University of David
The principle of least action has been the workhorse of theoretical physics for decades. Both its high versatility and prodigious efficiency largely make up for the weakness of its foundations. In its many implementations two different paths can be broadly discerned. The classical approach (in newtonian or relativistic geometry) encompasses mechanics and the theory of fields, and provides sound basis to statistical mechanics and thermodynamics. There are many ways to address the lagrangian specification (Morrison [19 ], Soper ), but the key is to proceed quickly to the phase space, endowed with a symplectic structure, where all the mathematical tools can be efficiently deployed (Hofer ). The various Einstein-Vlasov equations are an example of this approach (Choquet-Bruhat ). On the other hand quantum mechanics and the quantum theory of fields make also an intensive use of the principle of least action, as an hamiltonian or lagrangian is required as starting point. The two main differences are that the distinction between particles (matter fields) and ”force fields” (bosonic fields) is blurred, and that the basic axioms of quantum mechanics (such as summed up by Weinberg ) and the Wigner theorem open the way to a more direct analysis of the equations. It is the only theory that gives us some predictions for the physical characteristics of the particles and how they change but, if there is no need to aknowledge its power, we are still left with one the biggest enigma of modern physics : ”Where does the first quantization come from ?”. As both the classical and quantum approaches lead, through Poisson brackets and the likes, to Banach algebras, one way to answer this question is to circumvent the principle of least action and go straight to C*-algebra. It is roughly what is attempted with the algebraic quantum field theory (Halvorson ). One issue is that in the simplest of physical system (1 spinless particle) the set of observables is not a C*-algebra…and anyway one is still far away from understanding the axioms of quantum physics.