## Monday, August 29, 2016

### Dear Dr. B: How come we never hear of a force that the Higgs boson carries?

“Dear Dr. Hossenfelder,

First, I love your blog. You provide a great insight into the world of physics for us laymen. I have read in popular science books that the bosons are the ‘force carriers.’ For example the photon carries the electromagnetic force, the gluon, the strong force, etc. How come we never hear of a force that the Higgs boson carries?

Ramiro Rodriguez
Dear Ramiro,

The short answer is that you never hear of a force that the Higgs boson carries because it doesn’t carry one. The longer answer is that not all bosons are alike. This of course begs the question just how the Higgs-boson is different, so let me explain.

The standard model of particle physics is based on gauge symmetries. This basically means that the laws of nature have to remain invariant under transformations in certain internal spaces, and these transformations can change from one place to the next and one moment to the next. They are what physics call “local” symmetries, as opposed to “global” symmetries whose transformations don’t change in space or time.

Amazingly enough, the requirement of gauge symmetry automatically explains how particles interact. It works like this. You start with fermions, that are particles of half-integer spin, like electrons, muons, quarks and so on. And you require that the fermions’ behavior must respect a gauge symmetry, which is classified by a symmetry group. Then you ask what equations you can possibly get that do this.

Since the fermions can move around, the equations that describe what they do must contain derivatives both in space and in time. This causes a problem, because if you want to know how the fermions’ motion changes from one place to the next you’d also have to know what the gauge transformation does from one place to the next, otherwise you can’t tell apart the change in the fermions from the change in the gauge transformation. But if you’d need to know that transformation, then the equations wouldn’t be invariant.

From this you learn that the only way the fermions can respect the gauge symmetry is if you introduce additional fields – the gauge fields – which exactly cancel the contribution from the space-time dependence of the gauge transformation. In the standard model the gauge fields all have spin 1, which means they are bosons. That's because to cancel the terms that came from the space-time derivative, the fields need to have the same transformation behavior as the derivative, which is that of a vector, hence spin 1.

To really follow this chain of arguments – from the assumption of gauge symmetry to the presence of gauge-bosons – requires several years’ worth of lectures, but the upshot is that the bosons which exchange the forces aren’t added by hand to the standard model, they are a consequence of symmetry requirements. You don’t get to pick the gauge-bosons, neither their number nor their behavior – their properties are determined by the symmetry.

In the standard model, there are 12 such force-carrying bosons: the photon (γ), the W+, W-, Z, and 8 gluons. They belong to three gauge symmetries, U(1), SU(2) and SU(3). Whether a fermion does or doesn’t interact with a gauge-boson depends on whether the fermion is “gauged” under the respective symmetry, ie transforms under it. Only the quarks, for example, are gauged under the SU(3) symmetry of the strong interaction, hence only the quarks couple to gluons and participate in that interaction. The so-introduced bosons are sometimes specifically referred to as “gauge-bosons” to indicate their origin.

The Higgs-boson in contrast is not introduced by a symmetry requirement. It has an entirely different function, which is to break a symmetry (the electroweak one) and thereby give mass to particles. The Higgs doesn’t have spin 1 (like the gauge-bosons) but spin 0. Indeed, it is the only presently known elementary particle with spin zero. Sheldon Glashow has charmingly referred to the Higgs as the “flush toilet” of the standard model – it’s there for a purpose, not because we like the smell.

The distinction between fermions and bosons can be removed by postulating an exchange symmetry between these two types of particles, known as supersymmetry. It works basically by generalizing the concept of a space-time direction to not merely be bosonic, but also fermionic, so that there is now a derivative that behaves like a fermion.

In the supersymmetric extension of the standard model there are then partner particles to all already known particles, denoted either by adding an “s” before the particle’s name if it’s a boson (selectron, stop quark, and so on) or adding “ino” after the particle’s name if it’s a fermion (Wino, photino, and so on). There is then also Higgsino, which is the partner particle of the Higgs and has spin 1/2. It is gauged under the standard model symmetries, hence participates in the interactions, but still is not itself consequence of a gauge.

In the standard model most of the bosons are also force-carriers, but bosons and force-carriers just aren’t the same category. To use a crude analogy, just because most of the men you know (most of the bosons in the standard model) have short hair (are force-carriers) doesn’t mean that to be a man (to be a boson) you must have short hair (exchange a force). Bosons are defined by having integer spin, as opposed to the half-integer spin that fermions have, and not by their ability to exchange interactions.

In summary the answer to your question is that certain types of bosons – the gauge bosons – are a consequence of symmetry requirements from which it follows that these bosons do exchange forces. The Higgs isn’t one of them.

Thanks for an interesting question!

 Peter Higgs receiving the Nobel Prize from the King of Sweden. [Img Credits: REUTERS/Claudio Bresciani/TT News Agency]

Previous Dear-Dr-B’s that you might also enjoy:

1. I don't agree, don't completely. The Higgs boson IS of course the remaining degree of freedom after SSB (Spontaneous Symmetry Breaking) that the SM needs to give masses to electroweak bosons W⁺,W⁻ an Z⁰ BUT it does not mean that it carries no force at all...I think the issue is what force "means" here...If you consider the SM definition, bosons are FORCE carriers, even if you place apart the Higgs boson as "mass giver", its origin it is in the same mechanism that originates the W and Z degrees of freedom, so I would not take the Higgs apart. By definition at the fundamental level (I give up not fundamental bosons like mesons in QCD or alike), I mean. The Higgs field does interact electromagnetically in the sense it can decays into two photon pairs...And it interacts with color fields, otherwise we would not be able to produce the Higgs boson at the LHC. These are known facts. Now speculations: the Higgs sector of the SM can be "the portal" to other interactions, IF the Higgs boson (or any other particle more massive like it) can decay into "invisible modes"...This is what is called the hidden sector, the dark portal, or some other cool names...In principle, the Higgs particle could be sensitive to new interactions decaying into particles our current detectors can not see. These new "invisible" particles produce what is called the "invisible Higgs width" (very similar to the Z invisible width that was essential to confirm we have 3 light SM neutrino species), and this decay channel H--> nothing (in our detectors) would hint new quantum numbers/particle species and maybe shed light over the dark matter problem (and very unlikely, shed light on the dark energy problem -since the dark energy particle would have a mass too tiny for our LHC detectors).

2. J,

Well, yes, as everything it depends on the definitions of words, in this case on the definition of what is meant by "force". I believe I'm using the standard definition here, which is really the only way to answer this question. But if you want to use the word "force-carrier" in a different way, you'll come to different conclusions. I didn't distinguish between the Higgs-field and the boson in this post (I think it was enough to stomach as it is), but Higgs-the-field arguably isn't introduced for the same reason as the gauge-bosons, I hope we can at least agree on that. It's gauged, not a gauge field. Best,

B.

3. Thanks for the great post.

Is there any similarly simple explanation of symmetry types?

4. so what exactly is meant by force in this context?
I remember hearing about some kind of force mediated by electrons - maybe it is called the van der waals force, I don't know; I just remember it being there because the combined wave function of two electrons in orbit around two nuclei is not just the two wave functions added, but they combine in a way so that it is energetically favourable for the atoms to come closer together. Is, in this case, the electron a kind of force carrier?

5. David,

Yes, good point. I am only referring to the fundamental forces, of which there are four (too physicists' best present knowledge): the electromagnetic force, the strong and weak nuclear force, and gravity.

There are in addition also loads of 'emergent' ie not fundamental forces that are basically left-over forces at lower energies. The Van der Waals force is one of these, its origin is essentially electromagnetic + quantum mechanics. I don't think this force is exchanged by electrons, it's probably some kind of quasi-particle, but I'd have to look this up. Best,

B.

6. John,

Not sure what you mean by 'symmetry types'. Do you mean local vs global? Broken vs unbroken? Or do you mean the symmetry groups? Could mean all kinds of things. Best,

B.

7. "This of course begs the question just how the Higgs-boson is different, so let me explain."

Nice to see someone use "beg the question" properly. :-)

8. Brilliant reply. If you're an undergraduate: I would print this out.

9. How does the Higgs boson not carry a force? It's a spin-0 particle, it should produce a standard Yukawa interaction, attractive and short-range. A pair of electrons can exchange a Higgs boson, in addition to exchanging photons or Z bosons. How is this not a force? Have I missed something completely?

10. Will,

Yes, it does produce a Yukawa-interaction. Why isn't that referred to as a 'force' in the pop sci literature? I don't know. It's a question for a linguist, not for a physicist. As the current narrative goes, there are four fundamental forces - that's what my explanation is about. The alternative would have been to confuse everybody by saying actually there aren't four, and I'm not sure anybody would be helped by that.

In the end, as so often, it comes down to a matter of definition of words. But I think I'm using words in the most commonly used way here. Best,

B.

11. Leaving aside the war of words, there are serious proposals to measure 'Higgs-force' effects, e.g. arXiv:1601.05087 [hep-ph]. Note I'm not qualified to judge on the feasibility of these experiments.

12. Sabine,

You are using the white-collar "gauge boson" definition of "force". The everyman's blue-collar definition of force is "that which mediates an interaction between matter particles", and as such the Higgs (and any boson) qualifies. Indeed, as someone pointed out already, that the Higgs mediates an interaction is why we have been able to detect it at the LHC. What is so special about gauge bosons that make their "force" more "force-like" than other bosons that mediate physical interactions?

13. Orin,

This has very little to do with the color of my collar, and very much with the question I was asked. For better or worse the word 'force' has become used in the popular science literature in the way that I explained in my post, with there being four fundamental forces and so on. Now why that is so, I can't tell you.

One can debate back and forth if that's a good use of the word or isn't, but honestly I think it's a rather nonsensical discussion. Fact is, the Higgs is different from the gauge-bosons in the way that I explained, regardless of what you want or don't want to call a force. Do you really think it would help to clarify these matters by declaring the Higgs carries a fifth force, after the word 'fifth' force has become to mean a force that's not in the standard model? (You know, dilatons, chameleons and all that.) Best,

B.

14. Bee,

Unless you show that the graviton is a gauge boson of spin 1, I don't think you are being consistent with the description in your post :)

And yes, I do that that calling the Higgs a 5th force helps clarify the discussion for students. They can know intuitively what a force is: that which mediates interaction. They can't know intuitively what a gauge interaction is and what it is that separates it from another kind of interaction, not without graduate study at least. And as a general rule of thumb a good definition is one that aligns closely with our intuitions. This is why I called your definition "white-collar" -- it seems to exclude, unnecessarily, mere common folk from actually understanding what a darn "force" is, which in fact is something that pretty much everyone has direct experience about!

15. Orin,

Yes, you're right about the graviton. In my defense, I explicitly referred to the standard model in my post. To further obscure the matter, it is sometimes - quite reasonably - argued that gravity isn't a force. So, erm, where does this leave us then? With saying that physicists use the word 'force' in different ways but science writers (including me, sometimes) like to use it in a specific way that doesn't always make sense?

I have to admit that on some level I find this utterly frustrating, because it's a problem that simply doesn't exist on the level of equations. There's math-things that behave in certain ways and nobody really cares what they're called.

I can see that calling the Higgs a fifth force might make sense in class, but I think it would be a great confusion in the popular science literature, after telling people that there's four forces and a fifth one would be a great breakthrough and new physics and all that. I hope you see what I mean?

Best,

B.

16. Any sensible debate presupposes an agreement on the definitions of all words used in the debate by all participants in the debate. Otherwise, the "debate" is nonsense.

17. I agree it would be confusing for the student to suddenly hear about a fifth force, but only because not everyone is on board with my opinion on this. But I think if we all start now, things will be OK after some brief initial confusion. In my talks, I don't think I shock or confuse the students too badly, because the discovery of the Higgs boson is understood already to be a really big deal with a lot of popular media attention, so students are primed to accept that this big recent discovery represents the discovery of a fifth fundamental fundamental force of nature.

18. Dear B.

Hi again!
I agree with you, the Higgs boson IS different: there is no gauge symmetry (not at least in the SM) associated to it. Of course, it can be changed a little if we consider the Higgs field and the Higgs boson quanta as the pseudo-Nambu-Goldstone boson from a higher (gauge like) symmetry, like the little Higgs models in fact do.
I am happy to see how you went into the core of this non-trivial questions. Yes, the issue is what we mean by force. Force is in fact a definition, an axiom from physics. Giving up the wrong Aristotelian notion of Force (proportional to velocity!), we arrive to the current notion of force as something tied up to the change of the temporal rythm of change of velocity (Galileo, Newton, and current time). Quantum Mechanics and Quantum Field Theory have also changed or completed this vision, attaching to the notion of "fundamental force" the notion on "intermediary interaction". 2body forces are the only fundamental forces today. Force is F=ma=d(p)/dt=rate of change of linear momentum= mass x d(v)/dt= mass x rate of change of velocity with respect to time. Bosons=force carriers/fields, Fermions=matter particles/fields. BUT, as you say, scalar "fundamental" bosons or the hypothetical graviton (or even the higher spin particles of theories like hypergravity or superstrings) are just different to usual gauge boson fields. Today gravity is a pseudoforce (spacetime geometry, Einstein dixit). Higgs boson are yet a mystery, higher spin particles a dream (at least from experimental viewpoint). So, the answer to the question of Ramiro is of course dependent on what definition of force you use. Even worst, the lagrangian or hamiltonian picture of mechanics don't use a notion of "force". Exercise: how would you change the answer if we approach a lagrangian viewpoint?

19. Indeed, I think to remember a talk by a theorist that she gave, about to thinking about the Higgs as the fifth force. The criterion she used was based on couplings and/or self-couplings...I will see if I remember the talk. She did a great presentation...Let me think if I can find it as a pro in this question.

Note: perhaps we should count at the quantum level "forces" as couplings introduced via coupling into a lagrangian. With that definition, the Higgs does carry a force...

Let's see:

Strong force coupling ~1
Electromagnetic force ~e²~1/137
Fermi Constant ~10⁻⁵
Higgs coupling to Fermions (variable) but relative to itself it has a trilinear coupling about ~0.13 (SM prediction to be tested in the future).

Note (II): the Higgs mechanism just shifts the origin of mass to the origin of the yukawa coupling constants between a Higgs and fermion fields!!!!

NOte (III): gravity operates at other level of "force-ness" much weaker than all the rest but very important at bigger scales...

Question: is the same fundamental/quantum (gauge) force than quantum (non-gauge!) forces? Or course not! That is the key for the future physicists. We don't understand the Higgs field as a gauge field at current time beyond some BSM models! Take a preonic Higgs or composite, take a fundamental scalar with SUSY, it has consequences, or it should in phenomenology...The big thing about the Higgs-like particle is not said at all. See the 2012 higgsdependence conference, it makes sense that the Higgs boson force is "different". How? We don't even know. That is why we do and love Science :).

20. Hi Sabine - Excellent post, as usual! Congratulations on standing firm on the question you were answering and the language you are using. I think you got it exactly right from a pedagogical standpoint.

I would add only that the number of gauge bosons is completely determined by the symmetries: U(1) can only have one gauge boson, SU(2) can only have three, and SU(3) can only have eight. I'd always been very impressed that we observe exactly the number of gauge bosons predicted by the symmetry, which lends weight to the symmetry being exactly the right insight.

The number of gauge bosons, given a symmetry, is the dimension of the space of complex rotations described by that symmetry (at least for the known symmetries U(1), SU(2) and SU(3), which describe rotations in a complex space). I'll admit to being very strongly influenced by the geometrical interpretation of gauge symmetry. You were wise not to bring this into your answer, but I can't help it :) so here goes: At least at the "classical" field theory level, gauge bosons are geometrical connections exactly like the connections of General Relativity ("Cristoffel symbols"), the field tensor is curvature (it really is!), and the Lorentz force law is the geodesic equation. Of course this is not the geometry of spacetime, but the geometry of some weird abstract complex space that the wave function lives on, described via fiber bundles. But the framework is essentially the same as General Relativity. You can even put the space of your field theory on curved spacetime (fiber bundle over a curved manifold), write down the vacuum Einstein equation for that combination and BOOM! you get the Einstein equations in the presence of an electromagnetic field (in the case of U(1)). All very pretty and compelling for those of us who like this kind of thing. But, as Bruno Zumino once told me, the geometric interpretation not really useful.

One question: on spin-1 of the standard model forces vs. spin-2 gravity, doesn't that trace to the use of the metric (a 2-tensor) in General Relativity as the fundamental object vs. the vector (1-tensor) connections in the standard model? Hmm... What about the formulations of quantum GR based on the connection? Do they still say the graviton has spin 2?

21. "I have to admit that on some level I find this utterly frustrating, because it's a problem that simply doesn't exist on the level of equations. There's math-things that behave in certain ways and nobody really cares what they're called. "

So the subject of many of the comments here is language, not science. Now I see why you've added a philosopher to your team -- that's their specialty ;-)

22. Bee,

Do you think it is possible that the second metric of a massive gravity is the Higgs field ?

23. At this point I know too little at this point to even formulate an intelligent question, though I hope that may change with enough reading. But you used the expression "to bet the question" incorrectly. Many think it means "to demand that the question be answered," but it is the English translation of the logical error petitio principii, assuming that which one is trying to prove. I fully expect that is the only error I shall ever catch you in. Carry on!

24. Hi Sabine,

You've talked a bit about it above, but could you please explain how exactly a "fundamental force" is defined in common usage?
I am confused to why most sources talk of four "fundamental" forces when it is known that the electromagnetic and weak force are the same at high energies; doesn't this mean that there is only the electroweak force is fundamental?

As for gravity as you've mentioned, in GR it is not a force. Perhaps it is still considered an interaction as masses cause curvature which cause different geodesics..?

In any case, it is not clear why there are quoted four fundamental forces, not just electroweak and strong forces.

25. giralua,

Yes, 'begging the question' is the name of a logical fallacy. It is also simply a combination of three words in the English language that convey a certain meaning. In any case, the first sentence of my reply is the kind of circular argument that I thought the phrase was referring to, please correct me if I'm wrong.

26. A,

Well, that's what the above discussion with Orin was about. See, the current narrative that you'll read all over the internet and in every popular science book is that we have four fundamental forces: the electromagnetic force, the strong and weak nuclear force, and gravity. This refers, loosely speaking, to four symmetry principles that our current theories are based on. I think that this the 'common usage' of 'fundamental force' which you ask for and also the context in which I answered Ramiro's question.

Sure, gravity is an outsider in this because the symmetry isn't an internal one. If you (as several people have suggested to me) define force by a potential, then the Higgs carries a force, but gravity isn't a force, because it doesn't have a potential (at least not in general), it's a property of space-time. That nomenclature might make more sense in some regards, but it's not how the 'fundamental forces' have become used.

And, yes, if you believe in a theory of everything, then this means that fundamentally (at very high energies) there should be only one force. (To bind them all and so on ;)) Best,

B.

27. nicolas,

Well, the spin doesn't match. Maybe if you extract the scalar mode. But I don't know, I'm not a big fan of massive gravity.

28. Steve,

Yes, if you introduce 'gauge fields' in GR the same way as in the SM it's the connection that plays the role of the gauge-boson, not the metric (the difference being that now the internal and external indices both refer to space-time). The reason the graviton is spin-2 is that it isn't introduced by the minimal coupling, but by taking the perturbative limit of the Einstein-Hilbert action, which gives a wave-equation for a 'particle' that is the perturbation of the metric. It's this object that's called the graviton. (Or, if you look at the Lagrangian: the Lagrangians in the standard model are quadratic in the field-strength, but GR isn't quadratic in R, at least not in the normally used formulation.) Maybe another way to see it is in the coupling term in the perturbative limit, which is a contraction of the spin-two graviton and the stress-energy tensor.

Best,

B.

29. Sabine,

You began your very interesting post by replying, "The short answer is that you never hear of a force that the Higgs boson carries because it doesn’t carry one." This subsequently lead to a debate about what is or isn't a proper definition of a "force".

I submit that a short answer to Ramiro Rodriguez, which might have bypassed the "linguistic" debate about "force", would be, "Bosons are fundamental particles having integer spin (0,1,2). And, whereas all force-carrying particles are bosons, not all bosons are force carriers. The Higgs is not a force-carrying boson.

30. The Higgs boson is like a rolling stone. The former carries no force; the latter carries no moss.

31. Thanks, Sabine - your explanation starting with "Yes, if you introduce 'gauge fields' in GR the same way..." helps a lot. I understand about the metric causing spin 2 (loosely speaking). What I'm vague about is: does the connection approach suggest a "graviton" with a different spin, or do they also end up using the metric at the end? I'm not sure how to avoid a metric...

32. Steve,

I'm not sure who is "they"? The metric is an awkward object because if you're dealing with spinors (what you want to do anyway) you'd rather have tetrads. So you can combine the connection with tetrads and I think this pretty much works, except that, of course nobody really agrees whether these are the right variables. Maybe this helps. In any case, I don't think you'd redefine what is meant by 'graviton', either way you approach the quantization you should be able to get such an object at least approximately I would think. It's hard to see how you can avoid reproducing the perturbative limit effectively. That doesn't mean though that the variables that you have there are still good in the UV.

So, I think the answer to your question is no, the graviton is still a graviton.

Best,

B.

33. Steve,

I'm not sure who is "they"? The metric is an awkward object because if you're dealing with spinors (what you want to do anyway) you'd rather have tetrads. So you can combine the connection with tetrads and I think this pretty much works, except that, of course nobody really agrees whether these are the right variables. Maybe this helps. In any case, I don't think you'd redefine what is meant by 'graviton', either way you approach the quantization you should be able to get such an object at least approximately I would think. It's hard to see how you can avoid reproducing the perturbative limit effectively. That doesn't mean though that the variables that you have there are still good in the UV.

So, I think the answer to your question is no, the graviton is still a graviton.

Best,

B.

34. @Sabine, @giralua, @Philip Helbig

I think the war about "begging the question" is essentially over. Sure, it means "circular reasoning," but for decades now, its primary use has been "begs for the question to be asked." Language, especially English (we don't have an Academy!), just works that way; it changes via mistakes and mishearing. "An apron" was once "a napron." "Homing in" for targeting was a great phrase from radar engineers (and pigeon fanciers), but now, more than half the time, you'll hear "honing in." Why? Because "hone" means sharpen, so the phrase works perfectly well that way. Does anyone under 40 remember when "edgy" meant "nervous" instead of "avant garde"? My high school English teacher actually twisted my ear when I spoke of an animal's gender. "James," he said, "WORDS have gender. People have SEX." But that was 50 years ago, nowadays, people do have "gender" (and sex too!) and very useful it is; new genders seem to pop up weekly.

I know this post is off-topic for the blog in general, but I can't resist; I'm like the drunk under the lamppost -- I know a bit about language but even less about physics. Feel free to refer it to the Philosophy Department. BTW, yesterday's XKCD has some relevance to this discussion.

Jim

35. I tend to agree with Giralua (for what little my opinion is worth).

1. While the first sentence in your reply is somewhat circular, it is not the conclusion of an argument, so it is not a fallacious argument, so it does not "beg the question" in the technical sense.

2. For those who have never heard of the technical term "beg the question" it won't make any difference, but for those who have it is a bit jarring and distracting. Using "raises the question" instead of "begs the question" would not have this effect.

3. As the other Jim points out, English and all languages change over time, but personally I wish only improvements would occur, rather than meanings becoming opposite (e.g., "I couldn't care less" became "I could care less" in reference to something one does not care about). Evolution consists of more bad mutations than good ones, but this does not make bad mutations useful. So I believe in resisting such changes. (Of course I am losing badly.)

4. On the other hand, Dr. Helbig perceived it as a clever joke (which went over my head, as the saying goes), as was apparently your intention, so it is not an error and should be left as it is.

36. Yes, language changes. But that doesn't mean that every change is good. One needs a balance between no change at all (in which case we would still be speaking proto-Indo-European or whatever) and "anything goes", which would mean "nucular physics".

I think Steven Pinker strikes a good balance in his The Sense of Style. His other books (on psychology and the acquisition (as opposed to the use) of language (his technical specialty) are recommended as well.

37. The last word. :-) http://www.worldwidewords.org/qa/qa-beg1.htm

38. Hard sciences' nomenclature often begins before the target is fully characterized. "What is a force" is answerable in context. Context can be arbitrary.

"I have the perfect description!" (Non-classical gravitation, SUSY; economics, psychology; religion)
"Of what?"
"It depends."

Optical rotation arises from math plus empirical variables. It is useful but not meaningful. "Force" as a generality oozes and squirms.

39. What is a "planet"? A new definition was contrived so that schoolchildren could name them all without threat of future revision. Sadly, they now would have much more difficulty explaining why something is or isn't a planet. (You can nitpick my analogy; it isn't meant as a criticism of physics.)

40. Thanks, Sabine for the comment about tetrads. I've read through the tetrad formalism a few times over the years, but it has never transferred from "useful formalism" to "understanding what's going on geometrically". I suppose my not trying to actually calculate anything with tetrads or the connection representation is holding me back. For applications to quantum GR, I have Rovelli's and Vidotto's introduction to covariant loop quantum gravity on my reading list (I'd read some of Rovelli's introductory notes, but that was hard going). Are there any other resources you'd recommend?

41. Okay, since you have a well developed sense of humor, this one is just too damn good to pass on:

"I really liked his argument until he wrote that “Philosophers obsess over subtle ambiguities of language,” which pretty much sums up all that physicists hate about philosophy."

Based on this comment thread I would swear you all are a bunch of philosophers! I think we should all just start communicating in Lojban:

https://xkcd.com/191/

42. Nicely constructed (the initial answer), and aptly left with enough loose ends to follow-up (unlike the most) :).
Regarding a comment in the same: "...to cancel the terms that came from the space-time derivative, the fields need to have the same transformation behavior as the derivative, which is that of a vector, hence spin 1. "
Linking space-time derivative to internal space might not be that clear. Can it be made more clear through something like Pauli-Lubanski construction? Also, the issue of gauge dependence of spin in for a U(1) field is there.

43. But Wes, I miss Loglan!

44. Is it also true that the four fundamental forces are different because they are associated with conserved charges?

45. Ben: Conserved charges corresponding to different symmetries.

46. Ben,

As the inquisitor said, the charges belong to the symmetries. (Noether's theorem and all.)

47. Would one expect a similar phenomenon at the point where the strong and electroweak separated, similar to the symmetry breaking of the electrowwak? Is there a separate mechanism, perhaps, since one doesn't wind up with a local symmetry in this case? I'd never thought of this until reading your post this morning. Thanks and cheers.

48. Dear Dr. B,
Change of subject. Would you be able to provide some explanation of the developments going under the topic "ER=EPR"? Or if you don't have time, and know of a good exposition, would you post a link? I have read through Susskind's lecture here: http://lanl.arxiv.org/abs/1604.02589, but I feel there must be something else available that would be more appropriate to understand where this might lead. Thanks!

49. CGT,

I read that too... I can't say that I found it very convincing. Lots of words and pictures. I don't know of a good reference, sorry. I haven't really followed this. Maybe one of our readers has a suggestion? I have put this on my reading list, but I'm not sure how closely it's connected to ER=EPR (footnote 25 seems to indicate it's a somewhat stronger version). Best,

B.

50. Stipulated: ER=EPR wormhole-bound entangled particles. Said wormhole has free energy of opening and closing. If not, absent spontaneous wormholes must suffer selection rules for transition, limiting entanglement. No energy or other anomaly is observed for entangled particles' creation or collapse (re Pauli and neutrinos).

LIGO event GW150914 merged a 30 + 35 solar mass black hole binary, less 3 solar masses of binding energy flash-emitted, fully described by unamended GR. What of their still separated partners and contingent wormholes?

51. I don't think Susskind's lecture was meant to be very convincing. I think it was meant to be enticing.