Probability Theory on Token-Event Graphs

[maxitg]

The project goal

Path integral approach to quantum theory can be thought of as probability theory on histories. Token-event graphs might be a way to describe histories. However, we don't have any tools for computing probabilities on them, quantum mechanical or otherwise.

The goal for this project will be to develop such tools. Making the probabilities work the same way as they do in quantum mechanics might be difficult, but even developing any approach for computing probabilities, even if classical, would be helpful to advance our understanding.

Expected deliverables:

• Functions (properties) computing relative transition probabilities from one state to a group of other states.
• Research note describing the math behind the property and relation to physics, if any.

Motivations

To understand the potential connection to quantum theory, we need to understand token deduplication (as paths in path integral formulation can merge). This is part of the effort to understand token deduplication #349 (see the notebook attached to the issue). The other part (#649) is understanding token deduplication for the hypergraph substitution system (which allows new atoms to be created).

External prerequisites

• Familiarity with path integrals.
• Familiarity with graphs and graph algorithms.
• Experience with C++ or Wolfram Language.

SetReplace prerequisites

• Basic understanding of token-event graphs.
• Understanding of one of the implemented systems (such as MultisetSubstitutionSystem).

Difficulty

Developing any approach to probabilities would likely be a moderate difficulty project. The most challenging part would be designing algorithms on token-event graphs for computing probabilities which should not be too difficult for someone with algorithm design experience.

Making probabilities work the same as in quantum theory, however, is a very hard project, and we do not know if it's going to be possible.

Potential instructors

@maxitg

Existing work

• Research note describing local multiway systems and token-event graphs (called expression-event graphs in the note). There are some ideas on token deduplication in the end.
• Notebook describing a potential approach to deduplication. The probability theory will likely be based on graphs such as generated by TimelessTEG from this notebook.

Required future SetReplace features

TimelessTEG will need to be integrated into SetReplace, probably as an argument value for GenerateMultihistory.