Discrete Field Theory: Symmetries and Conservation Laws

Discrete ﬁeld theory: symmetries and conservation laws

M. Skopenkov

Abstract

We present a general algorithm constructing a discretization of a classical field theory from a Lagrangian. We prove a new discrete Noether theorem relating symmetries to conservation laws and an energy conservation theorem not based on any symmetry. This gives exact conservation laws for several discrete field theories: electrodynamics, gauge theory, Klein–Gordon and Dirac ones. In particular, we construct a conserved discrete energy-momentum tensor, approximating the continuum one at least for free fields. The theory is stated in topological terms, such as coboundary and products of cochains. Keywords: discrete field theory, discrete differential geometry, conservation law, Noether’s theorem 2010 MSC: 49M25, 49S05, 55N45, 81T25

Contents

1 Introduction 2 1.1 Quick start ...... 2 1.2 Background ...... 3 1.3 Main idea ...... 4 1.4 Statements ...... 5 1.5 Limitations ...... 7 1.6 Overview ...... 7

2 Examples 7 2.1 One-dimensional field theory ...... 7 2.2 Electrical networks ...... 10 2.3 Lattice electrodynamics ...... 14 2.4 Lattice gauge theory ...... 20 2.5 The Klein–Gordon field ...... 23 2.6 The Dirac field ...... 25

3 Generalizations 27 arXiv:1709.04788v3 [math-ph] 25 Jul 2021 4 Proofs 29 4.1 Basic results ...... 29 4.2 Integral conservation laws ...... 31 4.3 Identities ...... 33 4.4 Generalizations ...... 35 4.5 Proofs of examples ...... 38

5 Open problems 41

0The publication was prepared within the framework of the Academic Fund Program at the National Research University Higher School of Economics (HSE) in 2018-2019 (grant N18-01-0023) and by the Russian Academic Excellence Project “5-100”. The author has also received support from the Simons–IUM fellowship.

1 1 Introduction Dedicated to the last real scientists, which unlike merchants show both advantages and limitations of their theory.

This work is a try to build a general discrete field theory. This has the following motivation: • getting effective numeric algorithms for field theory; • putting field theory to a mathematically rigorous basis; • creating an alternative candidate for a fundamental field theory. Numerous discretizations of particular field theories are known [1,9, 10, 11, 14, 13, 15, 18]. Our aim is not to invent new discretizations, but to extract and study the best among the known ones. Discretizations exhibiting exact (not just approximate) conservation laws have been proved to be most successful for computational purposes [13]. This leads us to the following principles of discretization: • keep approximation of continuum theory; • keep conservation laws exact; • drop spatial symmetries easily. These principles have a built-in difficulty: we have to drop most continuous symmetries, but usually conservation laws are obtained just from such symmetries using the Noether theorem. We develop a new general method to get discrete conservation laws, simpler than the methods of [12, 13, 16, 17, 22]. The following basic warm-up results of discrete field theory are obtained in the present paper: • discretization of several field theories in a similar fashion keeping conservation laws exact (§2); • a new discrete Noether theorem relating symmetries to conservation laws (Theorems 1.2 and 3.3); • a new discrete energy conservation theorem not based on a symmetry (Theorems 1.3 and 2.2).

1.1 Quick start We start with an elementary and informal description of one result (Theorem 2.2), in the simplest unknown particular case. It is an energy conservation theorem for lattice electrodynamics; more precisely, for electrodynamics in 2 spatial and 1 time dimensions. For these small dimensions we just draw everything. The more realistic case of 3 spatial and 1 time dimensions is analogous; see §2.3. Recall briefly the energy conservation theorem in continuum electrodynamics (the Poynting theorem). Let 푥, 푦, 푡 be the Cartesian coordinates in space; see Figure1. Electric and magnetic fields are arbitrary smooth vector-valued functions Eì(푥, 푦, 푡) and Bì (푥, 푦, 푡) such that Eì ⊥ 푂푡 and Bì k 푂푡. The energy density and the energy flux 1 (ì2 + ì 2) ì × ì (the Poynting vector) are the functions 2 E B and E B. The Poynting theorem ì ì asserts that under Maxwell’s equations (where E =: (0, E푥, E푦) and B =: (B푡, 0, 0))

휕B 휕E푦 휕E 휕E 휕E푦 휕B 휕E푦 휕B 휕E 푡 + − 푥 = 푥 + = 푡 + = 푡 − 푥 = Figure 1: Cube 휕푡 휕푥 휕푦 0; 휕푥 휕푦 0; 휕푥 휕푡 0; 휕푦 휕푡 0; the following identity holds for each cube with the edges parallel to the coordinate axes:

∫ Eì2 + Bì 2 ∫ Eì2 + Bì 2 ∫ dA − dA = Eì × Bì dnì. 2 2

Here the cube is shown by dotted lines, and the faces over which a particular integral is taken are in bold. The ﬁrst two integrals mean the total energy contained in the same square in the 푂푥푦 plane at

2 two different moments of time 푡. The third integral means the total inward energy flux through the boundary between these two moments. Thus the equation means energy conservation. Let us discretize. Dissect the unit cube into 푁 × 푁 × 푁 equal cubes. By cubes we mean the latter cubes, by faces and edges — their faces and edges. A discrete electromagnetic field 퐹 is any

real-valued function on the set of faces. Informally, its values 퐹( ), 퐹( ), 퐹( ) discretize −B푡, E푦, E푥 respectively, depending on face direction. The well-known discrete Maxwell’s equations are

퐹( ) − 퐹( ) − 퐹( ) + 퐹( ) + 퐹( ) − 퐹( ) = 0; 퐹( ) − 퐹( ) − 퐹( ) + 퐹( ) = 0;

퐹( ) − 퐹( ) + 퐹( ) − 퐹( ) = 0; 퐹( ) − 퐹( ) + 퐹( ) − 퐹( ) = 0. (1) Here we sum the values of 퐹 at the faces of a particular cube (in the ﬁrst equation) or at the faces containing a particular edge (in the other equations), with appropriate signs. We write one equation per cube and one per nonboundary edge, and impose no boundary conditions. It’s time for our new deﬁnition. Let 푇 be the function on the set of nonboundary faces given by

1 푇 ( ) = 퐹( )· 퐹( ) + 퐹( )· 퐹( ) + 퐹( )· 퐹( ) 2 1 푇 ( ) = 퐹( )· 퐹( ) + 퐹( )· 퐹( ) 2 1 푇 ( ) = 퐹( )· 퐹( ) + 퐹( )· 퐹( ) 2 The value at a horizontal (respectively, vertical) face discretizes energy density (respectively, ﬂux). Proposition 2.9 asserts that under a natural choice of 퐹 we have uniform convergence as 푁 → ∞: ∫ ∫ ∫ 푇 ( ) 푁2 1 (ì2 + ì 2) , 푇 ( ) −푁2 ì × ì ì, 푇 ( ) 푁2 ì × ì ì, ⇒ 2 E B dA ⇒ E B dn ⇒ E B dn (2)

The desired discrete Poynting theorem (particular case of Theorem 2.2) asserts that assuming only Maxwell’s equations (1), we have the following identity for each nonboundary cube:

푇 ( ) − 푇 ( ) − 푇 ( ) + 푇 ( ) + 푇 ( ) − 푇 ( ) = 0. (3) Properties (2)–(3) are exactly what one requests from a discretization of energy density and ﬂux according to the above discretization principles; it is nontrivial to satisfy both properties simultaneously. A proof in pictures is in §4.1. And we proceed to a systematic discussion of discrete ﬁeld theory.

1.2 Background Discrete field theory is actually at least as old as the continuum one. In 1847 G. Kirchhoff stated the laws of an electrical network, which is in fact the simplest model of the theory; see §2.2. In the continuum limit, the laws approximate the Laplace equation; thus the model perfectly serves for numerical solution of the latter. Remarkable approximation theorems were proved by L. Lusternik [19], R. Courant–K. Friedrichs–H. Lewy [9] in 1920s and later generalized, e.g., in [7,6,3, 27]. Planar networks lead to the discretization of complex analysis having applications in statistical physics (e.g., obtained in 2010s by S.Smirnov et al. [6]) and even computer graphics [14]. Discrete field theory was closely related to topology from the youth of both subjects. The Kirchhoff laws are naturally stated in terms of the boundary and the coboundary operators; see §2.2 for an elementary introduction. Such formulation is usually attributed to H. Weyl; see [15, §1F, p. 31] for an elaborate historical survey. In 1930s G. de Rham established correspondence between these operators and the exterior derivative and its dual; see [1] for a survey and [26] for general philosophy. This lead to the above discrete Maxwell equations (1); see also §2.3 and [4, 15, 18, 25].

3 The next major step was done by A. Kolmogorov and J. Alexander in 1930s, who invented a product discretizing the exterior product in a sense. Kolmogorov commented that such discretization was his original motivation. The construction was soon modified by H. Whitney and others to give the now- famous cup-product [28]. The original product was anticommutative, whereas the cup-product was associative. One cannot get both properties simultaneously (this fact is crucial for rational homotopy theory). This reflects a general phenomenon that not all properties survive under discretization. We choose the associative cup-product as a discretization of the exterior product, in contrast to [14]. Later there appeared discrete models for other classical fields: e.g., Feynman checkerboard from 1940s and Regge calculus from 1960s for the Dirac and the gravitational field respectively; see [23] for an elementary introduction and survey of the former model. In 1970s F. Wegner and K. Wilson introduced lattice gauge theory as a computational tool for gauge theory describing all known interactions; see [20] or §2.4 for an elementary introduction and [10] for details. This culminated in determining the proton mass theoretically with an error < 2%. In 1980s A. Connes developed a formalism, dealing (to some extent) uniformly with continuous and discrete geometries [8]. Using it, A. Dimakis et al. discretized the Yang-Mills equations [11, Eq. (4.15)]. Corollary 2.3 extends their result by adding sources and the crucial unitarity constraint. Compare with the efforts put to achieve the gauge covariance in the remarkable survey [2, §9]. In 1990s J. Marsden et al. discretized basic general theorems of field theory: the Euler–Lagrange equations and the Noether theorem on a 2-dimensional grid; see [22, Eq. (5.2) and (5.7)], cf. [17, Eq. (60) and (69)], [16, Theorem 5.2.37], [12, Theorem 8.1, Corollary in §7]. These results extend the ones obtained earlier for 1-dimensional difference equations; see [16] for references. Discrete Euler–Lagrange equations in §1.4 are straightforward generalizations of the known ones; but Discrete Noether Theorem 1.2 is different. M. Kraus et al. have stepped beyond the Lagrangian formulation [17]. A discretization of hydrodynamics was introduced by E. Gawlik et al. in 2010s [13, §4]. They derived general Euler–Poincare equations and Kelvin–Noether theorem [13, §3]. Their approach was based on discretization of the diffeomorphism group, thus was applicable to rather specific class of models. In 2017 E. Mansfield et al. discussed conservation laws for finite-element approximations [21]. There was a folklore belief that no conserved discrete energy-momentum tensor exists in this framework. E.g., in 2016 D. Chelkak, A. Glazman, and S. Smirnov introduced a “halfway” conserved tensor [5, Corollary 2.12(1)], cf. [24]. Even the notion of a rank 2 symmetric tensor itself is hard to discretize [1, §7]. But in 2000s V. Dorodnitsyn disretized energy and momentum conservation in some particular cases [12, Example in §8], and finally in §1.1 and §2.3 we construct an exactly conserved discrete energy-momentum tensor, approximating the continuum one at least for free fields. Great success of discrete models forces to search for a general discretization method and even to build the whole field theory starting from discrete rather than continuous space and time.

1.3 Main idea We propose the following discretization algorithm for field theories: 1) take a continuum Lagrangian written in terms of exterior calculus operations from Table1; 2) replace the exterior calculus operations by cochain operations using Table1 literally; 3) get equations of motions/conservation laws from discrete Euler–Lagrange/Noether theorems. This idea is well-known but realization is new. In this subsection, in contrast to the rest of the paper, we assume familiarity with the basics of continuum field theory. Results of applying the algorithm to basic field theories are discussed in §2. The output discrete theories are usually simpler than the input continuum ones; knowledge of the latter is not required for understanding the former. All the output theories of §2 are known, but some obtained conservation laws are new. As a tool, we use discrete covariant differentiation (see §2.4 and [11]) and build a new discretization of tensor calculus involving non-antisymmetric tensors (see §2.3). This is done in terms of cochain operations from Table1, which appear naturally in examples and are defined easily. The algorithm provides conservation laws only for symmetries which are preserved by the discretization. Thus we usually guarantee charge conservation (based on the automatically preserved gauge symmetry) and energy-momentum conservation (not based on any symmetry in our setup).

4 Table 1: Correspondence between continuum and discrete notions Continuum Discrete Definition Algorithmic part I. Replacement in Lagrangian and action: differentiable manifold (spacetime) M simplicial or cubical complex 푀 1.1 with fixed vertices ordering 푘-form, R- or C푚×푛-valued ϕ 푘-cochain, R- or C푚×푛-valued 휙 1.1 exterior derivative d coboundary 훿 2.9 exterior product ∧ cup-product ⌣ 2.18 interior product y cap-product ⌢ 2.18 connection 1-form,Lie-algebra-valued A connection,not Lie-algebra-valued 퐴 2.17 curvature 2-form, Lie-algebra-valued F curvature, not Lie-algebra-valued 퐹 2.17 covariant exterior derivative DA covariant coboundary 퐷 퐴 2.20, 2.18 ♯ 푀 퐼푑 raising all indices sharp-operator (new notion) # 2.9 for = 푁 훾 훾 푀 퐼4 vector of the Dirac -matrices γ the Dirac 1-chain (new notion) 2.21 for = 푁 function on R or C푚×푛 (e.g., log or Tr) f the same function on R or C푚×푛 f — ∫ · 휖 spacetime integration of a 0-form MdV sum of the values of a 0-chain 2.2 Informal part II. Correspondence in equations of motion and conservation laws: codifferential, ♯-conjugated ♯δ♭ boundary 휕 2.9 ♯ ♯ ∗ ♭ 퐷∗ covariant codifferential, -conjugated DA covariant boundary 퐴 2.20, 2.18 ∗ interior product x cop-product (new notion) ⌢ 2.18 tensor product over 퐶∞(푀) ⊗ chain-cochain cross-product × 2.10 type (1, 1) tensor T type (1, 1) tensor (new notion) 푇 2.10 푘 ∫ h푇, 휋i 푀=퐼푑 integration of its -th component 휋T푘 flux (new notion) 푘 2.12 for 푁 푘 ∫ ϕ h휙, 휋i integration of a -form 휋 pairing 4.2

We stress that Part I of Table1 gives an algorithm, not just an analogy (as Part II). However putting a continuum Lagrangian to the required input form is not always possible and can be ambiguous: Example 1.1. The Lagrangian of continuum electrodynamics can be written as L[ϕ] = −♯dϕydϕ, where ϕ is a real-valued 1-form on R1,3 (vector-potential). The resulting discretization L[휙] = −#훿휙 ⌢ 훿휙 gives the known discrete Maxwell equations briefly recalled in §2.3. Example 1.2. The same continuum Lagrangian can be written as L[A] = −♯F[A]∗yF[A], where A = 푖ϕ is a u(1)-connection 1-form and F[A] = dA + A ∧ A = dA is the curvature 2-form on R1,3. Here A ∧ A = 0 identically because A assumes values in an Abelian Lie algebra. The resulting discretization is L[퐴] = −#퐹[퐴]∗ ⌢ 퐹[퐴], where 퐹[퐴] = 훿퐴 + 퐴 ⌣ 퐴. The discretization turns out to be different from Example 1.1 because 퐴 ⌣ 퐴 ≠ 0 and 퐹[퐴] ≠ 훿퐴 anymore. It is famous compact Abelian lattice gauge theory recalled in §2.4. So, depending on the choice of the input form of the Lagrangian, in Examples 1.1 and 1.2 we get two unequivalent discretizations of one continuum theory, both very useful in their own contexts. Remark 1.1. In Table1 we intentionally include no discretization for the Hodge star or products other than exterior, interior, tensor products. In all the examples, we have succeeded to avoid them. Continuum and discrete notations fit not that well. But both are commonly used in their own contexts (except a few new discrete objects, for which we keep the continuum notation in a different font).

1.4 Statements Let us precisely state the main new results in their simplest form. Formal deﬁnitions of some used notions and generalizations of the results to nontrivial connections are postponed until further sections.

Definition 1.1. A finite simplicial (respectively, cubical) complex is a finite set of simplices (respectively, hypercubes) in a Euclidean space of some dimension satisfying the following properties:

5 1) the intersection of any two simplices (respectively, hypercubes) from the set is either empty or their common face; 2) all the faces of a simplex (respectively, a hypercube) from the set belong to the set as well. Spacetime 푀 is an arbitrary finite simplicial or cubical complex with fixed vertices ordering. For a cubical complex, we require that the minimal and the maximal vertex of each 2-dimensional face are opposite. (Typical examples of spacetimes are a path with 푁 edges or an 푁 × 푁 grid with the dictionary order of vertices; see Definition 2.9.) The simplices/cubes of 푀 are called faces of 푀. A 푘-dimensional field or 푘-cochain is a real-valued function defined on the set of 푘-dimensional 푘 faces of 푀. Denote by 퐶 (푀; R) = 퐶푘 (푀; R) the set of all 푘-dimensional fields; see Remark 3.1 for comparison with the other definitions in literature. A Lagrangian is a function L : 퐶푘 (푀; R) → 푘 퐶0(푀; R). The action S : 퐶 (푀; R) → R is the sum of the values of the Lagrangian over all the vertices. A field 휙 ∈ 퐶푘 (푀; R) is on shell (i.e., lying on the shell given by the equations of classical stationary 휕 S[휙 + 푡Δ] = Δ ∈ 퐶푘 (푀 R) physics), if it is for the action functional, i.e., 휕푡 푡=0 0 for each ; . References to definitions of (co)boundary, chain-cochain cap- and cross-products are in Table1. Informally, a Lagrangian is local, if its value at a vertex depends only on the values of the field 휙 and the coboundary 훿휙 at the faces for which the vertex is maximal. Informally, partial derivatives with respect to 휙 and 훿휙 are fields of dimension 푘 and 푘 + 1 respectively, obtained by differentiating the Lagrangian as if 휙 and 훿휙 were independent variables. Formal definitions are in Definition 3.1. The following theorem is a straightforward generalizaion of known ones; cf. [22, Eq. (5.2)]. 푘 Theorem 1.1 (Discrete Euler–Lagrange equations). Let L : 퐶 (푀; R) → 퐶0(푀; R) be a local La- grangian. Then a field 휙 ∈ 퐶푘 (푀; R) is on shell, if and only if the following equation holds: 휕L[휙] 휕L[휙] 휕 + = 0. (4) 휕(훿휙) 휕휙 (Here a plus sign stands because the boundary operator 휕 for 푘 = 0 discretizes minus divergence.) A current is a 1-dimensional field 푗 ∈ 퐶1(푀; R). A current is conserved, if 휕 푗 = 0. The Noether theorem gives a conserved current for each continuous symmetry of the Lagrangian. 푘 Theorem 1.2 (Discrete Noether theorem). Let L : 퐶 (푀; R) → 퐶0(푀; R) be a local Lagrangian and 휙 ∈ 퐶푘 (푀; R) be a field on shell. The Lagrangian is invariant under an infinitesimal transformation Δ ∈ 퐶푘 (푀; R), i.e., 휕 L[휙 + 푡Δ] = , 휕푡 0 (5) 푡=0 if and only if the following current is conserved: 휕L[휙] 푗 [휙] = ⌢ Δ. (6) 휕(훿휙) This theorem is different from known discretizations of the Noether theorem in [12, 16, 17, 22]. Discrete spacetime has no continuous symmetries, but there is still a corresponding conserved tensor. Conserved tensors are defined in Definition 2.10. We emphasize that they are functions on faces of the Cartesian square 푀 ×푀 rather than of spacetime 푀 itself. We shall see that such functions appear naturally in examples in §2. 푘 Theorem 1.3 (Energy-momentum conservation). For each local Lagrangian L : 퐶 (푀; R) → 퐶0(푀; R) and each field 휙 ∈ 퐶푘 (푀; R) on shell we have the following conserved energy-momentum tensor: 휕L[휙] 휕L[휙] 푇 [휙] = × 훿휙 + × 휙. (7) 휕(훿휙) 휕휙 This theorem is completely new. An integral form of this conservation law on a grid is stated in §2.3 (see Theorem 2.2 sketched already in §1.1). In particular, to tensor (7) defined on 푀 × 푀 we assign a conserved quantity defined on the grid 푀 itself. In many examples, (6)–(7) approximate their continuum analogues; see Theorem 2.1, Propositions 2.9, 2.13, 2.15, and Remark 2.15.

6 After straightforward modification, these main results generalize to: - complex- or vector-valued fields: the real part of the rhs of (6) and (7) is conserved; - several interacting fields: one equation (4) per field; the sum of all currents (6) is conserved; - nonfree boundary conditions: equation (4) and conservation laws hold apart the boundary.

1.5 Limitations So far the proposed general discrete field theory has no applications (as a mathematical theory) and is not falsifiable (as a candidate for a fundamental physical theory). Most of the technical issues concern the discretization of energy conservation and tensor calculus: On one hand, the new notion of energy-momentum tensor (7) seems to be too abstract and too general. It discretizes not the continuum energy-momentum tensor precisely but a related object mapped to the latter; see Remark 2.9. Depending on a particular Lagrangian, (7) approximates either the nonsymmetric canonical energy-momentum tensor, or the symmetric Belinfante–Rosenfeld one, or even a nonconserved tensor; see Remark 2.11. On the other hand, discrete non-antisymmetric tensor calculus from §2.3 seems to be too restrictive: it includes only type (1, 1) tensors and only the trivial connection; integration is defined only on a grid. The way of further generalization is unclear: e.g., for lattice gauge theory from §2.4, a naive way to define a real gauge invariant energy-momentum tensor leads to a nonconserved tensor; cf. Remark 2.15. Approximation of continuum theories by discrete ones is not discussed at all, with the following two exceptions. First, for electrical networks the known approximation result is recalled in §2.2. Second, for the completely new discrete energy-momentum tensor the continuum limit is found in §2. Some other limitations are stated as open problems in §5.

1.6 Overview In §2 we give basic examples of discrete field theories. It contains an exposition of known results and a few new ones for nonspecialists; §2 is independent from §1. In §3 we state the main results in full generality. The only prerequisites for §3 are the definitions cited in Part I of Table1, Definitions 2.13, 2.15, and Remarks 2.15, 2.16. In §4 we prove the results of §§1–3. In §5 we state open problems. The paper is written in a mathematical level of rigor, i.e., all the definitions, conventions, and theorems (including corollaries, propositions, lemmas) should be understood literally. Theorems remain true, even if cut out from the text. The proofs of theorems use the statements but not the proofs of the other ones. Most statements are much less technical than the proofs; that is why the proofs are kept in a separate section. Remarks are informal and are not used elsewhere (hence skippable) unless the opposite is explicitly indicated. We tried our best to make the results accessible to nonspecialists and to minimize the background assumed from the reader. The required notions are introduced little by little in examples in §2.

2 Examples

2.1 One-dimensional field theory Toy model First we illustrate our main results in the trivial particular case of dimension 1. Consider a pipeline of 푁 identical pipes in series with sources at the two endpoints pumping incompressible fluid in and out; see Figure2 to the left. Let 푠 be the intensity of each source (measured in litres/second). The current 푗 (k) through k-th pipe (measured in litres/second) satisfies

• Mass conservation law: 푗 (1) = 푗 (N) = 푠 and 푗 (k + 1) = 푗 (k) for each k = 1,..., N − 1.

7 This just means that 푗 (k) = 푠 for k = 1,..., N. Throughout §2.1 we use bold font for pipe numbers. Formally, we define 푠 ∈ R to be a fixed number and the current to be a function 푗 : {1,..., N} → R satisfying the mass conservation. (There is no formal difference between symbols in different fonts.)

s 1 2 3 s + − 0 1 2 3 0 1 2 3 [휕 푗](푘) = 푗 (k) − 푗 (k + 1); [휓 ⌢ Δ](k) = 휓(k)Δ(푘 − 1); [훿휙](k) = 휙(푘) − 휙(푘 − 1); [휓 ⌣ Δ](k) = 휓(k)Δ(푘). Figure 2: A path with 3 edges viewed as a pipeline or an electrical network. The expressions for cochain operations on the path (where 푗, 휓 and 휙, Δ are functions on the sets of edges and vertices respectively) are given just to compare Propositions 2.1–2.2 and Theorems 1.1–1.2; cf. Deﬁnitions 2.9,2.18.

Let us state a least action principle for the toy model. A potential 휙 of the ﬂow is a function 휙 : {0, . . . , 푁} → R such that 휙(푘 − 1) − 휙(푘) = 푗 (k) for each 푘 = 1, . . . , 푁. Clearly, it satisﬁes • the Laplace equation: 휙(푘 + 1) − 2휙(푘) + 휙(푘 − 1) = 0 for each 푘 = 1, . . . , 푁 − 1; • the least action principle: among all functions on {0, . . . , 푁}, 휙 minimizes the functional

푁 1 ∑︁ (휙(푘) − 휙(푘 − 1))2 − 푠휙(0) + 푠휙(푁) 2 푘=1

The ﬁrst term is the total ﬂuid kinetic energy. The functional is the sum of the values of the function 푠, 푘 + if = 0 1  L[휙](푘) = (휙(푘) − 휙(푘 − 1))2 − 푠(푘)휙(푘), where 푠(푘) := 0, if 1 ≤ 푘 ≤ 푁 − 1 2  | {z } −푠, if 푘 = 푁. [훿휙](k)  Generalization

Such a “least action” formulation of the model has a straightforward generalization. The following deﬁnition is a particular case of Deﬁnition 3.1 below. A local Lagrangian L is a self-map of the set of all real-valued functions on {0, . . . , 푁} such that

L[휙](푘) = 퐿푘 (휙(푘), 휙(푘) − 휙(푘 − 1))

2 for some diﬀerentiable function 퐿푘 : R → R. The 2 arguments of 퐿푘 are denoted by 휙 and 훿휙. Set 휕L[휙] 휕L[휙] 휕퐿 (휙, 훿휙) { , . . . , 푁} → R, (푘) = 푘 휕휙 : 0 휕휙 : 휕휙 ; 휙=휙(푘), 훿휙=휙(푘)−휙(푘−1) 휕L[휙] 휕L[휙] 휕퐿 (휙, 훿휙) { ,..., } → R, ( ) = 푘 . 휕 훿휙 : 1 N 휕 훿휙 k : 휕 훿휙 ( ) ( ) ( ) 휙=휙(푘), 훿휙=휙(푘)−휙(푘−1) 휕L 휕L 휕L[휙] 푘 푠 푘 휕L[휙] 훿휙 We also set 휕(훿휙) (0) = 휕(훿휙) (N+1) = 0. E.g., in the toy model: 휕휙 ( ) = − ( ), 휕(훿휙) (k) = (k). The following obvious proposition is a particular case of Theorem 1.1 above. Proposition 2.1 (the Euler–Lagrange equation). Let L[휙] be a local Lagrangian. A function 휙 is Í푁 L[휙](푘) 푘 , . . . , 푁 stationary for the functional 푘=0 , if and only if for each = 0 we have 휕L 휕L 휕L (k) − (k + 1) + (푘) = 0. 휕(훿휙) 휕(훿휙) 휕휙 E.g., in the toy model above, the Euler–Lagrange equation is the Laplace equation. That model had a built-in conservation law, hidden after the least-action formulation. The following obvious proposition reveals conservation laws hidden in the Lagrangian; it is a particular case of Theorem 1.2.

8 Proposition 2.2 (the Noether theorem). If a local Lagrangian L[휙] is invariant under an inﬁnitesimal transformation Δ(푘), i.e., 휕 L[휙 + 푡Δ] = , 휕푡 0 푡=0 휙 Í푁 L[휙](푘) then for each stationary function for 푘=0 the following function is conserved, i.e. constant: 휕L[휙] 푗 (k) = (k)Δ(푘 − 1). 휕(훿휙)

E.g., in the above toy model, apart the endpoints, the Lagrangian is invariant under the transformation 휙 ↦→ 휙−푡, where 푡 ∈ R. The resulting Noether conserved function is exactly 푗 (k) = 휙(푘−1)−휙(푘).

Momentum conservation Let us state a less intuitive momentum conservation. The introduced discrete momentum tensor is a completely new object. First we give a heuristic motivation (cf. §2.2), then a precise definition. In the toy model above, momentum circulation is physically clear. The momentum of the fluid in the pipe k is proportional to 푗 (k). During time Δ푡, the volume proportional to 푗 (k)Δ푡 moves to the next pipe. Thus the momentum flux through the vertex 푘 per unit time is proportional to 푗 (k)2. (We ignore pressure and do not care of the proportionality constant because this is just a heuristic anyway.) Now consider a free field, i.e., L[휙](푘) = [훿휙](k)2 +푚2휙(푘)2, where 푚 ≥ 0. Let 휙 be a stationary function, i.e. just a function satisfying the equation 휙(푘 − 1) − (2 + 푚2)휙(푘) + 휙(푘 + 1) = 0 for each 0 < 푘 < 푁. One expects the following properties of the momentum flux 휎(푘) through a vertex 푘:

• 휎(푘) = 푗 (k)2 for 푚 = 0, i.e., for a linear potential 휙;

• 휎(푘) depends only on 휙(푘), 훿휙(k), 훿휙(k + 1), and is homogeneous quadratic in these values;

• 휎(푘) = const apart the endpoints, i.e., the momentum is conserved.

The simplest function 휎(푘) satisfying these properties is (we skip a direct checking) 1 휕L 1 휕L 휎(푘) = 훿휙(k + 1)훿휙(k) − 푚2휙(푘)2 = (k + 1)훿휙(k) − (푘)휙(푘). 2 휕(훿휙) 2 휕휙 Remark 2.1. A naive way to discretize the momentum flux would be to take the usual (continuum) momentum flux of a piecewise-linear extension of 휙. But the resulting quantity is not conserved in a reasonable sense. Our function 휎(푘) is very different from such naive “finite-element” discretization. For an arbitrary Lagrangian, the formula for 휎(푘) is not applicable literally but still suggestive. Since the formula involves the product of the values of 훿휙 at distinct edges, it is reasonable to view it as a “projection” of a more fundamental quantity defined on the Cartesian square of the pipeline.

[휓 × 휙](푘 × 푙) = 휓(푘)휙(푙). Figure 3: The Cartesian square of a path with 3 edges and the cross-product; see Deﬁnition 2.1

Deﬁnition 2.1. (This is a particular case of Deﬁnition 2.10.) The Cartesian square of a path with 푁 edges is the grid 푁 × 푁; see Figure3. The vertices of the grid have form 푘 × 푙, where 푘 and 푙 are vertices of the path. The 1 × 1 squares have form k × l, where k and l are edges.

9 For functions 휓, 휙 on the set of vertices (respectively, edges) of the path denote by 휓 × 휙 the function on the vertices (respectively, 1 × 1 squares) of the grid given by [휓 × 휙](푘 × 푙) = 휓(푘)휙(푙) (respectively, by [휓 × 휙](k × l) = 휓(k)휙(l)). A real-valued function on the disjoint union of the sets of vertices and 1 × 1 squares of the grid is a type (1, 1) tensor. (E.g., for the toy model, equation (7) gives the tensor equal 푠2 on each 1 × 1 square and vanishing on each nonboundary vertex.) A tensor 푇 is conserved, if for each 0 < 푘 < 푁 and 0 < 푙 ≤ 푁 the following equation holds:

푇 (푘 × 푙) − 푇 (푘 × (푙 − 1)) + 푇 (k × l) − 푇 ((k + 1) × l) = 0.

I.e., we have one equation per vertical nonboundary edge; see Figure3.

Remark 2.2. This is a well-known discretization of the Cauchy–Riemann equations [3, Eq. (2.2)], up to orientation. Thus tensor conservation means one half of the Cauchy–Riemann equations (for vertical edges only), like in [5, Corollary 2.12(1)], although our setup is very diﬀerent from theirs. The following obvious corollary of Proposition 2.1 is a particular case of Theorem 1.3.

Proposition 2.3 (Momentum conservation). Let L[휙] be a local Lagrangian and 휙 be a stationary Í푁 L[휙](푘) function for the functional 푘=0 . Then the tensor given by (7) is conserved. 푇 푘 1푇 (( + ) × ) − 1푇 (푘 × 푘) Define the flux of a tensor through a vertex by the formula 2 k 1 k 2 . E.g., for the free field, the flux of tensor (7) equals exactly 휎(푘). A tensor 푇 is symmetric, if 푇 (푘 ×푙) = 푇 (푙 × 푘) for all vertices or edges 푘, 푙. E.g., tensor (7) is symmetric essentially only for the free field (in spite of being a tensor on 1-dimensional spacetime). A conserved symmetric tensor has constant flux (this is a version of Proposition 2.7 below). E.g., for the toy model, the flux of tensor (7) is 푗 (k)2/2. The same toy model describes the electrical network of 푁 unit resistors in series (as well as many other systems); see Figure2 to the right. Now we switch entirely to the language of networks.

2.2 Electrical networks Basic model Consider an 푁 × 푁 grid of unit resistors; see Figure4. A standard problem is to ﬁnd currents in the grid, given the current sources at the boundary. It is solved using the following mathematical model.

2 v 3 d c 3 1 4 푓 2 푓 푔 4 a b 1 [휕 푗](푣) = − 푗 (1) − 푗 (2) + 푗 (3) + 푗 (4); [훿 푗]( 푓 ) = 푗 (1) + 푗 (2) − 푗 (3) − 푗 (4); [휙 ⌢ 휓](푣) = 휙(3)휓(3) + 휙(4)휓(4); [휙 ⌣ 휓]( 푓 ) = 휙(1)휓(2) − 휙(4)휓(3);

Figure 4: A 3 × 3 grid, boundary, coboundary, cap-, and cup-product (see Deﬁnitions 2.2, 2.4, 2.17)

Deﬁnition 2.2. Each of the 푁2 unit squares of the grid is called a face. Orient the boundary 휕 푓 of each face 푓 counterclockwise. Assume that the coordinate axes are parallel to the edges, and orient edges in the directions of the axes. A function on vertices/edges/faces is a real-valued function deﬁned on the set of vertices/edges/faces of the grid. A source 푠 is a function on vertices vanishing at all the nonboundary vertices. The current generated by the source 푠, or the current on shell, is the function on edges satisfying the two equations:

• the Kirchhoﬀ current law or charge conservation law: 휕 푗 = −푠;

• the Kirchhoﬀ voltage law in the case of unit resistances: 훿 푗 = 0.

10 Here the boundary 휕 푗 and the coboundary 훿 푗 of a function 푗 on edges are the functions on vertices and faces respectively given by the following formulae (see Figure4 to the middle and to the right): ∑︁ ∑︁ [휕 푗](푣) = 푗 (푒) − 푗 (푒), 푒 ending at 푣 푒 starting at 푣 ∑︁ ∑︁ [훿 푗]( 푓 ) = 푗 (푒) − 푗 (푒), 푒 oriented along 휕 푓 푒 oriented opposite to 휕 푓 for each vertex 푣 and face 푓 , where the sums are over edges 푒 containing 푣 and contained in 휕 푓 휖푠 Í 푠 푣 respectively. Hereafter an empty sum is set to be 0. Denote by := 푣 ( ) the sum over all vertices 푣 (the operator 휖 is deﬁned only for functions on vertices). The following existence and uniqueness result is well-known. Proposition 2.4. A current generated by a source 푠 exists, if and only if 휖푠 = 0. If a current generated by the source 푠 exists, then it is unique.

Remark 2.3. It could be more conceptual to write the Kirchhoﬀ voltage law in the form 훿푅 푗 = 0, where 푅 is a map between 1-chains and 1-cochains depending on the resistances. In our setup, chains and cochains are identiﬁed and the resistances equal to 1, hence 푅 is the identity map and is omitted.

Electrical potential Let us state a least-action principle for electrical networks. Throughout §2.2 푗 is a current on shell. Deﬁnition 2.3. An electrical potential 휙 is a function on vertices satisfying • the Ohm law in the case of unit resistances: 푗 = −훿휙. Here the coboundary 훿휙 is the function on edges given by the formula

[훿휙](푢푣) = 휙(푣) − 휙(푢), where 푢푣 denotes an oriented edge starting at 푢 and ending at 푣 hereafter. The following well-known existence and uniqueness result is straightforward. Proposition 2.5. For each current on shell there is a unique up to additive constant electrical potential.

The following properties of an electrical potential 휙 may serve as equivalent deﬁnitions: • the Laplace equation with the Neumann boundary condition: 휕훿휙 = 푠;

• the least action principle: among all the functions on vertices, 휙 minimizes the functional 1 ∑︁ ∑︁ S[휙] = (휙(푢) − 휙(푣))2 − 푠(푣)휙(푣) = 휖L[휙], where 2 edges 푢푣 vertices 푣 L[휙] = 1 훿휙 ⌢ 훿휙 − 푠 ⌢ 휙. 2

Here the cap-product ⌢ is deﬁned as follows; see Figure4 to the middle. Deﬁnition 2.4. Denote by max 푓 the vertex of a face 푓 or an edge 푓 having the maximal sum of the coordinates. Set max 푓 := 푓 , if 푓 is a vertex. The cap-product 휙 ⌢ 휓 of two functions 휙 and 휓 on faces (respectively, edges or vertices) is the function on vertices given by ∑︁ [휙 ⌢ 휓](푣) = 휙( 푓 )휓( 푓 ), 푓 : max 푓 =푣 where the sum is over faces (respectively, edges or vertices) 푓 such that max 푓 = 푣.

11 Magnetic field There is one more discrete field in an electrical network: the current 푗 generates a magnetic field. Definition 2.5. A magnetic field 퐹 (or magnetic flux through faces in the (0, 0, −1)-direction) generated by a current 푗 on shell is a function on faces satisfying the following equation apart the grid boundary: • the Ampère law in the case of unit-area faces: −휕퐹 = 푗. Here the boundary 휕퐹 is the function on edges given by the formula [휕퐹](푒) = 퐹( 푓 ) − 퐹(푔) for each pair of adjacent faces 푓 and 푔 such that 휕 푓 (respectively, 휕푔) is oriented along (respectively, opposite to) the common edge 푒; see Figure4 to the left. (The definition of [휕퐹](푒) for boundary edges 푒 is not required for this subsection and is postponed until §2.3.) The following well-known existence and uniqueness result is straightforward. Proposition 2.6. For each current on shell there is a unique up to additive constant magnetic field.

Throughout §2.2 the functions 휙 and 퐹 are an electrical potential and a magnetic field respectively. Remark 2.4. The pair (휙, 퐹) and − 푗 are discretizations of an analytic function and its derivative [6,3]. Definition 2.6. A magnetic vector-potential 퐴 of the field 퐹 is a function on edges such that 훿퐴 = 퐹. A magnetic vector-potential 퐴 has the following properties (proved similarly to the ones from §2.3): • the source equation: −휕훿퐴 = 푗 apart the grid boundary;

• gauge invariance: 퐴 + 훿푔 is a vector-potential of the same ﬁeld for any function 푔 on vertices;

• the least action principle: among all functions on edges, 퐴 minimizes S[퐴] = 휖L[퐴], where L[퐴] = 1 훿퐴 ⌢ 훿퐴 + 푗 ⌢ 퐴. 2

Energy and momentum Let us state energy and momentum conservation in an electrical network in a simple heuristic form. 휙 휓 휙, 휓 Í 휙 푓 휓 푓 For functions , on faces (respectively, edges or vertices), denote by h i = 푓 ( ) ( ) the sum over all faces (respectively, edges or vertices). The obvious identity h훿휙, 푗i = h휙, 휕 푗i implies • the Tellegen theorem or global energy conservation: h훿휙, 푗i + h휙, 푠i = 0. Now we study local conservation and the flow of energy. Energy flows in the direction of the Poynting vector, hence transversely to (not along) the resistors. This is why we define energy flow in a subdivision of the grid. The cross-product formula for the Poynting vector is then discretized directly. Definition 2.7. The doubling is the 2푁 × 2푁 grid with the vertices at vertices, edge midpoints, and face centers of the initial 푁 × 푁 grid. Orient all the edges still in the direction of the coordinate axes. The heat power 푊 is the function on the vertices 푣 of the doubling given by the formula ( −[훿휙](푒) 푗 (푒), if 푣 is the midpoint of an edge 푒; 푊 (푣) = 0, if 푣 is the center of a face or a vertex of the initial grid.

The Poynting vector or energy flux 푆 is the function on edges 푢푣 of the doubling, max 푢푣 = 푣, given by 훿휙 푒 퐹 푓 , 푢 푣 푒 푓 [ ]( ) ( ) and are the centers of a vertical edge and a face or vice versa;  푆(푢푣) = −[훿휙](푒)퐹( 푓 ), 푢 and 푣 are the centers of a horizontal edge 푒 and a face 푓 or vice versa;  0, 푢 or 푣 is a vertex of the initial grid.  12 The Lorentz force 퐿 is defined analogously to 푆, only 훿휙 is replaced by − 푗/2 (thus 퐿 = 푆/2 in our basic model). The magnetic pressure 푃 (or momentum flux of the magnetic field towards the edges in the normal direction) is the function on nonboundary vertices 푣 of the doubling given by the formula

퐹 푓 퐹 푓 , 푣 푓  ( ) ( )/2 if is the center of a face ;  푃(푣) = 퐹( 푓 )퐹(푔)/2, if 푣 is the midpoint of the common edge of faces 푓 and 푔;  0, if 푣 is a vertex of the initial grid.  Straightforward consequences of these deﬁnitions and the Kirchhoﬀ laws are:

• Energy conservation: 휕푆 − 푊 = 0.

• Momentum conservation for the magnetic ﬁeld: 훿푃 + 퐿 = 0 on those edges of the doubling which contain the face-centers of the initial grid.

In §2.3 we introduce a more conceptual form of the two laws, explaining the latter restriction. Now we state a less visual momentum conservation law for the electric ﬁeld. This is essentially [12, Example in §8]. One expects the following properties of the momentum ﬂux 휎(푒) across edges 푒 of the initial grid (the latter property being required by the discretization principles from §1):

• 휎(푒) equals the momentum ﬂux of a continuum electric ﬁeld across 푒, if the potential is linear;

• 휎(푒) depends only on the values of 훿휙 at the edges intersecting 푒 and is bilinear in these values;

• 훿휎 = 0 apart the grid boundary: the momentum ﬂux across the boundary of each face vanishes.

The simplest function 휎 satisfying these properties is deﬁned as follows; cf. Figure5 and Remark 2.1.

Definition 2.8. The momentum flux of the electric field across edges in the negative normal direction, or the electric part of the Maxwell stress tensor, is the pair 휎 = (휎1, 휎2) of functions on edges disjoint with the grid boundary given by the following formula for each 푘 = 1, 2:

푘+1 ( (−1) 훿휙(푢푢+)훿휙(푢푣) + 훿휙(푣푣+)훿휙(푢푣), if 푢푣 k 푂푥푘 ; 휎푘 (푢푣) = 2 훿휙(푢푣)훿휙(푢푣) − 훿휙(푣푣+)훿휙(푣−푣), if 푢푣 ⊥ 푂푥푘 , where 푢푢+, 푣−푣, 푣푣+ are the edges orthogonal to 푢푣 with the maximal vertices 푢+, 푣, 푣+; see Figure5.

푥 푥 2 2 푢+ 푣+ h i 휎 ( ) = − 1 훿휙( )· 훿휙( ) + 훿휙( )· 훿휙( ) 푣− 푣 푣+ 푢 푣 2 2

푢 푢+ 푣− h i 푥 휎 ( ) = − 1 훿휙( )· 훿휙( ) − 훿휙( )· 훿휙( ) 푥1 1 2 2

Figure 5: Notation in Deﬁnition 2.8 of discrete momentum ﬂux. The square 푢푣푣+푢+ is shown by dotted lines to the right. The edge at which a particular fucntion is evaluated is shown in bold.

Corollary 2.1 (Momentum conservation for the electric ﬁeld). (Cf. [12, Example in §8].) For each electric potential 휙 on shell we have 훿휎1 = 훿휎2 = 0 on each face not intersecting the grid boundary.

Remark 2.5. The function 휎푘 is the ﬂux (given by Deﬁnition 2.12) of the energy-momentum tensor 푇 [휙] = 훿휙 × 훿휙 L[휙] = 1 훿휙 ⌢ 훿휙 − 푠 ⌢ 휙 (given by Theorem 1.3 for the Lagrangian 2 ).

13 Approximation The basic network model indeed converges to a continuum one, as the grid becomes finer and finer. The continuum model is a homogeneous conducting plate defined as follows. Let I2 be the unit square, nì be the unit inner normal vector field on 휕I2 besides the corners, ∗ be the counterclockwise ( , 푘 푙 푙 1 if = ; rotation through 휋/2 about the origin (the Hodge star), δ푘푙 = δ := 푘 0, if 푘 ≠ 푙. A source s is a continuous function on 휕I2. The fields ìj,ϕ,F,W,Sì,Lì,P,L,σ generated by s are continuous scalar/vector/matrix fields on I2, being 퐶1 and satisfying the following conditions apart 휕I2: −∇ϕ = ì, = −∇ϕ · ì, ì = −∗∇ϕ · , L = 1 (∇ϕ)2 = 1 ϕ ϕ , j W j S F 2 2 d yd 휕ϕ 휕ϕ ì ì ì 1 1 2 ∗∇F = j, L =∗j · F, P = F · F, σ푘푙 = − δ푘푙 (∇ϕ) , 2 휕x푘 휕x푙 2 and the following boundary condition on 휕I2 besides the corners:

ìj · nì = s. ϕ + 푖 휕 ϕ = − In other words, F is an analytic function such that 휕nì s; the other fields are expressions in it. 2 2 Let the unit square I be dissected into 푁 equal squares. Given a source 푠푁 , define the fields 푗푁 , 휙푁 , 퐹푁 , 푊푁 , 푆푁 , 퐿푁 , 푃푁 , L푁 , 휎푁 on the resulting grid literally as above on the grid of size 푁 × 푁. Remark 2.6. It would be somewhat more conceptual to modify the above Ampère law for the resulting grid because the faces are not unit squares anymore. This leads just to normalization of the fields by powers of 푁. We avoid such modification for simplicity. Clearly, the continuum model has more symmetries than the discrete one: e.g., L is rotational- invariant whereas L푁 is not, at least in a naive sense; cf. [16, Definition 5.2.36]. Dissect each side of 휕I2 into 푁 + 1 (not 푁) equal segments called auxiliary segments. Write 푎 푥 푏 푥 푎 , 푏 푀 푎 푥 푏 푥 푁 푁 ( ) ⇒ 푁 ( ) for functions 푁 푁 on a set 푁 , if max푥∈푀푁 | 푁 ( ) − 푁 ( )| → 0 as → ∞. Theorem 2.1 . Let 휕 2 → R be a continuous source with ∫ = . (Approximation theorem) s: I 휕I2 s dl 0 2 2 Dissect I into 푁 equal squares and define a discrete source 푠푁 on the resulting grid by the formula ∫ 푠푁 (푣) := s dl, 푣−푣+

2 2 where 푣−푣+ ⊂ 휕퐼 is the arc formed by 1 or 2 auxiliary segments containing a vertex 푣 ∈ 휕퐼 . Take ì ì ì continuous ﬁelds j, ϕ, F, W, S, L, P, L, σ and discrete ones 푗푁 , 휙푁 , 퐹푁 , 푊푁 , 푆푁 , 퐿푁 , 푃푁 , L푁 , 2 휎푁 = (휎푁,1, 휎푁,2) generated by the sources. Assume that ϕ, F and 휙푁 , 퐹푁 vanish at the center of I and at one of the vertices or faces closest to the center respectively. Take 푟 > 0. Then on the set of all vertices 푣, edges 푒, faces 푓 , edge-midpoints 푒0, and face-centers 푓 0 at distance ≥ 푟 from 휕I2 we have: ∫ ∫ ì ì 2 0 0 0 0 ì ì 휙푁 (푣) ⇒ ϕ(푣), 푁 푗푁 (푒) ⇒ 푁 j · dl, 푁 푊푁 (푒 ) ⇒ W(푒 ), 푁푆푁 (푒 푓 ) ⇒ 2푁 S · dl, 푒 푒0 푓 0 ∫ ∫ 2 2 0 0 0 0 ì ì 푁 L푁 (푣) ⇒ L(푣), 퐹푁 ( 푓 ) ⇒ 푁 F dS, 푃푁 (푒 ) ⇒ P(푒 ), 푁퐿푁 (푒 푓 ) ⇒ 푁 L · dl, 푓 푒0 푓 0 ∫ 2 1 2 푁 휎푁,푘 (푒) ⇒ 푁 σ푘2 dx − σ푘1 dx as 푁 → ∞. 푒 The theorem is essentially known; it is easily deduced from highly nontrivial known results in §4.

2.3 Lattice electrodynamics A standard problem in electrodynamics is to find forces between given charges and currents. This is done in two steps: first the field generated by the charges and currents is computed, then — the action of the field upon them. For a discretization, continuum spacetime is replaced by a 4-dimensional grid.

14 Generation of the ﬁeld by the current 푑 퐼푑 푥 , 푥 . . . , 푥 푁 푑 Deﬁnition 2.9. The -dimensional grid 푁 is the hypercube 0 ≤ 0 1 푑−1 ≤ in R dissected 푑 into 푁 unit hypercubes. Fix the dictionary order of the grid vertices: set (푥0, 푥1 . . . , 푥푑−1) < (푦0, 푦1 . . . , 푦푑−1) if and only if 푥0 = 푦0,. . . , 푥푘−1 = 푦푘−1, and 푥푘 < 푦푘 for some 0 ≤ 푘 ≤ 푑 − 1. Denote by max 푓 (min 푓 ) the maximal (minimal) vertex of a face 푓 (on the grid, it is the vertex with the maximal (minimal) sum of the coordinates). 푘 퐼푑 Fix the following orientation of -dimensional faces of 푁 .A positively oriented basis in a face is formed by the 푘 vectors starting at the minimal vertex of the face, going along the edges of the face, and listed in the order opposite to the order of the endpoints. E.g., on the grid, the positively oriented basis in a 푑-dimensional face is (1/푁, 0,..., 0), (0, 1/푁, . . . , 0),..., (0, 0,..., 1/푁), as (1/푁, 0,..., 0) > (0, 1/푁, . . . , 0) > ··· > (0, 0,..., 1/푁).A 푘-dimensional face 푓 and a (푘 − 1)- dimensional face 푒 ⊂ 휕 푓 are cooriented (respectively, opposite oriented), if the ordered set consisting of the outer normal to 푒 in 푓 and a positive basis in 푒 is a positive (respectively, negative) basis in 푓 . The boundary 휕퐹 and the coboundary 훿퐹 of a function 퐹 on 푘-dimensional faces 푒 are the functions on (푘 − 1)- and (푘 + 1)-dimensional faces 푣 and 푓 respectively given by (see Figure6) ∑︁ ∑︁ [휕퐹](푣) = 퐹(푒) − 퐹(푒), 푒 cooriented with 푣 푒 oriented opposite to 푣 ∑︁ ∑︁ [훿퐹]( 푓 ) = 퐹(푒) − 퐹(푒). 푒 cooriented with 푓 푒 oriented opposite to 푓

휕퐹 훿퐹 푘 푘 푑 퐶푘 퐼푑 푘 < 푘 > 푑 Set = 0 and = 0 for ≤ 0 and ≥ respectively, and ( 푁 ; R) = {0} for 0 or . The Minkowski sharp operator # applied to a function 퐹 on 푘-dimensional faces 푓 , for 푘 > 0, is

(−1)푘−1퐹( 푓 ), if 푓 k (1, 0,..., 0),  [ 퐹]( 푓 ) =  | {z } # : 푑−1 zeroes  (−1)푘 퐹( 푓 ), if 푓 ⊥ (1, 0,..., 0). 

An electromagnetic vector-potential 퐴 generated by a current 푗 is a function on edges satisfying • The source equation: −휕#훿퐴 = 푗.

[휕퐹]( ) = 퐹( ) − 퐹( ) + 퐹( ) − 퐹( )

[훿퐹]( ) = 퐹( ) − 퐹( ) − 퐹( ) + 퐹( ) + 퐹( ) − 퐹( )

Figure 6: Boundary and coboundary (see Deﬁnition 2.9). A nonboundary 3-face (to the left) is shown again by dotted lines (to the right). The face at which a particular function is evaluated is in bold. The signs in the expression for 휕퐹 are diﬀerent from the ones in (1) because the latter depicts 휕#퐹.

Remark 2.7. We do not discuss conditions under which the vector-potential exists and is unique. The operator # is new. It is a discrete analogue of raising all indices in the metric of signature (+, −,..., −). We use it instead of a discrete Hodge star [25] to avoid working with the dual lattice, which would complicate the theory and its generalization to other spacetimes. For an arbitrary spacetime, the operators 휕 and 훿, as well as max and min (but not #) are defined analogously except that the dictionary order is replaced by the one fixed in Definition 1.1. The following 3 properties of an electromagnetic vector-potential 퐴 generated by a current 푗 immediately follow from the well-known identities 훿훿 = 0 and 휕휕 = 0; cf. (1):

• the Maxwell equations: 훿퐹 = 0 and −휕#퐹 = 푗, where 퐹 := 훿퐴 is the electromagnetic ﬁeld; • Gauge invariance: 퐴 + 훿푔 is generated by the same current 푗 for any function 푔 on vertices;

15 • Charge conservation: 휕 푗 = 0, if there exists a vector-potential generated by the current 푗. Corollary 2.2. An electromagnetic vector-potential 퐴 is generated by a current 푗, if and only if 퐴 is a stationary function for the functional S[퐴] = 휖L[퐴], where

L[퐴] = − 1 훿퐴 ⌢ 훿퐴 − 푗 ⌢ 퐴. 2 # Remark 2.8. Electrodynamics in linear nondispersive media is discretized analogously, only the Minkowski sharp operator is replaced by a linear operator depending on the media. To convince the reader that lattice electrodynamics is a realistic model, let us informally sketch 푑 퐼푑−1 a network model for it [18]. Set = 4. For each edge of the grid 푁 , take an oscillatory circuit consisting of one (nonconstant) current source, one unit capacitor, and as many unit-transformer coils as there are faces containing the edge; see Figure7 to the bottom-left. Join the obtained circuits in the shape of the grid, join the transformer cores in the shape of the 1-dimensional skeleton of the dual grid, join the capacitor dielectric cores in the shape of the 2-dimensional skeleton of the dual grid. We get an electric, a magnetic, and a dielectric network coupled together; a part is shown in Figure7. We conjecture that the integrals of appropriate currents and voltages over time intervals [푛, 푛 + 1], where 푛 ∈ Z, satisfy the discrete Maxwell equations above.

Figure 7: A network model for lattice electrodynamics; cf. [18]

Action of the field on the current The field acts on the current by the Lorenz force, which we are going to discretize now. The rest of §2.3 contains completely new notions and results (except the cross-product); cf. [4]. 퐼푑 퐼푑 푑 푑 Definition 2.10. Let 푁 × 푁 be the Cartesian square of the -dimensional grid. It is a 2 -dimensional 푒 푓 푒 푓 퐼푑 grid with the faces of the form × , where and are faces of 푁 of arbitrary dimension. 푞, 푞 푒 푓 퐼푑 퐼푑 A tensor of type ( 1), where = 1 or 0, is a function on all faces × of 푁 × 푁 such that dim 푓 − dim 푒 = 1 − 푞. The chain-cochain cross-product of fields 휙 and 휓 with dim 휙 − dim 휓 = 1 − 푞 is the tensor ( 휓(푒)휙( 푓 ), if dim 푒 = dim 휓 and dim 푓 = dim 휙; [휓 × 휙](푒 × 푓 ) = 0, if dim 푒 ≠ dim 휓 or dim 푓 ≠ dim 휙.

The boundary operator 휕 is the unique linear map between the spaces of type (1, 1) and (0, 1) tensors such that for each ﬁelds 휙, 휓 with dim 휙 = dim 휓 we have

휕(휓 × 휙) = 휕휓 × 휙 + 휓 × 훿휙

(cf. Deﬁnition 2.1 above and equation (22) below). A type (1, 1) tensor 푇 is conserved, if 휕푇 = 0.

16 The main motivation for this deﬁnition is that it satisﬁes the principles of discretization from §1, as we see later. Let us clarify the relation to continuum theory (this is not used elsewhere in the paper). (푞, ) É푑 퐶 (퐼푑 ) ⊗ 퐶 푝−푞+1(퐼푑 ) Remark 2.9. Equivalently, the set of type 1 tensors is 푝=0 푝 푁 ; R R 푁 ; R . Thus É푑 Ω푝 ( 푑)∗ ⊗ Ω푝−푞+1( 푑) 푞 ( 푑) it discretizes the space 푝=0 I I rather than the space T1 I of continuum type (푞, 1) tensors. (Here Ω푝 (I푑) denotes the set of C∞ 푝-forms on the unit hypercube I푑 and ⊗ denotes the tensor product over Ω0(I푑)). But the former space is mapped to the latter by the ‘contraction’ map

( 푚1...푚 푝 T , if 푞 = 0; 푚1...푚 푝 ↦→ 푘푚1...푚 푝 T푛1...푛푝−푞+1 푙푚2...푚 푝 1 푙 푚1...푚 푝 T − δ T푚 ...푚 , if 푞 = 1, 푝 > 0. 푘푚2...푚 푝 2푝 푘 1 푝 (Summation over repeating indices is understood.) Since no discretization of the image is available (at least for 푞 = 1), the discretization of the domain is proclaimed to be space of type (푞, 1) tensors. 푙 Here the role of the δ푘 -term is the same as in the Einstein tensor: it makes the ‘contraction’ map commute with certain codiﬀerentials when T has certain symmetry properties (i.e., ♯T is symmetric wrt interchanging 푚푝 and 푛푝 but antisymmetric wrt interchanging 푚푝 and 푚푞):

∗⊗ + ⊗ 휕L ⊗ ϕ + 휕L ⊗ ϕ ∈ É푑 Ω푝 ( 푑)∗ ⊗ Ω푝 ( 푑) −−−−−−−−−→d id id d É푑−1 Ω푝 ( 푑)∗ ⊗ Ω푝+1( 푑) 휕(dϕ) d 휕ϕ 푝=0 I I 푝=0 I I    y ‘contraction’y ‘contraction’y divergence 푙 ∈ 1( 푑) −−−−−−−→ 0( 푑). T푘 T1 I T1 I 휕L ⊗ ϕ + 휕L ⊗ ϕ 푙 Similarly, (7) discretizes 휕(dϕ) d 휕ϕ rather than the continuum energy-momentum tensor T푘 , 푚 ...푚 휕L 1 푝 휕L but the former is usually taken to the latter by the ‘contraction’ map. Here 휕( ϕ) := 휕( ϕ) . d d 푚1...푚푝 The former is conserved (i.e. taken to 0 by d∗ ⊗ id + id ⊗ d) regardless of symmetries of L[ϕ]. In contrast to continuum theory, type (0, 1) tensors are not 1-dimensional fields. 퐼푑 퐼푑 퐼2푑 Although 푁 × 푁 is naturally identified with 푁 , the boundary operator on tensors is not the 퐼2푑 퐼푑 퐼푑 퐼2푑 boundary operator on 푁 . To avoid confusion, we distinguish between 푁 × 푁 and 푁 below. 푞, 퐶푑+푞−1 퐼푑 퐼푑∗ 퐼푑∗ A type ( 1) tensor can be equivalently defined as an element of ( 푁 × 푁 ; R), where 푁 퐼푑 퐼푑∗ is the dual grid. Then the boundary operator on tensors is exactly the boundary operator on 푁 × 푁 . We avoid working with dual grids for simplicity and for easier generalization to arbitrary spacetimes. It would be more conceptual to restrict the domain of a tensor to a “neighborhood of the diagonal” 퐼푑 퐼푑 , 푒 푓 푒 푓 in 푁 × 푁 . E.g., type (0 1) tensors can be restricted to the set of faces × such that ⊂ : the values at the other faces do not contribute to integration. We avoid such restriction for simplicity. 퐼푑 퐼푑 푒 푓 The set of faces of 푁 × 푁 is naturally mapped to the set of faces of the doubling: to a face × assign the face of the doubling with the center at the midpoint of the segment joining the centers of 푒 and 푓 . Thus informally the values of a tensor are “sitting” on the faces of the doubling; in particular, the ones on the 2-dimensional faces are interpreted as off-diagonal components. Up to sign and factor 1/2, the fields 푊, 푆, 퐿, 푃 from §2.2 are “induced” by the latter map from 푗 × 훿휙, 퐹 × 훿휙, 푗 × 퐹, 퐹 × 퐹 respectively. These heuristic fields are now replaced by tensors.

Definition 2.11. Let 퐴 be a vector-potential generated by a current 푗, and 퐹 = 훿퐴. The Lorentz force , 퐿 푗 퐹 푒 푓 퐼푑 퐼푑 푒 푓 is the type (0 1) tensor = × . It has support on faces × ⊂ 푁 × 푁 such that dim =1, dim =2. The energy-momentum tensor, or stress-energy tensor, of the electromagnentic field (respectively, of both the field and the current) is the type (1, 1) tensor 푇0 = −#퐹 × 퐹 (respectively, 푇 = −#퐹 × 퐹 − 푗 퐴 푇0 푒 푓 퐼푑 퐼푑 푒 푓 × ). The tensor has support on 4-dimensional faces × ⊂ 푁 × 푁 such that dim = dim = 2. An immediate consequence of these definitions, Maxwell’s equations, and charge conservation is

• Energy and momentum conservation: 휕푇0 = 퐿 and 휕푇 = 0.

Remark 2.10. The latter is a particular case of Theorem 1.3 for the Lagrangian from Corollary 2.2. In contrast to 푇0, the tensor 푇 has no conserved continuum analogue.

17 0 푙 푙푚 More precisely, 퐿 and 푇 discretize the tensors j F푘푛 and −F F푘푛, but the latter two are taken to the continuum Lorenz force and energy-momentum tensor by the ‘contraction’ map from Remark 2.9. The formula for the discrete energy-momentum tensor 푇0 is even simpler than the continuum analogue. This is achieved at the cost of a rather subtle deﬁnition of discrete tensor integration below.

Integral conservation laws To make discrete tensors at all practical, we define their integration. This allows to get integral forms of the above conservation laws and to compare these tensors with their continuum analogues. The following construction works for any discrete field theory, not just electrodynamics, but only on the 퐼푑 푑 grid 푁 , where ≥ 2. In §1.1 (respectively, in Definition 2.8) we have actually applied the construction for 푑 = 3, 푘 = 0, and the tensor 푇0 (respectively, for 푑 = 2, 푘 = 1, 2, and the tensor 훿휙 × 훿휙). 푘 = , . . . , 푑 − 1 Let us introduce some notation. Let e푘 , where 0 1, be the vector of length 2 pointing in the direction of the axis 푂푥푘 . Each combination of such vectors with coefficients from the set , ,..., 푁 퐼푑 푓 {0 1 2 } is the center of a unique face of 푁 . We use the same notation for a face and its center. In particular, 푓 + e푘 denotes the face with the center at the point obtained from the center of 푓 by translation by the vector e푘 . The dimensions of 푓 and 푓 + e푘 are always different by 1.A hyperface 푑 퐼푑 is a ( − 1)-dimensional face of 푁 . Definition 2.12. A type (1, 1) tensor is partially symmetric, if 푇 (푒 × 푓 ) = 푇 ( 푓 × 푒) for each 푒 k 푓 (we set 푒 k 푓 , if 푒 and 푓 are vertices). For 푘 = 0, . . . , 푑 − 1, the 푘-th component of the flux of a partially symmetric tensor 푇 across a nonboundary hyperface ℎ ⊥ e푙 in the positive normal direction is (푇 푓 푓 푇 푓 푓 , ℎ 1 ∑︁ dim Pr( 푓 ,푘,푙)+푙+1 (( + e푙 − e푘 ) × ) + (( + e푙 + e푘 ) × ) if k e푘 ; h푇, ℎi푘 = (−1) · 2 푇 ( 푓 × 푓 ) − 푇 (( 푓 + ) × ( 푓 − )), ℎ ⊥ , 푓 : 푓 ⊂ℎ, 푓 3max ℎ; e푘 e푘 if e푘 푓 ke푘 for ℎke푘 where the sum is over faces 푓 of arbitrary dimension (we set 푓 ∦ e푘 , if 푓 is a vertex), and Pr( 푓 , 푘, 푙) is the orthogonal projection of 푓 to the linear span of all e푚 with min{푘, 푙} ≤ 푚 ≤ max{푘, 푙}. Assume that 푑 ≥ 2. Let 휋 be an oriented piecewise-linear hypersurface consisting of nonboundary hyperfaces. For each hyperface ℎ ⊂ 휋 denote ( +1, if the orientations of 휋 and ℎ agree, hℎ, 휋i = (8) −1, if the orientations of 휋 and ℎ are opposite.

The latter notation is also used, if 휋 and ℎ have any dimension 푝 > 0. The flux across 휋 is 푇, 휋 Í 푇, ℎ ℎ, 휋 푇 휕퐼푑 휕푇 푒 푓 푒, 푓 휕퐼푑 h i푘 := ℎ h i푘 h i. A tensor is conserved apart 푁 , if ( × ) = 0 for all faces ⊄ 푁 . Proposition 2.7 (Integral energy-momentum conservation). If a partially symmetric type (1, 1) tensor 퐼푑 푑 is conserved apart the boundary of the grid 푁 , where ≥ 2, then each component of the flux of the tensor across each closed oriented hypersurface consisting of nonboundary hyperfaces vanishes. Theorem 2.2 (Integral energy-momentum conservation for a free field). Let 푑 ≥ 2. If the Lagrangian 휙 훿휙 ⌢ 훿휙 푚2휙 ⌢ 휙 휙 퐶푘 퐼푑 is L[ ] = − # − and ∈ ( 푁 ; R) is on shell, then each component of the flux of tensor (7) across each closed oriented hypersurface consisting of nonboundary hyperfaces vanishes. Some particular cases of this theorem were established in [12, §8] by a different method. Remark 2.11. There are many other ways to define a tensor flux; we have chosen the simplest one. Our definition has the following informal motivation. Values of a tensor are “sitting” on the faces of the doubling; see the paragraph of Remark 2.9 before the last one. The flux across a hyperface is then the sum of these values over the faces adjacent to the hyperface from appropriate “side”. For nonconserved tensors an analogue of the Stokes formula holds; see Proposition 4.2. Similar results hold for 푑 = 1, only oriented hypersurfaces should be replaced by 0-chains. Unlike continuum theory, the 0-th component of the flux of the energy-momentum tensor 푇0 (see Definition 2.11) across a hyperface ℎ ⊥ (1, 0,..., 0) is not necessarily positive, thus cannot be interpreted as energy density. This is a higher order effect with respect to the discretization step 1/푁.

18 We use the notation h푇, 휋i푘 , with literally the same definition, even if 푇 is not partially symmetric. This makes no sense in discrete setup but is useful for the continuum limit; see Proposition 2.15. The energy-momentum tensor 푇 of both the field and the current (see Definition 2.11) is not partially symmetric. In a sense, it still approximates some continuum tensor, but the latter is not conserved. We know neither an integral conservation law nor a conserved continuum analogue for 푇. The energy-momentum tensor 푇0 is symmetric in a sense (after “raising an index”). In particular, we shall see that it approximates the symmetric Belinfante–Rosenfeld energy-momentum tensor rather than the nonsymmetric canonical energy-momentum tensor. In other field theories, e.g., for the Dirac field, the discrete energy-momentum tensor approximates the nonsymmetric canonical energy-momentum tensor rather than the Belinfante–Rosenfeld one; see Proposition 2.15. Let us illustrate analogy between tensor (7) and the continuum canonical energy-momentum tensor 휕L 휕ϕ 푙 푙 . T푘 = − δ푘 L 휕(휕ϕ/휕x푙) 휕x푘

푑 퐶0 퐼푑 퐶 퐼푑 Proposition 2.8. Let ≥ 2. Let a local Lagrangian L : ( 푁 ; R) → 0( 푁 ; R) be homogeneous quadratic in 휙 and 훿휙. Let 휙 be a 0-dimensional ﬁeld (not necessarily on shell) and 푇 be the energy- momentum tensor (not necessarily partially symmetric) given by (7). Then for each 0 ≤ 푘, 푙 < 푑 and each hyperface ℎ ⊥ e푙 having maximal vertex 푣 and disjoint with the grid boundary we have 휕 휙 휕 휙 푙 1 L[ ] L[ ] 푙 (−1) h푇, ℎi푘 = (푣 + e푙) + (푣 + e푙 − 2e푘 ) 훿휙(푣 − e푘 ) − δ L[휙](푣). 2 휕(훿휙) 휕(훿휙) 푘

Approximation The discrete energy-momentum tensor 푇0 indeed approximates the continuum one, as we show now. In continuum theory, an electromagnetic field is a continuous antisymmetric matrix field F푚푛 on the unit hypercube I푑. The (Belinfante–Rosenfeld) energy-momentum tensor of the field (for the metric of signature (+, −,..., −)) is the matrix field

푙 = − 푙푚 + 1 δ푙 푚푛 , T푘 F F푘푚 4 푘 F F푚푛 ( −F푚푛, if 푚 = 0 or 푛 = 0; where summation over repeating indices is understood and F푚푛 := F푚푛, if 푚 ≠ 0 and 푛 ≠ 0. Let I푑 be dissected into 푁 푑 equal hypercubes. Given an arbitrary discrete 2-dimensional field 퐹, 푇0 퐹 퐹 퐼푑 define the energy-momentum tensor = −# × on the resulting grid literally as on the grid 푁 . Remark 2.12. It is somewhat more natural to modify the definition of the operator # by the factor 푁2푘−푑 because the faces are not unit hypercubes anymore. This leads just to normalization of the energy-momentum tensor 푇0 by a power of 푁. We avoid such modification for simplicity.

푑 Proposition 2.9 (Approximation property). Let F푚푛 be a continuous electromagnetic field on I . 푑 푑 Dissect I into 푁 equal hypercubes and define a discrete 2-dimensional field 퐹푁 on faces 푓 of the resulting grid by the formula 퐹푁 ( 푓 ) := F푚푛 (max 푓 ), 푚 < 푛 , k 푓 푙 푇0 − 퐹 × 퐹 where the integers are determined by the conditions e푚 e푛 . Let T푘 and 푁 = # 푁 푁 be the continuous and discrete energy-momentum tensor respectively. Take 0 ≤ 푘, 푙 < 푑. Then on the 푑 set of all hyperfaces ℎ ⊥ e푙 not intersecting 휕I we have (under the notation before Theorem 2.1) 푙 푇0 , ℎ 푙 ℎ 푁 . (−1) h 푁 i푘 ⇒ T푘 (max ) as → ∞

Remark 2.13. Here the ﬁelds F푚푛 and 퐹푁 do not necessarily satisfy the Maxwell equations (and typically 퐹푁 cannot, even if F푚푛 does). Approximation of a smooth solution of the Maxwell equations by discrete ones, a standard question of computational electrodynamics, is not discussed in the paper.

19 2.4 Lattice gauge theory Classical gauge theory generalizes electrodynamics. It is a basis for quantum gauge theory describing all known interactions except gravity. The idea is simple, as shown by the following toy model; cf.[20].

Toy model Several cities are connected by roads in the shape of an 푀 × 푁 grid; see Figure8. Each city has its own type of goods in an unlimited quantity. E.g., city 푎 has apples and city 푏 has bananas. For two neighboring cities 푎 and 푏 an exchange rate 푈(푎푏) > 0 is fixed, e.g., 2 banana for an apple. The rate is symmetric, i.e., 푈(푏푎) = 푈(푎푏)−1: one gets back an apple for 2 banana. A cunning citizen can travel and exchange along a square 푎푏푐푑 to mul- tiply his initial amount of goods by a factor of 푈(푎푏)푈(푏푐)푈(푐푑)푈(푑푎). The total speculation profit is measured by the quantity ∑︁ S[푈] := log2(푈(푎푏)푈(푏푐)푈(푐푑)푈(푑푎)). Figure 8: Lattice gauge all faces 푎푏푐푑 theory on a 1 × 2 grid Here log2(푥) is chosen as a function vanishing at 푥 = 1 and positive for 푥 ≠ 1. The king can set exchange rates except those on the boundary of the grid. He sets them to minimize the quantity S[푈]. The resulting collection of rates is an Abelian gauge group field on shell. A gauge group field on shell is far from being unique. For an interior city, one can change the units of measurements, e.g., exchange dozens of apples instead of single ones. Such gauge transformation multiplies the rates for all the roads starting from the city by the same value but preserves S[푈]. A similar model on a 푑-dimensional grid (with an additional minus sign for each summand in S[푈] such that 푎푏푐푑 is parallel to (1, 0,..., 0)) is equivalent to lattice electrodynamics discussed in §2.3. This follows from Corollary 2.2, if one sets 퐴(푎푏) = log 푈(푎푏) and 푗 = 0; see also Remark 2.14.

Currents Now modify the model by introducing production of goods. For each pair of neighboring cities 푎 and 푏 fix a production rate 푗 (푎푏) ≥ 0: e.g., if 푎 has apples and 푏 has jam, then one produces 푗 (푎푏) units of jam from one apple. The rate is not at all symmetric: one cannot produce apples from jam. Assume that production always goes in the direction of the coordinate axes. There is a new way to profit: producing jam and exchanging back to apples, one multiplies the initial amount of apples by 푗 (푎푏)푈(푏푎). The total profit is now measured by the quantity 푆 푈, 푗 푆 푈 Í 푗 푎푏 푈 푏푎 푈 푆 푈, 푗 푗 [ ] = [ ] + 푎푏 ( ( ) ( ) − 1). A collection of rates minimizing [ ] for fixed is called generated by 푗. These rates may not exist, and the total profit can be negative. These rates satisfy the conservation law − 푗 (1)푈(1)−1− 푗 (2)푈(2)−1+푈(3)−1 푗 (3)+푈(4)−1 푗 (4) = 0 for each interior city 푣, where we use the notation from Figure4 to the middle (this law is a version of Corollary 2.3). This is a “gauge-invariant” equation, which coincides with the usual charge conservation 휕 푗 = 0 in the case when 푈 = 1 identically.

Non-Abelian gauge theory In non-Abelian gauge theory the goods become vectors and the rates become matrices. To catch the idea, one can start with the case when 푑 = 2, 푛 = 1, 퐺 = {푔 ∈ C : |푔| = 1}, and drop all #-operators.

Deﬁnition 2.13. Denote by C푚×푛 the set of matrices with complex entries having 푚 rows and 푛 columns. For 푢 ∈ C푚×푛 denote by 푢∗ ∈ C푛×푚 the conjugate transpose matrix. A gauge group 퐺 is a Lie group represented by unitary transformations of C푛.A gauge group ﬁeld 푈 푗 퐼푑 퐺 푛×푛 and a covariant current are functions on edges of 푁 assuming values in and C respectively.

20 The operator of parallel transport along an oriented path 휋 going along the edges and having no self-intersections is Ö 푈(휋) := 푈(푒)h푒,휋i, 푒 where the product is over all the edges 푒 of the path 휋, and h푒, 휋i = ±1 is given by (8). In particular, the trace Tr 푈(휕 푓 ) is a well-defined complex-valued function on 2-dimensional faces 푓 . A gauge group field 푈 generated by a covariant current 푗 is a stationary function for the functional (for fixed 푗) ∑︁ ∑︁ S[푈] = # (Re Tr 푈(휕 푓 ) − 푛) − Re Tr [ 푗∗(푒)푈(푒)]. (9) faces 푓 edges 푒 Since S[푈] is a continuous function on a compact set, we get the following existence theorem. Proposition 2.10. For each covariant current there exists a gauge group field generated by it. Now we state the Yang–Mills equation (necessary and sufficient for 푈 to be generated by 푗) and a conservation law. This is a new Corollary 2.3 extending [11, Eq. (4.15)]. It involves projection to certain tangent space of the Lie group 퐺. In gauge theory the role of the (co)boundary is played by the covariant (co)boundary, which is a “gauge covariant” operator equal the (co)boundary for 푈 = 1. Definition 2.14. Fix a gauge group field 푈. Let 푗 be a C푛×푛-valued function on edges. Its covariant 퐷∗ 푗 푛×푛 푣 boundary 퐴 is a C -valued function on vertices given by 퐷∗ 푗 푣 ∑︁ 푈 푒 −1 푗 푒 ∑︁ 푗 푒 푈 푒 −1. [ 퐴 ]( ) = ( ) ( ) − ( ) ( ) (10) 푒 ending at 푣 푒 starting at 푣 퐷∗ 퐹 푛×푛 푒 Denote by 퐴# the C -valued function on edges given by 퐷∗ 퐹 푒 ∑︁ 푈 푒 푈 휕 푓 푒 , [ 퐴# ]( ) = #( ( ) − ( − )) (11) 2-faces 푓 ⊃푒 where 휕 푓 − 푒 is the path starting at the vertex min 푒, consisting of the 3 edges of 휕 푓 − 푒, and ending 푒 퐷∗ 퐹 푑푐 푈 푑푎푏푐 푈 푑푓 푒푐 푈 푑푐 at max . E.g., in Figure8 we have [ 퐴# ]( ) = ( ) + ( ) − 2 ( ). 퐷∗ 푗 퐷∗ 퐹 So far the notations 퐴 and 퐴# should be viewed as indivisible. Separate conceptual defini- 퐴 퐹 퐷∗ tions of , , 퐴 are postponed until the end of §2.4, where (10)–(11) become easy propositions. 푛×푛 ∗ 푛×푛 Definition 2.15. The scalar product of 푢, 푣 ∈ C is h푢, 푣i := Re Tr [푢 푣]. Let 푇푢퐺 ⊂ C be the 퐺 푢 퐺 푛×푛 푇 퐺 linear subspace parallel to the tangent subspace to at a point ∈ . Let Pr푇푢퐺 : C → 푢 be the 푗 푒 푗 푒 푗 푒 . orthogonal projection and Pr푇푈 퐺 be the function on edges given by [Pr푇푈 퐺 ]( ) = Pr푇푈 (푒) 퐺 ( ) A 푗 퐷∗ 푗 covariant current is conserved, if 퐴Pr푇푈 퐺 = 0. Corollary 2.3. A gauge field 푈 generated by a covariant current 푗 satisfies the following equations: − 퐷∗ 퐹 푗 • the Yang–Mills equation: Pr푇푈 퐺 퐴# = Pr푇푈 퐺 ; 퐷∗ 푗 • Charge conservation law: 퐴Pr푇푈 퐺 = 0. Remark 2.14. The latter form of change conservation, different from the usual 휕 푗 = 0, reflects the fact that non-Abelian gauge fields are themselves charged. In contrast to continuum theory, this remains true even if 퐺 is Abelian (the reason is that the cup-product is non-Abelian; cf. Example 1.2). Also, 퐷∗ 푗 푗 퐷∗ 푗 푈 퐴 ≠ 0 in general: e.g., if vanishes on all edges except one, then 퐴 ≠ 0 whatever is. However, for the Abelian group 퐺 = {푒푖휙 : 휙 ∈ R} and 푑 = 2 the action can be modified so 휕 푗 푗 퐶 퐼2 that charge conservation returns to the form = 0 (here ∈ 1( 푁 ; R) is not a covariant current anymore): ∑︁ ∑︁ SAb [푈] = − 1 2 푈(휕 푓 ) + 푖 푗 (푒) 푈(푒). 2 arccos Re # log faces 푓 edges 푒 The range of 푈 must be restricted to {푒푖휙 : −휋/4 < 휙 < 휋/4} to keep the action single-valued and differentiable. The resulting theory is equivalent to lattice electrodynamics of §2.3, also with restricted SAb [푒푖휙] = 휖 − 1 훿휙 ⌢ 훿휙 − 푗 ⌢ 휙 휙 ∈ 퐶1(퐼2 ) |휙| < 휋/ range, because 2 # for 푁 ; R with 4.

21 Table 2: Products of (co)chains of dimension 0 and 1 (where 푎푏 denotes an edge with 푎 < 푏). dim 휙 = 1, dim 휓 = 0 dim 휙 = 0, dim 휓 = 1 dim 휙 = dim 휓 = 1 [휙 ⌣ 휓](푎푏) = 휙(푎푏) 휓(푏) [휙 ⌣ 휓](푎푏) = 휙(푎) 휓(푎푏) 휙 ⌣ 휓 is deﬁned in Figure4 휙 ⌢ 휓 푎푏 휙 푎푏 휓 푎 휙 ⌢ 휓 휙 ⌢ 휓 푏 Í 휙 푎푏 휓 푎푏 [ ]( ) = ( ) ( ) = 0 [ ]( ) = edges 푎푏:푎<푏 ( ) ( ) 휙 ⌢∗ 휓 휙 ⌢∗ 휓 푎푏 휙 푏 휓 푎푏 휙 ⌢∗ 휓 푏 Í 휙 푏푐 휓 푏푐 = 0 [ ]( ) = ( ) ( ) [ ]( ) = edges 푏푐:푐>푏 ( ) ( )

Connection and curvature Deﬁnition 2.16. Let 푔 and 휙 be 퐺- and C푛×푛-valued functions on vertices and 푘-faces respectively. The gauge transformation of 휙 by 푔 is the function 푔∗ ⌣ 휙 ⌣ 푔 on 푘-faces 푓 given by (cf. Table2)

[푔∗ ⌣ 휙 ⌣ 푔]( 푓 ) := 푔∗(min 푓 ) 휙( 푓 ) 푔(max 푓 ).

Corollary 2.4 (Gauge invariance). Each simultaneous gauge transformation of 푈 and 푗 by the same element 푔 preserves S[푈]. If 푈 is generated by 푗, then 푔∗ ⌣ 푈 ⌣ 푔 is generated by 푔∗ ⌣ 푗 ⌣ 푔. Definition 2.17. The unit gauge group field 1 equals the unit 푛 × 푛 matrix at each edge. For a gauge group field 푈, the connection (or gauge potential) is the C푛×푛-valued function 퐴[푈] = 푈 − 1. The curvature (or field strength) is the C푛×푛-valued function on the set of faces given by

퐹[푈](푎푏푐푑) := 푈(푎푏)푈(푏푐) − 푈(푎푑)푈(푑푐) for each face 푎푏푐푑 with the vertices listed counterclockwise starting from the minimal one; see Figure4. Remark 2.15. On a grid, a gauge group field 푈 is a gauge transformation of the unit gauge group field, if and only if the curvature 퐹[푈] vanishes (this is proved by a standard “homological” argument.) In contrast to continuum theory, the connection and curvature assume values not in the Lie algebra of the Lie group 퐺 but in certain other subsets of C푛×푛 approximating the Lie algebra in a sense. The fields 퐴 and 퐹 from §2.2–2.3 are neither connection nor curvature for no gauge group field. Analogously to Proposition 2.9, the tensor −Re Tr [#퐹∗ × 퐹] approximates the continuum Belinfante– Rosenfeld energy-momentum tensor. But the former is not conserved and even not gauge invariant. For a simplicial complex 푀 with fixed vertices ordering, the curvature is defined by the formula 퐹[푈](푎푏푐) = 푈(푎푏)푈(푏푐) − 푈(푎푐) for each face 푎푏푐 with the vertices listed in increasing order 푎 < 푏 < 푐. Proposition 2.11. There is the following expression for action (9): S[푈] = 휖 − 1 퐹∗ ⌢ 퐹 − 푗∗ ⌢ 푈 . Re Tr 2 # Such expression for S[푈] is the one given by the algorithm from §1.3 up to an additive constant.

Covariant differentiation The covariant (co)boundary is defined in terms of cochain products as follows; cf [11, §IV–V]. Particular cases of the definition shown in Table2 and in (10)–(11) are sufficient for all our examples. 퐶푘 퐼푑 푉 푘 Definition 2.18. Denote by ( 푁 ; ) the set of functions defined on the set of -dimensional faces 푉 푉 퐶푘 퐼푑 푉 and assuming values in a set . Here , and hence ( 푁 ; ), is a set, not necessarily a group. Denote by 푎 . . . 푏 the face 푓 such that min 푓 = 푎, max 푓 = 푏 (if such face 푓 exists, then it is unique by Definition 1.1). An ordered triple of faces 푎...푏,푏...푐 ⊂ 푎 . . . 푐 of dimensions 푘, 푙, 푘 + 푙 respectively is cooriented (repectively, opposite oriented), if the ordered set consisting of a positive basis in 푎 . . . 푏 and a positive basis in 푏 . . . 푐 is a positive (respectively, negative) basis in 푎 . . . 푐. Write ( +1, if 푎...푏,푏...푐,푎...푐 are cooriented, h푎, 푏, 푐i = −1, if 푎...푏,푏...푐,푎...푐 are oppositely oriented.

22 퐶푘 퐼푑 푝×푞 퐶푙 퐼푑 푞×푟 The cup-, cap-, and cop-product of functions Φ ∈ ( 푁 ; C ) and Ψ ∈ ( 푁 ; C ) are the C푝×푟 -valued functions on (푘 + 푙)-, (푘 − 푙)-, and (푙 − 푘)-dimensional faces respectively given by ∑︁ [Φ ⌣ Ψ](푎 . . . 푐) = h푎, 푏, 푐iΦ(푎 . . . 푏)Ψ(푏 . . . 푐); 푏: dim(푎...푏)=푘,dim(푏...푐)=푙 ∑︁ [Φ ⌢ Ψ](푏 . . . 푐) = h푎, 푏, 푐iΦ(푎 . . . 푐)Ψ(푎 . . . 푏); 푎: dim(푎...푐)=푘,dim(푎...푏)=푙 ∗ ∑︁ [Φ ⌢ Ψ](푎 . . . 푏) = h푎, 푏, 푐iΦ(푏 . . . 푐)Ψ(푎 . . . 푐), 푐: dim(푏...푐)=푘,dim(푎...푐)=푙 where the sums are over all the vertices such that there exist 3 faces 푎...푏,푏...푐 ⊂ 푎 . . . 푐. 퐶푘 퐼푑 푛×푛 For Φ ∈ ( 푁 ; C ), the covariant coboundary and the covariant boundary are respectively 푘 퐷 퐴Φ := 훿Φ + 퐴 ⌣ Φ − (−1) Φ ⌣ 퐴; (12) 퐷∗ 휕 ∗ ⌢ 퐴 ∗ 푘 퐴 ⌢∗ ∗ ∗. 퐴Φ := Φ + (Φ ) + (−1) ( Φ ) (13) Remark 2.16. For a simplicial complex 푀 the definition requires the following modifications (because a face is not determined by just the minimal and the maximal vertices anymore). Denote by 푎1푎2 . . . 푎푠+1 the 푠-dimensional face with the vertices 푎1 < 푎2 < ··· < 푎푠+1. The value h푎, 푏, 푐i and the “triality” of products is defined by the same formulae, only 푎 . . . 푏, 푏 . . . 푐, 푎 . . . 푐 are replaced by 푎1 . . . 푎푠푏, 푏푐1 . . . 푐푡, 푎1 . . . 푎푠푏푐1 . . . 푐푡 respectively, summation over 푏 is omitted, and summation over 푎 and 푐 is replaced by summation over all collections (푎1, . . . , 푎푠) and (푐1, . . . , 푐푡) respectively. The definition of the cup-product is equivalent to [28, (22.3)] but not [29, Chapter IX, §14, Eq. (7)]. Up to sign and factors interchange, the cop-product is the cap-product in the same complex but with reversed vertices ordering. The cap- and cop- products vanish for 푘 < 푙 and 푘 > 푙 respectively, and do not coincide for 푘 = 푙 ≠ 0. Usually both are denoted in the same way, which does not lead to a conflict until one identifies chains and cochains (hence the domains of the products). Since we have ∗ performed such identification, we need to introduce new notation ⌢ and new term “cop-product”.

Proposition 2.12. For each gauge group field 푈 we have 퐹 = 훿퐴 + 퐴 ⌣ 퐴, 퐷 퐴퐹 = 0, and (10)–(11). Remark 2.17. The results of §2.4 remain true for arbitrary spacetime, if one drops all #-operators. The section is intentionally written so that all the definitions, propositions, and corollaries (but not 퐼푑 necessarily the particular examples outside those environments) remain true, if 푁 is replaced by an 푀 퐼푑 푀 arbitrary cubical complex , the dictionary order on 푁 is replaced by the ordering on fixed in Definition 1.1, and all #-operators are dropped. For a simplicial complex 푀, the definitions should be modified according to Remarks 2.15–2.16. The proofs of the resulting generalizations are analogous, only for a simplicial complex each instance of the fourth vertex “푑” of a face 푎푏푐푑 is just removed, and a direct checking is used instead of Lemma 4.5.

2.5 The Klein–Gordon field The classical (not quantum!) Klein–Gordon field does not describe any real physical field but serves as an example for more realistic models. Corollaries 2.6, 2.9, 2.10 and Proposition 2.13 are new.

Basic model Definition 2.19. Fix a number 푚 ≥ 0 called particle mass. A complex-valued function 휙 on the set 퐼푑 푚 휕퐼푑 of vertices of 푁 is a Klein–Gordon field of mass , if the following equation holds apart 푁 : • the Klein–Gordon equation: −휕#훿휙 + 푚2휙 = 0. 휙 퐶0 퐼푑 Corollary 2.5. A function ∈ ( 푁 ; C) is a Klein–Gordon field, if and only if among all the functions 휕퐼푑 휙 휙 휖 휙 with the same values on 푁 , the function is stationary for the functional S[ ] = L[ ], where ∗ ∗ L[휙] = #훿휙 ⌢ 훿휙 − 푚2휙 ⌢ 휙 .

23 Here we impose a boundary condition, because the theory becomes trivial otherwise. The La- grangian L[휙] is globally gauge invariant, i.e., L[휙푔] = L[휙] for each 푔 ∈ C with |푔| = 1.

Corollary 2.6 (Charge, energy, momentum conservation). For a Klein–Gordon ﬁeld 휙 the current 푗 휙 훿휙∗ ⌢ 휙 푇 휙 훿휙∗ 훿휙 푚2휙∗ 휙 휕퐼푑 [ ] := −2Im(# ) and the tensor [ ] := 2Re[# × − × ] are conserved apart 푁 .

Approximation The resulting current 푗 [휙] and energy-momentum tensor 푇 [휙] indeed approximate continuum ones. In continuum theory, ϕ is a smooth complex-valued function deﬁned on I푑. (Hereafter smooth means 퐶1, and the derivative at the boundary 휕I푑 means a one-sided derivative.) The current and energy-momentum tensor of ϕ (for the metric signature (+, −,..., −)) are the vector and matrix ﬁelds

푙 휕푙 ∗ 푙 휕푙 ∗휕 푙 휕푛 ∗휕 푚2 ∗ , j = −2Im ϕ ϕ and T푘 = 2Re ϕ 푘 ϕ + δ푘 − ϕ 푛ϕ + ϕ ϕ ( +휕 ϕ, if 푛 = 0; 푛 휕 ϕ = 휕ϕ 휕푛ϕ = 푛 where summation over is understood, and we denote 푛 : 휕x푛 , : −휕푛ϕ, if 푛 ≠ 0. Proposition 2.13 (Approximation property). Let ϕ be a smooth complex-valued field on I푑, 푑 ≥ 2. 푑 푑 Dissect I into 푁 equal hypercubes and take the discrete field 휙푁 (푣) := ϕ(푣) on the vertices 푣 of the 푙 푙 푗 푗 [휙 ] resulting grid. Let j , T푘 be the continuous current and energy-momentum tensor. Define 푁 = 푁 , 푇푁 = 푇 [휙푁 ] by the same formulae as in Corollary 2.6 except that 푚 is replaced by 푚/푁. Take 푑 0 ≤ 푘, 푙 < 푑. Then on the set of all edges 푒 k e푙 and all hyperfaces ℎ ⊥ e푙 disjoint with 휕I , we have

푁 푗 푒 푙 푒 푙 푁2 푇 , ℎ 푙 ℎ 푁 . 푁 ( ) ⇒ j (max ) and (−1) h 푁 i푘 ⇒ T푘 (max ) as → ∞

Remark 2.18. The ﬁelds ϕ and 휙푁 are not necessarily Klein–Gordon ﬁelds (and typically 휙푁 cannot be such one, even ϕ is). In particular, 푗 [휙푁 ] and 푇 [휙푁 ] are not necessarily conserved.

Coupling to a gauge field Interaction with a gauge field is introduced by replacement of (co)boundary by covariant (co)boundary. 푈 퐶1 퐼푑 퐺 퐴 푈 퐹 Let ∈ ( 푁 ; ), = −1, be a gauge group field, the connection, and the curvature respectively. 퐶푘 퐼푑 1×푛 퐶푘 퐼푑 1×푛 푔 퐶0 퐼푑 퐺 Definition 2.20. The gauge transformation ( 푁 ; C ) → ( 푁 ; C ) by ∈ ( 푁 ; ) is the map 휙 ↦→ 휙 ⌣ 푔. 휙 퐶푘 퐼푑 1×푛 For ∈ ( 푁 ; C ) the covariant coboundary and the covariant boundary are respectively

푘 퐷 퐴휙 := 훿휙 − (−1) 휙 ⌣ 퐴; (14) 퐷∗ 휙 휕휙 푘 퐴 ⌢∗ 휙∗ ∗. 퐴 := + (−1) ( ) (15) 퐷 휙 푎푏 휙 푏 휙 푎 푈 푎푏 퐷∗ 휙 푏 Í 휙 푎푏 Í 휙 푏푐 푈 푐푏 E.g., [ 퐴 ]( ) = ( ) − ( ) ( ) and [ 퐴 ]( ) = 푎푏:푎<푏 ( ) − 푏푐:푏<푐 ( ) ( ) for 푘 = 0 and 1 respectively, where 푎푏 is an edge with 푎 < 푏 (these 2 cases are sufficient for our examples). 휙 퐶0 퐼푑 1×푛 휕퐼푑 A field ∈ ( 푁 ; C ) is a Klein–Gordon field interacting with the gauge field, if apart 푁 we have 퐷∗ 퐷 휙 푚2휙 • the Klein–Gordon equation in a gauge field: − 퐴# 퐴 + = 0. Remark 2.19. Definitions of a gauge transformation and covariant (co)boundary crucially depend on the set of field values (more precisely, on the representation of 퐺): compare (12)–(13) and (14)–(15). For 푛 = 1 there is a minor conflict of notation between these pairs of equations, cleared up by context. Informally, (14)–(15) mean the following. Think of the field value at a face 푒 as sitting at the maximal vertex max 푒. Then the covariant (co)boundary value at a face 푣 is defined just as the ordinary (co)boundary, but all the involved field values are parallelly transported to the maximal vertex max 푣.

24 휙 퐶0 퐼푑 1×푛 Corollary 2.7. A function ∈ ( 푁 ; C ) is a Klein–Gordon field interacting with a gauge group 푈 퐶1 퐼푑 퐺 휕퐼푑 field ∈ ( 푁 ; ), if and only if among all the functions with the same values on 푁 , the function 휙 is stationary for the functional S[휙, 푈] = 휖L[휙, 푈] for fixed 푈, where

L[휙, 푈] = 퐷 휙 ⌢ (퐷 휙)∗ − 푚2휙 ⌢ 휙∗ − 1 [ 퐹∗ ⌢ 퐹]. # 퐴 퐴 2 Re Tr # Remark 2.20. Using row-vectors 휙 rather than column-vectors is essential to make L[휙, 푈] a local Lagrangian with respect to the gauge group ﬁeld 푈 as well. The third summand in L[휙, 푈] can be dropped for ﬁxed 푈 but becomes essential for dynamic 푈 in Corollary 2.10.

Corollary 2.8 (Gauge invariance). The Lagrangian L[휙, 푈] from Corollary 2.7 is gauge invariant, 휙 ⌣ 푔, 푔∗ ⌣ 푈 ⌣ 푔 휙, 푈 휙 퐶0 퐼푑 1×푛 푈 퐶1 퐼푑 퐺 푔 퐶0 퐼푑 퐺 i.e., L[ ] = L[ ] for each ∈ ( 푁 ; C ), ∈ ( 푁 ; ), ∈ ( 푁 ; ). Corollary 2.9 (Charge conservation). For a Klein–Gordon field 휙 interacting with a gauge group 푈 푗 휙, 푈 휙∗ ⌣ 퐷 휙 퐶1 퐼푑 푛×푛 휕퐼푑 field the covariant current [ ] = −2 # 퐴 ∈ ( 푁 ; C ) is conserved apart 푁 , i.e., 퐷∗ 푗 [휙, 푈] 휕퐼푑 퐴Pr푇푈 퐺 = 0 apart 푁 . (Beware that the product of a column- and a row-vector is a matrix.) Corollary 2.10. A gauge group field 푈 is stationary for the functional S[휙, 푈] from Corollary 2.7 for 휙 퐶0 퐼푑 1×푛 푈 fixed ∈ ( 푁 ; C ), if and only if is generated by the covariant current from Corollary 2.9.

2.6 The Dirac field A classical (not quantum) Dirac field describes the wave function of an electron in quantum-mechanics (not quantum field theory). Our discretization is equivalent to [10, (5.19)] but not to [10, (5.55)]. In this subsection, the “topological” notation seems to be less clear than the original “coordinate” one [10], but we keep the former for sameness. Corollaries 2.12, 2.15, 2.16, and Proposition 2.15 are new.

Basic model Deﬁnition 2.21. Introduce the Dirac 훾-matrices (generators of the Cliﬀord algebra of R1,3):

1 0 0 0 0 0 0 1 0 0 0 −푖 0 0 1 0 훾0 0 1 0 0 , 훾1 0 0 1 0 , 훾2 0 0 푖 0 , 훾3 0 0 0 −1 . = 0 0 −1 0 = 0 −1 0 0 = 0 푖 0 0 = −1 0 0 0 0 0 0 −1 −1 0 0 0 −푖 0 0 0 0 1 0 0 훾 퐶 퐼4 4×4 훾 푒 훾푘 푒 푘 , , , The Dirac chain ∈ 1( 푁 ; C ) is given by ( ) = for each edge k e푘 , where = 0 1 2 3. 휓 퐶0 퐼4 4×1 푚 휕퐼4 A function ∈ ( 푁 ; C ) is a Dirac ﬁeld of mass , if the following equation holds apart : ∗ • the Dirac equation: 푖훾 ⌢ 훿휓 + 푖훾 ⌢ 훿휓 − 2푚휓 = 0.

Such form of the equation, with the Dirac chain appearing twice, is forced by the following variational principle and is a manifestation of lattice fermion doubling phenomenon. Set 휓¯ := 휓∗훾0. 휓 퐶0 퐼4 4×1 Corollary 2.11. A function ∈ ( 푁 ; C ) is a Dirac ﬁeld, if and only if among all the ﬁunctions 휕퐼4 휓 휓 휖 휓 with the same values on 푁 , the function is stationary for the functional S[ ] = L[ ], where

L[휓] = Re 휓¯ ⌢ (푖훾 ⌢ 훿휓 − 푚휓) .

Using column-vectors 휓 rather than row-vectors is essential to make the expression meaningful. 푑 퐼푑 The doubling of the -dimensional grid 푁 is defined analogously to Definition 2.7. 퐼4 Proposition 2.14. Consider a Dirac field on the doubling of 푁 . Then the restriction of the field to the 퐼4 initial grid 푁 besides the boundary satisfies the Klein–Gordon equation with twice larger mass. The Lagrangian L[휓] is globally gauge invariant: L[휓푔] = L[휓] for each 푔 ∈ C with |푔| = 1. In 5 the case 푚 = 0 there is also a symmetry L[푒푖훾 푡휓] = L[휓] for each 푡 ∈ R, where 훾5 := 푖훾0훾1훾2훾3.

25 Corollary 2.12 (Current, chiral current, energy, momentum conservation). For a Dirac ﬁeld 휓 the 휕퐼4 following current and tensor are conserved apart 푁 :

∗ 푗 [휓] = Re 휓¯ ⌣ 훾 ⌣ 휓 and 푇 [휓] = Re[(휓¯ ⌢ 푖훾) × 훿휓 − (훿휓¯ ⌢ 푖훾 + 2푚휓¯) × 휓].

푚 푗5 휓 휓¯ ⌣ 훾5훾 ⌣ 휓 휕퐼4 In the case when = 0 the current [ ] = Re is also conserved apart 푁 . Remark 2.21. Unlike continuum theory, 푗 [휓](푒) is not necessarily positive on edges 푒 k (1, 0, 0, 0) (because 휓 and 휓¯ are evaluated at distinct endpoints of 푒) and thus cannot be interpreted as probability. The tensor 푇 [휓] is not partially symmetric. Thus we know no integral form of its conservation.

Approximation The resulting current and energy-momentum tensor indeed approximate the continuum ones. In continuum theory, ψ:I4 → C4 is a smooth function. The current and the (canonical) energy- momentum tensor of ψ are the vector and matrix ﬁelds 푙 훾푙 푙 푖 훾푙 휕 푙 푖 훾푛휕 푚 , j = Re ψ¯ ψ and T푘 = Re ψ¯ 푘 ψ − δ푘 ( ψ¯ 푛ψ − ψψ¯ ) where summation over 푛 is understood. In what follows analogues of Remarks 2.12 and 2.18 apply.

Proposition 2.15 (Approximation property). Let ψ:I4 → C4 be a smooth function. Dissect I4 into 4 푁 equal hypercubes and define the discrete field 휓푁 (푣) := ψ(푣) on the vertices 푣 of the resulting grid. 푙 푙 Let j , T푘 be the continuous current and energy-momentum tensor. Define 푗푁 = 푗 [휙푁 ], 푇푁 = 푇 [휙푁 ] by the same formulae as in Corollary 2.12 except that 푚 is replaced by 푚/푁. Take 0 ≤ 푘, 푙 < 4. Then 4 on the set of all edges 푒 k e푙 and hyperfaces ℎ ⊥ e푙 not intersecting 휕I , we have

푙 푙 푙 푗푁 (푒) ⇒ j (max 푒) and (−1) 푁h푇푁 , ℎi푘 ⇒ T푘 (max ℎ) as 푁 → ∞.

Coupling to a gauge field 푈 퐶1 퐼4 퐺 푛 Definition 2.22. Let ∈ ( 푁 ; ) be a gauge group field. Assume that ≠ 4 to avoid notational 퐷 휓 휓 퐶푘 퐼4 4×푛 휓 conflict. The covariant coboundary 퐴 of ∈ ( 푁 ; C ) is defined literally as for ∈ 퐶푘 퐼4 1×푛 ( 푁 ; C ). Set ∗ ∗ ∗ 퐷¯ 퐴휓 = (훿휓 + 퐴 ⌣ 휓 ) . (16) 휓 퐶0 퐼4 4×푛 휕퐼4 A function ∈ ( 푁 ; C ) is a Dirac field interacting with the gauge field, if apart 푁 we have ∗ • the Dirac equation in a gauge field: 푖훾 ⌢ 퐷 퐴휓 + 푖훾 ⌢ 퐷¯ 퐴휓 − 2푚휓 = 0. 휓 퐶0 퐼4 4×푛 Corollary 2.13. A function ∈ ( 푁 ; C ) is a Dirac field interacting with a gauge group field 푈 퐶1 퐼4 퐺 휕퐼4 휓 ∈ ( 푁 ; ), if and only if among all functions with the same values on 푁 , the function is stationary for the functional S[휓, 푈] = 휖L[휓, 푈] for fixed 푈, where

L[휓, 푈] = 휓¯ ⌢ (푖훾 ⌢ 퐷 휓 − 푚휓) − 1 퐹∗ ⌢ 퐹 . Re Tr 퐴 2 # Corollary 2.14 (Gauge invariance). The Lagrangian L[휓, 푈] from Corollary 2.13 is gauge invariant.

Corollary 2.15 (Charge conservation). For a Dirac field 휓 interacting with a gauge field 푈, the 푗 [휓] −휓¯ ⌣ 푖훾 ⌣ 휓 ∈ 퐶1(퐼4 푛×푛) 퐷∗ 푗 [휓] covariant current = 푁 ; C is conserved, i.e., 퐴Pr푇푈 퐺 = 0 휕퐼4 푖푈 휕 푗 휓 , 푖푈 apart 푁 . In particular, its edgewise product with is conserved, i.e., h [ ] i = 0. Corollary 2.16. A gauge group field 푈 is stationary for the functional S[휓, 푈] from Corollary 2.13 휓 퐶0 퐼4 4×푛 푈 for fixed ∈ ( 푁 ; C ), if and only if is generated by the covariant current from Corollary 2.15.

26 3 Generalizations

In this section we state the main results of the paper in their full generality, i.e., for nontrivial connections and arbitrary spacetimes. The notions and the results from §1.4 are obtained in the particular case when the gauge group is trivial, i.e., 퐺 = {1}, and the fields assume values in R. Most of the results of §2 are obtained from these general results by substituting specific Lagrangians. 푑 퐼푑 푀 If one replaces the -dimensional grid 푁 by an arbitrary spacetime , then all notions from the middle column of Table1 except #, 훾, h푇, ℎi푘 are defined literally as in §2; see the right column for 훾 푇, ℎ 푀 퐼푑 definition numbers and Remarks 2.7,2.15,2.16. We do not use and do not define #, ,h i푘 for ≠ 푁 . Remark 3.1. To make the definitions of spacetime and fields more accessible to nonspecialists, we took the liberty to use equivalent definitions of some commonly used notions and to identify spaces connected by the unique fixed isomorphism. Caring for the convenience of specialists as well, now we compare Definition 1.1 with the other ones in literature. Often simplicial (or cubical) 푘-chains are defined in a more abstract way, as the elements of the linear space 퐶푘 (푀; R) generated by the 푘-dimensional faces of 푀 (with somehow fixed orientation); 푘 and 푘-cochains are defined as elements of the dual space 퐶 (푀; R). But space 퐶푘 (푀; R) comes with the obvious unique distinguished basis: the basis consists of all the 푘-dimensional faces; the orientation of the faces is determined by the order of their vertices in spacetime 푀 as specified in Definition 2.9; the faces are listed in the dictionary order with respect to the ordered lists of their 푘 vertices. The distinguished basis identifies both 퐶푘 (푀; R) and 퐶 (푀; R) with the set of real-valued functions defined on the set of 푘-dimensional faces, that is, 푘-cochains in the sense of Definition 1.1. Notice that this identification is not related to spacetime metric. Thus we do not distinguish between chains and cochains. Inserting the obvious isomorphism between their spaces in our formulae would give no advantage but would only complicate notation. However, to make notation compatible with the commonly used one, we sometimes switch between 푘 different notation 퐶 (푀; R) and 퐶푘 (푀; R) for the same object (in our setup). We do distinguish between row- and column-vectors. This makes clear, if the product of two vectors is a number or a matrix. Some of our results depend on the type of vectors used as field values. We do not assume that 푀 is a manifold. In fact, faces of 푀 of dimension > 2 have never appeared at all in the examples from §2 (except the identity 퐷 퐴퐹 = 0 which is anyway automatic). The whole ambient spacetime is not that important: think of an electric network lying on a table; is spacetime of the model 1-, 2-, 3- or 4-dimensional? This is why we avoid dual grids and the Hodge star. However dimension-like properties of 푀 like the average vertex degree are of course important.

Let us introduce some notation. For a vertex 푣 ∈ 푀 denote by e푣,푘 the set of all 푘-dimensional faces for which the maximal vertex is 푣. Fix the dictionary order of the set e푣,푘 with respect to the order of vertices. Denote by e푣,푘,푙 its 푙-th element. Denote by 푝 = 푝(푣, 푘) the number of faces in 푘 1×푛 e푣,푘 . Set 푞 = 푝(푣, 푘 + 1). For 휙 ∈ 퐶 (푀; C ) denote 휙(e푣,푘 ) := (휙(e푣,푘,1), . . . , 휙(e푣,푘,푝)). For 푚×푛 휕f 푛×푚 휕f 휕f 푖 휕f 푧 푧 푚×푛 f : C → R deﬁne : C → R by = ( ) − ( ) , where = ( 푘푙) ∈ C . For 휕푧 휕푧 푙푘 휕 Re 푧푘푙 휕 Im 푧푘푙 ( 푘 (−1) , if e + k (1, 0,..., 0), 푀 = 퐼푑 푔(푣, 푘, 푙) = 푣,푘 1,푙 푁 denote 푘−1 (−1) , if e푣,푘+1,푙 ⊥ (1, 0,..., 0).

Definition 3.1. (Cf. Definition 2.1.) A local Lagrangian is a differentiable function

푘 1×푛 1 푛×푛 L : 퐶 (푀; C ) × 퐶 (푀; C ) → 퐶0(푀; R) such that L[휙, 푈](푣) = 퐿푣 (휙(e푣,푘 ), [퐷 퐴[푈] 휙](e푣,푘+1)) (17) 퐿 (휙 , . . . , 휙 , 휙0 , . . . , 휙0 ) 푈 for some diﬀerentiable function 푣 1 푝 1 푞 not depending on . Deﬁne 휕 휕 L 푛×1 L 푛×1 ∈ 퐶푘 (푀; C ) and ∈ 퐶푘+1(푀; C ) 휕휙 휕(퐷 퐴휙)

27 Table 3: Partial derivatives of basic Lagrangians L[휙, 푈], L[휓, 푈], or L[푈] ∗ ∗ ∗ Lagrangian assumptions 퐿 (휙 , . . . , 휙 , 휙0 , . . . , 휙0 ) 휕L or 휕L 휕L 푣 1 푝 1 푞 휕휙 휕휓 휕(퐷 퐴휙) [ 푗 ⌢ 휙∗] 푗 ∈ 퐶 (푀 C1×푛) Í푝 푗 ( )휙∗ 푗 1 Re 푘 ; Re 푙=1 e푣,푘,푙 푙 0 휙 ⌢ 휙∗ Í푝 휙 휙∗ 휙 2 - 푙=1 푙 푙 2 0 퐷 휙 ⌢ (퐷 휙)∗ 푀 = 퐼푑 Í푞 푔(푣, 푘, 푙)휙0 (휙0)∗ 퐷 휙 3 # 퐴 퐴 푁 푙=1 푙 푙 0 2# 퐴 [휓 ⌢ 휓] 푀 = 퐼4 [휓∗훾0휓 ] 훾0휓 4 Re Tr 푁 Re Tr 1 1 2 0 휓 ⌢ 푖훾 ⌢ 퐷 휓 푀 퐼4 Í4 푖휓∗훾0훾푙−1휓0 푖훾0훾 ⌢ 퐷 휓 푖훾0훾 ⌣ 휓 5 Re Tr[ ( 퐴 )] = Re Tr 푙= 퐴 − 푁 1 1 푙 ∗ ∗ L[푈] 퐿 (푈, . . . , 푈 , 푈0 , . . . , 푈0 ) 휕L 휕L Lagrangian assumptions 푣 푝 1 푞 휕푈 휕(퐹[푈]) [ 푗∗ ⌢ 푈] 푗 ∈ 퐶 (푀 C푛×푛) Í푝 푗∗( )푈 푗 6 Re Tr 1 ; Re Tr 푙=1 e푣,1,푙 푙 0 [ 퐹∗ ⌢ 퐹] 푀 = 퐼푑 Í푞 푔(푣, , 푙)(푈0)∗푈0 퐹 7 Re Tr # 푁 Re Tr 푙=1 1 푙 푙 0 2#

by the formulae

휕L 휕퐿푣 (e푣,푘,푙) := (휙(e푣,푘 ), [퐷 퐴휙](e푣,푘+1)), (18) 휕휙 휕휙푙 휕L 휕퐿푣 (e + ) := (휙(e ), [퐷 휙](e + )). (19) 휕(퐷 휙) 푣,푘 1,푙 휕휙0 푣,푘 퐴 푣,푘 1 퐴 푙

A field 휙 ∈ 퐶푘 (푀; C1×푛) is on shell, if it is stationary for the functional S[휙, 푈] = 휖L[휙, 푈] for 푈 퐶1 푀 퐺 휙 퐶푘 푀 휓 퐶0 퐼4 4×푛 given fixed ∈ ( ; ). For ∈ ( ; R) or ∈ ( 푁 ; C ) the definition is analogous; in 푘 the former case L[휙, 1] is called a local Lagrangian 퐶 (푀; R) → 퐶0(푀; R).

Proposition 3.1. For ﬁxed 푗, each of the Lagrangians in Table3 to the left is local and the partial derivatives are given by the two columns to the right, under the assumptions in the third column.

푘 1×푛 1 푛×푛 Theorem 3.1 (the Euler–Lagrange equation). Let L : 퐶 (푀; C ) × 퐶 (푀; C ) → 퐶0(푀; R) be a local Lagrangian. Let 퐴 ∈ 퐶1(푀; C푛×푛) be a connection. Then 휙 ∈ 퐶푘 (푀; C1×푛) is on shell, if and only if 휕L[휙] ∗ 휕L[휙] ∗ 퐷∗ . 퐴 + = 0 (20) 휕(퐷 퐴휙) 휕휙 퐶1 푀 푛×푛 퐶 푀 휕L 퐶 푀 푛×푛 A local Lagrangian L : ( ; C ) → 0( ; R) and the partial derivatives 휕푈 ∈ 1( ; C ), 휕L 퐶 푀 푛×푛 휙 퐷 휙 휕(퐹[푈]) ∈ 2( ; C ) are defined analogously to Definition 3.1, only the fields and 퐴 are replaced by a gauge group field 푈 and the curvature 퐹[푈] respectively (notice that 퐹[푈] ≠ 퐷 퐴푈). A gauge group field 푈 is on shell, if it is stationary for the functional S[푈] = 휖L[푈] under the constraint 푈 ∈ 퐶1(푀; 퐺). For fixed 휙 ∈ 퐶푘 (푀; C1×푛), a local Lagrangian L[휙, 푈] in the sense of Definition 3.1 is a local Lagrangian in the sense of this paragraph (by the second paragraph of Remark 2.19).

1 푛×푛 Theorem 3.2 (the Euler–Lagrange equation). Let L : 퐶 (푀; C ) → 퐶0(푀; R) be a local La- grangian. Then a gauge group ﬁeld 푈 ∈ 퐶1(푀; 퐺) is on shell, if and only if

휕L[푈] ∗ 휕L[푈] ∗ Pr 퐷∗ + = 0. (21) 푇푈 퐺 퐴 휕(퐹[푈]) 휕푈

Theorem 3.3 (Noether’s theorem). If a local Lagrangian L[휙, 푈] satisfies (5) for some Δ ∈ 퐶푘 (푀; C1×푛) and 푈 ∈ 퐶1(푀; 퐺), then for each field 휙 on shell the edgewise scalar product of the covariant current ∗ L[ ] 푗 [휙, 푈] = 휕 휙,푈 ⌢ Δ with the gauge group field 푈 is conserved, i.e. 휕h 푗 [휙, 푈], 푈i = 0. 휕(퐷 퐴휙) A Lagrangian L[휙, 푈] is gauge invariant, if L[휙 ⌣ 푔, 푔∗ ⌣ 푈 ⌣ 푔] = L[휙, 푈] for each 휙 ∈ 퐶푘 (푀; C1×푛), 푈 ∈ 퐶1(푀; 퐺), 푔 ∈ 퐶0(푀; 퐺). For gauge invariant Lagrangians the numerous Noether currents are combined together as follows.

28 Theorem 3.4 (Charge conservation). If a local Lagrangian L[휙, 푈] is gauge invariant, then for each ﬁeld 휙 on shell and each gauge group ﬁeld 푈 the following covariant current is conserved:

휕L[휙, 푈] ∗ 휕L[휙, 푈] ∗ 푗 휙, 푈 ⌢ 휙 , 퐷∗ 푗 휙, 푈 . [ ] = = i.e., 퐴Pr푇푈 퐺 [ ] = 0 휕(퐷 퐴휙) 휕푈

Theorem 3.5 (Charge conservation). Let L[푈] = L0[푈] − Re Tr [ 푗∗ ⌢ 푈] be a local Lagrangian, 푛×푛 0 where 푗 ∈ 퐶1(푀; C ) is ﬁxed, L [푈] is gauge invariant and does not depend on 푗. Then for each 푈 푗 퐷∗ 푗 gauge group ﬁeld on shell the covariant current is conserved, i.e., 퐴Pr푇푈 퐺 = 0. The last three theorems are not completely obvious even if spacetime is a 1 × 1 grid. The crucial gauge invariance is usually guaranteed by the following result.

Proposition 3.2 (Gauge covariance, see [11]). For each 푈 ∈ 퐶1(푀; 퐺), Φ ∈ 퐶푘 (푀; C푛×푛), 휙 ∈ 퐶푘 (푀; C1×푛), 푔 ∈ 퐶0(푀; 퐺) we have:

퐴[푔∗ ⌣ 푈 ⌣ 푔] = 푔∗ ⌣ 퐴[푈] ⌣ 푔 + 푔∗ ⌣ 훿푔 (= 푔∗ ⌣ 퐴[푈] ⌣ 푔 − 훿푔∗ ⌣ 푔); 퐹[푔∗ ⌣ 푈 ⌣ 푔] = 푔∗ ⌣ 퐹[푈] ⌣ 푔; ∗ ∗ 퐷 퐴[푔∗⌣푈⌣푔] (푔 ⌣ Φ ⌣ 푔) = 푔 ⌣ (퐷 퐴[푈]Φ) ⌣ 푔; 퐷 퐴[푔∗⌣푈⌣푔] (휙 ⌣ 푔) = (퐷 퐴[푈] 휙) ⌣ 푔; 퐷∗ 푔∗ ⌣ ⌣ 푔 푔∗ ⌣ 퐷∗ ⌣ 푔 퐷∗ 휙 ⌣ 푔 퐷∗ 휙 ⌣ 푔. 퐴[푔∗⌣푈⌣푔] ( Φ ) = ( 퐴[푈]Φ) ; 퐴[푔∗⌣푈⌣푔] ( ) = ( 퐴[푈] ) All the Lagrangians in the left column of Table3 not involving 푗 are gauge invariant.

4 Proofs

4.1 Basic results First we prove the results of §1. We start with a heuristic elementary proof of the result of §1.1. Proof of identity (3). By deﬁnition twice the left-hand side of (3) equals + 퐹( ) 퐹( ) + 퐹( ) 퐹( ) + 퐹( ) 퐹( ) − 퐹( ) 퐹( ) − 퐹( ) 퐹( ) − 퐹( ) 퐹( ) | {z } | {z } | {z } | {z } 1 4 | {z } | {z } 2 3 5 6 − 퐹( ) 퐹( ) − 퐹( ) 퐹( ) + 퐹( ) 퐹( ) + 퐹( ) 퐹( ) | {z } | {z } | {z } 7 8 9 + 퐹( ) 퐹( ) + 퐹( ) 퐹( ) − 퐹( ) 퐹( ) − 퐹( ) 퐹( ) | {z } | {z } | {z } 10 11 12     퐹( )+퐹( ) 퐹( ) − 퐹( ) −퐹( ) + 퐹( ) +퐹( ) − 퐹( ) =   | {z } |{z} |{z} |{z} |{z}    1−4 푎+푏 9 푐+푑 12              + 퐹( ) − 퐹( ) − 퐹( ) +퐹( ) 퐹( ) + 퐹( ) − 퐹( ) −퐹( ) + 퐹( ) 퐹( )     |{z} | {z } |{z} |{z} |{z} | {z } |{z} |{z}  푎 푒   푏 푒 2   7 5   8              − 퐹( ) − 퐹 ( ) + 퐹( ) −퐹( ) 퐹( ) − 퐹( ) − 퐹( ) +퐹( ) − 퐹( ) 퐹( ) = 0.     |{z} | {z } |{z} |{z} |{z} | {z } |{z} |{z}  푐 푓   푑 푓 3   10 6   11  Here the terms labeled by letters cancel each other; the terms in square brackets vanish by (1). Now we prove the results of §1.4. Here the ﬁelds are R-valued and the connection 퐴 = 0.

29 푘 Lemma 4.1 (Lagrangian functional derivative). For a local Lagrangian L : 퐶 (푀; R) → 퐶0(푀; R) and arbitrary ﬁelds 휙, Δ ∈ 퐶푘 (푀; R) we have 휕L[휙 + 푡Δ] 휕L[휙] 휕L[휙] 휕L[휙] = + 휕 ⌢ Δ − (− )푘 휕 ⌢ Δ . 휕푡 휕휙 휕 훿휙 1 휕 훿휙 푡=0 ( ) ( )

Proof. Take a vertex 푣 ∈ 푀. Starting with (17)–(19), where 퐷 퐴휙 = 훿휙 because 퐴 = 0, then applying Deﬁnition 2.4, and ﬁnally the well-known ’integration by parts’ identity [28]

휕(휙 ⌢ 휓) = (−1)dim 휓 (휕휙 ⌢ 휓 − 휙 ⌢ 훿휓) (22) we get 휕L[휙 + 푡Δ] 휕 (푣) = 퐿 ([휙 + 푡Δ]( ), [훿휙 + 훿 푡Δ]( )) 휕푡 휕푡 푣 e푣,푘 e푣,푘+1 푡=0 푡=0 푝(푣,푘) ∑︁ 휕 휕 = 퐿푣 (휙(e푣,푘 ), 훿휙(e푣,푘+1)) [휙 + 푡Δ](e푣,푘,푙) 휕휙푙 휕푡 푙=1 푡=0 푝(푣,푘+1) ∑︁ 휕 휕 + 퐿푣 (휙(e푣,푘 ), 훿휙(e푣,푘+ )) [훿휙 + 훿 푡Δ](e푣,푘+ ,푙) 휕휙0 1 휕푡 1 푙=1 푙 푡=0 푝(푣,푘) 푝(푣,푘+1) ∑︁ 휕L[휙] ∑︁ 휕L[휙] = (e )Δ(e ) + (e + )훿Δ(e + ) 휕휙 푣,푘,푙 푣,푘,푙 휕(훿휙) 푣,푘 1,푙 푣,푘 1,푙 푙=1 푙=1 휕L[휙] 휕L[휙] = ⌢ Δ + ⌢ 훿Δ (푣) 휕휙 휕(훿휙) 휕L[휙] 휕L[휙] 휕L[휙] = + 휕 ⌢ Δ − (−1)푘 휕 ⌢ Δ (푣). 휕휙 휕(훿휙) 휕(훿휙)

푘 Lemma 4.2. Let 휙 ∈ 퐶푘 (푀; R). If 휖 [휙 ⌢ Δ] = 0 for each Δ ∈ 퐶 (푀; R), then 휙 = 0. 휙 휖 휙 ⌢ 휙 Í 휙 푓 2 Proof. Take Δ = . Then by Definition 2.4 we have 0 = [ ] = 푓 ( ) , where the sum is over all the 푘-dimensional faces 푓 of 푀. Thus 휙 = 0. Proof of the Euler–Lagrange Theorem 1.1. A field 휙 is on shell, if and only if for each field Δ we have 휕L[휙 + 푡Δ] 휕L[휙] 휕L[휙] 휕L[휙] 휕L[휙] 휕L[휙] = 휖 = 휖 + 휕 ⌢ Δ −(− )푘 휖휕 ⌢ Δ = 휖 + 휕 ⌢ Δ . 0 휕푡 휕휙 휕 훿휙 1 휕 훿휙 휕휙 휕 훿휙 푡=0 ( ) ( ) ( ) The latter two equalities follow from Lemma 4.1 and the obvious identity 휖휕 = 0 respectively. Since Δ is arbitrary, by Lemma 4.2 the resulting equation is equivalent to (4). Proof of the Noether Theorem 1.2. By Lemma 4.1 and Theorem 1.1 for a field 휙 on shell we get

휕L[휙 + 푡Δ] 휕L[휙] 휕L[휙] 휕L[휙] = + 휕 ⌢ Δ − (− )푘 휕 ⌢ Δ = −(− )푘 휕 푗 [휙]. 휕푡 휕휙 휕 훿휙 1 휕 훿휙 1 푡=0 ( ) ( ) Thus 푗 [휙] is a conserved current, if and only if the left-hand side vanishes. Proof of Theorem 1.3. By Theorem 1.1, Deﬁnition 2.10, and the known identity 휕휕 = 훿훿 = 0 we have 휕L[휙] 휕L[휙] 휕L[휙] 휕L[휙] 휕L[휙] 휕L[휙] 휕푇 [휙] = 휕 × 훿휙 − 휕 × 휙 = ×훿훿휙+휕 ×훿휙−휕 ×훿휙−휕휕 ×휙 = 0. 휕(훿휙) 휕(훿휙) 휕(훿휙) 휕(훿휙) 휕(훿휙) 휕(훿휙)

30 4.2 Integral conservation laws Now we prove the completely new results of the subsubsection “Integral conservation laws” of §2.3. 퐼푑 퐼푑 For that purpose we are going to integrate tensors defined on 푁 × 푁 over the faces of the doubling. For a vertex 푓 of the doubling, define 푓0, . . . , 푓푑−1 ∈ Z by the formula 푓 = 푓0e0 + · · · + 푓푑−1e푑−1. The face of the initial grid with the center 푓 is denoted by 푓 as well. Definition 4.1. Let 푇 be a partially symmetric type (1, 1) tensor, 푔 be a nonboundary hyperface of the doubling, e푙⊥푔, 푓=max 푔. The 푘-th component of the flux of 푇 across 푔 in positive normal direction is

−푇 (( 푓 − e ) × ( 푓 + e )), if 푙 ≠ 푘, 2 - 푓 , 2 - 푓 ;  푘 푙 푘 푙 !  Í 푇 (( 푓 + e푙 − e푘 ) × 푓 ), if 푙 ≠ 푘, 2 - 푓푘 , 2 | 푓푙; 푙+1+ 푓푚  1 min{푘,푙}≤푚≤max{푘,푙}  h푇, 푔i푘 = (−1) · 푇 ( 푓 × ( 푓 + e − e )), if 푙 ≠ 푘, 2 | 푓 , 2 - 푓 ; 2 푙 푘 푘 푙 −푇 (( 푓 + ) × ( 푓 − )), 푙 푘, | 푓 , | 푓  e푙 e푘 if ≠ 2 푘 2 푙;  푇 ( 푓 × 푓 ) − 푇 (( 푓 + e푙) × ( 푓 − e푘 )), if 푙 = 푘.  The flux across an oriented hypersurface 휋 consisting of nonboundary faces of the doubling is the sum of the fluxes across all the hyperfaces 푔 of 휋 with the coefficients h푔, 휋i given by (8). Let 퐿 be a type (0, 1) tensor, 푔 be a 푑-dimensional face of the doubling, 푓 = max 푔. Denote ( Í 퐿( 푓 × ( 푓 − )), | 푓 1 1+ 푓푚 e푘 if 2 푘 ; h퐿, 푔i푘 := (−1) 푚<푘 · 2 퐿(( 푓 − e푘 ) × 푓 ), if 2 - 푓푘 .

Proposition 4.1. The flux of a partially symmetric type (1, 1) tensor across a hyperface ℎ of the initial grid (see Definition 2.12) is the sum of fluxes across all the hyperfaces of the doubling contained in ℎ.

Proof. Compare the 푘-th components of the fluxes. Take e푙 ⊥ ℎ. Consider the 2 cases: 푙 = 푘 and 푙 ≠ 푘. For 푙 = 푘, the map 푔 ↦→ max 푔 is a 1–1 map between the set of hyperfaces of the doubling ℎ 퐼푑 ℎ ℎ contained in and the set of faces of the initial grid 푁 contained in and containing max . (Recall that the vertex max 푔 is identified with the face 푓 of the initial grid with the center at max 푔.) Since dim Pr( 푓 , 푘, 푘) = 0 = 푓푘 (mod 2), by Definitions 2.12 and 4.1 the case 푙 = 푘 follows. ( max 푔, if 2 - (max 푔)푘 For 푙 ≠ 푘, the map 푔 ↦→ is a 2–1 map between the set of hyperfaces max 푔 − e푘 , if 2 | (max 푔)푘 ; ℎ 푓 퐼푑 푓 ℎ 푓 ℎ 푓 of the doubling in and the set of faces of the initial grid 푁 such that ⊂ , 3 max , k e푘 . The contribution of a pair of hyperfaces mapped to the same face 푓 to the sum of fluxes is ! Í 푙+1+ 푓푚 1 (− ) min{푘,푙}≤푚≤max{푘,푙} 푇 (( 푓 + − ) × 푓 )+ 2 1 e푙 e푘 ! Í 푙+1+ ( 푓푚+δ푚푘 ) + 1 (− ) min{푘,푙}≤푚≤max{푘,푙} [−푇 (( 푓 + + ) × ( 푓 + − ))] = 2 1 e푘 e푙 e푘 e푘 = 1 (− )dim Pr( 푓 ,푘,푙)+푙+1 [푇 (( 푓 + − ) × 푓 ) + 푇 (( 푓 + + ) × 푓 )] 2 1 e푙 e푘 e푙 e푘 because 2 - 푓푘 and 2 | 푓푙 by the assumptions 푓 ⊂ ℎ ⊥ e푙 and 푓 k e푘 . Summation over all such pairs proves the case 푙 ≠ 푘. Now let us prove an analogue of the Stokes formula; cf. (3) and §4.1. For that we need a lemma. 푘 푓 푑 퐼푑 푓 퐶푘 퐼푑 Lemma 4.3. For each -dimensional face of the -dimensional grid 푁 denote by [ ] ∈ ( 푁 ; R) the field, which equals 1 at 푓 , and equals 0 at all the other faces. Then

Í ∑︁ ≤ ≤ 푓푚 휕[ 푓 ] = (−1) 0 푚 푙 ·([ 푓 − e푙] − [ 푓 + e푙]);

푙:2- 푓푙 Í ∑︁ ≤ ≤ 푓푚 훿[ 푓 ] = (−1) 0 푚 푙 ·([ 푓 − e푙] − [ 푓 + e푙]).

푙:2| 푓푙

31 Proof. This is a direct computation using Deﬁnition 2.9. It suﬃces to prove that 푓 and 푓 − e푙 are Í 푓 푓 cooriented, if and only if 2 | 0≤푚≤푙 푚. Assume that 2 | 푙; the opposite case is analogous. A positive basis in 푓 is the sequence formed by all the vectors e푚 such that 2 - 푓푚 in a natural order. A positive basis in 푓 − e푙 is obtained by insertion of e푙 into the sequence. Adding the outer normal to the former basis means adding e푙 at the beginning of the sequence instead. Since moving e푙 to the beginning of Í 푓 the sequence requires 0≤푚<푙 푚 (mod 2) transpositions, the lemma follows. Proposition 4.2 (the Stokes Formula). Let 0 ≤ 푘 < 푑 ≥ 2. For each partially symmetric type (1, 1) 푇 푑 푔 퐼푑 푇, 휕푔 휕푇, 푔 tensor and each -dimensional face of the doubling of 푁 we have h i푘 = h i푘 .

Proof. This is a direct computation; a technical difficulty is signs. Set 푓 = max 푔. Assume that 2 | 푓푘 ; the opposite case is discussed at the end of the proof. For any fields 휙 and 휓 denote 푇 (휓 × 휙) = Í 푇 푒 푓 휓 푒 휙 푓 휕푇 푒 푓 푇 푒 휕 푓 푇 훿 푒 푓 푒, 푓 ( × ) ( ) ( ). Then by Definition 2.10 we have ( × ) = ([ ] × [ ]) + ( [ ] × [ ]) and by Lemma 4.3 we have

휕푇 ( 푓 × ( 푓 − e푘 )) = 푇 ([ 푓 ] × 휕[ 푓 − e푘 ]) + 푇 (훿[ 푓 ] × [ 푓 − e푘 ]) Í ∑︁ ≤ ( 푓푚−δ푚푘 ) = (−1) 푚 푙 · [푇 ( 푓 × ( 푓 − e푘 − e푙)) − 푇 ( 푓 × ( 푓 − e푘 + e푙))]

푙:2- 푓푙−δ푘푙 Í ∑︁ ≤ 푓푚 + (−1) 푚 푙 · [푇 (( 푓 − e푙) × ( 푓 − e푘 )) − 푇 (( 푓 + e푙) × ( 푓 − e푘 ))] .

푙:2| 푓푙 Í It remains to show that here the 푙-th summand multiplied by (−1)1+ 푚<푘 푓푚 equals twice the difference 푙 of the fluxes across the two opposite hyperfaces of 푔 orthogonal to e푙 multiplied by (−1) . (The latter sign factor is required to get the right contribution of the two faces into the whole flux across 휕푔 0 0 in the positive normal direction; see Lemma 4.3 for 푘 = 푑). Denote 푓 = 푓 − e푙, 푘 = min{푘, 푙}, 0 푙 = max{푘, 푙}. Denote by 푔 + e푙/2 and 푔 − e푙/2 the hyperfaces of 푔 orthogonal to e푙 such that 0 max(푔 + e푙/2) = 푓 and max(푔 − e푙/2) = 푓 respectively. Consider the following 3 cases: 1) 푙 = 푘; 2) 푙 ≠ 푘 and 2 | 푓푙; 3) 푙 ≠ 푘 and 2 - 푓푙. 1+Í 푓 For 푙 = 푘 (hence 2 | 푓푘 = 푓푙) the 푙-th summands in the two sums multiplied by (−1) 푚<푘 푚 add up to Í Í 1+ 푓푚 ≤ ( 푓푚−δ푚푘 ) (−1) 푚<푘 (−1) 푚 푘 · [푇 ( 푓 × ( 푓 − 2e푘 )) − 푇 ( 푓 × 푓 )] + Í Í 1+ 푓푚 ≤ 푓푚 +(−1) 푚<푘 (−1) 푚 푘 · [푇 (( 푓 − e푘 ) × ( 푓 − e푘 )) − 푇 (( 푓 + e푘 ) × ( 푓 − e푘 ))] =

푓푘 +1 =(−1) · [푇 ( 푓 × 푓 ) − 푇 (( 푓 + e푘 ) × ( 푓 − e푘 ))] −

푓푘 −(−1) · [푇 (( 푓 − e푘 ) × ( 푓 − e푘 )) − 푇 (( 푓 − e푘 + e푘 ) × ( 푓 − 2e푘 ))] =

푘 푘+1+ 푓푘 =(−1) · (−1) · [푇 ( 푓 × 푓 ) − 푇 (( 푓 + e푘 ) × ( 푓 − e푘 ))] − 푘 푘+1+ 푓 0 0 0 0 0 −(−1) · (−1) 푘 · [푇 ( 푓 × 푓 ) − 푇 ( 푓 + e푘 ) × ( 푓 − e푘 ))] = 푘 푘 =(−1) 2h푇, 푔 + e푘 /2i푘 − (−1) 2h푇, 푔 − e푘 /2i푘 ; see Deﬁnition 4.1 applied for 푙 = 푘. We have found the contribution of the 푙-th summands for 푙 = 푘. 1+Í 푓 For 푙 ≠ 푘 and 2 - 푓푙 the 푙-th summand multiplied by (−1) 푚<푘 푚 is Í Í 1+ 푓푚 ≤ ( 푓푚−δ푚푘 ) (−1) 푚<푘 (−1) 푚 푙 · [푇 ( 푓 × ( 푓 − e푘 − e푙)) − 푇 ( 푓 × ( 푓 − e푘 + e푙))] = (∗) Í 1+ 0 ≤ ≤ 0 푓푚 = (−1) 푘 푚 푙 · [푇 ( 푓 × ( 푓 + e푙 − e푘 )) − 푇 (( 푓 − e푙 + e푙) × ( 푓 − e푙 − e푘 ))] = Í 푙 푙+1+ 0 ≤ ≤ 0 푓푚 =(−1) · (−1) 푘 푚 푙 · 푇 ( 푓 × ( 푓 + e푙 − e푘 ))− Í 0 푙 푙+1+ 0 ≤ ≤ 0 푓 0 0 −(−1) · (−1) 푘 푚 푙 푚 · [−푇 (( 푓 + e푙) × ( 푓 − e푘 ))] = 푙 푙 =(−1) 2h푇, 푔 + e푙/2i푘 − (−1) 2h푇, 푔 − e푙/2i푘 ; 푙 푘 푓 푓 푙 푘 푓 0 푓 0 see Deﬁnition 4.1 applied for ≠ , 2 | 푘 , 2 - 푙 and ≠ , 2 | 푘 , 2 | 푙 . Here (*) follows from (Í ∑︁ ∑︁ ≤ ≤ 푓푚, if 푘 < 푙; ∑︁ 1 + 푓 + ( 푓 − δ ) = 푘 푚 푙 = 푓 (mod 2), 푚 푚 푚푘 Í 푓 + , 푘 > 푙 푚 푚<푘 푚≤푙 푙<푚<푘 푚 1 if ; 푘 0≤푚≤푙0

32 where we used the conditions 2 - 푓푙 and 2 | 푓푘 to change the range of summation over 푚. 1+Í 푓 For 푙 ≠ 푘 and 2 | 푓푙 the 푙-th summand multiplied by (−1) 푚<푘 푚 is Í Í 1+ 푓푚 ≤ 푓푚 (−1) 푚<푘 (−1) 푚 푙 · [푇 (( 푓 − e푙) × ( 푓 − e푘 )) − 푇 (( 푓 + e푙) × ( 푓 − e푘 ))] = Í 0 ≤ ≤ 0 푓푚 =(−1) 푘 푚 푙 · [푇 (( 푓 + e푙) × ( 푓 − e푘 )) − 푇 (( 푓 − e푙) × ( 푓 − e푙 + e푙 − e푘 ))] = Í 푙 푙+1+ 0 ≤ ≤ 0 푓푚 =(−1) · (−1) 푘 푚 푙 · [−푇 (( 푓 + e푙) × ( 푓 − e푘 ))]− Í 0 푙 푙+1+ 0 ≤ ≤ 0 푓 0 0 −(−1) · (−1) 푘 푚 푙 푚 · 푇 ( 푓 × ( 푓 + e푙 − e푘 )) = 푙 푙 =(−1) 2h푇, 푔 + e푙/2i푘 − (−1) 2h푇, 푔 − e푙/2i푘 .

Summation of the expressions obtained in the three cases completes the proof in the case when 2 | 푓푘 . For 2 - 푓푘 the proof is analogous and starts from the evaluation of 휕푇 (( 푓 − e푘 ) × 푓 ). For 푙 = 푘 one ends up with an expression involving 푇 (( 푓 − e푘 ) × ( 푓 + e푘 )) rather than 푇 (( 푓 + e푘 ) × ( 푓 − e푘 )). But the latter two values are equal because 푇 is partially symmetric. Proof of Proposition 2.7. This follows directly from Propositions 4.1 and 4.2 because the closed hypersurface can be tiled by 푑-dimensional faces of the doubling. Proof of Theorem 2.2. Clearly, tensor (7) is partially symmetric for this particular Lagrangian L[휙]; cf. rows 2–3 of Table3. Thus the corollary follows directly from Theorem 1.3 and Proposition 2.7. Proof of Proposition 2.8. Consider the cases when 푙 ≠ 푘 and 푙 = 푘 separately. For 푙 ≠ 푘 the only nonvanishing contribution to the ﬂux of 푇 comes from the edge 푓 = 푣 − e푘 because 푓 ∦ e푘 otherwise. We have dim Pr( 푓 , 푘, 푙) = 1. Thus by Deﬁnition 2.12 and (7) we get the required expression (− )푙 h푇, ℎi = 1 (− )푙 (− )푙 [푇 ((푣 + ) × (푣 − )) + 푇 ((푣 + − ) × (푣 − ))] 1 푘 2 1 1 e푙 e푘 e푙 2e푘 e푘 1 휕L 휕L = (푣 + e푙) + (푣 + e푙 − 2e푘 ) 훿휙(푣 − e푘 ). 2 휕(훿휙) 휕(훿휙)

For 푙 = 푘 the contribution to the ﬂux comes from 푓 = 푣 and 푓 = 푣 − e푚 for each 푚 ≠ 푘. Thus " # 푘 1 푘 푘+1 ∑︁ (−1) h푇, ℎi푘 = (−1) (−1) 푇 (푣 × 푣) − 푇 ((푣 + e푘 ) × (푣 − e푘 )) + 푇 ((푣 − e푚) × (푣 − e푚)) 2 푚≠푘 " # 1 휕L 휕L ∑︁ 휕L = − (푣)휙(푣) − (푣 + e )[훿휙](푣 − e ) + (푣 − e )[훿휙](푣 − e ) 휕휙 휕(훿휙) 푘 푘 휕(훿휙) 푚 푚 2 푚≠푘 1 휕L 휕L = (푣 + e푘 ) + (푣 − e푘 ) 훿휙(푣 − e푘 ) 2 휕(훿휙) 휕(훿휙) " # 1 휕L ∑︁ 휕L − (푣)휙(푣) + (푣 − e )훿휙(푣 − e ) 휕휙 휕(훿휙) 푚 푚 2 푚 1 휕L 휕L = (푣 + e푘 ) + (푣 + e푘 − 2e푘 ) 훿휙(푣 − e푘 ) − L[휙](푣). 2 휕(훿휙) 휕(훿휙)

The latter equality is proved as follows. Since L is homogeneous quadratic, it follows that 휕퐿푣 휙 + 휕휙1 1 휕퐿푣 휙0 휕퐿푣 휙0 퐿 휙 , 휙0 , . . . , 휙0 . 휕L ⌢ 휙 휕L ⌢ 훿휙 휙 , 0 + · · · + 0 = 2 푣 ( 1 ) Hence + ( ) = 2L[ ] as required. 휕휙1 1 휕휙푑 푑 1 푑 휕휙 휕 훿휙 4.3 Identities For the sequel we need several identities for cochain operations, most of which are well-known. Definition 4.2. The pairing of fields 휙, 휓 ∈ 퐶푘 (푀; C푚×푛), where 푚 = 1 or 푚 = 푛, is defined by ∑︁ ∗ h휙, 휓i = Re Tr 휙( 푓 )휓∗( 푓 ) = 휖 Re Tr[휙 ⌢ 휓∗] = 휖 Re Tr[휙 ⌢ 휓∗]. 푘-dimensional faces 푓

33 1 1 1 푛×푛 Given 푈 ∈ 퐶 (푀; 퐺), denote by 퐶 (푀; 푇푈퐺) the set of all Δ ∈ 퐶 (푀; C ) such that Δ(푒) 푛×푚 belongs to the tangent space 푇푈(푒)퐺 for each edge 푒. For 휙 ∈ 퐶푘 (푀; C ), where 푚 = 1 or 푚 = 푛, denote 퐷ˇ ∗ 휙 퐷∗ 휙∗ ∗ 휕휙 푘 퐴 ⌢∗ 휙 휙 ⌢ 퐴. 퐴 = ( 퐴 ) = + (−1) + δ푚푛 · (23) 푚×푛 푛×푛 Lemma 4.4 (Nondegeneracy of the pairing). Let 휙 ∈ 퐶푘 (푀; C ), 휓 ∈ 퐶0(푀; C ), 휒 ∈ 푛×푛 퐶1(푀; C ). If h휙, Δi = 0 for each Δ ∈ 퐶푘 (푀; C푚×푛), then 휙 = 0. 휓, 퐶0 푀 푇 퐺 휓 If h Δi = 0 for each Δ ∈ ( ; 1 ), then Pr푇1퐺 = 0. 휒, 퐶1 푀 푇 퐺 휒 If h Δi = 0 for each Δ ∈ ( ; 푈 ), then Pr푇푈 퐺 = 0. Δ = 휙 = h휙, 휙i = Í [휙∗( 푓 )휙( 푓 )] = Í Í푚,푛 |휙 ( 푓 )|2 Proof. For the first assertion, take . Then 0 푓 Re Tr 푓 푖, 푗=1 푖 푗 . Thus 휙 = 0. 휒 휒, 휒 Í 휒 푒 , 휒 푒 For the third assertion, take Δ = Pr푇푈 퐺 . Then 0 = h Pr푇푈 퐺 i = 푒 h ( ) Pr푇푈 (푒) 퐺 ( )i = Í 휒 푒 , 휒 푒 푒 푒 hPr푇푈 (푒) 퐺 ( ) Pr푇푈 (푒) 퐺 ( )i, where the sums are over all edges , because Pr푇푈 (푒) 퐺 is an orthogonal , 푛×푛 휒 projection. Since the pairing h· ·i on C is nondegenerate, it follows that Pr푇푈 퐺 = 0. The second assertion is proved analogously. Lemma 4.5. In a cubical complex 푀, for each 푈 ∈ 퐶1(푀; 퐺) and Φ ∈ 퐶푘 (푀; C푛×푛) we have 푘 퐷 퐴Φ = 푈 ⌣ Φ − (−1) Φ ⌣ 푈; 퐹 = 푈 ⌣ 푈; 퐷ˇ ∗ ⌢ 푈 푘푈 ⌢∗ . 퐴Φ = Φ + (−1) Φ The two identities in the 1st column hold for a simplicial complex 푀 for 푘 = 0 and 푘 = 1 respectively. Proof. By Definitions 2.9 and 2.18 it follows that ∑︁ ∑︁ [훿Φ](푎 . . . 푐) = h푎, 푏, 푐iΦ(푏 . . . 푐) − (−1)푘 h푎, 푏, 푐iΦ(푎 . . . 푏) = 1 ⌣ Φ − (−1)푘 Φ ⌣ 1; 푏:dim(푎...푏)=1, 푏:dim(푎...푏)=푘, dim(푏...푐)=푘 dim(푏...푐)=1 ∑︁ ∑︁ ∗ [휕Φ](푏 . . . 푐) = h푎, 푏, 푐iΦ(푎 . . . 푐) + (−1)푘 h푏, 푐, 푑iΦ(푏 . . . 푑) = Φ ⌢ 1 + (−1)푘 1 ⌢ Φ, 푎:dim(푎...푏)=1, 푑:dim(푏...푑)=푘, dim(푎...푐)=푘 dim(푐...푑)=1 where 1 is the unit gauge group field and the sums are over the vertices such that there exist faces 푎...푏,푏...푐 ⊂ 푎 . . . 푐 or 푏...푐,푐...푑 ⊂ 푏 . . . 푑. Using (12)–(13), we get the required identities. Lemma 4.6. (Cf. [11]) For each 휙 ∈ 퐶푘 (푀; C푝×푞), 휓 ∈ 퐶푙 (푀; C푞×푟 ), 휒 ∈ 퐶푚 (푀; C푟×푠) we have 훿훿 = 0; 훿(휙 ⌣ 휓) = (훿휙) ⌣ 휓 + (−1)dim 휙휙 ⌣ 훿휓; (휙 ⌣ 휓) ⌣ 휒 = 휙 ⌣ (휓 ⌣ 휒); 휕휕 = 0; 휕(휙 ⌢ 휓) = (−1)dim 휓 (휕휙 ⌢ 휓 − 휙 ⌢ 훿휓); (휙 ⌢ 휓) ⌢ 휒 = 휙 ⌢ (휓 ⌣ 휒); ∗ ∗ ∗ ∗ ∗ ∗ 휖휕 = 0; 휕(휙 ⌢ 휓) = 휙 ⌢ 휕휓 + (−1)dim 휓−dim 휙훿휙 ⌢ 휓; 휙 ⌢ (휓 ⌢ 휒) = (휙 ⌣ 휓) ⌢ 휒; (휓∗ ⌢ 휙∗, 휙 ∗ if dim = 0; ∗ ∗ (휙 ⌣ 휓) = ∗ (휙 ⌢ 휓) ⌢ 휒 = 휙 ⌢ (휓 ⌢ 휒). 휓∗ ⌢ 휙∗, if dim 휓 = 0;

푛×푚 푘 푚×푛 1 For each 휙 ∈ 퐶푘+1(푀; C ), 휓 ∈ 퐶 (푀; C ), 푈 ∈ 퐶 (푀; 퐺), where 푚 = 1 or 푚 = 푛, we have dim 휙 퐷 퐴퐷 퐴휓 = −휓 ⌣ 퐹 + δ푚푛 · 퐹 ⌣ 휓; 퐷 퐴 (휙 ⌣ 휓) = 퐷 퐴휙 ⌣ 휓 + (−1) 휙 ⌣ 퐷 퐴휓; 퐷ˇ ∗ 퐷ˇ ∗ 휙 퐹 ⌢∗ 휙 휙 ⌢ 퐹 퐷ˇ ∗ 휙 ⌢ 휓 dim 휓 퐷ˇ ∗ 휙 ⌢ 휓 휙 ⌢ 퐷 휓 퐴 퐴 = − + δ푚푛 · ; 퐴 ( ) = (−1) ( 퐴 − 퐴 ); 휖 퐷ˇ ∗ 휙 , 푚 푛 휙 퐷ˇ ∗ 휙 ⌢∗ 휓 휙 ⌢∗ 퐷ˇ ∗ 휓 dim 휓−dim 휙퐷 휙 ⌢∗ 휓. Re Tr 퐴 = 0 if = and dim = 1; 퐴 ( ) = 퐴 + (−1) 퐴 푘 푚×푛 푙 푛×푛 푚×푚 푚×푛 1 For each 휙 ∈ 퐶 (푀; C ), 휓 ∈ 퐶 (푀; C or C ), 휒 ∈ 퐶푘+푙 (푀; C ), 푈 ∈ 퐶 (푀; 퐺), where 푚 = 1 or 푚 = 푛 (and 푙 = 1 for the identities in the 1st and 3rd column below), we have: 휒, 훿휙 휕 휒, 휙 휒, 휓 ⌣ 휙 휒∗ ⌢ 휓 ∗, 휙 퐷∗ 휓 휕 푈∗ 휓 h i = h i; h i = h( ) i; Re Tr 퐴 = Re Tr [ · ]; ∗ h휒, 퐷 휙i h퐷∗ 휒, 휙i h휒, 휙 ⌣ 휓i h(휓 ⌢ 휒∗)∗, 휙i 퐷∗ 휓 퐷∗ 휓. 퐴 = 퐴 ; = ; Pr푇1퐺 퐴 = 퐴Pr푇푈 퐺 In the 3rd column, “·” is the edgewise product, i.e., [푈∗ · 휓](푒) := 푈∗(푒)휓(푒) for each edge 푒.

34 Proof. The identities involving neither the cop-product nor covariant (co)boundary are well-known in the case when the functions assume values in a commutative ring; cf. [11]. Without the commutativity the proof is literally the same. Let us prove the remaining identities. For an ordered 4-ple of faces 푎...푏,푏...푐,푐...푑 ⊂ 푎 . . . 푑 write h푎, 푏, 푐, 푑i = +1, if the ordered set consisting of positive bases in 푎 . . . 푏, 푏 . . . 푐, 푐 . . . 푑 is a positive basis in 푎 . . . 푑. Otherwise write h푎, 푏, 푐, 푑i = −1. Clearly, h푎, 푏, 푐, 푑i = h푎, 푏, 푐ih푎, 푐, 푑i = h푎, 푏, 푑ih푏, 푐, 푑i. Thus by Deﬁnition 2.18 ∗ ∗ ∑︁ ∗ [휙 ⌢ (휓 ⌢ 휒)](푎 . . . 푏) = h푎, 푏, 푐i휙(푏 . . . 푐)[휓 ⌢ 휒](푎 . . . 푐) 푐:dim(푏...푐)=푘,dim(푎...푐)=푚−푙 ∑︁ = h푎, 푏, 푐ih푎, 푐, 푑i휙(푏 . . . 푐)휓(푐 . . . 푑)휒(푎 . . . 푑) 푐,푑:dim(푏...푐)=푘,dim(푐...푑)=푙,dim(푎...푑)=푚 ∑︁ = h푎, 푏, 푑ih푏, 푐, 푑i휙(푏 . . . 푐)휓(푐 . . . 푑) 휒(푎 . . . 푑) 푐,푑:dim(푏...푐)=푘,dim(푐...푑)=푙,dim(푎...푑)=푚 ∗ = [(휙 ⌣ 휓) ⌢ 휒](푎 . . . 푏).

Setting 푚 = 푘 + 푙, changing the notation 휒 to 휒∗, and applying the operator 휖 Re Tr, we obtain ∗ ∗ ∗ 푘 푚×푛 푚×푛 h(휓 ⌢ 휒 ) , 휙i = h휒, 휙 ⌣ 휓i. Taking 휓 = 퐴, 휙 ∈ 퐶 (푀; C ), 휒 ∈ 퐶푘+1(푀; C ), multiplying by (−1)dim 휙 = −(−1)dim 휒, adding the known identity h휕 휒, 휙i = h휒, 훿휙i (and for 푚 = 푛 also the 휒 ⌢ 휓∗ ∗, 휙 휒, 휙 ⌣ 휓 퐷∗ 휒, 휙 휒, 퐷 휙 known identity h( ) i = h i), and using (12)–(15), we get h 퐴 i = h 퐴 i. ∗ The formula for (휙 ⌢ 휓) ⌢ 휒 is proved analogously. 퐷ˇ ∗ 휙 ⌢ 휓 푚 푛 Next, the formula for 퐴 ( ) for a cubical complex and = follows from 푙 퐷ˇ ∗ 휙 ⌢ 휓 푙 푘−푙푈 ⌢∗ 휙 ⌢ 휓 푙 휙 ⌢ 휓 ⌢ 푈 (−1) 퐴 ( ) = (−1) (−1) ( ) + (−1) ( ) ∗ = (−1)푘 (푈 ⌢ 휙) ⌢ 휓 + (휙 ⌢ 푈) ⌢ 휓 − 휙 ⌢ (푈 ⌣ 휓) + (−1)푙 휙 ⌢ (휓 ⌣ 푈) 퐷ˇ ∗ 휙 ⌢ 휓 휙 ⌢ 퐷 휓, = ( 퐴 ) − 퐴 where we used Lemma 4.5 and the identities not involving (covariant) (co)boundary. Alternatively, the 퐷ˇ ∗ 휙 ⌢ 휓 훿 휙 ⌣ 휓 formula for 퐴 ( ) can be deduced from the formula for ( ) by pairing with an arbitrary ﬁeld Δ and applying Lemma 4.4 and the identities from the paragraph before the previous one; this works for a simplicial complex and for 푚 = 1 as well. 퐷 휙 ⌣ 휓 퐷ˇ ∗ 휙 ⌢∗ 휓 퐷 퐷 퐷ˇ ∗ 퐷ˇ ∗ The formulae for 퐴 ( ), 퐴 ( ), 퐴 퐴, 퐴 퐴 are proved analogously. Finally, for each vertex 푣 by Lemma 4.5 we have (where h푈, 휓i is the edgewise scalar product) 퐷∗ 휓 푣 휓∗ ⌢ 푈 푈 ⌢∗ 휓∗ ∗ 푣 ∑︁ 휓 푒 , 푈 푒 ∑︁ 휓 푒 , 푈 푒 휕 푈, 휓 푣 [Re Tr 퐴 ]( ) = Re Tr[ − ] ( ) = h ( ) ( )i − h ( ) ( )i = [ h i]( ); 푒:max 푒=푣 푒:min 푒=푣 ∗ ∗ 퐷∗ 휓 (( 휓)∗ ⌢ 푈 − 푈 ⌢ ( 휓)∗)∗ (휓∗ ⌢ 푈)∗ − (푈 ⌢ 휓∗)∗ 퐷∗ 휓. 퐴Pr푇푈 퐺 = Pr푇푈 퐺 Pr푇푈 퐺 = Pr푇1퐺 Pr푇1퐺 = Pr푇1퐺 퐴 휖 휖 퐷ˇ ∗ 휓 휖 퐷∗ 휓∗ 휖 휕 푈 휓 . Applying the operator we get Re Tr 퐴 = Re Tr 퐴 = Re Tr[ · ] = 0

4.4 Generalizations Now we proceed to the proof of the results of §3. The argument is parallel to that of §4.1. Proof of Proposition 3.1. This is a straightforward computation using the explicit expression for the 0 ∗ 0 function 퐿푣 given in the middle part of Table3. In row 5 we use the identity (훾 훾) = 훾 훾. Lemma 4.7 (Lagrangian functional derivative). For a local Lagrangian L : 퐶푘 (푀; C1×푛)×퐶1(푀; C푛×푛) → 푘 1×푛 1 퐶0(푀; R) and arbitrary ﬁelds 휙, Δ ∈ 퐶 (푀; C ), 푈 ∈ 퐶 (푀; 퐺) we have 휕L[휙 + 푡Δ, 푈] 휕L[휙, 푈] 휕L[휙, 푈] 휕L[휙, 푈] = + 퐷ˇ ∗ ⌢ Δ − (− )푘 퐷ˇ ∗ ⌢ Δ . 휕푡 Re Tr 휕휙 퐴 휕 퐷 휙 1 퐴 휕 퐷 휙 푡=0 ( 퐴 ) ( 퐴 ) 훿 휕 퐷 퐷ˇ ∗ Proof. This is proved literally as Lemma 4.1 with and replaced by 퐴 and 퐴 respectively, and 퐷ˇ ∗ 휙 ⌢ 휓 Re Tr applied to each summand. Instead of (22) use the formula for 퐴 ( ) from Lemma 4.6.

휙 휕S[휙+푡Δ,푈] 퐶푘 푀, 1×푛 Proof of Theorem 3.1. A ﬁeld is on shell, if and only if 휕푡 = 0 for each Δ ∈ ( C ). 푡=0 휖 퐷ˇ ∗ By Lemmas 4.7 and 4.4 this is equivalent to (20) because Re Tr 퐴 = 0 by Lemma 4.6. 1 푛×푛 Lemma 4.8 (Lagrangian functional derivative). For a local Lagrangian L : 퐶 (푀; C ) → 퐶0(푀; R) 1 1 and arbitrary ﬁelds 푈 ∈ 퐶 (푀; 퐺), Δ ∈ 퐶 (푀; 푇푈퐺) we have 휕L[푈 + 푡Δ] 휕L[푈] 휕L[푈] 휕L[푈] = + 퐷ˇ ∗ ⌢ Δ + 퐷ˇ ∗ ⌢ Δ . 휕푡 Re Tr 휕푈 퐴 휕 퐹 푈 퐴 휕 퐹 푈 푡=0 ( [ ]) ( [ ]) Proof. This is proved analogously to Lemma 4.1 with 휙 and 훿휙 replaced by 푈 and 퐹 = 훿퐴 + 퐴 ⌣ 퐴 퐷ˇ ∗ 휙 ⌢ 휓 (see Proposition 2.12), using the formula for 퐴 ( ) from Lemma 4.6 instead of (22), and 휕 휕 퐹[푈 + 푡Δ] = [훿(푈 + 푡Δ − ) + (푈 + 푡Δ − ) ⌣ (푈 + 푡Δ − )] 휕푡 휕푡 1 1 1 푡=0 푡=0 = 훿Δ + (푈 − 1) ⌣ Δ + Δ ⌣ (푈 − 1) = 퐷 퐴Δ.

푈 휕S[푈+푡Δ] Proof of Theorem 3.2. A gauge group ﬁeld is on shell, if and only if 휕푡 = 0 for each 푡=0 1 Δ ∈ 퐶 (푀, 푇푈퐺). By Lemmas 4.8, 4.6, and 4.4 this is equivalent to (21). Proof of Theorem 3.3. 휕h 푗 [휙, 푈], 푈i = 퐷∗ 푗 [휙, 푈] = 휕 L[휙 + 푡Δ, 푈] = This follows from Re Tr 퐴 휕푡 푡=0 0. Here the 1st equality is given by Lemma 4.6. The 2nd one is proved as in the proof of Theorem 1.2 훿 휕 퐷 퐷ˇ ∗ with , replaced by 퐴, 퐴, and Re Tr applied to each summand. The 3rd one is (5). Remark 4.1. If (5) holds in a subset of 푀, then the current h 푗 [휙, 푈], 푈i is conserved on the subset. Lemma 4.9 (Lagrangian functional derivative). For a local Lagrangian L : 퐶푘 (푀; C1×푛)×퐶1(푀; C푛×푛) → 푘 1×푛 1 푛×푛 퐶0(푀; R) and arbitrary ﬁelds 휙 ∈ 퐶 (푀; C ) and 푈, Δ ∈ 퐶 (푀; C ) we have 휕L[휙, 푈 + 푡Δ] 휕L[휙, 푈] 휕L[휙, 푈] 휕L[휙, 푈] = ⌢ 휙 ⌢ Δ = ⌢ 휙. 휕푡 Re Tr 휕 퐷 휙 and 휕푈 휕 퐷 휙 푡=0 ( 퐴 ) ( 퐴 ) Proof. Analogously to the proof of Lemma 4.1 using (14) and Lemma 4.6 we get

휕L[휙, 푈 + 푡Δ] 휕L[휙, 푈] 휕휙 휕L[휙, 푈] 휕(퐷 [ + Δ] 휙) = ⌢ + ⌢ 퐴 푈 푡 휕푡 Re Tr 휕휙 휕푡 휕 퐷 휙 휕푡 푡=0 ( 퐴 ) 푡=0 휕L[휙, 푈] 휕[훿휙 + 휙 ⌣ (푈 − 1 + 푡Δ)] = + ⌢ 0 Re Tr 휕 퐷 휙 휕푡 ( 퐴 ) 푡=0 휕L[휙, 푈] 휕L[휙, 푈] = Re Tr ⌢ (휙 ⌣ Δ) = Re Tr ⌢ 휙 ⌢ Δ . 휕(퐷 퐴휙) 휕(퐷 퐴휙) A local Lagrangian L[휙, 푈] is also local with respect to 푈 and does not depend on 퐹[푈]. Since L[ ] L[ ] Δ ∈ 퐶1(푀; C푛×푛) is arbitrary, by Lemmas 4.8 and 4.4 it follows that 휕 휙,푈 = 휕 휙,푈 ⌢ 휙. 휕푈 휕(퐷 퐴휙) Lemma 4.10 (Infinitesimal form of gauge invariance). For each gauge invariant differentiable function 푘 1×푛 1 푛×푛 0 L : 퐶 (푀; C ) × 퐶 (푀; C ) → 퐶0(푀; R) and each Δ ∈ 퐶 (푀, 푇1퐺) we have 휕 L[휙 + 푡휙 ⌣ Δ, 푈 + 푡퐷 Δ] = . 휕푡 퐴 0 푡=0 Proof. Since L[휙, 푈] is gauge invariant and differentiable, by Lemma 4.5 up to first order in 푡

L[휙, 푈] = L[휙 ⌣ exp(푡Δ), exp(−푡Δ) ⌣ 푈 ⌣ exp(푡Δ)] = L[휙 + 푡휙 ⌣ Δ, 푈 + 푡(푈 ⌣ Δ − Δ ⌣ 푈)] + 표(푡)

= L[휙 + 푡휙 ⌣ Δ, 푈 + 푡퐷 퐴Δ] + 표(푡) as 푡 → 0.

Diﬀerentiating with respect to 푡 and setting 푡 = 0, we get the required result.

36 1 Lemma 4.11 (Local covariant constants). For each 푈 ∈ 퐶 (푀; 퐺), 푔0 ∈ 푇1퐺, and each vertex 푣 there 0 is 푔 ∈ 퐶 (푀; 푇1퐺) such that 푔(푣) = 푔0 and [퐷 퐴푔](푢푣) = 0 for each neighbor 푢 of 푣.

Proof. Set 푔(푣) = 푔0, 푔(푢) = 푈(푢푣)푔(푣)푈(푣푢) at each neighbor 푢 of 푣, and let 푔 be arbitrary at the other vertices. By Lemma 4.5 we have [퐷 퐴푔](푢푣) = 푈(푢푣)푔(푣) − 푈(푢푣)푔(푣)푈(푣푢)푈(푢푣) = 0.

0 Proof of Theorem 3.4. Take an arbitrary vertex 푣 and 푔0 ∈ 푇1퐺. Let 푔 ∈ 퐶 (푀; 푇1퐺) be given by Lemma 4.11. Apply Lemma 4.10 for Δ = 푔. Since 퐷 퐴푔(푢푣) = 0 for each neighbor 푢 of 푣, we obtain that equation (5) holds at the vertex 푣 with Δ = 휙 ⌣ 푔 (notice that the connection in (5) does not depend on 푡). By Theorem 3.3, Remark 4.1, and Lemma 4.6, we have

휕L[휙, 푈] 휕L[휙, 푈] 휕 ⌢ 휙 ⌣ 푔 푈 푣 퐷ˇ ∗ ⌢ 휙 ⌢ 푔 푣 0 = Re Tr ( ) · ( ) = Re Tr 퐴 ( ) 휕(퐷 퐴휙) 휕(퐷 퐴휙) 휕L[휙, 푈] 휕L[휙, 푈] 휕L[휙, 푈] ∗ 퐷ˇ ∗ ⌢ 휙 ⌢ 푔 ⌢ 휙 ⌢ 퐷 푔 푣 퐷∗ ⌢ 휙 푣 푔 . = Re Tr 퐴 − 퐴 ( ) = Re Tr 퐴 ( )· 0 휕(퐷 퐴휙) 휕(퐷 퐴휙) 휕(퐷 퐴휙)

Here we used that [퐷 퐴푔](푢푣) = 0 for each edge 푢푣 containing 푣. Since the vertex 푣 and 푔0 ∈ 푇1퐺 ∗ L[ ] are arbitrary, by Lemma 4.4 it follows that Pr 퐷∗ 휕 휙,푈 ⌢ 휙 = 0. By Lemma 4.9 we have 푇1퐺 퐴 휕(퐷 퐴휙) L[ ] L[ ] 휕 휙,푈 ⌢ 휙 = 휕 휙,푈 = 푗 [휙, 푈]∗. By Lemma 4.6 we have 퐷∗ Pr 푗 [휙, 푈] = Pr 퐷∗ 푗 [휙, 푈] = 휕(퐷 퐴휙) 휕푈 퐴 푇푈 퐺 푇1퐺 퐴 0, i.e., the covariant current 푗 [휙, 푈] is conserved.

0 0 0 Proof of Theorem 3.5. Denote S[푈] = 휖L[푈] and S [푈] = 휖L [푈]. Take arbitrary Δ ∈ 퐶 (푀, 푇1퐺). By Lemmas 4.10 (with L[휙, 푈] replaced by L0[푈]) and 4.6 we get 휕 휕 S[푈 + 푡퐷 Δ] = (S0[푈 + 푡퐷 Δ] + h 푗, 푈 + 푡퐷 Δi) = + h 푗, 퐷 Δi = h퐷∗ 푗, Δi. 휕푡 퐴 휕푡 퐴 퐴 0 퐴 퐴 푡=0 푡=0

For a gauge group ﬁeld 푈 on shell the left-hand side vanishes, because 퐷 퐴Δ = 푈 ⌣ Δ − Δ ⌣ 푈 ∈ 퐶1 푀, 푇 퐺 푈 퐷∗ 푗, 퐶0 푀, 푇 퐺 ( 푈 ) is a possible variation of . Thus h 퐴 Δi = 0 for arbitrary Δ ∈ ( 1 ). By = 퐷∗ 푗 = 퐷∗ 푗 Lemmas 4.4 and 4.6 we get 0 Pr푇1퐺 퐴 퐴Pr푇푈 퐺 , as required. Proof of Proposition 3.2. Let us present the proof for a cubical complex. For a simplicial complex the argument is literally the same, only each instance of the fourth vertex “푑” is just removed. Since the group 퐺 consists of unitary matrices, for each edge 푢푣 and each face 푎푏푐푑 we have

퐴[푔∗ ⌣ 푈 ⌣ 푔](푢푣) = 푔∗(푢)푈(푢푣)푔(푣) − 1 = 푔∗(푢)(푈(푢푣) − 1)푔(푣) + 푔∗(푢)(푔(푣) − 푔(푢)) = [푔∗ ⌣ 퐴[푈] ⌣ 푔 + 푔∗ ⌣ 훿푔](푢푣); 퐹[푔∗ ⌣ 푈 ⌣ 푔](푎푏푐푑) = [푔∗ ⌣ 푈 ⌣ 푔](푎푏푐) − [푔∗ ⌣ 푈 ⌣ 푔](푎푑푐) = 푔∗(푎)푈(푎푏)푔(푏)푔∗(푏)푈(푏푐)푔(푐) − 푔∗(푎)푈(푎푑푐)푔(푐) = [푔∗ ⌣ 퐹[푈] ⌣ 푔](푎푏푐푑).

Now, using (14)–(15) and Lemma 4.6 we get

푘 ∗ ∗ 퐷 퐴[푔∗⌣푈⌣푔] (휙 ⌣ 푔) = 훿(휙 ⌣ 푔) − (−1) 휙 ⌣ 푔 ⌣ [푔 ⌣ 퐴[푈] ⌣ 푔 + 푔 ⌣ 훿푔] = (훿휙) ⌣ 푔 + (−1)푘 휙 ⌣ 훿푔 − (−1)푘 휙 ⌣ (푔 ⌣ 푔∗) ⌣ [퐴[푈] ⌣ 푔 + 훿푔]

= (퐷 퐴[푈] 휙) ⌣ 푔; ∗ 퐷∗ 휙 ⌣ 푔 휕 휙 ⌣ 푔 ∗ 푘 푔∗ ⌣ 퐴 푈 ⌣ 푔 훿푔∗ ⌣ 푔 ⌢∗ 휙 ⌣ 푔 ∗ 퐴[푔∗⌣푈⌣푔] ( ) = ( ) + (−1) [ [ ] − ] ( ) ∗ ∗ ∗ = 휕(푔∗ ⌢ 휙∗) + (−1)푘 [푔∗ ⌣ 퐴[푈] ⌣ 푔 − 훿푔∗ ⌣ 푔] ⌢ (푔∗ ⌢ 휙∗) ∗ ∗ ∗ ∗ ∗ = 푔∗ ⌢ 휕휙∗ + (−1)푘 훿푔∗ ⌢ 휙∗ + (−1)푘 (푔∗ ⌣ 퐴[푈] − 훿푔∗) ⌢ (푔 ⌢ (푔∗ ⌢ 휙∗)) ∗ 푔∗ ⌢∗ 휕휙∗ 푘 퐴 푈 ⌢∗ 휙∗ 퐷∗ 휙 ⌣ 푔 . = ( + (−1) [ ] ) = 퐴[푈]

The formulae involving Φ ∈ 퐶푘 (푀; C푛×푛) are proved analogously. Gauge invariance of the La- grangians not involving 푗 in Table3 is a straightforward consequence.

37 4.5 Proofs of examples Now we apply the general results of §3 to prove particular results of §2 (except those proved in §4.2). Proof of Corollary 2.1. This follows directly from Proposition 2.7 applied to the boundary hypersurface of a face and the tensor 푇 [휙] = 훿휙 × 훿휙, which is conserved by Theorem 1.3; cf. Remark 2.5.

Proof of Theorem 2.1. First let us prove the “convergence” of 퐹푁 to F. It is convenient to modify the grid slightly. Consider the auxiliary grid 푀 obtained by dissection of I2 into (푁 + 1)2 equal squares and its dual 푁 × 푁 grid 푀0 with the vertices at face-centers of 푀. Consider all the discrete fields in question as defined on 푀0 instead of the initial 푁 × 푁 grid; this does not affect approximation. 퐹0 푀 휕훿퐹0 휕 2 퐹0 휕 2 Let 푁 be the function on vertices of such that 푁 = 0 apart I and 푁 = F on I . The 퐹0 푀0 restriction of 푁 to nonboundary vertices can be considered as a function on faces of . Actually, it is 0 a magnetic field on 푀 generated by the source 푠푁 (in particular, it exists by Proposition 2.6). Indeed, 휕훿퐹0 the condition 푁 = 0 implies that it is a magnetic field generated by some source. The source is 푠 푣 푁 푁 퐹0 푣 퐹0 푣 exactly 푁 because for each boundary vertex of the initial × grid we have 푁 ( +) − 푁 ( −) = (푣 ) − (푣 ) = ∫ = 푠 (푣) 푣 , 푣, 푣 휕 2 F + F − 푣 푣 s dl 푁 , where − + are in the counterclockwise order along I . By − + 퐹0 퐹 푀0 푁 Propositions 2.4 and 2.6 the function 푁 − 푁 on faces of is a constant (depending on ). 푟 휕 2 퐹0 푣 푣 By [6, Proposition 3.3] on the set of vertices at distance ≥ from I , we have 푁 ( ) ⇒ F( ) as 푁 → ∞ 푓 푐 = ( 1 , 1 ) 퐹0 ( 푓 ) → (푐) = = . In particular, for one of the faces 푁 closest to : 2 2 we have 푁 푁 F 0 퐹 푓 푁 퐹0 퐹 푀 푁 ( 푁 ) as → ∞. Since 푁 − 푁 is a constant function on , it follows that ∫ 퐹 푓 퐹0 푓 푓 푁2 . 푁 ( ) ⇒ 푁 ( ) ⇒ F(max ) ⇒ F dS 푓

The convergence of 푗푁 = −휕퐹푁 follows immediately from the second part of [6, Proposition 3.3]. To prove the convergence of 휙푁 , join a vertex 푣 with the vertex 푢 closest to 푐 such that 휙푁 (푢) = 0 by a 푢푣 푗 휙 (푣) =Í h푢푣, 푒i 푗 (푒) ∫ ì· ì = ϕ(푣). shortest grid path . By the convergence of 푁 we get 푁 푒⊂푢푣 푁 ⇒ 푐푣 j dl The convergence of the other ﬁelds is a straightforward consequence. For instance, let 푒 = 푢푣 be a horizontal edge with the midpoint 푒0 and 푓 ⊃ 푒 be a face with the center 푓 0. Then ∫ ∫ ∫ ∫ 푁퐿 (푒0 푓 0) = 푁 푗 (푒)퐹 ( 푓 ) 퐹 ( 푓 ) 푁 ì · ì (푒0)ì(푒0)· 푁 ì = ∗ì(푒0) (푒0)· 푁 ì 푁 ì · ì, 푁 2 푁 푁 ⇒ 푁 2 j dl ⇒ F j 2 dl j F dl ⇒ L dl 푒 푒 푒0 푓 0 푒0 푓 0

2 ∫ 푁2휎 (푢푣) = − 푁 [훿휙(푢푣)2 − 훿휙(푣푣 )훿휙(푣 푣)] 1 휕ϕ (푣)2 − 1 휕ϕ (푣)2 = σ (푣) 푁 σ 1 − σ 2 , 푁,2 2 + − ⇒ 2 휕x 2 휕x 22 ⇒ 22 dx 21 dx 2 1 푒 as required (in the latter formula the notations 푣+ and 푣− from Deﬁnition 2.8 are used). Proof of Corollary 2.2. This follows from Theorem 1.1 and Proposition 3.1 for the particular case 휙, 푗 퐶1 퐼푑 푛 푈 퐷 휙 훿휙 when ∈ ( 푁 ; R), = 1, = 1, hence 퐴 = ; see rows 1 and 3 of Table3.

Proof of Proposition 2.9. First note that 퐹푁 ( 푓 ) = F푚푛 (max 푓 ) ⇒ F푚푛 (max ℎ) on the set of all pairs 푑 ( 푓 , ℎ) having common vertices, because F푚푛 is continuous on I , hence uniformly continuous. Consider the cases when 푙 = 푘 and 푙 ≠ 푘 separately. Assume that 푙 = 푘. For a 1- or 2-dimensional face 푓 ⊂ ℎ ⊥ e푘 we have dim Pr( 푓 , 푘, 푘) = 0. Thus

  푘 0 1  ∑︁ 0 ∑︁ 0  (−1) h푇 , ℎi푘 = −  푇 ( 푓 × 푓 ) − 푇 (( 푓 + e푘 ) × ( 푓 − e푘 )) 푁 2  푁 푁   푓 : 푓 ⊂ℎ, 푓 3max ℎ,dim 푓 =2 푓 : 푓 ⊂ℎ, 푓 3max ℎ,dim 푓 =1      1  ∑︁ ∑︁  =  #퐹푁 ( 푓 )퐹푁 ( 푓 ) − #퐹푁 ( 푓 + e푘 )퐹푁 ( 푓 − e푘 ) 2    푓 : 푓 ⊂ℎ, 푓 3max ℎ,dim 푓 =2 푓 : 푓 ⊂ℎ, 푓 3max ℎ,dim 푓 =1  " # " #  1 ∑︁ 푚푛 ∑︁ 푘푚 1 ∑︁ 푚푛 ∑︁ 푘푚 ⇒ F F푚푛 − F F푘푚 (max ℎ) = F F푚푛 − F F푘푚 (max ℎ) 2 4 푚,푛≠푘:푚<푛 푚≠푘 푚,푛 푚 푘 ℎ . = T푘 (max )

38 Assume that 푙 ≠ 푘. For a 2-dimensional face 푓 k e푘 , e푚, where 푚 ≠ 푘, 푙, we have dim Pr( 푓 , 푘, 푙) = 2 or 1 depending on if 푚 is between 푘 and 푙 or not. Thus

푙 0 1 ∑︁ dim Pr( 푓 ,푘,푙) (−1) h푇 , ℎi푘 = (−1) [#퐹푁 ( 푓 + e푙 − e푘 )) + #퐹푁 ( 푓 + e푙 + e푘 )] 퐹푁 ( 푓 ) 푁 2 푓 : 푓 ⊂ℎ, 푓 3max ℎ, dim 푓 =2, 푓 ke푘 ∑︁ min{푙,푚},max{푙,푚} ⇒ − sgn(푚 − 푘)sgn(푚 − 푙)F (max ℎ)Fmin{푘,푚},max{푘,푚} (max ℎ) 푚≠푘 ∑︁ = − F푙푚 (max ℎ)F (max ℎ) = T푙 (max ℎ). 푘푚 푘 푚≠푘 Proof of Proposition 2.11. Let 푎푏푐푑 be a face with the vertices listed in the order compatible with the positive orientation of its boundary (given by Deﬁnition 2.9), starting from the minimal vertex. Then

ReTr [#퐹∗(푎푏푐푑)퐹(푎푏푐푑)] = #ReTr [(푈(푎푏푐) − 푈(푎푑푐))∗(푈(푎푏푐) − 푈(푎푑푐))] = #ReTr [푈(푐푏푎푏푐) − 푈(푐푑푎푏푐) − 푈(푐푏푎푑푐) + 푈(푐푑푎푑푐)] = #ReTr [1 − 푈(푎푏푐푑푎) − 푈(푎푏푐푑푎)∗ + 1] = 2#(푛 − ReTr 푈(푎푏푐푑푎)).

Multiplying by −1/2 and summing over all the faces 푎푏푐푑, we get the required expression.

Proof of Proposition 2.12. By the formulas of Lemma 4.5 for 퐹 and for 퐷 퐴Φ in the case when 푈 = 1 and Φ = 퐴, we get 퐹 = (1 + 퐴) ⌣ (1 + 퐴) = 0 + 퐷0 퐴 + 퐴 ⌣ 퐴 = 훿퐴 + 퐴 ⌣ 퐴. By Lemma 4.5 and the associativity of the cup-product, 퐷 퐴퐹 = 푈 ⌣ 퐹 − 퐹 ⌣ 푈 = 푈 ⌣ (푈 ⌣ 푈) − (푈 ⌣ 푈) ⌣ 푈 = 0. By Lemma 4.5 and the 3rd column of Table2 we get (10). Let us prove (11). By Definition 1.1 for each 푓 ⊃ 푒 we have either min 푓 = min 푒 or max 푓 = max 푒. Consider a face 푓 = 푎푏푐푑 containing 푒 = 푎푏 such that min 푓 = 푎. Then 푈(푒) − 푈(휕 푓 − 푒) = 푈(푎푏) − 푈(푎푑푐푏) = (퐹(푎푏푐푑)∗푈(푏푐))∗. Applying # and summing the obtained expression over all such faces 푓 , we get (#퐹∗ ⌢ 푈)∗. Analogous sum over all the faces 푓 such that max 푓 = 푏 gives ∗ (푈 ⌢ #퐹∗)∗. Then Lemma 4.5 implies (11). Proof of Corollary 2.3. The Yang–Millsequation follows from Theorem 3.2, Propositions 2.11,2.12,3.1; see rows 6–7 of Table3 and Eq. (11). Proposition 3.2 and Theorem 3.5 imply charge conservation. Proof of Corollary 2.4. This follows directly from Propositions 2.11 and 3.2 (see line 7 of Table3) because Re Tr[ 푗∗ ⌢ 푈] is preserved under simultaneous gauge transformation of 푈 and 푗. Proof of Corollary 2.5. This follows from a version of Theorem 1.1 for complex-valued fields and nonfree boundary conditions and the case 푈 = 1, 푛 = 1 of Proposition 3.1; see rows 2–3 of Table3. Proof of Corollary 2.6. Since L[휙] is globally gauge invariant, it follows that (5) holds for Δ = 푖휙. By the versions of Theorems 1.2 and 1.3 for complex-valued fields and nonfree boundary conditions, it follows that the real parts of (6) and (7) are conserved apart the boundary, as required.

1 Proof of Proposition 2.13. Since ϕ is 퐶 , we get 푁 [훿휙푁 ](푒) ⇒ 휕푙ϕ(max 푒), 휙푁 (min 푒) ⇒ ϕ(max 푒), 푁 푗 푒 푁 훿휙∗ ⌢ 휙 푒 휕푙 ∗ 푒 푒 푙 푒 . 푁 ( ) = −2 Im[# 푁 푁 ]( ) ⇒ −2 Im ϕ (max )ϕ(max ) = j (max )

For 푣 :=max ℎ, by a version of Proposition 2.8 for C-valued ﬁelds and rows 2–3 of Table3, we get 푙 푁2 푇 , ℎ 푁2 훿휙∗ 푣 훿휙∗ 푣 훿휙 푣 (−1) h 푁 i푘 = Re # 푁 ( + e푙) + # 푁 ( + e푙 − 2e푘 ) 푁 ( − e푘 ) + h 2 i + 푁2δ푙 − 훿휙 ⌢ 훿휙∗ + 푚 휙 ⌢ 휙∗ (푣) 푘 # 푁 푁 푁2 푁 푁 휕푙 ∗휕 푣 푙 휕푛 ∗휕 푚2 ∗ 푣 푙 푣 . ⇒ 2Re ϕ 푘 ϕ ( ) + δ푘 − ϕ 푛ϕ + ϕ ϕ ( ) = T푘 ( ) Proof of Corollary 2.7. Drop the last term (not depending on 휙) from the Lagrangian L[휙, 푈]. Then by a version of Theorem 3.1 and rows 2–3 of Table3 the corollary follows.

39 Proof of Corollary 2.8. This follows directly from Proposition 3.2; see rows 2–3 of Table3. Proof of Corollary 2.9. This follows from Corollary 2.8, a version of Theorem 3.4 for nonfree boundary conditions, row 3 of Table3, and the formula for (휙 ⌣ 휓)∗ from Lemma 4.6. 휙 퐶0 퐼푑 1×푛 휙, 푈 푈 Proof of Corollary 2.10. For fixed ∈ ( 푁 ; C ) the Lagrangian L[ ] =: L[ ] from Corol- 푈 휕L[푈] 푗 휙, 푈 ∗ lary 2.7 is local with respect to . By Lemma 4.9 and row 7 of Table3 we get 휕푈 = [ ] 휕L[푈] 퐹∗ 푗 휙, 푈 푈 and 휕(퐹[푈]) = # , where [ ] is given by Corollary 2.9. Let 0 be stationary for the functional S[휙, 푈] = 휖L[푈]. By Theorem 3.2 푈0 satisfies the Yang–Mills equation from Corollary 2.3 with 푗 = 푗 [휙, 푈0]. Then again by Theorem 3.2 푈0 is stationary for S[푈] from Proposition 2.11, where 푗 = 푗 [휙, 푈0] is fixed (i.e., one keeps 푗 = 푗 [휙, 푈0] rather than 푗 = 푗 [휙, 푈] under a variation of 푈). Thus by Definition 2.13 푈0 is generated by 푗 [휙, 푈0]. The reciprocal assertion is proved analogously. Proof of Corollaries 2.11 and 2.13. Let us prove Corollary 2.13; 2.11 is a particular case. Drop the last term (not depending on 휓) from the Lagrangian L[휓, 푈]. By a version of Theorem 3.1, a field 휓 퐶0 퐼4 4×푛 휓 ∈ ( 푁 ; C ) is stationary for S[ ], if and only if the following expression vanishes:

∗ ∗ ∗ 퐷∗ 휕L + 휕L = 퐷∗ (−푖훾0훾 ⌣ 휓)+푖훾0훾 ⌢ 퐷 휓−2푚훾0휓 = 푖훾0훾 ⌢ 퐷¯ 휓+푖훾0훾 ⌢ 퐷 휓−2푚훾0휓. 퐴 휕(퐷 퐴휓) 휕휓 퐴 퐴 퐴 퐴

Left-multiplying by (훾0)−1, we get the Dirac equation in a gauge ﬁeld. Here the 1st equality is obtained by rows 4–5 of Table3 and the 2nd one follows from

퐷∗ 훾 ⌣ 휓 ∗ 휕 휓∗ ⌢∗ 훾∗ 퐴 ⌢∗ 휓∗ ⌢∗ 훾∗ 휓∗ ⌢∗ 휕훾∗ 훿휓∗ ⌢∗ 훾∗ 퐴 ⌣ 휓∗ ⌢∗ 훾∗ 훾 ⌢∗ 퐷¯ 휓 ∗, ( 퐴 ( )) = ( )− ( ) = − −( ) = −( 퐴 ) where we used the obvious identity 휕훾∗ = 0, equations (15)–(16), and Lemma 4.6. ∗ Proof of Proposition 2.14. Let the Dirac operator on the doubling act by 휕휓6 = 훾 ⌢ 훿휓 + 훾 ⌢ 훿휓 for each C4×1-valued field 휓 on the vertices of the doubling. Then the Dirac equation is 푖휕휓6 − 2푚휓 = 0. Applying the operator 푖휕6 +2푚 to the left-hand side and canceling the ±푖푚휕6 -terms we get 휕6 휕휓6 +4푚2휓 =0. It remains to prove the identity 6휕6휕 = −휕initial#훿initial, where 휕initial and 훿initial are the boundary and 퐼4 coboundary operators respectively on the initial grid 푁 . 푣 퐼4 훾푘 훾푙 훾푙 훾푘 푔푘푙 푘푙 푘0 Take a nonboundary vertex of 푁 . By the identity + = 2 := δ (4δ − 2) we get ∑︁3 푘 [6휕휕휓6 ](푣) = 훾 [6휕휓(푣 + e푘 )−휕휓 6 (푣 − e푘 )] 푘=0 ∑︁3 푘 푙 = 훾 훾 [휓(푣 + e푘 + e푙) − 휓(푣 + e푘 − e푙) − 휓(푣 − e푘 + e푙) + 휓(푣 − e푘 − e푙)] 푘,푙=0 ∑︁3 푘푘 = 푔 [휓(푣 + 2e푘 ) − 2휓(푣) + 휓(푣 − 2e푘 )] = −[휕initial#훿initial휓](푣). 푘=0 Remark 4.2. The actions from Corollaries 2.11 and 2.13 can be written as S[휓, 푈] = 휖L0[휓, 푈], where 0 1 ∗ L [휓, 푈] = Re Tr 휓¯ ⌢ (푖 퐷6 퐴휓 − 푚휓) − #퐹 ⌢ 퐹 ∗ 2 and 6퐷 퐴휓 := 훾 ⌢ 퐷 퐴휓 + 훾 ⌢ 퐷¯ 퐴휓 so that 6퐷0 =6휕. But in contrast to L[휓, 푈], the Lagrangian L0[휓, 푈] is nonlocal with respect to the gauge group field 푈. Proof of Corollary 2.12. Since L[푒−푖푡휓] = L[휓] for each 푡 ∈ R, we have (5) with Δ = −푖휓. Then by a version of Theorem 1.2 for complex-valued fields, row 5 of Table3 for 퐴 = 0, and the identity 휓∗ ⌢ 휙∗ = (휙 ⌣ 휓)∗ from Lemma 4.6 we have the conserved current h i h ∗i 푗 휓 휕L[휓] ⌢ 푖훾0훾 ⌣ 휓 ∗ ⌢ 푖휓 푖휓∗ ⌣ 푖훾0훾 ⌣ 휓 휓¯ ⌣ 훾 ⌣ 휓 . [ ] = Re 휕(훿휓) Δ = Re (− ) (− ) = Re (− ) = Re

The conservation of 푗5 [휓] is proved analogously, only take Δ = 푖훾5휓 and apply the identities (훾5)∗ = 훾5 and 훾5훾0 = −훾0훾5. The conservation of 푇 [휓] follows from Theorem 1.3, rows 4–5 of Table3, and the identity (훾0훾)∗ = 훾0훾.

40 1 Proof of Proposition 2.15. As ψ is 퐶 , we get ψ(min 푒) ⇒ψ(max 푒), 푁훿휓푁 (푒) ⇒ 휕푙ψ(max 푒), and 푙 푙 푙 푗푁 (푒) = Re[휓¯ 푁 ⌣ 훾 ⌣ 휓푁 ](푒) = Re ψ¯ (min 푒)훾 ψ(max 푒) ⇒ Re ψ¯ (max 푒)훾 ψ(max 푒) = j (max 푒).

For 푣 =max ℎ, by a version of Proposition 2.8 for vector-valued fields and rows 4–5 of Table3, (− )푙 푁h푇 , ℎi = 푁 [([−푖훾 훾 ⌣ 휓 ]∗(푣 + ) + [−푖훾 훾 ⌣ 휓 ]∗(푣 + − ))] 훿휓 (푣 − ) 1 푁 푘 2 Re 0 푁 e푙 0 푁 e푙 2e푘 푁 e푘 푁 푙 휓¯ ⌢ 푖훾 ⌢ 훿휓 푚 휓 푣 − δ푘 Re 푁 푁 − 푁 푁 ( ) 푖 훾푙 휕 푙 푖 훾푛휕 푚 푣 푙 푣 . ⇒ Re ψ¯ 푘 ψ − δ푘 ( ψ¯ 푛ψ − ψψ¯ ) ( ) = T푘 ( ) Proof of Corollary 2.14. This follows directly from Proposition 3.2; see rows 4,5,7 of Table3. Proof of Corollary 2.15. By Corollary 2.14 and Theorem 3.4 we get the conserved covariant current ∗ ∗ 푗 [휓] = 휕L[휓,푈] ⌢ 휓 = (−푖훾0훾 ⌣ 휓)∗ ⌢ 휓 = −휓¯ ⌣ 푖훾 ⌣ 휓. 휕(퐷 퐴휓) 퐷∗ 푗 [휓] = 퐷∗ 푗 [휓] = 퐺 = 푈(푛) −푖 · ∈ 푇 퐺 By Lemma 4.6 we get Pr푇1퐺 퐴 퐴Pr푇푈 퐺 0. Take so that 1 1 . Then 휕 푗 휓 , 푖푈 푖 퐷∗ 푗 휓 . h [ ] i = ReTr[(− ) 퐴 [ ]] = 0 휓 퐶0 퐼푑 4×푛 휓, 푈 푈 Proof of Corollary 2.16. For fixed ∈ ( 푁 ; C ) the Lagrangian L[ ] =: L[ ] from Corol- 푈 휕L[푈] 푗 휓 ∗ lary 2.13 is local with respect to . By Lemma 4.9 and row 7 of Table3 we get 휕푈 = [ ] and 휕L[푈] 퐹∗ 푗 휓 푈 휓, 푈 휖 푈 휕(퐹[푈]) = # , where [ ] is given by Corollary 2.15. Let be stationary for S[ ] = L[ ]. By Theorem 3.2 푈 satisfies the Yang–Mills equation from Corollary 2.3 with 푗 = 푗 [휓]. Again by Theorem 3.2 푈 is stationary for S[푈] from Proposition 2.11, i.e., 푈 is generated by 푗 = 푗 [휓]. The reciprocal assertion is proved analogously.

5 Open problems

• Expand the suggested discretization algorithm to:

– quantum field theories via path integral formalism; – general relativity via discretizing the raising index operator ♯ for nonflat spacetimes; – hydrodynamics via discretizing the fluid energy-momentum tensor.

• Extend the suggested discretization algorithm to involve the following conservation laws:

– energy conservation in nontrivial connection via making the cross-product gauge invariant; – angular momentum conservation via discretizing the radius vector; – integral-form energy conservation in general complexes via discretizing tensor integration.

• Prove the conservation of the discrete covariant chiral current. Generally, is the covariant current from Theorem 3.3 times 푖 conserved for each gauge invariant Lagrangian satisfying (5)?

• Prove analogous conservation laws in statistical field theory. E.g., is the expectation of a covariant current conserved, if the gauge group field is random with the probability density proportional to the exponential of the action from Definition 2.17?

• Apply the discretization algorithm to characteristic classes to obtain invariants of piecewise- linear homeomorphisms or rational homotopy type.

• Constuct a “second-generation” discretization algorithm for ﬁeld theories, in which not only spacetime, but also the set of ﬁeld values becomes discrete; e.g., as in the Feynman checkerboard.

• Prove that the discussed discrete ﬁeld theories approximate continuum ones in a sense. Even no analogue of Theorem 2.1 for planar graphs with faces not being inscribed is known [6, 27].

41 • State and prove a “reciprocal Noether theorem” giving a symmetry of the continuum limit for each discrete conservation law.

• Find an experimentally measurable quantity in our discretization not converging to the continuum counterpart; this would make the discretization falsiﬁable against the continuum theory.

Conclusions We have introduced a new general discretization algorithm for field theories (see §1.3), in many cases leading to both approximation of continuum theory and exact conservation laws. The latter are produced by a new discrete Noether theorem relating them to continuum symmetries (Theorems 1.2 and 3.3). Compared to known results, the new Noether theorem is simple enough to write the resulting conservation laws explicitly as one-line formulae (using only standard topological notation) in numerous examples. Since discrete spacetime has no continuous symmetries, exact energy conservation is obtained separately by a different method not based on a symmetry, extending [12] (Theorems 1.3 and 2.2). For that purpose, a new discretization of tensor calculus involving non-antisymmetric tensors is applied (see §2.3), although it has serious limitations (see §1.5). Approximation of continuum theory is established in many examples (Theorem 2.1, Propositions 2.9, 2.13, 2.15, and Remark 2.15), some of them slightly extending the known results on electrical networks [6] (see Theorem 2.1) and gauge theory [11] (see Corollary 2.3). In particular, the new conserved discrete energy-momentum tensor approximates the continuum one at least for free fields. Thus for a variety of field theories, the algorithm achieves the principles of discretization from §1.

Acknowledgements. The author is grateful to E.Akhmedov, L.Alania, D.Arnold, A.Bossavit, V.Buchstaber, D.Chelkak, M.Chernodub, M.Desbrun, F.Günther, I.Ivanov, M.Kraus, N.Mnev, F.Müller- Hoissen, S.Pirogov, P.Pylyavskyy, A.Rassadin, R.Rogalyov, I.Sabitov, P.Schröder, I.Shenderovich, B.Springborn, A.Stern, S.Tikhomirov, S.Vergeles for useful discussions.

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