c, States, Wavefunctions, Entropy, and the Arrow Without Time
A rigorous map for advanced students: causal structure, reduced states, unitary evolution, coarse entropy, and the disciplined use of the reality quotient.
Thesis
It isn’t that time flows; closedness fails. The “arrow” you feel is statistical: coarse-grained entropy in a reduced state tends to rise as boundaries lose hold and information integrates.
1) Causal scaffold: what c actually fixes
The speed of light c sets light cones and admissible influences. A “state” is best thought of on a spacelike slice (a foliation). Any update, inference, or interaction must respect causality: no superluminal signalling; influences propagate within light cones (≤ c).
2) States, microstates, and entities (indices first)
- State s (the ellipse): boundary + labels + constraints (a hack-closed rulebook).
- Microstate ω: complete snapshot of everything inside s, all at once, under those labels (spatial/combinatorial, not a timeline).
- Entity X: a labeled degree of freedom (aperture) inside s.
- Distribution
: predictor’s beliefs on the microstates of s as seen through the aperture X.
3) Wavefunctions: unitary vs. readout
On an isolated state, the quantum state evolves unitarily; von Neumann entropy stays constant. Entropy rises when you reduce (trace out environment), measure, or coarse-grain—exactly the operations that define practical, hack-closed states.
For a subsystem, , correlations spread only within causal cones; the reduced entropy of A can increase as entanglement and mixing reach it (≤ c).
4) Entropy: where it lives and why the arrow appears
Entropy is a number on the distribution over microstates for a fixed state s:
Fine-grained (fully labeled, isolated) entropy is constant; coarse-grained entropy typically drifts upward because we blur labels, ignore couplings, and allow weak exchange. That drift—not a fluid called time—is the felt arrow.
5) Absolute information bounds that include c
Bekenstein bound (energy E, radius R):
Black-hole entropy (area law; ):
These tie entropy to energy, size, and fundamental constants—c limits how much “arrangement capacity” a finite state can admit.
6) Speed limits: how fast reconfiguration can happen
Quantum speed limits constrain minimal time to evolve between distinguishable states:
In relativistic settings, feasible energy densities and causal propagation (≤ c) further bound the rate at which reduced subsystems can mix, entangle, and thus increase coarse entropy.
7) Collapse, locality, and “no drama”
Operationally, “collapse” is sample + update. A realized microstate yields an observable via a map:
Knowledge updates elsewhere respect light cones; there is no superluminal signal. Observable consequences propagate ≤ c.
8) Reality quotient placed carefully
Reports are per-entity, per-state:
A is a local scalar from the realized microstate; summarizes predictor concentration and idea magnitude for the same (X,s). Because information and interaction reach X only within causal cones, the rate at which R (and S) can change is indirectly bounded by c and available energy/resources.
9) Classroom instantiation (Money Lab)
- Fix the state
: rubber-band hull + rules (participants, legal trades, pricing).
- A microstate
is the entire ledger at once; an entity X is one labeled degree of freedom.
- Operational “time”: count unbiased reconfiguration steps inside fixed s. The distribution over microstates mixes; coarse
tends not to decrease (fluctuations allowed). If you move the band or rules, you created a new state; reset the clock.
10) Red lines and quick repairs
- Do not assign entropy to “outside” or to the center; entropy lives on a specified state with specified labels.
- Entity ≠ microstate. Entity = window; microstate = everything (inside s) all at once.
- Arrow ≠ law; arrow = boundary condition + coarse-graining + mixing (causally bounded by ≤ c).
- “Collapse” = sample + update; mythos is welcome in prose, not in the math.
Board-ready lines
Microstate = everything, all at once; entity = a window.
Entropy lives on distributions.
c fixes the causal pace; the arrow appears because closedness fails.