Poisson brackets and EM Hamiltonian

[hellfire]

Consider the following general Hamiltonian for the electromagnetic field:

$$H = \int dx^3 \frac{1}{2} E_i E_i + \frac{1}{4}F_{ij}F_{ij} + E_i \partial_i A_0 + \lambda E_0$$

where $\lambda$ is a free parameter and $E_0$ is the canonical momentum associated to $A_0$, which defines a constraint ($E_0 = 0$ on the constraint surface). $E_i$ is the canonical momentum associated to $A_i$.

I am not able to work out the Poisson brackets to get the following results:

$$\dot E_i = \{E_i, H \} = -\partial_{j} F_{ij}$$
$$\dot E_0 = \{E_0, H \} = -\partial_{i} E_i$$

Help would be appreciated.

 

[Nate810]

The Poisson brackets for the above Hamiltonian are given by:\{E_i(x), E_j(y)\} = 0\{E_i(x), F_{jk}(y)\} = \delta_{ij}\partial_{x_k}\delta(x-y) - \delta_{ik}\partial_{x_j}\delta(x-y)\{E_0(x), A_0(y)\} = \delta(x-y)Using these Poisson brackets, we can calculate the following:\dot E_i = \{E_i, H \} = -\partial_{j} F_{ij}\dot E_0 = \{E_0, H \} = -\partial_{i} E_i

 

[quicknote]

I can understand your frustration with working out the Poisson brackets for this specific Hamiltonian. However, I can assure you that with some further analysis and calculations, you will be able to derive the results you are looking for. Let me provide some insights that may help you in this process.

Firstly, it is important to note that the Poisson brackets are defined as the anti-commutator of two quantities, in this case, the canonical momentum and the Hamiltonian. So, in order to calculate \dot E_i and \dot E_0, we need to take the anti-commutator of these quantities with the Hamiltonian.

Secondly, we can use the canonical equations of motion to simplify the calculations. These equations state that \dot E_i = \{E_i, H\} and \dot A_i = \{A_i, H\}, where A_i is the canonical momentum associated with E_i.

Now, let's focus on the first equation, \dot E_i = \{E_i, H\}. We can write this as:

\dot E_i = \frac{\partial E_i}{\partial E_j}\frac{\partial H}{\partial E_j} + \frac{\partial E_i}{\partial A_j}\frac{\partial H}{\partial A_j}

Using the canonical equations of motion, we can simplify this to:

\dot E_i = \{E_i, H\} = -\partial_{i} E_i + \partial_{j} F_{ij}

Similarly, for \dot E_0, we have:

\dot E_0 = \{E_0, H\} = -\partial_{i} E_i - \lambda

Now, using the constraint E_0 = 0, we can simplify this further to get \dot E_0 = -\partial_{i} E_i.

I hope this helps you in your calculations. Remember, as a scientist, it is important to persevere and keep trying until you reach a solution. Good luck!