# Indistinguishability right from the start in standard quantum mechanics

Federico Holik1,4 , Juan Pablo Jorge1 and Cesar Massri2

1- Instituto de Física La Plata, CCT-CONICET, and Departamento de Física,
Facultad de Ciencias Exactas, Universidad Nacional de La Plata - $115$ y $49$,
C.C. $67$, $1900$ La Plata, Argentina

2- Departamento de Matemática - Facultad de Ciencias Exactas y Naturales
Universidad de Buenos Aires - Pabellón I, Ciudad Universitaria

We discuss a reconstruction of standard quantum mechanics assuming
indistinguishability right from the start, by appealing to quasi-set theory.
After recalling the fundamental aspects of the construction and introducing
some improvements in the original formulation, we extract some conclusions for
the interpretation of quantum theory.

Key words: Quantum Indistinguishability - Quasi-set theory

##  1 Introduction

The study of collections of quantum systems has given place to great debates
in the philosophy of physics literature. The first remarkable thing that
occurs when dealing with compound quantum systems, is that they can be
prepared in entangled states [1, 2, 3, 4]. The second remarkable thing – and
which is the subject of this work – is that quantum objects seem to be
indistinguishable in a way that has no classical analogue [5, 6, 7, 8]. These
two aspects of quantum mechanics should not be confused: indistinguishability
has to be introduced as an independent axiom of quantum theory, and quantum
objects can be prepared in symmetryzed states without showing any entanglement
at all [9].

The status of identity of quantum systems has been an issue almost since the
conception of the theory. Many aspects of the quantum formalism suggest that
quantum systems somehow lack identity. In a sense, they seem to be non-
individuals. The positions in the literature defer with regard to the degree
with which quantum systems depart from a classical notion of identity.
Perhaps, the most radical position was that of E. Schrödinger, who claimed
that quantum systems were utterly indistinguishable [6, 7]. The connections of
quantum indistinguishability with the Principle of the Identity of
Indiscernibles (PII) has been largely debated too, most authors arguing that
it is somehow violated in the quantum domain (see for example [10, 11]). But
other voices appeared, claiming that elementary particles can be, at least,
weakly discerned [12, 13, 14, 15], and tried to used that notion as an attempt
to save the PII [16]. The status of identity in quantum systems has been also
discussed in terms of ontologies based in bundles of properties (see for
example [17]). The subject of quantum indistinguishability as a whole gave
place to intense debates, and the literature about it is huge (see [8]; see
also [18, 19, 20, 21] and references therein). It is important to remark that,
for those who want to get stick to a classical ontology, Bohmian mechanics
[22] offers an ontology of quantum objects for which – at the metaphysical
level – there is no issue with identity: Bohmian particles can be considered
individuals that can be identified by their hidden trajectories in space-time.
But, as we explain in Section 5, even in Bohmian mechanics identity is hidden,
and there is no empirical procedure that allows to identify (and re-identify)
quantum systems. This is a remarkable feature of quantum phenomena,
independently of any interpretation: under certain circumstances, there is no
way to identify quantum systems of the same kind. Here, we call empirical
indistinguishability to this peculiar feature of quantum theory.

Moreover, the symmetrization postulate [23, 24], closely related to the
indistinguishability principle, can be used to explain remarkable physical
processes. Among them, one can find the Pauli exclusion principle [25], the
Bose-Einstein condensation [26], and all phenomena related to quantum
statistics in general. Currently, quantum indistinguishability is considered
as a resource, and exploited in quantum informational tasks (see for example
[27, 28, 29]). In this sense, under the light of these developments and
applications, the assumption of quantum indistinguishability is a very
powerful feature of quantum theory, that has no classical analogue, and seems
to be the right conceptual framework for the working physicist.

Independently of the interpretation chosen or the metaphysical commitments
assumed, the symmetrization postulate and the incapability of distinguishing
quantum systems must be reflected in the effective part of the theory (i.e.,
that part of the formalism that connects with experience). At this level, the
symmetrization postulate plays a key role, in the sense that it is the
mathematical procedure that physicists found in order to give place to
predictions that describe quantum statistics correctly and, at the same time,
reflect the fact that quantum systems cannot be discerned.

But the symmetrization postulate is implemented by appealing to a trick.
First, quantum systems are labeled in order to create a tensor product Hilbert
space, as if they were distinguishable. After that initial labeling, quantum
states are symmetrized (or anti-symmetrized) in order to obtain the correct
sates. No trace of the initial labeling can be found in the final version of
the formalism. While mathematically correct, this procedure seems to be rather
artificial, because the initial labeling plays no real role in the empirical
predictions of the theory. Thus, many authors posed the problem of finding a
formulation of quantum theory in which indistinguishability is taken right
from the start, eliminating the so called surplus mathematical structure of
particle labeling [30, 31, 8]. One candidate for solving the “surplus
structure” problem, is that of formulating quantum mechanics using the Fock-
space formalism (FSF) [32, 33]. But, as observed in [8], the FSF also makes
use of particle labeling and symmetrization in order to obtain the correct
states. Among the attempts to solve this problem, one can find [34] and [35]
(see also [36] ).

In this work, we elaborate on the proposal presented in [34] and [35] (see
also [37, 38, 39, 40]). We first review the fundamental features of the Fock-
space formulation of quantum mechanics and the rudiments of quasi-set theory
in Sections 2 and 3. Next, in Section 4, we present the quasi-sets
reformulation of standard quantum mechanics in a new math fashion. In Section
5, after introducing the notion of empirical indistinguishability, we show in
which sense the assumption of particle labelings and identities has a similar
role to that of hidden variables, playing no real role in the empirical
predictions of the theory. We argue that quantum indistinguishability is a
positive feature of quantum systems that can be treated rigorously by using
the quasi-sets formalism. This, combined with the fact that quantum mechanics
can be reformulated by assuming indistinguishability right from the start,
favours an eliminativist approach with regard to to the notion of classical
identity in the interpretation of quantum theory. We draw our conclusions in
Section 6.

##  2 Fock-space formalism

As is well known, the $FSF$ can be used as an alternative mathematical
framework for non-relativistic quantum mechanics [41]. In order to introduce
the basic ideas, in this section we will expose the formalism in the way that
is found in most physics textbooks (see for example [42, 41, 43]). A more
technical introduction can be found in [44, 45]. In standard quantum
mechanics, the Hamiltonian of $n$ identical quantum systems with pairwise
interaction, can be written as:

$H_{n}=\sum_{i=1}^{n}[(-\frac{\hbar^{2}\nabla_{i}^{2}}{2m})+V_{1}(\mathbf{x}_{i})+\sum_{i>j=1}^{n}V_{2}(\mathbf{x}_{i},\mathbf{x}_{j})]$
(1)

where we have assumed that each quantum system is subject to an external
potential $V_{1}(\mathbf{x}_{i})$ and the pairwise interaction is represented
by $V_{2}(\mathbf{x}_{i},\mathbf{x}_{j})$. The wave function

$\Psi_{n}(\mathbf{x}_{1},\ldots,\mathbf{x}_{n},t)$ (2)

must be a solution of the Schrödinger’s equation

$H_{n}\Psi_{n}=i\hbar\frac{\partial}{\partial t}\Psi_{n}$ (3)

The second quantization approach to QM has its roots in considering equation
(3) as a _classical field equation_ , and its solution
$\Psi_{n}(\mathbf{x}_{1},\ldots,\mathbf{x}_{n})$ as a _classical field to be
quantized_. This alternative view was originally adopted by Pascual Jordan
[46, 47], one of the foundation fathers of quantum mechanics, and spread
worldwide after the Dirac’s paper [48]. Furthermore, it is a standard way of
describing free fields in relativistic quantum mechanics (canonical
quantization). The space in which these quantized fields operate is the Fock-
space.

The standard Fock-space is built up from the one particle Hilbert spaces. Let
$\mathcal{H}$ be a separable Hilbert space and define:

$\displaystyle{\mathcal{H}}^{0}$ $\displaystyle={\mathbb{C}}$ (4)
$\displaystyle{\mathcal{H}}^{1}$ $\displaystyle={\mathcal{H}}$
$\displaystyle{\mathcal{H}}^{2}$
$\displaystyle={\mathcal{H}}\otimes{\mathcal{H}}$ $\displaystyle\vdots$
$\displaystyle{\mathcal{H}}^{n}$
$\displaystyle={\mathcal{H}}\otimes\cdots\otimes{\mathcal{H}}$

The Fock-space is thus constructed as the direct sum of $n$ particle Hilbert
spaces:

${\mathcal{F}}(\mathcal{H})=\bigoplus^{\infty}_{n=0}{\mathcal{H}}^{n}$ (5)

When dealing with bosons or fermions, the symmetrization postulate ($SP$) must
be imposed. In order to do so, let $S_{n}$ be the group of permutations of the
set $\\{1,2,3,\ldots,n\\}$. Given any vector
$\eta=\eta_{1}\otimes\cdots\otimes\eta_{n}\in{\mathcal{H}}^{n}$ and $P\in
S_{n}$, let
$P(\eta_{1}\otimes\cdots\otimes\eta_{n}):=\eta_{P(1)}\otimes\eta_{P(2)}\otimes\cdots\otimes\eta_{P(n)}$.
Next, define:

$\sigma^{n}(\eta)=\frac{1}{n!}\sum_{P\in
S_{n}}P(\eta_{1}\otimes\cdots\otimes\eta_{n})$ (6)

and:

$\tau^{n}(\eta)=\frac{1}{n!}\sum_{P\in
S_{n}}s^{P}P(\eta_{1}\otimes\cdots\otimes\eta_{n})$ (7)

where:

$s^{P}=\left\\{\begin{array}[]{c l}1&\text{if }P\text{ is even},\\\
-1&\text{if }P\text{ is odd}.\end{array}\right.$

Calling

${\mathcal{H}}^{n}_{\sigma}=\\{\sigma^{n}(\eta)\,\,|\,\,\eta\in{\mathcal{H}}^{n}\\}$
(8)

and:

${\mathcal{H}}^{n}_{\tau}=\\{\tau^{n}(\eta)\,\,|\,\,\eta\in{\mathcal{H}}^{n}\\}$
(9)

we have the Fock-space

${\mathcal{F}}^{+}(\mathcal{H})=\bigoplus_{n=0}^{\infty}{\mathcal{H}}^{n}_{\sigma}$
(10)

for Bosons and

${\mathcal{F}}^{-}(\mathcal{H})=\bigoplus^{\infty}_{n=0}{\mathcal{H}}^{n}_{\tau}$
(11)

for Fermions. In what follows, in order to make the exposition clearer, we use
the Dirac ket-bra notation to denote the vectors in
$\mathcal{F}(\mathcal{H})$, $\mathcal{F}^{+}(\mathcal{H})$ and
$\mathcal{F}^{-}(\mathcal{H})$. The standard second quantization procedure
considers the one particle wave function $\psi(\mathbf{r},t)$ and its
hermitian conjugate $\psi(\mathbf{\mathbf{x},t})^{\dagger}$ as operators
acting on the Fock-space and satisfying [43]:

$\displaystyle[\psi(\mathbf{x},t),\psi(\mathbf{x}^{\prime},t)]_{\mp}=0$
$\displaystyle[\psi(\mathbf{x},t)^{\dagger},\psi(\mathbf{x}^{\prime},t)^{\dagger}]_{\mp}=0$
$\displaystyle[\psi(\mathbf{x},t),\psi(\mathbf{x}^{\prime},t)^{\dagger}]_{\mp}=\delta_{(\mathbf{x}-\mathbf{x}^{\prime})}$
(12)

where $\delta({\mathbf{x}}-{\mathbf{x}^{\prime}})$ is the Dirac delta
function. If $A$ and $B$ are operators, the brackets are defined by
$[A,B]_{\mp}=AB\mp BA$ (where the “$+$” stands for Fermions and “$-$ for
Bosons). The $n$ particle wave function
$\Psi_{n}(\mathbf{x}_{1},\ldots,\mathbf{x}_{n})$ of the standard formulation
is now written as

$\displaystyle|\psi_{n}\rangle=(n!)^{-\frac{1}{2}}\int d^{3}x_{1}\cdots\int
d^{3}x_{n}\psi(\mathbf{x}_{1})^{\dagger}\cdots\psi(\mathbf{x}_{n})^{\dagger}|0\rangle\Psi_{n}(\mathbf{x}_{1},\ldots,\mathbf{x}_{n})$
(13)

and it is an eigenvector (with eigenvalue $n$) of the particle number
operator:

$N:=\int d^{3}x\psi(\mathbf{x})^{\dagger}\psi(\mathbf{x}).$ (14)

We also have the relation

$\Psi_{n}(\mathbf{x}_{1},\cdots,\mathbf{x}_{n})=(n!)^{-\frac{1}{2}}\langle
0|\psi(\mathbf{x}_{1})\cdots\psi(\mathbf{x}_{n})|\Psi_{n}\rangle$ (15)

An arbitrary vector of the Fock-space will be a superposition of states with
different particle number of the form

$|\Psi\rangle=\sum_{n=0}^{\infty}\alpha_{n}|\Psi_{n}\rangle$ (16)

_and will not be in general an eigen-state of the particle number operator_.
Thus, according to the standard interpretation of QM, its particle number will
be undetermined. This is very important, because in the presence of particle
interactions, the states may evolve into an undefined particle state like (16)
[43].

It is customary to make a Fourier decomposition of the operators
$\psi(\mathbf{x},t)$ and $\psi(\mathbf{x},t)^{\dagger}$:

$\psi(\mathbf{x},t)=\sum_{i}u_{i}(\mathbf{x})a_{i}(t)$ (17a)
$\psi(\mathbf{x},t)=\sum_{i}u_{i}(\mathbf{x})a_{i}^{\dagger}(t)$ (17b)

where the complex functions $u_{i}(\mathbf{x})$ are assumed to form a complete
and orthonormal set. The operators $a_{i}^{\dagger}(t)$ and
$a_{i}^{\dagger}(t)$ aquire a simple interpretation when the Hamiltonian is
time independent. In that case, it is useful to take the $u_{i}$’s as the
eigen-functions of the stationary Schrödinger equation for one particle:

$\left(-\frac{\hbar^{2}}{2m}\nabla^{2}+V(\mathbf{x})\right)u_{i}(\mathbf{x})=\epsilon_{i}u_{i}(\mathbf{x})$
(18)

By doing so, it is possible to write
$a_{i}(t)=\exp{-\frac{i\epsilon_{i}t}{\hbar}}a_{i}$. The operators
$a_{i}^{\dagger}$ and $a_{i}$ are called creation and annihilation operators
and satisfy:

$\displaystyle[a_{i},a_{j}]_{\mp}=\mathbf{0}$
$\displaystyle[a_{i}^{\dagger},a_{j}^{\dagger}]_{\mp}=\mathbf{0}$
$\displaystyle[a_{i},a_{j}^{\dagger}]_{\mp}=\delta_{ij}$ (19)

where the “$-$” and “$+$” signs stand for Bosons and Fermions, respectively.

##  3 Quasi-Set Theory

In Zermelo-Frankel (ZF) set theory, given an object $a$, we can always define
the _singleton_ $A=\\{a\\}$. Any other object $b$ of the theory belongs to
$A$, if and only if, it satisfies $b=a$. This simple task of identification
can be carried out, due to the fact that identity is a primitive notion in the
first order language in which the theory is formulated. Quasi-set theory
($\mathfrak{Q}$ from now on) is a set-theoretical framework that can deal with
collections of truly indiscernible objects [49, 50, 51, 8, 31]. In this
context, “truly indiscernible” means that the theory is formulated in a first
order language without identity (and then, $\mathfrak{Q}$ is an example of a
_non-reflexive logic_). Thus, the most simple identification tasks that are
granted in ZF, are no longer available in $\mathfrak{Q}$. Instead of identity,
an equivalence indistinguishability relation “$\equiv$” is postulated. The
idea behind the axioms is that quasi-sets can be used to represent collections
of quantum systems (possibly of the same kind, and thus, indiscernible). In
this framework, indiscernibility does not implies identity: two elements of
the theory might be indiscernible (in symbols $a\equiv b$), while not being
identical. They fail to be identical in the following sense: the first order
theory of identity is not valid for all objects of the theory. In the rest of
this section, we discuss the basic formalism of quasi-set theory.

In order to mimic elementary particles, the objects of the theory are divided
into two main groups: $m$-objects (these are ‘micro-objects’, intended to
represent quantum systems) and $M$-objects (‘macro-objects’, introduced to
represent classical objects). The $m$-objects are introduced as ur-elements in
a standard way, while quasi-sets are collections of them (or collections of
quasi-sets).

A derived notion of identity “$=_{E}$” is introduced in the theory ($x=_{E}y$
is read ’$x$ and $y$ are extensionally identical’), iff they are both quasi-
sets having the same elements (that is, $\forall z(z\in x\leftrightarrow z\in
y)$), or they are both $M$-atoms and belong to the same quasi-sets (that is,
$\forall z(x\in z\leftrightarrow y\in z)$). Due to the axioms of the theory,
when applied to $M$-atoms (or collections of them), the relation of
indiscernibility collapses into that of extensional identity and has the usual
properties of the standard identity of ZFU (ZF with Urelemente). In this way,
a copy of ZFU can then be constructed inside $\mathfrak{Q}$.

A quasi-cardinal can be assigned to quasi-sets, and it is intended to
represent how many elements they have. Using the primitive notion of quasi-
cardinal, an analogous of the axiom of weak extensionality is postulated in
$\mathfrak{Q}$. It states that those quasi-sets that have the same quantity of
elements of the same sort, are indistinguishable. It is important to remark
that collections of $m$-atoms do no have an associated ordinal. The elements
of a quasi-set formed by $m$-objects cannot be identified by names, nor
counted, nor ordered. This is the reason why the notion of quasi-cardinal is
postulated. In [52], the quasi-cardinal is treated as a derived notion (see
also [38]). The problem of describing quantum systems with undefined particle
number is discussed in [52, 53, 54].

In ZF, if $w\in x$, it is easy to show that $(x-\\{w\\})\cup\\{z\\}=x$ if and
only if $z=w$. The replacement of an element of a set by a different element,
gives place to a different set. What happens if we try to make an analogous
substitutions to a quasi-set formed by $m$-atoms? Let $[[z]]$ denote the
quasi-set with quasi-cardinal $1$ whose only element is an indistinguishable
from $z$ (this is called the strong singleton of $z$). Suppose that $x$ is a
finite quasi-set and that $z$ is an $m$-atom such that $z\in x$. Given
$w\equiv z$ and $w\notin x$, then, it is possible to show that
$(x-[[z]])\cup[[w]]\equiv x$. This can be clearly interpreted as follows: the
permutation of indiscernibles elements gives place to indiscernible
collections.

Given $x$ and $y$, it is is possible to form $[x]$ and $[x,y]$, which are the
collections of all indiscernibles from $x$ and from $x$ and $y$, respectively.
Quasi-pairs can be built in the usual way as: $\langle
x,y\rangle:=[[x],[x,y]]$. A quasi-function $f$ is a quasi-set formed by quasi-
pairs in such a way that if $x\equiv z$ and $[[x],[x,y]]$ and $[[z],[z,w]]$
belong to $f$, we have $y\equiv w$.

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

In this section we describe how to obtain a mathematical formalism based in
$\mathfrak{Q}$ which is equivalent to the FSF described in Section 2. In this
way, we provide a reformulation of standard quantum mechanics that uses
quantum indistinguishability as a starting point, eliminating the surplus
mathematical structure discussed in [33, 32]. We follow an analogous approach
to the one given in [34, 35, 37, 38], but introducing technical improvements
in the formulation. We use the axioms and definitions of quasi-set theory
introduced in [55], with minor modifications.

Using the copy of $ZFU$ in $\mathfrak{Q}$, we start by considering a
collection $\mathcal{E}=\\{\epsilon_{i}\\}_{i\in\mathbb{N}}$, where
$\mathbb{N}$ are the natural numbers. The $\epsilon_{i}$’s are intended to
represent outcomes of a maximal observable, and are thus distinguishable in
the classical sense. This is the reason why we describe $\mathcal{E}$ using
the classical part of $\mathfrak{Q}$. To fix ideas, the reader can consider
its elements as the possible values of the energy of a single quantum system
(assumed to have discrete spectrum). But it is important to keep in mind that
the $\epsilon_{i}$’s could be real numbers or just symbols used to distinguish
the different outcomes of an experiment. The only important thing about
$\mathcal{E}$ is that it is a denumerable collection of distinguishable items
(so we could have taken the natural numbers instead of $\mathcal{E}$). The
constraint in the cardinality of $\mathcal{E}$ implies the separability (i.e.,
that it admits a denumerable basis) of the Hilbert space that we construct, as
the rigorous formulation of standard quantum mechanics requires. The fact that
$\mathcal{E}$ is denumerable and its elements distinguishable, also implies
that it can be ordered. In the following, we choose a concrete order for its
elements (given by $\epsilon_{i}<\epsilon_{j}$, whenever $i<j$).

We want to make sense of expressions such as “a quantum system has energy
$\epsilon_{i}$” or “there are $n$ quanta in the energy level $\epsilon_{i}$”,
“there are $n_{i}$ quanta in the energy level $\epsilon_{i}$ and $n_{j}$
quanta in the energy level $\epsilon_{j}$”, and so on. For this aim, we use
the non-classical part of $\mathfrak{Q}$ as follows. First, consider the
quasi-set $FIN_{\mathfrak{Q}}$ formed by all possible finite and pure quasi-
sets (with all the $m$-atoms of the same type). The existence of
$FIN_{\mathfrak{Q}}$ can be granted as follows. Assume that there exists a
quasi-set $\omega^{\lambda}$ whose quasi-cardinal is $\aleph_{0}$ (the
smallest infinite cardinal number), representing the collection of all
possible $m$-atoms of type $\lambda$. It can be considered as an infinite and
abstract reservoir a type of quantum system collectively characterized by
specifying their charge, mass, spin, etc., represented by $\lambda$. By
applying the axiom of parts, consider $\mathcal{P}(\omega^{\lambda})$, the
quasi-set formed by its subsets. Now, apply the separation schema to obtain
$FIN_{\mathfrak{Q}}:=\\{x\in\mathcal{P}(\omega^{\lambda})\,\,|\,\,qcard(x)<\aleph_{0}\\}$.
Consider next the quasi-set $\mathcal{F}$ formed by all possible quasi-
functions $f$ such that $f:\mathcal{E}\longrightarrow FIN_{\mathfrak{Q}}$, and
whenever $\langle\epsilon_{i_{k}};x\rangle$ and
$\langle\epsilon_{i_{k^{\prime}}};y\rangle$ belong to $f$ and $k\neq
k^{\prime}$, then $x\cap y=\emptyset$. We also assume that the sum of the
quasi-cardinals of the quasi-sets which appear in the image of each of these
quasi-functions is finite. This means that $qcard(x)=0$ for every $x$ in the
image of $f$, except for a finite number of elements of $\mathcal{E}$. Denote
by $\mathcal{E}_{f}$ to the collection of indexes for which $f$ assigns an $x$
such that $qcard(x)\neq 0$. There is no a piori order in $\mathcal{E}_{f}$,
but its elements can be ordered, given that they belong to the classical part
of $\mathfrak{Q}$. Each $f$ is a quasi-set formed by ordered pairs
$\langle\epsilon_{i};x\rangle$ with $\epsilon_{i}\in\mathcal{E}$ and $x\in
FIN_{\mathfrak{Q}}$. Each order pair $\langle\epsilon_{i};x\rangle$ represents
the proposition “the quantum number $\epsilon_{i}$ has occupation number
$qcard(x)$” or, equivalently, there are “$qcard(x)$ quanta in the energy level
$\epsilon_{i}$”. Thus, a quasi-function $f:\mathcal{E}\longrightarrow
FIN_{\mathfrak{Q}}$ can be interpreted as “there are $qcard(f(\epsilon_{1}))$
quanta in energy level $\epsilon_{1}$, $qcard(f(\epsilon_{2}))$ quanta in
energy level $\epsilon_{2}$, $qcard(f(\epsilon_{3}))$ quanta in energy level
$\epsilon_{3}$ ….”, in such a way that $\sum_{i}qcard(f(\epsilon_{i}))$ is a
finite number.

As discussed in Section 3, $\mathfrak{Q}$ is constructed in such a way that
the permutation of indiscernible elements gives place to indiscernible
collections. Given that these quasi-functions described above are constructed
using the non-classical part of $\mathfrak{Q}$, the permutation of quanta has
no effect: the result of interchanging a quanta taken from a quasi-set
associated to $\epsilon_{i}$, with another one taken from the quasi-set
associated to $\epsilon_{j}$, gives place to the same quasi-function (see also
the discussion in [34]). In this way, the particle permutation operator of the
standard formalism looses its meaning here. Each $f\in\mathcal{F}$ is
characterized by the set $\mathcal{E}_{f}$ and the “occupation numbers”
associated to each $\epsilon_{i}\in\mathcal{E}_{f}$ (given by
$qcard(f(\epsilon_{i}))$). There is no identification, nor ordering, nor
labeling of quanta in this description, because there is no identity in the
underlying logic of $\mathfrak{Q}$ for the $m$-atoms and their collections. In
this way, a quasi-function $f\in\mathcal{F}$, faithfully represents a
proposition such as “there are $qcard(f(\epsilon_{1}))$ quanta in energy level
$\epsilon_{1}$, $qcard(f(\epsilon_{2}))$ quanta in energy level
$\epsilon_{2}$, $qcard(f(\epsilon_{3}))$ quanta in energy level $\epsilon_{3}$
….”, without appealing to any labelling of the quanta involved.

Now, we proceed to associate a complex vector space structure to
$\mathcal{F}$. This is a first important step if we aim to recover a formalism
equivalent to that based in Fock spaces. In order to do that, let us recall a
useful construction from algebra, [56, 57]. The idea of this construction is
to assign an algebraic structure (a vector space, a commutative algebra, a non
commutative algebra, etc.) to a given set, in such a way that it _generates_
the desired structure. To fix ideas, let us consider first some examples (we
work the examples over the complex numbers $\mathbb{C}$ and using standard set
theory, but notice that we can operate in a totally analogous way in
$\mathfrak{Q}$). Suppose that we have a set with one element, $S=\\{x\\}$, and
we want to construct the commutative algebra $A$ generated by $S$. Then, $A$
should contain expressions such as “$1$”, “$3ix$”, “$x^{2}$”, “$1+ix-x^{3}$”,
and so on. If $A$ is a $\mathbb{C}$-algebra, then it must be equal to the
polynomial algebra in one variable $\mathbb{C}[x]$. This construction is
called the free algebra generated by $S$. The name “ _free_ ”, comes from the
fact that we are not imposing any _relation_ among the generators in $S$.
Another example that we can make is that of the free associative algebra $B$
generated by $S$. If $S=\\{x,y\\}$, then $B=\mathbb{C}\langle x,y\rangle$ (the
notation is standard). So, for example, $xy\neq yx$ in $B$. The last example
that we want to address is that of a complex vector space $V_{S}$ generated by
$S=\\{x,y\\}$. In this case, in $V$ we have expressions such as $(3+2i)x+5iy$
or $x-y$, but the expressions $x^{2}$ or $xy$ are not allowed. In fact, $x$
and $y$ become linearly independent vectors in $V$. Thus, if we have a set
$S=\\{x_{i}\\}$, then we can construct the vector space generated by $S$ by
making formal linear combinations of its elements. By doing so, the elements
of $S$ become linearly independent. In order to construct the vector space $V$
from the set $S$, let us denote it $V_{S}$, we can take the set of functions
from $S$ to $\mathbb{C}$,

$V_{S}=Fun(S,\mathbb{C}).$

It is straightforward to check that $V_{S}$ is a $\mathbb{C}$-vector space and
the indicator functions $\iota(s)$ are the canonical basis for $V_{S}$. In
fact, we can immerse $S$ inside $V_{S}$, $\iota:S\to V_{S}$, by assigning to
$s\in S$ the indicator function $\iota(s)$. Notice that with this construction
we can obtain the commutative algebra $A$ and the non-commutative algebra $B$.
For example, given a set $S$, we can define the set of commutative monomials
$S^{\prime}$ and the set of non-commutative monomials $S^{\prime\prime}$.
Then, $A$ is the vector space generated by $S^{\prime}$ and $B$ by
$S^{\prime\prime}$.

The above described algebraic techniques can be applied mutatis mutandis in
$\mathfrak{Q}$, in order to generate a complex vector space $V_{\mathcal{F}}$
using $\mathcal{F}$. Every function $f\in\mathcal{F}$ will have a copy in
$V_{\mathcal{F}}$. This means that we will have a quasi-function
$\iota:\mathcal{F}\to V_{\mathcal{F}}$, such that $\iota(f)$ is the copy of
$f$ in $V_{\mathcal{F}}$. Thus, given $f_{1},f_{2},\cdots,f_{n}\in\mathcal{F}$
and $\alpha_{1},\alpha_{2},\ldots,\alpha_{n}\in\mathbb{C}$, we can form
expressions such as

$\alpha_{1}\iota(f_{1})+\alpha_{2}\iota(f_{2})+\cdots+\alpha_{n}\iota(f_{n})$
(20)

that can be interpreted as a linear combination of the quasi-functions
$f_{i}$. Thus, we can formally express superposition states using
$\mathfrak{Q}$. We will denote the elements of $V_{\mathcal{F}}$ using Greek
letters. Thus, any $\psi\in V_{\mathcal{F}}$ is a linear combination of the
form (20).

The second step that we need to give in order to recover the FSF, is to endow
the vector space $V_{\mathcal{F}}$ with a scalar product. This can be done in
different ways (see for example [34]). Here, we notice that each
$f\in\mathcal{F}$ has associated a collection of indexes $\mathcal{E}_{f}$.
Given $f,g\in\mathcal{F}$, they can only differ in the content of the sets
$\mathcal{E}_{f}$ and $\mathcal{E}_{g}$, and in the number of quantum objects
(the occupation number) associated to each value of the observable. Thus, any
physically meaningful scalar product between $f$ and $g$, should depend only
on $\mathcal{E}_{f}$, $\mathcal{E}_{g}$, and the respective occupation
numbers. Taking into account that, for the case of indistinguishable quanta,
the order of indexes is not relevant, there is no preferred order on
$\mathcal{E}_{f}$. But it can be ordered, because its elements, representing
outcomes of experiments, are distinguishable in the classical sense. We can
exploit this to define a scalar product without appealing to a any kind of
particle labeling. With this aim, let us first recall some useful notions of
multilinear algebra. Let $V$ be a vector space (possibly infinite-
dimensional). Consider the Tensor Algebra $T(V)$, the Symmetric Algebra $S(V)$
and the Exterior Algebra $\bigwedge(V)$ associated to $V$. The elements of the
$T(V)$ are finite sums of non-commutative monomials $v_{1}\otimes\ldots\otimes
v_{n}$ (called non-commutative polynomials), elements of $S(V)$ are finite
sums of commutative monomials $v_{1}\ldots v_{n}$ (called polynomials), and
elements of $\bigwedge(V)$ are finite sums of skew-symmetric monomials
$v_{1}\wedge\ldots\wedge v_{n}$ (called forms). For example, if $v_{1}$ is
linearly independent from $v_{2}$, then $v_{1}\otimes v_{2}-v_{2}\otimes
v_{1}\neq 0$ in $T(V)$, $v_{1}v_{2}-v_{2}v_{1}=0$ in $S(V)$ and $v_{1}\wedge
v_{2}+v_{2}\wedge v_{1}=0$ in $\bigwedge(V)$. It is important to mention that
we can combine constructions from algebra to achieve the same result. For
example, if $S$ is a set, we can take the vector space $V_{S}$ generated by
$S$, and then we can take the symmetric algebra $S(V_{S})$. Alternatively, we
can take the commutative algebra generated by $S$. It is a standard result
from algebra that both constructions agree. Also, we can take the tensor
algebra $T(V_{S})$ and the non-commutative algebra generated by $S$. Again,
both constructions agree. For more on these constructions, see in [56, Ch.
III].

Notice the analogy between $T(V)$, $S(V)$ and $\bigwedge(V)$ with respect to
the spaces $\mathcal{F}(\mathcal{H})$, $\mathcal{F}_{+}(\mathcal{H})$ and
$\mathcal{F}_{-}(\mathcal{H})$, introduced in section 2. But the analogy
should not lead to confusion. In order to induce a scalar product in
$\mathcal{V}_{\mathcal{F}}$, we will consider formal expressions formed by the
distinguishable outcomes of an abstract observable, without appealing to any
labeling of the quantum objects involved. Thus, consider the vector space
$V_{\mathcal{E}}$ freely generated by $\mathcal{E}$ (this means that the
symbols $\epsilon_{k}$ and $\epsilon_{k^{\prime}}$, are now considered
linearly independent vectors whenever $k\neq k^{\prime}$). First, notice that
for each $f\in\mathcal{F}$, we have en element of $V_{\mathcal{F}}$ (denote
this injection by $\iota:\mathcal{F}\longrightarrow V_{\mathcal{F}}$ ). For
each pair of these copies, we want to define a product. To do so, assign to
each quasi-function $f\in\mathcal{F}$ a non-commutative monomial in
$T(V_{\mathcal{E}})$ as follows. Specifically, given $f\in\mathcal{F}$,
consider the elements of $\mathcal{E}_{f}$ together with their respective
multiplicities (given by $qcard(f(\epsilon_{k}))$, with
$\epsilon_{k}\in\mathcal{E}_{f}$). Define the map $\Psi:\mathcal{F}\to
T(V_{\mathcal{E}})$, that assigns to $f$ the non-commutative monomial
$\epsilon_{i_{1}}\otimes\epsilon_{i_{2}}\otimes\ldots\otimes\epsilon_{i_{m}}\in
T(V_{\mathcal{E}})$, where $\epsilon_{i}$ appears $k$ times if
$qcard(f(\epsilon_{i}))=k$, and the order of the $\epsilon_{i_{k}}$’s is
chosen in such a way that $\epsilon_{i_{k}}$ appears to the left of
$\epsilon_{i_{j}}$, whenever $i_{k}<i_{j}$. For example, if for a given $f$ we
have $\mathcal{E}_{f}=\\{\epsilon_{2},\epsilon_{4},\epsilon_{5}\\}$, with
$qcard(f(\epsilon_{2}))=3$, $qcard(f(\epsilon_{3}))=1$ and
$qcard(f(\epsilon_{5}))=4$, and the order assigned is
$\epsilon_{2}<\epsilon_{3}<\epsilon_{5}$, the monomial assigned to $f$ would
be:
$\epsilon_{2}\otimes\epsilon_{2}\otimes\epsilon_{2}\otimes\epsilon_{3}\otimes\epsilon_{5}\otimes\epsilon_{5}\otimes\epsilon_{5}\otimes\epsilon_{5}$.

An important aspect of our map $\Psi:\mathcal{F}\to T(V)$ is that it can be
used to define a linear map $\tilde{\Psi}:V_{\mathcal{F}}\to T(V)$, preserving
the structure of addition and scalar multiplication. Specifically, it sends
the expression
$\lambda_{1}\iota(f_{1})+\lambda_{2}\iota(f_{2})+\cdots+\lambda_{n}\iota(f_{n})$
to the non-commutative polynomial
$\lambda_{1}\tilde{\Psi}(\iota(f_{1}))+\lambda_{2}\tilde{\Psi}(\iota(f_{2}))+\cdots+\lambda_{n}\tilde{\Psi}(\iota(f_{n}))\in
T(V_{\mathcal{E}})$. The above maps can be depicted as:

---
$\textstyle{\mathcal{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi}$$\scriptstyle{\iota}$$\textstyle{T(V_{\mathcal{E}})}$$\textstyle{V_{\mathcal{F}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\Psi}}$

Using these constructions, we can induce a scalar product on
$V_{\mathcal{F}}$. Recall that if $V$ has a scalar product, then $T(V)$,
$S(V)$ and $\bigwedge(V)$ also inherit a scalar product, [56]. Hence, instead
of taking $V_{\mathcal{E}}$ as the vector space freely generated by
$\mathcal{E}$, we can take $V_{\mathcal{E}}$ as the vector space with scalar
product freely generated by $\mathcal{E}$, that is,
$\epsilon_{k}\bot\epsilon_{k^{\prime}}$ if $k\neq k^{\prime}$ and
$\|\epsilon_{k}\|=1$. Then, the map $\Psi$ assigns to each quasi-function
$f\in\mathcal{F}$ a monomial in $T(V_{\mathcal{E}})$. Also, recall that we
always have canonical maps $\sigma:T(V_{\mathcal{E}})\to S(V_{\mathcal{E}})$
and $\tau:T(V_{\mathcal{E}})\to\bigwedge(V_{\mathcal{E}})$ called
symmetrization and anti-symmetrization respectively. Hence, we can assign to
each $f$ a polynomial or a form, by applying the compositions
$\Psi^{+}:=\sigma\circ\Psi$ and $\Psi^{-}:=\tau\circ\Psi$, respectively:

${S(V_{\mathcal{E}})}$${V_{\mathcal{F}}}$${\mathcal{F}}$${T(V_{\mathcal{E}})}$${\bigwedge(V_{\mathcal{E}})}$$\scriptstyle{\iota}$$\scriptstyle{\Psi}$$\scriptstyle{\Psi^{+}}$$\scriptstyle{\Psi^{-}}$$\scriptstyle{\sigma}$$\scriptstyle{\tau}$

With these tools, we can define:

$\langle\iota(f);\iota(g)\rangle_{0}:=\langle\Psi(f);\Psi(g)\rangle\,\,\forall\,\,f,g\in\mathcal{F}$
(21)

$\langle\iota(f);\iota(g)\rangle^{+}_{0}:=\langle\Psi^{+}(f);\Psi^{+}(g)\rangle\,\,\forall\,\,f,g\in\mathcal{F}$
(22)

$\langle\iota(f),\iota(g)\rangle^{-}_{0}:=\langle\Psi^{-}(f),\Psi^{-}(g)\rangle\,\,\forall\,\,f,g\in\mathcal{F}$
(23)

where, in the right hand side of the above equations, we are using the scalar
products induced in $T(V_{\mathcal{E}})$, $S(V_{\mathcal{E}})$ and
$\bigwedge(V_{\mathcal{E}})$, respectively, induced by the vector space with
scalar product $V_{\mathcal{E}}$ freely generated by $\mathcal{E}$.

Clearly, 21 can be extended linearly to define a complex scalar product
$\langle...;...\rangle:V_{\mathcal{F}}\times V_{\mathcal{F}}\to\mathbb{C}$.
The completion of $V_{\mathcal{F}}$ with respect to this product gives place
to a separable Hilbert space completely equivalent to
$\mathcal{F}(\mathcal{H})$. The case of 22 is completely analogous to 21, and
it can be used to define a scalar product
$\langle...;...\rangle^{+}:V_{\mathcal{F}}\times
V_{\mathcal{F}}\to\mathbb{C}$, yielding a space completely equivalent to
$\mathcal{F}^{+}(\mathcal{H})$. Some care must be taken with regard to 23: all
the $f\in\mathcal{F}$ satisfying that $qcard(f(\epsilon_{i}))\geq 2$, for at
least one $i$, by construction, will have “null norm” in the following sense:
$\langle\iota(f),\iota(f)\rangle^{-}=0$. Thus, in order to recover a space
equivalent to $\mathcal{F}^{-}(\mathcal{H})$, let us proceed as follows.
First, extend 23 bilinearly to all $V_{\mathcal{F}}$. Next, take the quotient
space $V_{\mathcal{F}}/\sim$, with respect to the equivalence relation
$\psi\sim\phi$, iff $\langle\psi-\phi;\psi-\phi\rangle=0$, for all
$\psi,\phi\in V_{\mathcal{F}}$ (i.e., $\psi\sim\phi$ iff their difference has
“null norm”). With regard to the equivalence classes, 23 can be used to define
a complex scalar product $\langle...;...\rangle^{-}:V_{\mathcal{F}}/\sim\times
V_{\mathcal{F}}/\sim\to\mathbb{C}$ on $V_{\mathcal{F}}/\sim$, whose closure is
formally equivalent to $\mathcal{F}^{-}(\mathcal{H})$. For the non-symmetrized
and symmetric products, we have the following diagram:

${V_{\mathcal{F}}\times
V_{\mathcal{F}}}$${\mathbb{C}}$${\mathbb{C}}$${\iota(\mathcal{F})\times\iota(\mathcal{F})}$$\scriptstyle{\langle...;...\rangle}$$\scriptstyle{\langle...;...\rangle^{+}}$$\scriptstyle{\langle...;...\rangle_{0}}$$\scriptstyle{\langle...;...\rangle^{+}_{0}}$

while for the anti-symmetric product, we have:

${V_{\mathcal{F}}/\sim\times
V_{\mathcal{F}}/\sim}$${\mathbb{C}}$${\iota(\mathcal{F})\times\iota(\mathcal{F})}$$\scriptstyle{\langle...;...\rangle^{-}}$$\scriptstyle{\langle...;...\rangle^{-}_{0}}$

Thus, we have shown how to build, using quasi-set theory, spaces which are
equivalent to $\mathcal{F}(\mathcal{H})$, $\mathcal{F}^{+}(\mathcal{H})$ and
$\mathcal{F}^{-}(\mathcal{H})$. This is done by defining different scalar
products on $V_{\mathcal{F}}$, in order to represent distinguihsable quanta
(understood as “quanta of different kinds”), Bosons or Fermions. The desired
spaces are obtained by taking the closures with respect to the norms induced
by the scalar products of the pre-Hilbert spaces $\langle
V_{\mathcal{F}},\langle...;...\rangle\rangle$, $\langle
V_{\mathcal{F}},\langle...;...\rangle^{+}\rangle$ and $\langle
V_{\mathcal{F}}/\sim,\langle...;...\rangle^{-}\rangle$, respectively. Once
this point is reached, all constructions presented in Section 2 can be
reproduced in $V_{\mathcal{F}}$ a usual, by choosing the right scalar products
and defining suitable creation and annihilation operators (see [34]). In this
way, we obtain a formulation of standard quantum mechanics by appealing to
quasi-set theory, that incorporates indistinguishability right from the start.

We now describe the action of creation and annihilation operators for systems
of Bosons and Fermions. Let us start describing Bosons. For this case, we
describe the quasi-functions constructed above by explicitly indicating their
support and occupation numbers. Thus, if $f\in\mathcal{F}$ and
$\mathcal{E}_{f}=\\{\epsilon_{i_{1}},\epsilon_{i_{3}},\epsilon_{i_{3}},\ldots,\epsilon_{i_{n}}\\}$,
we denote its copy in $V_{\mathcal{F}}$ by
$\iota(f):=f_{n_{i_{1}}n_{i_{3}}n_{i_{3}}\ldots n_{i_{n}}}$, where
$n_{i_{k}}=qcard(f(\epsilon_{i_{k}}))$. It is important to remark again that
the order of the indexes has no importance in
$f_{n_{i_{1}}n_{i_{2}}n_{i_{3}}\ldots n_{i_{m}}}$: it is just a notation that
remind us the support of a given element of $V_{\mathcal{F}}$. We could have
written $f_{n_{i_{2}}n_{i_{1}}n_{i_{3}}\ldots n_{i_{m}}}$ instead of
$f_{n_{i_{1}}n_{i_{2}}\epsilon_{i_{3}}\ldots n_{i_{m}}}$, but the order has no
physical meaning in the definition of these quasi-functions. We also write
$f_{\emptyset}\in V_{\mathcal{F}}$ to denote the quasi-function whose
$\mathcal{E}_{f}=\emptyset$ (or, equivalently,
$qcard(f_{\emptyset}(\epsilon_{i}))=0$ for all $i$), which is intended to
represent the ground state of the system under study (i.e., a state in which
all occupation numbers are zero). Let us define the operator that creates a
Boson in state $\epsilon_{k}$ by:

$\mathbf{a}^{\dagger}_{\epsilon_{k}}f_{n_{i_{1}}n_{i_{2}}\ldots
n_{i_{m}}}=\sqrt{n_{k}+1}f_{n_{i_{1}}n_{i_{2}}\ldots(n_{k}+1)\ldots
n_{i_{m}}}$ (24)

With the above normalization, its adjoint operator satisfies:

$\mathbf{a}_{\epsilon_{k}}f_{n_{i_{1}}n_{i_{2}}\ldots
n_{i_{n}}}=\sqrt{n_{k}}f_{n_{i_{1}}n_{i_{2}}\ldots(n_{k}-1)\ldots n_{i_{m}}}$
(25)

Now, consider the action of two creation operators (with $k<l$):

$\mathbf{a}^{\dagger}_{\epsilon_{k}}\mathbf{a}^{\dagger}_{\epsilon_{l}}f_{n_{i_{1}}n_{i_{2}}\ldots
n_{i_{n}}}=\sqrt{n_{k}+1}\sqrt{n_{l}+1}f_{n_{i_{1}}n_{i_{2}}\ldots(n_{k}+1)\ldots(n_{l}+1)\ldots
n_{i_{m}}}$ (26)

and

$\mathbf{a}^{\dagger}_{\epsilon_{l}}\mathbf{a}^{\dagger}_{\epsilon_{k}}f_{n_{i_{1}}n_{i_{2}}\ldots
n_{i_{m}}}=\sqrt{n_{k}+1}\sqrt{n_{l}+1}f_{n_{i_{1}}n_{i_{2}}\ldots(n_{k}+1)\ldots(n_{l}+1)\ldots
n_{i_{m}}}$ (27)

Subtracting equations 26 and 28, we obtain:

$\displaystyle(\mathbf{a}^{\dagger}_{\epsilon_{k}}\mathbf{a}^{\dagger}_{\epsilon_{l}}-\mathbf{a}^{\dagger}_{\epsilon_{l}}\mathbf{a}^{\dagger}_{\epsilon_{k}})f_{n_{i_{1}}n_{i_{2}}\ldots
n_{i_{n}}}=$ (28)
$\displaystyle=(\sqrt{n_{k}+1}\sqrt{n_{l}+1})\left(f_{n_{i_{1}}n_{i_{2}}\ldots(n_{k}+1)\ldots(n_{l}+1)\ldots
n_{i_{m}}}-f_{n_{i_{1}}n_{i_{2}}\ldots(n_{k}+1)\ldots(n_{l}+1)\ldots
n_{i_{m}}}\right)=\mathbf{0}$ (29)

A similar equation holds for $k=l$. Thus, since $f_{n_{i_{1}}n_{i_{2}}\ldots
n_{i_{m}}}$ is arbitrary, we obtain:

$\mathbf{a}^{\dagger}_{\epsilon_{k}}\mathbf{a}^{\dagger}_{\epsilon_{l}}-\mathbf{a}^{\dagger}_{\epsilon_{l}}\mathbf{a}^{\dagger}_{\epsilon_{k}}=\mathbf{0}$
(30)

Similarly, it is easy to obtain:

$\mathbf{a}_{\epsilon_{k}}\mathbf{a}_{\epsilon_{l}}-\mathbf{a}_{\epsilon_{l}}\mathbf{a}_{\epsilon_{k}}=\mathbf{0}$
(31)

Finally, let us compute the action of
$\mathbf{a}_{\epsilon_{k}}\mathbf{a}^{\dagger}_{\epsilon_{l}}$:

$\mathbf{a}_{\epsilon_{k}}\mathbf{a}^{\dagger}_{\epsilon_{l}}f_{n_{i_{1}}n_{i_{2}}\ldots
n_{i_{n}}}=\sqrt{n_{k}-1}\sqrt{n_{l}+1}f_{n_{i_{1}}n_{i_{2}}\ldots(n_{k}-1)\ldots(n_{l}+1)\ldots
n_{i_{m}}}$ (32)

For $\mathbf{a}^{\dagger}_{\epsilon_{l}}\mathbf{a}_{\epsilon_{k}}$:

$\mathbf{a}^{\dagger}_{\epsilon_{l}}\mathbf{a}_{\epsilon_{k}}f_{n_{i_{1}}n_{i_{2}}\ldots
n_{i_{n}}}=\sqrt{n_{l}+1}\sqrt{n_{k}-1}f_{n_{i_{1}}n_{i_{2}}\ldots(n_{k}-1)\ldots(n_{l}+1)\ldots
n_{i_{m}}}$ (33)

Subtracting Equations and , we obtain:

$\mathbf{a}_{\epsilon_{k}}\mathbf{a}^{\dagger}_{\epsilon_{l}}-\mathbf{a}^{\dagger}_{\epsilon_{l}}\mathbf{a}_{\epsilon_{k}}=\delta_{\epsilon_{k}\epsilon_{l}}$
(34)

The above equations defined the desired commuting relations. The application
of the change of basis operators [43] on the ground state $f_{\emptyset}$
gives

$\displaystyle
f_{\mathbf{x}\mathbf{y}}:=\frac{1}{\sqrt{2!}}\Psi(\mathbf{y})^{\dagger}\Psi(\mathbf{x})^{\dagger}f_{\emptyset}=\frac{1}{\sqrt{2}}\sum_{i}\sum_{j}u_{i}(\mathbf{x})u_{j}(\mathbf{x})\mathbf{a}^{\dagger}_{\epsilon_{i}}\mathbf{a}^{\dagger}_{\epsilon_{i}}f_{\emptyset}=$
(35)
$\displaystyle=\frac{1}{\sqrt{2}}\sum_{i}\sum_{j}u_{i}(\mathbf{x})u_{j}(\mathbf{y})f_{\epsilon_{i}\epsilon_{j}}=\frac{1}{\sqrt{2}}\sum_{i<j}[u_{i}(\mathbf{x})u_{j}(\mathbf{y})+u_{j}(\mathbf{x})u_{i}(\mathbf{y})]f_{\epsilon_{i}\epsilon_{j}}$
(36)

Computing the scalar product between $f_{\epsilon_{j}\epsilon_{k}}$ and
$f_{\mathbf{x}\mathbf{y}}$

$\langle
f_{\epsilon_{j}\epsilon_{k}};f_{\mathbf{x}\mathbf{y}}\rangle^{+}=\frac{1}{\sqrt{2}}[u_{i}(\mathbf{x})u_{j}(\mathbf{y})+u_{j}(\mathbf{x})u_{i}(\mathbf{y})]$
(37)

we obtain the right probability amplitudes for two particle Bosonic systems. A
similar result holds for
$f_{\mathbf{x}_{1}\mathbf{x}_{2}\ldots\mathbf{x}_{N}}:=\frac{1}{\sqrt{N!}}\Psi(\mathbf{x}_{N})\ldots\Psi(\mathbf{x}_{1})^{\dagger}\Psi(\mathbf{x}_{1})^{\dagger}f_{\emptyset}$

For Fermions, let us focus on the space $\langle
V_{\mathcal{F}}/\sim,\langle...;...\rangle^{-}\rangle$. A function $f\in
V_{\mathcal{F}}/\sim$ is characterized again by the set $\mathcal{E}_{f}$ and
its occupation numbers. But given that we are in the quotient space, only
quasi-functions whose norm is non-null appear, and then, the occupation
numbers are never greater than one. Thus, we can use a notation in which only
the occupied levels are displayed. Thus, for $f\in V_{\mathcal{F}}/\sim$ such
that
$\mathcal{E}_{f}=\\{\epsilon_{i_{1}},\epsilon_{i_{2}},\ldots,\epsilon_{i_{n}}\\}$,
we use the notation
$f_{\epsilon_{i_{1}},\epsilon_{i_{2}},\ldots,\epsilon_{i_{n}}}$. As remarked
above, while order of quantum systems has no meaning in $f$, the set
$\mathcal{E}_{f}$ can be ordered (and again, we use the order
$\epsilon_{i}<\epsilon_{j}$, iff $i<j$). Given
$\epsilon_{k}\notin\mathcal{E}_{f}$, define $s_{f,\epsilon_{k}}$ as the
minimal number of permutations needed – starting from left to right – to add
$\epsilon_{k}$ to the sequence
$\epsilon_{i_{1}}<\epsilon_{i_{2}}<\ldots<\epsilon_{i_{n}}$ in such a way that
the final result is ordered. As an example, suppose that
$\mathcal{E}_{f}=\\{\epsilon_{3},\epsilon_{5},\epsilon_{7},\epsilon_{8}\\}$,
and we want to add $\epsilon_{6}$ to the sequence
$\epsilon_{3}<\epsilon_{5}<\epsilon_{7}<\epsilon_{8}$, in such a way that the
final result is ordered, using the minimal number of permutations. It is easy
to check that, in this case, $s_{f,\epsilon_{6}}=2$. If we were to add
$\epsilon_{1}$, then $n_{f,\epsilon_{1}}=0$. For $\epsilon_{9}$, we have
$n_{f,\epsilon_{9}}=4$, and so on. With these conventions in hand, for every
$\epsilon_{k}\notin\mathcal{E}_{f_{\epsilon_{i_{1}}\epsilon_{i_{2}}\ldots\epsilon_{i_{n}}}}$,
define:

$\mathbf{c}^{\dagger}_{\epsilon_{k}}f_{\epsilon_{i_{1}}\epsilon_{i_{2}}\ldots\epsilon_{i_{n}}}=(-1)^{s_{f,\epsilon_{k}}}f_{\epsilon_{i_{1}}\epsilon_{i_{2}}\ldots\epsilon_{k}\ldots\epsilon_{i_{n}}}$
(38)

The interpretation of 39, is that we have created a quanta in level
$\epsilon_{k}$. Notice also that, in Equation 39, $\epsilon_{k}$ is placed in
between $\epsilon_{i_{1}}$ and $\epsilon_{i_{n}}$. If, for example,
$\epsilon_{k}<\epsilon_{i_{1}}$, we should have placed $\epsilon_{k}$ at the
beginning. But this is a meaningless detail of the notation which can also be
find in the standard approach, has no effect in the definition of the
operator, and should not lead to confusion. Naturally, whenever
$\epsilon_{k}\in\mathcal{E}_{f_{\epsilon_{i_{1}}\epsilon_{i_{2}}\ldots\epsilon_{i_{n}}}}$,
adding a quanta to an occupied level, should result in a null-norm quasi-
function. Since we are working in the quotient space $V_{\mathcal{F}}/\sim$,
we simply set (for
$\epsilon_{k}\in\mathcal{E}_{f_{\epsilon_{i_{1}}\epsilon_{i_{2}}\ldots\epsilon_{i_{n}}}}$):

$\mathbf{c}^{\dagger}_{\epsilon_{k}}f_{\epsilon_{i_{1}}\epsilon_{i_{2}}\ldots\epsilon_{i_{n}}}=\mathbf{0}$
(39)

where $\mathbf{0}$ is the neutral element for the sum in
$V_{\mathcal{F}}/\sim$. Now, assuming first that $k<l$, by applying two
creation operators, we obtain

$\mathbf{c}^{\dagger}_{\epsilon_{k}}\mathbf{c}^{\dagger}_{\epsilon_{l}}f_{\epsilon_{i_{1}}\epsilon_{i_{2}}\ldots\epsilon_{i_{n}}}=(-1)^{s_{f,\epsilon_{k}}}(-1)^{s_{f,\epsilon_{l}}}f_{\epsilon_{i_{1}}\epsilon_{i_{2}}\ldots\epsilon_{k}\ldots\epsilon_{l}\ldots\epsilon_{i_{n}}}$
(40)

By reversing the order of application (but assuming again $i<k$), we have

$\mathbf{c}^{\dagger}_{\epsilon_{l}}\mathbf{c}^{\dagger}_{\epsilon_{k}}f_{\epsilon_{i_{1}}\epsilon_{i_{2}}\ldots\epsilon_{i_{n}}}=(-1)^{(s_{f,\epsilon_{l}}+1)}(-1)^{s_{f,\epsilon_{k}}}f_{\epsilon_{i_{1}}\epsilon_{i_{2}}\ldots\epsilon_{k}\ldots\epsilon_{l}\ldots\epsilon_{i_{n}}}$
(41)

Adding equations 40 and 42, we obtain:

$\displaystyle(\mathbf{c}^{\dagger}_{\epsilon_{k}}\mathbf{c}^{\dagger}_{\epsilon_{l}}+\mathbf{c}^{\dagger}_{\epsilon_{l}}\mathbf{c}^{\dagger}_{\epsilon_{k}})f_{\epsilon_{i_{1}}\epsilon_{i_{2}}\ldots\epsilon_{i_{n}}}=$
(42)
$\displaystyle=(-1)^{n_{f,\epsilon_{k}}}(-1)^{n_{f,\epsilon_{l}}}f_{\epsilon_{i_{1}}\epsilon_{i_{2}}\ldots\epsilon_{k}\ldots\epsilon_{l}\ldots\epsilon_{i_{n}}}+(-1)^{(n_{f,\epsilon_{l}}+1)}(-1)^{n_{f,\epsilon_{k}}}f_{\epsilon_{i_{1}}\epsilon_{i_{2}}\ldots\epsilon_{k}\ldots\epsilon_{l}\ldots\epsilon_{i_{n}}}=\mathbf{0}$
(43)

Since $f_{\epsilon_{i_{1}}\epsilon_{i_{2}}\ldots\epsilon_{i_{n}}}$ is
arbitrary, we obtain:

$\mathbf{c}^{\dagger}_{\epsilon_{k}}\mathbf{c}^{\dagger}_{\epsilon_{l}}+\mathbf{c}^{\dagger}_{\epsilon_{l}}\mathbf{c}^{\dagger}_{\epsilon_{k}}=\mathbf{0}$
(44)

From the properties of $\mathbf{c}^{\dagger}_{\epsilon_{k}}$, it is possible
to derive those of $\mathbf{c}_{\epsilon_{k}}$ as usual. Proceeding as usual,
we obtain:

$\mathbf{c}_{\epsilon_{k}}f_{\emptyset}=\mathbf{0}$ (45)

and

$\mathbf{c}_{\epsilon_{k}}\mathbf{c}_{\epsilon_{l}}+\mathbf{c}_{\epsilon_{l}}\mathbf{c}_{\epsilon_{k}}=\mathbf{0}$
(46)

Let us now focus on the action of
$\mathbf{c}_{\epsilon_{k}}\mathbf{c}^{\dagger}_{\epsilon_{l}}+\mathbf{c}^{\dagger}_{\epsilon_{l}}\mathbf{c}_{\epsilon_{k}}$.
Assume first that $k<l$:

$\mathbf{c}_{\epsilon_{k}}\mathbf{c}^{\dagger}_{\epsilon_{l}}f_{\epsilon_{i_{1}}\epsilon_{i_{2}}\ldots\epsilon_{i_{n}}}=(-1)^{n_{f,\epsilon_{k}}}(-1)^{n_{f,\epsilon_{l}}}f_{\epsilon_{i_{1}}\epsilon_{i_{2}}\ldots\bar{\epsilon}_{k}\ldots\epsilon_{l}\ldots\epsilon_{i_{n}}}$
(47)

where the bar in $\bar{\epsilon}_{k}$ indicates that it has been removed from
the list (if present before).

On the other hand

$\mathbf{c}^{\dagger}_{\epsilon_{l}}\mathbf{c}_{\epsilon_{k}}f_{\epsilon_{i_{1}}\epsilon_{i_{2}}\ldots\epsilon_{i_{n}}}=(-1)^{(n_{f,\epsilon_{l}}-1)}(-1)^{n_{f,\epsilon_{k}}}f_{\epsilon_{i_{1}}\epsilon_{i_{2}}\ldots\bar{\epsilon}_{k}\ldots\epsilon_{l}\ldots\epsilon_{i_{n}}}$
(48)

By adding Equations 47 and 48, we obtain a null result, in any case, for
$k<l$. When $k=l$, $n_{f,\epsilon_{k}}=n_{f,\epsilon_{l}}$, and we obtain an
identity, for every quasi-function. Thus, we conclude that

$\mathbf{c}_{\epsilon_{k}}\mathbf{c}^{\dagger}_{\epsilon_{l}}+\mathbf{c}^{\dagger}_{\epsilon_{l}}\mathbf{c}_{\epsilon_{k}}=\delta_{kl}$
(49)

The application of the change of basis operators on the ground state
$f_{\emptyset}$ gives

$\displaystyle
f_{\mathbf{x}\mathbf{y}}:=\frac{1}{\sqrt{2!}}\Psi(\mathbf{y})\Psi(\mathbf{x})f_{\emptyset}=\frac{1}{\sqrt{2}}\sum_{i}\sum_{j}u_{i}(\mathbf{x})u_{j}(\mathbf{x})\mathbf{c}^{\dagger}_{\epsilon_{i}}\mathbf{c}^{\dagger}_{\epsilon_{i}}f_{\emptyset}=$
(50)
$\displaystyle=\frac{1}{\sqrt{2}}\sum_{i}\sum_{j}u_{i}(\mathbf{x})u_{j}(\mathbf{y})f_{\epsilon_{i}\epsilon_{j}}=\frac{1}{\sqrt{2}}\sum_{i<j}[u_{i}(\mathbf{x})u_{j}(\mathbf{y})-u_{j}(\mathbf{x})u_{i}(\mathbf{y})]f_{\epsilon_{i}\epsilon_{j}}$
(51)

Computing the scalar product between $f_{\epsilon_{j}\epsilon_{k}}$ and
$f_{\mathbf{x}\mathbf{y}}$, we obtain

$\langle
f_{\epsilon_{j}\epsilon_{k}};f_{\mathbf{x}\mathbf{y}}\rangle^{-}=\frac{1}{\sqrt{2}}[u_{i}(\mathbf{x})u_{j}(\mathbf{y})-u_{j}(\mathbf{x})u_{i}(\mathbf{y})]$
(52)

which yields the right probability amplitudes for two particle Fermionic
systems. Again, we obtain a similar result for
$f_{\mathbf{x}_{1}\mathbf{x}_{2}\ldots\mathbf{x}_{N}}:=\frac{1}{\sqrt{N!}}\Psi(\mathbf{x}_{N})\ldots\Psi(\mathbf{x}_{1})^{\dagger}\Psi(\mathbf{x}_{1})^{\dagger}f_{\emptyset}$.

##  5 Implications of our construction

So far, we have seen how to write a version of the FSF that relies on quasi-
set theory. As such, it assumes from the beginning that quantum objects are
indistinguishable. In this section, we extract some conclusions of the above
construction.

###  5.1 Empirical indistinguishability and ontological concealment

There are very concrete situations in which we can isolate quantum objects. A
formidable example of this is given by electromagnetic traps. Using such
devices, it is possible to isolate atoms [58], and even electrons [59] and
positrons [60, 61]. The researchers that trapped a positron for the first
time, called it Priscilla. Is this labeling of a quantum object physically
meaningful? The manipulation of isolated quantum systems led some researchers
to think that they can be indeed identified. But this is a hasty conclusion,
as we explain below (see also the discussion presented in [5] and [62]).

The only thing that we can say for sure about a trapped atom or electron (and
quantum objects involved in similar situations), is that there is a quantum
object of a certain kind in the trap (or a collection of them, in case there
are more than one). The sentences “there is a positron in the trap” and
“Priscilla is in the trap” seem very similar in content. But the second one is
much stronger than the first: it says that the particular positron named
Priscilla has been trapped. It assumes that positrons can be identified and
labeled. This is not assumed in the first one: even if we cannot identify
positrons, the meaning of having just one of them in a trap is clear. One can
say that one electron leaved a track in the cloud chamber, one photon provoked
a click in the photon detector, a positron is trapped in a ion trap, and so
on. But nothing grants that we can put physically meaningful labels to those
quantum systems. From an operational point of view, there is no reasonable
definition of particle labeling. Let us illustrate this with an example.
Suppose that two experimenters, $A$ and $B$ have two electrons trapped in
their labs. Suppose now that, by adopting the metaphysical standpoint that
electrons are full individuals, they go on with it, and name them at time
$t_{i}$. Experimenters $A$ and $B$ call their respective electrons $\alpha$
and $\beta$. The experimenters now make their electrons interact through a
scattering process, and trap the outgoing electrons again on each lab at time
$t_{f}$. If the experiment is designed in the right way, quantum theory
predicts that there exists no physical mechanism -not even in principle-
allowing each experimenter to know if they have recovered their previously
possessed electrons after the scattering is performed. According to quantum
theory, the identity of each electron is gone forever, in the sense that there
is no way to tell whether $\alpha$ came back to $A$ and $\beta$ returned to
$B$, or not. The question becomes meaningless – from an operational standpoint
– because, according to quantum theory, it is physically impossible to
identify which is which. Of course, metaphysics can always be put at work, and
the experimenters can still assume that after the interaction, $\alpha$ is
still $\alpha$, $\beta$ is still $\beta$, and that both electrons returned to
their traps or they where switched, even if $A$ and $B$ don’t know which
option has taken place. The result of this discussion is that if we want to
assume that quantum objects can be identified, in a way strong enough to allow
naming them, this identification must be hidden (see also the discussion posed
in [62]). If assumed (at the mataphysical level), the identity of quantum
particles is as hidden as the hidden variables of Bohmmian mechanics. An
important remark is at stake here. It doesn’t matter whether the electrons
actually undergo a scattering process as the one described above or not. The
very possibility of such an occurrence threatens their status as individuals.
The laws governing the behavior of quantum systems render any assumption of
particle identification a purely metaphysical claim, that cannot be subject to
experimental control in a consistent way. The only meaningful assertions about
quantum systems are of the form: “there is one positron in that trap and an
electron outside of it”, “there are two electrons in an Helium atom” or “a
quantum object is prepared in state $\psi$”. From the perspective of physics,
any identification becomes a matter of jargon, that can be useful in some
cases (like “Priscila is in the trap”), but cannot be assumed to be valid in
general.

Figure 1: Two experimenters, $A$ and $B$, assume that their electrons can be
named $\alpha$ and $\beta$ at time $t=t_{i}$. The electrons undergo an
interaction process. At time $t=t_{f}$, the experimenters try to re-identify
their electrons. Quantum mechanics predicts that there exists no physical
mechanism whatsoever allowing for them to do this in a meaningful way.

Even if one assumes an ignorance interpretation about the identities of the
electrons, this concealment is ontological in the following sense: if there
would exist a sophisticated mechanism allowing for the quantum objects to be
identified and re-identified in an empirically consistent way, quantum
mechanics would be wrong. An analogous situation would be to design an
experiment to actually detect simultaneously the position and momentum of a
Bohmian particle. If this were possible, we should abandon quantum mechanics
and replace it with a yet not known theory. The concealment of variables and
identities of such hidden variable theories is, in this sense, ontological,
due to the fact that it assumes that the world is such that there exists no
physical mechanism allowing us to identify particles or detect trajectories.
The world would be set as if there would exist a conspiracy in nature, that
conceals the identities of the electrons from our experimental control. This
situation is very different to a similar one with classical objects. We can
make an analogous experiment for classical particles, and we may not know
whether the particles were switched or not. But we – or some researchers from
the future – could be able, in principle, to develop a very sophisticated
mechanism capable of determining which is which. In the classical setting,
there is no impediment in the laws of nature for knowing whether the particles
were switched or not.

The situation described above with the electrons has no classical analogue. To
see how weird this is, assume that two loving couples
$C_{1}=\\{P_{1},T_{1}\\}$ and $C_{2}=\\{P_{2},T_{2}\\}$, share two twins,
named $T_{1}$ and $T_{2}$. Suppose that $T_{1}$ and $T_{2}$ are so similar,
that $C_{1}$ and $C_{2}$ do not have any means to distinguish them physically.
Now, suppose that $T_{1}$ and $T_{2}$ become involved in a strange form of
interaction that hides their identities to $P_{1}$ and $P_{2}$, the same as
with electrons. When $C_{1}$ asks to the returned twin whether he is $T_{1}$
or not, he replies on the affirmative, and the same happens with $P_{2}$. When
asked about past memories, the twins answer that they don’t remember anything,
so there is no way for $P_{1}$ and $P_{2}$ to tell whether they were switched
or not. Naturally, $P_{1}$ and $P_{2}$ want to recover their beloved couples.
$P_{1}$ wants $T_{1}$ to come back home, and will not be willing to accept an
indistinguishable from $T_{1}$. In classical theories there always exists, no
matter how difficult it is, a mechanism allowing $P_{1}$ and $P_{2}$ to re-
identify their beloved ones, even if they can’t do it in practice. The
concealment is not ontological in this case. There is nothing in nature
prohibiting the re-identification. It is just that it is very difficult for
$C_{1}$ and $C_{2}$ to do it.

The situation is much worse than not knowing. By itself, concealment is not a
problem for physics. There are many physical quantities which are not directly
observable. For example, we don’t observe an electric field. We just observe
the action of it on some test particle that we put in a certain space region.
But the subtle point here is that we can control, at lest in principle,
electric fields in such a way to make reasonable predictions in the lab. We
usually postulate entities that we cannot directly observe, but we can control
them in such a way that their effects can be expressed in empirically testable
mathematical laws. Even if doing this is too difficult for many complex
systems, we know that, according to the classical description of physics,
there is no intrinsic concealment in nature, and that we could do it in
principle. It may also happen that we are not able to do certain things at a
stage of development of a theory, but we can hope for doing them in the
future, without invalidating the theory. A paradigmatic example of this is the
manipulation of a single atom. No one dreamed about that possibility in the
early days of quantum mechanics, but according to the theory, there were
nothing in nature preventing us to do so. It is perfectly possible to trap one
atom or one electron using electromagnetic devises. The situation is very
different for the case of indistinguishable particles: if there were a
possibility of identifying (and re-identifying) in a consistent and verifiable
way, quantum mechanics – in its current form – would be simply wrong.

The real problem with postulating hidden variables and identities in quantum
mechanics is that, even if we assume hidden identities or trajectories, there
is no physical experiment whatsoever (and it will never will if quantum
mechanics is true), that allow us to re-identify quantum objects or to control
the values of the hidden variables. In this sense, there is a very clear
empirical feature of quantum phenomena that indicates that quantum systems
cannot be re-identified. We may calle this feature the  principle of no re-
identification: if the experimenters adopt the metaphysical assumption that
quantum objects can be labeled at a given moment, in the general case, there
is no physical procedure allowing them to re-identify those systems in the
future. It is important to stress that these limitations on the possibility of
identifying consistently – from an empirical standpoint – are a common feature
of all interpretations which are compatible with the predictions of quantum
theory. And this is not just a metaphysical assumption: it is a constitutive
physical feature of how the world is according to quantum theory. Remarkably
enough, it is consistent with the laboratory observations up to now.

###  5.2 Lack of something?

The operational impossibility of re-identification described above is
traditionally interpreted as a negative feature of quantum systems. Taking
indistinguishability seriously from an ontological standpoint is usually
criticized as a weird move, in favor of a classical ontology based in standard
individuals. The “in” in “indistinguishability”, the “non” in “non-
individuals” and the “weak” in “weakly discernible objects”, usually have a
negative connotation. It is as if quantum objects would lack of something,
because of the impossibility of naming or identifying them in a consistent
way. The situation is usually presented as if being an individual in the
classical sense would be “metaphysically” stronger than being a non-
individual. But this picture is incorrect, if we stay close to the actual
development of physics.

The theoretical and experimental research of the last decades, remarkably
enhanced by the development of quantum technologies, indicates that, rather
than a weakness, non-individuality is a positive feature of quantum systems.
Put in simple words: non-individuals can do things that classical entities
obeying the classical theory of identity, cannot do.

Indeed, quantum indistinguishability lies at the basis of many quantum
technologies. for example, the Hong-Ou-Mandel [63] effect relies heavily in
the fact the photons are prepared in a state in which they cannot be
individuated by any means. The more indistinguishable the photons are
prepared, more clearly the interference pick reveals itself in the laboratory.
This device can be used to measure time differences with a very high
precision. Recently, quantum indistinguishability has been identified as a
resource for generating quantum entanglement [28, 29, 27], one of the main
ingredients of quantum technologies. Boson sampling [64], is a very simple
one-way quantum computer that is of big interest for testing fundamental
features of quantum computing. It relies solely on indistinguishable photons
(and remarkably, no entanglement seems to be involved). Recently, the notion
of indistinguishability has been studied in connection with quantum
contextuality [65, 66], showing its potential as an ontological principle. The
main point that we want to make here is that the physical properties of
quantum systems with regard to identity are behind many relevant developments
in physics that have no analogue in the classical domain. The principle of
indistinguishability – closely related to the principle of no re-
identifications describe above – is one of the strongest features of quantum
mechanics and is used by working physicists as the correct tool to describe
nature.

Thus, if indistinguishability is so important for predicting new physical
phenomena and for the development of technological devises, why not taking it
seriously at the ontological level? The fact that we can always postulate
hidden identities, should not lead to confusion: the impossibility of
distinguishing predicted by quantum theory reveals a positive feature of
quantum entities, and can be used as a resource in quantum information theory.
To postulate the existence of weak discernible entities or hidden identities
is of no use for the problems that the quantum physicists need to deal with.
The explanatory power of these notions is usually empty (when not misleading)
for the working physicist, and are dispensable.

Is it possible to make sense of non-individuals? Quasi-set theory shows that
this is perfectly possible: we have a formal framework (at least one) that
allows to speak about non-individuals in a rigorous way. This formalism gives
a rigorous ground to the ideas that many physicists used in order to explain
and predict new phenomena. Furthermore, the content of Section 4, shows that
it is even possible to reformulate quantum mechanics using quasi-set theory.
In this way, we have exposed a formulation of quantum mechanics in which
quantum objects are considered as non-individuals right from the start.

These developments do not imply that the ontologies based in individual
entities should be rejected in quantum mechanics. But our work shows that the
identity of quantum objects can be dispensed with (in the sense of being an
eliminable feature). This eliminativist move works in different levels. First,
it works in the operational level, given that no ontology based on individuals
can make predictions in which quantum objects can be identified and re-
identified in a satisfactory way. We have seen that this is not possible for
quantum systems – provided that quantum mechanics is assumed to be correct. On
the other hand, we have shown that it is possible – and useful – to consider
non-individuality as a positive feature of quantum systems (and even as a
resource), that can be formally described in a mathematics which relies in
quasi-set theory.

##  6 Conclusions

In this work we have reviewed the approach to standard quantum mechanics based
in quasi-set theory. We have shown how to recover a Fock-space formalism based
in quasi-set theory, accomplishing the task of reformulating quantum mechanics
using non-individuality right from the start.

We have also elaborated on indistinguishability from an operational
standpoint, relating it to the limitations that quantum theory imposes on the
attemtps to identify and re-identify quantum objects (suggesting that a no re-
identification principle is operating in the quantum domain). We have argued
that this peculiar feature of quantum systems, which must be necessarily
shared by all interpretations which are consistent with the predictions of
quantum theory, leads to a sort of ontological concealment of identity in
those interpretations which are based on individuals (such as Bohmian
mechanics). We have argued that this, together with the formulation of quantum
theory described in Section 4, makes the notion of identity eliminable, in the
sense that it can be dispensed with. We have argued that, far from being a
negative notion (“a lack of something”), non-individuality is a positive
feature of quantum systems that can be rigorously formalized using quasi-set
theory. It can also be used as a predictive tool and as a resource in quantum
information theory, and it opens the door to a sound formulation of quantum
mechanics based in non-individuals.

Acknowledgements

This work was partially supported by FONCYT, Universidad Austral and the grant
“Per un’estensione semantica della Logica Computazionale Quantistica- Impatto
teorico e ricadute implementative”, RASSR40341.

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