3 ms·

Ah, you're looking for charge I think. https://en.wikipedia.org/wiki/Charge_conservation https://en.wikipedia.org/wiki/Charge_conservation The whole thing spi

by tobmlt 9y ago

Ah, you're looking for charge I think. https://en.wikipedia.org/wiki/Charge_conservation https://en.wikipedia.org/wiki/Charge_conservation The whole thing spins my head around differently every time I try to learn more but I hope I am helping by pointing this out. "The symmetry that is associated with charge conservation is the global gauge invariance of the electromagnetic field"

danbruc 9y ago
No, I really meant what I said, gauge symmetries are non-physical redundancies in gauge theories. Gauge fixing is the act of choosing one of the equivalent alternatives for carrying out calculations, gauge transformations allow switching between those alternatives. The first paragraph of the Wikipedia article on gauge fixing [1] actually says pretty much exactly this, unfortunately I just looked at it seconds ago. In the physics of gauge theories, gauge fixing (also called choosing a gauge) denotes a mathematical procedure for coping with redundant degrees of freedom in field variables. By definition, a gauge theory represents each physically distinct configuration of the system as an equivalence class of detailed local field configurations. Any two detailed configurations in the same equivalence class are related by a gauge transformation, equivalent to a shear along unphysical axes in configuration space. Most of the quantitative physical predictions of a gauge theory can only be obtained under a coherent prescription for suppressing or ignoring these unphysical degrees of freedom. And because you mentioned the Aharonov–Bohm effect in your other comment, further down at the end of the gauge freedom section you can find a statement that the Aharonov-Bohm effect does not enable experimentally distinguishing different gauges either. Not until the advent of quantum field theory could it be said that the potentials themselves are part of the physical configuration of a system. The earliest consequence to be accurately predicted and experimentally verified was the Aharonov–Bohm effect, which has no classical counterpart. Nevertheless, gauge freedom is still true in these theories. For example, the Aharonov–Bohm effect depends on a line integral of A around a closed loop, and this integral is not changed by A → A + ∇ψ. I also think you are wrong when you state in your other comment that charge conservation is not exact. I am pretty sure it is exact and you are mixing up CPT-symmetry and the violations of its components with the gauge symmetry of QED. [1] https://en.wikipedia.org/wiki/Gauge_fixing https://en.wikipedia.org/wiki/Gauge_fixing
- tobmlt 9y ago
  Good point about charge conjugation symmetry vs "charge conservation"- I was totally mixing up one for the other! Thanks (a lot) for driving that home. For anyone who might follow this - C in CPT - is charge conjugation - the swap of a particle for its anti-particle. Certainly not the same thing as charge conservation itself. Thanks for that correction. I tossed around a lot of stuff I shouldn't have. Okay, so I think I get you now. I was basically working from the point of view that "gauge matters" because the current 4 vector is conserved only if the E&M Lagrangian is gauge invariant. Hence charge conservation is inextricably linked with gauge symmetry - it "matters" to the derivation of the physical dynamics. But, yes, to your mathematical description of gauge fixing above... And to not being able to experimentally distinguish gauges. Phase differences, after all, are not gauge differences. Whatever the worth of the Aharonov–Bohm effect, it is not going to matter here. Good point. I get excited about this stuff. To your question: "Why would a part we want to get rid of determine the part we are really interested in" I'd answer: Why do we "want to get rid of" it? We cannot detect which gauge we are in, and we can twice confirm that it doesn't matter which one we are in precisely because current four vector is conserved. The redundant descriptions of the Lagrangian are the way to ensure physical descriptions of reality. I think this is a matter of what we like or don't like, not physics or math. But hey, sometimes those are clues! Gauge symmetries are non-physical redundancies... which are essential to our mathematical description of real physics. Maybe they are a clue that there is a better, undiscovered, way to do business? Maybe.
  - danbruc 9y ago
    I'd answer: Why do we "want to get rid of" it? I'd counter with why would you want additional degrees of freedom in your theory that don't correspond to something real? To pick up my silly example, why would you use complex temperatures to do the math if the physical temperature is real and only has one and not two degrees of freedom? I can only think of two sensible scenarios. The first one is that you have a theory with real temperatures but you can make working with that theory easier if you switch to complex temperatures for the calculations in witch case the redundancies are purely a mathematical tool. The second one is that you suspect that temperatures are actually complex and you just haven't figure out how to detect the imaginary component in experiments. Neither option seems to be true for gauge theories - as far as I know there are no alternative equivalent theories that don't have gauge symmetries and are just harder to work with mathematically and I also think nobody or at least not many believe that gauge redundancies correspond to yet undiscovered physical things.

by tobmlt 9y ago

danbruc 9y ago

No, I really meant what I said, gauge symmetries are non-physical redundancies in gauge theories. Gauge fixing is the act of choosing one of the equivalent alternatives for carrying out calculations, gauge transformations allow switching between those alternatives. The first paragraph of the Wikipedia article on gauge fixing [1] actually says pretty much exactly this, unfortunately I just looked at it seconds ago. In the physics of gauge theories, gauge fixing (also called choosing a gauge) denotes a mathematical procedure for coping with redundant degrees of freedom in field variables. By definition, a gauge theory represents each physically distinct configuration of the system as an equivalence class of detailed local field configurations. Any two detailed configurations in the same equivalence class are related by a gauge transformation, equivalent to a shear along unphysical axes in configuration space. Most of the quantitative physical predictions of a gauge theory can only be obtained under a coherent prescription for suppressing or ignoring these unphysical degrees of freedom. And because you mentioned the Aharonov–Bohm effect in your other comment, further down at the end of the gauge freedom section you can find a statement that the Aharonov-Bohm effect does not enable experimentally distinguishing different gauges either. Not until the advent of quantum field theory could it be said that the potentials themselves are part of the physical configuration of a system. The earliest consequence to be accurately predicted and experimentally verified was the Aharonov–Bohm effect, which has no classical counterpart. Nevertheless, gauge freedom is still true in these theories. For example, the Aharonov–Bohm effect depends on a line integral of A around a closed loop, and this integral is not changed by A → A + ∇ψ. I also think you are wrong when you state in your other comment that charge conservation is not exact. I am pretty sure it is exact and you are mixing up CPT-symmetry and the violations of its components with the gauge symmetry of QED. [1] https://en.wikipedia.org/wiki/Gauge_fixing https://en.wikipedia.org/wiki/Gauge_fixing

tobmlt 9y ago
Good point about charge conjugation symmetry vs "charge conservation"- I was totally mixing up one for the other! Thanks (a lot) for driving that home. For anyone who might follow this - C in CPT - is charge conjugation - the swap of a particle for its anti-particle. Certainly not the same thing as charge conservation itself. Thanks for that correction. I tossed around a lot of stuff I shouldn't have. Okay, so I think I get you now. I was basically working from the point of view that "gauge matters" because the current 4 vector is conserved only if the E&M Lagrangian is gauge invariant. Hence charge conservation is inextricably linked with gauge symmetry - it "matters" to the derivation of the physical dynamics. But, yes, to your mathematical description of gauge fixing above... And to not being able to experimentally distinguish gauges. Phase differences, after all, are not gauge differences. Whatever the worth of the Aharonov–Bohm effect, it is not going to matter here. Good point. I get excited about this stuff. To your question: "Why would a part we want to get rid of determine the part we are really interested in" I'd answer: Why do we "want to get rid of" it? We cannot detect which gauge we are in, and we can twice confirm that it doesn't matter which one we are in precisely because current four vector is conserved. The redundant descriptions of the Lagrangian are the way to ensure physical descriptions of reality. I think this is a matter of what we like or don't like, not physics or math. But hey, sometimes those are clues! Gauge symmetries are non-physical redundancies... which are essential to our mathematical description of real physics. Maybe they are a clue that there is a better, undiscovered, way to do business? Maybe.
- danbruc 9y ago
  I'd answer: Why do we "want to get rid of" it? I'd counter with why would you want additional degrees of freedom in your theory that don't correspond to something real? To pick up my silly example, why would you use complex temperatures to do the math if the physical temperature is real and only has one and not two degrees of freedom? I can only think of two sensible scenarios. The first one is that you have a theory with real temperatures but you can make working with that theory easier if you switch to complex temperatures for the calculations in witch case the redundancies are purely a mathematical tool. The second one is that you suspect that temperatures are actually complex and you just haven't figure out how to detect the imaginary component in experiments. Neither option seems to be true for gauge theories - as far as I know there are no alternative equivalent theories that don't have gauge symmetries and are just harder to work with mathematically and I also think nobody or at least not many believe that gauge redundancies correspond to yet undiscovered physical things.