Wilson Action of Lattice Gauge Fields with An Additional Term from Noncommutative Geometry 1 0 Jian Dai†, Xing-Chang Song 0 2 Theoretical Group, Department of Physics, Peking University, Beijing, P.R.China, 100871 n †Room 2082, Building 48, Peking University, Beijing, P. R. China, 100871 a J E-mail: [email protected], [email protected] 6 2 January 23th, 2001 1 v 4 8 1 Abstract 1 0 1 Differential structure of lattices can be defined if the lattices are treated as models of noncommuta- 0 tive geometry. The detailed construction consists of specifying a generalized Dirac operator and a wedge / h t product. Gauge potential and field strength tensor can be defined based on this differential structure. - p When aninner productis specifiedfor differentialforms,classicalactioncanbe deducedfor lattice gauge e h fields. BesidesthefamiliarWilsonactionbeingrecovered,anadditionalterm,relatedtothenon-unitarity : v of link variables and loops spanning no area, emerges. i X r a PACS: 02.40.Gh, 11.15.Ha Key words: Lattice gauge theory, Wilson action, noncommutative geometry, Dirac operator 0 I Introduction Our work gains inspiration from three sources. First, exterior algebra, the foundation of differential forms on differentiable manifolds, has an intimate relation with Clifford algebra, the root of spinor [1]. More- over, exterior differential together with metric on differential manifolds can be realized strikingly on the spinor bundles by Dirac operator under A. Connes’ operator theoretic construction of classical geometry [2]. Physically speaking the same results, fermions provide representations for differential structures as well as metric structures of manifolds [3][4]. Connes generalized this scenario to noncommutative algebra settings and introduced noncommutative geometry(NCG) [5]. Second, a lattice provides a marvelous model for NCG developed by Dimakis and Mu¨ller-Hoissen(D&M) [6]. D&M geometry on discrete sets is essentially cohomologic description of broken lines; there is no non-commutativity at the level of 0-step broken lines which are just functions on these sets, while the non-commutativity is characterized by that the ordering to combine two broken lines not all of 0-step can not be exchanged. Third, a proper lattice Dirac operator, with or without root in NCG, more or less being attempted to solve the problem of chiral fermion in lattice field theory(LFT), was pursued by different authors [7][8][9]. WedevisedaDiracoperator D onan-dimensionallattice recently withthefollowing properties: i)Adopting Connes’ distance formula, weprovethattheinduced metric of D recovers Euclideangeometry onthislattice, when n = 1,2 [10]; ii) D is a Fredholm operator of index zero[11], i.e. an operator subjected to a “geometric square-root” condition D2 ∼ 1 [12]; iii) A differential structureon the lattice definedby reduction of calculus within the formalism of D&M can be realized onto spinor space upon this lattice through D, what’s more this representation is Junk-free [13](see [14] for a mathematical reference); iv) Under a specific matrix rep- resentation, D possesses a staggered Dirac operator interpretation, if D is slightly modified to fit a “physical square-root” condition D2 = ∆ where ∆ is the lattice laplacian [13]. Till now we just explored properties of this operator on a lattice without presenting of gauge fields. In this contribution, gauge coupling is added within the conventional approach in Connes’ geometry. We will show that the unitarity of representation of gaugegroupsresultsfromthecompatibility ofinvolution ofcovariant differentialandgaugetransformations. The classical action of lattice gauge fields is calculated. Resulting from the differential structure defined by us, an additional term, which will vanish under the usual assumption of unitarity of link variables in LFT, appears aside to the familiar Wilson action for lattice gauge fields. The losing of unitarity of link variables 1 will be showed having effect on the induced metric on lattices. Other authors’ works with similar purpose deserve to be referred here [15][16], though due to the difference of differential structure definitions they did not notice the additional term. This paper is organized as following. In Section II, necessary notations and formalism of differential forms on a lattice are introduced for describing lattice gauge fields, and then field strength tensor is calculated. In Section III, a inner product on differential forms is defined and the classical action of lattice gauge fields, which differs from the conventional Wilson action by a term vanishing if link variables are unitary, is calculated. Finally we discuss the significance of non-unitary link variables and the mathematical rigidity of our work in Section IV. II Differential Forms, Connection and Curvature on Lattices Notations A n-dimensional lattice can be regarded as a discrete abelian group Zn, the direct product of N n cyclic groups Z , in which N is an integer being large enough. The elements in Z can be written as N N 0,1,...,N−1andthemultiplicationisjusttheadditionmoduloN. LetAbethealgebraofcomplexfunctions on Zn. A collection of delta functions on Zn is defined to be a subset of A: ǫx(y) = Πn δxi,∀x,y ∈ Zn N N i=1 yi N where xi,yi are the ith components of x,y respectively. The algebraic structure of A can be expressed as ǫxǫy = Πn δxiyi, ǫx = 1 where 1 (x) = 1, since all ǫx form a natural basis for A. Shift operators i=1 x A A P acting on A is defined by (T±f)(x) = f(x±µˆ),∀x ∈ Zn,∀f ∈ A or equivalently T±ǫx = ǫx∓µˆ, where µˆ µ N µ denotes the unit vector along µ-axis. Introduce formal partial derivatives by ∂± := T±−1 with 1 being the µ µ trivial action on A. End (V) refers to C-linear endomorphism algebra on a linear space V, thus T±,1,∂± C µ µ are all elements in End (A). C II.1 Differential Structure on Lattices Based on the observation that (∂±f)(x) are two completely independent numbers with x being fixed on a µ lattice, we introduce differential of a function f in A in this way n df = (∂+fχµ +∂−fχµ) X µ + µ − µ=1 2 in which χµ form a basis of first-order differential forms. Thisintuition gains its rigidity underthefollowing ± construction. Acanbeextendedtobeagradeddifferentialalgebra(Ω(A),d)bytheconstructionrulesshown below: i)Ω(A) = ⊕∞ Ωk(A), Ω0(A) = A and the elements in Ωp(A) are referred as p-order differential k=0 forms or just p-forms; ii)Ωp(A)·Ωq(A) = Ωp+q(A); iii)d : Ωk(A) → Ωk+1(A),k = 0,1,... is a linear map satisfying graded Leibnitz rule and nilpotent rule d(ω ω′) = d(ω )ω′+(−)pω d(ω′),∀ω ∈ Ωp(A),ω′ ∈ Ω(A), p p p p d2 = 0; iv)1 is the unit of Ω(A). To make the construction be consistent, some conditions on χµ have to be A ± imposed, χµf = (T±f)χµ (1) ± µ ± {χµ,χν} = 0,{χ−µ,χ−ν} = 0,{χµ,χ−ν}= δµνdχµ = δµνdχ−ν (2) for all µ,ν = 1,2,...,n. We refer [17][13] for a detailed account of these statements. Eq.(1) is the fundamen- tal non-commutativity on lattices, which indicates that functions is no longer commutative with differential forms, while the “error” of this non-commutativity is a shift by one lattice spacing, hence vanishing under continuum limit. The geometric interpretation of above construction is that p-forms correspond to linear combinations of p-stepped broken lines on Zn. For example, let n = 1 and consider χ . It is easy to show N + that χ = N−1ǫxdǫx+1, thus χ can be interpreted as combination of 1-step line segments from x to x+1 + Px=0 + with coefficients all being one. Next we formulate a noncommutative exterior differential algebra (Λ(A),d) by introducing an equivalent relation dχµ ∼ 0 onto (Ω(A),d), s.t. Λ(A) ∼= Ω(A)/ ∼. This definition avoids the potential ambiguity ± of applying wedge product directly, which results from non-commutative relation Eq.(1). Equivalently and practically, Λ(A) can be defined to be a subset of Ω(A) together with a projection Π such that Λ(A) = Π(Ω(A)), in which Π is defined as 1 Π(f χµf χνf ):= f (Tsf )(TsTtf )(χµχν −χνχµ) = f (Tsf )(TsTtf )χµ∧χν (3) 0 s 1 t 3 2 0 µ 1 µ ν 3 s t t s 0 µ 1 µ ν 3 s t for all f ∈ A,α = 0,1,2, s,t ∈ {+,−}, µ,ν = 1,2,...,n. Note that Π(f χµ ·f χν) = f (Tsf )χµ ∧χν 6= α 0 s 1 t 0 µ 1 s t −(Ttf )f χµ ∧ χν = Π(f χν · f χµ) generally, therefore ω ∧ ω is not necessarily equal to zero for a ν 0 1 s t 1 t 0 s (1) (1) 3 generic 1-form ω . The notion of wedge product makes sense by the projection Π, and (Λ(A),d) will be (1) referred as a differential structure on Zn. N II.2 Representation of Differential Structure, Dirac-Connes Operator on the Lattice A“fermion” representation for (Λ(A),d) is developed in this subsection. Introducea spinorspace H = C2n, s and let H be a finite dimensional Hilbert space A⊗H under the conventional definition of inner product s (ψ1,ψ2) = Px∈ZnP2i=n1ψ1i(x)ψ2i(x). A is represented on H by π : π(f) = f ⊗1s, thus H is turned out to be a left free A-module in mathematical literature. Now extend π to be the “fermion” representation on µ End (H) by specifying π(χ ) and applying algebraic homomorphism rule. First, define C ± π(χµ) = iηµ,ηµ = T±⊗Γµ (4) ± ± ± µ ± µ Γ are generators of Clifford algebra of 2n-dimensional Euclidean space, thus satisfy that ± {Γµ,Γν} = 0,{Γµ,Γν} = 0,{Γµ,Γν} = δµν1 ,µ,ν = 1,2,...,n + + − − + − s as well as that (Γµ)† = Γµ. Accordingly, ± ∓ {ηµ,ην} = 0,{ηµ,ην}= 0,{ηµ,ην} = δµν1⊗1 ,µ,ν = 1,2,...,n + + − − + − s which is the representation of Eq.(2). Since (T±)† = T∓, there is (ηµ)† = ηµ. Eq.(1) is realized as µ µ ± ∓ ηµπ(f)= π(T±f)ηµ and π(df) = i[D,π(f)] where ± µ ± n µ µ D := (η +η ) X + − µ=1 D is a so-called Dirac-Connes operator in NCG. One can check that the relation D2 = n1 ⊗ 1 holds, s which is the condition for a Fredholm operator of index zero. Consequently, (H,D) forms a Fredholm module on A. Besides, it is obvious that D is hermitian, D† = D. Second, Eq(3) is implemented by defining the product of two conjoint ηµ,ην to be wedge product, thus a formal noncommutative A-linear s t rule ηµ∧(ηνf)= ηµ∧((Ttf)ην) =(ηµ(Ttf))∧ην = ((TsTtf)ηµ)∧ην follows. s t s ν t s ν t µ ν s t II.3 Gauge Fields on Lattices Let Ck be the color space H upon which gauge group G is represented as R : G → Aut (H ). Directly c C c product H with H to form a new A-module H˜ = H ⊗H. π(Λ(A)) acts on H˜ as 1 ⊗ω,∀ω ∈ π(Λ(A)) c c c 4 to which we will still use the symbol ω. Let Λ˜(A) = End (H )⊗π(Λ(A)) consisting of differential forms C c valued in End (H ) which are still referred as differential forms without introducing confusion. A sequence C c of concepts in gauge theory can be developed following a conventional routine. Define connection 1-form to be n A = i (A+ηµ +A−ηµ) ∈ Λ˜(A) X µ + µ − µ=1 in which gauge potential A± act trivially on spinor space H . A is required to be anti-hermitian, A† = −A, µ s hence satisfying that A− = T−(A+†). Let covariant differential be ∇ := iD+A, so ∇† = −∇. Introduce µ µ µ parallel transport by U± = 1+A±, then µ µ n ∇ = i (U+ηµ +U−ηµ) X µ + µ − µ=1 and U− = T−(U+†) (5) µ µ µ NoteimportantlythatU± isreferredaslinkvariablesinphysicalliteratureandthat,however,nounitarityas µ aprescriptionisforcedonU± inthisarticle. Curvatureorfield strength tensorisdefinedtobeF = −∇∧∇= µ −i{D,A} −A∧A. Itis easy tocheck thatF† = F andthatBianchi identityholdsforF, namelyi[∇,F] = ∧ ∧ 0. Even if the gauge group is abelian, there is still an A2 term in curvature due to non-commutativity. The detailed form of F can be computed using either gauge potentials A± or parallel transports U±. After a µ µ sheetofpaper’salgebra, onewillreachthatF = (F++ηµ∧ην +F+−ηµ∧ην +F−+ηµ∧ην +F−−ηµ∧ην) µν µν + + µν + − µν − + µν − − P in which 1 1 F++ = ((∂+A+−∂+A+)+(A+(T+A+)−A+(T+A+))) = (U+(T+U+)−U+(T+U+)) (6) µν 2 µ ν ν µ µ µ ν ν ν µ 2 µ µ ν ν ν µ 1 1 F+− = ((∂−A+−∂−A+)+(A+(T+A−)−A−(T−A+))) = (U+(T+U−)−U−(T−U+)) (7) µν 2 µ ν ν µ µ µ ν ν ν µ 2 µ µ ν ν ν µ and F−−, F−+ can be given by +↔- in F++, F+−. We point out that the exchange of ± is a symme- µν µν µν µν tryinourformalismwhichwewillusebroadlybelow,andthatF++ = −F++,F−− = −F−−,F−+ = −F+−. µν νµ µν νµ µν νµ Gauge transformations are 0-forms valued in R(G), denoted by g. If connect 1-form transforms affinely as A′ =gAg−1+ig[D,g−1], then covariant differential and curvature transform adjointly as ∇′ = g∇g−1,F′ = gFg−1. Involution of covariant differential and curvature is compatible with any gauge transformation g, 5 i.e. ∇′† = −∇′,F′† = F′, if [∇,g†g] = 0. Consequently, g†g equals to unit up to an overall scalar factor; R becomes a unitary representation if this factor equals to one. Nevertheless, in our understanding, the unitarity of representation R does not imply necessarily the unitarity of link variables U±. µ III Inner Product on Differential Forms and Classical Action of Gauge Fields An inner product of differential forms (,) :Λ˜(A)⊗Λ˜(A) → C has to be specified, such that classical action for gauge fields on this lattice can be written as 1 1 S[U]= (F,F) = (∇2,∇2) (8) 2(tr 1 ) 2(tr 1 ) s s s s First we introduce a hermitian structure h,i : Λ˜(A)⊗Λ˜(A) → End (H )⊗End (A), ω˜ ⊗ω˜′ 7→ tr (ω˜†ω˜′) C c C s for all ω˜,ω˜′ ∈ Λ˜(A) where tr is trace on End (H ). Then (ω˜,ω˜′) := tr Sphω˜,ω˜′i where Sp is the trace on s C s c End (A) and tr is the trace on End (H ). Gauge invariance is guaranteed naturally under this definition C c C c of S[U]. To simplify the calculation of S[U], we rewrite F to be F = (F++ηµ ∧ην +F−−ηµ ∧ην +2F+−ηµ ∧ην) X µν + + µν − − µν + − µν Consider a metric tensor  hη+µ ∧η+ν,η+λ ∧η+ρi hη+µ ∧η+ν,η+λ ∧η−ρi hη+µ ∧η+ν,η−λ ∧η−ρi  g :=  hηµ ∧ην,ηλ ∧ηρi hηµ ∧ην,ηλ ∧ηρi hηµ ∧ην,ηλ ∧ηρi  (2)  + − + + + − + − + − − −     hηµ ∧ην,ηλ ∧ηρi hηµ ∧ην,ηλ ∧ηρi hηµ ∧ην,ηλ ∧ηρi   − − + + − − + − − − − −  Due the +↔- symmetry, hηµ ∧ην,ηλ ∧ηρi= hηµ ∧ην,ηλ ∧ηρi, hηµ ∧ην,ηλ ∧ηρi = −hηµ ∧ην,ηλ ∧ηρi, − − + + + + − − − − + − + + + − hηµ ∧ ην,ηλ ∧ ηρi = hηµ ∧ ην,ηλ ∧ ηρi, hηµ ∧ ην,ηλ ∧ ηρi = −hηµ ∧ ην,ηλ ∧ ηρi. Therefore, only − − − − + + + + + − + + + − − − hηµ ∧ην,ηλ ∧ηρi, hηµ ∧ην,ηλ ∧ηρi, hηµ ∧ην,ηλ ∧ηρi, hηµ ∧ην,ηλ ∧ηρi, hηµ ∧ην,ηλ ∧ηρi need to be + + + + + + + − + + − − + − + − + − − − computed. Some algebra is needed to show 1 hηµ ∧ην,ηλ ∧ηρi = (tr 1 )(δµλδνρ −δµρδνλ)1 ⊗1 + + + + 4 s s c 1 hηµ ∧ην,ηλ ∧ηρi = (tr 1 )δµλδνρ1 ⊗1 + − + − 4 s s c hηµ ∧ην,ηλ ∧ηρi = hηµ ∧ην,ηλ ∧ηρi = hηµ ∧ην,ηλ ∧ηρi = 0 + + + − + + − − + − − − 6 Consequently,  (δµλδνρ−δµρδνλ) 0 0  1 g(2) = 4(trs1s) 0 δµλδνρ 0 1c⊗1    0 0 (δµλδνρ −δµρδνλ)    Now apply the result of g , (2) hF,Fi = (T−T+(F+−†F+−)hηµ ∧ην,ηµ′ ∧ην′i+ X µ ν µν µ′ν′ + − + − µνµ′ν′ T−T−(F++†F++)hηµ ∧ην,ηµ′ ∧ην′i+T+T+(F−−†F−−)hηµ ∧ην,ηµ′ ∧ην′i) µ ν µν µ′ν′ + + + + µ ν µν µ′ν′ − − − − 1 = (tr 1 ) (2T−T+(F+−†F+−)+T−T−(F++†F++)+T+T+(F−−†F−−)) (9) 2 s s X µ ν µν µν µ ν µν µν µ ν µν µν µν Substitute detailed expressions in Eqs.(6)(7) into (9), and notice anti-hermitian condition (5), hF,Fi =W +(tr 1 )S s s in which the symbols are defined in the following way: Wilson term 1 W = − (tr 1 ) (F−+ +F+−+F++ +F−−) 4 s s X µν µν µν µν µ6=ν 1 1 F∓± = (P∓±+P±∓)−1 ⊗1,F±± = (P±± +P±±)−1 ⊗1,µ 6= ν µν 2 µν νµ c µν 2 µν νµ c P−+ = U−(T−U+)(T−T+U+)(T+U−),P+− = P−+(+ ↔ −),µ 6= ν µν µ µ ν µ ν µ ν ν µν µν P−− = U−(T−U−)(T−T−U+)(T−U+),P++ = P−−(+ ↔ −),µ 6= ν µν µ µ ν µ ν µ ν ν µν µν and an additional term 1 1 S = (S−++S−++S+−+S+−+S+++S+++S−−+S−−)+ (S++ +S−−−S+−−S−+) 8 X µν νµ µν νµ µν νµ µν νµ 4X µ µ µ µ µ6=ν µ S∓± = Π∓± −1 ⊗1,S±± = Π±±−1 ⊗1,µ 6= ν µν µν c µν µν c Π−+ = U+(T+U−)(T−T+U+)(T+U−),Π+− = Π−+(+ ↔ −),µ 6= ν µν ν ν µ µ ν µ ν ν µν µν Π−− = U−(T−U−)(T−T−U+)(T−U+),Π++ = Π−−(+ ↔ −),µ 6= ν µν ν ν µ µ ν µ ν ν µν µν S∓± = Π∓±−1 ⊗1,S±± = Π±±−1 ⊗1 µ µ c µ µ c 7 Π−+ = U−(T−U+)U+(T+U−),Π+− = Π−+(+ ↔ −) µ µ µ µ µ µ µ µ µ Π−− = U−(T−U+)U−(T−U+),Π++ = Π−−(+ ↔ −) µ µ µ µ µ µ µ µ µ The geometric interpretations of Pst(x), Πst (x), Πst(x) are Wilson-loop operators of parallel transports µν µν µ illustrated as P∓±(x): x→ (x±νˆ) → (x∓µˆ±νˆ) → (x∓µˆ) → x µν P±±(x): x→ (x±νˆ) → (x±µˆ±νˆ) → (x±µˆ) → x µν Π∓±(x) : x→ (x±νˆ) → (x∓µˆ±νˆ)→ (x±νˆ)→ x µν Π±±(x) : x→ (x±νˆ) → (x±µˆ±νˆ)→ (x±νˆ)→ x µν where µ 6= ν, and Π∓±(x) : x→ (x±µˆ) → x→ (x∓µˆ)→ x µ Π±±(x) : x→ (x±µˆ) → x→ (x±µˆ)→ x µ The Wilson-loops producing Pst(x) span fundamental plaquettes, while those producing Πst (x) and Πst(x) µν µν µ span no area. Involutive properties can be checked as (P∓±)† = P±∓,(P±±)† = P±±,µ 6= ν µν νµ µν νµ (Π∓±)† = Π∓±,(Π±±)† = Π±±,µ 6=ν µν µν µν µν (Π∓±)† = Π±∓,(Π±±)† = Π±± µ µ µ µ Collect these results and calculate S[U] in Eq.(8) giving S[U] = S [U]+S [U] W NU S is of the form of standard Wilson action for lattice gauge fields in LFT up to a normalization factor W S [U] = − tr F(f.p.) W X c f.p. where f.p. refers to fundamental plaquette and F(f.p.) equals to one Fst(x) whose (−)sµˆ,(−)tνˆ span this µν fundamental plaquette; while, 1 1 S [U]= Sp(tr S) = (tr S)(x) NU 2 c 2 X c x∈Zn and will vanish if parallel transport U± satisfy unitarity (U±)†U± = 1 ⊗1. µ µ µ c 8 IV Discussions IV.1 Non-unitary Link Variables After the tedious calculation in the last section, we show that an additional term gaining contributions from those Wilson-loops spanning no area has to be added to the classical action of gauge fields on the lattice, if nounitarity of link variables isassumed. Hereweillustrate thegeometric significanceof non-unitaryparallel transports on Connes’ distance by an one-dimensional example. Connes’ distance on Zn can be defined by d (x,y) = sup{|f(x)−f(y)| :k [D,π(f)] k≤ 1},∀x,y ∈ Zn. Let n = 1,k = 1, then D,∇ take on the forms D like  0 T+   0 U+T+  D = ,∇ = i n=1 n=1  T− 0   U−T− 0      In [18][10], it is showed that d (x,y) = |x−y|. Now consider d (x,y), and one can check that Dn=1 (−i)∇n=1 k[(−i)∇,π(f)] k= sup{|U+|2(x)|∂+f|2(x) :x∈ Z}, noticing the anti-hermitian condition Eq.(5); therefore, if U+ is unitary, d = d , else then (−i)∇n=1 Dn=1 1 1 1 d (x,x+k)= + +...+ (−i)∇n=1 |U+(x)| |U+(x+1)| |U+(x+k−1)| for all x∈ Z,k = 1,2,..., namely non-unitary link variables will modify induced metric on lattices. IV.2 Mathematical Rigidity Till now we do not apply Connes’ formalism in a restrictive manner. In fact in [2], Rennie showed that what are recovered from Connes’ axioms for commutative algebras are necessarily spinC-manifolds. Hence, a lattice is outside to be a rigid model of Connes’ formalism being applied to commutative algebras, unless additional structures like a real structure[19] or equivalently a charge conjugation in physics jargons is considered. As one aspect of this contradiction in our understanding, most constructions of Dirac-Connes operatorsondiscretesetsincludingoursfailtofulfillfirstorder axiomwhichrequires[[D,f],g] = 0,∀f,g ∈ A. However, it can be shown that the “error” caused by this violation is proportional to the lattice spacing. In fact, introduce a to be lattice spacing and still consider the previous one-dimensional example with T± → T±/a in the expression of D and ∂± → (T±−1)/a, there is n=1  0 (∂+f)(∂+f′)T+  [[D,π(f)],π(f′)] = a (10)  (∂−f)(∂−f′)T− 0    9