# Lorentz and Gauge invariance of EM

• AHSAN MUJTABA
In summary, the conversation discusses the concept of local gauge invariance in electrodynamics and its role in QED and QCD. It is mentioned that this principle may be a fundamental law of nature and that the Lagrangians for both theories can be derived from it. The topic of Lorentz invariance of fields is also mentioned, with a suggested resource for further reading being Tong's notes on QFT.
AHSAN MUJTABA
I have been reading the book of Chris Quigg, Gauge theories, Chapter 3, sec 3.3 in which he explains how local rotations transform wave function and variations in Schrodinger equation forces us to introduce the electromagnetic interaction between the particles. I need a bit deep concept of the ideas of gauge variance and invariance in electrodynamics. Kindly share some relevant material if possible as well.

Delta2
We may postulate an interaction between the electron and the EM field (as is done in classical EM). But, in QED, we may instead demand local gauge invariance. Neither the Dirac Lagrangian nor the EM Lagrangian is locally gauge invariance, but in combination with a relevant interaction term the total Lagrangian is.

This raises the possibility that local gauge invariance may be a fundamental law of nature.

And, when we apply the same principle of local gauge invariance to the quark model, we get the Lagrangian for QCD.

That means that, rather than having separate and arbitrary interaction postulates for QED and QCD, both theories may arise from the demand for local gauge invariance.

You might also take a look at Yang-Mills theory.

Last edited:
Thanks that helps.
I also need some details on Lorentz invariance of fields. TIA

AHSAN MUJTABA and etotheipi
Thanks a lot

## 1. What is Lorentz invariance in the context of electromagnetism?

Lorentz invariance refers to the principle that the laws of electromagnetism remain the same for all observers moving at constant velocities relative to each other. This means that the equations describing electromagnetic phenomena, such as Maxwell's equations, are valid in all inertial reference frames.

## 2. What is Gauge invariance in electromagnetism?

Gauge invariance is a mathematical symmetry in electromagnetism that ensures the physical predictions of the theory are independent of the choice of gauge. A gauge transformation is a mathematical transformation that does not affect the physical properties of the system, but only the mathematical representation of the system.

## 3. How are Lorentz and Gauge invariance related in electromagnetism?

Lorentz and Gauge invariance are closely related in electromagnetism. The principle of Lorentz invariance ensures that the laws of electromagnetism are the same for all observers, while Gauge invariance ensures that the mathematical representation of these laws is independent of the choice of gauge. Together, these principles form the foundation of the theory of electromagnetism.

## 4. Why is Lorentz and Gauge invariance important in electromagnetism?

Lorentz and Gauge invariance are important in electromagnetism because they allow for a consistent and unified understanding of electromagnetic phenomena. These principles ensure that the laws of electromagnetism are the same for all observers and that the mathematical representation of these laws is independent of the choice of gauge, making the theory more robust and accurate.

## 5. Are there any experimental tests for Lorentz and Gauge invariance in electromagnetism?

Yes, there are several experimental tests for Lorentz and Gauge invariance in electromagnetism. One example is the Michelson-Morley experiment, which demonstrated that the speed of light is the same for all observers, regardless of their relative motion. Another example is the Aharonov-Bohm effect, which shows that the electromagnetic potential can have physical effects even in regions where the electric and magnetic fields are zero, thus confirming the principle of Gauge invariance.

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