Consider the real line $\mathbb{R}$ and the one-dimensional lattice $L = \{nh\}$ on the line with $h> 0$ and $n \in \mathbb{Z}$. To every element $q \in Q = \{L,R\}^\star$ we can associate a unique point of $L$ by starting at the origin $0$ and from left to right interpreting $q$ as a series of instructions on how to move a pointer. For example $RLRRRL$ would correspond to $2h$ and $LLR$ to $-h$. Thus we obtain a map
$s : Q \rightarrow L$
Hence the "space" $L$ is obtained by forgetting the history, the generation process of each element of $Q$ and only remembering the operational result, the "point". In this case the point is determined by the difference between the number of occurrences of $R$ and $L$ in the element $q$. Note that $s^{-1}(l)$ for $l \in L$ is infinite and can be thought of as containing "cycles" of arbitrary large amplitude. We can define the "energy" $E(l)$ of a point $l \in L$ relative to a $q$ such that $s(q) = l$ as the number of times $l$ is passed by $q$ until settling at $l$.
The same construction can be carried over to $\mathbb{R}^m$ and $m$-dimensional lattices. There is also an affine version in which $Q$ is given a groupoid structure. As $h \rightarrow 0$ $Q$ becomes the path groupoid and $L$ becomes ordinary Euclidean space. However what is important is that in the discrete case space is constructed out of an algebraic structure (a free semigroup).
Given a subset $\Psi \subset Q$ consider $s(\Psi) \subset L$. It makes some sense to associate probabilities to $p \in s(\Psi)$ based on $E(p)$. There are also generalization of momentum and the wave function that could be made. If a $p$ has near 1 probability then this means that most of the elements in $\Psi$ cycle past and rest finally at $p$, possibility with arbitrarily large orbits - hence with high energy and "momentum".
Another interesting approach: Bassi et al. (2013), Models of wave-function collapse, underlying theories, and experimental tests.
https://arxiv.org/abs/1204.4325
No comments:
Post a Comment