as computed in the toy model is not a completely arbitrary functional of its argument . For example, is real and not complex, and it is local in the sense that (to any finite order in ) it consists of a finite sum of powers of the field and its derivatives, all evaluated at the same point.
Why should this be so? Both of these turn out to be general features (so long as only massive degrees of freedom are integrated out) which are inherited from properties of the underlying physics at higher energies:
Since is constructed to reproduce this time evolution of the full theory, it must be real in order to give a Hermitian Hamiltonian as is required by unitary time evolution1.
That is, heavy particles may be produced so long as they are then re-destroyed sufficiently quickly. Such virtual production is possible because the Uncertainty Relations permit energy not to be precisely conserved for states which do not live indefinitely long. A virtual state whose production requires energy non-conservation of order therefore cannot live longer than , and so its influence must appear as being local in time when observed only with probes having much smaller energy. Similar arguments imply locality in space for momentum-conserving systems. (This is a heuristic explanation of what goes under the name operator product expansion [156, 41] in the quantum field theory literature.)
Since it is the mass of the heavy particle which sets the scale over which locality applies once it is integrated out, it is which appears with derivatives of low-energy fields when is written in a derivative expansion.
© Max Planck Society and the author(s)