Q-spaces and the Foundations of Quantum Mechanics

Abstract

Our aim in this paper is to take quite seriously Heinz Post's claim that the non-individuality and the indiscernibility of quantum objects should be introduced right at the start, and not made a posteriori by introducing symmetry conditions. Using a different mathematical framework, namely, quasi-set theory, we avoid working within a label-tensor-product-vector-space-formalism, to use Redhead and Teller's words, and get a more intuitive way of dealing with the formalism of quantum mechanics, although the underlying logic should be modified. We build a vector space with inner product, the Q-space, using the non-classical part of quasi-set theory, to deal with indistinguishable elements. 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. Thus, this paper can be regarded as a tentative to follow and enlarge Heinsenberg's suggestion that new phenomena require the formation of a new "closed" (that is, axiomatic) theory, coping also with the physical theory's underlying logic and mathematics.

Non-reflexive Logical Foundation for Quantum Mechanics

“…In some cases, as those referred to in the introduction to the paper, ≡ is really important, and identity, for some authors lose, at least in part, its meaning. However, here, we shall employ ZF R basically as a classical set theory, although one could try to handle ZF R, as a strict non-reflexive logic, to eliminate identity in connection with quantum particles (see, for example, [17] and also [1,10,15,23]). Therefore, one may say that ZF R is a kind of 'classical non-reflexive' logic.…”

Remarks on the Theory of Quasi-sets

“…Such collections can be used also to express semantic aspects of predicates which have not a well-defined extension (see [13]), 2 so as to enable a formal discussion on certain kinds of vagueness that appear in the quantum domain [11]. Recently, Domenech et al began a research program to express a quantum mechanics taking quasi-set theory as the background language [6,7], thus beginning the construction of a non-reflexive quantum mechanics. 3 In the final section, we shall speak a little bit about QFT and other developments.…”

“…Thus, we have modified the pair axiom to that one presented at section 3 and corrected the discourse concerning the way of expressing the limitations of the concept of identity accordingly. 6 All of this will be discussed below.…”

“…Concerning the possible applications of the theory for interpreting m-atoms as quantum objects and for the development of a quantum mechanics based on Q (as presented in [6]), nothing must be changed, for the modification in the pair axiom does not cause any new result; on the contrary, it avoids some undesirable counter-intuitive results as for instance that given two qsets A and B such that A ⊂ B and being all the elements of A indistinguishable m-atoms, then A × A = B × B, so as that, being x ≡ y ≡ z ≡ w, then 12 Replacement Axioms and the qset version of the Axiom of Choice may be taken exactly as in [12]. 13 But see [12, p.293].…”

Quantifiers and the Foundations of Quasi-Set Theory

“…So, within Q, we can express that 'permutations are not observable', without necessarily introducing symmetry postulates, and in particular to derive 'in a natural way' the quantum statistics (see Krause et al 1999;French & Krause 2006, chap.7). Further applications to the foundations of quantum mechanics can be seen in Domenech et al 2008. …”