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Poincar´esymmetry. Although we replace the continuous G⊲ P coordinate dependence of our companion paper’s fields— free σ f(x)—with a discrete lattice vertex dependence—f[n]— π Poincar´esymmetry is yet preserved. More practically minded, discrete algorithms derived {n} from our theory will conserve linear and angular energy- momentum (as well as the other Noether currents of the FIG. 1: A depiction of the discrete principal bundle Standard Model) to machine precision. In effect, our (P,π, n) with its free left G-action, and a choice of section theory is designed for computation, rather than for pen σ ∈ Γ(P ). The latter is defined such that its composition and paper analysis. with the projection π recovers the identity map on {n}, i.e. Our approach to this problem is prompted by the ob- π ◦ σ = id. servation that, in spacetime-continuum theories, inde- pendent spacetime coordinates—acting as a Poincar´e- n = (nt,nx,ny,nz) and corresponding edges, faces, cells, set—readily admit models that conserve energy and and hypercells. For simplicity, P is taken to have a trivial momentum. Such models are the workhorse of con- product structure with respect to its Poincar´estructure temporary physics; even (GR) [4], group—that is, with its dynamical metric, has non-dynamical, Poincar´e- P = n G+ transformable spacetime coordinates. On the other hand, { }× 3,1 + (3) to construct a discrete, Poincar´e-invariant theory, depen- = (nt,nx,ny,nz) R ⋊ SO (3, 1) . dent dynamical variables are the prerequisite targets for · { }× + Poincar´egroup actions.2 We denote by G the connected subgroup of the Poincar´e This view—on the necessity of dependent Poincar´e-sets group, and we correspondingly call P the discrete prin- + in discrete theories—was promoted in a 1986 speech of cipal G -bundle. Here a point p P is specified by n ∈ n T.D. Lee’s. His speech concluded, “I suggest that this p = ( ,g), which represents a vertex assignment in the base manifold, and a choice g G+ in the group fiber: discrete formulation might be more fundamental.”[7] We ∈ µ proceed with this suggestion in mind. µ α αβ g = (Λ , ϕν ) = exp γαP + ωαβM (4) ν ν   R α αβ UNGAUGED LATTICE 5-VECTOR THEORY for some γα,ωαβ . P and M are, respectively, the translation{ and} Lorentz∈ transformation generators of the Poincar´eLie algebra. Note that we have already sep- Having introduced 5-vector field theory on a continu- arated the discrete lattice of the base manifold from the ous Cartesian background in our companion paper, we continuous Poincar´e-sets that exist ‘above’ the lattice, in now explore a discrete setting for this theory. We be- the fibers. gin with a discussion of the discrete principal Poincar´e We define the free left G+-action ⊲ on an arbitrary bundle, defined with a hypercubic lattice base space. We point p = (n,g) P as then develop an ungauged discrete 5-vector theory, which ∈ is counterpart to a scalar theory in flat spacetime. In g′ ⊲p = g′ ⊲ (n,g) = (n,g′ g) (5) · later sections, we evolve this effort into a gravitational · • where denotes the usual group operation of G+ theory by introducing a lattice Poincar´egauge field. • ′ ′ ′ ′µ σ ′ σ g g = (Λ , ϕ ) (Λ, ϕ)= Λ σΛ ν , ϕ σΛ ν + ϕν , • • (6) Mathematical Preliminaries  with identity element g = (1, 0), and where g−1 G+ e ∈ The discrete principal bundle and its structure group is given by g−1 = (Λ, ϕ)−1 = Λ−1, ϕΛ−1 . (7) To set the stage, we define a trivial principal bun- − dle with a discrete ‘base manifold’, which we denote In particular, we will employ the following faithful, n 3 non-unitary 5 5 matrix representation ρ of the Poincar´e (P, π, )—see Fig. 1. The base manifold is taken × to be{ a hypercubic} lattice, with integer-labeled vertices group: Λµ 0 (Λ, ϕ) Λ ν → ≡ ϕν 1    2 µ 0 Such a conclusion seems inevitable after a study of the varia-  δν ρ :  (1, 0) Λ0 (8) tional complex detailed in Refs. [5],[6]—though we have made  → ≡ 0 1    no attempt at a proof.  −1 µ 0 3 As a colloquialism, we will continue to refer to the hypercubic −1 Λ−1 (Λ ) ν (Λ, ϕ) −1 σ . lattice base space as a ‘base manifold’, error that it is.  → ≡ ϕσ(Λ ) ν 1  −    3

Lastly, on our trivial bundle P , we note that the pro- the following left action ⊲ of the Poincar´egroup:7 jection map π : P n is simply given by: →{ } φ(g⊲p)= g⊲ φ(p)=(Λ, ϕ) ⊲ φ(p) π(p)= π (n,g) = n. (9) · Λµ 0 φν · = ν  · " ϕν 1# · " φ # (14) The associated fibers—the solder and matter fields— µ ν and their group transformations Λ ν φ = ν . "φ + ϕν φ # For the solder field of our theory, we define a G+- We note that “G+-equivariance” is inversely defined8 equivariant, vector-space-valued function—(namely, a for the matter field φ relative to the more conventional 5 5 matrix-valued function)—denoted e, which is de- equivariance of the solder field, defined in Eq. (11). fined× at each point of the principal bundle:4 We further define the 5-vector φ’s antiparticle—the e : P Fe twisted 5-vector → a 0 (10) ˜ e φ : P Fφ˜ p e(p) = µ (p) . → 7→ · eµ 1 ˜ (15)   p φ(p) = φ˜µ φ˜ (p) 7→ · Here, we define G+-equivariance in the usual way: h i —as well as its G+-equivariant right group action: e(g⊲p)= e(p) ⊳g−1. (11) φ˜(g⊲p)= φ˜(p) ⊳gT = φ˜(p) ⊳ (Λ, ϕ)T e 5 We refer to the field as the f¨unfbein, and we define the µ 6 Λ ν ϕν right action ⊳ in Eq. (11) as follows: = φ˜ν φ˜ (16) · · " 0 1 # e −1 e −1 h i (p) ⊳g = (p) ⊳ (Λ, ϕ) ˜ν µ ˜ ˜ν = φ Λ ν φ + ϕν φ . −1 a 0 µ 0 eµ Λ ν h i = For ease of notation, we leave the transposes of com- · eµ 1 · ϕν 1 (12) " # " # ponents implicit in the above expressions—i.e., (φ˜ν )T , a −1 µ µ T T e (Λ ) 0 (Λ ) and (ϕν ) . = µ ν . ν −1 µ −1 σ " eµ(Λ ) ν ϕσ(Λ ) ν 1 # − The symmetry generators For our matter field, we define a G+-equivariant, vector-space-valued function φ defined at each point of the principal bundle: Because P has a trivial product structure, we can glob- ally separate the ‘vertical’ and ‘horizontal’ components of ∗ φ : P Fφ its tangent bundle T P (and cotangent bundle T P ). We → note that, like P , T P = n + aˆ T G+ has a half- φµ (13) { 2 } × p φ(p) = (p) . discrete, half-continuous structure. Here, we denote by 7→ · φ n aˆ n n " # + 2 the edge linking vertices and +ˆa. We emphasize thata ˆ t,ˆ x,ˆ y,ˆ zˆ represents not a unit tangent vector We define φ to take values in the vector space of 5- on a spacetime∈ { manifold,} but a lattice direction. The lat- vectors, which we define by their transformation under tice edges themselves form the for horizontal vector fields on P . Similarly, 1-forms on the lattice are, in a sense, simpler than they are on a continuous manifold:

4 The choice of the notation F for the target of a group-equivariant function derives from the correspondence of such a function with the section of an associated bundle P with fiber F . F is re- F 7 quired to be vector-space-valued to enable its use in the discrete Drawing an analogy between φµ and the partial derivative ∂µφ , which will be defined in the gauged version of scalar theory, we note that the translation group action on of our theory. the scalar component of the 5-vector is seemingly a first-order 5 Note that we define our solder field as a zero-form—a function of truncation of its scalar theory counterpart: p ∈ P —rather than as a one-form with arguments in the tangent µ φ(x) → φ(x + ϕ) ≈ φ(x)+ ϕµ∂ φ(x)+ ··· bundle TP . Still, we note that the lattice index a appears in (the vierbein components of) the f¨unfbein, which demonstrates how it can be associated to a function over TP , if desired. 8 The inverse group actions on the matter and solder fields encode 6 At times we suppress the p-dependence of our fields—e.g. the familiar dichotomy of the ‘active’ and ‘passive’ notions of a a eµ ≡ eµ (p)—for compactness of notation. spacetime symmetry transformations. 4

They are single-valued along each edge and their values Several notes are in order regarding this discrete ac- need not (indeed, cannot) vary smoothly from one edge tion. First, we have employed the lattice difference oper- to the next. ators10 + + We recall that the Poincar´eLie algebra g = Tge G is α αβ ∼ ± n ±a 1 n n n specified by 10 generators P ,M satis- da f[ ] = S f[ ]= f[ aˆ] f[ ] , α6=β∈{0,1,2,3} · ± − ± ± − fying the following Lie brackets:  (20)   α β qP , P y =0 to construct the matrix lattice operator dm in Eq. (18). Second, we have, for now, forgone any notion of a qM αβ, P µy = ηαµP β ηβµP α − volume measure—(e.g., d4x√ g). We regard the La- qM αβ,M µν y = ηαµM βν + ηβν M αµ ηαν M βµ ηβµM αν − − − grangian as a real number at each vertex of the lattice (17) base manifold; neighboring vertices are simply separated ± b b by a dimensionless integer: 1. In particular, da x = δa. where we define ηµν to have signature ( +++). Though We use the Einstein summation convention to sum over − our theory in this section is as yet ungauged, we note repeated lattice indices, that in later sections we shall define a g+-valued connec- tion 1-form, whose pullback to the base manifold takes its X Y a = X Y a = XaY bη , a · a ab (21) values along each edge of the lattice. We delay this dis- ·a∈{t,x,y,z} a,b∈{t,x,y,z} cussion for now, focusing instead on global gauge trans- X X formations. and Latin indices may be raised and lowered with the With our mathematical technology in place, we now Minkowski metric ηab of signature ( +++). Greek − turn to a Lagrangian description of the physics. indices may be raised and lowered with the metric n a n b n gµν [ ] ηabeµ [ ]eν [ ], although it will become clear in the≡ later gauged theory that the vierbein is best re- Physics of the Ungauged Discrete Action garded as Lorentz-valued in the ungauged theory, such that g [n]= η n. µν µν ∀ Let us derive the equations of motion (EOM) from the We define the ungauged lattice 5-vector theory’s dis- action in Eq. (18). At this point we leave our theory crete action on P as follows:9 ungauged, which leads us to make the following ungauged assumptions for the f¨unfbein solder field: S = L[n] · n X (a) e[n] is constant at every n, so that its finite differ- ˜ T ± L[n] = φ[n]e [n]dme[n]φ[n] ences vanish (d e[n] = 0); · a a + b 0 ν µ eµ eµ ηab d e φ e n = φ˜ φ˜ − a ν (b) [ ] transforms under global Poincar´egauge trans- · 0 − 2 · " 1 # " db m # "eν 1# " φ # formations; and h i − = φ˜µg φν φ˜µe ad+φ φ˜ d−(e bφν )+ m2φ˜ φ , µν − µ a 0 − 0 b ν 0 0 (c) e[n] is a dynamical variable (though its EOM will (18) turn out to be indeterminate in the ungauged the- ory). where we have defined the gauge-invariant quantities: With these assumptions for the solder field, we find n n n µ n φ0[ ] = φ[ ]+ eµ[ ]φ [ ] the variation of our action by applying the discrete Euler · µ (19) φ˜0[n] = φ˜[n]+ eµ[n]φ˜ [n]. operators of our matter fields, as defined by Ref. [6]: · ∂L[n] E L n −m q( [ ]) = S n m , (22) · m ∂q[ + ] X 9 We note that this action assumes the fields φ[n] and e[n] are in fact defined on the base manifold as functions of n—and not as functions of p ∈ P . As usual in Yang-Mills theory, these definitions may be facilitated by the pullback of a section of P — 10 The lattice shift operator Sm is defined such that σ ∈ Γ(P )—to {n}. While important, such mathematical rigor m will not be required in this work, though the reader may consult S f(n)g(n) = f(n + m)g(n). [8] or [9]. From hereon, we shall follow the more conventional description of gauge theories in physics, by abstracting away the We shall always include parentheses if a lattice operator—i.e., ±a ± principal bundle underlying the physics. As we did for fields on {S , da }—acts on more than just the object immediately to its P , we shall often use the notation f ≡ f[n], suppressing the right, except in matrix expressions—as in Eq. (18). We further lattice coordinate for the field f, when evaluated at n. note that shift operators commute: SaSb = SbSa. 5

where the sum over m includes all lattice vertices. We We apply these generators to L[n] as we would differential define the following gauge-invariant quantities operators, and we observe that their prolongations trans- form our matrix fields wherever those fields appear— 2 n − b n ν n ( ) = m φ0[ ] d (eν [ ]φ [ ]) ± ◦ · − b even under a difference operator da . In particular, each 2 ˜ n − b n ˜µ n ( ) = m φ0[ ] + d (eµ [ ]φ [ ]) of these matrix generators—and their appropriate nega- • · b (23) b n ν n + n tions and transposes, as determined by the group actions ( )a = ηabeν [ ]φ [ ] da φ0[ ] ◦◦ · − of Eqs. (12)-(16)—applied to the fields of the Lagrangian b ˜µ + ˜ ( )a = ηabe [n]φ [n] + d φ0[n], in situ, generate the same transformations as the follow- •• · µ a · ing prolonged vector fields, respectively: and derive our EOM from L[n], as follows:

n a n n α α α 0= Eφ˜σ (L[ ]) = eσ [ ]( )a + eσ[ ]( ) pr[P ]= φ [n]∂ n + φ˜ [n]∂ ∂ n ◦◦ ◦ φ[ ] φ˜[n] eα[ ] n n − 0= Eφ˜(L[ ]) = ( ) X h i ◦ αβ σ α βν β αν n a n n pr[M ]= φ [n](δ η δ η )∂ ν n 0= Eφσ (L[ ]) = eσ [ ]( )a + eσ[ ]( ) σ σ φ [ ] •• • (24) n − 0= E (L[n]) = ( ) X h (28) φ ˜σ n α βν β αν • + φ [ ](δσ η δσ η )∂φ˜ν [n] 0= E (L[n]) = φ˜σ[n]( )+ φσ[n]( ) − eσ α βσ β ασ ◦ • n n σ σ eσ[ ](δν η δν η )∂eν [ ] 0= Ee a (L[n]) = φ˜ [n]( )a + φ [n]( )a. − − σ ◦◦ •• a n α βσ β ασ eσ [ ](δν η δν η )∂e a[n] . We see that all of our EOM are solved when: − − ν i ( ) = ( ) = ( )a = ( )a =0. (25) ˜ ◦ • ◦◦ •• Given the Poincar´einvariance of the terms φeT and The EOM of ungauged 5-vector theory are analogous eφ—evident from Eqs. (12), (14) and (16)—L[n] is in- to the EOM of scalar field theory in flat spacetime. We variant under Poincar´etransformations. In particular: note that both φ0 and φ˜0 obey discrete Klein-Gordon equations on shell: pr[P α](L)=0 αβ (29) 0 = (d2 m2)φ [n] = (d2 m2)φ˜ [n], (26) pr[M ](L)=0. − 0 − 0 2 ab + − where d η da db . Despite this identical behavior, the Eq. (29) demonstrates that our 10 infinitesimal Poincar´e ≡ ˜ ( ) and ( )a EOM for φ have an important sign differ- symmetries are variational symmetries of the discrete 5- • •• ence with respect to ( ) and ( )a. The internal dynam- vector Lagrangian. We may therefore, in principle, pur- ◦ ˜ ◦◦ ics of the components of φ are distinctly opposite those sue our theory’s conservation laws via the canonical dis- ˜ of φ, a fact which supports φ’s interpretation as the an- crete Noether procedure. φ tiparticle of . The canonical Noether procedure for discrete conserva- We observe that our solder field’s 20 degrees of freedom tion laws—detailed in Appendix A—is often more chal- (DOF) are indeterminate in the EOM of Eq. (25). This is lenging to compute than its continuous counterpart; in- to be expected in an ungauged theory—inasmuch as the deed, it is often easier to discover such conservation laws ‘dynamics’ of ηµν are ‘indeterminate’ in a flat-spacetime 11 by inspection. [10] Nevertheless, the canonical linear mo- scalar theory. The solder field’s indeterminacy will be mentum Noether current can be shown to be trivial— lifted, and gravity introduced, when we add gauge cur-

vature to our Lagrangian. a Let us now describe the infinitesimal Poincar´esymme- AP α =0 (30) try of our discrete theory. The matrix generators of the Poincar´egroup are given by: —mirroring the canonical linear momentum of continu- ous 5-vector theory in our companion paper.12 0 0 [P α]µ = Fortunately, Eq. (67) of our companion paper’s [1] con- σ · α · "δσ 0# tinuous 5-vector theory provides a natural point of depar- (27) ture for discovering these nontrivial currents by inspec- δαηβµ δβηαµ 0 [M αβ]µ = σ − σ . σ · 0 0 · "  #

12 We have not pursued the discrete canonical Noether angular mo- 11 The indeterminacy of e does not hinder the simulation of our mentum here, but we suspect that it, too, would be trivial. As ungauged theory; after all, the solder field may simply be chosen discussed in [1], this triviality comports with Noether’s second to take a nonsingular constant value ∀ n. theorem. 6 tion. We soon find: We shall proceed in two steps: (i) We will first explore + ¯aα n the tools of necessary to define a gauged 0 =da T [ ] matter Lagrangian LM [n], and derive its 5-vector EOM— the lattice equivalent of a scalar field theory evolving in =d+ eα ηbc S−a e aφ˜µ d−φ · a b µ c 0 a static curved spacetime. (ii) We will then define a pure · "    gauge Lagrangian LG[n] and recover a discrete analog of −a − a µ ˜ Einstein’s equations. We thereupon discover a discrete + φ˜ S d (e φ ) φ φ (31) 0 c µ − ↔ energy-momentum conservation law of gauged 5-vector  #   theory. + ¯aαβ n  0 =da L [ ]

+ α b n ¯aβ n β b n ¯aα n =da e bx [ aˆ]T [ ] e bx [ aˆ]T [ ] . Mathematical Preliminaries for LM · " − − − # These conservation laws may be verified by the re- Building upon our earlier development, we introduce peated application of: the tools of gauge theory required to define a connection on our discrete principal Poincar´ebundle P . 1. our EOM as expressed in Eq. (25);

2. the discrete Leibniz rule: The Poincar´egauge field

± ± ± d f1[n]f2[n] = d f1[n] f2[n aˆ]+ f1[n]d f2[n]; a a ± a We begin by defining two structures on our principal     (32) G+-bundle that follow the conventional development of a gauge theory: 3. the ungauged solder field assumption, which im- ± b n g+ g+ plies that da eµ [ ] = 0; and 1. a -valued connection one-form ω TP, ; and

4. the relation 2. a section σ : n n,g n , such that π σ = id. 7→ σ( ) ◦ ± ∓a ∓ We denote the pull-back of this connection = σ∗ω, da S = da . (33) ◦ such that is itself a g+-valued one-form thatA is evalu- ated alongA the edges—(the ‘tangent bundle’)—of the lat- Validation of d+L¯aαβ[n] additionally requires setting a 13 n + ¯aα n tice. In particular, we let a[ ] denote the value of the da T [ ] = 0 and noting the (α β) symmetry of A n aˆ α ¯aβ n ↔ connection one-form along the oriented link + 2 , and e aT [ ]; this symmetry is easily discovered by using n n n µ ˜µ we note that −a[ +ˆa]= a[ ]. We define a[ ] to the EOM to substitute for φ (or φ ) in terms of φ0 (or A −A A ˜ be a dynamical 5 5 matrix-valued field: φ0). × Having described the Lagrangian, EOM, symmetries, α αβ a[n] spanR [P ] , M (34) and conservation laws of discrete 5-vector theory, we now A ∈ α6=β∈{0,1,2,3}    introduce its Poincar´egauge field. with matrices [P α] and M αβ defined in Eq. (27). We furthermore define the G+-valued comparator matrix   GAUGED LATTICE 5-VECTOR THEORY a[n] = exp( a[n]), (35) U · A n aˆ n n −1 We now gauge our 5-vector theory to discover a lat- along + 2 and note that −a[ +ˆa]= a[ ] . U µ n U tice gauge theory of gravity. To this end, we introduce More concretely, we define Γ νa[ ] asan so(3, 1)-valued a dynamical Poincar´egauge field and affine connection, one-form and ǫνa[n] as a t(4)-valued one-form, which which are used to define discrete covariant derivatives form the Lorentz and translation components, respec- and gauge curvature. tively, of the g+-valued gauge field [n]: Aa We first observe that, unlike canonical gauge theories, µ 0 µ 0 the ungauged Lagrangian of Eq. (18) is already locally Γ νa exp Λ νa a[n] = [n] a[n] = [n]. Poincar´einvariant. It might appear, therefore, that our A · " ǫνa 0# −−→ U · " ϕνa 1# task is already done. We quickly note, however, that (36) because the conservation laws of Eq. (31) require that ± b da eµ = 0, these conservation laws do not hold when the solder field e[n] undergoes local Poincar´etransforma- tions. Our motivation, then, is to gauge our preceding 13 Here, we adopt the discrete exterior calculus (DEC) [11] notion of theory so as to restore energy-momentum conservation a one-form on a hypercubic lattice as taking values on its edges, in a locally invariant theory. or links. 7

We note that the comparator ±a[n] transforms as Pursuing a lattice analog of the U4 Riemann-Cartan usual to enforce local Poincar´esymmetryU in our lattice spacetime described in [12], we assume that the affine b n gauge theory: connection Γ ca[ ] is metric-compatible, as follows: φ[n] g[n]φ[n] 0 =D+g [n] → a bc ˜ n ˜ n n T n i n j n n φ[ ] φ[ ]g[ ] (37) =gij [ +ˆa] exp( Γ ba[ ]) exp( Γ ca[ ]) gbc[ ]. → · − − − [n] g[n] [n]g[n aˆ]−1 (39) U±a → U±a ± where g[n],g[n aˆ] G+ are local gauge transforma- Here, we have defined the covariant derivative of the tions at{ their respective± } ∈ lattice vertices. lattice metric, and set it to vanish. We observe that, µ n + while Γ νa[ ] so (3, 1) is a Lorentz-Lie-algebra-valued ∈ b n R 1-form, the affine connection Γ ca[ ] gl(4, ) is a 1-form The affine connection n aˆ ∈ R4×4 on the link + 2 that may assume any value in . When written in terms of the vierbein, Eq. (39) is a We have already encountered aspects of our 5-vector quadratic matrix expression of the form theory that depart from the canonical development of −T −1 −T −1 Yang-Mills gauge theories. In particular, our ungauged (XA) η(XA) = B ηB , (40) Lagrangian L[n] of Eq. (18) already enjoys local Poincar´e X b n A b n B b n symmetry—a feature of gauge theories usually accom- where exp(Γ ca[ ]), eµ [ +ˆa], eµ [ ], and ≡ ≡ ≡ b n plished only after the introduction of a gauge field. For η ηµν . It is apparent from Eq. (40) that Γ ca[ ] is metric-compatible≡ if and only if Y SO(3, 1) such that this reason, the Poincar´egauge field we have introduced ∃ ∈ above merely formalizes the description of this local sym- Y = B−1XA. (41) metry. As we previously noted, however, the conserved cur- Y is therefore naturally identified with the comparator rents in Eq. (31) reflect only the global Poincar´esym- of our Lorentz gauge field—Y = exp(Γµ [n])—and we n νa metry of L[ ]. To render a theory with local symmetry accordingly require that and conservation laws simultaneously, we are compelled µ n b n b n µ n ν n to introduce another gauge field beyond Γ νa[ ] in our exp(Γ ca[ ]) = eµ [ ] exp(Γ νa[ ])e c[ +ˆa]. (42) 5-vector Poincar´egauge theory—the affine connection · b n b n Γ ca[ ]. Although at first Γ ca[ ] may have appeared to intro- Although the affine connection is a core feature of duce new DOF into our theory, the definitional relation GR, it appears in our Poincar´egauge theory as some- of Eq. (42) ensures that this is not the case. thing of a ‘noncanonical’ feature. Nonetheless, we are led to develop the affine connection by the desire to relax +e Parallel transport and the covariant derivative the da = 0 assumption of our ungauged theory while preserving energy-momentum conservation; in particu- ± ± lar, we will jointly apply the affine and Poincar´egauge We now seek Da —a covariant generalization of da to connections to define a covariant derivative such that be derived from the parallel transport of our fields along +e Da = 0. the links of our base manifold. To guide our effort, we We adapt the affine connection formalism of [12, 13] first recall that the covariant derivative of GR varies with to our present discrete theory. We first promote the flat the tensorial rank of the objects it differentiates: lattice metric ηab to its gauged counterpart: σφ = ∂σφ µ ν ∇ · (43) gab[n] = ηµν e [n]e [n]. (38) µ µ µ ν · a b σφ = ∂σφ + σν φ · ∇ · { } Latin (lattice) indices may be raised and lowered with µ etc., where the affine connection σν may be defined to gab, and Greek (Poincar´e) indices may be raised and low- enforce metric-compatibility:15 { } ered with ηµν . The vertical Poincar´efiber itself evidently 14 τ τ has no notion of metric curvature; as we shall see, the 0= σgµν = ∂σgµν σµ gτν σν gµτ . (44) curvature we will associate with the Poincar´efibers will ∇ −{ } −{ } be that of a gauge theory, which reflects the anholonomy We presently adapt σ for our lattice gauge theory. of the Poincar´egauge connection. ∇

15 µ We employ the notation { σν } for GR’s symmetric affine con- 14 Here we see why the vierbein was best regarded as Lorentz-valued nection to distinguish it from our discrete lattice theory’s gauge in the ungauged theory, such that gµν [n]= ηµν . fields. 8

T To form a discrete analog of σ from our gauge fields, In Eq. (50), the comparators ±a and ±a effect the we must define the parallel transport∇ of our matter parallel transport e∓a of our shiftedU matterU fields. In and solder fields along lattice links; furthermore, such keeping with the field-specific notion of µ in GR—as ± ∇ a parallel transport must be consistent with the group- in Eq. (43)—we have defined Da to act on each field of equivariant (g⊲p) actions of Eqs. (12)-(16). 5-vector theory according to its own lattice and Poincar´e We begin by defining a covariant lattice shift operator index rank. m m m ± S˚ = e− S (45) Having defined Da for the matter and solder fields of · 5-vector theory, we briefly explore a few extensions of this m · where the notation e− f[n + m] is used to denote the definition. To take the covariant derivative of components n m n ˜ a parallel transport of a generic field f from + to . We of the above fields (e.g. φ or eµ ), we simply treat the correspondingly define the discrete covariant derivative: components as if they were Poincar´e-transformed within + ˜ e ± n ˚±a 1 n their G -representation (e.g. φ or ), as in Eq. (47). To Da f[ ] = S f[ ]. (46) · ± − take the covariant derivative of a product of fields, we We may now reinterpret Eq. (42) as defining the van- act on each field according to its own gauge transforma- ishing covariant derivative of our vierbein: tion. As a consequence, the covariant derivative obeys a 16 + b n covariant discrete Leibniz rule: 0 = Da eµ [ ] (47) b n c n ν n b n = exp(Γ ca[ ])eν [ +ˆa] exp( Γ µa[ ]) eµ [ ]. ± ± ˚±a ± − − Da (f1f2) = (Da f1)S f2 + f1Da f2. (51) Eqs. (45)-(47) implicitly define the parallel transport of our vierbein. To preserve metric-compatibility, it is ap- As a further consequence, invariant expressions like φ0 lattice + + parent that the indices of our solder field must have trivial covariant derivatives—Da φ0 = da φ0. In- transform appropriately under parallel transport—even deed, parallel transport is trivial for any expression whose though we omit their transformation under a Poincar´e indices are all contracted;17 parallel transport is likewise gauge transformation, as in Eq. (12). We regard this nontrivial for any expression with an uncontracted (lat- distinction between gauge transformation and parallel tice or Poincar´e) index. e transport as a defining property of the solder field; is We summarize in Table I the covariant shifts that arise in a sense defined to have vanishing covariant derivative. from our definitions of parallel transport throughout this We therefore generalize Eq. (47) to define solder-field- section. compatibility as follows:

±e ±a ∓a ±a 0 =Da Q[n] S˚ Q[n] = e S Q[n] · −1 (48) = U [n] ⊲ e[n aˆ] ⊳ [n] e[n] φ[n] ±a[n]φ[n aˆ] · ± ±a ± U±a − U ± · φ˜[n] φ˜[n aˆ] [n]T h i ± U±a where we denote by (the non-calligraphic) Ua a new, e n n e n n −1 e n R [ ] U±a[ ] [ aˆ] ±a[ ] = [ ] lattice-indexed GL(5, )-valued comparator: ± U −1 b[n] ±a[n] b[n aˆ] ±a[n + ˆb] b 0 b 0 U n U n U n± U n ˆ −1 Γ ca exp Λ ca −b[ ] ±a[ ] −b[ aˆ] ±a[ b] Aa[n] = [n] Ua[n] = [n]. U U U ± U − · 0 · 0 n n n n ˆ −1 · " 0# −−→ · " 1# Ub[ ] U±a[ ]Ub[ aˆ]U±a[ + b] ± (49) U [n] U [n]U [n aˆ]U [n ˆb]−1 −b ±a −b ± ±a − The group action symbols ⊲ and ⊳ in Eq. (48) serve to TABLE I: For convenience in deriving our EOM, we emphasize the proper ordering of matrix multiplications. summarize the covariant shifts that have been defined (or We have omitted the translation degrees of freedom in implied) throughout the preceding section. the definitions of Aa and Ua in Eq. (49), a choice that we will revisit in our later discussion. Given the G+-equivariant (g⊲p) actions of Eqs. (12)- (16), and the covariant derivative of our solder field de- fined in Eq. (48), the self-consistent parallel transport of our matter fields is uniquely determined. The covariant 16 ± ± ˜ For clarity, we note that—unlike da and Da —an index on a shift derivatives of φ and φ are straightforwardly defined in operator (S±a or S˚±a) or parallel transport operator (e±a) is not agreement with these group-equivariant actions as fol- to be summed over unless it appears in an otherwise summed lows: expression. 17 We note that the scalar quantity φ of a 5-vector transforms non- ± n n n n trivially under parallel transport. In this sense, it can be un- Da φ[ ] = ±a[ ] ⊲ φ[ aˆ] φ[ ] · ± U ± − derstood to have an ‘implicit’ Poincar´eindex, as did the scalar (50) α ± ˜ n h ˜ n T n ˜ n i components of the P generator in Eq. (27). On the other hand, Da φ[ ] = φ[ aˆ] ⊳ ±a[ ] φ[ ] . · ± ± U − φ0 is completely invariant. h i 9

Physics of the Gauged Matter Action In the continuous limit, our matter fields thus recover the curved-spacetime dynamics of a scalar field in GR: We are at last prepared to gauge the matter La- µ 2 n 18 0= ∂µ m φ. (57) grangian of Eq. (18). We define LM [ ] as follows: ∇ −   n ˜ n eT n e n n Having successfully derived the gauged EOM for the LM [ ] =φ[ ] [ ]Dm [ ]φ[ ] b · matter fields of lattice 5-vector theory, we now pursue a + b 0 ν µ eµ eµ gab Da eν φ the dynamics of its gauge and solder fields. b = φ˜ φ˜ − · 0 − 2 · " 1 # " Db m # "eν 1# " φ # h i − =φ˜µη φν φ˜µe aD+φ φ˜ D−(e bφν )+ m2φ˜ φ . µν − µ a 0 − 0 b ν 0 0 Mathematical Preliminaries for LG (52) The gauge field curvature ± ± By simply promoting ηab to gab, and da to Da , we have adopted the principle of minimal coupling to form the We begin by exploring the curvature of the Poincar´e matter Lagrangian. In the final equality above, we have gauge field. We note that there are no known fundamen- noted that g e ae b = η . ab µ ν µν tal forces associated with the translation components of Let us promptly derive the matter field EOM of our the Poincar´egroup—which might, for example, give rise gauged theory, employing the gauge covariant Euler op- to longitudinal gravity waves. Although in principle our erators ˚E (L [n]) for q φ˜σ, φ,˜ φσ, φ , as defined in Ap- q M theory could accommodate such a force, we shall follow pendix B. We discover that∈{ } the example of GR and omit these degrees of freedom b ˚ n ⊛⊛ n ⊛ from our definition of curvature—as we did from our def- 0= Eφ˜σ (LM [ ]) = ( )σ + eσ[ ]( ) inition of Aa[n] in Eq. (49). ˚ n ⊛ 0= Eφ˜(LM [ ]) = ( ) We therefore select our gauge curvature to capture only b (53) the Lorentz gauge field’s anholonomy. We find that the 0= ˚Eφσ (LM [n]) = ( )σ + eσ[n]( ) b ∗ ∗ ∗ formalism of Einstein-Cartan (EC) gravity [14–17] natu- 0= ˚E (L [n]) = ( •)• • φ bM ∗ rally accommodates the fields of our theory. In EC grav- • ity, the vierbein—rather the metric—plays a central role. where we have symbolized the following expressions: b The affine connection in EC gravity is also free to admit 2 − b µ torsion—Γb [n] =Γb [n]—a feature that comports with (⊛) = m φ0 D (eµ φ ) ca ac · − b the distinct roles6 of the affine connection’s lower indices 2 − b µ ( ) = m φ˜0 + D (e φ˜ ) ∗ · b µ (54) in Eq. (42). · µ a + (⊛⊛•)σ = ησµφ eσ Da φ0 For the purposes of our lattice theory, we therefore en- · − ˜µ a + ˜ deavor to define a Wilson loop [18] for the Lorentz com- ( )σ = ησµφ + eσ Da φ0. •∗ •∗ · ponents of the gauge field that recovers the EC action in Our gauged EOM are therefore satisfied when: the continuous limit. We follow several previous efforts [19–25] that similarly define a gravitational lattice curva- (⊛) = ( ) = (⊛⊛) = ( ) =0. (55) ture, although we modify the foregoing gauge theories to ∗ σ ∗ ∗ σ • •• suit the ‘standard’ (or ‘fundamental’) Lorentz represen- We immediately note that both φ0 and φ˜0 obey a dis- tation of our gauge field. Following these references, we crete analog of the curved-spacetime Klein-Gordon equa- employ the Ne’eman-Regge-Trautman (NRT) formalism tion: for the EC action [26, 27]. In the continuous limit, we would like the discrete 0= D−gbad+ m2 φ = D−gbad+ m2 φ˜ . (56) b a − 0 b a − 0 gauge Lagrangian LG to recover the following NRT La- h i h i grangian: + + We have reduced Da φ0 to its equivalent da φ0 in the − 1 above expression (as we have for φ˜0). However, D can- abcd α β µ σν b L = ǫ ǫµναβ e ce R η not be reduced in the same way, given its operation on · 8 d σab e (58) gbad+φ , an expression with an uncontracted lattice in- a b µ σν a 0 = eµ eν R σabη dex b that parallel transports nontrivially via the affine 2 b connection Γ ca. where the second equality follows from the identity [19]

ǫabcdǫ eα eβ =2e(e ae b e ae b) (59) µναβ c d µ ν − ν µ 18 b and the antisymmetry of the Riemann , defined by We employ the notation LM to distinguish this matter La- grangian from LM in Eq. (66), which includes a volumetric mea- µ µ µ λ sure factor. R (x) = ∂ Γ (x)+Γ (x)Γ σ|b](x). (60) σab · [a σ|b] λ[a 10

In the above, ǫ denotes the Levi-Civita symbol, and19 Physics of the Gauged Total Action

µ e = det [e a]= det[gab]= √ g. (61) · − · − We are at last prepared to fully define gauged 5-vector In this formulation of EC gravity,p the solder field and the theory. We define our total Lagrangian to be Poincar´egauge field are assumed to be independent. We L[n] = κLM [n]+ LG[n] (65) furthermore observe that the translation degrees of free- · dom of the gauge field are omitted from this definition, where as desired. LM [n] = e[n]LM [n] To form a discrete analog of Eq. (60), we define the · (66) discrete Riemann tensor by a Wilson loop: LG[n] = e[n]LG[n] · b µ [n] =Λµ [n]Λλ [n +ˆa]Λσ [n + ˆb]−1Λτ [n]−1 L n L n νab · λa σb τa νb with M [ ] defined in Eq. (52)b and G[ ] defined in · (62) Eq. (63). We have introduced a constant of proportion- ality κb between our matter and gauge Lagrangians. where we use the notation Λµ [n] exp(Γµ [n]) of νa νa We now proceed to derive our gauge and solder field Eq. (36), and we recall that Λµ [n]−1≡= exp( Γµ [n]). νa νa EOM. This effort is assisted by a few preliminary obser- Λµ [n] is appropriately viewed as a Lorentz− compara- νa vations. First, we note an identity: tor—a ‘subfield’ of a[n]. We furthermore note that, if U σ a desired, Eq. (42) may be substituted into Eq. (62) to ex- ∂(eσ )e = ee a, (67) µ − press  [n] in terms of the affine connection Γb [n]. νab ca a relation that may be derived from the well known ex- Following Eq. (58), therefore, we define a discrete ana- pression δg = gg δgab. log of the Einstein-Cartan Lagrangian, as follows: ab Second, we− note that a measure of care is required n 1 abcd α n β n µ n σν to derive EOM for our gauge field while preserving its LG[ ] = ǫ ǫµναβe c[ ]e [ ] [ ]η · 8 d σab group-theoretic properties. If we derived the EOM for e[n] an arbitrary component of Γµ , for example, we would = e a[n]e b[n]µ [n]ησν (63) νa 2 µ ν σab be forced to exogenously constrain the field’s evolution e[n] to the Lorentz Lie algebra. To avoid this complication, = R[n]. · 2 we may solve for the EOM of the coefficients of the gauge field’s generators, as described in [29]. In particular, we As above, e[n] = det (eµ [n]), and we have defined R[n], · a may decompose our gauge field into its Lorentz genera- a discrete analog· of the Ricci scalar curvature. The tors: second equality of Eq. (63) follows after substituting µ νσ σ νµ µ n µ n n στ µ Eq. (59), as well as the identity νabη = νbaη . Λ νa[ ] = exp(Γ νa[ ]) = exp(ωστa[ ][M ] ν ) (68) As discussed in [20, 22], we have not defined a lattice Lorentz curvature by the traditional method of a Yang- where we have used brackets to denote the matrix Tr n n n ˆ −1 n −1 Lorentz generators, as in Eq. (27). We then derive our Mills theory— Λa[ ]Λb [ +ˆa]Λa[ + b] Λb[ ] — n because such a trace over group indices would reduce to EOM with respect to the coefficients ωµνa[ ]. For exam- µ νab  ple, we note that: R νabRµ in the continuous limit. In this sense, Eq. (63) does not define a canonical gauge theory, since it requires µ n στ µ λ n ∂Λ νa[ ] [M ] λΛ νa[ ] if a = b the vierbein in its construction. = (69) ∂ωστb[n] 0 otherwise. Before proceeding, we verify that Eq. (63) recovers ( Eq. (58) in the continuous limit. We follow [28] Eq. (8.7), Lastly, we consider how to apply the gauge covari- µ ˚ applying the BCH formula to σab to find: ant Euler operator Eq to the gauge field comparators in µ LG[n]. As expressed in Eq. (62), the comparators of the µ  exp ∂aΓb ∂bΓa + [Γa, Γb] Wilson loop can be regarded as ‘based at’ n and its near- σab ≈ − σ µ (64) est neighbors— n, n +ˆa, n + ˆb —and not at n +ˆa + ˆb. µ   { } δσ + ∂aΓb ∂bΓa + [Γa, Γb] + ˚E L n ≈ − σ ··· When calculating ωστc ( G[ ]), therefore, we will paral-   lel transport each comparator of the Wilson loop to n From the first equality of Eq. (63), it is evident that the by a zero- or single-link path only. See Table I for ex- 0th-order contribution to R[n] vanishes in the continuous plicit definitions of our comparators’ parallel transports. limit, due to the contraction of symmetric and antisym- In simple cases, these transports produce appealingly in- metric indices in δµησν ǫ . The 1st-order contribution σ µναβ tuitive results. For example: recovers Eq. (58), as desired. ea [n]= [n +ˆa] Ua Ua ˚−aµ n µ n S σab[ ]= σb,−a[ ] (70) 19 eµ inverse Although we will continue to refer to a as the vierbein, µ µ a ˚−b n  n it is perhaps a more natural geometric object than eµ . S σab[ ]= σ,−ba[ ]. 11

˚ n We are now in position to derive our gauge and solder To simplify the EOM of Eωαβc (L[ ]) in Eq. (71), we αβ µ field EOM: substitute Eq. (27) for [M ] τ and again use the vierbein to freely exchange and rearrange indices, yielding: 0= ˚E (L[n]) = e φ˜σ (⊛)+ φσ( ) eσ ∗ n σ • o b µ νσ /c[b d]a a[b d]/c 0= ˚E a (L[n]) = ee κL + L + ee  η   eσ a M G µ νba 0= g /ca + g /c,−a − (77) n o c[b d]a a[b d]/c κe φ˜σD+φ + φ˜ D−φσ = g/  + S˚−a g  . − b a 0 b 0 a /ca a/c 1 n o   0= ˚E (L[n]) = eηνσ ωαβc 2 Lastly, we observe that although the Lorentz DOF of αβ µ /c b τ a /c τ [M ] eµ e  + e eν  · τ ν σ/cb µ σ/c,−a the solder and gauge fields now evolve nontrivially, the    EOM for eσ in Eq. (71) remains indeterminate. This is to a /c µ /c b µ αβ τ e eν  + eµ e  [M ] . be expected, given our omission of the translation DOF − µ τa/c ν τ,−b/c σ  from the definition of gauge curvature.   (71) Having derived the gauged EOM, we now briefly ex- In the last expression, we denote the index c with a slash amine the conservation laws of our ‘noncanonical’ gauge to indicate that it is not summed over. theory. Although a full exposition of our gauged conser- These EOM may be considerably simplified. Noting vation laws is deferred to future work, we derive a few that immediate results. We first note that the ungauged theory’s conserved ˜µ ˜ LM = φ (⊛⊛)µ + φ0(⊛) (72) linear momentum T¯aα in Eq. (31) was ‘symmetrized’ by subtracting (φ φ˜). This made it possible to construct µ ↔ aαβ vanishes on shell,b we may contract η e with ˚E b (L[n]) ¯ σµ a eσ a conserved L . However, it is in Eq. (71) to discover a discrete analog of Einstein’s easily verified that an ‘unsymmetrized’ linear momentum equations: T aα is also conserved in the ungauged theory:

ν c µ 1 µ ˜σ + ˜ − σ e aeµ  gabR = κe aηµσ φ D φ0 + φ0D φ . + aα n νcb − 2 b b 0 =da T [ ]   h (73)i + α bc −a a ˜µ − ˜ −a − a µ =da e bη S eµ φ dc φ0 + φ0S dc (eµ φ ) . · " # Eq. (73) can be reexpressed in the familiar notation of      GR:20 (78)

G [n]= κT˚ [n] (74) ab ab T aα of Eq. (78) is essentially the ungauged counterpart of T˚ in Eq. (75), although lattice-specific aspects of where ab their expressions distinguish them. Nonetheless, in the n ν n c n µ n continuous, ungauged (g η, Γ 0) limit: Rab[ ] =e a[ ]eµ [ ] [ ] · νcb → → 1 Gab[n] =Rab[n] gab[n]R[n] · − 2 αβ b ca ˚ cα · η eβ g Tab T . (79) ˚ n µ n ˜σ n + n ˜ n − σ n ≈ Tab[ ] =e a[ ]ηµσ φ [ ]D φ0[ ]+ φ0[ ]D φ [ ] . · b b h (75)i In this sense, T˚ab appearing on the right hand side of Eq. (74) behaves like the familiar energy-momentum ten- We note that by allowing the exchange of lattice and sor of GR. On the other hand, R = 0 leads to an in- Lorentz indices, [ab] 6 dicial asymmetry in T˚ab. This asymmetry in the energy- n c n momentum tensor is a typical feature of EC gravity. Rab[ ]=  acb[ ]. (76) Furthermore, we should expect that, as in previous expositions of continuous EC gravity [12, 13, 30], the covariant divergence of T˚ab will be nonvanishing in our 20 We use the notation T˚ to distinguish the gauged energy- discrete gauged theory. Rather than attempting to define momentum from its ungauged counterpart T —as appears in a conservation law from T˚ab as defined in Eq. (75), we Eq. (31). T¯ and T˚¯ denote the energy-momenta symmetrized instead ‘gauge’ the symmetrized tensor T¯aα from Eq. (31) ˜ over (φ ↔ φ). of the ungauged theory. Substituting from the EOM of 12

Eq. (55), we calculate as follows: particular, our discrete lattice theory describes the coex- istence of a ‘digital’ absolute universal Newtonian clock + ˚¯aα n + α bc ˚−a a ˜µ − nt , and an ‘analog’ relativistic local Einsteinian clock Da T [ ] =Da e bg S eµ φ Dc φ0 · e{t [}n].  n   a We may view our theory as a natural consequence of + φ˜ S˚−a D−(e aφµ) φ φ˜ 0 c µ − ↔ an axiom and an experimental fact:   o  a α bc ˜ − − µ ˜µ + − • =eµ e bg φ0 Da , Dc φ + φ Da , Dc φ0 Axiom: The universe is discrete.   • Fact: Its laws have Poincar´esymmetry.  φ φ˜ .   − ↔ (80) +  The discrete principal G -bundle we have constructed is In the continuous limit, this conservation law takes the an instinctive mathematical model for this axiomatic uni- form verse. However, we ascribe no uniqueness to the discrete principal bundle in this regard. + ˚¯aα n a α bc ˜ µ ν ˜µ b ˜ Da T [ ] eµ e bg φ0R νacφ + φ S ac∂bφ0 φ φ It is worth dwelling in some detail on the nature of the ≈ − ↔ discrete lattice we have defined. Its ‘dimensionality’ is h i (81)  only determined by the number of edges linking a vertex b to its neighbors. Furthermore, there is nothing within where Cartan’s S ac is defined by the an- tisymmetric part of the affine connection: the hypercubes of the lattice—it is not a structure em- bedded in some larger continuous manifold; the lattice in b b b S ac =Γ ac Γ ca. (82) our theory represents the entirety of the universe. In this · − spare conception, the lattice is better regarded as a data Eq. (81) recovers a result similar to Eq. (4.39) of [12]. structure that contains a countable data set. It can be In EC gravity, the gauged energy-momentum tensor is deformed, and its information can be stored in any man- conserved up to second-order derivatives—a remainder ner desired, so long as its oriented graph and the data of that, due to its ‘non-locality’, is nonetheless consistent its fields are preserved. In this sense, our lattice theory with the equivalence principle of GR. [13] shares the deformability characteristic of the holographic We lastly observe that the conservation law in Eq. (80) principle [31, 32]. admits local Poincar´etransformations, as we originally A description of the universe as a lattice—more pre- hoped for our gauged 5-vector theory. cisely, as a discrete principal bundle—raises more ques- tions than it answers. Taking the notion of a discrete uni- verse seriously, one is compelled to ask: How do the fun- CONCLUSIONS AND QUESTIONS damental forces regulate the sizes of lattice ‘pixels’—that µ n is, the values of e a[ ]? How many pixels comprise the Having developed lattice 5-vector theory and demon- various structures of an atom? Can the oriented graph strated its apparent compatibility with the formalism of of the lattice evolve? Are there viable lattice geometries Einstein-Cartan gravity, we now reflect upon the physical beyond the hypercubic lattice, and how would their pre- and mathematical implications of this theory. We allow dictions differ? How can we observe the effects of the ourselves considerable latitude in this discussion, and of- lattice, and at what scales would these effects be visible? fer more speculative thoughts in the hope of spurring This last question raises the possibility that lattice ef- further extensions of this work in the community. fects might be observed on large scales in astronomical data. For example, a pixelated universe would presum- ably vary in its ‘resolution’ [n], which could be esti- Physical Considerations mated via a local average of theR volumetric measure e[n]:

The foregoing Poincar´elattice gauge theory appears to −1 n m have significant implications for our interpretation of the N [ ] = e[ ] m n . (83) R · | − |≤N physical universe. In our judgment, the most consequen- h i tial suggestion of this work is the possibility that the laws In ‘low-resolution’ sectors of a discrete universe, a con- of physics can be faithfully rendered on a discrete lattice. tinuous gravitational theory such as GR would predict Our redefinition of Poincar´esymmetry is central to this different outcomes from experiment. For example, a dis- evolution of our physical intuition. crete theory would modify predictions for the radial pro- We also observe the modified notions of time and space file of a galaxy’s rotational speed. appearing in our theory. As discussed in our companion Another ‘anomaly’ of a low-resolution part of the uni- paper, 5-vector theory apparently draws upon both con- verse would appear as a pressure in a continuous physical ceptions of time asserted by Newton and Einstein. In model. Suppose an evacuated volume of spacetime were 13 surrounded by a gas. If the evacuated volume were in- matter of the Standard Model. However, it is not clear stead comprised of relatively few pixels, the entropic ori- how a discrete spinorial gauge theory of gravity would gin of its supposed low pressure would be nullified; after couple to a scalar particle like the Higgs boson. Reinter- all, there would be few lattice vertices for the surround- preting the scalar Higgs field as a 5-vector, on the other ing gas to occupy. Similarly, the expansion of a single hand, naturally prescribes such a coupling. µ n lattice pixel via the evolution of e a[ ] would seemingly Because of the matter-coupling defined in gauged 5- require less energy than the expansion of an analogous vector theory, we have been able to employ a Poincar´e volume of spacetime. gauge field whose Lorentz components are in the stan- These observations invite the appealing speculation dard representation. Cartan’s frame field therefore finds that the supposed effects of dark matter and dark energy natural purchase in the solder field of 5-vector theory, could be reinterpreted as the tell-tales of a discrete uni- and serves as a map between the standard Lorentz gauge verse. The uncountability of spacetime in the ‘volumes’ field and the affine connection, as in Eq. (42). Indeed, of lattice hypercubes might lead in continuum models the standard Lorentz representation, embodied in the in- µ n to the residual matter and energy that have been esti- dex µ of e a[ ], comports with our notion of the met- µ ν mated to comprise 95% of the known universe [33]. ric gab = ηµν e ae b as measuring distances in a pseudo- We also note that,∼ if desired, a cosmological constant Riemannian spacetime. can be introduced in a discrete EC gravity formalism via In this sense, we may speculate as to the finiteness of the vierbeins, as in [19]. a quantized version of our lattice theory of gravity. On We may further consider 5-vector theory from a quan- the one hand, the lattice has countably many discrete tum theoretical point of view. We note that path integral data points. On the other hand, our solder field may [34] formulations of quantum field theories are better- nonetheless gauge transform to an infinitesimal UV scale. defined on a lattice than in continuous spacetime. There- This duality of discrete and continuous scales requires fore, we hope that our discrete, Poincar´e-symmetric the- further study. ory, once quantized, might admit unitary time evolution. Although we will not pursue them here in any detail, We may also consider fluctuations in eµ[n]—the trans- fermionic extensions of the 5-vector are apparently de- lation components of the solder field—from a quantum fined without obvious additional complication. Following mechanical perspective. Examining the expression the technique of 5-vector theory, the spacetime deriva- tives (∂µψ) of a Dirac fermion would be ‘internalized’ µ φ = φ0 eµφ within ψ—say, as ψµ. For example, a 20-vector ψ could − (84) µσ b + incorporate the four components of the Dirac fermion and = φ0 eµη eσ db φ0, − the 16 spacetime derivatives of these components. Such one notes that quantum fluctuations in eµ might appear a higher dimensional Poincar´erepresentation would nat- as fluctuations in the ‘location’ of the field φ[n]. In a urally incorporate both spinorial and standard Lorentz sense that only a fully quantized theory would make rig- transformations in a single field. orous, longitudinal gravitational fluctuations in 5-vector One prediction of 5-vector theory is that seemingly theory appear to give rise to an effect reminiscent of scalar particles—such as the Higgs boson—have distinct Heisenberg uncertainty. antiparticles of identical mass. Thus, there would exist We recall that the translation components eµ remained a twisted 5-vector that would presumably correspond to indeterminate in the EOM of Eq. (71), and that we omit- the negative frequency modes of the scalar Higgs field. ted the translation DOF from our definition of gauge A more complete set of gauge particles could also be curvature in Eq. (63). It should be noted that, when added to 5-vector theory without any obvious complica- simulating 5-vector theory classically, these DOF pose tion. This would involve expanding the structure group no obstacle; they can be set to an arbitrary constant of the discrete principal Poincar´ebundle. For example, a value—e.g., zero—without penalty. Should longitudinal discrete universe with the four fundamental forces might gravity waves ever be observed, it would be natural to be described by the following discrete principal bundle: extend our theory to describe a ‘translation curvature’. + We do not attempt to understand the apparent absence P = n G SU(3) SU(2) U(1). (85) · { }× × × × of such a force, but we note that prior work on the role of symmetry breaking in gravity—as described in [35], for example—appears to offer an attractive framework for Mathematical Considerations exploring this matter. In some ways, lattice 5-vector theory seems more nat- A noncanonical gauge theory ural than previous lattice gauge theories of gravity. In the main, such theories have focused on spinorial repre- Throughout our construction of gauged 5-vector the- sentations of the Lorentz gauge field, presumably due to ory, we have attempted to highlight its differences from a desire to produce a theory that couples to the fermionic a canonical Yang-Mills gauge theory. While the presence 14 of the solder field in our definition of gauge curvature is pr[v](L[n]) = 0, the M-tuple A is determined by the already striking, the most important difference appears higher Euler operators, as follows: to be the already-local symmetry of the ungauged La- N ˜ grangian. The very purpose of gauge fields is modified in a ia +1 1 I ℓ I,a n A = (S ) Q E ℓ (L[ ]) . (87) our theory; rather than restore local symmetry to the La- #I +1 − u ℓ=1 #I≥0 grangian, the gauge fields serve to restore local symmetry X X   to conservation laws. To systematically discover the conservation laws of 5- We furthermore note that the conservation laws of our vector theory via the discrete Noether procedure, the theory, expressed in Eqs. (31) and (80), were not derived “least” index that appears in the Lagrangian L[n] should from a Noether procedure; we intuited them, primar- be n. For our present purposes, since d− appears in ≥ b ily from the development of continuous 5-vector theory Eq. (18), this requires that we shift the ‘base point’ of our in our companion paper. We hope that a mathemati- Lagrangian forward. One may correspondingly denote cal formalism that accommodates the noncanonical fea- i = aˆ = tˆ+ˆx +ˆy +ˆz tures of our gauge theory would suggest its own Noether- · (88) · a like procedure—and thereby more systematically gener- X ate the conservation laws we have identified. and define the shifted Lagrangian Li as follows:

˜ T Li[n] =φ[n + i]e [n + i]dme[n + i]φ[n + i] · A discrete gauge-theoretic variational calculus µ ν =φ˜ [n + i]gµν [n + i]φ [n + i] φ˜µ[n + i]e a[n + i]d+φ [n + i] (89) We have employed a few elements of novel mathemati- − µ a 0 cal machinery in this paper—most crucially, the discrete ˜ n i − b n i ν n i φ0[ + ]db (eν [ + ]φ [ + ]) covariant Euler operator ˚E defined in Appendix B. ˚E es- − q q + m2φ˜ [n + i]φ [n + i]. sentially defines a new variational derivative for discrete 0 0 gauge theories. However, we do not systematically de- The Noether procedure is then performed on Li[n], as in velop a gauge covariant lattice variational calculus. Such Eq. (30). a mathematical effort—which might extend the tools of [6] and [8], for example, to a gauged lattice—might sub- stantially deepen the results we have derived here. APPENDIX B

In this appendix, we define the discrete gauge covari- APPENDIX A ant Euler operator ˚Eq used to derive our gauged EOM in Eq. (53). We first recall the (non-covariant) discrete We include an appendix to outline the tools of dis- Euler operator defined in Eq. (22): crete variational calculus necessary to pursue the canon- ∂L[n] ical Noether procedure in a discrete ungauged theory. E L n −m q( [ ]) = S n m An introduction of notation is helpful here. For a sys- · m ∂q[ + ] i X tem of M independent variables x and N dependent −m n ℓ { } ∂ S L[ ] variables u , we denote a (k 0)-order multi-index J = (90) ≥ ∂q[n] by J (j ,...,j ), where 1 j M. We let #J de- m  ≡ 1 k ≤ i ≤ X note the order (i.e., length) of the multi-index J, where ∂ SmL[n] = . any repetitions of indices are to be double-counted. We ∂q[n] let ( I ) I!/[J!(I J)!] when J I, and 0 otherwise. We m  J ≡ \ ⊆ X define I!= ˜i1! ˜iM ! , where ˜ia denotes the number of In the second equality, we have noted that the shift op- ··· occurrences of the integer a in multi-index I. I J denotes erator can be brought within ∂/∂q[n + m] so long as q is  \ the set-difference of multi-indices, with repeated indices shifted accordingly. In the last equality, and simply out treated as distinct elements of the set. of preference, we have used the sum over all m to flip the Let us define the higher Euler operators as in [6]: sign of the shift operator. The covariant Euler operator ˚Eq is now straightfor- I L n J n −I ∂ [ ] wardly generalized from Eq. (90) as follows: E ℓ (L[ ]) = S . (86) u · J ℓ n I · I J ! ∂u [ + ] m X⊇ ∂ S˚ L[n] ˚E (L[n]) = v q · ∂q[n] For a vertical variational symmetry of a Lagrangian · m  n X (91) L[ ], we may derive its canonical conservation law ∂ e−mSmL[n] DivA = 0 from the total homotopy operator of [6]. = . ∂q[n] For v defined by its characteristics Qℓ and satisfying m  X 15

m m This definition employs the notations S˚ and e− used [10] P. E. Hydon, Difference Equations by Differential Equa- to define the discrete covariant derivative in Eq. (46). tion Methods, Cambridge Monographs on Applied and Intuitively, ˚E shifts the Lagrangian and then parallel Computational No. 27 (Cambridge Univer- q sity Press, 2014). transports it back to n for differentiation. Since [11] M. Desbrun et al., arXiv preprint math/0508341 (2005). 48 m [12] F. W. Hehl et al., Reviews of Modern Physics , 393 S = L[n]= S L[n], · (92) (1976). · n m [13] F. W. Hehl, in Cosmology and Gravitation: Spin, X X Torsion, Rotation, and Supergravity, NATO Advanced ˚Eq can be viewed as parallel transporting each term of Study Institutes Series, edited by P. G. Bergmann and the action S back to n. V. De Sabbata (Springer US, Boston, MA, 1980) pp. 5– 61. A rigorous demonstration of the viability of the opera- ˚ [14] E. Cartan, Annales scientifiques de l’cole normale su- tor Eq would require a robust discussion of the variational prieure 40, 325 (1923). complex on the discrete principal bundle—extending the [15] E. Cartan, Annales scientifiques de l’cole normale su- variational complex of [6], for example—which falls be- prieure 42, 17 (1925). yond the scope of our present efforts. [16] T. W. B. Kibble, Journal of 2, 212 (1961). However, we observe that ˚Eq satisfies a minimal re- [17] D. W. Sciama, Reviews of Modern Physics 36, 463 quirement: im D± ker ˚E . That is, a ⊂ q (1964). [18] K. G. Wilson, Physical Review D 10, 2445 (1974). ˚ ± n [19] L. Smolin, Nuclear Physics B 148, 333 (1979). Eq Da Q( , qi)=0 (93) ◦ [20] C. Mannion and J. Taylor, Physics Letters B 100, 261 h i (1981). for any function Q of the lattice coordinates n and { } [21] K. Kondo, Progress of Theoretical Physics 72, 841 dependent fields qi q. (1984). { } ∋ [22] P. Menotti and A. Pelissetto, Annals of Physics 170, 287 (1986). 35 ACKNOWLEDGMENTS [23] P. Menotti and A. Pelissetto, Physical Review D , 1194 (1987). [24] H. W. Hamber and R. M. Williams, Physical Review D Thank you to Professor Nathaniel Fisch for his early 81 (2010), 10.1103/PhysRevD.81.084048. encouragement of this effort. Thank you to Yuan Shi, [25] S. Catterall et al., The European Physical Journal whose intuitions helped spur this work. And thank you Plus 127 (2012), 10.1140/epjp/i2012-12101-4, arXiv: to Sebastian Meuren, for several helpful discussions. This 0912.5525. [26] Y. Ne’eman and T. Regge, Physics Letters B 74, 54 research was supported by the U.S. Department of En- (1978). ergy (DE-AC02-09CH11466). [27] A. Trautman, Bull. Acad. Polon. Sci., s´er. sci. math, astr. et phys. 20 (1972). [28] J. B. Kogut, Reviews of Modern Physics 51, 659 (1979). [29] A. Ipp and D. M¨uller, Eur. Phys. J. C 78, 884 (2018). [30] A. Trautman, Encyclopedia of Mathematical Physics 2, [1] A. S. Glasser and H. Qin, Lifting Spacetime’s Poincar´e 189 (2006). Symmetries (in preparation, 2019). [31] G. t’ Hooft, arXiv:gr-qc/9310026 (1993), arXiv: gr- [2] E. Noether and M. A. Tavel, Transport Theory and Sta- qc/9310026. tistical Physics 1, 186 (1971), arXiv: physics/0503066. [32] L. Susskind, Journal of Mathematical Physics 36, 6377 [3] C. N. Yang and R. L. Mills, Physical Review 96, 191 (1995), arXiv: hep-th/9409089. (1954). [33] N. Jarosik et al., The Astrophysical Journal Supplement [4] A. Einstein, Relativity: The Special and General Theory Series 192, 14 (2011). (Project Gutenberg, 2009—orig. 1916). [34] R. P. Feynman, Rev. Mod. Phys. 20, 367 (1948). [5] P. J. Olver, Applications of Lie Groups to Differential [35] D. K. Wise, J. Phys.: Conf. Ser. 360, 012017 (2012). Equations, 2nd ed., Graduate Texts in Mathematics, Vol. 107 (Springer-Verlag New York, 1993). [6] P. E. Hydon and E. L. Mansfield, Foundations of Com- putational Mathematics 4, 187 (2004). [7] T.D. Lee, Journal of Statistical Physics 46 (1987). [8] D. Bleecker, Gauge Theory and Variational Principles, Global Analysis (Addison-Wesley Publishing Co., Read- ing, MA, 1981). [9] F. Schuller, “Local representations of a connection on the base manifold: Yang-Mills fields - Lec 22,” https: //www.youtube.com/watch?v=KhagmmNvOvQ&t=1s& list=PLPH7f_7ZlzxTi6kS4vCmv4ZKm9u8g5yic&index=23 (2015).