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Ulm 2004 – scientific programme

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MP: Theoretische und Mathematische Grundlagen der Physik

MP 2: Symmetrien,Integrabilit
ät und Quantisierung

MP 2.4: Fachvortrag

Monday, March 15, 2004, 15:15–15:40, SR 2203

Replacing the algebraic basis {Q,P,1} of quantum mechanics by the Lie algebra of SO(1,2) — •Hans Kastrup — DESY, Theorie Gruppe, Notkestr. 85, 22603 Hamburg

Quantization of the canonical pair “angle” and “action” variables ϕ and I has been a controversial problem since 1926. The associated classical phase space Sϕ,I = {ϕ mod 2π,I>0} has the global topology S1 × R+ of a simple cone and cannot be quantized in the usual manner, namely in terms of the nilpotent Weyl group. The appropriate quantizing group for Sϕ,I is the simple group SO(1,2). The basic “canonical” variables on Sϕ,I are h0 = Ih1= I cosϕ and h2=−I sinϕ, the Poisson brackets {hi,hj }ϕ,I of which obey the Lie algebra of SO(1,2). Quantization of Sϕ,I is implemented by replacing the hj by the 3 self-adjoint generators Kj of a positive discrete series irreducible unitary representation of SO(1,2) or one of its covering groups. The usual canonical annihilation and creation operators are a = (K0+k)−1/2Ka+ = K+ (K0+k)−1/ 2K± =K1 ± i K2, where k is the (“Bargmann”) index labeling the representation. Then Q=(a++a)/√2P= i(a+a)/√2. Illustrative Example: Harmonic oscillator. (Ref.: quant-ph/0307069)

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