Spin-One Klein Gordon Equation

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Discussion Overview

The discussion centers on the spin-one Klein-Gordon equation, specifically its formulation and the associated conserved current. Participants explore the definitions and differences between the Klein-Gordon equation and the Proca equation, as well as the implications of these equations in the context of fields with definite mass.

Discussion Character

  • Technical explanation, Debate/contested

Main Points Raised

  • One participant asks for clarification on the spin-one Klein-Gordon equation and the formula for the conserved current.
  • Another participant states that the Klein-Gordon equation for spin-one can be expressed as $$(\square - m^2)A^\mu=0$$, suggesting this is a straightforward application of the general form of the equation.
  • A later reply identifies the equation as the Proca equation, indicating a specific context for spin-one fields.
  • Another participant provides a technical correction, noting that the Proca equation is actually ##(\square - m^2)A^\mu = \partial^\mu \partial^\nu A_\nu## and emphasizes that the Klein-Gordon equation applies to any field with a definite mass.
  • This participant also mentions that the Proca equation requires the additional condition ##\partial_\nu A^\nu=0## to be satisfied.

Areas of Agreement / Disagreement

Participants express differing views on the definitions and implications of the Klein-Gordon and Proca equations, indicating that there is no consensus on the precise formulation or conditions required for these equations.

Contextual Notes

There are unresolved aspects regarding the definitions of the equations and the conditions under which they apply, particularly concerning the additional constraints for the Proca equation.

Jogging-Joe
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TL;DR
The spin zero Klein Gordon equation is commonly discussed. How about the spin one Klein Gordon Equation?
What is the spin one Klein Gordon Equation? What is the formula for the conserved current, i.e. the electric current density four-vector?
 
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I don't know what do you understand by the spin-one KG equation...
But the KG equation is just ##(\square - m^2)\phi=0##.
So I would say that the KG equation for spin-one is just $$(\square - m^2)A^\mu=0$$
 
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Gaussian97 said:
I don't know what do you understand by the spin-one KG equation...
But the KG equation is just ##(\square - m^2)\phi=0##.
So I would say that the KG equation for spin-one is just $$(\square - m^2)A^\mu=0$$
Yes, and it's called Proca equation.
 
Well, technically the Proca equation is ##(\square - m^2)A^\mu = \partial^\mu \partial^\nu A_\nu##.
Klein-Gordon equation should (as far as I know), be fulfilled by any field with definite mass, since it's the quantum version of ##p^2 + m^2 = 0##.
To obtain the Proca equation you need the extra condition ##\partial_\nu A^\nu=0##.
 
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