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Thanks for the explanation. If you have the time, can you also explain _why_ would we be multiplying "each point of the universe with a different complex number of modulo 1?" What does it mean in physical reality; why multiply points with any number at all?

Not GP, or a physicist, but my understanding is that the different number at each point you multiply with represents a degree of freedom at each point of spacetime, and in this degree of freedom is where the electromagnetic field lives.

To find out what kind of symmetry a field (as described by a Lagrangian) admits. It's called gauge symmetry: https://en.wikipedia.org/wiki/Gauge_theory

It's a complex number for QED but in general, it's a unitary matrix: a rotation which preserves the magnitude of a wavefunction (a complex vector).

In physics, symmetries play a fundamental role.

The simple answer is "because we can". In general, physicists have found that we should write down the most general mathematical theory compatible with what's observed. A famous example of this is Einstein's cosmological constant -- I'll leave that one to wikipedia [1] since I'm a particle physicist and not an expert on GR.

In the case of gauge theory, the idea that we should consider the most general case has been well proven. As GP pointed out, all observable phenomena ultimately depend only on the absolute modulus of the field Ψ, so a theoretical physicist naturally wonders, what happens if you allow its complex phase to vary. Turns out nothing interesting happens if you apply a global phase, but if you allow the phase to vary at every point in spacetime, it ends up breaking the theory. That is, unless you include an additional field at every point in spacetime that precisely cancels out the change induced by the gauge freedom.

In other words, the motivation is that we can't simply look and "see" whether or not there is a locally varying phase on the wavefunction Ψ, since we only can measure |Ψ|^2. So we have to assume there is, until proven otherwise. Since a local phase would imply the existence of an extra field to cancel it out, we can indirectly check for this scenario by looking for the corresponding field. As pointed out by GP, in the case of a U(1) gauge, it turns out there is such a field, and electromagnetism (and all of its laws) exactly fit the bill.

There are other "unmeasurable" symmetries you could apply to the wave function as well, beyond just a complex phase. SU(2) is a Lie group symmetry which would mean that the measurable properties of certain tuples of fields (Ψ, ϕ) are indistinguishable under a sort of complex-valued rotation of Ψ->ϕ and ϕ->Ψ. Again, if you assume a such symmetry is locally varied at every point in spacetime, you end up requiring not one but three new fields to cancel out the effects on the SM Lagrangian. It turns out that the vector bosons W+, W-, and Z, which mediate weak nuclear forces exactly fit the bill.

[1] - https://en.wikipedia.org/wiki/Cosmological_constant

Wow you blew my mind. You need to consider creating YouTube videos about this.

And at the risk of muddying the original point, it turns out that strictly speaking the neither the neutral Z boson nor the electromagnetic photon can be said to come from SU(2) or U(1). Instead, each field is a linear combination of some abstract fields (namely the neutral W0 and the weak hypercharge B bosons) from the unitary gauge induced by the compound symmetry SU(2)xU(1). This is because when both symmetries SU(2) and U(1) are present, there are different ways you can "mix" the two into the Lagrangian. The mixing that gives us the photon and the Z0 boson is known because of the experimental confirmation of these particles. This is what is meant when it is said that electromagnetism and weak neuclear forces are united in a higher-energy theory as a single "electroweak" force.

Thank you!

Thanks for these comments. This simplified a whole bunch of things I had questions about.

This is great, you perfectly bridged the gap between having been 25 once and vaguely remembering the math terms mentioned and bringing it back to reality. Really epic how this (at least to me) abstract and arcane math not only explains the behaviors of the subatomic particles, but even predicts the existence of more and their related forces.

The work of theoretical physicists always seemed like the other side of an ocean of math away, but thanks to this explanation it feels like I can at least make out a lighthouse in the distance.

> That is, unless you include an additional field at every point in spacetime that precisely cancels out the change induced by the gauge freedom.

Could you elaborate on this a bit? To a layperson this sounds like a hack. "Things get screwy when you screw with them, UNLESSSSS we add a magic thing that undoes our work". Well yeah.

It turns out that the magic thing we have to add, exactly fits the observations we have of electromagnetism. That's an indication that we screw with the theory in the same way that nature does, and in the end that's the goal of physics: understand the way nature behaves.

It's just a mathematical tool of curiosity that we have found very useful.

Here's an analogy: You find a box, and you can't see inside it. You have no reason to think there's something inside it. But also, boxes have stuff in them sometimes. So, you shake the box, and hear something clinking around. Therefore, you infer there's something in the box.

Somebody next to you says "This sounds like a hack. There was a box and you had to go shake it until it started making sounds that it wasn't making before. UNLESSSSS we now magically have to agree that there's something in the box."

It's a perfectly reasonable question, and I'm just turning your words on you in good faith :)

To take the technical discussion a bit further, it's exactly this kind of reasoning that led to the discovery of the Higgs boson. Strictly speaking, it's impossible for gauge bosons (that's what the particles are called that show up when you add these locally-varying symmetries) to have nonzero mass. The photon and gluons (from the SU(3) strong force) are massless, but the W and Z bosons are VERY massive. This was a big problem with the Standard Model; the vector gauge bosons had every property expected from the gauge theory, except for this one point about their mass, which was experimentally incontrovertible.

That is, until Brout/Englert/Higgs came along. They said "Yeah the vector bosons must be massless UNLESSSSSSSS you assume there's this magic additional field that couples to every particle's mass, in which case it perfectly cancels out all the problems and allows the W and Z bosons to be heavy". It took 50 years but we found that particle eventually.

You're a really great writer on this topic. If you're not already, and you have the time, you should seek out ways to do this in a way that has more reach. Thank you!

You know I tend to agree with the previous poster, not that it's a hack, but that I've always felt its a bit backward reasoning. You say the physics should be invariant under curvature of the field, but it isn't unless you add another field to cancel it. But you might as well have said from the start that the field is curved by another field and we need to consider that in our derivatives that describe our local physics, making them covariant. The explanation that "it makes sense that the fields are locally gauge invariant" always seemed a bit constructed after the fact so to speak.

The argument isn't against physicists inventing new fields or interactions to fit data. The argument is about why you can motivate it as "something that has to be added out of pure logic" :)

Why shouldn't you be able to? Hypothesizing and then seeing if it leads to contradictions is a fundamentally useful technique.

It makes a lot more sense if you have the historical context though. When this is taught the context is left out and then it feels more like mathemagics to me and probably others as well. Maybe there are good books that explain this from a historical point of view, while still teaching the theory, but they must be pretty rare.

My comment (the GG..GGP) was an (over)reaction to the posted article. The article presents the new approach like a very "natural" explanation, but there are already some other "natural" explanations.

In a Physics degree, the order is quite historical. Like one full course for the three first next items, and all the other together. I'm not sure if there is a book with all of them, you probably need 4 or 5 books.

1) Non-Quantum Non-Relativistic Electromagnetism

2) Quantum Non-Relativistic Electromagnetism

3) Non-Quantum Relativistic Electromagnetism

4) Quantum Relativistic Electromagnetism

5) By the way, you can interpret the Quantum Relativistic Electromagnetism as a U(1) symmetry. (my comment)

6) It looks like a good idea. Let's use other groups to explain other known forces: the weak and strong force. (The G...GP comment about SU(2) and SU(3).)

7) ???

[See note 1]

For some reason, popular science articles love to show something almost magical and prefer to present something like the "5)". It makes it easier to hide the math and use hand waving.

Also, Physicist working in physic particle also believe that "5)" and "6)" are the correct approach, and the other are just useful for teaching and for historical reasons. But to discover "7)" it's better to think about some weird new symmetry group [2].

For examples, a few years ago, it was popular to think the next step "7)" was using a new group SU(5) that combines SU(2) and SU(3). The problems is that the experiments gave different results than then new proposed theory, not too bad but like 1% off. I still remember my professor talking about how great was SU(5) and how the experiment disagree, and he looked heartbroken because he really liked SU(5).

[1] You should add some material about the historical discovery of the weak and strong forces between "5)" and "6)".

[2] Other's prefer superstrings for "7)", there are other approach, but all are weird.

I don't think it's backwards. We started out with this big sweeping and simple statement that (seems to) holds true for physical reality, but it doesn't account for everything. The statement allows for two realities, one where there is one single global phase, and another where each point has its own local phase.

The first option might be true, no real way to measure it and that's sort of it, no explanation for all the other stuff going on. The second option is a little more complex, it requires an extra construct to make it work. And apparently when we do the math and work out this construct, it exactly maps to things we can measure in reality, that were not explained by the other simpler option.

So it's not backwards because it's simply the first full match in a depth first search through the possible realities that follow from this wave function theory.

A really simple example is voltage. What does it even mean if one cable is on a potential of 5 V? It's always compared with the Ground voltage since the voltage is always a difference between 2 electric potentials. That means you could add a constant to each potential and nothing would change. So in this case not gauging would be quite hacky... (This example has nothing to do with the phase though, but just to illustrate. Almost always when you measure something, some gauging is at least implicitly involved.)

So it turns out this happens quite often that there is some kind of constant that can be divided out. In case of particle physics a whole framework has been developed out of it that has really close relations to Lie group theory. (The experimentally confirmed parallels are just astonishing with group generators and elements corresponding to interaction particles and the normal particles.)

But there is an important distinction between global and local symmetries. Global symmetries, when you apply the same change to the field at every point, are physically meaningful, not so local symmetries, when you apply [potentially] different changes at different points.

For example the laws of physics are time translation invariant, i.e. it does not matter what point in time you call t equals zero which essentially means that the equations do not contain time but only time differences so that you can add the same constant to all your times and nothing changes as the constant cancels out when calculating a difference between two times.

Its the same with voltage, where you put your reference potential does not matter but this is again a global symmetry and you have to use the same reference potential everywhere, you will obviously get nonsense if you use different reference potentials at different points.

Local symmetries on the other hand are kind of defects in the mathematics of physical theories, they are the expression of redundancies in the mathematical description. Say you want to describe the orientation - but not the strength - of the magnetic field on earth and for simplicity lets assume the earth is flat and the magnetic field parallel to the surface, then you could do this by associating a two dimensional vector with each point on earth that describes the tip of a compass needle placed at that point.

But there is a problem, there are longer and shorter compass needles but that is irrelevant for the orientation of the magnetic field, the length of the vector does not actually matter. You could multiply this vector field with a different constant at every point, i.e. independently change the length of all the compass needles and you would still describe the same magnetic field orientation across earth. What you do to fix this is to declare that all fields are physically equivalent if they only differ by a constant factor - which may depend on the point - at each point.

The other solution is to use a better mathematical representation without the redundancy, instead of compass needle tip vectors you use the bearing angle and take the compass needle length out of the equation to begin with. Problem solved. Now you can no longer change the field value, i.e. the angle, at every point independently and still describe the same physical situation. Also note that there is still a global symmetry, you can still change all the angles by the same constant, you are free to pick which direction you label with angle zero, which is again physically meaningful and expresses that space is isotropic, i.e. there is no preferred direction and the equation therefore do not depend on the direction but only on the angle between directions.

And you can do the same thing with the known physical laws, you can for example get rid of the U(1) symmetry - which should better be called a redundancy - in quantum electro dynamics. The price you have to pay is that the resulting equations lack some other properties often considered desirable, for example they are no longer obviously local.

That's a surprisingly intuitive illustration. Thank you!

Nitpicking: IIRC Adding 5V everywhere is like multiplying Ψ by expt(i 5V t cte), where t is the time and cte is a constant that may involve c, ℏ, e, and perhaps a number, but I'm too lazy to lookup.

Looking at my comment, it says "multiply each point of the universe by a different complex" it's actually a "event" like in relativity, i.e. (ct, x, y z). This is an easy case where the function you use to multiply does not depend on x, y, z, but only on t.

>To a layperson this sounds like a hack. "Things get screwy when you screw with them, UNLESSSSS we add a magic thing that undoes our work". Well yeah.

What you describe as a "hack" is actually the empirical (look/probe and see what happens) nature of physics as a science.

It seems like a hack because there's no a priori theoritical reason to do it. But physics is not based on verifying empirically a set of a priori rules (who would give them?), but by building theories and rules by empirically looking at things at seeing what model fits, and if a changed model fits better - we then verify those empirically with more experiments.

And this is not "Things get screwy when you screw with them, UNLESSSSS we add a magic thing that undoes our work", but more like a reverse engineering session:

"Behavior X appears to be described by this formula with parameter p. What if we changed the parameter to -p? Hmm, the results would still be consistent with X, if only there was an additional factor v in the formula.

Would this (-p,v) combo buy us anything over our previous (p)?

Wow, yeah, v would then perfectly match the behavior we see for this other thing Y too. So (-p, v) seems to describe both X and Y, whereas before with p we could only describe X.

You are excellent at explaining this stuff. You're bringing some clarity to things that typically sound like abstract gobbledygook to me. We need more people who can bridge that gap.

I don't know about quantum mechanics, but when we talk about space we should be free to add a quantity to the whole universe (like adding 1 to the x coordinate of everything) because this just shifts the whole universe, or accordingly, shifts the origin - the (0, 0, 0) point - in the opposite direction.

The origin is set at an arbitrary point so this "space shift invariance" is saying that it doesn't matter what point we set for the origin (and mathematically this corresponds to the conservation of momentum - see Noether's theorem[0])

Hmm maybe the "zero" for the quantum states is arbitrary, so you should be able to add anything to it for the whole universe, and this merely changes the zero state in the opposite direction.. and since this should be a conversation law, pretty sure this is equivalent to the conservation of electric charge

https://en.wikipedia.org/wiki/Noether's_theorem

I may be starting to get it. The magic numbers we add to everything or multiply everything with really represents just a change in the viewpoint, origin of the observer. And because it should be possible to change the "location" of the observer (say the voltage we take to be 0 volts) and still get the same theory to hold up, we can discover that it can only hold up if we assume the existence of some new field. Something like this?

Yes.

*conservation law

Based on my reading, the particle physics explanation/theory is precisely the current consensus. And, given my research into electrodynamics in order to understand the propagation of EM wave fronts in antenna design, I also think the article has some merit (which is, at its core, a call for some experiments).

Given that "light" is fundamentally a electromagnetic wave and its propagation in spacetime is constant, and this results in time slowing down when you go faster to maintain this property, it isn't unreasonable to hypothesize a more fundamental basis here.

Personally, I think adding in the time component will be essential to completing this puzzle but all in all it makes for an avenue of investigation which is interesting.

Note that light isn't particularly special in this context, what is special (very special) is the speed of light. Other things travel at the speed of light, indeed everything massless is forced to travel at the speed of light. Other things that travel at the speed of light include gravitational waves and gluons*.

Basically "the speed of light" should be called "the speed of massless things" or possibly "the speed of causality" or something. We just call it "the speed of light" because light is the first thing we discovered that travels at this special speed.

*the star is because everything about quantum-chromodynamics is terrible so gluons don't really ever exist as particles themselves. If they did they would travel at the speed of light.

I’ve found that this way of phrasing it helps people click: Speed of light is infinite*. What we call speed of light is the speed of reality.

* not actually infinite, because reality itself propagates at a finite speed

(I’d love to know if this is wrong - this is my best attempt to make sense of it from college classes)

It's always helped me to think of c as the speed of causality. It's the speed at which one thing causes another

>Personally, I think adding in the time component will be essential to completing this puzzle but all in all it makes for an avenue of investigation which is interesting.

Not a physicist, but I've often wondered if the basis of QFT got off on the wrong foot by making time a privileged coordinate instead of a quantum operator like it does for position.

You just get into a twist of not being able to renormalize if you create a dependency chain like that. The same reason quantum gravity is such a problem - gravitons emit gravitons..

While I sympathize with your unease about making time a privileged coordinate, even in conventional quantum mechanics an operator for time seems difficult. What would that operator measure? The time at which a given object "is"? The whole point of physics is to describe the dynamic nature of reality, parametrized by time.

Speaking of which, time(-of-arrival) measurements in quantum mechanics have recently attracted quite some interest: The classic Copenhagen formalism doesn't seem to give an answer here (or at least not a unique one – it depends on how you perform the calculation). Meanwhile, Bohmian mechanics does seem to make a precise prediction. It will be interesting to see what experiments will yield.

This is absolutely an amazing explanation of local gauge symmetry I've ever seen. Thank you for writing this up. Gems like this is the reason I love HN!

> but the article is full of technical terms that you really don't want to know

The article looks a lot like word salad... too many specific technical terms for the average person to manage, yet lacking in the specifics that would be needed by someone capable of understanding their theory.

If it's written for their audience, then the only audience they seem to be targeting is average people who won't understand their assertions may be a load of crap.

Or is there someone here with a deep understanding of these topics that would care to chime in?

The article is based on a paper which is linked at the very end. Here's my take:

https://news.ycombinator.com/item?id=27944642

(Warning: Rather technical and full of math)

I have been really interested in general relativity and watching youtube videos for the past few months. I understood all the words (I would not have a year ago.) I think it does have enough detail to understand what they're saying. There's a lot missing, but I assume you'd go to the paper for that.

But I think the article is well written. It essentially says, "We think this interesting thing is true. It has these nice properties, and should be provable/falsifiable. Please help us prove it!"

Is this A you speak of the electromagnetic vector potential?

I'm not the person you're replying to, but yes A definitely is the electromagnetic vector potential (from classical electrodynamics). In gauge theory A tells you how to relate the phase of Ψ at nearby points in space/time.

Now that may purely a choice of convention for Ψ at different points in space/time (a choice of "gauge" in the jargon), but where it gets interesting is if your successive nearby points in space/time trace out a closed loop. If your A is such that the phase of Ψ ends up different as a result of going round the loop, you have an electromagnetic field!

The more correct technical term is "electromagnetic four potential", or "four vector".

A is quantum, the electromagnetic potential is classical, so they are not really the same thing.

Related question: is it settled which one is "real"? A or B?

https://en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effect#P...

Both?

Well I'm glad that's settled!

Actually since it's a spacetime potential, it's a 4-vector valued. So it's related to both the (three-)vector potential A and the scalar potential V.

If I understand you correctly, you claim that electromagnetism follows immediately from quantum mechanics because Born's rule exhibits a U(1) symmetry? That doesn't seem right to me.

It's not only the Born's rule. All the equations have a similar symmetry. (You may have ∂Ψ=VΨ, or Ψ∂Ψ*+∂ΨΨ*+ΨΨ*. See the real examples in the link in Wikipedia. But you never have something like ΨΨ+Ψ*Ψ* that mix the number of times that the linear and the conjugate version appears.)

Also, it's not so immediate, because you must be stubborn enough to think that a global obvious symmetry "must" be extended to a local symmetry. And in any case, it took like 40 years a few brilliant persons to discover it.

So A/electromagnetism is a natural consequence of the universe having a free (complex modulo 1) parameter/field in addition to just the wavefunction Ψ. ?

"I'm not sure how compatible is it with the proposal in the article discussed here."

Right. It seems to me that both approaches make sense. Perhaps with some cleaver yet-to-be-determined math both ideas can finally be mated.

I've never been convinced that the æther doesn't exist. Sure, it's been long debunked in the luminiferous æther sense but as the article points out "...the aether hypothesis was abandoned, and to this day, the classical theory of electromagnetism does not provide us with a clear answer to the question in which medium electric and magnetic fields propagate in vacuum." It is this aspect of the abolition of the æther that has always worried me.

For starters, any new model of the æther would have to exhibit Lorentz-invariant properties. Then there's the matter of vacuum permittivity ε0 and vacuum permeability μ0 to consider as the speed of light/aka 'electromagnetism' is directly linked to these physical constants via the expression c = 1/(μ0 ε0)^0.5. If one constant were to change then so too would the others including α Sommerfeld's fine structure constant, RK the von Klitzing constant, and Z0 the vacuum (free space) impedance, etc., etc. (Anyway, one would expect them to change—not that we'd ever know as we'd likely not exist if they did). ;-)

But I digress a little. We know that ε0 and μ0 have actual non-zero values and cannot be equated out (as we sort of tried to do in the days when we expressed electromagnetism in cgs units). In essence, physical constants ε0 and μ0 are absolutely intrinsic to electromagnetism, and whilst I cannot prove the fact, it seems to me they would be just as intrinsic to any new definition of the æther. Moreover, similar reasoning makes me think that QFT, ZPE/Zero-point energy/quantum vacuum state, ε0 and μ0 are all inextricably linked to GR.

It seems to me that whilst matters such as whether the spacetime manifold is Ricci-flat, etc. are extremely important principally from the perspective that when properly dovetailed into any new theory they'll provide proof thereof—are secondary to the proposition (note, I'm not saying they're secondary aspects of physics, only that they're secondary to the initial proposition).

Moreover, an equally important question to ask is why the constants ε0 and μ0 have the values they do given the quantum vacuum state, etc. Of course, the same logic applies to both α and c. Finally, we base just about everything on c it being the fundamental immutable constant (despite the the perennial emphasis/importance of α ≈ 1/137). The question is, is it in fact so, or is it that underlying physics first determines ε0 and μ0 and thus these constants could be considered more fundamental to any new formulation of the æther than that of c, it being the consequential resultant of the properties of those constants. (Heresy I know, but it would seem to make sense to view c in this context if or when we end up with new definition for the æther.)

My personal take on the question of space is that because of Lorentz contraction there is always an observer that sees the universe as a flat pancake. There is no true "distance" between any particles, and therefore no space. Every interaction between particles is because these particles actually collided for some specific hypothetical observer.

Want to try ELI15?

I'd be satisfied with ELHighschool

It's very hard! I tried to simplify as much as possible and use as many small lies as possible, but keep the explanation faithful to the central idea.

For example, I used that if k is a complex number with modulus 1, and k* is the conjugate, then kk*=1. It's a subject from a course of the first years of the university of a technical degree and perhaps a high school. Adding a detailed explanation in the middle makes the explanation too long. Also, it's a very important part of the correct technical complete version of the explanation, it's not a side comment or a metaphor.

I think it could be explained better, or with different tradeoff to make it easier to understand. Anyway, it's my best effort with my personal taste.

If you tell me the part that confused you, I can try to explain that part more. (And if you provide some personal background, like age range and what you studied, I can try to tailor it more.)

>_In particles physics ... very natural

Er, I'm not sure that things like dressed particles [1] and off-shell matter [2] violating E=mc^2 [3] could be described as even remotely "natural".

Perhaps they are valid theories, but "natural" certainly isn't an appropriate description of most of what unavoidably follows from particle assumptions.

[1] https://en.m.wikipedia.org/wiki/Dressed_particle

[2] https://en.m.wikipedia.org/wiki/On_shell_and_off_shell

[3] https://en.m.wikipedia.org/wiki/Virtual_particle#Properties

Virtual particles are basically just a mathematical trick to make doing calculations easier in perturbative quantum field theory. You shouldn't take them too seriously.

If you do the calculations in another way (e.g. by discretizing stuff on a lattice) no virtual particles appear but your calculations become a lot harder.

They are still a trick because they are used to propagate a particle exactly where it needs to go in order to exert the force of the field. So the field is everywhere, but is pretty much invisible except for when virtual particles mediate it.. No efficient simulation on a computer could operate in such a manner. It's explanatory but not a constructive proof. I can't build an efficient simulation of our universe based on the virtual particle paradigm.

They don't just become harder, they become open problems -- open problems with a million dollar bounty on them! There is no known non-perturbative quantization of the Yang-Mills theory.

http://www.claymath.org/millennium-problems/yang–mills-and-m...