# A glance beyond the quantum model

1. Feb 2, 2010

### DrChinese

I wanted to follow up on a couple of specific points that were raised in another thread, and felt it would be better to split the discussion off here. The references for the discussion are:

A glance beyond the quantum model, Navascues and Wunderlich (2009)

"One of the most important problems in Physics is how to reconcile Quantum Mechanics with General Relativity. Some authors have suggested that this may be realized at the expense of having to drop the quantum formalism in favor of a more general theory. However, as the experiments we can perform nowadays are far away from the range of energies where we may expect to observe non-quantum effects, it is difficult to theorize at this respect. Here we propose a fundamental axiom that we believe any reasonable post-quantum theory should satisfy, namely, that such a theory should recover classical physics in the macroscopic limit. We use this principle, together with the impossibility of instantaneous communication, to characterize the set of correlations that can arise between two distant observers. Although several quantum limits are recovered, our results suggest that quantum mechanics could be falsified by a Bell-type experiment if both observers have a sufficient number of detectors. "

...And a recent comment on the above (by a PF member no less :)

Comment on "A glance beyond the quantum model", Peter Morgan (2010)

"The aim of "A glance beyond the quantum model" [arXiv:0907.0372] to modernize the Correspondence Principle is compromised by an assumption that a classical model must start with the idea of particles, whereas in empirical terms particles are secondary to events. The discussion also proposes, contradictorily, that observers who wish to model the macroscopic world classically should do so in terms of classical fields, whereas, if we are to use fields, it would more appropriate to adopt the mathematics of random fields. Finally, the formalism used for discussion of Bell inequalities introduces two assumptions that are not necessary for a random field model, locality of initial conditions and non-contextuality, even though these assumptions are, in contrast, very natural for a classical particle model. Whether we discuss physics in terms of particles or in terms of events and (random) fields leads to differences that a glance would be well to notice. "

----------

Of interest - and there has been recent discussion about several of these points - are the following:

a) Can you speak of particles without discussing the associated fields?

b) Are the fields themselves discrete or continuous?

c) It the correspondence between the macroscopic world and the microscopic world fundamental? Can we recover certain classical concepts - such as "no-signaling principle" or the introduced idea of "macroscopic locality" when a large number of particles are involved and our measurement devices fail to resolve discrete particles?

d) Are their low-energy Bell-type experiments that can set limits on the unification of quantum theory and gravity?

e) Anything else you might think of from the above...

2. Feb 2, 2010

### Peter Morgan

a) Not easily, but one can talk about measurement results (which one can say are properties of the measured systems, even though the systems themselves are not seen) in a finite-dimensional Hilbert space context, without introducing the Schrodinger equation or a quantum field. Indeed, quantum information lives very happily in this regime and often is thought to be fundamental.

b) One can work with either discrete or continuous mathematics. QFT on a lattice has certainly taught us stuff. Universality makes it difficult to make a categorical statement here, because discrete structure at the Planck scale would presumably be washed out at the scales at which we have any present hope of making measurements.

c) I think I take the Correspondence Principle to be a methodological requirement that comes from social issues in science. If one wants a new theory to get a running start, it helps to be able to point out how it is the same as and how it is different from the theories we currently take to be empirically effective. If one can show that the new theory can legitimately adopt a lot of the empirical effectiveness of an existing theory, it leaves less to do to establish how the new theory is better. The Correspondence Principle was used with stunning effectiveness by the founders of the new quantum theory in the late 20s to constrain quantum theory, so it seems worthwhile to attempt the same sort of approach now.

From a Correspondence Principle point of view, given that our current best theories are QFT and GR, it makes sense to stick with a field approach of some sort in attempts to construct new theories. The choice of an effective mathematical structure is important. Part of my enthusiasm for random fields is that they are a powerful generalization of classical differentiable fields that introduces the concept of probability in a mathematically correct way (whereas differentiable fields don't sit well with the measure theory), and which can be presented in a way that is very closely parallel to quantum fields. Retrospectively, a random field formalism might make a Correspondence Principle approach more possible.

However, this does not, to my more-or-less empiricist approach, entail that the world is continuous, only that it can be useful to use continuum mathematics, always supposing that we can get finite answers out, somehow.

d) Dunno. When I critiqued Navascués' and Wunderlich's assumptions I was careful not to get into the actual purpose of their paper because I know not much about QG.

e) Even less to say. Except that I believe the published version of the Navascués and Wunderlich paper is free to access on Proc. Roy. Soc. A, at http://rspa.royalsocietypublishing.org/content/466/2115/881" [Broken], and is preferable, from the point of view of my Comment, because they introduced classical fields into their paper only after the current arXiv version.

Last edited by a moderator: May 4, 2017
3. Feb 2, 2010

### strangerep

I've been wondering about a similar question in a different context,
namely the problem of IR divergences in QED, and correct identification
of the asymptotic dynamics, and hence also the asymptotic fields.

(I'm not sure whether what follows is tangential to your focus in this
thread, but I guess you'll tell me if so. :-)

In brief, I find it interesting that considering charged particles
together with their Coulomb fields seems to solve the IR problem in
QED in a much more physically satisfactory way than the typical
textbook treatments. This (imho) lends credence to the proposition that
it's better to treat particles together with their entourage of associated
fields, although such composite dressed entities are of course nonlocal.

For those who want more detail, here's an extract from a summary I've
been writing up for myself about it...

-----------------------------

Textbook treatments of the infrared (IR) divergences in quantum
electrodynamics (QED) typically introduce a small fictitious photon
mass to regularize the integrals. Allowing this mass to approach zero,
it becomes necessary to sum physically measurable quantities, such as
the cross sections for electron scattering, over all possible
asymptotic states involving an infinite number of soft photons, yielding
the so-called "inclusive" cross section.

The IR divergences are thus dealt with by restricting attention only to these
"IR-safe" quantities such as the inclusive cross section. However, various
authors have expressed dissatisfaction with this state of affairs in which
the cross sections become the objects of primary interest rather than the
S-matrix. The seminal paper of Chung (Chu) showed how one may
dress the asymptotic electron states with an operator familiar from the
Glauber theory of photon coherent states, thereby eliminating IR divergences
in the S-matrix to all orders for the cases he considered.

In a series of papers, Kibble (Kib1,Kib2,Kib3,Kib4) provided a much
more extensive (and more rigorous) development of Chung's idea, solving
the dynamical problem to show that IR divergences are eliminated by
dressing the asymptotic electron states by coherent states of soft
photons. Kibble constructed a very large nonseparable state space, within
which various separable subspaces are mapped into each other by the
S-matrix, but there is no stable separable subspace that is mapped into
itself.

less cumbersome treatment involving modification of the asymptotic
condition and a new space of asymptotic states which is not only
separable, but also relativistically and gauge invariant. They were
able to derive Chung's formulas without the laborious calculations of
Kibble, yet also obtained a more satisfactory generalization to the
case of arbitrary numbers of charged particles and photons in the
initial and final states.

KF emphasized the role of the nonvanishing interaction of QED at
asymptotic times as the source of the problems. This inconvenient fact
means that QED's asymptotic dynamics is not governed by the usual
free Hamiltonian $$H_0$$, so perturbative approaches starting from such
free states are singular (a so-called "discontinuous" perturbation).
Standard treatments rely on the unphysical fiction of adiabatically
switching off the interaction, but KF wished to find a more physically
satisfactory operator governing the asymptotic dynamics.

Much earlier, Dirac (Dir55) took some initial steps in
constructing a manifestly gauge-invariant electrodynamics. The dressing
operator he obtained is a simplified version of those mentioned above
involving soft-photon coherent states, but he did not
address the IR divergences in this paper. Neither Chung, Kibble, nor
Kulish and Faddeev cite Dirac's paper, and the connection between explicit
gauge invariance and resolution of the IR problem did not emerge until
later. (Who was the first to note this??) In 1965 Dirac noted (Dir65, Dir66)
that problems in QED arise because the full gauge-invariant Hamiltonian is
typically split into a "free" part $$H_0$$
and an "interaction" part $$H_I$$ which are not separately
gauge-invariant. Indeed, Dirac's original 1955 construction had
resulted in an electron together with its Coulomb field, which is
clearly a more physically correct representation of electrons at
asymptotic times: a physical electron is always accompanied by its
Coulomb field.

More recently, Bagan, Lavelle and McMullan (BagLavMcMul-1, BagLavMcMul-2)
("BLM" hereafter) and other collaborators have developed these ideas
further, applying them to IR divergences in QED, and also QCD in which
a different class of so-called "collinear" IR divergence occurs. (See
also the references therein.) These authors generalized Dirac's
construction to the case of moving charged particles. Their dressed
asymptotic fields include the asymptotic interaction, and they show
that the on-shell Green's functions and S-matrix elements for these
charged fields have (to all orders) the pole structure associated with
particle propagation and scattering.

--------------------------------------------
Bibliography for the above:

\bibitem{BagLavMcMul-1}
E. Bagan, M. Lavelle, D. McMullan,
"Charges from Dressed Matter: Construction",
(Available as hep-ph/9909257.)

\bibitem{BagLavMcMul-2}
E. Bagan, M. Lavelle, D. McMullan,
"Charges from Dressed Matter: Physics \& Renormalisation",
(Available as hep-ph/9909262.)

\bibitem{Bal} L. Ballentine,
"Quantum Mechanics -- A Modern Development",
World Scientific, 2008, ISBN 978-981-02-4105-6

\bibitem{Chu}
V. Chung,
"Infrared Divergences in Quantum Electrodynamics",
Phys. Rev., vol 140, (1965), B1110.

\bibitem{Dir55}
P.A.M. Dirac,
"Gauge-Invariant Formulation of Quantum Electrodynamics",
Can. J. Phys., vol 33, (1955), p. 650.

\bibitem{Dir65}
P.A.M. Dirac,
Phys. Rev., vol 139, (1965), B684-690.

\bibitem{Dir66}
P.A.M. Dirac,
"Lectures on Quantum Field Theory",
Belfer Graduate School of Science, Yeshiva Univ., NY, 1966

\bibitem{Dol}
J. D. Dollard,
"Asymptotic Convergence and the Coulomb Interaction",
J. Math. Phys., vol, 5, no. 6, (1964), 729-738.

\bibitem{Kib1}
T.W.B. Kibble,
"Coherent Soft-Photon States \& Infrared Divergences. I.
Classical Currents",
J. Math. Phys., vol 9, no. 2, (1968), p. 315.

\bibitem{Kib2}
T.W.B. Kibble,
"Coherent Soft-Photon States \& Infrared Divergences. II.
Mass-Shell Singularities of Green's Functions",
Phys. Rev., vol 173, no. 5, (1968), p. 1527.

\bibitem{Kib3}
T.W.B. Kibble,
"Coherent Soft-Photon States \& Infrared Divergences.
III. Asymptotic States and Reduction Formulas.",
Phys. Rev., vol 174, no. 5, (1968), p. 1882.

\bibitem{Kib4}
T.W.B. Kibble,
"Coherent Soft-Photon States \& Infrared Divergences.
IV. The Scattering Operator.",
Phys. Rev., vol 175, no. 5, (1968), p. 1624.

J. R. Klauder \& B. Skagerstam,
"Coherent States -- Applications in Physics \& Mathematical Physics",
World Scientific, 1985, ISBN 9971-966-52-2

"Asymptotic Conditions and Infrared Divergences in Quantum
Electrodynamics",
Theor. Math. Phys., vol 4, (1970), p. 745

-----------------------------

4. Feb 2, 2010

### strangerep

It's been a while since I looked at your papers, and I don't remember the point about
"random fields ... introducing the concept of probability in a mathematically correct way".
Could you elaborate on the details of this point, and/or give specific places in your
earlier papers where you discuss this, please?

5. Feb 2, 2010

### Peter Morgan

I found your discussion of IR divergences interesting, and I've bookmarked it both for the discussion and for the references, but I regret that I can't speak to it at this point, except to say that I've never seen anything that makes dressed particles look conceptually simple enough (or, more specifically, algebraically simple enough -- though that's not a conceptual direction one necessarily has to take).
I've had considerable trouble getting this across to anyone, although it seems clear as day to me, so I'm happy to try again. If one introduces a path integral approach for particles (though the same fact can be expressed in Hamiltonian formalisms), the path integral is dominated by nowhere differentiable paths (I've just seen this cited from Reed & Simon, which I don't have, but it ties in with my understanding of Hamiltonian methods). This works OK for particle trajectories, but notoriously, people have trouble making path integral methods rigorous in the field context, where there are more infinite limits to be taken. In the field context, I would say that no-one has really adequate mathematical control of the procedure, although some people are happy to say that renormalization is adequate mathematical control.

For a quantum field, some mathematical control (but not enough) is achieved by defining the quantum field to be an operator-valued distribution, not an operator-valued field, so that to construct an operator one has to "average" the quantum field over a finite region. As we consider smaller regions, the variance of such operators diverges, so if we try to talk about the quantum field at a point we find that, more-or-less, we would always observe either +infinity or -infinity, which isn't a good start for constructing a differentiable function. For the vacuum state of a free quantum field, the two point correlation function $$\left\langle 0\right|\widehat{\phi}(x)\widehat{\phi}(y)\left|0\right\rangle$$ is finite for $$x-y$$ non-zero, but diverges as $$x\rightarrow y$$, which is to say that the variance $$\left\langle 0\right|\widehat{\phi}(x)^2\left|0\right\rangle$$ is infinite. It's also to say that the correlation coefficient between the observed values at x and y is finite/infinity=zero, if we relinquish decent control of what limits we're taking. For interacting fields this only gets much worse, of course.

For classical fields, when we introduce the classical probability density $${\normalfont exp}(-\beta H(\phi))$$ of a thermal state we also find ourselves working with classical fields that are nowhere differentiable. I shouldn't say that there's no other way to deal with the situation, it can be managed, but random fields do deal with it pretty well, without introducing anything relatively exotic such as nonstandard analysis, for example.

I've taken it to be useful to consider random fields because they can be presented as random-variable-valued distributions, or even as mutually commutative operator-valued distributions, which are close enough to quantum fields to make comparison of random fields and quantum fields very interesting. In comparison of classical differentiable fields with quantum fields it's hard to know how to start. Part of why this is good to do is that it does give a new way to think about quantum fields, even if the more ambitious hopes I have for my program don't work out.

6. Feb 2, 2010

### strangerep

Yep... standard stuff so far. $$\widehat{\phi}(x)$$ are not operators, therefore applying
them to a state vector is technically illegal. They must be smeared with test functions to
give bonafide operators. OK.

introduce the concept of probability in a "mathematically correct way".
My perception of your random fields (or should we say "Lie fields"?)
is as an inf-dim Lie algebra parameterized by spacetime points,
as we discussed a while back. But obviously I'm missing some crucial
connection between this and probability. I need you to be more explicit/expansive
on this point if I'm to understand...

7. Feb 2, 2010

### Peter Morgan

Perhaps a more abstract approach? A set of operators $$\{\widehat{\phi}_{f_i}\}$$ generates a *-algebra (to which we add an operator 1, which acts as a multiplicative identity in the *-algebra). A state $$\omega(\widehat{A})$$ over the *-algebra is positive on any operator of the form $$AA^\dagger$$, $$\omega(AA^\dagger)\ge 0$$, and $$\omega(1)=1$$, which allows us to use the GNS-construction of a Hilbert space. We take $$\omega(\widehat{A})$$ to be the expected value associated with the random variable A, corresponding to the operator $$\widehat{A}$$. The sample space associated with A is the set of eigenvalues of $$\widehat{A}$$, and the probability density in the state $$\omega$$ can be written as $$P(x)=\omega(\delta(\widehat{A}-x.1))$$. From this we can generate the characteristic function associated with that probability density as a Fourier transform $$\widetilde{P}(\lambda)=\omega(exp(i\lambda\widehat{A}))$$.

All that is standard QM, albeit not in elementary terms. When we introduce joint observables $$\widehat{A}$$ and $$\widehat{B}$$, the difference between QM and random fields is only whether they always commute, which they do not for QM, but they do for a random field. In the random field case, the function $$\widetilde{P}(\lambda,\mu)=\omega(exp(i\lambda\widehat{A}+i\mu\widehat{B}))$$ is a joint characteristic function, whereas it is not (in general) the Fourier transform of a positive function in the QM case (unless $$[\widehat{A},\widehat{B}]=0$$, which will be the case if the two operators are constructed using only quantum field operators associated with mutually space-like regions).

I hope this is at an appropriate level and helpful? I'm not sure it's an answer even if the level is OK, in which case sorry.

I realize now that I should also note that IMO a random field and a quantum field are better considered as indexed by smooth functions on space-time, not indexed by space-time points. I find it helpful to think of the index functions as "window functions", which is the name this concept is given in signal processing. Learning to work intuitively with the concept of operator-valued distributions took me several years, but it seems obvious enough by now that I have trouble explaining. Sorry.

8. Feb 2, 2010

### strangerep

I thought I was conversant with the algebraic approach,
but what is your $\delta$ in the above expression?

9. Feb 2, 2010

### meopemuk

Hi strangerep,

Thank you very much for the interesting review of IR divergences and references. I am very interested in combining these ideas with the dressed particles approach of Greenberg and Schweber. Do you have any suggestions?

Eugene.

10. Feb 3, 2010

### strangerep

I'm wrestling with related questions, but it's too soon for me to say anything.
(And it would probably be too speculative for Physics Forums anyway. :-)

Perhaps after you've had a look through the referenced papers we could
discuss further in a separate thread, or privately.

11. Feb 3, 2010

### Peter Morgan

Hee! It's a Dirac delta, perhaps too quick and dirty as a way to construct a probability density. It's also, formally, the inverse Fourier transform of the characteristic function that follows,
$$\omega(exp(i\lambda\widehat{\phi}_f))= \omega(\sum_{k=0}^{\infty}\left[\frac{(i\lambda\widehat{\phi}_f)^k}{k!}\right])$$.
Except, urp, that there should be a factor of $2\pi$. That seems a better way to introduce it. The expected values of $\widehat{\phi}_f^k$ are the moments of the vacuum state's probability density over $\widehat{\phi}_f$, giving us the characteristic function, which we can formally inverse Fourier transform to obtain the probability density. In practice, one constructs the characteristic function as a scalar function of $\lambda$, which for the free field vacuum would be a Gaussian, which inverse Fourier transforms into a Gaussian probability density.

It does seem that what you have to say about IR divergences and dressed particles is pretty vague at the moment. A "discuss these papers" thread would seem reasonable to me, however, and it's generally a helpful discipline to pay attention to how speculative what we're doing is and to look for ways to rein it in. Indeed, I think that the path to my getting papers into journals is very much about that process, partly because anything that looks speculative is often picked on by referees as a reason to reject a paper that they only have general misgivings about. If you make no speculations, the referee's rejection letter is generally much more helpful, because they have to engage more with the paper to give a clear reason to reject it. At a grosser level, which I have often visited, editors can spot speculative from about a light-year away, so one then doesn't get as far as relatively more detailed feedback from a referee. Getting papers published is just making the speculation look well reasoned --- not getting rid of it, which IMO often makes for a boring paper.

12. Feb 3, 2010

### Peter Morgan

A citation is possible, page 119 of Itzykson & Zuber, Section 3-1-2, eq. (3-63) does exactly this (in a 1980, McGraw-Hill paperback edition; I don't know whether there are substantially different editions, which is why I'm over-specifying).

13. Feb 3, 2010

### strangerep

OK, so let's see if I now understand what your random fields are...

Your random fields (and their noncommutative quantum generalization) are basically
a generalization of certain concepts in classical statistical mechanics. Actually, let me
quote some stuff from the draft book of Neumaier & Westra, arXiv:0810.1019v1,
that (I think) relates to this way of looking at things...

(This is from their sect 1.2...)
But (iiuc) a difference between this approach and yours is that, whereas classical
quantities f are normally interpreted as functions over phase space (hence the
Liouville measure above), your random fields are just over 4D spacetime (or rather
over a space of test functions over 4D spacetime). (?)

So I'm now trying to follow your criticism of Navascues and Wunderlich more carefully...
But... in the online version (arXiv:0907.0372), which is all I have access to right now,
I can't relate your quotes to their section numbering. I also can't find the mention
of "continuous fields"? Is the Proc Roy Soc version different from the online version,
and you were commenting on the former?

14. Feb 3, 2010

### Peter Morgan

Yes, my definition and constructions are manifestly Lorentz and translation invariant. The usual definition is Lorentz and translation invariant, but not manifestly so. The usual phase space approach, is only possible if only mass shell components of a test function contribute. The restriction to only a single mass shell (if that's what is wanted) is implemented in my approach by the inner product having a delta function restriction to the mass shell.
The Proc. Roy. Soc. A version is different, which I didn't discover until after I submitted my comment. The Proc. Roy. Soc. A version is available for free, I think because of the Royal Society's anniversary celebrations. Go to http://dx.doi.org/" [Broken]. [You won't find any citation to the arXiv version in my paper, the arXiv administration added it to the arXiv abstract, in their wisdom.]

Last edited by a moderator: May 4, 2017
15. Feb 3, 2010

### strangerep

At the end of your comment paper, you say:

In this one paragraph, you say "requires a vigorous condemnation" but then say
"this comment does not touch Navascués’ and Wunderlich’s argument". So... you're not
actually arguing against the essential results of NW's paper? But only the "little" flaw
of using the phrase "continuous fields" rather than "random fields" ?

Last edited by a moderator: May 4, 2017
16. Feb 4, 2010

### Peter Morgan

Mixed messages indeed. I was cross at Navascués’ and Wunderlich’s assumptions, not at their argument. When I saw your quote, "requires a vigorous condemnation", I thought you might have quoted me out of context, because I thought I surely must have mentioned that it was their assumptions that require a vigorous condemnation, but I see that I was vigorous at their whole paper.

Getting the politeness right in a critical comment apparently evaded me, but I do think these are serious Physicists, running with a respectable idea, that we might look at what the Correspondence Principle might tell us about the Planck scale and beyond. I think it's a good idea to do that in principle, but I'm telling them that "Oh, I wouldn't start from there". That could be boring.

What saves their paper, I think, and what made it possible for me to make my comment constructive, I hope, is that they introduce classical continuous fields. They do it half-heartedly, and they might even have been made to introduce fields because the referee said, "well, but what about fields?", but they do it. This is a potentially significant development, because the ways in which classical fields might be used to model Physics is underdeveloped. Good math approaches to QFT take it almost for granted that QFT is about
fields, not about particles, particularly because of the Unruh effect, since about 20 years ago, say.

Very few Physicists, however, take the obvious next step, which is that in that case we'd better find ways to talk about fields instead of about particles. A notable exception is Art Hobson, whose web-site has available a copy of the paper of his that I cite. He's concerned with how to teach QFT, and proposes to do it by emphasizing a field perspective. Andrei Khrennikov, who is a Mathematical Physicist who turned seriously to foundations of Physics about ten years ago, takes a similar line, but his mathematical methods and mine are very different. 't Hooft's approach is also similar but different, as also for Wolfram. Elze and Wetterich are two other hard Mathematical Physicists who are developing entirely different formalisms. Khrennikov, Elze, and Wetterich are developing fairly traditional stochastic differential equation methods, 't Hooft and Wolfram are developing finite automata models, in which the statistics are generated by simulation; I'm the only one, to my knowledge, who is seriously developing an algebraic presentation of random fields. A friend, Ken Wharton, is developing a view based on classical fields, with me trying to persuade him that he has to introduce probability in a mathematically decent way, and him dragging his heels, perfectly reasonably, because he doesn't like a metaphysics that includes probability. If I have a metaphysics, it is a metaphysics of statistics and ensembles rather than of probability, with my being content with a relatively loose, somewhat post-positivist relationship between observed statistics and the mathematics of probability, but we've been negotiating this fine point for a while. All this move to fields, and lattices, has more-or-less started to happen in the last ten years (although there's also Stochastic Electrodynamics, dating from the 60s, and Nelson's approach, too, from the 70s, but these are arguably problematic because they are preoccupied with fermions being particles, bosons being fields, and these programs, although I believe always continuing, have had significant hiatuses).

As far as all these different approaches are concerned, I consider that mine has the most to gain from comparison with QFT, in a Correspondence Principle sort of way, because I can even show that a free complex quantum field is empirically equivalent to a free random field, so I'm especially happy to see NW talking about CP. Nonetheless, I would only claim that my approach gives a useful counterpoint to stochastic or lattice methods, not that my approach is correct. I wish not to claim that the world is continuous rather than discrete, for example. A random field, properly speaking, is only an indexed set of random variables, it is only associated with a continuous space-time if we specifically take the index set to be the Schwarz space of functions on space-time (or some other well enough controlled function space on space-time).

So why should NW be pulled up for this rather than someone else, given that almost no-one pays any attention to how their use of particle-talk conditions their thinking? Their fault, I suppose, is that they mention fields so glibly, without thinking about how rich the seam is, and proceed with a discussion that would make almost no sense if they tried to accommodate both particle and field ways of thinking. At the very least, their conclusions would have to be hedged with a statement such as "if we think only in terms of particles, ...".

NW's discussion of Bell inequalities is similarly conventional. If they did the job properly, they would know that the flow of ideas surrounding Bell inequalities has been shifting dramatically over the last 10 years, with roots that go back to about 1980. The relative significances of contextuality and of locality are gradually being teased out more and more clearly. Anybody trying to talk about Bell inequalities should at least acknowledge those currents, and again they should either accommodate the various possibilities or explicitly hedge their conclusions.

As the last sentence of the abstract says, "Whether we discuss physics in terms of particles or in terms of events and (random) fields leads to differences that a glance would be well to notice." Perhaps I might add, even more facetiously, "or at least what is not noticed ought to be mentioned", but that would go far enough that I imagine the editors would have sent it back to me unrefereed. As it is, my comment is with referees; I hope they see that my criticism is constructive.

17. Feb 4, 2010

### strangerep

When I first read NW's sentence where they mention "continuous fields"
my first thought was "what _precisely_ do they mean by that phrase"?
(Such pedantic detail becomes important in discussions about
"introducing probability in a mathematically correct way"...)

So let me ask you the question...

You've explained in earlier posts what you mean by a random field
(i.e., an inf-dim commutative *-algebra, with basis elements indexed
from a space of well-behaved functions over spacetime, such that
state functionals over this algebra make sense).
What then are your definitions of the phrases "classical field"
and "continuous field" ?

18. Feb 4, 2010

### Peter Morgan

Including their context, NW say, "Now, in order to establish a connection with classical physics, Alice and Bob should not regard these intensities as fluxes of discrete particles, but rather as continuous fields". I think from this they mean continuous fields simpliciter, although I suppose that if they were given a second try and were more careful, they might say differentiable or possibly second differentiable fields. The question that I think is impossible for anyone except them to answer is "what would they say if they were being careful?". If they took the kind of criticism I have raised seriously, would they retreat to a space of functions defined in terms of measurability, or would they keep to a space defined in terms of continuity or differentiability?

If NW take the kind of criticism I have raised seriously, then they could fairly easily retreat to a point that would be recognizably quite similar to mine, in that they would introduce some kind of mathematics that would allow probability to be introduced decently. I don't much care whether they retreat to a stochastic processes approach to introducing probability or to an algebraic approach, but I would hope that they might recognize that a classical continuous field is not as good a starting point for constructing a meaningful Correspondence Principle as a classical probabilistic mathematical structure of some kind would be. I don't know, of course, how embarrassed they will feel at being called out in this way, and how annoyed they will be at whatever embarrassment they may feel, which could easily affect how they respond.

I sent a copy of my Comment to Navascués, who is the corresponding author, last Wednesday. I haven't heard back from them, but I consider them to be within their rights not to respond at any point. I'm curious whether, supposing my comment is accepted by Proc. Roy. Soc. A, they might not exercise their right to reply in Proc. Roy. Soc. A. Even if Proc. Roy. Soc. A were to publish my Comment, I think there is a good chance that it will die from lack of attention. I think there is a move of the zeitgeist towards fields, enough that the editors sent my Comment to referees, but, even if I'm right that the zeitgeist is moving, 2010 may still be too early on the wave for there to be a serious debate. I think, in any case, that the state of development of the mathematics, understanding, and presentation of random fields is currently too fragmentary to persuade anyone who doesn't want to be persuaded that there is a curious and novel alternative to existing interpretations of QFT/QM here in prospect.

Thanks for pushing me on this, Strangerep. I hope I'm getting better at arguing the case, thinking of new, more interesting ways of discussing the issues. I hope I'm not just getting better at rambling on and on. Time will tell, if no-one else does.

19. Feb 4, 2010

### strangerep

Or they might just wonder "what on earth is this guy on about?".

I now find the whole particle-vs-field debate quite bizarre, since the message I take
from both (modern) classical mechanics and (modern) quantum theory is
that the important thing is (tensor products of) irreducible representations
of certain groups. Field theoretic stuff is then just the inf-dim version of this.

BTW, if you can find the time, you might be interested in some of the other
non-obvious threads woven through that book of Neumaier & Westra which I
mentioned earlier. One such thread concerns the not-well-known similarities
between classical and quantum physics, as an improved way of unifying the two.
(But one must read quite a lot, and not get turned off by math, to follow it.)

Well, er, yes, you do have a tendency for that, if I may say.

Case in point: you didn't answer the other part of my question about
how you define "classical field" and "continuous field".... (?)

20. Feb 5, 2010

### Peter Morgan

If they don't get it at all, if they don't feel embarrassed because they think my criticisms are nonsense, I have more work to do, as always.
OK, but then to get from a symmetry group to probabilities of given measurement results, I suppose one needs a positive linear form of some kind, which gets us to a particular von Neumann algebra, right? Do we give an abstract presentation of the symmetry group in terms of commutation relations between unbounded operators in a *-algebra, or do we give an abstract presentation of a C*-algebra of bounded operators? If we do the latter, then we have work to do to make contact with the empirically successful presentation of QFT as states over an unbounded algebra, which lead fairly directly to S-matrices and to cross-sections, etc. One little remarked aspect of QFT is the importance of the vacuum projection operator, which is used everywhere to construct projection operators, corresponding to probabilities of events in detectors. The use of the vacuum projection operator as part of the algebra of observables makes it seriously nonlocal, but this is a vital part of the empirical success of QFT.

I think we have to ask whether there are some vector spaces that support representations of a given symmetry group (so, modules) that seem more natural than other modules, given that we have to make contact with classically presented information about experimental results. I think it's a reasonable question whether particular internal symmetry groups emerge as a mathematically natural consequence of some other mathematics. A place to look, in the first instance, is in the geometry of Minkowski space, its tangent manifold, etc., perhaps torsion with a trivial metric connection. In the second instance, if Minkowski space is not enough, one might look at the geometry of higher dimensional spaces, ... , but then the naturalness of the construction becomes gradually more remote. One wants the simplest structure that is explanatory and that is consistent with experiment. I won't bore you with more speculations.

I'm not certain that gauge groups are necessarily a feature of future theories. I particularly worry that fermion fields are not a happy mathematical definition because they are supposed to be operator-valued distributions, but only sesquilinear forms of the fermion field at a point are gauge invariant observables. Since gauge invariance is specifically tied to this improper structure, I would prefer to introduce interactions in a mathematically better defined way. I would prefer to work with the observables associated with fermion fields instead of working with the fermion fields. Which I have (lots of) work to make happen.
I didn't focus on the Neumaier & Westra, but I have now downloaded it. I didn't notice this when it came out on arXiv, which surprises me. It's long enough to keep me out of trouble.
If I give you a rope to hang me with, ... . I use PF for conversation about developing ideas, so it's inevitably more prolix than an attempt at a published paper.
Briefly, solutions of a differential equation, so a first or second differentiable function of position in Minkowski space. So of course no general covariance.

I'm feeling worn out by a week or so of intensively being here on PF. Time, soon, to go back to the longer wave of calculations instead of so much speculation. Thanks for the various reality checks, Strangerep. Good luck with your own work.