand the Mackey-Piron-Soler approach, according to bhobba, gets finite and infinite dimensional quantum mechanics. The other "reformulation" I know that can get infinite dimensional QM is the Leifer-Spekkens http://arxiv.org/abs/1107.5849. That paper only deals with the finite dimensional case, but in the comments here he says the extension to the infinite dimensional should be straightforward http://mattleifer.info/2011/08/01/the-choi-jamiolkowski-isomorphism-youre-doing-it-wrong/.

 

[bhobba]

What does it even mean for an inherently probabilistic theory to be (non-)contextual?

Non contextuality means the probability measure does not depend on what resolution of the identity a projection operator is part of. Its a very natural condition to impose mathematically because its simply expressing basis independence - you would not expect a measure to depend of your basis - after all that is a pretty basic property of vector spaces - the important geometric stuff like length or angle doesn't depend on basis. But physically it has profound implications because a resolution of the identity corresponds to an actual measurement apparatus.

Its the key ingredient in Gleason's proof - there are others such as the strong superposition principle - but that is the key one.

In the geometric approach to QM you start with a logistic then show its observables (one can define things like observables, states, even probability measures on observables, etc in a logistic - see Chapter 3 of Varadarajan) are isomorphic to a Hilbert space. One then uses Gleason to show that probability measure is the Born rule.

The issue though is when you model something using it. Are all the things that determine how the system behaves observables? There may be hidden variables and they may be contextual. In BM for example the pilot wave is explicitly contextual:
http://philsci-archive.pitt.edu/3026/1/bohm.pdf

Don't necessarily agree with that link saying BM is not using hidden variables - but non contextual ones. The pilot wave is hidden - you can't ever directly observe it.

But of course Dymystifyer is the expert on BM around here - not me.

Thanks
Bill

 

[bhobba]

As I understand it, the non-contextuality in Gleason's theorem seems to have nothing to do with hidden variables, since it is just about measures on states in Hilbert space.

You are correct - it doesn't SEEM to. But if there are contextual variables, hidden or otherwise, then Gleason's breaks down.

Gleason is basically a stronger version of Kochen-Speker - in fact Kochen-Speker is a simple corollary to Gleason.

Thanks
Bill

 

[bhobba]

However, I am not sure if Hardy's formulation of QM works for infinite dimensional systems like a particle in space.

Personally in discussing foundational issues I stick to finite vector spaces.

I view the infinite dimensional case via the Rigged Hilbert Space formalism where the dual is simply introduced for mathematical convenience. Here the dual I am referring to is the dual to the space of all sequences of finite length. In the weak convergence of that space any linear functional is the limit of a sequence of the space the functionals are defined on.

Thanks
Bill

 

[kith]

I don't understand this issue very well. As I understand it, the non-contextuality in Gleason's theorem seems to have nothing to do with hidden variables, since it is just about measures on states in Hilbert space.

Well, let's look at KS first. I am not sure if I am on the right track but my current understanding is this (following Peres).

KS says that if we have a Hilbert space of dimension N ≥ 3, we cannot find a function v which consistently assigns a probability of 0 and 1 to all projectors acting on this Hilbert space. Namely, if we combine a projector with N-1 other commuting projectors with \sum_i P_i=1 we cannot have \sum_i v(P_i)=1 for all choices of the N-1 projectors.

Let's say we have an observable Q with possible experimental outcomes a, b and c and associated projectors P_a, P_b and P_c (these commute and sum to the identity). In order to assign a true but hidden value to our quantity we could assign a probability v of 0 or 1 to each of the projectors. Let's say that the true value of Q is b, so v(P_a)=v(P_c)=0 and v(P_b)=1.

KS now implies that there exists a similar observable R where the probabilities either don't sum to one or are inconsistent with previously assigned probabilities. The situation could be something like this: Q and R share the projector P_a but instead of P_b and P_c, R is associated with P_{\beta} and P_{\gamma}. Consistency with other observables Q&#039;, Q&#039;&#039;, ... forces us to either choose v(P_{\beta}) = v(P_{\gamma}) = 0 or v(P_a)=1. So either the sum of probabilities for R is equal to 0 or the sum for Q is equal to 2. Both options imply that v isn't a probability in the first place, so this kind of hidden variable assignment is ruled out.

The problem goes away if the probability associated with P_a depends on the context, i.e when we allows different probabilities v(P_a; P_b,P_c)=0 and v(P_a; P_{\beta}, P_{\gamma})=1 for different observables Q and R.

 

[kith]

Perhaps that is why Hardy's derivation doesn't work for continuous variables.

Has it really been shown that it doesn't work or did he simply restrict his discussion to the finite / countably infinite case?

 

[kith]

Gleason is basically a stronger version of Kochen-Speker - in fact Kochen-Speker is a simple corollary to Gleason.

I haven't followed the logic of the proofs in detail but I tend to agree. KS says that the non-contextual probability distribution cannot look a certain way while Gleason says how exactly it has to look.

 

[atyy]

Well, let's look at KS first. I am not sure if I am on the right track but my current understanding is this (following Peres).

KS says that if we have a Hilbert space of dimension N ≥ 3, we cannot find a function v which consistently assigns a probability of 0 and 1 to all projectors acting on this Hilbert space. Namely, if we combine a projector with N-1 other commuting projectors with \sum_i P_i=1 we cannot have \sum_i v(P_i)=1 for all choices of the N-1 projectors.

Let's say we have an observable Q with possible experimental outcomes a, b and c and associated projectors P_a, P_b and P_c (these commute and sum to the identity). In order to assign a true but hidden value to our quantity we could assign a probability v of 0 or 1 to each of the projectors. Let's say that the true value of Q is b, so v(P_a)=v(P_c)=0 and v(P_b)=1.

KS now implies that there exists a similar observable R where the probabilities either don't sum to one or are inconsistent with previously assigned probabilities. The situation could be something like this: Q and R share the projector P_a but instead of P_b and P_c, R is associated with P_{\beta} and P_{\gamma}. Consistency with other observables Q&#039;, Q&#039;&#039;, ... forces us to either choose v(P_{\beta}) = v(P_{\gamma}) = 0 or v(P_a)=1. So either the sum of probabilities for R is equal to 0 or the sum for Q is equal to 2. Both options imply that v isn't a probability in the first place, so this kind of hidden variable assignment is ruled out.

The problem goes away if the probability associated with P_a depends on the context, i.e when we allows different probabilities v(P_a; P_b,P_c)=0 and v(P_a; P_{\beta}, P_{\gamma})=1 for different observables Q and R.

Thanks! I found this explanation by Spekkens which seems to match what you wrote http://arxiv.org/abs/quant-ph/0406166v3: "Traditionally, a noncontextual hidden variable model of quantum theory is one wherein the measurement outcome that occurs for a particular set of values of the hidden variables depends only on the Hermitian operator associated with the measurement and not on which Hermitian operators are measured simultaneously with it. For instance, suppose A,B and C are Hermitian operators such that A and B commute, A and C commute, but B and C do not commute. Then the assumption of noncontextuality is that the value predicted to occur in a measurement of A does not depend on whether B or C was measured simultaneously. The Bell-Kochen-Specker theorem shows that a hidden variable model of quantum theory that is noncontextual in this sense is impossible for Hilbert spaces of dimension three or greater."

Intuitively, this seems like a generalization of the idea that canonically conjugate observables like momentum and position do not have simultaneous existence, since it depends on non-commuting observables, even if only indirectly.

 

[atyy]

Has it really been shown that it doesn't work or did he simply restrict his discussion to the finite / countably infinite case?

Originally I thought that his method did not work for the infinite dimensional case, because axiom five which is the distinction between finite dimensional classical and quantum theories, clearly holds for classical continuous variables. However, according to his discussion of the issue in section 9 of http://arxiv.org/abs/quant-ph/0101012, his axioms still rule out classical continuous variables, because of axiom 3: "A system whose state is constrained to belong to an M dimensional subspace (i.e. have support on only M of a set of N possible distinguishable states) behaves like a system of dimension M."

So I guess it is unknown whether his axioms work or not for the continuous case.

 

[kith]

However, according to his discussion of the issue in section 9 of http://arxiv.org/abs/quant-ph/0101012, his axioms still rule out classical continuous variables, because of axiom 3: "A system whose state is constrained to belong to an M dimensional subspace (i.e. have support on only M of a set of N possible distinguishable states) behaves like a system of dimension M."

This sounds a bit unphysical to me. Such a finite-dimensional subspace of phase space is a set of measure zero (loosely speaking a collection of delta functions). It seems strange to say that classical statistical mechanics violates this axiom because such sets of measure zero are not considered physical there anyway. So I would say that this axiom is irrelevant in classical statistical mechanics because there are no physical subspaces with a smaller dimension than the whole phase space.

Another interesting thought in section 9 is that superpositions smooth out the discontinuities of a possibly quantized space which may reduce the discomfort associated with this notion.

 

[atyy]

This sounds a bit unphysical to me. Such a finite-dimensional subspace of phase space is a set of measure zero (loosely speaking a collection of delta functions). It seems strange to say that classical statistical mechanics violates this axiom because such sets of measure zero are not considered physical there anyway. So I would say that this axiom is irrelevant in classical statistical mechanics because there are no physical subspaces with a smaller dimension than the whole phase space.

Another interesting thought in section 9 is that superpositions smooth out the discontinuities of a possibly quantized space which may reduce the discomfort associated with this notion.

Should there be a natural distinction between classical and quantum mechanics for continuous variables? For position and momentum, there's Bohmian mechanics which is a classical way of viewing quantum mechanics. There's also Montina's arument http://arxiv.org/abs/0711.4770 that hidden variables for a finite dimensional quantum system must be continuous, if the dynamics are Markovian.

So I tend to think of quantum mechanics as a very good effective theory, because the true underlying variables are usually much more inconvenient.