Ivancevic_Applied-Diff-Geom

624 Applied Differential Geometry: A Modern IntroductionHermitian form∫〈ψ|ψ ′ 〉 = ψψ ′ (−g) 1/2 d m+1 qQNote that ̂f (4.148) are symmetric operators ̂f = ̂f ∗ in E Q , i.e., 〈 ̂fψ|ψ ′ 〉 =〈ψ| ̂fψ ′ 〉. However, the space E Q gets no physical meaning in RM.As was mentioned above, the function H T (4.145) need not belong tothe quantum algebra A, but a polynomial function H T can be quantizedas an element of the enveloping algebra A by operators ĤT (4.150). Thenthe quantum constraint (4.146) serves as a relativistic quantum equation.Let us again consider a massive relativistic charge whose relativisticHamiltonian isIt defines the constraintH = 12m gµν (p µ − eA µ )(p ν − eA ν ).H T = 1 m 2 gµν (p µ − eA µ )(p ν − eA ν ) − 1 = 0. (4.163)Let us represent the function H Tfunctions of momenta,(4.163) as symmetric product of affineH T = (−g)−1/4m(p µ − eA µ )(−g) 1/4 g µν (−g) 1/4 (p ν − eA ν ) (−g)−1/4− 1.mIt is quantized by the rule (4.150), where(−g) 1/4 ◦ ̂∂ α ◦ (−g) −1/4 = −i∂ α .Then the well–known relativistic quantum equation(−g) −1/2 [(∂ µ − ieA µ )g µν (−g) 1/2 (∂ ν − ieA ν ) + m 2 ]ψ = 0. (4.164)is reproduced up to the factor (−g) −1/2 .4.12 Symplectic Structures on Fiber BundlesIn this section, following [Lalonde et al. (1998); Lalonde et al. (1999);Lalonde and McDuff (2002)], we analyze general symplectic structures onfiber bundles. We first discuss how to characterize Hamiltonian bundles andtheir automorphisms, and then describe their main properties, in particularderiving conditions under which the cohomology of the total space splits as