# Reconstruction of Quantum Particle Statistics: Bosons, Fermions, and
Transtatistics

Nicolás Medina Sánchez<EMAIL_ADDRESS>University of
Vienna, Faculty of Physics, Vienna Center for Quantum Science and Technology
(VCQ), Boltzmanngasse 5, 1090 Vienna, Austria  University of Vienna, Vienna
Doctoral School in Physics, Boltzmanngasse 5, 1090 Vienna, Austria  Borivoje
Dakić<EMAIL_ADDRESS>University of Vienna, Faculty of Physics,
Vienna Center for Quantum Science and Technology (VCQ), Boltzmanngasse 5, 1090
Vienna, Austria  Institute for Quantum Optics and Quantum Information (IQOQI),
Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria.

###### Abstract

Identical quantum particles exhibit only two types of statistics: bosonic and
fermionic. Theoretically, this restriction is commonly established through the
symmetrization postulate or (anti)commutation constraints imposed on the
algebra of creation and annihilation operators. The physical motivation for
these axioms remains poorly understood, leading to various generalizations by
modifying the mathematical formalism in somewhat arbitrary ways. In this work,
we take an opposing route and classify quantum particle statistics based on
operationally well-motivated assumptions. Specifically, we consider that a)
the standard (complex) unitary dynamics defines the set of single-particle
transformations, and b) phase transformations act locally in the space of
multi-particle systems. We develop a complete characterization, which includes
bosons and fermions as basic statistics with minimal symmetry. Interestingly,
we have discovered whole families of novel statistics (dubbed transtatistics)
accompanied by hidden symmetries, generic degeneracy of ground states, and
spontaneous symmetry breaking– effects that are (typically) absent in ordinary
statistics.

##  I Introduction

The concept of identical particles was introduced by Gibbs in 1902 Gibbs
(2010) as an alternative to solve the problem related to the extensitivity of
entropy, the so-called _Gibbs paradox_. According to Gibbs, a system consists
of identical particles if its physical magnitudes are invariant under any
permutation of its elements. Bose has put forward this idea in quantum
mechanics in his derivation of Planck’s law of blackbody radiation Bose
(1924). This was further developed by Dirac Dirac and Fowler (1926) and
Heisenberg Heisenberg (1926), who formulated the well-known _symmetrization
postulate_ : physical states must be symmetric in such a way that the exchange
of particles does not give any observable effect. Put in the standard language
of wavefunctions, if the state of, say, two particles is given by
$\psi(x_{1},x_{2})$, then

$\psi(x_{2},x_{1})=e^{i\varphi}\psi(x_{1},x_{2}).$ (1)

Applying the particle swap twice trivially reveals $e^{i\varphi}=\pm 1$. This
is the origin of two types of particle statistics: _bosons_ (symmetric) and
_fermions_ (antisymmetric).

Another approach to explain the origin of quantum statistics is the
_topological_ argument Leinaas and Myrheim (1977); Laidlaw and DeWitt (1971).
Namely, the exchange symmetry is directly related to the continuous movement
of particles in a physical (configuration) space, which implies that only
bosonic and fermionic phases are allowed, given that the number of spatial
dimensions is three or greater. In lower dimensions, one gets fractional
phases and anyonic statistics Wilczek (1982).

The third common way of addressing the question of particle statistics is to
take the _algebraic (field)_ approach Weinberg (1995), i.e., by postulating
the set of canonical relations

$[a_{i},a_{j}^{\dagger}]_{\pm}=\delta_{ij}\openone,$ (2)

where $\pm$ stands for (anti)commutator of operators (for fermions and bosons,
respectively). Starting with an assumption of a unique vacuum state, one can
build the multi-particle state space (Fock space) for two types of particle
statistics.

While these approaches agree at the level of ordinary statistics (bosons and
fermions), all of them have been criticized for their _ad hoc_ nature Messiah
and Greenberg (1964); Mirman (1973); van Enk (2019). This leaves the door open
for various generalizations, many of which resort to somewhat arbitrary
assumptions added to the quantum formalism. Earliest work along these lines
dates back to Gentile and his attempt to interpolate between two statistics
Gentile j. (1940), and since then, we have seen dozens of generalized and
exotic statistics, such as parastatistics Green (1953), quons and intermediate
statistics Greenberg (1991, 1999); Lavagno and Narayana Swamy (2010); Fivel
(1990), infinite statistics Greenberg (1990); Medvedev (1997), generalizations
of fractal and topology-dependent statistics CHEN  _et al._ (1996);
Polychronakos (1999); Cattani and Bassalo (2009); Surya (2004), ewkons
Hoyuelos and Sisterna (2016) modifications of statistics due to quantum
gravity Swain (2008); Balachandran  _et al._ (2001); Baez  _et al._ (2006),
non-commutative geometry Arzano and Benedetti (2009) and others Maslov (2009);
Trifonov (2009); Bagarello (2011); Niven and Grendar (2009).

###  I.1 Operational approach and particle statistics

So far, exotic statistics have never been observed in nature. This situation
can be interpreted at least in two ways: we need more sophisticated and
precise experiments, or (some) generalizations are in collision with basic
laws of physics (believed to hold universally). An excellent example of the
latter point is a question of the parity superselection rule (PSR) for
fermions derived from the impossibility of discriminating a $2\pi$-rotation
from the identity in three-dimensional space Wightman (1995). One may wonder
how to apply this reasoning in a more abstract scenario, such as fermions
occupying some discrete degrees of freedom (e.g., energy) where no notion of
rotation (_a priori_) exists. An elegant study was provided in a recent work
Johansson (2016) based on techniques from quantum information, showing that a
PSR violation would allow for superluminal communication. Thus, the parity
superselection rule can be derived from a more basic law, i.e., the _no
signaling_ principle Ghirardi (2013). Such an approach to physical theories
(from physical laws to mathematical formalism) resembles Einstein’s original
presentation of special relativity. In that case, a concise set of physical
postulates, namely the covariance of physical laws and the constancy of the
speed of light in all frames of reference, paved the way for the formalism of
Lorentz transformations. In the realm of quantum foundations, the application
of this methodology was particularly successful. With the pioneering work of
Hardy Hardy (2001), the field of _operational reconstructions_ of quantum
theory Hardy (2001); Dakić and Brukner (2011); Chiribella  _et al._ (2011);
Masanes and Müller (2011); Dakić and Brukner (2016); Höhn and Wever (2017) was
established where one recovers the abstract machinery of Hilbert spaces
starting from a set of information-theoretic axioms. Considering the
significance of identical particles in quantum information processing (such as
in linear optical quantum computing Knill  _et al._ (2001)), it becomes
evident that utilizing this operational approach holds significant potential
to derive particle statistics based on physically grounded assumptions. Rather
than exploring possible modifications of the existing formalism, a more
constructive approach may begin by defining a typical quantum experiment and
addressing straightforward physical questions. For example, how do we define
particle (in)distinguishability from an experimental standpoint? Is it
possible to establish a clear operational differentiation between various
types of identical particles, and if so, how do we characterize the
corresponding mathematical formalism? Our work can be understood as an attempt
to answer these questions. Along these lines, promising research studies
appeared in the context of the symmetrization postulate Goyal (2019), anyonic
statistics Neori (2016), quantum field theory D’Ariano and Perinotti (2014);
Eon  _et al._ (2022) and identical particles in the framework of generalized
probabilistic theories D'Ariano  _et al._ (2014); Dahlsten  _et al._ (2013).

###  I.2 Reconstruction, mathematical foundations and summary of the results

Following the instrumentalist approach of Hardy Hardy (2001), we study
identical quantum particles in an operationally well-defined setup composed of
laboratory primitives, such as preparations, transformations, and measurements
(see Fig. 1). Our starting point is a single quantum particle which we assume
is an ordinary quantum particle described by standard formalism and unitary
dynamics. This appears rather natural, as a single quantum particle is
insensitive to statistics. We introduce a typical apparatus for a single-
particle transformation described by a unitary channel on $d$-modes ($d\times
d$ unitary matrix) and a set of $d$ detectors at the output. For such a fixed
circuit, we investigate the scenario with multiple identical particles at the
input and analyze the probability of detecting them after the transformation.
Detectors can register only particle numbers but cannot distinguish them;
thus, indistinguishability is built in from the beginning. As we shall see,
the Fock-space structure will naturally arise as an ambient space for multi-
particle states. Two central mathematical ingredients will thus figure
prominently in our reconstruction of particle statistics:

  * 1)

unitary group $U(d)$ describing single-particle transformations, and

  * 2)

the Fock space structure encompassing multi-particle states.

Paired with the locality assumption (i.e., phase transformations acting
locally in Fock space), these two elements will determine how particles are
organized in multiparticle states. Mathematically, the problem concerns the
classification of representations of the $U(d)$ group in Fock space subjected
to locality constraint. We found a one-to-one correspondence to the well-
studied mathematical problem of characterizing completely-positive sequences
Bump and Diaconis (2002); Aissen  _et al._ (1952); Davydov (2000); Borger and
Grinberg (2015). This, in turn, provided us with a _complete categorization of
particle statistics_ based on integral polynomials. To be more precise, a list
of integers

$[q_{0},q_{1},\dots]_{\pm},$ (3)

defines a type of particle statistics, provided that $Q_{\pm}(x)=\sum_{s}(\mp
1)^{s}q_{s}x^{s}$ are polynomials with all negative ($+$) or positive ($-$)
roots. We coin the term _transtatistics_ for this generalized statistics. This
is a natural generalization of ordinary statistics into two types: fermionic-
like $[\dots]_{-}$ (_transfermions_) and bosonic-like $[\dots]_{+}$
(_transbosons_), and to the best of our knowledge, was not presented in the
literature. Ordinary statistics is the simplest possibility (degree-one)
$[1,1]_{\pm}$ with multiparticle Fock state being completely specified by
irreducible representations (IR) of $U(d)$. On the other hand, general
transtatistics requires additional quantum numbers to identify states of
indistinguishable particles; thus, _hidden symmetries_ McIntosh (1959) and
_new degrees of freedom_ emerge exclusively from these types of particles. We
discuss further physical consequences by analyzing the thermodynamics of non-
interacting gases. In doing so, we find an interesting inclusive degeneracy of
ground-states followed by _spontaneous symmetry breaking_ Peierls (1991),
which (usually) does not exist in ordinary statistics.

Symmetry is central to our reconstructions. In particular, the $U(d)$ symmetry
of single-particle transformations is uniquely related to ordinary statistics
and transtatistics. This also brings the main difference to other generalized
statistics, which rely on different symmetries. Apart from the foundational
relevance, our findings apply to quantum information and quantum many-body
physics. Concretely speaking, transtatistics brings novel theoretical models
for non-interacting identical particles. The latter is relevant for studying
strongly-correlated quantum systems (see Cirac  _et al._ (2021) and references
therein), many of which are reducible to non-interacting models of
indistinguishable particles Schultz  _et al._ (1964). Therefore, one may find
new integrable models among strongly interacting quantum systems reducible to
our non-interacting model. On the quantum information side, quantum statistics
is essential in complexity theory and intermediate quantum computing models,
such as in boson sampling Aaronson and Arkhipov (2011). In this respect, our
classification is relevant as it may lead to the discovery of new intermediate
computational models. These points are only summarized here and will be
discussed in more detail in the last section of the manuscript.

##  II Operational setup for indistinguishable particles

Figure 1: Operational setup. a) Quantum circuit represented by
$U(d)$-transformation for a single particle (example of $d=4$ is shown). For
many particles injected into the setup, detectors can register their number
only. This incorporates the notion of indistinguishability. b) Disconnected
(independent) phase gates acting on particles locally, in individual modes.

The operational framework for indistinguishable particles is illustrated in
Fig. 1$a)$. The apparatus consists of $d$ modes into which particles can be
injected, followed by a transformation $g$ and a set of $d$ detectors that
register particles after the transformation. The transformation $g$ is fixed
and independent of particle number at the input and particle-statistics type.
One can think of this transformation as a quantum circuit composed of
elementary gates, such as beam-splitters and phase shifters used in quantum
linear optics to produce a general unitary transformation $g\in U(d)$ on $d$
modes Pan  _et al._ (2012), where $U(d)$ is the set of $d\times d$ unitary
matrices. As long as just one particle is injected into the setup, e.g., in
mode $i$, the $j$th detectors will register the particle with the probability
$p_{j}=|g_{ji}|^{2}$ with $g_{ji}$ being the matrix element of $g$. In other
words, $g$ represents standard complex unitary dynamics of a single quantum
particle with $d$ levels (modes). The critical question to be answered is what
will happen if more than one particle is injected into such apparatus? To
formalize the situation, there are three points to be addressed in the first
place:

  * i)

We shall determine the ambient Hilbert space describing the multi-particle
system,

  * ii)

we have to find the corresponding representation of transformations (as
defined by the $U(d)$ group) in such a space, and

  * iii)

finally, determine the Born rule to calculate probabilities of detection
events.

To identify the Hilbert space of many particles, we use the fact that
particles are indistinguishable, i.e., detectors can register only particle
numbers (how many particles land in a particular detector without
distinguishing them). Thus the overall measurement outcome is described by a
set of numbers $(n_{1},n_{2},\dots,n_{d})$, with $n_{k}=0,1,2,\dots$. This
outcome fully specifies the physical configuration; thus, we associate to it
the measurement vector $\ket{n_{1},\dots n_{d}}$ such that the Born rule gives
detection probabilities

$p_{n_{1},\dots,n_{d}}=|\braket{n_{1},\dots,n_{d}}{\psi_{g}}|^{2},$ (4)

where $\ket{\psi_{g}}$ is the state of the system after the transformation
$g$. From here, we directly see that $\ket{\psi_{g}}\in\mathcal{F}_{d}$
resides in a Fock space defined as a span over number states, i.e.,

$\mathcal{F}_{d}=\mathrm{span}\\{\ket{n_{1},n_{2},...,n_{d}}~{}|~{}n_{k}=0,1,2,\dots
p\\}.$ (5)

Here $span$ denotes the complex linear span (hull) of basis vectors. Since
outcomes $(n_{1},n_{2},\dots,n_{d})$ are perfectly distinguishable, vectors
$\ket{n_{1},n_{2},...,n_{d}}$ form an orthonormal set. We introduced the
possibility of there being a maximal occupation number $p\in\mathbb{N}$, which
is the _generalized Pauli exclusion principle_. As we shall see, $p=1$ will
correspond to fermionic statistics, while bosons are associated with the case
$p=+\infty$. At this stage, $p$ is characteristic of statistics and is kept as
an integer parameter (possibly infinite). Note that the Fock space in (5)
shall not be _a priori_ identified with the standard (textbook) Fock space
constructed as a direct sum of particle sectors. Our Fock space is an ambient
Hilbert space for multi-particle states naturally emerging from operational
considerations and the measurement postulate defined in (4). Note also that
the Fock space in (5) is of the tensor product form, i.e.,
$\mathcal{F}_{d}=\mathcal{F}_{1}^{\otimes d}$.

Now, we shall find an appropriate unitary representation of $g\in U(d)$ in the
ambient space $\mathcal{F}_{d}$, i.e. $\Delta_{d}:U(d)\mapsto GL(\mathcal{F})$
such that

$\ket{\psi_{g}}=\Delta_{d}(g)\ket{\psi_{in}},$ (6)

with $\Delta_{d}(g)$ being unitary representation and
$\ket{\psi_{in}}\in\mathcal{F}_{d}$ is some input state to the circuit in Fig.
1$a)$. For example, $\ket{\psi_{in}}=\ket{1,1,0,\dots,0}$ represents the input
state of two particles injected in mode $1$ and $2$. In general,
$\ket{\psi_{in}}$ may involve the superposition of number states.
Representation $\Delta_{d}$ is reducible in general, and the _group character_
completely determines its decomposition into irreducible (IR) sectors Fulton
and Harris (2004), that is, a function defined over the elements of the group

$\chi_{d}(g)=\mathrm{Tr}(\Delta_{d}(g)),\quad\forall g\in U(d).$ (7)

As we shall see, the irreducible decomposition of Fock space (5) will be in
one-to-one correspondence to the type of particle statistics. So, the group
character will be our main object of interest.

###  II.1 Locality assumption

To evaluate character on $U(d)$ group, recall that any unitary matrix can be
diagonalized, i.e., $g=StS^{\dagger}$, with
$t=\mathrm{diag}[x_{1},\dots,x_{d}]\in T_{d}=U(1)\times\dots\times U(1)$ being
an element of the maximal torus (also known as the phase group) with
$x_{k}=e^{i\theta_{k}}\in U(1)$. Therefore, the character of $U(d)$ is
entirely specified by the character evaluated on $T_{d}$, that is,
$\chi_{d}(StS^{\dagger})=\mathrm{Tr}\Delta_{d}(StS^{\dagger})=\mathrm{Tr}\Delta_{d}(t)=\chi_{d}(t)$
(i.e., class function), thus it effectively becomes a function of phase
variables, i.e., $\chi_{d}(\vec{x})=\chi_{d}(x_{1},\dots,x_{d})$.

Consider the case of a single-mode ($d=1$) with the Fock space
$\mathcal{F}_{1}=\mathrm{span}\\{\ket{n}|~{}n=0,1,\dots,p\\}$ on which the
group $U(1)$ acts with representation $\Delta_{1}(x)$, with $x=e^{i\theta}$.
We can think of $\Delta_{1}(x)$ representing a simple device providing a phase
shift to the state of a single particle placed in a mode. We can now consider
the collection of $d$ such devices disconnected from each other and operating
independently in separate modes, as illustrated in Fig. 1$b)$. These
transformations form the phase group $T_{d}$ acting in the entire Fock space,
and given their operational independence, it appears natural to assume the
following.

######  Assumption 1 (Locality).

_The action of the phase group $T_{d}$ in Fock space is local, i.e.,_

$\Delta_{d}(\vec{x})=\Delta_{1}(x_{1})\otimes\dots\otimes\Delta_{1}(x_{d}),$
(8)

_for $\vec{x}\in T_{d}$._

By taking the trace of the last equation, one gets

$\chi_{d}(\vec{x})=\prod_{k=1}^{d}\chi_{1}(x_{k}),$ (9)

with $\chi_{1}(x)=\mathrm{Tr}(\Delta_{1}(x))$ being the single-mode character.
One can also go in the reversed direction, i.e., starting with the character
factorization in (9), we may derive the tensor factorization in (8), which
follows from general character theory Fulton and Harris (2004).

Assumption 1 is our central assumption. We see that the single-mode character
$\chi_{1}$, a function of a single variable, entirely specifies the character
of the whole $U(d)$. The problem then simplifies, and our goal is to determine
the most general form of $\chi_{1}(x)$ such that $\chi_{d}(\vec{x})$ in (9) is
a valid character of $U(d)$.

###  II.2 Generalized number operator and conserved quantities

What follows from Assumption 1 and factorization given in (9) is that the
single-mode character $\chi_{1}(x)$ completely specifies the character of the
whole $U(d)$ and consequently determines the decomposition of Fock space into
IR sectors. Note that the action of the single-mode phase transformation
$x=e^{i\theta}\in U(1)$ can be seen as an instance of the Hamiltonian
evolution. Thus we can write $\theta=\epsilon t/\hbar$, where $\epsilon$ is
the single-particle energy associated with this mode. With this, the
representation of the phase transformation becomes
$\Delta_{1}(e^{i/\hbar\epsilon t})=e^{i/\hbar\hat{H}t}$, where $\hat{H}$ is
the single-mode Hamiltonian (generator of phase). From the invariance under
$(2\pi)$-rotations, i.e., $e^{i(\theta+2\pi)}=e^{i\theta}$, we conclude that
all eigenvalues of $\hat{H}$ are integer multiples of $\epsilon$, that is,
$\hat{H}=\epsilon\tilde{N}$ with $\tilde{N}$ being the operator with integer
eigenvalues. This defines the _generalized number operator_ or _excitation
operator_ $\tilde{N}$. This work will consider only the case $\tilde{N}\geq
0$. Without loss of generality, we can assume $U(1)$ action to be number
preserving, thus

$\tilde{N}=\sum_{n=0}^{p}f_{n}\ket{n}\bra{n},$ (10)

with $f_{n}$ being non-negative integers. Note that $\tilde{N}$ is in general
different from the standard number operator
$\hat{N}=\sum_{n=0}^{p}n\ket{n}\bra{n}$. The two will coincide only if
$f_{n}=n$, and as we shall see, this happens only in the case of ordinary
statistics.

Finally, we can write the single-mode character
$\chi_{1}(e^{i\theta})=\mathrm{Tr}(e^{i\theta\tilde{N}})$ as

$\chi_{1}(x)=\sum_{s=0}^{+\infty}a_{s}x^{s}=x^{f_{0}}+x^{f_{1}}+\dots+x^{f_{p}},$
(11)

with $a_{s}$ being a non-negative integer. Mathematically speaking, the
formula above is the decomposition of $\chi_{1}$ into irreducible
representations of $U(1)$. For fermions, we have $\chi_{1}^{(-)}(x)=1+x$,
while for bosons $\chi_{1}^{(+)}(x)=1+x+x^{2}+\dots=\frac{1}{1-x}$.

For the case of $d$ modes, the action of the phase group in (8) becomes

$\Delta_{d}(\vec{x})=e^{i\theta_{1}\tilde{N}_{1}+\dots+i\theta_{d}\tilde{N}_{d}},$
(12)

where
$\tilde{N}_{k}=\openone^{\otimes(k-1)}\otimes\tilde{N}\otimes\openone^{\otimes(d-k)}$
are generators of local phases. The vector $\vec{x}=\theta(1,1,\dots,1)^{T}\in
T_{d}$ corresponds to the scalar $d\times d$ matrix $e^{i\theta}\openone_{d}$
commuting with all $U(d)$ matrices, thus the operator

$\tilde{N}=\sum_{k=1}^{d}\tilde{N}_{k},$ (13)

is a conserved quantity (Casimir operator) and represents the total number of
excitations. We can also write (12) as being generated by the following
Hamiltonian

$\hat{H}=\sum_{k=1}^{d}\epsilon_{k}\tilde{N}_{k},$ (14)

where $\theta_{k}=\epsilon_{k}t/\hbar$ and $\epsilon_{k}$ is are the single-
particle energies.

##  III Particle statistics and its classification

###  III.1 On exchange symmetry

In the $1$st quantization approach, particle statistics are classified via the
exchange of particles and symmetrization postulate as given in equation (1).
However, this method does not apply to the Fock-space approaches simply
because there is no particle label (they are indistinguishable). A partial
solution to this problem is to introduce _permutation of modes operator_ Stolt
and Taylor (1970)

$\Delta_{d}(\sigma)\ket{n_{1},n_{2},\dots,n_{d}}=\ket{n_{\sigma(1)},n_{\sigma(2)},\dots,n_{\sigma(d)}},$
(15)

for some permutation $\sigma\in S_{d}$ of $d$ elements. In this way, the
permutation group acts in Fock space and plays the same role as the exchange
of particles in the $1$st-quantized picture. For ordinary statistics, we have
the usual sign change, i.e.,
$\Delta_{d}(\sigma)\ket{1,1,\dots,1}=(\pm)^{\sigma}\ket{1,1,\dots,1}$, where
$(..)^{\sigma}$ denotes the parity of permutation ($+1$ for bosons and
$(-1)^{\sigma}$ for fermions). Nevertheless, the permutation of modes is only
a discrete subgroup of the group of single-particle transformation, thus
insufficient for the whole physical picture. For example, in our case, it is
the subgroup of the unitary group, i.e., $S_{d}<U(d)$. But it can also be a
subgroup of some other group, such as an orthogonal group, in which case one
gets parastatistics Ryan and Sudarshan (1963); Stoilova and der Jeugt (2008).
Therefore, to fully understand how different types of particles integrate into
multi-particle states in Fock space, one must study transformation properties
under the action of the whole group of single-particle transformations. This
work concerns $U(d)$ as our premise is that standard unitary quantum mechanics
governs the physics of one particle.

###  III.2 Physical consequences

Figure 2: Hong-Ou-Mandel effect Hong  _et al._ (1987). a) Boson bunching, and
b) fermion antibunching. See main text for details.

To illustrate how single-particle transformations affect the physical behavior
of indistinguishable particles, take an example of two particles entering the
$50/50$ beamsplitter (BS) at different ports (modes), as shown in Fig. 2. The
beam-splitter is defined via unitary matrix
$u_{bs}=\frac{1}{\sqrt{2}}\begin{pmatrix}1&1\\\ 1&-1\end{pmatrix}$. Now, if
particles are bosons, then the input state is
$\ket{1,1}=a_{1}^{\dagger}a_{2}^{\dagger}\ket{0,0}$, where
$a_{1(2)}^{\dagger}$ are bosonic ladder operators associated to two different
modes (ports of BS). The output state (after BS) is given by
$\frac{1}{2}(a_{1}^{\dagger}+a_{2}^{\dagger})(a_{1}^{\dagger}-a_{2}^{\dagger})\ket{0,0}=\frac{1}{\sqrt{2}}(\ket{2,0}-\ket{0,2})$.
We see that bosons exit the BS bunched together, and this is the well-known
Hong-Ou-Mandel effect Hong  _et al._ (1987). In contrast, if particles were
fermions, the calculation remains the same but with fermionic ladder operators
$a_{1/2}^{\dagger}$, thus we have the output state
$\frac{1}{2}(a_{1}^{\dagger}+a_{2}^{\dagger})(a_{1}^{\dagger}-a_{2}^{\dagger})\ket{0,0}=-a_{1}^{\dagger}a_{2}^{\dagger}\ket{0,0}=-\ket{1,1}$.
This means that fermions exit the BS antibunched (in different ports). These
two complementary behaviors can be deduced from the decomposition of the Fock
space (5) into IR sectors of the $U(2)$ group and action of the $u_{bs}$
element. In the case of two bosons, $U(2)$ reduces into three-dimensional
subspace $\mathrm{span}\\{\ket{2,0},\ket{1,1},\ket{2,0}\\}$ (bosonic IR) which
encompasses the bunching effect. For two fermions, we have the one-dimensional
IR spanned by $\\{\ket{1,1}\\}$ (fermionic IR), directly resulting in
fermionic antibunching.

###  III.3 Particle statistics

As explained at the beginning of this section, the group of single-particle
transformations determines the physical behavior of non-interacting
indistinguishable particles, and different types of particle statistics arise
due to the Fock space’s $U(d)$-IR decomposition. Therefore, what we mean by
classification of particle statistics is a _classification of all possible
ways_ the Fock space (5) decomposes into IR sectors, i.e.

$\mathcal{F}_{d}=\bigoplus_{\lambda}c_{\lambda}\mathcal{V}_{\lambda},$ (16)

where $\mathcal{V}_{\lambda}$ is an $U(d)$-IR. These are indexed Fulton and
Harris (2004) by a partition (Young diagram)
$\lambda=(\lambda_{1},\dots\lambda_{d})$ with
$\lambda_{1}\geq\dots\geq\lambda_{d}$, and $c_{\lambda}\in\mathbb{N}_{0}$ is
the frequency of the IR. Now, recall that the character of a representation
completely determines its decomposition into IR sectors. A well-known fact
from representation theory is that IRs of $U(d)$ have _Schur polynomials_
$s_{\lambda}(\vec{x})$ as characters (see Appendix A for definition). Thus,
equation (16) translates to decomposition of character (9) into Schur-
polynomials, i.e.

$\prod_{k=1}^{d}\chi_{1}(x_{k})=\sum_{\lambda}c_{\lambda}s_{\lambda}(\vec{x}),~{}~{}c_{\lambda}\in\mathbb{N}_{0}.$
(17)

We see that the single-mode character $\chi_{1}(x)$ _completely specifies
particle statistics_ (in the sense of definition (16)) and this is a direct
consequence of our locality assumption 1.

To clarify the point, we provide examples of bosonic and fermionic statistics.
For fermions, the maximal occupation number is $p=1$, thus the single-mode
character in (11) reduces to $\chi_{1}^{(-)}(x)=1+x$. For $d$-modes, character
(9) can be expanded as

$\displaystyle\chi_{d}^{(-)}(\vec{x})$ $\displaystyle=$
$\displaystyle\prod_{k=1}^{d}(1+x_{k})=1+(x_{1}+\dots+x_{d})+$
$\displaystyle+$
$\displaystyle(x_{1}x_{2}+\dots+x_{d-1}x_{d})+\dots+x_{1}x_{1}\dots x_{d}.$

Written in terms of Schur-polynomials, this equation reads

$\displaystyle\chi_{d}^{(-)}(\vec{x})$ $\displaystyle=$ $\displaystyle
s_{(0,0,\dots,0)}(\vec{x})+s_{(1,0,\dots,0)}(\vec{x})+s_{(1,1,\dots,0)}(\vec{x})+\dots$
(19) $\displaystyle+$ $\displaystyle s_{(1,1\dots,1)}(\vec{x}).$

This expansion corresponds to the decomposition of the Fock space
$\mathcal{F}_{d}=\bigoplus_{N=1}^{d}\mathcal{V}_{-}^{(N)}$ into fermionic
irreducible subspaces $\mathcal{V}_{-}^{(N)}$ associated with particle
sectors.

Similarly, for the case of bosons and $p=+\infty$, equation (11) reads
$\chi_{1}^{(+)}(x)=1+x+x^{2}+\dots=\frac{1}{1-x}$. For $d$ modes, (9) reads

$\displaystyle\chi_{d}^{(+)}(\vec{x})$ $\displaystyle=$
$\displaystyle\prod_{k=1}^{d}\frac{1}{1-x_{k}}=1+(x_{1}+\dots+x_{d})$
$\displaystyle+$
$\displaystyle(x_{1}^{2}+x_{1}x_{2}+x_{2}^{2}+\dots+x_{d-1}x_{d}+x_{d}^{2})$
$\displaystyle+$
$\displaystyle(x_{1}^{3}+x_{1}^{2}x_{2}+x_{1}x_{2}^{2}+x_{2}^{3}+\dots+x_{d-1}x_{d}^{2}+x_{d}^{3})\dots$

or written in terms of bosonic Schur polynomials

$\displaystyle\chi_{d}^{(+)}(\vec{x})$ $\displaystyle=$ $\displaystyle
s_{(0,0,\dots,0)}(\vec{x})+s_{(1,0,\dots,0)}(\vec{x})+s_{(2,0,\dots,0)}(\vec{x})$
(21) $\displaystyle+$ $\displaystyle s_{(3,0,\dots,0)}(\vec{x})+\dots$

Again, this corresponds to the decomposition of the Fock space
$\mathcal{F}_{d}=\bigoplus_{N=1}^{+\infty}\mathcal{V}_{+}^{(N)}$ into bosonic
irreducible subspaces $\mathcal{V}_{+}^{(N)}$ associated with particle
sectors.

An important remark is in order about the single-particle sector
$\mathcal{F}_{d}^{(1)}=\mathrm{span}\\{\ket{n_{1},n_{2},...,n_{d}}~{}|~{}\sum_{k}n_{k}=1\\}$
which is $d$-dimensional. This subspace is associated with the character
$s_{(1,0,\dots,0)}(\vec{x})=x_{1}+\dots+x_{d}$, same for bosons and fermions,
i.e. we have
$\mathcal{F}_{d}^{(1)}=\mathcal{V}_{+}^{(1)}=\mathcal{V}_{-}^{(1)}$. This is
consistent with the fact that the quantum physics of one particle is
insensitive to the type of statistics. This also agrees with our operational
setup in Fig. 1$a)$, which was defined through $d\times d$ unitary matrices
acting in the space of one particle. Such representation is called the
_standard_ or _defining_ representation.

###  III.4 Partition theorem and general statistics

Generally, not all $U(1)$-characters
$\chi_{1}(x)=\sum_{s\in\mathbb{N}_{0}}a_{s}x^{s}$ induce a valid
$U(d)$-character in (9). To see this, take a simple example of
$\chi_{1}(x)=1+x^{2}$. For two-modes equation (9) reads
$(1+x_{1}^{2})(1+x_{2}^{2})=s_{(0,0)}(x_{1},x_{2})+s_{(2,0)}(x_{1},x_{2})+s_{(2,2)}(x_{1},x_{2})-s_{(1,1)}(x_{1},x_{2})$
and we have a negative expansion coefficient $c_{(1,1)}<0$, which contradicts
$c_{\lambda}\geq 0$ in equation (16).

Next, suppose that the first $k$ coefficients in the single-mode character
expansion are zero. Then, we can always write
$\chi_{1}(x)=\sum_{s=k}^{+\infty}a_{s}x^{s}=x^{k}\sum_{s=0}^{+\infty}a_{s+k}x^{s}=x^{k}\tilde{\chi}_{1}(x)$.
For the general $d$ mode character in (9), we will have

$\chi_{d}(\vec{x})=(x_{1}\dots x_{d})^{k}\tilde{\chi}_{d}(\vec{x}).$ (22)

The term $(x_{1}\dots x_{d})^{k}=(\det g)^{k}$ equals determinant of a unitary
matrix $g$ with eigenvalues $x_{1},\dots x_{d}$. From here, we recognize that
$\chi_{d}$ and $\tilde{\chi}_{d}$ are equivalent up-to-determinant. Therefore,
without loss of generality, we will assume $a_{0}>0$.

The problem of classifying all single-mode characters that induce valid
representation of $U(d)$ involves non-trivial mathematics. Luckily, we found
an equivalent formulation to the well-studied combinatorial problem of
characterizing completely-positive sequences Bump and Diaconis (2002); Aissen
_et al._ (1952); Davydov (2000); Borger and Grinberg (2015). Details are
provided in the Appendix B together with the proof of our main theorem:

######  Theorem 1 (Partition).

For $\chi_{1}(x)=\sum_{s\in\mathbb{N}_{0}}a_{s}x^{s}$ with $a_{0}>0$, a
symmetric function $\prod_{k=1}^{d}\chi_{1}(x_{k})$ is a $U(d)$ character for
all $d\in\mathbb{N}$ if and only if the generating function is of the form

$\chi_{1}(x)=\frac{Q_{-}(x)}{Q_{+}(x)},$ (23)

where $Q_{\pm}(x)$ is an integral polynomial with all positive (negative)
roots. Furthermore $Q_{+}(0)=1$.

In other words, $Q_{\pm}(x)=c_{\pm}\prod_{i}(1\mp\alpha_{i}x)$ are polynomials
with integer coefficients, where $\alpha_{1}>\alpha_{2}>\dots>0$, $c_{+}=1$
and $c_{-}\in\mathbb{N}$. From here, it follows that $Q_{\pm}(x)$ is a
polynomial with all non-zero coefficients.

Note that we are interested only in _irreducible_ statistics, i.e., the Fock
space cannot be factorized as a tensor product
$\mathcal{F}=\mathcal{F}_{1}\otimes\mathcal{F}_{2}$, with $\mathcal{F}_{1/2}$
being associated with different particle types. Therefore, the character of
irreducible statistics cannot be factorized as
$\chi_{1}(x)=\mu_{1}(x)\nu_{1}(x)$, with $\mu_{1}(x)$ and $\nu_{1}(x)$ being
of the type (23). Thus equation (23) for irreducible statistics is either
$\chi_{1}=Q_{-}$ or $\chi_{1}=1/Q_{+}$. We conclude that statistics is of two
kinds, i.e., _fermionic-like_ $[\dots]_{-}$ and _bosonic-like_ $[\dots]_{+}$
specified by

$Q_{\pm}(x)=\sum_{s=0}^{\mathrm{deg}[Q_{\pm}]}(\mp
1)^{s}q_{s}x^{s}:=[q_{0},q_{1},\dots]_{\pm},~{}~{}q_{s}\in\mathbb{N},$ (24)

with $Q_{\pm}(x)$ being irreducible polynomials over integers satisfying
conditions in (23). The corresponding single-mode characters are $Q_{-}(x)$
and $1/Q_{+}(x)$, respectively. This classification naturally generalizes
ordinary statistics, and we term it _transtatistics_ with two possible types:
_transfermions_ (type $[\dots]_{-}$) and _transbosons_ (type $[\dots]_{+}$).
Here $\mathrm{deg}[Q_{\pm}]$ is the degree of $Q_{\pm}(x)$ to which we also
refer as _order of statistics_. Order $0$ is a trivial case, thus we assume
$\mathrm{deg}[Q_{\pm}]\geq 1$. For $[\dots]_{-}$ statistics, the generalized
Pauli principle applies with $p=Q_{-}(1)-1<+\infty$ being the maximal number
of particles per mode, while for $[\dots]_{+}$ we have $p=+\infty$. From now
on, we shall use the label $[q_{0},q_{1},\dots]_{\pm}$ to refer to a
particular type of particle statistics.

Note that one can find the eigenvalues of the excitation operator $\tilde{N}$
defined in (10) by solving the following equation

$x^{f_{0}}+x^{f_{1}}+\dots+x^{f_{p}}=\left(Q_{\pm}(x)\right)^{\mp 1}.$ (25)

##  IV Irreducible particle sectors: bosons and fermions

Ordinary statistics is order-one statistics of the type $[1,1]_{\pm}$. To
answer what makes bosons and fermions special in the whole family of
generalized statistics classified in (24), we introduce the following
assumptions:

######  Assumption 2 (Irreducibility).

_All symmetries of the system of indistinguishable particles are determined by
the $U(d)$ group._

Assumption 2 essentially states that the Fock space decomposes into $U(d)$-IR
sectors without multiplicity; thus, no additional symmetries (conserved
quantities) are present in the system. We show now that only ordinary
statistics has this property.

We start with a general single-mode character
$\chi_{1}(x)=\sum_{s=0}^{+\infty}a_{s}x^{s}$. Character equation (9) for
$d$-modes can be expanded as follows

$\displaystyle\chi_{d}(\vec{x})$ $\displaystyle=$ $\displaystyle
a_{0}^{d}+a_{0}^{d-1}a_{1}(x_{1}+\dots+x_{d})+W(\vec{x})$ $\displaystyle=$
$\displaystyle
a_{0}^{d}s_{(0,0,\dots,0)}(\vec{x})+a_{0}^{d-1}a_{1}s_{(1,0,\dots,0)}(\vec{x})+W(\vec{x}),$

where $W(\vec{x})$ is the symmetric function that contains quadratic and
higher-order terms in variables $\vec{x}=(x_{1},\dots,x_{d})^{T}$. Since Schur
polynomials of degree $l$ form the basis in the space of $l$-degree symmetric
polynomials, the constant and linear terms in the equation (IV) are already
IR-decomposed. Because assumption 2 requests no multiplicities, we have
$a_{0}=0,1$ and $a_{1}=0,1$.

Now we turn to concrete cases. For transfermions, the single-mode character
reads

$\displaystyle Q_{-}(x)$ $\displaystyle=$ $\displaystyle
c_{-}\prod_{i}(1+\alpha_{i}x)$ $\displaystyle=$ $\displaystyle
c_{-}+c_{-}\left(\sum_{i}\alpha_{i}\right)x+c_{-}\left(\sum_{i<j}\alpha_{i}\alpha_{j}\right)x^{2}+\dots$
$\displaystyle=$ $\displaystyle a_{0}+a_{1}x+a_{2}x^{2}+\dots$

with $\alpha_{1}>\alpha_{2}>\dots>0$ and $c_{-}\in\mathbb{N}$. This is
consistent with the previous analysis of (IV) only if $c_{-}=a_{0}=1$ and
$\sum_{i}\alpha_{i}=a_{1}=1$. For the quadratic coefficient in (IV) we have
$a_{2}=\sum_{i<j}\alpha_{i}\alpha_{j}=\frac{1}{2}(\sum_{i}\alpha_{i})^{2}-\frac{1}{2}\sum_{i}\alpha_{i}^{2}=\frac{1}{2}-\frac{1}{2}\sum_{i}\alpha_{i}^{2}\in\mathbb{N}_{0}$
because $Q_{-}$ is an integral polynomial. This is possible only if
$\alpha_{1}=1$ and $\alpha_{2}=\dots=0$. Thus, we recover the fermionic
character $\chi_{1}(x)=1+x$.

For the case of transbosons, we have the single-mode character

$\displaystyle 1/Q_{+}(x)$ $\displaystyle=$ $\displaystyle
1/\prod_{i}(1-\alpha_{i}x)$ $\displaystyle=$ $\displaystyle
1+\left(\sum_{i}\alpha_{i}\right)x+\left(\sum_{i}\alpha_{i}^{2}+\sum_{i<j}\alpha_{i}\alpha_{j}\right)x^{2}+\dots$
$\displaystyle=$ $\displaystyle a_{0}+a_{1}x+a_{2}x^{2}+\dots$

By the same analysis as for transfermions, we conclude $\sum_{i}\alpha_{i}=1$.
For the quadratic term in (IV), we have
$a_{2}=\sum_{i}\alpha_{i}^{2}+\sum_{i<j}\alpha_{i}\alpha_{j}=\frac{1}{2}(\sum_{i}\alpha_{i})^{2}+\frac{1}{2}\sum_{i}\alpha_{i}^{2}=\frac{1}{2}+\frac{1}{2}\sum_{i}\alpha_{i}^{2}\in\mathbb{N}_{0}$.
Again, this is satisfied only if $\alpha_{1}=1$ and $\alpha_{2}=\dots=0$. Thus
$\chi_{1}(x)=\frac{1}{1-x}$ and we recover the bosonic character. This
concludes that only bosonic and fermionic statistics are consistent with the
assumption 2.

For ordinary statistics, the excitation operator in (10) coincides with the
standard number operator. The Casimir operator in (13) becomes the total
number of particles which is a conserved quantity linked with $N$-particle
sectors

$\mathcal{F}_{d}^{(N)}=\mathrm{span}\\{\ket{n_{1},n_{2},...,n_{d}}~{}|~{}\sum_{k}n_{k}=N\\}.$
(29)

These are also $U(d)$-IR sectors associated with the standard bosonic
(fermionic) subspaces $\mathcal{V}_{\pm}^{(N)}$.

It is worth pointing out that only in the case of ordinary statistics is the
solution to the equation (25) for spectrum $f_{n}$ of the excitation operator
$\tilde{N}$ non-degenerate (in this case $f_{n}=n$). In all other cases,
degeneracy necessarily appears. This follows from the fact that coefficients
in the polynomial $Q_{\pm}(x)$ are all non-zero, and at least one of them is
$2$ or greater (otherwise, all coefficients are equal to $1$, and we have
ordinary statistics). Given this, at least one expansion coefficient on the
right-hand side of (25) is $2$ or greater. Thus at least two $f_{n}$ numbers
on the left-hand side of (25) are the same.

##  V Hidden symmetry and transtatistics

We learned from the previous analysis that multiplicities in the Fock space
decomposition (16) will necessarily appear for all transtatistics apart from
bosonic and fermionic. These multiplicities cannot be resolved without
additional, so-called _hidden symmetry_ , present in the system McIntosh
(1959). The latter is typically identified as a higher symmetry of the
Hamiltonian required to fully resolve the degeneracy of the energy spectrum
(sometimes called ‘accidental’ degeneracy). The classic example is the
degeneration of the spectrum of the hydrogen atom not captured by the
rotational symmetry ($\mathrm{SO}(3)$ group) of the Hamiltonian but requires a
higher (hidden) symmetry for resolution, which is the $\mathrm{SO}(4)$ group
Pauli Jr (1926). In our case, the situation is similar; the multiplicities in
the Fock space decomposition are in one-to-one correspondence with the
degeneration of the Hamiltonian
$\hat{H}=\sum_{{}^{k}=1}^{d}\epsilon_{k}\tilde{N}_{k}$ defined in (14)
(generator of the $U(d)$ action). The total energy is given by

$E=\sum_{k=1}^{d}\epsilon_{k}f_{k},$ (30)

with $f_{k}$ being the eigenvalues of the excitation operator $\tilde{N}$
defined in (10). As long as this operator is non-degenerate, the energy
spectrum $E$ is well-resolved with the set of quantum numbers
$(f_{k_{1}},\dots,f_{k_{d}})$. Nevertheless, we have seen that this happens
only in the case of ordinary statistics. For all other cases, degeneracy in
spectrum $f_{n}$ necessarily appears, which is to be resolved by different
quantum numbers unrelated to the $U(d)$ group. Without the specification of
these numbers, the representation of $U(d)$ in Fock space remains unspecified,
defined only up to IR-multiplicity.

We will study these effects in detail for the first non-trivial case beyond
ordinary statistics, i.e., the order-one statistics $[1,q]_{\pm}$, with
$q\in\mathbb{N}$. To make the analysis more accessible, we will separate the
notation for transbosons (type $[1,\beta]_{+}$) and transfermions (type
$[1,\alpha]_{-}$) with $\alpha,\beta\in\mathbb{N}$. The reason why we set the
first coefficient in $[\dots]_{\pm}$ to be $1$ is because we will restrict our
analysis only to the case of a unique vacuum state $\ket{0}^{\otimes d}$. To
be more precise, we will study the cases in which the only invariant state
under $U(d)$ is a vacuum state. This is possible only if the first coefficient
in the single-mode character $\chi_{1}(x)=\sum_{s=0}^{+\infty}a_{s}x^{s}$ is
set $a_{0}=1$ (see discussion around equation (IV)).

To begin with, take an example of transfermions $[1,\alpha]_{-}$ with
$\alpha=2$. In this case, the singe-mode Fock space is three-dimensional
(follows from $\chi_{1}(x)=1+2x=1+2e^{i\theta}$), and the maximal occupation
number is $p=\alpha=2$. For the case of two modes, the character reads
$\chi_{2}(x_{1},x_{2})=1+2(x_{1}+x_{2})+4x_{1}x_{2}=s_{(0,0)}(x_{1},x_{2})+2s_{(1,0)}(x_{1},x_{2})+2^{2}s_{(1,1)}(x_{1},x_{2})$.
Thus, the Fock space decomposes into fermionic multiplets of the size
$\alpha^{N}=2^{N}$, for $N=0,1,2$. This exponential growth of multiplicity is
generic to order-one transtatistics. It is formalized in the following theorem
(see Appendix C for proof)

######  Theorem 2.

Fock-spaces for $[1,\alpha]_{+}$ and $[1,\beta]_{-}$ decompose into IR sectors
as

$\displaystyle\mathcal{F}_{d}$
$\displaystyle=\bigoplus_{N=0}^{d}\alpha^{N}\mathcal{V}_{-}^{(N)},~{}~{}\alpha\in\mathbb{N},$
(31) $\displaystyle\mathcal{F}_{d}$
$\displaystyle=\bigoplus_{N=0}^{+\infty}\beta^{N}\mathcal{V}_{+}^{(N)},~{}~{}\beta\in\mathbb{N},$
(32)

where $\mathcal{V}_{-}^{(N)}$ and $\mathcal{V}_{+}^{(N)}$ are the fermionic
and bosonic IRs ($N$-particle sectors for ordinary statistics), respectively.

In the next section, we will build the concrete ansatz to identify auxiliary
quantum numbers to resolve the degeneracy in (31)-(32). Based on this, we will
construct the $U(d)$ representation in Fock space.

###  V.1 Hidden quantum numbers

We start with transfermions $[1,\alpha]_{-}$ for some $\alpha\geq 2$. For this
case, the single-mode character reads $\chi_{1}(x)=1+\alpha x$ with
$x=e^{i\theta}\in U(1)$. The single-mode Fock space is
$(\alpha+1)$-dimensional
$\mathcal{F}_{1}=\mathrm{span}\\{\ket{n}~{}|~{}n=0,1,2,\dots,p=\alpha\\}$.
Equation (25) reads

$x^{f_{0}}+x^{f_{1}}+\dots+x^{f_{p}}=1+\alpha x,$ (33)

with the solution $f_{0}=0$ and $f_{n}=1$ for $n=1,\dots,\alpha$. The
generator of $U(1)$ action $\Delta_{1}(e^{i\theta})=e^{i\theta\tilde{N}}$ is
the excitation operator defined in (10) and in our case, it acts is as follows

$\tilde{N}\ket{n}=\begin{cases}0&n=0,\\\
+1\ket{n}&n=1,\dots,\alpha.\end{cases}$ (34)

Given this, one can re-interpret the single-mode states $\ket{n}$ for $n\geq
1$ as _de facto_ being the single-particle excitations distinguished by some
auxiliary degree of freedom with $\alpha$ values. Therefore, we can introduce
decomposition $n=k+z$ with $k=0,1$ being the ‘real’ occupation number of the
fermionic type and $z=0,\dots\alpha^{k}-1$ as an auxiliary quantum number
accounting for degeneracy. Having this, the formula (34) takes the standard
form, i.e $\tilde{N}\ket{k+z}=k\ket{k+z}$. Now, to separate degrees of freedom
captured by $k$ and $z$ quantum numbers, we introduce the mapping

$L_{1}\ket{k+z}=\begin{cases}\ket{0}_{F}&k=0,\\\
\ket{1}_{F}\otimes\ket{z}_{A}&k=1,\end{cases}$ (35)

where $\ket{k}_{F}$ is the ordinary fermionic number state with $k=0,1$, while
$\ket{z}_{A}$ (with $z=0,\dots\alpha-1$) is a new degree of freedom emerged
solely from the _statistics type_. The ansatz straightforwardly generalizes to
the $d$-mode Fock space. We define

$L_{d}\ket{n_{1},\dots,n_{d}}=\mathcal{T}L_{1}^{\otimes
d}\ket{n_{1},\dots,n_{d}},$ (36)

where $\mathcal{T}$ is the shift operator needed to separate degrees of
freedom, i.e., to shift all auxiliary states to the right. For example,
$\mathcal{T}\ket{k_{1}}\ket{z_{1}}\ket{k_{2}}\ket{z_{2}}=\ket{k_{1},k_{2}}_{F}\otimes\ket{z_{1},z_{2}}_{A}$.
To fully clarify the mapping in (36), let
$\ket{n_{1},\dots,n_{d}}=\ket{k_{1}+z_{1},\dots,k_{d}+z_{d}}$, where again,
$k_{s}=0,1$ and $z_{s}=0,\dots,\alpha^{k_{s}}-1$. We form the ordered list
$(z_{s_{1}},\dots,z_{s_{N}})$ for which $k_{s_{r}}=1$, i.e. the list of all
non-zero fermionic excitations. Here
$N=d-(\delta_{0,n_{1}}+\dots+\delta_{0,n_{d}})$ is the total number of them.
Then, the equation (36) reads

$L_{d}\ket{n_{1},\dots,n_{d}}=\ket{k_{1},\dots,k_{d}}_{F}\otimes\ket{z_{s_{1}},\dots,z_{s_{N}}}_{A}$
(37)

where $\ket{k_{1},\dots,k_{d}}_{F}$ is the ordinary $N$-particle fermionic
state, with the auxiliary label of particles
$\ket{z_{s_{1}},\dots,z_{s_{N}}}_{A}$. This brings us precisely to the
decomposition in (31), which can also be written as

$\mathcal{F}_{d}=\bigoplus_{N=0}^{d}\mathcal{V}_{-}^{(N)}\otimes\mathcal{H}_{A}^{\otimes
N},$ (38)

where $\mathcal{H}_{A}=\mathrm{span}\\{\ket{z}|~{}z=0,\dots\alpha-1\\}$ is the
auxiliary space. Given this factorization, it is clear that $U(d)$ acts only
in the fermionic part $\mathcal{V}_{-}^{(N)}$, while $\mathcal{H}_{A}^{\otimes
N}$ remains untouched. The additional $U(\alpha)$ group acting in the space
$\mathcal{H}_{A}^{(N)}$ can be added to resolve degeneracy completely. Now,
for an element $g\in U(d)$, let the standard action on the fermionic number
state is $\Delta_{d}^{(F)}(g)\ket{k_{1},\dots,k_{d}}_{F}$. This induces the
action $\Delta_{d}(g)$ in the Fock space (5) as

$\Delta_{d}(g)=L_{d}^{-1}\left(\Delta_{d}^{(F)}(g)\otimes\openone_{A}\right)L_{d},$
(39)

where $L_{d}$ is the mapping given in(36). With this, we have defined the
action of $U(d)$ in the Fock space.

In complete analogy, we provide an ansatz for transbosons of $[1,\beta]_{+}$
type with $\beta\geq 2$. In this case, we have the single mode character
$\chi_{1}(x)=\frac{1}{1-\beta x}$ with $x=e^{i\theta}\in U(1)$ and the single-
mode Fock space is infinte-dimensional
$\mathcal{F}_{1}=\mathrm{span}\\{\ket{n}~{}|~{}n=0,1,2,\dots\\}$. As before,
we shall solve equation (25)

$x^{f_{0}}+x^{f_{1}}+x^{f_{2}}+\dots=\frac{1}{1-\beta x}.$ (40)

It is convenient to write the particle number $n$ in the form

$n=1+\beta^{2}+\dots+\beta^{k-1}+z=\frac{\beta^{k}-1}{\beta-1}+z,$ (41)

with $k=0,1,2,\dots$ and $z=0,\dots,\beta^{k}-1$. Having this notation, the
solution to (40) is simple, i.e., $f_{\frac{\beta^{k}-1}{\beta-1}+z}=k$. We
have the following action of the excitation operator $\tilde{N}$

$\tilde{N}\ket{\frac{\beta^{k}-1}{\beta-1}+z}=k\ket{\frac{\beta^{k}-1}{\beta-1}+z}.$
(42)

Here $k$ represents the ‘new’ occupation number of the bosonic type, while $z$
is an auxiliary quantum number. Since $z=0,\dots,\beta^{k}-1$ counts all
possible states associated to $k$ bosonic excitations, it is convenient to
write $z$ in the $\beta$-base, i.e.
$z=z_{k-1}\beta^{k-1}+z_{k-2}\beta^{k-2}+\dots+z_{0}\beta^{0}:=z_{k-1}z_{k-2}\dots
z_{0}$, where $z_{s}=0,\dots,\beta-1$ are the digits. With this, we can
introduce the mapping

$L_{1}\ket{\frac{\beta^{k}-1}{\beta-1}+z}=\begin{cases}\ket{0}_{B}&k=0,\\\
\ket{k}_{B}\otimes\ket{z_{k-1}z_{k-2}\dots z_{0}}_{A}&k>0,\end{cases}$ (43)

where $\ket{k}_{B}$ is the ordinary bosonic Fock (number) state with
$k=0,1,2,\dots$, while $\ket{z_{k-1}z_{k-2}\dots z_{0}}_{A}$ (with
$z_{s}=0,\dots,\beta-1$) is associated to the statistics degree of freedom.
The generalization to the $d$-mode Fock state is as for the case of
transfermions, i.e., we use the same equation (36). In this case, we have

$L_{d}\ket{n_{1},\dots,n_{d}}=\ket{k_{1},\dots,k_{d}}_{B}\otimes\ket{z_{s_{1}},\dots,z_{s_{N}}}_{A},$
(44)

where $\ket{k_{1},\dots,k_{d}}_{B}$ is the ordinary bosonic number state with
$k_{s}=0,1,2,\dots$, while $\ket{z_{s_{1}}\dots z_{s_{N}}}_{A}$ comes from
type of statistics. The action of $U(d)$ is introduced in complete analogy to
the fermionic case and equation (39).

###  V.2 Is hidden symmetry an ordinary internal symmetry?

We may question if the hidden quantum numbers introduced in the previous
section are related to some genuine degree of freedom emerging from the type
of statistics. Could these numbers be associated with the standard internal
degrees of freedom, such as spin? For example, degeneration in (30) could be
potentially explained by the argument that energy is spin-independent, and
then, transatistics may be just an ordinary (fermionic of bosonic) statistics
where $U(d)$ affects only external degrees of freedom (such as modes
represented by paths of particles in Fig. 1$a$). However, this argument cannot
be well-aligned with the Fock-state decomposition in (31)-(32), even though
only multiplets of ordinary statistics appear in decomposition. This is due to
the dimension discrepancy between ordinary statistics and transtatistics. To
see this, suppose that we deal with ordinary fermions with $d$ real degrees of
freedom ($d$ modes on which $U(d)$ acts) and some internal degree of freedom
(e.g., spin) with $z=0,\dots,\alpha-1$ values, which is unaffected by $U(d)$.
The overall dimension of the single-particle space is $\alpha d$; hence the
dimension of the fermionic Fock space is $2^{\alpha d}$. This starkly
contrasts the dimension $\alpha^{d}$ of the transfermionic Fock space for
$[1,\alpha]_{-}$ type. As we shall see, this dimension discrepancy will
differentiate the thermodynamics of non-interacting systems of ordinary and
transtatistics. The latter will be accompanied by the effect of generic
spontaneous symmetry breaking absent in ordinary statistics.

Note that analogy to, e.g., spin degree of freedom discussed here is only
possible for order-one statistics. For higher-order statistics, no (obvious)
similarities can be concluded. We will discuss this point later.

###  V.3 Relation to thermodynamics

To study thermodynamics, we consider the single-particle energy spectrum
$\epsilon_{1},\dots,\epsilon_{d}$, where $\epsilon_{k}$s represent the
energies associated with different modes. This situation is similar to the one
discussed in Section II.2. In that section, we examined the unitary evolution
generated by the Hamiltonian given in equation (14). However, the system is in
contact with a thermal bath in the present case. The thermodynamical
quantities (e.g., for canonical ensemble) can be derived from the partition
function $Z_{d}(\beta)=\mathrm{Tr}e^{-\beta\hat{H}}$, and its explicit form
follows directly from the form of $H$, i.e.,

$Z_{d}(\beta)=\prod_{k=1}^{d}Z_{1}(e^{-\beta\epsilon_{k}})=\prod_{k=1}^{d}\chi_{1}(e^{-\beta\epsilon_{k}})$
(45)

with $\beta=1/k_{B}T$ being the Boltzmann factor.

The physical relevance of character $\chi$ can also be understood through
thermodynamics Balantekin (2001). This is because we can get the partition
function from the character via Wick’s rotation, that is,
$it/\hbar\epsilon_{k}\rightarrow-\beta\epsilon_{k}$. In this respect, the
product form of equation (45) arises directly from our central assumption 1,
i.e., the overall partition function can be expressed as a product of
individual partition functions (associated with individual modes). This aligns
with the expected behavior for independent systems, such as a set of
independent modes. Therefore, assumption 1 is in one-to-one correspondence
with the independence in a thermodynamical sense. The formula (45) trivially
holds for ordinary statistics (bosons and fermions) Ashcroft and Mermin
(1976).

When the system is capable of exchanging excitations (particles) with a
reservoir, we can analyze its behavior using the grand canonical partition
function $\mathcal{Z}_{d}=\mathrm{Tr}e^{-\beta(\hat{H}-\mu\tilde{N})}$. In
this expression, $\tilde{N}$ represents the excitation operator as defined in
equation (10), and $\mu$ corresponds to the chemical potential (variable
conjugated to $\tilde{N}$). The explicit form of the grand canonical partition
function $\mathcal{Z}_{d}$ is as follows

$\mathcal{Z}_{d}=\prod_{k=1}^{d}Z_{1}(e^{-\beta(\epsilon_{k}-\mu)})=\prod_{k=1}^{d}\chi_{1}((e^{-\beta\epsilon_{k}-\mu)}).$
(46)

###  V.4 Thermodynamics of ideal gasses and spontaneous symmetry breaking

Figure 3: Mean particle number for ordinary (blue and green) and generalized
(orange and red) statistics.

Let us examine the thermodynamical properties of a non-interacting system for
general order-one statistics $[1,q]_{\pm}$ with $q\in\mathbb{N}$. Ordinary
statistics is recovered for $q=1$. We consider a grand-canonical ensemble
defined by a set of single-particle energies $\epsilon_{1},\dots,\epsilon_{d}$
associated with different modes. The system is described by a equilibrium
state $\rho=\frac{1}{\mathcal{Z}_{d}}e^{-\beta(\hat{H}-\mu\tilde{N})}$, where
$\mathcal{Z}_{d}$ is the grand-canonical partition function defined in (46).
All thermodynamical quantities can be evaluated from the grand canonical
potential $\Omega=-\frac{1}{\beta}\log\mathcal{Z}_{d}$. For example,
$N=\frac{\partial\Omega}{\partial\mu}$ gives the mean particle number. For the
case of transtatistics $[1,q]_{\pm}$, we get

$N=\sum_{i}n_{i}=\sum_{i}\frac{1}{\frac{1}{q}e^{\beta(\epsilon_{i}-\mu)}\pm
1}.$ (47)

This expression reduces to the Fermi-Dirac and Bose-Einstein distributions for
$q=1$. The plots for $n_{i}$ in (47) for various statistics are presented in
Fig. 3. For the fermionic-type statistics, equation (47) reduces to the Fermi-
Dirac distribution $n_{i}=\theta(\mu-\epsilon_{i})$ at zero temperature for
all $q$. For the bosonic type, the mean number diverges at the values of
energy $\epsilon=\mu+\frac{1}{\beta}\log q$ when the Bose-Einstein
condensation occurs. In the classical limit of $\beta(\epsilon-\mu)\gg 1$, the
formula (47) reduces to the standard Maxwell-Boltzmann distribution, i.e.,
$n_{i}\approx qe^{-\beta(\epsilon_{i}-\mu)}$, where the factor $q$ appears as
the degeneracy factor. The same factor appears in the classical limit for
standard quantum gasses with $q=2s+1$ coming from spin $s$ (see, for example,
Chapter 8.3. in Salinas (2001)). This is because the energy is independent of
spin, and thus, the energy spectrum degenerates.

Note that the chemical potential $\mu$ in the formulas above is temperature
dependent. To be more precise, the standard approach to thermodynamics of
ideal gasses is to keep total particle number $N$ as a fixed parameter and
then invert (47) to calculate chemical potential $\mu=\mu(N,T)$ as a function
of a total number of particles and temperature Ashcroft and Mermin (1976).
Given this, one can introduce a simple change of variables
$\mu\rightarrow\mu-\frac{1}{\beta}\log q$, and the formula (47) would reduce
to one for ordinary statistics. This means that solution for the chemical
potential for order-one transtatistics is

$\mu_{q}=\mu_{q=1}-k_{B}T\log q,$ (48)

where $\mu_{q=1}$ is the chemical potential of ordinary statistics. What
follows is that almost all thermodynamical quantities (e.g., mean energy, heat
capacity, etc.) remain the same as in the case of ordinary statistics for
arbitrary $q$. Nevertheless, the entropy will change. To see this, note that
$S=-\beta\Omega+\beta\langle E\rangle-\beta\mu N$, thus the shift of
$-k_{B}T\log q$ in the chemical potential introduces a change in the entropy,
i.e.

$S_{q}=S_{q=1}+k_{B}N\log q.$ (49)

The entropy of ordinary statistics $S_{q=1}$ vanishes at $T=0$; hence, a
residual entropy of $k_{B}N\log q$ remains at zero temperature for all $q>1$.
This is consistent with the fact that fermionic (bosonic) $N$-particles IRs in
the Fock space decomposition (31)-(32) appear $q^{N}$ times; therefore, the
ground state is $q^{N}$ times degenerate. This degeneration is known to result
in _residual entropy at zero temperature_ and is associated with _spontaneous
symmetry breaking_ Peierls (1991), here present for transtatistics. This is
one of the main differences compared to ordinary quantum gasses exhibiting
non-degenerate ground states.

##  VI Discussion and outlook

###  VI.1 Statistics of higher order

Here we briefly analyze some of the technical and conceptual difficulties that
arise when dealing with statistics of higher order. As an illustration, we
take the example of statistics of order two $[1,q_{1},q_{2}]_{\pm}$. A simple
inspection shows that polynomial $Q_{\pm}(x)=1\mp q_{1}x+q_{2}x^{2}$ has non-
negative (positive) roots for $q_{1}^{2}>4q_{2}$. To see how the Fock space
decomposes in some simple cases, consider transfermions $[1,q,1]_{-}$ and the
corresponding two-mode character

$\displaystyle\chi_{2}(x_{1},x_{2})$ $\displaystyle=$
$\displaystyle(1+qx_{1}+x_{1}^{2})(1+qx_{2}+x_{2}^{2})$ (50) $\displaystyle=$
$\displaystyle s_{(0,0)}(x_{1},x_{2})+qs_{(1,0)}(x_{1},x_{2})+$
$\displaystyle=$
$\displaystyle(q^{2}-1)s_{(1,1)}(x_{1},x_{2})+s_{(2,0)}(x_{1},x_{2})+$
$\displaystyle+$ $\displaystyle
qs_{(2,1)}(x_{1},x_{2})+s_{(2,2)}(x_{1},x_{2}).$

The IR characters $s_{(2,1)}$ and $s_{(2,2)}$ that are not fermionic nor
bosonic type show-up in the decomposition. This is a typical feature that
appears for any higher-order statistics. In turn, finding Fock space’s
decomposition for general $d$ modes, such as one provided for order-one
statistics in (31)-(32), is more difficult. Next, the dimension of the single-
mode Fock space is $q+2$, and the maximal occupation number is $p=q+1$. The
solution to the single-mode character equation (25)

$x^{f_{0}}+\dots+x^{f_{q+1}}=1+qx+x^{2}$ (51)

is $f_{0}=0$ and $f_{q+1}=2$, while $f_{n}=1$ for $n=1,\dots,q$. Recall that
these are the eigenvalues of the excitation operator $\tilde{N}$ in (10), and
as we see, we have three distinct values $k=0,1,2$. Again, we have
degeneration of the spectrum, but resolving it is a more delicate issue than
for the case of order-one statistics we have presented in Section V. This is
partially because a clear interpretation is missing. For example, we may try
to label the single-mode Fock states with two quantum numbers, $k=0,1,2$ (for
excitations), and one auxiliary number $z_{k}$, to account for degeneracy. As
before, we have $\tilde{N}\ket{k+z_{k}}=k\ket{k+z_{k}}$, with $z_{k}=0$ for
$k=0,2$, while for $k=1$ we have $z_{k}=0,\dots,q$. This appears paradoxical
because degeneracy is present for one excitation but disappears for two.
Unfortunately, due to such issues in categorizing hidden quantum numbers and
the involvement of ‘non-standard’ IRs, the analysis becomes significantly more
complicated, and we leave it for future investigations.

###  VI.2 Relation to other generalized statistics

An obvious question is if and how our statistics classified in (24) differs
from other generalized statistics presented in the literature. Of course, we
are not in the position to exhaustively compare but rather analyze the most
common cases. The first remark is that the main difference is due to the
underlying symmetries. Our classification relies on the $U(d)$ group, while in
most of the cases, other generalized statistics is based on a different group.
Take an example of fractal statistics Wilczek (1982); Leinaas and Myrheim
(1977) where topological defects and representation of braid groups
Fredenhagen  _et al._ (1989) play the central role. We can have, for example,
the action of $2\pi$-rotation, leaving a non-trivial phase. This contrasts the
$2\pi$-periodicity, essential to derive the integer spectrum for the
excitation operator in our equation (10). This suggests that we speak of
different kinds of particle statistics due to the involvement of different
symmetry groups. On the other hand, the recent work of Bernevig and Haldane
(2008) suggests that fractal statistics can be phrased in terms of Jack
polynomials, which generalize Schur polynomials (our primary tool to classify
statistics). This relationship is worth looking into in the future.

Very similar holds for many generalized statistics related to deformed
canonical commutation relations. Take an example of $q$-deformations (quons)
with $a_{i}a^{\dagger}_{j}-qa^{\dagger}_{j}a_{i}=\delta_{ij}\openone$
Greenberg (1991). However, $q$-deformations introduce new symmetries even at
the level of a single particle, i.e., $q$-deformed $U(d)$ Jimbo (1985) group,
while our statistics is directly paired to the $U(d)$ symmetry. Still, some
comparison might be possible for the order-one statistics, where our ansatz of
Section V provides the means to construct the algebra of creation and
annihilation operators and evaluate the corresponding commutation relations.

Finally, the question is how our generalization is related to parastatistics
Green (1953). As already pointed out, the group behind the parastatistics is
different Ryan and Sudarshan (1963); Stoilova and der Jeugt (2008). This leads
to the different Fock space decomposition, i.e., for parastatistics of order
$p$, we have HARTLE and TAYLOR (1969); Stoilova and Van der Jeugt (2020)

$\displaystyle\mathcal{F}_{\text{parab}}$
$\displaystyle=\bigoplus_{l(\lambda)\leq p}\mathcal{V}_{\lambda}$ (52)
$\displaystyle\mathcal{F}_{\text{paraf}}$
$\displaystyle=\bigoplus_{l(\lambda^{\prime})\leq p}\mathcal{V}_{\lambda},$
(53)

where the sum runs over Young diagrams $\lambda$ (parabose case) or
$\lambda^{\prime}$ (parafermi case) of the length $l(\lambda)$ (number of
rows). Here $\lambda^{\prime}$ is the conjugated diagram of $\lambda$ and
$\mathcal{V}_{\lambda}$ is an $U(d)$-IR associated to $\lambda$. This
decomposition contains no multiplicities and thus is compatible with our
classification only for the case of ordinary statistics.

###  VI.3 Some open questions and applications

The broad range of possibilities for generalized statistics introduced here
leaves many interesting open questions and potential for applications.
Firstly, an open question is what is more on the physical side (compared to
new effects already discussed) that tanstatistics brings. As we already
discussed, there are technical difficulties with higher-order statistics,
mainly in the context of hidden quantum numbers. Nevertheless, we may study
thermodynamics directly by the ansatz defined in section V.4. One has to
calculate partition functions given in (45) for more general characters. In
this case, a simple shift of the chemical potential in (48) will not reduce
thermodynamical quantities to ones given by ordinary statistics as it happens
for order-one statistics. Given this, we can expect other novel physical
effects to appear.

The next exciting point to analyze is the application of our method to
diagonalize solid-state Hamiltonians, such as it is done for spin-chains via
spin-fermion mapping (Jordan-Wigner transformation) Schultz  _et al._ (1964).
For example, the transfermionic Fock space for $[1,\alpha]_{+}$ is isomorphic
to $(\mathbb{C}^{(\alpha+1)})^{\otimes d}$ which is suitable to study higher
dimensional spin chains. In complete analogy to the spin-fermion mapping, one
can expect to find other integrable many-body Hamiltonians that reduce to our
non-interacting model.

An interesting perspective on our results comes from the quantum computational
complexity of quantum statistics. Namely, it is well-known that the non-
interacting bosons are computationally hard to simulate Aaronson and Arkhipov
(2011), while non-interacting fermions are not Terhal and DiVincenzo (2002).
One can ask a similar question here, i.e., what is the computational power of
the non-interacting model for transtatistics? Any answer to it is relevant and
may find applications in quantum computing.

Finally, on the speculative side, an interesting idea of applying generalized
statistics in the context of dark-matter modeling was recently presented
Hoyuelos and Sisterna (2022). The main point is to study thermodynamics and
the effects of the negative relation between pressure and energy density,
emphasized in many existent dark energy candidates. Our methods provide a
direct way to calculate thermodynamical properties of transtatistics and thus
might be worthy of investigating relations to dark-matter models.

###### Acknowledgements.

The authors thank S. Horvat, J. Morris and Č. Brukner for their helpful
comments. This research was funded in whole, or in part, by the Austrian
Science Fund (FWF) [F7115] (BeyondC). For the purpose of open access, the
author(s) has applied a CC BY public copyright licence to any Author Accepted
Manuscript version arising from this submission.

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##  Appendix A Schur polynomials

The linear representations of the general linear group $GL(n,\mathbb{C})$ and
its maximally compact subgroup $U(n)$ are unambiguously identified by _Schur
polynomials_ as their characters Fulton and Harris (2004). These families of
polynomials appear in different regions of mathematics, from pure
combinatorics to algebraic geometry. This is why several standard definitions
exist depending on the specific context they are discussed. Here we present
them in the combinatorial definition to emphasize the combinatorial of our
operational reconstruction of particle statistics. Other methods, such as the
classical (determinant) definition, are standardly found in the literature
Stanley and Fomin (1999).

Let $\lambda=(\lambda_{1},\lambda_{2},\dots)$ be an integer partition with
$\lambda_{1}\geq\lambda_{2}\geq\dots$, usually represented with a _Young
diagram_ (see Figure 4). The total number of boxes in a diagram is denoted by
$|\lambda|=\lambda_{1}+\lambda_{2}+\dots$, and the partition length (the
number of rows) is labeled by $l(\lambda)$.

Figure 4: Young diagram for $\lambda=(6,3,1)$. The filling of the diagram
defines one SSYT.

The _semistandard_ Young tableau (SSYT) of shape $\lambda$ is a _filling_ of
the boxes in the Young diagram with positive integers such that the entries
weakly increase along each row and strictly increase down each column. For
given a SSYT $T$, we define the _type_ of $T$, i.e.,
$\alpha(T)=(\alpha_{1},\alpha_{2},\dots)$ with $\alpha_{i}(T)$ being the
number of repetitions of the number $i$ in $T$. For example, for the SSYT
given in Fig. 4, we have $\alpha(T)=(1,4,3,2)$. For a set of variables
$x_{1},x_{2},\dots$ we define

$x^{T}\equiv x_{1}^{\alpha_{1}(T)}x_{2}^{\alpha_{2}(T)}\dots.$ (54)

######  Definition 1.

Let $\lambda$ be a partition. The _Schur function_
$s_{\lambda}(x_{1},\dots,x_{d})$ in $d\geq l(\lambda)$ variables associated
with $\lambda$ is a homogeneous symmetric polynomial of degree $|\lambda|$
defined as:

$s_{\lambda}(x)=\sum_{T\in\mathrm{SSYT}_{\lambda}(d)}x^{T},$ (55)

where the sum runs over all semistandard Young tableaux (SSYTs) of shape
$\lambda$ with the filling from the set $\\{1,\dots,d\\}$.

The common examples are the bosonic Schur functions (_homogeneous symmetric
polynomials_)

$s_{(n,0,\dots,0)}(x_{1},\dots,x_{d})=\sum_{k_{1}+\dots+k_{d}=n}x_{1}^{k_{1}}\dots
x_{d}^{k_{d}},~{}~{}~{}~{}k_{s}=0\dots d,$ (56)

associated with partitions having a single row of size $n$ ($n$-particle
sector), and the fermionic Schur functions (_elementary symmetric
polynomials_)

$s_{(1,\dots,1,0,\dots,0)}(x_{1},\dots,x_{d})=\sum_{1\leq
i_{1}<i_{2}\dots<i_{n}\leq d}x_{i_{1}}x_{i_{2}}\dots x_{i_{n}},$ (57)

associated with partitions having a single column of size $n$ ($n$-particle
sector). Notably, these bosonic and fermionic partitions are conjugate to each
other. Two partitions are said to be conjugate if they can be obtained from
each other by interchanging rows and columns in their Young diagrams.

##  Appendix B Proof of main (partition) theorem

Before proceeding with the proof, we must introduce some basic definitions and
important known theorems.

######  Definition 2.

We call $f(x)=\sum_{n}a_{n}x^{n}$ the generating function of sequence
$\\{a_{n}\\}$.

######  Definition 3.

A matrix $(A_{ij})_{i,j\in I}$ is called Töplitz if $A_{i,j}=a_{i-j}$.

######  Definition 4.

A sequence of real numbers $\\{a_{n}\\}_{n\in\mathbb{Z}}$ is totally positive
if and only if all the minors of the Töplitz matrix
$(a_{i-j})_{i,j\in\mathbb{Z}}$ are non-negative. For sequences defined only on
non-negative integers $\\{a_{n}\\}_{n\in\mathbb{N}_{0}}$, we assume the
extension $a_{n}=0$ for $n<0$. Generating function $f(x)=\sum_{n}a_{n}x^{n}$
is called totally positive if and only if $\\{a_{n}\\}$ is totally positive.

######  Proposition 1 (Bump and Diaconis (2002)).

Let $f(x)=\sum_{n}a_{n}x^{n}$ and
$D_{d-1}^{\lambda,\mu}(f)=\det(a_{\lambda_{i}-\mu_{j}-i+j})_{1\leq i,j\leq d}$
with $\lambda$ and $\mu$ being a pair of partitions of possibly different
integers. Then every Töplitz minor of matrix $(a_{i-j})_{i,j\in\mathbb{Z}}$ is
of the form $D_{d-1}^{\lambda,\mu}(f)$. Furthermore, we have

$D_{d-1}^{\lambda,\mu}(f)=\int_{U(d)}\Phi_{n,f}(g)\overline{s_{\lambda}(g)}s_{\mu}(g)dg$
(58)

with $\Phi_{d,f}=f(x_{1})...f(x_{d})$ where $x_{1},...,x_{d}$ are the
eigenvalues of $g$, and $\int_{U(d)}\dots dg$ is the Haar measure integral.

######  Proposition 2 (Aissen  _et al._ (1952)).

The sequence $\\{a_{n}\\}_{n\in\mathbb{N}_{0}}$ with $a_{0}=1$ is totally
positive if and only if it is generated by a function $f(x)$ of the form

$f(x)=e^{\gamma x}\frac{\Pi_{i}(1+\alpha_{i}x)}{\Pi_{i}(1-\beta_{i}x)}$ (59)

with $\alpha,\beta,\gamma\geq 0$ and $\sum\alpha_{i},\sum\beta_{i}$
convergent.

The last proposition is the most prominent result for characterizing totally-
positive sequences. We can prove now the following simple lemma (an almost
equivalent statement was presented in Borger and Grinberg (2015); Davydov
(2000)).

######  Lemma 1.

An integral series $f(x)=\sum_{n=0}^{\infty}a_{n}x^{n}$ with $a_{0}>0$ is
totally positive if and only if it is of the form

$f(x)=\frac{g(x)}{h(x)},$ (60)

for some integral polynomials $g(x)$ and $h(x)$ with $h(0)=1$, such that all
complex roots of $g(x)$ are negative and all those of $h(x)$ are positive real
numbers.

###### Proof.

A sequence $\\{a_{n}\\}_{\mathbb{N}_{0}}$ is totally positive if and only if
$\\{ra_{n}\\}_{\mathbb{N}_{0}}$ is totally positive for $r>0$. We set
$r=1/a_{0}$ and by the Proposition 2, we get that

$f(x)=a_{0}e^{\gamma x}\frac{\Pi_{i}(1+\alpha_{i}x)}{\Pi_{i}(1-\beta_{i}x)}$
(61)

Because $\sum_{i}(\alpha_{i}+\beta_{i})<+\infty$, the function $f$ is
meromorphic in $|x|\leq 1$ with a finite number of poles and zeros inside the
unit disc. By the theorem of Salem Salem (1945), this function is rational,
i.e., of the form $g(x)/h(x)$. It is not difficult to show that for rational
function $g(x)/h(x)=\sum_{n\in\mathbb{N}_{0}}a_{n}x^{n}$ with $a_{n}$
integers, polynomials $g(x)$ and $h(x)$ are relatively prime with $h(0)=1$
(see exercise $2(a)$ in Chapter 4 of Stanley (2011)). ∎

Having these in mind, we can write down the proof of the Partition theorem 1.
For clarity, we repeat the statement.

######  Theorem 1 (Partition).

For $\chi_{1}(x)=\sum_{s\in\mathbb{N}_{0}}a_{s}x^{s}$ with $a_{0}>0$, a
symmetric function $\prod_{k=1}^{d}\chi_{1}(x_{k})$ is a $U(d)$ character for
all $d\in\mathbb{N}$ if and only if the generating function is of the form

$\chi_{1}(x)=\frac{Q_{-}(x)}{Q_{+}(x)},$ (62)

where $Q_{\pm}(x)$ is an integral polynomial with all positive (negative)
roots. Furthermore $Q_{+}(0)=1$.

###### Proof.

Clearly $\\{a_{n}\\}$ is integral because $\chi_{1}(x)$ is a character of
$U(1)$. The character
$\chi_{d}(x_{1},...,x_{d})=\prod_{k=1}^{d}\chi_{1}(x_{k})$ is a class function
over $U(d)$ and $x_{1},...,x_{d}$ are variables on the maximal torus in
$U(d)$, i.e., $x_{k}=e^{i\theta_{k}}$. For any symmetric function
$f(x_{1},...,x_{d})$, the notation $f(g)$ assumes $f$ being evaluated at
eigenvalues $x_{1},\dots,x_{d}$ of matrix $g\in U(d)$. Now, as a consequence
of the Littlewood-Richardson rule, the symmetric function
$\chi_{d}(x_{1},...,x_{d})$ is Schur-positive (expands in non-negative
coefficients over Schur polynomials) if and only if the product
$s_{\mu}(x_{1},...,x_{d})\chi(x_{1},..,x_{d})$ is also Schur-positive. Using
this, we expand $s_{\mu}(x_{1},...,x_{d})\chi(x_{1},...,x_{d})$ in Schur
polynomials as

$s_{\mu}(x_{1},...,x_{d})\chi_{d}(x_{1},...,x_{d})=\sum_{\lambda}c_{\lambda}s_{\lambda}(x_{1},...,x_{d})$
(63)

where the sum runs over Young diagrams $\lambda$ and $c_{\lambda}\geq 0$.
Since Schur polynomials are orthogonal under the Haar measure

$\langle
s_{\lambda},s_{\mu}\rangle=\int_{U(d)}\overline{s_{\lambda}(g)}s_{\mu}(g)dg=\delta_{\lambda,\mu},$
(64)

we get the expression

$c_{\lambda}=\int_{U(d)}\chi_{d}(g)\overline{s_{\lambda}}(g)s_{\mu}(g)dg$ (65)

Using the fact that character is of factorization form
$\chi_{d}(x_{1},\dots,x_{d})=\prod_{k=1}^{d}\chi_{1}(x_{k})$, the conditions
of the Proposition 1 are met and we can rewrite the previous expression as

$c_{\lambda}=\det(a_{\lambda_{i}-\mu_{j}-i+j})_{1\leq i,j\leq
d}=D_{d-1}^{\lambda,\mu}(\chi_{1}),$ (66)

where $\\{a_{n}\\}$ is a sequence that generates the single-mode character
$\chi_{1}(x)$. Schur-positivity $c_{\lambda}\geq 0$ thus reads

$D_{d-1}^{\lambda,\mu}(\chi_{1})=\det(a_{\lambda_{i}-\mu_{j}-i+j})_{1\leq
i,j\leq d}\geq 0,$ (67)

which is by Proposition 1 equivalent to the condition that all minors of the
Töplitz matrix $(a_{i-j})_{i,j\in\mathbb{Z}}$ are non-negative. This means
$\\{a_{n}\\}$ is totally positive. $\\{a_{n}\\}$ is also an integral sequence
with $a_{0}>0$, therefore the main result follows by Lemma 1. ∎

##  Appendix C Fock space decomposition for order-one statistics

Suppose two sets of variables $x_{1},...,x_{n}$ and $y_{1},...,y_{m}$ with
$m\leq n$. The Cauchy identities Stanley and Fomin (1999) are the following

$\displaystyle\prod_{i=1}^{n}\prod_{j=1}^{m}(1+x_{i}y_{j})$ $\displaystyle=$
$\displaystyle\Sigma_{l(\lambda)\leq
m}s_{\lambda^{\prime}}(x_{1},...,x_{n})s_{\lambda}(y_{1},...,y_{m})$ (68)
$\displaystyle,\prod_{i=1}^{n}\prod_{j=1}^{m}\frac{1}{1-x_{i}y_{j}}$
$\displaystyle=$ $\displaystyle\Sigma_{l(\lambda)\leq
m}s_{\lambda}(x_{1},...,x_{n})s_{\lambda}(y_{1},...,y_{m})$ (69)

where $l(\lambda)$ is the length of the diagram (number of rows), and
$\lambda^{\prime}$ is the conjugate partition of $\lambda$. Using these
identities, the proof of Theorem 2 is as follows.

###### Proof.

For statistics $[1,\alpha]_{-}$ and $[1,\beta]_{+}$ we have the following
$d$-mode characters (defined in (9))

$\displaystyle\chi^{-}_{d}(\vec{x})$ $\displaystyle=$
$\displaystyle\prod_{i=1}^{d}(1+\alpha x_{i}),$ (70)
$\displaystyle\chi^{+}_{d}(\vec{x})$ $\displaystyle=$
$\displaystyle\prod_{i=1}^{d}\frac{1}{1-\beta x_{i}}.$ (71)

Now, we apply the first Cauchy identity by setting $m=1$ and $\alpha=y_{1}$
and we get

$\displaystyle\chi^{-}_{d}(\vec{x})$ $\displaystyle=\Sigma_{l(\lambda)\leq
1}s_{\lambda}(\alpha)s_{\lambda^{\prime}}(x_{1},...,x_{d})$ (72)
$\displaystyle=\Sigma_{l(\lambda)\leq
1}\alpha^{|\lambda|}s_{\lambda^{\prime}}(x_{1},...,x_{d}).$ (73)

Similarly, we apply the second Cauchy identity by setting $m=1$ and
$\beta=y_{1}$ and we get

$\displaystyle\chi^{+}_{d}(\vec{x})$ $\displaystyle=\Sigma_{l(\lambda)\leq
1}s_{\lambda}(\beta)s_{\lambda}(x_{1},...,x_{d})$ (74)
$\displaystyle=\Sigma_{l(\lambda)\leq
1}\beta^{|\lambda|}s_{\lambda}(x_{1},...,x_{d}).$ (75)

∎