Lessons learned

2.4 Lessons learned

It is clear that the kind of discussion given for the toy model can be performed equally well for any other system having two well-separated energy scales. There is a number of features of this example which also generalize to these other systems. It is the purpose of this section to briefly list some of these features.

2.4.1 Why are effective Lagrangians not more complicated?

$ℒe ff$ as computed in the toy model is not a completely arbitrary functional of its argument $θ$ . For example, $ℒeff$ is real and not complex, and it is local in the sense that (to any finite order in $1∕mR$ ) it consists of a finite sum of powers of the field $θ$ 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:

Reality:

The reality of $ℒeff$ is a consequence of the unitarity of the underlying theory, and the observation that the degrees of freedom which are integrated out to obtain $ℒe ff$ are excluded purely on the grounds of their energy. As a result, if no heavy degrees of freedom appear as part of an initial state, energy conservation precludes their being produced by scattering and so appearing in the final state.

Since $ℒ eff$ 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 .

Locality:

The locality of $ℒeff$ is also a consequence of excluding high-energy states in its definition, together with the Heisenberg Uncertainty Relations. Although energy and momentum conservation preclude the direct production of heavy particles (like those described by $χ$ in the toy model) from an initial low-energy particle configuration, it does not preclude their virtual production.

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 $ΔE ∼ M$ therefore cannot live longer than $Δt ∼ 1∕M$ , 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 $M$ of the heavy particle which sets the scale over which locality applies once it is integrated out, it is $1∕M$ which appears with derivatives of low-energy fields when $ℒeff$ is written in a derivative expansion.