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In a rigorous fashion, how does one define the Hamiltonian of QFT as

$$\hat{H}(t) = \int d^3x \hat{\mathcal{H}}(x, t)$$

For now I'm ignoring the fact that $\hat{\mathcal{H}}$ itself may be ill-defined since it involves product of distributions. Let's assume that $\hat{\mathcal{H}}$ is a properly defined operator-valued distribution.

First how does one take the integral of a distribution like this? While one could take something resembling an integral as $\hat{\mathcal{H}}[\phi]$ if $\mathcal{H}$ had compact support and $\phi = 1$ on that support, there's no guarantee that this is possible in general. Is it possible for instance to define the operator-valued distribution as the limit of a sequence of operator-valued functions and take something akin to

$$\hat{H}(t) = \lim_{n\to \infty} \int d^3x \hat{\mathcal{H}}_n(x,t)$$

assuming that the limit can commute with the integral. Also, what would be the integral used here, is it one of those projection-valued measure?

Alternatively, can I define it as

$$\hat{H}(t) = \lim_{n\to \infty} \hat{\mathcal{H}}[\phi_n]$$

for $\phi_n$ a test function equal to $1$ on a compact support of ever increasing size?

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Here's the big picture:

  • In many cases (nonabelian chiral gauge theories, like the Standard Model), we don't yet know how to define the Hamiltonian rigorously at all.

  • In most of the cases where we do know how to define the Hamiltonian rigorously, the definition involves replacing continuous space with a discrete (and finite) lattice.

  • Only in exceptional cases (like models with quadratic Lagrangians) do we know how to define things rigorously without resorting to a spatial lattice. Because this only works in exceptional (usually boring) cases, I won't address it here.

To be clear, when I say "define the Hamiltonian" here, I really mean "define the whole QFT, including giving a well-defined expression for the Hamiltonian in terms of the field operators."

In a lattice-based Hamiltonian formulation of QFT, space has a finite number of points, so the concerns expressed in the question disappear. Time remains continuous. One way to approach the formulation is to use the usual canonical quantization process, starting with a classical Lagrangian in which space has already been discretized (so spatial gradients are replaced by finite differences, and integrals are replaced by sums). The messiness of the formulation depends on the types of fields involved:

  • The formulation is straightforward when only scalar fields are involved.

  • When gauge fields are involved, it becomes relatively straightforward in the temporal gauge (in which the gauge field $A_\mu$ has no $\mu=0$ component) if we use elements of the Lie group (instead of the Lie algebra) to represent the gauge field. One drawback of this formulation is that the resulting Hamiltonian is never only quadratic in the gauge field, not even for a $U(1)$ gauge field like the EM field. This is an obstacle to closed-form calculations. By the way, this "compact $U(1)$" version of electrodynamics automatically includes magnetic monopoles (if space is 3-dimensional), which decouple in the continuum limit.

  • When Dirac spinor fields are involved, the formulation becomes discouragingly messy, but it's doable.

  • When chiral (Weyl) fermions are involved, in the generic case where their interactions with the gauge fields are not invariant under spatial reflection, we don't even know how to do it yet.

There is also a rigorous version of the "path integral" formulation in which spacetime is replaced by a discrete lattice. Similar comments apply in this case, and the lattice Hamiltonian formulation can be recovered by considering the continuous-time limit. This is an alternative approach to working out the details of the lattice Hamiltonian formulation, and the result can be checked against the canonical-quantization approach mentioned above. They should agree with each other, at least modulo differences that should become negligible in the continuum limit.

Manual calculations are rarely done using an explicit lattice-based formulation, because it quickly becomes prohibitively messy. However, understanding how a model can be rigorously defined on a lattice is still valuable, because whenever we run into trouble with naive continuous-space manipulations (like divergent Feynman integrals, etc), we can retrace our steps starting from a well-defined lattice formulation to understand exactly what went wrong and how to fix it — at least conceptually.

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