underlying spacetime. The Clifford algebra of spacetime used in physics thus has more structure than . It has in addition a set of preferred transformations – Lorentz transformations. Whether complexification is necessary to begin with depends in part on conventions used and in part on how much one wants to incorporate straightforwardly, but complexification is most often necessary in quantum mechanics where the spin representation of the Lie algebra sitting inside the Clifford algebra conventionally requires a complex Clifford algebra. For reference, the spin Lie algebra is given by This is in the convention, hence fits in . The Dirac matrices were first written down by Paul Dirac when he was trying to write a relativistic first-order wave equation for the electron, and give an explicit isomorphism from the Clifford algebra to the algebra of complex matrices. The result was used to define the Dirac equation and introduce the Dirac operator. The entire Clifford algebra shows up in quantum field theory in the form of Dirac field bilinears. The use of Clifford algebras to describe quantum theory has been advanced among others by Mario Schönberg, by David Hestenes in terms of geometric calculus, by David Bohm and Basil Hiley and co-workers in form of a hierarchy of Clifford algebras, and by Elio Conte et al. Computer vision Clifford algebras have been applied in the problem of action recognition and classification in computer vision. Rodriguez et al. propose a Clifford embedding to generalize traditional MACH filters to video (3D spatiotemporal volume), and vector-valued data such as optical flow. Vector-valued data is analyzed using the Clifford Fourier Transform. Based on these vectors action filters are synthesized in the Clifford Fourier domain and recognition of actions is performed using Clifford correlation. The authors demonstrate the effectiveness of the Clifford embedding by recognizing actions typically performed in classic feature films and sports broadcast television. Generalizations While this article focuses on a Clifford algebra of a vector space over a field, the definition extends without change to a module over any unital, associative, commutative ring. Clifford algebras may be generalized to a form of degree higher than quadratic over a vector space. See also Algebra of physical space, APS Cayley–Dickson construction Classification of Clifford algebras Clifford analysis Clifford module Complex spin structure Dirac operator Exterior algebra Fierz identity Gamma matrices Generalized Clifford algebra Geometric algebra Higher-dimensional gamma matrices Hypercomplex number Octonion Paravector Quaternion Spin group