Bosons and Fermions: Indistinguishable Particles with Intrinsic Spin

If we imagine a Hamiltonian, H^(r1,r2), describing two identical particles at positions r1 and r2 and then we interchange the particles, the Hamiltonian will be unaffected, i.e. H^(r1,r2)=H^(r2,r1). If we introduce an exchange operator P^r1,r2 such that P^r1,r2H^(r1,r2)=H^(r2,r1)P^r1,r2=H^(r1,r2)P^r1,r2, we see that they commute, or [P^r1,r2,H^(r1,r2)]=0. We know then that P^r1,r2andH^(r1,r2) have common eigenfunctions. We can then easily show that the eigenfunctions of the exchange operator must be either even or odd. Experiments show that odd exchange symmetry corresponds to half-integer spin particles called fermions, while even exchange symmetry corresponds to integer spin particles called bosons. The notes then discuss the implications of the new postulate and then presents the Heitler–London theory and the Heisenberg exchange Hamiltonian which has been so successful in predicting molecular structure.

Cut-and-join structure and integrability for spin Hurwitz numbers

Spin Hurwitz numbers are related to characters of the Sergeev group, which are the expansion coefficients of the Q Schur functions, depending on odd times and on a subset of all Young diagrams. These characters involve two dual subsets: the odd partitions (OP) and the strict partitions (SP). The Q Schur functions $$Q_R$$QR with $$R\in \hbox {SP}$$R∈SP are common eigenfunctions of cut-and-join operators $$W_\Delta $$WΔ with $$\Delta \in \hbox {OP}$$Δ∈OP. The eigenvalues of these operators are the generalized Sergeev characters, their algebra is isomorphic to the algebra of Q Schur functions. Similarly to the case of the ordinary Hurwitz numbers, the generating function of spin Hurwitz numbers is a $$\tau $$τ-function of an integrable hierarchy, that is, of the BKP type. At last, we discuss relations of the Sergeev characters with matrix models.

Random walk on a sphere and on a Riemannian manifold

A random walk on a sphere consists of a chain of random steps for which all directions from the starting point are equally probable, while the length a of the step is either fixed or subject to a given probability distribution p(a). The discussion allows the fixed length a or given distribution p(<x), to vary from one step of the chain to another. A simple formal solution is obtained for the distribution of the moving point after any random walk ; the simplicity depends on the fact that the individual steps commute and therefore have common eigenfunctions. Results are derived on the convergence of the eigenfunction expansion and on the asymptotic behaviour after a large number of random steps. The limiting case of diffusion is discussed in some detail and compared with the distribution propounded by Fisher (1953). The corresponding problem of random walk on a general Riemannian manifold is also attacked. It is shown that commutability does not hold in general, but that it does hold in completely harmonic spaces and in some others. In commutative spaces, complete analogy with the method employed for a sphere is found.