Why the non-locality is bad?

Question

Suppose that in order to cancel some divergences of the given quntum field theory  I have to introduce the non-local counter-term of the form

$$\Delta S = \int d^{4}x\frac{1}{\square}A(x),$$

where $\square$ is d'Alembert operator and $A(x)$ is some polynomial. People often say that non-local counter-terms are bad for theory. But now I didn't meet the clear explanation why.

Do You know it?

Comments

Because it makes the whole theory non local, no?

So far we managed to describe the experimental data with the local theories and this is often pronounced as a "proof "of the absence of non-local interactions in Nature :-)

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@VladimirKalitvianski : but do these counter-terms spoil some general properties of the quantum field theory, such as causality, positive definiteness of vacuum, unitarity? If yes, how to argue this?

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I am not good in this subject, but when you google the internet, you obtain many sources discussing causality and other issues. I do not say "non-locality is bad" since even a local theory may be cast into "non local" form (see eq. (24), for example).

Answer 1

Of course causality is at risk of breaking down as soon as there is non-locality in time. And since in a relativistic theory there is no clear-cut distinction between space and time, causality is at risk as soon as the theory is non-local.

There is this ancient article from the The Great Soviet Encyclopedia (1970-1979)  which puts it very well:

Microcausality condition

a requirement that the causality condition (which states that cause must precede effect) be satisfied down to an arbitrarily small distance and time interval. The microcausality condition usually refers to distances ≲ 10^{-16} cm and to times ≲ 10^{-24} sec.

It is shown in the theory of relativity that the assumption of the existence of physical signals that propagate with a velocity greater than the velocity of light leads to violation of the causality requirement. Thus, the microcausality condition prohibits the propagation of signals at a velocity greater than the velocity of light “in the small”.

In quantum theory, where operators correspond to physical quantities, the microcausality condition requires the interchangeability of any operators that pertain to two points of space-time if these points cannot be linked by a light signal. This interchangeability means that the physical quantities to which these operators correspond can be precisely determined independently and simultaneously. The microcausality condition is important in quantum field theory, especially in the dispersion and axiomatic approaches; these approaches are not based on specific model concepts of interaction and therefore can be used for direct verification of the microcausality condition. In the most highly developed branch of quantum field theory — quantum electrodynamics — the microcausality condition has been experimentally verified for distances ≲ 10^{-15} cm (and, correspondingly, for times ≲ 10^{-25} sec).

The violation of the microcausality condition would make it necessary to radically alter the method of describing physical processes and to reject the dynamic description used in modern theories, in which the state of a physical system at a given moment of time (the effect) is determined by the states of the system at preceding times (the cause).

On the other hand, non-locality need not necessarily produce a bad theory, it just means that things get much more subtle. In fact many people have and are arguing that quantum gravity will have to be a non-local theory. Certainly string theory is non-local around the string scale, due to the finite extension of the fundamental string. we are just talking about this in the thread How is causality encoded in string theory?

As discussed there, string theory is an example of a theory that is non-local at high energy, and still okay.