Reduced source, readout, and target

Reduced source, readout, and target

The reduced source space

Throughout, $S$ denotes a reduced source space. By this we mean the space obtained from whatever raw description one began with after quotienting by every redundancy that has been declared physically inert: local gauge transformations, coordinate reparametrizations, relabelings of identical constituents, and any further identification that the modeling commitments of the problem require. The point of insisting on this is not fastidiousness. If $S$ still carried gauge copies, then $p$ would appear to lose information whenever it merely forgot a gauge label, and every theorem below would be vacuous or misleading: one would be proving obstructions that could be dissolved by a change of gauge. By working on the reduced space we guarantee that two distinct points of $S$ are genuinely distinct physical configurations, so that a fiber of $p$ containing two points is a genuine collapse of physical distinctions and not a bookkeeping artifact.

We write $\sim_p$ for the equivalence relation on $S$ induced by $p$, so $s\sim_p s'$ iff $p(s)=p(s')$ iff $s$ and $s'$ lie in a common fiber. The readout is lossy, or noninjective, when some fiber contains more than one point. No topological, smooth, or measurable structure is required for the results of Section 3, Section 4, Section 5, Section 6, Section 7, Section 8; those are statements about maps of sets, and their strength derives from that generality. Structure is imposed only in Section 9, Section 10, Section 11, where the source space is a moduli space of connections and the target values are spectra.

Targets

That $a_Q$ is a well-defined map on all of $S$ is a substantive standing assumption, and it is worth saying what it does and does not import. It asserts that the question has a determinate answer for each reduced source state , that once the physical configuration is fixed, so is the correct answer. It does not assert that the answer is computable, that it is known, or that it is accessible from any particular data. Indeed, the entire content of the paper is the analysis of when the answer, though determinate as a function on $S$, fails to be a function of $p(s)$. If a purported target does not have a determinate answer per reduced source state , if, say, the answer depends on further facts not encoded in $S$ , then it is not a target on $S$ in the sense of Definition 2.2, and the source space must be enlarged before the present analysis applies.