5 ms·

Which seems quite strange to me. I assume we would be really happy to find theories equivalent to existing field theories but without the gauge redundancies so

Which seems quite strange to me. I assume we would be really happy to find theories equivalent to existing field theories but without the gauge redundancies so that we have a one-to-one mapping between physical states and mathematical states. Why would a part we want to get rid of determine the part we are really interested in?

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

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