Grassmann algebra

In the contraction of any wedge product the position of the upper and lower indices generates two types of terms in Grassmann algebra [26] (i) pure contraction terms where the upper and lower contraction indices appear on the same component of the wedge product, and (ii) transvection terms, where the upper and lower contraction indices appear on different components of the wedge product. To illustrate, we consider the contraction of the wedge product between A and A ... 

The connected stmcture of the CSE has also been explored by Yasuda [23] using Grassmann algebra, by Kutzelnigg and Mukheijee [27] using a cumulant version of second-quantized operators, and by Herbert and Harriman [30] using a diagrammatic technique. 

In fact, the algorithm will be clear to anyone who understands Chapter 2 Ref. [1] and the pages following p. 26 about Grassmann algebra. In the following, numbers refer to steps in the algorithm. 

X, xAa1 6 G and , a1 [f 6 G. The interior multiplications a11 and J a, decrease the rank of tensors in G and G, respectively. It is obvious that the exterior multiplications a A and A a1 and the interior multiplications a1 [ and J a may be formalized as the familiar creation and annihilation operators on the direct V i and adjoint i spaces, respectively, except that now the operators and their adjoints do not have the same domain. Specifically (a A)t =Jai. These concepts are part of the general theory of Dual Grassmann Algebras /67/. Some other relevant results are ... 

The general definition and properties of the Clifford algebras may be seen in [34], We simply mention here that Cl(i ) is an associative real algebra acting upon the elements of R and the vectors of 1C, in relation with the Grassmann algebra AE. 

These numbers do not obey all of the laws of the algebra of complex numbers. They add like complex numbers, but their multiplication is not commutative. The general rules of multiplication of n-dimensional hypercomplex numbers were investigated by Grassmann who found a number of laws of multiplication, including Hamilton s rule. These methods still await full implimentation in physical theory. 

In pure mathematics, the discrete/continuum dichotomy plays a central role. Justus Grassmann according to Lewis (1977) had already in 1844 pointed to the fundamental distinction between the algebraic approach based on the discontinuous unit, and geometric conceptions which originate from the idea of a point from which a continuous line is generated. His son Hermann Grassmann developed his classification of mathematical fields based on this distinction. 

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