I guess this may be more of a mathematical than a physics question, but it comes down to physical interpretations, so I'm posting it here.
In classical Quantum Mechanics, we can define a state $\left| \psi \right\rangle$ to represent some probability amplitude over all of space. Specifically, it corresponds to a square-integrable function $\psi:\mathbb{R}^3 \rightarrow \mathbb{C}$. This state may vary in time, or one may take the view that the state remains constant and operators on the Hilbert space vary in time. (Schrodinger vs. Heisenberg)
To do a classical real scalar field, a state $\left| \Psi \right\rangle$ represents a functional probability amplitude over the possible field configurations: specifically, it corresponds to a functional $\Psi: \mathbb{R}^{\mathbb{R}^3} \rightarrow \mathbb{C}$. We could again take either the Schrodinger or Heisenberg picture here. (is this right so far?)
Most QFT introductions jump straight into fields over Minkowski space $\mathbb{M}^4$. This is where I get confused. It seems our field states still correspond to fields over spatial coordinates in $\mathbb{R}^3$ that vary in time. In $\mathbb{M}^4$, this would be saying that, given coordinates $(t,\mathbf{x})$, each constant-time slice $t=t_0$ has a field state associated with it. It seems to me, though, that choosing the slices and setting up the Hilbert spaces on each slice to get the states breaks Lorentz covariance. The alternative to me is to treat a state as being a functional $\Psi: \mathbb{R}^{\mathbb{M}^4} \rightarrow \mathbb{C}$, which is a probability amplitude over possible field configurations on the entire $\mathbb{M}^4$. However, this doesn't fit the math as far as I can tell.
What am I doing wrong? Or am I way off the mark? And what books/references can I find to better understand the formal/mathematical underpinnings of states in QFT?
Best Answer
As correctly pointed out by Daniel Sank in the comment section, the key to understanding the state space in quantum field theory is the realization that it contains information about the excitations of operator-valued functions (quantum fields) of spacetime. The latter consists of one time and three spatial coordinates (at least in the context of the standard model). Note that unlike in non-relativistic quantum mechanics, there is no longer a position operator, hence space and time are treated on an equal footing, as can be understood from the fact that we seek to deal Poincaré invariant theories. In order to achieve this, there is also the possibility of instead promoting time to an operator, as is done in string theory, but this is a different story.
As mentioned above, states in quantum field theory encode information about the excitations of quantum fields. By quantizing these fields, one introduces ladder operators acting on the ground state at each point in spacetime. The ground state is usually denoted as $|0\rangle$ and corresponds to the vacuum, while excited states represent particles. In the case of a non-interacting theory, the Hilbert space is simply a Fock space, while in the interacting case, the construction of the state space is a highly non-trivial problem (for more detail see the answers to this question by Arnold Neumaier and me). The problem of choosing slices in spacetime and therefore breaking Lorentz invariance does not arise, the formalism can be written down in a completely covariant way.
The amplitudes one looks at are specified by the particle content of the in- and out-states. A typical example is that of particle decay: one has to calculate the amplitude for an in-state with one field excitation and an out-state with several excitations, not necessarily of the same field.