# Entropic Phase Maps in Discrete Quantum Gravity

## Abstract

**:**

## 1. Introduction

#### 1.1. Path Summation in Quantum Gravity

#### 1.2. Path Summation Rudiments

#### 1.3. Effects of Gravity

#### 1.4. Motivation for Entropic Phase Maps

## 2. Discrete Causal Theory

#### 2.1. Causal Metric Hypothesis

#### 2.2. Classical Theory

**Definition**

**1.**

**directed set**$(D,\prec )$ is a set D equipped with a binary relation ≺. A

**morphism**from a directed set $(D,\prec )$ to a directed set $({D}^{\prime},{\prec}^{\prime})$ is a set map $f:D\to {D}^{\prime}$ such that $f\left(x\right){\prec}^{\prime}f\left(y\right)$ whenever $x\prec y$. The

**category of directed sets**$\mathcal{D}$ is the category whose objects are directed sets and whose morphisms are morphisms of directed sets. A

**subobject**of a directed set $(D,\prec )$ is a directed set $({D}^{\prime},{\prec}^{\prime})$, where ${D}^{\prime}$ is a subset of D, and where ${\prec}^{\prime}$ is a subset of ≺ consisting of relations between pairs of elements of ${D}^{\prime}$. The

**causal dual**of a directed set $(D,\prec )$ is the directed set $(D,{\prec}^{*})$, where $x{\prec}^{*}y$ if and only if $y\prec x$.

**Definition**

**2.**

**multidirected set**$(M,R,i,t)$ consists of a set of elements M, a set of relations R, and

**initial**and

**terminal element maps**$i:R\to M$ and $t:R\to M$. A

**morphism**from a multidirected set $(M,R,i,t)$ to a multidirected set $({M}^{\prime},{R}^{\prime},{i}^{\prime},{t}^{\prime})$ consists of a

**map of elements**${f}_{\mathrm{ELT}}:M\to {M}^{\prime}$ and a

**map of relations**${f}_{\mathrm{REL}}:R\to {R}^{\prime}$, such that ${f}_{\mathrm{ELT}}\left(i\left(r\right)\right)={i}^{\prime}\left({f}_{\mathrm{REL}}\left(r\right)\right)$ and ${f}_{\mathrm{ELT}}\left(t\left(r\right)\right)={t}^{\prime}\left({f}_{\mathrm{REL}}\left(r\right)\right)$ for each r in R. The

**category of multidirected sets**$\mathcal{M}$ is the category whose objects are multidirected sets and whose morphisms are morphisms of multidirected sets. A

**subobject**of a multidirected set $(M,R,i,t)$ is a multidirected set $({M}^{\prime},{R}^{\prime},{i}^{\prime}{,}^{\prime}t)$, where ${M}^{\prime}$ and ${R}^{\prime}$ are subsets of M and R, respectively, and where ${i}^{\prime}$ and ${t}^{\prime}$ are the restrictions of i and t to ${R}^{\prime}$. The

**causal dual**of a multidirected set $(M,R,i,t)$ is the multidirected set $(M,R,t,i)$.

**Definition**

**3.**

**chain**in a multidirected set $(M,R,i,t)$ is a sequence of relations $...,{r}_{k},{r}_{k+1},...$ such that $t\left({r}_{k}\right)=i\left({r}_{k+1}\right)$. The

**past**of an element x of $(M,R,i,t)$ is the set of all elements w in M such that there exists a chain ${r}_{0},...,{r}_{N}$ with $i\left({r}_{0}\right)=w$ and $t\left({r}_{N}\right)=x$. The

**future**of x is the set of all elements y in M such that there exists a chain ${r}_{0},...,{r}_{N}$ with $i\left({r}_{0}\right)=x$ and $t\left({r}_{N}\right)=y$. An

**antichain**in $(M,R,i,t)$ is a subset σ of M with no chain connecting any pair of its elements, distinct or otherwise. The

**past relation set**${R}^{-}\left(x\right)$ of an element x in M is the set of all relations r in R such that $t\left(r\right)=x$. The

**future relation set**${R}^{+}\left(x\right)$ of x is the set of all relations r in R such that $i\left(r\right)=x$. The

**relation set**$R\left(x\right)$ of x is the union ${R}^{-}\left(x\right)\cup {R}^{+}\left(x\right)$.

**Definition**

**4.**

- 1.
**Binary axiom:**Classical spacetime may be modeled as a directed set $D=(D,\prec )$, whose elements represent events, and whose relations represent causal relationships between pairs of events.- 2.
**Generalized measure axiom:**D is equipped with a set function μ from the power set $\mathcal{P}\left(D\right)$ of D to the extended real numbers $\mathbb{R}\cup \{\infty \}$, which assigns finite positive values to nonempty finite subsets of D, and infinite values to infinite subsets of D.- 3.
**Countability:**D is countable.- 4.
**Star finiteness:**For every element x of D, the star $St\left(x\right)=\left\{x\right\}\cup R\left(x\right)$ of x is finite.- 5.
**Acyclicity:**D possesses no cycles, i.e., sequences of relations ${x}_{0}\prec ...\prec {x}_{N}$ with ${x}_{0}={x}_{N}$.

#### 2.3. Relation Space

**Definition**

**5.**

- 1.
- The
**induced relation**≺ on R is defined by setting ${r}_{0}\prec {r}_{1}$ if and only if $t\left({r}_{0}\right)=i\left({r}_{1}\right)$. - 2.
- The directed set $\mathcal{R}\left(M\right)=(R,\prec )$ is called the
**relation space over**M.

**Theorem**

**6.**

**Proof.**

**Theorem**

**7.**

**Proof.**

#### 2.4. Quantum Theory

**Definition**

**8.**

**transition**in the category $\mathcal{D}$ of directed sets is a monomorphism $\tau :D\to {D}^{\prime}$, embedding its

**source**D into its

**target**, ${D}^{\prime}$, as a proper, full, originary subobject. Here, “proper" means that $\tau \left(D\right)$ has nontrivial complement in ${D}^{\prime}$, “full" means that $\tau \left(x\right)\prec \tau \left(y\right)$ in ${D}^{\prime}$ if and only if $x\prec y$ in D, and “originary" means that the isomorphic image $\tau \left(D\right)$ of D in ${D}^{\prime}$ contains its own past.

**Definition**

**9.**

**proper, full, originary co-relative history**$h:{D}_{i}\Rightarrow {D}_{t}$ is an equivalence class of transitions $\tau :{D}_{i}\to {D}_{t}$, where two transitions τ and ${\tau}^{\prime}$ are equivalent if and only if there exists an automorphism β of ${D}_{t}$ mapping $\tau \left({D}_{i}\right)$ onto ${\tau}^{\prime}\left({D}_{i}\right)$. The common source ${D}_{i}$ of the transitions representing h is called the

**cobase**of h, and the common target ${D}_{t}$ of these transitions is called the

**target**of h.

**Definition**

**10.**

**kinematic scheme**is a pair $\mathbb{S}=(\mathcal{K},\mathscr{H})$, where $\mathcal{K}$ is a class of directed sets, and $\mathscr{H}$ is a class of co-relative histories between pairs of members of $\mathcal{K}$ satisfying the following properties:

- 1.
**Accessibility**: If D is in $\mathcal{K}$, then there exists a sequence of co-relative histories in $\mathscr{H}$ terminating at D.- 2.
**Hereditary property**: $\mathcal{K}$ is closed under the formation of proper, full, originary subobjects.

## 3. Entropy and the Second Law of Thermodynamics

#### 3.1. Entropy

**Definition**

**11.**

**entropy system**$(S,\mathsf{\Pi},\mu )$ consists of a set S, a set $\mathsf{\Pi}:={\left\{{P}^{\alpha}\right\}}_{\alpha \in A}$ of partitions ${P}^{\alpha}$ of S for some index set A, strictly partially ordered by refinement, and a family μ of measures ${\mu}^{\alpha \beta}$ on the quotient sets ${S}^{\beta}$, one for each relation ${P}^{\alpha}\prec {P}^{\beta}$ in Π. Each such relation induces an

**entropy quadruple**$(S,{P}^{\alpha},{P}^{\beta},{\mu}^{\alpha \beta})$. The

**entropy**of a member V of ${P}^{\alpha}$ is ${e}^{\alpha \beta}\left(V\right):=log{\mu}^{\alpha \beta}\left({V}^{\beta}\right)$, where ${V}^{\beta}\subset {S}^{\beta}$ is the image of V under the quotient map $S\to {S}^{\beta}$, and where $log\infty $ is understood to mean ∞.

#### 3.2. The Second Law

#### 3.3. Discrete Causal State Spaces

**Definition**

**12.**

**terminal state**of τ. If ${\mathcal{R}}^{n}\left({\mathrm{\Delta}}^{\tau}\right)$ is a nonempty antichain, then the

**order**$\mathrm{ord}\left({\mathrm{\Delta}}^{\tau}\right)$ of ${\mathrm{\Delta}}^{\tau}$ is n.

**Definition**

**13.**

- 1.
- The
**nth-degree terminal state**${T}^{n}\left(D\right)$ of D is the subobject of D consisting of all elements connected to a maximal element of D by a chain of length at most n, together with all relations in such chains. - 2.
- The
**nth-degree initial state**${I}^{n}\left(D\right)$ of D is the subobject of D constructed by deleting all non-minimal elements of ${T}^{n}\left(D\right)$ from D, together with all relations in D terminating at such elements. - 3.
- The
**nth-degree transition**${\tau}_{D}^{n}:{I}^{n}\left(D\right)\to D$ associated with D is the inclusion map ${I}^{n}\left(D\right)\to D$.

**Definition**

**14.**

**nth-order state space**${\mathbb{D}}^{n}$ is the set of all isomorphism classes of countable star finite acyclic directed sets Δ such that ${\mathcal{R}}^{n}\left(\mathrm{\Delta}\right)$ is a nonempty antichain. The

**finite-order state space**$\mathbb{D}$ is the disjoint union ${\coprod}_{n=0}^{\infty}{\mathbb{D}}^{n}$, and the

**(total, countable, acyclic) state space**$\overline{\mathbb{D}}$ is the set of all isomorphism classes of countable acyclic directed sets, which may be viewed as limits of sequences in $\mathbb{D}$.

**Definition**

**15.**

- 1.
- The
**nth-order superset state space**${\mathbb{D}}_{\mathrm{SUP}}^{n}$ is the set of full, originary co-relative histories $\eta :{\mathrm{\Delta}}^{*}\Rightarrow {{\mathrm{\Delta}}^{\prime}}^{*}$. where Δ is a member of ${\mathbb{D}}^{n}$ and ${\mathrm{\Delta}}^{\prime}$ is a member of $\mathbb{D}$. Its elements are called**superset microstates.**The corresponding**finite-order superset state space**${\mathbb{D}}_{\mathrm{SUP}}$ and**(total, countable, acyclic) superset state space**$\overline{{\mathbb{D}}_{\mathrm{SUP}}}$ are defined in the obvious ways. - 2.
- The
**nth-order labeled state space**${\mathbb{D}}_{LAB}^{n}$ is the set of complete labelings of members Δ of ${\mathbb{D}}^{n}$, where two labelings of Δ are considered to be equivalent if they are related by an element of $\mathrm{Aut}\left(\mathrm{\Delta}\right)$. Its elements are called**labeled microstates.**The corresponding**finite-order labeled state space**${\mathbb{D}}_{LAB}$ and**(total, countable, acyclic) labeled state space**$\overline{{\mathbb{D}}_{LAB}}$ are defined in the obvious ways. - 3.
- The
**nth-order symmetry state space**${\mathbb{D}}_{\mathrm{SYM}}^{n}$ is the set of partial labelings of members Δ of ${\mathbb{D}}^{n}$ induced by applying elements of $\mathrm{Aut}\left(\mathrm{\Delta}\right)$ to arbitrary initial labelings of the subsets $\tilde{\mathrm{\Delta}}$ of Δ not fixed by $\mathrm{Aut}\left(\mathrm{\Delta}\right)$. Its elements are called**symmetry microstates.**The corresponding**finite-order symmetry state space**${\mathbb{D}}_{\mathrm{SYM}}$ and**(total, countable, acyclic) symmetry state space**$\overline{{\mathbb{D}}_{\mathrm{SYM}}}$ are defined in the obvious ways.

**Definition**

**16.**

- 1.
- Add or delete an isolated element.
- 2.
- Add or delete a relation between two elements.

**absolute distance**$d(M,{M}^{\prime})$ between M and ${M}^{\prime}$ is the minimal number of elementary operations required to convert M to ${M}^{\prime}$, if this number is finite. Otherwise, $d(M,{M}^{\prime})=\infty $.

**Definition**

**17.**

- 1.
- The
**directed distance**${d}_{\mathbb{S},D}({T}^{n}\left(D\right),\mathrm{\Delta})$ between ${T}^{n}\left(D\right)$ and Δ in $\mathbb{S}$ with respect to D is the minimal length of chains $x\left(D\right)\prec x\left({D}_{1}\right)\prec ...\prec x\left({D}_{N}\right)$ in $\mathcal{M}\left(\mathbb{S}\right)$, where ${T}^{n}\left({D}_{N}\right)=\mathrm{\Delta}$. - 2.
- The
**undirected distance**${\ell}_{\mathbb{S},D}({T}^{n}\left(D\right),\mathrm{\Delta})$ between ${T}^{n}\left(D\right)$ and Δ in $\mathbb{S}$ with respect to D is the minimal length of undirected paths $x\left(D\right),x\left({D}_{1}\right),...,x\left({D}_{N}\right)$ in $\mathcal{M}\left(\mathbb{S}\right)$ with initial element $x\left(D\right)$ and terminal element $x\left({D}_{N}\right)$, where ${T}^{n}\left({D}_{N}\right)=\mathrm{\Delta}$.

#### 3.4. Multiplicities and Entropies

**Definition**

**18.**

**superset multiplicity**${\mu}_{\mathrm{SUP}}^{n}\left(\mathrm{\Delta}\right)$ of a finite state Δ is the number of co-relative histories $\eta :{\mathrm{\Delta}}^{*}\Rightarrow {{\mathrm{\Delta}}^{\prime}}^{*}$, where the complement of the image of ${\mathrm{\Delta}}^{*}$ under any transition representing η has cardinality n. The nth

**superset entropy**${e}_{\mathrm{SUP}}^{n}\left(\mathrm{\Delta}\right)$ of Δ is $log{\mu}_{\mathrm{SUP}}^{n}\left(\mathrm{\Delta}\right)$.

**Definition**

**19.**

**lexicographic superset entropy system**.

**Definition**

**20.**

**labeled multiplicity**${\mu}_{\mathrm{LAB}}\left(\mathrm{\Delta}\right)$ of a state Δ of cardinality K is $K!/\left|\mathrm{Aut}\right(\mathrm{\Delta}\left)\right|$. The

**labeled entropy**${e}_{\mathrm{LAB}}\left(\mathrm{\Delta}\right)$ of Δ is $log{\mu}_{\mathrm{LAB}}\left(\mathrm{\Delta}\right)=logK!-log\left|\mathrm{Aut}\left(\mathrm{\Delta}\right)\right|$.

**Definition**

**21.**

**labeled entropy system**.

**Definition**

**22.**

**symmetry multiplicity**${\mu}_{\mathrm{SYM}}\left(\mathrm{\Delta}\right)$ of a finite state Δ is $\left|\mathrm{Aut}\right(\mathrm{\Delta}\left)\right|$. The

**symmetry entropy**${e}_{\mathrm{SYM}}\left(\mathrm{\Delta}\right)$ of Δ is $log{\mu}_{\mathrm{SYM}}\left(\mathrm{\Delta}\right)=log\left|\mathrm{Aut}\left(\mathrm{\Delta}\right)\right|$.

**Definition**

**23.**

**symmetry entropy system**.

**Definition**

**24.**

- 1.
- The
**state automorphism group**of τ is $\mathrm{Aut}\left({\mathrm{\Delta}}^{\tau}\right)$. - 2.
- The
**relative extension group**${E}^{{\tau}_{1}{\tau}_{2}}$ of $({\tau}_{1},{\tau}_{2})$ is the subgroup of $\mathrm{Aut}\left({\mathrm{\Delta}}^{{\tau}_{1}}\right)$ of automorphisms of ${\mathrm{\Delta}}^{{\tau}_{1}}$ that extend to automorphisms of ${\mathrm{\Delta}}^{{\tau}_{2}}$. - 3.
- The
**relative symmetry multiplicity**${\mu}_{\mathrm{SYM}}^{{\tau}_{1}{\tau}_{2}}$ of $({\tau}_{1},{\tau}_{2})$ is $|\mathrm{Aut}\left({\mathrm{\Delta}}^{{\tau}_{1}}\right)|-|{E}^{{\tau}_{1}{\tau}_{2}}|$. - 4.
- The
**relative symmetry entropy**${e}_{\mathrm{SYM}}^{{\tau}_{1}{\tau}_{2}}$ of $({\tau}_{1},{\tau}_{2})$ is $log{\mu}_{\mathrm{SYM}}^{{\tau}_{1}{\tau}_{2}}$.

**Definition**

**25.**

- 1.
- The
**initial entropy**${e}_{i}^{{\tau}_{i}}\left(h\right)$ of h with respect to ${\tau}_{i}$ is $e\left({\mathrm{\Delta}}^{{\tau}_{i}}\right)$. - 2.
- The
**terminal entropy**${e}_{t}^{{\tau}_{t}}\left(h\right)$ of h with respect to ${\tau}_{t}$ is $e\left({\mathrm{\Delta}}^{{\tau}_{t}}\right)$. - 3.
- The
**relative entropy**${e}^{{\tau}_{i}{\tau}_{t}}\left(h\right)$ of h with respect to the pair $({\tau}_{i},{\tau}_{t})$ is $e\left({\mathrm{\Delta}}^{{\tau}_{t}}\right)-e\left({\mathrm{\Delta}}^{{\tau}_{i}}\right)$.

**Definition**

**26.**

- 1.
- The nth
**initial entropy**${e}_{i}^{n}\left(h\right)$ of h is $e\left({T}^{n}\left({D}_{i}\right)\right)$. - 2.
- The nth
**terminal entropy**${e}_{t}^{n}\left(h\right)$ of h is $e\left({T}^{n}\left({D}_{t}\right)\right)$. - 3.
- The nth
**relative entropy**${e}^{n}\left(h\right)$ of h is $e\left({T}^{n}\left({D}_{t}\right)\right)-e\left({T}^{n}\left({D}_{i}\right)\right)$.

## 4. Entropic Phase Maps

#### 4.1. Examples of Phase Maps

#### 4.2. Interference Effects

#### 4.3. Objections and Alternatives

**Definition**

**27.**

- 1.
- The
**state-level Lagrangian quantity ${\mathcal{L}}^{{\tau}_{i}{\tau}_{t}}\left(h\right)$**of h with respect to the pair $({\tau}_{i},{\tau}_{t})$ is the number of elementary operations necessary to convert ${\mathrm{\Delta}}^{{\tau}_{i}}$ to ${\mathrm{\Delta}}^{{\tau}_{t}}$. - 2.
- The
**history-level Lagrangian $\mathcal{L}$**is the functional assigning to each co-relative history h the number of elementary operations involved in converting ${D}_{i}$ to ${D}_{t}$, i.e., the number of elements and relations added to ${D}_{i}$ by h.

**Definition**

**28.**

- 1.
- The
**state-level action quantity ${\mathcal{S}}^{\left\{{\tau}_{ik}\right\},\left\{{\tau}_{tk}\right\}}\left(\gamma \right)$**along γ with respect to the pair of sequences of transitions $\left\{{\tau}_{ik}\right\}=\{{\tau}_{i0},...,{\tau}_{iN}\}$ and $\left\{{\tau}_{tk}\right\}=\{{\tau}_{t0},...,{\tau}_{tN}\}$ is the sum$${\mathcal{S}}^{\left\{{\tau}_{ik}\right\},\left\{{\tau}_{tk}\right\}}\left(\gamma \right)=\sum _{k=0}^{N}{\mathcal{L}}^{{\tau}_{ik}{\tau}_{tk}}\left({h}_{k}\right)$$ - 2.
- The
**history-level action $\mathcal{S}$**is the functional assigning to each chain γ the number of elementary operations involved in converting ${D}_{i0}$ to ${D}_{tN}$, i.e., the number of elements and relations added to ${D}_{i0}$ by the sequence of co-relative histories ${h}_{0},...,{h}_{N}$.

#### 4.4. Summary and Conclusions

## Acknowledgments

## Conflicts of Interest

## References

- Feynman, R. Space-Time Approach to Non-Relativistic Quantum Mechanics. Rev. Mod. Phys.
**1948**, 20, 367. [Google Scholar] [CrossRef] - Bombelli, L.; Lee, J.; Meyer, D.; Sorkin, R. Space-Time as a Causal Set. Phys. Rev. Lett.
**1987**, 59, 521. [Google Scholar] [CrossRef] [PubMed] - Finkelstein, D. Space-Time Code. Phys. Rev.
**1969**, 184, 1261. [Google Scholar] [CrossRef] - Finkelstein, D. “Superconducting” Causal Nets. Int. J. Theor. Phys.
**1988**, 27, 473–519. [Google Scholar] [CrossRef] - Knuth, K.H.; Bahreyni, N. A potential foundation for emergent space-time. J. Math. Phys.
**2014**, 55, 112501. [Google Scholar] [CrossRef] - Ambjorn, J.; Dasgupta, A.; Jurkiewicz, J.; Loll, R. A Lorentzian cure for Euclidean troubles. Nucl. Phys. B Proc. Suppl.
**2002**, 106, 977–979. [Google Scholar] [CrossRef] - Markopoulou, F. Quantum Causal Histories. Class. Quantum Gravity
**2000**, 17, 2059. [Google Scholar] [CrossRef] - Rovelli, C. Quantum Gravity. In Cambridge Monographs on Mathematical Physics; Cambridge University Press: Cambridge, UK, 2004. [Google Scholar]
- Thiemann, T. Modern Canonical Quantum General Relativity. In Cambridge Monographs on Mathematical Physics; Cambridge University Press: Cambridge, UK, 2007. [Google Scholar]
- D’Ariano, G.M.; Perinotti, P. Derivation of the Dirac Equation from Principles of Information Processing. Phys. Rev. A
**2014**, 90, 062106. [Google Scholar] [CrossRef] - Finster, F. Causal Fermion Systems: An Overview. In Quantum Mathematical Physics; Springer: Berlin, Germany, 2016. [Google Scholar]
- Finster, F. The Continuum Limit of Causal Fermion Systems: From Planck Scale Structures to Macroscopic Physics. In Fundamental Theories of Physics; Springer: Berlin, Germany, 2016. [Google Scholar]
- Chen, H.; Sasakura, N.; Sato, Y. Emergent Classical Geometries on Boundaries of Randomly Connected Tensor Networks. arXiv, 2016; arXiv:1601.04232. [Google Scholar]
- Dribus, B.F. Discrete Causal Theory: Emergent Spacetime and the Causal Metric Hypothesis; Springer: Berlin, Germany, 2017. [Google Scholar]
- Dribus, B.F. On the Foundational Assumptions of Modern Physics. In Questioning the Foundations, the Frontiers Collection; Springer: Berlin, Germany, 2015; pp. 45–60. [Google Scholar]
- Dribus, B.F. On the Axioms of Causal Set Theory. arXiv, 2013; arXiv:1311.2148. [Google Scholar]
- D’Ariano, G.M.; Chiribella, G.; Perinotti, P. Quantum Theory From First Principles; Cambridge University Press: Cambridge, UK, 2017. [Google Scholar]
- Knuth, K.H. Information-based Physics: An observer-centric foundation. Contemp. Phys.
**2014**, 55, 12–32. [Google Scholar] [CrossRef] - Verlinde, E. On the origin of gravity and the laws of Newton. J. High Energy Phys.
**2011**, 4, 29. [Google Scholar] [CrossRef] - Kleitman, D.J.; Rothschild, B.L. Asymptotic Enumeration of Partial Orders on a Finite Set. Trans. Am. Math. Soc.
**1975**, 205, 205–220. [Google Scholar] [CrossRef] - Moore, C. Comment on “Space-Time as a Causal Set”. Phys. Rev. Lett.
**1988**, 60, 655. [Google Scholar] [CrossRef] [PubMed] - Bombelli, L.; Lee, J.; Meyer, D.; Sorkin, R. Bombelli et al. Reply to Comment on “Space-Time as a Causal Set”. Phys. Rev. Lett.
**1988**, 60, 656. [Google Scholar] [CrossRef] [PubMed] - Hawking, S.W.; King, A.R.; McCarthy, P.J. A new topology for curved space-time which incorporates the causal, differential, and conformal structures. J. Math. Phys.
**1976**, 17, 174–181. [Google Scholar] [CrossRef] - Malament, D.B. The class of continuous timelike curves determines the topology of spacetime. J. Math. Phys.
**1977**, 18, 1399–1404. [Google Scholar] [CrossRef] - Martin, K.; Panangaden, P. A Domain of Spacetime Intervals in General Relativity. Commun. Math. Phys.
**2006**, 267, 563–586. [Google Scholar] [CrossRef] - Bombelli, L.; Meyer, D. Origin of Lorentzian geometry. Phys. Lett. A
**1989**, 141, 226–228. [Google Scholar] [CrossRef] - Parrikar, O.; Surya, S. Causal topology in future and past distinguishing spacetimes. Class. Quantum Gravity
**2011**, 28, 155020. [Google Scholar] [CrossRef] - Myrheim, J. Statistical Geometry. Available online: https://cds.cern.ch/record/293594/files/197808143.pdf (accessed on 30 June 2017).
- Hooft, G. Quantum Gravity: A Fundamental Problem and some Radical Ideas. In Recent Developments in Gravitation; Springer: New York, NY, USA, 1978; pp. 323–345. [Google Scholar]
- Ahmed, M.; Dodelson, S.; Greene, P.B.; Sorkin, R. Everpresent Λ. Phys. Rev. D
**2004**, 69, 103523. [Google Scholar] [CrossRef] - Bombelli, L.; Henson, J.; Sorkin, R. Discreteness without symmetry breaking: A theorem. Mod. Phys. Lett. A
**2009**, 24, 2579–2587. [Google Scholar] [CrossRef] - Harary, F.; Norman, R.Z. Some Properties of Line Digraphs. Rediconti del Circolo Matematico di Palermo
**1960**, 9, 161–168. [Google Scholar] [CrossRef] - Major, S.A.; Rideout, D.; Surya, S. Spatial Hypersurfaces in Causal Set Cosmology. Class. Quantum Gravity
**2006**, 23, 4743–4751. [Google Scholar] [CrossRef] - Surya, S. Directions in Causal Set Quantum Gravity. In Recent Research in Quantum Gravity; Dasgupta, A., Ed.; Nova Science Publishing Incorporated: Hauppauge, NY, USA, 2012. [Google Scholar]
- Sorkin, R. Expressing entropy globally in terms of (4D) field-correlations. J. Phys. Conf. Ser.
**2014**, 484, 012004. [Google Scholar] [CrossRef] - Sorkin, R.; Yazdi, Y. Entanglement Entropy in Causal Set Theory. arXiv, 2016; arXiv:1611.10281v1. [Google Scholar]
- Bender, E.A.; Robinson, R.W. The Asymptotic Number of Acyclic Digraphs II. J. Comb. Theory Ser. B
**1988**, 44, 363–369. [Google Scholar] [CrossRef] - Rideout, D.; Sorkin, R. Classical sequential growth dynamics for causal sets. Phys. Rev. D
**2000**, 61, 024002. [Google Scholar] [CrossRef] - Sorkin, R. Toward a Fundamental Theorem of Quantal Measure Theory. Math. Struct. Comput. Sci.
**2012**, 22, 816–852. [Google Scholar] [CrossRef] - Isham, C. Quantum Logic and the Histories Approach to Quantum Theory. J. Math. Phys.
**1994**, 35, 2157. [Google Scholar] [CrossRef] - Isham, C. Topos Theory and Consistent Histories: The Internal Logic of the Set of all Consistent Sets. Int. J. Theor. Phys.
**1997**, 36, 785. [Google Scholar] [CrossRef] - Isham, C. Quantising on a Category. Found. Phys.
**2005**, 35, 271–297. [Google Scholar] [CrossRef] - Isham, C. Topos Methods in the Foundations of Physics. In Deep Beauty: Understanding the Quantum World through Mathematical Innovation; Halvorson, H., Ed.; Cambridge University Press: Cambridge, UK, 2011. [Google Scholar]
- Penrose, R. Cycles of Time; Vintage Books: New York, NY, USA, 2010. [Google Scholar]
- Benincasa, D.M.T.; Dowker, F. Scalar Curvature of a Causal Set. Phys. Rev. Lett.
**2010**, 104, 181301. [Google Scholar] [CrossRef] [PubMed] - Glaser, L. A closed form expression for the causal set D’Alembertian. Class. Quantum Gravity
**2014**, 31, 5007. [Google Scholar] [CrossRef] - Aslanbeigi, S.; Saravani, M.; Sorkin, R. Generalized Causal Set d’Alembertians. arXiv, 2014; arXiv:1403.1622. [Google Scholar]

**Figure 1.**In a fixed spacetime background, the Lagrangian $\mathcal{L}$ “chooses” the classical path ${\gamma}_{\mathrm{CL}}$ via Hamilton’s principle; in a background independent theory, different paths imply different spacetimes.

**Figure 2.**${\mathbb{R}}^{3+1}$ partitioned via sequences of spatial sections $\left\{{\sigma}_{k}\right\}$ and $\left\{{\sigma}_{k}^{\prime}\right\}$; evolutionary pathways defined by $\left\{{\sigma}_{k}\right\}$ and $\left\{{\sigma}_{k}^{\prime}\right\}$. Both pathways share the same “limit history” ${\mathbb{R}}^{3+1}$.

**Figure 4.**Induced relation between relations ${r}_{0}$ and ${r}_{1}$ in a directed set D; global view of $\mathcal{R}\left(D\right)$.

**Figure 5.**Cauchy surface $\sigma $ in a globally hyperbolic manifold X, intersected by two causal curves; maximal antichain $\sigma $ in a directed set D, permeated by two chains.

**Figure 6.**Four co-relative histories sharing a common cobase with two elements x and y and one relation $x\prec y$; morphisms (transitions) representing these co-relative histories.

**Figure 7.**Positive sequential kinematic scheme ${\mathbb{S}}_{\mathrm{PS}}$ (first four generations); gray nodes show the four co-relative histories from Figure 6; thickened edges illustrate a co-relative kinematics.

**Figure 8.**Portion of $\mathcal{M}\left({\mathbb{S}}_{\mathrm{PS}}\right)$ illustrating the permeability problem; corresponding portion of $\mathcal{R}(\mathcal{M}\left({\mathbb{S}}_{PS}\right))$ showing an impermeable maximal antichain.

**Figure 9.**Setup for deriving Equation (4): $\gamma ={\gamma}^{-}\bigsqcup r$ and $\mathsf{\Theta}\left(\gamma \right)=\mathsf{\Theta}\left({\gamma}^{-}\right)\theta \left(r\right)$.

**Figure 10.**Sequence of co-relative histories in ${\mathbb{S}}_{\mathrm{PS}}$; terminal states indicated by dark nodes and edges; “new elements” added by each co-relative history indicated by arrows.

**Figure 11.**Partitions of state space; conventional state spaces exhibit regions of very different sizes; state space inducing an “inverse second law of thermodynamics”.

**Figure 13.**Curve in state space along which entropy increases; map from a linearly ordered set into an entropy quadruple, showing no discernible second law.

**Figure 14.**History D and terminal state $\mathrm{\Delta}$; causal atomic resolution of $\mathrm{\Delta}$; superset microstate of $\mathrm{\Delta}$; labeled microstate of $\mathrm{\Delta}$; symmetry microstate of $\mathrm{\Delta}$.

**Figure 15.**22 of the 96 superset microstates of ${\mathrm{\Delta}}_{7}$ given by adding one prehistorical element.

**Figure 17.**Sequential growth process from Figure 10; region of ${\mathbb{D}}_{\mathrm{SYM}}^{1}$ through which this process moves; abstract view of the process.

**Figure 18.**Sequence of co-relative histories ${h}_{k}^{\prime}$ suggestive of gravitational collapse.

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Dribus, B.F. Entropic Phase Maps in Discrete Quantum Gravity. *Entropy* **2017**, *19*, 322.
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Dribus BF. Entropic Phase Maps in Discrete Quantum Gravity. *Entropy*. 2017; 19(7):322.
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**Chicago/Turabian Style**

Dribus, Benjamin F. 2017. "Entropic Phase Maps in Discrete Quantum Gravity" *Entropy* 19, no. 7: 322.
https://doi.org/10.3390/e19070322