Many-Particle Systems in Mathematics and Physics

Franco Strocchi

Mathematical Description of a Physical System Mathematical Description of a Quantum System The Quantum Particle Quantum Dynamics. The Schrodinger Equation Examples Quantum Mechanics and Stochastic Processes.

Emmanuel Meeron

H.-K. Chan

It is commonly said that exchanging two identical particles is equivalent to rotating one of them by one revolution. The present note shows that, in this equivalence, both the exchange and the rotation are purely mathematical operations—symbolic manipulations of the wave function, made during the design phase of an experiment. Neither occurs in the physical spacetime in which the experiment runs. While in general, “exchange” and “rotation” refer to physical processes, in the context of this equivalence, they do not. The key observation is that either one can return the system to its original state perfectly—which is possible only for a mathematical operation, not for a real physical process. At best, it is an idealized physical process—the limiting case of a real one. However, in this equivalence, those two mathematical operations cannot even be considered as limiting cases, because their very definitions are purely mathematical. A physical process—real or idealized—is a function of time. Neither exchange (as particle label-swap) nor rotation (as frame coordinate transformation) has the time variable. The very meaning of “process” is the set of intermediate steps. Therefore: no time, no process. They are proposed mathematical properties of possible physical worlds. Mathematical properties—once well-defined or proved—are always true within their formal system. However, those properties can be inapplicable in some worlds. Therefore, it is possible to use physical experiment to test whether they apply in this physical reality. But they are still not physical processes within any particular spacetime. While Euclidean geometry is mathematical, the statement “this world follows Euclidean geometry if we define straight lines as paths of light” is physical, because it is possible to be falsified by experiment. Still, the geometric assertions never refer to any physical processes. As the operations are purely mathematical, the equivalence itself is likewise mathematical. Moreover, since either operation can exist without the other, the equivalence is not a logical necessity, but a convention.

Paul C. Martin, Julian Schwinger

Franck Laloë

This chapter gives a summary of the mathematical formalism used in quantum mechanics, with a short bibliography put directly at its end. It should be seen as a complement, to be used by readers who wish to know more than what has been recalled about the mathematical tools in the other chapters. Some results are given without explicit proofs; they can be found for instance in Chapters II and IV of [1] . Many quantum mechanics textbooks introduce its general formalism in a more complete way, for instance Chapter VII of [2] , Chapter 3 of [3] , or Chapter 2 of [4] . We first summarize the general formalism of quantum mechanics for any physical system (§11.1), with the Dirac notation; we then study how this formalism treats the grouping of several physical systems into one single quantum system (§11.2); finally, we study a few simple special cases (§11.3), for instance a single particle in an external potential, with or without spin; the reader who prefers wave functions to more abstract reasonings in spaces of states may begin with this section. General physical system The general formalism of quantum mechanics applies to all physical systems, whether they contain a single particle, many particles of various sorts, one or several fields, etc. Quantum space of states The physical state of a system at each time is defined in quantum mechanics by a state vector which, in Dirac notation, is written ∣ ψ 〈 – or ∣ ψ(t) 〈 if one wishes to make the time dependence explicit.