Field arena

In plain terms

Everything else in this glossary concerns systems with finitely many outcomes, while the working theory of the physical world is quantum field theory, which has infinitely many modes and no limit on the number of quanta. A reconstruction that can't even gesture at fields would be a reconstruction of a toy, and the field arena is that gesture made precise, deliberately at finite resolution.

A mode is one independent way a field can vibrate, like a single note on a string, and a quantum is one unit of excitation of a mode, the field-theoretic version of a particle. The field arena keeps finitely many modes and lets each hold at most some fixed number of quanta, and on the space of quantum states built from those configurations the canon of field theory is re-proved as exact finite theorems. Free propagators, which describe how a disturbance travels when nothing interacts, oscillate at the right frequency. The effects of an interaction are priced linearly in the strength of the coupling. There's a genuine light cone, a limit on how fast an influence can spread, with an explicit bound on the velocity. Thermal states satisfy the KMS condition, the exact condition that characterises equilibrium at a given temperature.

This is breadth, and it's labelled as such, as it's the image of standard physics inside the programme's arena and not the frontier of the reconstruction.

In CSD

The field arena is the stage for the programme's continuous-variable track. It's the space of quantum states over configurations of finitely many modes with a fixed cap on the quanta per mode, and its role in the programme is honestly bounded. It's the isolated-piece image, as it shows that the structure of effective field theory survives at finite cutoff on exactly the kind of arena the reconstruction runs on, and no continuum limit and no renormalisation-group flow is claimed.

The bridge theorems tie it back to the record layer. Observations on the arena are pairings with a density, dynamics act by kicks, observables supported on disjoint sets of modes commute exactly, and the Lieb-Robinson light cone bounds how fast a record cell in the fibre can be steered from afar. The arena stopped being a special case in October 2026, as the construction behind it now works for any finite way of labelling configurations rather than only for mode configurations, so the field arena and the state space the rest of the programme works in are one construction read two ways. That's what let a single two-wing experiment be stated on one arena instead of on two spaces that were only believed to be the same.

Mathematically

FieldArena K N is the projective space over configurations of K modes at cutoff N. Mode-local subalgebras are unital star-subalgebras supported on their blocks, free and interacting drives are exact matrix exponentials, and the two-point and four- point functions close into Wick's pairing sum at explicit thresholds.

The commutator bound decays factorially in graph distance with velocity constant 2e^2 times the coupling norm, and composite arenas compose by mode concatenation, with no-signalling exact for all states, entangled included. GenArena is the arena over any finite index with decidable equality, and FieldArena K N is that at the configuration space, so unitVec, dmVec, arenaDM, arenaObs and arenaKick now read over any index with every earlier statement unchanged. SupportedOn stays mode-indexed, as its content is that an operator leaves the modes outside a region alone, which needs the index to be a space of functions.

Related entries

• 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.
• 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.
• No exact finite-dimensional CCRIn a world with only finitely many distinguishable states, the founding equation of quantum mechanics can't hold exactly, and one line of arithmetic proves it while CSD says precisely where the equation gives way.
• 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.
• Interaction priceSwitching on an interaction between field modes makes local quantities spread at a rate proportional to the coupling, and the corpus proves the rate is exact, at most twice the coupling times the elapsed time, and at least a fixed fraction of it.