Why does gauge fixing break gauge symmetry?

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Discussion Overview

The discussion revolves around the concept of gauge fixing in gauge theories, exploring why it is said to break gauge symmetry and how it affects the mathematical analysis of physical models. Participants examine the implications of gauge choices in both classical and quantum contexts, touching on the nature of gauge symmetry and its physical manifestations.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants propose that fixing a gauge simplifies the mathematical analysis of models, allowing for easier handling of equations in both classical and quantum gauge theories.
  • Others argue that gauge fixing is akin to choosing a coordinate system, which can facilitate problem-solving by reducing the number of variables involved.
  • One participant asserts that gauge symmetry is not a "real" symmetry, suggesting that it merely allows for the identification of equivalent states through gauge transformations, rather than relating different physical states.
  • There are claims that while gauge choices do not have direct physical manifestations, gauge invariance does, particularly in systems with nontrivial topology, leading to quantization phenomena.
  • A later reply questions the meaning of "state" in this context, seeking clarification on how states relate to descriptions of physical systems and the implications of symmetry in this relationship.

Areas of Agreement / Disagreement

Participants express differing views on the nature and implications of gauge fixing and gauge symmetry. There is no consensus on the interpretation of gauge symmetry as a "real" symmetry, and the discussion remains unresolved regarding the implications of these concepts.

Contextual Notes

Participants highlight the dependence of gauge fixing on the choice of gauge and the potential complexity introduced by different gauges, indicating that the discussion may be limited by varying interpretations of gauge symmetry and its manifestations.

TimeRip496
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By fixing a gauge (thus breaking orspending the gauge symmetry), the model becomes something easier to analyse mathematically, such as a system of partial differential equations (in classical gauge theories) or a perturbative quantum field theory (in quantum gauge theories), though the tractability of the resulting problem can be heavily dependent on the choice of gauge that one fixed.
https://terrytao.wordpress.com/2008/09/27/what-is-a-gauge/

What do you mean by this? As in why does gauge fixing made the model easier to analyse? Isnt gauge fixing something like choosing a coordinate?
 
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TimeRip496 said:
Isnt gauge fixing something like choosing a coordinate?
Right, and choosing a coordinate system can make things easier. As an example from classical mechanics, you can reduce the six position variables of the Kepler problem (three coordinates per object) to two variables by going to the center of mass system and choosing one coordinate to be orthogonal to the motion of the objects.
 
Gauge symmetry is not a real symmetry. A symmetry relates different states, gauge invariance allows you to identify states being related by a gauge transformation as the same state. You choose the gauge which is easiest to work with, for example in electrostatics we always use Coulomb gauge because Lorenz gauge would just make everything incredibly complicated.

There are no physics manifestations of your gauge choice. However, there are physical manifestations of gauge invariance. It appears in systems with nontrivial topology and results in quantization of things like magnetic charge and the hall conductance. You can use topological invariants to look at these systems.
 
radium said:
Gauge symmetry is not a real symmetry. A symmetry relates different states, gauge invariance allows you to identify states being related by a gauge transformation as the same state. You choose the gauge which is easiest to work with, for example in electrostatics we always use Coulomb gauge because Lorenz gauge would just make everything incredibly complicated.

There are no physics manifestations of your gauge choice. However, there are physical manifestations of gauge invariance. It appears in systems with nontrivial topology and results in quantization of things like magnetic charge and the hall conductance. You can use topological invariants to look at these systems.
When you say state, what do you really mean? Do you mean description of a system like rotation, translation, etc? And what do you mean by symmetry relate different state?
 

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