Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 619system on the subspace N of T ∗ Q given by the equationH T = g µν ∂ µ H∂ ν H − 1 = 0. (4.145)To perform geometrical quantization of NRM, we give geometrical quantizationof the cotangent bundle T ∗ Q and characterize a quantum relativisticHamiltonian system by the quantum constraintĤ T ψ = 0. (4.146)We choose the vertical polarization on T ∗ Q spanned by the tangent vectors∂ α . The corresponding quantum algebra A ⊂ C ∞ (T ∗ Q) consists of affinefunctions of momentaf = a α (q µ )p α + b(q µ ) (4.147)on T ∗ Q. They are represented by the Schrödinger operatorŝf = −ia α ∂ α − i 2 ∂ αa α − i 4 aα ∂ α ln(−g) + b, (g = det(g αβ )) (4.148)in the space C ∞ (Q) of smooth complex functions on Q.Note that the function H T (4.145) need not belong to the quantumalgebra A. Nevertheless, one can show that, if H T is a polynomial ofmomenta of degree k, it can be represented as a finite compositionH T = ∑ if 1i · · · f ki (4.149)of products of affine functions (4.147), i.e., as an element of the envelopingalgebra A of the Lie algebra A [Giachetta et. al. (2002b)]. Then it isquantizedH T ↦→ ĤT = ∑ îf 1i · · · ̂f ki (4.150)as an element of A. However, the representation (4.149) and, consequently,the quantization (4.150) fail to be unique.The space of relativistic velocities of RM on Q is the tangent bundle T Qof Q equipped with the induced coordinates (t, q i , ˙q α ) with respect to theholonomic frames {∂ α }. Relativistic motion is located in the subbundle W gof hyperboloids [Mangiarotti and Sardanashvily (1998); Mangiarotti andSardanashvily (2000b)]g µν (q) ˙q µ ˙q ν − 1 = 0 (4.151)