You can define quantum mechanics on a cantor set, but in order for it to be nontrivial, it needs to be a Levy quantum mechanics, not a Gaussian quantum mechanics, in that it will be the quantum analog of a Levy process, not a Brownian motion, as the ordinary Schrodinger equation is.
To define Schrodinger quantum mechanics, you take the continuum limit of a nearest neighbor amplitude random walk. To do this, I will first remind you of the standard imaginary time map between random walks and quantum mechanical systems. When you have a stochastic process on a discrete space in discrete time, you have a transition operator:
$$ \rho_j(t+1) = \sum_i \rho_i(t) K_{i\rightarrow j} $$
where $K_{i\rightarrow j} = K_{ij} $ is a stochastic matrix:
$$ \sum_j K_{ij} = 1 $$
These stochastic matrices generically have a stationary distribution, which I will call $\rho^0$. I will assume that this stationary distribution obeys detailed balance, or in mathemtical jargon, that it is the "reversing measure" for K:
$$ \rho^0_i K_{ij} = \rho^0_j K_{ji}$$
This says that the transitions between states i and j balance in equilibrium separately from any other transitions. The stationary distribution for random walk on a graph obeys detailed balance and it is ${1\over D(i)}$ where D is the degree of the vertex.
When you take the continuous time limit, you make K equal to the identity plus an infinitesimal transition rate, and the stochastic equation becomes:
$$ {d\over dt} \rho_j = \sum_i \rho_i R_{ij} $$
And you still have a stationary distribution $\rho0$ for the continuous time case. Now you can define a symmetric H from the continuous time random process:
$$ H_{ij} = {1\over \sqrt{\rho^0_i}} R_{ij} \sqrt{\rho^0_j} $$
and the detailed balance condition gives you symmetry of H. You then can define the imaginary time continuation as a standard quantum mechanical unitary time evolution, generated by this Hamiltonian. This is the most abstract form of Wick continuation.
If you do this process on a random walk whose limit is a Brownian motion, you get ordinary Schrodinger quantum mechanics. If you do the same process on a random walk which takes steps of size s according to a distribution:
$$ P(s) \propto {1\over s^{1+\alpha}}$$
Where $0<\alpha<2$, you get Levy quantum mechanics.
So to define quantum mechanics on a Cantor set, all you need is an appropriate stochastic motion. The ordinary Brownian motion fails to have a limit, it just stays still on the Cantor set--- it ends up fully localized. But the Levy process generalizes just fine.
The Cantor set can be defined as all base 3 numbers with digits which are all 0 or 2. A discrete approximation is truncating this at N digits. Define a random walk on this graph by toggling a digit between 0 to 2 at digit position k with a rate which goes as:
$$ e^{-ak} $$
where $a>0$. If you take the limit of continuous time, timesteps of size $\epsilon$, and $a= {A\over \epsilon}$, you get a hop which is a power law in size (since it is an exponetial distribution on exponentially shrinking sizes, and this is a powerlaw in the size), and the continuum limit is Levy quantum mechanics restricted to the Cantor set.
This is related to the question of localizing Dirac Fermions, since the |k| dispersion relation is Levy. You don't localize Levy particles with a local potential, unlike normal Schrodinger particles. This was the subject of this question: How to localize the massless fermions in Dirac materials? .
This post imported from StackExchange Physics at 2015-03-30 13:26 (UTC), posted by SE-user Ron Maimon
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