Interaction price

In plain terms

A field can be thought of as a collection of modes, independent patterns of vibration, like the separate notes a string can sound, and a quantity is local if it concerns only a particular set of modes. When the modes don't interact, the dynamics keeps every local quantity confined to its own modes for ever, and interactions spread them, so a quantity that started out concerning one mode comes to depend on others as time passes.

That qualitative statement is folklore, and the quantitative question is the interesting one, which is at what rate an interaction buys this spreading, and whether the rate you can prove as an upper limit is also the rate that actually occurs.

The corpus answers with a sandwich. On one side there's an upper bound, as after one period of the interacting drive every local quantity stays within twice coupling- times-time of some exactly local quantity, where the distance between two quantities means the largest difference they could show on any state, so spreading is priced linearly in the coupling strength and the elapsed time. On the other side there's a lower bound on an explicit two-mode example, as a computable commutator, a quantity measuring how far two operations are from being interchangeable, forces every exactly local quantity to sit at least a comparable linear distance away.

Between the two bounds, "costs at most" becomes "costs exactly", up to a constant factor lying between one over pi and two.

In CSD

CSD's field track, the part of the programme that treats fields, uses the interaction price as its quantitative locality backbone. It feeds the interacting corrections to the propagator, the description of how disturbances travel through the field. It feeds the renormalised couplings, effective interaction strengths that don't depend on the cutoff, the artificial limit on resolution used to keep calculations finite. And it feeds the coarse-graining budget at the level of channels, where the whole error of a renormalisation step, a step that deliberately discards fine detail, is the interaction's price, because the free part of the dynamics is carried across exactly.

The lower bound matters because an upper bound on its own leaves open whether the price is ever actually paid. The explicit witness shows that it is, so the sharpness of the bound is a theorem rather than a hope.

Mathematically

The theorem price_linear_attained states that on the two-mode witness pair, and for tau lambda in the window from zero to pi (both bounds are hypotheses of the theorem and not decoration, since the sine is only monotone there), the distance from the interacting Heisenberg observable to the subalgebra supported on the free modes satisfies tau lambda / pi <= dist <= 2 tau lambda.

The lower bound runs through the commutator functional, as any operator supported on the free modes commutes with a disjoint probe, so one computed commutator bounds the distance to the whole subalgebra. The commutator entry has modulus 2 sin(tau lambda / 2) exactly, and the Jordan inequality converts the sine to the linear form. Historically the upper bound stood alone with its sharpness explicitly unclaimed, and the witness construction retired that caveat.

Related entries

• Field arenaTake finitely many field modes, cap the number of quanta each can hold, and the whole toolbox of quantum field theory, propagators, light cones, Wick's theorem and thermal states, reappears as exact theorems on one finite arena.
• Lieb-Robinson boundEven without relativity, influence in a quantum system with local interactions has a speed limit, and on CSD's finite field arena that limit is a theorem with the velocity written out.
• Record light coneRecords are safe from distant meddling by theorem, as anything that happens outside the light cone can shift the outcome written into the arena's fibre by no more than a bound that collapses factorially with distance.
• IsolationBeing isolated isn't scene-setting, as it's the variable the whole theory turns on, and quantum mechanics is what isolated pieces of the arena look like from outside.
• Wick's theoremThe bookkeeping rule at the heart of quantum field theory, which reduces any correlation of field operators to a sum over pairings, survives a finite cutoff exactly, and the threshold at which it fails is part of the theorem.