# Presenting Nonreflexive Quantum Mechanics: Formalism and Metaphysics

Décio Krause Partially supported by CNPq. Jonas R. B. Arenhart

(Research Group in Logic and Foundations of Science
Department of Philosophy
Federal University of Santa Catarina
(Forthcoming in Cadernos de Hist ria e Filosofia da Ci ncia, Campinas - SP,
Brazil)
October 2015)

###### Abstract

Nonreflexive quantum mechanics is a formulation of quantum theory based on a
non-classical logic termed nonreflexive logic (a.k.a. ‘non-reflexive’). In
these logics, the standard notion of identity, as encapsulated in classical
logic and set theories, does not hold in full. The basic aim of this kind of
approach to quantum mechanics is to take seriously the claim made by some
authors according to whom quantum particles are non-individuals in some sense,
and also to take into account the fact that they may be absolutely
indistinguishable (or indiscernible). The nonreflexive formulation of quantum
theory assumes these features of the objects already at the level of the
underlying logic, so that no use is required of symmetrization postulates or
other mathematical devices that serve to pretend that the objects are
indiscernible (when they are not: all objects that obey classical logic are
individuals in a sense). Here, we present the ideas of the development of
nonreflexive quantum mechanics and discuss some philosophical (mainly
metaphysical) motivations and consequences of it.

##  1 Introduction

Indiscernibility of physical systems (or physical objects) appear already in
classical physics. For instance, in 1808 John Dalton, the founder of modern
atomism, claimed that there would be no differences among chemical elements of
a same kind: “we may conclude that the ultimate particles of all homogeneous
bodies are perfectly alike in weight, figure, etc.” [6, p.142-3]. In 1900, Max
Planck, still reasoning within the scope of classical physics, made the
fundamental move to quantum mechanics by assuming that in distributing an
integral number $N$ of energy elements over $C$ states, the indiscernibility
of the elements should be taken into account. Planck’s formula

$Z=\frac{(N+C-1)!}{N!(C-1)!}$

shows that the division by $N!$ makes the trick. Some years before, the so-
called Gibbs paradox appeared in classical statistical mechanics showing that
indiscernible particles are not only part of the physics, but also of the
philosophical problem concerning the proper understanding of how an
appropriate metaphysics for physical theories would look like. Indeed, should
we take the natural approach suggested by the Sackur-Tetrode formula for the
entropy of an ideal gas and accept that nature made particles
indistinguishable and that there is no more talking about the topic? Or should
we adopt a minimalist approach and concede that indistinguishability may help
us in drawing the line between classical and quantum particles (see French and
Krause [10, chap.2])? The first route would blur the difference between
classical and quantum particles, both being absolutely indistinguishable in
some cases. In the second case, we may argue that classical particles are
different from their quantum counterparts for they may be considered to be
individuals and distinguishable in some sense, while the matter is not so
easily settled for quantum particles, which, due to their indiscernibility,
may be taken as some kind of non-individuals.

The word ‘non-individuals’ has an historical use. Several forerunners of
quantum theory have referred to quantum objects this way, mainly in the sense
that quantum particles “are not individuals”, as suggested by Schr dinger [21,
p.206]. But the term ‘non-individual’ may confuse the reader in suggesting
that these entities would be something different from that which we can refer
to or speak about. In fact, we don’t know in precise terms what they are. Each
theory gives its own characteristics to these entities (B. Falkenburg traces
the ‘metamorphoses’ of the term ‘particle’ from classical physics to the most
advanced physical theory of today [8, Chap.6]). Being non-individuals does not
prevent quantum particles from being able to be isolated by some time, but
that they cannot be regarded as individuals in the standard sense of the word,
as something that have an identity and can be identified in other times even
if mixed with others of similar species. As it is well known, this does not
happen with quantum particles. It is in this sense that we say that all
protons are absolutely indiscernible, so are all electrons, all neutrons, and
do on. Hence, all atoms of the same chemical isotope, made from protons,
neutrons and electrons, are also indiscernible. Without this hypothesis,
chemistry does not work, and quantum physics does not work either.

We shall take the second approach mentioned earlier and concede that there is
a significant difference between quantum and classical particles. In classical
mechanics we can at least in principle follow well-defined trajectories for
each particle and by adding a reasonable assumption concerning their
impenetrability we may grant that some form of spatio-temporal principle of
individuality holds. When we turn to quantum mechanics, on the other hand,
things get much more complicated; in fact, for quantum particles there is no
trajectory in the classical sense, unless a hidden-variables approach is
followed, which has its own shortcomings; the impenetrability assumption is
also unavailable, for it may happen that particles are in entangled states.
More than that, quantum mechanics encapsulates indistinguishability in the
statistical treatment of aggregates of particles through the famous Bose-
Einstein and Fermi-Dirac statistics. These are seen as grounding the fact that
in some cases nothing whatsoever may distinguish quantum particles, for as is
famously remarked, permutations of particles do not count in this statistics
as giving rise to different states.

One of the difficulties with this fact — quantum indistinguishability as
obtained through the statistics of the theory — concerns precisely the
relationship between the experimental fact that is being expressed and its
mathematical formulation. Experimentally, it is found that there are
situations in which one cannot distinguish between two particles of the same
kind, as the theory predicts; mathematically, though, since the formalism of
the theory is constructed within classical set theory using a part of
functional analysis, the indistinguishability must be brought into the theory
through the assumption of some symmetry conditions that grant us the desired
effect. In a standard set theory, let us emphasize, whenever we have two
objects, they are necessarily discernible: they are distinct. Now, from a
philosophical point of view the assumption of these symmetry conditions seems
to be at odds with the idea of genuinely indistinguishable entities, for it
seems to be a mechanism to mask the fact that the entities dealt with are not
really indistinguishable, since they belong to classical logic and set theory
(see the discussions in Krause and Arenhart [14]).

The difficulty involving standard mathematics and logic and their relation to
‘genuine’ indistinguishability can be expressed in a nutshell as follows: on
the one hand, orthodox quantum mechanics says that there are situations
according to which we cannot discern the quanta; on the other hand, within
classical mathematics, we represent this fact using symmetry postulates, whose
aim is to mask the fact that in standard logic and mathematics one can always
know that wherever there are two items, they are discernible by purely logical
means inside set theory. We may even opt for a realistic interpretation
according to which, even conceding indistinguishability, the particles remain
there with their hidden properties, and even in this case there is no way to
tell them apart. So, concerning this situation, three options seem to be
available: (i) either the problem concerns the relationship between the
entities dealt with by QM and standard mathematics, so that those entities may
not sit comfortably together with assumptions made in classical mathematics,
or (ii) there is some kind of hidden variable which may confer individuality
to quantum entities, but, being hidden, is not taken into account by the
standard formalism, or (iii) the quanta must be treated mathematically as
certain entities which are discernible but their discernibility cannot be
expressed in qualitative terms by the vocabulary of the theory. Perhaps the
last two may be seen as just one option, but nothing in principle prevents us
from distinguishing them. In fact, (ii) allows that in principle we could
extend the formalism to accommodate the hidden variables accounting for the
individuality, something (iii) in this reading does not allow for; in this
last case the individuality is there but the theory does not allow us to
express how it is constituted.

All these options present their own challenges; after briefly discussing items
(ii) and (iii) in sections 2 and 3, we turn to the first one. This is perhaps
the place to say that we are not claiming that QM violates standard
mathematics and logic; our point is that it seems that this theory can be
formulated in different and non equivalent ways, most of them within these
‘classical’ frameworks. In saying that, we are not assuming that there is just
one QM, we take it as a fact that there are different formulations of
different theories which are empirically equivalent. What we intent to do is
to explore an alternative approach and look for the (at least conceptual)
gains we may have. As we shall see in section 4, there is an alternative
formulation/construction of quantum mechanics that incorporates non-
individuality right from the start and makes use of this fact in order to
develop the formalism of the theory. It is our goal here to render this
formulation more easily accessible and clear. Once that is done, the
philosophical view that quantum particles are not individuals (along with some
simple logical theory of quasi-set) is seen as giving rise to fruitful
developments of the theory itself, as an instance of a kind of reflexive
equilibrium procedure: the standard mathematical formulation and experiments
suggest that quantum particles are not individuals; however, that mathematical
apparatus does not reflect that non-individuality, so, we use that fact in
order to develop a mathematical formalism that takes non-individuality into
account; this formalism, now, gives rise to another version of quantum
mechanics. We discuss this view in section 5.

##  2 Indistinguishability as a quantum phenomenon

To better understand the reasons why quantum particles are considered to be
absolutely indistinguishable and how that absolute individuality is supposed
to lead to their non-individuality, we may start by comparing quantum and
classical statistics. This is the standard trope on this issue, and we believe
it is a good strategy to be followed, because since we have no access to the
particles themselves, it seems that their collective behavior may give us some
clues as to their identity and (in)distinguishability (see French and Krause
[10, chap.4]).

The first point to be noticed is that there is some sense in which both
classical and quantum particles are taken to be indistinguishable: when they
share all their intrinsic (or state independent) properties. Sometimes it is
common to call particles sharing all their intrinsic properties ‘identical
particles’. However, this is just an abuse of language to say that the
particles are of the same kind. The similarities among quantum and classical
stop there, however. The main feature behind the claim that classical
particles are individuals after all and that they are not really
indistinguishable in the same sense that quantum particles are said to be
concerns their collective behavior; the way that classical statistical
mechanics describes aggregates of classical particles differs significantly
from the way quantum mechanics describes aggregates of quantum particles.

Classical particles behave collectively according to Maxwell-Boltzmann
statistics (MB). Let us illustrate it by considering the particular case in
which there are two particles, labeled $1$ and $2$, to be distributed in two
different states, $A$ and $B$. Then, according to MB statistics, we can have
the following cases:

  1. 1.

A(1) and A(2);

  2. 2.

B(1) and B(2);

  3. 3.

A(1) and B(2);

  4. 4.

A(2) and B(1).

Assuming the equiprobability hypothesis — the claim that none of these cases
has more chances to occur than the others (a claim not without its
difficulties, see French and Krause [10, chap. 2]) — implies that all these
possibilities are assigned the same weight, that is, $\frac{1}{4}$, and
despite being considered as indistinguishable in the sense discussed above,
permutations of particles, as it happens in the cases 3 and 4, are counted as
giving rise to distinct possibilities. Here lies the key where the alleged
difference between classical and quantum particles concerning identity and
individuality matters.

The fact that permutations should bee seen as giving rise to different
situations is generally taken to witness for their individuality. In fact,
even though the particles share all their intrinsic properties, this simple
fact seems to speak in favor of some individuality being manifested by the
particles as they are permuted. Their individuality accounts for the
difference in cases 3 and 4. Another way to put it is in terms involving state
descriptions as describing a possible world: a permutation of particles in
case 3 leads to a distinct possible world, one in which it is particle 2 that
is in A and 1 in B. What accounts for the differences in the two worlds? Not
the properties of the particles, assuming them indiscernible. So, it must be
the case that they have an individuality intrinsic to them that is responsible
for this possibility. That individuality can be understood in terms of some
underlying substratum, a haecceity, or even more simply in terms of the
trajectories (spatio-temporal location) with the impenetrability assumption of
which we have already spoken about earlier (see, for instance, Ladyman [16]
for a discussion in those terms).

In quantum mechanics, on the other hand, nothing like this happens. To keep
with the case of two particles labeled once again $1$ and $2$ distributed in
states $A$ and $B$, we have two cases, one for bosons and other for fermions.
For the bosons, the following possibilities obtain:

  1. 1.

A(1) and A(2);

  2. 2.

B(1) and B(2);

  3. 3.

A(*) and B(*).

Again assuming equiprobability for each of these cases, each of them will have
the same probability of $\frac{1}{3}$ to occur. Notice that in case 3 we have
employed $*$ instead of the particles labels; that accounts for the
distinctive feature of quantum statistics: the fact that permutations do not
give rise to a distinct situation. The $*$ then means one particle in $A$ and
one in $B$. The same idea can be used for Fermi-Dirac statistics, the one
employed for fermions:

  1. 1.

A(*) and B(*).

For fermions, as for bosons, permutations change nothing, but differently from
bosons, the fermions obey the famous Pauli Exclusion Principle, which roughly
speaking grants that no two fermions can occupy the same state; thus there is
only one option in this particular case. The great novelty of quantum
statistics comes from the fact that permutations do not give rise to distinct
states. In terms of the description involving possible worlds above, a
permutation does not give rise to a distinct world. Contrarily to the
classical case, there is nothing in the particles to ground the fact that the
situation before the permutation is distinct from the situation after the
permutation. Thus, they are the same situation. As a consequence, there is no
intrinsic individuality principle doing that job.

Obviously, if that kind of reasoning is going to ground the claim that quantum
particles have no individuality and are really indistinguishable, as we wish
it to do, then it must be recognized that our language is inadequate to
express that fact, for we began by labeling the particles, which amounts to
individuate and identify them (something we cannot do for absolutely
indistinguishable items), and then we were forced to ‘forget’ the labels and
place an $*$ in their place to neutralize the fact that if a label was allowed
in the counting we would have intuitively the same classical statistics given
by Maxwell-Boltzmann. Now, this is precisely the conundrum between the
“nature” of quantum particles and the language we use to talk about them. To
deal more properly with this topic, let us see how the formalism of quantum
mechanics deals with the statistics. This will help us to appreciate the
relevance of the alternative formulation of quantum mechanics we introduce
later.

##  3 The standard way to deal with indistinguishable objects in QM

We would like to emphasize that in standard approaches to the formalism of
quantum mechanics, given inside some classical set theory, quanta are
individuals (as entities represented either by sets or by ur-elements) whose
individuality is veiled by symmetry principles that are added by hand. In
fact, the standard formalism of quantum mechanics masks the individuality of
the particles through the use of a mathematical trick: one begins with
labelled individuals (particles 1 and 2, say) and by convenient manipulation,
a clever postulate mimics the behavior of indiscernible non-individuals, say
by means of symmetric and anti-symmetric vectors/functions. The price to be
paid for the use of such a manœuvre is mainly philosophical, since physics
works for all practical purposes: one wants to be committed with an ontology
of non-individuals, but in the end, all we are left with are the same old
individuals we started with, clothed as non-individuals by the symmetry
postulates asserting the invariance of their permutability. Let us see
precisely how it is done, for when we know what we wish to avoid, it is then
easier to fully appreciate what is gained by the use of the
$\mathfrak{Q}$-spaces introduced below. Also, this way of approaching quantum
mechanics, through the use of labels (we don’t know how to proceed without
them within such standard mathematical frameworks) is one of the signs of
individuality.

Consider the quantum statistics mentioned above. To each particle $1$ and $2$
we associate a Hilbert space $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$
respectively. If the particles are indiscernible, these spaces are in fact the
same for both. For the system composed by both particles we associate the
tensor product $\mathcal{H}_{1}\otimes\mathcal{H}_{2}$. As is usual, we write
a typical vector of this space by $|\psi_{1}\rangle\otimes|\psi_{2}\rangle$,
or simply $|\psi_{1}\rangle|\psi_{2}\rangle$ for short. Then, the
possibilities available for the particles $1$ and $2$ in states $A$ and $B$
are stated as the following:

  1. 1.

$|\psi_{1}^{A}\rangle|\psi_{2}^{A}\rangle$

  2. 2.

$|\psi_{1}^{B}\rangle|\psi_{2}^{B}\rangle$

  3. 3.

$\frac{1}{\sqrt{2}}(|\psi_{1}^{A}\rangle|\psi_{2}^{B}\rangle\pm|\psi_{2}^{A}\rangle|\psi_{1}^{B}\rangle)$.

We are here simplifying the notation: the third case splits in two, according
to whether we deal with bosons or fermions; for bosons, we have cases 1-3 with
a plus sign in the last one (symmetric vector), for fermions we only have the
third possibility, now with the minus sign (anti-symmetric vector). What
really matters is that to obtain the correct statistics one must employ
symmetric and anti-symmetric vectors. This is the first part of the trick:
label the particles and write the appropriate states, that is, symmetrical for
bosons and anti-symmetrical for fermions. This means in particular that the
asymmetrical states $|\psi_{1}^{A}\rangle|\psi_{2}^{B}\rangle$ and
$|\psi_{2}^{A}\rangle|\psi_{1}^{B}\rangle$, like the ones employed in
classical statistics, are not taken in consideration.111They constitute
surplus formal structures, according to a terminology introduced by M.
Redhead, that is, elements of the formalism corresponding to nothing in the
real world (see [23, p.25]). Notice that the use of these states allows us to
distinguish between the particles: we are able to attribute specific states to
each one of them, although indiscernible (this is one way to substantiate the
claim that quantum particles are indiscernible individuals; see French and
Krause [10, chap.4] for the details).

But how can we make sure that only the states with the appropriate symmetry
types are available? That is the second part of the trick, and it consists in
postulating that only the appropriately symmetrized states obtain. The famous
Indistinguishability Postulate (IP) does just this job:

$\langle\psi_{12}|\hat{O}|\psi_{12}\rangle=\langle
P\psi_{21}|\hat{O}|P\psi_{21}\rangle.$ (1)

What this principle says, roughly speaking, is that the result of a
measurement before and after a permutation of the labels of particles results
in the same quantity, so, permuting the labels amounts to no physical
difference at all. In fact, a more technical reading of IP says that only
observables compatible with the permutation operators $P$ (the ones whose
purpose is to permute the labels given to the particles) are allowed. This
amounts to the following relation:

$[\hat{O}P,P\hat{O}]=\hat{O}P-P\hat{O}=0$ (2)

As one can check, only symmetrical and anti-symmetrical states satisfy these
conditions. There is a whole discussion on what are the grounds to postulate
IP and to accept it, and the different readings of this principle that may be
compatible with it; in one of them, for example, which makes it weaker, the
asymmetric states are not banned, but remain simply inaccessible to the
particles (as we mentioned previously). In this last case, the particles may
be regarded as individuals of some kind, but we shall not enter into these
discussions here (but see French and Krause [10, chap.4] for further
discussions). We just remark that this possibility shows that quantum
particles can be considered as individuals too! (But, as we are trying to
show, this is not the better way to see them).222In fact, quantum physics also
can (and is) be developed within a standard set theory which comprises
individuals; as we have already remarked, physics works fine, but the
philosophical problems turn to light.

So, this is in short how the trick is done in the standard formalism of
quantum mechanics: to grant that permutations of particles do not give rise to
different states, that is, to grant that the particles are absolutely
indiscernible, one begins by labeling them in the first place, and then, by
using only certain symmetrized vectors and by imposing a condition that only
these vectors represent physical states, we achieve the desired result.

Each of these steps has its own difficulties for the philosopher interested in
a commitment with ‘real’ indistinguishable entities, in particular for a
defense of a metaphysics of nonindividuals. Really, to begin with, by assuming
that there is some sense in labeling particles, we may consider them to have
well defined identity conditions, for one of them was labeled $1$ and _the
other_ was labeled $2$. This idea runs counter to the whole enterprise being
pursued, that is, that the particles must be really indiscernible. As they are
generally understood, labels seem to demand a semantic theory committed with
individuals and identity.

Here it would be interesting to comment that even the Fock space formalism
(which will be deal with below) is not completely free from initially attached
labels, as Redhead and Teller have claimed (see [23]). In fact, in their usual
construction, they begin with labeled Hilbert spaces (spaces $\mathcal{H}_{1}$
and $\mathcal{H}_{2}$, as we have been discussing); so, even if in the final
presentation of a Fock space the labels are not mentioned, we need labels in
Hilbert spaces for the usual constructions of the Fock spaces. The trick is
not, after all, dispensed with! This criticism justifies our approach in the
next section to construct Fock spaces inside a nonreflexive system of logic in
which none of the mentioned tricks is required. We begin with indiscernible
objects and build from there.

Note: — in fact, for $n$ indistinguishable particles, one takes the same
Hilbert space $n$ times. But this does not eliminate the trick, for we still
work with vectors of the form $|n_{1},n_{2},\ldots,n_{k}\rangle$, representing
a state with $n_{i}$ quantum particles with a certain eigenvalue in a certain
state (which would attest only to the quantity of them but not their
individuality). In pursuing the details, however, we need to go back to sets
and individuals. There is no scape within standard mathematics!

##  4 The $\mathfrak{Q}$-spaces

In order to avoid the (even hidden) assumption that we need to start with
individuals, we introduce the notion of $\mathfrak{Q}$-spaces. We begin with
an outline of quasi-set theory, the framework within which we develop our
nonreflexive quantum mechanics.

###  4.1 Basic quasi-set theory $\mathfrak{Q}$

Quasi-set theory is a ZFU-like first-order set theory comprising two kinds of
atoms:333We work with atoms because it seems to be more intuitive to regard
physical objects not as sets. As is well known, atoms are entities that can
belong to sets, but which have no elements (in the sense of the membership
relation), yet they could be composed of other entities in a mereological
sense. The problem is that a quantum mereology was not developed yet –
regarding mereology, see [22]; concerning some problems related to a quantum
mereology, see [12]. the M-atoms, which behave like ZFU atoms and are denoted
in the language with the help of a unary predicate $M$, and the m-atoms, which
satisfy the unary predicate $m$. For the last kind of atoms identity is not
defined, for according to our intuitive interpretation of the theory, we
attribute to them the role of quantum particles: they must be not only
indistinguishable, but more than that, identity statements must not make sense
for them. To grant that both goals are achieved, that is, that there is one
kind of objects that can be related by an indistinguishability relation and be
such that identity and difference statements do not make sense for them, we
must in $\mathfrak{Q}$ depart from some of the original ZFU features. Let us
check briefly how it can be done (for more details, see [10, chap.7], [11]).

Note: — before we begin with the theory, let us make another general remark.
Why to advance the thesis that the notion of identity is senseless for the
objects denoted by the m-atoms? The problem is that, if we assume that they
obey the traditional theory of identity of classical logic (and set theory),
once we have two of them, they are necessarily different. Well, in this case,
they must have some difference, given by a property (since we are, as usual,
avoiding to suppose notions of substratum)! Well, either the physical theory
presents this property or not. In the first case, the entities could not be
indiscernible; in the second case, the difference would be hidden in the
underlying logic, but even so it needs to be regarded as existing. Then we
have again two cases: either logic (or set theory) shows us the property or it
does not. In both cases the entities turn to be individuals — we remark that
even in standard mathematics there are entities (real numbers, say) that are
different but whose difference cannot be described by the theory (an example
would be the two least elements of two disjoint sets of real numbers)
according to a given well-order, which can be assumed to exist in the presence
of the axiom of choice, but which cannot be described by a formula of the
theory of sets).

Let us begin with the underlying logic of $\mathfrak{Q}$: it is first-order
logic _without_ identity. From a syntactical point of view, the postulates of
this logic are the same as those of classical logic, but semantically they are
understood differently. To understand this restriction one needs only to
remember that classical semantics is developed inside a classical set theory
like ZFC. If this kind of semantics were employed to the underlying logic of
$\mathfrak{Q}$, then we would be committing ourselves with precisely those
features of classical logic we want to avoid, namely, that identity makes
sense for every object and indistinguishability relations collapse in
identity. Quantifier must be understood with some reservation: $\forall
x\varphi(x)$ is to be not read as ‘for each object of the domain’, which would
entail their identification, but simply as ‘for all objects of the domain’,
meaning precisely this: for all objects of the domain. Similarly, $\exists
x\varphi(x)$ means that some object of the domain satisfies the formula
$\varphi(x)$, and we need not to identify it, say by a name; for details, see
[1],[2].

Besides atoms, the theory is designed to grant that we can build collections.
By definition, a quasi-set (q-set for short) is something that is not an atom,
and we distinguish them for they obey a primitive unary predicate $Q$; that
is, $Q(x)$ indicates that $x$ is a q-set. Such collections are assumed to be
formed in stages, as it happens in the classical cumulative hierarchy; for
some of these collections, in some step m-atoms may be involved as member, but
for others it may happen that no m-atom appear in any step of the formation of
the q-set. To designate this last kind of q-set we introduce the primitive
predicate $Z$, denoting what we call a classical q-set, or simply a set, and
they turn to be copies of the sets of ZFU. Technically speaking, sets are
collections with no m-atoms in their transitive closure. We call the M-atoms
and sets classical things of the theory. When restricted to the classical
things, $\mathfrak{Q}$ allows us to develop all the standard mathematics that
can be developed inside ZFU.

Identity must be introduced by definition. This fact allows us to manufacture
an identity relation restricted to q-sets and M-atoms only, but not holding
among m-atoms (to fit the intuitive interpretation we have attributed them).
The definition is the following:

###### Definition 4.1

$x=y:=(Q(x)\wedge Q(y)\wedge\forall z(z\in x\leftrightarrow z\in
y))\vee(M(x)\wedge M(y)\wedge\forall z(x\in z\leftrightarrow y\in z))$.

According to the definition, equal q-sets are those that have the same
members, and equal M-atoms are those that belong to the same q-sets. If there
are m-atoms involved, since the notion of identity does not make sense to
them, it may be the case that we are unable to effectively know whether two
q-sets are identical, for we have no means to identify their elements. But
things here work in the conditional: if two q-sets have the same elements,
then they are identical and reciprocally.444We recall Bertrand Russell’s
definition of pure mathematics (and $\mathfrak{Q}$ is the pure counterpart of
our quantum mechanics) as the class of propositions of the form ‘$p$ implies
$q$’, given in his The Principles of Mathematics.

One can easily prove from this definition that identity is reflexive, and the
substitution law is postulated as an axiom; then, identity (where it holds)
has both properties that characterize it in first-order systems. What about
indistinguishability? We postulate that it has the properties of an
equivalence relation (i.e. it is reflexive, symmetric and transitive), but it
is not compatible with the membership relation, that is, from $x\equiv y$ and
$x\in z$ we cannot always derive that $y\in z$; this is done in order to grant
that identity and indistinguishability do not collapse in the same concept.
For the other properties $P$ of the language, though, we have that if $x\equiv
y$ and $P(x)$, then $P(y)$. Also, for the classical things, identity and
indistinguishability are equivalent.

The construction of q-sets proceeds in much the same way as in ZFU: by the
iterated application of well-known set theoretical operations as power set,
union, cartesian product, among others. One difference that is worth
mentioning concerns the notion of unordered pair; in $\mathfrak{Q}$ we cannot
proceed as in the usual set theories, postulating that for every $x$ and $y$
there is a collection of objects equal to either $x$ or $y$, and the reason is
simple: for m-atoms identity is not defined, so that to restrict ourselves to
the classical definition would amount to the impossibility of pairs of
m-atoms, something undesirable in a theory of collections. To mend the
situation, the theory encompasses an unordered pair axiom stating that for
every two items $x$ and $y$ there is a q-set $z$ containing them as elements
(as in the classical case, there may be other elements in $z$ as well). We
then apply the separation axiom (which has its quasi-set theoretical version
similar to the classical formulation) to a q-set $z$ and separate a ‘pair’, in
fact, which is the q-set of the elements of $z$ containing only those elements
that are indistinguishable from either $x$ or $y$. This collection is denoted
$[x,y]_{z}$; if $x\equiv y$ holds, then we write simply $[x]_{z}$ and call it
the weak singleton of $x$. Notice that, intuitively speaking, the quantity of
elements in weak pairs and weak singletons may be higher than 2 and 1
respectively, for $z$ may contain many elements indistinguishable from $x$
and/or $y$.

From unordered pairs we can then define a weak version of ordered pairs as
follows, which stands for the q-set of the items indistinguishable from $x$
and from $y$ that belong to $z$:

######  Definition 4.2 (Weak ordered pair)

$\langle x,y\rangle_{z}:=[[x]_{z},[x,y]_{z}]_{z}$

Of course, any binary relation (or q-relation as we shall see) involving only
m-atoms is symmetric for $\langle x,y\rangle_{z}=\langle y,x\rangle_{z}$.
Then, the definition of cartesian product follows as usual:

######  Definition 4.3 (Cartesian Product)

$z\times w:=[\langle x,y\rangle_{z\cup w}:x\in z\wedge y\in w]$

If there are only m-atoms involved, then $z\times w=w\times z$ for all $z$ and
$w$, as is easy to see. The theory has also a union axiom which works as in
standard set theories; so, we shall use the symbols $\cup$ and $\cap$ taken
then from granted from now on.

######  Definition 4.4 (Quasi-relation)

A q-set $R$ is a binary quasi-relation between q-sets $z$ and $w$ if its
elements are weak ordered pairs of the form $\langle x,y\rangle_{z\cup w}$,
with $x\in z$ and $y\in w$.

Now, for quasi-functions we cannot proceed as usual, for the classical concept
of function uses the notion of identity. To generalize the usual concept,
then, and allow that it applies also for q-sets of m-atoms, we require that a
mapping does not attribute different kinds of values to indistinguishable
arguments. Thus, we have:

######  Definition 4.5 (Quasi-function)

$f$ is a quasi-function among q-sets $A$ and $B$ if and only if $f$ is quasi-
relation between $A$ and $B$ such that for every $u\in A$ there is a $v\in B$
such that if $\langle u,v\rangle\in f$ and $\langle w,z\rangle\in f$ and
$u\equiv w$ then $v\equiv z$.

What this definition grants us is that a quasi-function maps indistinguishable
elements to indistinguishable elements. For classical objects, both quasi-
relations and quasi-functions coincide with the classical definitions. With
some restrictions, one can also define the concepts of injection, surjection
and bijection, but we shall not present those definitions here (see [10, chap.
7]).

Another important concept in $\mathfrak{Q}$ concerns the attribution of
cardinality to q-sets. Since collections having m-atoms as elements cannot be
well-ordered (for that concept also presupposes identity), one cannot proceed
in the usual ways to grant that every q-set will have a cardinal number
associated to it, for example, through the attribution of ordinals following
the well-known von Neumann definition. The usual strategy has been to adopt a
primitive concept of quasi-cardinal which generalizes the classical notion of
cardinal: in the classical part of quasi-set theory one builds cardinals, as
usual, and then, with the primitive quasi-function $qc$ one attributes
cardinals to every q-set, making sure that classical sets will have as their
quasi-cardinal (the cardinal attributed by $qc$) precisely the same cardinal
attributed to them in the classical part of the theory.

The notion of quasi-cardinal also allows us to obtain in $\mathfrak{Q}$
another interesting concept: the strong singleton. As we have seen, given any
object $x$ whatever we may obtain the weak singleton of $x$, a q-set which in
the case of $x$ being an m-atom may have quasi-cardinal greater than 1. The
strong singleton of $x$ relative to a q-set $w$, denoted by $\llbracket
x\rrbracket_{w}$ is simply a q-set containing an element that is
indistinguishable from $x$ which belong to $w$ and whose quasi-cardinal is
precisely 1. With the resources of $\mathfrak{Q}$ however, if $x$ is an m-atom
one cannot deduce that $x$ is precisely the only element of $\llbracket
x\rrbracket_{w}$, for identity must be used for that. So, we can reason within
$\mathfrak{Q}$ that we may have in $\llbracket x\rrbracket_{w}$ an m-atom
having determinate properties, without being able to specify which particular
one it is.

The following theorem expresses the invariance by permutations in
$\mathfrak{Q}$, a result we shall employ below555Recall the motivations for
our development of quantum mechanics inside $\mathfrak{Q}$: we want to use the
features of non-individuals that are encapsulated in quasi-set theory to
somehow obtain quantum theory:

######  Theorem 4.1 (Unobservability of Permutations)

Let $x$ be a finite q-set such that $\neg(x=[z]_{t})$ for some $t$ and let $z$
be an m-atom such that $z\in x$. If $w\in t$, $w\equiv z$ and $w\notin x$,
then there exists $\llbracket w\rrbracket_{t}$ such that

$(x-\llbracket z\rrbracket_{t})\cup\llbracket w\rrbracket_{t}\equiv x$

In words: two indiscernible elements $z$ and $w$, with $z\in x$ and $w\notin
x$, expressed by their strong-singletons $\llbracket t\rrbracket_{t}$ and
$\llbracket w\rrbracket_{t}$, are ‘permuted’ and the resulting q-set remains
indiscernible from the original one. The hypothesis that $\neg(x=[z]_{t})$
grants that there are in $t$ indiscernible elements from $z$ which do not
belong to $x$ (for details and the proof of the Unobservability of
Permutations theorem, see [10, pp.295,296]).

These are the central notions that will be important in our development of
$\mathfrak{Q}$-spaces, to which we now turn below.

###  4.2 Quasi-functions

From now on, we shall follow [7], where further details can be seen. We begin
with a q-set of real numbers $\epsilon=\\{\epsilon_{i}\\}_{i\in I}$, where $I$
is an arbitrary collection of indexes, denumerable or not. Since it is a
collection of real numbers, which may be constructed in the classical part of
$\mathfrak{Q}$ we have that $Z(\epsilon)$. Intuitively, the elements
$\epsilon_{i}$ represent the eigenvalues of a physical observable $\hat{O}$,
that is, they are the values such that
$\hat{O}|\varphi_{i}\rangle=\epsilon_{i}|\varphi_{i}\rangle$, with
$|\varphi_{i}\rangle$ the corresponding eigenstates. The fact that the
observables are Hermitian operators grants us that the eigenvalues are real
numbers, thus we are justified in assuming $\epsilon$ to be a set of real
numbers. Consider then the quasi-functions
$f:\epsilon\longrightarrow\mathcal{F}_{p}$, where $\mathcal{F}_{p}$ is the
quasi-set formed of all finite and pure quasi-sets (that is, finite quasi-sets
whose only elements are indistinguishable m-atoms). Each of these $f$ is a
q-set of ordered pairs $\langle\epsilon_{i},x\rangle$ with
$\epsilon_{i}\in\epsilon$ and $x\in\mathcal{F}_{p}$. From $\mathcal{F}_{p}$ we
select those quasi-functions $f$ which attribute a non-empty q-set only to a
finite number of elements of $\epsilon$, the image of $f$ being $\emptyset$
for the other cases. We call $\mathcal{F}$ the quasi-set containing only these
quasi-functions. Then, the quasi-cardinal of most of the q-sets attributed to
elements of $\epsilon$ according to these quasi-functions is $0$. Now,
elements of $\mathcal{F}$ are quasi-functions which we read as attributing to
each $\epsilon_{i}$ a q-set whose quasi-cardinal we take to be the occupation
number of this eigenvalue. We write these quasi-functions as
$f_{\epsilon_{i_{1}}\epsilon_{i_{2}}\ldots\epsilon_{i_{m}}}$. According to the
given intuitive interpretation, the levels
$\epsilon_{i_{1}}\epsilon_{i_{2}}\ldots\epsilon_{i_{m}}$ are occupied. We say
that if the symbol $\epsilon_{i_{k}}$ appears $j$-times, then the level
$\epsilon_{i_{k}}$ has occupation number $j$. For example, the notation
$f_{\epsilon_{1}\epsilon_{1}\epsilon_{1}\epsilon_{2}\epsilon_{3}}$ means that
the level $\epsilon_{1}$ has occupation number $3$ while the levels
$\epsilon_{2}$ and $\epsilon_{3}$ have occupation numbers $1$. The levels that
do not appear have occupation number zero. Another point to be remarked is
that since the elements of $\epsilon$ are real numbers, we can take the
standard ordering relation over the reals and order the indexes according to
this ordering in the representation
$f_{\epsilon_{i_{1}}\epsilon_{i_{2}}\ldots\epsilon_{i_{m}}}$. This will be
important when we consider the cases for bosons and fermions.

The quasi-functions of $\mathcal{F}$ provide the key to the solution to the
problem of labeling states. In fact, since we use pure quasi-sets as the
images of the quasi-functions, there is simply no question of indexes for
particles, for all that matters are the quasi-cardinals representing the
occupation numbers. To make it clear that permutations change nothing, one
needs only to notice that a quasi-function is a q-set of weak ordered pairs
(see definitions 4.2 and 4.5). Taking two of the pairs belonging to some
quasi-function, let us say $\langle\epsilon_{i},x\rangle$,
$\langle\epsilon_{j},y\rangle$, with both $x$ and $y$ non-empty, a permutation
of particles would consist in changing elements from $x$ with elements from
$y$. But, by the unobservability of permutations theorem 4.1, what we obtain
after the permutation is a q-set indistinguishable from the one we began with.
Remember also that a quasi-function attributes indistinguishable images to
indistinguishable items; thus, the indistinguishable q-set resulting from the
permutations will also be in the image of the same eigenvalue. To show this
point precisely, we recall that by definition $\langle\epsilon_{i},x\rangle$
abbreviates $[[\epsilon_{i}],[\epsilon_{i},x]]$,666We are leaving aside the
subindices in this notation. and an analogous expression holds for
$\langle\epsilon_{j},y\rangle$. Also by definition, $[\epsilon_{i},x]$ is the
collection of all the items indistinguishable from $\epsilon_{i}$ or from $x$
(taken from a previously given q-set). For this reason, if we permute $x$ with
$x^{\prime}$, with $x\equiv x^{\prime}$ we change nothing for
$[\epsilon_{i},x]\equiv[\epsilon_{i},x^{\prime}]$. Thus, we obtain
$\langle\epsilon_{i},x\rangle\equiv\langle\epsilon_{i},x^{\prime}\rangle$ and
the ordered pairs of the ‘permuted’ quasi-function will be indiscernible (the
same if there are no m-atoms involved). Thus, the permutation of
indistinguishable elements does not produce changes in the quasi-functions.

###  4.3 A Vector Space Structure

Now, we wish to have a vector space structure to represent quantum states. To
do that, we need to define addition and multiplication by scalars. Before we
go on, we must notice that we cannot define these operations directly on the
q-set $\mathcal{F}$, for there is no simple way to endow it with the required
structure; our strategy here is to define $\star$ (multiplication by scalars)
and $+$ (addition of vectors) in a q-set whose vectors will be quasi-functions
from $\mathcal{F}$ to the set of complex numbers $\mathbb{C}$. Let us call $C$
the collection of quasi-functions which assign to every $f\in\mathcal{F}$ a
complex number. Once again, we select from $C$ the sub-collection $C_{F}$ of
quasi-functions $c$ such that every $c\in C_{F}$ attributes complex numbers
$\lambda\neq 0$ for only a finite number of $f\in\mathcal{F}$. Over $C_{F}$ we
can define a sum and a product by scalars in the same way as it is usually
done with functions as follows.

###### Definition 4.6

Let $\gamma$ $\in\mathcal{C}$, and $c$, $c_{1}$ and $c_{2}$ be quasi-functions
of $C_{F}$, then

$(\gamma\star c)(f):=\gamma(c(f))$ $(c_{1}+c_{2})(f):=c_{1}(f)+c_{2}(f)$

The quasi-function $c_{0}\in C_{F}$ such that $c_{0}(f)=0$ for every
$f\in\mathcal{F}$ acts as the null element for the sum operation. This can be
shown as follows:

$(c_{0}+c)(f)=c_{0}(f)+c(f)=0+c(f)=c(f),\forall f.$ (3)

With both the operations of sum and multiplication by scalars defined as above
we have that $\langle C_{F},\mathbb{C},+,\star\rangle$ has the structure of a
complex vector space, as one can easily check. Some of the elements of $C_{F}$
have a special status though; if $c_{j}\in C_{F}$ are the quasi-functions such
that $c_{j}(f_{i})=\delta_{ij}$ (where $\delta_{ij}$ is the Kronecker symbol),
then the vectors $c_{j}$ are called the basis vectors, while the others are
linear combinations of them. For notational convenience, we can introduce a
new notation for the q-functions in $C_{F}$; suppose $c$ attributes a
$\lambda\not=0$ to some $f$, and $0$ to every other quasi-function in
$\mathcal{F}$. Then, we propose to denote $c$ by $\lambda f$. The basis quasi-
functions will be denoted simply $f_{i}$, as one can check. Now,
multiplication by scalar $\alpha$ of one of these quasi-functions, say
$\lambda f_{i}$ can be read simply as $(\alpha\cdot\lambda)f_{i}$, and sum of
quasi-functions $\lambda f_{i}$ and $\alpha f_{i}$ can be read as
$(\alpha+\lambda)f_{i}$. What about the other quasi-functions in $C_{F}$? We
can extend this idea to them too, but with some care: if, for example $c_{0}$
is a quasi-function such that $c_{0}(f_{i})=\alpha$ and
$c_{0}(f_{j})=\lambda$, attributing $0$ to every other quasi-function in
$\mathcal{F}$, then $c_{0}$ can be seen as a linear combination of quasi-
functions of a basis; in fact, consider the basis quasi-functions $f_{i}$ and
$f_{j}$, (this is an abuse of notation, for they are representing quasi-
functions in $C_{F}$ that attribute 1 to each of these quasi-functions). The
first step consists in multiplying them by $\alpha$ and $\lambda$,
respectively, obtaining $\alpha f_{i}$ and $\lambda f_{j}$ (once again, this
is an abuse, for these are quasi-functions in $C_{F}$ that attribute the
mentioned complex numbers to $f_{i}$ and to $f_{j}$). Now, $c_{0}$ is in fact
the sum of these quasi-functions, that is, $c_{0}=\alpha f_{i}+\lambda f_{j}$,
for this is the function which does exactly what $c_{0}$ does. One can then
extend this to all the other quasi-functions in $C_{F}$ as well.

###  4.4 Inner Products

The next step in our construction is to endow our vector space with an inner
product. This is a necessary step for we wish to calculate probabilities and
mean values. Following the idea proposed in [7], we introduce two kinds of
inner products, which lead us to two Hilbert spaces, one for bosons and
another for fermions. We begin with the case for bosons:

###### Definition 4.7

Let $\delta_{ij}$ be the Kronecker symbol and
$f_{\epsilon_{i_{1}}\epsilon_{i_{2}}\ldots\epsilon_{i_{n}}}$ and
$f_{\epsilon_{i^{\prime}_{1}}\epsilon_{i^{\prime}_{2}}\ldots\epsilon_{i^{\prime}_{m}}}$
two basis vectors (as discussed above), then

$f_{\epsilon_{i_{1}}\epsilon_{i_{2}}\ldots\epsilon_{i_{n}}}\circ
f_{\epsilon_{i^{\prime}_{1}}\epsilon_{i^{\prime}_{2}}\ldots\epsilon_{i^{\prime}_{m}}}:=\delta_{nm}\sum_{p}\delta_{i_{1}pi^{\prime}_{1}}\delta_{i_{2}pi^{\prime}_{2}}\ldots\delta_{i_{n}pi^{\prime}_{n}}.$
(4)

Notice that this sum is extended over all the permutations of the index set
$i^{\prime}=(i^{\prime}_{1},i^{\prime}_{2},\ldots,i^{\prime}_{n})$; for each
permutation $p$,
$pi^{\prime}=(pi^{\prime}_{1},pi^{\prime}_{2},\ldots,pi^{\prime}_{n})$.

For the other vectors, the ones that can be seen as linear combinations in the
sense discussed above, we have:

$(\sum_{k}\alpha_{k}f_{k})\circ(\sum_{k}\alpha^{\prime}_{k}f^{\prime}_{k}):=\sum_{kj}\alpha_{k}^{\ast}\alpha^{\prime}_{j}(f_{k}\circ
f^{\prime}_{j}),$ (5)

where $\alpha^{\ast}$ is the complex conjugate of $\alpha$. Now, let us
consider fermions. As remarked above in page 4.2, the order of the indexes in
each $f_{\epsilon_{i_{1}}\epsilon_{i_{2}}\ldots\epsilon_{i_{n}}}$ is
determined by the canonical ordering in the real numbers. Thus, we define
another “$\bullet$” inner product as follows, which will do the job for
fermions:

###### Definition 4.8

Let $\delta_{ij}$ be the Kronecker symbol and
$f_{\epsilon_{i_{1}}\epsilon_{i_{2}}\ldots\epsilon_{i_{n}}}$ and
$f_{\epsilon_{i^{\prime}_{1}}\epsilon_{i^{\prime}_{2}}\ldots\epsilon_{i^{\prime}_{m}}}$
two basis vectors, then

$f_{\epsilon_{i_{1}}\epsilon_{i_{2}}\ldots\epsilon_{i_{n}}}\bullet
f_{\epsilon_{i^{\prime}_{1}}\epsilon_{i^{\prime}_{2}}\ldots\epsilon_{i^{\prime}_{m}}}:=\delta_{nm}\sum_{p}\sigma_{p}\delta_{i_{1}pi^{\prime}_{1}}\delta_{i_{2}pi^{\prime}_{2}}\ldots\delta_{i_{n}pi^{\prime}_{n}}$
(6)

where: $\sigma_{p}=1$ if p is even and $\sigma_{p}=-1$ if p is odd.

This definition can be extended to linear combinations as in the previous
case.

###  4.5 Fock spaces using $\mathfrak{Q}$-spaces

We begin with a definition to simplify the notation. For every function
$f_{\epsilon_{i_{1}}\epsilon_{i_{2}}\ldots\epsilon_{i_{n}}}$ in $\mathcal{F}$,
we put:

$\alpha|\epsilon_{i_{1}}\epsilon_{i_{2}}\ldots\epsilon_{i_{n}}):=\alpha
f_{\epsilon_{i_{1}}\epsilon_{i_{2}}\ldots\epsilon_{i_{n}}}$

Note that this is a slight modified version of the standard notation. We begin
with the case of bosons.

Suppose a normalized vector $|\alpha\beta\gamma\ldots)$, where the norm is
taken from the corresponding inner product. Let $\zeta$ stand for an arbitrary
collection of indexes. We define
$a_{\alpha}^{\dagger}|\zeta)\propto|\alpha\zeta)$ in such a way that the
proportionality constant satisfies
$a_{\alpha}^{\dagger}a_{\alpha}|\zeta)=n_{\alpha}|\zeta)$. From this it will
follow, as usual, that:

$((\zeta|a_{\alpha}^{\dagger})(a_{\alpha}|\zeta))=n_{\alpha}.$

###### Definition 4.9

$a_{\alpha}|\ldots n_{\alpha}\ldots):=\sqrt{n_{\alpha}}|\ldots
n_{\alpha}-1\ldots)$

On the other hand

$a_{\alpha}a_{\alpha}^{\dagger}|\ldots
n_{\alpha}\ldots)=K\sqrt{n_{\alpha}+1}|\ldots n_{\alpha}\ldots),$

where $K$ is a proportionality constant. Applying $a_{\alpha}^{\dagger}$
again, we have

$a_{\alpha}^{\dagger}a_{\alpha}a_{\alpha}^{\dagger}|\ldots
n_{\alpha}\ldots)=K^{2}\sqrt{n_{\alpha}+1}|\ldots n_{\alpha}+1\ldots).$

Using the fact that $a_{\alpha}^{\dagger}a_{\alpha}|\zeta)=n_{\alpha}|\zeta)$,
we have that:

$(a_{\alpha}^{\dagger}a_{\alpha})a_{\alpha}^{\dagger}|\ldots
n_{\alpha}\ldots)=\sqrt{n_{\alpha}+1}K|\ldots n_{\alpha}+1\ldots).$

So, $K=\sqrt{n_{\alpha}+1}$.

Then, we have:

###### Definition 4.10

$a_{\alpha}^{\dagger}|\ldots n_{\alpha}\ldots):=\sqrt{n_{\alpha}+1}|\ldots
n_{\alpha}+1\ldots).$

From this definition, with additional computations, we obtain:
$(a_{\alpha}a_{\beta}^{\dagger}-a_{\beta}^{\dagger}a_{\alpha})|\psi)=\delta_{\alpha\beta}|\psi)$.
In our language, this means the same as

$[a_{\alpha};a_{\beta}^{\dagger}]=\delta_{\alpha\beta}I.$

In an analogous way, it can be shown that:

$[a_{\alpha};a_{\beta}]=[a_{\alpha}^{\dagger};a_{\beta}^{\dagger}]=0.$

So, the bosonic commutation relation are the same as in standard Fock space
formalism.

For fermionic states we use the antisymmetric product $\bullet$. We begin by
defining the creation operator $C_{\alpha}^{\dagger}$:

###### Definition 4.11

If $\zeta$ is a collection of indexes of non-null occupation numbers, then
$C_{\alpha}^{\dagger}:=\alpha|\zeta)$

If $\alpha$ is in $\zeta$, then $|\alpha\zeta)$ is a vector of null norm. This
implies that $(\psi|\alpha\zeta)=0$, for every $\psi$. It follows that systems
in states of null norm have no probability of being observed. Furthermore,
their addition to another vector does not contribute with any observable
difference. To take the situation into account, we define:

###### Definition 4.12

Two vectors $|\phi)$ and $|\psi)$ are similar if the difference between them
is a linear combination of null norm vectors. We denote similarity of $|\phi)$
and $|\psi)$ by $|\phi)\cong|\psi)$.

Using the definition of $C_{\alpha}^{\dagger}$ we can describe what is the
effect of $C_{\alpha}$ over vectors:

$(\zeta|C_{\alpha}:=(\alpha\zeta|.$

Then, for any vector $|\psi)$,

$(\zeta|C_{\alpha}|\psi)=(\alpha\zeta|\psi)=0$

for $\alpha\in\zeta$ or $(\psi|\alpha\zeta)=0$. Then, if $|\psi)=|0)$, then
$(\zeta|C_{\alpha}|0)=(\alpha\zeta|0)=0$. So, $C_{\alpha}|0)$ is orthogonal to
any vector that contais $\alpha$, and also to any vector that does not contain
$\alpha$, so that it is a linear combination of null norm vectors. So, we can
put by definition that $\vec{0}:=C_{\alpha}|0)$. In an analogous way, if
$\sim\alpha$ denotes that $\alpha$ has occupation number zero, then we can
also write $C_{\alpha}|(\sim\alpha)\ldots)=\vec{0}$, where the dots mean that
other levels have arbitrary occupation numbers.

Now, using our notion of similar vectors, we can write
$C_{\alpha}|0)\cong\vec{0}$ and $C_{\alpha}|(\sim\alpha)\ldots)\cong\vec{0}$.
The same results are obtained when we use $\cong$ and the sign of identity. By
making $|\psi)=|\alpha)$, we have
$(\zeta|C_{\alpha}|\alpha)=(\alpha\zeta|\alpha)=0$ in every case, except when
$|\zeta)=|0)$. In that case, $(0|C_{\alpha}|\alpha)=1$. Then, it follows that
$C_{\alpha}|\alpha)\cong 0$. In an analogous way, we obtain
$C_{\alpha}|\alpha\zeta)=\cong|(\sim\alpha)\zeta)$ when $\alpha\notin\zeta$.
In the case $\alpha\in\zeta$, $|\alpha\zeta)$ has null norm, and so, for every
$|\psi)$:

$(\alpha\zeta|C_{\alpha}^{\dagger}|\psi)=(\alpha\zeta|\alpha\psi)=0.$

It then follows that

$(\psi|C_{\alpha}|\alpha\zeta)=0,$

so that $C_{\alpha}|\alpha\zeta)$ has null norm too.

Now we calculate the anti-commutation relation obeyed by the fermionic
creation and annihilation operators. We begin calculating the commutation
relation between $C_{\alpha}$ and $C_{\beta}^{\dagger}$. We do that by
studying the relationship between $|\alpha\beta)$ and $|\beta\alpha)$. Let us
consider the sum $|\alpha\beta)+|\beta\alpha)$. The product of this sum with
any vector distinct from $|\alpha\beta)$ is null. For the product with
$|\alpha\beta)$ we obtain
$(\alpha\beta|[|\alpha\beta)+|\beta\alpha)]=(\alpha\beta||\alpha\beta)+(\alpha\beta||\beta\alpha)$.
By definition, this is equal to
$\delta_{\alpha\alpha}\delta_{\beta\beta}-\delta_{\alpha\beta}\delta_{\beta\alpha}+\delta_{\alpha\beta}\delta_{\alpha\alpha}-\delta_{\alpha\alpha}\delta_{\beta\beta}$.
This is equal to $1-0+0-1=0$.

The same conclusion holds if we multiply the sum $|\alpha\beta)+|\beta\alpha)$
by $(\beta\alpha|$. It then follows that $|\alpha\beta)+|\beta\alpha)$ is a
linear combination of null norm vectors, which we denote by $|nn)$, so that

$|\alpha\beta)=-|\beta\alpha)+|nn).$

Given that, we can calculate

$C_{\alpha}^{\dagger}C_{\beta}^{\dagger}|\psi)=|\alpha\beta\psi)=-|\beta\alpha|\psi)+|nn)=-C_{\beta}^{\dagger}C_{\alpha}^{\dagger}|\psi)+|nn).$

From this it follows that
$\\{C_{\alpha}^{\dagger};C_{\beta}^{\dagger}\\}|\psi)=|nn)$. We do not lose
generality by setting
$\\{C_{\alpha}^{\dagger};C_{\beta}^{\dagger}\\}|\psi)=0$. In an analogous way
we conclude:

$\\{C_{\alpha};C_{\beta}\\}|\psi)=0.$

Now we calculate the commutation relation between $C_{\alpha}$ and
$C_{\beta}^{\dagger}$. There are some cases to be considered. We first assume
that $\alpha\neq\beta$. If $\alpha\notin\psi$ or $\beta\in\psi$ then

$\\{C_{\alpha};C_{\beta}^{\dagger}\\}|\psi)\approx\vec{0}.$

If $\alpha\in\psi$ and $\beta\notin\psi$, assuming that $\alpha$ is the first
symbol in the list of $\psi$, then
$\\{C_{\alpha};C_{\beta}^{\dagger}\\}|\psi)=C_{\alpha}|\beta\psi)+C_{\beta}^{\dagger}|\psi(\sim\alpha))\cong-|\beta\psi(\sim\alpha))+|\beta\psi(\sim\alpha))=\vec{0}$.

Now, if $\alpha=\beta$ and $\alpha\in\psi$, then
$\\{C_{\alpha};C_{\alpha}^{\dagger}\\}|\psi)=C_{\alpha}|\alpha\psi)+C_{\alpha}^{\dagger}|\psi(\sim\alpha))\cong\vec{0}+|\psi)=|\psi)$.

If $\alpha=\beta$ and $\alpha\notin\psi$, then
$\\{C_{\alpha};C_{\alpha}^{\dagger}\\}|\psi)=C_{\alpha}|\alpha\psi)+C_{\alpha}^{\dagger}|\psi(\sim\alpha))\cong|\psi)+\vec{0}=|\psi)$.

In any case, we recover
$\\{C_{\alpha};C_{\alpha}^{\dagger}\\}|\psi)\cong\delta_{\alpha\beta}|\psi)$.
So, we can put

$\\{C_{\alpha};C_{\alpha}^{\dagger}\\}=\delta_{\alpha\beta}.$

It then follows that the commutation properties in $\mathfrak{Q}$-spaces are
the same as in traditional Fock spaces.

Using this formalism, we can adapt all the developments done in [18, Chap.7]
and [19, Chap.20] for the number occupation formalism. But here, contrary to
what happens in these books, no previous (even unconscious) assumptions about
the individuality of quantum objects is taken into account.

##  5 Foundational and metaphysical remarks

Now that a non-standard version of quantum mechanics was sketched inside
quasi-set theory, what are the main philosophical lessons that one can draw
from this whole enterprise? Are we supposed to substitute traditional quantum
mechanics by its nonreflexive version? The answer to the last question is an
obvious NO, or not necessarily. What is illustrated by this construction is
only that one can take a version of non-individuality seriously and, building
from there, to develop the formalism of quantum mechanics anew, without
labeling and individuality. We remark once again that in using standard
mathematics (standard set theory), we need to assume that the represented
objects are individuals at the start, and need to make some mathematical
tricks in order to keep them with something that mimic non-individuality. Once
indiscernible non-individuals are essential to quantum mechanics, our move
illustrates that one can assume non-individuality and indistinguishability as
central in a formalism and, along with some set theoretical tools compatible
with such non-individuality (i.e. quasi-set theory), derive a version of the
formalism of the theory.

This kind of move illustrates how one may profitably approach the cooperative
work between the study of foundations of science and metaphysics. Metaphysics
may be seen as contributing positively to the foundations of a scientific
theory like quantum mechanics, by suggesting that due to the main features of
the theory, we are dealing with a new category of entities, the non-
individuals. Metaphysicians are entitled to describe the main metaphysical
features that entities qualified as non-individuals could have. By searching
for the logic that adequately describes those entities we are led to the
development of $\mathfrak{Q}$-spaces. Quantum mechanics, on its own side,
contributes to the development of metaphysics. By being our most successful
theory, and by putting so much weight on indiscernible entities, it forces
metaphysicians to develop an adequate notion of non-individuals and
indiscernible entities compatible with the description provided by quantum
mechanics.

What we aim now is to present how we understand this kind of collaboration
between science and metaphysics in a more general picture. The following may
be seen as a rudimentary approach to how a productive relation between science
and metaphysics may be developed which does not focus only on science (such as
Ladyman and Ross do in [17]) and also does not rely so heavily on what
metaphysicians have already achieved (as it seems to be the case of French’s
approach in [9, chap.3]). We intend to present how a really collaborative
effort to understand quantum metaphysics, for instance, may proceed. This is
not achieved by adopting neither a ‘physics first’ approach, and nor by
providing a ‘metaphysics first’ approach. Both physics and metaphysics must be
cooperatively involved in the enterprise. The ‘dialectics’ of such a
cooperation may be roughly described as a case of a process aiming at a state
of Reflective Equilibrium. Let us see briefly how.

We begin, as usual, with an actually available formulation of our scientific
theory; in this case, quantum mechanics. In general there is an informal
interpretation of the theory. Anyway, the mathematics of the theory may give
us some clues as to how the entities being described behave. In general, we
simply talk about the entities described by the theory by transferring our
everyday categories of stable objects, properties, relations and things in
general to the domain being described by the scientific theory. What our most
successful theories in the beginning of the twentieth century have shown us is
that this transference is not always justified. General relativity, for
instance, demanded for a revision in our concepts of space and time, while
quantum mechanics demanded a revision of many other concepts as well, such as
stable objects possessing properties and having well defined identity
conditions (it is enough to recall some no-go theorems like Kochen-Specker’s
and Bell’s). That such is the case is, as is well-known, the main contention
of Ladyman and Ross in [17, Chap.1]: metaphysicians have not yet taken
seriously the lessons from modern physics.

However, after perceiving that a scientific theory demands a revision in some
fundamental concepts, what are metaphysicians supposed to do? In the case of
quantum mechanics, in the specific case of non-individuals, how can we
rigorously describe the situation? Well, we must first recognize that the
mathematical apparatus employed to describe the theory encapsulates some of
the items requiring revision, like labeling and identity. So, how should we
proceed? Just as suggested by French [9] with his viking approach to
metaphysics, we may check what the metaphysicians can say about the new kind
of entities. Have they already articulated a theory about the behavior of
those entities? If they have, do those entities, thus categorized, mesh well
with what the scientific theory says about the world? In the case of quantum
entities, it seems, as we have argued in sections 3 and 2, the metaphysical
ideas of non-individuals, of indiscernible classes of entities, do not fit
very well with some of the assumptions that are encoded in the standard
formalism of the theory. So, in this case, the search for an alternative
logical foundations is a first required step.

Our suggestion is that we could then use the information brought by the
scientific theory to develop a new logic, a logic that already encodes the
basic information about the entities dealt with. Nonreflexive logics do just
that, as we have briefly sketched in the beginning of section 4. So, we use
the actual formulation of a theory in order to discover what are the main
features of the entities being dealt with. In case those features demand a
revision of some of our most general categories, then we have at least two
options: either (i) we reformulate the theory in order to keep the traditional
categories as much as possible (as it happens for instance, in Bohmian
mechanics) while still retaining empirical adequacy, or else (ii) we try to
take those novelties seriously, at face value as it were, and look for the
best place to implement them. Given that the idea of non-individuals, as we
have been dealing with them, is a general category, related with identity, it
seemed that the underlying logic was the most appropriate place to introduce
that category. Also, that change of view proved to be very fruitful: we could
not only account for the non-individuality of quantum particles, but also show
how that non-individuality is able to generate a formalism of quantum
mechanics anew! Another way to account for the novelties, still concerning
this second case, is not to change the formalism or the logic, but to keep the
mathematics of the theory intact and provide for an interpretation of the
formalism that somehow accounts for the novelties of the theory. One of such
options is now known as wave function realism. The main idea behind this
interpretation seems to be that the wave function of the mathematical
formalism is the reality of which the theory is about. In this case, the very
concept of ordinary space is to be completely redefined, along with the
concept of three dimensional objects. Even though this is an interesting
approach, we shall not discuss it in what follows, keeping our focus on the
notion of non-individuality (for more on wave function realism, see the
discussion in the collection by Alyssa Ney and David Albert [20], [13]).

So, by reformulating the formalism of Fock spaces in terms of
$\mathfrak{Q}$-spaces, we are doing much more than merely paying lip service
the a revisionary form of metaphysics: we show, somehow, that the concept of a
non-individual is fundamental in at least one understanding of quantum
mechanics. This is different from, for instance, the claim that some kind of
paraconsistent logic may be useful for our understanding of quantum
superpositions (as advanced in [5]; see also the discussion in [3]). In this
case, one interprets a physical fact in logico-metaphysical terms. Our claim
is that the nonreflexive foundations of quantum mechanics developed here go
one step further: we do not merely take the claims of quantum theory and re-
interpret them in a non-classical logic system, but we begin with the non-
classical system and inside it we develop the scientific theory itself. So, by
reading some of the main features of the entities dealt with by the theory, we
are able to use now those very features as the cornerstones for a new
formulation of the theory, one that takes those features as central. This move
closes a kind of circle: from a standard formulation of the theory we obtain
some metaphysical feature of the entities. By laboring over those features, we
develop a metaphysical approach to them that is rigorous enough to be
encapsulated by a system of logic. Finally, by employing that system of logic
we are able to develop a formalism that is able to express the theory.

Perhaps we could describe the kind of procedure developed here with the help
of an analogy with the situation in foundations of mathematics, as presented
by Kunen [15, p.191]. In discussing the foundations of mathematics, Kunen
claims that logic must be developed twice: we begin with our intuitions and
intuitive rules of inference and, based solely on them, develop an informal
set theory. Obviously, some logic is required in order to develop a set
theory, and in the case of standard set theories this logic is informal
classical logic. Now, inside the set theory we have just developed, we can
begin anew and develop, in a completely rigorous way, _using set theoretical
tools_ , our very system of logic. Now, inside such a set theory can we make
sure that a reasonable model theory, for instance, is available. So, logic
must be built twice: first, as an informal system, capable of grounding the
development of a set theory and such that it does not presuppose set
theoretical concepts in its formulation. Then, inside the just developed set
theory, we can develop, as a set theoretical entity, logic itself. In the
second case, the tools of set theory are clearly available, and rigorous
research is possible (for a discussion of that move concerning nonreflexive
logic, see [2] and [1]).

A similar situation happens in the metaphysical foundations of quantum
mechanics we have advanced. We begin with our intuitive conceptual apparatus,
good enough for our everyday working of prediction and development of the
theory. Perhaps that apparatus comprises a minimal constructive logic (as
suggested by da Costa [4], such a logic would have the minimum for our actual
development of science). With that apparatus we can develop logic and the
mathematics required for quantum mechanics. Now, as we have discussed, quantum
mechanics seems to demand a revision of the apparatus we began with.777Again,
see the discussion in [14] for further information on those issues. Logic must
be built again. Now, however, differently from Kunen’s description of the
situation in mathematics, a new system of logic must be developed, not inside
a classical set theory, but totally new, from scratch, taking into account the
lessons from science. This is what we have attempted to provide here in the
case of non-individuals. Obviously we do not require that every field of
knowledge take nonreflexive logic into account; no, our everyday activities
are rooted on our everyday experiences, for which identity does not seem to be
eliminable, at least by this time. That can be interpreted in two ways: either
nonreflexive logics takes the place of classical logic everywhere, with the
proviso that it is the ‘classical’ part of $\mathfrak{Q}$ that is being used,
or else one accepts a pluralism in logic, claiming that distinct fields of
knowledge demand distinct systems of logic. That is a interesting debate, but
we shall not face it now.

Suffice to say, for now, that the construction of $\mathfrak{Q}$-spaces is
only a beginning to the study of the non-reflexive foundations of quantum
mechanics. Nonreflexive logics still present philosophical challenges and
advance some interesting problems in the foundations of quantum mechanics. It
is up to us to develop those systems and discover whether the system can face
the challenges.

## References

  * [1] Arenhart, J. R. B., Semantic analysis of non-reflexive logics. Logic Journal of the IGPL, 22(4), pp.565-584, 2014.
  * [2] Arenhart, J. R. B., Krause, D., Quantifiers and the foundations of quasi-set theory, Principia, 13(3), pp. 251-268, 2009.
  * [3] Arenhart, J. R. B. and Krause, D. Contradictions, quantum mechanics, and the square of oppositions. Forthcoming in Logique et Analyse, 2015.
  * [4] da Costa, N. C. A., Ensaio sobre os fundamentos da l gica. $3^{rd}$ ed. S o Paulo: Hucitec, 2008.
  * [5] da Costa, N. C. A. and de Ronde, C. 2013. The Paraconsistent Logic of Superpositions. Foundations of Physics 43: 854-858.
  * [6] Dalton, J. 1808. A new system of chemical philosophy. London: S. Russell.
  * [7] Domenech, G., Holik, F. and Krause, D., Q-spaces and the foundations of quantum mechanics, Foundations of Physics 38 (11), pp.969-994, 2008.
  * [8] Falkenburg, B. Particle Metaphysics: A Critical Account of Subatomic Reality. Berlin-Heidelberg: Springer-Verlag.
  * [9] French, S., The Structure of the World. Metaphysics and Representation. Oxford: Oxford Un. Press, 2014.
  * [10] French, S. and Krause, D., Identity in Physics: A Historical, Philosophical, and Formal Analysis, Oxford:Oxford University Press 2006.
  * [11] French, S., Krause, D., Remarks on quasi-set theory, Studia Logica, 95(1-2) pp. 101-124, 2010.
  * [12] Krause, D., On a calculus of non-individuals: ideas for a quantum mereology. In Dutra, L. H. de A.; Luz, A. M. (orgs.), Linguagem, Ontologia e A o. Florian polis: NEL/UFSC, v. 10, Cole o Rumos da Epistemologia. pp. 92-106, 2011.
  * [13] Krause, D. and Arenhart, J. R. B., Entangling and disentangling realism and wave function. Essay Review of The Wave Function. Essays on the metaphysics of quantum mechanics, edited by Alyssa Ney and David Z. Albert (Oxford: Oxford Un. Press, 2013). Metascience 23 (3): 467-73, 2014.
  * [14] Krause, D., Arenhart, J. R. B., Individuality, Quantum Physics and a Metaphysics of Nonindividuals: the role of the formal. To appear in Alexandre Guay and Thomas Pradeau (eds.) Individuals Across the Sciences, Oxford Un. Press, 2015, pp. 61-80
  * [15] Kunen, K. The Foundations of Mathematics. London: College Publications, 2009.
  * [16] Ladyman, J., On the identity and diversity of objects in a structure. Proceedings of the Aristotelian Society Supplementary Volume LXXXI, pp.23-43, 2007.
  * [17] Ladyman, J., Ross, D., Everything Must Go: Metaphysics Naturalized. Oxford: Oxford Un. Press, 2007.
  * [18] Mattuck, R. D., A Guide to Feynman Diagrams in the Many-Body Problem. 2nd. ed. New York: McGraw-Hill, 1967. (Reprinted by Dover Pu., 1992).
  * [19] Merzbacher, E., Quantum Mechanics. 2nd.ed., New York: John Wiley.
  * [20] Ney, A., Albert, D., (eds.) The Wave Function: Essays on the Metaphysics of Quantum Mechanics. Oxford: Oxford Un. Press, 2013.
  * [21] Schr dinger, E. What is an elementary particle? In Schr dinger, E. Science, Theory and Man. London: George Allen and Unwin Ltd., pp. 193-223.
  * [22] Simons, P. Parts: a Study in Ontology. Oxford: Clarendon Press, 1987.
  * [23] Teller, P. An Interpretive Introduction to Quantum Field Theory. Princeton: Princeton Un. Press, 1995.