That is not some special property of the BMS group. It also happens with Lorentz in the Poincare group and for rotations within the Euclidean group. The semi-direct product is not some mysterious property of BMS.Kartik PrabhuScience people on twitter! This is great: https://twitter.com/igalia/status/1539736505467510784
Now we need better authoring tools for writing math on the web.Kartik Prabhu
Yes, I know how to do that, which is why I asked a much more specific question.Kartik PrabhuSo here’s why what I asked earlier is useful for physics. I don’t think I need to explain why the Lorentz group is useful; relativity and all that. Its double cover is SL(2,C) which is useful for handling spinors (as some would say “fermions”). I’ll be a bit vague though.
In relativity, the kinds of “particles” (really fields) one wants are representations of the Poincare group i.e. spacetime translations and Lorentz. This was solved a long time ago by Wigner with a really ingenious construction. I’ll skip over unitary/faithfulness etc…
The translations are “easily” represented in Fourier space: so instead of a function of spacetime position x you have a function of momentum p (note p and x are Minkowski 4-vectors). Now, we need to represent the Lorentz transformations.
So here’s Wigner’s idea: pick a momentum p and find the largest subgroup of SL(2,C) which leaves this p invariant; that’s the little group L_p of the momentum you chose. Now you can act on this p by Lorentz transformations to get a whole bunch of other momenta.
If you act by anything in L_p it just gives you back p; so you will get something different if you act by SL(2,C)/L_p i.e. Lorentz transformations “not in” L_p. This quotient space is called the orbit space O_p of the chosen p.
In our case O_p is a manifold, and has a measure invariant under SL(2,C). So to construct the representations (wavefunctions/fields) you start with functions on the orbit O_p and integrate them over the orbit. Sounds like fancy nonsense so examples:
1) If you choose p to be a future timelike momentum, then L_p = SU(2) i.e. rotations; and O_p = H3 the Riemannian hyperboloid sitting in the future light cone. And the wavefunctions are functions on H3; these are the usual wavefunctions of massive fields in momentum space.
2,3) If you choose p to be null or spacelike you get a similar story with O_p being future light cone or the one-sheeted Lorentzian hyperboloid. The wavefunctions are then for massless or tachyonic fields.
In each case, we had an “obvious” invariant measure on the orbit space O_p since they all sit nicely as submanifolds of the Minkowski spacetime. This measure is obtained as a “quotient” of the Haar measure on SL(2,C); not sure how explicitly!
So what’s the big deal, we know all the cases we want, right? Well, enter General Relativity, where there is no symmetry. But there is an asymptotic symmetry group, which is BMS not Poincare, if you care about gravitational radiation and memory effect.
The analog of the 4-momentum p is now the supermomentum, which is a function on a 2-sphere not a 4-vector. But you can still play the same little group game as Wigner; with some very important subtleties about topology on infinite-dimensional groups like BMS.
This was worked out a long time ago by P. J. McCarthy e.g. https://doi.org/10.1098/rspa.1972.0157 You get more weird little groups.
1) One little group you get is a simple double cover of U(1), whose orbit space is H3 × S2
2) Another one is little group Z2, whose orbit space is H3 × RP3, this is just the Lorentz group (no double cover)
In each case an invariant measure on the orbit space is guaranteed to exist due to math (McCarthy does this in an appendix) but is there a nice formula for these measures?Kartik Prabhu
I have looked at that paper/notes before, but I am not sure I understand it. If you look at the last formula in section 1.3 on pg. 3, the k and k’ are in SU(2) and the r is a real radial coordinate. So that measure looks like it is 7-dimensional which seems wrong.Kartik Prabhu
It will. But that is also not explicitly given anywhere in terms of some nice coordinates.Kartik Prabhumath/physics question (cc:@johncarlosbaez)
Take the group SL(2,C), the double cover of the identity-connected component of the Lorentz group. As a manifold this group is homeomorphic to R3 × S3, where R3 are Lorentz boosts and S3 the group SU(2) (double cover of rotations).
You can also view the R3 as a Riemannian hyperbolic space H3. Now H3 has a natural Lorentz invariant measure, and S3 has a natural rotation invariant measure. So the left-invariant Haar measure on SL(2,C) should be some positive function F times the measures on H3 and S3.
Is there an explicit expression for this Haar measure in term of some/any coordinates on H3 and S3?
(Asking here because I’m tired of the condescension on math/stack overflow towards explicit examples)Kartik PrabhuTired of slap-fest? Why not read something new about infrared scattering in QFT and quantum gravity: https://arxiv.org/abs/2203.14334
The usual formulation of scattering for massless fields (which escape to null infinity) is beset by divergences in the infrared (IR), i.e, the S-matrix amplitudes diverge at zero frequency. This zero frequency effect is directly related to the physical memory effect; so we cannot just wish this IR divergence away. At null infinity, this memory effect labels an infinite-number of Hilbert spaces all of which are unitarily-inequivalent!
In QED with massive scalars, the Faddeev-Kulish construction by gives a nice way to stitch these Hilbert spaces together, by “dressing” all the in/out massive states with memory to have a fixed value of infinitely-many charges at spatial infinity which is conserved in scattering.
If you use a massless scalar source, the “dressing” becomes too singular — so singular in fact that the energy flux through null infinity diverges! Massless charged scalars do not wear a shirt to the red carpet!
But then GR shows up and slaps all the dressings in the face! The only state in GR which has fixed charges at spatial infinity is the Minkowski vacuum state! Mayhaps GR will win an Oscar someday too…
The issue really is trying to shoehorn all the quantum states into a single Hilbert space; quantum states, in the algebraic sense, are perfectly well-defined. So we suggest that we should try to define scattering also in the algebraic formulation.Kartik PrabhuCan someone explain to me what the hell UV/IR mixing means?Kartik Prabhu
That list sounds like you are in your own bubble, like the string theorist bubble you complain of. There is a lot of important (and fantastic IMO) work by Wald (and collaborators), Zoupas, Hollands, Ishibashi; hell I’ll even put my own name in there and include @PhysicistBeKartik Prabhu
That is quite misleading! Abhay’s work on asymptotics at null infinity stands completely on its own without reference to any LQG stuff. In fact, it actually would be a nice project if someone wants to figure out the LQG version of asymptotic quantization!Kartik Prabhu
It looks like Harvard has a very good PR department. The relation between memory, symmetries and infrared sectors were noted a long time ago in mathematical GR, especially by Ashtekar. High-energy folks simply ignored this; and now credit everything to Strominger. ¯\_(ツ)_/¯
Since I’m a scientist, I’ll cite my sources (note the dates)
1) A. Ashtekar, Asymptotic quantization of the gravitational field, Phys. Rev. Lett. 46 (1981) 573.
2) A. Ashtekar and K. S. Narain, Infrared Problems in Quantum Field Theory and Penrose’s Null Infinity, Syracuse University Report, Presented at the VIth International Conference on Mathematical Physics (1981).
3) A. Ashtekar, Asymptotic Quantization: Based On 1984 Naples Lectures, Monographs and Textbooks in Physical Science. Bibliopolis, Naples, Italy, 1987.
4) E. T. Newman and R. Penrose, Note on the Bondi-Metzner-Sachs Group, J. Math. Phys. 7 (1966) 863.Kartik PrabhuFinally, managed to do something with twistors https://arxiv.org/abs/2111.00478 ! 17 year-old self would be very happy :D.
Turns out you cannot impose the twistor equation at null infinity. But, you can impose three of the six components of it! You get an infinite-dimensional space of spinor solutions, BMS twistors.
Then, you can take two of these BMS twistors and generate a complex BMS symmetry at null infinity!
Not sure why this works but seems pretty neat.Kartik Prabhu
The universal structure is whatever structure (boundary, manifold/smoothness structure, certain fields like the metric) which are the same for all solutions you want to consider. The symmetries are the things which preserve this universal structure. A few examples:
1) Say you have a scalar field in Minkowski. What is the universal structure for all solutions? Well, it is the background Minkoowski spacetime; so the symmetries are Poincare. But if you look only at certain solutions, like rotationally-symmetric ones, then there is additional universal structure namely the rotational Killing field; so the symmetries are smaller than Poincare.
2) At null infinity, the universal structure is given by the usual conformal structure in the Penrose completion; the symmetries are BMS. If you choose to look at some smaller subset of asymptotically flat solutions, like stationary ones then the symmetries are reduced to Poincare instead of BMS.
3) At a finite null surface, you have much less universal structure, since a lot of quantities do not fall off and are not the same for all solutions. Consequently, you get more symmetries since you have less universal structure to preserve and so you can do more transformations. Again, if you look at a subclass of solutions like Killing horizons, there is more universal structure (the horizon Killing field) and the symmetries are also reduced.Kartik Prabhu
Those are not even continuous vector fields and don’t even preserve the manifold structure of the spacetime. So not sure why they are “symmetries”. If you are allowed those kinds of vector fields then might as well take a delta function vector field and call it “ultra soft hair”!Kartik Prabhu
I don’t know what “soft hair” means and what it has to do with supertranslations.
Null infinity is “less” symmetric than a horizon; it has a smaller symmetry group since null infinity has much more structure due to everything falling-off.
Supertranslations do not change the geometry, they are diffeomorphisms and just like any diffeomorphism they do not change the local geometry.Kartik Prabhu
I am not sure what “the problem of quantization” is!?
If we are given a non-relativistic theory, we do not ask whether a unique “relativization” of the theory exists. Similarly, we do not ask for a unique “general relativization” of Newtonian gravity. We also do not ask for a “moleculization” of Navier-Stokes equations to “derive” that water is made up of discrete molecules.
Usually, these are questions phrased in the reverse order, i.e., “how to get a non-relativistic or continuum” limit in certain special situations from a more fundamental theory.
So why is one asking for the existence and uniqueness of the “quantization” of some classical phase space?Kartik PrabhuHow to find asymptotic symmetries and their charges? Looky here
“Before rewriting the quantum theory in a new form, we generalize the above considerations to a collection of n decoupled, time-independent harmonic oscillators of frequencies ω1,…,ωn.”
WTF does that mean for the election?Kartik PrabhuHot off the press — more spatial infinity stuff; symmetries and charges; with Ibrahim Shehzad : https://arxiv.org/abs/1912.04305Kartik PrabhuNew paper on angular momentum of Einstein-Maxwell at null infinity with Beatrice Bonga and Alexander M. Grant. Go get it!
anything violating the uniqueness theorems would be fair game: asymptotics, higher dimensions, modified gravity… But getting an explicit counter-example seems pretty hard. Would be nice to have some local existence proof though not sure how to even approach that!Kartik Prabhu
btw: for Einstein-Maxwell the electric potential being constant is a result by Carter in some old-ish Les Houches publication that I can’t find right now.
I also expect static to imply that the electric flux/area element is constant, but I really have no idea how to show that without using global results like Birkhoff’s theorem or uniqueness (in the stationary case). Somehow our guts know about global results that our brains don’t!Kartik Prabhu
The electric potential (which is the analog of surface gravity) is indeed constant as I showed in my thesis (Sec. 3 Theorem 1). The electric flux E.n is more analogous to the area element/ Noether charge which need not be constant.Kartik PrabhuSome casual reading on causal diamonds for the weekend: https://arxiv.org/abs/1908.00017Kartik Prabhu