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).