Another intriguing answer came from JGWeissman. Apparently, as we learn new physics, we tend to discard inconvenient versions of old formalisms. So electromagnetic potentials turn out to be “more true” than electromagnetic fields because they carry over to quantum mechanics much better. I like this answer because it seems to be very well-informed!
I don’t like this explanation- while potentials are useful calculation tools both macroscopically and quantum mechanically, fields have unique values whereas potentials have non-unique values. It’s not clear to me how to compare those two benefits and decide if one is “more true.”
The alternative way to look at it: if you only knew E&M, would you talk in terms of four-vector potentials or in terms of fields? Most of the calculations for complicated problems are easier with potentials (particularly for magnetism), but the target is generally coming up with the fields from those potentials. Similarly, most calculations in QM are easier with the potentials (I’ve never seen them done with fields, but I imagine it must be possible- you can do classical mechanics with or without Hamiltonians), but the target is wavefunctions or expectation values.
So it’s not clear to me what it means to choose potentials over fields, or vice versa. The potentials are a calculation trick, the fields are real, just like in QM the potentials are a calculation trick, and the wavefunction is real. They’re complementary, not competing.
I don’t like this explanation- while potentials are useful calculation tools both macroscopically and quantum mechanically, fields have unique values whereas potentials have non-unique values. It’s not clear to me how to compare those two benefits and decide if one is “more true.”
You can just as easily move to a different mathematical structure where the gauge is “modded out”, a “torsor”. Similarly, in quantum mechanics where the phase of the wavefunction has no physical significance, rather than working with the vectors of a Hilbert space, we work with rays (though calculational rules in practice reduce to vectors).
There are methods of gaugeless quantization but I’m not familiar with them, though I’d definitely like to learn. (I’d hope they’d get around some of the problems I’ve had with QFT foundations, though that’s probably a forlorn hope.)
I don’t like this explanation- while potentials are useful calculation tools both macroscopically and quantum mechanically, fields have unique values whereas potentials have non-unique values. It’s not clear to me how to compare those two benefits and decide if one is “more true.”
Immediate thought: Why not just regard the potentials as actual elements of a quotient space? :)
So it’s not clear to me what it means to choose potentials over fields, or vice versa. The potentials are a calculation trick, the fields are real, just like in QM the potentials are a calculation trick, and the wavefunction is real. They’re complementary, not competing.
Are you familiar with the Aharonov-Bohm effect? My understanding is that it is a phenomenon which, in some sense, shows that the EM potential is a “real thing”, not just a mathematical artifact.
I am and your understanding is correct for most applications. I don’t think it matters for this question, as my understanding is that the operative factor behind the Aharonov-Bohm effect is the nonlocality of wavefunctions.* Because wavefunctions are nonlocal, the potential formulation is staggeringly simpler than a force formulation. (The potentials are more real in the sense that the only people who do calculations with forces are imaginary!)
You still have gauge freedom with the Aharonov-Bohm effect- if you adjust the four-potential everywhere, all it does is adjust the phase everywhere, and all you can measure are phase differences.
Although, that highlights an inconsistency: if I’m willing to accept wavefunctions as real, despite their phase freedom, then I should be willing to accept potentials are real, despite their gauge freedom. I’m going to think this one over, but barring any further thoughts it looks like that’s enough to change my mind.
*I could be wrong: I have enough physics training to speculate on these issues, but not to conclude.
[edit] It also helps that Feynman, who certainly knows more about this than I do, sees the potentials as more real (I suppose this means ‘fundamental’?) than the fields.
wavefunctions as real, despite their phase freedom,
Heh. It gets worse. Typically one is taught that the wavefunction is defined up to a global constant. You might have thought that the difference in phase between two places would at least be well defined. This is true, so long as you stick to one reference frame. A Galilean boost will preserve the magnitude everywhere, but add a different phase everywhere.
I don’t like this explanation- while potentials are useful calculation tools both macroscopically and quantum mechanically, fields have unique values whereas potentials have non-unique values. It’s not clear to me how to compare those two benefits and decide if one is “more true.”
The alternative way to look at it: if you only knew E&M, would you talk in terms of four-vector potentials or in terms of fields? Most of the calculations for complicated problems are easier with potentials (particularly for magnetism), but the target is generally coming up with the fields from those potentials. Similarly, most calculations in QM are easier with the potentials (I’ve never seen them done with fields, but I imagine it must be possible- you can do classical mechanics with or without Hamiltonians), but the target is wavefunctions or expectation values.
So it’s not clear to me what it means to choose potentials over fields, or vice versa. The potentials are a calculation trick, the fields are real, just like in QM the potentials are a calculation trick, and the wavefunction is real. They’re complementary, not competing.
You can just as easily move to a different mathematical structure where the gauge is “modded out”, a “torsor”. Similarly, in quantum mechanics where the phase of the wavefunction has no physical significance, rather than working with the vectors of a Hilbert space, we work with rays (though calculational rules in practice reduce to vectors).
There are methods of gaugeless quantization but I’m not familiar with them, though I’d definitely like to learn. (I’d hope they’d get around some of the problems I’ve had with QFT foundations, though that’s probably a forlorn hope.)
Immediate thought: Why not just regard the potentials as actual elements of a quotient space? :)
Are you familiar with the Aharonov-Bohm effect? My understanding is that it is a phenomenon which, in some sense, shows that the EM potential is a “real thing”, not just a mathematical artifact.
I am and your understanding is correct for most applications. I don’t think it matters for this question, as my understanding is that the operative factor behind the Aharonov-Bohm effect is the nonlocality of wavefunctions.* Because wavefunctions are nonlocal, the potential formulation is staggeringly simpler than a force formulation. (The potentials are more real in the sense that the only people who do calculations with forces are imaginary!)
You still have gauge freedom with the Aharonov-Bohm effect- if you adjust the four-potential everywhere, all it does is adjust the phase everywhere, and all you can measure are phase differences.
Although, that highlights an inconsistency: if I’m willing to accept wavefunctions as real, despite their phase freedom, then I should be willing to accept potentials are real, despite their gauge freedom. I’m going to think this one over, but barring any further thoughts it looks like that’s enough to change my mind.
*I could be wrong: I have enough physics training to speculate on these issues, but not to conclude.
[edit] It also helps that Feynman, who certainly knows more about this than I do, sees the potentials as more real (I suppose this means ‘fundamental’?) than the fields.
Heh. It gets worse. Typically one is taught that the wavefunction is defined up to a global constant. You might have thought that the difference in phase between two places would at least be well defined. This is true, so long as you stick to one reference frame. A Galilean boost will preserve the magnitude everywhere, but add a different phase everywhere.