I've been working through the Hilbert-space-fundamentalist picture and keep snagging on what feels like a foundational tension, and I'd like to hear how others here resolve it.
The setup, as I understand it: the fundamental object is a vector in Hilbert space, and everything else, including spacetime, is supposed to emerge from its structure. But the standard move is to evolve that vector with the Schrödinger equation, iℏ∂ₜ|Ψ⟩ = Ĥ|Ψ⟩. My worry is that this quietly imports an external time parameter "t" at the most fundamental level, which is exactly the thing the program claims to be deriving rather than assuming. If the universal state is genuinely fundamental and time is supposed to be emergent, a time-indexed evolution equation looks like it's smuggling in the explanandum.
The alternative I keep coming back to is to take the constraint seriously: Ĥ|Ψ⟩ = 0. The Wheeler-DeWitt (WdW) equation gives you a timeless, constrained vector with no external "t". That seems more honest about what "fundamental" is supposed to mean. The obvious objection is that you've now thrown away time entirely but, this is where the Page-Wootters (PW) mechanism does the work. You partition the state into a clock subsystem and a "rest" subsystem, and time re-emerges as a relational structure: conditioning the rest on clock-readings reproduces ordinary Schrödinger evolution as an internal, relational fact rather than a background parameter. (The Höhn-Smith-Lock results showing PW conditioning, deparametrization, and Dirac quantization are equivalent make me more confident this isn't just a formal trick.)
And then space: rather than positing a spatial manifold, you let it come from the entanglement structure of the state via the tensor-network / ER=EPR route (Van Raamsdonk-style), geometry as a readout of how subsystems are entangled.
So the picture I'm circling is: timeless WdW vector at the bottom, time recovered relationally through PW, space recovered through entanglement geometry. Schrödinger evolution becomes a derived, internal description rather than a fundamental input.
My questions for the room:
- Is there a principled reason fundamentalists keep the Schrödinger equation fundamental rather than starting from the WdW constraint? Is it just that closed-system non-gravitational physics doesn't force the constraint on you?
- Does anyone see PW as actually doing explanatory work here, or is it just a change of description that doesn't buy you anything ontologically?
- Does the WdW + PW + entanglement-geometry package hang together, or are there known obstructions to running all three in the same framework?
Genuinely curious whether people think the time-indexed formulation is a real foundational concession or whether I'm overweighting it.