Realization states, physical equivalence, and readout

Realization states, physical equivalence, and readout

The realization space and its reduction

Let $\Omega$ be a set of admissible realization states. Depending on the application, an element of $\Omega$ may be a microscopic configuration, a field configuration, a geometric structure, a solution together with boundary data, or a complete realization package specified by the theory in question. Nothing below depends on which of these it is; what matters is that the theory has committed to $\Omega$ as its description of what there is.

A theory does not, however, treat every distinction available in its own description as a physical distinction. Let $\simeq_{\mathrm{phys}}$ be the equivalence relation on $\Omega$ generated by all transformations that the theory declares physically redundant. Typically this includes gauge transformations and coordinate changes, in the cases where such transformations alter only the representation of a state and not the state itself. The declaration is part of the theory’s content, not a mathematical consequence of it; the mathematics below takes it as given.

The passage from $\Omega$ to $S$ is not a technical convenience; it is the step that makes the rest of the paper meaningful. A difference between two elements of $\Omega$ that is removed by $\simeq_{\mathrm{phys}}$ is, by the theory’s own declaration, not a difference in the world. If a readout map forgets such a difference, it has forgotten a redundancy, not a physical structure, and no obstruction worth the name has been exhibited. Every claim of lost source structure made below is therefore a claim about $S$, never about $\Omega$.

The readout and its induced map

Let $$\Pi\colon\Omega\longrightarrow T\tag{2}$$ be the readout map, whose values are the data the theory treats as public. Throughout we adopt the convention $$T=\mathop{\mathrm{Im}}\Pi,\tag{3}$$ so that $\Pi$ is surjective onto its codomain. This is a normalization, not an assumption: readout values that no realization state produces play no role, and retaining them would only clutter statements about fibers with vacuous cases.

We assume the readout is physically invariant, that is, invariant under physical equivalence: $$\omega\simeq_{\mathrm{phys}}\omega' \quad\Longrightarrow\quad \Pi(\omega)=\Pi(\omega').\tag{4}$$ This is the minimal coherence requirement on a readout: if two descriptions describe the same physical situation, they must yield the same public data. A map violating (4) would let one read off gauge-dependent information, which the theory has declared meaningless.

By the universal property of the quotient (1), condition (4) yields a unique map $$p\colon S\longrightarrow T, \qquad \Pi=p\circ q,\tag{5}$$ and $p$ is surjective because $\Pi$ is. From this point onward, all source–readout comparisons are made between $S$ and $T$ by means of $p$. Gauge and coordinate redundancy have been removed before any question of non-equivalence is raised.

By (3) every fiber $F_{t}$ is nonempty, and the fibers partition $S$ into precisely the $\sim_{p}$-classes.

It is worth fixing the vocabulary that the rest of the paper will keep separate. Equality in $\Omega$ is identity of descriptions. Physical equivalence $\simeq_{\mathrm{phys}}$ is the theory’s declared redundancy. Readout equivalence $\sim_{p}$ is indistinguishability by public data. These are three different relations, and they are related by inclusion in one direction only: equality refines $\simeq_{\mathrm{phys}}$, and $\simeq_{\mathrm{phys}}$ (transported to $S$ as equality) refines $\sim_{p}$. The content of this paper lies in the second refinement being strict, and in what that strictness does and does not entail.