Ivancevic_Applied-Diff-Geom

1050 Applied Differential Geometry: A Modern IntroductionNow, motor learning occurring in the cerebellum can be observed usingfunctional MR imaging, showing changes in the cerebellar action potential,related to the motor tasks (see, e.g., [Mascalchi et. al. (2002)]). To accountfor these electro–physiological currents, we need to add the source termJ i (t)q i (t) to the affine Hamiltonian action (6.80), (the current J i = J i (t)acts as a source J i A i of the cerebellar electrical potential A i = A i (t)),S aff [q, p, J] =∫ toutt indτ [ p i ˙q i − H aff (q, p) + J i q i] ,which, subsequently gives the cerebellar path integral with the action potentialsource, coming either from the motor cortex or from other subcorticalareas.Note that the standard Wick rotation: t ↦→ it (see [Klauder (1997);Klauder (2000)]), makes all our path integrals real, i.e.,∫∫D[w, q, p] e i S aff [q,p]W−−−→ick D[w, q, p] e − S aff [q,p] ,while their subsequent discretization gives the standard thermodynamicpartition functions,Z = ∑ je −wjEj /T , (6.82)where E j is the energy eigenvalue corresponding to the affine HamiltonianH aff (q, p), T is the temperature–like environmental control parameter, andthe sum runs over all energy eigenstates (labelled by the index j). From(6.82), we can further calculate all statistical and thermodynamic systemproperties (see [Feynman (1972)]), as for example, transition entropy S =k B ln Z, etc.6.3.10 Path Integrals via Jets: Perturbative QuantumFieldsRecall that an elegant way to make geometrical path integrals rigorous is toformulate them using the jet formalism. In this way the covariant Hamiltonianfield systems were presented in [Bashkirov and Sardanashvily (2004)].In this subsection we give a brief review of this perturbative quantum fieldmodel.Let us quantize a Lagrangian system with the Lagrangian L N (5.316)on the constraint manifold N L (5.314). In the framework of a perturbative