Observer-Equivariance: Objectivity as Invariance Under Perspective Change

Observer-Equivariance: Objectivity as Invariance Under Perspective Change

By Gustaf Ullman

Modern physics is saturated with perspective-laden structure: reference frames, gauges, coordinate charts, foliations, clock choices, operational access, and observer-dependent decompositions. Yet the usual philosophical picture of “objectivity” is often described as if objectivity were achieved by removing perspective. This is not how physics actually works.

The guiding idea of Observer-Equivariance (OE) is simple to state: physical objectivity is not the absence of perspective but invariance (or descent) under permitted changes of perspective. OE makes explicit what is tacit in ordinary practice: the formalism of physics already binds a perspective variable.

Contents

1. The core idea

In physics, objectivity is normally secured by showing that a quantity or statement is invariant (or appropriately covariant) under a specified class of transformations. OE treats this not as an incidental technique but as the organizing principle:

OE thesis. Objectivity is the content that survives admissible changes of perspective.

This immediately shifts the central question from “how do we remove the observer?” to: what is preserved under legitimate changes of perspective?

2. The span: phenomenology, perspectives, structure

OE uses a minimal ontological scaffold: a domain of perspectives and two projections. 

Phen  ←  O  →  S
        Φ     p
  • \(O\): the domain of observer-perspectives (primitive level)
  • \(S\): structural content (the world of physics)
  • \(\mathrm{Phen}\): phenomenal content (the world of experience)
  • \(p: O \to S\): projection to structure
  • \(\Phi: O \rightharpoonup \mathrm{Phen}\): partial phenomenal projection (not defined everywhere)

This is intended as a dual-aspect stance (in the broad Russellian-monist neighborhood): physics and phenomenology are not two substances, but two projections from a common ground. Importantly, \(\Phi\) is allowed to be partial. The framework explicitly permits that there may be regimes where structural continuity is meaningful while phenomenal continuity is not.

3. The Quinean diagnosis: the silent bound variable

OE is easy to misread as “adding consciousness to physics.” The intended starting point is the reverse: it begins by noticing what physics already presupposes.

Physics textbooks constantly use bound-variable clauses such as: “for any inertial frame,” “up to gauge,” “for any coordinate chart,” “for any observer with this operational access.”

Formally, these are instances of an implicit quantification over admissible perspectives and transformations. OE makes that quantification explicit and treats it as part of the ontology rather than as disposable prose.

4. Monad and time: traversal in \(O\)

OE separates (i) structural time internal to models in \(S\) from (ii) experienced time as directed actualization. A “monad” is represented as a trajectory through perspective space:

\[ \gamma : I \to O, \] where \(I\) is an ordered index (an interval/category capturing directedness).

The point is not to deny time in physics. It is to locate the directionality of lived time in the traversal \(\gamma\), rather than in a global external time parameter in \(S\). This makes it structurally natural that fundamental law might be timeless (constraint form) while experience remains directed.

5. Groupoids, symmetry, equivariance, descent

Mathematically, one models \(O\) as a groupoid (or category) whose morphisms represent admissible perspective changes. A symmetry group \(G\) (or symmetry groupoid) acts on \(O\). “Objective content” is what is invariant under this action, or what descends to an appropriate quotient.

A recurring technical motif is descent: objective content is what can be consistently identified across related perspectives. There is a substantive choice between:

  • global descent: demand consistency across all related perspectives;
  • overlap-based descent: demand consistency only on operational overlap (shared access).

6. Derived results (brief)

6.1 Uncertainty from translation symmetry in \(O\)

If a sector of \(O\) carries an abelian translation symmetry \(A\) represented unitarily on \(L^2(A)\), then Fourier duality and Plancherel–Parseval yield canonical uncertainty relations between conjugate generators. The novelty is not the inequality itself but the interpretive placement: uncertainty is read as a constraint emerging from symmetry in perspective space.

6.2 Born density as a conditional invariance result

The Born rule is treated conditionally: specify invariance constraints on event weights (locality, gauge invariance, invariance under unitary two-cell mixing) and show that these constraints single out the quadratic density \(|\psi|^2\) (under mild regularity conditions).

6.3 Timeless structural law from temporal perspective-invariance

If temporal standpoints are included among admissible perspectives, then invariance can push law away from “evolution in external time” and toward constraint form. Schematically, one is led to a Wheeler–DeWitt-like posture: \[ H\psi = 0, \] with time reappearing as relational correlation within \(S\), while experienced directedness is carried by \(\gamma\).

6.4 Quantum-correlated perspectives (QRF-motivated)

A natural enrichment replaces classical \(O\) by a quantizable/relational \(\widehat{O}\), where “changes of perspective” can be modeled as correspondences (spans) \[ o_1 \leftarrow c \rightarrow o_2. \] Classical \(O\) is then recovered as a decohered limit.

7. Surprises: what OE yields without building it in

A Lakatos-style evaluative criterion is whether a research program is progressive: once its core commitments are fixed, it generates consequences that were not inserted as ad hoc patches. In OE, three surprises were central.

7.1 Quantum reference frames: quantizable perspectives

OE was developed with classical perspectives. QRF entered as a stress test: if physics is quantum and observers are within physics, why should the “observer side” remain permanently classical? The outcome was not “patch the framework,” but “the framework strongly motivates quantizable perspectives.” In that sense, the move \(O \to \widehat{O}\) is a consequence of the core commitments.

7.2 Wheeler–DeWitt-style timelessness from temporal perspective-invariance

OE began as a thesis about perspective and objectivity, not as a thesis about timelessness. But once temporal standpoints are treated as perspectival variables, invariance pushes law toward constraint form rather than external-time evolution. This is not a post-hoc interpretation; it is what one gets by applying the core criterion to temporal standpoint-change.

7.3 Descent on overlap as a principled diagnosis in black-hole contexts

Applied to black holes, OE suggests that some paradoxes arise from demanding global consistency where only overlap-based consistency is operationally licensed. “Descent on overlap” appears as the natural operational reading of the framework’s objectivity criterion, not as a black-hole-specific invention.

8. OE as a bridge: mind, neuroscience, and physics

Debates about consciousness often stall because philosophers of mind (who take phenomenal facts seriously) and neuroscientists/physicists (who emphasize structural and functional description) talk past each other. OE proposes a different entry point: start from physics, not introspection.

The Quinean diagnosis (physics already binds a perspective variable) makes the span diagram a stable shared object:

  • \(\Phi\) takes phenomenal content seriously without turning it into an external intrusion into physics.
  • \(p\) targets structural content—exactly what physics and neuroscience study.
  • Neither projection is defined as ontologically prior; neither is reduced to the other.

This does not solve the “hard problem.” It changes the conversation: disagreements are relocated to explicit questions about the nature of \(O\), the status and domain of \(\Phi\), and the descent conditions connecting perspectives—rather than being fought as slogans (“reductionism vs dualism vs eliminativism”).

9. Black holes as a stress test

The AMPS firewall paradox is often framed as a clash between unitarity, locality, and equivalence principles under a global-consistency demand. OE suggests a different diagnosis: some paradoxes arise from demanding global identifications beyond operational overlap.

9.1 Descent on overlap

Consistency requirements between perspectives are imposed only where operational overlap exists (shared accessible correlations). Outside overlap, identifications are treated as gauge-like and are not required to match.

9.2 Breakdown of classical distinctness (QRF-motivated)

Combining overlap-based descent with the possibility of quantum-correlated perspectives suggests a further point: near a horizon, the assumption that “Alice” and “Bob” are classically distinct perspectives may break down. The question “what does Alice see separately from what Bob sees?” may become ill-posed in the same structural sense that “where is the particle?” is ill-posed for a superposed state relative to incompatible contexts.

9.3 Structural transport through a Planck regime (programmatic)

OE cleanly separates three continuities: structural continuity via \(p\), identity continuity via a functor \(q:\widehat{O}\to\mathrm{Id}\), and phenomenal continuity via \(\Phi\) (which may be undefined). This allows a programmatic possibility: structural content is transformed and re-encoded into late-time exterior structure (e.g. into correlations accessible in Hawking radiation), identity is preserved, while \(\Phi\) may be undefined through a Planck-scale locus.

10. Ethical Equivariance (brief extension)

Ethical Equivariance applies an invariance constraint to normative judgment: normative evaluations should be invariant under permutations of persons that preserve morally relevant structure. The Golden Rule is the two-role special case.

This is not a derivation of “ought” from physics. It is a formal constraint: mere indexicality (which person I am) is not morally relevant, unless the permutation changes morally relevant structure.

11. Limits and open problems

OE is a scaffold, not a completed physical theory. It does not (yet) provide direct empirical predictions, and it does not solve the combination problem in philosophy of mind. It also does not uniquely specify the nature of \(O\) beyond its projections.

Open problems that currently delimit the program:

  1. Specification of \(O\): minimal axioms on \(O\) (as groupoid/category) that recover familiar physical symmetry structure in \(S\).
  2. Descent criteria: formalize when global descent is licensed vs overlap-based descent (stack/sheaf-like conditions over operational domains).
  3. Quantized perspectives: explicit models of \(\widehat{O}\) and their relation to QRF formalisms; identification of a decohered classical limit.
  4. Born-rule assumptions: clarify regularity conditions and operational meaning of “two-cell mixing invariance.”
  5. Timeless structure: make precise the relation between constraint-form law and emergent relational time within \(S\).
  6. Partial \(\Phi\): principled criteria for when \(\Phi\) should be expected to fail to be defined, and how this interacts with identity continuity.

Closing remark. OE is an attempt to make explicit what physics already relies on: a bound perspective variable and an invariance/descent criterion for objectivity. The payoff is not a new set of slogans, but a cleaner way to state what counts as objective structure and to map where additional assumptions are required—especially in regimes where global identifications are operationally suspect.

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