Boundedness of quantum observables?

A. Neumaier

I don't like the C^*-algebraic foundations of quantum mechnaics since it assumes that every observable must be bounded and self-adjoint.

But most physical observables are not bounded.

This came up in another thread, from which I quote some context:


You are only missing implicitly understood embellishments. In full precision:

The integral of F_12(x) with a real, smooth hat function of narrow support is - by conventional standards - an observable whose support is a bounded region of space-time, but has continuous spectrum, hence is not bounded, and therefore not observable according to your definition.

Fredrik

My take on this (which may still be a bit naive, since I haven't yet reached the point where I can say that I really understand the algebraic approach), is that it depends on what exactly we mean by "observable". If this term refers to an operator that can be associated in a meaningful way with a "measurable quantity" (a concept that we don't really define), or if it refers to a generator of a one-parameter group of symmetries of the theory, then we need to allow the operators to be unbounded. The most obvious example is momentum. On the other hand, if "observable" refers to an equivalence class of measuring devices (as described on the first few pages of Araki), I see no reason why we shouldn't let a C*-algebra be the structure that represents the set of observables mathematically.

Since P is unbounded, it can't be a member of the C*-algebra of operators that correspond to observables. But is that a problem? Can't we use P to construct bounded operators that correspond to more realistic measuring devices? P corresponds to a measuring device that puts the particle in a momentum eigenstate (a concept that has issues of its own), but the bounded operators constructed from P would correspond to (for example) measuring devices that only confine the value of the momentum to a specific interval.

I admit that I haven't thought this through to the end.

One thought that occurs to me is that maybe the C*-algebra stuff is the best way to define observables when we intend to use a Hilbert space, and something else (that includes unbounded operators) is the best way to define them when we intend to use a rigged Hilbert space. But that's another thing I don't fully understand yet.

A. Neumaier

My primary requirement is that the usage should not be too different from what physicists in general call observables. Of course it entails always an idealization since nobody can measure all values in an unbounded domain of results, or even all real numbers in a bounded domain. But before the advent of axiomatic field theory, basic observables were position, momentum, angular momentum, energy, force, torque, etc., all of them unbounded. The axiomatists changed the terminology for purely technical reasons, which is unacceptable and will never find general agreement, I think.

Most of physics is phrased in terms of differential equations involving unbounded observables. Most of physics become unexpressible or clumsy to express when phrased in terms of bounded operators only. No commutation relations, no continuity equation, no field equations, no Noether theorem....


Thus we'd have two different notions of observables - the theoretical physicists one and the mathematical physicist's one. Unfortunate and unnecessary.

So you introduce a multiplicity of different momentum variables - one for each interval bounding the measuring range. Thus scales with a maximum weight of 120kg and scales with a maximum weight of 150kg would measure different observables.

Sounds strange and is against Occam's razor.

In both cases, one can represent the unbounded observables on the nuclear space, take its completion if the Hilbert space is needed, and the dual space if more singular objects are encountered.

So I think the right mathematical setting should be an inner product space on which all observables of interest are defined (this common domain exists in all applications I am aware of), and its closure in various topologies depending on what one wants to do on the technical level.

DarMM

To be honest A. Neumaier, I don't really have a disagreement. My initial post in the other forum was perhaps a bit trite. The Algebraic approach takes the bounded self-adjoint observables to correspond to measuring equipment, not observables. Without playing silly language games, an observable is precisely what you stated, a self-adjoint operator.

So momentum is an observable and is represented by an unbounded self-adjoint operator. Machines which measure momentum are represented by bounded self-adjoint operators. This is a fundamental point which I carelessly glossed over. In Algebraic QFT, the C*-algebra is the algebra of observables in the sense of what can be observed by actual equipment, not in terms of what are physical quantities.

The interesting thing is that the theory can be developed and that certain points become clearer when you formulate the theory this way. For example it clears up some points in QFT in curved spacetime and other areas, because it describes pure and mixed states in a more unified way.

Of course there are many cases where it is more cumbersome, e.g. analysis of the stress-energy tensor, field equations, e.t.c.

(By the way, the canonical commutation relations fail to make sense in four-dimensions. Something which is related to wave-function renormalisation.)

DarMM

I should also say that there is still disagreement what exactly the algebra of bounded observables in a region represents. All that is certain is that all the information in a QFT is contained in them. That is you can reconstruct the QFT from these objects. (Again that statement has caveats, since AQFT is more general than field theory and contains relativistic quantum systems which are not field theories.)

A. Neumaier

OK, this sounds better. Though it is stiill very idealized actual equipment, since you still need to be able to measure arbitrarily large momentum, fields in the neighborhood of arbitrary points (e.g., the center of the sun)., and all sorts of bounded self-adjoint operators for which nobody would be able to define even a thought experiment for measuring it.

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.

But it seems to me that the relevant unbounded observables always have a common domain on which they are true linear self-mappings, so that one could work instead with the algebra of linear self-mappings of this domain. Resolvents and exponentials would then live in closures of dense subalgebras of this algebra under appropriate topologies. Do you know of any work in that direction?

Pure and mixed states were already unified in von Neumann's book. How can things be even more unified?


Can you substantiate this? We don't know of any rigorous interactive 4D quantum field theory. So how can the CCR be known to fail? At least, the asymptotic fields are free if there is a mass gap, and they must satisfy canonical commutation relations.

Fredrik

I can't imagine that this is possible, at least not with a theory that resembles QM. To define QM properly, we need to choose a "set of things" in the real world, and a mathematical structure that can represent it. The starting point of the algebraic approach is that the "things in the real world" are equivalence classes of measuring devices, and that a C*-algebra is what represents the set of such equivalence classes mathematically. What alternatives do we have? The Hilbert space approach? The starting point of that is that equivalence classes of idealized preparation procedures are represented by the 1-dimensional subspaces of a complex separable Hilbert space.

This only shifts the weight from "measurement" to "preparation", so the allusion to measurement is just as strong in the Hilbert space approach, but perhaps better hidden. And you know what, the self-adjoint operators on this Hilbert space are still going to correspond to equivalence classes of measuring devices.

A. Neumaier

To define quantum mechanics (or any other physical theory) properly, we only need to define the calculus and then say how to relate the quantities that can be calculated from quantum mechanical models to the stuff experimental physicists talk about.

The idealization that goes into the interpretation used in your description is only a didactical trick to make definitions a bit easier to swallow. (How do you justify the C^*-algebra axiom ||A^*a||=||a||^2 from measurement??)

In experimental physics, measurement is a very complex thing - far more complex than your ''definition'' suggests. To measure the distance between two galaxies, the mass of the top quark, or the Lamb shift - just to mention three basic examples - can never be captured by the idealistic measurement concept in your definition.

In each case, one assembles a lot of auxiliary information and ultimately calculates the measurement result from a best fit of a model to the data. Clearly the theory must already be in place in order to do that. (We don't even know what a top quark should be whose mass we are measuring unless we have a theory that tell us this.)

And the Lamb shift (one of the most famous real observables in the history of quantum mechanics) is not even an observable in your sense!

We proceed as in the modern account of the oldest of the physical sciences: Euclidean geometry, where (on laboratory scales) there is consensus about how theory and reality correspond:

We develop a theory that simply gives a precise formal meaning to the concepts physicists talk about. This is pure math, in case of geometry consisting of textbook linear algebra and analytic geometry. The identification with real life is done _after_ having the theory (though the theory and the nomenclature was _developed_ with the goal to enable this identification in a way consistent with tradition):

For geometry, by declaring anything in real life resembling an ideal point, line, plane, circle, etc., to be a point, line, plane, circle, etc., if and only if it can be assigned in an approximate way (determined by the heuristics of traditional measurement protocols, whatever that is) the properties that the ideal point, line, plane, circle, etc., has, consistent to the assumed accuracy with the deductions from the theory. If the match is not good enough, we can explore whether an improvement can be obtained by modifying measurement protocols (devising more accurate instruments or more elaborate error-reducing calculation schemes, etc.) or by modifying the theory (to a non-Euclidean geometry, say, which uses the same concepts but assumes slightly different properties relating them.

For quantum mechanics, by declaring anything in real life resembling an ideal photon, electron, atom, molecule, crystal, ideal gas, etc., to be a photon, electron, atom, molecule, crystal, ideal gas, etc., if and only if it can be assigned in an approximate way (determined by the heuristics of traditional measurement protocols, whatever that is) the properties that the ideal photon, electron, atom, molecule, crystal, ideal gas, etc., has, consistent to the assumed accuracy with the deductions from the theory. If the match is not good enough, we can explore whether an improvement can be obtained by modifying measurement protocols (devising more accurate instruments or more elaborate error-reducing calculation schemes, etc.) or by modifying the theory (to a hyper quantum mechanics, say, which uses the same concepts but assumes slightly different properties relating them.

This identification process is fairly independent of the way measurements are done, as long as they are capable to produce the required accuracy for the matching.

Then, having established informally that the theory is an appropriate model for the physical aspects of reality, one can study the measurement problem rigorously on this basis:One declares that a real instrument (in the sense of a complete experimental arrangement including the numerical postprocessing of raw results that gives the final result) performs a real measurement of an ideal quantity if modeling the real instrument as a macroscopic quantum system (with the properties assigned to it by statistical mechanics/thermodynamics) predicts raw measurements such that, in the model, the numerical postprocessing of raw results that gives the final result is in sufficient agreement with the value of the ideal quantity in the model. Thus measurement analysis is now a scientific activity like any other rather than a philosophical prerequisite for setting up a consistently interpreted quantum mechanics.

DarMM

This is sensible.

The existence of such a domain basically follows from the Wightman axioms, I'll call it [tex]D[/tex]. I have never seen a study of the mappings of this domain itself. Getting a workable specification of this domain is an open problem in axiomatic quantum field theory. A subset of this domain and its mappings are well studied. This domain are the states generated by polynomials of the fields acting on the vacuum and a study of its mappings is found in papers by Borchers, Ruelle, Dixmier and Haag. It is called [tex]D_{0}[/tex].

It is a long standing conjecture that:
[tex]D = D_{0}[/tex]
However this is not proven. At one time it was a major goal of axiomatic quantum field theory.

By placing them in their appropriate context mathematically and unifying how they appear in physics. This would take a long time to explain, but I'm more than happy to do so if you wish. It goes well beyond what von Neumann did, but of course grew out of it.


Sorry, it fails for fields that aren't free. Even though no 4D interacting quantum field theory has been constructed, we know enough to know the commutation relations break down. Basically any estimate on the fields requires that (if they exist) they are singular enough as distributions that no time zero field exists and these are needed for the canonical commutation relations. For example if the quantum field is [tex]\phi(t,\b{x})[/tex], then the time zero field is:
[tex]\phi(0,\b{x}) = \int{\delta\left( t \right)\phi(t,\b{x})} dt[/tex]
You can see that this requires quite a non-singular distribution. The field has to be an operator not only after smearing with a test-function, but even after a smearing with a temporal delta function.
Since the fields grow more singular with increasing dimension and are more singular with interactions (in path integral language the path measure is supported on more singular fields), estimates on 4D fields imply that the time zero fields do not exist.
However the field [tex]\phi_{r}[/tex] given by [tex]\phi = Z\phi_{r}[/tex], where [tex]Z[/tex] is an infinite constant in vague terms (more precise terms available if you want them) does have a time-zero field. It is of course the wave-function renormalisation.

A. Neumaier

Yes, please, but in a new thread so that I don't lose the oversight....

But isn't the renormalized field the physical field, and hence the one that should appear in the Wightman formulation?

DarMM

My explanation was very poor, let me try again. If in constructing the field theory one has to perform infinite wavefunction renormalization, then the resulting field will have no well defined restriction to a single time-slice which will cause the canonical commutation relations to fail.

For example [tex]\phi^{4}_{4}[/tex], the theory is poorly defined when written on Fock space with the Hamiltonian:
[tex]\int{d^{3}x \frac{\pi_{0}^{2}}{Z} - Z \nabla \phi_{0} \cdot \nabla \phi_{0} + m^{2}\phi_{0}^{2}+ \lambda\phi_{0}^{4}}[/tex]
Where [tex]\phi_{0}[/tex] is the free field.

To make it well-defined one puts in a cutoff [tex]\kappa[/tex] and gives [tex]Z,m[/tex] and [tex]\lambda[/tex] dependence on the cutoff. (I know you know all this, I'm just setting things up)

If done correctly one will find that the Hamiltonian and its ground state [tex]\Omega[/tex] no longer have divergences. However the Hamiltonian will no longer converge to an operator in Fock space and [tex]\Omega_{\kappa}[/tex] will no longer converge to an element of the Fock space.

To resolve this problem, one moves to the algebra of operators and treats [tex]\Omega_{\kappa}[/tex] as a state on that operator algebra. [tex]\Omega_{\kappa}[/tex] will then have a limit as a state on the algebra, but the Hilbert space you can construct from [tex]\Omega_{\infty}[/tex] is disjoint from Fock space.

In the new Hilbert space the true physical field [tex]\phi_{p}[/tex], will have a well-defined Hamiltonian given by:
[tex]\int{d^{3}x : \pi_{p}^{2} - \nabla \phi_{p} \cdot \nabla \phi_{p} + m^{2}\phi_{p}^{2}+ \lambda\phi_{p}^{4} :}[/tex]

([tex]::[/tex] indicates Wick ordering with respect to the vacuum of this Hilbert space)

This Hamiltonian acts on the correct Hilbert space with no divergences. However this field will possess no time-slice localisation and cannot obey the canonical commutation relations.

The axiomatic field theory way of viewing this would be to say that the Hamiltonian defined as a function of the physical field on the correct Hilbert space, is finite and well defined. However we only know how to work with Fock space. So to obtain/construct the correct field and Hilbert space we must use a cutoff approximation on Fock space as a starting point and use renormalization to take the correct limit out of Fock space. The type of renormalizations required tell you something about the real theory. In the case of wavefunction renormalization, the fact that the Fock space approximation needs a cutoff diverging term in the kinetic piece indicates that the real Hilbert space is one which does not support the canonical commutation relations.

A. Neumaier

This far things are clear to me. But isn't your last sentence rigorously proved only in space-time dimension <4?

Your statement reminds me of the nonrigorous discussion of Dyson's intermediate representation in
TS Walhout,
Similarity renormalization, Hamiltonian flow equations, and Dyson's intermediate representation,
Phys. Rev. D 59, 065009 (1999)
http://arxiv.org/pdf/hep-th/9806097

But why this should follow rigorously was my question. It seems not to follow in dimension <4. But How can you give a rigorous argument in 4D when H hasn't even been constructed?



Well, you claimed that several times, but now you are more cautious and only say: it indicates. So what is conjecture and what is proved? (where?)

DarMM

I'm not sure I understand, so tell me if this doesn't answer your question. [tex]H[/tex] doesn't need to be completely constructed in order to answer this question. You will see the failure of the canonical commutation relations explicitly when you try to construct the theory. It is directly the origin of certain difficulties.

Maybe my English wasn't clear. I'm not using indicate to say "it is suggested". Replace "indicate" with "tells you" and my meaning should be clearer. Basically certain renormalizations or what order they appear at tell you things about the real Hilbert space. Sorry about that.

For references Hepp is probably the best read if you can get it:
Hepp K 1969 Theorie de la Renormalisation (Berlin: Springer)

In their first monograph Glimm and Jaffe make some comments:
Glimm J and Jaffe A 1972 Boson quantum field models London 1971, Mathematics of Contemporary Physics (London) pp 77–143

It's explicitly dealt with in:
Zavialov O I and Sushko V N 1973 Statistical Physics and Quantum Field Theory ed N N Bogoliubov (Moscow: Nauka)

You might find the papers of O Yu Shvedov interesting, he treats this stuff in the case of infinite discrete degrees of freedom. Not quite QFT, but half between QFT and QM. Makes some things clearer with out a lot of the technicalities.

Also try any papers where they try to construct a 4D QFT, for example Schrader's.
It was also first "discovered" very early on by the axiomatic field theory community unfortunately they never brought out a paper on it and the closest you'll get in early material is a brief reference in "PCT, Spin and Statistics, and all that" on page 101.

I should say that the canonical commutation relations only fail in the sense of the formula they are usually expressed in.

For example it is still true that:
[tex] \left[\phi(x,t), \pi(y,s) \right] = D(x-y, t-s)[/tex]
For some function [tex]D(x-y, t-s)[/tex]. It's just that [tex]D(x-y, t-s)[/tex] has singularities too strong when [tex] t \rightarrow s[/tex] that no Wightman field could satisfy the equation in that limit.

A. Neumaier

Let's postpone this until you climbed the second and third rung of your Phi^4_d ladder - then I'll ask again whatever the references you pointed out leave open.

Which one?

So one just has worse than tempered distributions?

yoda jedi

there are various ongoing initiatives.



.

A. Neumaier

Maybe. But to make your remark useful, please point to some references, preferably online ones.

Hurkyl

Have y'all considered that while an operator like X isn't bounded, other operators like arctan(X) are?

I really don't think you lose anything by laying the foundations via C*-algebras. Once you have a C*-algebra, you can use calculus (or other means) to construct additional algebras, if you so desire.