Talk:Representation theory of the Lorentz group/Archive 1

Archive 1

My article

I didn't write this, I just moved material here which was written by Linas, whom I have called upon to write his own article the way he sees fit. I will not contribute any further to this article on representations of the Lorentz group.---CH (talk) 03:12, 17 July 2005 (UTC)

Thank you, Chris. It looks like it turned into a marvelous article over the years. User:Linas (talk) 19:04, 30 November 2013 (UTC)

a bunch of stuff

the mood struck me, so I added a bunch of stuff to this article. This treatment comes from my memory of a QFT by Ryder, and well, I think it looks pretty sloppy. I guess somewhere we should find the represenation theory of the rotation group, upon which this is predicated. A longer exposition on how to form field equations for a given representation would be nice too. And uh, I guess... since the Lorentz group is noncompact, can I expect more than a discrete set of representations? I dunno. Help me out. -lethe ^{talk +} 12:36, 5 March 2006 (UTC)

Z/2?

In the article it mentions "for each j in Z/2, one has the (2j+1)-dimensional spin-j representation spanned by the spherical harmonics with j as the highest weight." in the section of finding representations. By Z/2 the author means ... -1/2, 0, 1/2, 1, 3/2, 2, .... But isn't Z/2 the group of integers modulo 2? In which case, the author is saying that the possible representations are labelled by the group {0,1}?

128.120.51.98 21:14, 4 January 2007 (UTC)kiwidamien

Z\2Z

You're thinking of the quotient group Z\2Z. --30apr2008

Finite-dimensional representations

There was a sentence that " The Lorentz group has no unitary representation of finite dimension, except for the trivial representation (where every group element is represented by 1). " This is simply not true, so I removed it. For example, the mapping that takes -Id to -1, and evertyhing else to 1 is a 1 dimensional representation. It's certainly not faithful, but it's unitary (and irreducible as it's one-dimensional). — Preceding unsigned comment added by Goens (talk • contribs) 14:48, 15 July 2012 (UTC)

This article is an advanced piece that goes directly to infinite-dimensional representations of the Lorentz group. Today I added the reference by Paerl that discusses both finite and infinite dimensional representation. Unless there is a protest, I intend to preface the infinite dimensional material with the finite.Rgdboer (talk) 04:53, 19 January 2010 (UTC)

As the Lorentz group has six real dimensions there are more elementary ideas that are useful. May I suggest the biquaternion#Algebraic properties representation of the Lorentz group.Rgdboer (talk) 00:14, 8 July 2010 (UTC)

su(2) - sl(2;c)

First, on the above section: " The Lorentz group has no unitary representation of finite dimension, except for the trivial representation (where every group element is represented by 1). "

I recognize that scentence. It's from Weinbergs "The Quantum Theory of Fields". It refers to the identity component (the proper ortochronous transformations) of the Lorentz group. It is a true statement from a reliable source, and should be put back because it is of some importance. Perhaps with the qualification that it is about the identity component.

Then there is this su(2) issue. The algebra su(2) is not the complexification of the algebra of the rotation group, so(3). The algebra su(2) is a real algebra isomorphic so so(3). What is actually used is the complexification of su(2), namely sl(2;C). For this latter issue, see e.g. Brian C. Hall, "Lie Groups, Lie Algebras, and representations; An Elementary Introduction".

One might add that this error in terminology exist (explicitly or implicitly) in pretty much every physics book there is. Typically it is itroduced at the same time the "ladder operators" are defined. One makes a complex change of basis of generators of su(2) and lands in sl(2;C). It is not the same thing as the different conventions regarding the different definitions of a Lie Algebra (a factor of i). YohanN7 (talk) 23:03, 12 September 2012 (UTC)

There is more that is pretty much backwards.

The assignment of J and K as pseudovectors and vectors respectively looks suspect.

The representations of the algebra sl(2;c) (and hence those of su(2)) do not stand in one-to-one correspondence with representation of the rotation group SO(3)

The section "Full Lorentz group" seems to get the meaning of irreducibility backwards in places. If a representation of the restricted group happens to be irreducible, then it is certainly irreducible under the full group. On the contrary, a representation may be ireducible under the full group, but not irreducable when restricted.

In particular, the (m,n) representation is in general not irreducible (under the restricted Lorentz group). A process entirely analogous to the Clebsch-Gordan decomposition can be applied to the ones that aren't irreducible.

I am probably going to attempt an edit ragarding these points (if nobody objects), a minimal rewrite + addition of the fact that the finite dimensional representations are never unitary. YohanN7 (talk) 11:03, 14 September 2012 (UTC)

So the striked out text above isn't entirely correct either. It should, of course, be this:

In particular, the (m,n) representation is in general not irreducible under the subgroup SO(3). A Clebsch-Gordan decomposition can be applied to the ones that aren't irreducible.

This results in an (m,n)-representation having SO(3)-invariant subspaces of sizes m+n, m+n-1, ..., |m-n| where each occurs exactly once. These subspaces don't mix under rotations but they mix under boosts. An example is given by the vector representation (1/2,1/2) which splits into J=0 (1-dimensional, e.g. time-component of EM vector potential A) and J=1 (3-dimensional, e.g. space components of A).

The assignment of J and K as a pseudovector and vector is pointless here. (Full O(3)) or full Lorentx group is needed for that, the adjoint action of SO(3) will not tell.) YohanN7 (talk) 10:11, 17 September 2012 (UTC)

Definition of representation

I wonder if this article should mention that some of the representations are projective representations and don't necessarily meet the full definition of a representation, i.e. not always$D(A)D(B)=D(AB)$ for $A,B$ elements of the group, $D$ the representation: there could be multiplication by a phase factor. Or would this be obvious to somebody with the necessary background to read this article? Count Truthstein (talk) 21:49, 17 January 2013 (UTC)

What is actually presented (in the finite dimensional case) is representations of the Lie algebra, not of the group itself. They are representations in the true sense. YohanN7 (talk) 15:18, 19 January 2013 (UTC)

Reps "of the Lie algebra, not of the group itself" are of little use for applications, because we need to transform quantities across reference frames. Count Truthstein is right, the concept of projective representation has some physical implications: see Spin-½#Complex_Phase. Incnis Mrsi (talk) 16:06, 19 January 2013 (UTC)

Projective representations and representations with phase factors don't seem to mean exactly the same things in math and physics. The former is a representation (in math), and the latter is a "lift" (in math) of the former where phase factors are introduced. These phase factors need to be such that the associative law still holds if I understand this correctly. Near the identity, this will work automatically by exponentiation of the Lie algebra reps (all phase factors are 1). Either way, the article should absolutely somehow address these issues. YohanN7 (talk) 17:03, 20 January 2013 (UTC)

2013-02-13 Edit

Hi!

I changed quite a bit in "Finding representations".

• Corrected main formula. Formerly it said so(3,1) = su(2) + su(2) which is just plain wrong.
• The main thrust used to be to use representations of SO(3) as a basic building block, or at least present things that way. It doesn't work, because there are more reps of su(2) than come from SO(3).
• I emphasized a bit the distinction between groups and algebras (so that rotation group (SO(3)), su(2), and sl(2;C)) are different things. [From previous version: "su(2) is the complexification of the rotation algebra" - just hideous]
• Mention of how to get group reps (as opposed to algebra), and that this can result in projective representations,

I did retain one thing though. The example of spherical harmonics is still there as a representation of SU(2). To make this work logically, one has first to get an su(2) rep, and then proceed from there. I don't really like it. YohanN7 (talk) 17:11, 13 February 2013 (UTC)

• I removed references to spherical harmonics too. Reason: In part because it was like going over the bridge for water, and in part because it wasn't correct, at least not with only the classical spherical harmonics which work only for integer spin, (which is the only case covered in the Wikipedia linked article).
• Removed too the fact that SU(2) is simply connected. True but irrelevant.
• Relevant fact not yet in article: sl(2,C) is the universal covering group of the Lorentz group.
• Reinstated an old remark that the irreps are never unitary.

The logic is this: The known irreps of su(2) give all of the irreps of sl(2;C). These, in turn, give all those of so(3;1)C which then finally restrict to so(3;1). All reps (irreducible or not) are then direct sums of the irreps. Group representations, possibly projective, may be obtained by exponentiation.

The former article (as per 2013-02-12) gave the impression that spherical harmonics and SO(3) are sufficient building material for the so(3;1) irreps. This is just not the case. The spherical harmonics are a beautiful illustration of the Lie Group SO(3). In addition, it is true that, given a representation of the Lorentz group, one may restrict it to SO(3). It just doesn't cover all the cases when one tries to go the other way. YohanN7 (talk) 19:48, 13 February 2013 (UTC)

I rewrote the section Full Lorentz group completely.

• The previous wording was a bit awkward. "...not only is this not an irreducible representation, it is not a representation at all,..."
• More importantly, it was not entirely correct as it stood. The (m,n)+(n,m) representations do not automatically include parity, it must be specified separately what constitutes the parity inversion representative.
• Time reversal is now discussed along the same lines.
• The terminology "vector" and "pseudo-vector" is now introduced alongside the equations that motivates it. (I removed it from Finding representations) YohanN7 (talk) 00:13, 14 February 2013 (UTC)

Inline citations now in place. Main math source is Hall (ref section), an easy introductory text on Lie groups, algebras and reps. Main physics source is Weinberg (ref section), a not-quite-so-easy text on QFT that covers a lot of the more advanced concepts when it comes to projective representations (Section 2.7 + Appendix B). YohanN7 (talk) 08:45, 14 February 2013 (UTC)

2013-02-13 Planned Edits

So, I have already done quite a bit. I'll wait a while before I proceed with more, but here is a list (in order of priority) of what I plan to include. Comments, both "yay" and "nay" are appreciated.

• Explicit so(3;1) matrices in the standard representation.
• The Lie algebra so(3;1) (i.e. the commutation relations among the above matrices).
• A clear description of the connection between finite dimensional representations of a Lie group and that of its Lie algebra. (The exponential mapping)
• The topic of simple connectedness, and therefore the failure of the exponential mapping to yield a proper representation.
• How a projective representation is still obtained, and why it is useful
• A nontrivial example of how this works out using Clifford algebras. YohanN7 (talk) 00:38, 14 February 2013 (UTC)

• Another slight problem: The electromagnetic vector potential, which is a 1-form, lives in this rep. In the literature, depending on how precise the presentation wants to be, the EM potential either is or is not a 4-vector. In contexts where they want to be precise, they point out that the EM potential is not a 4-vector precisely because it does not transform under the (1/2,1/2) representation. The transformation rule is actually A -> ΛA + the gradient of an arbitrary function. Therefore, I think the example is a particularly bad choice. YohanN7 (talk) 11:16, 14 February 2013 (UTC)

  Yes, I agree about the vector potential. From the theoretical point of view, it is the connection in a U(1) gauge theory, not a (co)vector field at all. Incnis Mrsi (talk) 07:40, 16 February 2013 (UTC)

So what would you say is a good example? I can see nothing wrong with simply using the coordinates x^μ of events in spacetime, except that it wouldn't be as "nifty" the other examples. YohanN7 (talk) 17:42, 16 February 2013 (UTC)

The four-momentum is the best we could invent. The wavefunction of a massive vector boson is another reasonable choice, though. Incnis Mrsi (talk) 19:11, 16 February 2013 (UTC)

Done. Besides, when I gathered enough courage I'll move the examples section so that it comes directly after "Finding representations". Done. YohanN7 (talk) 13:04, 19 February 2013 (UTC) Examples ought to appear early. YohanN7 (talk) 20:13, 16 February 2013 (UTC)

• So, I added a section Induced representations. This is really general representation theory, but the induced transformations ΠAΠ^-1 on End(V) are a bit special for the Lorentz group since it is doubly connected: Projective reps on V become reps proper on End(V). Besides, it partly anwers the question "What are projective (here double valued) reps good for?" YohanN7 (talk) 18:51, 16 February 2013 (UTC)
• Go all the way in defining group reps, i.e. show the formula for "defining Π along a path" Done. In the process I removed this: "This can be compared to the situation with SO(3), so(3), su(2), and SU(2). The irreducible representation of the latter three all stand in one-to-one-correspondence with each other, because su(2) and so(3) are isomorphic and SU(2) is simply connected, but only those representations of so(3) coming from representation of su(2) with odd dimension (integer spin) lift via the exponential mapping to an actual representation of SO(3)." I don't think the analogy is bad, but it is very marginally simpler than the example at hand. This means saved space. I'd like to use it for the following:
• The map exp:(so(3;1)->SO(3;1)⁺ is certainly not one-to-one, it is for most g∈SO(3;1)⁺ many-to-one. Is it onto? For pure rotations it is onto, and I believe it's onto for pure boosts as well. Every proper orthocronous LT can be written as a pure boost times a pure rotation. This doesn't immediately answer the question because e^{i(θ · J + ζ · K)} ≠ e^{i(θ · J)}e^{i(ζ · K)} because J and K do not commute. I'd appreciate help here. YohanN7 (talk) 23:14, 16 February 2013 (UTC)