Boundedness of quantum observables?

[A. Neumaier]

But what I want to do is pull this discussion away from some silly textbook prejudices people have to situations where it really matters. For example to QFT or quantum gravity: that is where these issues really show their theeth, not in standard QM.

It is against the rules of PF to hijack a thread whose goal is something different.

I started this thread and want to discuss here only that part of QM which has a rigorous mathematical foundation. This includes QFT only as far as it has been rigorously constructed, and excludes quantum gravity unless you can offer a construction of its dynamics - i.e., a rigorous proof of solvability of its dynamical equations.

If you want to discuss the boundedness question on the looser level of theoretical physics, you should open your own thread.

 

[dextercioby]

No. What you emphasized by bold characters holds iff the Hamiltonian is self-adjoint.

The self-adjointness of an operator with mixed spectrum is a tricky business, because the spectral equation for the operator no longer has solutions in the original Hilbert space. That's why i mentioned the restricting condition of purely discrete spectrum.

For in this case it is the infinitesimal generator of a 1-parameter group exp(itH), which is a bounded operator defined on the full Hilbert space, and psi(t)=exp(-it/hbar H)psi(0) is a well-defined solution of the Schroedinger equation for every psi(0) in the Hilbert space.

You'd be surprised to know (in case you didn't already) that the trick with the Stone's theorem and its converse can very well be carried to distribution spaces, simply because there's a spectral theorem for unitary operators as well (see Gelfand's book).

So if psi(0) doesn't live in the h-space because of the continuous spectrum, psi(t) defined the way you did won't live either.

No. Most of quantum mechanics is done with time-independent Hamiltonians - for example the whole of quantum chemistry. If all states were eigenstates, chemical reactions would be impossible!

The bolded one I agree with. The structure of matter (atomic physics, molecular physics, chemical bond) is indeed time-independent. But the last one implies that quantum chemistry doesn't cover chemical reactions. Then what fundamental theory does explain chemical reactions ?

 

[Careful]

No. What you emphasized by bold characters holds iff the Hamiltonian is self-adjoint.

For in this case it is the infinitesimal generator of a 1-parameter group exp(itH), which is a bounded operator defined on the full Hilbert space, and psi(t)=exp(-it/hbar H)psi(0) is a well-defined solution of the Schroedinger equation for every psi(0) in the Hilbert space.

Thus the statement holds in _all_ well-defined and time-reversal invariant quantum theories.

But not a single realistic theory satisfies your conditions! Even the harmonic oscillator Hamiltonian is not a self-adjoint operator on the whole of Hilbert space. What we mean with self-adjoint extensions of unbounded symmetric operators A is that there exists a B such that B* = B and A < B, but B is of course not everywhere defined. Therefore, Stone's theorem does not hold and one can safely relegate it to the trashbin. This is actually a key insight in my book which should not be taken lightly.

No. Most of quantum mechanics is done with time-independent Hamiltonians - for example the whole of quantum chemistry. If all states were eigenstates, chemical reactions would be impossible!

Sure, that's why it doesn't work. The linear time picture of QFT is not commensurable with the nonlinear time picture of GR.

You are too obsessed with simple theorems which allow for nice structures. There exist more general structures which are still within control, you know.

Careful

 

[Careful]

It is against the rules of PF to hijack a thread whose goal is something different.

I started this thread and want to discuss here only that part of QM which has a rigorous mathematical foundation. This includes QFT only as far as it has been rigorously constructed, and excludes quantum gravity unless you can offer a construction of its dynamics - i.e., a rigorous proof of solvability of its dynamical equations.

If you want to discuss the boundedness question on the looser level of theoretical physics, you should open your own thread.

There is no loose level of unbouded operators! Nothing of QFT nicely fits within the limited mathematical tools you are using; even free QFT requires tools which go beyond the theorems you are quoting and this is certainly the case for interacting QFT. If you have a rigorous construction of interacting QFT within the C* language I would love to see it. It would be an (impossible) breakthrough, say for QED ? Moreover, even QED is not rigorously solved yet, why would this then be the case for quantum gravity?

But on the other hand if you wish to study/discuss things which do not appear in nature, then I will withdraw from this thread. I actually thought that you still were a bit interested in that... since after all this is still PHYSICS forums.

 

[A. Neumaier]

The self-adjointness of an operator with mixed spectrum is a tricky business, because the spectral equation for the operator no longer has solutions in the original Hilbert space. That's why i mentioned the restricting condition of purely discrete spectrum.

But my statement is nevertheless correct (Hille-Yosida theorem). It has nothing to do with distributions or the existence of eigenvectors.

The bolded one I agree with. The structure of matter (atomic physics, molecular physics, chemical bond) is indeed time-independent.

... whereas you were saying in #68 that ''the whole quantum dynamics and everything going around in the (quantum) world is driven by time-dependent Hamiltonians.''

Please be consistent, else it is frustrating to discuss with you.

But the last one implies that quantum chemistry doesn't cover chemical reactions.

No. Quantum chemistry does cover chemical reactions, and my statement assumes that.

But quantum chemistry would not cover chemical reactions if your interpretation that all states must be eigenstates of the Hamiltonian were valid.

 

[A. Neumaier]

But on the other hand if you wish to study/discuss things which do not appear in nature, then I will withdraw from this thread.

Yes please. After all, there is a separate forum for quantum gravity.

By the way, most of what appears in nature (with exception only of the biggest and the tiniest) is described by quantum chemistry, which exclusively works in the Hilbert space setting.

I actually thought that you still were a bit interested in that... since after all this is still PHYSICS forums.

After all, mathematical physics and quantum physics on the Hilbert space level are also physics, though you think little of them.

 

[Careful]

Yes please. After all, there is a separate forum for quantum gravity.

By the way, most of what appears in nature (with exception only of the biggest and the tiniest) is described by quantum chemistry, which exclusively works in the Hilbert space setting.

Sure, because they use some glorified multiparticle formalism with bounded potentials from below and above. Nothing surprising about that... but ok I will withdraw myself

 

[dextercioby]

But my statement is nevertheless correct (Hille-Yosida theorem). It has nothing to do with distributions or the existence of eigenvectors.

Spoken like a true mathematician. (No irony here). I get your point, hopefully you've gotten mine, even though we may probably still disagree. :P

... whereas you were saying in #68 that ''the whole quantum dynamics and everything going around in the (quantum) world is driven by time-dependent Hamiltonians.''

Please be consistent, else it is frustrating to discuss with you.

Yes, you're kind of right with the lack of consistency, though someone putting both my statements one to the other would still understand that I'm making a difference between the quantum statics (time-independent Hamiltonians) and quantum dynamics (time-dependent ones).

No. Quantum chemistry does cover chemical reactions, and my statement assumes that.

But quantum chemistry would not cover chemical reactions if your interpretation that all states must be eigenstates of the Hamiltonian were valid.

My interpretation is that all physical states of systems with time-independent hamiltonians must be eigenstates (either in HS space or in one of its possible extensions) of the hamiltonian. .

With this part I'm pretty sure of being consistent throughout the thread.

What I reiterated is substantially different than what you made of my statements (and which is bolded in the quote).

 

[A. Neumaier]

someone putting both my statements one to the other would still understand that I'm making a difference between the quantum statics (time-independent Hamiltonians) and quantum dynamics (time-dependent ones).

But this is _very_ different from how the terms are used by everyone else.

Time-independent Hamiltonians describe both the static aspects (equilibrium) and the dynamic aspects (motion) of a quantum system.

My interpretation is that all physical states of systems with time-independent hamiltonians must be eigenstates (either in HS space or in one of its possible extensions) of the hamiltonian. .

Increasing the size of your statements doesn't make them less invalid.

Probably you meant: ''Stationary states of systems with time-independent Hamiltonians must be normalized eigenstates of the Hamiltonian.'' This is a correct statement, but it is about a very small subset of states, namely only the stationary ones.

 

[strangerep]

I started this thread and want to discuss here only that part of QM which has a rigorous mathematical foundation. [...]

Re-reading your original post in this thread, it's still rather vague (to me anyway)
what issue/question you intend for this thread. It seems to be wandering all over
the place.

Now that the disruptive element has left the room, would you perhaps re-state
your focus/question of this thread more precisely (assuming further discussion
is still desired) ?

 

[A. Neumaier]

Re-reading your original post in this thread, it's still rather vague (to me anyway) what issue/question you intend for this thread. It seems to be wandering all over the place.

would you perhaps re-state your focus/question of this thread more precisely (assuming further discussion is still desired) ?

It is difficult to keep a thread focused...

I took partially inconsistent comments from DarMM about unbounded observables in the C^* algebra approach to rigorous field theory as my starting point.

The intended goal was to discuss the limitations of C^* algebras in this regard, and what the possible alternatives are.

 

[A. Neumaier]

Apparently there's some work in the field of <algebras of unbounded operators> as this review article (and the quoted bibliography) shows:
http://arxiv.org/abs/0903.5446

Thanks. This is a nice paper that I didn't know before. I need to read it more carefully.

Well, on more careful reading I found it a bit disappointing. It sacrifices the product of unbounded operators completely!

But I want a concept that covers the algebra of differential operators on Schwartz space, which is the right space on which the physical observables for QM of one degree of freedom act. Here the product is always well-defined.

 

[Careful]

The intended goal was to discuss the limitations of C^* algebras in this regard, and what the possible alternatives are.

In that case, your comments regarding my contribution are not what they seemed at first sight.

 

[A. Neumaier]

In that case, your comments regarding my contribution are not what they seemed at first sight.

You completely removed the basis of the discussion, dropping the existence of a definite inner product, referring to what is needed for quantum gravity, so that nothing is left but speculation. A discussion can lead nowhere when there is no common ground on which the participants agree.

I want to keep _all_ structure that theoretical physicists use when discussing ordinary quantum mechanics - the definite inner product, the unitarity of exp(iA) for the traditional observables, the unbounded spectrum of the Hamiltonian, but to drop the shackles of C^*-algebra, which was imposed for mathematical, not physical considerations.

This is quite different from what you propose - to drop most of the structure that gives sensible restrictions to QM, and allows the application of powerful mathematics. This almost killed the purpose of the thread - so I protested when you announced that it is ineed your goal to move the topic away form where it was.

Thus the two things can hardly be discussed in a single thread, unless all connections to my interests in this thread are lost. If you open a new thread about observable in indefinite spaces (or whatever), we can discuss your interests there.

 

[Careful]

You completely removed the basis of the discussion, dropping the existence of a definite inner product, referring to what is needed for quantum gravity, so that nothing is left but speculation.

No speculation, operators in Krein space have been rigorously studied as well as spectral decompositions and so on. It is just much less known obviously.

I want to keep _all_ structure that theoretical physicists use when discussing ordinary quantum mechanics - the definite inner product, the unitarity of exp(iA) for the traditional observables, the unbounded spectrum of the Hamiltonian, but to drop the shackles of C^*-algebra, which was imposed for mathematical, not physical considerations.

That are indeed very limited goals (which do not even suit QFT). However, this was not clear from what you wrote before, I am not a mind reader you know.

 

[A. Neumaier]

I want to keep _all_ structure that theoretical physicists use when discussing ordinary quantum mechanics - the definite inner product, the unitarity of exp(iA) for the traditional observables, the unbounded spectrum of the Hamiltonian, but to drop the shackles of C^*-algebra, which was imposed for mathematical, not physical considerations.

That are indeed very limited goals (which do not even suit QFT).

They suit _all_ QFTs whose existence is currently known, and they were necessary for proving their existence.

The main reason why I want to keep these restrictions is precisely because I want to understand the most interesting open case, QED, from a rigorous point of view.

Dropping structure that is present would only rob one of mathematical tools, and thus make the goal - the rigorous construction of QED - even harder than necessary.

 

[Careful]

They suit _all_ QFTs whose existence is currently known, and they were necessary for proving their existence.

You mean, all these which are not realized in nature

The main reason why I want to keep these restrictions is precisely because I want to understand the most interesting open case, QED, from a rigorous point of view.

I explained you 20 times why you will never succeed in constructing a physical theory with those limited tools, but I am afraid that we are in a circle where I offer evidence which has not been fully worked out yet and where you offer nothing at all so far.

Dropping structure that is present would only rob one of mathematical tools, and thus make the goal - the rigorous construction of QED - even harder than necessary.

Again, entirely false... the mathematical tools roughly stay the same. The computations just become a bit more elaborate but that was to be expected, no?

 

[A. Neumaier]

The main reason why I want to keep these restrictions is precisely because I want to understand the most interesting open case, QED, from a rigorous point of view.

I explained you 20 times why you will never succeed in constructing a physical theory with those limited tools, but I am afraid that we are in a circle where I offer evidence which has not been fully worked out yet and where you offer nothing at all so far.

Nothing in your book or your discussions here on PF has anything to do with a rigorous construction of QED. Nothing in your book carries even the slightest tint of rigor. Therefore, repeating your reasons another 20 times will not convince me of their relevance.

The evidence about QED from a rigorous point of view - ''which has not been fully worked out yet'' - is as empty as your promise to stay out of this thread. Therefore this will be my final reply to you here. What you say has nothing to do with the topic under discussion.

 

[Careful]

Nothing in your book or your discussions here on PF has anything to do with a rigorous construction of QED.

Sure, nothing which has been done so far has anything to do with that. Moreover, saying that nothing in my book is rigorous is a tremendous lie, everything which is written out there isn't less rigorous than standard textbook QM or general relativity on the level of Robert Wald. But I know, you have never understood the rigor of unbouded operators.

The evidence about QED from a rigorous point of view - ''which has not been fully worked out yet'' - is as empty as your promise to stay out of this thread.

Likewise is your babbling about rigorous techniques for quantum physics.

Therefore this will be my final reply to you here. What you say has nothing to do with the topic under discussion.

Indeed, there is no topic of physical relevance. Let us fight elsewere, will we?

 

[dextercioby]

Up until this dispute between prof Neumaier and Careful, the thread makes a useful reading though.

I hope all parties agree that there's no 100% mathematically rigorous theory of quantum mechanics and quantum field theory in flat 4 Minkowski space-time and that work can still be done to achieving it, of course, if somebody is still interested in it and has not migrated towards strings and quantum gravity.

I think your dispute comes from the fact that there seems to be almost a void intersection between prof. Neumaier;s intentions/expectations and Careful's work part of which is published on arxiv.

 

[A. Neumaier]

Up until this dispute between prof Neumaier and Careful, the thread makes a useful reading though.

Yes. I am sorry to have responded at all to Careful's posts. Our views are too different to result in a productive exchange.

I hope all parties agree that there's no 100% mathematically rigorous theory of quantum mechanics and quantum field theory in flat 4 Minkowski space-time and that work can still be done to achieving it, of course, if somebody is still interested in it and has not migrated towards strings and quantum gravity.

Currently there is no mathematically rigorous of a causal interacting and Poincare invariant quantum field theory in 4D (as defined by the Wightman axioms); neither is there a proof that no such object exists.

Thus it is a legitimate and highly challenging endeavor to try to settle this question one way or another.

One particular such question was selected by Arthur Jaffe (former president of the International Association of Mathematical Physics, and former president of the American Mathematical Society) and Edward Witten (surely one of the most influential physicists) as one of the Clay Millennium Problems. Their paper http://www.claymath.org/library/MPP.pdf#page=114 describes (in terms also accessible to the nonspecialist, familiar with quantum mechanics though) why solving the problem would be a major breakthrough.

 

[DarMM]

To return to A. Neumaier's original discussion about C*-algebras, the C*-algebra structure is chosen for mathematical convenience rather than physical convenience.
In the C*-algebraic approach there are no field equations, in fact there are no fields, e.t.c. Hence the C*-algebraic approach, although equivalent, is vastly different in appearance from the standard physical approach. This is why (even though there are currently many attempts) it isn't viewed as a good route for mathematically constructing a theory.

However it has a few advantages. Firstly, the connection between quantum mechanics and Kolmolgorov probability become extremely obvious, leading to the observation that many theorems in quantum mechanics are simply noncommutative versions of those in probability. Secondly, it becomes extremely easy to characterise different kinds of states, something which makes it useful for QFT in curved spacetime.

Personally I would view algebraic field theory as an alternative way of writing quantum field theory (just like the path integral) that makes certain general theoretical features more obvious. I wouldn't view it as a replacement for the standard formulation. Even in rigorous field theory it isn't viewed like this.

 

[A. Neumaier]

To return to A. Neumaier's original discussion about C*-algebras, the C*-algebra structure is chosen for mathematical convenience rather than physical convenience.

Personally I would view algebraic field theory as an alternative way of writing quantum field theory (just like the path integral) that makes certain general theoretical features more obvious. I wouldn't view it as a replacement for the standard formulation. Even in rigorous field theory it isn't viewed like this.

Do you use ''algebraic field theory'' synonymous with the C^* algebra approach?

I do not deny that C^* algebras are useful for some purposes - only that they are appropriate as foundations.

Isn't the algebra of all continuous linear mappings on Schwartz space also a mathematically very convenient object, containing all polynomials in p and q and the exponentials exp(ixp) and exp(ikq)? Isn't there an abstract characterization of its properties that generalizes to more complex situations?

 

[kote]

I won't pretend that I can follow all of this thread, but I'd like to point out that philosophers studying the foundations of QFT have preferred the AQFT approach due to its mathematical rigor. Some of you might be interested in http://www.princeton.edu/~hhalvors/aqft.pdf. Then again, philosophers care less about maintaining all of the common language of physics if they think they can come up with something more consistent.

From the title of this Chapter, one might suspect that the subject is some idiosyncratic approach to quantum field theory (QFT). The approach is indeed idiosyncratic in the sense of demographics: only a small proportion of those who work on QFT work on algebraic QFT (AQFT). However, there are particular reasons why philosophers, and others interested in foundational issues, will want to study the "algebraic" approach. ...

So, philosophers of physics have taken their object of study to be theories, where theories correspond to mathematical objects (perhaps sets of models). But it is not so clear where "quantum field theory" can be located in the mathematical universe. In the absence of some sort of mathematically intelligible description of QFT, the philosopher of physics has two options: either find a new way to understand the task of interpretation, or remain silent about the interpretation of quantum field theory.1

It is for this reason that AQFT is of particular interest for the foundations of quantum field theory. In short, AQFT is our best story about where QFT lives in the mathematical universe, and so is a natural starting point for foundational inquiries.

 

[Careful]

I won't pretend that I can follow all of this thread, but I'd like to point out that philosophers studying the foundations of QFT have preferred the AQFT approach due to its mathematical rigor. Some of you might be interested in http://www.princeton.edu/~hhalvors/aqft.pdf. Then again, philosophers care less about maintaining all of the common language of physics if they think they can come up with something more consistent.

As far as I see, there is no dynamics in this paper; that is where the difficulty resides, not in the kinematics (Fock space is dandy fine even if you regard it as a rigged Hibert space like I do). I just glanced at it, but isn't the essential idea roughly not the following ? We consider any open set (with compact closure) of spacetime and limit ourselves to wave functions and operators with support on this open set. Since the extend of this open set is finite, the typical momentum we want to consider is inversely proportional to the extend of it. Hitherto, the representations of the Poincare algebra on such states are nice operators (no problems with unboundedness and so on in the high UV). The virtues are the same as in the lattice approaches, but you do not break manifest Lorentz covariance because you consider all possible coverings. Is that what it is about ?

 

[DarMM]

You are falling over semantics and miss the point again. What people call physical states is a matter of agreement, it is a social construct without any deeper meaning. Even at this level, you constantly use the distributional states by means of the Fourier transform, still you wish to expell them to the margins. Let me turn the game around and you show us a Hilbert space construction which is adequate for QFT. I have given plenty of positive arguments against Hilbert space, so you show now a positive argument pro Hilbert space. Then, we will talk.

Although I am not certain how this is a response to the original point, I'll attempt a response.

My original statement was that there are well-defined states in the free-particle case, it is simply that none of them are eigenstates of the Hamiltonian. There are still several other observables for which they are eigenstates. I don't need distributional states to make any of this valid.

As for showing you a Hilbert space that works, there is Fock space for free theories, for free electromagnetism there is the Coherent space or Strocchi spaces, for interacting theories there is Glimm space, Osterwalder space, e.t.c. For QFT in curved spacetime there are several spaces available, Wald spaces ,e.t.c. For thermal states there are Boltzmann Hilbert spaces.

 

[DarMM]

Then what you say totally disagrees with what's axiomatized through the Schroedinger equation. The physical states are solutions of the SE. If the Hamiltonian is time-independent (true for the free particle in the Schroedinger picture), then the physical state must have the form

$$\Psi_{\mbox{physical}}(t) = \psi_E \exp{\frac{t p^2}{2mi\hbar}}$$

where \psi_E is an eigenfunction of the free-particle hamiltonian, a member of S'(R^3), so the whole physical state becomes a tempered distribution, thus contradicting the physical state postulate.

It is correct that the physical states are solutions of Schrödinger's equation. However what you have written down is the stationary state equation. This is simply the equation a state must satisfy to be an eigenstate of the Hamiltonian. For example a linear combination of two such states is certainly a state (principle of superposition) but doesn't satisfy that equation.

 

[dextercioby]

It is correct that the physical states are solutions of Schrödinger's equation. However what you have written down is the stationary state equation. This is simply the equation a state must satisfy to be an eigenstate of the Hamiltonian. For example a linear combination of two such states is certainly a state (principle of superposition) but doesn't satisfy that equation.

Yes, my statement was a little imprecise. The formula in your quote is for a generic vector from the basis of the linear space of physical states. So any (normalized) physical state of a quantum system with time-independent Hamiltonian operator in the Schroedinger picture should be a linear combination of vectors of that form.

 

[A. Neumaier]

Yes, my statement was a little imprecise. The formula in your quote is for a generic vector from the basis of the linear space of physical states. So any (normalized) physical state of a quantum system with time-independent Hamiltonian operator in the Schroedinger picture should be a linear combination of vectors of that form.

But in this form you original complaint abouit the massive Galilei particle is invalid since its nonexistenrt eigenstates cannot form a basis of the Hilbert space. The states, however, are still made of linear combinations of an arbitrary basis.

 

[dextercioby]

My complaint was about the fact the axioms for state description (#1 in the set I posted in the other thread) and state dynamics (#4 in the set I posted in the other thread) lead to the rebuttal of a free massive particle moving freely in R^3.

The quote below is from post #56 of this very thread.

If i better think about it, we've got conflicts in the Hilbert space axiomatization as well*. It turns out that, if one accepts/postulates that physical quantum states are described by unit rays in a complex separable Hilbert space, then the free massive Galilean particle doesn't exist, as it has no physical states, as follows from solving the Schroedinger equation (which is also postulated, of course). So the probabilistic interpretation a\ la Born of the free Galiean particle is not defined, as the probability to find this particle along the whole real axis is infinite.

 

[DarMM]

My complaint was about the fact the axioms for state description (#1 in the set I posted in the other thread) and state dynamics (#4 in the set I posted in the other thread) lead to the rebuttal of a free massive particle moving freely in R^3.

The quote below is from post #56 of this very thread.

This is incorrect. For the free particle there are no eigenstates of the Hamiltonian. However every state evolves under the time evolution operator, or in different words, satisfies the Schrödinger equation.

For example, the Schrödinger equations is:
$$\iota \frac{\partial}{\partial t}\Psi = H \Psi$$

This equation can have well defined solutions even if there is no function satisfying $$H \Psi = E \Psi$$, with $$E \in \mathbb{R}$$.

 

[dextercioby]

Apparently you don't understand my point. The SE you posted is a first order differential equation for the variable Psi (t). The question is: what mathematical assumptions do you make about the searched for solution of the PDE/ODE, other than differentiability in the argument "t", if the H operator is a hamiltonian of a free particle ? More precisely, in what space/set of functions do you search for solutions of this equation ?

 

[DarMM]

Apparently you don't understand my point. The SE you posted is a first order differential equation for the variable Psi (t). The question is: what mathematical assumptions do you make about the searched for solution of the PDE/ODE, other than differentiability in the argument "t", if the H operator is a hamiltonian of a free particle ? More precisely, in what space/set of functions do you search for solutions of this equation ?

$$\mathcal{L}^{2}\left(\mathbb{R}^{3}\right)$$, so that you have a normalised probability density.

 

[dextercioby]

Ok, so do you find a solution in that space for the Schroedinger equation for a nonrelativistic massive particle ?

 

[DarMM]

Ok, so do you find a solution in that space for the Schroedinger equation for a nonrelativistic massive particle ?

I'm not sure where this is going, but yes you can. For example:
$$\Psi\left( x, t \right) = \int{e^{-p^{2}}e^{-\iota \frac{p^{2}}{2m}t} e^{\iota p x} dp}$$

At any time this function is an element of $$\mathcal{L}^{2}\left(\mathbb{R}^{3}\right)$$ and satisfies Schrödinger's equation.