Total vs extended action on constrained dynamics

Question

Studying the electromagnetic hamiltonian dynamics, I used the extended formalism (after finding all constraints using the primary hamiltonian, also following the Dirac's recipe) to calculate the equations of motion and eventually find the gauge transformations. With the following extended hamiltonian (where the total one is achieved by ignoring the last term)

$$ \mathscr{H}_{\text{ext}}=\frac{1}{4}F^{i j} F_{i j} -\frac{1}{2}\pi^{i}\pi_{i} -A_0 \partial_i \pi^{i} + j^{\mu}A_{\mu} + \lambda_1 \pi^0 +\lambda_2 \partial_{i}\pi^{i} $$

I got the following results for the E.O.M.

\begin{align} \dot{A}_0&=\lambda_1 \\ \dot{A}_i&=-\pi_i - \partial_i \lambda_2 + \partial_i A_0 \\ \dot{\pi}^0&=\partial_i \pi^i -j^0 \\ \dot{\pi}^i&=-j^i + \partial_j F^{j i} \end{align}

and for the gauge transformations

\begin{align} \delta_{\epsilon_1}A_0&=\epsilon_1 \\ \delta_{\epsilon_2}A_{i}&=\partial_i \epsilon_2 \end{align}

Now, up to this point I would think that the gauge transformation parameters were completely arbitrary, as a great part of the literature suggests, but that is of course not the case since it is well known that the gauge freedom of the electromagnetic field takes the form $\delta A_\mu=\partial_\mu \Lambda$, for arbitrary $\Lambda$.

This made me go into more deep considerations on the matter, and on an attempt of following the hint given by Henneaux (on his book Quantization of gauge systems) that the total hamiltonian is the (secondary) gauge fixed version of the extended one, I imposed that $\lambda_2=0$, then, absorbing the new terms inserted by a gauge transformation on $\lambda_2$ and imposing $\delta \lambda_2=0$ I obtained the right condition on the parameters, i.e., $\epsilon_1=\dot{\epsilon_2}$. The problem is that I don't quite understand why this would formally be generally correct and neither have found examples on the literature.

So, as expressed by Pitts on his paper A First Class Constraint Generates Not a Gauge Transformation, But a Bad Physical Change: The Case of Electromagnetism, authors don't really carry these simple calculations on and as far as I can see treat the subject by different points of view, most of them based on the Dirac's conjecture and the assumptions that lead to it. Rothe, for instance, derives on his book Classical and Quantum Dynamics of Constrained Hamiltonian Systems (for point particle mechanics, which shouldn't be the problem here) that on the extended formalism the gauge parameters should be arbitrary (of which, as I understood so far, the electromagnetic case is a counterexample). Sundermeyer doesn't even address the problem.

To sum up, my question is: is the 'gauge fixing' of the extended hamiltonian as I described a general way for finding the most general relation between the parameters? Why? And, ignoring the cloudy parts of the total vs extended debate, what can be undoubtedly said about the hamiltonian method (on this context)?

Right now what I suspect is: the correct equations of motion are derived by the total hamiltonian (that is, $\dot F=\{F,H_T\}$ instead of $\dot F=\{F,H_{\text{ext}}\}$ as leaded me to the presented results) and the gauge transformations are generated by a nicely tuned combinations of all first class constraints, and not by each individually nor by ignoring the secondary ones.

Sorry if I couldn't make myself clear. I'd appreciate any insights, I've been strugling on this for quite some time and couldn't get the answers from the math alone.

Answer 1

I don't want to step into the murky waters of the total vs. extended dynamics here either, except to say that the dynamics generated by the extended Hamiltonian involves arbitrary gauge transformations generated by the non-dynamical (= specified by hand) fields that you call $\lambda_1$ and $\lambda_2$ while evolution, while the total Hamiltonian is more "conservative," not allowing these things.

But the other question you raise is a lot more interesting, piercing into the heart of what separates the Lagrangian and Hamiltonian approaches; an abyss so deep that few have looked into its heart. Which is all just an excuse to say that it is best not too have too high a hope for my fumbling attempts at an explanation.
The basic point is this: on a single spatial slice, a field and its time-derivative are essentially two independent fields, since you can't figure out the derivative if I give you the value of a function at one point. The difference between the Lagrangian and Hamiltonian approaches is in how they deal with this fact. In the Lagrangian approach, the variables we specify on an initial slice are fields and their time-derivatives. So, in an action the standard EM gauge transformation $\delta A_\mu = \partial_\mu \lambda$ is perfectly fine.
In the Hamiltonian approach, however, you aren't allowed to specify time-derivatives but only variables and conjugate momenta. Suddenly, $\delta A_0 = \dot{\lambda}$ is looking a lot more fishy, especially if we want to write down the extended Hamiltonian where the gauge transformation happens along the path and the transformation parameter is to be treated as a (non-dynamical) field. So, we write down two fields $\lambda_1$ and $\lambda_2$ which act as the two gauge transformation parameters. Note that because, on a spatial slice fields and time-derivatives are independent, we don't really need to relate the two.

While it's not that interesting here, this is the reason that the BRST charge has the form $$Q = \int i \Pi_{\bar{c}} \Pi^0 - c \partial_i \Pi_i.$$ The two ghosts replace the two $\lambda$s, and since the BRST charge acts on a time-slicce at a time we need two fields to specify the gauge transformation on the fields.
As for why it's $\Pi_{\bar{c}}$ in the first term, the formalism (for reasons I haven't really understood) remembers that the parameter that generates the $A_0$ transformation is a momentum-like variable (because it's a time-derivative).