Vladimir KISIL
School of Mathematics, University of Leeds,
Leeds LS2 9JT, UK
and
Odessa National University,
Odessa, UKRAINE
Email: kisilv@maths.leeds.ac.uk
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 H^{n}. The De DonderWeyl 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
groupa nilpotent step two Lie group with multidimensional
centre. In the onedimensional case an element of the Galilean group
is (s_{1},¼,s_{n},x,y_{1},¼,y_{n}) with corresponding Lie
algebra described by the nonvanishing commutators:
For the field theory it worth to consider Clifford valued
representations induced by the Clifford valued ``characters''
e^{2p(e1h1 s1+¼+enhn sn)} of the
centre, where e_{1}, ... 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:

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, 353373; math.CV/9806150;
MR 2000b:81044; Zbl 923.22003.

I.V. Kanatchikov, Precanonical quantization and the Schr\"odinger wave
functional, Phys. Lett. A, V.283 (2001), no. 12, 2536; hepth/0012084;
MR
2002d:81119.