A No-Go Theorem For The Compatibility Between Involution Of First Order Differential On Lattice And Continuum Limit

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1 A No-Go Theorem For The Compatibility Between Involution Of First Order Differential On Lattice And Continuum Limit Jian Dai, Xing-Chang Song Theory Group, Department of Physics, Peking University Beijing, P. R. China, May 6th, 2001 Abstract We prove that differentials of lattice coordinate functions x µ must be algebraically independent of their involutive conjugate, else no correct continuum limit exists. PACS: Gh, Ha Keywords: Involution, lattice differential, bimodule, continuum limit, No-Go theorem 0

2 To treat lattice as a simple noncommutative geometry (NCG) model has given fruitful results for lattice field theory. In fact, with discrete differential calculus, Wilson action for lattice gauge field was recovered by Dimakis et al [1]; axial anomaly in U(1) lattice gauge theory was analyzed [2][3]. All noncommutativity of lattice differential can be traced back to a bimodule structure equation dx µ f =(T µ f)dx µ, f A (1) where A is the algebra of complex functions on lattice and dx µ form a module basis for bimodule Ω 1 (A) of one forms. With lattice constant a and coordinate functions x µ, Eq.(1) can be presented into a deformation form [x µ,dx ν ]=aδ µν dx µ (2) Eq.(2) is a special case in the category of differential calculi over commutative associative algebras [4]. There is a canonical involution on A defined by pointwise complex conjugation. However, attempt to extend this involution onto Ω 1 (A) based on structure equation (2) will encounter inconsistency. In fact, if this involution is required to be an antihomomorphism, then it is easy to check that a in Eq.(2) has to be pure imaginary, providing x µ, dx µ are selfadjoint under this involution. In this short note, we make the above observation rigid in mathematics by proving the following nonexistence theorem for the antihomomorphic involution on Ω 1 (A). Theorem 1 (No-Go Theorem) Let A be the algebra of all complex functions on a d-dimensional hypercubic lattice 1

3 with x µ,µ =1, 2,..., d being coordinate functions valued in integers and M be a bimodule generated by ξ µ,µ=1, 2,..., d over A subjected to a structure equation ξ µ x ν =(x ν + δ µν a ν )ξ µ, µ, ν =1, 2,..., d (3) where a µ are positive real scalars. Suppose there is no two-sided ideal in A annihilating M. Then there does not exist :Ω 1 (A) Ω 1 (A) satisfied the following conditions simultaneously: 1)Antihomomorphism. λ = λ, (f + f ) = f + f, (ff ) = f f, (v + v ) = v + v, (fv) = v f, (vf) = f v, λ C, v, v M, f,f A; 2)Direct-product Property. For a fixed µ, ξ µ = f(x µ )ξ µ where f(x µ ) is a generic function varying only along µ-direction. 3)Continuum Limit. When max({a µ }) 0, x µ x µ (p) = O(a µ ), (ξ µ ξ µ )/ξ µ = O(a µ ) where p is any point on the lattice, the norm is l -norm on A and the formal division by ξ µ is the operation to single out the left coefficient of ξ µ. Proof: Assume that there exists such a. SinceM is generated algebraically by ξ µ, condition 2) makes good sense, i.e. ξ µ is algebraically dependent on ξ µ. Follow condition 2), the proof is reduced to 1-dimensional case, i.e. we do not consider the complexity introduced by the topology of high dimension. Rewrite structure equation (3) for d =1asξx (x + a)ξ = 0. Applying on both sides, and noting condition 1), one gets that x ξ ξ (x + a) =0 (4) Condition 3) can be described as x = x + af (x),ξ = ξ + ag(x)ξ (5) 2

4 in which F is the restriction of an analytic function on real numbers and G is the restrictions of a bounded functions on real numbers. Substitute expansion Eq.(5) into Eq.(4), and notice the facts that F is real analytic and that no two-sided ideal of A annihilates M. One reaches (1 + ag(x))(2 + F (x + a) F (x)) = 0 (6) However due to boundedness of G and analyticity of F, Eq.(6) fails to be an identity when a is small enough. Therefore, the theorem follows. In [1], the above problem is avoided by defining to be homomorphism instead. This solution can not be generalized to the case where A becomes noncommutative. In [5], consistent involution on abelian discrete groups, with lattice being taken as a special case, is defined to be f (g) =f( g); it violates the requirement of correct continuum limit. In [6], dx µ and dx µ are algebraically independent generators of first order differential forms; this so-called nearest symmetric reduction has also been mentioned as an example in [7]. Acknowledgements This work was supported by Climb-Up (Pan Deng) Project of Department of Science and Technology in China, Chinese National Science Foundation and Doctoral Programme Foundation of Institution of Higher Education in China. References [1] A. Dimakis, F. Müller-Hoissen, T. Striker, Non-commutative differential calculus and lattice gauge theory, J. Phys. A: Math. Gen. 26(1993)

5 [2] M. Lüscher, Topology and the axial anomaly in abelian lattice gauge theories, Nucl. Phys. B538(1999) , hep-lat/ [3] T. Fujiwara, H. Suzuki, K. Wu, Non-commutative Differential Calculus and the Axial Anomaly in Abelian Lattice Gauge Theories, Nucl. Phys. B569(2000) ,hep-lat/ ;ibid, Axial Anomaly in Lattice Abelian Gauge Theory in Arbitrary Dimensions, Phys. Lett. B463(1999)63-68, heplat/ ; ibid, Application of Noncommutative Differential Geometry on Lattice to Anomaly, IU-MSTP/37, hep-lat/ [4] H. C. Baehr, A. Dimakis, F. Müller-Hoissen, Differential Calculi on Commutative Algebras, MPI-PhT/94-83, hep-th/ [5] B. Feng, Differential Calculus on Discrete Groups and its Application in Physics, MS Disertation (1997, Peking University). [6] J. Dai, X-C. Song, Noncommutative Geometry of Lattice and Staggered Fermions, Phys. Lett. B508(2001) , hep-th/ [7] A. Dimakis, F. Müller-Hoissen, Differential Calculus and Discrete Structures, GOET-TP 98/93, in the proceedings of the International Symposium Generalized Symmetries in Physics (ASI Clausthal, 1993), hep-th/