# Hamiltonian of Massive Spin 1

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1. Oct 12, 2016

### hbar340

1. The problem statement, all variables and given/known data
Starting from the proca lagrangian $$L=-\frac14 F_{uv}F^{uv}+\frac12 m^2 A_uA^u$$

2. Relevant equations
$$H=\sum p_i\dot{q_i}-L$$

3. The attempt at a solution
$$L=-\frac14F_{uv}F^{uv}+\frac12m^2A_uA^u\rightarrow\partial_uF^{uv}+m^2A^v=0$$
$$p^i=\frac{\partial L}{\partial (\partial_0 A_i)}=-F^{0i}$$
$$H=p^i\partial_0A_i+\frac{1}{4}F_{uv}F^{uv}-\frac12m^2\left(A_o^2-\vec A^2\right)$$
$$=p^i(\partial^iA_0-p^i)+\frac12(p^i)^2+\frac14 F_{ij}F^{ij}-\frac12m^2\left(A_o^2-\vec A^2\right)$$
$$=p^i\partial^iA_0-\frac12(p^i)^2+\frac{1}{4}F_{ij}F^{ij}-\frac12m^2\left(A_o^2-\vec A^2\right)$$
$$EOM: **A_0=-\frac{1}{m^2}\partial_ip^i**$$
$$H=p^i\partial^i\left(-\frac{1}{m^2}\partial_ip^i\right)-\frac12 (p^i)^2+\frac14 F_{ij}F^{ij}+\frac12m^2\vec A^2-\frac{\left(\partial_ip^i\right)^2}{m^2}$$

I am not sure how to get positive values in this, as H should be positive up to a total derivative

2. Oct 17, 2016

### Staff: Admin

Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.

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