9 Elements and observables

Vectornaut edited this page 2025-02-25 21:32:12 +00:00

Table of Contents

• Separation
• General thoughts
• Contrasts
• Unification
• Questions related to variables and linked constraints

Separation

General thoughts

• In physics, there's a long and successful tradition of thinking about states and observables as distinct and dual to each other.
  • Specifying consistent values for enough observables picks out a state.
  • Elements correspond to state space building blocks.
• Elements don't interact directly with each other, and constraints don't interact directly with each other either. In other words, the graph of direct interactions among elements and constraints is bipartite.

Contrasts

Elements	Observables
Naturally seen as geometric objects	Naturally seen as quantities
It always makes sense to ask whether a point lies on an element	It rarely makes sense to ask whether a point lies on an observable
No notion of valence; can interact with any number of constraints	A valence can be usefully and naturally assigned to many key observables
An assembly consisting entirely of elements seems meaningful and potentially useful	An assembly consisting entirely of observables seems hard to interpret

To do: Expand and clarify list of contrasts.

Unification

In some physics settings, states can at least resemble observables.

• A random variable is a function on a measurable space, while a probability distribution is a measure. When the measurable space is finite, however, you can express every probability distribution as a function too, which multiplies the counting measure.
• In a finite-dimensional quantum system, states can be expressed as density operators.

Can we realize constraints as geometric objects? For example:

• A real angle constraint might be seen as a pair of infinitesimal ribbons crossing at a fixed angle along a circle
• An imaginary angle constraint might be seen as a circle that passes perpendicularly through a pair of spheres, with a fixed cross ratio.

Maybe being subject to a constraint can then be seen as a kind of incidence.

To do: Explore ideas for unifying elements and observables in spite of the contrasts described above.

Questions related to variables and linked constraints

• Could a Scalar be an element? I.e., a "free-floating" real number?
  • Scalars differ from elements like spheres and points in fundamental ways.
    • They don't correspond to sets of points.
    • It's hard to imagine showing them in the 3D display view.
  • Geometric algebra suggests some potential similarities.
    • Since Clifford algebras are unital, they do each contain a field of scalars.
    • If we eventually have reason to do algebraic operations on elements like spheres and points, their algebraic structure trait could be shared with scalars.
• What is/should be the mechanism for making numerous angles identical be (say)? By experience, using a bunch of equality constraints and relying on transitivity becomes cumbersome and a bit hard to "see what's going on". Those interfaces that allow one to have "named quantities" and then use those "named quantities" as the values of other parameters have felt more understandable, and easier to manipulate.
  • From a user's perspective, I like the idea of promoting regulator set points from real numbers to expressions that can include variables.
  • If every set point is an affine-linear combination of variables, I think we can enforce the resulting relations between set points using basically the same mechanism that we currently use to freeze entries of representation vectors.
• What is the status of a "second-order" constraint, e.g "these two observables are equal"? Are they just additional Constraints, even though their slots are other Constraints? If we use the "filter Constraints by an Element" metaphor, and we select a plane P and it shows us the three angle Constraints it is involved in and one of those angle Constraints \theta is further constrained to be equal to another angle \rho that does not involve P, does that equality constraint show up? Does \rho itself show up?
  • If we promote set points to expressions that can involve variables, then the variable \rho will appear in the expression for the set point \theta.