User talk:YohanN7/Representation theory of the Lorentz group

Change to Representation theory of the Lorentz group
I want to make some chages to Representation theory of the Lorentz group. Some issues are mentioned (by me) at Talk:Representation theory of the Lorentz group.

I have written a draft for the section Representation theory of the Lorentz group.

Another yet so far unaddressed problem is that the (m,n) representations do not yield representationd of the group, only the algebra when either m or n is half-integer.


 * I'm a bit at sea on the actual content (representation theory, Lie groups and many concepts behind quantum physics), so I'll make only surface comments:
 * The existing section Properties of the (m,n) irrep is brief and opaque in the extreme, and this is a far more accessible replacement.
 * The existing section makes a statement about the angular momentum operator that does appear in the replacement. Should something similar be included?
 * Yes it should. It's there now, but it's (too) implicit. The SO(3)-invariant subspaces have definite spin.
 * The terms "rep" and "irrep" occur in isolated spots in the article. While the meaning seems obvious through inference, I don't think these should appear in section headings, and if used in the text at all should be introduced.
 * I agree.
 * The last two paragraphs seem to make claims of some results in quantum physics follow from representation theory and some very limited constraints. Firstly, these results would follow with or without representation theory; representation theory may be a powerful means to uncover results, and if so, that is how it should be presented.
 * Yes, probably. Weinberg (reference) actually arrives at the Dirac eqn, the Maxwell eqn, the Klein-Gordon eqn, and others, from general principles (like Lorentz invariance and the unitarity of the S-matrix) and particular representations of the Lorentz algebra. Every "derivation" of these equations is more or less ad hoc. Weinbergs prescription is by far the "least" ad hoc version I have ever seen. But agreed, the last two section need work.
 * The last parapgraph in particular leaves too much hanging in the air, in a way that the reader would not know how to go about fleshing out an understanding of the claim. That the Dirac equation can be derived from the constraint of parity invariance alone (aside from the obvious relativistic requirement of being under the full Lorentz group) seems improbable: there must be further constraints not stated here. The charged fermions are surely not the only fields that satisfy the parity invariance constraint.
 * It's living inside the (1/2,1/2) (1/2,0)+(0,1/2) representation that is crucial. Parity invariance then implies that the particles are fermions. Then there is the "cluster decomposition principle" stating basiclly that distant experiments don't interfere with each other. Or, in other words, the S-matrix factorizes; Sab,cd = SabScd for processes a->b and c->d at large distances from each other implying that the Hamiltonian must commute with itself at spacelike differences. This is the principle of causality (derived from the cluster decomposition principle). This is enough to arrive at the Dirac equation.
 * Surely not. At fermions, yes. But you need another constraint to distinguish quarks from electrons. The quarks do not satisfy the Dirac equation.
 * I'm missing something. I'll check Weinberg (vol 2) + Greiner's QCD book tonight. Unfortunately, I don't know much about QCD, but I should be able to find this part out given time. Hopefully it's something simple like charge conjucation symmetry that makes the difference.
 * So, the quarks live in the same LT representation as electrons? (I understand that there is the additional symmetry of SU(3)-invariance, but the particles must have a definite behavior under pure Lorentz tranfromations.)
 * B t w, the equations derived this way aren't the usual wave equations. They are Heisenbergs equations of motion for the field (-operators). It "just happens" that these are identical in form with the corresponding "classical" wave equations in some cases. It's true for electrons (Dirac), and scalar particles (Klein-Gordon) and (unless I remember incorrectly) the massive vector particles (Proca). In other cases, they differ from the wave equations.
 * The Klein-Gordon equation is supposed to hold for every component in every massive field. Moreover (as I remember from Weinberg), when considering massless spin 1 and spin 2 (graviton) particles, the Maxwell equations and the Riemann-Christoffel curvature tensor pop up plus the need for gauge invariance (photon field) and "something like general covariance" (graviton field).


 * I have taken the liberty of making minor changes: disambiguation, use of minus (−) and n-dash (–) at places, spelling, minor language tweak. I have made no change to the symbols for groups; these appear to be inconsistent in WP; the non-italic "SO(3,1)" might predominate.
 * Thanks. B t w, is there a way to check the spelling in the editor?
 * Nope. The editor is part of your browser. You may be able to get a spell-checker plug-in.
 * Damned! Actually, I can spell. I just forget how to do it too often;)
 * In all, your text is far more interpretable and complete than as it stands in the article. I would hope others with more than my gloss of understanding in the field will comment, but that can wait until after you make the substitution. — Quondum 06:14, 23 September 2012 (UTC)


 * A thought: Diracs original approach was pretty nifty too. Take Einsteins energy-momentum relation. Then "canonically" replace E and p with quantum operators. Then operate on a state with at least two components (spin up + down)->Four components + the Dirac matrices in a Clifford Algebra - and the Dirac equation. Note here that you don't need even Lorentz invariance explicitly because it's already there (Einsteins energy-momentum relation)!


 * I like derivations from clearly stated minimal principles. But I'm out of my depth on any of the treatments (I like Clifford algebras, but multiparticle field theory is beyond me; the interpretation of the Dirac equation in geometric algebra rather than using his matrices is very pretty).


 * Thanks for the input. I'll go through the bullets carefully beginning tomorrow. You have made it clear that I either need to flesh out or cut down in the last two sections. I'll try first to flesh out.


 * Most needed additions:
 * Link to the S-matrix + short explanation of the requirement of unitarity (invariance of  under LT)
 * Link to principle of causality + infer [H(x),H(x')] = 0 for (x-x')2 spacelike.
 * Of course, sources, Mostly Weinberg. It can't be wrong to stick to the same source. There is certainly nothing controversial here (except that I might put things incorrectly). YohanN7 (talk) 11:32, 23 September 2012 (UTC)

Field equations from Lorentz invariance
I have failed to mention something of some importance. The equations are the free field equations. No interactions are present.

Given the (1/2,0)+(0,1/2), spin 1/2, and general considerations, then parity invariance is actually enough to derive the free field Dirac eqation,

The assumption (1/2,0)+(0,1/2) leads to an matrix equation given roughly by UJ=SU and VJ=SV where J are the "usual" spin 1/2 rep of the rotation group and S are the Pauli matrices. [It is here that the special representation enters the picture.] U(V) is built up from the coefficient function u(p,s) (v(p,s)) of the field creation (annahilation) operator evaluated at zero momentum p. The spin z-component is denoted s.

Apply Schurs lemma -> u(0,-1/2), u(0,+1/2), v(0,-1/2) and v(0,-1/2) (four column vectors) are found with all together four unknown constants. Parity invariance will fix these. I.e. we have spinors at zero momenta (up to a constant). Arbitrary momenta are reached by an LT in a standard way. [The spacetime dependence of u(v) is exp((+/-)ipx))].

Then consider the commutator of the fields. The fields are essentially a momentum integral over the coefficient functions multiplying the creation and annahilation operators, u(p,s)a'(p,s)exp(-ipx)) and v(p,s)a(p,s)exp(ipx)) respectively.

We have [ψl, ψ'l'] = an integral over p ( = 0 if causality is to hold for spacelike distances)

A little algebra (2 pages) shows that this happens only if the free field Dirac equation holds for the field operator ψ. YohanN7 (talk) 18:57, 23 September 2012 (UTC)

I don't know what to do with the quarks yet, but I see it as concievable that a free non-interacting quark actually does satisfy the free Dirac equation (if it lives in the (1/2,0)+(0,1/2) rep together with its antiquark). There appears to be no escape from this. After all, such a quark would be just an electron, but with a different mass and a different charge under. It couldn't behave differently from and electron under LT.

Aha!!

Here is the Lagrangian from Quantum chromodynamics:

\mathcal{L}_\mathrm{QCD} = \bar{\psi}_i\left(i \gamma^\mu (D_\mu)_{ij} - m\, \delta_{ij}\right) \psi_j - \frac{1}{4}G^a_{\mu \nu} G^{\mu \nu}_a $$ This gives exactly the free field Dirac equation with G = 0 and D = ordinary partial derivative! YohanN7 (talk) 19:57, 23 September 2012 (UTC)


 * Hmm. All way over my head, of course, and doing a lot of handwaving. That Dμ hides a whole pile of coupling constants and vector potentials, and even if you set the vector potential(s) to zero, you still get the self-interaction via the electromagnetic field. So perhaps it is the field equation of a neutrino if you treat it as "free field". I must say I don't really buy the power of inference at this point. But I can see what you're driving at. — Quondum 20:52, 23 September 2012 (UTC)


 * There are no interactions when one talks about a free particle. There are no self-interaction terms in the original interaction either. (Self-interaction via the EM field isn't self-interaction by definition.) All coupling constants can be set to 0. Then Lagrangian above reduces to that of the free electron. They are identical. It's another thing that there are no free quarks in reality. The important thing is that they would satisfy the free Dirac equation.


 * One should probably think of the actual home of the quark wave function as a tensor product of a "color freedom vector space" [an SU(3)-rep] and the vector space associated to the (1/2)+(1/2) representation of SO(3;1). The neutrino lives in a different representation.


 * At the moment the article is in a complete mess, and it is getting too big. So please don't read now. I'll continue tomorrow. Thanks for you help so far. It's highly appreciated!!!!
 * /Johan YohanN7 (talk) 22:38, 23 September 2012 (UTC)


 * A word of caution about developing consequences and additional assumptions for a given context (being physics). All this about physics has been a bit of a diversion from the topic. Remember that "Representation theory of the Lorentz group" is pure math (no physics). Developing "consequences" too far might not be within the scope of the title.
 * I would have assumed that a theory of representations could not lead to conclusions that are not mathematically implicit in the group's action (in this case a Lorentz transformation) anyway. I see from Representation of a Lie group that a representation of a group is a group homomorphism from that group to the automorphism group of a vector space. This definition seems to allow some freedoms: allowable vector spaces that would be compatible with this definition for any given group might not be unique. Also, given a vector space, such a representation may not be unique (up to isomorphism), and the list suggests not. Perhaps you can see what I'm getting at: if there are distinct representations, choice should be addressed. I see a list of representations, which suggests that these are enumerable. The interpretation at this level should be addressed before physics implications (esp. with additional assumptions) detailed, and then not in an ad hoc manner. Or perhaps I'm just too weak in the area to be getting a handle on it. — Quondum 04:07, 24 September 2012 (UTC)


 * Yes, it's too much diversion. I tried to see how short I could make it. It became too long so I stopped in the middle.
 * The (m,n) representations are unique uo to isomorphism of representations. These are the only finite dimensional irreps.
 * New reps (not ireps) can be formed from the irreps, e.g. (m,n)+(p,q), (m,n)*(p,q) [direct sum, direct product], and also (m,n)' [dual rep]. These are general constructions in rep theory. Sometimes there are connections between reps constructed this way, sometimes not. The main point is that SO(3;1) has the complete reducibility property, saying that every rep is isomorphic to a direct sum of the irreps. [complete reducibility property <-> so(3;1) is semisimle].
 * You can't get conclusions from a representation alone. Thats of course correct. Extra assumption are necessary. The simplest assumtion is that of a quantum spin j particle, meaning that jz can only take on the (2j+1) distinct values j,j-1,...,-j. A measurement, or an application of the z-component of the angular momentum operator Jz to a state must give one of the allowable values of Jz. If the states of such a particle transform under an (m,n) irrep then the allowable states must be confined to a subspace which is irreducible under the action of SO(3) - otherwise the spin j of the particle would depend on the basis chosen.
 * General requirements of quantum field theory, including the causality principle, and a prescribed behavior under space inversions, time reversal (in fill Lorentz group) leads to explicit constraints. Examples include free field equations for particles with mass. The free klein-Gordon equation, the free Dirac equation, and others arise in this way.
 * When considering massless particles other restrictions arise. These correspond to gauge degrees of freedom. En example is the electromagnetic potential A. This is living in the 4-vector representation (1/2,1/2), but it is not quite a 4-vector. Using the general framework one can obtain the equation [Lorentz gauge condition here].
 * As far as being "ad hoc" goes, you are right. Any attempt to go deeper without going on for 10 pages, will necessarily be ad hoc. It's better to link the concepts. All I really need to have explicit is the principle of causality. There is no need to prove statements. Since everything here can be properly sourced, we don't need handwaving either.
 * the spin-statistics theorem must be mentioned (linked).
 * The CPT theorem must be mentioned (linked).


 * With ingredient from the above list, I think I can make the last two sections in the proposal short and to the point. [Both CPT and spin-statistics follow from general considerations.] YohanN7 (talk) 12:21, 24 September 2012 (UTC)

An unpolished draft
I'm quite happy with current version, except for the polishing and lack of inline refs.

One thing that worries me is that it is extremely hard not to abuse terminology. QFT is even worse than representation theory in this respect. Operators and the thingies they operate on are used interchangeable and the same notation and terminology is used. Like this:

Consider the tensor fμν ...

There is a lot that is implicitly said here.

/Johan YohanN7 (talk) 16:17, 24 September 2012 (UTC)

Nearly done
I plan one more section: Reducing tensor products (m,n) ⊗ (p,q). Then I'll polish the notation. Oh, yes, references + inline refs.

After that I think I'll be bold! I'm already very seriously out of my depth.

Is there a point in first asking in the math + physics projects talk pages? (I fear that I might be met with silence, or worse - endless discussions with somebody who actually knows too much or too little about the subject.)

If I put this is I'll cause minimal damage since I'll replace a only sentence or two (with about a million new ones). YohanN7 (talk) 21:23, 26 September 2012 (UTC)

Now, on my part, only polishing is left. YohanN7 (talk) 20:04, 27 September 2012 (UTC)

Feedback
The article is too expository and too much like a textbook, and probably too big are among the comments.

I agree.
 * It needs edits to change tone in parts.
 * It needs to be shorter without losing facts.
 * Some general facts (e.g Tensor products, Direct products) that are now outlined could be reduced to links.
 * The QFT section could be removed as whole (or better, be placed in a separate article).
 * The whole section could be split in two: "Finite-dimensional representations" ans "Applications".

On the other hand... (I'll play the devils advocate, i.e. my own):
 * The Lorentz group is the most important group there is besides perhaps GL(n;C) and Sn. We live and breathe inside (1/2,1/2).
 * It it vital to physical sciences. Physicists use it, so the content should be correct and provide basic facts. Today's article does not.
 * I actually never give any proofs except for how to construct the representations. There is Wiki's "no proof policy" of course, but it allows exceptions. There is a construction in the present article (incorrect) which is probably copied from some likewise incorrect textbook. Physics textbooks tend to have serious problems with so(3), su(2), sl(2;C) and sl(3;1). Their use of them has passed the thin line of merely abusing notation in my view. Since finding the irreps is there now, I think it should stay there, but corrected.
 * Building representations using representations is something basic.
 * As far as mere length, I don't see a problem if it is packed with useful info. I appeal here to the fact that the Lorentz group is ubiquitus.
 * For the QFT part. Without it there would be no applications of the theory. I also think that it is quite interesting. It represents my interpretation of Steven Weinbergs view of the rôle of special relativity in QFT.
 * Looking at the section titles below, it's hard to say that anything except the QFT could be left out at all.
 * As an example of what is really really really too much of a schoolbook, have a look at this: Rotation matrix. I don't think I'm even close;) Still, that article is rated B. Lorentz transformations are rotations too, and ,as opposed to regular rotations, deserve a bit of treatment. Escpecially in the form of representations.
 * As I browse math forums, questions like "How do I construct the SU(2) representation of the Lorentz Group using that SU(2)×SU(2)≅SO(3,1)?" pop up. [The missing Answer is: You can't because SU(2)×SU(2)≅SO(3,1) isn't true.] The proposed Lorentz page would does serve a purpose here.
 * Actually, math forums often refer to Wikipedia, even the more exclusive ones, like Math Overflow. There are plenty of reasons that Wikipedia's LT rep page should be correct and somewhat detailed.

Done playing the devils advocate. I need concrete suggestions. Perhaps concrete suggestions per subsection. YohanN7 (talk) 13:00, 28 September 2012 (UTC)

Devils advocate again. NB. "The Lorentz group of theoretical physics has a variety of representations, corresponding to particles with integer and half-integer spins in quantum field theory" is the current "lead of the lead". A section outlining this surely can't be out of place. YohanN7 (talk) 14:54, 28 September 2012 (UTC)

..., and again. This is a post from the person who seem to have originally written the section on finite-dimensional representations: Talk:Representation theory of the Lorentz group. He/she doesn't seem to be around anymore, but there were visions in the direction I have taken. YohanN7 (talk) 17:03, 28 September 2012 (UTC)

= Suggestions per section = 1 Properties of the finite dimensional representations

1.1 Terminology
 * Must be changed. The whole section can be placed at the bottom of the article, but linked from the top.
 * Proper use of representation: π:g->Gl(V) is a representation. Standard use: (π,V) is a representation. Standard Abuse: V is a representation, or π(V) is a representation.
 * Lie algebra convention. The physics convention is used (in the original too). That is exp(itX) is in the group if and only if X is in the algebra. In math, X<->iX. This applies to commutators too.
 * Lorentz group = O(3,1). This is a convention too. Could as well use O(1,3). Different signatures of metric.
 * SO(3;1) = subgroup of O(3;1) with unit determinant. Physics lingo: "proper"
 * SO+(3;1) = Identity component. Physics lingo: "proper" and "ortochronous". SO(3;1) -> SO+(3;1) everywhere.
 * Complex representation (π,V) <-> V is complex.
 * Complex Lie algebra g <-> g = gC
 * Complex group <-> Its Lie algebra is complex. [Note: I didn't write "it's"!]
 * Real linear representation (π,V) <-> π(aX) = aπ(X). for a real. Always true.
 * Complex linear representation (π,V) <-> π(cX) = cπ(X) for c complex. Not always true.
 * Holomorphic representation (π,V) <-> π([cX,Y]) = c[π(X),π(Y)] for c complex. Not always true for complex linear reps. Not used, can and should be skipped.
 * Generator (physics lingo) = basis element of Lie algebra.


 * Possibly links to "invariance", "covariance" and "conserved quantities". YohanN7 (talk) 16:44, 28 September 2012 (UTC)

1.2 General properties

1.2.1 Isomorphisms

1.2.2 Semisimplicity
 * Move note on simplicity of so(3;1) here. Add note (link) on classification of semisimple algebras in conjuction with so(3;1)C = so(4,C).

1.2.3 Building representations
 * This should be linked: Representation theory. I can rinse out a lot, but I want to keep definitions spelled out. EDIT: Yes, defs must be kept. the above mentioned article is not complete w r t what is needed here. Also, it hints that for quotient reps to exist, the subspace mapped to the identity must be stable under the group action. I'd say that this is not correct. Needs to be checked. YohanN7 (talk) 19:54, 28 September 2012 (UTC)

1.2.3.1 Complexification

1.2.3.2 Direct sums

1.2.3.3 Tensor products

1.2.3.4 Other constructs
 * Quotient rep. (Can't find in Wikipedia) V = vector space. Q = V/~ = Quotient space. Let q:V->Q; q(v) = [v]. If G is a group and g an element of G, then g'[v] = [gv] defines a group action. (This is useful for the QED section (Equivalence relations on H come from exchange symmetry.)
 * Dual reps are defined by (being a little naïve here) by π*(X) = -π(X)tr (algebra) and Π*(g) = [Π(g-1))tr (group). They always exist (if V is finite-dimensional) and act on the dual space V*. Not found on Wiki either. I believe that (1/2,0) and (0,1/2) are dual reps. Must check this last statement.
 * New concept: Equivalence of representations should be linked or defined. YohanN7 (talk) 18:55, 28 September 2012 (UTC)

1.2.3.5 Subrepresentations
 * I have used incorrect terminology. A subrepresentation is restriction in the other end, i.e. a subspace of the representation space V, not the restriction to a subgroup or to a subalgebra. I use it in the latter sense in the later sections. (No proper subrepresentations exist, because we are talking about irreps.) YohanN7 (talk) 19:42, 28 September 2012 (UTC)

1.2.3.6 The (m,n) representations
 * Move note on simplicity of so(3;1) up to "Semisimplicity". YohanN7 (talk) 15:56, 29 September 2012 (UTC)

1.2.4 Group vs Lie algebra representations

1.2.4.1 Projective representations
 * Exemplify by demonstrating that a projective rep has a useful interpretation on quantum states.

1.2.5 Induced representations

1.3 Properties of the (m,n) representations

1.3.1 Properties of general finite dimensional representations
 * There is a serious error as it stands. The calculation can be done at the level of su(2), but before one passes back to SO(3;1), one must recognize what are actually sl(2;C)⊗sl(2;C) (= (m,n)) combinations first.
 * A general N-tensor representation is given by (1/2,1/2)⊗(1/2,1/2)⊗...⊗(1/2,1/2) (N times). By reducing this one will find (A,B) reps with A = N/2, N/2 - 1, ... and B = N/2, N/2 - 1, .... Thus if A + B is an integer, then (A,B) will be found (taking N = max(2A,2B)). If A + B is a half an odd integer, then (A,B) can be formed by taking the tensor product of the spin 1/2 representation and a tensor, [(1/2,0)⊕(0,1/2)]⊗(A,B), where A + B is integer.
 * By combining the above (which is the whole point), it follows that every finite-dimensional representation of sl(3;1) is either a tensor or a spinor-tensor. YohanN7 (talk) 12:20, 29 September 2012 (UTC)

1.4 Extensions to the full Lorentz group (New)
 * The (m,n) representation can be extended to O(3;1) if and only if m = n. This follows from βJβ-1=J and βKβ-1=-K <-> βBβ-1=A, where β is the representative of space inversion. If A ≠ B then (A,B)⊕(B,A) can be extended to an irreducible representation of o(3;1). [It is not irreducible as a representation of so(3;1).] YohanN7 (talk) 12:36, 29 September 2012 (UTC)

1.3.2 Quantum mechanics
 * A further (and better) example is the Hamiltonian H and the momentum operator P which transform as the time and space components of a 4-vector.

1.3.3 Classical electrodynamics.
 * (New section) I will have a look into David Jackson's book. The classical EM field tensor, the stress-energy-momentum tensor and the angular momentum "thingie" (forgot what it is called) should provide nice classical examples using a just sentence or two + good links.

1.4 Consequences of Lorentz invariance
 * This->The in first sentence. ... the weak interaction->the theory of the weak interaction.

1.4.1 Quantum field theory

1.4.1.1 The Hilbert space
 * A typical element of Hilbert space will look like A|α>⊗|β>⊗...⊗|ς> ⊕ ... ⊕ B|α>⊗|β>⊗...⊗|ς>, where α,β, etc. are complete sets of quantum numbers, |α>, |β>, etc. are single particle states, and A,B are constants. Not all of these states are particularly meaningful. The meaningful states will exhibit certain exchange symmetries and are subject to normalization in most computations.

1.4.1.2 Linear operators Hilbert space
 * The creation and annihilation operators constitute a basis in the following sense: Every operator is expressible as an infinite double sum of multidimensional 4-momentum integrals over polynomials in a, a* multiplied by coefficients depending on the momenta. Yuck! I'll give the equation explicitely with a page ref. to Weinberg. YohanN7 (talk) 17:39, 29 September 2012 (UTC)

1.4.1.3 Transformation of single-particle states
 * ..."set of labels" -> "complete set".
 * In this section the "master equation" from which all results here can be deduced should be explicitly given.
 * Perhaps also give the formula where the infinite-dimensional reps enter the stage.

1.4.1.4 Transformation of multi-particle states
 * Emphasize that "all that is needed" are the constructions from the purely math part plus representations of the symmetric group Sn stemming from exchange symmetries. Other "degrees of freedom" associated with gauge symmetry will enter in a multiplicative fashion with a formalism analogous to that of LT.

1.4.1.5 Transformation of linear operators
 * Perhaps display the equation showing how a,a* transform.

1.4.1.6 Fields
 * Display formula for general field.

1.4.1.7 The S-matrix
 * Display sample formula for S-matrix element. YohanN7 (talk) 17:55, 29 September 2012 (UTC)

1.4.1.8 Causality

1.4.1.9 Free field equations and gauge principles

1.4.1.10 Other consequences

2 Notes

3 References

I'll let the article "rest" for a couple of days. YohanN7 (talk) 10:46, 28 September 2012 (UTC)

= Quick note about the representation draft =

There's line about so(3;1) being simple, followed by "That is, so(3;1) can not be decomposed into a direct sum of simple pieces." I'm not sure my intuition about semisimple algebras applies here, but I don't think that's a correct characterization of simplicity. For rings, a simple ring R is characterized by having no nontrivial homomorphic images. However a simple ring, (and I believe a simple Lie algebra) can still be semisimple, whence it decomposes into simple pieces. Rschwieb (talk) 21:36, 30 September 2012 (UTC)
 * All simple Lie algebras are semisimple by convention. One-dimensional algebras are not simple (by convention again). What is meant in the draft is that so(3;1) is semisimple and simple. [This is the key to so(3;1) having no unitary finite-dimensional irreps.] I'll try to improve on clarity in that section.
 * The characterization of simplicity is probably the quirk here. The statement is likely to be correct (I have no proof myself, but there are mentions of this (without proof) in reliable sources.) A correct statement would probably be "There is no real linear Lie algebra isomorphism h:so(3;1)->g1⊕g2...gn where n>1 and the gi are all nontrivial. YohanN7 (talk) 10:42, 1 October 2012 (UTC)
 * This should do the trick: "That is, so(3;1) cannot be decomposed into a direct sum of more than one simple summands." YohanN7 (talk) 16:05, 1 October 2012 (UTC)

edit conflict
 * OK, I can spell out my concern here a little better. A ring R is called directly irreducible if, when written as a direct sum of two rings S and T, one of S or T is zero. Now, simple rings (=only trivial two-sided ideals) are certainly directly irreducible, but the converse is false. There are many local rings which are nonsimple, but cannot be decomposed into rings. I believe the same holds for Lie algebras.
 * About the suggestion "There is no real linear Lie algebra isomorphism h:so(3;1)->g1⊕g2...gn where n>1 and the gi are all nontrivial.: This is an accurate statement of direct irreducibility of so(3;1), but it is not an accurate description of its simplicity. The right idea for simplicity is: If A is a simple algebra, every nonzero algebra homomorphism of A is injective.
 * What you have there is OK, but I'm going to have to alter the conjunction "That is" to something else, because it is conveying the wrong idea.
 * Rschwieb (talk) 16:11, 1 October 2012 (UTC)

Your suggested edit is a great improvement, but I tweaked it a little more. Really, there is no need to say "simple summands" here. It cannot be written as a sum of two nonzero algebras at all. Rschwieb (talk) 16:13, 1 October 2012 (UTC)
 * Thanks for pointing this out. It didn't occur to me that I accidentally proposed an incorrect definition of "simple". I didn't even know about its definition (and meant to leave it undefined in this article). I would never have caught that myself. YohanN7 (talk) 20:22, 1 October 2012 (UTC)
 * Ah, now I remember. This is probably not standard terminology, but anyway: A Lie algebra g is indecomposable if it has no nontrivial ideals. It is simple if it is indecomposable and dim(g) > 1. This is from Brian C. Halls book.
 * Related things: Semisimple = direct sum of simple algebras. Reductive = direct sum of indecomposable algebras. None of them are needed in the present context though. I use semisimple <-> Complete reducibility property, which seems appropriate in the context of representations. YohanN7 (talk) 20:43, 1 October 2012 (UTC)
 * I see that in Hall's book... very strange choice of terms, but I can definitely agree that an associative semisimple ring is simple iff indecomposable. I bet the same thing holds for semisimple Lie rings. I appreciate the reference, so that I can get familiar with that book. Rschwieb (talk) 00:24, 2 October 2012 (UTC)

= Discussion from Quondum's talk page =


 * Representation theory of the Lorentz group

Hi!

I have some problems with Representation theory of the Lorentz group. There are several issues, some of which appear on the articles talk page, but I have at least begun on a proposal for improvements. I'd be very happy if you could take a look at what I have written so far at User:YohanN7/Representation theory of the Lorentz group. It's not in mint condition yet, but I don't want to spend too much time on something that might be reverted without comment.

I glanced through your talk page and saw that you play around with C++. That's kind of my area. Math and physics are more of a hobby project, but I'm not completely lost.

I'd appreciate any comments and suggestions on how to proceed. I'd even appreciate if you said "just drop it dude". (Would save me time.) If you respond, please do it on the proposals talk page.

Best regards, Johan YohanN7 (talk) 02:13, 23 September 2012 (UTC)


 * Have done. Though much of the content is Greek to me, it seems a worthwhile change. You could also drop a brief request for a glance at Wikipedia talk:WikiProject Mathematics or Wikipedia talk:WikiProject Physics once you have done if there are too few watchers on the page. Maths and physics are also essentially a hobby to me... — Quondum 06:45, 23 September 2012 (UTC)
 * Hi!
 * I have bumped into something I can't understand. It's probably easy, and doesn't need to go into the article, but I want to understand it anyway. Reference desk/Mathematics. Thanks for the edit! (Subtle little things...)
 * Best regards, Johan YohanN7 (talk) 17:22, 26 September 2012 (UTC)
 * I think this is a good question to post at www.math.stackexchange.com Rschwieb (talk) 17:46, 26 September 2012 (UTC)
 * Yup, a more specialized audience will probably help. I am still for from being able to properly interpret the question, and WP doesn't have a huge number of suitably specialized people watching. — Quondum 17:59, 26 September 2012 (UTC)
 * I have learnt something significant today. Hyphens and minus signs are different creatures indeed. I honestly had no idea. Next problem: How do I produce them using my keyboard? YohanN7 (talk) 21:13, 26 September 2012 (UTC)
 * Hyphen, mimus, n-dash, m-dash; all different. At the bottom of the edit window, you should find a toolbar with various characters (you may need to go to Special:Preferences to set some editor settings, I'm not sure.) Under Insert you'll find n-dash (–), m-dash (—), minus (−) and a middot (·). Under Math and logic you'll find minus again, and a dot operator (⋅), which is not the same as middot, but looks almost the same. That's what I use; it's a bit clumsy. Otherwise I use a copy-and-paste from a small text document that I've set up with a list of useful Unicode characters. You can also type them in long-hand; the site converts them when displaying the page, e.g.   produces – &mdash; &middot; &times;. — Quondum 22:31, 26 September 2012 (UTC)
 * New draft! Not polished tough. Coding in binary is less error prone than LaTex;) Asked User:Rschwieb for comments too. I don't plan more contentwise - at least not before it is accepted in some form. YohanN7 (talk) 20:40, 27 September 2012 (UTC)
 * I've done a copyedit mainly for format (not 100% yet); I may have misinterpreted some things. Some symbols (e.g. A, B, p) seem to have unbold and bold forms and this not made clear. You should go through it to see that I haven't inadvertently made it wrong. As to LaTeX, this kind of article (dealing extensively with  groups) will often do better with very little.
 * I am not qualified to comment, but I am somewhat concerned that this seems have become an exposition/textbook rather than a reference. Something that becomes too rambling loses its value as a reference. WP should not be explaining extensively. — Quondum 06:54, 28 September 2012 (UTC)
 * Thanks for your feedback. I take it seriously. I responded here User talk:YohanN7/Representation theory of the Lorentz group. YohanN7 (talk) 10:50, 28 September 2012 (UTC)
 * And thanks for the edits. [I was aware of some of it, like the inappropriate version of ε, but not all of it. Sometimes I just don't know where to find the correct character. [Not to mention aligning equations with LaTex. One of them looks awful.]] YohanN7 (talk) 16:02, 28 September 2012 (UTC)
 * The LaTeX you'll learn. You may want to read Help:Displaying a formula. Alignment of equations is fairly straightforward .  My playing with formatting is a bit premature: the higher-level consideration are more important, to focus on format too early is wasting effort. You should be focusing on content at this point, not format. — Quondum 16:48, 28 September 2012 (UTC)
 * Hi! A new version on its way. Its status is, at the moment, being in no mans land, i.e, I haven't rewritten all I intend to rewrite yet. It should be clear though where the thing is heading. I have tried to minimize the expository tone, especially in the pure math section. It should sound less as a textbook now. I have not removed any definitions, even though most all of them are linked.


 * Much more trimming away of remarks of the sort "It should be noted that ...", or definitions and the like will make it well nigh impossible to read the material, because there are subtle points. Please note that I mean read, not understand. I don't expect anybody to "understand", but I hope that some reader (if only one) will be able to see something useful and perhaps even new to him.


 * The physics section is now (hopefully on its way to become) more tightly knit with the math part. I plan to decorate most sections with an equation or two illustrating what is going on. The general ref is Weinbergs QFT, Vol. 1. /Johan YohanN7 (talk) 20:21, 30 September 2012 (UTC)


 * I'm having problems with the "nowrap" template and have removed a few. I'll put them back once I figured them out, because they are obviously needed. YohanN7 (talk) 16:16, 1 October 2012 (UTC)


 * The typical problems with nowrap are when the text contains "=" or "|". I generally use , where "..." can include pretty much anything except "|", which can be included as   (see !). — Quondum 16:35, 1 October 2012 (UTC)
 * Spot on. Yes, it has been "=" and "|". Now I understand the mysterious occurrences of "1". Thanks! One more thing. I have tried to find a way to make LaTex display in a smaller size in a couple of big equations. I can't make it work (and haven't spent too much time on it either to be fair). I tried \small and \tiny inside \text and it compiles fine. I have an old standalone LaTex editor, TeXnicCenter I cranked up. I'll read a manual tomorrow. YohanN7 (talk) 21:31, 1 October 2012 (UTC)

By the way, the editor interface is being messed about with. I find that in Special:Preferences, editing tab, you now have to switch off the "Enable enhanced editing toolbar" option to get the charinsert bar that I was talking about for all the useful math characters. — Quondum 00:47, 2 October 2012 (UTC)
 * Hi again. I've filled in some stuff and rinsed out other stuff. The math part is probably nearly complete - as far as I can take it (see comments on Rschwieb's page). The physics need a couple of more equations to show explicitly what is going on, but not much more is planned on my part. Would you say that it is possible at all for an interested layman with decent background (like QM + abstract algebra + typical senior undergrad math) to "follow" what is outlined? Any suggestions are appreciated. The Gamma matrices article is linked. It would take a little bit more to arrive at Clifford algebra. But, I should probably provide one or more steps showing how the gammas appear. It is basically just a little linear algebra.
 * For the physics part, the Weinberg QFT book (vol 1) is the major reference. [I'd say it is in a league of its own. There are practically no errors at all in the 600 pages. It isn't the easiest book in the world though.] The equations are from there. Wigner's fundamental 1935 paper should be referenced too (the Wigner rotation is from there). The math formulas and statements can mostly be sourced to Brian C. Halls book. [This book is easy, but drops in quality in the later chapters because he tries to keep things too simple. Plenty of small errors.]
 * Is it still too much of exposition and textbook style? The physics part is admittedly a little bit of an exposition, but it is, after all, an application of the math. As such it must step down a little bit at least from just plain statements, or a collection of links. What it arrives at in the end (field equations) might be what the original author of the finite dimensional section had in mind. There is post in the talk page indicating this. YohanN7 (talk) 23:27, 2 October 2012 (UTC)
 * I'm moving this discussion to the talk page. It is on my watch list, I just don't get email notifications of changes to it, which is fine. — Quondum 05:58, 4 October 2012 (UTC)


 * The above discussion has been moved from Quondum's talk page.

The simple answer is that there is far too much content for a single article. The application to physics (quantum mechanics) will have to be separated out into a separate article, which would the have to be linked to from here. You'll have to reframe in your mind what an article should seek to achieve: it should not be where someone comes to learn about something, but rather to determine the details described by the topic, and to give links to related topics. This is how I see the distinction between a textbook and a reference. So I suggest that you try to entirely separate out the application to physics (leaving only a link), with a view to creating a separate article (which will have challenges of its own). The maths that will be left behind will still be daunting. I'm not quite at the level you asked about; my QM is weak (I've never studied the math of QM, just picked up some general principles), abstract alogebra is quite new to me, representation theory I only happen on when you asked me, and even the specific groups involved are not generally known to me. I can only tell you this: I'll need a lot of study to follow what you're actually saying. It may make sense to have something that states the things I could follow (and does not particularly try to justify how it gets there).

I have been able to gather (from Representation theory) that a representation is essentially a set of matrices (or more generally, linear transformations) such that (in this context) there is a group isomorphism between them (with matrix multiplication) and a group of interest. Here we are interested in the Lorentz group, of which I have a reasonable understanding. If you wish me to understand the topic better, you firstly need to connect the Lorentz group to the representation. AFAICT, the representation is simply the set of Lorentz transformations between orthogonal bases. These are orthogonal transformations given the quadratic form. These transformations are usually considered to be passive, meaning that they only change the choice of basis are, in the sense of tensors, all the same linear transform: the identity transform. It is evident that this is not what is meant; the statement that all physics equations remain invariant under a Lorentz transformation is saying that a rotation or boost applied to a physical system cannot be distinguished. A passive transformation is saying that the quantities involved are tensors (i.e. that any observer will see the same system adjusted for his point of view), the active transformation is saying that the equations are isotropic (i.e. any "preferred" frame moves with the system if we rotate/boost it).

Representation theory suggests that the vector space on which the representing linear transformations act must be chosen, and are probably pretty arbitrary. This is where I come unstuck. We need not be dealing with R4, but rather a Hilbert space. I think, if nothing else, this choice and how we are constrained by the group in this choice should be highlighted by the article Representation theory of the Lorentz group. I imagine that, given a tensor field, the vector space concerned would be the space of all possible fields in R4. This is clearly a configuration space of an infinite number of dimensions. If we constrain these fields to piecewise-continuity, the number of dimensions may be countable. Anyway, this is the sort of vector space that I start thinking of in the context, but this is not even mentioned. Do you agree that there may be a major gap between the title and the content of the article? — Quondum 07:18, 4 October 2012 (UTC)
 * I agree with pretty much everything.
 * About major gap between title and article: The set of all tensor fields on R4 certainly contain representations of the Lorentz group. They are tensor products of (possibly infinite) representations of function space and finite dimensional representations. One can think of the action on a tensor field as
 * The components (functions of spacetime) undergo transformations under a possibly infinite dimensional representations. They become other functions of the same type (representation)
 * The components are shuffled around.
 * The set of all spinor fields on R4 should be a nice example. The Lorentz group does impose constraints in some representations. Using the same example, the Dirac equation (which follow) imposes these constraints on the full space of spinor fields. This is one major point of the physics part. Passing to the set of linear operators (I call them induced reps, compare adjoint rep) on the set of all tensor and spinor fields (or combinations thereof), Quantum Fields, changes nothing in principle. This is what the QFT part outline.
 * There is one gap though, if one looks at my parts only. The infinite reps on function space isn't there, but that part is covered well in the original article.
 * I'll work on splitting all up into two pieces (and cutting down on the pieces). YohanN7 (talk) 10:54, 4 October 2012 (UTC)
 * You could say that the point of view I have is that the topic of representation theory of the Lorentz group is much bigger than the theory of the group itself, and that of its Lie algebra, both mathematically and in terms of application. It enters "everywhere" (meaning pretty much in every equation) in physical theories obeying special relativity.
 * It is not an area of current research (as far as I know) in mathematics, but it is used heavily on a day to day basis in physics. This is my motive for a substantial article.
 * Slightly off topic: The Clifford algebra Cl(1;3,R) seems to contain five representations (irreps under the full group, but not under SO(3;1)+), two (0,0), two (1/2,1/2), one 6-dimensional (which combination of irreps?). These are the bilinear covariants - a topic (amazingly) missing entirely. This is stuff I want to get into the Dirac algebra article some day. YohanN7 (talk) 11:39, 4 October 2012 (UTC)
 * As to whether a finite dimensional representation of the algebra (or the group) is a set of matrices, this is true only after a basis for the representation space has been chosen. In Lorentz group there is an example where the elements of the algebra are differential operators on functions on R 4< . In classical field theory, the elements in a representation are matrices containing differential operators. Compare the classical Dirac equation written out in full.
 * As far as connections to the Lorentz group go, this calls for more content yet (in the math part). A Lie algebra representation can (sometimes) be exponentiated to a representation of the group. [Differential operators (vector fields in manifold theory) can be exponentiated too.] One usually works in the algebra setting because it is so much easier (because it's a vector space.)
 * Splitting into Reps of the Lorentz algebra and Reps of the Lorentz group would be truly horrible.
 * Passive vs active transformations deserves a mention. They are isomorphic in a sense; compactly, <$\pi$(X)y|A|π(X)x> = .
 * Quote:"It should not be where someone comes to learn about something, but rather to determine the details described by the topic, and to give links to related topics." Exactly! YohanN7 (talk) 12:26, 4 October 2012 (UTC)

= Discussions from Rschwieb's talk page =

Hi!

I was led here from Quondums talk page. He has given me valuable input to improvements of Representation theory of the Lorentz group that I'm working on. I am concentrating on finite-dimensional representations.

There is a first complete draft: User:YohanN7/Representation theory of the Lorentz group.

This draft is complete as far as content goes, according to what I believe it should contain. It's far from complete with regards to polishing the appearance, uniform notation and so on, but I think that it is in readable condition.

I need some (a lot!) input from a real mathematician. I'll be needing input from a real physicist as well, but that is less urgent, since the level of rigor in the QFT section is zero. The (primarily) mathematical sections are intended to be rigorously correct though.

I'm not yet ready to post a general review request in the math & physics projects, but I'll do that as soon as I've "prettyfied". (It would be a petty if such problems became the main focus.) I'd be extremely happy if you could ha a look if and when you get the time.

I'll post that question about irreducibility on the forum you recommended tomorrow. I have a similar question regarding reduction of tensor products too.

Best regards, Johan YohanN7 (talk) 20:34, 27 September 2012 (UTC)


 * I'm really interested in seeing what you come up with, but I have to ask you something fast. Is it OK if th article you are working on never makes it to mainspace? I'm not sure if it would be accepted by the community. If this is just for your personal pleasure (I can appreciate that kind of work!) then there's no issue. If you really want to put it up, then we need to find out what kind of features of independent interest would "sell" the article into the mainspace. In any case, I'll help you however I can. I'm not very good at representation theory, however I am very intersted in thinking about it. I'm in a good position to pick it up quickly. Rschwieb (talk) 20:57, 27 September 2012 (UTC)


 * R, please note that this is to replace an existing section in an existing article. The existing section content is inadequate. But I've been cautioning Johan that what has been produced is too expository; while some expansion will be acceptable, too much will meet resistance. — Quondum 07:02, 28 September 2012 (UTC)


 * Thanks for the input. I responded here User talk:YohanN7/Representation theory of the Lorentz group. YohanN7 (talk) 10:48, 28 September 2012 (UTC)
 * I had no idea it was existing :) That's a great foodhold then! Rschwieb (talk) 13:01, 28 September 2012 (UTC)
 * A quick one. If (G,V) is a finite dimensional representation, H is a linear subspace of V, then V/H is a representation too by setting g[v] = [gv] (with all sorts of notational abuse). Right or wrong? (I'd say it's right.) YohanN7 (talk) 21:21, 28 September 2012 (UTC)
 * No, not for a general subspace H. The subspace H has to be additionally G-invariant for the quotient to be a G representation in the natural way. Check for the second occurrence of the word "invariant" in this pdf for example. Rschwieb (talk) 13:49, 30 September 2012 (UTC)
 * Thanks! I spelled it all out and can see now what I missed at first. The reference seems nice and concise b t w. YohanN7 (talk) 15:17, 30 September 2012 (UTC)
 * New versions of User:YohanN7/Representation theory of the Lorentz group on its way. I left some comments on Quondum's page. /Johan YohanN7 (talk) 20:25, 30 September 2012 (UTC)
 * The pure math section is now quite near what I want it to be. The physics section is not quite there yet, there are a couple of more equations that I want to add, but I hope that the math and physics sections are beginning to tell a coherent story.
 * One thing that is missing in its entirety is how to go from representations of Lie algebras to representations of groups. I don't think that this is (in theory) difficult. It's a matter of exponentiating matrices: But, there are things I'm uncertain of (and/or just can't remember at the moment), so I'll wait with that. [To what extent do we have a one-to-one mapping? Near the identity it is guaranteed. The map exp:so(3)->SO(3) is onto, as I recall. Etc.] Perhaps the "Wigner rotation" from the physics section will be of help for the full O(3;1). There, the exp map certainly isn't one-to-one.
 * Naturally, there will be some errors. Remember that i just an amateur. I'd be happy if you could take one more look at it. YohanN7 (talk) 22:31, 2 October 2012 (UTC)

I don't know enough about representation theory to be a whole lot of help. I know that the exponential map from $$\mathfrak{sl}_2(\mathbb{R})\rightarrow SL_2(\mathbb{R})$$ is not onto, but I have no idea about the case of $$\mathfrak{so}_2(\mathbb{R})\rightarrow SO_2(\mathbb{R})$$. It looks like you've developed a lot of good content, but I have to comment that it is a lot of content... is there any way you will be able to distill some of it? Q might have some helpful input along those lines... Rschwieb (talk) 02:04, 3 October 2012 (UTC)
 * One possible way of shortening it would be to leave the QFT part out for now as a whole. Then, at a later stage (hopefully) do the short section + main article trick.
 * The math part is harder to cut down though. I think that the math section sticks to the point and gives the facts and the appropriate links. If the links had been somewhat "uniform", further reduction would be possible. I do admittedly flesh out and actually (attempt to) prove the formula for the (m,n) irreps.
 * Leaving out the QFT would be ok, but I'd be a little bit sad about it. Huge chunks of the foundations of QFT follow from special relativity and particular finite-dimensional representations of the Lorentz group. The wiki articles on QFT mainly use the old-fashioned approach to QFT (canonical quantization) basically which amounts to quantizing classical fields. B t w, I just managed to get in the final formula in that section, so that it is "logically" somewhat coherent now. I'll try to find a physicist willing to cast an eye on it. Do you know anyone you think I should contact? I'm a little afraid of writing in any of the project pages yet, just because the size of the thing. It 'is a problem.
 * I'll be able to find out about SO(3) if needed. I have another book (excellent, except for all typos) by Wulff Rossman, where I know i can find it. YohanN7 (talk) 18:17, 3 October 2012 (UTC)
 * I think your content will boil down to something very good. I see you've been meticulously filling out all the details, but I think in the end you're going to have to sacrifice a lot of that. With the limited space you have, and the limited attention span of most readers, you're going to want to pick out the most profound and useful ideas and present them, but not necessarily with complete rigor. Maybe you can make a second draft that's an "elevator pitch", and then you can see what is really important in the page, to you :) Rschwieb (talk) 00:55, 4 October 2012 (UTC)
 * I'll split it into two parts for a start. Should we move this discussion to User talk:YohanN7/Representation theory of the Lorentz group? Q has moved the coresponding stuff from his talk page to there. YohanN7 (talk) 12:53, 4 October 2012 (UTC)
 * Feel free to transfer the discussion elsewhere at any point, and please include a link directly there so I don't get lost :) Rschwieb (talk) 14:21, 4 October 2012 (UTC)

= Notes to myself (or anybody) =

For the new draft (math): How about a table listing all relevant constructions with formulas in one column, ie. duals, sums, products, complexification, induced on End(V), quotients, etc (rows) + links (column) and remarks (column). Should cover about one screen (at most).

Error in expression for K (+->-).

Quotient representations examples: exterior algebra and symmetric algebra. They should correspond (physics part) to a fermionic and a bosonic Hilbert space respectively.

For the new draft (physics): Cluster decomposition principle <-> The S-matrix factorizes for "distant" processes? At least one implication (->) is correct. The r h s is intuitively correct and easier (shorter) to describe. The l h s is mathematically amenable (yields connected part SC of the S-matrix). The causality principle is not precisely equivalent to commuting Hamiltonian densities, but close enough [Weinberg]. I use one too many concepts, two should go.

Link Lorentz invariance.

Just thoughts out of curiosity: The Bargmann-Wigner equations (week-old article) should follow too from general principles. (Anti?)symmerization of (1/2,1/2)⊗(1/2,1/2)⊗(1/2,1/2)⊗...⊗(1/2,1/2) (N times)??

The trivial representation isn't all that lame, see Lorentz scalar. Forming Lorentz scalars (e.g a Lagrangian) from general spinor-tensor fields put's the 3j-symbols (6-j, etc) in a context. Trickier: forming Hamiltonians (should transform as 0-component of 4-vector) or Hamiltonian densities (00-component of (at best) symmetric tensor). This is the reason Lagrangians are the primary objects used and the Hamiltonian is derived (via Legendre transform). YohanN7 (talk) 16:29, 4 October 2012 (UTC)

Oops. Positronium state expression is wrong. Move integration measure past |vac>.

A couple of DAGGERS, a MATHCAL and the MASTER EQUATION.

= Playing with format = Unicode template (new)       : &#x27E8;&#x27E9; Raw unicode chars (new)      : &#x27E8;x|y&#x27E9;

Unicode template (deprecated) : &#2329;&#x232A; Raw unicode chars (deprecated): &#x2329;x|y&#x232A;

Interesingly, only the last combo works correctly on my computer. The unicode template gets the "bra" wrong, even for deprecated chars. The new ones don't work at all, except when I use Cambria math for sans serif (which is pretty awful.)


 * They all work for me except the deprecated bra inside the unicode. So for now best use raw deprecated, no unicode template. BTW, the angles (&amp;#x3008, &amp;#x3009) are not great, they tend to get replaced by others and spacing is bad. — Quondum 14:03, 7 October 2012 (UTC)

Bra-ket
Is this bra-ket, 〈x|x〉is preferable to ? At any rate, this ket, |x⟩ shows as a |x followed by a box. I have just downloaded purchased the bulkiest unicode font I could find. YohanN7 (talk) 19:09, 6 October 2012 (UTC)
 * Um, yes it is preferable, if your font supports it. Since it is a standard character on the Wikipedia editing interface, I would have expected it to be standard on most systems. I have had one person mention the same problem. Try to play with font selection on your browser, perhaps selecting Cambria as your serif font, though the problem is probably with the sans serif font. If you select MathJax here, how does $$|x\rangle$$ appear (remember to refresh this after saving the MathJax selection)? What font does it use on your browser, and does it generate the same Unicode character? — Quondum 22:37, 6 October 2012 (UTC)
 * At least I know what's going on now. http://www.unicode.org/reports/tr25/ says
 * "Deprecated Delimiters. The angle brackets formerly aliased as “bra” and “ket”, U+2329〈 LEFT-POINTING ANGLE BRACKET and U+232A 〉RIGHT-POINTING ANGLE BRACKET, are now deprecated for use with mathematics because their canonical equivalence to CJK angle brackets is Unicode Support for Mathematics Unicode Technical Report #25  17 likely to result in unintended spacing problems when used in mathematical formulae. Instead one should use U+27E8 and U+27E9, respectively."


 * Old braket &#x2329;&#x232A;
 * New braket &#x27E8;&#x27E9;


 * I can see the old version, but not the new. I guess both me and my computer are deprecated. I knew about me, but the computer is only 7 years. Time to purchase more fonts! I may have to switch to the old version temporarily in the draft. Does it display differently than the new? YohanN7 (talk) 09:33, 7 October 2012 (UTC)


 * Cambria math (actually free with Powerpoint 2007 viewer) displays correctly, but I still need to find a working sans serif font. YohanN7 (talk) 10:17, 7 October 2012 (UTC)


 * I see the new symbols were added in Unicode 3.2 (dated 2002-03-27). Many fonts do not support these extensions yet, but it seems the older (deprecated) symbols are supported by some fonts that do not yet support the newer ones. Ouch.
 * My Firefox browser displays them all (and the old and new appear identical) with Candara selected as my sans serif font, but I (very strongly) suspect that it is finding the character in another font set. MSWord certainly simply pulls in Cambria Math for the purpose. I would suggest simply using the deprecated versions for now, and not worry about the semantic subtleties. At a future date when fonts generally provide wider Unicode support, some bot could in principle go through locating and potentially replacing them if it is deemed an offense to the pristineness of WP.  With something like WP, I'd think that the actual coding used is not too critical as long as the display is faithful. The kludge |x> is visually too distorted for me to consider it acceptable, but |x&#x232A; is fine (I'm not sure about the mentioned "spacing problems" though). Another thing you may wish to try (if this displays correctly for you is  : &#x27E8;&#x27E9;; the template is supposed to enhance compatibility with Unicode. — Quondum 11:27, 7 October 2012 (UTC)
 * I'll have to go with the old chars for now. YohanN7 (talk) 13:27, 7 October 2012 (UTC)

= So, what do you think? =

I've done the following:
 * Split into pure math and pure physics.
 * Latexed some formulas.
 * Added explicit matrices.
 * Added a nontrivial example.
 * Added connection between algebra and group reps.
 * Introduced buzz-words links whenever I could. (Central charges and superselection rules. Love those!, The Baker-Campbell-Hausdorff formula.)

The math part makes up 6 screens on my computer. This is very much compared to the attention the subject gets in current mathematical research of today (zero). It is very little compared to the importance (usefulness) of the Lorentz group representation in physics. Thus six pages strike a nice balance i m o.

The one thing that could be shortened is the actual proof of the (m,n)-representations. (This is the only proof.) I'd object to that because the representations are incorrectly inaccurately given (worded) in the majority of existing documentation (published or on the net) on the subject. (This includes the present article, and the article on the Lorentz group.) Example: I don't know of any QM book mentioning that we step outside of su(2) when the "ladder operators" are constructed. This is not to say that I haven't made mistakes myself here.

The nontrivial example, which does use stuff from later sections, is meant to put (a part of) the Dirac algebra in a simple representation-theoretic setting. It arrives at an almost explicit representation of Spin(3), which makes me suspicious. Those kinds of things are supposed to be extremely messy (and require a Ph.D) to describe. The section also introduces spinors. Of course, I don't attempt to define them, but it will leave the impression that spinors are thingies (vectors) that transform under either of two reps of the Lorentz group.

The connection between Lie algebra reps and Lie group reps present both the abstract exp map, and the pushforward, (via links) and the matrix exponential and the simple formula to actually compute the pushforward. Discussion about issues like the exp not being one-to-one is ignored. YohanN7 (talk) 19:48, 10 October 2012 (UTC)

Yuck! There are several issues with that ubiquitous factor of i. The physicists sacrifice a lot by insisting on Hermitean matrices. YohanN7 (talk) 13:41, 11 October 2012 (UTC)


 * Under The standard representation you give the commutation properties. There is an erroneously repeated index (μ) in the last term. The whole expression looks very like the Kulkarni–Nomizu product, so it may be worth finding if they are directly related. The commutator [,] normally generates a difference of two products, so the discrepancy should be explained. Just below that the expansion of X=θ⋅J+ξ⋅K appears to have indexing errors. As you should have noticed, aside from trivia such as these, you have long ago passed my level of expertise in this, and I find it thoroughly daunting. This will probably be the same for almost all undergrads; it's not clear to me what the target audience is. I'm sure you're going to run into resistance on this aspect. — Quondum 21:29, 11 October 2012 (UTC)
 * I'm pretty much of the same sentiment that Q is. This has a lot of awesome information, (which I will probably save as a PDF because I like it!) but I'm still wondering how this is going to be boiled down for use in the mainspace. Have you formulated the "elevator pitch" yet? I am interested in seeing what you have singled out as the most important stuff. Rschwieb (talk) 13:05, 12 October 2012 (UTC)
 * I'll first correct the errors I know of, Then physically separate the math and physics and then think about it for a few days. R, could you describe the "elevator pitch"? I'm not sure I understand. Q, could you mention a section or two that you find daunting? Also, are they daunting compared to the present article om infinite dimensional reps? — Preceding unsigned comment added by YohanN7 (talk • contribs) 18:28, 12 October 2012 (UTC)