The principles of Newtonian and quantum mechanics. The need for Planck's constant, \(h\) (Q2763690)

The aim of the book is to expose the mathematical machinery underlying Newtonian mechanics, as well as semiclassical and classical non-relativistic quantum mechanics. By semi-classical mechanics, the author means the particle mechanics which was evolved following Bohm's interpretation of quantum mechanics. These three disciplines are here developed from a single mathematical object, the Hamiltonian flow as an abstract group. Apart from the contributions of the Bohmian School, which started from the later half of the last century, the topics discussed in the book were developed mostly during the last two decades. In fact, the main thesis of the book centres round the notion of metaplectic representation of symplectic group and Gromov's non-squeezing theorem which were obtained in the middle of the eighties. As such, this is the first book (at least in the reviewer's opinion) on these topics.NEWLINENEWLINENEWLINEThe book begins with a review of the basic principles of Newtonian and quantum mechanics, with particular emphasis on the Bohmian formulation of the quantum motion of particles. This is followed by the presentation of Newtonian mechanics from the symplectic point of view, with special attention paid to the Poincaré-Cartan form.NEWLINENEWLINENEWLINEThe next chapter is devoted to a thorough study of the symplectic group \(Sp(n)\), in particular, to its topological properties leading to Gromov's non-squeezing theorem (``symplectic camel''). It states that the action of symplectic transformation on phase space has a rigidity. This result is thought to be somehow akin to the uncertainty principle in quantum mechanics. Further, the author shows that an arbitrary volume-preserving diffeomorphism cannot be approximated by a sequence of symplectic mappings. Based on the properties of ``symplectic camel'', the author proposes a quantization scheme for phase space.NEWLINENEWLINENEWLINEAction and phase are the topics of the next chapter. These can be most easily apprehended by using Poincaré-Cartan integral invariants. Here the author also introduces the important notion of generating function, and shows that the gain in action is related to the phase of a Lagrangian manifold.NEWLINENEWLINENEWLINEThe most mathematically advanced chapters in the book, ``Semi-classical mechanics'' and ``The metaplectic group'' are devoted to symplectic geometry in relation to the phase space. Here one of the central themes is the metaplectic representation of symplectic group. The symplectic group has a double covering which can be realized as a group of unitary operators on \(L^2(\mathbb{R})\), the metaplectic group \(Mp(n)\). Each element of \(Mp(n)\) is the product of two quadratic Fourier transforms. Noting that the spin group is the double covering of orthogonal group, the metaplectic group is the spin covering of symplectic group.NEWLINENEWLINENEWLINEThe last chapter is entitled as ``Schrödinger's equation and metatron''. Metatron is the entity whose motion is governed by Bohm's equation derived from the wave function. The author shows that the metaplectic representation yields an algorithm allowing one to calculate the solutions of Schrödinger equation from classical trajectories and conversely. Classical trajectories can be recovered from the wave function, since both are derived from the classical Hamiltonian flow. This is demonstrated explicitly only for quadratic Hamiltonian.NEWLINENEWLINENEWLINEThough the material pertaining to physics is naive, the style of presentation is elegant (at least to physicists), perhaps due to the fact that the ``Bourbakian rigor mortis causa'' is avoided. The occasional humour, rare in scientific literature, is quite enjoyable. Added to it, the historical remarks are extra bonus to serious readers.NEWLINENEWLINENEWLINEHowever, the consequences, both physical and mathematical, of the quantum potential in Bohmian mechanics are not yet clear. Heisenberg's uncertainty principle is universal, in the sense that its limit depends neither on the potential nor on the nature of particle. In fact, it arises out of the process of observation due to the existence of Planck constant \(h\). As a parameter \(h\) in the Schrödinger equation tends to 0, there arises a strong singularity, and the equation loses its meaning in this limit. Only for those physical quantities for which \(h\) appears on the both sides of the equation, one gets the same results from classical and quantum mechanics, e.g. line splitting in anomalous Zeeman effect etc, or for quantities for which quantum nature is implicitly assumed, e.g. magnetic susceptibility etc.