Meaning of Square of four vector potential

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SUMMARY

The square of the four vector potential, expressed as (\phi,A)^2=\phi ^2 - A^2, is a Lorentz invariant that has distinct physical interpretations based on its timelike or spacelike nature. When the invariant is timelike, it corresponds to the scalar potential in a frame where the vector position is zero. Conversely, if the invariant is spacelike, it represents the magnitude of the vector potential in a frame where the scalar potential is zero. The interpretation of this invariant can become complex unless the Lorentz gauge is employed, as gauge conditions are frame-dependent.

PREREQUISITES
  • Understanding of four vector potential in electromagnetism
  • Familiarity with Lorentz invariance and its implications
  • Knowledge of gauge theories, specifically Lorentz gauge
  • Basic concepts of scalar and vector potentials in physics
NEXT STEPS
  • Research the implications of Lorentz invariance in electromagnetism
  • Study the properties and applications of the Lorentz gauge
  • Explore the relationship between scalar and vector potentials in different frames
  • Investigate the role of gauge transformations in field theories
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Physicists, particularly those specializing in electromagnetism and field theories, as well as students seeking to deepen their understanding of gauge invariance and the four vector potential.

merrypark3
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The square of the four vector potential.

(\phi,A)^2=\phi ^2 - A^2

What's the physical meaning of this lorentz invariant?
 
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If the invariant is timelike then it would be the scalar potential in a frame where the vector position was 0, and if the invariant is spacelike then it would be the magnitude of the vector potential in a frame where the scalar potential is 0. However, unless you are using the Lorentz gauge your gauge condition is frame variant so interpreting it becomes a little strange.
 
hi merrypark3! :smile:

you can add any constant vector to the potential without changing the physics …

so you can make that invariant anything you like! :wink:
 

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