I don’t know the order that things were discovered historically, but if it helps, here’s how I understood the motivation of gauge theory when it was first introduced in undergrad; we started with basic electromagnetism and wound up with the idea “Given a quantum particle with nonzero electric charge, you can just pick what phase its wavefunction has, however you want, arbitrarily, and you can choose it independently for every point in spacetime”. Here goes:
Classically, there’s an electric field E (a vector at each point of spacetime) and a magnetic field B (a vector at each point of spacetime).
Gauss’s law for magnetism says ∇⋅B=0, which turns out to be exactly equivalent to the claim “we can write B in the form B=∇×A“ where A is some other field (vector at each point of spacetime) called “vector potential”. As it turns out, there are many different A’s that yield the same B, and they’re all equally correct.
Likewise, the Maxwell-Faraday Equation is exactly equivalent to the claim “we can write E in the form E=–∇ϕ–∂A/∂t, where ɸ is some scalar field (real number at each point of spacetime) called “scalar potential”. Again, as it turns out, there are many different ɸ’s that are equally valid.
This is progress! By working with ɸ & A instead of E & B, we only have to calculate four components instead of six, and we only need to think about two Maxwell’s equations instead of four.
(In special relativity, it turns out that ɸ & A come together into the four components of the “electromagnetic 4-potential”.)
Gauge freedom says we can pick any scalar field Λ (real number at each point of spacetime), and replace (ϕ,A) with (ϕ−∂Λ/∂t,A+∇Λ), and it doesn’t change anything about the real world. You still get the same E & B.
So far we’re still in classical electromagnetism.
Now we switch to single-particle nonrelativistic semiclassical quantum mechanics (i.e., E & B are still classical fields, but there’s a quantum charged particle). It turns out that there’s still gauge freedom, but you also need to multiply the wavefunction of the charged particle by a position-dependent phase factor:
(ϕ,A,ψ)↦(ϕ−∂Λ∂t,A+∇Λ,eiqΛ/ℏψ)
where ψ is the wavefunction of our particle with charge q.
And now it’s obvious that you have complete freedom to mess with the time&position-dependent phase of ψ however you want, just by choosing an appropriate Λ.
Thanks! One place where I struggle with this idea is that people go around saying things like “Given a quantum particle with nonzero electric charge, you can just pick what phase its wavefunction has”. I don’t know how to think of an electron having a wavefunction whose phase I can pick. The wavefunctions that I know assign amplitudes to configurations, not to particles; if I have a wavefunctionover three-electron configurations then I don’t know how to “choose the phase” for each electron, because a three-particle wave-functions doesn’t (in general) factor as a product of three one-particle wave-functions.
I can make guesses about what it means to “choose electron phases” when there’s multiple electrons, but if Vanessa’s right (and in my experience she usually is about this sort of thing), then my initial guess was wrong.
(I appreciate how you gave the equation for how a one-particle wavefunction ψ changes with a change in gauge, but it precisely dodges the case I was unclear about; what I was unclear on is how the wavefunction changes with the gauge when a configuration comprises two or more charged particles. Overall you’ve given quite a lovely demonstration of how the standard story manages to leave me unclear on this key point :-p.)
That said, your comment seems to me like fine exposition about how one might stumble across the electromagnetic 4-potential, and discover that it has gauge freedom. Thanks again!
My recollection is that in nonrelativistic N-particle QM you would have a wavefunction ψ(r1,r2,…,rN) (a complex number for every possible choice of N positions in space for the N particles), and when you change gauge you multiply that that wavefunction by eiq1Λ(r1)/ℏeiq2Λ(r2)/ℏ⋯eiqNΛ(rN)/ℏ. I think this is equivalent to what Vanessa said, and if not she’s probably right :-P
That was my original guess! I think Vanessa suggested something different. IIUC, she suggested
ψ(r1,r2,…,rN)↦N∏k=1(eiqkΛ(rk)/ℏψ(r1,r2,…,rN))
which has N factors of the wavefunction, instead of 1.
(You having the same guess as me does update me towards the hypothesis that Vanessa just forgot some parentheses, and now I’m uncertain again :-p. Having N factors of the wavefunction sure does seem pretty wacky!)
(...or perhaps there’s an even more embarassing misunderstanding, where I’ve misunderstood physicist norms about parenthesis-insertion!)
I think it’s just one copy of the ψ—I don’t think Vanessa intended for ψ to be included in the Π product thing here. I agree that an extra pair of parentheses could have made it clearer. (Hope I’m not putting words in her mouth.)
I don’t know the order that things were discovered historically, but if it helps, here’s how I understood the motivation of gauge theory when it was first introduced in undergrad; we started with basic electromagnetism and wound up with the idea “Given a quantum particle with nonzero electric charge, you can just pick what phase its wavefunction has, however you want, arbitrarily, and you can choose it independently for every point in spacetime”. Here goes:
Classically, there’s an electric field E (a vector at each point of spacetime) and a magnetic field B (a vector at each point of spacetime).
Gauss’s law for magnetism says ∇⋅B=0, which turns out to be exactly equivalent to the claim “we can write B in the form B=∇×A“ where A is some other field (vector at each point of spacetime) called “vector potential”. As it turns out, there are many different A’s that yield the same B, and they’re all equally correct.
Likewise, the Maxwell-Faraday Equation is exactly equivalent to the claim “we can write E in the form E=–∇ϕ–∂A/∂t, where ɸ is some scalar field (real number at each point of spacetime) called “scalar potential”. Again, as it turns out, there are many different ɸ’s that are equally valid.
This is progress! By working with ɸ & A instead of E & B, we only have to calculate four components instead of six, and we only need to think about two Maxwell’s equations instead of four.
(In special relativity, it turns out that ɸ & A come together into the four components of the “electromagnetic 4-potential”.)
Gauge freedom says we can pick any scalar field Λ (real number at each point of spacetime), and replace (ϕ,A) with (ϕ−∂Λ/∂t,A+∇Λ), and it doesn’t change anything about the real world. You still get the same E & B.
So far we’re still in classical electromagnetism.
Now we switch to single-particle nonrelativistic semiclassical quantum mechanics (i.e., E & B are still classical fields, but there’s a quantum charged particle). It turns out that there’s still gauge freedom, but you also need to multiply the wavefunction of the charged particle by a position-dependent phase factor:
(ϕ,A,ψ)↦(ϕ−∂Λ∂t,A+∇Λ,eiqΛ/ℏψ)
where ψ is the wavefunction of our particle with charge q.
And now it’s obvious that you have complete freedom to mess with the time&position-dependent phase of ψ however you want, just by choosing an appropriate Λ.
Thanks! One place where I struggle with this idea is that people go around saying things like “Given a quantum particle with nonzero electric charge, you can just pick what phase its wavefunction has”. I don’t know how to think of an electron having a wavefunction whose phase I can pick. The wavefunctions that I know assign amplitudes to configurations, not to particles; if I have a wavefunctionover three-electron configurations then I don’t know how to “choose the phase” for each electron, because a three-particle wave-functions doesn’t (in general) factor as a product of three one-particle wave-functions.
I can make guesses about what it means to “choose electron phases” when there’s multiple electrons, but if Vanessa’s right (and in my experience she usually is about this sort of thing), then my initial guess was wrong.
(I appreciate how you gave the equation for how a one-particle wavefunction ψ changes with a change in gauge, but it precisely dodges the case I was unclear about; what I was unclear on is how the wavefunction changes with the gauge when a configuration comprises two or more charged particles. Overall you’ve given quite a lovely demonstration of how the standard story manages to leave me unclear on this key point :-p.)
That said, your comment seems to me like fine exposition about how one might stumble across the electromagnetic 4-potential, and discover that it has gauge freedom. Thanks again!
My recollection is that in nonrelativistic N-particle QM you would have a wavefunction ψ(r1,r2,…,rN) (a complex number for every possible choice of N positions in space for the N particles), and when you change gauge you multiply that that wavefunction by eiq1Λ(r1)/ℏeiq2Λ(r2)/ℏ⋯eiqNΛ(rN)/ℏ. I think this is equivalent to what Vanessa said, and if not she’s probably right :-P
That was my original guess! I think Vanessa suggested something different. IIUC, she suggested
ψ(r1,r2,…,rN)↦N∏k=1(eiqkΛ(rk)/ℏψ(r1,r2,…,rN))
which has N factors of the wavefunction, instead of 1.
(You having the same guess as me does update me towards the hypothesis that Vanessa just forgot some parentheses, and now I’m uncertain again :-p. Having N factors of the wavefunction sure does seem pretty wacky!)
(...or perhaps there’s an even more embarassing misunderstanding, where I’ve misunderstood physicist norms about parenthesis-insertion!)
I think it’s just one copy of the ψ—I don’t think Vanessa intended for ψ to be included in the Π product thing here. I agree that an extra pair of parentheses could have made it clearer. (Hope I’m not putting words in her mouth.)