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amathematician
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To complete Stefan's answer, notice that the isomorphism of unital associative algebras from Cl(V,q) to the algebra obtained by appropriate deformation quantization of the exterior algebra of V is classical (and should perhaps be better known). This isomorphism is due to Chevalley and Riesz, see:

  1. C. Chevalley, The Algebraic Theory of SpinorsThe Algebraic Theory of Spinors, Columbia University Press, 1954.

  2. C. Chevalley, The construction and study of certain important algebrasThe construction and study of certain important algebras, Math. Soc. Japan 1955, in: collected works, Vol. 2, Springer, Berlin, 1997.

  3. M. Riesz, Clifford algebras and spinorsClifford algebras and spinors, eds. E. F. Bolinder, and P. Lounesto, Fundamental Theories of Physics 54, Springer, 1993.

Another classical reference for this (which stresses the deformation quantization aspect) is:

  1. F. A Berezin, M. S. Marinov, Particle spin dynamics as the grassmann variant of classical mechanicsParticle spin dynamics as the grassmann variant of classical mechanics, Ann. Phys. 104 (1977), 336 – 362.

The Chevalley-Riesz isomorphism globalizes to manifolds as shown in:

  1. F. F. Voronov, Quantization on supermanifolds and an analytic proof of the Atiyah- Singer index theoremQuantization on supermanifolds and an analytic proof of the Atiyah- Singer index theorem, J. Soviet Math. 64 (1993) 4, 993 – 1069.

  2. W. Graf, Differential forms as spinorsDifferential forms as spinors, Annal. I.H.P. Phys. Théor. 29 (1978) 1, 85 – 109.

The bundle obtained by "vertical" deformation quantization of the exterior bundle of a pseudo-Riemannian manifold is usually called the Kahler-Atiyah bundle (it is isomorphic to the Clifford bundle as a bundle of unital associative algebras), see for example:

E. Kähler, Der innere DifferentialkalkülDer innere Differentialkalkül, Rend. Mat. 3-4 (1960) 21, 425 – 523.

Working with this bundle instead of the Clifford bundle provides the so-called Kahler-Atiyah (a.k.a. "geometric algebra") approach to spin geometry, see the following Math Overflow question and references therein:

What's "geometric algebra"?

To complete Stefan's answer, notice that the isomorphism of unital associative algebras from Cl(V,q) to the algebra obtained by appropriate deformation quantization of the exterior algebra of V is classical (and should perhaps be better known). This isomorphism is due to Chevalley and Riesz, see:

  1. C. Chevalley, The Algebraic Theory of Spinors, Columbia University Press, 1954.

  2. C. Chevalley, The construction and study of certain important algebras, Math. Soc. Japan 1955, in: collected works, Vol. 2, Springer, Berlin, 1997.

  3. M. Riesz, Clifford algebras and spinors, eds. E. F. Bolinder, and P. Lounesto, Fundamental Theories of Physics 54, Springer, 1993.

Another classical reference for this (which stresses the deformation quantization aspect) is:

  1. F. A Berezin, M. S. Marinov, Particle spin dynamics as the grassmann variant of classical mechanics, Ann. Phys. 104 (1977), 336 – 362.

The Chevalley-Riesz isomorphism globalizes to manifolds as shown in:

  1. F. F. Voronov, Quantization on supermanifolds and an analytic proof of the Atiyah- Singer index theorem, J. Soviet Math. 64 (1993) 4, 993 – 1069.

  2. W. Graf, Differential forms as spinors, Annal. I.H.P. Phys. Théor. 29 (1978) 1, 85 – 109.

The bundle obtained by "vertical" deformation quantization of the exterior bundle of a pseudo-Riemannian manifold is usually called the Kahler-Atiyah bundle (it is isomorphic to the Clifford bundle as a bundle of unital associative algebras), see for example:

E. Kähler, Der innere Differentialkalkül, Rend. Mat. 3-4 (1960) 21, 425 – 523.

Working with this bundle instead of the Clifford bundle provides the so-called Kahler-Atiyah (a.k.a. "geometric algebra") approach to spin geometry, see the following Math Overflow question and references therein:

What's "geometric algebra"?

To complete Stefan's answer, notice that the isomorphism of unital associative algebras from Cl(V,q) to the algebra obtained by appropriate deformation quantization of the exterior algebra of V is classical (and should perhaps be better known). This isomorphism is due to Chevalley and Riesz, see:

  1. C. Chevalley, The Algebraic Theory of Spinors, Columbia University Press, 1954.

  2. C. Chevalley, The construction and study of certain important algebras, Math. Soc. Japan 1955, in: collected works, Vol. 2, Springer, Berlin, 1997.

  3. M. Riesz, Clifford algebras and spinors, eds. E. F. Bolinder, and P. Lounesto, Fundamental Theories of Physics 54, Springer, 1993.

Another classical reference for this (which stresses the deformation quantization aspect) is:

  1. F. A Berezin, M. S. Marinov, Particle spin dynamics as the grassmann variant of classical mechanics, Ann. Phys. 104 (1977), 336 – 362.

The Chevalley-Riesz isomorphism globalizes to manifolds as shown in:

  1. F. F. Voronov, Quantization on supermanifolds and an analytic proof of the Atiyah- Singer index theorem, J. Soviet Math. 64 (1993) 4, 993 – 1069.

  2. W. Graf, Differential forms as spinors, Annal. I.H.P. Phys. Théor. 29 (1978) 1, 85 – 109.

The bundle obtained by "vertical" deformation quantization of the exterior bundle of a pseudo-Riemannian manifold is usually called the Kahler-Atiyah bundle (it is isomorphic to the Clifford bundle as a bundle of unital associative algebras), see for example:

E. Kähler, Der innere Differentialkalkül, Rend. Mat. 3-4 (1960) 21, 425 – 523.

Working with this bundle instead of the Clifford bundle provides the so-called Kahler-Atiyah (a.k.a. "geometric algebra") approach to spin geometry, see the following Math Overflow question and references therein:

What's "geometric algebra"?

Source Link
amathematician
amathematician
  • 149
  • 1
  • 4

To complete Stefan's answer, notice that the isomorphism of unital associative algebras from Cl(V,q) to the algebra obtained by appropriate deformation quantization of the exterior algebra of V is classical (and should perhaps be better known). This isomorphism is due to Chevalley and Riesz, see:

  1. C. Chevalley, The Algebraic Theory of Spinors, Columbia University Press, 1954.

  2. C. Chevalley, The construction and study of certain important algebras, Math. Soc. Japan 1955, in: collected works, Vol. 2, Springer, Berlin, 1997.

  3. M. Riesz, Clifford algebras and spinors, eds. E. F. Bolinder, and P. Lounesto, Fundamental Theories of Physics 54, Springer, 1993.

Another classical reference for this (which stresses the deformation quantization aspect) is:

  1. F. A Berezin, M. S. Marinov, Particle spin dynamics as the grassmann variant of classical mechanics, Ann. Phys. 104 (1977), 336 – 362.

The Chevalley-Riesz isomorphism globalizes to manifolds as shown in:

  1. F. F. Voronov, Quantization on supermanifolds and an analytic proof of the Atiyah- Singer index theorem, J. Soviet Math. 64 (1993) 4, 993 – 1069.

  2. W. Graf, Differential forms as spinors, Annal. I.H.P. Phys. Théor. 29 (1978) 1, 85 – 109.

The bundle obtained by "vertical" deformation quantization of the exterior bundle of a pseudo-Riemannian manifold is usually called the Kahler-Atiyah bundle (it is isomorphic to the Clifford bundle as a bundle of unital associative algebras), see for example:

E. Kähler, Der innere Differentialkalkül, Rend. Mat. 3-4 (1960) 21, 425 – 523.

Working with this bundle instead of the Clifford bundle provides the so-called Kahler-Atiyah (a.k.a. "geometric algebra") approach to spin geometry, see the following Math Overflow question and references therein:

What's "geometric algebra"?