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- Homework Statement
- How do I simplify the Proca lagrangian for a spin-1 field

- Relevant Equations
- L = -(1/16*pi) * ( ∂^(μ)A^(ν) - ∂^(ν)A^(μ))(∂_(μ)A_(ν) - ∂_(ν)A_(μ)) + 1/(8*pi) * (mc/hbar)^2* A^ν A_ν

Hello,

I'm trying to figure out where the term (3) came from. This is from a textbook which doesn't explain how they do it.

∂_μ(∂L/(∂(∂_μA_ν)) = ∂L/∂A_ν (1)

L = -(1/16*pi) * ( ∂^(μ)A^(ν) - ∂^(ν)A^(μ))(∂_(μ)A_(ν) - ∂_(ν)A_(μ)) + 1/(8*pi) * (mc/hbar)^2* A^ν A_ν (2)

Here is Eq (1) the Euler-Lagrange equation and Eq (2) is the lagrangian for a vector field. In the textbook they just state the term

∂_μ(∂L/(∂(∂_μA_ν)) = -1/(4*pi)*(∂^(μ)A^(ν) - ∂^(ν)A^(μ)) (3)

Where does the term -1/(4*pi) come from, and how do I cancel out the rest of the term so that the equation becomes

∂_μ(∂^(μ)A^(ν) - ∂^(ν)A^(μ)) + (mc/hbar)^2* A^ν (4)

I'm trying to figure out where the term (3) came from. This is from a textbook which doesn't explain how they do it.

∂_μ(∂L/(∂(∂_μA_ν)) = ∂L/∂A_ν (1)

L = -(1/16*pi) * ( ∂^(μ)A^(ν) - ∂^(ν)A^(μ))(∂_(μ)A_(ν) - ∂_(ν)A_(μ)) + 1/(8*pi) * (mc/hbar)^2* A^ν A_ν (2)

Here is Eq (1) the Euler-Lagrange equation and Eq (2) is the lagrangian for a vector field. In the textbook they just state the term

∂_μ(∂L/(∂(∂_μA_ν)) = -1/(4*pi)*(∂^(μ)A^(ν) - ∂^(ν)A^(μ)) (3)

Where does the term -1/(4*pi) come from, and how do I cancel out the rest of the term so that the equation becomes

∂_μ(∂^(μ)A^(ν) - ∂^(ν)A^(μ)) + (mc/hbar)^2* A^ν (4)