# Jordan Algebraic Formulation of Quantum Mechanics
and
The Non-commutative Landau Problem

Tekin Dereli1,2111 E.mail<EMAIL_ADDRESS>, Ekin Sıla Yörük1222 E.mail:
<EMAIL_ADDRESS>
( 1 Department of Physics, College of Sciences, Koç University, 34450 Sarıyer,
Istanbul, Turkey
2 Faculty of Engineering and Natural Sciences, Maltepe University, 34857
Maltepe, Istanbul,Turkey

)

We present a Jordan algebraic formulation of the non-commutative Landau
problem coupled to a harmonic potential. To achieve this, an alternative
formulation of the Hilbert space version of quantum mechanics is presented.
Using this construction, the Hilbert space corresponding to the non-
commutative Landau problem is obtained. Non-commutative parameters are then
described in terms of an associator in the Jordan algebraic setting. Pure
states and density matrices arising from this problem are characterized. This
in turn leads us to the Jordan-Schrödinger time-evolution equation for the
state vectors for this specific problem.

##  1 Introduction

In the algebraic formulations of quantum mechanics, observables play a central
role. The main motivation for such approaches is to formulate quantum
mechanics in terms of essential ingredients only; which are the observables,
states, expectation values and their time evolution. For instance, it is
possible [1] to take quantum mechanical observables as the self-adjoint part
of a $C^{*}$-algebra $\mathcal{A}$ and states as positive linear functionals
on $\mathcal{A}$. Then the real number that a state assigns to an observable
will be the expectation value of that observable in that state333See [2], [3],
[4] for deeper formulations of quantum mechanics.. However, it turns out that
the $C^{*}$-algebra one uses in such approaches usually allows for non-self-
adjoint entities. Consequently, to start with, a more restrictive structure is
necessary to have a consistent description of the algebra of observables.

The Jordan algebraic formulation [5] adopted here is based on the fact that
quantum mechanical observables form an algebra $\mathcal{A}$ under the non-
associative Jordan product:

$A\bullet B=\frac{1}{2}(AB+BA).$

This algebra qualifies as a real Jordan algebra of operators since the Jordan
product is commutative and satisfies the Jordan identity

$(A\bullet B)\bullet A^{2}=A\bullet(B\bullet A^{2}).$

This last relation is equivalent to power-associativity of $\mathcal{A}$ when
$\mathcal{A}$ is formally real [5]. Furthermore, a Jordan algebra is called
special if it comes from the Jordan product in an associative algebra, which
is the case in present definition of Jordan algebra of observables. Since the
algebras arising in quantum mechanics are typically infinite dimensional, it
is of interest to develop a structure theory compatible with the infinite
dimensional case. This idea is initiated by [6], where the direct analogues of
$C^{*}$-algebras that are called $JB$-algebras are studied. $JB$-algebras are
Jordan algebras with a Banach space structure. They satisfy the following
conditions:

$\displaystyle\|A^{2}\|=\|A\|^{2},$
$\displaystyle\|A^{2}-B^{2}\|\leq\max\\{\|A^{2}\|,\|B\|^{2}\\},$
$\displaystyle\|A\bullet B\|\leq\|A\|\|B\|.$

The first condition, also known as the $C^{*}$ condition, is a consequence of
the second (positivity condition) and the third conditions (an analogue of
Banach space axiom). Therefore, the observables, which can be considered as
the self-adjoint part of a $C^{*}$-algebra, form a $JC$-algebra, or in other
words, a special $JB$-algebra. At this point, we can say that if one wants to
formulate quantum mechanical problems using the essential ingredients only, it
is important to write down a formulation using the Jordan algebraic approach.
Also, we note that all $JB$-algebras, with the exception of
$H_{3}(\mathbb{O})$, can be seen as the self-adjoint part of a
$C^{*}$-algebra, so they can be represented on a complex Hilbert space [6]. We
refer to [7], [8], [9], [10] for more detailed work on the exceptional
character of $H_{3}(\mathbb{O})$, and Jordan algebras.

Apart from Jordan algebras, we also would like to note other non-associative
structures which found applications in physical problems [11], [12], [13],
[14], [15], [16]. The associative product used in quantum mechanics do not
apply to some exotic systems such as magnetic monopoles, which involve non-
associative algebras. It turns out that if we consider dynamics of electrons
in uniform distributions of magnetic charge, certain non-associative
deformations of quantum mechanics and gravity in three dimensions are needed.
The quantitative framework for non-associative quantum mechanics in this
setting exhibits new effects compared to ordinary quantum mechanics with
sourceless magnetic fields. On the other hand, the non-associativity of
translations in a quantum system with a background magnetic field has received
renewed interest in association with topologically trivial gerbes over
$\mathbb{R}^{n}$. Then the non-associativity arises when trying to lift the
translation group action on the 1-particle system to a second quantized
system, described by a 3-cocycle of the group $\mathbb{R}^{n}$ with values in
the unit circle $S^{1}$.

In this paper, our main interest will be the Landau problem that involves the
motion of an electron interacting with a constant, uniform background magnetic
field. The minimal coupling of the electron charge density to an external
magnetic field is described in terms of the electromagnetic vector potential
$\mathbfcal{A}$ and leads to the Hamiltonian in the symmetric gauge

$H=\frac{1}{2m}\left({\bf\it
p}-\frac{e}{c}\mathbfcal{A}\right)^{2}=:\frac{1}{2m}{\bf\it P}^{2},$

where ${\bf\it P}$ denotes kinetic momenta. Thus, the Landau problem will
relate to the quantum mechanics of a charged particle on the transverse
$xy$-plane in the presence of a constant, uniform magnetic field present along
the $z$-axis. In metals, electrons occupy Landau energy levels given by

$E_{n}=\hbar\omega\left(N+\frac{1}{2}\right),$

where the Larmor frequency $\omega_{L}=\frac{eB}{mc}$ and
$N=n+\frac{m-|m|}{2}$. The infinitely degenerate Landau levels are labeled by
two quantum numbers; the radial quantum number $n\geq 0$, and the magnetic
quantum number $m$ with range $m\geq-n$. The energy spectrum is bounded from
below: $N\geq 0$.

From the physical point of view, the Landau problem turned out to be one of
the most engaging problems in quantum mechanics. It is inspired by ideas in
diverse areas of physics such as quantum optics and condensed matter physics.
Since it finds numerous applications in different areas of physics, the Landau
problem is still being studied extensively in the literature. In particular,
the first author et al. [17] showed recently that a Newton-Hooke duality
exists between the $2D$ Hydrogen atom and the Landau problems via the Tits-
Kantor-Koecher construction of the conformal symmetries of the Jordan algebra
of real symmetric $2\times 2$ matrices. Here we seek a further application of
the Jordan algebraic approach to the non-commutative Landau problem.

Non-commuting spatial coordinates and fields can be seen in actual physical
situations [18]. Landau models and their quantum Hall limit have become an
instrument of intense research activity as a physical realization of the
simplest example of non-commutative geometries [19], [20], [21], [22], [23],
[24], [25], [26], [27], [28]. Given an electron moving in a plane in the
presence of a constant external magnetic field ${\bf\it B}$ perpendicular to
that plane, the spectrum of the quantized theory gives infinitely degenerate
Landau levels, with separation $\mathcal{O}\left(\frac{eB}{mc}\right)$. The
limit $B\rightarrow\infty$ or $m\rightarrow 0$ projects onto the lowest Landau
level. On each of the projected Landau level, one obtains a non-commuting
algebra for the space coordinates giving the center of the cyclotron motion,

$[X,Y]=-\frac{i\hbar c}{eB}\neq 0.$

We note that similar structures also arise in the non-perturbative string
theory or relativistic quantum theories of gravity. A common expectation for
the purpose of unifying gravity with quantum mechanics involves replacing the
classical geometry of spacetime by a non-commutative quantum geometry. The
main motivation of this paper follows from the proof below that the non-
commutativity of spatial coordinates that involve only individual Landau
levels can be cast as non-associativity of observables based on a
corresponding Jordan algebraic formulation.

In what follows, we formulate the non-commutative Landau problem coupled to a
harmonic potential in the Jordan algebraic approach we give below. It turns
out that, in this problem, one obtains a pair of commuting Hamiltonians, which
can be written in terms of two pairs of mutually commuting creation and
annihilation operators of oscillator type. These operators further generate
two mutually commuting von Neumann algebras. Based on the Hilbert-Schmidt
operator formulation of the non-commutative Landau problem, it is possible to
write down Jordan algebraic formulas. At this point, we take the associative
case as our guideline and point out parallels to the associative case whenever
possible. It turns out that this problem at hand results as a composite
system. The Jordan algebraic approach has the particular advantage that, given
the density matrix of the composite system, one can identify the state vectors
corresponding to the subsystems. We accomplish this by defining a trace
partially on the ambient space. Finally at the end, we are able to derive a
Schrödinger-like time evolution equation that involves Jordan algebraic
associators for this specific problem. This approach we think is something new
with further possible applications.

##  2 Hilbert-Schmidt Operator Formulation

It turns out that given a pure state $\psi$ of a $C^{*}$-algebra
$\mathcal{A}$, one can construct a Hilbert space $\mathscr{H}$, and write down
a representation between $\mathcal{A}$ and the collection of bounded operators
on $\overline{\mathcal{A}/\mathcal{I}_{\psi}}$, where $\mathcal{I}_{\psi}$ is
defined as follows:

$\mathcal{I}_{\psi}=\\{A\in\mathcal{A}:\psi(A^{*}A)=0\\}.$

This construction lies at the heart of the celebrated the Gelfand-Naimark-
Segal (GNS) theory. But this method is not suitable for the Jordan algebraic
approach. Therefore, one demands another Hilbert space construction, which is
based on traces of operators instead of their states.

From now on, let us assume that $\mathcal{A}$ is the algebra
$\mathcal{B}(\mathscr{H})$ of all bounded operators on $\mathscr{H}$, which is
well-known to be a von Neumann algebra. As a result of this, there exists a
semi-finite, faithful, normal trace function on $\mathcal{A}$ [29]. We will
then denote positive, trace class elements by $\mathcal{A}_{1}$, and those
$A\in\mathcal{A}$ with $tr(A^{*}A)<\infty$ by $\mathcal{A}_{2}$. One can check
that $\mathcal{A}_{2}$ is an ideal in $\mathcal{A}$. In particular, for all
$A,B\in\mathcal{A}_{2}$, we have

$\langle A,B\rangle=trA^{*}B,\ \ \ \ \ \|A\|_{tr}=\sqrt{trA^{*}A}.$

Inspired by the GNS construction, one obtains the Hilbert space of Hilbert-
Schmidt operators in the form

$\hat{\mathscr{H}}=\overline{\mathcal{A}_{2}}^{tr}.$

Since $\mathcal{A}_{2}$ is an ideal, and $\mathcal{A}$ is associative,
$\mathcal{A}$ can be represented over $\mathcal{A}_{2}$ by the multiplication
operator which is defined by

$\pi(A)B=AB$

for all $A\in\mathcal{A}$, $B\in\mathcal{A}_{2}$. Thus, $\pi$ can be extended
to a faithful representation of $\mathcal{A}$ on $\hat{\mathscr{H}}$ [30]. If
we take a state $\psi$ represented by a density matrix
$\rho\in\mathcal{A}_{1}$, then there is at least one $B\in\mathcal{A}_{2}$
such that $\rho=BB^{*}$ holds. At this point, it is worth mentioning that the
expectation value of an observable $A\in\mathcal{A}$ is given by

$\psi(A)=tr\rho A=trBB^{*}A=trB^{*}AB=\langle B|A|B\rangle.$

Next let $U:\mathscr{H}\xrightarrow{}\mathscr{H}$ be a bounded, unitary
operator. If we take any vector $B\in\mathcal{A}_{2}$, then $BU$ becomes a
vector in $\hat{\mathscr{H}}$ describing the same vector as $B$ does.
Therefore, the representation of states in $\hat{\mathscr{H}}$ is not unique,
more precisely the states in $\hat{\mathscr{H}}$ are represented by right
unitary orbits defined by

$BUU^{*}B^{*}=BB^{*}=\rho.$

On the other hand, every Hilbert-Schmidt operator $B\in\mathcal{A}_{2}$ with
nonzero trace leads to a density matrix

$\rho_{B}:=\frac{BB^{*}}{tr(BB^{*})}.$

To emphasize the difference between $\mathscr{H}$ and $\hat{\mathscr{H}}$,
note that now, every (pure and mixed) state is represented by at least one
vector in $\hat{\mathscr{H}}$. It turns out that $\mathscr{H}$ and
$\hat{\mathscr{H}}$ are connected in the following algebraic sense [30]

$\hat{\mathscr{H}}=\overline{\mathscr{H}\otimes\mathscr{H}^{*}}^{tr}.$

We will now restrict this formalism to the physical system of quantum harmonic
oscillator, which turns out to be very crucial in the formulation of the
Landau problem. It is then necessary to focus on the application of Hilbert-
Schmidt operators, which can be viewed as the collection of bounded operators
acting on the configuration space

$\mathscr{H}:=span\left\\{|n\rangle:\frac{1}{\sqrt{n!}}(a^{\dagger})^{n}|0\rangle\right\\}_{n=0}^{\infty}.$

Furthermore, this space is isomorphic to the boson Fock space
$\mathcal{F}=\\{|n\rangle\\}_{n=0}^{\infty}$, where the annihilation and
creation operators obey the Fock algebra $[a,a^{\dagger}]=1$. Observe that the
physical states of the system represented on $\hat{\mathscr{H}}$, the set of
Hilbert-Schmidt operators, in other words the set of all bounded operators on
$\mathscr{H}=L^{2}(\mathbb{R})$ equipped with the scalar product

$\langle
X|Y\rangle=tr(X^{*}Y)=\sum_{k}{\langle\phi_{k}|X^{*}Y\phi_{k}\rangle},$

where $\\{\phi_{k}\\}_{k=0}^{\infty}$ is an orthonormal basis of
$\mathscr{H}$. Moreover, the basis vectors of $\hat{\mathscr{H}}$ are given by

$\phi_{nl}:=|\phi_{n}\rangle\langle\phi_{l}|,\ \ \ n,l=0,1,2,\dots\ .$

Here, we should note that $\mathscr{H}\otimes\mathscr{H}^{*}$ is a pre-Hilbert
space containing all finite sums of the form

$\sum_{j,k=0}^{n}\lambda_{jk}|\phi_{j}\rangle\otimes|\phi_{k}\rangle^{*},$

where a basis of $\overline{\mathscr{H}\otimes\mathscr{H}^{*}}^{tr}$ is

$\\{|\phi_{j}\rangle\otimes|\phi_{k}\rangle^{*}\\}_{j,k=0}^{\infty}.$

Setting
$|\phi_{j}\rangle\otimes|\phi_{k}\rangle^{*}=|\phi_{j}\rangle\langle\phi_{k}|$,
$\hat{\mathscr{H}}$ admits an orthonormal basis
$\\{\phi_{jk}\\}_{j,k=0}^{\infty}$ such that

$\phi_{jk}:=|\phi_{j}\rangle\langle\phi_{k}|.$

Note that $\hat{\mathscr{H}}$ can also be described as the Hilbert space of
square integrable functions on $(-\infty,\infty)$ by

$\hat{\mathscr{H}}=\left\\{\psi(x_{1},x_{2}):\psi(x_{1},x_{2})\in\mathcal{B}(\mathscr{H}),\
tr(\psi(x_{1},x_{2})^{\dagger}\psi(x_{1},x_{2}))<\infty\right\\}.$ (1)

Recall that $\hat{\mathscr{H}}$ is also the set of bounded operators,
described in (1), acting on the configuration space $\mathscr{H}$. Clearly, a
general element in this space can be written as

$|\psi\rangle=\sum_{n,m=0}^{\infty}c_{n,m}|n,m\rangle,$

where $\\{|n,m\rangle:=|n\rangle\langle m|\\}_{n,m=0}^{\infty}$ is a basis of
$\hat{\mathscr{H}}$ satisfying the inner product condition

$\langle n_{1},m_{1}|n_{2},m_{2}\rangle=tr[(|n_{1}\rangle\langle
m_{1}|)^{\dagger}|n_{2}\rangle\langle
m_{2}|]=\delta_{n_{1},n_{2}}\delta_{m_{1},m_{2}}.$

##  3 Jordan Algebraic Formulation

###  3.1 Hilbert Space Construction

In this section, we will present a Jordan algebraic GNS-type construction. But
this time, our development will be based on the JB algebra of observables. The
idea that a non-associative algebra of observables equipped with a tri-cyclic
trace leading to a Hilbert space construction is very fruitful. For more
details on this construction and its applications, we especially recommend the
recent work [31]. It turns out that there is an analogue of von Neumann
algebras in the context of JB algebras, which are called as JBW algebras [32].
One can work with normal, faithful, semi-finite traces on a JBW algebra
$\mathcal{A}$ [33]. Trace on a Jordan algebra is defined as a weight and it
satisfies the cyclicity condition

$tr(A\bullet(B\bullet C))=tr((A\bullet B)\bullet C).$

Also for positive elements $A,B\in\mathcal{A}$, and positive real numbers
$\lambda$, we have

$tr(A+B)=trA+trB,$ $tr\lambda A=\lambda trA.$

Similarly, we will then denote positive, trace class elements by
$\mathcal{A}_{1}$, and those $A\in\mathcal{A}$ with $tr(A^{2})<\infty$ by
$\mathcal{A}_{2}$. One can check that $\mathcal{A}_{2}$ is a Jordan ideal in
$\mathcal{A}$. Trace induces a bilinear, symmetric, real scalar product on
$\mathcal{A}_{2}$, and in particular, we have

$\langle A|B\rangle=tr(A\bullet B),\ \ \ \ \ \|A\|_{tr}=\sqrt{trA^{2}}.$

Similar to the associative case, closure with respect to this norm yields a
real Hilbert space

$\hat{\mathscr{H}}_{J}=\overline{\mathcal{A}_{2}}^{tr}.$

Next we aim to write down a Jordan representation from $\mathcal{A}$ to this
Hilbert space. Note that $\mathcal{A}_{2}$ is a Jordan module since for all
$\omega\in\mathcal{A}_{2}$, and $A,B\in\mathcal{A}$, we have [32], [34]

$A\bullet\omega=\omega\bullet A,$
$A^{2}\bullet(A\bullet\omega)=A\bullet(A^{2}\bullet\omega),$
$2A\bullet(B\bullet(A\bullet\omega))+(B\bullet A^{2})\bullet\omega=2(A\bullet
B)\bullet(A\bullet\omega)+A^{2}\bullet(B\bullet\omega).$

We may then define

$\pi_{J}(A):\mathcal{A}_{2}\xrightarrow{}\mathcal{A}_{2},\ \ \
\pi_{J}(A)B=A\bullet B$

for all $A\in\mathcal{A},B\in\mathcal{A}_{2}$. It is easy to check that
$\pi_{J}$ satisfies the following properties

$\pi_{J}(A^{2})\bullet\pi_{J}(A)=\pi_{J}(A)\bullet\pi_{J}(A^{2}),$
$2\pi_{J}(A)\bullet\pi_{J}(B)\bullet\pi_{J}(A)+\pi_{J}(B\bullet
A^{2})=2\pi_{J}(A\bullet B)\bullet\pi_{J}(A)+\pi_{J}(A^{2})\bullet\pi_{J}(B).$

Therefore, it becomes a Jordan representation. Moreover, due to the continuity
of multiplication, $\pi_{J}$ can be extended to whole space
$\hat{\mathscr{H}}_{J}$.

At this point, we should stress the relationship between
$\hat{\mathscr{H}}_{J}$ and $\hat{\mathscr{H}}$. It turns out that
$\hat{\mathscr{H}}_{J}$ is the real subspace of self-adjoint Hilbert-Schmidt
operators embedded in $\hat{\mathscr{H}}$ [35]. Turning back to the problem of
quantum harmonic oscillator, $\hat{\mathscr{H}}_{J}$ is a real span of
following three family of operators

$|\phi_{j}\rangle\langle\phi_{j}|,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
$\frac{1}{2}(|\phi_{j}\rangle\langle\phi_{k}|+|\phi_{k}\rangle\langle\phi_{j}|),$
(2)
$\frac{i}{2}(|\phi_{j}\rangle\langle\phi_{k}|-|\phi_{k}\rangle\langle\phi_{j}|),$

where $j>k$, and $j,k$ ranges from $0$ to $\infty$. Note that these elements
form a basis for $\hat{\mathscr{H}}_{J}$ since they are self-adjoint. Here,
$\\{\phi_{k}\\}_{k=0}^{\infty}$ is an orthonormal basis for
$L^{2}(\mathbb{R})$ which can be written in terms of Hermite polynomials in
the position space representation.

In a Jordan algebra, one can define the associator in the form

$[A,B,C]:=(A\bullet B)\bullet C-A\bullet(B\bullet C).$

Our aim is to find a connection between the commutator defined over the
associative algebra $\mathcal{B}$ and the associator over $\mathcal{A}$, which
is the self-adjoint part of $\mathcal{B}$.

######  Proposition 1.

Every self-adjoint element $H\in\mathcal{A}$ can be written as a finite sum of
commutators in the form

$H=i\sum_{j=1}^{N}[{H_{L}}_{j},{H_{R}}_{j}],$ (3)

where ${H_{L}}_{j}$, and ${H_{R}}_{j}$ are self-adjoint operators in
$\mathcal{A}$.

###### Proof.

Let $Aut^{0}(\mathcal{A})$ be the connected $1-$component of the Banach Lie
group of all automorphisms of $\mathcal{A}$. Let $A$ be a self-adjoint element
in $\mathcal{A}$. We define an operator $U$ as $U=\exp(iA)$, which is clearly
a unitary operator. By [36], $Aut^{0}(\mathcal{A})$ is known to be
algebraically generated by the symmetries in $\mathcal{A}$. Moreover by
Theorem 1.6 of [36], each unitary operator can be written as a finite product
of symmetries in $\mathcal{A}$. Combining these facts, we deduce that $U\in
Aut^{0}(\mathcal{A})$. Armed with this, we can now apply Theorem 2.5 of [36]
to obtain

$U=\exp\left(\sum_{j=1}^{n}[L_{a_{j}},L_{b_{j}}]\right),$

where $a_{j},b_{j}\in\mathcal{A}$ for all $j$, and $L_{a}b:=a\bullet b$ is the
left multiplication operator. Consequently, we have

$\exp\left(iA-\sum_{j=1}^{n}[L_{a_{j}},L_{b_{j}}]\right)=1.$

Our next goal is to verify that

$\exp\left(t\left(iA-\sum_{j=1}^{n}[L_{a_{j}},L_{b_{j}}]\right)\right)$

is a one parameter family for $t\in\mathbb{R}$. First, this operator depends
continuously on $t$. Second, when $t=0$, the operator reduces to the identity
operator. Lastly, for any $t,s\in\mathbb{R}$, it follows from the properties
of the exponential of an operator that

$\exp\left(\left(t+s\right)\left(iA-\sum_{j=1}^{n}[L_{a_{j}},L_{b_{j}}]\right)\right)=\exp\left(t\left(iA-\sum_{j=1}^{n}[L_{a_{j}},L_{b_{j}}]\right)+s\left(iA-\sum_{j=1}^{n}[L_{a_{j}},L_{b_{j}}]\right)\right)$
$=\exp\left(t\left(iA-\sum_{j=1}^{n}[L_{a_{j}},L_{b_{j}}]\right)\right)\exp\left(s\left(iA-\sum_{j=1}^{n}[L_{a_{j}},L_{b_{j}}]\right)\right).$

By the uniquness of generators of one-parameter families of operators (see
Stone’s theorem [37]), we may now conclude that

$A=i\sum_{j=1}^{n}[L_{a_{j}},L_{b_{j}}].$

And (3) follows. ∎

###  3.2 Example: Non-commutative Landau Problem

Let $\mathcal{B}$ be the $C^{*}$-algebra of operators on $\mathscr{H}$ with
self-adjoint part $\mathcal{A}$, which is considered as the Jordan algebra of
observables. Now we consider the motion of an electron in the $xy$-plane under
the influence of a constant magnetic field pointing along the positive $z$
direction in the symmetric gauge
$\mathbfcal{A}=\left(-\frac{B}{2}y,\frac{B}{2}x\right)$. Furthermore, we will
assume the presence of a harmonic potential. This problem is described by the
following Hamiltonian

$H_{\theta}=\frac{1}{2m}\left(p_{x}+\frac{eB}{2c}y\right)^{2}+\frac{1}{2m}\left(p_{y}-\frac{eB}{2c}x\right)^{2}+\frac{m\omega^{2}}{2}(x^{2}+y^{2}),$

where $\theta=-\frac{\hbar c}{eB}$ is the non-commutative parameter,
$\omega_{L}=\frac{eB}{mc}$ is the Larmour frequency. Position and momentum
operators satisfy the following nonzero commutation relations of the non-
commutative Heisenberg algebra

$[x,y]=i\theta,\ \ \ [x,p_{x}]=[y,p_{y}]=i\hbar.$ (4)

Next, we define new operators in $\mathcal{B}$, corresponding to complex
variables, related to chiral decomposition of non-commutative Landau problem.
It is worth remarking that our approach here is reminiscent of the
introduction of complex coordinates in the Landau problem pertaining to the
Bloch waves quasi-periodicity on a torus (see [38]). Precisely, we have

$z=\frac{1}{\sqrt{2}}(x+iy),\ \ \ \overline{z}=\frac{1}{\sqrt{2}}(x-iy).$

Corresponding derivation operators are computed to be

$\partial_{z}=\frac{1}{\sqrt{2}}(\partial_{x}-i\partial_{y}),\ \ \
\partial_{\overline{z}}=\frac{1}{\sqrt{2}}(\partial_{x}+i\partial_{y}).$

These newly defined operators can be shown to satisfy the following
commutation relations in $\mathcal{B}$

$[\partial_{z},z]=[\partial_{\overline{z}},\overline{z}]=1,$
$[\partial_{z},\overline{z}]=[\partial_{\overline{z}},z]=[z,\overline{z}]=[\partial_{z},\partial_{\overline{z}}]=0.$

We set two pairs of creation and annihilation operators in $\mathcal{B}$

$A_{+}=\xi\frac{\overline{z}}{2}+\frac{i}{\xi\hbar}p_{z},\ \ \
A_{+}^{\dagger}=\xi\frac{z}{2}-\frac{i}{\xi\hbar}p_{{\overline{z}}},$
$A_{-}=\xi\frac{z}{2}+\frac{i}{\xi\hbar}p_{{\overline{z}}},\ \ \
A_{-}^{\dagger}=\xi\frac{\overline{z}}{2}-\frac{i}{\xi\hbar}p_{z},$

where

$\xi=\sqrt[4]{\frac{m^{2}\omega^{2}/\hbar^{2}+m^{2}{\omega_{L}}^{2}/4\hbar^{2}}{1-(m\omega_{L}\theta/2)+(m^{2}\omega^{2}\theta^{2}/16)+(m^{2}{\omega_{L}}^{2}\theta^{2}/64)}}.$

The nonzero commutation relations between these operators are

$[A_{+},A_{+}^{\dagger}]=1,\ [A_{-},A_{-}^{\dagger}]=1.$

Finally, defining $\Omega_{\pm}:=\Omega\pm\frac{\tilde{\omega}_{L}}{2}$, where

$\Omega=\sqrt{\omega^{2}+\frac{\omega_{L}^{2}}{4}-\frac{m\omega_{L}\omega^{2}\theta}{2}-\frac{m\omega_{L}^{3}\theta}{8}+\left(\omega^{2}+\frac{\omega_{L}^{2}}{4}\right)\left(\left(\frac{m\omega\theta}{4}\right)^{2}+\left(\frac{m\omega_{L}\theta}{8}\right)^{2}\right)},$
$\tilde{\omega}_{L}=\omega_{L}\left(1-\left(\frac{\omega_{L}}{4}+\frac{\omega^{2}}{\omega_{L}}\right)m\theta\right),$

one can write $H_{\theta}$ as follows [39], [27]

$H_{\theta}=\hbar\Omega_{+}\left(N_{+}+\frac{1}{2}\right)+\hbar\Omega_{-}\left(N_{-}+\frac{1}{2}\right),\
\ \ N_{\pm}=A_{\pm}^{\dagger}A_{\pm}.$ (5)

Let

$H_{\pm}=\hbar\Omega_{\pm}\left(N_{\pm}+\frac{1}{2}\right).$

We define the operator $H_{+}\otimes\mathbb{I}+\mathbb{I}\otimes H_{-}$ to be
the infinitesimal generator of the strongly continuous one-parameter unitary
group $e^{itH_{+}}\otimes e^{itH_{-}}$. Thus by Stone’s theorem,
$H_{+}\otimes\mathbb{I}+\mathbb{I}\otimes H_{-}$ is self-adjoint. Using that,
(5) can be written as

$H_{\theta}=H_{+}\otimes\mathbb{I}_{\hat{\mathscr{H}}_{J,-}}+\mathbb{I}_{\hat{\mathscr{H}}_{J,+}}\otimes
H_{-}.$

Here $\hat{\mathscr{H}}_{J,+}\otimes\hat{\mathscr{H}}_{J,-}$ is the real
Jordan-Hilbert space for this problem described in (2), and
$\mathbb{I}_{\hat{\mathscr{H}}_{J,\pm}}$ are the corresponding identity
operators.

######  Theorem 1.

Nonzero commutation relations in $\mathcal{B}$, given in $(4)$, can be written
as associator relations in $\mathcal{A}$ as follows

$[y_{L},x,y_{R}]=\theta,\ \ \ [{p_{x}}_{L},x,{p_{x}}_{R}]=\hbar,\ \ \
[{p_{y}}_{L},y,{p_{y}}_{R}]=\hbar,$ (6)

where

$y_{L}={p_{x}}_{L}={p_{y}}_{L}=4H_{\theta},$
$y_{R}=\left(\frac{1}{2\hbar\Omega_{-}}-\frac{1}{2\hbar\Omega_{+}}\right)x-\left(\frac{1}{\xi^{2}\hbar^{2}\Omega_{+}}+\frac{1}{\xi^{2}\hbar^{2}\Omega_{-}}\right)p_{y},$
${p_{x}}_{R}=\left(\frac{\xi^{2}}{4\Omega_{+}}+\frac{\xi^{2}}{4\Omega_{-}}\right)x+\left(\frac{1}{2\hbar\Omega_{+}}-\frac{1}{2\hbar^{2}\Omega_{-}}\right)p_{y},$
${p_{y}}_{R}=\left(\frac{\xi^{2}\hbar}{4\Omega_{+}}+\frac{\xi^{2}\hbar}{4\Omega_{-}}\right)y+\left(\frac{1}{2\Omega_{-}}-\frac{1}{2\Omega_{+}}\right)p_{x}.$

###### Proof.

In the case of a special Jordan algebra, an associator can be expressed as a
double commutator

$[A,B,C]=\frac{1}{4}[B,[A,C]].$ (7)

In order to find the operators explicitly, one needs to write $x,y,p_{x}$, and
$p_{y}$ in terms of newly defined creation and annihilation operators. It
turns out that

$x=\frac{1}{\sqrt{2}\xi}(A_{-}+A_{+}+A_{+}^{\dagger}+A_{-}^{\dagger}),$
$y=\frac{i}{\sqrt{2}\xi}(-A_{-}+A_{+}-A_{+}^{\dagger}+A_{-}^{\dagger}),$
$p_{x}=\frac{i\xi\hbar}{2\sqrt{2}}(-A_{-}-A_{+}+A_{+}^{\dagger}+A_{-}^{\dagger}),$
$p_{y}=\frac{\xi\hbar}{2\sqrt{2}}(-A_{-}+A_{+}+A_{+}^{\dagger}-A_{-}^{\dagger}).$

The following commutation relation can be written for $y$

$\left[\frac{H_{\theta}}{\Omega_{+}},A_{+}+A_{+}^{\dagger}\right]+\left[\frac{H_{\theta}}{\Omega_{-}},-A_{-}-A_{-}^{\dagger}\right]$
$=\hbar([A_{+}^{\dagger}A_{+},A_{+}]+[A_{+}^{\dagger}A_{+},A_{+}^{\dagger}]+[A_{-}^{\dagger}A_{-},-A_{-}]+[A_{-}^{\dagger}A_{-},-A_{-}^{\dagger}])$
$=\hbar(-A_{+}+A_{+}^{\dagger}+A_{-}-A_{-}^{\dagger})=i\sqrt{2}\xi\hbar y.$

So we have,

$i\left[H_{\theta},\left(\frac{1}{2\hbar\Omega_{-}}-\frac{1}{2\hbar\Omega_{+}}\right)x-\left(\frac{1}{\xi^{2}\hbar^{2}\Omega_{+}}+\frac{1}{\xi^{2}\hbar^{2}\Omega_{-}}\right)p_{y}\right]=y.$

Similarly, one can check that

$i\left[H_{\theta},\left(\frac{\xi^{2}}{4\Omega_{+}}+\frac{\xi^{2}}{4\Omega_{-}}\right)x+\left(\frac{1}{2\hbar\Omega_{+}}-\frac{1}{2\hbar^{2}\Omega_{-}}\right)p_{y}\right]=p_{x},$
$i\left[H_{\theta},\left(\frac{\xi^{2}\hbar}{4\Omega_{+}}+\frac{\xi^{2}\hbar}{4\Omega_{-}}\right)y+\left(\frac{1}{2\Omega_{-}}-\frac{1}{2\Omega_{+}}\right)p_{x}\right]=p_{y}.$

Using (7) and the commutation relation provided for $y$, $p_{x}$ and $p_{y}$,
associator relations (6) follow. ∎

###  3.3 Density Matrices in the JBW-algebra Setting

It turns out that one needs a notion of density matrix in the JBW-algebra
$\mathcal{A}$ in order to describe the state of a quantum system. For the
purpose of applications to JBW-algebras, let us recall the following result
from [40], [41].

######  Theorem.

Let $\mathcal{A}$ be a commutative, order unit algebra that is the dual of a
base norm space $V$ such that the multiplication in $\mathcal{A}$ is
separately $w^{*}$\- continuous. Then for each $a\in\mathcal{A}$ and each
$\epsilon>0$ there are orthogonal projections $p_{1},p_{2},\dots,p_{n}$ in the
$w^{*}$-closed subalgebra generated by $a$ and $1$, and scalars
$\lambda_{1},\lambda_{2},\dots,\lambda_{n}$ such that

$\|a-\sum_{i=1}^{n}\lambda_{i}p_{i}\|<\epsilon.$

The set of finite linear combinations of orthogonal projections is norm dense
in $\mathcal{A}^{**}$, here $\mathcal{A}^{**}$ denotes the bidual of
$\mathcal{A}$, which is a JBW-algebra.

We will assume in Theorem 3 that any element can be written as a finite linear
combination of orthonormal projection operators. In particular, for any
$B\in\hat{\mathscr{H}}_{J}$ we have

$B=\sum_{i=1}^{n}\lambda_{i}P_{i},$ (8)

where $\lambda_{i}$’s are real scalars, and $P_{i}$’s denoting the orthonormal
operators. And for a general element, we use the fact that it can be
approximated in norm by linear combinations of orthonormal projections. So,
the general case follows from the norm continuity of multiplication.

######  Definition 1.

A self-adjoint operator $\rho\in\mathcal{A}$ is a density matrix if $\rho$ is
non-negative and $tr(\rho)=1$.

######  Theorem 2.

Assume that $\rho$ is a density matrix on $\hat{\mathscr{H}}_{J}$, then the
map $\Phi_{\rho}:\mathcal{A}\xrightarrow{}\mathbb{R}$ given by

$\Phi_{\rho}(A)=tr(\rho\bullet A)$

is a family of expectation values.

###### Proof.

In order to show this, we need to verify the following three properties of
$\Phi$.
(i) $\Phi_{\rho}(\mathbb{I})=1$,
(ii) $\Phi_{\rho}(A)$ is real for all $A\in\mathcal{A}$, and $\Phi_{\rho}(A)$
is non-negative whenever $A$ is non-negative.
(iii) For any sequence $A_{n}\in\mathcal{A}$, if

$\lim_{n\rightarrow\infty}\|A_{n}\bullet\psi-A\bullet\psi\|=0$

holds for all $\psi\in\hat{\mathscr{H}}_{J}$, then $\Phi_{\rho}(A_{n})$
converges to $\Phi_{\rho}(A)$.
Note that (i) is immediate from the definition of a density matrix. Since
$A\in\mathcal{A}$ is self-adjoint, $tr(\rho\bullet A)$ is real by definition.
Let $B\in\hat{\mathscr{H}}_{J}$ be the non-negative, self-adjoint square root
of $\rho$. It follows that $tr(A\bullet(B\bullet B))=tr(B\bullet(A\bullet
B))\geq 0$, since $B\bullet(A\bullet B)\in\hat{\mathscr{H}}_{J}$ is self-
adjoint and non-negative. This verifies condition (ii). Finally, assume
$A_{n}\bullet\psi$ converges in norm to $A\bullet\psi$ for all
$\psi\in\hat{\mathscr{H}}_{J}$. Then, $\|A_{n}\bullet\psi\|$ is bounded for
each fixed $\psi$. By the principle of uniform boundedness, there is a
constant $C$ such that $\|A_{n}\|\leq C$. Let $P_{i}$’s denote the orthonormal
projection operators introduced in (8), we have

$|\langle P_{i},(B\bullet(A_{n}\bullet B))\bullet P_{i}\rangle|=|\langle
P_{i},(A_{n}\bullet(B\bullet P_{i}))\bullet B\rangle|$ $=|\langle(B\bullet
P_{i}),A_{n}\bullet(B\bullet P_{i}))\rangle|\leq C\|B\bullet P_{i}\|^{2},$
$\lim_{m\rightarrow\infty}\sum_{i=1}^{m}\|B\bullet
P_{i}\|^{2}=\lim_{m\rightarrow\infty}\sum_{i=1}^{m}\langle B\bullet
P_{i},B\bullet P_{i}\rangle=\lim_{m\rightarrow\infty}\sum_{i=1}^{m}\langle
P_{i},\rho\bullet P_{i}\rangle=tr(\rho)<\infty.$

Since $A_{n}\bullet(B\bullet P_{i})$ converges to $A\bullet(B\bullet P_{i})$
for each $i$, by dominated convergence

$tr(B^{2}\bullet A)=tr(B\bullet(A\bullet
B))=\lim_{m\rightarrow\infty}\sum_{i=1}^{m}\langle P_{i},(B\bullet(A\bullet
B))\bullet P_{i}\rangle$
$=\lim_{m\rightarrow\infty}\lim_{n\rightarrow\infty}\sum_{i=1}^{m}\langle
P_{i},(B\bullet(A_{n}\bullet B))\bullet P_{i}\rangle$
$=\lim_{m\rightarrow\infty}\sum_{i=1}^{m}\langle P_{i},(B\bullet(A\bullet
B))\bullet P_{i}\rangle=\lim_{n\rightarrow\infty}tr(B\bullet(A_{n}\bullet
B))=\lim_{n\rightarrow\infty}tr(B^{2}\bullet A_{n}).$

∎

For a normal state $B\in\hat{\mathscr{H}}_{J}$, the expectation value of an
observable $A\in\mathcal{A}$ is given by

$\Phi_{B}(A)=\langle B,A\bullet B\rangle=tr(B\bullet(A\bullet
B))=tr(B^{2}\bullet A)=tr(\rho\bullet A),$

where $\rho=B^{2}$ is the corresponding density matrix.

######  Definition 2.

A density matrix is called a pure state if it can be written as a projection
operator in $\mathcal{A}$, otherwise it is called a mixed state.

Now, we can revisit the non-commutative Landau problem, where the Hilbert
space for the composite system is the Hilbert tensor product
$\hat{\mathscr{H}}_{J,+}\otimes\hat{\mathscr{H}}_{J,-}$ of the Hilbert spaces
$\hat{\mathscr{H}}_{J,+}$ and $\hat{\mathscr{H}}_{J,-}$ describing the
subsystems. Assuming $\rho$ is a density matrix on
$\hat{\mathscr{H}}_{J,+}\otimes\hat{\mathscr{H}}_{J,-}$, there exists a unique
density matrix $\rho_{+}$ on $\hat{\mathscr{H}}_{J,+}$ with the property that
$tr(\rho_{+}\bullet
A)=tr(\rho\bullet(A\otimes\mathbb{I}_{\hat{\mathscr{H}}_{J,-}}))$ for all
$A\in\mathcal{B}(\hat{\mathscr{H}}_{J,+})$. Similarly, there exists a unique
density matrix $\rho_{-}$ on $\hat{\mathscr{H}}_{J,-}$ with the property that
$tr(\rho_{-}\bullet
B)=tr(\rho\bullet(\mathbb{I}_{\hat{\mathscr{H}}_{J,+}}\otimes B))$ for all
$B\in\mathcal{B}(\hat{\mathscr{H}}_{J,-})$. Since, the state of the first
system is independent of the state of the second system, the density matrix
$\rho$ is of the form $\rho=\rho_{+}\otimes\rho_{-}$ for this problem.

######  Theorem 3.

A pure state is being represented in
$\hat{\mathscr{H}}_{J,+}\otimes\hat{\mathscr{H}}_{J,-}$ by $\pm
P_{0,+}\otimes\pm P_{0,-}$, where $P_{0,+}$ and $P_{0,-}$ are idempotent
elements on $\hat{\mathscr{H}}_{J,+}$ and $\hat{\mathscr{H}}_{J,-}$,
respectively.

###### Proof.

To prove our claim, we consider a pure state $\rho=\rho_{+}\otimes\rho_{-}$,
given by an idempotent $P_{0,+}\otimes P_{0,-}$. Without loss of generality,
we can take $P_{0,+}$ as the first basis vector of $\hat{\mathscr{H}}_{J,+}$,
and $P_{0,-}$ as the first basis vector of $\hat{\mathscr{H}}_{J,-}$. We
denote the other basis vectors as $P_{i,\pm}$. Let $B$ be the square root of
$\rho$. Then we have

$B=\left(\sum_{i=0}^{n}\lambda_{i}P_{i,+}\right)\otimes\left(\sum_{i=0}^{n}\mu_{i}P_{i,-}\right).$

Using this, $\rho$ can be written as

$\rho=B^{2}=\left(\sum_{i=0}^{n}\lambda_{i}^{2}P_{i,+}\right)\otimes\left(\sum_{i=0}^{n}\mu_{i}^{2}P_{i,-}\right),$

Taking traces of both sides, we obtain

$\left(\sum\lambda_{i}^{2}\right)\left(\sum\mu_{i}^{2}\right)=1.$

Finally, we have

$P_{0,+}\otimes
P_{0,-}=\left(\sum_{i=0}^{n}\lambda_{i}^{2}P_{i,+}\right)\otimes\left(\sum_{i=0}^{n}\mu_{i}^{2}P_{i,-}\right).$

The above formula clearly implies that $\lambda_{0}=\pm 1$, $\mu_{0}=\pm 1$,
and $P_{i,\pm}=0$ for $i>0$. ∎

Now, we will define new observables before stating our next theorem giving the
time evolution equation for the state vectors of non-commutative Landau
problem. Let

$X_{+}=C_{1_{+}}(A_{+}^{\dagger}+A_{+}),\ \ \ \ \
X_{-}=C_{1_{-}}(A_{-}^{\dagger}+A_{-}),$

be the position operators corresponding to subsystems. And

$P_{+}=iC_{2_{+}}(A_{+}^{\dagger}-A_{+}),\ \ \ \ \
P_{-}=iC_{2_{-}}(A_{-}^{\dagger}-A_{-}),$

be the corresponding momentum operators, where

$C_{1_{+}}=\sqrt{\frac{\hbar}{2m\Omega_{+}}},\ \ \ C_{2_{+}}=\sqrt{\frac{\hbar
m\Omega_{+}}{2}},\ \ \ C_{1_{-}}=\sqrt{\frac{\hbar}{2m\Omega_{-}}},\ \ \
C_{2_{-}}=\sqrt{\frac{\hbar m\Omega_{-}}{2}}.$

######  Theorem 4.

Given a state vector $v_{+}\otimes
v_{-}\in\hat{\mathscr{H}}_{J,+}\otimes\hat{\mathscr{H}}_{J,-}$, the time
evolution equation can be written as follows

$\frac{d}{dt}(v_{+}\otimes
v_{-})=\frac{-4}{\hbar}((([L_{S_{1_{+}}},L_{R_{1_{+}}}]+[L_{S_{2_{+}}},L_{R_{2_{+}}}])v_{+}\otimes
v_{-})+(v_{+}\otimes([L_{S_{1_{-}}},L_{R_{1_{-}}}]+[L_{S_{2_{-}}},L_{R_{2_{-}}}])v_{-})),$

where $H_{+}=i[R_{1_{+}},S_{1_{+}}]+i[R_{2_{+}},S_{2_{+}}]$,
$H_{-}=i[R_{1_{-}},S_{2_{-}}]+i[R_{2_{-}},S_{2_{-}}]$,

$R_{1_{+}}=-\frac{\Omega_{+}}{4}(X_{+}A_{+}+A_{+}^{\dagger}X_{+}),$
$R_{2_{+}}=-\frac{\Omega_{+}}{4}(2C_{1_{+}}P_{+}A_{+}+2C_{1_{+}}A_{+}^{\dagger}P_{+}),$
$S_{1_{+}}=P_{+}A_{+}+A_{+}^{\dagger}P_{+},$
$S_{2_{+}}=2C_{1_{+}}A_{+}^{\dagger}A_{+},$
$R_{1_{-}}=-\frac{\Omega_{-}}{4}(X_{-}A_{-}+A_{-}^{\dagger}X_{-}),$
$R_{2_{-}}=-\frac{\Omega_{-}}{4}(2C_{1_{-}}P_{-}A_{-}+2C_{1_{-}}A_{-}^{\dagger}P_{-}),$
$S_{1_{-}}=P_{-}A_{-}+A_{-}^{\dagger}P_{-},$
$S_{2_{-}}=2C_{1_{-}}A_{-}^{\dagger}A_{-},$

###### Proof.

Let us first see that $H_{+}=i[R_{1_{+}},S_{1_{+}}]+i[R_{2_{+}},S_{2_{+}}]$,
$H_{-}=i[R_{1_{-}},S_{2_{-}}]+i[R_{2_{-}},S_{2_{-}}]$. We will only show the
first equality, since two subsystems are independent identical copies. Note
that

$\displaystyle\frac{-i}{C_{1_{+}}C_{2_{+}}}[X_{+}A_{+}+A_{+}^{\dagger}X_{+},P_{+}A_{+}+A_{+}^{\dagger}P_{+}]$
$\displaystyle=[2A_{+}^{\dagger}A_{+}+A_{+}^{2}+{A_{+}^{\dagger}}^{2},{A_{+}^{\dagger}}^{2}-A_{+}^{2}]$
$\displaystyle=[2A_{+}^{\dagger}A_{+},{A_{+}^{\dagger}}^{2}]-[2A_{+}^{\dagger}A_{+},A_{+}^{2}]+2[A_{+}^{2},{A_{+}^{\dagger}}^{2}]$
$\displaystyle=2(A_{+}^{\dagger}[A_{+},{A_{+}^{\dagger}}^{2}]-[A_{+}^{\dagger},A_{+}^{2}]A_{+}+[A_{+},{A_{+}^{\dagger}}^{2}]A_{+}+A_{+}[A_{+},{A_{+}^{\dagger}}^{2}])$
$\displaystyle=4{A_{+}^{\dagger}}^{2}+4A_{+}^{2}+8A_{+}^{\dagger}A+4\mathbb{I}.$

Similarly,

$\displaystyle\frac{-i}{C_{2_{+}}}[P_{+}A_{+}+A_{+}^{\dagger}P_{+},A_{+}^{\dagger}A]$
$\displaystyle=[{A_{+}^{\dagger}}^{2},A_{+}^{\dagger}A_{+}]-[A_{+}^{2},A_{+}^{\dagger}A_{+}]$
$\displaystyle=-[A_{+}^{\dagger}A_{+},A_{+}^{\dagger}]A_{+}^{\dagger}-A_{+}^{\dagger}[A_{+}^{\dagger}A_{+},A_{+}^{\dagger}]+[A_{+}^{\dagger}A_{+},A_{+}]A_{+}+A_{+}[A_{+}^{\dagger}A_{+},A_{+}]$
$\displaystyle=-2{A_{+}^{\dagger}}^{2}-2A_{+}^{2}.$

Combining these, we get the required result. The time evolution equation for
density matrices in a $C^{*}$-algebra is given by

$\dot{\rho}=-\frac{i}{\hbar}[H,\rho],$ (9)

where $H$ denotes the Hamiltonian. In the case of Jordan algebras, inner
derivations are given in terms of the associator, which acts as a derivation
on the second argument [32]. Assuming $H=i[R,S]$, where $R$ and $S$ are self-
adjoint, (9) transforms into the following equation

$\dot{\rho}=\frac{-4}{\hbar}[R,\rho,S]=\frac{-4}{\hbar}[L_{S},L_{R}]\rho,$

where $L_{S}$ and $L_{R}$ are left multiplication operators with $S$ and $R$,
respectively. This amounts to the Jordan-algebraic evolution equation for the
density matrices. All density matrices can be written in the form
$\rho=B^{2}$, for some $B\in\mathcal{A}_{2}$. Taking the derivative of both
sides

$\dot{\rho}=2B\bullet\dot{B}.$

On the other hand, we also have

$\dot{\rho}=\frac{-4}{\hbar}[R,\rho,S]=\frac{-4}{\hbar}[R,B^{2},S]=\frac{-4}{\hbar}2B\bullet[R,B,S].$

Comparing them, one obtains

$\dot{B}=\frac{-4}{\hbar}[R,B,S]=\frac{-4}{\hbar}[L_{S},L_{R}]B.$

Let us remark that this equation can be extended to whole space
$\hat{\mathscr{H}}_{J}$ by

$\dot{v}=\frac{-4}{\hbar}[R,v,S]=\frac{-4}{\hbar}[L_{S},L_{R}]v.$

Next, let $v_{+}\otimes v_{-}$ be an element of
$\hat{\mathscr{H}}_{J,+}\otimes\hat{\mathscr{H}}_{J,-}$ corresponding to the
density matrix $\rho_{+}\otimes\rho_{-}$, and let
$H_{\theta}=H_{+}\otimes\mathbb{I}_{\hat{\mathscr{H}}_{J,-}}+\mathbb{I}_{\hat{\mathscr{H}}_{J,+}}\otimes
H_{-}$ be the Hamiltonian, where
$H_{+}=i[R_{1_{+}},S_{1_{+}}]+i[R_{2_{+}},S_{2_{+}}]$,
$H_{-}=i[R_{1_{-}},S_{2_{-}}]+i[R_{2_{-}},S_{2_{-}}]$. Based on these, we may
deduce that

$\frac{d}{dt}(\rho_{+}\otimes\rho_{-})=-\frac{i}{\hbar}[H_{+}\otimes\mathbb{I}_{\hat{\mathscr{H}}_{J,-}}+\mathbb{I}_{\hat{\mathscr{H}}_{J,+}}\otimes
H_{-},\rho_{+}\otimes\rho_{-}]$ $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \
=-\frac{i}{\hbar}([H_{+}\otimes\mathbb{I}_{\hat{\mathscr{H}}_{J,-}},\rho_{+}\otimes\rho_{-}]+[\mathbb{I}_{\hat{\mathscr{H}}_{J,+}}\otimes
H_{-},\rho_{+}\otimes\rho_{-}])$ $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
=-\frac{i}{\hbar}(([H_{+},\rho_{+}]\otimes\rho_{-})+(\rho_{+}\otimes[H_{-},\rho_{-}])).$

Using the discussion above, we may replace $-i[H,\rho]$ with
$-4[L_{S_{1}},L_{R_{1}}]v-4[L_{S_{2}},L_{R_{2}}]v$ to arrive at the formula

$\frac{d}{dt}(v_{+}\otimes
v_{-})=\frac{-4}{\hbar}((([L_{S_{1_{+}}},L_{R_{1_{+}}}]+[L_{S_{2_{+}}},L_{R_{2_{+}}}])v_{+}\otimes
v_{-})+(v_{+}\otimes([L_{S_{1_{-}}},L_{R_{1_{-}}}]+[L_{S_{2_{-}}},L_{R_{2_{-}}}])v_{-})).$

Finally, the solution to this Jordan-Schrödinger equation can be written as

$v_{+}(t)\otimes
v_{-}(t)=e^{\frac{-4t}{\hbar}([L_{S_{1_{+}}},L_{R_{1_{+}}}]+[L_{S_{2_{+}}},L_{R_{2_{+}}}])}v_{+}(0)\otimes
e^{\frac{-4t}{\hbar}([L_{S_{1_{-}}},L_{R_{1_{-}}}]+[L_{S_{2_{-}}},L_{R_{2_{-}}}])}v_{-}(0).$

∎

##  4 Concluding Remarks

In this work, for a two dimensional non-commutative space, a Jordan algebraic
formulation of the Hamiltonian associated to the motion of an electron under a
uniform magnetic field with a confining harmonic potential has been provided.
The main result is given in Theorem 1, and the non-commutative parameter
$\theta$ is written in terms of an associator in the Jordan algebra of
observables. Secondly, a general understanding for the expectation value of an
observable is provided in terms of the Jordan algebraic setting. In addition,
the pure states are characterized for this specific problem. Lastly, a Jordan
algebraic time evolution equation for the state vectors in the constructed
Hilbert space and its solution are presented.

##  5 Acknowledgement

T.D. thanks The Turkish Academy of Sciences (TÜBA) for partial support.

##  6 Data Availability

Data sharing is not applicable to this article.

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Numerical Simulation of Collinear Capillary-Wave Turbulence

Evgeny Kochurin1∗, Guillaume Ricard2, Nikolay Zubarev1,3, Eric Falcon2
1Institute of Electrophysics, Ural Division, Russian Academy of Sciences, 106
Amundsen Street, 620016 Ekaterinburg, Russia

2Université de Paris, Univ. Paris Diderot, MSC Laboratory, UMR 7057 CNRS, F-75
013 Paris, France

3P.N. Lebedev Physical Institute, Russian Academy of Sciences 119991, 53
Leninskij prospect, Moscow, Russia

*E-mail<EMAIL_ADDRESS>
###### Abstract

We report on direct numerical simulation of quasi-one-dimensional
bidirectional capillary-wave turbulence. Although nontrivial three-wave and
four-wave resonant interactions are absent in this peculiar geometry, we show
that an energy transfer between scales still occurs concentrated around the
linear dispersion relation that is broadened by nonlinearity. The wave
spectrum displays a clear wave number power-law scaling that is found to be in
good agreement with the dimensionally prediction for capillary-wave turbulence
involving four-wave interactions. The carried out high-order correlation
analysis (bicoherence and tricoherence) confirms quantitatively the dominant
role of four-wave quasi-resonant interactions. The Kolmogorov-Zakharov
spectrum constant is also estimated numerically. We interpret our results as
the first numerical observation of anisotropic capillary-wave turbulence in
which four-wave interactions play a dominant role.

Introduction. Wave turbulence occurs when nonlinear waves interact with each
other. This phenomenon is ubiquitous: oceanographic waves, plasma waves in the
solar wind, nonlinear optical waves, elastic waves, or gravitational waves [1,
2]. A weakly nonlinear theory (or weak turbulence theory) predicts
analytically the wave spectra in various domains involving nonlinear waves [1,
2, 3, 4, 5]. Such Kolmogorov-Zakharov’s (KZ) spectra can be also obtained by
dimensional analysis [6], an approach allowing also the spectrum predictions
in hitherto unexplored wave systems, such as electrohydrodynamic [7, 8],
magnetohydrodynamic [9, 10, 11], or hydroelastic wave turbulence [12, 13].
Wave turbulence has been mainly studied for isotropic systems, such as
gravity-capillary waves on the surface of a fluid (see reviews [14, 15, 16,
17]). KZ spectrum has been notably confirmed for isotropic capillary wave
turbulence both numerically [18, 19, 20, 21, 22, 23, 24] and experimentally
[14, 25]. Anisotropic wave turbulence (i.e. when there is a preferred
direction of wave propagation) is much less studied. In this case, nonlinear
coherent structures can strongly affect the wave spectrum [26, 27, 28],
whereas the leading order of nonlinear wave interactions can be modified (e.g
five-wave interactions in bidirectional gravity waves [29]).

Here, we focus on anisotropic capillary-wave turbulence, when the waves
propagate along one direction on the fluid surface (quasi-1D geometry). We
perform direct numerical simulation of the cubic nonlinear equations of the
corresponding hamiltonian. We observe that four-wave interactions play a
dominant role in this peculiar geometry, leading to an efficient energy
transfer between scales. The power-law wave spectrum found numerically is in
good agreement with the prediction that we obtained by dimensional analysis
for four-wave interactions. This latter differs from the usual KZ spectrum of
isotropic capillary weak turbulence involving three-wave interactions. Note
that for capillary waves, additional spatial symmetry imposes four-wave
interactions at the leading order [30]. Pure capillary-wave turbulence can be
reached experimentally either in low-gravity environment [31], or at the
interface of two immiscible fluids of close densities [32, 30], but has not
been performed in a quasi-1D geometry to the best of our knowledge.

Theoretical background – We consider pure capillary waves on the surface of a
fluid. The dispersion relation of linear waves then reads

$\omega(k)=\left(\frac{\sigma}{\rho}\right)^{1/2}k^{3/2},$ (1)

with $\omega$ the angular frequency, $k$ the wave number, $\sigma$ the surface
tension, and $\rho$ the mass density of the fluid. For weakly nonlinear waves,
energy transfers between waves occur due to a $N$-wave resonant interaction
process. The dominant nonlinear wave interaction corresponds to the number $N$
defined as the minimal integer for which the $N$-wave resonant conditions are
both satisfied

$\begin{split}\omega(k_{1})\pm\omega(k_{2})\ldots\pm\omega(k_{N})&=0{\rm\
,}\\\ k_{1}\pm k_{2}\ldots\pm k_{N}&=0{\rm\ .}\\\ \end{split}$ (2)

For pure capillary waves in 1D, there is no three-wave resonance interaction:
only $k_{1}=k_{2}=k_{3}=0$ is solution of (2) for $N=3$. No nontrivial four-
wave resonance exists: $k_{1}=k_{2}$ and $k_{3}=k_{4}$ is solution for $N=4$
but does not lead to energy exchange between modes in a $2\leftrightarrow 2$
interaction process. These can be easily proved graphically (e.g. see [2]).
Although exact resonance does not exists, a set of quasi-resonances still
exists due to the nonlinear spectral widening. Indeed, for finite wave
amplitudes, the widening of the dispersion relation authorizes approximated
resonance conditions [i.e. (2) with $\approx 0$ instead of $=0$]. A 1D
capillary-wave turbulence regime is thus expected due to quasi-resonances.

Let us now obtain the wave-elevation spectrum prediction of weak turbulence by
dimensional analysis following [6]. The wave elevation power spectrum is
defined as $S(k)=|\eta_{k}|^{2}$, where $\eta_{k}$ is the spatial Fourier
transform of the wave elevation $\eta(x)$ at the location $x$. For a capillary
wave system involving $N$-wave interactions, we assume

$S(k)=CP^{\frac{1}{N-1}}\left(\frac{\sigma}{\rho}\right)^{X}k^{Y}{\rm\ ,}$ (3)

where $C$ is the Kolmogorov-Zakharov constant, $k$ the wave number modulus,
and $P$ a constant energy flux to small scales (direct energy cascade). $X$
and $Y$ are obtained dimensionally, and reads

$X=\frac{3}{2(1-N)},\quad Y=d-5+\frac{1-2d}{2(N-1)}{\rm\ ,}$ (4)

where $d$ is the dimension in the physical space of the energy density $E_{k}$
($E_{k}$ has dimension $L^{6-d}T^{-2}$, $[S(k)]=L^{5-d}$, and
$[P]=L^{5-d}T^{-3}$). $S(k)$ is related to the spectral energy density (per
unit surface and mass density) by $E(k)=(\sigma/\rho)k^{2}S(k)$. Let us now
define $D$ as the dimension of Fourier space. In the case of anisotropic
geometry, as here, $d$ and $D$ are not equal to each other. Here we focus on a
bidirectional propagation of waves in a quasi-1D geometry, one has thus $D=1$,
and $d=2$ because the wave energy is distributed in the 2D space despite of
the collinear geometry.

Assuming that the leading order process is three-wave interaction, (3)-(4)
with $d=2$, $N=3$ leads to the prediction of the anisotropic ($D=1$)
capillary-wave turbulence spectrum

$\displaystyle S(k)$ $\displaystyle=C^{{\rm 3w}}_{{\rm
1D}}P^{1/2}\left(\frac{\sigma}{\rho}\right)^{-3/4}k^{-15/4}{\rm\ ,}$ (5)
$\displaystyle S(\omega)$ $\displaystyle=\frac{2}{3}C^{{\rm 3w}}_{{\rm
1D}}P^{1/2}\left(\frac{\sigma}{\rho}\right)^{1/6}\omega^{-17/6}{\rm\ .}$ (6)

Equation (6) being obtained from (5) with $S(k)dk=S(\omega)d\omega$. Equation
(5) is similar to the well known Zakharov-Filonenko spectrum of isotropic
capillary wave turbulence ($d=D=2$, $N=3$) [3], since the dimensional analysis
is independent of $D$. Equations (5)-(6) are obtained under the assumption of
the dominant influence of three-wave interactions.

At the next order (four-wave interactions), the anisotropic ($D=1$) capillary-
wave turbulence spectrum reads from (3)-(4) with $d=2$, $N=4$

$\displaystyle S(k)$ $\displaystyle=C^{{\rm 4w}}_{{\rm
1D}}P^{1/3}\left(\frac{\sigma}{\rho}\right)^{-1/2}k^{-7/2}{\rm\ ,}$ (7)
$\displaystyle S(\omega)$ $\displaystyle=\frac{2}{3}C^{{\rm 4w}}_{{\rm
1D}}P^{1/3}\left(\frac{\sigma}{\rho}\right)^{1/3}\omega^{-8/3}{\rm\ .}$ (8)

Note that the predictions of (7)-(8) involving four-wave interactions differ
from the ones of (5)-(6) obtained for $N=3$. The difference in the $k$-power-
law exponent is roughly 7%: $-15/4=-3.75$ ($N=3$) vs. $-7/2=-3.5$ ($N=4$).
Since (4) does not depend on $D$, (8) is in agreement with the solution
derived theoretically for four-wave interactions of isotropic capillary waves
[30]. The aim of this work is to perform direct numerical simulation of
anisotropic capillary-wave turbulence with high accuracy. We show that a
capillary-wave turbulence regime is observed in this peculiar 1D bidirectional
geometry, once a high enough level of pumping is reached. We show that the
$k$-scaling of the wave elevation spectrum is very close to the prediction of
(7) for the four-wave interaction case.

Model equations – Consider the irrotational potential flow of an ideal
incompressible fluid of infinite depth. The numerical model used here is based
on the well-known hamiltonian equations of motion of an ideal fluid with the
free surface obtained by Zakharov et al. [3, 4]. In this approach, the
original three-dimensional equations of hydrodynamics are transformed to
weakly nonlinear equations that directly describe the dynamics of the free
surface in two spatial coordinates $x$ and $y$. Here, we neglect gravity and
only the dynamics of capillary waves is considered.

The Hamiltonian of the system under study in dimensionless units ($\rho=1$,
$\sigma=1$) reads

$\begin{split}H&=\frac{1}{2}\int\left(\sqrt{1+(\nabla\eta)^{2}}-1\right)d\textbf{r}\\\
&+\frac{1}{2}\int\left(\psi\hat{k}\psi+\eta\left[|\nabla\psi|^{2}-(\hat{k}\psi)^{2}\right]\right)d\textbf{r}\\\
&+\frac{1}{2}\int\eta(\hat{k}\psi)\left[\hat{k}(\eta\hat{k}\psi)+\eta\nabla^{2}\psi\right]d\textbf{r}\
{\rm\ ,}\\\ \end{split}$ (9)

with $d\textbf{r}=dxdy$. The function $\psi(x,y,t)$ is the value of the
velocity potential on the boundary of the liquid,
$\nabla=(\partial_{x},\partial_{y})$ is the nabla operator, and $\hat{k}$ is
the integral operator defined in Fourier space as
$\hat{k}f_{\textbf{k}}=|\textbf{k}|f_{\textbf{k}}$. The equations of the
boundary motion are obtained by taking the variational derivatives:

$\frac{\partial\eta}{\partial t}=\frac{\delta
H}{\delta\psi},\qquad\frac{\partial\psi}{\partial t}=-\frac{\delta
H}{\delta\eta}.$ (10)

The equations (10) of the boundary motion are applicable for any symmetry of
the surface perturbations. In the quasi-1D geometry they can be simplified.
Since, the functions $\eta_{\textbf{k}}$ and $\psi_{\textbf{k}}$ depend only
on $x$-component of the wave vector, the absolute value of k is given by
$k=|\textbf{k}|=|k_{x}|$, and the nabla operator is reduced to the partial
derivative $\nabla\to\partial_{x}$. In such geometry, we can integrate the
right-hand side in (9) with respect to $y$ and introduce the normalized
hamiltonian as, $H_{1D}=H/L_{y}$, where $L_{y}$ is the characteristic length
of the system in the direction normal to the wave propagation.

For a complete description of the turbulent motions, we add to the equations
(10) the terms responsible for the pumping and dissipation of energy. As a
result, the cubically nonlinear equations (10) of the boundary motions take
the form

$\eta_{t}=\hat{k}\psi-(\eta\psi_{x})_{x}-\hat{k}(\eta\hat{k}\psi)+\hat{k}(\eta\hat{k}[\eta\hat{k}\psi])\\\
+\frac{1}{2}(\eta^{2}\hat{k}\psi)_{xx}+\frac{1}{2}\hat{k}(\eta^{2}\psi_{xx})+\hat{D}_{k}\eta\
{\rm\ ,}$ (11)
$\psi_{t}=\eta_{xx}-\frac{1}{2}\left[(\psi_{x})^{2}-(\hat{k}\psi)^{2}\right]\\\
-(\hat{k}\psi)\hat{k}(\eta\hat{k}\psi)-(\eta\hat{k}\psi)\psi_{xx}+F(x,t)+\hat{D}_{k}\psi{\rm\
.}$ (12)

The operator $\hat{D}_{k}$ is responsible for the effect of viscosity, in
$k$-space it is defined as,

$\hat{D}_{k}=-\nu_{0}(|k|-k_{d})^{2},\quad|k|\geq
k_{d};\quad\hat{D}_{k}=0,\quad|k|\leq k_{d}.$

Here $\nu_{0}$ is a constant, and $k_{d}$ is the wave number determining the
spatial scale at which the energy dissipation occurs. The term responsible for
the pumping of energy is defined as follows,

$F(x,t)=a_{0}\sum_{i=1}^{4}\sin(f_{i}t)\cos(k_{i}x+\phi_{i}),$ (13)

where $a_{0}$ is a constant defining the forcing strength, $k_{i}=i$ are the
forcing wave numbers, $f_{i}$ and $\phi_{i}$ are the random numbers normally
distributed in the interval $[0,2\pi]$. The forcing term thus corresponds to a
superimposition of few oscillating stationary waves each of random amplitudes
and phases in the following range of the forcing scale: $k_{i}=1$, 2, 3 and 4.
To stabilize the numerical scheme, we retain only the linear term in the
expansion of the boundary curvature in (12), see [33] for more details.

Figure 1: (Color online) Normalized hamiltonian $H_{1D}$ vs time. Blue solid
line corresponds to forcing amplitude $a_{0}=2$, and black line to a decaying
simulation (no forcing $a_{0}=0$). Inset: PDF of wave height rescaled by its
standard deviation (blue dots), red dashed line corresponds to Gaussian fit;
the inset shows the temporal evolution of the wave height $\eta(t)$ at $x=0$.

Simulation results – The numerical integration scheme for the solution of the
system of (11)–(12) in time is based on the fourth-order explicit Runge-Kutta
method with the step $dt$. The spatial derivatives and integral operators are
calculated using the pseudo-spectral methods with the total number harmonics
$\mathcal{N}$. The simulations were performed in a periodic 1D box of size
$L_{x}=2\pi$ with the parameters: $x\in[0,L_{x}]$, $\mathcal{N}=2048$,
$dt=2.5\ 10^{-5}$, $a_{0}=2$, $k_{i}=1$, 2, 3, and 4, $k_{d}=500$, and
$\nu_{0}=10$.

We present first numerical simulations corresponding to a continuous energy
pumping over time. Figure 1 shows the temporal evolution of the normalized
hamiltonian $H_{1D}$ (the total energy of the system). It first increases with
time and then reaches a quasi-stationary state for $t\geq 2000$ (see blue
solid line). Within this quasi-stationary state, the wave height $\eta(t)$ at
fixed location is found to be erratic (see inset of Fig. 1), and its
probability density function (PDF) computed over $t\in[2000,5000]$ is close to
Gaussian distribution (see inset of Fig. 1). The time-averaged value of the
wave steepness in the quasi-stationary state is
$\left\langle\sqrt{\int_{L_{x}}|\partial\eta(x,t)/\partial
x|^{2}dx/L_{x}}\right\rangle\simeq 0.36$, $\langle\cdot\rangle$ denoting a
temporal averaging. Although the latter is not weak, the distribution is found
to be symmetric (Skewness
$Sn\equiv\langle\eta^{3}\rangle/\langle\eta^{2}\rangle^{3/2}\simeq 0.01$),
whereas the extreme events are less likely than for a Gaussian distribution
(Kurtosis $\langle\eta^{4}\rangle/\langle\eta^{2}\rangle^{2}\simeq 2.65<3$)
underlying an unexpected symmetric light-tailed distribution. Indeed, contrary
to gravity or gravity-capillary waves where nonlinearities sharpen the crests
and flatten the troughs (i.e. $Sn>0$), highly symmetric distribution is
observed here (as for isotropic capillary-wave turbulence for 3-wave [23] or
4-wave interactions [30] for comparable wave steepness, but with light tails
due to nonlinear effects.

_Dispersion relation –_ Figure 2 shows the full space-time Fourier power
spectrum $S(k,\omega)$ of the wave elevation $\eta(x,t)$ in the stationary
state ($t\in[2000,5000]$). We observe that the most of the energy injected at
low wave numbers is transferred within a large range of wave numbers, and is
concentrated around the linear dispersion relation (1) as expected for weak
capillary-wave turbulence. Energy is also present out of this law
corresponding to a spectral widening due to nonlinear effects. Since no wave
resonant interaction is authorized in this 1D system (see above), this
broadening is consistent with an energy transfer due to quasi-resonance (see
below). No signature of coherent structures appears on this nonlinear
dispersion relation such as bound waves as reported routinely in isotropic
gravity or gravity-capillary wave turbulence [34, 35, 36]. No deviation from
(1) is also observed due to nonlinear corrections [37, 35].

Figure 2: (Color online) Space-time Fourier power spectrum $S(k,\omega)$ of
wave elevation. Black dashed line corresponds to the linear dispersion
relation of (1), and white solid lines show a nonlinear frequency broadening:
$\omega(k)\pm\delta_{\omega}$ with $\delta_{\omega}=40$.

_Spatial spectrum –_ Figure 3 shows the time-averaged spatial-power spectrum
$S(k)$ of the wave height $\eta(x,t)$ in the stationary state. A clear power-
law scaling is observed on more than one decade in $k$. The best fit is
$S(k)\sim k^{-3.5\pm 0.1}$ which is closer to the prediction of (7) than to
the one of (5). The compensated spectra are shown in the inset of Fig. 3.
Indeed, the wave turbulence spectrum $S(k)$ is better approximated by
$k^{-7/2}$ ($N=4$) than by $k^{-15/4}$ ($N=3$) within the inertial range
$5<k<180$. Thus, our direct numerical simulations suggest that four-wave
quasi-resonant interactions are involved for 1D capillary-wave turbulence. Let
us prove it below indirectly.

Figure 3: (Color online) Time-averaged spatial spectrum $S(k)$ of wave
elevation in the quasi-stationary state. Black bullets correspond to the
harmonics pumped. Solid line corresponds to (7), dashed line to (5). Inset:
compensated spectra $S(k)k^{15/4}$ and $S(k)k^{7/2}$ vs. $k$.

_Quasi-resonances –_ High-order correlation analysis is now performed to
confirm quantitatively the influence of four-wave quasi-resonant interactions
in a consistent way with the above results from the wave spectrum (second-
order correlation). As explained above, the three-wave resonance conditions
(2) are not satisfied exactly in 1D, but the occurrence of quasi-resonances
due to nonlinear spectral widening is still possible. Write the three-wave
quasi-resonance condition in $\omega$,

$|(k_{1}+k_{2})^{3/2}-k_{1}^{3/2}-k_{2}^{3/2}|\leq\delta_{\omega}{\rm\ ,}$
(14)

where $\delta_{\omega}$ is the nonlinear wave frequency shift, usually
estimated by nonlinear spectral widening as in Fig. 2. The condition for the
four-wave quasi-resonance reads

$|(k_{1}+k_{2}-k_{3})^{3/2}-k_{1}^{3/2}-k_{2}^{3/2}+k_{3}^{3/2}|\leq\delta_{\omega}{\rm\
.}$ (15)

To check the existence of 3-wave quasi-resonant interactions satisfying (14),
we compute the normalized third-order correlation in $k$ of the wave height
(the so-called bicoherence) [38, 39]

$B(k_{1},k_{2})=\frac{|\langle\eta^{*}_{k_{1}}\eta_{k_{2}}\eta_{k_{1}+k_{2}}\rangle|}{\sqrt{\langle|\eta_{k_{1}}|^{2}\rangle\langle|\eta_{k_{2}}\eta_{k_{1}+k_{2}}|^{2}\rangle}}{\rm\
,}$ (16)

where $\ast$ denotes the complex conjugate. To highlight 4-wave quasi-resonant
interactions satisfying (15), we compute the fourth-order correlation of the
wave height (or tricoherence) [40]

$T(k_{1},k_{2},k_{3})=\frac{|\langle\eta^{*}_{k_{1}}\eta^{*}_{k_{2}}\eta_{k_{3}}\eta_{k_{1}+k_{2}-k_{3}}\rangle|}{\sqrt{\langle|\eta_{k_{1}}\eta_{k_{2}}|^{2}\rangle\langle|\eta_{k_{3}}\eta_{k_{1}+k_{2}-k_{3}}|^{2}\rangle}}{\rm\
.}$ (17)

Figure 4: (Color online) Bicoherence (a) and tricoherence (b) of the Fourier
transform of the wave elevation estimated from (16) and (17), resp. The
tricoherence is shown for the fixed value $k_{3}=20$. White lines correspond
to (14) in (a) [resp. (15) in (b)] with $\delta_{\omega}=40$, i.e. to the
boundaries of authorized 3-wave [resp. 4-wave] quasi-resonances.

Due to the peculiar normalizations used in (16)-(17), the functions
$B(k_{1},k_{2})$ and $T(k_{1},k_{2},k_{3})$ range from 0 (no correlation) to 1
(perfect correlation). $B(k_{1},k_{2})$ and $T(k_{1},k_{2},k_{3})$ are
computed in the quasi-stationary regime, and are displayed in Fig. 4. Figure
4(a) shows that three-wave quasi-resonances are indeed present but their
number is very small: most of the quasi-resonant modes are located at low wave
numbers near forcing ones within the area bounded by (14) (white line). Figure
4(b) shows a completely different situation. We can see a large number of
quasi-resonant modes within the area bounded by (15) (white lines) as well as
degenerated 4-wave exact resonances at $k_{1}=k_{3}$, and $k_{2}=k_{3}$.
Quantitatively, the ratio between the sum of tricoherence values (at fixed
$k_{3}$) excluding trivial resonances and the bicoherence ones is
$r=\int\int\frac{T(k_{1},k_{2},20)}{B(k_{1},k_{2})}dk_{1}dk_{2}\simeq 1.6$.
Taking into account all possible values of $k_{3}$ in the inertial range leads
to $r\simeq 70$. Thus, the energy transfers are mostly due to four-wave quasi-
resonant interactions.

Figure 5: (Color online) Spectrum coefficient $\gamma^{{\rm 4w}}(P)$ versus
energy dissipation rate $P$ (red markers). Solid line corresponds to the fit
$0.43P^{1/3}$. Inset: compensated Kolmogorov-Zakharov’s constants versus $P$
for a 3-wave (black square) or 4-wave (red circle) process.

_Kolmogorov-Zakharov’s constant –_ One can finally estimate the spectrum
constant $C^{{\rm 4w}}_{{\rm 1D}}$ of (7) from simulation series in decaying
wave turbulence. To wit, after $t=1800$ from the beginning of the simulations,
we suddenly stop the pumping. The system then slowly loses energy from this
wave turbulence regime (see black line in Fig. 1). Rewriting the weak
turbulence spectrum prediction of (7) as $S(k)=\gamma^{{\rm 4w}}k^{-7/2}$
where the coefficient $\gamma^{{\rm 4w}}$ depends on the energy dissipation
rate $P$ and on the constant $C^{{\rm 4w}}_{{\rm 1D}}$ as $\gamma^{{\rm
4w}}(P)=C^{{\rm 4w}}_{{\rm 1D}}P^{1/3}$ (since $\sigma=\rho=1$). Figure 5
shows the numerically obtained dependence of $\gamma^{{\rm 4w}}$ with $P$
during the decay ($P$ being estimated from the solid black line in Fig. 1, and
$\gamma^{{\rm 4w}}$ from the decaying spectrum). By fitting this coefficient
with a $P^{1/3}$ dependence, we infer Kolmogorov-Zakharov’s constant $C^{{\rm
4w}}_{{\rm 1D}}$ as $0.43\pm 0.04$. This value could be compared to the
theoretical constant that can, in principle, be computed [30]. Although 3-wave
quasi-resonance interactions are not the dominant process, we perform an
analogous analysis for a 3-wave process leading to the numerical value for
$C^{{\rm 3w}}_{{\rm 1D}}=5.9\pm 0.06$. The inset of Fig. 5 shows the
compensated values of the Kolmogorov-Zakharov’s constants for the two
interaction processes.

_Scale separation –_ Weak turbulence theory assumes a timescale separation
between the linear timescale $\tau_{{\rm lin}}=1/\omega$, and the nonlinear
time, which reads dimensionally $\tau_{{\rm nl}}^{{\rm 4w}}\sim
P^{-2/3}(\sigma/\rho)^{1/2}k^{-1/2}$ for capillary waves ($N=4$, $d=2$,
$\forall D$). This means a nonlinearity parameter $\tau_{{\rm lin}}/\tau_{{\rm
nl}}^{{\rm 4w}}\sim P^{2/3}(\sigma/\rho)^{-1}k^{-1}\ll 1$. As nonlinearity
increases with $P$, breaking of weak turbulence is thus expected to occur for
$P>P_{c}=k^{3/2}(\sigma/\rho)^{3/2}$ ($P_{c}$ is independent on $N$). The
maximum value of $P$ used here is few orders of magnitude less than this
critical flux regardless the value of $k$, and thus satisfies indirectly the
expected scale separation.

_Conclusion –_ Although resonant wave interactions are forbidden in this
geometry, we show that anisotropic (quasi-1D) pure capillary waves display
small-scale wave turbulence due to four-wave quasi-resonant interactions, as
highlighted by the wave spectrum scaling and high-order correlations analysis.
An unexpected symmetric light-tailed wave distribution is also observed. We
hope that our study will trigger future investigations, notably to better
understand the large-scale dynamics (larger than the forcing scale) of
collinear wave turbulence such as the inverse cascade or the statistical
equilibrium.

Acknowledgments – E.K. thanks partial support by Russian Science Foundation
project No. 19-71-00003. E.F. thanks partial support of the French National
Research Agency (ANR Dysturb, project No. ANR-17-CE30-0004), and of the Simons
Foundation/MPS No. 651463-Wave Turbulence notably for the mission of E.K. in
Paris, France. Software tool development for numerical simulation was
partially supported by RFBR, project No. 20-38-70022.

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