Universal gauge-invariant cellular automata
UUniversal gauge-invariant cellular automata
Pablo Arrighi
Université Paris-Saclay, CNRS, LMF, 91190 Gif-sur-Yvette, FranceIXXI, Lyon
Marin Costes ! ENS Paris-Saclay, CNRS, LMF, 91190 Gif-sur-Yvette, France
Nathanaël Eon
Aix-Marseille Université, Université de Toulon, CNRS, LIS, Marseille, France
Abstract
Gauge symmetries play a fundamental role in Physics, as they provide a mathematical justificationfor the fundamental forces. Usually, one starts from a non-interactive theory which governs ‘matter’,and features a global symmetry. One then extends the theory so as make the global symmetry intoa local one (a.k.a gauge-invariance). We formalise a discrete counterpart of this process, known asgauge extension, within the Computer Science framework of Cellular Automata (CA). We prove thatthe CA which admit a relative gauge extension are exactly the globally symmetric ones (a.k.a thecolour-blind). We prove that any CA admits a non-relative gauge extension. Both constructions yielduniversal gauge-invariant CA, but the latter allows for a first example where the gauge extensionmediates interactions within the initial CA.
Theory of computation → Models of computation
Keywords and phrases
Cellular automata, Gauge Invariance, Universality
Digital Object Identifier
Acknowledgements
The authors would like to thank Guillaume Theyssier for asking us the questionwhether any CA admits a gauge extension. This publication was made possible through the support ofthe ID
Symmetries are an essential concept, whether in Computer science or in Physics. In this paperwe explore the Physics concept of gauge symmetry by taking it into the rigorous, ComputerScience framework of Cellular Automata (CA). Implementing gauge-symmetries withinCA may prove useful in the fields of numerical analysis; quantum simulation; and digitalPhysics—as these are constantly looking for discrete schemes that simulate known Physics.Quite often, these discrete schemes seek to retain the symmetries of the simulated Physics;whether in order to justify the discrete scheme as legitimate or as numerically accurate(e.g. by doing the Monte Carlo-counting right [9]). More specifically, the introduction ofgauge-symmetries within discrete-time lattice models has proven useful already in the fieldof Quantum Computation, where gauge-invariant Quantum Walks and Quantum CellularAutomata [1] provide us with concrete digital quantum simulation algorithms for particlePhysics. These come to complement the already existing continuous-time lattice modelsof particle Physics [7, 12]. Another field where this has played a role is Quantum errorcorrection [10, 11], where it was noticed that gauge-invariance amounts to invariance undercertain local errors. This echoes the fascinating albeit unresolved question of noise resistancewithin Cellular Automata [8, 15]. © Pablo Arrighi, Marin Costes and Nathanaël Eon;licensed under Creative Commons License CC-BY 4.042nd Conference on Very Important Topics (CVIT 2016).Editors: John Q. Open and Joan R. Access; Article No. 23; pp. 23:1–23:14Leibniz International Proceedings in InformaticsSchloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany a r X i v : . [ n li n . C G ] F e b In [13] the authors study G –blind cellular automata, where G is a group of permutationacting on the state space of cells. Blind cellular automata are globally symmetric under G , i.e. the global evolution commutes with the application of the same g ∈ G at once onevery cell. They show the surprising result that any CA can be simulated by such a globallysymmetric CA, when G is the symbol permutations. Globally symmetric CA are thereforeuniversal. Local symmetry, aka gauge symmetry, is way more stringent however: a different g x can now be chosen for every cell x . Still, in this paper, we prove that any CA can beextended into a gauge-invariant CA. Gauge-invariant CA are therefore universal.From a Physics perspective one usually motivates the demand for a certain gaugesymmetry, from an already existing global symmetry. From a mathematical perspective,the gauge field that then gets introduced for that purpose is often seen as a connectionbetween two gauge choices at neighbouring points. This raises questions however, becausethere is no immediate reason why a gauge symmetry should necessarily arise from an alreadyexisting global symmetry (one could ask for a certain ad hoc gauge symmetry from scratch).Nor is there an immediate reason why a gauge field should necessarily be interpretable asa connection (a gauge field could be made to hold absolute instead of relative informationabout gauge choices).In this paper, we prove an original result relating these two folklore perspectives about gaugetheories. Namely, we prove that the CA that admit relative gauge extension are exactlythose that have the corresponding global symmetry in the first place.Although the gauge field was initially introduced in order to obtain gauge symmetry,it allows for new dynamics. Amongst those dynamics, one could ask for the matter fieldto influence the dynamics of the gauge field, as is the case in Physics. In this paper, weprovide a first a Gauge-invariant CA where this is happening. This CA is obtained througha non-relative gauge extension. We leave it as an open question whether the same can beachieved though a relative gauge extension.The present work builds upon two previous papers by a subset of the authors, which laiddown the basic definitions of gauge-invariance for CA and provided a first set of examples, inboth the abelian [2] and the non-abelian [3] cases. Sec. 2 first recalls these basic definitions,but it also formalises the notions of general and relative gauge extensions, which were stillmissing. Sec. 3 shows that CA admit a relative gauge extension if and only if they are globallysymmetric. Sec. 4 shows that any CA admits a general gauge extension. Sec. 5 draws theconsequences upon universality. Sec. 6 provides a first example of a gauge-extended CAwhose gauge field is sourced by the matter field. Cellular automata (CA) consist in an array of identical cells, each of which may take one in afinite number of possible states. The whole array evolves in discrete time steps by iterating afunction F . Moreover this global evolution F is shift-invariant (it acts everywhere the same)and causal (information cannot be transmitted faster than some fixed number of cells pertime step). Let us make this formal. ▶ Definition 2.1 (Configuration) . A configuration c over an alphabet Σ and a space Z d is afunction that associates a state to each point: c : Z d −→ Σ The set of all configurations will be denoted C . . Arrighi, M. Costes and N. Eon 23:3 A configuration should be seen as the state of the CA at a given time. We use theshort-hand notation c x = c ( x ) for x ∈ Z d and c | I for the configuration c restricted to the setI—i.e. c : I −→ Σ—for I ⊆ Z d . The association of a position and its state is called a cell.The way to describe a global evolution F that is causal, is via the provision of a localrule. A local rule takes as input a configuration restricted to x + N , and outputs the nextvalue of the cell x , i.e. f : Σ N −→ Σ, where N is a finite subset of Z d referred to as ‘theneighbourhood’. Applying f at every position x simultaneously, implements F . ▶ Definition 2.2 (Cellular automata) . The CA F having neighbourhood N and local function f : Σ N −→ Σ is the function F : C −→ C such that for all x ∈ Z d , F ( c ) x = f ( c | x + N ) . We sometimes denote by c t,x the value of a cell at position x and time t , where c t +1 = F ( c t ). Global symmetry
We say that a CA is globally symmetric whenever its global evolution is invariant under theapplication of the same alphabet permutation at every position at once. Globally symmetricCA are also known as G -blind CA [13] with G a group of permutations over Σ. ▶ Definition 2.3 (Globally symmetric) . Let F : Σ Z n → Σ Z n be a CA and G be a groupof permutations over Σ . For all g ∈ G , let γ g denote its application at every positionsimultaneously: γ g ( c ) i = g ( c i ) . We say that F is globally G -symmetric if and only if, for any g in G , we have F ◦ γ g = γ g ◦ F . Local/gauge symmetry
We say that a CA is locally symmetric whenever its global evolution is invariant underthe application of a local permutation at every position. The first difference with globallysymmetric CA is the permutation is now allowed to differ from one position to the next.The second difference is that the permutation is now allowed to act on the surrounding cells.Locally symmetric CA are referred to as gauge-invariant CA [2, 3, 4]. ▶ Definition 2.4 (Local gauge-transformation group) . Let g be a permutation over Σ (2 s +1) d ,with s ∈ N . We denote by g x : C −→ C the function that acts as g on the cells at [[ x − s, x + s ]] d ,and trivially everywhere. A local gauge-transformation group G is a group of bijections over Σ (2 s +1) d , such that for any g, h ∈ G and any x ̸ = y ∈ Z d , g x ◦ h y = h y ◦ g x . This permutation condition makes it irrelevant to consider which local gauge-transformationgets applied first, so that the product g x h y be defined. The condition is decidable, checkingit over the [[ − s, +2 s ]] d suffices. ▶ Definition 2.5 (Gauge-transformation) . Consider G a group of local gauge-transformations.A gauge-transformation is then specified by a function γ : Z d −→ G . It is interpreted asacting over C as follows: γ ( c ) = ( Y x ∈ Z d γ x )( c ) , where γ x is short for γ ( x ) x . We denote by Γ the set of gauge-transformations. C V I T 2 0 1 6
Notice how an element γ ∈ Γ may be thought of as a configuration over the alphabet G —with g x the state at x .Gauge-invariant CA are ‘insensitive’ gauge-transformations: performing γ before F amounts to performing some γ ′ after F . ▶ Definition 2.6 (Gauge-invariant CA) . Let F be a CA, G be a local gauge-transformationgroup, and Γ be the corresponding set of gauge-transformations. F is Γ -gauge-invariant ifand only if there exists a CA Z over the alphabet G , such that for all γ ∈ Γ : Z ( γ ) ◦ F = F ◦ γ. The reason why γ ′ must result from a CA Z , instead of being left fully arbitrary, is because F is deterministic, shift-invariant and causal—from which it follows that γ ′ , if it exists, canbe computed deterministically, homogeneously and causally from the γ applied before. Thus,the above is demanding a weakened commutation relation between the evolution F andthe set of gauge-transformations Γ. In practice in Physics Z is often the identity, makinggauge-invariance a commutation relation. This will be the case in our constructions. In Physics, one usually begins with a theory that explains how matter freely propagates, i.e.in the absence of forces. This initial theory solely concerns the ‘matter field’, and is not gauge-invariant. For instance, the Dirac equation, which dictates how electrons propagate, is not U (1)–gauge-invariant. Next, one enriches the initial theory with a second field, the so-calledgauge field, so as to make the resulting theory gauge-invariant. For instance, the case of theelectron, U (1)–gauge-invariance is obtained thanks to the addition of the electromagneticfield. The resulting theory can still account for the free propagation of the matter field, butthe presence of the gauge field also allows for richer behaviours, e.g. electromagnetism. Quitesurprisingly three out of the four fundamental forces can be introduced mathematically, andthereby justified by gauge symmetry requirements, through this process of ‘gauge extension’.But when is it the case that a theory is a gauge extension of another, exactly? In Physicsthis is left informal. One of the contributions of this paper is the provide a first rigorousdefinition of the notion of gauge extension, and of its relative subcase, in the discrete contextof CA. General gauge extension
A gauge extension must simulate the initial CA, extend the required gauge-transformations,and achieve gauge-invariance overall: ▶ Definition 2.7 (gauge extension) . Let F be a CA over alphabet Σ . Let Γ be a gauge-transformation group. Let Λ be a gauge field alphabet. A gauge extension of ( F, Γ) is a tuple ( F ′ , Γ ′ ) with F ′ a CA over alphabet Σ × Λ and Γ ′ an extended local gauge-transformationgroup, such that : (Simulation) there exists ϵ ∈ Λ such that F ′ simulates one step of F when the gauge fieldvalue is set ϵ everywhere. In other words for any c ∈ Σ Z d , there exists e ′ ∈ Λ Z d , F ′ ( c, e ) = ( F ( c ) , e ′ ) where e is the constant gauge field configuration ( x ϵ ) . . Arrighi, M. Costes and N. Eon 23:5 (Extension) Γ ′ extends Γ : for any γ ′ ∈ Γ ′ , there exists γ ∈ Γ such that for any c, λ ∈ Σ Z d × Λ Z d , there exists λ ′ ∈ Λ Z d , γ ′ ( c, λ ) = ( γ ( c ) , λ ′ ) (1)(Gauge-invariance) F ′ is Γ ’-gauge-invariant.We used here the fact that (Σ × Λ) Z d is the same as Σ Z d × Λ Z d Notice that when the gauge field does not evolve in time, we can rewrite the simulationcondition as F ′ ( c, e ) = ( F ( c ) , e ). Then F ′ intrinsically simulates F , in the usual way,whenever the gauge field is set to e .Intuitively, the gauge field’s role is to keep track of which gauge-transformation gotapplied where, so as to hold enough information to insure gauge-invariance. There aredifferent ways to do this; for instance one could indeed store the ‘gauge’ at each point, i.e.which gauge-transformation has happened at the specific point. But one could be moreparsimonious and store just the ‘relative gauge’, i.e. which gauge-transformation relates thatwhich has happened at every two neighbouring points. Relative gauge extension
The standard choice in the Physics literature is to place the gauge field between the mattercells only—i.e. on the links between two cells. This choice of layout is sometimes referred toas the ‘quantum link model’ [6, 14]. The mathematical justification for this choice, is preciselythat the gauge field may be interpreted as relative information between neighbouring mattercells. Geometrically speaking, it may be understood as a ‘connection’ relating two closeby‘tangent spaces’ on a manifold.Our previous definition of general gauge extensions does allow for such relative gaugeextensions as a particular case, up to a slight recoding, as shown in Fig-1, i.e. the link modelis simulated by transferring the value of a gauge field on a link, to the vertex at the tip ofthe link.
NS EW xy (a) The link model layout. . .
W SN E xy (b) . . . encoded in the general gauge extension layout. Figure 1
Capturing the link model used for relative gauge extensions with the general definition.
The following specialises the previous, mathematical notion of gauge extension, to therestricted way in which it is understood in Physics: ▶ Definition 2.8 (Relative gauge extension) . Given a CA F and a local gauge-transformationgroup Γ of radius s = 0 , we say that a gauge extension ( F ′ , Γ ′ ) is relative when: (a) the gauge field is positioned on the links ( x, x + e d ) , where e d takes values in { (1 0 0 . . . ) , (0 1 0 . . . ) , . . . } . (b) the gauge field takes values in G —i.e. Λ = G C V I T 2 0 1 6 (c) for every position x the gauge-transformations Γ ′ act both on the matter field at x according to g x ∈ G , and on the gauge fields of its links, as follows: ( g x ( a ) ( x − e d ,x ) = g x ◦ a ( x − e d ,x ) g x ( a ) ( x,x + e d ) = a ( x,x + e d ) ◦ g − x (2)Thus relative extension keeps track of the difference of gauge between two neighbouring cells.The above definitions were given for the Z d grid, in order to establish the notion of gaugeextension in full generality. The next section, however, will be given just in one dimension( d = 1) for clarity. We have established it in arbitrary dimension d in a private manuscript. From a Physics perspective, the gauge symmetry one seeks to impose usually comes froman already existing global symmetry. We show here that there is an equivalence betweenbeing globally G -symmetric and having a gauge extension with respect to G a subgroup ofthe permutations of Σ. ▶ Theorem 3.1 (Global symmetry and relative gauge extension) . Let F be a CA over alphabet Σ and G a subgroup of the permutations of Σ . Then the following two properties are equivalent: (i) F is globally G -symmetric (ii) ( F, G ) admits a relative gauge extension ( F ′ , G ′ ) with the identity for the gauge fieldevolution. Proof. (i ⇒ ii) Let f be the local rule of F with radius r . Let F ′ be a CA of radius r over theextended configurations—containing a gauge field in between neighboring cells—such thatthe gauge field evolution is the identity and the local rule f ′ for the evolution of the matterfield is defined as follows: f ′ ( c − r , a ( − r +1 , − r ) , ..., c , a (0 , , ..., c r ) = f ◦ − ri = − a ( i,i +1) ( c − r ) ...a ( − , ( c − ) c a − , ( c ) ... ◦ i = r a − i − ,i ) ( c r ) (3)where a is the gauge field and ◦ − ji = − a ( i,i +1) ( c − j ) = a ( − , ◦ . . . ◦ a ( − j +1 , − j +2) ◦ a ( − j, − j +1) ( c − j ) ◦ ji =1 a − i,i +1) ( c j ) = a − , ◦ . . . ◦ a − j − ,j − ◦ a − j − ,j ) ( c j ) .f ′ is defined so as to apply f on a configuration that is on the same gauge basis—i.e. whererelative gauge-transformations are cancelled and every cell is looked through the eyes of thegauge at position 0.We shall now prove that ( F ′ , G ′ )—with G ′ defined through Eq.(2)—is a relative gaugeextension of ( F, G ).The fact that this extension is relative is immediate from the definition. We thereforeneed to prove that this extension has the 3 required properties from definition 2.7. . Arrighi, M. Costes and N. Eon 23:7 (Simulation)
The fact that F ′ simulates F when the gauge field is the identity is immediatefrom the definition of f ′ . (Extension) Because G ′ is defined through definition 2.8, it is also immediate that itverifies the extension property. (Gauge-invariance) For any γ ′ ∈ Γ ′ —where Γ ′ is built from G ′ through definition 2.5—wewill check that γ ′ ◦ F ′ = F ′ ◦ γ ′ .For j between 1 and r , we apply the gauge-transformation on the inputs ( c and a )—usingEq.(2)—and obtain by simple computation the following : ◦ ji =1 (cid:2) γ i ◦ a ( i − ,i ) ◦ γ − i − (cid:3) − γ j ( c j ) = γ ◦ h ◦ ji =1 a − i − ,i ) ( c j ) i ◦ − ji = − (cid:2) γ i +1 ◦ a ( i,i +1) ◦ γ − i (cid:3) γ j ( c j ) = γ ◦ h ◦ − ji = − a ( i,i +1) ( c j ) i where γ i ∈ G . Therefore using this and Eq.(3) we have that( F ′ ◦ γ ′ ( c, a )) = f ◦ γ ◦ − ri = − a ( i,i +1) ( c − r ) ...a ( − , ( c − ) c a − , ( c ) ... ◦ i = r a − i − ,i ) ( c r ) (4)where γ is here applied to every element of the tuple. Since F is globally G -symmetric,we have that f ◦ γ = γ ◦ f and therefore ( F ′ ◦ γ ′ ( c, a )) = γ ◦ ( F ′ ( c, a )) which finishesto prove that F ′ ◦ γ ′ = γ ′ ◦ F ′ through translation invariance and because the gauge fieldevolution is the identity.(ii ⇒ i) Suppose that ( F ′ , G ′ ) is a relative gauge extension of ( F, G ), we shall prove that F is globally G -symmetric. Let c be a configuration and e denote the empty configurationof the gauge field. For any local gauge-transformation g , we write γ g the global gauge-transformation applying g everywhere— g denotes both the element of G and G ′ dependingon the context: γ g ◦ F ′ ( c, e ) = γ g ( F ( c ) , a ) (gauge extension)= ( γ g ( F ( c )) , γ g ( a )) (gauge-transformation)and F ′ ◦ γ g ( c, e ) = F ′ ( γ g ( c ) , γ g ◦ e ◦ γ − g ) (gauge-transformation)= F ′ ( γ g ( c ) , e )= ( F ( γ g ( c )) , a ′ )where a and a ′ can be any gauge field configuration depending on F ′ . The G ′ -gauge-invarianceof F ′ give us γ g ◦ F ′ ( c, e ) = F ′ ◦ γ g ( c, e ) and thus γ g ( F ( c )) = F ( γ g ( c )) . Therefore F is globally G -symmetric. ◀ This theorem proves useful when looking for relative gauge extensions: first search fora global symmetry. The construction will be used in Sec. 5 to prove that relative gaugeextensions of CA are universal.
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We now prove that any CA can be intrinsically simulated by a gauge-invariant one, withrespect to any gauge-transformation group, of any radius. The construction of this sectionuses non-relative gauge extensions, but it allows us to get rid of the prior requirements thatthere be a global symmetry or that the gauge-transformations be of radius 0. ▶ Theorem 4.1 (Every CA admits a gauge extension) . For any CA F and gauge-transformationgroup Γ there exists for some gauge field alphabet a gauge extension ( F ′ , Γ ′ ) . Furthermorethe local rule of F ′ acts as the identity over the gauge field. Proof.
We give here a constructive proof for any CA over Z d .Let F be a CA of radius s ′ and G be a local gauge-transformation group of radius s . Wedenote r the highest radius between s and s ′ . In the following we will consider neighbourhoods R kx = [ x − k × r, x + k × r ] d of each point x ∈ Z d , with [ a, b ] = { n ∈ Z | a ≥ n ≥ b } .First we choose G as gauge field alphabet and define the effect of a global gauge-transformation γ x , which applies g ∈ G around x according to γ x ( a ) x = g ◦ a x . where a denotes a gauge field configuration. The definition is so that the gauge field simply keepstrack of every gauge-transformation applied around x . For any other cell of the gauge field, γ x has no impact. This condition along with the extension property of 2.7 fully defines thenew gauge-transformation group G ′ .Next we define a new local rule f ′ over the neighbourhood R ′ x . The definition belowjust states that the local rule applies Q i ∈ R x a − i to undo all previous gauge-transformations,computes the evolution of f , and finally reapplies all the gauge-transformations. I.e. f ′ (cid:0) c | R x , a | R x (cid:1) = (cid:16) Y i ∈ R x a i ◦ f | R x (cid:0) Y i ∈ R x a − i ( c | R x ) | R x (cid:1) , a x (cid:17) where f | R x denotes the function from R x to R x which calculates the temporal evolution ofour automaton.We can rewrite this local rule globally, using the notation a to denote either the gaugefield or a gauge-transformation which applies a x around each position x : F ′ ( c, a ) x = (cid:16) a ◦ F ◦ a − ( c ) , a (cid:17) x Let us check that ( F ′ , Γ ′ ) is a gauge extension: (Simulation) When the gauge field is the identity f ′ acts the same as f over the matterfield, and as the identity over the gauge field. (Extension) We used this property to define G ′ . (Gauge-invariance) For any γ ′ ∈ Γ ′ —where Γ ′ is built from G ′ through definition 2.5.We must check that γ ′ ◦ F ′ = F ′ ◦ γ ′ . We reason globally to simplify notations: F ′ ◦ γ ′ ( c, a ) x = F ′ ( γ ( c ) , γ ( a )) x (Extension)= (cid:16) γ ( a ) ◦ F ◦ γ ( a ) − ( γ ( c )) , γ ( a ) (cid:17) x (Definition of F ′ )= (cid:16) γ ◦ a ◦ F ◦ a − ◦ γ − ◦ γ (cid:0) c (cid:1) , γ ( a ) (cid:17) x (Commutation 2.4)= (cid:16) γ ◦ a ◦ F ◦ a − (cid:0) c (cid:1) , γ ( a ) (cid:17) x (Commutation 2.4) . Arrighi, M. Costes and N. Eon 23:9 γ ′ ◦ F ′ ( c, a ) x = γ ′ (cid:16) a ◦ F ◦ a − ( c ) , a (cid:17) x (Definition of F ′ )= (cid:16) γ ◦ a ◦ F ◦ a − (cid:0) c (cid:1) , γ ( a ) (cid:17) x (Extension) ◀ In [13], the authors prove that for any alphabet Σ containing 2 symbols or more, thereexists an intrinsically universal globally G -symmetric cellular automaton on Σ Z , where G is the group of all permutations of σ . The proof involves an extension which encodes theinformation in the structure of the configuration rather than the states, the idea being thata global transformation will conserve the structure—thus the information. Combining thisresult and Th. 3.1, we can easily prove the following corollary: ▶ Corollary 5.1 (Relative gauge-invariant cellular automata are universal) . For any alphabet Σ with | Σ | ≥ , and any cellular automaton F on Σ Z , there exists a G ′ -gauge-invariant CA F ′ which intrinsically simulates F , with G ′ the extended gauge-transformation based on G .Moreover, F ′ arises as the relative gauge extension of a CA. Proof.
Let F ′′ be a globally G -symmetric CA on Σ Z that intrinsically simulates F using[13, Theorem 1]. From theorem 3.1, ( F ′′ , G ) admits a relative gauge extension ( F ′ , G ′ ) withthe evolution of the gauge field being the identity. Thus F ′ intrinsically simulates F ′′ , fromwhich it follows that F ′ is a G ′ -gauge-invariant CA which intrinsically simulates F . ◀ Such result is interesting on two accounts: (i) it shows that universality only requiresrelative gauge information and does not need any absolute information stored in the gaugefield; (ii) it shows that relative gauge extensions, which are the ones usually appearing inPhysics, are universal. Still, the universality of gauge-invariant CA is an even more directcorollary of Th. 4.1. With that construction we can just pick any universal CA F , any localtransformation group G of any radius, and gauge-extend F into F ′ . F ′ acts trivially on thegauge-field in this construction, it thus intrinsically simulates F and is therefore universal. In both the construction of Th. 3.1 and Th. 4.1, the evolution rule of the gauge field to theidentity, meaning that it does not evolve with time. It is often the case in Physics that afurther twist is then introduced, so that the the matter field now influences the gauge field.We wish to do the same and find a gauge-extended CA whose gauge field influences thematter field, and whose matter field backfires on the gauge field.We use here the general definition of a gauge extension (Def. 2.7) to search a gaugeextension F ′ of a non-gauge invariant CA F . Without loss of generality, F ′ = ( F ′ , F ′ ), where F ′ takes ( c, a ) as input and returns the matter field after one time-step, and F ′ does thesame for the gauge field. We impose that the gauge (resp. matter) field be sourced by thematter (resp. gauge) field, in the strongest possible manner, i.e. we ask for F ′ (resp. F ′ ) tobe injective in its first (resp. second) parameter.We begin by choosing the alphabet Σ = { , , } and the space Z and we denote by c li and c ri respectively the left and the right part of the cell. In the following definitions weconsider that all the additions and all the subtractions are modulo 3. C V I T 2 0 1 6 b cb + c b - c i + 1 ii − i + 2 tt + 1 Figure 2
The local rule of F
We define the initial automaton by the local rule: F ( c ) i = ( c li − − c ri , c li + c ri +1 ), cf. Fig.2.We consider a local group of gauge-transformation containing three elements, namely: G = { σ , σ , σ } where σ i is the function of radius 0 that adds i to each part of the cell.We can check that F is not gauge-invariant for Γ (as defined from G ), by considering aconfiguration c which associates (1 ,
1) to position i and (0 ,
0) to all other positions. Let γ bea gauge-transformation which applies σ over i , γ ( c ) is then the fully empty configuration e .Since F preserves emptiness we have: F ◦ γ ( c ) = γ ( c ) = e But when we apply F to c we obtain non-empty cells in i + 1 and i −
1, this contradicts thegauge-invariance definition. This idea is illustrated in sub-Figs 3a and 3b.We now provide a gauge extension for ( F, Γ). We begin by choosing the gauge fieldalphabet Λ = Σ and placing the gauge field between each cell.Next we engineer the injective influence of the gauge field over the matter field in thesimplest possible way. We simply add, to each sub-cell of the matter field, the value of thenearest sub-cell of the gauge field during each evolution. See Fig. 4b.Finally we extend gauge-transformations to the gauge field (Fig. 4a) and choose theevolution of the gauge field (green cells of Fig. 4b) to make sure that F ′ is a reversible gauge-invariant dynamics. Notice that the figures where chosen so that all cases are given. Theproof of gauge-invariance for this example is given in appendix-A, and can be visually seenfrom Figs. 3c and 3d where a gauge-transformation does not impact the overall dynamics.Overall, starting from a CA F we have defined a gauge extension F ′ which features astrong interaction between the gauge and the matter field. In the world of CA this is thefirst example of the kind [5, 3]. Building this example required the choice of a very specificextension of the gauge-transformation over the gauge field (cf fig.4a) so as to obtain gauge-invariance whilst preserving reversibility and injectivity. Under a relative gauge extensionthis extension of the gauge-transformation is forced upon us, it seems hard to find such anexample.Notice that since the gauge field is sourced by the matter field it typically does notremain empty during the evolution. Thus F ′ can only simulate F for one time step. Thismay seem strange from a mathematical point of view, as we may expect from an extensionthat it preserves the original dynamics over several steps, too. But in Physics the initial nongauge-invariant theory is indeed used to inspire a more complex dynamics, which enrichesand ultimately diverges from the original one. . Arrighi, M. Costes and N. Eon 23:11 (a) F over c (b) F over γ ( c ) (c) F ′ over ( c, e ) (d) F ′ over γ ( c, e ) Figure 3
Space-time representation of F and F ′ over the same initial configurations c and γ ( c ),where c is the configuration at the bottom line of 3a 3c. The values 1 and 2 are respectivelyrepresented by orange and red, while 0 is just an empty cell. Only the matter field is representedhere. a − k a − k − k a a c l c r c l c r c l + k c r + k c l + k c r + k σ k σ k c e γ ( c ) (a) The new gauge-transformation group Γ ′ a + c l a + a + c r a a c l ... ... c r ... c l + c r + a c l − c r + a ... cF ′ ( c ) (b) The new local rule f ′ Figure 4
Description of the gauge extension ( F ′ , Γ ′ ). Green circles represent the gauge field andblack rectangles the matter field. In order to obtain a gauge-invariant theory, starting from a non-gauge-invariant one, theusual route is to extend the theory by means of a gauge field. As discussed in the introduction,the gauge field usually turns out to be a connection between gauge choices at neighbouringpoints, but there is no immediate reason why this should be the case. In the first part, weformalised, in the framework of Cellular Automata (CA), the notions of gauge extension andrelative gauge extension. The latter forces the gauge field to act as a connection.
C V I T 2 0 1 6 (a)
Evolution of F (b) Evolution of F ′ Figure 5
Evolution of F and its gauge extension F ′ over 9000 temporal step Again in Physics one usually starts from a theory featuring a global symmetry, before‘making it local’ through the gauge extension. Again there is no immediate reason why thisshould be the case. In our framework, we were able to establish a logical relation betweenglobal symmetry and relative gauge-invariance. Namely we proved that the CA that admit arelative gauge extension are exactly those that have the corresponding global symmetry. Tothe best of our knowledge, no continuous equivalent of that theorem exists in the literature;perhaps the discrete offers better opportunities for formalisation.We also proved that any CA can be extended into a gauge-invariant one. Thus, gauge-invariant CA are universal. Two different constructions were provided. The first constructionuses the gauge field to store, at each location, the value of the gauge-transformation whichthe matter field has undergone at that location, thereby allowing for the action of thetransformation to be counteracted. This path uses a non-relative gauge extension. Thesecond construction puts together the fact that any CA can be made globally-symmetric[13], with the fact that any globally-symmetric CA admits a relative gauge extension. Thus,relative gauge-extended CA are universal.Whilst the introduction of the gauge field is initially motivated by the gauge symmetryrequirement, the gauge field ends up triggering new, richer behaviours as it influences thematter field. However, in order for it to mediate the interactions within the matter field, asis the case in Physics, it should be the case that the matter field also influences the gaugefield—and back. In this paper, we provided a first example of a gauge-extended CA whosematter field injectively influences gauge field, whilst preserving reversibility. This was donethrough a general gauge extension, we leave it open whether this can be achieved througha relative gauge extension. The difficulty here is that relative gauge extensions seem tostore just the minimal amount of information required for gauge-invariance, and any furtherinfluence upon the gauge field runs the risk of jeopardising that.This difficulty can be circumvented in the quantum setting: the Quantum CellularAutomaton of [1] arises from a relative gauge extension, and yet features and a gauge fieldwhich is ‘sourced’ by the matter field. The construction directly yields a quantum simulationalgorithm for one-dimensional quantum electrodynamics. This should serve us a reminderthat whilst this work is theoretical, it is not merely of theoretical interest. Gauge extensionsis exactly what one needs to do in order to capture physical interactions within discretequantum models. This may lead for instance to digital quantum simulation algorithms,with improved numerical accuracy, as fundamental symmetries are preserved throughout thecomputation. . Arrighi, M. Costes and N. Eon 23:13
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A Proof of gauge-invariance for Sec. 6
In order to prove that the example illustrated in Fig-4 is gauge-invariant, we will show that γ ′ ◦ F ′ = F ′ ◦ γ ′ for any γ ′ ∈ Γ ′ . It is sufficient to prove this locally, we do so using thenotations of the figure and we denote by f ′ and g ′ the local application of the evolution anda gauge-transformation: f ′ ◦ g ′ ( c l , a , a , c r ) = c l + c r + a + k ,a + c l − k ,a + a + c r − k − k ,c l − c r + a + k = g ′ ◦ f ′ ( c l , a , a , c r ) C V I T 2 0 1 6
Therefore F ′ is Γ ′′