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friend
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Are any of the non-division algebras similar to the grassmann algebra?
friend said:Are any of the non-division algebras similar to the grassmann algebra?
friend said:But, perhaps there is a natural reason why one should use grassmann or Cl(7) numbers. It seems that there is a Cayley-Dickson type construction in quantum theory. There are the reals used for probability, and complex numbers used in wavefunction, and quaternions used in SU(2) QFT, and octonions used in SU(3) QFT. These are all the division algebras. But can this patern be iterated into the non-division algebras, and do they have application in physics as well? One would then keep iterating and applying these higher dimensional algebras until you get to the Cl(7) numbers where you'd be describing SUSY, right?
friend said:But perhaps this begs a deeper question as to why only the division algebras find expression in physics. Could it be because the non-division algebras cannot be established over the reals?
The Grassmann algebra (Λ(V),g) endowed with this product is
denoted by Cℓ(V,g) or Cℓp,q, the Clifford algebra associated with V ≃ Rp,q, p + q = n.
Non-division algebra is a type of algebra in which not all elements have multiplicative inverses. This means that not every element can be divided by another element within the algebra.
Yes, non-division algebra has various applications in fields such as physics, engineering, and computer science. It is used to model systems that do not have a division operation, such as quantum mechanics and computer graphics.
Grassmann algebra is a type of non-division algebra that extends the concepts of vector spaces and linear transformations to include the notion of multiplication. It is often used in geometric algebra to solve problems in physics and engineering.
Traditional algebra follows the rules of commutativity and associativity, while grassmann algebra does not. In addition, grassmann algebra introduces the concept of anti-commutativity, meaning that the order of multiplication matters. It also includes the concept of exterior product, which generalizes the cross product in three-dimensional space.
Grassmann algebra is commonly used in computer graphics to represent rotations, translations, and other transformations in three-dimensional space. It is also used in physics to describe quantum mechanics and relativity, as well as in engineering for problems involving electromagnetism and fluid dynamics.