Fock space from Q-sets for indistinguishable particles

G. DOMENECH, F. HOLIK Y D. KRAUSE

Gdansk, Polonia

Congreso; 9th Biennial IQSA Meeting "Quantum Structures Brussels-Gdansk '08"; 2008

Indistinguishable particles —like protons, electrons and the like— seem to violatesome characteristic properties of the classical identity relation. E. Schr¨odingercalled attention on the questions this poses to the notion of individual. In hiswords: “[...] we have [...] been compelled to dismiss the idea that [...] a particle isan individual entity which retains its ‘sameness’ forever. Quite the contrary, we arenow obliged to assert that the ultimate constituents of matter have no ‘sameness’at all.”In spite of these claims, in the quantum mechanical treatment of indistinguishableparticles, one first labels the particles and then imposes a symmetrizationcondition. Here we join the position that indistinguishability must be addressed bythe formalism “right at the start” and not made a posteriori by introducing symmetryconditions. Using a different mathematical framework, namely, quasi-settheory, we avoid working within a label-tensor-product-vector-space-formalism.To do so, we build a vector space with inner product to deal with indistinguishableelements. These vectors refer only to occupation numbers like the usualFock space formalism. Operators —which are intended to represent observablequantities— are obtained as combinations of creation and annihilation operators,in order to also avoid particle indexation in the expression of observables.Regarding experimental predictions, our formulation is equivalent to that ofstandard quantum mechanics. But when subjected to careful analysis, the conceptualdifference is important specially with respect to interpretational issues.(trabajo completo enviado para publicación)