# Biorthogonal Renormalization

Elisabet Edvardsson  Department of Physics, Stockholm University, AlbaNova
University Center, 106 91 Stockholm, Sweden  J Lukas K König  Department of
Physics, Stockholm University, AlbaNova University Center, 106 91 Stockholm,
Sweden<EMAIL_ADDRESS>Marcus Stålhammar  Nordita, KTH Royal
Institute of Technology and Stockholm University, Hannes Alfvéns väg 12,
SE-106 91 Stockholm, Sweden

###### Abstract

The biorthogonal formalism extends conventional quantum mechanics to the non-
Hermitian realm. It has, however, been pointed out that the biorthogonal inner
product changes with the scaling of the eigenvectors, an ambiguity whose
physical significance is still being debated. Here, we revisit this issue and
argue that this choice of normalization is of physical importance. We
illustrate in which settings quantities such as expectation values and
transition probabilities depend on the scaling of eigenvectors, and in which
settings the biorthogonal formalism remains unambiguous. To resolve the
apparent scaling ambiguity, we introduce an inner product independent of the
gauge choice of basis and show that its corresponding mathematical structure
is consistent with quantum mechanics. Using this formalism, we identify a
deeper problem relating to the physicality of Hilbert space representations,
which we illustrate using the position basis.

  *

##  1 Introduction

The foundations of quantum mechanics rely on the Hermiticity constraint, which
ensures that all operators related to physical observables have real spectra.
However, recent years have marked a paradigm shift as the study of non-
Hermitian Hamiltonians has intensified greatly, both from an experimental and
a theoretical point of view [1, 2]. These operators serve as effective
descriptions of systems subject to, e.g., dissipation or gain and loss, and
are fundamentally different from their Hermitian counterparts, partly because
they have complex spectra and different sets of left and right eigenvectors.
Consequently, non-Hermitian operators display many physical features that have
no Hermitian counterparts; arguably the most prominent and well-studied have
been the breakdown of the bulk-boundary correspondence [3, 4, 5, 6] and the
appearance of deficiencies at which both eigenvalues and eigenvectors
coalesce, the so-called exceptional points [7, 8, 9, 10,
sayyadRealizingExceptionalPoints2022a, 11, 12]. Additional examples include
the extended 38-fold symmetry classification [13, 14, 15], and the physical
consequences of the respective symmetries [16, 17, 18, 19, 20], where parity-
time symmetry comprise one well studied case. While parity-time-symmetric
operators can replace their Hermitian counterparts in an equivalent
formulation of quantum mechanics, due to their capacity of hosting real
eigenvalue spectra [21, 22], they are today understood as effective
descriptions of optical systems where the symmetry reflects a balance between
gain and loss [23].

The relaxation of the Hermiticity constraint has fundamental consequences on
the underlying mathematical framework of the theory. As an example, the
previously mentioned different sets of left and right eigenvectors,
$\\{\ket{L_{n}}\\}$ and $\\{\ket{R_{n}}\\}$, respectively, are no longer
individually orthogonal. Instead, they are _biorthogonal_ , i.e.,
$\innerproduct{L_{n}}{R_{m}}\propto\delta_{nm}$, and the notion of inner
product has to be modified in order to make connections to, e.g., probability
and projections. To fulfill this purpose, the non-Hermitian community is
mainly employing what is called the _biorthogonal inner product_ [24]. This
inner product has several benefits and is of physical relevance as it may
predict the (dis)appearance of boundary states in lattice models and can thus
be used to formulate the biorthogonal bulk-boundary correspondence for non-
Hermitian systems [4, 5, 25].

Despite its range of successful applications, there is an ambiguity in
defining the biorthogonal inner product, see, e.g., Ref. [26]. The
normalization condition used in the theory developed in Ref. [24], henceforth
referred to as the biorthogonal formalism, leaves a degree of freedom in how
to pick the eigenvectors of the Hamiltonian; if $\ket{R_{n}}$ is multiplied by
some number $c_{n}\in\mathbb{C}$, $\ket{L_{n}}$ can simply be rescaled by
$1/c^{*}_{n}$ and still satisfy the normalization condition. Such a change of
basis alters the biorthogonal inner product, making it apparent that its
definition depends on the choice of eigenbasis. In situations where a single
Hamiltonian is considered, this is not a problem as the choice of scaling of
the eigenvectors merely determines how the states and obsrvables should be
represented [27], but it becomes problematic when studying different
Hamiltonians and comparisons between results are desired, for example through
a shared position representation. Examples include previous works in the
biorthogonal bulk-boundary correspondence [4, 5, 25], where the expectation
value of the operator $\outerproduct{e_{n}}{e_{n}}$, with $\ket{e_{n}}$
denoting the vector represented by
$\begin{pmatrix}0&\dots&0&1&0&\dots&0\end{pmatrix}^{T}$, is shown to be of
physical importance despite its actual meaning being affected by how states
are represented. This means that one needs to be careful in these and similar
situations, as it is important that the physical meaning of the quantities
remains the same when making comparisons between systems with different
Hamiltonians. Similar problems are expected when studying the position
representation of wave functions in the continuum case, as is done in, e.g.,
Ref. [28], where the Berry connection is computed. It is further argued in
Ref. [28] that when computing expectation values of position and momentum
operators, the conventional Hermitian definition of expectation values is
preferable over the notion stemming from the biorthogonal formalism.

To address the problems above, we revisit the biorthogonal formalism of non-
Hermitian quantum mechanics in this work. We outline the potential problems
and ambiguities of the formalism, with a particular focus on the biorthogonal
inner product, and explain when they are of physical relevance. To eliminate
these problems, we formulate a more general inner product in terms of an inner
product matrix $G$ – similar to what is done in Ref. [29] – which is
independent of the gauge choice of basis of the eigenvectors, leaving
quantities such as the expectation value invariant under physically irrelevant
choices. While Ref. [29] discusses general matrices, we instead focus on
specifying $G$ and argue that one particular such choice is favorable. Our
work is of relevance in non-Hermitian physics as it explicitly suggests a
simple, basis independent extension of the biorthogonal formalism, compatible
with conventional quantum mechanics, that can directly be applied to physical
setups. The topic of uniqueness has also been explored in the context of a
metric operator formulation in Ref. [30].

The outline of the article is as follows. We start in Sec. 2 by giving a short
introduction to the biorthogonal formalism, discuss apparent problems with the
formalism and when they are of physical importance. We especially focus on the
biorthogonal inner product and its dependence on a gauge choice of basis
vectors, which directly comprises the motivation for our work. In Sec. 3 we
set out to resolve these problems. In particular, we define a new, basis
independent, inner product and investigate its physical and mathematical
properties. We sort out which problems in the biorthogonal formalism can be
resolved in this way, and which remain. Sec. 4 is devoted to a discussion
about the result. We conclude in Sec. 5 by summarizing our most important
results and suggesting future research directions within the field.

##  2 Background and Motivation

We start by introducing important concepts and present the main motivations
for our work. In Sec. 2.1, we give a brief survey the biorthogonal formalism
outlined in Ref. [24]. In Sec. 2.2, we identify potential issues with the
definition of the corresponding inner product and point out what problems may
arise from it, while we show in Sec. 2.3 why these issues have not caused
problems in previous studies.

The operators studied in this work are assumed to be linear, finite
dimensional and free from eigenvalue degeneracies; in particular they are non-
defective. The notations and other conventions set in this section will be
used throughout the rest of the work, unless otherwise specified.

###  2.1 The Biorthogonal Inner Product

Take as starting point two sets of vectors, denoted
$\left\\{\ket{R_{n}}\right\\}$ and $\left\\{\ket{L_{n}}\right\\}$, that both
span $\mathbb{C}^{N}$, but that do not necessarily consist of orthogonal
vectors. Assume further that these two sets are biorthogonal to one another
and that the vectors are scaled according to

$\innerproduct{L_{m}}{R_{n}}=\delta_{mn}.$ (1)

These sets can be used to introduce an inner product in the following way. For
each vector

$\ket{\alpha}=\sum_{n}a_{n}\ket{R_{n}},$ (2)

define an _associated vector_ ,

$\ket{\tilde{\alpha}}=\sum_{n}a_{n}\ket{L_{n}}.$ (3)

Then the _biorthogonal inner product_ , denoted by $(\cdot,\cdot)_{B}$, is
defined in the following way:

$\left(\ket{\alpha},\ket{\beta}\right)_{B}=\innerproduct{\tilde{\alpha}}{\beta}.$
(4)

Assuming that $\ket{\beta}=\sum_{n}b_{n}\ket{R_{n}}$, Eq. (4) becomes

$\left(\ket{\alpha},\ket{\beta}\right)_{B}=\sum_{n}a_{n}^{*}b_{n}.$ (5)

As is argued in Ref. [24], this is a valid, positive definite, inner product
and it is of particular use in systems described by non-Hermitian
Hamiltonians; the left and right eigenvectors of such a Hamiltonian form two
biorthogonal sets, and can thus be used to define a biorthogonal inner
product. The vector space $\mathbb{C}^{N}$, in which the vectors
$\ket{\alpha}$ and $\ket{\beta}$ live, together with the biorthogonal inner
product, forms a Hilbert space, which is denoted by $\mathcal{H}_{B}$.
Physical states will be represented by vectors in $\mathcal{H}_{B}$ and
observables by operators acting on the space.

As in the Hermitian case, the biorthogonal inner product can be used to
compute probabilities, and the transition probability between states
represented by $\ket{\alpha}$ and $\ket{\beta}$ is given by

$p_{\alpha\rightarrow\beta}=\frac{\innerproduct{\tilde{\alpha}}{\beta}\innerproduct{\tilde{\beta}}{\alpha}}{\innerproduct{\tilde{\alpha}}{\alpha}\innerproduct{\tilde{\beta}}{\beta}}.$
(6)

This is a number between $0$ and $1$ and, as argued in Refs. [24, 27], any
choice of biorthogonal basis can be used to derive a consistent probability
theory. Since the notion of probability exists, it is possible to also define
expectation values. In the biorthogonal framework, the expectation value of an
operator $Q$ in a state represented by the vector $\ket{\alpha}$ is introduced
as,

$\langle
Q\rangle=\frac{\left(\ket{\alpha},Q\ket{\alpha}\right)_{B}}{\left(\ket{\alpha},\ket{\alpha}\right)_{B}}=\frac{\matrixelement{\tilde{\alpha}}{Q}{\alpha}}{\innerproduct{\tilde{\alpha}}{\alpha}}.$
(7)

Any operator $Q$ can be written in the form

$Q=\sum_{mn}q_{mn}\outerproduct{R_{m}}{L_{n}},$ (8)

with $q_{mn}\in\mathbb{C}$. The operators for which the numbers $q_{mn}$ form
a Hermitian matrix are called _biorthogonally Hermitian_. For these operators,
the expectation value given by Eq. (7) is always real, and hence these
operators are taken to correspond to observables and vice versa.

###  2.2 A Physically Relevant Ambiguity

The normalization condition in Eq. (1) leaves a degree of freedom in the
choice of eigenvectors of a non-Hermitian Hamiltonian. If
$\left\\{\ket{R_{n}}\right\\}$ and $\left\\{\ket{L_{n}}\right\\}$ denote right
and left eigenvectors of a non-Hermitian Hamiltonian that satisfies Eq. (1),
any other sets on the form $\left\\{c_{n}\ket{R_{n}}\right\\}$,
$\left\\{(c_{n}^{*})^{-1}\ket{L_{n}}\right\\}$, with $c_{n}\in\mathbb{C}$,
will also satisfy Eq. (1). Thus, the biorthogonal inner product can in
principle be defined using any of these sets of eigenvectors, and still
satisfy the biorthonormality condition Eq. (1). As is stated in Ref. [24, 27],
this is not a problem when considering a single closed system, since the
physical state can be represented by a different vector, leading to the same
predictions. When comparing different systems to each other this might,
however, matter. For example, it is common to consider a family of
Hamiltonians that depend on some parameter $\gamma$ describing a lattice model
of size $N$. This situation is studied in a non-Hermitian context in e.g.
Refs. [4, 25, 3, 5]. The biorthogonal inner product, and thus how physical
properties of the system are evaluated, will depend both on $\gamma$ and $N$,
but also on the choice of eigenvectors of $H(\gamma,N)$.

When studying lattice models, the vector $\ket{e_{n}}$ is typically taken to
represent the $n$th site in the lattice, for all values of $\gamma$ and $N$.
This means that a physical meaning is assigned to the vector $\ket{e_{n}}$
without reference to an inner product. The question is now how this fits with
the biorthogonal formalism. To investigate this, it is natural to study the
consequences of a rescaling of the eigenvectors of the Hamiltonian for a fixed
vector $\ket{\alpha}$. Take the vectors $\ket{\alpha}$ and $\ket{\beta}$ as in
Sec. 2.1, consider the previous sets of eigenvectors
$\left\\{\ket{R_{n}}\right\\}$ and $\left\\{\ket{L_{n}}\right\\}$, and then
introduce another set of eigenvectors as
$\left\\{\ket{R_{n}^{\prime}}\right\\}=\left\\{c_{n}\ket{R_{n}}\right\\}$ and
$\left\\{(c_{n}^{*})^{-1}\ket{L_{n}}\right\\}$. As noted earlier, this
transformation gives another basis that is normalized as
$\innerproduct{L_{m}^{\prime}}{R_{n}^{\prime}}=\delta_{mn}$, just like the
previous basis for which $\innerproduct{L_{m}}{R_{n}}=\delta_{mn}$. The two
states $\ket{\alpha}$ and $\ket{\beta}$ can be expressed in either basis,
using the coefficients

$\ket{\alpha}=\sum_{n}a_{n}\ket{R_{n}}=\sum_{n}\frac{a_{n}}{c_{n}}\ket{R_{n}^{\prime}}\,\,\,\,\textrm{and}\,\,\,\,\ket{\beta}=\sum_{n}b_{n}\ket{R_{n}}=\sum_{n}\frac{b_{n}}{c_{n}}\ket{R_{n}^{\prime}}.$
(9)

The associated vector $\ket{\tilde{\alpha}^{\prime}}$, corresponding to this
new basis, is different from $\ket{\tilde{\alpha}}$ defined in Eq. (3), and
reads

$\ket{\tilde{\alpha}^{\prime}}=\sum_{n}\frac{a_{n}}{c_{n}}\ket{L_{n}^{\prime}},$
(10)

which means that the inner product defined by the new basis, denoted by
$(\cdot,\cdot)_{B^{\prime}}$, is given by

$(\ket{\alpha},\ket{\beta})_{B^{\prime}}=\innerproduct{\tilde{\alpha}^{\prime}}{\beta}=\sum_{n}\frac{a_{n}^{*}b_{n}}{|c_{n}|^{2}}.$
(11)

Comparing Eqs. (5) and (11), we conclude that in general

$(\ket{\alpha},\ket{\beta})_{B}\neq(\ket{\alpha},\ket{\beta})_{B^{\prime}},$
(12)

i.e., the inner product between two vectors is not kept invariant under a
rescaling of the eigenvectors. The probability of measuring the energy $E_{n}$
of a particle whose state is represented by the vector $\ket{\alpha}$
corresponds in the two different inner products to $p_{n}$ and
$p_{n}^{\prime}$ to,

$\displaystyle p_{n}$
$\displaystyle=\frac{\innerproduct{L_{n}}{\alpha}\innerproduct{\tilde{\alpha}}{R_{n}}}{\innerproduct{\tilde{\alpha}}{\alpha}\innerproduct{L_{n}}{R_{n}}}=\frac{\absolutevalue{a_{n}}^{2}}{\sum_{m}\absolutevalue{a_{m}}^{2}},$
(13) $\displaystyle p_{n}^{\prime}$
$\displaystyle=\frac{\innerproduct{L_{n}^{\prime}}{\alpha}\innerproduct{\tilde{\alpha}^{\prime}}{R_{n}^{\prime}}}{\innerproduct{\tilde{\alpha}^{\prime}}{\alpha}\innerproduct{L_{n}^{\prime}}{R_{n}^{\prime}}}=\frac{\absolutevalue{a_{n}}^{2}\absolutevalue{c_{n}}^{-2}}{\sum_{m}\absolutevalue{a_{m}}^{2}\absolutevalue{c_{m}}^{-2}},$
(14)

which means that the probability depends on the choice of $c_{n}$. The same
thing holds true for expectation values. The expectation values of the
Hamiltonian in the state represented by the vector $\ket{\alpha}$ using the
two different inner products read

$\displaystyle\langle H\rangle$
$\displaystyle=\frac{\sum_{n}E_{n}\absolutevalue{a_{n}}^{2}}{\sum_{n}\absolutevalue{a_{n}}^{2}},\mbox{\quad
and\quad}$ (15) $\displaystyle\langle H\rangle^{\prime}$
$\displaystyle=\frac{\sum_{n}E_{n}\absolutevalue{a_{n}}^{2}\absolutevalue{c_{n}^{-2}}}{\sum_{n}\absolutevalue{a_{n}}^{2}\absolutevalue{c_{n}^{-2}}}.$
(16)

This means that in general the expectation values $\langle H\rangle$ and
$\langle H\rangle^{\prime}$ differ even for biorthogonally Hermitian
operators. This clearly leads to interpretational challenges when turning, for
example, to the lattice models described previously.

We illustrate this using the concrete Hamiltonian

$H(\gamma)=\begin{pmatrix}0&1+\gamma&&\\\ 1-\gamma&\ddots&\ddots&\\\
&\ddots&\ddots&1+\gamma\\\ &&1-\gamma&0\end{pmatrix},$ (17)

which describes the Hatano-Nelson model with open boundary conditions [31].
The lattice model associated with this Hamiltonian is shown in Fig. 1.

Figure 1: An illustration of the lattice model corresponding to the Hatano-
Nelson Hamiltonian in Eq. (17). As $\gamma\neq 0$, the right and left hopping
amplitudes are different, resulting in a non-Hermitian Hamiltonian
description.

The left and right eigenvectors of $H(\gamma)$ can be used to define a
biorthogonal inner product in several different ways according to the above
reasoning. Here, we choose the following three options:

  1. 1.

Fix $\innerproduct{R_{n}}{R_{n}}=1$ and choose $\ket{L_{n}}$ such that
$\innerproduct{L_{n}}{R_{n}}=1$.

  2. 2.

Fix $\innerproduct{L_{n}}{L_{n}}=1$ and choose $\ket{R_{n}}$ such that
$\innerproduct{L_{n}}{R_{n}}=1$.

  3. 3.

For each $\gamma$ and $n$, fix $\innerproduct{R_{n}}{R_{n}}$ to a random
number between $0$ and $1$, and then pick $\ket{L_{n}}$ such that
$\innerproduct{L_{n}}{R_{n}}=1$.

The expectation values of $H(\gamma)$ in the state represented by
$\ket{e_{1}}+\ket{e_{2}}$ in the different inner products are shown in Fig. 2
as a function of $\gamma$.

Figure 2: Absolute value of the expectation value of $H$ as a function of
$\gamma$ in the state represented by $\ket{e_{1}}+\ket{e_{2}}$ using the
biorthogonal inner product with three different normalization conditions. The
inner product used in panel (a) is normalized according to
$\innerproduct{R_{n}}{R_{n}}=1$, in panel (b) according to
$\innerproduct{L_{n}}{L_{n}}=1$, while in panel (c), the quantity
$\innerproduct{R_{n}}{R_{n}}$ takes random values. It should be noted that,
despite being similar in shape, the graphs in panel (a) and (b) indeed differ
from each other. The qualitative difference between the three panels indicates
that the spare degrees of freedom caused by how eigenvectors can be rescaled
in the biorthogonal formalism can affect physically relevant quantities.

This shows that choosing different inner products can significantly change the
qualitative shape of the curve, and that one has to be careful when making
comparisons between different systems.

###  2.3 Expectation Values in Eigenstates

We have shown that the scaling of eigenvectors does affect physically relevant
quantities like the energy expectation value when we fix a vector rather than
a state. It is important to note, however, that if $\ket{\alpha}$ is an
eigenvector of the Hamiltonian, the expectation value in the different inner
products will be the same. Let
$\ket{\alpha}=r\ket{R_{n}}=\frac{r}{c_{n}}\ket{R_{n}^{\prime}}$ be an
eigenvector of the Hamiltonian. Then

$\langle
H\rangle^{\prime}=\frac{E_{n}|r|^{2}|c_{n}|^{-2}}{|r^{2}||c_{n}|^{-2}}=\frac{|r|^{2}E_{n}}{|r^{2}|}=E_{n}=\langle
H\rangle.$ (18)

This holds also for a general operator $Q$, as,

$\langle
Q\rangle^{\prime}=\frac{\matrixelement{\tilde{\alpha}^{\prime}}{Q}{\alpha}}{\innerproduct{\tilde{\alpha}^{\prime}}{\alpha}}=\frac{\matrixelement{L_{n}}{Q}{R_{n}}|r|^{2}|c_{n}|^{-2}}{\innerproduct{L_{n}}{R_{n}}|r|^{2}|c_{n}|^{-2}}=\langle
Q\rangle.$ (19)

This is important, as it explains why most results in the literature are not
affected by the ambiguity described in Sec. 2.2. Examples include the
expectation values of $\Pi_{n}=\outerproduct{e_{n}}{e_{n}}$ computed in Refs.
[5, 4, 25], where they are used to formulate the biorthogonal bulk-boundary
correspondence, and Berry connections [28, 32, 33], which are both computed
solely from the eigenstates of the respective systems. It is however important
to stress that even though the expectation value of $\Pi_{n}$ in an eigenstate
is unaffected by this problem, the physical meaning of its constituents, i.e.,
the vectors $\ket{e_{n}}$ is unclear and can depend on the choice of the
corresponding eigenvectors. This becomes problematic when moving away from
eigenstates, where the meaning of $\Pi_{n}$ depends on the choice of basis and
the representation of vectors, such that its physical interpretation breaks
down.

Although it is common to study properties of eigenvectors of the Hamiltonian
in non-Hermitian systems, there is little work examining the superposition of
eigenvectors. Such superpositions are central to a theory being quantum, and
are regularly studied in the Hermitian case. The time evolution of a particle
put on a specific lattice site is an example. Similar studies in non-Hermitian
systems are desirable, but if these are to be carried out using the
biorthogonal formalism, we have shown that problems may arise. Instead, such
future work requires a formalism that does allow for a consistent comparison
of different systems.

In summary, we have established that the representations of physical
observables and states in the biorthogonal formalism depend on a choice of
basis vectors and that this yields the following two problems:

###### Problem 1

Given a family of Hamiltonians $H(\gamma)$, the Hilbert space will change with
$\gamma$ and physical states will thus be represented by different vectors
depending on the choice of $\gamma$. Since the Hilbert space also depends on
the choice of scaling of the eigenvectors of the Hamiltonian, comparing the
physics of systems described by different Hamiltonians becomes difficult.

###### Problem 2

When studying non-Hermitian lattice models where Hamiltonians take the form of
tight-binding matrices, the physical meaning of lattice position is implicitly
assigned to the vectors $\ket{e_{n}}$. As the Hilbert space changes with the
Hamiltonian and when choosing different eigenstates, the physical state that
the vector $\ket{e_{n}}$ represents will also change. Thus the physical
meaning of the vectors $\ket{e_{n}}$ will change even though they seem
suitable for making physical predictions, e.g., predicting gap closings.

Recall that Problem 2 is an instance of a larger class of problems related to
any physical meaning that comes with the representation of a Hamiltonian. In
the context of real-eigenvalued Hamiltonians, this problem has been discussed
using a metric operator formalism [30]. We now turn to how to solve these
problems.

##  3 Basis Independent Inner Product

In this section, we address Problems 1 and 2, and investigate if they can be
solved by considering an inner product different from the one given by Eq.
(4). In Secs. 3.1-3.3 we consider Problem 1. We first introduce the inner
product formalism that avoids the ambiguity discussed in the previous section
(Sec. 3.1). We then study the notion of observables and the corresponding
mathematical structure, and compare these to earlier interpretations (Sec.
3.2). We go on to show that this inner product allows for representations of
states to be mapped between different Hilbert spaces (Sec. 3.3). Finally we
treat Problem 2 and the corresponding issues related to the physical meaning
of position vectors $\ket{e_{k}}$ in Sec. 3.4.

###  3.1 Reformulation

The scaling ambiguity related to Problem 1 can be solved in several ways. One
way would be to specify some eigenbasis that is to be used to construct
associated states, say by requiring a fixed value of
$\innerproduct{R_{n}}{R_{n}}$. Another way is to modify the inner product in
such a way that it does not depend on the gauge choice of eigenbasis. Here we
will pursue the latter. When constructing such an inner product, we impose the
following constraints:

###### Constraint 1

The inner product should admit a probabilistic interpretation.

###### Constraint 2

The inner product should have the standard inner product as its Hermitian
limit.

###### Constraint 3

The inner product should be uniquely determined by the Hamiltonian.

Since a Hamiltonian admits several different inner products that satisfy the
first two constraints, there can be multiple choices that also satisfy the
third one. To choose between those, we require:

###### Constraint 4

The inner product should be a natural choice.

To achieve the above, we find it beneficial to describe the inner product by a
matrix $G$, such that it reads
$\left(\ket{\alpha},\ket{\beta}\right)_{G}\coloneqq\matrixelement{\alpha}{G}{\beta}$.
Given a non-defective Hamiltonian, any set of left eigenvectors
$\left\\{\ket{L_{n}}\right\\}$ is linearly independent and spans the vector
space. Thus, the matrix $G$ can be expanded as,

$G=\sum_{mn}g_{mn}\outerproduct{L_{m}}{L_{n}}.$ (20)

Specifying the inner product corresponds to determining the constants
$g_{mn}$. For the right eigenvectors of the Hamiltonian to be interpreted as
stationary states, we require

$\matrixelement{R_{k}}{G}{R_{l}}\propto\delta_{kl},$ (21)

for all $k,l$. This implies

$g_{kl}\innerproduct{R_{k}}{L_{k}}\innerproduct{L_{l}}{R_{l}}\propto\delta_{kl},$
(22)

which means that $g_{kl}=0$ if $k\neq l$. Thus the matrix of the inner product
takes the form

$G=\sum_{n}g_{n}\outerproduct{L_{n}}{L_{n}}.$ (23)

This is similar to the form of the metric described in Refs. [24, 34]. To
achieve a well-defined and positive definite inner product, $G$ must be
Hermitian and all $g_{n}>0$, which additionally means it is invertible.
Consequently, $G$ is in fact a Gram matrix. The particular case when all
$g_{n}=1$ corresponds to the biorthogonal inner product in the biorthogonal
formalism introduced in Ref. [24], which is not invariant under rescaling of
the eigenvectors and thus failing to fulfill Constraint 3.

Let us now address the points Constraints 1-4 above, starting with Constraint
1. A physical state $\Psi$ is defined by its probability amplitudes that when
squared give the probabilities for different measurement outcomes. Assuming
the state $\Psi$ has probability amplitudes $c_{n}$ corresponding to
measurement of the energies $E_{n}$, the representation of $\Psi$ in the
Hilbert space generated by the inner product defined by Eq. (23), becomes

$\ket{\Psi}=\sum_{n}c_{n}\frac{\ket{R_{n}}}{\sqrt{\left(\ket{R_{n}},\ket{R_{n}}\right)_{G}}},$
(24)

where the fact that $\innerproduct{L_{m}}{R_{n}}\propto\delta_{mn}$ ensures
that $c_{n}$ can be computed from

$c_{n}=\frac{\left(\ket{R_{n}},\ket{\Psi}\right)_{G}}{\sqrt{\left(\ket{R_{n}},\ket{R_{n}}\right)_{G}}}.$
(25)

This means that, for any choice of constants $g_{n}$, a state described by a
set of probability amplitudes can be represented by a vector. Only the
representation of this state depends on the choice of $g_{n}$.

We emphasize that the inner product defined by Eq. (23) keeps the notion of
stationarity. As right eigenvectors are orthogonal in this inner product,
$\left(\ket{R_{n}},\ket{R_{m}}\right)_{G}\propto\delta_{mn}$, transition
probabilities between two different eigenstates vanish.

Let us now address the remaining constraints. Constraint 3 reduces to that the
inner product should be independent of the scale of $\ket{L_{n}}$ and
$\ket{R_{n}}$ (which the biorthogonal formalism generally is not, as shown in
Sec. 2). There are many inner products that satisfy this, so we now turn to
Constraint 4 and choose

$G=\sum_{n}\frac{\innerproduct{R_{n}}{R_{n}}}{\absolutevalue{\innerproduct{L_{n}}{R_{n}}}^{2}}\outerproduct{L_{n}}{L_{n}}.$
(26)

That this is a natural choice, comes from the fact that the Hamiltonian can be
written in terms of projectors $P_{n}$,

$H=\sum_{n}E_{n}P_{n},$ (27)

where $P_{n}$ is an operator projecting onto the $n$th eigenstate. More
precisely, these operators satisfy $P_{m}P_{n}=\delta_{mn}P_{n}$ and can be
represented in terms of eigenstates of $H$ as
$P_{n}=\outerproduct{R_{n}}{L_{n}}/\innerproduct{R_{n}}{L_{n}}$. In terms of
projectors, the matrix $G$ can be written as

$G=\sum_{n}P_{n}^{\dagger}P_{n}.$ (28)

It is therefore a natural construction of a Hermitian positive definite
matrix, given a Hamiltonian. The corresponding representation of a state with
probability amplitudes $c_{n}$ is with this particular choice of $G$ given by

$\ket{\Psi}=\sum_{n}c_{n}\frac{\ket{R_{n}}}{\sqrt{\innerproduct{R_{n}}{R_{n}}}}.$
(29)

The inner product Eq. (26) is independent of the scaling of eigenvectors and a
natural choice given a specific Hamiltonian. Furthermore, in the Hermitian
limit, the matrix of the inner product becomes the identity matrix,
$G=\mathbf{1}$, which means that the inner product reduces to the standard one
when the considered system is Hermitian. Thus, the choice of

$g_{n}=\frac{\innerproduct{R_{n}}{R_{n}}}{\absolutevalue{\innerproduct{L_{n}}{R_{n}}}^{2}}$
(30)

fulfills Constraints 1-4. This is the inner product which we consider for the
remainder of this work.

To illustrate the benefit of using this inner product instead of the one used
in the biorthogonal formalism, Fig. 3 shows the expectation value of the
Hamiltonian in Eq. (17) in the state represented by the vector
$\ket{e_{1}}+\ket{e_{2}}$, computed using the inner product defined in Eq.
(26). Just a in Fig. 2, this is done for three different choices of
eigenvectors, but contrary to what is seen in Fig 2, the expectation values
displayed in Fig. 3 shows no dependence on the choice of normalization of the
eigenvectors of the Hamiltonian.

Figure 3: Expectation value in the inner product in Eq. (26) of the
Hamiltonian in the state represented by the vector $\ket{e_{1}}+\ket{e_{2}}$
as a function of $\gamma$ for three different choices of eigenvectors. Here,
the expectation value does not change with the scaling of the eigenvectors,
leaving an unambiguous interpretation of quantities computed from the inner
product.

Thus, this provides a consistent way of comparing results for different
$\gamma$.

Lastly, it is important to note that when only eigenstates are considered, the
inner product defined in Eq. (23) gives the same result as the biorthogonal
inner product. As we have noted before, this is why the inner product
ambiguity has not disrupted previous work.

###  3.2 Observables and Self-Adjoint Operators

So far, we have mainly been discussing the representation of states, but the
representation of observables is equally important. In standard quantum
mechanics, observables are represented by Hermitian operators, motivated by
their real spectra. When considering the Hilbert space corresponding to the
inner product defined by Eq. (26), this notion has to be appropriately
changed. A natural generalization is comprised of operators that are self-
adjoint with respect to this inner product, since such operators also have
purely real eigenvalues. The adjoint $Q^{\star}$ of some operator $Q$ is
defined by the action on arbitrary vectors via

$\left(\ket{\alpha},Q^{\star}\ket{\beta}\right)_{G}:=\left(Q\ket{\alpha},\ket{\beta}\right)_{G},$
(31)

which means that the adjoint operator can be written as

$Q^{\star}=G^{-1}Q^{\dagger}G.$ (32)

The self-adjoint operators are thus operators satisfying

$Q=Q^{\star}=G^{-1}Q^{\dagger}G,$ (33)

with $G$ the matrix of the corresponding inner product. Thus, observables in
the Hilbert space corresponding to the inner product $(\cdot,\cdot)_{G}$ are
represented by operators satisfying Eq. (33). When considering the Hermitian
limit, i.e., when $G$ converges to the identity operator, Eq. (33) corresponds
exactly to the notion of Hermitian operators with respect to the standard
inner product. Thus, identifying observables in this way not only provides a
notion from which the concepts of Hermitian quantum mechanics can be retrieved
as a special case, but also constitutes a natural basis independent extension
of the notion of biorthogonal Hermiticity introduced in Ref. [24].

Before we continue, we make a note that together with the operator norm
induced by the inner product, the adjoint in Eq. (31) defines a
C$\star$-algebra which is the same structure that underlies traditional
quantum mechanics [35, 36]. Importantly, it provides the probability
interpretation, as also explicitly shown in Sec. 3.1, as well as the concepts
of states and observables. States in this formalism arise as positive, linear
functionals of unit norm, which for our purposes reduces to vectors normalized
in the norm induced by the new inner product. The notion of observables is
directly related to the definition of adjointness, as observables in a
C$\star$-algebra correspond to operators that are self-adjoint.

###  3.3 Mapping States and Observables Between Different Hilbert Spaces

Suppose that a Hamiltonian can be represented by two operators $H_{1}$ and
$H_{2}$ on the Hilbert spaces $\mathcal{H}_{G_{1}}$ and $\mathcal{H}_{G_{2}}$,
respectively. Note that $H_{1}$ and $H_{2}$ share eigenvalues as they
represent the same observable. Denote the respective eigenvectors by
$\left\\{\ket{R_{n}^{(1)}}\right\\}$, $\left\\{\ket{L_{n}^{(1)}}\right\\}$ and
$\left\\{\ket{R_{n}^{(2)}}\right\\}$, $\left\\{\ket{L_{n}^{(2)}}\right\\}$,
normalized according to,

$\ket{R_{n}^{(1/2)}}=\frac{\ket{R_{n}^{(1/2)}}}{\sqrt{\innerproduct{R_{n}^{(1/2)}}{R_{n}^{(1/2)}}}},\quad\innerproduct{L_{n}^{(1)}}{R_{n}^{(1)}}=\innerproduct{L_{n}^{(2)}}{R_{n}^{(2)}}.$
(34)

This gives rise to two different inner products, $G_{1}$ and $G_{2}$, and thus
two different representations of states and observables. The representations
have to be compatible for the theory to be consistent; there has to exist a
map from $\mathcal{H}_{G_{1}}$ to $\mathcal{H}_{G_{2}}$ (and the other way
around), preserving observables and states. Now we show that the map
$\mathcal{T}:\mathcal{H}_{G_{1}}\rightarrow\mathcal{H}_{G_{2}}$, defined as,

$\mathcal{T}\ket{R_{n}^{(1)}}=\ket{R_{n}^{(2)}},$ (35)

fulfills exactly that. As $\ket{R_{n}^{(1/2)}}$ spans $\mathcal{H}_{G_{1/2}}$,
$\mathcal{T}$ is invertible and it is thus straight-forward to show,

$\bra{L_{n}^{(1)}}\mathcal{T}^{-1}=\bra{L_{n}^{(2)}},\quad
G_{1}=\mathcal{T}^{\dagger}G_{2}\mathcal{T}.$ (36)

Consider the transformation of states first. Suppose that there is a state
represented by the vector $\ket{\alpha}\in\mathcal{H}_{G_{1}}$, defined as

$\ket{\alpha}=\sum_{n}c_{n}\ket{R_{n}^{(1)}}.$ (37)

If $E_{n}$ is the eigenvalue corresponding to $\ket{R_{n}^{(1)}}$, then the
probability of measuring the energy $E_{n}$ is given by $|c_{n}|^{2}$
according to Eq. (29). $\mathcal{T}$ acts on $\ket{\alpha}$ as

$\mathcal{T}\ket{\alpha}=\sum_{n}c_{n}\ket{R_{n}^{(2)}},$ (38)

where $\mathcal{T}\ket{\alpha}\in\mathcal{H}_{G_{2}}$. Since $H_{1}$ and
$H_{2}$ share eigenvalues, and $E_{n}$ thus is the eigenvalue corresponding
also to $\ket{R_{n}^{(2)}}$, the probability of measuring the energy $E_{n}$
is still $|c_{n}|^{2}$, meaning that $\mathcal{T}$ preserves the probability
notion, and hence the states.

Let us now turn to the case of operators. Let $Q$ be a representation of an
observable on $\mathcal{H}_{G_{1}}$. By definition, $Q$ is self-adjoint in
$\mathcal{H}_{G_{1}}$ with respect to the inner product $G_{1}$, meaning that

$G_{1}Q=Q^{\dagger}G_{1}.$ (39)

The operator $\mathcal{T}$ acts on $Q$ as
$\mathcal{T}:Q\mapsto\mathcal{T}Q\mathcal{T}^{-1}$. We show that
$\mathcal{T}Q\mathcal{T}^{-1}$ is an observable on $\mathcal{H}_{G_{2}}$.
Acting with $G_{2}$ from the left yields,

$\begin{split}G_{2}\mathcal{T}Q\mathcal{T}^{-1}&=(\mathcal{T}^{\dagger})^{-1}G_{1}\mathcal{T}^{-1}\mathcal{T}Q\mathcal{T}^{-1}=(\mathcal{T}^{\dagger})^{-1}Q^{\dagger}G_{1}\mathcal{T}^{-1}\\\
&=(\mathcal{T}^{\dagger})^{-1}Q^{\dagger}\mathcal{T}^{\dagger}(\mathcal{T}^{\dagger})^{-1}G_{1}\mathcal{T}^{-1}=(\mathcal{T}Q\mathcal{T}^{-1})^{\dagger}G_{2},\end{split}$
(40)

which means that $\mathcal{T}Q\mathcal{T}^{-1}$ is self-adjoint in the inner
product $G_{2}$ and thus represents an observable in $\mathcal{H}_{G_{2}}$.
Importantly, $Q$ and $\mathcal{T}Q\mathcal{T}^{-1}$ represent the same
observable in $\mathcal{H}_{G_{1}}$ and $\mathcal{H}_{G_{2}}$, respectively,
since $Q$ and $\mathcal{T}Q\mathcal{T}^{-1}$ have the same eigenvalues and
expectation values. The latter can be seen by making a direct calculation of
the expectation value of $Q$ in the state represented by $\ket{\alpha}$ in
$\mathcal{H}_{G_{1}}$, which gives

$\frac{\matrixelement{\alpha}{G_{1}Q}{\alpha}}{\matrixelement{\alpha}{G_{1}}{\alpha}}=\frac{\matrixelement{\alpha}{\mathcal{T}^{\dagger}G_{2}\mathcal{T}Q\mathcal{T}^{-1}\mathcal{T}}{\alpha}}{\matrixelement{\alpha}{\mathcal{T}^{\dagger}G_{2}\mathcal{T}}{\alpha}},$
(41)

where the right hand side corresponds exactly to the expectation value of
$\mathcal{T}Q\mathcal{T}^{-1}$ in the state represented by
$\mathcal{T}\ket{\alpha}$ in $\mathcal{H}_{G_{2}}$.

We can therefore conclude that under a transformation $\mathcal{T}$, the
expected transformation of vectors and operators, i.e.,

$\mathcal{T}:Q\mapsto\mathcal{T}Q\mathcal{T}^{-1},\,\,\,\,\text{and}\,\,\,\,\,\
\mathcal{T}:\ket{\alpha}\mapsto\mathcal{T}\ket{\alpha}.$ (42)

preserves the notions of states and observables in different Hilbert spaces.
This also extends to representatives of the Hamiltonian;
$\mathcal{T}H_{1}\mathcal{T}^{-1}=H_{2}$, and hence the map $\mathcal{T}$ also
induces representations on different Hilbert spaces; the two notions are
equivalent.

To conclude, the above shows that for inner products of the form of Eq. (26),
there exist well-defined linear transformations that map the representations
of both states and observables from one Hilbert space to another.
Consequently, this allows us to interpret the meaning of states and
observables in various different Hilbert spaces. In combination with the
reasoning and results derived in Secs. 3.1 and 3.2, this provides a solution
to Problem 1.

###  3.4 The Physical Meaning of Vectors

As Secs. 3.1-3.3 have dealt with Problem 1, we now turn to Problem 2 and the
physical meaning of the vectors $\ket{e_{k}}$. Concretely, we want to answer
the question of whether or not the physical meaning of vectors $\ket{e_{k}}$,
i.e., the physical state that the vector $\ket{e_{k}}$ represents, can be
transferred from one Hilbert space to another. It is clear that the inner
product defined in Eq. (26) does not preserve the meaning of these vectors,
and the question is if it is possible to do it by choosing a different inner
product. As previously argued, this question is of relevance in several
physical setups, where quantities related to the expectation value of the
projection operator $\Pi_{k}=\outerproduct{e_{k}}{e_{k}}$ are computed, e.g.,
to predict gap closings and the existence of boundary states in lattice
models.

To investigate this, recall that the norm and relation to other states and
operators are central to the notion of a quantum state. It is therefore
natural to check if the norm of $\ket{e_{k}}$ and the overlap between
$\ket{e_{k}}$ and $\ket{e_{l}}$ can be preserved when the Hilbert space is
changed. Consider the two Hamiltonians $H_{1}$ and $H_{2}$ of dimension $N$
and let them have eigenstates $\ket{R_{n}^{(1)}},\ket{L_{n}^{(1)}}$ and
$\ket{R_{n}^{(2)}},\ket{L_{n}^{(2)}}$, respectively. Define the inner product
matrices

$G_{1}=\sum_{n}g_{n}^{(1)}\outerproduct{L_{n}^{(1)}}{L_{n}^{(1)}},\,\,\,\,\text{and}\,\,\,\,\,G_{2}=\sum_{n}g_{n}^{(2)}\outerproduct{L_{n}^{(2)}}{L_{n}^{(2)}},$
(43)

and let the vectors $\ket{e_{k}}$ be given by

$\ket{e_{k}}=\sum_{n}c_{kn}^{(1)}\ket{R_{n}^{(1)}}=\sum_{n}c_{kn}^{(2)}\ket{R_{n}^{(2)}},$
(44)

Assume now that the set of $g_{n}^{(1)}$ is given. The question posed above
then boils down to whether or not it is possible to choose the constants
$g_{n}^{(2)}$ such that, for every $k,l\in\mathbb{Z}_{+}$,

$\displaystyle\matrixelement{e_{k}}{G_{1}}{e_{l}}$
$\displaystyle=\matrixelement{e_{k}}{G_{2}}{e_{l}}.$ (45)

Inserting Eq. (44) yields,

$\displaystyle\sum_{n}g_{n}^{(1)}|c_{kn}^{(1)}\innerproduct{L_{n}^{(1)}}{R_{n}^{(1)}}|^{2}$
$\displaystyle=\sum_{n}g_{n}^{(2)}|c_{kn}^{(2)}\innerproduct{L_{n}^{(2)}}{R_{n}^{(2)}}|^{2},\quad
l=k,$ (46)
$\displaystyle\sum_{n}g_{n}^{(1)}\left(c_{kn}^{(1)}\right)^{*}c_{ln}^{(1)}|\innerproduct{L_{n}^{(1)}}{R_{n}^{(1)}}|^{2}$
$\displaystyle=\sum_{n}g_{n}^{(2)}\left(c_{kn}^{(2)}\right)^{*}c_{ln}^{(2)}|\innerproduct{L_{n}^{(2)}}{R_{n}^{(2)}}|^{2},\quad
l\neq k,$ (47)

Eq. (46) corresponds to a system of equations whose size equals the dimension
of the Hamiltonian, meaning that given a set of $N$ constants $g_{n}^{(1)}$,
this system defines the set of $N$ constants $g_{n}^{(2)}$. Thus, it is
possible to preserve the norm of all position vectors $\ket{e_{k}}$ by
choosing the inner product appropriately. Eq. (47) yields a system of order
$N^{2}$ equations with only $N$ constants to choose. Consequently, the overlap
between the vectors $\ket{e_{k}}$ and $\ket{e_{l}}$ cannot in general be
preserved between different representations and there seem to be properties of
these vectors that inherently depend on the considered Hamiltonian. In other
words, _within this framework, the vector $\ket{e_{n}}$ cannot consistently
represent the same physical state when the Hamiltonian is changed_; its
physical meaning changes with the inner product.

This means that Problem 2 cannot be solved in its entirety by choosing a
different inner product. We thus argue that the desired choice of inner
product in non-Hermitian systems is the one presented in Eq. (26), providing a
basis independent notion allowing for the mapping of representations of
vectors between different Hilbert spaces, solving Problem 1.

##  4 Discussion

The problems with the biorthogonal formalism, listed in Sec. 2.3, indicate
that conceptual and fundamental difficulties may arise when applying it in
certain situations and setups; it is inconvenient in practice to work with an
inner product that changes with the scaling of the eigenvectors of the
Hamiltonian. As seen in Secs. 3.1-3.3, Problem 1 can be solved by describing
the inner product using an inner product matrix. However, in contrast to Ref.
[29], we argue that it is important to specify the coefficients of the matrix
$G$, denoted $g_{n}$ in Eq. (26), such that the inner product can be used in a
simple way in physical setups. This provides a motivation for explicitly
defining an inner product satisfying Constraints 1-4. Constraint 1 has its
origins in physics, while Constraints 2-4 are aimed at making the formulation
straightforward to apply, and their combination hence allows us to define an
inner product that is unique for a given Hamiltonians, without compromising
its practicality.

We claim in Sec. 3.1 that the biorthogonal inner product can be recovered by
choosing a particular case of the inner product given by Eq. (26). As the
latter is expressed in terms of a matrix, and the former in terms of
associated vectors, such a comparison is not obviously apparent. However, the
inner product Eq. (26) can be re-written in terms of associated vectors,
facilitating such a comparison. Given a vector
$\ket{\alpha}=\sum_{n}c_{n}(\sqrt{\innerproduct{R_{n}}{R_{n}}})^{-1}\ket{R_{n}}$,
the associated vector reads

$\ket{\tilde{\alpha}}=G\ket{\alpha}=\sum_{n}c_{n}\frac{\sqrt{\innerproduct{R_{n}}{R_{n}}}}{\innerproduct{R_{n}}{L_{n}}}\ket{L_{n}}.$
(48)

Importantly, and contrary to the associated vector in Eq. (3), this is
independent of the choice of eigenvectors of $H$. The biorthogonal formalism
is recovered when choosing
$\innerproduct{R_{n}}{R_{n}}=\innerproduct{R_{n}}{L_{n}}=1$. Using an inner
product matrix has several advantages as it makes it easier to discuss several
different inner products simultaneously. In addition, the matrix $G$ itself
contains information about the system. For example, as stated in Sec. 3.2, we
see that all observables, the self-adjoint operators in our formalism, have
real eigenvalues. In fact they correspond to operators that are pseudo-
Hermitian with respect to $G$ [12].

Addressing Problem 2, we showed in Sec. 2.3 that this cannot be solved by
modifying the inner product by any choice of $g_{n}$; in the biorthogonal
setting the vectors $\ket{e_{n}}$ do not have meaning on their own without
reference to an inner product. By extension the same holds true for all
vectors; the vector space itself cannot describe physics without an inner
product. We could leave it at that and say that it is pointless to discuss the
physical meaning of the vectors $\ket{e_{n}}$, but since the biorthogonal
bulk-boundary correspondence relies on information about gap closings provided
by the expectation values of $\outerproduct{e_{n}}{e_{n}}$, their physical
significance cannot be disregarded. Furthermore, the experimental realization
of the biorthogonal bulk-boundary correspondence is often done using classical
systems, the mechanical system described in Ref. [6] comprising a concrete
example. In such systems the equations of motion can be rewritten as a
Schrödinger equation of the form

$i\frac{d}{dt}\mathbf{x}=H\mathbf{x},$ (49)

where the vector $\mathbf{x}$ is not a position vector but instead contains
both positions and velocities. Thus $\mathbf{x}$ contains physical quantities,
while the matrix $H$ can be non-Hermitian and can therefore be thought of as a
non-Hermitian Hamiltonian. In the particular case of Ref. [6] the mechanical
system maps to the non-Hermitian SSH-chain, for which a biorthogonal
description implies that the vectors $\ket{e_{n}}$ do not have a physical
meaning, cf., Eqs. (46) and (47). However, since the vector $\mathbf{x}$ has
physical meaning in the system, the vector $\ket{e_{n}}$ here clearly should
have physical meaning of its own, without reference to an inner product. This
implies that even though the biorthogonal expectation value can be used to
find a bulk-boundary correspondence in these systems, it is uncertain whether
such an inner product can be used for other things. There does not seem to be
a one-to-one-correspondence between the physical quantities in biorthogonal
quantum mechanics and the physical quantities in the mechanical systems mapped
to non-Hermitian Hamiltonians. This suggests that biorthogonal quantum
mechanics, even though it manages to predict gap closings in classical systems
via the biorthogonal bulk-boundary correspondence, cannot be used to
understand all the phenomena of these classical systems.

Even though the present study has mainly discussed discrete lattice models, it
is apparent that similar interpretational problems can arise in the continuous
case. In the context of wave functions, quantities such as
$\innerproduct{\mathbf{r}}{\psi}$ are usually referred to, which are non-
trivial in a biorthogonal framework since the vectors $\ket{\mathbf{r}}$
suffer from the same interpretational problems as the vectors $\ket{e_{k}}$.
That a biorthogonal inner product is not always suitable for describing the
physics of a system is further supported by Ref. [28], where it is explicitly
stated that they prefer to use the expectation value used in Hermitian physics
and not its biorthogonal counterpart.

##  5 Conclusion and Outlook

In this work, we have studied the mathematical foundations of non-Hermitian
quantum mechanics, focusing on problems arising from the definition of the
biorthogonal inner product in the biorthogonal formalism. As pointed out in,
e.g., Ref. [26], the inner product leaves a scaling degree of freedom in how
to choose the eigenvectors, a degree of freedom that we show affects
physically relevant quantities including expectation values and transition
probabilities. We have explained in some detail when this spare degree of
freedom is important and when it is physically irrelevant. This clarifies why
the biorthogonal formalism can be successfully applied in certain setups,
e.g., when studying the biorthogonal bulk-boundary correspondence or when
computing the Berry connection.

We have resolved this problem by defining a new, generalized, and basis
independent inner product and by showing that one specific choice is favorable
from a physical point of view. We have explained how notions such as
probability, observables and states should be modified, and shown that our
formalism reduces to that of conventional quantum mechanics in the Hermitian
limit. Furthermore, we have seen that the generalized inner product allows us
to translate the meaning of states between different Hilbert spaces in a
consistent manner.

We have discussed the position representation and physical consequences
related to the vectors $\ket{e_{n}}$, whose physical meaning changes when the
inner product changes. Unfortunately, this problem remains in the inner
product formalism; simply changing the inner product does not allow for the
physical meaning of vectors to be kept when mapped to different Hilbert
spaces. We have concluded that a modified inner product can be used to
_translate_ between different representations corresponding to the same
physics, but that it is not sufficient to allow _the same_ vector as a
representative of a physical state in different Hilbert spaces. We expect
similar problems for other representations that carry physical meaning.

A natural next step following this work would be to study the physical meaning
of $\ket{e_{n}}$: Are there other methods that can be used in an attempt to
retain a unified physical meaning of $\ket{e_{n}}$ in different Hilbert
spaces, or is this generally impossible? The former would open up for a
unification of the biorthogonal bulk-boundary correspondence and non-Hermitian
quantum mechanics, while the latter indicates that the formalism itself is
what implicitly assigns the physical meaning of $\ket{e_{n}}$, which on its
own also would require further study.

The Hamiltonian operators considered in this work were assumed to be free from
eigenvalue degeneracies. In particular, no exceptional points appear in their
spectrum of the operators. Just as for the biorthogonal formalism, it is
unclear how the formalism developed in this work extends to include defective
operators. Given the attention of properties of exceptional points in the
community, such an extension would comprise an additional way to study these
exotic objects and increase the understanding of the fundamentals of non-
Hermitian physics.

_Note added_ : After initial submission we were made aware of a line of
research that addresses overlapping problems, see e.g. Ref. [37].

## Acknowledgements

We thank Lukas Rødland, Emil J. Bergholtz and Jonas Larson for stimulating and
fruitful discussions. We thank Eddy Ardonne for useful discussions and
comments on the manuscript. E.E. and J.L.K.K. are supported by the Swedish
Research Council (VR) and the Knut and Alice Wallenberg Foundation. J.L.K.K.
is furthermore supported by the project Dynamic Quantum Matter (2019.0068) of
the Knut and Alice Wallenberg Foundation.

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