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I tried to explain what a superselection sector is which is less easy than one would think…of course what I would really like to do is to explain some DHR-superselection theory and there one can of course cite the mathematical definition without any further explanation of the origins :-)
Thanks for starting this!
Am a bit in a hurry, so just very briefly to thoughts:
Should the term “irreducible representation” appear somewhere in the entry?
Is it true without further qualification that there cannot be states in QFT that are not electron number eigenstates?
Should the term “irreducible representation” appear somewhere in the entry?
Definitly, I’ll add that as soon as I come up with an explanation that I like :-)
Is it true without further qualification that there cannot be states in QFT that are not electron number eigenstates?
Now that’s a really tricky question, instead of trying to answer that I would rather write “a naive example from QM would be”… To begin with, it’s less than easy to explain what a particle even is in AQFT (not to mention that some people say that there cannot be such a concept as “particle”).
(not to mention that some people say that there cannot be such a concept as “particle”).
Yeah, so we should eventually give asome precise statement here. I suppose it is imagined that in the QFT describing some fermion there should be an operator of the kind $\sum_{\vec k} \int_X a^\dagger_k a_k$ or the like whose eigenstates one would want to call “electron number eigenstates”. I think whether or not its Eigenstates are superselection sectors depends on the details of the theory, for notably on the details of the interaction terms.
… in the QFT describing some fermion there should be an operator of the kind…
That’s where the problem starts :-) An electron should be a localized eigenstate of some number operator, which does not exist, thanks to the Reeh-Schlieder theorem, in AQFT. I strongly agree that this “paradox” should be explained as precise as possible. Maybe we get to clean up some of the misunderstandings of the string wars in this way…(Ok, that may appear to be megalomaniac, let’s see how the story unfolds).
I think whether or not its Eigenstates are superselection sectors depends on the details of the theory, for notably on the details of the interaction terms.
I’m embarrassed to admit that I haven’t thought about this before. Is there a concrete model where this can be shown?
An electron should be a localized eigenstate of some number operator,
Why do you require it to be localized?
Anyway, we don’t need to write this entry now, I need to concentrate on something else..
Maybe we get to clean up some of the misunderstandings…
I hope that’s what we are doing!
I think whether or not its Eigenstates are superselection sectors depends on the details of the theory, for notably on the details of the interaction terms.
I’m embarrassed to admit that I haven’t thought about this before. Is there a concrete model where this can be shown?
I don’t mean anything profound. Just that if the theory has an interaction term that does not preserve the number of particle of some species (does not commute with their number operator) then that particle number cannot be a superselection label.
Maybe to clarify the thing about the electron number:
think of a simple “second quantized” QFT as considered in solid state physics, say describing particles and/or quasiparticles (phonons) in a chunk of metal with periodic boundary conditions. Then there are well defined and common particle numer oiperators. Whether or not they commute with the Hamiltonian depends on the kind of interaction term.
Why do you require it to be localized?
A detector should be modelled by a positive operator, the first attempt I would try would be to define a detector to be a localized positive operator that does not detect anything in the vacuum state (the detector stands one day in a lab and does not detect anything). Reeh-Schlieder says that this does not work, the next best thing is a quasi-localized positive operator (that is one that can be approximated in norm by localized ones). But that’s only the first step in the program to even define what a particle is and what the particle content of a theory is and how to deduce this from intrinsic properties of the Haag-Kastler net…
Anyway, we don’t need to write this entry now, I need to concentrate on something else..
Sure, I’d like to do the definintion of the DHR-category first, and then try to understand the proof of Michael Müger in the appendix of the Halvorson paper…
Then there are well defined and common particle number operators. Whether or not they commute with the Hamiltonian depends on the kind of interaction term.
Ok, but I am unaware of any models where you can actually create electric charges out of nowhere (I mean create a positive charge without creating a negative charge).
Ok, but I am unaware of any models where you can actually create electric charges out of nowhere (I mean create a positive charge without creating a negative charge).
Sure, so electron minus positron number should be preserved in QED. But not each separately. But I haven’t really thought of this in detail, maybe it does not make much sense.
But in my simpler example: certainly in standard solid state QFT models, for instance phonon number is certainly not preserved. And the systems are traditionally modeled as finite with periodic boundary conditions. Of course they won’t be modeled by Haag-Kastler net, but at best by Euclidean nets. But clearly this are examples of QFT.
Urs said:
But clearly this are examples of QFT.
I think I understand the part where free quantum fields are infinite collections of harmonic oscillator and how this picture can be used to model a solid state body with periodic boundary conditions…but
…clearly this is an example of QFT…
…overextends me. I’m really not sure if - and if, where - the analogy of a QFT model of a non-relativistic system with infinite degrees of freedom (solid state physics) and that of a relativistic QFT model (on the continuum or on a grid) breaks down. Seems to be a good anchor for me to extend my understanding a little bit.
Ian said:
…it states that each superselection rule asserts that some Hermitian operator in quantum theory is not an observable …
Sorry, I don’t get it, what is the connection? Do you have a reference I could look into? Or, of course, we could try to discuss this here…
I created a page DHR superselection theory, including the definition of transportable endomorphism, so that the next step could be the exposition of the “Category of Localized Transportable Endomorphisms” of the Halvorson-Müger paper.
Tim,
remember, all I was commenting on was your remark on the entry superselection theory that an example for superselection sectors is electron number.
You state this without specifying a formal framework that makes the statement precise, and so it is a bit unfair that you criticize my questioning of your statement by pointing out that I didn’t give you a precise framework :-)
I think maybe what you meant to say is that electric charge constitutes superselection sectors. You don’t mean to say that the number of electrons in a chunk of space in the real world is a well defined number (classical observable), do you?
Hm, it seems that I am wrong. Or at least not using terminology correctly. Just a second…
Okay, I think the point that caused some trouble, at least on my side, is the term “electron number” as a technical term. I said that it’s electron minus positron number that is superselectred, but apparently I am not following standard terminology with this.
In any case, it’s the observable given in Minkowski space by the expression
$\int_{\Sigma_3} \bar \psi \gamma^0 \psi$for $\psi$ the electron field (instead of half the time-Fourier series of this).
I see that in the old article by Streater this is discussed as electron lepton number superselection .
Ian wrote:
…it states that each superselection rule asserts that some Hermitian operator in quantum theory is not an observable …
Tim asked:
Sorry, I don’t get it, what is the connection? Do you have a reference I could look into? Or, of course, we could try to discuss this here…
For instance page 782 of Streater’s old survey.
Urs said:
You state this without specifying a formal framework that makes the statement precise, and so it is a bit unfair that you criticize my questioning of your statement by pointing out that I didn't give you a precise framework :-)
I accept the criticism, I'm simply not sure how to come up with something better...(my last statements are me trying to figure out what would be a better example, not me criticizing the criticism).
For instance page 782 of Streater's old survey.
The link takes me to Andrew's homepage...?
Edit: do you mean The C*-algebra approach to quantum field theory?
Sorry, typo in the URL. Try this
http://iopscience.iop.org/0034-4885/38/7/001/pdf/0034-4885_38_7_001.pdf#page=782
(on top of the usual typos introduced by my fingers, there is now also a key broken on my keyboard and from time to time it starts inserting characters on its own accord. Need to get a new machine. But always hate to switch systems, as it wastes my time…)
Thanks for the references, I skimmed chapter 11 in “Quantum Paradoxes” and am reading the Streater paper. That’s a really nice one! From the abstract:
This suggests that field theory will be solved and used to describe elementary particles within the next few years.
Yes yes yes, predictions are a tough business, escpacially those about the future :-)
…it states that each superselection rule asserts that some Hermitian operator in quantum theory is not an observable …
Somehow I read “Hamilonian” instead of Hermitian = self-adjoint. But the implication “superselection sectors exist $\Rightarrow$ there are selfadjoint operators that are not observable” is certanly true. But I think that there may be selfadjoint operators that are not observable independently from the existence of superselection sectors, but I don’t know if someone constructed a Haag-Kastler net with unobservable selfadjoint operators that are reduced by the superselection sectors. I remember reading this statement in the form of a conjecture in Haag’s book.
But the implication "superselection sectors exist there are selfadjoint operators that are not observable" is certanly true. But I think that there may be selfadjoint operators that are not observable independently from the existence of superselection sectors...
…could the notion of nonobservable selfadjoint operators be used to formulate a hidden variable theory…?
I don’t know, I don’t see any connection…the “hidden variable theories” try to get rid of the probablisistic aspects of quantum measurements, while nonobervable selfadjoint operators in this context are operators such that there cannot be a device that measures the observable represented by the operator. I don’t know any good example of the latter situation, but e.g. AQFT has no lengthscale. If effects described by quantum field theory have a smallest lengthscale (like the “diameter” of the electron) than any observable that is sharper localized than this lengthscale would be unobservable :-) (That’s speculation only).
...the "hidden variable theories" try to get rid of the probablisistic aspects of quantum measurements...
If effects described by quantum field theory have a smallest lengthscale (like the "diameter" of the electron) than any observable that is sharper localized than this lengthscale would be unobservable
Actually, they also aim to explain things like entanglement which appear to violate causality.
One long term goal of mine is to explain Bell’s theorem and the maximal violation of Bell’s inequalities in AQFT (in the vacuum state), in order to show that there is no contradiction between entanglement and causality.
Isn’t this a bit like what string theory is all about?
The kindergarten introduction to string theory that I know is about the quantization of a relativistic bosonic string, the “length” of the string is an “instrinsic length scale” of the theory, I guess that that’s what you are alluding to?
One long term goal of mine is to explain Bell's theorem and the maximal violation of Bell's inequalities in AQFT (in the vacuum state), in order to show that there is no contradiction between entanglement and causality.
the "length" of the string is an "instrinsic length scale" of the theory, I guess that that's what you are alluding to?
Actually, they also aim to explain things like entanglement which appear to violate causality.
Nonsense. At the level of measurements there are no viaolations of causality and the level of wave function is not a problem as it is not measurable. So EPR is just a paradox, not a violation.
So EPR is just a paradox, not a violation.
The paper
discusses some of the aspects of Einstein microcausality and the violation of Bell’s inequalities, for example it explains that in the relativistic vacuum state Bell’s inequalities are maximally violated (see e.g. References on Reeh-Schlieder theorem). So, in AQFT you don’t even need “particles” :-)
One important paragraph from this paper is this:
Commonly, physicists say that theories violating Bell’s inequalities are “nonlo- cal”; yet, here are fully local models maximally violating Bell’s inequalities. This linguistic confusion is probably so profoundly established by usage that it cannot be repaired, but the reader should be aware of the distinct meanings of these two uses of “local”. The former refers to nonlocalities in certain correlations (in certain states), while the latter refers to the commensurability of observables localized in spacelike separated spacetime regions. So the former is a property of states, while the latter is a property of observable algebras. The results discussed above estab- lish the generic compatibility of the former sort of “nonlocality” with the latter kind of “locality”. The wary reader should always ascertain which sense of “local” is being employed by a given author.
Link:
Yet More Ado About Nothing: The Remarkable Relativistic Vacuum State
PS: Back when I was a physics graduate student, I was fascinated by the quantum vacuum. I spent days, months, (years?) pondering it while wishing I was smarter. One of my favorites:
Bell thus insisted upon the importance of "experiments of the type proposed by Bohm and Aharanov, in which the settings are changed during the flight of the particles." In such a "timing experiment," the locality condition would then become a consequence of Einstein's causality, preventing any faster-than-light influence.
Ian wrote:
The resultant experiment violates Bell’s inequalities and thus, given the above conclusion, appears to violate causality.
As Summers points out, the word “locality” has different technical meanings, and mixing them up leads to lots of trouble. The same is true for “causality”. In axiomatic quantum field theory there are at least 2 famous meanings of “causality”: Einstein causality (commutativity of spacelike separated observables) and the diamond property (the C*-algebra of observables in any set is equal to that of its causal completion). Neither of these is contradicted by the fact that the vacuum state violates Bell’s inequalities!
Aspect’s experiment refers specifically to Einstein causality as the quote above points out. Nevertheless, my point is that Aspect’s wording in the above quote allows for the possible interpretation that a violation (in his example) of the inequalities violates Einstein causality (I should have added the ’Einstein’).
Aside from all that, from a philosophical point of view, specifically regarding Aspect’s third experiment described above, I feel as if we’ve used a lot of mathematical abstraction and shoe-horned physical reasoning over the years in order to make the results more palatable to us. But the fact remains that that specific experiment, to an empiricist, gives the impression of superluminal signaling. As Aspect says further along in the same paper,
In this experiment, switching between the two channels occurs about each 10 ns. Since this delay, as well as the lifetime of the intermediate level of the cascade (5 ns), is small compared to L/c (40 ns), a detection event on one side and the corresponding change of orientation on the other side are separated by a spacelike interval.
The italics are mine. The key word here, to me, is corresponding. If the detection event on the one side and the change in orientation on the other are connected in some way and they are spacelike separated, then the possibility has to exist that Einstein causality is violated. While there are certainly causal interpretations for this, they are simply that: interpretations (otherwise we wouldn’t have all these conferences on the foundations of QM every year - no one would have anything to talk about :)).
We as human beings just seem to be hard-wired to want things to be causal. I tend to actually do the same - and I want a causal explanation. But none of the existing ones are empirically satisfying. They all seem like trying to hammer a square peg into a round hole rather than going back to the beginning and trying to mill a new peg that fits.
If the detection event on the one side and the change in orientation on the other are connected in some way and they are spacelike separated, then the possibility has to exist that Einstein causality is violated.
There are mathematical models that explain this experiment and obey Einstein causality, so I don’t follow the “then the possibility has to exist”-part. This possibility exists, of course, but it is not a necessary conclusion of Aspect’s experiments.
While there are certainly causal interpretations for this, they are simply that: interpretations (otherwise we wouldn’t have all these conferences on the foundations of QM every year - no one would have anything to talk about :)).
While I’m mostly interested in connecting axiomatic QFT frameworks to experiments, that does not imply that I think that everything about QM and it’s interpretation is said and done, of course, I don’t :-) I simply can’t think about everything…
I feel as if we’ve used a lot of mathematical abstraction and shoe-horned physical reasoning over the years in order to make the results more palatable to us.
That’s one possible approach to an interpretation of QM: Common sense of humans is of no help, therefore we have to start with some mathematical axioms and derive the interpretation in terms of “common sense” aka “classical language” as an approximation. That’s for example the program of Roland Omnès in his book “The Interpretation of Quantum Mechanics”. It’s the adjoint of the empiricist’s approach :-)
There are mathematical models that explain this experiment and obey Einstein causality, so I don’t follow the “then the possibility has to exist”-part. This possibility exists, of course, but it is not a necessary conclusion of Aspect’s experiments.
Ah, but that’s just it. They’re mathematical models. Tom Moore makes a really interesting observation in one of his textbooks about physical theories: Most physical theories have layers with experimental results overlaying “physical reality” (if you believe in such things). A conceptual layer sits “on top” of this and a mathematical layer then sits “on top” of the conceptual layer. QM seems to be missing the conceptual layer (or, as I like to say, a single, consistent, agreed-upon conceptual layer).
I simply can’t think about everything…
You can’t?!?! What’s wrong with you? Just kidding, of course. ;-)
That’s for example the program of Roland Omnès in his book “The Interpretation of Quantum Mechanics”. It’s the adjoint of the empiricist’s approach :-)
LMAO, that’s funny. I’m familiar with Omnès, though it has been 7 or 8 years since I’ve read him.
I know I keep harping on the “empiricist” thing, but maybe that’s the engineer in me. “Reality” is experiential. As soon as a mathematical model becomes your best “conceptual”/experiential explanation, you immediately need to consider the ontological status of the mathematical objects/structures themselves (which is a whole other can of worms).
Tom Moore makes a really interesting observation in one of his textbooks about physical theories: Most physical theories have layers with experimental results overlaying “physical reality” (if you believe in such things).
Although you recommended his textbooks to me on your blog in April, I did not have the opportunity to look at them yet - which one do I need to look up the observation you mention?
I think I don’t understand the semantic of “overlaying” in the sentence above, or that of “ontological status” later on, but in case I come up with any interesting question to ask I’ll do that over at your blog, if that is Okay with you (may take another couple of months, though).
I simply can’t think about everything.
I wrote that because from time to time I feel the need to stress that “I am mostly interested in X” does not mean that I endorse the opinion that
X is the most important topic of the world so that
the most intelligent people of the world work on X so that
if you don’t like X proves that you are simply too stupid to understand 1. and 2. and
any theories that X is built upon are completely understood and anyone doing research there is simply too stupid to understand that
etc.
Maybe it is not necessary to do that here on the nForum.
Although you recommended his textbooks to me on your blog in April, I did not have the opportunity to look at them yet - which one do I need to look up the observation you mention?
That would be Six Ideas That Shaped Physics, Unit Q: Particles Behave Like Waves (ISBN 0-07-239713-6). For a nice overview of his philosophy, check out his support website. In the book he gives a figure to demonstrate what he’s talking about (you’ll see what I mean by “overlaying” in the figure). Again, it is an introductory text, but the idea behind it (all six “volumes”) is to give students a sophisticated understanding right off the bat. But on top of that, it presents a whole philosophy of physics (apparently unintentionally - I spoke to him about it recently and he hadn’t even realized he was doing it). Yet it is an approachable text that I’ve used at a variety of levels. I do have one or two beefs with certain things here and there, but I’ve been using it for close to a decade now with excellent results.
I ordered a copy.
Again, it is an introductory text…
Reading an introductory text from time to time is certainly not beneath me, quite on the contrary, it is part of what Terry Tao calls relearn your field.
I assume I made it clear, but in case I didn’t, Unit Q serves as a portion of a full-year intro to physics course (as opposed to an intro to QM). So the students I use this with are usually freshman or sophomores. I’ve written some supplements to it including an extension that introduces my own derivation (based on Moore’s) of the time-dependent Schrödinger equation (he does the time-independent one).
To really get a sense of his philosophy, I’d also recommend the first few chapters of Unit C (the first one: ISBN 0-07-229152-4).
Happy reading!
added pointer to
Interesting to see that the cutting-edge development of hologrpahic entanglement entropy now makes use of the core of classical AQFT: the DHR superselection theory
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