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Selene Routley
Selene Routley
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A global gauge symmetry is one where the configuration space is a simple Cartesian product (i.e. a trivial fiber bundle) of the set of physically distinct equivalence classes and a redundant parameter, as with your difference between two values example. If the physical description is a Lagrangian description, then this is where Noether's theorem comes to the fore and identifies conserved quantities, one for each such redundant parameter. The gauge group, i.e. group of symmetries, affects all equivalence classes (fibers) equally. Subtraction of a constant potential from an electrostatic potential is such a symmetry, and a huge advance for Corvid Civilization, as it lets crows sit on high tension powerlines and happily shoot the breeze together, discussing their latest thoughts on gauge theories, comfortable in the knowledgeand declaring that "Nevermore!" shall we fear the global addition of 22kV to the electrostatic potential changes nocan change the physics of the system we belong to.

However, usually when physicists speak of a gauge theory, they mean one where the symmetry group can act in a more general way, with a different group member acting at each point on the configuration space. The corresponding fiber bundle is no longer trivial. Although you wanted a simpler example than electrodynamics, I don't think there is one. The phase added to the electron wavefunction can be any smooth function of co-ordinates, and the extra terms that arise from the Leibniz rule applied to the derivatives in the wavefunction's equation of motion (Dirac, Schrödinger) are exactly soaked up into the closed part of the EM potential one-form. Incidentally, as an aside, I always like to visualize EM potential in Fourier space, which we can do with reasonable restrictions (e.g. a postulate that we're only going to think about tempered distributions, for example), because the spatial part of the redundant part of the four-potential is then its component along the wavevector (i.e thought of as a 3-vector), and only the componentscomponent normal to the wavevector matters physically: it is the only part that survives $A\mapsto \mathrm{d} A = F$.

Lastly I should mention that gauge / fiber bundle notions are also useful when we artificially declare equivalence classes of configurations grounded on the needs of our problem, even if there is a physical difference between equivalence class members. One of the loveliest examples of this way of thinking is Montgomery's "Gauge Theory of the Falling Cat". We study equivalence classes of cat configuration that are equivalent modulo proper Euclidean isometry to formulate a cat shape space, which, in the standard treatment where the cat is thought of as a two-section robot with twist-free ball-and-socket joint turns out to be the real projective plane $\mathbb{RP}^2$. The whole configuration space is then a fiber bundle with the shape space $\mathbb{RP}^2$ as base and the group $SO(3)$ defining orientations as fiber. The cat can flip whilst conserving angular momentum using cyclic deformations of its own shape owing to the curvature of the connexion that arises from the notion of parallel transport that is implied by angular momentum conservation.

A global gauge symmetry is one where the configuration space is a simple Cartesian product (i.e. a trivial fiber bundle) of the set of physically distinct equivalence classes and a redundant parameter, as with your difference between two values example. If the physical description is a Lagrangian description, then this is where Noether's theorem comes to the fore and identifies conserved quantities, one for each such redundant parameter. The gauge group, i.e. group of symmetries, affects all equivalence classes (fibers) equally. Subtraction of a constant potential from an electrostatic potential is such a symmetry, and a huge advance for Corvid Civilization, as it lets crows sit on high tension powerlines and happily shoot the breeze together, discussing their latest thoughts on gauge theories, comfortable in the knowledge that the global addition of 22kV to the electrostatic potential changes no physics of the system.

However, usually when physicists speak of a gauge theory, they mean one where the symmetry group can act in a more general way, with a different group member acting at each point on the configuration space. The corresponding fiber bundle is no longer trivial. Although you wanted a simpler example than electrodynamics, I don't think there is one. The phase added to the electron wavefunction can be any smooth function of co-ordinates, and the extra terms that arise from the Leibniz rule applied to the derivatives in the wavefunction's equation of motion (Dirac, Schrödinger) are exactly soaked up into the closed part of the EM potential one-form. Incidentally, as an aside, I always like to visualize EM potential in Fourier space, which we can do with reasonable restrictions (e.g. a postulate that we're only going to think about tempered distributions, for example), because the spatial part of the redundant part of the four-potential is then its component along the wavevector (i.e thought of as a 3-vector), and only the components normal to the wavevector matters physically: it is the only part that survives $A\mapsto \mathrm{d} A = F$.

Lastly I should mention that gauge / fiber bundle notions are also useful when we artificially declare equivalence classes of configurations grounded on the needs of our problem, even if there is a physical difference between equivalence class members. One of the loveliest examples of this way of thinking is Montgomery's "Gauge Theory of the Falling Cat". We study equivalence classes of cat configuration that are equivalent modulo proper Euclidean isometry to formulate a cat shape space, which, in the standard treatment where the cat is thought of as a two-section robot with twist-free ball-and-socket joint turns out to be the real projective plane $\mathbb{RP}^2$. The whole configuration space is then a fiber bundle with the shape space $\mathbb{RP}^2$ as base and the group $SO(3)$ defining orientations as fiber. The cat can flip whilst conserving angular momentum using cyclic deformations of its own shape owing to the curvature of the connexion arises from the notion of parallel transport that is implied by angular momentum conservation.

A global gauge symmetry is one where the configuration space is a simple Cartesian product (i.e. a trivial fiber bundle) of the set of physically distinct equivalence classes and a redundant parameter, as with your difference between two values example. If the physical description is a Lagrangian description, then this is where Noether's theorem comes to the fore and identifies conserved quantities, one for each such redundant parameter. The gauge group, i.e. group of symmetries, affects all equivalence classes (fibers) equally. Subtraction of a constant potential from an electrostatic potential is such a symmetry, and a huge advance for Corvid Civilization, as it lets crows sit on high tension powerlines and happily shoot the breeze together, discussing their latest thoughts on gauge theories, and declaring that "Nevermore!" shall we fear the global addition of 22kV to the electrostatic potential can change the physics of the system we belong to.

However, usually when physicists speak of a gauge theory, they mean one where the symmetry group can act in a more general way, with a different group member acting at each point on the configuration space. The corresponding fiber bundle is no longer trivial. Although you wanted a simpler example than electrodynamics, I don't think there is one. The phase added to the electron wavefunction can be any smooth function of co-ordinates, and the extra terms that arise from the Leibniz rule applied to the derivatives in the wavefunction's equation of motion (Dirac, Schrödinger) are exactly soaked up into the closed part of the EM potential one-form. Incidentally, as an aside, I always like to visualize EM potential in Fourier space, which we can do with reasonable restrictions (e.g. a postulate that we're only going to think about tempered distributions, for example), because the spatial part of the redundant part of the four-potential is then its component along the wavevector (i.e thought of as a 3-vector), and only the component normal to the wavevector matters physically: it is the only part that survives $A\mapsto \mathrm{d} A = F$.

Lastly I should mention that gauge / fiber bundle notions are also useful when we artificially declare equivalence classes of configurations grounded on the needs of our problem, even if there is a physical difference between equivalence class members. One of the loveliest examples of this way of thinking is Montgomery's "Gauge Theory of the Falling Cat". We study equivalence classes of cat configuration that are equivalent modulo proper Euclidean isometry to formulate a cat shape space, which, in the standard treatment where the cat is thought of as a two-section robot with twist-free ball-and-socket joint turns out to be the real projective plane $\mathbb{RP}^2$. The whole configuration space is then a fiber bundle with the shape space $\mathbb{RP}^2$ as base and the group $SO(3)$ defining orientations as fiber. The cat can flip whilst conserving angular momentum using cyclic deformations of its own shape owing to the curvature of the connexion that arises from the notion of parallel transport that is implied by angular momentum conservation.

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Selene Routley
Selene Routley
  • 88.5k
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These calculations very often depend only on the difference between two values, not the concrete values themselves. You are therefore free to choose a zero to your liking. Is this an example of gauge invariance in the same sense as the graduate examples above?

Yes indeed it is, in the most general definition of gauge invariance, it's what physicists call a global gauge invariance. More on that below.

If I had to write a one sentence answer to your title, it would be this:

Gauge invariance is the well definedness of physical law under a quotent map that condenses a configuration/ parameter space/ co-ordinates for a physical system into a set of equivalence classes of physically equivalent configurations.

This is in the same sense that, for example, the coset product is well defined under the map that quotients away a group's normal subgroup. The physics of a configuration is independent of the choice of equivalence class member.

In its barest terms, gauge invariance is simply an assertion that there is redundancy in a mathematical description of a physical system. Otherwise put, the system has a symmetry, an invariance with respect to a group of transformations.

A global gauge symmetry is one where the configuration space is a simple Cartesian product (i.e. a trivial fiber bundle) of the set of physically distinct equivalence classes and a redundant parameter, as with your difference between two values example. If the physical description is a Lagrangian description, then this is where Noether's theorem comes to the fore and identifies conserved quantities, one for each such redundant parameter. The gauge group, i.e. group of symmetries, affects all equivalence classes (fibers) equally. Subtraction of a constant potential from an electrostatic potential is such a symmetry, and a huge advance for Corvid Civilization, as it lets crows sit on high tension powerlines and happily shoot the breeze together, discussing their latest thoughts on gauge theories, comfortable in the knowledge that the global addition of 22kV to the electrostatic potential changes no physics of the system.

However, usually when physicists speak of a gauge theory, they mean one where the symmetry group can act in a more general way, with a different group member acting at each point on the configuration space. The corresponding fiber bundle is no longer trivial. Although you wanted a simpler example than electrodynamics, I don't think there is one. The phase added to the electron wavefunction can be any smooth function of co-ordinates, and the extra terms that arise from the Leibniz rule applied to the derivatives in the wavefunction's equation of motion (Dirac, Schrödinger) are exactly soaked up into the closed part of the EM potential one-form. Incidentally, as an aside, I always like to visualize EM potential in Fourier space, which we can do with reasonable restrictions (e.g. a postulate that we're only going to think about tempered distributions, for example), because the spatial part of the redundant part of the four-potential is then its component along the wavevector (i.e thought of as a 3-vector), and only the components normal to the wavevector matters physically: it is the only part that survives $A\mapsto \mathrm{d} A = F$.

There are two things I believe you should take from the EM example:

  1. Even though practically it leads to quite a bit of further complexity, conceptually, it is only a small jump from your simple global gauge symmetric example; we simply allow the symmetries to act locally instead of acting on all configuration space points equally;

  2. Taking a lead from the experimentally real electromagnetism, we postulate that this gauge invariance might be relevant more generally, and so we look its presence in other physical phenomena. This is nothing more than a deed motivated by a hunch. Experimentally, we find that this is a fruitful thing to do. In physics, there is no deeper insight than experimental results.

Lastly I should mention that gauge / fiber bundle notions are also useful when we artificially declare equivalence classes of configurations grounded on the needs of our problem, even if there is a physical difference between equivalence class members. One of the loveliest examples of this way of thinking is Montgomery's "Gauge Theory of the Falling Cat". We study equivalence classes of cat configuration that are equivalent modulo proper Euclidean isometry to formulate a cat shape space, which, in the standard treatment where the cat is thought of as a two-section robot with twist-free ball-and-socket joint turns out to be the real projective plane $\mathbb{RP}^2$. The whole configuration space is then a fiber bundle with the shape space $\mathbb{RP}^2$ as base and the group $SO(3)$ defining orientations as fiber. The cat can flip whilst conserving angular momentum using cyclic deformations of its own shape owing to the curvature of the connexion arises from the notion of parallel transport that is implied by angular momentum conservation.