I Eigenvalue degeneracy in real physical systems

[A. Neumaier]

This is exactly what I asked.

No. You had asked the following:

Now, if we were to simulate all (observable in real world) physical systems, we would need to know whether the eigenvalues of all Hermitian operators that correspond to the real physical systems are distinguishable. Otherwise, our "supercomputer" would be unable to determine, which eigenstate the system falls into after measurement. In particular, it is true when all the operators are represented by non-degenerate matrices.

Are there (or have there been observed) real-world physical systems known to have indistinguishable eigenvalues?

it's the issue of uncomputability of spectra.

The reference to the real world assumes real measurements of real systems. They are never known to infinite precision hence the question of uncomputability of the spectra is irrelevant - it would be the inaccurate spectrum of an operator that is inaccurate anyway, and would apply only to the idealized situation, since the measurement is not of the Copenhagen type, so errors in the simulations don't matter.

Simulations are approximate also, by their very nature - so who cares about uncomputability? Already $e^x$ is uncomputable for most $x$ - since one needs an infinite time to get the exact answer. A simulation only uses approximations to everything. This eliminates all problems of uncomputabiliy.

In cases where the Born rule applies (e.g., scattering events) one has an integral spectrum (highly degenerate but with a priori known projectors).

In other cases, for example when measuring enrgies through spectra, one has a discrete energy spectrum where energy differences are measured as spectral lines (with a width computable only by using more detailed models), etc.

If you pose the wrong question you shouldn't expect to get answers to what you had in mind.

 

[rubi]

Well, not necessary. That's at least what I am familiar with. And by the way. it's more of a problem with exact computation of operator spectra, which is impossible, than with interpretations of QM.

This is exactly what I asked. In other words, it's the issue of uncomputability of spectra.

Well, this problem only appears in Copenhagen-style interpretations, where the collapse is an essential part of the dynamics. This view is out of fashion today anyway. However, it could be in principle resolved by what I've written in posts #17 and #19.

There is no such recipe for a general measurement.

Unless there is such a recipe, the projective dynamics is ill-defined. Different choices of projectors will lead to different predictions. Just look at these extreme examples:
1. We could choose the projector onto the whole Hilbert space, since it certainly projects onto the measured eigenspace. This is equivalent to having no projection postulate at all and it can't explain for instance the quantum Zeno effect. (Note: I assume a model that explicitely does not include decoherence and uses the projection postulate instead!)
2. We could choose a very narrow projector. This might remove a part of the wave-function might later become important. For example if we perform a filtering in a Stern-Gerlach experiment and somehow the filtered electrons are led back into the beam, this will impact the results of the experiment and this impact wouldn't be reflected in our description, since we have removed the filtered electrons from the picture.

It is therefore crucial in a quantum theory with projection postulate to know, which projector must be choosen and a canonical choice would be to take the measurement uncertainty. The dynamics is ill-defined if you don't supply such a choice. Of course, this does not apply to theories without projection postulate, but the OP is specifically interested in projective dynamics.

The Born rule is well-defined (through aprecise specification of the meaning of '''measurement'') only for interpreting the results of collision experiments, i.e., the S-matrix elements.Born originally had it only in the form of a law for predicting the result of collisions, and it is verifiable in these situations.

Later it was abstracted into the modern form by on Neumann, who introduced an ''ideal'' measurement without aclear meaning - so that only the conformance to the rule ''defines'' whether a particular measurement is ''ideal''. - Almost none is. Neither photodetection nor electron detection works as claimed by the rule.

The Born rule and the projection postulate are two different things. You can have the Born rule without having the projection postulate.

For the interpretation of real measurmeents one uses instead sophisticated models of Lindblad type that predict the dynamics of the state and the probabilities of the outcomes.

I am aware of that. It's just not what the OP asked for. His question is specifically about projective dynamics. When someone asks a question about the Bohr model, telling him that it is outdated and he should really be considering quantum mechanics, wouldn't be an appropriate answer either.

Simulations are approximate also, by their very nature - so who cares about uncomputability? Already $e^x$ is uncomputable for most $x$ - since one needs an infinite time to get the exact answer. A simulation only uses approximations to everything. This eliminates all problems of uncomputabiliy.

Uncomputability is much worse than the fact that computations are approximate. If the predictions of a theory can't be computed in principle, then it's questionable whether the theory is a scientific theory at all. Computability theory is a part of the foundations of mathematics. It's not just an engineering topic. The exponential function is a computable function.

 

[A. Neumaier]

The Born rule and the projection postulate are two different things. You can have the Born rule without having the projection postulate.

True, but ErikZorkin was explicitly interested in the projection version:

Otherwise, our "supercomputer" would be unable to determine, which eigenstate the system falls into after measurement.

So I wonder which reality he wants to simulate - since almost no part of reality satisfies the postulate!

When someone asks a question about the Bohr model, telling him that it is outdated and he should really be considering quantum mechanics, wouldn't be an appropriate answer either.

It would be fully appropriate if he'd first discuss the Bohr model and then ask which real life atoms had two Bohr orbits with the same radius. it is exactly this kind of question that was asked.

 

[ErikZorkin]

I think you totally misunderstand the term "computability". e^x is computable

 

[ErikZorkin]

It is therefore crucial in a quantum theory with projection postulate to know, which projector must be choosen and a canonical choice would be to take the measurement uncertainty. The dynamics is ill-defined if you don't supply such a choice. Of course, this does not apply to theories without projection postulate, but the OP is specifically interested in projective dynamics.

Well, to be honest, I am starting to see so many flaws in this framework that I'd better to look for other interpretations maybe. Let me generalize a bit. I feel that the major problem is with the spectral decomposition. What alternatives are there that are more suited for practice and more computable?

 

[A. Neumaier]

I think you totally misunderstand the term "computability". e^x is computable

To arbitrary finite precision only, not exactly. Matters of computability in your sense don't matter in physics, only in the foundations of computer science.

In practical issues (including all simulation) it is completely irrelevant.

We don't even know whether solutions of the Navier-Stokes equations exist for natural initial conditions - let alone whether they are computable. Nevertheless physicsist in the airplane industry routinely compute solutions of interest using a precision of 16 decimal digits only in their computation - and they get results of a quality that we trust entering an airplane and expect exiting it at the destination.

That's the real world.

 

[A. Neumaier]

the major problem is with the spectral decomposition. What alternatives are there that are more suited for practice and more computable?

In th POVM approach you only need the condition $\sum_k P_k^*P_k=1$, which poses no diffiulties at all.

 

[ErikZorkin]

To arbitrary finite precision only, not exactly. Matters of computability don't matter in physics, only in the foundations of computer science.

In practical issues (including all simulation) it is completely irrelevant we don't even know whether solutions of the Navier-Stokes equations exist for natural initial conditions - let alone whether they are computable. Nevertheless physicsist in the airplane industry routinely compute solutions of interest - to a quality that we trust entering an airplane and expect exiting it at the destination.

That's the real world.

I'd like to avoid such a discussion to be honest.

In th POVM approach you only need the condition $\sum_k P_k^*P_k=1$, which poses no diffiulties at all.

I sympathize with this approach. But some subtleties, such as Neumark's theorem, get me worried. After all, how can you even claim that POVMs themselves are computable? I've googled a bit and found some approaches, but they don't seem to be recognized solutions. Seems that you substitute one uncomputable apparatus with another.

 

[A. Neumaier]

I sympathize with this approach. But some subtleties, such as Neumark's theorem, get me worried. After all, how can you even claim that POVMs themselves are computable? I've googled a bit and found some approaches, but they don't seem to be recognized solutions. Seems that you substitute one uncomputable apparatus with another.

In real life you fit free parameters in a model of the $P_k$ to the available data. If done correctly, this gives a description of the real apparatus with the usual $O(N^{-1/2})$ accuracy for the resulting parameters. More is not needed for probabilistic modeling.

In a simulation, you would simply define the apparatus by specifying a family of $P_k$s that does what you want it to do. Thus you have complete control over everything of computational relevance.

By the way, I am a math professor with a chair in computational mathematics. I know a lot about simulation in practice!

 

[ErikZorkin]

In a simulation, you would simply define the apparatus by specifying a family of PkP_ks that does what you want it to do. Thus you have complete control over everything of computational relevance.

I do sympathize with this framework as it (correct me if I am wrong) allows avoiding usage of spectral theorem. But:

However, POVMs can't resolve the mathematical computability issue that ErikZorkin brought up, since they can always be seen as PVMs on a larger Hilbert space, so if they could resolve the issue, then the issue with the PVMs would also be resolved, which is apparently impossible. I think the physical resolution is what I have written in posts #17 and #19 and of course it can also be formulated using POVMs.

However, and I would like to encourage rubi to clarify, just the mere equivalence between POVMs and PVMs might not be the issue. It's spectral decomposition, that leads to troubles. It turns out spectral theorem is only computable in approximate manner, whence we might drop off important content of the final state as pointed out by rubi, or if we know the multiplicity of eigenvalues in advance, which is speculative.

 

[A. Neumaier]

I do sympathize with this framework as it (correct me if I am wrong) allows avoiding usage of spectral theorem.

Nothing needs correction, except your interpretation of Naimak's theorem.

That you can simulate a POVM in a - different, nonphysical - Hilbert space doesn't have any practical relevance. POVMs work not because of Naimark's theorem but because of agreement with experiments.

 

[atyy]

However, POVMs can't resolve the mathematical computability issue that ErikZorkin brought up, since they can always be seen as PVMs on a larger Hilbert space, so if they could resolve the issue, then the issue with the PVMs would also be resolved, which is apparently impossible. I think the physical resolution is what I have written in posts #17 and #19 and of course it can also be formulated using POVMs.

I think Naimark's equivalence between POVMs and PVMs depending on small or large Hilbert spaces only applies to the Born rule part of the observables, not the collapse (I don't think the projection postulate exists for continuous variables).

However, I do agree with you that POVMs are not the solution, since the question asked by the OP can be stated for discrete variables. It needs some generalization for continuous variables, but the discrete version is not misleading.

 

[ErikZorkin]

I think Naimark's equivalence between POVMs and PVMs depending on small or large Hilbert spaces only applies to the Born rule part of the observables, not the collapse (I don't think the projection postulate exists for continuous variables).

However, I do agree with you that POVMs are not the solution, since the question asked by the OP can be stated for discrete variables. It needs some generalization for continuous variables, but the discrete version is not misleading.

Actually, I was thinking of a simple discrete example in the first place, not even continuous. Take Stern-Gerlach experiment, for instance. There, you can easily demonstrate the degeneracy problem. If the beam splitting is solely undetectable, how can you "project" your state correctly? I thought POVMs could at least describe rigorously what an approximate measurement is (in terms of measuring range, not exact value). Which is, at least for me, a reminiscent of post #17. Because POVMs describe what the final state is explicitly without application of spectral decomposition, as far as I understand.

 

[atyy]

Actually, I was thinking of a simple discrete example in the first place, not even continuous. Take Stern-Gerlach experiment, for instance. There, you can easily demonstrate the degeneracy problem. If the beam splitting is solely undetectable, how can you "project" your state correctly? I thought POVMs could at least describe rigorously what an approximate measurement is (in terms of measuring range, not exact value). Which is, at least for me, a reminiscent of post #17. Because POVMs describe what the final state is explicitly without application of spectral decomposition, as far as I understand.

For discrete observables, POVMs and the old-fashioned projection rule are equivalent, depending on how big of a Hilbert space one chooses to work with. I think rubi gave you the answer back around post #17? The generalization of the von Neumann rule that includes degenerate spaces is called the Luders rule http://arxiv.org/abs/1111.1088v2.

 

[ErikZorkin]

For discrete observables, POVMs and the old-fashioned projection rule are equivalent, depending on how big of a Hilbert space one chooses to work with. I think rubi gave you the answer back around post #17? The generalization of the von Neumann rule that includes degenerate spaces is called the Luders rule http://arxiv.org/abs/1111.1088v2.

Thanks for the hint!

Well, rubi gave a good answer, but then he himself pointed out some difficulties with it. See post #52

 

[atyy]

Thanks for the hint!

Well, rubi gave a good answer, but then he himself pointed out some difficulties with it. See post #52

They are not real difficulties, as long as the variable is discrete. Within Copenhagen, what counts as a "measurement" is subjective. So we can always take the Luders rule and add any unitary operation to it, and count the (Luders rule + unitary operation) as the "measurement".

However, it should be said that if one considers the spirit of Copenhagen to be a "smaller Hilbert space" view, in the sense that a sensible interpretation of the wave function of the universe is not available, then POVMs are more fundamental than projection measurements. http://mattleifer.info/wordpress/wp-content/uploads/2008/11/commandments.pdf

 

[ErikZorkin]

They are not real difficulties, as long as the variable is discrete

Well, discrete means all distinct? Because otherwise, we are in trouble. If we simply approximate the spectrum and projections, we might drop off something important.

By the way, what about other interpretations? Does, say, Bohmian pilot wave interpretation also suffer from spectral decomposition?

 

[atyy]

Well, discrete means all distinct? Because otherwise, we are in trouble. If we simply approximate the spectrum and projections, we might drop off something important.

The projection postulate only holds for discrete variables. If a position measurement is made, the state after that cannot be a position eigenstate, because the position eigenstate is not a valid state (not square integrable).

By the way, what about other interpretations? Does, say, Bohmian pilot wave interpretation also suffer from spectral decomposition?

I didn't quite understand the spectral decomposition problem. I was referring to the collapse rule needing an additional assumption to become defined (the instrument and measurement operators, as can be seen for the generalized collapse rule for POVMs).

 

[ErikZorkin]

I didn't quite understand the spectral decomposition problem.

Spectral decomposition is uncomputable. You can't even DEFINE eigenvalues/vectors/spaces/projections. Only in approximate format or if it is known beforehand that eigenvalues are distinct.

 

[atyy]

Spectral decomposition is uncomputable. You can't even DEFINE eigenvalues/vectors/spaces/projections. Only in approximate format or if it is known beforehand that eigenvalues are distinct.

Really, could you give a reference?

 

[ErikZorkin]

Really, could you give a reference?

Sure.

 

[atyy]

Sure.

But one just needs to know how many distinct eigenvalues there are. Presumably this should be known from experiment.

 

[ErikZorkin]

Presumably this should be known from experiment.

No.

Take an example of Stern-Gerlach experiment and suppose that beam splitting is completely undetectable (that it's beyond Planck scale, for instance -- the process of collapse still happens but we have no idea about the final state). See the image:

 

[atyy]

No.

Take an example of Stern-Gerlach experiment and suppose that beam splitting is completely undetectable (that it's beyond Planck scale, for instance -- the process of collapse still happens but we have no idea about the final state). See the image:

Then just omit the spin variable.

 

[ErikZorkin]

Then just omit the spin variable.

Eh, what?

And by the way, how can you deduce the number of eigenvalues from the experiment? It's a mathematical property of the operator.

 

[atyy]

Eh, what?

If there is no spin, then you won't get any splitting in the Stern-Gerlach experiment. So there will be no splitting in your theory, and no observed splitting, and all will be well.

And by the way, how can you deduce the number of eigenvalues from the experiment? It's a mathematical property of the operator.

OK, I don't know. In all practical cases, we don't seem to have a problem. For example, whether one needs a degeneracy or not depends on how good the experimentalist is. For example, in non-relativistic QM, the simplest Hamiltonians have lots of degeneracy. But then experiments get better, and one sees splittings, so one adds terms or uses better computations to get theory and experiment to match, eg. Zeeman, Lamb shift etc.

 

[ErikZorkin]

If there is no spin, then you won't get any splitting in the Stern-Gerlach experiment.

There is spin and there is splitting! They are just not detectable.

OK, I don't know. In all practical cases, we don't seem to have a problem.

Well, only because physicists don't care about math much. It is a fact that spectrum is uncomputable. Rubi was right, the only thing you can measure is that an eigenvalue lies in so or so range. You can't try to measure something, that you postulated mathematically, but what it uncomputable. It's nonsense. I am simply looking for a consistent explanation that is used in physics. Classical spectral decomposition and projection postulate are simply wrong when it comes to real experiments.

 

[ErikZorkin]

so one adds terms or uses better computations

What's that? "Adjusting" operators to better match with the reality (i.e. remove degeneracy)?

 

[bhobba]

What's that? "Adjusting" operators to better match with the reality (i.e. remove degeneracy)?

The value assigned to an observational outcome is entirely arbitrary.

The link I gave before explains clearly what's going on with POVM's etc etc:
https://www.physicsforums.com/threads/the-born-rule-in-many-worlds.763139/page-7

The value assigned to the elements of the POVM are irrelevant hence degeneracy is irrelevant and easily avoided - or not - it doesn't matter. The POVM is the important thing.

Thanks
Bill

 

[bhobba]

Well, only because physicists don't care about math much.

Why anyone would believe that has me beat. Some physicists have won Fields medals.

Thanks
Bill

 

[ErikZorkin]

Well, in the Stern-Gerlach experiment, the outcomes are quite certain, right? It's either spin up or down.

 

[bhobba]

Well, in the Stern-Gerlach experiment, the outcomes are quite certain, right? It's either spin up or down.

Spin up and spin down do not appear in the observable - you must assign it a number. That number is entirely arbitrary - it could be 1 and 0, 1 and 2, 1 and -1 or 1 an 1 - in which case you have degeneracy.

Thanks
Bill

 

[ErikZorkin]

As far as I remember, not all degenerates may be removed.

 

[bhobba]

As far as I remember, not all degenerates may be removed.

Hmmmm. Theoretically it should be possible - but likely not in a natural way.

Thanks
Bill

 

[ErikZorkin]

Theoretically it should be possible

How, for example?

 

[bhobba]

How, for example?

Read the link I gave.

An observation is a mapping to a POVM. The value assigned is entirely arbitrary.

An observation/measurement with possible outcomes i = 1, 2, 3 ... is described by a POVM Ei such that the probability of outcome i is determined by Ei, and only by Ei, in particular it does not depend on what POVM it is part of.

Thanks
Bill

 

[atyy]

What's that? "Adjusting" operators to better match with the reality (i.e. remove degeneracy)?

Yes. https://en.wikipedia.org/wiki/Fine_structure

 

[ErikZorkin]

Yes. https://en.wikipedia.org/wiki/Fine_structure

Nice example! But, again, it's not always possible to split the eigenvalues, as far as I understand.

An observation is a mapping to a POVM.

I sympathize with POVM approach as it (seemingly) avoids direct use of spectral decomposition (or?). Apparently, there has been a bit of discussion as to how it really addresses the question in the first place. For instance, POVM don't seem suitable for discrete variables.

 

[bhobba]

I sympathize with POVM approach as it (seemingly) avoids direct use of spectral decomposition (or?). Apparently, there has been a bit of discussion as to how it really addresses the question in the first place. For instance, POVM don't seem suitable for discrete variables.

I think you mean continuous variables.

Yes there is an issue - but it really needs a thread of its own.

That said continuous values don't ever actually occur.

Thanks
Bill

 

[vanhees71]

This is a phantastic example for the strange inability of mathematicians and physicists to communicate with each other. I must admit, I lost track what's the problem with "degeneracy" here. There is no such problem from a physicist's point of view, because it's all well defined in terms of observables (discribed by self-adjoint operators of Hilbert space) and states (described by a self-adjoint trace-class operator with trace 1, the statistical operator):

Any complete set of independent compatible observables $A_j$ ($j \in \{1,2,\ldots,n\}$) defines via the common (generalized) eigenvectors of the corresponding self-adjoint operators $\hat{A}_j$ (that are pairwise commuting by definition), denoted by $|a_1,\ldots,a_n \rangle$. Here the the components of the tupel $(a_1,\ldots,a_n) \in \mathbb{R}^n$ can run over discrete and/or continuous domains.

If the state of the system is represented by the statistical operator $\hat{\rho}$ the probability/probability density to simultaneously find the values $(a_1,\ldots,a_n)$ measuring the observables $A_j$ is given by (Born's rule):
$$P_{\rho}(a_1,\ldots,a_n)=\langle a_1,\ldots,a_n|\hat{\rho}|a_1,\ldots,a_n \rangle.$$
Everything else follows from the standard rules of probability theory. If I measure only von observable, e.g., $A_1$, I just have to sum/integrate over all other $a_2,\ldots,a_n$ over the corresponding spectrum, i.e.,
$$P_{\rho}(a_1)=\sum_{a_2,\ldots,a_n} P_{\rho}(a_1,\ldots,a_n).$$
This refers to measuring precisely the observable $A_1$. Of course in this case the eigenspaces of $\hat{A}_1$ alone are degenerate, because you need the other independent compatible observables to completely define the orthonormalized basis vectors (up to phase factors for each basis vector of course, but these cancel always out in physical results that are all dictated by the above given probability measure according to Born's rule). QT as a mathematical theory is consistent, as "weird" as some physicists and mathematicians seem to think it might be ;-))!

If you measure something not precisely you can envoke more complicated descriptions in terms of POVMs. Strictly speaking that you must do in any case if you measure an observable in the continuous part of the spectrum, because then you necessarily must use an apparatus with finite resolution, i.e., you can make, e.g., a position measurement of a particle, only up to a certain resolution determined by a the technology available for the measurement apparatus (which can be as fine as you like but not absolutely exact ever).

Now you can of course make this into a precise and well-defined mathematical formalism, and this is nice too, but one should not forget this not that difficult to understand physics and "metrological" constraints, which are very clear to everybody who as ever done experiments (and be it as a theoretician only in the standard lab practice mandatory to attend for every physics student wherever physics is taught).

 

[ErikZorkin]

I think you mean continuous variables.

? POVMs are problematic when coupled with continuous variables? I thought in exactly the opposite way. Nevertheless, I don't care about continuous variables. The key problem of this story is: you can't computationally define the operator's spectrum, let alone eigenstates/projections. That's it. There is certainly a way to avoid such a treatment.

 

[bhobba]

I don't care about continuous variables. The key problem of this story is: you can't computationally define the operator's spectrum, let alone eigenstates/projections.

Then there is no issue.

QM has nothing to do with the practicalities of computing eigenvalues any more than classical mechanics has anything to do with the practicalities of numerically solving a differential equation. It leas to interesting things like the butterfly - but has nothing to do with the theories validity.

Thanks
Bill

 

[ErikZorkin]

Then there is no issue.

I am pretty sure! I thought I almost found an answer (POVMs), but then I got a bit concerned about the difficuleties as pointed out by rubi. Forgive me my ignorance, but do you apply spectral theorem when dealing the POVM--based QM?

 

[bhobba]

I am pretty sure! I thought I almost found an answer (POVMs), but then I got a bit concerned about the difficuleties as pointed out by rubi. Forgive me my ignorance, but do you apply spectral theorem when dealing the POVM--based QM?

No. Please, please read the link I gave. The spectral theorem only applies to what's called resolutions of the identity which are disjoint POVM's.

Thanks
Bill

 

[atyy]

Nice example! But, again, it's not always possible to split the eigenvalues, as far as I understand.

I think as long as we have experiment as a guide, it is ok. The theorem you cite assumes finite dimensions and fails for infinite dimensions. For finite dimensions, the degeneracy cannot be greater than the dimensionality. Let the dimensionality be N. Then we guess the number of eigenvalues m to range between 1 .. N. For each guess, we compute the predictions. Although we will never know with certainty which m is correct, we just take the provisional answer to be the one that matches observations most closely.

This will be fine, because in practice, even for non-degenerate Hamiltonians, our ability to do a brute-force diagonalization is already insufficient. For typical Hamiltonians in condensed matter, we quickly run out of electrons in the universe that can do the computation, eg. http://fqxi.org/data/essay-contest-files/Swingle_fqxi2012.pdf.

 

[ErikZorkin]

QT as a mathematical theory is consistent

What's QT? Quantum mechanics or QFT? First of all, the very formal foundation of QM is not done yet. Second, you can't prove its consistency. Because your axiomatic system would necessarily include arithmetic whose consistency can't be proven in arithmetic alone.

 

[bhobba]

First of all, the very formal foundation of QM is not done yet.

Von Neumann sorted that out ages ago. Dirac's elegant formulation is now rigorous since Rigged Hilbert Spaces have been worked out.

Please, please read a good book on QM such as Ballentine.

Thanks
Bill

 

[atyy]

First of all, the very formal foundation of QM is not done yet.

Formal foundations of QM are done, eg. http://arxiv.org/abs/1110.6815 (p9), which can be generalized to continuous variables, partial example http://arxiv.org/abs/0706.3526 (Eq 3, 4).

 

[bhobba]

Because your axiomatic system would necessarily include arithmetic whose consistency can't be proven in arithmetic alone.

Godels theorem is irrelevant in this context. Its as consistent as any other physical theory ie as consistent as geometry, arithmetic etc etc.

Thanks
Bill

 

[ErikZorkin]

Von Neumann sorted that out ages ago. Dirac's elegant formulation is now rigorous since Rigged Hilbert Spaces have been worked out.

Please, please read a good book on QM such as Ballentine.

Thanks
Bill

Well, the axioms of QM are far from what's called a formal mathematical axiom. cf. axioms of ZFC