An active transformation of a physical system implies its motion. A passive transformation of the system is a change in the method of describing it. An analogy transformation means transition to another physical system, similar in some respect to the original one. In some cases transformations of one type may imitate those of another. The operation of particle permutation in quantum mechanics implies a passive transition to describing the same state of the system by a different method of introducing particle numbering. The problems of transitions from describing identical particles to describing different ones and that of explaining the probabilistic meaning of wave functions statistically are touched upon.
An active transformation of a physical system implies its motion. A passive transformation of the system is a change in the method of describing it. An analogy transformation means transition to another physical system, similar in some respect to the original one. In some cases transformations of one type may imitate those of another. The operation of particle permutation in quantum mechanics implies a passive transition to describing the same state of the system by a different method of introducing particle numbering. The problems of transitions from describing identical particles to describing different ones and that of explaining the probabilistic meaning of wave functions statistically are touched upon.
If a system in atomic physics contains a number of particles of the same kind, e.g. a number of electrons, the particles are absolutely indistinguishable one from another. No observable change is made when two of them are interchanged … A satisfactory theory ought, of course, to count any two observationally indistinguishable states as the same state and to deny that any transition does occur when two similar particles exchange places. ( Dirac, 1958, p. 207 ) Introduction In our contribution to this volume we deal with discrete symmetries: these are symmetries based upon groups with a discrete set of elements (generally a set of elements that can be enumerated by the positive integers). In physics we find that discrete symmetries frequently arise as ‘internal’, non-spacetime symmetries. Permutation symmetry is such a discrete symmetry, arising as the mathematical basis underlying the statistical behaviour of ensembles of certain types of indistinguishable quantum particle (e.g. fermions and bosons). Roughly speaking, if such an ensemble is invariant under a permutation of its constituent particles (i.e. permutation symmetric) then one doesn't ‘count’ those permutations which merely ‘exchange’ indistinguishable particles; rather, the exchanged state is identified with the original state. This principle of invariance is generally called the ‘indistinguishability postulate’ (IP), but we prefer to use the term ‘permutation invariance’ (PI).
In classical physics, the question whether there are subsets of identical particles is not particularly interesting. After all, whether the particles are identical or not, we can always track each of them at any time. As we shall see, in quantum mechanics identical particles are indistinguishable because we cannot track trajectories, and this fact induces some important symmetry properties on the total wave function describing the system of particles under consideration. Generally speaking, a symmetry is an effect of an invariance, that is, the system does not change if one effects certain transformations (see Ch. 8). In this case, the system is not modified under exchange of two identical particles. Moreover, such a symmetry yields a most interesting connection between spin and statistics. After a short introduction to the significance of statistics in quantum mechanics (in Sec. 7.1), we define the permutation operator and discuss the symmetry properties of the wave function of N identical particles (Sec. 7.2). In Sec. 7.3 we show that quantum particles, differently from classical particles (that obey the Maxwell–Boltzmann statistics), may be subject to two different statistics, namely the Fermi–Dirac statistics for fermions and the Bose–Einstein statistics for bosons. In Sec. 7.4 we treat the so-called exchange interaction and, finally, in Sec. 7.5 we consider some applications of the two statistics. Statistics and quantum mechanics Identical particles are those particles whose Hamiltonian is symmetric under exchange of one particle with the other.
Symmetries play a central role in physics. They dictate what one can change in a physical system without affecting any of its properties. You might have encountered symmetries like translational symmetry, where a system remains unchanged if it is spatially translated by an arbitrary distance. A system with rotational symmetry, however, is invariant under rotations. Some symmetries, like the ones mentioned above, give information about the structure of the system. Others have to do with the more fundamental physical framework that we adopt. An example for this is the invariance under Lorentz transformations in relativistic physics. Other types of symmetries can be even more subtle. For example, it is rather self-evident that physics should remain unchanged if we exchange two identical point-like particles. Nevertheless, this fundamental property that we call statistical symmetry gives rise to rich and beautiful physics. In three spatial dimensions it dictates the existence of bosons and fermions. These are particles with very different quantum mechanical properties. Their wave function acquires a +1 or a -1 phase, respectively, whenever two particles are exchanged. A direct consequence of this is that bosons can actually occupy the same state. In contrast, fermions can only be stacked together with each particle occupying a different state. When one considers two spatial dimensions, a wide variety of statistical behaviours is possible. Apart from bosonic and fermionic behaviours, arbitrary phase factors, or even non-trivial unitary evolutions, can be obtained when two particles are exchanged (Leinaas and Myrheim, 1977).
“Nothing changes” is used for three different operations in the treatment of identical particles. The operations are not equivalent, and the sentence changes meaning as it moves among them. Passive relabeling, A^'=P_12 A, changes only mathematical labels. In the standard quotient description, one physical situation is represented twice. The labels were never observables, and they supply no evidence that “electron A” and “electron B” are independently identifiable individuals. That does not forbid every metaphysics of primitive individuality. It makes such individuality empirically idle. Exchange symmetry is a different fact. For a state already in the fermionic subspace, P_12 ψ=-ψ. The isolated global sign is not itself a new pointer reading. Antisymmetry still constrains the allowed state space and has observable consequences, including exclusion. Active physical exchange is a process. A Hamiltonian can carry excitations along paths. Hong–Ou–Mandel and the Neder two-electron experiment show exchange-statistics interference in joint detection. Tschernig and collaborators directly measured the particle-exchange phase of photons by interfering a two-photon state with its physically permuted version. A braid is not a relabeling. In standard three-dimensional treatments the familiar sectors for observed elementary particles are bosonic and fermionic; parastatistics remains a formal complication. Anyonic statistics beyond +1/-1belong to braid structure in two-dimensional configuration spaces, typically effective quasiparticles, and do not reclassify ordinary electrons. The note draws two boundaries and refuses both overreaches. The same final invariant state does not imply the same physical process. A different process with no surviving record does not imply an additional present physical distinction. A difference of history counts as a present physical distinction only insofar as it remains written in the state, in a record, or in an observable relation. When no such mark survives, the final state does not contain an additional empirically discriminated distinction between the histories. This does not make the processes identical, and it does not make either process unoccur. This note introduces no modification of identical-particle quantum mechanics. Its claim is about inference. A statement valid for relabeling, for a symmetry transformation, or for physical transport cannot be carried from one operation to another without re-establishing what remains invariant. What this note is not It is not a correction of quantum mechanics. It is not a claim that electrons have hidden individual histories. It is not a claim that every metaphysics of primitive thisness is logically impossible. It is not a result about ordinary electrons in three dimensions ceasing to be fermions.