Talk:Mathematical formulation of quantum mechanics

Mature physical theories are formulated in mathematical or quantitative language.
Quantum mechanics represents a departure from the languages used in previous physical theories.

The nature of that departure should be better explained in the introduction.CSTAR 00:36, 23 Nov 2004 (UTC)

I have added some more. Charles Matthews 07:01, 23 Nov 2004 (UTC)

The introduction, before the ToC (not to be confused with the WoT) should be able to stand on its own. Why split off a section on old-quantum theory, when there is already an article on it? 17:04, 23 Nov 2004 (UTC)

Oops and I forgot to sign the comment above ... or something. CSTAR 17:50, 23 Nov 2004 (UTC)

I certainly don't agree with discussion of the photoelectric effect here; it's not quantum mechanics, in the sense that the mechanics only came around 1925. So I think I agree with the comment above (?). By all means write something about the 'old' theory in another article. But the continuities there seem to me (not an expert) to be in the physics, while in the mathematical formulation there was more like a break. Charles Matthews 17:44, 23 Nov 2004 (UTC)

The problem is that CSTAR wants to be very specific about the state of physics prior to the break, which forces you to discuss a bunch of stuff irrelevant to the mathematics of the new quantum theory.

There is no article on the old quantum theory in the form that it needs to be in, including Plank's quanta, Einstein's photons, Bohr's atom, and the Bohr-Sommerfeld and Sommerfeld-Wilson-Ishiwara quantizations. The article old quantum theory currently redirects to Bohr model which is taking the part for the whole. The Bohr model was the first phenomenological model of the atom, but later Bohr himself (and others) developed a more precise mathematical formulation based on allowing only closed orbits in phase space enclosing an area equal to an integer multiple of planck's constant. That is as far as classical mechanics could take you, and is the right point of comparison with Schroedinger's wave mechanics. So we are not at the point where you can write the introduction in the form you want, CSTAR.

I can do one of two things: either expand the section here and then refactor it into a new article leaving a summary, or go off and write the article on the old quantum mechanics first, and then come back and summarize it here.

What I want is not a full discussion of the photoelectric effect, but maybe a few words about where Planck's constant came from and how Einstein used it to invent photons. You can then point out that neither Planck nor Einstein took wave-particle duality seriously, and it took a physicist of the younger generation (de Broglie) to do that.

— Miguel 18:46, 2004 Nov 23 (UTC)

The introduction now is certainly a lot better. Your changes address my concerns. Thanks. CSTAR 19:05, 23 Nov 2004 (UTC)

It was I who made old quantum theory a redirect. It obviously can be an article in its own right, perhaps under a more 'professional' name. Charles Matthews 20:29, 23 Nov 2004 (UTC)

the name old quantum theory is what you find in physics textbooks. There can be more 'professionally-named' articles about the subtopics, such as Bohr model and Bohr-Sommerfeld quantization. — Miguel 20:41, 2004 Nov 23 (UTC)

Epistemology and separability

The assertion about the relation between separability of H and sufficiency of countably many observations is interesting, but may require more elucidiation. However, how is it related to epistemology? I assume one could argue that there are sequences of observations A_i which ultimately distinguish a pair of states. The problem with such a claim is that it will be inevitably subject to much critical examination.CSTAR 20:38, 24 Nov 2004 (UTC)

The relationship to spistemology is that we can only have access to a finite amount of information, hence a finite amount of experiments. A countable number is necessary to reason mathematically about finitely many experiments. That is exactly the point of separability in the theory of stochastic processes, except that nobody introduces it in that way so it sounds like a completely unmotivated assumption made for mathematical convenience.

You can remove the mention of epistemology if you think it will lead to too much discussion (on second thought, I agree it will), but leave there the statement about countability. I don't know in how much more detail we can get in this article before having to spin off other pages. — Miguel 21:00, 2004 Nov 24 (UTC)

Ab-initio

Whazzat? Do you mean derivation from first principles? This is an expression so often used by physicists (and other scientists) so maybe it should have a separate article? How is this different from derivation from axioms? Note that I put in a wikilink to phenomenology (science) to explain its (correct) use in this article. A similar link is needed here, I think.CSTAR 16:12, 25 Nov 2004 (UTC)

While we're at it, I want a separate phenomenology (physics) article. There is much to be said about the scope, goals and level of success of phenomenology in different branches of physics. High-energy physics phenomenology is broad, deep and successful enough to be an independent branch of physics in its own right. — Miguel 18:38, 2004 Nov 25 (UTC)

Whoops. I reistated the old link. Sorry, I though you made a mistake. Fui yo que me equivoqué. I will revert.18:51, 25 Nov 2004 (UTC)

There are about 200,000 google hits for ab-initio calculation. It does mean "from first principles", but it is, in fact, customary in atomic and molecular and solid-state physics. — Miguel 16:34, 2004 Nov 25 (UTC)

I don't dispute that it's use is customary in certain areas of physics. But for example the difference between a derivation from first principles and a derivation within an axiomatic system requires some explanation — it's certainly not obvious to me. These are important cultural nuances (for example, differences between the culture of physicists and mathematicians) which an encyclopedia has to deal with and explain to be useful as a mirror of not only knowledge, but culture.CSTAR 16:55, 25 Nov 2004 (UTC)

Can you explain what you have in mind about the difference between from first principles and within an axiomatic system? I grok not.

Physics is not axiomatic, so people such as von Neumann talk about postulates to weasel out of that fact.

A calculation from first principles or ab initio seems to mean with "no" experimental input.

I anxiously await your elucidacion of this philosophical tangle.

— Miguel 17:13, 2004 Nov 25 (UTC)

This is a very tentative response

Physics is not axiomatic. Agreed, but there is such a thing as axiomatic physics, which consists of experimentally testable assertions as axioms in a formal theory and deriving everything else purely by rigorous mathematics (in principle formalizable within a formal logical system).

A derivation from first principles makes reasonable guesses about what should be true in nature. For example the relative sizes of things (where a mathematically rigorous calculation would be purely asymptotic), or excluding certain cases as being physically unreasonable.

Now this distinction does require more elaboration, but my purpose is to provide a plausible argument as to why these two concepts are different.CSTAR 17:33, 25 Nov 2004 (UTC)

On a different matter:

I put in a link to ab initio which states

in sciences (especially physics and chemistry): from first principles. A calculation is said to be "ab initio" (or "from first principles"") if it only assumes basic and established laws and does not assume the validity of further assumptions such as models.

That's OK except further assumptions such as models should be changed to further assumptions includeing additional models.CSTAR 17:33, 25 Nov 2004 (UTC)

I added the only experimental input is the values of fundamental constants. — Miguel 18:02, 2004 Nov 25 (UTC)

Measurement

What happened to the caveat that the operator was assumed to have pure point spectrum? What about measurement for operators (such as position or momentum for a free particle) with continuous spectrum? CSTAR 04:14, 26 Nov 2004 (UTC)

Measurements of position or momentum result not in point values but in ranges. The collapse is represented by the orthogonal projector associated to the observed range by the spectral theorem. (This is the interpretation of the projector-valued measure in the spectral theorem)

However, if we can't write down the measurement postulate without assuming a discrete spectrum maybe we should think of a different way to discuss measurement. Something like this: in the Heisenberg picture, if the system is in state <math>\left|\psi\right\rangle<math> and observables <math>A_i<math> are measured at times <math>t_i

<math>\left\langle\psi\mid A_n(t_n)\cdots A_1(t_1)\mid\psi\right\rangle<math>

where the <math>A_n(t_n)<math> are Heisenberg operators.

I'm not sure, though. Von Neumann's postulate has always seemed suspect to me. Moreover, I think a discussion of quantum measurement belongs in interpretation of quantum mechanics rather than here. We already have a complete description of the mathematical formalism of quantum mechanics. I was wondering what ever happened to the canonical commutation relations and the usual Schrödinger equation, but IMHO those topics belong in quantization or relation between classical and quantum mechanics, or classical limit of quantum mechanics.

I would personally remove the measurement discussion from this article, but I hesitate to do it because that's a very radical step to take. But it would save us a lot of mathematical and philosophical heasaches. In any case, quantum measurement is an active research area, not a postulate of quantum theory (even if von Neumann thought he could wrap it all up neatly in his book). — Miguel 05:23, 2004 Nov 26 (UTC)

Actually, there is already an article on measurement in quantum mechanics. I think we should remove all detailed discussion of measurement from this article and just link to that. Sooner or later I'll have to go and pitch in on quantum measurement. The current article is very limited and does not really go beyond von Neumann. That is neglecting 70 years of development of the theory. —Miguel 05:34, 2004 Nov 26 (UTC)

By the way, after collpse one would have to rescale the "collapsed" state vector to unit length, which makes the mathematical description of collapse all the more awkward, as it is now orthogonal projection followed by rescaling to unit norm. Can we not allow non-normalized states and density matrices and divide by <math>\left\langle\psi\mid\psi\right\rangle<math> or <math>\operatorname{tr}\rho<math> in the expected value of an observable? Also, we should maybe be more explicit and say that states are really equivalence classes of vectors in H, even though physicists cringe at the mention of equivalence classes. — Miguel 05:53, 2004 Nov 26 (UTC)

Well the article on quantum operation is a more comprehensive discussion of general measurement (not just projective measurements). There is an intermediate discsuuion in quantum logic and quantum statistical mechanics. The quantum operation operation is completely equivalent to the relative state approach.CSTAR 06:02, 26 Nov 2004 (UTC)

Actually, there is already an article on measurement in quantum mechanics

It's not very good.CSTAR 06:05, 26 Nov 2004 (UTC)

I totally agree. Let's finish this one and then move to that one ;-) — Miguel 06:12, 2004 Nov 26 (UTC)

I would personally remove the measurement discussion from this article, but I hesitate to do it because that's a very radical step to take.

Yes, it would be very radical and no, I don't think it should be removed. It could remain as is, with enough pointers to other articles.CSTAR 06:17, 26 Nov 2004 (UTC)

Where the article is going

I should really just let you guys write what you're going to write.

This sounds like there is some criticism (besides the issue of formalism) that you decided to keep to yourself on second thought. I'd still like to know what you had in mind. — Miguel 18:46, 2004 Nov 26 (UTC)

Not really. If this is going to be a long article, that's OK. It seems a little strange to have the major discussion of the physics of 1925-1930 in an article about its mathematics. I'm not familiar with the WP articles on Dirac and so on; so I was just going to wait a little. Charles Matthews 19:38, 26 Nov 2004 (UTC)

I have begun to worry about the length of the article, too, and I wouldn't mind moving most of the references to physics and mathematics before 1925 to a different article. I wouldn't call what is in this article a major discussion of the physics, though. In any case, to me the physics and the mathematics of quantum mechanics are inseparable: you cannot really understand one without the other (well, you can understand the mathematics on its own, but then it's meaningless). I also suspect that most readers of this page would expect a discussione of the physical interpretation.

I personally like to discuss the history of a subject first, not last, but would you prefer to see the mathematical structure section be the first after the TOC?

If I had to strip this article to its bare bones, I would leave it at the first paragraph of the introduction and the mathematical structure section, moving eveyrthing else to other article and leaving behind very short pointers. In that case I would rename this article "Hilbert-space formulation of quantum mechanics", because the C^*-algebra formulation needs an article of its own.

—Miguel 20:06, 2004 Nov 26 (UTC)

My personal approach is quite historicist (within mathematics). So it doesn't worry me, really. In the end the site should contain these 'stories'. Charles Matthews

The length of the article doesn't concern me at all. And I don't like the idea of changing the pre ToC material by addition or deletion. That section looks good to me.

The C*-algebra approach really does require a more extensive article (explaining some of the concepts, particularly how it relates to Jordan's view of QM.) Certianly if Quantum logic has its own article then the C*-algebra approach to QM should also have its own article.

I also like the historicist perspective. Unfortunately, that's very hard to get right.CSTAR 20:35, 26 Nov 2004 (UTC)

I have expanded 1.3 Later developments, and would like to use that as a replacement for 3 Other formulations and 4 The problem of measurement. That is, can we agree to remove sections 3 and 4 or merge their content with appropriate other WP articles? — Miguel 01:20, 2004 Nov 27 (UTC)

"Formalism"

I did want to make one point: I object to formalism in relation to mathematics. It obscures a distinction: it can mean a notation, or I suppose it can mean the result of a formal development of a topic. There is a big problem in relation to mathematics/physics (I think) in that 'formal meaning' means opposite things depending on who you are. 'Mathematical formulation' implies to me that bra-ket notation is not just a formalism, but something with a very definite semantic content. Dirac's delta function was a formally-introduced symbol; then we had what you could call 'mathematical formulation of generalised functions' and δ(x) is a Schwartz distribution with a definite semantics. I realise it depends which side of the fence you stand, whether this is an issue or not. Given the article's title I feel that 'formalism' could be banished.

Charles Matthews 09:32, 26 Nov 2004 (UTC)

The article formalism does not address this, but it would help me if I could read an article about it. —Miguel 16:00, 2004 Nov 26 (UTC)

I put in a bullet in formalism. This probably should be expanded in an article such as scientific formalism.CSTAR 18:03, 26 Nov 2004 (UTC)

Point taken. I think we should address this concern. I tend to melange the two, but I am aware of the distinction and am also picky about it when I notice.CSTAR 14:20, 26 Nov 2004 (UTC)

Bargmann Segal

I thought one of the points of the Bargmann's paper was to show that the (symmetrized) Fock Representation was unitarily equivalent to the Bargmann Segal representation. Am I confused about something? CSTAR 04:13, 27 Nov 2004 (UTC)

OK I guess one shouldn't interpret your statement as asserting that these representations are inequivalent. I misread it.CSTAR 04:38, 27 Nov 2004 (UTC)

Precisely. — Miguel 04:56, 2004 Nov 27 (UTC)

The article now looks pretty good.

I really want to yank the measurement and other formulations sections. I think the first is actually a physical, not mathematical, problem, and that the second is superseded by the later developments section. —Miguel 06:50, 2004 Nov 28 (UTC)

I might start an article on Bargmann Segal. It fits naturally with the article Stone-von Neumann theorem; unfortunately, the notation I used in my notes on Bargmann Segal aren't quite consistent with the notation used in the Stone-von Neumann article (even though I wrote most of that one). I may have to do a little work to make them consistent. Geometric quantization (Weyl-Maslov-Hormander-Voros-Guilleman-Sternberg Quantization) might be another useful article although I have nothing written on those.CSTAR 06:22, 28 Nov 2004 (UTC)

C*-algebraic formulation

I think I'm going to concentrate on the C^*-algebraic quantization now. —Miguel 06:50, 2004 Nov 28 (UTC)

Ah. That sounds more interesting, although I'm not sure where one should begin. Jordan Algebras? Or part 6 of Boguliubov, Axiomatic Quantum Fied Theory?CSTAR 06:58, 28 Nov 2004 (UTC)

The weyl relations and the GNS construction?

Anyway, you seem to know more than I do about this stuff. — Miguel 15:21, 2004 Nov 28 (UTC)

I do NOT particularly wish to see that developed in this article, though, but in a separate one. Given the large number of formulations listed under later developments, singling out quantum logic and C^* algebras seems wrong. Moreover, people would expect to find the Schroedinger/Heisenber/Dirac/von Neumann formulation under the current title. — Miguel 20:31, 2004 Nov 28 (UTC)

I have no objection. Go ahead and delete those subsections. This is legacy from previous versions.CSTAR 20:44, 28 Nov 2004 (UTC)

Well I was bold and deleted them.CSTAR 22:04, 28 Nov 2004 (UTC)

Of course there is the wikipedia article on GNS construction.

Yes, but that is not C-star algebraic quantization. — Miguel 20:31, 2004 Nov 28 (UTC)

Whatever you do, call it C*-algebraic quantization. C^*-algebra looks weird and I have been consistently using C*-algebra.CSTAR 20:44, 28 Nov 2004 (UTC)

It would be nice to start a C*-algebraic quantization article with a comphrehensible account of Heisenberg's matrix mechanics or at least refer to such an account. In the past, when I tried reading these accounts in various places, my eyes glaze over. These accounts also are supposed to reflect the Zeitgeist of operationalism of which Heisenberg was apparently a follower.

There is also a brief account of matrix mechanics in Alain Connes' book, Noncommutative Geometry (pp 33-39) which I have looked at but there too, my eyes glaze. Anyway Connes tries to bring in groupoids, but of course these weren't around for Heisenberg to play around with. CSTAR 17:31, 28 Nov 2004 (UTC)

Old quantum mechanics

It would be nice to find out how in the world Heisenberg stumbled upon matrix mechanics. I just don't see how that would be likely. I can understand how Schrödinger came up with wave mechanics. That's a pretty natural consequence of de Broglie's ideas, but how did Heisenberg come up with matrix mechanics? Phys 05:37, 29 Nov 2004 (UTC)

Another picture?

I noticed this in the article

closely related to the Schrödinger picture where time is not a parameter but an operator. Here, time, t and energy (which is not the Hamiltonian, which is a function of position and momenta) are canonically conjugate operators which commute with position and momenta. Here, we look at the rigged Hilbert space and insist that the valid states are those which satisfy (H-E)|?>=0. Physical observables are operators which commute with (H-E).

O.K. I'm stumped. Notice that in this "new" picture, the operator E is by the uniqueness of the CCRs unitarily equivalent to a multiple of 1/i d/dt. In particilar it has spectrum R.

You could make E be a bounded operator, or bounded below, and make T ~ id/dE. That would solve your problem. — Miguel 20:13, 2004 Nov 29 (UTC)

My confusion comes from the remark

Here, time, t and energy (which is not the Hamiltonian, which is a function of position and momenta) are canonically conjugate operator

If time T and energy E are canonically conjugate operators — which I take to mean [T,E] = r I, that forces E to have a spectrum R.

I suppose in operator-theoretic terms, one could take a compression of E, T to some appropriate subspace H and get a physically realizable (e.g. non-negative) energy operator proj_H E proj_H on H. CSTAR 20:34, 29 Nov 2004 (UTC)

How about x and p for a particle in a box? Isn't x bounded, p unbounded and [x,p]=i hbar? — Miguel 20:47, 2004 Nov 29 (UTC)

It depends on what you mean by [x,p]=i hbar. The eqn

holds for functions φ which are zero near the boundary of the box. However, the operator x is not essentially self-adjoint on that domain (it has many self-adjoint extensions depending on the bdary conditions). So the commutation relations cannot be integrated to get them in Weyl form for which uniqueness holds.

I am aware of the essential self-adjointness issue, but you can fix that with (for instance) periodic boundary conditions. I do not know what a compression is in a technical sense. — Miguel 21:04, 2004 Nov 29 (UTC)

No, you can't! Then you have functions on a circle and x is not really well-defined even as a classical variable (it is defined modulo 2pi) — Miguel 21:06, 2004 Nov 29 (UTC)

Well yes it does have many self-adjoint extensions (periodic, Dirichlet,Neumann, Robin etc). But for any of these self-adjoint extensions the eqn

Correction. Yes it does have many self-adjoint extensions, corresponding to the fact that the space of square-intregrable distributional solutions of

form a 1-dimensional space. To talk about Dirichlet,Neumann bdary conditions for a first order operator doesn't make too much sense.

no longer holds in any reasonable way, since for all them the extensions have pure point spectrum. Compression of an operator T to a subspace H just \ means proj_H T proj_H .CSTAR 21:13, 29 Nov 2004 (UTC)

I guess in that sense, the compression to the kernel of H-E leads to E being bounded below. The quantization of systems with constraints is a thorny issue, though. — Miguel 21:19, 2004 Nov 29 (UTC)

In that case, what I said in the previous paragraph is true: one could take a compression (in some sense) of x and p on L²(R) to L²(TheBox) .

Ooops, I meant the operator p. x is essentially self adjoint there.CSTAR 21:03, 29 Nov 2004 (UTC)

Hmmm, you can formulate classical nonrelativistic mechanics by adding t and E as variables satisfying CCRs and another unphysical parameter s. Classically, s parameterizes the trajectory of the system in x-t space, and the equation H(p,q)=E becomes a constraint (in the sense of Dirac) whose Poisson brackets generates changes in the parameter s.

The Schroedinger representation of that is no different from the ordinary schroedinger representation of the CCR, it is just applied to a larger phase space with coordinates (q,p,t,E) instead of (q,p).

That is the problem: the Schroedinger representation of the CCR is part of the problem of quantization and has nothing to do with pictures of dynamics or the overarching mathematical framework of quantum mechanics.

— Miguel 20:13, 2004 Nov 29 (UTC)

Also the set of ψ satisfying (H-E)|ψ>=0 is not a vector space, or a direct sum of vector spaces so this is an odd kind of condition. Of course anything is possible. CSTAR 15:01, 29 Nov 2004 (UTC)

SOrry for the triplicate. My crippleware version of Opera really sucks.CSTAR 15:03, 29 Nov 2004 (UTC)

Whatever the merits

I don't think this particular graf on E, t belongs here in thiis article, maybe in a separate article about issues in canonical quantization.CSTAR 21:36, 29 Nov 2004 (UTC)

I agree. It is not wrong, but I have argued above what that is not really part of the overarching mathematical framework. Similarly, I do think that measurement is also not part of the mathematical framework. Von Neumann thought he could dispatch it with a postulate, but 70 years of research into quantum optics, hidden variables, decoherent histories, quantum cosmology and quantum information theory have shown him wrong. — Miguel 06:14, 2004 Nov 30 (UTC)

ready to move on

I am pretty happy with the article at this point. I think I am going to move on to quantization or something like that. — Miguel 07:05, 2004 Nov 30 (UTC)

Relative state interpretation

It's really equivalent to the mathematical machinery of quantum operations (via the Stinespring-Choi-Kraus represntation theorm).CSTAR 15:47, 30 Nov 2004 (UTC)

Symmetries

This is still an introductory article. Given that, why does it include the following sentence in the symmetries section

Gauge symmetries are not exactly and are dealt with in a more complicated manner using BRST.

This is actually about field theory - yank it from the article, or put it in the "later developments" section. — Miguel 21:07, 2004 Dec 11 (UTC)

Are gauge symmetries explained anywhere in the article? CSTAR 05:16, 11 Dec 2004 (UTC)

Decomposition of reps of symmetries

What's all this stuff about irreducibles? One is interestd in the

primary decomposition = factor decomposition = central decomposition = decomposition into multiples of irreducible representations (in type I case)

not so much of the group of symmetries as of the group of symmetries + algebra of local observables which it normalizes.

CSTAR 03:19, 27 Dec 2004 (UTC)

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