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Schlagwörter:
Mathematical Physics, math-ph,General Relativity and Quantum Cosmology, gr-qc,High Energy Physics - Theory, hep-th,Mathematics, Mathematical Physics, math.MP
Zusammenfassung:
We formulate a notion of group Fourier transform for a finite dimensional Lie
group. The transform provides a unitary map from square integrable functions on
the group to square integrable functions on a non-commutative dual space. We
then derive the first order phase space path integral for quantum mechanics on
the group by using a non-commutative dual space representation obtained through
the transform. Possible advantages of the formalism include: (1) The transform
provides an alternative to the spectral decomposition via representation theory
of Lie groups and the use of special functions. (2) The non-commutative dual
variables are physically more intuitive, since despite the non-commutativity
they are analogous to the corresponding classical variables. The work is
expected, among other possible applications, to allow for the metric
representation of Lorentzian spin foam models in the context of quantum
gravity.
