Postulates for the formal core of quantum mechanics

[A. Neumaier]

For a completely specified set of postulates appropriate for modern quantum mechanics (fully compatible but in detail differing from post #128) see my Postulates for the formal core of quantum mechanics from my theoretical physics FAQ. If you want to discuss these, please do so in a separate thread.)

The postulates 1-6 in your FAQ are indeed standard QT, and I subscribe to them. I've only one question: Why is it sufficient to define the stat. op. as hermitean? I always thought it must be (essentially) self-adjoint.

I also don't think that you have to change anything concerning relativistic QFT. Of course, observables must be defined by (gauge-independent) observables in terms of the quantum fields, and indeed, a particle interpretation is valid only in a very specific limited sense of asymptotic free Fock states.

I'll answer this in this thread, later today.

 

[A. Neumaier]

The postulates 1-6 in your FAQ are indeed standard QT, and I subscribe to them. I've only one question: Why is it sufficient to define the stat. op. as hermitean? I always thought it must be (essentially) self-adjoint.

In A2, I assumed that rho(t) is a Hermitian, positive semidefinite, linear trace class operator. But every Hermitian trace-class operator is self-adjoint. So this doesn't need to be assumed.

I also don't think that you have to change anything concerning relativistic QFT. Of course, observables must be defined by (gauge-independent) observables in terms of the quantum fields, and indeed, a particle interpretation is valid only in a very specific limited sense of asymptotic free Fock states.

As I mentioned in the prelude to my postulates in the FAQ, I describe nonrelativistic quantum theory in the Schroedinger picture. In the relativistic case, one needs to add axioms about the Poincare group (which changes A1 and needs assumptions about the representation) and causal commutation relations, and change to the Heisenberg picture (which drastically changes A2 and A5) if one wants to remain covariant. Moreover, MI must be modified to hold in every Lorentz frame. Finally, the current setting does not account for superselection rules, which are present in QED and QCD. So the whole thing would look quite different.

I prefer to base the interpretation of QFT solely on the Wightman axioms. These are valid on the level of rigor of theoretical physics for every relativistic QFT, in particular for the gauge invariant local observables of QED and QCD; thus these axioms are physically plausible. (The constructive question is wide open from a mathematical point of view. In 4 dimensions, no interacting QFT satisfying it has been constructed with mathematical rigor, but there also exists no no-go theorem for them.)
QFT in the Wightman form accounts for all states with total charge zero, and hence probably for our universe. Charged states (relevant for S-matrix computations) need a different representation of the same observable algebra, and gauge fields need an extension of the framework in a form that is not yet decided in the literature.

For QFT based on field expectations and correlation functions only, I find the postulates discussed in my FAQ inadequate, and only the less demanding interpretation indicated in this post is appropriate.

 

[vanhees71]

As I mentioned in the prelude to my postulates in the FAQ, I describe nonrelativistic quantum theory in the Schroedinger picture. In the relativistic case, one needs to add axioms about the Poincare group (which changes A1 and needs assumptions about the representation) and causal commutation relations, and change to the Heisenberg picture (which drastically changes A2 and A5) if one wants to remain covariant. Moreover, MI must be modified to hold in every Lorentz frame. Finally, the current setting does not account for superselection rules, which are present in QED and QCD. So the whole thing would look quite different.

Ok, maybe for any attempt to be mathematically more rigorous than the standard physicists' treatment there may be a difference, but formally A1 fully holds in QFT. Of course to construct the concrete application of this axiomatic formalism you have to assign the operators you talk about in your axioms a concrete meaning. As well as you need the Poincare group to find the observable algebra in the relativistic case you need the Galilei group in the non-relativistic case. Often one uses a shortcut and starts just with the Heisenberg algebra for position and momentum and then "hand-waves" the way to the standard Hamiltonians we all love to present in the QM 1 lecture. That's of course legitimate to get to the physics as fast as possible without confusing the students with too much formalism, but a solid foundation starts from the symmetry group of spacetime in both the non-relativistic and the (special-) relativistic case.

As I stressed before, I do not understand which difference the choice of the picture makes at all (modulo mathematical trouble a la Haag's theorem). Of course, if you like to stay manifestly covariant and formulate relativistic QFT as a local realization of the Poicare group, the Heisenberg picture is most convenient, but it's not principally different from the Schrödinger picture. A nice (physicists') treatment of different formalisms for non-relatistic QFT (usual operator formalism, Schrödinger functional, and path integral) is given in

Hatfield, QFT of Point Particles and Strings

What has to be changed (even drastically) to A2 and A5? The state is represented by $\hat{\rho}$ and also expectation values are calculated from it the same in both relativistic QFT and non-relativistic QM.

MI is fulfilled in standard QFT due to the microcausality condition (local observable-operators commute at space-like separation of their arguments).

Could you comment more on the superselection-rule thing? Isn't this implied by the symmetry principles you base the theory on (e.g., in non-relativistic QT there's a superselection rule forbidding superpositions of states with different mass, while such a rule does not follow in relativistic QFT)? In both cases the angular-momentum superselection rule follows from the representation of the rotation group, which is contained in both the Galileo and the Poincare groups, etc. So what else do I have to postulate concerning the selection rules.

I prefer to base the interpretation of QFT solely on the Wightman axioms. These are valid on the level of rigor of theoretical physics for every relativistic QFT, in particular for the gauge invariant local observables of QED and QCD; thus these axioms are physically plausible. (The constructive question is wide open from a mathematical point of view. In 4 dimensions, no interacting QFT satisfying it has been constructed with mathematical rigor, but there also exists no no-go theorem for them.)
QFT in the Wightman form accounts for all states with total charge zero, and hence probably for our universe. Charged states (relevant for S-matrix computations) need a different representation of the same observable algebra, and gauge fields need an extension of the framework in a form that is not yet decided in the literature.

I'm not very familiar with axiomatic QFT. I guess one has to live with the quite unsatisfactory state of QFT as it is, except somebody finds a solution of these problems. I think the pragmatic way is to just take QFT as defined by the calculations done to compare to experiment. It's amazing, how successful a concept can be without being mathematically rigorous.

For QFT based on field expectations and correlation functions only, I find the postulates discussed in my FAQ inadequate, and only the less demanding interpretation indicated in this post is appropriate.

I think that's also valid for non-relativistic QT. That's what QT has to predict about real-world observations and that's indeed how it is used in all applications to real-world experiments.

 

[A. Neumaier]

you need the Galilei group in the non-relativistic case.

Not in quantum information theory and in may simple examples of quantum systems. Thus it is inappropriate to enforce it in the axioms, which I therefore refrain from.

the Heisenberg picture is most convenient, but it's not principally different from the Schrödinger picture.

It is principally different, as one in the Schroedinger picture has no natural way to talk about time correlation functions, which are essential in statistical mechanics.

What has to be changed (even drastically) to A2 and A5? The state is represented by $\hat{\rho}$ and also expectation values are calculated from it the same in both relativistic QFT and non-relativistic QM.

No change is needed in the Schroedinger picture. But as mentioned again and again it only gives a single-time formalism.

Could you comment more on the superselection-rule thing? Isn't this implied by the symmetry principles you base the theory on (e.g., in non-relativistic QT there's a superselection rule forbidding superpositions of states with different mass, while such a rule does not follow in relativistic QFT)?

The situation is complicated and not very well understood. A large part of the infrared problems are due to these issues. In QED they are somewhat under control but not in QCD. This is a big topic and I don't have time to start a discussion on it. (I should have finished all these discussions on PF, including writing the summarizing insight articles, a week ago. Our summer term starts on Tuesday and I still haven't finished my preparations...)

In both cases the angular-momentum superselection rule follows from the representation of the rotation group, which is contained in both the Galileo and the Poincare groups, etc. So what else do I have to postulate concerning the selection rules.

Everything should follows in QFT from the representation theory of certain infinite-dimensional Lie groups, but this works well only in 1+1 dimension. In 1+3 dimensions many known techniques from representation theory fail because in the known approaches things diverge too violently.

I'm not very familiar with axiomatic QFT. I guess one has to live with the quite unsatisfactory state of QFT as it is, except somebody finds a solution of these problems. I think the pragmatic way is to just take QFT as defined by the calculations done to compare to experiment. It's amazing, how successful a concept can be without being mathematically rigorous.

The success points to that there must be a rigorous basis behind it. It is like Dirac's use of distributions, where it took 18 years before the mathematical justification was found. QFT is considerably harder than distributions, so it takes longer to settle the issues. Until then the right way is to begin with the axioms and complement them by less rigorous arguments. This is how the LSZ formulas indispensable in moder QFT were discovered, and this is how I think QFT should be always (but unfortunately never was) presented.

Statistical mechanics can start with Hilbert spaces, unitary dynamics for operators, density operators for Heisenberg states, the definition of

(EX)$~~~~~\langle A\rangle:=\mbox{tr}~\rho A$

as mathematical framework, and the following rule for interpretation, call it (SM) for definiteness:

the practice of statistical mechanics says:
''Upon measuring a Hermitian operator $A$, the measured result will be approximately $\bar A=\langle A\rangle$, with an uncertainty at least of the order of $\sigma_A=\sqrt{\langle (A-\bar A)^2\rangle}$. If the measurement can be sufficiently often repeated (on an object with the same or sufficiently similar state) then $\sigma_A$ will be a lower bound on the standard deviation of the measurement results.''
Everything deduced in statistical mechanics about macroscopic properties follows [... and one] has a completely self-consistent setting in which the classical situation is simply the limit of a huge number of particles.

I think that's also valid for non-relativistic QT. That's what QT has to predict about real-world observations and that's indeed how it is used in all applications to real-world experiments.

Yes, and it works in the Heisenberg picture and in the Schroedinger picture, without change. That's why I want to make a fresh and clean start with just these properties. They are nice, simple, and nontechnical, as it should be for postulates. From this point of view, even the postulates in my FAQ are out of date.

 

[vanhees71]

I always thought that your axioms are the general framework for any quantum model. Of course, if you look at systems like spins only, you have a finite-dimensional Hilbert space and need only the representations of SO(3) (restricted to one of the finite-dimensional irreps.).

Your ansatz of reformulating the standard intro of QFT sounds very interesting, but of course, one has not enough time for such fundamental questions :-((. Our semester dates are a bit more human in Germany :-)). Our summer semester starts on April 13.

Just one more thought: In QED the asymptotic free states are something like charged particles and photons, but not the naive "plane-wave" ones we use in our usual handwaving presentation but something I'd call "dressed plane waves", where dressed means a charged particle together with a "cloud" of soft photons, i.e., a kind of coherent state and coherent states for the photons. In the usual hand-waving perturbative formalism one does the same thing in a pretty hidden way by using the usual soft-photon ("ladder") resummation techniques (see Weinberg QT of Fields, vol I). There's a nice paper by Kulish and Faddeev and a four-paper series by Kibble on the alternative method defining the correct asymptotic states exactly.

For QCD it's of course very much more complicated. There the asymptotic states are not quarks and gluons but hadrons (most probably including states not yet observed like glue balls), but these are outside of the realm of perturbative methods.

 

[A. Neumaier]

In QED the asymptotic free states are something like charged particles and photons, but not the naive "plane-wave" ones we use in our usual handwaving presentation but something I'd call "dressed plane waves", where dressed means a charged particle together with a "cloud" of soft photons, i.e., a kind of coherent state and coherent states for the photons.

Yes, and the modes of the coherent states fall into many inequivalent superselection sectors; this can be seen by trying to constructthe Bogoliubov transformations transforming them into each other - they turn out to be ill-defined (divergent if one uses a cutoff and tries to remove them. Thus a Hilbert space based approach needs many Hilbert spaces, one for each superselection sector. A unified description needs C^*-algebra techniques. But this gets soon very technical.

Maybe during the summer I have again some time for discussion. Until then I'll be only sporadically here on PF.
Thanks to all who contributed to the sometimes heated discussions on quantum weirdness and their foundations.
Good bye!

 

[Mentz114]

Maybe during the summer I have again some time for discussion. Until then I'll be only sporadically here on PF.
Thanks to all who contributed to the sometimes heated discussions on quantum weirdness and their foundations.
Good bye!

Au Revoir !

 

[A. Neumaier]

[... postulates (EX) and (SM) ...]

Yes, and it works in the Heisenberg picture and in the Schroedinger picture, without change. That's why I want to make a fresh and clean start with just these properties. They are nice, simple, and nontechnical, as it should be for postulates. From this point of view, even the postulates in my FAQ are out of date.

I recommend alternative foundations based upon the postulates (EX) and (SM) that I had formulated. They apply to measuring both macroscopic variables (as expectations with error bars) and pure eigenstates of an operator $A$ with eigenvalue $\alpha$ (where $\bar A=\alpha$ and $\sigma_A=0$, capture far better the quantum mechnaical practice, and are much easier to state than Born's rule, especially if one compares it with the complicated form of Born's rule needed in the applications. Born's rule is derivable from these postulates in the special cases where it fully applies. See Section 10.5 of http://arxiv.org/abs/0810.1019. .

Your ansatz of reformulating the standard intro of QFT sounds very interesting, but of course, one has not enough time for such fundamental questions :-((.

As you can see from my online book, distributed over the years I had at least some time for that. My task this year is to bring the less well developed parts of this book to publication quality, so that it can be published next year (by de Gruyter).

Comments (in new threads, or by email) by anyone on the arXiv version of the book are very welcome, as I can use them in preparing the final version.

 

[A. Neumaier]

I always thought that your axioms are the general framework for any quantum model.

They are a framework for modeling both small quantum systems where only their relevant degrees of freedom are modeled and huge quantum systems where it would be absurd to assume the existence of a material outside observer. Thus they transcend the traditional foundations of quantum mechanics in a way that is extremely natural, that matches both microscopic and macroscopic reality without having to split it into a quantum and a classical part, and that is free of dubious philosophical ingredients such as knowledge or observers at times before the human civilization existed.

See this post for how the single particle case arises as an extension of the statistical interpretation to the limit $N=1$, leading to my thermal interpretation of quantum mechanics.