Gauge Invariance - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

Physics for IIT JAM, UGC - NET, CSIR NET

Created by: Akhilesh Thakur

Physics : Gauge Invariance - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

The document Gauge Invariance - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev is a part of the Physics Course Physics for IIT JAM, UGC - NET, CSIR NET.
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Gauge Invariance in Classical Electrodynamics

Maxwell's equation

Gauge Invariance - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

suggests that there is a vector potential fulfilling

Gauge Invariance - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

The magnetic field is unchanged if one adds a gradient of an arbitrary scalar field Λ:

Gauge Invariance - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

Similar in line, the Maxwell equation

Gauge Invariance - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

suggests that there is a scalar potential V fulfilling

Gauge Invariance - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

In this case one can add a time derivative of an arbitrary scalar field Λ to the scalar potential V

Gauge Invariance - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

without changing the electric field. 

To summarize this in a covariant notation: The field-strength tensor

Gauge Invariance - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

with

Gauge Invariance - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

is unchanged under a 'gauge transformation'

Gauge Invariance - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

with Λ(x) being an arbitrary function. 
The same electrodynamics can be described by many different four-vector potentials. This is what is meant by GAUGE INVARIANCE of classical electrodynamics. 

The two Maxwell equations from above are then rewritten as

Gauge Invariance - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

The two remaining Maxwell equations

Gauge Invariance - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

and

Gauge Invariance - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

can be written in the compact form

Gauge Invariance - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

with the electromagnetic current being

Gauge Invariance - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev


CONSEQUENCES 

1) The electromagnetic current is conserved:

Gauge Invariance - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

2) The time derivative of the electric field in the fourth Maxwell equation guaranteeing local charge conservation leads also to the prediction of electromagnetic waves:

Gauge Invariance - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

In the absence of external electromagnetic currents and using the Lorentz gauge

Gauge Invariance - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

one obtains for each compoenent of the four-potential (identified with the photon field) a Klein-Gordon equation for a massless particle:

Gauge Invariance - Electromagnetic Theory, CSIR-NET Physical Sciences Physics Notes | EduRev

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