4 From Classical Mechanics to Quantum Field Theory. A Tutorial
of a (non-zero) complex number^3. Thus the physical space of state is notHitself,
by the projective Hilbert spacePHwhich, despite not being a vector space any-
more, has a rich geometric structure that will be investigated in Subsect. 1.2.2.
More specifically, we will show how the Hermitean form onH, projects down to
PH, letting it to inherit both a symplectic and a Riemannian structure which
make it a K ̈ahler manifold.
In the last decades, more and more attention has been dedicated to phenomena
and applications in QM for which the geometric structure of the space of states
plays a fundamental role. This is the case of of the Aharonov-Bohm effect[ 1 ]and
other problems connected to adiabatic phases[4; 29]as well as other topics that
have recently caught the attention of the researchers, such as entanglement[7;
14 ]or tomography[ 26 ]. Geometric structures are also key ingredients to un-
derstand how one can start from a classical system and develop a procedure to
“quantize” it.
The discussion about the procedure which one can use to pass from the classical
to the quantum description of a given physical system accompanied the birth
of QM theory and cannot abstract from a discussion about what the principal
structures are that define what a classical or a quantum theory is. Since then,
different methods of quantization, that allow to find the quantum counterpart of a
classical physical problem, have been proposed and developed, which put emphasis
on different aspects of the quantum theory and use either analytical or algebraic
or geometrical techniques. The second part of this volume will focus on the logic
and algebraic structuresof quantum mechanics.
Following the seminal paper of Dirac[ 10 ], one may start by assuming that
quantum states are represented by wave-functions, i.e. -say in the coordinate
representation- by square integrable functions over the classical configuration man-
ifold Q= {q ≡ (q 1 ,···,qn)}, taken usually asRn: H =L^2 (Q)={ψ(q):
‖ψ(q)‖ 2 <∞}. Quantum observables are self-adjoint operators onH,sotohave
a real spectrum and admitting a spectral decomposition. This is the analogue of
what we do in a classical context, in which the space of states is given by the
phase spaceT∗Q={(p, q)≡(p 1 ,···,pn;q 1 ,···,qn)}and observables are real
(regular, usuallyC∞) functions on it^4 :f(p, q)∈R. Notice that both the space of
self-adjoint operators onHand of real regular functions onT∗Qare vector spaces,
actually algebra, on which we might want to assign suitable topologies.
(^3) This holds for the case of a pure state. In the case of a mixed state, the physical content is
encoded in a collection ofklinearly independent states, defined up to the action of the unitary
group 4 U(k). A pure state is recovered whenk=1.
WheneverT∗Q=R^2 n,(p, q) represents coordinates in a local chart.