What does the EM gauge freedom have to do with U(1)

  • Thread starter hideelo
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In summary: I think the symmetry you are talking about is more analogous to rotational symmetry, where at each point in spacetime I can choose any point on the real line with no change in the corresponding equation of motion. I also think that you are missing something because this symmetry is not possible in classical electrodynamics.
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hideelo
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I know that we have a free parameter in the relativistic potential for electromagnetism. I also know that we can introduce this as a scalar field ψ which gives this free parameter. I understand that this can be related to U(1) by mapping ψ: ---> e^iψ, which is the U(1) group. It just seems a little forced. I mean, sure you can map every scalar field into U(1) but why do it. It seems to me that the symmetry here is more analagous to some one dimensional translational symmetry since at each point in spacetime I can choose any point on the real line with no change n the corresponding equation of motion. Am I missing something?
 
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In classical electrodynamics, we couple the EM potential to a 4-current via ##-A_\mu J^\mu## and gauge invariance of the Lagrangian implies that ##\partial_\mu J^\mu =0##, i.e., the current is conserved. We can simply take the gauge group to be ##(\mathbb{R},+)## with no ill consequences.

In quantum electrodynamics, we must introduce a separate quantum field for the matter degrees of freedom, e.g., the electron, with kinetic term ## \bar{e} \gamma^\mu \partial_\mu e## (##e## is a spinor, ##\gamma^\mu## are Dirac matrices). The gauge transformation on the electron field is ##e\rightarrow U e## and the kinetic term transforms as

$$ \bar{e} \gamma^\mu \partial_\mu e \rightarrow \bar{e} U^\dagger U \gamma^\mu \partial_\mu e + \bar{e} U^\dagger \gamma^\mu (\partial_\mu U) e. $$

The second term is canceled by including the coupling of the electron current to the gauge field ## - \bar{e} \gamma^\mu A_\mu e##. From the first term, the Lagrangian will only be invariant if ##U^\dagger U = 1##, which means that ##U## is a unitary transformation. So our gauge group must be ##U(1)## rather than ##(\mathbb{R},+)##.
 
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fzero said:
In classical electrodynamics, we couple the EM potential to a 4-current via ##-A_\mu J^\mu## and gauge invariance of the Lagrangian implies that ##\partial_\mu J^\mu =0##, i.e., the current is conserved. We can simply take the gauge group to be ##(\mathbb{R},+)## with no ill consequences.

In quantum electrodynamics, we must introduce a separate quantum field for the matter degrees of freedom, e.g., the electron, with kinetic term ## \bar{e} \gamma^\mu \partial_\mu e## (##e## is a spinor, ##\gamma^\mu## are Dirac matrices). The gauge transformation on the electron field is ##e\rightarrow U e## and the kinetic term transforms as

$$ \bar{e} \gamma^\mu \partial_\mu e \rightarrow \bar{e} U^\dagger U \gamma^\mu \partial_\mu e + \bar{e} U^\dagger \gamma^\mu (\partial_\mu U) e. $$

The second term is canceled by including the coupling of the electron current to the gauge field ## - \bar{e} \gamma^\mu A_\mu e##. From the first term, the Lagrangian will only be invariant if ##U^\dagger U = 1##, which means that ##U## is a unitary transformation. So our gauge group must be ##U(1)## rather than ##(\mathbb{R},+)##.
First of all, thanks for answering. Secondly, I won't say I understand everything you are talking about, but at least I know what it is that I am missing.
 

Related to What does the EM gauge freedom have to do with U(1)

What is the EM gauge freedom?

The EM gauge freedom refers to the ability to choose a different gauge for the electromagnetic field without affecting its physical properties or behavior. This freedom arises because the electromagnetic field does not have any physical observable quantities that are affected by changes in the gauge.

What is U(1) in relation to the EM gauge freedom?

U(1) is a mathematical group that represents the symmetry of the electromagnetic field. This symmetry arises because the electromagnetic field is invariant under local phase transformations, which are represented by U(1). The EM gauge freedom is closely related to this symmetry, as the choice of gauge corresponds to a specific representation of U(1).

Why is the EM gauge freedom important?

The EM gauge freedom is important because it allows us to simplify the mathematical description of the electromagnetic field. By choosing a particular gauge, we can eliminate unnecessary degrees of freedom and make calculations easier. It also helps us understand the underlying symmetry of the electromagnetic field and its relationship to other physical phenomena.

How does the EM gauge freedom affect the behavior of the electromagnetic field?

The EM gauge freedom does not affect the physical behavior of the electromagnetic field. This means that the choice of gauge does not change the values of measurable quantities, such as electric and magnetic fields, or the outcomes of experiments. It only affects the mathematical representation of the field.

Can the EM gauge freedom be extended to other gauge theories?

Yes, the concept of gauge freedom can be applied to other gauge theories, such as the strong and weak nuclear forces. These theories also have symmetries that give rise to gauge freedom, and the choice of gauge is important in simplifying their mathematical descriptions. However, the specific details of gauge freedom may differ for each theory.

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