Hamiltonian of Massive Spin 1

[hbar340]

Homework Statement

Starting from the proca lagrangian $$L=-\frac14 F_{uv}F^{uv}+\frac12 m^2 A_uA^u$$

Homework Equations

$$H=\sum p_i\dot{q_i}-L$$

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

 

[mark726]

. I think the issue may be in the calculation of A_0. Can someone please help me figure out where I went wrong?

There are a few issues with this solution. First, the equation for A_0 is incorrect. It should be A_0 = -1/m^2 * p^i, not -1/m^2 * partial_i p^i. This can be seen by taking the derivative of A_0 with respect to time and using the EOM for A_0.

Second, the expression for H is not correct. The term p^i * partial^i * A_0 should be -p^i * partial_i * A_0, as seen in the original lagrangian. Additionally, the last term - (partial_i p^i)^2/m^2 should be (partial_i p^i)^2/m^2. This results in a final expression for H of:

H = -p^i * partial_i * A_0 - 1/2 * (p^i)^2 + 1/4 * F_ij * F^ij + 1/2 * m^2 * A^2 + (partial_i p^i)^2/m^2

This expression is positive, as desired. The total derivative can be ignored since it does not affect the overall positivity of H.