We construct a new discrete analog of the Dirac-Kähler equation in which some key geometric aspects of the continuum counterpart are captured. We describe a discrete Dirac-Kähler equation in the intrinsic notation as a set of difference equations and prove several statements about its decomposition into difference equations of Duffin type. We study an analog of gauge transformations for the massless discrete Dirac-Kähler equations.
We discuss a discrete analogue of the Dirac-Kähler equation in which chiral properties of the continual counterpart are captured. We pay special attention to a discrete Hodge star operator. To build one a combinatorial construction of double complex is used. We describe discrete exterior calculus operations on a double comlex and obtain the discrete Dirac-Kähler equation using these tools. Self-dual and anti-self-dual discrete inhomogeneous forms are presented. The chiral invariance of the massless discrete Dirac-Kähler equation is shown. Moreover, in the massive case we prove that a discrete Dirac-Kähler operator flips the chirality.
Discrete models of the Dirac-Kähler equation and the Dirac equation in the Hestenes form are discussed. A discrete version of the plane wave solutions to a discrete analogue of the Hestenes equation is established.
ABSTRACT. Two discrete models of Yang-Mill equations are constructed in the space ~ for some matrixvalued Lie group. A gauge-invariant discrete model is examined.KEY WORDS: Yang-Mills equations, discrete model, gauge invariance, gauge transformation.We construct two discrete analogs of Yang-Mills equations in the space Rn for some matrix-valued Lie group G. We define a discrete analog of the operator of exterior covariant differentiation d~ by using the formalism described in [1] and apply this operator to construct a model satisfying the gauge invariance principle. To specify a more exact discrete analog of the exterior differentiation operator d, we consider another model of Yang-Mills equations, which formally coincides with the previous one, but the equations are not gauge-invariant. The author of [2] considered two-dimensional models of the classical Yang-MiUs equations and passed to discrete analogs, which, however, neglected the invariance requirements. Our approach is close to that of [2], but removes this shortcoming.If the initial manifold is of even dimension (n = 4), then the Yang-Mills theory can be regarded as a nonlinear generalization of Hodge theory [3]. A similar result is obtained in this paper for discrete models.w Connection, Curvature, and Gauge Transformations Let M be a smooth Riemannian manifold. Consider a product bundle P = M x G over this manifold, where G is a matrix-vedued Lie group. Generally, all assertions made in this section are valid for an arbitrary vector bundle P, but for constructing the discrete model, we only use the product bundle. Both the Lie group G and the manifold M are differentiable structures, hence, the bundle P is a differentiable manifold. Therefore, we can consider the tangent bundle TP and the cotangent bundle T*P over P.A connection is specified by a 1-form w E T*P taking values in the Lie algebra g of the group G (see, e.g., [4, p. 176]). Let (x, g), x E M, and g E G be local coordinates of the bundle P. Then w has the form = g-dg + g-lAg, where A = Z A (x) odx"a~lz and the A~ are the basis of the Lie algebra 9-The differential 1-form A, which takes values in 9, is called a connection form and the functions A~(x), connections. Consider a transformation of the coordinates of the bundle P with respect to which the 1-form w is invariant. Let (x, g) be mapped into (x', g') under this transformation. We shall consider only transformations that do not change the coordinate covering of the manifold, i.e., such that x = x' andThe invariance of the 1-form w means thatAny transformation (2) satisfying condition (3) induces a certain transformation law for the connection form A. Taking into account the fact that dg r = dhg + hdg and dhh -1 + hdh -1 = 0, we obtain A' = hdh -1 + bah -1.
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