Talk:Symmetry in quantum mechanics

To add in time

• particle exchange,
• isospin,
• C, P, T, symmetries, CP violation, the CPT theorem,
• Lorentz violation,
• which quantities/operations are conserved/preserved in which fundamental interactions,
• matter/antimatter,
• some things like QFT gauge theories and supersymmetry are already mentioned but should be expanded on,
• expand on color charge symmetry?

I don't know enough about the scale invariance/renormalization group, topological conservation laws, and instantons, so will leave that for those inclined and knowledgeable. M∧Ŝc²ħεИ_τlk 07:01, 5 June 2013 (UTC)

I know this is out of nowhere, but to be crystal clear on the article's scope in case anyone comes along in the future and questions it: this article should clarify all the jazz about generators and groups and the links between operators, parameters, representations... It should certainly

• not just list symmetries,
• not state formulae without some basic outlining derivations (they are allowed and should be included to illuminate connections between things),
• not just duplicate all the formulae elsewhere, but collate them in a very careful way showing the similarities transcending space/time translations, boosts, rotations, then similarly for all the discrete symmetries and quantum numbers, then similarly for all gauge fields and their potentials and symmetries, etc.

To exemplify - some of the equations in the section angular momentum operator#Angular momentum as the generator of rotations make little sense to me. I find the versions in the article now much more comprehensible. Statements like

"let $R({\hat {n}},\phi )$ be a rotation operator, which rotates any quantum state about axis ${\hat {n}}$ by angle $\phi$."

and

"In simpler terms, the total angular momentum operator characterizes how a quantum system is changed when it is rotated."

carry little information (so what if a "quantum state is rotated" - what does that physically correspond to?). Textbooks say the exact same thing and I hate it. It's so much clearer to actually say what is meant from the start: "the position coordinates are rotated and the transformed wavefunction is a function of the new coordinates". One encounters the non-relativistic wavefunction before Dirac notation and the abstract generality of quantum states, and it should stay that way.

The derivation here Momentum operator#Derivation from infinitesimal translations is a complete stumbling block to follow from the start - it kicks off with an unfamiliar equation and Dirac notation, unfortunately for the layman.

Basic Taylor expansions of a multivariable function is the minimum amount of mathematics required and this is what the derivations I wrote start with.

Anyway - enough of me talking. Apologies for giving the wikiproject a second lecture on this. M∧Ŝc²ħεИ_τlk 15:02, 8 June 2013 (UTC)

"...and it should stay that way": I don't want to be contrary, but insisting on introducing anything nonrelativistic first is going to forever hamper understanding and keep quantum mechanics arcane, and I'm not the only one who thinks that. I'm not disagreeing with the scope as you describe it, though.

Unfortunate choice of example on rotation – while I agree with your point, you are describing a passive transformation (change of coordinates), not an active one (rotation of a state). The two are not equivalent (classically, uniform rotation of coordinates does nothing, uniform rotation of the state causes everything to fly apart). Or am I wrong? — Quondum 16:07, 8 June 2013 (UTC)

What?

"insisting on introducing anything nonrelativistic first is going to forever hamper understanding and keep quantum mechanics arcane"

What's wrong with starting with the non-relativistic case then generalizing to the relativistic case after? How many treatments on QM are there placing RQM before non-RQM?

And for the sake of losing some generality, analysis with a wavefunction as a function is easier and more familiar than abstract elements of Hilbert spaces, which is why a treatment of functions, even if limited in scope, should come first. Also compare the number of people who have seen Dirac notation (presumably lots) to those who understand what it means (physics undergrads and above). Unsurprisingly there have been numerous complaints on QM talk pages that dislike any and all jumps into Dirac notation. Sometimes it may be essential, but here it isn't.

And what do you mean by a (classical?) uniform rotation of the state? If you mean rotating every consistent of the classical system all by exactly the same angle, then why should that cause everything to fly apart? There is a description of this at Poincaré group, though you would have seen the article. "Rotating state vectors" may be in the textbooks but it's unhelpful phrasing. M∧Ŝc²ħεИ_τlk 17:15, 8 June 2013 (UTC)

I'm sorry I started a debate on details; I think the main point is one we agree on: that a phrasing like "rotating state vectors" just doesn't say enough to tell you what it means. I also agree that to jump into more difficult notation and concepts is to be avoided if reasonably possible. — Quondum 17:53, 8 June 2013 (UTC)

Section ordering and general introduction

As it stands now, the first section after the lead is "Overview of Lie group theory". I have nothing against Lie groups, but I would suggest that a better way to organize the article would be to start with an intuitive explanation of what's implied by Noether's theorem. Even experimentalists like myself get a lot of mileage out of symmetry arguments, and I have no clue what a Lie group is =p. I'll give a shot at this later, going roughly from the treatment in Sakurai. a13ean (talk) 23:32, 5 June 2013 (UTC)

Had a try... M∧Ŝc²ħεИ_τlk 06:47, 6 June 2013 (UTC)

Looking better. I'll also try my hand when I have a chance, but a few points that I find particularly useful are

• Degeneracies and symmetries: ie for rotational invariance we know that since [D(R), H]==0, [Ji,H]=0, so from an initial state any states you can get by applying some combination of the Ji's must have the same energy. Good quantum numbers, etc. Similarly, if H commutes with parity then any eigenfunction that's odd under parity is at least two-fold degenerate, and so on.
• Brief mention of Bloch's Theorem
• Brief mention of symmetry considerations in solid state physics
• Time reversal invariance and real eigenfunctions.

a13ean (talk) 17:12, 6 June 2013 (UTC)

A good list: the commutators with the Hamiltonian and degeneracies are obviously missing. Bloch's theorem and other instances from condensed matter physics would make a very valuable section: an important subject for real-life exemplifications which would remove bias from fundamental QM theory and plain applications in particle physics. I'll try more later too. M∧Ŝc²ħεИ_τlk 05:46, 8 June 2013 (UTC)

Choice of article title

This article, as it stands, captures a lot of details about individual symmetries, and would be better described by the title Symmetries in quantum mechanics. The distinction is that with the existing title, one expects an explanation of the role of symmetry in quantum mechanics, and with the title as I've suggested one expects a list-like approach, detailing each symmetry (which is what it appears to be trying to do). Symmetry (physics) already covers the general picture pretty well, even with regard to quantum mechanics. — Quondum 13:51, 8 June 2013 (UTC)

That was the original title... but Teply suggested to use the singular and not plural on my talk page here. I don't see much problem either way. If there is a broad consensus to change the title, I don't mind. M∧Ŝc²ħεИ_τlk 15:02, 8 June 2013 (UTC)

Before anyone says "THERE ARE ERRORS"...

Yes, there may well be.

The whole article needs to solidify the conventions used, as well as fixing more headache-beating notations for what means what. Some conventions are already decided, but others like which Minkowski metric needs to be chosen and used throughout.

The worst nightmare is the letter D, which could mean the Wigner matrices in the context of spin matrices (or possibly representations for the Wigner matrices themselves?), representations of group elements, or representations of generators. Perhaps something like:

$D$ for the representations of group elements,

${\bar {D}}$ for the representations of generators,

would be clearer.

The next nightmare is, what are A and B? If J and K are generators, so must be A and B.

According to E. Abers, A and B start off as generators, then after some wishy-washy use of the term "representations" (with reference to what??), then finally states D^{(a, b)}(J) are still the angular momentum operators (rotation generators) and D^{(a, b)}(K) the boost generators. However - the D^{(a, b)} notation refers to a representation of the boost and rotation generators, not the generators themselves. I followed E. Abers to be on the safe side. Representation theory of the Lorentz group is vague on what A and B actually are. My consensus is that A and B are simply generators, while D^{(a, b)}(K) and D^{(a, b)}(J) are irreducible reps that happen to be able to be expressed in terms of the generators A and B, instead of the reps of generators D(A) and D(B), i.e. D^{(a, b)}(K) = −i[D(A) − D(B)] where the reps are just the identity transformations? Still looking for refs which make it crystal clear.

For sake of clarity, all expressions using hats will be converted to LaTeX, even if they're inline.

Will fix ERRORS soon. M∧Ŝc²ħεИ_τlk 14:01, 22 June 2013 (UTC)

I'll choose

• the Minkowski metric (+,−,−,−),
• ${\bar {D}}$ for reps of generators. The overbar is not to be confused for complex conjugation, which is denoted by a star.

For the record, the above stroked out post (on the Lorentz group irreps) has been resolved, by means of Weinberg's books on QFT (vol 1) - at least the expressions for A and B are correct now. They were already given in representation of the Lorentz group but that article is equation-dense and difficult to follow, so didn't immediately recognize... M∧Ŝc²ħεИ_τlk 21:42, 27 June 2013 (UTC)

C.N. Yang (1957)

A quote from C.N. Yang's Nobel lecture in 1957 [1]:

"It was however not until the development of quantum mechanics that the use of the symmetry principles began to permeate into the very language of physics. The quantum numbers that designate the states of a system are very often identical with those that represent the symmetries of the system. It indeed is scarcely possible to overemphasize the role played by the symmetry principles in quantum mechanics. To quote two examples: The general structure of the periodic table is essentially a direct consequence of the isotropy of Coulomb's law. The existence of the antiparticles -- namely the positron, the anti-proton and the anti-neutron -- were theoretically anticipated as consequences of the symmetry of physical laws with respect to Lorentz transformations..."

The article needs to do much more to explain how this close association between symmetries and states arises -- what it is in the framework of quantum mechanics specifically that makes this happen. Jheald (talk) 18:58, 1 July 2013 (UTC)

Of course it does. Thanks very much for the reference to Yang. Feel free to add it. M∧Ŝc²ħεИ_τlk 21:01, 1 July 2013 (UTC)

Irreducible/Reducible

I made a tweak of the definition in the article, effectively not defining what "reducible" means. The reason is that not irreducible isn't necessarily the same thing as reducible (as plain English suggests). It is true for semi-simple groups, but not in general. YohanN7 (talk) 18:22, 7 November 2013 (UTC)

Thanks for the refs to Hall and other tweaks.

You may be interested in the now-an-article irrep, which I tried to change from a redirect to an article, and failed. User:R.e.b. has introduced the concept of "indecomposable" as different to "irreducible". Seems like the term is confused in places (not just me). M∧Ŝc²ħεИ_τlk 05:42, 8 November 2013 (UTC)

"Indecomposable" refers (in Halls book) to the Lie algebra itself, while irreducible (universally) refers to reps. Technicalities about the terminology: Suppose that there is an invariant subspace U of a rep. Then the orthogonal complement of U may or may not be an invariant subspace itself. In the former case, the rep is reducible (to U and its complement), but not in the latter case. If the space breaks up completely (by repeatedly finding invariant subspaces and their invariant complements), then the rep is completely reducible. If every rep of a group/algebra breaks up this way, it has the complete reducibility property. Semisimple Lie algebras all have the complete reducibility property.

I have quickly glanced the new article, and I suggest that the term indecomposable should be avoided because it isn't standard, and apparently has multiple meanings. YohanN7 (talk) 14:46, 10 November 2013 (UTC)

When you say "Then the orthogonal complement of U may or may not be an invariant subspace itself.", is this to do with linear independence of basis vectors between the spaces?

Are you asking about what the orthogonal complement means? If so, yes, the orthogonal complement to a subspace is the set of vectors orthogonal to that subspace. This implies linear independence. What I wrote may be better presented without reference to "orthogonal" though, since it makes it necessary to choose an inner product.

B t w, did you know that there is an algebra named after you? User:YohanN7/Gamma matrices YohanN7 (talk) 21:15, 10 November 2013 (UTC)

I'm going to copy and paste some of this thread over to talk:Irreducible representation, so others can post on the more relevant page. Hope you don't mind. Note R.e.b. actually wrote indecomposable without adding citations to support the terminology, and where I added references which do in fact use the term "irrep".

As for this article I would agree to keep things as simple as possible, and only use "irrep" where needed.M∧Ŝc²ħεИ_τlk 17:04, 10 November 2013 (UTC)

I know what orthogonal complement means, but not so much about invariant subspaces... M∧Ŝc²ħεИ_τlk 21:22, 10 November 2013 (UTC)

An subspace is invariant or stable under the action of a rep if no element of a rep can take a vector in the subspace out of it. It is irreducible if it contains no smaller invariant subspaces. Naturally, there are the standard mathematical qualifiers like nontrivial and such splattered around in the definitions in the obvious way, including or excluding the whole space and the zero subspace as appropriate. YohanN7 (talk) 23:00, 10 November 2013 (UTC)

Combining rotations and boosts

What is this?

$\mathbf {M} ={\begin{pmatrix}0&K_{1}&K_{2}&K_{3}\\-K_{1}&0&J_{3}&-J_{2}\\-K_{2}&-J_{3}&0&J_{1}\\-K_{3}&J_{2}&-J_{1}&0\\\end{pmatrix}}\,,$

and correspondingly, the boost and rotation parameters are collected into another antisymmetric four-dimensional matrix ω, with entries:

$\omega _{0a}=-\omega _{a0}=\varphi {\hat {a}}_{a}\,,\quad \omega _{ab}=\theta \varepsilon _{abc}{\hat {n}}_{c}\,,$

or explicitly in the standard form:

${\boldsymbol {\omega }}={\begin{pmatrix}0&\varphi {\hat {a}}_{1}&\varphi {\hat {a}}_{2}&\varphi {\hat {a}}_{3}\\-\varphi {\hat {a}}_{1}&0&\theta {\hat {n}}_{3}&-\theta {\hat {n}}_{2}\\-\varphi {\hat {a}}_{2}&-\theta {\hat {n}}_{3}&0&\theta {\hat {n}}_{1}\\-\varphi {\hat {a}}_{3}&\theta {\hat {n}}_{2}&-\theta {\hat {n}}_{1}&0\\\end{pmatrix}}\,,$

YohanN7 (talk) 02:01, 11 November 2013 (UTC)

It should be clear that M is the generator of boosts and rotations, collecting the generators in Representation theory of the Lorentz group#Explicit formulas into a single matrix, which corresponds to relativistic angular momentum (if you haven't seen that article, feel free to nitpick on the talk page there). Similarly, ω collects the parameters into one matrix. This is from the book by E. Abers.

You should know this. But apparently it isn't clear so what's missing? M∧Ŝc²ħεИ_τlk 09:52, 11 November 2013 (UTC)

No, I don't know this, and I doubt that it is "standard". What is the point in displaying a matrix of matrices (the M)? This will confuse the reader because the supermatrix is of no help algebraically. He already has a couple of levels of abstraction to grasp. Even the fact that M^μν is a matrix will be confusing to him since it looks like we are talking about components on the surface of things. The ω makes a little more sense if you remove the hats.

But, ..., I am changing my mind now. The supermatrix is of help if you explain it better, and point out its feature and virtues (antisymmetry all the way) and that the entries of ω are a priori "forced" to be antisymmetric because any symmetric part would drop out anyhow. YohanN7 (talk) 12:49, 11 November 2013 (UTC)

The expression:

$\omega _{\alpha \beta }M^{\alpha \beta }=2(\theta {\hat {\mathbf {n} }}\cdot \mathbf {J} +\varphi {\hat {\mathbf {a} }}\cdot \mathbf {K} )$

is in a number of pdfs floating around (in the article and google) and at least some books (such as T. Ohlsson (2011) p.7-8), and it's also exactly what you have on your page (aside from different letters). It should also be clear that the hats on n and a and their components indicate they are unit vectors, not operators or anything else. Then again... it looks like each component is a unit vector, but it should be clear from the context. I haven't had much of a chance to see how standard what I wrote above is through more refs.

I believe the expression (slightly wrong as it stands, two theta) above should go into the article.

For the matrix ω, yes, it looks like each component is a unit vector. There is no way that this is standard, but that doesn't matter if you define things, like if you somewhere write ${\hat {\mathbf {a} }}=({\hat {a}}_{1},{\hat {a}}_{2},{\hat {a}}_{3})$. YohanN7 (talk) 10:34, 17 November 2013 (UTC)

Copy/paste typo... The expressions $\theta {\hat {\mathbf {n} }}=\theta ({\hat {n}}_{1},{\hat {n}}_{2},{\hat {n}}_{3})$ and $\varphi {\hat {\mathbf {a} }}=\varphi ({\hat {a}}_{1},{\hat {a}}_{2},{\hat {a}}_{3})$ are included, but shall rewrite as you suggest. M∧Ŝc²ħεИ_τlk 11:05, 17 November 2013 (UTC)

Also, this matrix expression is nothing compared to the other "levels of abstraction" the reader has to grasp, so I can't see what that's relevant.

But what do you use it for? (The M matrix.) And how? Not matrix multiplication from what I can see. It confuses me at any rate. YohanN7 (talk) 10:34, 17 November 2013 (UTC)

Nothing to do with matrix multiplication. I thought it was just to collect the boost and rotation generators into one generator for boosts/rotations in spacetime, analogous to the energy and 3-momentum operators into the four-momentum operator as the generator of spacetime translations, then these are used in the commutation relations and boost/rotation operators etc. for the Lorentz/Poincaré group. The matrices seem like more trouble than the space they take up (lots), so the explicit matrices may as well be deleted, but not the component definitions which are correct (aside from typos and sign conventions). M∧Ŝc²ħεИ_τlk 11:05, 17 November 2013 (UTC)

The analogy with 4-momentum looks fine. How about "By analogy with ..., define the (cartesian (to distinguish from spherical)) tensor operator M = ..."? YohanN7 (talk) 17:11, 17 November 2013 (UTC)

Sure. M∧Ŝc²ħεИ_τlk 20:03, 17 November 2013 (UTC)

However, what seems more commonly discussed in RQM and is not in this article are the similar-looking expressions for the transformations of Dirac bispinors in terms of the spin angular momentum like this (such as A. Wachter (2011) p. 123):

$\omega \sigma _{\alpha \beta }{(I_{n})}^{\alpha \beta }\,,\quad \sigma _{\alpha \beta }={\frac {i}{2}}[\gamma _{\alpha },\gamma _{\beta }]\,,\quad I_{n}={\begin{pmatrix}0&0&0&0\\0&0&u_{3}&-u_{2}\\0&-u_{3}&0&u_{1}\\0&u_{2}&-u_{1}&0\end{pmatrix}}$

where (using Wachter's unfortunate notation) I_n is the rotation matrix around a unit vector u and ω the rotation angle, instead of the one above for M. In the book by W. Greiner (1990) ch. 16 there are similar expressions, in which he includes boosts along the 0i and i0 entries for i = 1,2,3 of I_n, but tends to give concrete values of indices rather than general matrix expressions. M∧Ŝc²ħεИ_τlk 08:01, 17 November 2013 (UTC)

The I_n^αβ of above are (at least, should be, don't know Wachter) reps of the Lie algebra of the Lorentz group for spin n/2. Expressions like the one above belong in the article too. They should be on general form (with or without all indices, like in I_n^αβ_μν), the restriction to spin 1/2 isn't necessary (or desirable). YohanN7 (talk) 10:34, 17 November 2013 (UTC)

Agreed, we want the formula for any spin, but the spin-1/2 case should be mentioned. I don't have time now though. M∧Ŝc²ħεИ_τlk 11:05, 17 November 2013 (UTC)

For reference,

$\phi _{r}(x)\rightarrow \phi _{r}(x)+{\frac {1}{2}}\omega _{\mu \nu }(I^{\mu \nu })_{rs}\phi _{s}(x)$ when the ω_μν are infinitesimal.YohanN7 (talk) 12:27, 17 November 2013 (UTC)

Yes, very often the first order expansion is given (this is already the case for the non-relativistic time/space/rotation operators). M∧Ŝc²ħεИ_τlk 20:03, 17 November 2013 (UTC)

Real irreducible representations and spin

I appreciate the effort to keep things simple, but I believe that some care needs to be taken so that the math is right. For instance, expressions like,

The irreducible representations of D(K) and D(J), in short "irreps", can be used to build to spin representations of the Lorentz group. Defining new generators:

$\mathbf {A} ={\frac {\mathbf {J} +i\mathbf {K} }{2}}\,,\quad \mathbf {B} ={\frac {\mathbf {J} -i\mathbf {K} }{2}}\,,$

so A and B are simply complex conjugates of each other, it follows...:

is the reason I began working on the Lorentz rep article in the first place. This here is easy to read, and the reader may well think he understand what he is reading, but he doesn't.

I am taking the above as an example. Physics books are all, almost without exception, horrible HORRIBLE at explaining group theory adequately, and the example above is quite typical. It's something pulled out of the hat. It sounds good, but explains little.

I like the general approach in the article (as I have said before), but simplicity and volume shouldn't come at the price of moderate mathematical rigor.

My advice would be to take math (and even mathematical terminology, provide a translation table) as a baseline. A reader should be able to leave the article with ,for instance, at least some understanding of the interconnections between su(2), SU(2), so(3), SO(3), so(3;1), SO(3;1)⁺, etc, and SL(2;C) for that matter since spin is involved, even if it makes the article a little "harder" to read (and write). YohanN7 (talk) 02:49, 11 November 2013 (UTC)

I agree with everything you say. But please advance a specific way to rewrite the part you quote from the article, similarly for the other yucky bits. M∧Ŝc²ħεИ_τlk 09:52, 11 November 2013 (UTC)

Point taken. One technical issue is that you are not defining new generators. You leave the Lie algebra so(1,3) here since it is a real Lie algebra, and hence not closed under multiplication by i. In the sequel, there are more similar points. If you want something very explicit (far too explicit for the article), check out that old proposal I wrote for the L rep article. You know where to find it, and it comes with bugs and w/o warranty. YohanN7 (talk) 13:02, 11 November 2013 (UTC)

Yes. According to Ohlsson they A and B are simply operators, which makes sense. , since in physics generators tend to be operators corresponding to conserved quantities (as we know: energy for time evolution, momentum for translation, angular momentum for rotation). Then of course D(J) and D(K) are the representations of the J and K operators (irrelevant). M∧Ŝc²ħεИ_τlk 20:03, 17 November 2013 (UTC)

Not sure what you mean here. YohanN7 (talk) 21:38, 17 November 2013 (UTC)

J and K A and B are operators. Or not? In any case I removed the statement that they are generators. M∧Ŝc²ħεИ_τlk 21:42, 17 November 2013 (UTC)

The J and K are of course operators and generators of (reps of) the Lorentz group. The A and B are operators, but they not generators of the Lorentz group. YohanN7 (talk) 22:29, 17 November 2013 (UTC)

Actually, we need to be more precise than this. There is a desperate need for a translation table between math and physics. YohanN7 (talk) 22:37, 17 November 2013 (UTC)

Every single edit today has been riddled with mistyped comments... I meant A and B. Anyway, OK. M∧Ŝc²ħεИ_τlk 22:36, 17 November 2013 (UTC)

Nice idea, let's expand on this below. M∧Ŝc²ħεИ_τlk 16:59, 18 November 2013 (UTC)

Thank you very much, YohanN7, for your criticisms. Beyond the simple facts the physics books are really horrible when they (try to) treat the irreps of the Lorentz group (and its covering group). I say this though I'm a theoretical physicist who works more like an engineer with Feynman diagrams... What is largely missing in the textbooks is the concept of complexification. Also, the unitarian trick of Weyl is poorly explained, sometimes with only two or three words. (It is not that thing with making a rep of a finite/compact group unitary. Weyl himself used this wording for something different, see ch. VIII sec. 11. of his "Classical groups") I never understood it until I discovered the roughly 40 pages in Naimark's and Stern's book on group representations. Complexification of vector spaces, real forms of Lie algebras and of Lie groups are great concepts and useful to really understand the constructions of the irreps. The book of Jeevanjee would be a good starting point. (I started preparing a seminal text treating the irreps of the Lorentz group but it's a hard work.) Stefan Neumeier (talk) 12:35, 17 November 2013 (UTC)

Excellent, we can do with a theoretical physicist around here. I don't have the books you mention (I have Weyl's "QM and Group Theory" (or is it "Group Theory in QM"?) somewhere though.) Is Unitarian trick providing an adequate explanation of the trick?

Perhaps we can beef up quite a few existing articles with the right stuff. When it comes to representations, the concepts of tensor products, direct sums, complexification, duals, quotients, etc aren't coherently described anywhere. YohanN7 (talk) 13:33, 17 November 2013 (UTC)

By all means Stefan Neumeier and YohanN7 don't hold back. Feel free to correct/cut out anything you think is irrelevant. Later I'll have a look for some of the books Stefan Neumeier mentioned. M∧Ŝc²ħεИ_τlk 20:03, 17 November 2013 (UTC)

Translation table between physics and mathematics terminology (generators, operators, reps, irreps...)

What would it look like? Let's present it below. M∧Ŝc²ħεИ_τlk 16:59, 18 November 2013 (UTC)

For internal usage we should define what we mean by a representation. "Representation" is probably one of the most (mis)used word (both i math and physics). For a framework, we should explicitly assume a Hilbert space H (the elements being quantum states), and the set of linear operators on it, End(H).

In our case, a representation then could be a subset of End(H) endowed with the appropriate structures. Here we need to be precise. Some would mean that a representation is a subset of H itself. A formalist might argue that the representation is the map from the abstract group or algebra to End(H). Neither is wrong, but we should be precise by what we mean.

Generators are of course elements of a Lie algebra, but here it becomes murky. Should we let generators be elements of representations of Lie algebras? Logically, yes, since the reps are as much Lie algebras as the abstract Lie algebra itself. Also, when plural is used, it is often meant a complete set of basis vectors for the Lie algebra.

There's much more to define, but lets cook this table slowly. YohanN7 (talk) 17:27, 18 November 2013 (UTC)

And now I got a brilliant idea: Make a picture with the Hilbert space, the operators, and the groups/algebras. If you come up with just something here to start with, we can find a way to make it very illustrative. YohanN7 (talk) 17:32, 18 November 2013 (UTC)

Not sure how to make pictures of Hilbert spaces, operators, groups, algebras, at the moment. But there is a diagram in Ohlsson's book p.7 figure 1.2, would that sort of thing be useful? M∧Ŝc²ħεИ_τlk 18:01, 18 November 2013 (UTC)

Google wouldn't let me see Ohlsson's book. For visualizing our thingies, consider the Hilbert space, the groups, etc a sets. A rep sitting inside the operators on Hilbert space is a subset of all operators. The invariant subspaces of that rep will be a subset of the Hilbert space. Etc. A set can be represented by a box/circle/cloud-like thing or whatever looks good and conveys the right message. Arrows for maps can't be wrong. YohanN7 (talk) 19:27, 18 November 2013 (UTC)

Sorry the link didn't work, have you tried going to google books and typing "relativistic quantum physics"? One of the first books which appears should be Ohlsson's. I'll make a quick SVG to upload soon.

Anyway, what you were saying about Venn-like diagrams with arrows is actually what Ohlsson's fig 1.2 is like for classifying Lorentz groups. M∧Ŝc²ħεИ_τlk 21:46, 18 November 2013 (UTC)

Yeah, something like that, but much beefier with more and larger "components". I'm thinking about a screen filling thing that clarifies where various notions "live" and how they are interconnected. In addition, it should relate to our terminology. Lie algebras, Lie groups, the Hilbert space, and the operators are four natural "big blobs" in the picture I have in mind. "Sub-blobs" are reps (of various dimensions) inside the operator blob, their invariant subspaces inside the Hilbert space. Natural maps are the "exp" maps between algebras and groups (in several places) and the π (or D) for reps. YohanN7 (talk) 15:52, 19 November 2013 (UTC)

Supersymmetry

So, being in a speculative mode, I fleshed out a bit. I can (and will in due time) back up all with refs, except the last statement:

"A massive high spin relatively stable particle is inherently inert, in that it would be difficult to balance the relativistic momentum and total angular momentum in particle reactions in a Feynman diagram."

I made this up, but think that it is true. Does someone know of a ref where this is discussed? If not, it has to be scrapped. At any rate, it needs reformulation. In essence, "difficult to balance..." -> extremely small cross section. (This is why we don't see it.) YohanN7 (talk) 17:45, 11 November 2013 (UTC)

I can only apply rough heuristics to this, but it sounds like you might be assuming that few modes of interaction implies small cross-section. In only takes the right (possibly exotic) particles to interact with. Oxygen is chemically inert until you mix it with a reducing agent... I'd suggest removing it as OR, and if you do com across this is some guise, insert it then. — Quondum 19:23, 11 November 2013 (UTC)

Remove it, but where do you see the O or the R in OR? But let it be for a couple of days plz. Somebody knowledgeable of the subject might know.

(It probably fits better into the WP:BS category than the WP:OR, but I don't think it is even BS. It is by no means the worst problem with this or any article.) YohanN7 (talk) 19:54, 11 November 2013 (UTC)

I've never seen such an argument, i.e, that it is phase space factors and conservation constraints, not strength of interaction, used to justify the non-interacting nature of higher spin particles. It is possible to construct massive interacting higher spin theories, e.g., [2] and [3]. String theory has massive particles of arbitrarily high spin, although those aren't seen because we don't live at Planck scale. --Mark viking (talk) 20:34, 11 November 2013 (UTC)

Ok, I'll remove it. Thanks! YohanN7 (talk) 00:36, 12 November 2013 (UTC)

Missing topics

With a general title like this, there should also be discussion about symmetries typically encountered in condensed matter systems and chemistry. General discussion about symmetry breaking is also needed. Most likely, discussion about such topics already exist somewhere in WP, and could be just summarized here. Jähmefyysikko (talk) 08:26, 27 February 2024 (UTC)