Ivancevic_Applied-Diff-Geom

Technical Preliminaries: Tensors, Actions and Functors 85The field Euler–Lagrangian equation∂ q L ∂qϕ i = L ϕ i,in Hamiltonian form gives a pair of field canonical equations 9˙ϕ i = [ϕ i , H], ˙π i = [π i , H]. (2.56)(3) Once we have a satisfactory description of fields and motions, we canperform the Feynman quantization 10 of classical equations [Feynmanand Hibbs (1965)], using the same action A[w] as given by (6.230),but now including all trajectories rather than just the extremal one.Namely, to get the probability amplitude 〈f|i〉 of the system transitionfrom initial state i(w(t 0 )) at time t 0 to final state f(w(t 1 )) at time t 1 ,9 Here the field Poisson brackets are slightly generalized in the sense that partialderivatives ∂ are replaced with the corresponding variational derivatives δ, i.e.,„ δA δB[A, B] field =δq k − δA «δBδp k δp k δq k .10 Recall that quantum systems have two modes of evolution in time. The first, governedby Schrödinger equation:i ∂ |ψ〉 = Ĥ |ψ〉 ,∂t(where Ĥ is the Hamiltonian (energy) operator, i = √ −1 and is Planck’s constantdivided by 2π (≡ 1 in natural units)), describes the time evolution of quantum systemswhen they are undisturbed by measurements. ‘Measurements’ are defined as interactionsof the system with its environment. As long as the system is sufficiently isolated fromthe environment, it follows Schrödinger equation. If an interaction with the environmenttakes place, i.e., a measurement is performed, the system abruptly decoheres i.e.,collapses or reduces to one of its classically allowed states.A time–dependent state of a quantum system is determined by a normalized, complex,wave psi–function ψ = ψ(t), that is a solution of the above Schrödinger equation. InDirac’s words, this is a unit ‘ket’ vector |ψ〉 (that makes a scalar product ‘bracket’ 〈, 〉with the dual, ‘bra’ vector 〈ψ|) , which is an element of the Hilbert space L 2 (ψ) witha coordinate basis (q i ). The state ket–vector |ψ(t)〉 is subject to action of the Hermitianoperators (or, self–adjoint operators), obtained by the procedure of quantization ofclassical mechanical quantities, and whose real eigen–values are being measured. Quantumsuperposition is a generalization of the algebraic principle of linear combination ofvectors.The (first) quantization can be performed in three different quantum evolution pictures,namely Schrödinger (S)–picture, in which the system state vector |ψ(t)〉 rotatesand the coordinate basis (q i ) is fixed; Heisenberg (H)–picture, in which the coordinatebasis rotates and the state vector is fixed; and Dirac interaction (I)–picture, in whichboth the state vector and the coordinate basis rotate.