Gauge Fields as Geometry of Information Transport: Deriving Maxwell and Yang--Mills Equations in Quantum Cellular Automata

Abstract In standard quantum eld theory, gauge elds are introduced by demanding local invariance of a Lagrangian under a prescribed Lie group, and are geometrically interpreted as connections on principal bre bundles. In a universe described by a quantum cellular automaton (QCA), however, the microscopic ontology is a discrete network of nite-dimensional quantum systems updated by causal local unitaries. In such a discrete ontology there is no preferred global reference frame for internal degrees of freedom at dierent lattice sites; each cell only has access to a local Hilbert-frame. This work develops a framework in which gauge elds arise as translation protocols between local information frames on a QCA. Starting from three assumptions(i) a strictly causal, translation-invariant QCA on a regular lattice; (ii) local frame independence of physical predictions; and (iii) a minimal-distortion principle for information transportwe show: 1. Local redundancy in the choice of internal basis enforces the introduction of link variables as parallel-transport operators between neighbouring cells, with the standard lattice-gauge transformation law. Abelian phase redundancy yields a U(1) connection; internal multi-component redundancy yields non-Abelian G -connections. 2. In the continuum limit of a Dirac-type QCA, the requirement that the discrete dynamics be covariant under local frame changes is equivalent to minimal coupling of a Dirac eld to a gauge potential Aµ , and the gauge curvature is obtained as the logarithm of elementary plaquette holonomies. 3. Imposing that the dynamics of these link variables minimize a local, gauge-invariant, quadratic information-distortion functional uniquely selects, under mild regularity and isotropy assumptions, an eective action which reduces in the continuum to the Maxwell or YangMills action. 4. The microscopic gauge coupling can be expressed in terms of ratios of informationtransport rates between internal and inter-site channels within the QCA, connecting running couplings to changes in eective connectivity and time-step at dierent probing scales. In this picture, fundamental interactions are not extra entities added to matter elds, but constraints on how information can be transported consistently across a discrete network of local frames. Gauge curvature measures the failure of information-parallel transport to be path-independent, and the gauge eld equations arise as EulerLagrange equations of an information-distortion functional dened on QCA link variables. Keywords: Quantum Cellular Automata; Gauge Invariance; Wilson Loops; Information Transport; Maxwell Equations; YangMills Theory; Lattice Gauge Theory 1 Introduction & Historical Context From the perspective of continuous eld theory, the introduction of gauge elds typically follows the "gauge principle": given a free Lagrangian with global symmetry group G , promote the symmetry to a spacetime point-dependent local symmetry G(x) , and restore invariance by introducing gauge potential Aµ(x) and covariant derivative Dµ=∂µ−igAµ . For G=U(1) , this yields electromagnetic elds; for G=SU(2) ×U(1) , SU(3) , etc., yields weak and strong interactions in the Standard Model. Geometrically, gauge potential is a connection one-form on a principal bundle, and curvature Fµν corresponds to the curvature two-form of the connection. In the non-perturbative regime, discretization of gauge theory was pioneered by Wilson's lattice gauge theory: spacetime is discretized into lattice sites and links, gauge elds are group elements Ux,µ ∈G valued on links, curvature is given by group element products on closed Wilson loops, and in the continuum limit the Wilson action converges to YangMills action. Lattice gauge theory not only provided the rst controlled description of low-energy behavior in strong interactions but also supplies a natural discrete structure for quantum simulation. 1 On the other hand, Quantum Cellular Automata (QCA) model the physical universe as a collection of nite-dimensional quantum systems on discrete lattice sites, globally updated by local unitary operators at discrete time steps. Reviews by Arrighi, Farrelly, et al. show that QCA can unify numerous discrete spacetime quantum models, including quantum walks, lattice eld theories, and partially discretized quantum eld theories, and can emerge Dirac, Weyl, and Maxwell equations in appropriate continuum limits. Furthermore, Arnault et al. demonstrated that discrete-time quantum walks (DTQW) can realize U(1) and non-Abelian discrete gauge theories with strict lattice gauge invariance, simulating Dirac matter coupling to electromagnetic/YangMills elds in the continuum limit. These results indicate that introducing gauge elds in discrete frameworks is not dicult; the diculty lies in their ontological status. In traditional constructions, even on lattices, gauge elds are often viewed as "numerical tools for simulating a given continuous gauge eld theory." In the discrete ontology perspective of QCA, however, lattice sites and links are the most fundamental "universe primitives," with no deeper continuous eld as reference. From this perspective, natural questions arise:  Can gauge elds be understood as a geometric constraint on information transport in QCA networks, rather than additional "elds"?  Can gauge symmetry emerge from "redundancy of local information reference frames" and "consistency of information transport," rather than as an a priori symmetry assumption? This paper aims to provide such an interpretation from the QCA perspective. Specically: 1. Viewing each cell's internal Hilbert space Hx as a local information reference frame, assume physical predictions are insensitive to the choice of internal basis at each point; 2. Prove that to maintain unitary and causally consistent information transport without global basis, one must necessarily introduce "parallel-transport operators" Ux,µ as connection elds between lattice sites, with transformation law identical to lattice gauge theory; 3. In the smooth continuum limit, combining locality, isotropy, and unitary evolution constraints, characterize the minimal-distortion action for such connection elds, proving its continuum limit uniquely selects Maxwell/YangMills action; 4. Using the previously proposed principle of conservation of information rate, relate gauge coupling constant to ratios of QCA internal/external information channel rates, thereby giving microscopic information-theoretic-geometric interpretation to charge and coupling constants. In this framework, gauge elds are no longer "auxiliary objects articially introduced to maintain Lagrangian invariance under a local symmetry group," but "geometric costs that must be paid to allow information to be stably transmitted and dierent observers to reach consistent descriptions in a discrete universe without global basis." Gauge curvature characterizes the extent to which information-parallel transport is no longer path-independent. 2 Model & Assumptions This section presents the QCA model, local reference frame redundancy and information transport principles, and denes gauge structure as the minimal geometric object satisfying these principles. 2 2.1 Basic Structure of Quantum Cellular Automata Consider a d -dimensional regular lattice Λ = aZd with lattice spacing a and time step ∆t . At each lattice site x∈Λ , place a nite-dimensional Hilbert space Hx∼ =CNint , called internal degrees of freedom. The global Hilbert space is H=O x∈Λ Hx. One evolution step of the QCA is given by a unitary operator G:H → H satisfying the following conditions: 1. Causality: there exists nite radius R such that G 's action on local algebra Ax at some lattice site x depends only on degrees of freedom within radius R neighborhood of x ; 2. Translation invariance: there exists a family of translation operators Ta such that G commutes with Ta ; 3. Local decomposition: G can be written as nite-depth local quantum circuit G=QℓGℓ , where each Gℓ acts on nitely many adjacent lattice sites. Given internal dimensionality and locality constraints, one can construct Dirac-type QCA whose continuum limit yields Dirac equation. This paper does not depend on a specic construction, but assumes existence of such a class of "free matter QCA" whose single-particle eective Hamiltonian is H0=X x,µ ψ†(x)Kµψ(x+aˆµ)+h.c.+X x ψ†(x)Mψ(x), where ψ(x)∈CNint , Kµ, M are xed matrices, and in the continuum limit H0 corresponds to some Lorentz-invariant free eld (such as Dirac eld). 2.2 Local Information Reference Frame and Gauge Redundancy In discrete universe ontology, there is no external "God's-eye-view" global reference frame. Each cell can only access its own internal Hilbert space Hx 's local basis choice. For each lattice site x , choose an orthonormal basis {|ei(x)⟩}Nint i=1 , then state vector's coordinate representation is ψx=X i ψi x|ei(x)⟩. Local basis choice is arbitrary: for any Vx∈G⊂U(Nint) , performing local transformation |ei(x)⟩ 7→ X j (Vx)ji|ej(x)⟩, ψx7→ Vxψx, physical predictions should remain unchanged. The set {Vx}x∈Λ constitutes local gauge group G=Y x∈Λ Gx, Gx∼ =G. This embodies "redundancy of local information reference frame." Depending on internal structure and physical context, consider two typical cases: 1. G=U(1) : only global phase redundancy, corresponding to charge U(1) gauge; 2. G=SU(Nc) or more general compact Lie group: internal degrees of freedom have "color" or "avor" multi-component structure, corresponding to non-Abelian gauge group. Subsequent discussion unies using general compact Lie group G , with Abelian case as special case. 3 2.3 Information Transport and Denition of Connection Field In QCA, spatial propagation of information is realized by coupling between adjacent cells. In idealized description xing some global basis, free Hamiltonian H0 contains terms ψ†(x)Kµψ(x+aˆµ), whose physical meaning is "hopping amplitude from x to x+aˆµ ." However, from ontology perspective without global basis, ψ(x) and ψ(x+aˆµ) are represented in respective local bases and cannot be directly added. To describe such cross-cell information transport, a "paralleltransport operator" Ux,µ must be introduced between lattice sites, mapping Ux,µ :Hx−→ Hx+aˆµ. Denition 1 (Connection eld/link variable). Given gauge group G , a family of operators {Ux,µ} is called connection eld of the QCA if for each link (x, x +aˆµ) , Ux,µ ∈G⊂U(Hx,Hx+aˆµ), and under local gauge transformation {Vx} satises Ux,µ 7→ U′ x,µ =Vx+aˆµUx,µV† x. Under this denition, QCA matter Hamiltonian containing connection eld naturally writes as Hmatter[U] = X x,µ ψ†(x)KµUx,µψ(x+aˆµ)+h.c.+X x ψ†(x)Mψ(x), formally identical to "inserting group element on link" in lattice gauge theory, but here Ux,µ is geometric quantity necessarily introduced by local reference frame redundancy and information transport principle, not a trick "for discretizing some continuous gauge eld theory." 2.4 Information Distortion and Principle of Gauge Field Action Connection elds are not arbitrary degrees of freedom. Physical evolution of QCA should transport information as "gently" as possible, i.e., minimizing distortion in parallel transport while maintaining unitarity and causality. For this, introduce the following principle: Principle A (Local information distortion minimization). Among all connection eld dynamics compatible with given matter Hamiltonian Hmatter[U] and satisfying local gauge invariance, actual physical evolution corresponds to trajectories extremizing some local gaugeinvariant functional Sgauge[U] . To embody locality and isotropy, Sgauge[U] should depend only on parallel-transport operator products on minimal closed loops (plaquettes), i.e., Wilson loops W□= trU□, U□=Ux,µUx+aˆµ,νU† x+aˆν,µU† x,ν, and converge in continuum limit to some integrable local action density L(Fµν) . Standard lattice gauge theory shows that under second-derivative and locality assumptions, the unique gauge-invariant quadratic form in continuum limit is precisely YangMills action tr(FµνFµν) . This paper reinterprets this conclusion as "under local reference frame redundancy and information distortion minimization principles, eective action for connection eld necessarily reduces to Maxwell/YangMills action." 3 Main Results (Theorems and Alignments) This section presents core theorems and explains their correspondence with existing theories. 4 3.1 Theorem 1 (Local Basis Redundancy ⇒ Gauge Connection Transformation Law) Let G be a compact Lie group, G⊂U(Nint) some unitary representation. Consider a given free QCA, introducing local basis transformation Vx∈G at each lattice site x , and assume matter Hamiltonian Hmatter[U] is physically equivalent under G=QxGx . If requiring that under arbitrary local transformation Vx , cross-lattice transition amplitude ψ†(x)KµUx,µψ(x+aˆµ) is physically invariant, then there exists and is unique a family of link operators Ux,µ ∈G whose transformation law under G is Ux,µ 7→ Vx+aˆµUx,µV† x. This transformation law is completely consistent with link variable transformation in lattice gauge theory. 3.2 Theorem 2 (Minimal Coupling in Free QCA Continuum Limit) Let a class of QCA without connection eld have single-particle eective Hamiltonian in longwavelength limit yielding free Dirac Hamiltonian H0=Zddx ψ†(x)−iγ0γi∂i+mγ0ψ(x), satisfying translation, rotation, and discrete CPT symmetry. Applying Theorem 1's framework to this QCA, introducing G -valued connection eld Ux,µ on links, and requiring evolution covariant under local gauge transformation. Then in smooth limit where a, ∆t→0 and Ux,µ →I , singleparticle eective Hamiltonian of Hmatter[U] is equivalent to H=Zddx ψ†(x)−iγ0γiDi+mγ0ψ(x), Dµ=∂µ−igAµ(x), where Aµ(x)∈g is Lie algebra-valued gauge potential, with relation to Ux,µ : Ux,µ = exp (−igaAµ(x)) + O(a2). In particular, for G=U(1) yields charged DiracMaxwell coupling, for non-Abelian G yields DiracYangMills minimal coupling. This result is consistent with DTQW model conclusion that "Diracgauge eld coupling emerges in continuum limit." 3.3 Theorem 3 (Local Information Distortion Functional ⇒ Maxwell / Yang Mills Action) Let Sgauge[U] be action dened on link variables, satisfying: 1. Locality: Sgauge can be written as sum of local functions of each nite Wilson loop; 2. Gauge invariance: invariant under Ux,µ 7→ Vx+aˆµUx,µV† x ; 3. Isotropy: invariant under action of discrete rotation group on lattice; 4. Smooth limit: when Ux,µ values near identity, action density is positive denite quadratic form of Fµν with no derivatives higher than second order. 5 Then in continuum limit a→0 , Sgauge[U] is equivalent to Sgauge → −1 2Zdd+1xtr (FµνFµν) + O(a2), where gauge curvature Fµν =∂µAν−∂νAµ−ig[Aµ, Aν], reduces to Maxwell action for G=U(1) , yields YangMills action for non-Abelian G . This conclusion is consistent with continuum limit of Wilson action. 3.4 Theorem 4 (Information-Theoretic Characterization of Gauge Coupling Constant) In a QCA universe satisfying conservation of information rate v2 ext +v2 int =c2, let each cell have Nch channels available for external information transport (links), with coordination number z . Denote rint = eective rate of internal phase update per time step total information rate , rhop = eective rate of inter-lattice hopping per time step total information rate , then under natural normalization, gauge coupling g and dimensionless ne structure constant α=g2/(4π) can be written as g2∝rint z Nch rhop , α ∝rint z Nch rhop . When probing scale changes lead to changes in eective coordination number and channel number, α will "run" accordingly, thereby giving discrete geometric mechanism for renormalization group ow in QCA. This result connects the previously proposed "universal conservation of information rate" with gauge coupling constant. 4 Proofs This section provides proof outlines for the above theorems, with detailed derivations in appendices. 4.1 Proof of Theorem 1: Gauge Connection Transformation Law Without connection eld, matter Hamiltonian writes as H0=X x,µ ψ†(x)Kµψ(x+aˆµ)+h.c.+X x ψ†(x)Mψ(x). Under local basis transformation ψ(x)7→ Vxψ(x) , cross-lattice term becomes ψ†(x)Kµψ(x+aˆµ)7→ ψ†(x)V† xKµVx+aˆµψ(x+aˆµ). If Vx varies with x , this term's matrix structure generally changes, breaking QCA's translation invariance and form of original coupling matrix Kµ . To maintain physical equivalence of "matter connection system" after local transformation, link variable Ux,µ must be inserted in cross-lattice term, requiring after transformation ψ†(x)KµUx,µψ(x+aˆµ)7→ ψ†(x)KµU′ x,µψ(x+aˆµ), 6 i.e., there exists some U′ x,µ making new and old Hamiltonian density forms consistent. Writing local transformation explicitly: ψ†(x)KµUx,µψ(x+aˆµ)7→ ψ†(x)V† xKµUx,µVx+aˆµψ(x+aˆµ). To rewrite as ψ†(x)KµU′ x,µψ(x+aˆµ) , need KµU′ x,µ =V† xKµUx,µVx+aˆµ. Assuming Kµ invertible in representation space, then U′ x,µ =K−1 µV† xKµUx,µVx+aˆµ. To make U′ x,µ still take values in G and maintain translation-invariant form of Kµ , natural requirement is that Kµ commutes with G 's representation, making K−1 µV† xKµ=V† x . This can be satised by choosing Kµ scalar on internal space or compatible with G 's representation. Thus obtaining U′ x,µ =Vx+aˆµUx,µV† x, the standard transformation law of lattice gauge theory. Since Vx arbitrary and G compact group, can prove this law uniquely maintains Hamiltonian structure and local translation symmetry given H0 , obtaining Theorem 1. 4.2 Proof of Theorem 2: Minimal Coupling in Continuum Limit Consider unied representation of Abelian case G=U(1) and non-Abelian case. Let Ux,µ = exp (−igaAµ(x)) , Aµ(x)∈g. In long-wavelength limit, assume ψ(x) varies slowly on lattice, can write ψ(x+aˆµ) = ψ(x) + a∂µψ(x) + O(a2) . Cross-lattice term is ψ†(x)KµUx,µψ(x+aˆµ) = ψ†(x)Kµexp (−igaAµ(x)) ψ(x) + a∂µψ(x) + O(a2). Expanding to O(a) : ψ†(x)Kµψ(x) + aψ†(x)Kµ∂µψ(x)−igaψ†(x)KµAµ(x)ψ(x) + O(a2). For continuum limit of free QCA, existing results show there exists choice such that X µ ψ†(x)Kµ∂µψ(x)→ψ†(x)γ0γi∂iψ(x), and mass term given by Pxψ†(x)Mψ(x) . On this basis, absorbing −igaψ†KµAµψ term into derivative, can replace spatial derivative with covariant derivative ∂µ7→ Dµ=∂µ−igAµ(x), obtaining Diracgauge eld minimal coupling form. Time direction coupling can be obtained through similar construction for QCA's time update operator. In actual DTQW/QCA literature, strictly lattice gauge-invariant quantum walks have been demonstrated, whose continuum limit yields Dirac eld coupled Hamiltonian with external electromagnetic eld; interpreting "manually inserted" connection eld therein as above construction yields precise statement of Theorem 2. 7 4.3 Proof of Theorem 3: Continuum Limit of Information Distortion Functional By Principle A, Sgauge[U] should be constructed from Wilson loops on each minimal plaquette. For a plaquette □ dene U□=Ux,µUx+aˆµ,νU† x+aˆν,µU† x,ν. Requiring Sgauge invariant under Ux,µ →Vx+aˆµUx,µV† x means Sgauge must be constructed from tr(U□) and conjugate. Simplest local isotropic functional is Wilson action SW[U] = X □ β Nint Re trI−U□, where β related to coupling constant g . For Ux,µ ≈exp(−igaAµ) expanding in small a limit, obtain U□= exp −iga2Fµν(x) + O(a3), see Appendix A for BCH expansion calculation. Thus I−U□= iga2Fµν (x) + 1 2g2a4F2 µν(x) + O(a6), real part of trace to O(a4) is Re trI−U□=1 2g2a4trF2 µν(x)+O(a6). Approximating P□ as Rdd+1x/ad+1 , obtain SW[U]→βg2 2Nint Zdd+1xtrFµνFµν, which is YangMills action. Reduces to Maxwell action for G=U(1) . Conversely, can prove that given above four conditions, any other local gauge-invariant quadratic form in continuum limit only adds higher-derivative corrections to this action, thus Theorem 3 holds. 4.4 Proof of Theorem 4: Coupling Constant and Information Rate In previous work, conservation of information rate principle states: for any local excitation, its external group velocity vext and internal state evolution velocity vint satisfy v2 ext +v2 int =c2, where c is light-cone velocity of QCA. Internal state evolution velocity can be characterized by spectral width and phase rotation rate of local Hamiltonian, external group velocity given by inter-lattice transition amplitude and coordination number. In QCA with gauge eld, part of internal phase evolution is carried by "rotation" of gauge connection, corresponding from particle perspective to coupling with gauge eld. Denoting rint =v2 int c2, rext =v2 ext c2, rint +rext = 1, can decompose rint into gauge-related part rgauge and "bare internal degrees of freedom" part rbare . In a QCA with z nearest neighbors, Nch transport channels per link, eective inter-link information transport capacity increases with zNch ; to maintain total rate conservation, rgauge relative share should decrease accordingly. Natural scaling relation is g2∝rgauge z Nch rext . In weak coupling limit, rgauge is dominant part of rint , can approximate rgauge ≈rint , thus obtaining Theorem 4's relation. When probing energy scale rises, QCA's eective lattice spacing a and visible coordination number z change, leading to scaling changes in g2 and α , thus corresponding to renormalization group ow in continuous eld theory. 8 5 Model Apply This section gives several concrete models, demonstrating how the above general framework is implemented in specic QCA and corresponds to familiar physical theories in continuum limit. 5.1 U(1) Gauge Field in One-Dimensional Dirac QCA Consider one-dimensional Dirac-type QCA, whose free evolution can be written as split-step quantum walk form: at each lattice site has two-component spin ψ(x) = (ψ↑, ψ↓)T , time update given by alternating rotation and conditional shift. Appropriately choosing rotation angle and conditional shift, can prove continuum limit yields one-dimensional Dirac equation. On this basis, introduce U(1) phase Ux,±= exp[−igaA±(x)] on each link, corresponding to left/right transport channels respectively. Requiring full-step evolution covariant under local U(1) phase transformation ψ(x)7→ eiα(x)ψ(x) , obtain Ux,+7→ eiα(x+a)Ux,+e−iα(x), Ux,−7→ eiα(x−a)Ux,−e−iα(x). This is precisely Abelian special case of Theorem 1. Expanding continuum limit, Ux,±≈ exp[−igaAx(x)] , yields Diracelectromagnetic minimal coupling. Full-ensemble Wilson loops give discrete curvature of one-dimensional electric eld, action adopting Wilson-type function yields Maxwell action. 5.2 Two-Dimensional QCA and Non-Abelian SU (2) Gauge Field On two-dimensional lattice, consider QCA with internal space Nint = 4 , spin and "color" each occupying two components. Let gauge group G=SU(2) act on color space, internal Hamiltonian Kµ and M scalar for color space. Introduce link variables Ux,µ ∈SU(2), Ux,µ = exp[−igaAa µ(x)Ta], where Ta are SU(2) generators. Local gauge transformation ψ(x)7→ Vxψ(x) , Vx∈SU(2) acting only on color space, causes link variables to transform according to Theorem 1's rule. Choosing Wilson action to establish gauge eld dynamics, continuum limit yields standard equations for SU(2) YangMills eld. This construction echoes Arnault et al.'s work on "quantum walks and non-Abelian discrete gauge theory," which explicitly demonstrated discrete-time quantum walks with exact discrete U(N) gauge invariance, yielding YangMillsDirac coupling in continuum limit. 5.3 Electromagnetic Field from Information Geometry Perspective In above QCAgauge structure, electric and magnetic elds can correspond to the following information geometric quantities: 1. Electric eld Ei : phase growth rate dierence on same link between adjacent time steps, characterizing desynchronization degree of local clocks. Discretely can write Ei(x)∼1 ga∆targ hUx,i(t+ ∆t)U† x,i(t)i. 2. Magnetic eld Bi : phase of Wilson loop on minimal plaquette in spatial plane, characterizing incompatibility degree of parallel transport on spatial circular path. Discretely can write Bi(x)∼1 ga2arg U□(x), where □ is planar plaquette perpendicular to i direction. 9