Hyperquantum physics lecture notes

Some of Dr. Miranda Sierra's notes for 8.940 Intro to Hyperquantum Physics

Hyperquantum computation limits in a causal volume (P = NP)

‘t Hooft’s holographic principle tells us that all of the information about a region is encoded on a surface bounding that region and the maximum amount of information is given by Bekenstein’s bound (the number of bits ~ A / ℓp² where ℓp is the Planck length).

As time progresses, a causal region of radius R increases at the speed of light c such that the region and its surface area grow as

\(\begin{eqnarray} R & \sim & c \; t\\ A & \sim & (c \; t)^{2} \end{eqnarray}\)

where t is the elapsed time from some initial state.

The horizon states (the ~ ℓp² sized bits on the surface of the causal region) can only increase in discrete units of ~ ℓp². The time-evolution adds a qubit that is a superposition of a 1 or a 0 — effectively a series binary choices that map out the “many worlds” of every possible future of the initial causal region.

As this is always a choice between 1 and 0 for each additional ~ ℓp² sized area, the the tree of all possible histories is fundamentally a binary tree and the tree is always a balanced one. After time t, we have N ~ c² t² / ℓp² horizon states and Yehaya’s algorithm (HyperQuantum Binary Search, HQBS) was proven to be an o(log N) algorithm for finding a particular state.

Now consider a computer (Turing machine) verifying a solution to an NP problem in polynomial time of order k. If the solution is valid, the computer turns on a light¹; otherwise it does not. If the clock steps of the computer are ~ δt then at the time of the solution we have a region of size R and horizon area A

\(\begin{eqnarray} R & \sim & c \; \delta t \; n^{k}\\ A & \sim & (c \; \delta t)^{2} \; n^{2k}\\ N & \sim & 2^{n^{2k}} \end{eqnarray}\)

where N represents the size of the possible solution space.

Using Yehaya’s algorithm (HQBS) to search the horizon states for the one encoding the computer with the light on is then of complexity

\(\log N \sim n^{2k}\)

implying the existence of a deterministic algorithm that solves the original problem in polynomial time (i.e. complexity class P). Since this approach will work for any problems in NP (verifiable in polynomial time with order k), all of them are in P (solvable in polynomial time with order 2k)². Therefore P = NP.

Dirac equation and horizon entropy

Consider a fluctuation in the horizon entropy ΔS due to a particle of mass m moving Δx (4-vector). Via Bousso’s covariant entropy bound (Bousso 2003) we have ΔS ~ m Δx. Via the fluctuation theorem:

\(\frac{P(|\Delta S|)}{P(-|\Delta S|)} = e^{\Delta S}\)

Performing a Wick rotation and taking the logarithm

\(\log P(\Delta x) - \log P(-\Delta x) = - i m \Delta x\)

Taking the limit as Δx → 0

\(\frac{1}{P( x)} \frac{d}{d x} P( x) = - i m\)

So that

\((i \frac{d}{dx} - m) P(x) = 0\)

which is the Dirac equation in one dimension³.

1. In the technical proof, this is actually a heater such that the temperature of the system T is different in the possible worlds where the computer’s random input was a solution and is at a maximum for that quantum history which found it first due to fluctuations in the fundamental clock time ST (e.g. Susskind 2014). The heuristic derivation presented here still captures the main points.

2. Of course, this is not entirely a free lunch because the factors out front would be on the order of the area of the causal region measured in Planck units.

3. This heuristic derivation ignores spin. Taking into account the horizon Grassman number fields in the fluctuation theorem results in the probability function being a probability amplitude ψ(x) and introduces spinors. However, the relationship between fluctuations in horizon entropy and the probabilistic nature of quantum fields holds.