Talk:Particle physics and representation theory

There are a number of items that I put on this page that I'm not 100% sure about. For example, I'm not sure exactly what Wigner's Theorem is or how it fits in. I also may be misstating the facts about color and flavour symmetry, and I may be oversymplifying the classification of representations of the Poincare group. Steve

What does it mean for a particle to "lie in" a representation of G? A representation is just a homomorphism from G to the Hilbert space, so is this saying that the action of G on the vector representing the particle is nontrivial, or something else? - 72.58.19.66 00:58, 8 May 2006 (UTC)

I think the "lie in" is clearer now. Steve

second the comment below re clean-up required. a representation is a homomorphism from G to the, in this case, bounded operators on a Hilbert space. "lie in" doesn't seem like very good language. in algebra, it is called the orbit of a given |p₀>. Mct mht 05:21, 8 February 2007 (UTC)

sloppy

This article is very vague about a number of points and I would recommend that it be put on status to require "clean up" by an expert. The most pressing need is the need for a few equations to eliminate some of the hand waving. There are also factual errors in the text. For example, SU(3) is the gauge of the strong force, but SU(2) (left) is the gauge of the weak force. They are distinct gauge groups. SU(2) (left) is weakly violated (and hence is only an approximate symmetry; SU(3) is an exact symmetry.

I don't understand those "factual errors". Yes, SU(2) (left) is the gauge of the weak force, but does the article say (or imply) otherwise? The article does say that color SU(3) is an example of an exact symmetry, which is true. So exactly which sentence(s) of the article are you disputing? Steve

General picture: the reference to Weinberg's book seems to be more appropriate in Wigner's theorem. Poincaré group is believed to be a subgroup of G only in a flat space (which is not the case for our Universe). Also, representations of Poincaré group are characterized by mass squared and some other quantum number (spin, helicity) which depends on the sign of $m^{2}$. In particular, there are massless representations with continuous "spin". It also seems appropriate to change SU(2) to SO(3) in the "hypothetical example" section. Perhaps, we should merge Eightfold_way_(physics) with this article? --Terminus0 (talk) 01:29, 8 November 2009 (UTC)

I tried to make those corrections to the article, they're all good points. Thanks! I'm skeptical about the merge, I think of this article as a general physical idea, and Eightfold way, Georgi-Glashow model, Wigner's classification, isospin, etc. as offshoots of it. But I don't feel too strongly, it could work either way. --Steve (talk) 05:29, 8 November 2009 (UTC)

General picture: 'Let G be the symmetry group of the universe -- that is, the set of symmetries under which the laws of physics are invariant' -- this is too vague. We should try to say what is the set-up -- at least what is G and what does G acts on (you refer to lagrangian formulation and continuous symmetries?). By 'law's of physics' you mean these which can be written as equations in some coordinates I suppose. —Preceding unsigned comment added by Karol8 (talk • contribs) 14:30, 16 May 2010 (UTC)

I added a few words to suggest a "coordinate-free" interpretation. Do you think that helps? --Steve (talk) 19:03, 17 May 2010 (UTC)

Antiscientific

I don't think so. That might be missing the point of Karol's comment. The problem isn't the coordinates. The article summarily introduces an abstract, formal concept as science, then carries on as if this concept grounds, and explains the universe. This introduction is done by postulate. Before we can talk about the arrangement of facts, the argument must be capable of distinguishing truth. I will try to flesh out some grounds for working this out without an expert. Hard to guess how much detail is normal to other people. Let me know. Onward. This is a science page. Let me field two basic tests of scientific process.

1. The sciences take exactly one thing for granted: the physical universe.

2. Philosophical coherence is irrelevant to scientific inquiry. Particles won't change course to obey me.

I think the problem is here, at the head of the article, but perhaps we can start with the extrapolations and work backward to some kind of agreement. I would like to begin with this: "As an example of what an approximate symmetry means, suppose we lived inside an infinite ferromagnet, with magnetization in some particular direction. An experimentalist in this situation would find not one but two distinct types of electrons".

... unless she doesn't. There is no such thing as the result of an experiment without the experiment itself. In this article, we know the results of pretend experiments in pretend universes. This is explicitly antiscientific. Do you see how the thrust of the argument relies on the fact that the experiment cannot be conducted at all? As a reader, I am to infer the philosophically desired conclusion, as a physical truth. But I can make up whatever universe I want! I am free to disagree that the "expected" properties follow.* None of this noodling informs reality or the results of experiment.

It is easy to replace these sentences with a scientific asssertion. Tell the truth. E.g., "we have no idea what else would be true in such a universe." This correctly reseats the elements of argument. Now we can freely append, "but suppose in addition we hypothesize that there are exactly two kinds of electrons...", and explain why we have decided to keep these particular properties in our hypothetical example. Again, it is obvious that fixing the mistake guts the argument. The presumed relationship --governing framework gives physical deductions without checking-- does not exist.

The use of "experimentalist" is ugly and misleading. First, there is no such thing, just ordinary people checking their assertions. Second, the argument cheats on both sides: on the one hand, the reader is to infer that these ideas are grounded in empirical verification, on the other hand, verification is dismissed as unnecessary to the conclusion. In science, we check. In general, this article presupposes that the formal inferences are absolutely true, and hunting for ways to illustrate these truths. Such a position is unassailable because it is unreasoned, and unreasonable (begging the question).

• About the existence of truths to which no facts can be, or are, brought to bear, I am free to have any opinion I wish. KingSleepy (talk) 08:21, 15 January 2014 (UTC)

I think the "target audience" of this page is a reader who:

(A) Already understands some basic laws of physics, including aspects of electromagnetism, quantum mechanics, and special relativity -- what someone might learn in the first two or three semesters of college physics;

(B) Not only understands these laws but accepts that these are true laws of the universe.

You seem to be grasping at a basic question: Why is there such a thing as a "thought experiment"? This is not such a stupid question: After all, what's the point of a thought experiment, when you can't possibly know what would happen in an experiment that you're not actually performing?

Well, here is an answer: When there is a shared premise that some laws of physics are correct (e.g., (A-B) above), a thought experiment can help you understand the consequences of those laws.

Do you agree?

Anyway, it might be a good idea to put a disclaimer at the top of the article: "This article assumes that the reader already understands basic quantum mechanics, special relativity, and representation theory".

When it's possible, I think it's good to write physics articles that are understandable to someone who knows no physics whatsoever, and I think it's good to write physics articles that justify why we believe that these laws are true from an empirical basis (to convince the skeptical conspiracy-theorists). But I don't think either of those two goals is realistically possible in this particular article, or necessary. --Steve (talk) 16:23, 15 January 2014 (UTC)

which postulate

From the first paragraph, "This postulate states that each particle is an irreducible representation of the symmetry group of the universe." To which postulate is this referring? Mathchem271828 05:35, 6 January 2007 (UTC)

error in maths

There appears to be some sort of error, giving the message "Failed to parse (<math_output_error>): \{g|p_0\rangle|g\in G\}." in the general picture section. I don't know what its supposed to say so can't fix it. Sheepe2004 (talk) 12:57, 20 May 2009 (UTC)

Seems to be fixed, see discussion at Wikipedia:Village pump (technical)/Archive 134#Strange math bug. --Steve (talk) 22:29, 20 May 2009 (UTC)

Remarks

I think the "General picture" paragraph should be rewritten (I would have done it myself hadn't my English sucked so much). It should be stressed that the space of states of a quantum theory is a projective Hilbert space PH. That is not a minor issue, since it follows that symmetries of the system, which form a group PU(H), are, in general, automorphisms of PH (i.e. bijections of PH that preserve |<v/|v|,w/|w|>|).

Now, there is a theorem, let's call it Wigner's theorem, which tells us that all automorphisms of PH are induced by unitary or antiunitary tranformations of H. So, given a Lie group G of symmetries (say Poincare or Galileo's group for example) what we want is to classify projective representations of G (that is continuous homomorphisms from G to PU(H)). The point is that doing that may give us a clue about what the space H should be.

It would be very nice if one could lift the projective representation to a unitary (G → U(H)) one, since those are much more well studied. There are two kind of obstructions that may arise. The first is that G may not be simply connected so that it is not even sure if there is a continuous map from G to U(H) associated to each representation. This problem is solved by passing to the universal over of G. The second problem in lifting projective representations is a cohomological one, which may be stated for the Lie algebra of G: every projective rapresentation of G determines an element a of the second cohomology (with real coefficients) H²(g,R) of the Lie algebra g of G, and this representation can be lifted to a unitary one iff a = 0. This is called Bergmann's theorem. This second obstruction is very important since: for the Poincare group we have H²(p,R) = 0 so that all projective representations of the (double cover of the) Poincare group can be lifted to unitary ones, which is not the case for the Galilean group for example.

About the unitary representations of the (double cover of the) Poincare group, today it is done using the so called Mackey machine. The treatment in the first book about QFT of Weinberg uses a kind of ad hoc method which, if memory doesn't fail me, has the problem of not giving a practical description of H (which for irreducible rep., using the Mackey machine, one directly sees is a space of square integrable functions on orbits in momentum space) making it more difficult to link this description to the the usual one with relativistic wave functions in position space (which is a kind of Fourier transform with some issues to deal with).

Good references on theese subjects are Simms Lie groups and quantum mechanics; Varadarajan Geometry of Quantum Theory (last 5 chapters); Varadarajan "Introduction to supersymmetry for mathematicians" (the very nice outline in the first chapter); and last but not last the beautiful "Quantum field theory, a tourist guide for mathematicians" by Folland. Skyhc (talk) 12:58, 17 May 2010 (UTC)

I tried to start correcting some of these errors you pointed out, but I can't get very far because I don't know enough and don't have time to learn it. Of course if you try editing it yourself I will try to improve your English (which actually sounds fine to me). I will also be attempting to keep it comprehensible for physics undergraduates who don't know much math beyond basic linear algebra and maybe group theory, who are I think much of the audience for this article. --Steve (talk) 19:02, 17 May 2010 (UTC)

I strongly second Steve above. If you're knowledgeable, which sounds to me like you certainly are, you should not hesitate to edit. Mct mht (talk) 02:01, 21 January 2011 (UTC)

This article is in a pretty bad shape. It shoud be hidden. Jtico (talk) 19:56, 19 January 2011 (UTC)