Quantum indistinguishability

F. HOLIK

Utrecht

Conferencia; Coloquium Geschiedenis en Grondslagen; 2008

Institute for History and Foundations of Science (Utrecht University)

The concept of individuality in quantum mechanics shows radical differences from the concept of individuality in classical physics, as E. Schrödinger pointed out in the early steps of the theory. Regarding this fact, it has been suggested that quantum mechanics does not possess its own language because it makes use of the functional analysis based on set theory, and sets may be regarded as " [...] collections of definite and separate objects of our intuition or our thought ". Nevertheless, it is possible to represent the idea of quantum indistinguishability with a first order language using quasiset theory Q and then reobtain, using Q, the whole formalism of quantum mechanics. In this talk we intend to show the new formalism and defend its advantages. First we build a vector space with inner product, the Q-space, to represent the states of indistinguishable quanta (and also of distinguishable ones). Vectors in Q-space refer only to occupation numbers and permutation operators act as the identity operator on them, reflecting in the formalism the fact of unobservability of permutations. Operators that represent observable quantities are built as combinations of creator and annihilation operators, resembling the Fock-space formalism, in order to avoid particle indexation in their expressions. Creation and annihilation operators which act on Q-space obey the usual commutation and anticommutation relations for bosons and fermions respectively, and this means that our construction is equivalent to that of the Fock-space formulation of quantum mechanics. Thus, we can recover the n-particles wave equation of the standard theory.