The Grassmann is Always Greener . . .: Fermions and Spins in Clifford
Algebra
Matthew R. Francis
Abstract
Typically, the pseudoclassical path integral technique used in quantum
field theory uses so-called Grassmann numbers to describe the
anticommuting fermionic fields. Likewise, matrix methods are used
to describe quantum spins. However, both the Grassman algebra and
matrix algebra is are easily formulated in the Clifford algebra, an
incredibly powerful mathematical toolbox that encompasses complex algebra,
tensors, differential geometry, and so on ad nauseum.
In this talk, I will begin by defining what Clifford algebra _is_,
focusing less on the abstract formalism and more on how we can use it in
field theory. This includes the "second quantized" formalism and
lattices, amongst others. Depending on the time, I may also make some
connections with relativistic systems, which are useful in conformal field
theory, and group-theoretic techniques.
The mathematical rigor of this talk will be low; I will send you to the
literature or to Professor Kosowsky's class next fall for more information
about the underlying formalism. However, many of the topics will be
technical; I hope to be able to define the terms sufficiently well for
those who do not have a strong background in field theory, but if I do
not, please stop me and ask questions.
Some of the topics I discussed are, alas, unpublished, so until I get my notes into LaTeX form, they will stay that way. However, here are a few relevant papers; note that "Geometric Algebra" (or GA) is synonymous with Clifford algebra.
- John W. Negele and Henri Orland, Quantum Many-Particle Systems (Perseus Books, 1988).
An essential text on path integral techniques and many-body systems in general, including the Grassmann algebra topics. For more information about that, however, the classic text is . . .
- F. A. Berezin, The Method of Second Quantization (Academic Press, 1966).
Berezin, as far as I understand, established a lot of the techniques for Grassmann calculus that we use today.
- C. Doran, A. Lasenby, and S. Gull, "Grassmann Mechanics, Multivector Derivatives and Geometric Algebra", in Spinors, Twistors, Clifford Algebras, and Quantum Deformations (Kluwer Academic, 1993).
The opening shot in reformulation of the Grassmann language in GA form, and also the first paper (as far as I can tell) to propose the idea of multivector-valued Lagrangians, which are necessary for the path-integral treatment of fermionic systems. Also found at
- A. Lasenby, C. Doran, and S. Gull, "Grassmann Calculus, Pseudoclassical Mechanics, and Geometric Algebra", J. Math. Phys. 34, 3683 (1993).
Far more detailed than the previous paper, this paper develops the tools for real (as opposed to complex) Grassmann algebras. There is also a scheme, not followed through in any kind of satisfactory detail, for writing path integrals.
- A. Lasenby, C. Doran, and S. Gull, "A Multivector Derivative Approach to Lagrangian Field Theory", Found. Phys. 23, 1295 (1993).
More on the general theory of Lagrangians. The emphasis here is on Lorentz-invariance, but the techniques are generally applicable.
- See also the references in my previous talk, especially noting the introductory-level treatises on Clifford algebra.