Vladimir KISIL
School of Mathematics, University of Leeds,
Leeds LS2 9JT, UK
Odessa National University,
E-mail: kisilv@maths.leeds.ac.uk

Clifford algebras from symmetries of quantum field theory

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,,sn,x,y1,,yn) 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'' e2p(e1h1 s1++enhn sn) of the centre, where e1, ... en 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.


  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.