8 ms·

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 o

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 https://en.wikipedia.org/wiki/Cosmological_constant

cominous 5y ago
Wow you blew my mind. You need to consider creating YouTube videos about this.
cshimmin 5y ago
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.
- trenchgun 5y ago
  Thank you!
- jasonwatkinspdx 5y ago
  Thanks for these comments. This simplified a whole bunch of things I had questions about.
- tinco 5y ago
  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.
vehementi 5y ago
> 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.
- Denvercoder9 5y ago
  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.
- cshimmin 5y ago
  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.
phkahler 5y ago
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.

cominous 5y ago

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

cshimmin 5y ago

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.

trenchgun 5y ago
Thank you!
jasonwatkinspdx 5y ago
Thanks for these comments. This simplified a whole bunch of things I had questions about.
tinco 5y ago
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.

vehementi 5y ago

> 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.

Denvercoder9 5y ago
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.
cshimmin 5y ago
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.

phkahler 5y ago

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.