# How does gauge invariance determine the nature of electromagnetism?

• joneall
In summary, Krauss's statement is better understood within quantum mechanics and the full story is revealed in relativistic QFT. The principle can also be made clear from non-relativistic physics using the Schrödinger equation and Noether's theorem. By making the global symmetry "local", the gauge invariance leads to a coupled system of equations with the electromagnetic charge and current densities expressed in terms of the Schrödinger field. This can easily be extended to the relativistic case using the Klein-Gordon or Dirac equation. The formulas can be easily typed using LaTeX code in the forum provided.
joneall
Gold Member
In his book, "The greatest story ever told", Lawrence Krauss states: "Gauge invariance ... completely determines the nature of electromagnetism."

My question is simple: How?

I have gone back thru the math. Gauge invariance allows us to use the Lorenz gauge with the vector and scalar potentials (I find these equations easier to visualize than the relativistic ones with tensors), which when used with Maxwell's equations (the ones with sources) to discover that the potentials obey wave equations with sources. So the wave equations + the Lorenz gauge are equivalent to Maxwell's equations. It seems to me this is a long way from Krauss's statement.

Or maybe I am expecting too much.

(Btw, is there a page somewhere showing how to code formulas on this forum?)

You are correct, that is a long way from Krauss’ statement. In fact, you can derive Maxwell’s equations from the gauge invariance, see: https://aapt.scitation.org/doi/abs/10.1119/1.17279?journalCode=ajp as well as the original paper that this simplified version is based on.

One key thing to recognize is the importance of Noether’s theorem. Any invariance of the Lagrangian leads to a conserved quantity. In the case of the EM gauge invariance the corresponding conserved quantity is the charge. The conservation of charge itself can be used to derive half of Maxwell’s equations.

anorlunda and vanhees71
Krauss's statement can be better understood within quantum mechanics than within classical physics. The full story is revealed in relativistic QFT only, but the principle can also be made clear from non-relativistic physics.

$$\mathrm{i} \hbar \partial_t \psi(t,\vec{x})=-\frac{\hbar^2}{2m} \Delta \psi(t,\vec{x}).$$
This equation can be derived from Hamilton's principle, using the Lagrangian
$$\mathcal{L}=\mathrm{i} \hbar \psi^* \partial_t \psi -\frac{\hbar^2}{2m} (\vec{\nabla} \psi^*) \cdot (\vec{\nabla} \psi).$$
Here ##\psi^*## and ##\psi## can be varied independently, and it's easy to see that variation with respect to ##\psi^*## leads to the free-particle Schrödinger equation.

Now the Lagrangian (and thus the action) is invariant under global transformations,
$$\psi(t,\vec{x}) \rightarrow \psi'(t,\vec{x})=\exp(-\mathrm{i} q \alpha) \psi(t,\vec{x}), \quad \psi^*(t,\vec{x}) \rightarrow \psi^{\prime *}(t,\vec{x})=\exp(\mathrm{i} q \alpha) \psi(t,\vec{x}).$$
Then Noether theorem tells us that there's a conservation law,
$$\partial_t \rho+\nabla \cdot \vec{j}=0$$
with
$$\rho=q|\psi|^2, \quad \vec{j} = -q \frac{\mathrm{i} \hbar}{2m} (\psi^* \vec{\nabla} \psi-\psi \vec{\nabla} \psi^*).$$
Now the general idea of "local gauge symmetries" is to make such a symmetry "local". By this we ask, how we can extend the above symmetry to get a symmetry for space-time dependent parameters, ##\alpha \rightarrow \alpha(t,\vec{x})##. As it turns out, to this end we simply have to introduce a scalar and a vector field ##\Phi## and ##\vec{A}## and make everywhere, where there's a partial derivative wrt. ##t## and ##\vec{x}##
$$\partial_t \rightarrow \partial_t + \mathrm{i} q \Phi, \quad \vec{\nabla} \rightarrow \vec{\nabla}+\mathrm{i} q \vec{A}.$$
Then, with space-time dependent ##\alpha##,
$$(\partial_t + \mathrm{i} q \Phi) \psi'(t,\vec{x})=\exp(-\mathrm{i} q \alpha) [\partial_t\psi + (\mathrm{i} \Phi - \mathrm{i} q \partial_t \alpha) \psi],$$
and this stays invariant when we also transform the scalar potential by
$$\Phi \rightarrow \Phi'=\Phi+\partial_t \alpha.$$
In the same way we can show that we must transform the vector potential by
$$\vec{A} \rightarrow \vec{A}+\vec{\nabla} \alpha.$$
The new Lagrangian then gives the free-Schrödinger-field Lagrangian augmented by the interaction with the electromagnetic field.

To also get Maxwell's equations you have to add the corresponding free-field Lagrangian, which in the end gives a coupled system of equations with the electromagnetic charge and current densities expressed in terms of the Schrödinger field ##\psi##.

All this can of course easily made relativistic by starting from the Klein-Gordon or the Dirac equation and following exactly the same steps.

The formulas are best typed in LaTeX code. It's easy in the forum:

https://www.physicsforums.com/help/latexhelp/

Last edited:
Dale

## 1. What is gauge invariance?

Gauge invariance is a principle in physics that states that the physical laws and equations should remain unchanged when certain transformations, known as gauge transformations, are applied. These transformations do not affect the physical observable quantities, but rather the mathematical representation of the system.

## 2. How does gauge invariance relate to electromagnetism?

Gauge invariance plays a crucial role in determining the nature of electromagnetism. It is the underlying principle that allows us to describe and understand the behavior of electromagnetic fields and their interactions with charged particles.

## 3. What is the gauge freedom in electromagnetism?

The gauge freedom in electromagnetism refers to the fact that there are multiple ways to mathematically describe the same physical system. This is due to the fact that gauge transformations can be applied to the equations without changing their physical meaning. This freedom allows us to choose the most convenient mathematical representation for a given problem.

## 4. How does gauge invariance affect the equations of electromagnetism?

Gauge invariance imposes constraints on the equations of electromagnetism, specifically Maxwell's equations. These constraints ensure that the equations are consistent and do not produce unphysical results. They also allow for the inclusion of important physical phenomena, such as the conservation of charge and the existence of electromagnetic waves.

## 5. Why is gauge invariance important in understanding the fundamental forces of nature?

Gauge invariance is a fundamental principle that is present in all of the fundamental forces of nature, not just electromagnetism. It allows us to formulate consistent and accurate theories of these forces, and has played a crucial role in the development of modern physics. Without gauge invariance, our understanding of the fundamental forces would be incomplete.

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