Clifford algebras from symmetries of quantum field theory

Abstract:
Mathematical formalism of quantum mechanics uses complex numbers in order to provide a unitary infinite dimensional representation of the Heisenberg group Hⁿ. The De Donder-Weyl formalism for classical fields theories [2] in a similar way requires Clifford numbers for their quantisation. It was recently realised [1] that the appearance of Clifford algebras is induced by the Galilean group-a nilpotent step two Lie group with multidimensional centre. In the one-dimensional case an element of the Galilean group is (s₁,¼,s_n,x,y₁,¼,y_n) with corresponding Lie algebra described by the non-vanishing commutators:

[X, Yj] = Sj.

For the field theory it worth to consider Clifford valued representations induced by the Clifford valued ``characters'' e^{2p(e₁h₁ s₁+¼+e_nh_n s_n)} of the centre, where e₁, ... e_n are imaginary units spanning the Clifford algebra. The associated Fock spaces was described in [1]. There are important mathematical and physical questions related to the construction, which deserve careful considerations.

References:

1. J. Cnops and V.V. Kisil, Monogenic Functions and Representations of Nilpotent Lie Groups in Quantum Mechanics, Math. Methods Appl. Sci., V.22 (1999), no. 4, 353-373; math.CV/9806150; MR 2000b:81044; Zbl 923.22003.
2. I.V. Kanatchikov, Precanonical quantization and the Schr\"odinger wave functional, Phys. Lett. A, V.283 (2001), no. 1-2, 25-36; hep-th/0012084; MR 2002d:81119.