# 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