4 ms·

The 'mathematical states' here are descriptions of nature. A symmetry in the description doesn't correspond to identified physical states, it corresponds to a c

by Retra 9y ago

The 'mathematical states' here are descriptions of nature. A symmetry in the description doesn't correspond to identified physical states, it corresponds to a constraint on the description. For instance, if you want to prove that something is unique, you have to show how all the different ways of describing it lead to the same conclusions.

danbruc 9y ago
A symmetry in the description doesn't correspond to identified physical states [...] As I said before, I am not a physicist, but I am pretty sure that you are wrong here. Unless you are a physicist and you are sure about that. My understanding is that gauge symmetries are a misnomer and should better be called gauge redundancies because they essentially establish equivalence classes of mathematical states corresponding to identical physical states. This is not the same kind of symmetry as for example the Poincaré symmetry of special relativity. Let me illustrate my understanding with a silly example that would almost certainly not actually work out mathematically but it should get the point accross. Say all you have in your mathematical toolbox are complex numbers and you come up with a theory of thermodynamics where temperatures are described by complex numbers. Everything works out nicely besides that there is no discernable physical difference between all the temperatures differing only by their imaginary parts. So you declare that there is a gauge symmetry in your theory of thermodynamics and all temperatures only differing by their imaginary parts are actually the same physical temperature. This theory has translations along the imaginary axis as a gauge symmetry.
- 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.
- tobmlt 9y ago
  Okay I can't resist - this stuff is just too cool. Here is my take on some of this stuff; assume grains of salt throughout as I am an engineer of sorts. (And sorry if this is old hat - and that the direction I go in strays pretty far, at times, from the original question about gauge theory actually having to do with physical things) It's funny that charge conservation, noted below as a consequence of gauge symmetry, is not exact. See CP symmetry violation and "all that". Furthermore there are various notions of internal symmetries... in engineer talk, perhaps gauge symmetry is a symmetry of parameterization - in math talk the base space of a fibre bundle "parameterizes" (in a coordinate free way) a larger space. --Gauge symmetry is a freedom in how exactly this is done (shout out to your comment about equivalence classes - totally agree). (For example the space time manifold showing how to put a vector space at every location in the manifold via tangent space constructions, or similar) I get lost sometimes.. Anyway, looking at part of the Wikipedia charge conservation article we read that: "...an electromagnetic field is unchanged when the scalar and vector potential are shifted by the gradient of an arbitrary scalar field." In quantum mechanics the scalar field is equivalent to a phase shift in the wavefunction of the charged particle..." This shift in the Lagrangian (to call it a different way) is gauge symmetry (U(1)) again. Interestingly, the phase shift can occur even if the space is "E&M flat" (no detectable field) and this shift IS observable: https://en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effect https://en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effect -A purely topological effect due to the fact that loops (around the solenoid in the experiment) in the space cannot be shrunk to a point! (not without leaving "the space" that is) Any phase shift is in fact a manifestation of Weyl's "path dependent length" which he'd speculated about to Einstein. (Einstein shot it down, but it was reborn as phase) But wait, path dependent lengths (generalized path length shift == phase shift) mean we have some, admittedly abstract, notion of curvature! This is the connection (ahem) between E&M theories and General Relativity. (and even closer connection with E&M-like, but non-abelian (Yang Mills) ) The potential, A, in electrodynamics plays a role analogous to that of the connection coefficient (Christoffel) in differential geometry. The tensor field strength in electrodynamics is analogous to curvature! (luckily for those who like their minds un-exploded E&M "happens" in flat space.) Once you get to Yang Mills, the commutator of the potential has to be added to the derivative of the Faraday tensor to get the total field strength. This is just the same as in General Relativity, where the curvature is the derivative of the connection plus its commutator. I guess what I am saying is, depending on how you look at it, the field theories are sort of shot-through with gauge symmetry, to both quantifiable effect and mathematical purpose, and it is neat.

by Retra 9y ago

danbruc 9y ago

A symmetry in the description doesn't correspond to identified physical states [...] As I said before, I am not a physicist, but I am pretty sure that you are wrong here. Unless you are a physicist and you are sure about that. My understanding is that gauge symmetries are a misnomer and should better be called gauge redundancies because they essentially establish equivalence classes of mathematical states corresponding to identical physical states. This is not the same kind of symmetry as for example the Poincaré symmetry of special relativity. Let me illustrate my understanding with a silly example that would almost certainly not actually work out mathematically but it should get the point accross. Say all you have in your mathematical toolbox are complex numbers and you come up with a theory of thermodynamics where temperatures are described by complex numbers. Everything works out nicely besides that there is no discernable physical difference between all the temperatures differing only by their imaginary parts. So you declare that there is a gauge symmetry in your theory of thermodynamics and all temperatures only differing by their imaginary parts are actually the same physical temperature. This theory has translations along the imaginary axis as a gauge symmetry.

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.
tobmlt 9y ago
Okay I can't resist - this stuff is just too cool. Here is my take on some of this stuff; assume grains of salt throughout as I am an engineer of sorts. (And sorry if this is old hat - and that the direction I go in strays pretty far, at times, from the original question about gauge theory actually having to do with physical things) It's funny that charge conservation, noted below as a consequence of gauge symmetry, is not exact. See CP symmetry violation and "all that". Furthermore there are various notions of internal symmetries... in engineer talk, perhaps gauge symmetry is a symmetry of parameterization - in math talk the base space of a fibre bundle "parameterizes" (in a coordinate free way) a larger space. --Gauge symmetry is a freedom in how exactly this is done (shout out to your comment about equivalence classes - totally agree). (For example the space time manifold showing how to put a vector space at every location in the manifold via tangent space constructions, or similar) I get lost sometimes.. Anyway, looking at part of the Wikipedia charge conservation article we read that: "...an electromagnetic field is unchanged when the scalar and vector potential are shifted by the gradient of an arbitrary scalar field." In quantum mechanics the scalar field is equivalent to a phase shift in the wavefunction of the charged particle..." This shift in the Lagrangian (to call it a different way) is gauge symmetry (U(1)) again. Interestingly, the phase shift can occur even if the space is "E&M flat" (no detectable field) and this shift IS observable: https://en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effect https://en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effect -A purely topological effect due to the fact that loops (around the solenoid in the experiment) in the space cannot be shrunk to a point! (not without leaving "the space" that is) Any phase shift is in fact a manifestation of Weyl's "path dependent length" which he'd speculated about to Einstein. (Einstein shot it down, but it was reborn as phase) But wait, path dependent lengths (generalized path length shift == phase shift) mean we have some, admittedly abstract, notion of curvature! This is the connection (ahem) between E&M theories and General Relativity. (and even closer connection with E&M-like, but non-abelian (Yang Mills) ) The potential, A, in electrodynamics plays a role analogous to that of the connection coefficient (Christoffel) in differential geometry. The tensor field strength in electrodynamics is analogous to curvature! (luckily for those who like their minds un-exploded E&M "happens" in flat space.) Once you get to Yang Mills, the commutator of the potential has to be added to the derivative of the Faraday tensor to get the total field strength. This is just the same as in General Relativity, where the curvature is the derivative of the connection plus its commutator. I guess what I am saying is, depending on how you look at it, the field theories are sort of shot-through with gauge symmetry, to both quantifiable effect and mathematical purpose, and it is neat.