Dirac’s Algorithm for Constrained Systems from the perspective of Order Theory

Abstract Dirac based his theory of constrained systems on Linear Algebra foundations. It is a brackets-algebraic consistency procedure with multiple outcomes, including new constraints dropping out and redeclaring brackets becoming necessary (Dirac brackets). This procedure has not yet been edited, however, to caution about and remove scaffolding structures that turn out to not in general be brackets-algebraically consistent. We perform this task here. Our main innovation is moreover demonstration that substantial progress can be made from placing Dirac’s Algorithm on Linear Algebra and Order Theory foundations. For chains, lattices, posets, and digraphs abound therein For instance, in the simpler version of the algorithm, its iterations form a chain, of which its Dirac brackets updating steps are a subchain. Its consistent algebraic structures, meaningful notions of weak equality, of appended Hamiltonians and of observables form bounded lattices. Many key notions – such as Dirac’s extended Hamiltonian or Dirac observables – are identified as extrema of these lattices, cementing their permanence. Others are however revealed to be but simplest examples of middles. By this, e.g. Kuchar observables are in general to be replaced by a poset of algebraicially-consistent middling A-observables.

In the harder – path dependent – (previously called bifurcating or branching) version of the algorithm, moreover,
iterations and Dirac brackets types become digraph-valued. What previously was a lattice of consistent constraint subalgebraic structures now becomes a competing lattice, described overall by a semi-lattice, with weak equalities, appended Hamiltonians and observables following suit. Order Theory conceptualization thus remains both lucid and under control within this harder case. Such Order-Theoretic considerations furthermore transcend to extended variants and to Temporal Relationalism implementing variants. And to the Generalized Lie Algorithm: a vast generalization of scope in which to apply Dirac’s insights from constrained dynamical systems to wherever Lie Theory is applicable.

Article dedicated to the memory of Niall o’Murchadha (see the Acknowledgments section if visiting here in this regard).

Based on material presented at the 2021 Summer school on Combinatorial and Topological Applications to Fundamental Physics.

13/10/2025. Proposition 2 is true, but the proof given above is nonsense.

1. The word “commutative” went missing from the working. This was pointed out in October 2025 by K. Everard. As in the result being alluded to is that the product of 2 commutative projectors is itself a projector.
2. But I then remembered working out some years ago that the Dirac projectors that can be abstracted from each Dirac bracket are not in general commutative. So the domain of validity of the proof by projectors does not include the Dirac bracket.
3. Now as to an alternative proof making no mention of projectors. By Sniatycki 1974, Dirac brackets are Poisson. One can then just as well form a Dirac bracket from ‘a Poisson bracket that so happens to be the Dirac bracket obtained from some previous Poisson bracket’!

As an advanced notification, I will have some time for projectors (Dirac and more basic) in Lent 2026. I will tidy up this preprint then. Either add material to my Dirac projector note or write a second note. Write an extra chapter for my Linear Mathematics book. And co-write a separate note with K. Everard on some finer details of commuting projectors.