Path summation offers a flexible general approach to quantum theory, including quantum gravity. In the latter setting, summation is performed over a space of evolutionary pathways in a history configuration space. Discrete causal histories called
acyclic directed sets offer certain advantages over similar models appearing in the literature, such as causal sets. Path summation defined in terms of these histories enables derivation of discrete Schrödinger-type equations describing quantum spacetime dynamics for any suitable choice of algebraic quantities associated with each evolutionary pathway. These quantities, called
phases, collectively define a
phase map from the space of evolutionary pathways to a target object, such as the unit circle
, or an analogue such as
or
. This paper explores the problem of identifying suitable phase maps for discrete quantum gravity, focusing on a class of
-valued maps defined in terms of “structural increments” of histories, called
terminal states. Invariants such as
state automorphism groups determine multiplicities of states, and induce families of natural entropy functions. A phase map defined in terms of such a function is called an
entropic phase map. The associated dynamical law may be viewed as an abstract combination of Schrödinger’s equation and the second law of thermodynamics.
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