Does the vacuum energy problem of quantum field theory only occur in the Hamiltonian approach, or also in the path integral approach and in AQFT?

Question

In a standard QFT class, you're being indoctrinated that there is the "infinite vacuum energy density problem". (This is sometimes paraphrased as the "cosmological constant problem", which is in my opinion a misnomer since the calculation that calculates a finite value for the vacuum density and relates it to the cosmological constant is ill-founded and dubious.)

The argument goes as follows: When defining the Hamiltonian $H$ from the Lagrangian density $\mathcal{L}$, we first define the Lagrangian function $L := \int \operatorname{d}^3 x \mathcal{L}$ and the momenta $\pi(x) := \frac{\partial L}{\partial \dot \phi(x)}$ and finally the Hamiltonian $H := \int \operatorname{d}^3 x \pi \dot \phi - L$.

We then promote all the quantities that occured to operators and calculate the decomposition of $H$ into creation and annihilation operators $a^\dagger$, $a$.

Then, the lecturer makes a big fuss (prototypical example) about the Hamiltonian being of the form $\int \operatorname{d}^3 p E_p \frac{1}{2} \left( a^\dagger a + a a^\dagger \right)$ = $\int \operatorname{d}^3 p E_p \left( a^\dagger a + \delta(0) \right)$ and the $\delta(0)$ being an infinite contribution to the vacuum energy density. (I always found the argument fishy because we can't know beforehand if some quantisation strategy will succeed or not, so we should rather start with a quantised Hamiltonian and ask whether its classical limit is the theory we started with, which can be answered with yes.)

My question now is: Does a similar problem also occur in the Feynman path integral approach, or in any approach from algebraic quantum field theory?

Answer 1

In algebraic QFT, the Hamiltonian is always well-defined, and the vacuum state is an eigenstate of zero energy. Thus no such terms arise - they are eliminated by careful renormalizations before a continuum limit is taken.

In the path integral approach, the problem is still present, hidden in the renormalization prescriptions for making perturbative sense of the contributions to the path integral. Note that infinities must be cancelled to get finite results, and as $\infty-\infty$ is undetermined there remain finite renormalizations that can be fixed by a number of different recipes that (should) give equivalent parameterizations of the same manifold of renormalized field theories.