My friend John Moffat has a finite quantum field theory that, I think, deserves more attention than it gets.
The theory is nonlocal (then again, so is quantum physics to begin with). However, it does not violate causality. So its nonlocality is a mathematical curiosity, not a physical impossibility.
The essence of the theory is present in the form of its “nonlocal field operator”. Given, e.g., a scalar field in the form \(\phi(x),\) the field is transformed as
$$\tilde\phi(x)=\int d^4x’f(x-x’)\phi(x’).$$
Now if we just used the Dirac delta-function \(f(x-x’)=\delta^4(x-x’),\) we’d get back \(\phi(x).\) But what if we use some other function, the only restriction being that \(f(x)\) must be an entire function, which is to say, unambiguously defined without poles or singularities over the entire complex plane?
Well, then, assuming again that \(f(x)\) is an entire function, we can integrate iteratively in parts, until we arrive at an expression in the form,
$$\tilde\phi(x)={\cal F}(\partial_x)\phi(x),$$
where \({\cal F}(\partial_x)\) is a derivative operator, typically some power series in the form \(\lambda_i\partial_x^i\), acting on \(\phi(x).\)
Why is this good for us? Because this field redefinition can suppress high-energy divergences in the theory, essentially doing away with the need for renormalization, which, of course, is a Big Claim indeed but I think John’s theory works.
John’s first substantive papers on this topic were titled Finite quantum field theory based on superspin fields (J. W. Moffat, Phys. Rev. D 39, 12 (1989)) and Finite nonlocal gauge field theory (J. W. Moffat, Phys. Rev. D 41, 4 (1990)). Unfortunately these papers predate arxiv.org so only the paywalled versions are available. They are beautiful papers that deserve more recognition. More recently, John wrote another paper on the subject, collaborating with a student. One of these days, I’m hoping to spend some time myself working a bit on John’s theory because I believe it has merit: The theory appears to remain causal despite the nonlocal operator, and by doing away with the need for renormalization, it makes canonical quantization almost trivially possible. I keep wondering if there is, perhaps, a catch after all, but if that’s the case, I have yet to find it.