Boundedness of quantum observables?

[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.

 

[A. Neumaier]

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.

If the conflict is only with your axioms, it is much more likely that your axioms are faulty than that something is now nonexistent that has been considered to be respectable by everyone except you...

 

[dextercioby]

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.

Alright, so you chose a function from the domain of selfadjointness. Alright, good start.

Question:
Do you have anything against the fact that for time-independent Hamiltonians, the time dependence of the solution of the SE can be factored out, it is nothing but a phase factor and can totally be removed from a discussion on whether the solution to the SE bears physical relevance or not ?

 

[Fra]

I apologize fur just jumping in, I admitt I didn't follw neither this entire discussion nor it's history in other threads.

I prefer to have the foundations free from allusion to measurement. The latter should be a derived many-particle process to be analyzed by the statistical mechanics of the equipment interacting with the observed system.

I personally doesn't understand why a foundations of mesurement theory, should avoid the measurement.

The idea to see the observer and it's environment, from a different perspective (and apply some complex system or stat mech) seems to me a way to not take the meaning of an intrinsic measurement theory seriously, it seems like you seek an "external view" w/o measurement, where the "intrinsic view" with measurement is explained.

The deep issue I have with that general approach is that your "explanation" in terms of this external view, is effectively introducting some sort of superobserver or alternatively, some level of realism that is IMO against the very spirit of seeking a "measurment theory".

I've always seen the conceptual heart of measurement theory in the context of science as the intent that all we shall do is try to describe what outcome we expect of nature in response our measurement. "We" and "our" are a bit unprecise here and the refined version I see is that "the observer can only have an EXPECTATION as to how a fellow subsystem will respond to measurement". Measurement theory merely should try to understand HOW this expectaiton is constructed/computed from the observers state, and how it's REVISED/UPDATED in the event that new unexpected information arrives. Ie. it does NOT explain WHAT new information that arrivees, it only explains the "logic of information update", the logic of making an optimal correction.

It seems to me that if you seek a foundation that doesn't take this process to be importance at the foundational level, then you are probably against what I call the spirit of intent behind measurement theory in the first place?

Ie. you seek som structural realism, or mathematical truth that has removed the observer notion from the fundamental picture?

Am I wrong in my impression? I'm curious to find elaboration as to what you mean by foundations of measurement hteory without allusion to measurement. It's sounds funny to me.

About bounded operators, it seems natural to me that if one takes the measurement and representation serious, then given any observer, the assumption that no observer can hold and store infinite information seems to me to equally suggest that although this bound is relative to the observer, given a definite observer, there is some bound?

Any other notiong of observable, one may question as to wether it's useful for physics?

My objection to QM as it stands, is that it consider the measurement as some silly projection. Ie. it considers only the communication channel. It ignores at the founding level the possibility of evaporating transmitters and saturated receivers, and how this must deform the effective communication. Ie. I think we need to take into acount not only the commmunication but also information processing and representation and coding at the observer. Ie. the "internal structure of the observer".

(Which in the case of QG, translates to matter, which is an open issue)

But to try to remove the observer and measurement, seems to me to be a step in the wrong direction?

/Fredrik

 

[A. Neumaier]

I personally don't understand why a foundations of mesurement theory, should avoid the measurement.

I am talking about the foundations of quantum mechanics, not that of measurement theory.

A foundation of measurement theory must assume quantum mechnaics and show how the complex quantum systems called measurement devices effect what is called a measurement of another quantum system.

But this is a difficult question of statistical mechanics, not a matter of philosophy.

Any other notion of observable, one may question as to wether it's useful for physics?

There is an established, very useful notion of observable as ''densely defined, self-adjoint operator on a Hilbert space''.The widespred use of it is proof of its usefulness, although it is not fully adequate to describe what experimentalist call an observable.

 

[A. Neumaier]

the fact that for time-independent Hamiltonians, the time dependence of the solution of the SE can be factored out, it is nothing but a phase factor and can totally be removed from a discussion on whether the solution to the SE bears physical relevance or not ?

This is not a fact but a severe misunderstanding.

Try to do it for the state mentioned in the post you answered.

 

[Fra]

I am talking about the foundations of quantum mechanics, not that of measurement theory.

A foundation of measurement theory must assume quantum mechnaics and show how the complex quantum systems called measurement devices effect what is called a measurement of another quantum system.

But this is a difficult question of statistical mechanics, not a matter of philosophy.

Ah ok. Perhaps I missed the context; I just jumped in.

So you are talking about finding a rigorous mathematical foundation of QM as it stands, right? Not talking about how to solve open issues that relates to unification and QG? (which may need REVISION of QM)

If so, I see your stance.

Yes that's a worthy goal on it's own. But it sounds like a pure mathematical project.

I know some mathematicians that work on fidning rigorous formalizations of the often "sloppy" mathematics that physicists has come up with, but without changing the physics. From a mathematicians views is often that physicists derivations is more like an informal argument than formal proof.

/Fredrik

 

[A. Neumaier]

So you are talking about finding a rigorous foundation of QM as it stands, right? Not talking about how to solve open issues that relates to unification and QG? (which may need REVISION of QM)

Yes. QM as far as it is based on a Hilbert space view.

 

[Fra]

Yes. QM as far as it is based on a Hilbert space view.

Ok, then I see. My confusion of context.

/Fredrik

 

[dextercioby]

This is not a fact but a severe misunderstanding.

Try to do it for the state mentioned in the post you answered.

Ok, then I'll present with my reasoning and kindly ask you to spot my flaws of any kind:

<Assume the the system is described by a function \Psi (x,t) on a TVS call it \mathcal{V}.

We would like to solve the following PDE to find the general solution \Psi (x,t) subect to the condition

\int \limits_{\mathbb{R}}\Psi^{*}(x,t) \Psi (x,t) \, dx \, =1 \, , \, \forall t\in\mathbb{R} (1).

The PDE sounds like

\frac{\partial\Psi (x,t)}{\partial t} = \frac{1}{i\hbar} \frac{(-i\hbar)^2}{2m} \frac{\partial^2 \Psi (x,t)}{\partial x^2}.

subject to an initial condition \Psi (x,t=0) = \psi (x).

The key assumption we're making is that \Psi (x,t) = \psi (x) V(t) which when plugged in the PDE written above gives

\psi (x) \frac{d V(t)}{dt} = \frac{i\hbar}{2m} V(t) \frac{d^2 \psi (x)}{dx^2}

A manipulation of the above equation leads to

\frac{2m}{i\hbar} \frac{1}{V(t)} \frac{d V(t)}{dt} = \frac{1}{\psi (x)} \frac{d^2 \psi (x)}{dx^2} = K

K is a real constant with dimension of 1/area.

So

V(t) = \exp\left(\frac{i\hbar K}{2m}t\right)

, a phase factor (I assume the condition V(t=0)=1, so the integration constant is 0) and the other ODE is

\frac{d^2 \psi (x)}{dx^2} = K \psi (x)

which, compared to the spectral equation for the free Hamiltonian

-\frac{\hbar^2}{2m} \frac{d^2 \psi (x)}{dx^2} = E \psi (x)

gives

K = -\frac{2mE}{\hbar^2} \leq 0

for E\geq 0

So one finds complex exponential solutions for the x variable as well which are not normalizable as per (1).

What am I doing wrong?

 

[A. Neumaier]

The key assumption we're making is that \Psi (x,t) = \psi (x) T(t)

V(t) = \exp\left(\frac{i\hbar}{2m}t\right), a phase factor. [/tex]

-\frac{\hbar^2}{2m} \frac{d^2 \psi (x)}{dx^2} = E \psi (t)

for E\geq 0

What am I doing wrong?

You have been sloppy in each of the four lines quoted above.

In the first line, you should have V(t) in place of T(t), in view of what follows.

In the second line, you lost both an integration constant and the constant K

In the third line, you write psi(t) in place of psi(x).

In the fourth line, you introduce an extra assumption without saying so.

After correcting these items (it would have paid off to go through all arguments after typing them, and have them checked for correctness), you get a correct derivation of _some_ solutions, namely all those that satisfy the assumption introduced in the first line above.

But since you didn't _prove_ this assumptions, your derivation tells nothing at all about solutions that do not satisfy it. There are lots of them, since by the superposition principle, any linear combination of the solutions you constructed is again a solution. You can easily convince yourself that these will usually not have the separable form you assumed.

 

[dextercioby]

You have been sloppy in each of the four lines quoted above.

I apologize for being sloppy. It's been corrected now.

In the fourth line, you introduce an extra assumption without saying so.[...] you get a correct derivation of _some_ solutions, namely all those that satisfy the assumption introduced in the first line above.

OK.

But since you didn't _prove_ this assumptions, your derivation tells nothing at all about solutions that do not satisfy it.

Is separability the assumption you need me to prove ? Or the E>0 one ?

There are lots of them, since by the superposition principle, any linear combination of the solutions you constructed is again a solution. You can easily convince yourself that these will usually not have the separable form you assumed.

I have obtained a basis on the space of all solutions. I claim that if separability of time dependence is assumed and E>0 (which can be proven), a basis of solutions is given by the e^(iAx)e^(iBt) , A,B in R, nonzero. From these you can form linear combinations which would again be solutions of the initial equation.

 

[A. Neumaier]

Is separability the assumption you need me to prove ? Or the E>0 one ?

You can prove neither, since they don't follow from the time-dependent Schroedinger equation. The fact that you had to assume the former but thought this could be done without losing any solution is your real mistake.

I have obtained a basis on the space of all solutions.

Well, how did you convince yourself of this fact? Please give the argument, so that I can correct you.

I claim that if separability of time dependence is assumed and E>0 (which can be proven), a basis of solutions is given by the e^(iAx)e^(iBt) , A,B in R, nonzero. From these you can form linear combinations which would again be solutions of the initial equation.

Even that is, with the arguments given so far, only a claim and not something proved.

But the space of the solutions is the space of all solutions, not only the separable ones
that your last statement claimed to classify. Thus even if this claim had been proved, you are still far away from having a proof that you know all solutions (and that none of them is normalizable - which you claimed but which the example of DarMM showed to be faulty).

 

[dextercioby]

You can prove neither, since they don't follow from the time-dependent Schroedinger equation. The fact that you had to assume the former but thought this could be done without losing any solution is your real mistake.

So you're challanging my separation of variables t and x. Why would this particular case be any different than the tons of cases in which PDE's in mathematical physics are solved
through this method ? In other words, help me understand why this 2 variable separation is problematic, and the one in the heat conduction equation for the 1d case is not.

Can you provide me with a solution of the SE for 1D case in which t and x are not separated ? (DarMM's solution is separable and accepting it would have to find other flaws in my argument than separability of variables).

Thank you

Daniel

p.s. if separability is assumed, E>0 follows.

 

[A. Neumaier]

So you're challenging my separation of variables t and x. Why would this particular case be any different than the tons of cases in which PDE's in mathematical physics are solved
through this method ? In other words, help me understand why this 2 variable separation is problematic, and the one in the heat conduction equation for the 1d case is not.

Heat conduction is not different. You only get separable solutions from the separable ansatz, and there are lots of nonseparable solutions (namely most linear combinations of separable ones).

What is missing in your argument is the discussion why you should get _all_ solutions as linear combinations of separable ones. So please try to prove this instead of just copying a textbook template without thinking yourself. If you fail, produce your attempt, so that I can see what you are having difficulties with.

Can you provide me with a solution of the SE for 1D case in which t and x are not separated ? (DarMM's solution is separable and accepting it would have to find other flaws in my argument than separability of variables).

DarMM's solution is normalizable but not separable. The time-dependent exponential depends also on p, so it can't be factored out.

if separability is assumed, E>0 follows.

How? It is impossible to correct you unless you explain the reasons why you think your claims are true.

 

[dextercioby]

Hi, Arnold, you're right. DarMM's solution is a valid solution of the SE. I used Fourier transformations (which are allowable on L^(RxR,dx)) and got the general solution, to which the function written by DarMM is only a particular case.

Now I'm bothered by the fact that my separation of variables leads me to non-normalizable solutions and how these are related to my Fourier analysis which apparently stays in the Hilbert space.

Any clue ?