Is Schrödinger's Cat
Object-Oriented? - Chapter 7




VII. Multimedia, Parallel Universes
And
Path Integrals.

In some OO design techniques[1-3] it is common to utilize state transition diagrams. These diagrams show the state space of an object, display events causing a transition from a state to another, and characterize the new state. Hypermedia is an interactive media in which the user can affect the state of a system by some action such as clicking the mouse on a selected object. In this section we briefly introduce path integrals, an alternative representation of quantum mechanics, and suggest some analogies between hypermedia and path integrals.

A very useful mathematical representation of quantum mechanics is via path integrals[6,8] Suppose one creates an object, say, myVector3D that belong to the class TVector3D, at time t1 in some state | s1 >, where s1 consists of the variables that define the state (say, the three Cartesian coordinates of the myVector3D object). As previously discussed, the state vector has, associated with it, an abstract vector space where all states can be represented. In our example, the state space is just a (usual Euclidean) three-dimensional space. As time evolves, the state vector changes and this can be pictured as the motion of the point representing the current vector's state in the state space.

In QP, due to the "probabilistic" nature of the state vectors [i.e., quantum state vectors only give probabilities for the object, when measured, to be in given states], objects created at (s1, t1) and annihilated at (s2, t2) can make the transition from their initial to their final state in many (sometimes, infinite) possible ways (paths). See Fig. 5. In contrast, in classical mechanics there is just a single possible path (that which minimizes certain function called the action related to the Hamiltonian of the system). In QP all paths are possible but each one is weighted by its action. In other words, the action on a given path is a measure of the probability of that path. Using the path integral formulation of QP, one obtains identical results as using the alternative operator representation sketched in previous sections (i.e., the two representation are equivalent).

Figure 5. Given two events s1 and s2 at times t1 and t2, such as the position of a moving ball at two distinct times, in classical mechanics there is just a single path (i.e., trajectory) which could join both points. This is the path that minimizes the action. In contrast, in QP all paths are allowed with some probability (i.e., each path is weighted by the path's action). In hypermedia, say, an interacting book, we could move from the beginning to the end of the story in many ways. For example, in each page, by clicking on some objects in some order we "choose" some trajectory in the "story" space.

One controversial interpretation of QP (many-universe interpretation)[9] says if something physically can happen, it does, in some universe. In other words, each of the very many paths in the state space do occur in some universe. Then, physical reality would consist of a collection of universes (each running all possible scenarios of a play). In fact, this many-world model is approximated in hypermedia. In hypermedia[10], there is an information/event (state) space for a story one wishes to tell (e.g., an interacting book). The space can have any number of dimensions. Actual events are represented as points in the space. Navigable paths are curves in this space linking various events together. These curves intersect at specific events, and the user can, in principle, explore the full space. In hypermedia (and virtual reality) navigation in the event space is usually manual. Yet, provided with a model (Hamiltonian), a simulation engine either could select a single path in the information space (classical physics), or assign probabilities to each path (quantum model).




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