One of the canonical prescriptions to quantize a classical theory is to reproduce the *-Algebra of functions (functionals) on $M$ on a complex Hilbert Space.
In relativistic quantum mechanics I saw the so called equal time second quantization, that is one usually substitute the field Poisson bracket with Commutator/Anticommutator for a Bosonic/Fermionic system. That's fine if only we are doing a relativistic field theory.
The Poisson bracket we start with in the classical regime:
$$\{F(x,t),G(y,t)\}^{(3)} = \int_{\Sigma_t}d^3z \left( \frac{\delta F(x)}{\delta \phi(z)}\frac{\delta G(y)}{\delta \pi(z)} - \frac{\delta F(x)}{\delta \pi(z)}\frac{\delta G(y)}{\delta \phi(z)}\right) . $$
Where we have chosen a spacelike hypersurface $\Sigma_t$ with $x^0=t = constant$
Quantization imposes the canonical Poisson brackets $$ \{\pi(x,t),\pi(y,t)\}^{(3)} = 0$$ and $$ \{\pi(x,t),\phi(y,t)\}^{(3)} = \delta(x - y) $$ idem for $\phi$.
First of all how does a functional transform under a general and a Lorentz transformation?
This is well defined in non relativistic field theory, but I am not sure it is in relativistic field theory, as a Lorentz transformation is not a canonical transformation (we are not using the extended Hamiltonian formalism)
How can I check that equal time second quantization is well defined? I mean we are implicitly choosing a particular observer and an hypersurface of constant t. What if I define the bracket (and then quantize) on a different spacelike hypersurface connected with the first one by a Lorentz transformation?
Because Lorentz transformation shouldn't be canonical transformation, so they will not preserve the Hamiltonian formalism which decouples space and time. Moreover two inertial observers will obtain different values for their Poisson brackets and different quantization procedures.
Why do we quantize in a formulation which isn't clearly Lorentz invariant? (I know there are other methods, but how is this allowed?) Shouldn't have we used the extended Hamiltonian formalism which is Lorentz covariant and in which Lorentz transformation are canonical transformations?
Thanks!
