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 S_ϕ,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 S_ϕ,I is the simple group SO^↑(1,2). The basic “canonical” variables on S_ϕ,I are h₀ = I,  h₁= I cosϕ and h₂=−I sinϕ, the Poisson brackets {h_i,h_j }_ϕ,I of which obey the Lie algebra of SO^↑(1,2). Quantization of S_ϕ,I is implemented by replacing the h_j by the 3 self-adjoint generators K_j 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)^−1/2K₋, a⁺ = K₊ (K₀+k)^{−1/ 2}, K_± =K₁ ± i K₂, where k is the (“Bargmann”) index labeling the representation. Then Q=(a⁺+a)/√2, P= i(a⁺−a)/√2. Illustrative Example: Harmonic oscillator. (Ref.: quant-ph/0307069)