#  Momentum Gauge Fields and Non-Commutative Space-Time

Eduardo Guendelman<EMAIL_ADDRESS><EMAIL_ADDRESS>Department of Physics, Ben-Gurion University of the Negev, Beer-Sheva, Israel
Frankfurt Institute for Advanced Studies, Giersch Science Center, Campus
Riedberg, Frankfurt am Main, Germany
Bahamas Advanced Studies Institute and Conferences, 4A Ocean Heights, Hill
View Circle, Stella Maris, Long Island, The Bahamas  Douglas Singleton
<EMAIL_ADDRESS>Physics Department, California State University
Fresno, Fresno, CA 93740

###### Abstract

In this work we present a gauge principle that starts with the momentum space
representation of the position operator
(${\hat{x}}_{i}=i\hbar\frac{\partial}{\partial p_{i}}$) rather than starting
with the position space representation of the momentum operator
(${\hat{p}}_{i}=-i\hbar\frac{\partial}{\partial x_{i}}$). This extension of
the gauge principle can be seen as a dynamical version of Born’s reciprocity
theory which exchanges position and momentum. We discuss some simple examples
with this new type of gauge theory: (i) analog solutions from ordinary gauge
theory in this momentum gauge theory, (ii) Landau levels using momentum gauge
fields, (iii) the emergence of non-commutative space-times from the momentum
gauge fields. We find that the non-commutative space-time parameter can be
momentum dependent, and one can construct a model where space-time is
commutative at low momentum but becomes non-commutative at high momentum.

##  I Gauge theory in momentum space

Gauge theories have been one of the central ideas of theoretical physics in
the past hundred years straumann ; abers . The Standard Model of particle
physics, which describes all known non-gravitational interactions, is a gauge
theory sm ; RelativisticQuantumMechanicsandRelatedTopics  and general
relativity can be viewed as a gauge theory lasenby . It is very important to
emphasize the central role of Professor Steven Weinberg in the development and
applications of the gauge principle in the construction of what we call now
the Standard Model. Professor Weinberg was also very active into the issue of
extending the Standard Model, exploring the ideas of axions w-axion ,
supersymmetry w-susy , string theory and cosmological issues w-cosmo . Much of
Professor Weinberg’s work dealt with symmetries in physics and their
applications. In this work we present a new extension of the gauge symmetry
principle, by extending the usual gauge symmetry to momentum space.

In the standard formulation of a gauge theory one starts with a space-time
dependent matter field $\Psi(x)$ which satisfies some matter field equation
(e.g. Schrödinger equation, Klein-Gordon equation, Dirac equation) and
requires that this matter field satisfy a local phase symmetry of the form
$\Psi(x)\to e^{-i\lambda(x)}\Psi(x)$. The gauge function, $\lambda(x)$, can
depend on space and time. Along with this local phase symmetry of the matter
field, one needs to introduce the kinetic momentum/gauge covariant derivative
$p_{i}\to p_{i}-eA_{i}(x)$ or $\frac{\partial}{\partial
x_{i}}\to\frac{\partial}{\partial x_{i}}-ieA_{i}(x)$, where the vector
potential obeys $A_{i}(x)\to
A_{i}(x)-\frac{1}{e}\frac{\partial\lambda(x)}{\partial x_{i}}$. This standard
construction is done in position space: the matter field, $\Psi$ is a function
of position, the momentum operator is given as a derivative of position
($p_{i}=-i\frac{\partial}{\partial x_{i}}$ and we take $\hbar=1$), and the
vector potential and gauge function are functions of space and time
coordinates.

However, quantum mechanics can be carried out in momentum space as well with
the matter field being a function of momentum, $\Psi(p)$, and the position
operator being given by $x_{i}=i\frac{\partial}{\partial p_{i}}$. In this
construction the momentum operator is just multiplication by $p_{i}$ just as
the position operator in position space is multiplication by $x_{i}$. The
momentum space gauge transformation of the matter field should be

$\Psi(p)\to e^{-i\eta(p)}\Psi(p)~{}.$ (1)

The equivalent of the generalized position/gauge covariant derivative is

$x_{i}\to x_{i}-gC_{i}(p)~{}~{}~{}~{}{\rm
or}~{}~{}~{}~{}\frac{\partial}{\partial p_{i}}\to\frac{\partial}{\partial
p_{i}}+igC_{i}(p).$ (2)

We have used $x_{i}=i\frac{\partial}{\partial p_{i}}$, $g$ is some momentum-
space coupling, and $C_{i}(p)$ is a momentum-space gauge function which must
satisfy

$C_{i}(p)\to C_{i}(p)+\frac{1}{g}\frac{\partial\eta(p)}{\partial p_{i}}.$ (3)

Finally one can construct a momentum-space field strength tensor which is
invariant under just (3), namely

$G_{ij}=\frac{\partial C_{i}}{\partial p_{j}}-\frac{\partial C_{j}}{\partial
p_{i}}.$ (4)

This is the $p_{i}p_{j}$ component of the momentum gauge field, field strength
tensor. It is the analog of $x_{i}x_{j}$ component of the standard gauge
field, field strength tensor $F_{ij}=\frac{\partial A_{i}}{\partial
x_{j}}-\frac{\partial A_{j}}{\partial x_{i}}$. The 4-vector version of the
standard gauge potential and field strength tensor are $A_{i}\to A_{\mu}$ and
$F_{ij}\to F_{\mu\nu}$. One needs to make a similar 4-vector/4-tensor
extension for the momentum gauge field and associated field strength tensor
via $C_{i}(p)\to C_{\mu}(p)$ and $G_{ij}\to G_{\mu\nu}$. The momentum
generalized gauge field and field strength tensor are reminiscent of the Berry
connection and Berry curvature berry1984 , where the Berry connection/Berry
“gauge” field is the function of some parameter, that is not necessarily
position. Here $C_{i}(p)$ and $G_{ij}(p)$ are Berry connections and Berry
curvatures that are specifically functions of momentum.

One can ask about the units of the momentum coupling, $g$ and momentum gauge
field, $C_{\mu}$, relative to the standard coupling, $e$, and standard gauge
field, $A_{\mu}$. From $p_{i}\to p_{i}-eA_{i}(x)$ and $x_{i}\to
x_{i}-gC_{i}(p)$ one sees that $eA_{i}(x)$ has units of momentum while
$gC_{i}(p)$ has units of position. This leaves two options for the units of
$g$ and $C_{\mu}$. First one can choose for $g$ to have the same units as $e$
and then the units of $C_{\mu}$ would be the units of $A_{\mu}$ multiplied by
$\frac{[position]}{[momentum]}=\frac{[time]}{[mass]}$. Second one can choose
for $C_{\mu}$ and $A_{\mu}$ to have the same units and in this case the units
of $g$ would be the units of $e$ again multiplied by the same factor
$\frac{[position]}{[momentum]}=\frac{[time]}{[mass]}$.

One can ask if there is some deeper connection or condition between the
standard coupling $e$ and momentum coupling $g$, perhaps something like the
Dirac quantization condition dirac  between electric and magnetic charge. One
idea might be to take the option above where $e$ and $g$ have the same units
and then via Born reciprocity require the couplings to be exchangeable i.e.
$e\leftrightarrow g$. We leave this question for future work.

The above discussion shows that one can easily construct a momentum-space
analog of the canonical position-space gauge procedure. There are two
questions this raises: (i) What physical use/significance would this momentum
gauge field construction have? (ii) Why is this momentum gauge field
construction not as common as the standard gauge field construction? The first
question will be addressed in following sections, but here we will address the
second question. The answer may lie in the asymmetric way in which the
momentum and position operators appear the in simplest, free particle
Hamiltonian. For a non-relativistic object of mass $m$ this Hamiltonian is

$H=\frac{1}{2m}(p_{1}^{2}+p_{2}^{2}+p_{3}^{2})=-\frac{\hbar^{2}}{2m}\left(\frac{\partial^{2}}{\partial
x^{2}_{1}}+\frac{\partial^{2}}{\partial
x^{2}_{2}}+\frac{\partial^{2}}{\partial
x^{2}_{3}}\right)=-\frac{\hbar^{2}}{2m}\nabla^{2}~{}.$ (5)

The Hamiltonian in (5) is suited for the covariant derivative $p_{i}\to
p_{i}-eA_{i}(x)$ or $\frac{\partial}{\partial
x_{i}}\to\frac{\partial}{\partial x_{i}}-ieA_{i}(x)$ but there is no room, nor
use for the momentum space version in (2). However, a more symmetric starting
point would be to consider the non-relativistic simple harmonic oscillator
Hamiltonian

$H=\frac{1}{2m}(p_{1}^{2}+p_{2}^{2}+p_{3}^{2})+\frac{m\omega^{2}}{2}(x_{1}^{2}+x_{2}^{2}+x_{3}^{2})\to\frac{1}{2}(p_{1}^{2}+p_{2}^{2}+p_{3}^{2})+\frac{1}{2}(x_{1}^{2}+x_{2}^{2}+x_{3}^{2})~{}.$
(6)

In the last step we have chosen the mass and frequency of the oscillator as
$m=1$ and $\omega=1$. Looking at the last form in (6) one sees a symmetry
between momentum and position of $p_{i}\leftrightarrow x_{i}$. This symmetry
provides an argument to have the gauge principle apply not only the momentum
via $p_{i}\to p_{i}-eA_{i}$, but also to the position via $x_{i}\to
x_{i}-gC_{i}$. One can argue for the naturalness of
$\frac{p_{i}^{2}}{2m}+\frac{m\omega^{2}}{2}x_{i}^{2}$ over just
$\frac{p_{i}^{2}}{2m}$ by pointing to the quantum field theory (QFT) vacuum,
which can be viewed as a collection of harmonic oscillators zee , so that
having both the momentum and position terms in the Hamiltonian is more natural
than have only momentum or only position.

In the above we have exchanged the roles of the position and momentum
operators in the usual construction of a gauge theory. The momentum gauge
fields provides for a dynamical model of Born reciprocity born1949  which
exchanged the position and momentum operators as ${\hat{x}}\to{\hat{p}}$ and
${\hat{p}}\to-{\hat{x}}$, which would imply that for every ordinary gauge
field there should be a corresponding momentum gauge field. For example the
Hamiltonian in (6) is invariant under this swap of position and momentum in
units where $m=1$ and $\omega=1$. The minus sign in the momentum to position
transformation keep the standard form of the position-momentum commutator
under this change i.e. $[{\hat{x}},{\hat{p}}]=i\hbar$ is invariant under
${\hat{x}}\to{\hat{p}}$ and ${\hat{p}}\to-{\hat{x}}$. The Hamiltonian in (6)
is also invariant under the transformation ${\hat{x}}\to{\hat{p}}$ and
${\hat{p}}\to{\hat{x}}$, but this would then change the sign of the position-
momentum commutator $[{\hat{x}},{\hat{p}}]=-i\hbar$. Nevertheless this would
still lead to the same uncertainty principle since $\Delta x\Delta
p=\frac{1}{2}\left|\langle[{\hat{x}},{\hat{p}}]\rangle\right|$. The main
difference between the model we lay out here and that in reference born1949
is that we have included a momentum gauge field (3) and momentum field
strength tensor (4), thus making possible a dynamical model of Born
reciprocity.

##  II Connection to non-commutative space-time

###  II.1 Constant non-commutativity parameter

In this subsection we point out the connection of the above momentum gauge
theory with non-commutative geometry, by which we mean coordinates obeying

$[x_{i},x_{j}]=i\Theta_{ij}~{},$ (7)

where $\Theta_{ij}$ is an anti-symmetric, constant rank-2 tensor. A review of
non-commutative field theory can be found in Noncommutativefieldtheory , and
interesting applications of non-commutative geometry to modifications on the
hydrogen atom spectrum can be found in NoncommutativeHydrogenatom . There are
also work that looks at how non-commutative geometry may cure the
singularities found in black holes and other solutions in general relativity
piero-2009 . The construction from the previous section leads exactly to this
kind of non-commutativity between the coordinates. We begin with equation (2)
and define a generalized, gauge invariant coordinate
$X_{i}=x_{i}-gC_{i}(p)=i\partial_{p_{i}}-gC_{i}(p)$. In its first form this
looks like coordinate translation by $gC_{i}(p)$. Calculating the commutator
of $X_{i}$ and $X_{j}$ gives

$[X_{i},X_{j}]=igG_{ij}~{},$ (8)

with the momentum-space field strength $G_{ij}$ defined in (4). Equation (8)
is of the form (7) with $\Theta_{ij}=gG_{ij}$.

The result in (8) is reminiscent of the non-commutativity of the covariant
derivative for regular, minimally coupled fields,
$\pi_{i}=p_{i}-eA_{i}(x)=-i\partial_{x_{i}}-eA_{i}(x)$. Calculating the
commutator of $\pi_{i}$ with $\pi_{j}$ gives

$[\pi_{i},\pi_{j}]=ieF_{ij}=ie\epsilon_{ijk}B^{k}~{},$ (9)

where
$B^{k}=\frac{1}{2}\epsilon^{kij}(\partial_{x_{i}}A_{j}-\partial_{x_{j}}A_{i})=\frac{1}{2}\epsilon^{kij}F_{ij}$
is the regular magnetic field. Comparing (8) with (9) one can define a
momentum gauge field “magnetic field” as ${\cal
B}^{k}=\frac{1}{2}\epsilon^{kij}(\partial_{p_{i}}C_{j}-\partial_{p_{j}}C_{i})=\frac{1}{2}\epsilon^{kij}G_{ij}$.
This in turn defines the non-commutation parameter of the spatial coordinates
on the right hand side of (8) to be constant only if the momentum “magnetic”
field is constant.

One can easily arrange for such a constant “magnetic” field solution via

$C^{0}=0~{}~{}~{},~{}~{}~{}C^{i}=\frac{1}{2}\epsilon^{ijk}{\cal B}^{j}p^{k}$
(10)

with ${\cal B}^{j}$ being a constant. Taking the curl of (10), using momentum
derivatives, and doing index gymnastics yields
$\epsilon^{lmi}\partial^{p^{m}}C^{i}={\cal B}^{l}$ – one gets a constant
“magnetic” field. This gives a constant non-commutative tensor
$\Theta_{ij}=gG_{ij}=g\epsilon_{ijk}{\cal B}^{k}$ i.e. in this way one
recovers a constant non-commutative parameter which is the most common
assumption in the literature Noncommutativefieldtheory ;
NoncommutativeHydrogenatom .

A fully 4-vector version of the spatial coordinate non-commutativity in (7) is
accomplished by promoting the 3-Latin indices to 4-Greek indices giving

$[x_{\mu},x_{\nu}]=i\Theta_{\mu\nu}~{},$ (11)

where $\Theta_{\mu\nu}$ is an anti-symmetric 4-tensor. In conjunction with
(11) the 4-tensor version of (8) becomes

$[X_{\mu},X_{\nu}]=igG_{\mu\nu}~{},$ (12)

In order to get a constant $\Theta_{\mu\nu}$ for a component with one space
index (e.g. $\mu=i$) and one time index (i.e. $\nu=0$) we need to have a
constant momentum gauge field, “electric” field. This is accomplished by
selecting the momentum gauge field as

$C^{0}=-{\cal E}^{j}p^{j}~{}~{}~{}~{};~{}~{}~{}~{}C^{j}=0$ (13)

The momentum gauge “electric” field is given by
$G_{0i}=\partial_{p^{0}}C_{i}-\partial_{p^{i}}C_{0}={\cal E}^{i}$ which is the
sought after constant momentum gauge field “electric” field. Using equations
(11) and (12) this gives the connection between the non-commutativity
parameter and momentum gauge field electric field of
$\Theta_{0i}=gG_{0i}=g{\cal E}_{i}$.

###  II.2 Variable non-commutativity parameter

In the previous subsection we looked at momentum gauge field configuration
with constant “magnetic” and constant “electric fields” in equations (10) and
(13) respectively. In this subsection we examine momentum gauge field
configurations which are variable. These variable momentum gauge fields then
imply a varying of the non-commutativity parameter via the connection
$\Theta_{uv}\propto G_{\mu\nu}$.

We first write down two common, ordinary gauge field solutions which have
gauge fields that vary with space and time and then construct the varying
momentum gauge field analogs. The two ordinary gauge field solutions we
consider are a plane wave and a static points charge. The Lagrange density for
standard gauge fields is ${\cal L}_{F}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$ with
$F^{\mu\nu}=\partial^{x^{\mu}}A^{\nu}-\partial^{x^{\nu}}A^{\mu}$. The
equations of motion from ${\cal L}_{F}$ are

$\partial_{x_{\mu}}(\partial^{x_{\mu}}A^{\nu}-\partial^{x_{\nu}}A^{\mu})=4\pi
J^{\nu}(x)\to\partial_{x_{\mu}}\partial^{x_{\mu}}A^{\nu}=4\pi
J^{\nu}(x)\to\Box_{x}A^{\nu}=4\pi J^{\nu}(x)~{},$ (14)

with $J^{\nu}(x)$ being a conserved 4-current coming from some matter source,
and $\Box_{x}$ is the d’Alembertian with respect to the time-position
coordinates. In the last line we have taken the Lorenz gauge
$\partial_{x_{\mu}}A^{x_{\mu}}=0$. Let us look at two common solutions to
(14): the plane wave vacuum solution and the point charge solution.

  * •

In vacuum ($J^{\nu}=0$) (14) has the solution $A^{\nu}\propto e^{i(px-
Et)}\varepsilon^{\nu}\delta(p^{2}-E^{2}/c^{2})$ where the $\delta$-function
enforces the mass shell condition $\frac{E^{2}}{p^{2}}=c^{2}$ and
$\varepsilon^{\nu}$ is the polarization vector.

  * •

For a point charge at rest one has the current
$J^{\nu}=(q\delta^{3}(r),0,0,0)$, which has the solution $A^{0}=\frac{q}{r}$
and ${\vec{A}}=0$, since
$\nabla_{x}^{2}\left(\frac{1}{r}\right)=4\pi\delta(r)$ .

We now examine how the above plays out for the momentum gauge fields. The
momentum gauge field Lagrange density is ${\cal
L}_{G}=-\frac{1}{4}G_{\mu\nu}G^{\mu\nu}$ with
$G^{\mu\nu}=\partial^{p_{\mu}}C^{\nu}-\partial^{p_{\nu}}C^{\mu}$. The
equations of motions that follow from this Lagrange density are

$\partial_{p_{\mu}}(\partial^{p_{\mu}}C^{\nu}-\partial^{p_{\nu}}C^{\mu})=4\pi{\cal
J}^{\nu}(p)\to\partial_{p_{\mu}}\partial^{p_{\mu}}C^{\nu}=4\pi{\cal
J}^{\nu}(p)\to\Box_{p}C^{\nu}=4\pi{\cal J}^{\nu}(p)~{},$ (15)

with ${\cal J}^{\nu}(p)$ being a 4-current matter source that is a function of
$p$, and $\Box_{p}$ is the d’Alembertian with respect to energy-momentum. In
the last expression we use the momentum space equivalent of the Lorenz gauge
$\partial_{p_{\mu}}C^{p_{\mu}}=0$. The current conservation in momentum space
reads $\partial_{p_{\mu}}{\cal J}^{p_{\mu}}=0$,

We now repeat the two types of solutions listed above for the standard gauge
theory, but for the momentum gauge theory.

  * •

In vacuum (${\cal J}^{\nu}=0$) (15) has solution $C^{\nu}\propto e^{i(px-
Et)}\varepsilon^{\nu}\delta(x^{2}-c^{2}t^{2})$ where the $\delta$-function
enforces the light-cone condition $\frac{x^{2}}{t^{2}}=c^{2}$, and
$\varepsilon^{\nu}$ is the polarization vector.

  * •

The momentum gauge equivalent of the charge at rest is given by ${\cal
J}^{\nu}=(g\delta^{3}(p),0,0,0)$, with $C^{0}=\frac{g}{p}$ and ${\vec{C}}=0$
since $\nabla_{p}^{2}\left(\frac{1}{p}\right)=4\pi\delta(p)$.

Notice that the point source in momentum space, that is ${\cal
J}^{\nu}=(g\delta^{3}(p),0,0,0)$ is a totally homogeneous solution in
coordinate space, since it is concentrated at zero momentum, which means
indeed the assumption of a totally homogeneous state. More generally, it is
interesting to observe that any current of the form ${\cal
J}^{\nu}=(f(p),0,0,0)$ , with ${\cal J}^{0}=f({\vec{p}})$ being $p^{0}$
independent, will satisfy the current conservation law of
$\partial_{p_{\mu}}{\cal J}^{\mu}=0$. Doing a Fourier transformation on this
to coordinate space yields $x_{\mu}{\tilde{\cal J}}^{\mu}=0$, where
${\tilde{\cal J}}^{\mu}$ is the Fourier transformation of ${\cal J}^{\mu}$.
The equivalent statements for a a regular 4-source would be
$J^{\nu}=(f({\vec{x}}),0,0,0)$, which satisfies the conservation law
$\partial_{x_{\mu}}J^{\mu}=0$ or Fourier transforming to momentum space
$p_{\mu}{\tilde{J}}^{\mu}=0$.

One can construct other conserved current sources for momentum gauge fields
that satisfy $x_{\mu}{\cal{\tilde{J}}}^{\mu}=0$. Starting with any 4-vector
$V^{\mu}$, we construct
${\cal{\tilde{J}}}^{\mu}=V^{\mu}-x^{\mu}V^{\nu}x_{\nu}/x^{2}$ which is easily
seen to satisfy $x_{\mu}{\cal{\tilde{J}}}^{\mu}=0$.

As a final comment the equation of motion for, $A_{\mu}$, given in (14) leads
to a propagator in momentum space that is proportion to
$\propto\frac{1}{p^{2}+i\epsilon}$. The imaginary infinitesimal term
$i\epsilon$ is a convergence factor to deal with the divergence as $p\to 0$.
In turn the momentum gauge field equation of motion, $C_{\mu}$, given in (15),
leads to a position space propagator proportional to $\propto\frac{1}{x^{2}}$.
Here we have not inserted a factor of $i\epsilon$. This is because the limits
$p\to 0$ and $x\to 0$ are physically different. The $p\to 0$ limit is the
infrared/low energy limit which is dealt with by inserting a convergence
factor of $i\epsilon$ which is taken to zero at the end. The $x\to 0$ limit is
the ultraviolet/high energy limit which is dealt with using the
renormalization procedure if the theory turns out to be renormalizable, or by
introducing a cut off if the theory turns out to be non renormalizable.

##  III Generalized Landau Levels

In this section we work on the case of generalized Landau levels with a
particle of mass $m$ in a constant ordinary magnetic field and constant
momentum “magnetic” field. We take both the ordinary and momentum magnetic
field to point in the $3/z$-direction. We want to take these magnetic fields
and minimally couple them to the free particle in equation (6). Applying
minimal coupling for both coordinate gauge fields and momentum gauge fields,
leads to $p_{i}\to p_{i}-eA_{i}$, and $x_{i}\to x_{i}-gC_{i}$. Having a
constant, ordinary magnetic field and a constant, momentum magnetic fields in
the $3/z$-direction can be obtained in the symmetric gauge with $A_{1}$ and
$A_{2}$ given by,

$A_{0}=0~{}~{},~{}~{}A_{1}=-\frac{1}{2}By~{}~{},~{}~{}A_{2}=\frac{1}{2}Bx~{},$
(16)

and with $C_{1}$ and $C_{2}$ also in the symmetric gauge given by,

$C_{0}=0~{}~{},~{}~{}C_{1}=-\frac{1}{2}{\cal
B}p_{y}~{}~{},~{}~{}C_{2}=\frac{1}{2}{\cal B}p_{x}~{},$ (17)

The constant values of the ordinary magnetic field and momentum magnetic field
from (16) and (17) are $B$ and ${\cal B}$ respectively.

So the equation of motion for the double gauged harmonic oscillator reads,

$\displaystyle H$ $\displaystyle=$
$\displaystyle\frac{1}{2m}\left(p_{x}+\frac{eBy}{2}\right)^{2}+\frac{1}{2m}\left(p_{y}-\frac{eBx}{2}\right)^{2}$
(18) $\displaystyle+$ $\displaystyle\frac{m\omega^{2}}{2}\left(x+\frac{g{\cal
B}p_{y}}{2}\right)^{2}+\frac{m\omega^{2}}{2}\left(y-\frac{g{\cal
B}p_{x}}{2}\right)^{2}+\frac{p_{z}^{2}}{2m}+\frac{m\omega^{2}}{2}z^{2}$

or (we drop the part of the Hamiltonian associated with the kinetic energy and
harmonic oscillator in the $z$-direction)

$H=\left(1+\frac{(gm\omega{\cal
B})^{2}}{4}\right)\left(\frac{p_{x}^{2}}{2m}+\frac{p_{y}^{2}}{2m}\right)+\left(1+\frac{(eB)^{2}}{4m^{2}\omega^{2}}\right)\frac{m\omega^{2}}{2}(x^{2}+y^{2})+L_{z}(-g_{1}B+g_{2}{\cal
B})~{}.$ (19)

Here $L_{z}=xp_{y}-yp_{x}$, this the angular momentum in the $z$-direction.
and $g_{1}=\frac{e}{2m}$ and $g_{2}=\frac{gm\omega^{2}}{2}$ are the coupling
strengths of the angular momentum to the coordinate magnetic field $B$ and the
momentum magnetic field ${\cal B}$ respectively.

The above results can be compared with the formulation of non-commutative
quantum mechanics Noncommutativequantummechanics  for the case of an harmonic
oscillator potential, and the results agree with those in
Noncommutativequantummechanics , if the identification of the non-commutative
parameter is made according to expression (8).

The coupling between $B$ and $L_{z}$ is exactly what one has from the standard
analysis of Landau levels. The coupling between $L_{z}$ and ${\cal B}$ is a
new feature arising from the momentum gauge fields, but the two coupling terms
to $L_{z}$ have a dual symmetry between the regular magnetic field, $B$, and
momentum gauge “magnetic” field, ${\cal B}$.

The first term in (19) shows that the system has now developed a new,
effective mass given by

$m_{eff}=\frac{m}{1+\frac{(gm\omega{\cal B})^{2}}{4}}~{}.$ (20)

The effective mass depends on the momentum “magnetic” field and is always less
than $m$ i.e. $m_{eff}<m$. In addition the second terms in (19) implies a new
effective frequency. Taking into account the effective mass in (20) to write
this second term in the form $\frac{m_{eff}\omega_{eff}^{2}}{2}$ gives a new
effective frequency of

$\omega_{eff}=\omega\sqrt{\left(1+\frac{(gm\omega{\cal
B})^{2}}{4}\right)\left(1+\frac{e^{2}B^{2}}{4m^{2}\omega^{2}}\right)}~{}.$
(21)

Note that in the effective frequency and the effective mass above, both
momentum (i.e. ${\cal B}$) and coordinate magnetic (i.e. $B$) fields
contribute.

One can define an effective magnetic field as

$B_{eff}=\frac{-g_{1}B+g_{2}{\cal B}}{\sqrt{g^{2}_{1}+g^{2}_{2}}}~{},$ (22)

so that the coupling of the $z$-component of angular momentum to the two
magnetic fields, ${\cal B}$ and $B$ in (19) can be written as
$\sqrt{g^{2}_{1}+g^{2}_{2}}B_{eff}L_{z}$. One can also define a generalized
magnetic field orthogonal to $B_{eff}$ via

$B_{nc}=\frac{g_{1}{\cal B}+g_{2}B}{\sqrt{g^{2}_{1}+g^{2}_{2}}}~{}.$ (23)

The subscripts $nc$ stand for “non-coupling” since $B_{nc}$, unlike $B_{eff}$,
does not couple to $L_{z}$.

The definition of the two generalized magnetic fields in (22) and (23) is
mathematically identical to the definition of the $Z^{0}$ boson and photon in
the Standard Model abers ; sm ; RelativisticQuantumMechanicsandRelatedTopics .
Further from (23) , (22) we can define an analog of the “Weinberg angle” via
the definition

$cos(\theta_{mixing})=\frac{g_{1}}{\sqrt{g^{2}_{1}+g^{2}_{2}}}~{}.$ (24)

Putting all of the above together the total Hamiltonian is then,

$H=\frac{1}{2m_{eff}}(p_{x}^{2}+p_{y}^{2})+\frac{1}{2}\omega_{eff}^{2}m_{eff}(x^{2}+y^{2})+\sqrt{g^{2}_{1}+g^{2}_{2}}B_{eff}L_{z}$
(25)

Note that $B_{eff}$ couples to the angular momentum, while $B_{nc}$ does not.
This is similar to the Standard model where the $Z^{0}$ has a mass term while
the photon remains massless.

Following abers-2  one can define creation/annihilation operators in terms of
$p_{x},p_{y}$ and $x,y$ as

$\displaystyle x$ $\displaystyle=$
$\displaystyle\sqrt{\frac{\hbar}{2\omega_{eff}m_{eff}}}\left(a_{1}+a^{\dagger}_{1}\right)~{}~{}~{};~{}~{}~{}y=\sqrt{\frac{\hbar}{2\omega_{eff}m_{eff}}}\left(a_{2}+a^{\dagger}_{2}\right)$
$\displaystyle~{}~{}~{}{\rm and}~{}~{}~{}$ (26) $\displaystyle p_{x}$
$\displaystyle=$ $\displaystyle
i\sqrt{\frac{\hbar\omega_{eff}m_{eff}}{2}}\left(a^{\dagger}_{1}-a^{\dagger}_{1}\right)~{}~{}~{};~{}~{}~{}p_{y}=i\sqrt{\frac{\hbar\omega_{eff}m_{eff}}{2}}\left(a^{\dagger}_{2}-a^{\dagger}_{2}\right)~{}.$

The creation and annihilation operators obey the usual relationship
$[a_{i},a^{\dagger}_{j}]=\delta_{ij}$. With these definitions we find
$L_{z}=xp_{y}-yp_{x}=i\hbar(a_{1}a_{2}^{\dagger}-a_{2}a_{1}^{\dagger})$ and
the Hamiltonian in (25) becomes
$H=\hbar\omega_{eff}(a^{\dagger}_{1}a_{1}+a^{\dagger}_{2}a_{2}+1)+i\hbar\sqrt{g^{2}_{1}+g^{2}_{2}}B_{eff}(a_{1}a_{2}^{\dagger}-a_{2}a_{1}^{\dagger})$.
The first two terms can be seen to be the normal 2D harmonic oscillator. The
third term looks a like a coupling between the generalized magnetic field and
the angular momentum in the $z$ direction.

##  IV Momentum dependent non-commutativity parameter

In this section we examine two simple examples where the non-commutativity
parameter, $\Theta_{\mu\nu}$, is not a constant but depends on the momentum.
Recently, other authors momentumdependenttheta  have considered momentum
dependent non-commutative parameters. However, in this work the inspiration is
quite different as it exploits some geometry in momentum space. Also the non-
commutativity parameter in momentumdependenttheta  depends on both momentum
and position, while our in our construction below the non-commutativity
parameter depends only on momentum, which is closer to the energy-momentum
dependence of masses and couplings in QFT that one finds from the
renormalization group.

The examples we choose are the momentum gauge field version of a capacitor and
solenoid, with the momentum gauge fields being piece-wise constant in
different momentum ranges, leading to different, $\Theta_{\mu\nu}$’s in these
different ranges.

###  IV.1 Capacitor-type momentum electric field configuration

The standard, infinite parallel plate capacitor has a 4-current source of

$J^{\nu}=(f(z),0,0,0)~{}~{}{\rm
with}~{}~{}f(z)=\sigma[\delta(z+a)-\delta(z-a)]$ (27)

This source represents two infinite planes of surface charge $\pm\sigma$
placed perpendicular to the $z$-axis at $z=\mp a$. This source gives an
electric field of

$E_{z}=4\pi\sigma~{}~{}{\rm for}~{}~{}-a\leq z\leq a~{}~{}~{}{\rm
and}~{}~{}~{}E_{z}=0~{}~{}{\rm for}~{}~{}|a|\leq|z|~{},$ (28)

i.e. non-zero between the planes and zero outside the planes.

The momentum gauge field analog of this standard capacitor system has a
constant momentum “electric” field similar to that in equation (13), but it
should be restricted in momentum rather than position as is the case in
equation (28). Actually for the momentum gauge field system we want the
inverse of the above standard capacitor – we want the momentum “electric”
field to be zero between the planes (i.e at small momentum) and non-zero
outside the planes (i.e at large momentum). The capacitor-like configuration
for the momentum gauge fields that we want has a 4-current source of

${\cal J}^{\nu}=(f(p),0,0,0)~{}~{}{\rm
with}~{}~{}f(p)=\Sigma[\delta(p_{z}+p_{a})+\delta(p_{z}-p_{a})].$ (29)

The planes are symmetrically placed at $p_{z}=\pm p_{a}$ and, in contrast to
the sources for the standard capacitor in (27), the momentum planes now have
the same “surface charge”, $\Sigma$. This same “surface charge” set up leads
to a momentum “electric” field in the $p_{z}$ direction given by

$\displaystyle{\cal E}_{z}$ $\displaystyle=$ $\displaystyle
4\pi\Sigma~{}~{}{\rm for}~{}~{}p_{z}\geq p_{a}~{}~{},~{}~{}{\cal
E}_{z}=-4\pi\Sigma~{}~{}{\rm for}~{}~{}p_{z}\leq-p_{a},$ $\displaystyle{\rm
and}~{}~{}~{}{\cal E}_{z}$ $\displaystyle=$ $\displaystyle 0~{}~{}{\rm
for}~{}~{}-p_{a}\leq p_{z}\leq p_{a}.$ (30)

The momentum “electric” field of (IV.1) is zero between the plates and non-
zero outside the plates, which is the inverse of the standard capacitor (28).

The reason for building our momentum gauge field capacitor system as the
inverse of the normal capacitor is due to the connection between the non-
commutativity parameter, $\Theta_{\mu\nu}$ and the momentum gauge field
tensor, $G_{\mu\nu}$, as given equations (11) and (12) i.e.
$\Theta_{\mu\nu}=gG_{\mu\nu}$. We want to have a normal position-position
commutator (i.e. $[X_{\mu},X_{\nu}]=0$) for momenta near zero (i.e. for
$-p_{a}\leq p_{z}\leq p_{a}$) but we want non-commutative space-time effects
for large momenta i.e. we want $\Theta_{\mu\nu}\propto G_{\mu\nu}\neq 0$ for
large momenta, $|p_{a}|\leq|p_{z}|$. This is different from the usual non-
commutative space-time approach where the non-commutative parameter is “turned
on” for all momentum. Here the non-commutativity, at least for the
$\Theta_{0i}$ components, is turned on only for $z$-momentum magnitude
satisfying $|p_{a}|<|p_{z}|$.

###  IV.2 Current sheet-type momentum magnetic field

In this subsection we carry out a similar construction as in the preceding
subsection, but for the space/space components of $\Theta_{\mu\nu}$ and
$G_{\mu\nu}$. In this case the standard gauge field system we want to build a
momentum gauge field analog of is two infinite plane sheet currents located at
$z=\pm a$. These current sheets are symmetrically placed on the $z$-axis
around $z=0$. The explicit surface currents are

${\vec{K}}=\pm J{\bf{\hat{y}}}~{}~{}~{}{\rm at}~{}~{}~{}z=\mp a$ (31)

This leads a regular magnetic field of

${\vec{B}}=4\pi J{\bf{\hat{x}}}~{}~{}~{}{\rm for}~{}~{}-a\leq z\leq
a~{}~{}~{}{\rm and}~{}~{}~{}{\vec{B}}=0~{}~{}{\rm for}~{}~{}|a|\leq|z|$ (32)

i.e. the magnetic field is a non-zero constant between the sheets and zero
outside the sheets.

The momentum gauge field analog of this is two momentum gauge field current
sheets at the momentum planes, $p_{z}=\pm p_{a}$. These planes are symmetric
around the origin through the $p_{z}$-axis. Explicitly the “momentum” current
sheets are

${\vec{\cal K}}={\cal J}{\bf{\hat{y}}}~{}~{}~{}{\rm at}~{}~{}~{}p_{z}=\pm
p_{a}$ (33)

Note that here we have the currents in the same direction, rather than
opposite direction as for the regular gauge field current sheets of (31). The
reason for this is the same as for the momentum gauge field, capacitor-like
system of the preceding subsection: we want the non-commutativity parameter to
be zero for momentum in the range $-p_{a}\leq p_{z}\leq p_{a}$ and we want a
non-zero non-commutativity parameter for momentum in the range
$|p_{a}|\leq|p_{z}|$. Putting this all together the momentum gauge field
“magnetic” field is

$\displaystyle{\vec{\cal B}}$ $\displaystyle=$ $\displaystyle 4\pi{\cal
J}{\bf{\hat{x}}}~{}~{}{\rm for}~{}~{}p_{a}\leq p_{z}~{}~{}~{}~{}{\rm
and}~{}~{}~{}~{}{\vec{\cal B}}=-4\pi{\cal J}{\bf{\hat{x}}}~{}~{}{\rm
for}~{}~{}p_{z}\leq-p_{a}$ $\displaystyle{\rm and}~{}~{}{\vec{\cal B}}$
$\displaystyle=$ $\displaystyle 0~{}~{}~{}~{}{\rm for}~{}~{}-p_{a}\leq
p_{z}\leq p_{a}~{}.$ (34)

The momentum gauge “magnetic” field is a non-zero, constant outside the
current sheets and zero between the current sheets. This implies that the
space/space non-commutativity parameter, $\Theta_{ij}$, is zero for momenta in
the range $-p_{a}\leq p_{z}\leq p_{a}$, while for large magnitude momenta
(i.e. $|p_{a}\leq|p_{z}|$) the space/space component
$\Theta_{yz}=gG_{yz}=g\epsilon_{yzx}{\cal B}_{x}=\pm g{\cal B}$ is a non-zero
constant. Both this simple example and the example from the preceding
subsection show that one can construct non-commutative space-times where the
non-commutativity only “turns” on at some large enough momentum, rather than
being on all the time.

##  V Summary and conclusions

In this paper we have studied the formulation of the gauge principle in
momentum space, or energy-momentum space in the relativistic case. Instead of
only starting with the momentum operator and introducing a covariant momentum
as $p_{i}\to p_{i}-eA_{i}(x)$ we also considered the position operator and
introduced a covariant position as $x_{i}\to x_{i}-gC_{i}(p)$. The preference
for having only the covariant momentum and not the covariant position comes
from the fact that in general one starts with a free Hamiltonian (5) which has
only momentum dependence. However a more symmetric treatment, motivated by the
fact that the QFT vacuum can be seen as a collection of oscillators, leads to
a Hamiltonian of the form given in (6) which then calls for both covariant
momentum and covariant position.

We presented several simple examples of this momentum formulation of the gauge
principle, showing that one could construct momentum gauge field analogs to
plane wave solutions, point charge solutions, and to Landau levels. All these
examples are underpinned by a dual symmetry, exchange symmetry or reciprocity
born1949  between momentum and position, namely ${\hat{x}}\to{\hat{p}}$ and
${\hat{p}}\to-{\hat{x}}$ (or also ${\hat{x}}\to{\hat{p}}$ and
${\hat{p}}\to{\hat{x}}$) which then relates the regular gauge fields to the
momentum gauge fields. A criticism of this momentum formulation of the gauge
principle is whether or not it has any concrete physical application or use.
In this regard we mention that the model presented here is similar to Born’s
reciprocity theory born1949  which Born had hoped would play a role in the
theory of elementary particle. The new feature here is that our version of
Born’s reciprocity is dynamical since we have introduced momentum gauge fields
(3) and momentum field strength tensors (4). We will explore physical
consequences of this idea in future work.

One potentially interesting application of this momentum gauge theory is that
it naturally lead to non-commutative geometry as given in equations (8) and
(12). This non-commutativity of space-time has been studied previously as a
way to extend QFT Noncommutativefieldtheory , as a way to test for extensions
to QED NoncommutativeHydrogenatom , and as a way to deal with the
singularities of general relativity piero-2009 . The non-commutativity of
these works rests on non-trivial space-time commutators of the form (11) where
the non-commutativity parameter, $\Theta_{\mu\nu}$ is a constant. In our
formulation, since the non-commutativity parameter is a momentum gauge field,
field strength tensor, $gG_{\mu\nu}$, it can vary with momentum, since the
momentum gauge field, $C_{\mu}$ can vary with momentum. In section IV we
constructed a very simple system, based on the infinite charge sheets and
infinite current sheets of introductory E&M, where the non-commutativity
parameter, $gG_{\mu\nu}$, would only turn on when the magnitude of the
momentum became large enough i.e. when the momentum satisfied
$|p_{a}|\leq|p_{z}|$ for some large, fixed $p_{a}$. This could have
interesting consequences since one could have commutative space-time below
$p_{a}$ that turns into non-commutative space-time above $p_{a}$.

These are examples to show that the non-commutativity parameters can be
screened by “charges” in the infrared or ultraviolet regions. This screening
has been studied here by introducing external momentum currents, but they
could also rise from quantum fluctuation, as in ordinary gauge theories where
coupling constants are screened in the infrared (i.e. in QED) or ultraviolet
(i.e. in QCD) regions. Another subject that could be studied is the
possibility of coordinate gauge fields and momentum gauge field mixing and/or
oscillating. This is suggested for example by the result that minimally
coupling both coordinate gauge fields and momentum gauge fields produces a
very specific linear combination (22) that couples to the angular momentum of
matter. Thus after integrating out the matter, we should be left with
coordinate gauge fields and momentum gauge field mixing and/or oscillating.
There are also no obstacles to considering non-Abelian momentum gauge fields.
Furthermore, a connection between momentum gauge fields and curved momentum
space can be established, where the momentum gauge field appear from a higher
dimensional curved momentum space from a Kaluza Klein mechanism KKmomentum .
This momentum space Kaluza-Klein approach could be further extended and could
provide additional insights into higher dimensional theories. Lastly the
present authors have worked on other ways to modify the gauge principle with
non-vector gauge fields dual-gauge-2 ; scalar-gauge ; oldscalar-gauge  or by
gauging a dual symmetry, dual-gauge . Also the “Curtright generalized gauge
fields” presents yet another way to generalize the gauge principle
curtrightgauge . However, the present way of modifying the gauge principle
that can have an additional symmetry principle underlying it namely the
exchange symmetry or Born reciprocity where ${\hat{x}}\to{\hat{p}}$ and
${\hat{p}}\to-{\hat{x}}$ or ${\hat{x}}\to{\hat{p}}$ and
${\hat{p}}\to{\hat{x}}$ i.e. the role of momentum and position are exchanged .

Motivated from the Born reciprocity, we may suspect that the simultaneous
existence of momentum and coordinate gauge fields could have important
consequences, for example in momentumandconfigurationgaugefieldstogether  it
is found that the simultaneous momentum-like Coulomb solution for momentum
gauge fields, given by $C^{0}=\frac{g}{p}$ and ${\vec{C}}=0$, together with
the regular configuration space Coulomb solution, given by $A^{0}=\frac{e}{r}$
and ${\vec{A}}=0$, can be related to the generation of an emergent spacetime.

##  VI Acknowledgements

We would like to thank FQXi - Foundational Questions Institute and COST Action
CA18108 - Quantum gravity phenomenology in the multi-messenger approach (QG-
MM) for financial support and to Fabian Wagner for very interesting
discussions.

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# Electron transport in a ferromagnetic/normal/ferromagnetic tunnel junction
based on the surface of a topological insulator

Jian-Hui Yuan  The corresponding author, E-mail<EMAIL_ADDRESS>Yan
Zhang  The department of Physics, Guangxi medical university, Nanning,
Guangxi, 530021, China  Jian-Jun Zhang  Ze Cheng  School of Physics, Huazhong
University of Science and Technology, Wuhan 430074, China

###### Abstract

We theoretically study the electron transport properties in a
ferromagnetic/normal/ferromagnetic tunnel junction, which is deposited on the
top of a topological surface. The conductance at the parallel (P)
configuration can be much bigger than that at the antiparallel (AP)
configuration. Compared P  with  AP configuration, there exists a shift of
phase which can be tuned by gate voltage. We find that the exchange field
weakly affects the conductance of carriers for P configuration but can
dramatically suppress the conductance of carriers for AP configuration. This
controllable electron transport implies anomalous magnetoresistance in this
topological spin valve, which may contribute to the development of spintronics
. In addition, we find that there is a Fabry-Perot-like electron interference.

Topological insulator, Electronic transport, Ferrimagnetic

###### pacs:

72.80.Sk, 73.40.-c, 75.50.Gg

##  I Introduction

The concept of a topological insulator (TI) dates back to the work of Kane and
Mele, who focused on two-dimensional (2D) systems 1. There has been much
recent interest in TIs, three-dimensional insulators with metallic surface
states protected by time reversal invariance ${[1-25]}$. Its theoretical
${[2]}$ and experimental ${[3]}$ discovery has accordingly generated a great
deal of excitement in the condensed matter physics community. In particular,
the surface of a three-dimensional (3D) TI, such as Bi2Se3 or Bi2Te3 ${[4]}$,
is a 2D metal, whose band structure consists of an odd number of Dirac cones,
centered at time reversal invariant momenta in the surface Brillouin zone
${[5]}$. This corresponds to the infinite mass Rashba model ${[6]}$, where
only one of the spin-split bands exists. This has been beautifully
demonstrated by the spin- and angle-resolved photoemission spectroscopy
${[7,8]}$. Surface sensitive experiments such as angle-resolved photoemission
spectroscopy (ARPES) and scanning tunneling microscopy (STM) ${[9,10]}$ have
confirmed the existence of this exotic surface metal, in its simplest form,
which takes a single Dirac dispersion. Recent theoretical and experimental
discovery of the two dimensional (2D) quantum spin Hall system ${[11-18]}$ and
its generalization to the TI in three dimensions ${[19-21]}$ have established
the state of matter in the time-reversal symmetric systems.

The time-reversal invariant TI is a new state of matter, distinguished from a
regular band insulator by a nontrivial topological invariant, which
characterizes its band structure ${[11]}$. Currently, most works focus on
searching for TI materials and novel transport properties. To my knowledge,
the fabrication of such TI-based nanostructure is still a challenging task.
Usually such structures are fabricated by utilizing the split gate and etching
technique ${[22]}$. On the other hand, the 3D TIs are expected to show several
unique properties when the time reversal symmetry is broken ${[23-25]}$. This
can be realized directly by a ferromagnetic insulating (FI) layer attached to
the 3D TI surface. One remarkable feature of the Dirac fermions is that the
Zeeman field acts like a vector potential: the Dirac Hamiltonian is
transformed as $\sigma\cdot\textbf{k}\longrightarrow\sigma\cdot(\textbf{k+H})$
by the Zeeman field H ${[26]}$. This is in contrast to the
Schr$\ddot{o}$dinger electrons in conventional semiconductor heterostructures
modulated by nanomagnets ${[27-29]}$.

In this work, we study the electron transport properties in a ferromagnetic
(F)/normal/N)/ ferromagnetic (F) tunnel junction, which is deposited on the
top of a topological surface. Ferromagnetic Permalloy electrodes are formed by
electron-beam lithography (EBL) followed by thermal evaporation; a second EBL
step establishes contact to the Permalloy via Cr /Au electrodes ${[30]}$. As
shown in Fig.1, the FI is put on the top of the TI to induce an exchange field
via the magnetic proximity effect. The easy axis of a FI stripe is usually
along its length direction and thus either in parallel (P) or antiparallel
(AP) with the $+y$ axis. We find that the conductance at the P configuration
can be much bigger than that at the AP configuration. Compared P  with  AP
configuration, there exists a shift of phase which can be tuned by gate
voltage. We find that the exchange field weakly affects the conductance of
carriers for P configuration but can dramatically suppress the conductance of
carriers for AP configuration. This controllable electron transport implies
anomalous magnetoresistance in this topological spin valve, which may
contribute to the development of spintronics. Compared with the conventional
F/N/F tunneling based on two dimensional electron gas (2DEG), the result
implies the existence of Fabry-Perot-like electron interference in F/N/F based
on the TI. In Sec. II , we introduce the model and method for our calculation.
In Sec. III, the numerical analysis to our important issues is reported.
Finally, a brief summary is given in sec. IV.

##  II  model and method

Figure 1:  Schematic illustration of the device. Top: Schematic diagram of two
ferromagnetic barriers on the topological surface divided by a gate electrode
at a distance $L$. Bottom: The magnetization directions of adjacent FI stripes
are parallel (P) in the configuration and antiparallel (AP) in the
configuration.

Now, let us consider a F/N/ F tunnel junction which is deposited on the top of
a topological surface where a gate electrode is attached to the ferromagnetic
material. The ferromagnetism is induced due to the proximity effect by the
ferromagnetic insulators deposited on the top as shown in Fig. 1. We assume
that the initial magnetization of  FI stripes in the region I is aligned with
the +y axis. In an actual experiment, one can use a magnet with very strong
(soft) easy axis anisotropy to control the ferromagnetic material. Thus we
focus on charge transport at the Fermi level of the surface of TIs, which is
described by the 2D Dirac Hamiltonian

$\displaystyle
H=\upsilon_{F}\sigma\cdot\textbf{p}+\sigma\cdot\textbf{M}+V{(x)},$ (1)

where $\sigma$ is Pauli matrices ,
$\textbf{M}=M_{y}(x)=M_{0}(\Theta(-x)+\gamma\Theta(x-L))$ is the effective
exchange field and $V(x)=U_{g}\Theta(x)\Theta(L-x)+V_{g}\Theta(x-L)$ is the
gate voltage, where $\gamma=+1$ ($-1$) corresponds to the P (AP)
configurations of magnetization and $\Theta(x)$ is the Heaviside step
function.

Because of the translational invariance of the system along $y$ direction, the
equation $H\Psi(x,y)=E\Psi(x,y)$ admits solutions of the form
$\Psi(x,y)=(\Psi_{1}(x),\Psi_{2}(x))^{T}\exp(ik_{y}y)$. We set
$\hbar=\upsilon_{F}=1$ in the following. Then, with the above Hamiltonian, the
wave function in the whole system is given by

$\displaystyle\Psi_{1}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{lllll
}\exp(ik_{x_{1}}x)+r\exp(-ik_{x_{1}}x),&\quad x<0,\\\ \\\
a\exp(iq_{x}x)+b\exp(-iq_{x}x),&\quad 0<x<L,\\\ \\\
t\exp(ik_{x_{2}}(x-L)),&\quad x>L,\end{array}\right.$ (7)

$\displaystyle\Psi_{2}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{lllll
}\alpha^{+}\exp(ik_{x_{1}}x)+r\alpha^{-}\exp(-ik_{x_{1}}x),&\quad x<0,\\\ \\\
a\beta^{+}\exp(iq_{x}x)+b\beta^{-}\exp(-iq_{x}x),&\quad 0<x<L,\\\ \\\
t\alpha\exp(ik_{x_{2}}(x-L)),&\quad x>L,\end{array}\right.$ (13)

where $k_{x_{1}}=E\cos\theta_{F_{1}}$, $q_{x}=(E-U_{g})\cos\theta$ and
$k_{x_{2}}=(E-V_{g})\cos\theta_{F_{2}}$ are wave vectors in region I, region
II and region III, $\alpha^{\pm}=\pm\exp(\pm i\theta_{F_{1}})$,
$\beta^{\pm}=\pm\exp(\pm i\theta)$ and $\alpha=\exp(i\theta_{F_{2}})$. The
momentum $k_{y}$ conservation should be satisfied everywhere such as
$k_{y}=E\sin\theta_{F_{1}}-M_{0}=(E-U_{g})\sin\theta=(E-V_{g})\sin\theta_{F_{2}}-\gamma
M_{0}$. Also, $r$ and $t$ are reflection and transmission coefficients,
respectively. Continuities of the wave function $\Psi$ at $x=0$ and $x=L$ are
$\Psi(0^{-})=\Psi(0^{+})$ and $\Psi(L^{-})=\Psi(L^{+})$, respectively. We find
that the transmitted electron coefficient $t_{\gamma}$ is given by

$\displaystyle
t_{\gamma}=\frac{2\cos\theta_{F_{1}}\cos\theta\exp(-ik_{x_{2}}L)}{s_{1,\gamma}\cos(q_{x}L)+is_{2,\gamma}\sin(q_{x}L)},$
(14)

with $s_{1,\gamma}=\cos\theta(\exp(i\theta_{F_{2}})+\exp(-i\theta_{F_{1}}))$
and
$s_{2,\gamma}=i\sin\theta(\exp(-i\theta_{F_{1}})-\exp(i\theta_{F_{2}}))-\exp(i(\theta_{F_{2}}-\theta_{F_{1}}))-1$.
Then

$\displaystyle
T_{\gamma}=|t_{\gamma}|^{2}\Re(\cos\theta_{F_{2}}/\cos\theta_{F_{1}}),$ (15)

where the factor $\Re(\cos\theta_{F_{2}}/\cos\theta_{F_{1}})$ is due to
current conservation. In the linear transport regime and for low temperature,
we can obtain the conductance $G$ by introducing it as the electron flow
averaged over half the Fermi surface from the well-known Landauer-Buttiker
formula ${[25,31,32]}$

$\displaystyle G_{\gamma}\sim
1/2\int_{-\pi/2}^{\pi/2}T_{\gamma}(E_{F},E_{F}\cos\theta_{F_{1}})\cos\theta_{F_{1}}d\theta_{F_{1}}.$
(16)

##  III Results and Discussions

Figure 2:  Gate voltage dependence of the conductances with a  P ($\gamma=1$)
and AP ($\gamma=-1$) configuration in the two cases: (a) $V_{g}/E_{F}=0$ and
(b) $V_{g}/E_{F}=2$. The values of the other parameters are $E_{F}=0.1$ and
$\eta=0.5$

For convenience we express all quantities in dimensionless units by means of
the length of the basic unit $L$ and the energy $E_{0}=\hbar v_{F}/L$. For a
typical value of $L=50$ nm and the Bi2Se3 material $v_{F}=5\times 10^{5}$ m/s,
one has $E_{0}=6.6$ meV. We set the energy of electron $E=E_{F}$ and also
define the value $\eta$ with the form $\eta=M_{0}/E_{F}$ in our calculation.

In Fig.2, we show gate voltage dependence of the conductances with a  P
($\gamma=1$) and AP ($\gamma=-1$) configuration in the two cases: (a)
$V_{g}/E_{F}=0$ and (b) $V_{g}/E_{F}=2$. The value of the other parameter is
$E_{F}=0.1$ and $\eta=0.5$. The presence of quantum modulation are seen in
these two figures. We can see an oscillation of the electrical conductance
with a period of $\pi$ when the voltage $U_{g}$ is larger than $E_{F}$. The
conductance at the P configuration can be much bigger than that at the AP
configuration. We find that a minimum of conductance at the P configuration
corresponds to a maximum of conductance at the AP configuration [see in fig.2
(a)] when the voltage $U_{g}$ is larger than $E_{F}$. In Fig. 2(b), a similar
tendency to Fig. 2(a) is seen. In distinct contrast to Fig.2(a), a minimum of
conductance at the P configuration here corresponds to a maximum of
conductance at the AP configuration [see in fig.2 (b)]. That is to say, there
exists a shift of $\pi$-phase. To understand these results intuitively, we
consider that the gate voltage $U_{g}$ is larger than the Fermi energy
$E_{F}$. For the given Fermi energy $E_{F}=0.1$, the condition $U_{g}\gg
E_{F}$ is easily satisfied. In this limit we have $\theta\rightarrow 0$ and
hence the transmission probability
$T_{\gamma}\sim(2\cos^{2}\theta_{F_{1}}/(1+\cos\theta_{F_{1}}\cos\theta_{F_{2}}-\cos(2U_{g}L)\sin\theta_{F_{1}}\sin\theta_{F_{2}})\Re(\cos\theta_{F_{2}}/\cos\theta_{F_{1}})$.
For $\gamma=1$ and $V_{g}/E_{F}=0$ (or $2$), we find the
$\theta_{F_{1}}\equiv\theta_{F_{2}}$ (or $-\theta_{F_{2}}$), and thus
$T_{\gamma}\sim\cos^{2}\theta_{F_{1}}/(1-\cos^{2}(U_{g}L+\delta)\sin^{2}\theta_{F_{1}})$
where $\delta=0$ (or $\pi/2$) corresponds to $V_{g}/E_{F}=0$ (or $2$). Thus
the phase difference between $V_{g}/E_{F}=0$ and $V_{g}/E_{F}=2$ is given by
$U_{g}L$. We find $G_{\gamma}\propto\cos^{2}(U_{g}L)$ for $V_{g}/E_{F}=0$ but
$G_{\gamma}\propto\sin^{2}(U_{g}L)$ for $V_{g}/E_{F}=2$. When $U_{g}L$ is
equal to the half period of $\pi$, a minimum of conductance will appear for
$V_{g}/E_{F}=0$ but a maximum of conductance will appear for $V_{g}/E_{F}=2$.
When $U_{g}L$ is equal to the period of $\pi$, a maximum of conductance will
appear for $V_{g}/E_{F}=0$ but a minimum of conductance will appear for
$V_{g}/E_{F}=2$. Furthermore, we find that $G_{\gamma}$ oscillates between
$2/3$ and $1$ for $\gamma=1$. For $\gamma=-1$ and $V_{g}/E_{F}=0$ (or $2$),
there is a similar tendency to the case of $\gamma=1$. We can see that
$G_{\gamma}$ is suppressed obviously by the strength of the effective exchange
field . Nevertheless, there exists a shift of $\pi$-phase because of the
factor $\cos(2U_{g}L)$.

Figure 3:  Gate voltage dependence of the conductances with a  P ($\gamma=1$)
and AP ($\gamma=-1$) configuration for four different values $\eta=0,0.2,0.5$,
and $0.8$. The solid lines are for $V_{g}/E_{F}=0$ while the dashed lines are
for $V_{g}/E_{F}=2$. The value of the other parameter is $E_{F}=0.1$.

In order to observe the effect of the exchange field $\eta$ on the
conductance, in Fig.3 we show the gate voltage dependence of the conductances
with a  P ($\gamma=1$) and AP ($\gamma=-1$) configuration for four different
values $\eta=0,0.2,0.5$, and $0.8$. The solid lines are for $V_{g}/E_{F}=0$
while the dashed lines are for $V_{g}/E_{F}=2$. The value of the other
parameter is $E_{F}=0.1$. A similar tendency to Fig. 2 is seen in Fig. 3. It
is easily seen that the exchange field $\eta$ weakly affects the conductance
of carriers for $\gamma=1$ but profoundly influences the conductance of
carriers for $\gamma=-1$. For $\gamma=-1$, $G_{\gamma}$ is suppressed
obviously by increasing the value $\eta$. Due to current conservation, the
factor $\Re(\cos\theta_{F_{2}}/\cos\theta_{F_{1}})$ must be real and then we
have $\sin\theta_{F_{1}}=\pm\sin\theta_{F_{2}}+2\eta$ where sign + (or -)
corresponds to $V_{g}/E_{F}=0$ (or 2). We can see
$2\eta-1\leq\sin\theta_{F_{1}}\leq 1$ and
$2\eta-1\leq\sin(\mp\theta_{F_{2}})\leq 1$ where sign - (or +) corresponds to
$V_{g}/E_{F}=0$ (or 2). Thus we find the ranges of the angle-allowable
$\theta_{F_{1}}$ and $\theta_{F_{2}}$ depend on $\eta$. The transmission is
nonzero only for $\theta_{F_{1}}$ and $\theta_{F_{2}}$ in these ranges and
vanishes for $\eta\geq 1$. The number of channels decreases with increasing of
$\eta$, so we can see that $G_{\gamma}$ dramatically decreases with the
increase of $\eta$ for $\gamma=-1$. Noting that the $\eta\geq 1$ for
$\gamma=-1$, the conductance of carriers is forbidden, which implies anomalous
magnetoresistance in this topological spin valve.

Figure 4:  Gate voltage dependence of the conductances with a  P ($\gamma=1$)
and AP ($\gamma=-1$) configuration for three different values $E_{F}=0.1,1.0$,
and $5.0$. In (a) and (b), the $V_{g}$ is set as $V_{g}/E_{F}=0$ while in (c)
and (d) the $V_{g}$ is set as $V_{g}/E_{F}=2$. The value of the other
parameter is $\eta=0.5$.

In Fig. 4, we show the gate voltage dependence of the conductances with a  P
($\gamma=1$) and AP ($\gamma=-1$) configuration for three different values
$E_{F}=0.1,1.0$, and $5.0$. In (a) and (b), the $V_{g}$ is set as
$V_{g}/E_{F}=0$ while in (c) and (d) the $V_{g}$ is set as $V_{g}/E_{F}=2$.
The value of the other parameter is $\eta=0.5$. For $E_{F}=0.1$, we can see
that the $\pi$ periodicity appears. However, the $\pi$ periodicity is broken
for $E_{F}=1$ (or $5$) because the condition $U_{g}\gg E_{F}$ is not satisfied
for the smaller $U_{g}$. Nevertheless, we get the $\pi$ periodicity of
conductance again by choosing a bigger $U_{g}$ for the bigger $E_{F}$.
Furthermore, we find that the minimum of the conductance will appear when the
gate voltage arrives at a certain value. It is easily seen that the minimum of
the conductance shifts to the right with increasing of the Fermi energy. The
larger the Fermi energy is, the smaller the minimum of the conductance is.
This phenomena is very obvious for the  P ($\gamma=1$) configuration [see in
figs.4 (a) and (d)]. From Figs.4 and 5, we find that the conductance at the
parallel (P) configuration can be much bigger than that at the antiparallel
(AP) configuration. However it may be not satisfied for the larger Fermi
energy when the gate voltage is not bigger enough. We find that there is a
Fabry-Perot-like electron interference in the F/N/F tunnel junction, which is
deposited on the top of a topological surface. The two ferromagnetic
electrodes and the barrier can compose a Fabry-Perot resonator ${[33,34]}$.
The transmitted electron waves in this resonator can be reflected by the two
ferromagnetic electrodes. The electron waves undergo multiple reflections back
and forth along the resonator between the two ferromagnetic electrodes. The
conductance oscillations are caused by the interference of electron waves
among the modes of the channel-allowable. When the gate voltage $U_{g}$ is
larger than the Fermi energy $E_{F}$, the round trip between the two
ferromagnetic electrodes adds a further phase change $\delta\sim 4\pi/\lambda$
where the Fermi wavelength $\lambda\sim 2\pi/U_{g}$ because of the value
$\theta\sim 0$. When the round trip between the two ferromagnetic electrodes
is equal to the a multiple of wavelength, the quantum interference happens.
This implies that the oscillation period is equal to $\triangle U_{g}=\pi$.

Figure 5:  Gate voltage dependence of the conductances with a  P ($\gamma=1$)
and AP ($\gamma=-1$) configuration for three different values $E_{F}=1.0,5.0$,
and $10.0$. In (a) , the $V_{g}$ is set as $V_{g}/E_{F}=0$ while in (b)
$V_{g}$ is set as $V_{g}/E_{F}=2$. The value of the other parameter is
$\eta=0.5$.

In order to investigate the solution of the standard electron described by the
Schr$\ddot{o}$dinger equation with a parabolic band structure, we consider a
2DEG in (x,y) plane with a magnetic field B in the z direction as described in
Refs.[26-28]. Thus we can fabricate a F/N/F tunneling based on the 2DEG. We
can apply all the relevant quantities in dimensionless units, which are the
same with the Ref. 28. So we can define the value $\Delta=M_{0}=B$ where
$M_{0}$ is the the effective exchange field corresponding to a F/N/F tunneling
based on the TI and $B$ is the magnetic field corresponding to a F/N/F
tunneling based on  2DEG. Nevertheless, we ignore the splitting of energy
induced by the spin of electron. As described in Fig.1, we set the left
electrode potential $V_{1}=0$. For the electron with parabolic spectrum,
${E_{F}}\sin{\theta_{{F_{1}}}}$ in Eq.(6) should be replaced by
$\sqrt{2{E_{F}}}\sin{\theta_{{F_{1}}}}$. Then the continuity of the wave
function gives the transmission coefficient

$\displaystyle{t_{\gamma}}=\frac{{2{\kern 1.0pt}{k_{1}}{\kern
1.0pt}{q_{x}}}}{{-{q_{x}}{\kern 1.0pt}({k_{1}}+{k_{3}})\cos{\kern 1.0pt}{\kern
1.0pt}({q_{x}}{\kern 1.0pt}L){\kern 1.0pt}+i{\kern 1.0pt}{\kern
1.0pt}({k_{1}}{k_{3}}+q_{x}^{2})s{\kern 1.0pt}in{\kern 1.0pt}({q_{x}}{\kern
1.0pt}L){\kern 1.0pt}{\kern 1.0pt}}},$

and transmission probability
${T_{\gamma}}={\left|{{t_{\gamma}}}\right|^{2}}{\kern 1.0pt}{\kern
1.0pt}{\mathop{\rm\Re}\nolimits}({k_{3}}/{k_{1}})$ where
${k_{1}}=\sqrt{2{\kern 1.0pt}{\kern 1.0pt}({E_{F}}{\kern 1.0pt}-{V_{1}}){\kern
1.0pt}{\kern 1.0pt}}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}\cos{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\theta_{{F_{1}}}}$,
${q_{x}}=\sqrt{2{\kern 1.0pt}{\kern 1.0pt}({E_{F}}{\kern 1.0pt}-{V_{2}}){\kern
1.0pt}{\kern 1.0pt}}\cos{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}\theta$ and
${k_{3}}=\sqrt{2{\kern 1.0pt}{\kern 1.0pt}({E_{F}}{\kern 1.0pt}-{V_{3}}){\kern
1.0pt}{\kern 1.0pt}}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}\cos{\kern
1.0pt}{\kern 1.0pt}{\theta_{{F_{3}}}}$. We can easily see that
${T_{\gamma}}\equiv 0$ when the Fermi energy is smaller than the right
electrode voltage ${V_{3}}$ , which is different from the case of Dirac band
structure. In Fig.6, we show the wave vector $k_{y}$ dependence of the
transmission probability with a  P ($\gamma=1$) and AP ($\gamma=-1$)
configuration for Dirac electrons shown in (a) and the standard electrons
shown in (b). The values of the other parameter are $\Delta=0.5$,
$V_{2}=U_{g}=0$, $V_{3}=V_{g}$ and $k_{x}=k_{x_{1}}=k_{1}=2$. We can find that
transmission is significantly more pronounced for Dirac electrons than for the
usual electrons. Compared with the  P ($\gamma=1$) configuration , it can be
seen that the channel of electron transporting from the left electrode to the
right electron is suppressed especially for the standard electron. It is
easily seen that the variety of the tunneling conductance of F/N/F with the
change of the gate voltage is obviously different between the Dirac electron
and standard electron [see in Fig.7]. For the standard electron, the
conductance will decrease monotonously with the increase of the gate voltage
because of evanescent wave modes. However, we can see that the conductance
with a $\pi$ periodicity appears as the gate voltage is large enough for the
Dirac electron. On the one hand, Dirac confined electron exhibits a jittering
motion called Zitterbewegung , originating from the interference of states
with positive and negative energy. On the other hand, the transmitted Dirac
electron waves in this resonator can be reflected by the two ferromagnetic
electrodes one after another, so the phase interference will appear. As a
result, this implies the existence of Fabry-Perot-like electron interference
in a F/N/F tunneling based on the TI.

Figure 6:  Wave vector $k_{y}$ dependence of the transmission probability with
a  P ($\gamma=1$) and AP ($\gamma=-1$) configuration for the Dirac electrons
shown in (a) and the standard electrons shown in (b). The values of the other
parameter are $\Delta=0.5$, $V_{2}=U_{g}=0$, $V_{3}=V_{g}$ and
$k_{x}=k_{x_{1}}=k_{1}=2$.

Figure 7:  Gate voltage $U_{g}$ dependence of the conductance with a  P
($\gamma=1$) and AP ($\gamma=-1$) configuration for the Dirac electrons and
the standard electrons. The values of the other parameter are $\eta=0.5$,
$V_{3}=V_{g}=0$ and $E_{F}=1.0$.

##  IV Conclusion

In summary, we have theoretically investigated transport features of Dirac
electrons on the surface of a three-dimensional TI under the modulation of a
exchange field provided by an FI stripes. We find that the conductance at the
P configuration can be much bigger than that at the AP configuration. Compared
P  with  AP configuration, there exists a shift of phase which can be tuned by
gate voltage. We find that the exchange field weakly affects the conductance
of carriers for P configuration but can dramatically suppress the conductance
of carriers for AP configuration. This controllable electron transport implies
anomalous magnetoresistance in this topological spin valve, which may
contribute to the development of spintronics .

Acknowledgments: This work was supported by the National Natural Science
Foundation of China under Grants No. 10174024 and No. 10474025.

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# Isomorphism of Skew-Holomorphic Harmonic Maass-Jacobi Forms and Certain Weak
Harmonic Maass Forms

Ranveer Kumar Singh

###### Abstract

Recently Bringmann, Raum and Richter generalised the definition of Jacobi
forms and Skoruppa’s skew-holomorphic Jacobi forms by intertwining with
harmonic Maass forms. We prove the isomorphism of the Kohnen’s plus space
analogue of harmonic Maass forms of weight $k-1/2$ for $\Gamma_{0}(4m)$ and
the space of these skew-holomorphic harmonic Maass-Jacobi forms of weight $k$
and index $m$ for $k$ odd and $m=1$ or a prime.

## 1 Introduction

Let $k$ be an integer, and $m$ a positive integer. Let
$M_{k-\frac{1}{2}}\left(\Gamma_{0}(4m)\right)$ and
$M_{k-\frac{1}{2}}^{!}\left(\Gamma_{0}(4m)\right)$ be the space of holomorphic
and weakly holomorphic modular forms respectively, of weight $k-\frac{1}{2}$
for $\Gamma_{0}(4m).$ Let $M_{k-\frac{1}{2}}^{+}\left(\Gamma_{0}(4m)\right)$
be the Kohnen’s plus space which is a subspace of
$M_{k-\frac{1}{2}}\left(\Gamma_{0}(4m)\right)$ defined as

$M_{k-\frac{1}{2}}^{+}\left(\Gamma_{0}(4m)\right):=\left\\{f\in
M_{k-\frac{1}{2}}\left(\Gamma_{0}(4m)\right)|\leavevmode\nobreak\
c_{f}(n)=0\text{ unless }(-1)^{k}n\equiv 0,1\bmod 4m\right\\}.$

Eichler and Zagier systematically developed the theory of Jacobi forms in [5].
They also proved that the Kohnen’s plus space of holomorphic modular forms of
weight $k-1/2$ for $\Gamma_{0}(4m)$ is isomorphic to the space of holomorphic
Jacobi forms of weight $k$ and index $m$ for $k$ even and $m=1$ or a prime.
They did so by decomposing the Fourier expansion of Jacobi forms in terms of
theta series and showing that the coefficients transform like modular forms.
For $k$ odd, Skoruppa showed that the plus space is isomorphic to the space of
a new kind of Jacobi form which is real analytic in variable $\tau$ and
holomorphic in $z$. These forms were called the skew-holomorphic Jacobi forms
[6, 7].

In 1920, Ramanujan, in his last letter to Hardy [10] listed 17 functions which
he called mock theta functions. These functions led to huge developments in
the theory of automorphic forms. After about eight decades, Zwegers [9] in
2002 came up with a systematic framework for mock theta functions and linked
it to harmonic Maass forms. Motivated by this, the theory of Jacobi forms was
generalised to harmonic Maass-Jacobi forms by Bringmann and Richter [2]. With
this generalisation, Cho defined the analogue of Kohnen’s plus space for
harmonic Maass forms and proved Zagier type isomorphism between the plus space
of harmonic Maass forms and the space of harmonic Maass-Jacobi form [8]. To be
precise they proved the isomorphism of plus space of harmonic Maass forms and
the space of vector valued harmonic Maass forms.

Recently the theory of skew holomorphic Jacobi forms was generalised to
include harmonic properties with respect to some differential operator by
Bringmann, Raum and Richter [3]. Several different generalisations and
structure results have been proved in [3], but we will only consider a
subspace of the generalisation containing Skoruppa’s skew-holomorphic Jacobi
forms. The aim of the present article is to prove the isomorphism of the plus
space of harmonic Maass forms and skew-holomorphic harmonic Maass-Jacobi forms
combining the results of [3, 8], in the spirit of Skoruppa. To describe the
main results, we first introduce notations and definitions.

## 2 Preliminaries

###  2.1 Harmonic Maass Forms

In this section, we will briefly review harmonic Maass forms. The reader is
referred to [4] for details.
Let $\mathbb{H}$ denote the usual upper half space. We write
$\tau\in\mathbb{H}$ as $\tau=u+iv$ and $z\in\mathbb{C}$ as $z=x+iy$. Put
$q=e(\tau)=e^{2\pi i\tau}$ and $\zeta=e(z)=e^{2\pi iz}.$ Define

$\Gamma_{0}(N)\coloneqq\left\\{\begin{pmatrix}a&b\\\ c&d\\\
\end{pmatrix}\in\mathrm{SL}_{2}(\mathbb{Z}):c\leavevmode\nobreak\
\equiv\leavevmode\nobreak\ 0\leavevmode\nobreak\
(\text{mod}\leavevmode\nobreak\ N)\right\\}.$

Define the weight-$k$ $(\in\mathbb{R})$ hyperbolic Laplacian

$\Delta_{k}\coloneqq-v^{2}\left(\frac{\partial^{2}}{\partial
u^{2}}+\frac{\partial^{2}}{\partial
v^{2}}\right)+ikv\left(\frac{\partial}{\partial u}+i\frac{\partial}{\partial
v}\right)=-4v^{2}\frac{\partial}{\partial\tau}\frac{\partial}{\partial\bar{\tau}}+2ikv\frac{\partial}{\partial\bar{\tau}}.$

We define Harmonic Maass forms following Bruiner and Funke [1].

######  Definition 2.1.

Let $k\in\frac{1}{2}\mathbb{Z}$. A smooth function (in real sense)
$f:\mathbb{H}\rightarrow\mathbb{C}$ is called a weight-$k$ harmonic Maass form
on $\Gamma_{0}(N)$ ($4|N$ if $k\in\frac{1}{2}\mathbb{Z}\setminus\mathbb{Z}$)
if

  1. (i)

For all $\begin{pmatrix}a&b\\\ c&d\\\ \end{pmatrix}\in\Gamma_{0}(N)$ and
$\tau\in\mathbb{H}$, we have

$f\left(\frac{az+b}{cz+d}\right)=\begin{cases}(cz+d)^{k}f(\tau)&\text{if
$k\in\mathbb{Z}$}\\\
(\frac{c}{d})\varepsilon_{d}^{-2k}(cz+d)^{k}f(\tau)&\text{if
$k\in\frac{1}{2}+\mathbb{Z}.$}\\\ \end{cases}$

Here $\left(\frac{c}{d}\right)$ is the Jacobi symbol and
$\varepsilon_{d}=\sqrt{\left(\frac{-1}{d}\right)}$. Here $\sqrt{.}$ denotes
the principal branch of square root.

  2. (ii)

$\Delta_{k}(f)=0$.

  3. (iii)

There exists a polynomial $P_{f}(\tau)\in\mathbb{C}[q^{-1}]$ such that
$f(\tau)-P_{f}(\tau)=O(e^{-\varepsilon v})$ as $v\rightarrow\infty$ for some
$\varepsilon>0$. Similar conditions hold at other cusps.

If the third condition in the above definition is replaced by
$f(\tau)=O(e^{\varepsilon v})$, then $f$ is said to be a harmonic Maass form
of _manageable growth_. Space of harmonic Maass forms of weight $k$ is denoted
by $H_{k}(\Gamma_{0}(N))$ and that of harmonic Maass forms of manageable
growth is denoted by $H_{k}^{!}(\Gamma_{0}(N))$.

######  Remark 2.2.

One easily sees that $M_{k}^{!}(\Gamma_{0}(N))\subset
H_{k}(\Gamma_{0}(N))\subset H_{k}^{!}(\Gamma_{0}(N)).$

$f\in H_{k}^{!}(\Gamma_{0}(N))$ has a Fourier expansion [1] of the form

$f(\tau)=f(u+iv)=\sum\limits_{n>>-\infty}c_{f}^{+}(n)q^{n}+c_{f}^{-}(0)v^{1-k}+\sum\limits_{\begin{subarray}{c}n<<\infty\\\
n\neq 0\end{subarray}}c_{f}^{-}(n)\Gamma(1-k,-4\pi nv)q^{n}.$ (1)

where $\Gamma(s,z)$ is the incomplete gamma function defined as

$\Gamma(s,z)\coloneqq\int\limits_{z}^{\infty}e^{-t}t^{s}\frac{dt}{t}.$ (2)

The notation $\sum\limits_{n>>-\infty}$ means
$\sum\limits_{n=\alpha_{f}}^{\infty}$ for some $\alpha_{f}\in\mathbb{Z}$.
$\sum\limits_{n<<\infty}$ is defined similarly. If $f\in H_{k}(\Gamma)$ then

$\qquad
f(\tau)=f(u+iv)=\sum_{n>>-\infty}c_{f}^{+}(n)q^{n}+\sum_{n<0}c_{f}^{-}(n)\Gamma(1-k,-4\pi
nv)q^{n}.$ (3)

We call

$f^{+}(\tau)=\sum\limits_{n>>-\infty}c_{f}^{+}(n)q^{n}$

the _holomorphic part_ of $f$ and

$f^{-}(\tau)=c_{f}^{-}(0)v^{1-k}+\sum\limits_{\begin{subarray}{c}n<<\infty\\\
n\neq 0\end{subarray}}c_{f}^{-}(n)\Gamma(1-k,-4\pi nv)q^{n}$

the _nonholomorphic part_ of $f$.

###  2.2 Vector Valued Harmonic Maass Forms

Let $\mathrm{Mp}_{2}(\mathbb{R})$ be the metaplectic two-fold cover of
$\mathrm{SL}_{2}(\mathbb{R})$ consisting of elements of the form $(A,\varphi)$
where $A=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in\mathrm{SL}_{2}(\mathbb{R})$
and $\varphi:\mathbb{H}\rightarrow\mathbb{C}$ is a holomorphic function such
that $\varphi^{2}(\tau)=c\tau+d$. The operation in
$\mathrm{Mp}_{2}(\mathbb{R})$ is defined as

$(A,\varphi)(B,\psi)=(AB,\varphi(B\tau)\psi(\tau)).$

Let $\mathrm{Mp}_{2}(\mathbb{Z})$ be the inverse image of
$\mathrm{SL}_{2}(\mathbb{Z})$ under the covering map. One can show that
$\mathrm{Mp}_{2}(\mathbb{Z})$ is generated by
$\widetilde{T}:=\left(\begin{pmatrix}1&1\\\ 0&1\end{pmatrix},1\right)$ and
$\widetilde{S}:=\left(\begin{pmatrix}0&-1\\\
1&0\end{pmatrix},\sqrt{\tau}\right)$.

Let $V$ be a rational vector space over $\mathbb{Q}$ with a non-degenerate
quadratic form $Q$. Let $(x,y)=Q(x+y)-Q(x)-Q(y)$ be the associated bilinear
form of signature $\left(p,q\right).$ Let $L\subset V$ be an even lattice with
dual $L^{*}$. We denote the standard basis elements of the group algebra
$\mathbb{C}\left[L^{*}/L\right]$ by $\mathfrak{e}_{\gamma}$ for $\gamma\in
L^{*}/L$. Let $\varrho_{L}$ be the Weil representation of
$\mathrm{Mp}_{2}(\mathbb{Z})$
$\operatorname{on}\mathbb{C}\left[L^{*}/L\right],$ defined by

$\begin{array}[]{l}\varrho_{L}(\widetilde{T})\left(\mathfrak{e}_{\gamma}\right):=e(Q(\gamma))\mathfrak{e}_{\gamma}\\\
\varrho_{L}(\widetilde{S})\left(\mathfrak{e}_{\gamma}\right):=\frac{e\left(\left(q-p\right)/8\right)}{\sqrt{\left|L^{*}/L\right|}}\sum\limits_{\delta\in
L^{*}/L}e(-(\gamma,\delta))\mathfrak{e}_{\delta}.\end{array}$

######  Definition 2.3.

Let $k\in\frac{1}{2}\mathbb{Z}.$ A holomorphic function
$f:\mathbb{H}\rightarrow\mathbb{C}\left[L^{*}/L\right]$ is called a weakly
holomorphic modular form of weight $k$ and type $\varrho_{L}$ for the group
$\mathrm{Mp}_{2}(\mathbb{Z})$ if it satisfies:

  1. 1.

$f(M\tau)=\varphi(\tau)^{2k}\varrho_{L}(M,\varphi)f(\tau)$ for all
$(M,\varphi)\in\mathrm{Mp}_{2}(\mathbb{Z})$

  2. 2.

$f$ has a Fourier expansion of the form

$f(\tau)=\sum_{\gamma\in L^{*}/L}\sum_{n\in\mathbb{Z}+Q(\gamma)\atop
n\gg-\infty}c_{f}(\gamma,n)e(n\tau)\mathfrak{e}_{\gamma}.$

The space of these $\mathbb{C}\left[L^{*}/L\right]$ -valued weakly holomorphic
modular forms is denoted by $M_{k,\varrho_{L}}^{!}$. Similarly we define
holomorphic modular forms of weight $k$ and type $\varrho_{L}$ and denote the
corresponding space by $M_{k,\varrho_{L}}.$

######  Definition 2.4.

A smooth function $f:\mathbb{H}\rightarrow\mathbb{C}\left[L^{*}/L\right]$ is
called a harmonic Maass form of weight $k$ and type $\varrho_{L}$ for the
group $\mathrm{Mp}_{2}(\mathbb{Z})$ if it satisfies:

  1. 1.

$f(M\tau)=\varphi(\tau)^{2k}\varrho_{L}(M,\varphi)f(\tau)$ for all
$(M,\varphi)\in\mathrm{M}\mathrm{p}_{2}(\mathbb{Z})$;

  2. 2.

$\Delta_{k}f=0$;

  3. 3.

There is a polynomial $P_{f}(\tau)=\sum_{\gamma\in
L^{*}/L}\sum_{n\in\mathbb{Z}+Q(\gamma)\atop-\infty\ll n\leq
0}c_{f}^{+}(\gamma,n)e(n\tau)\mathfrak{e}_{\gamma}$ such that
$f(\tau)=P_{f}(\tau)+O\left(e^{-\varepsilon v}\right)$ as $v\rightarrow\infty$
for some $\varepsilon>0$.

We denote by $H_{k,\varrho_{L}}$ the space of these
$\mathbb{C}\left[L^{*}/L\right]$-valued harmonic Maass forms. One can
similarly define harmonic Maass forms of weight $k$ and type $\varrho_{L}$ of
manageable growth by replacing Condition 3 above by $f(\tau)=O(e^{\varepsilon
v})$. The space of such forms is denoted by $H_{k,\varrho_{L}}^{!}.$

######  Remark 2.5.

It can easily be seen that $M_{k,\varrho_{L}}^{!}\subset
H_{k,\varrho_{L}}\subset H_{k,\varrho_{L}}^{!}.$

Define the following subspace of $H_{k-\frac{1}{2}}^{!}(\Gamma_{0}(4m))$

$H_{k-\frac{1}{2}}^{!+}(\Gamma_{0}(4m))\coloneqq\\{f\in
H_{k-\frac{1}{2}}^{!}(\Gamma_{0}(4m))\leavevmode\nobreak\
|\leavevmode\nobreak\ c_{f}^{\pm}(n)=0\leavevmode\nobreak\
\text{unless}\leavevmode\nobreak\ (-1)^{k}n\leavevmode\nobreak\
\equiv\leavevmode\nobreak\ 0,1(\text{mod}\leavevmode\nobreak\ 4m)\\}.$

Similarly, one can define $H^{+}_{k-\frac{1}{2}}(\Gamma_{0}(4m))$. Note that
$M_{k-\frac{1}{2}}^{+}(\Gamma_{0}(4m))\subset
M_{k-\frac{1}{2}}^{!+}(\Gamma_{0}(4m))\subset
H_{k-\frac{1}{2}}^{+}(\Gamma_{0}(4m))\subset
H_{k-\frac{1}{2}}^{!+}(\Gamma_{0}(4m)).$

Now for some $m\in\mathbb{Z}_{>0}$, suppose
$(L^{*}/L,Q)=(\mathbb{Z}/2m\mathbb{Z},Q)$ where $Q(\gamma)=\gamma^{2}/4m$ for
$\gamma\in L^{*}/L$. In this case the level of $L$ is $4m$ and we have
$p-q\equiv 1(\text{mod}\leavevmode\nobreak\ 8).$ One easily sees that for this
lattice and quadratic form, the Weil representation is given by

$\begin{array}[]{l}\varrho_{m}(\widetilde{T})\mathfrak{e}_{\ell}:=e\left(\frac{\ell^{2}}{4m}\right)\mathfrak{e}_{\ell}\\\
\varrho_{m}(\widetilde{S})\mathfrak{e}_{\ell}:=\frac{1}{\sqrt{2im}}\sum\limits_{\ell^{\prime}(\bmod
2m)}e\left(-\frac{\ell\ell^{\prime}}{2m}\right)\mathfrak{e}_{\ell^{\prime}}.\end{array}$

With these notations, given an $f\in
H_{k-\frac{1}{2}}^{!+}\left(\Gamma_{0}(4m)\right)$ we define a
$\mathbb{C}\left[L^{*}/L\right]$-valued function
$F=\sum_{\gamma\in\mathbb{Z}/2m\mathbb{Z}}F_{\gamma}\mathfrak{e}_{\gamma}$ by

$F_{\gamma}(\tau):=\frac{1}{s(\gamma)}\sum_{n\in\mathbb{Z}}c_{f}(n,y/4m)q^{n/4m}$

where

$c_{f}(n,y):=\begin{cases}c_{f}^{+}(n)+c_{f}^{-}(n)\Gamma\left(\frac{3}{2}-k,4\pi|n|y\right)&\text{if
$n\neq 0$}\\\ c_{f}^{+}(n)+c_{f}^{-}(n)v^{\frac{3}{2}-k}&\text{if $n=0$}\\\
\end{cases}$

and $s(\gamma)=1$ if $\gamma\equiv 0,m\bmod 2m,$ and 2 otherwise. Cho proved
the following Theorem [8].

######  Theorem 2.6.

If $k$ is odd and $m=1$ or a prime, then the map $f\mapsto F$ defines an
isomorphism of $H_{k-\frac{1}{2}}^{!+}\left(\Gamma_{0}(4m)\right)$ onto
$H^{!}_{k-\frac{1}{2},\varrho_{L}}$.

######  Remark 2.7.

The isomorphism of Theorem 2.6 restricts to the following isomorphisms (cf.
Remark 2,(2) of [8]):

  1. 1.

$H_{k-\frac{1}{2}}^{+}\left(\Gamma_{0}(4m)\right)\simeq
H_{k-\frac{1}{2},\varrho_{L}}$,

  2. 2.

$M_{k-\frac{1}{2}}^{!+}\left(\Gamma_{0}(4m)\right)\simeq
M^{!}_{k-\frac{1}{2},\varrho_{L}}$,

  3. 3.

$M_{k-\frac{1}{2}}^{+}\left(\Gamma_{0}(4m)\right)\simeq
M^{+}_{k-\frac{1}{2},\varrho_{L}}$.

###  2.3 Skew-Holomorphic Harmonic Maass Jacobi Forms

We mostly follow the notation of [3] for this section. Let
$k,m\in\mathbb{Z},m>0$. Let
$\Gamma^{J}=\textrm{SL}_{2}(\mathbb{Z})\ltimes\mathbb{Z}^{2}$ be the Jacobi
modular group. Define the following skew slash operator: For
$A=\left(\begin{pmatrix}a&b\\\ c&d\end{pmatrix},(\lambda,\mu)\right)$

$\begin{array}[]{l}\left(\left.\Phi\right|_{k,m}^{\text{sk}}A\right)(\tau,z):=\quad\Phi\left(\frac{a\tau+b}{c\tau+d},\frac{z+\lambda\tau+\mu}{c\tau+d}\right)(c\bar{\tau}+d)^{1-k}|c\tau+d|^{-1}e^{2\pi
im\left(-\frac{c(z+\lambda\tau+\mu)^{2}}{c\tau+d}+\lambda^{2}\tau+2\lambda
z\right)}\end{array}$ (4)

Define the differential operator - the _skew Casimir element_ by

$\mathcal{C}_{k,m}^{\text{sk}}=-\frac{iv^{2}}{\pi
m}v^{\frac{1}{2}-k}\circ\frac{\partial}{\partial\overline{\tau}}\circ
v^{k-\frac{1}{2}}L_{m},$ (5)

where

$L_{m}\coloneqq 8\pi
im\frac{\partial}{\partial\tau}-\frac{\partial^{2}}{\partial z^{2}}$

is the heat operator.

######  Definition 2.8.

A function $\Phi:\mathbb{H}\times\mathbb{C}\rightarrow\mathbb{C}$ is a skew-
holomorphic harmonic Maass-Jacobi form of weight $k$ and index $m$ if $\Phi$
is real-analytic in $\tau\in\mathbb{H}$ and holomorphic in $z\in\mathbb{C}$,
and satisfies the following conditions:

  1. (i)

For all $A\in\Gamma^{J}$,
$\left(\left.\Phi\right|_{k,m}^{\text{sk}}A\right)=\Phi$;

  2. (ii)

$\mathcal{C}_{k,m}^{\text{sk}}(\Phi)=0$;

  3. (iii)

$\varphi(\tau,z)=O(e^{\varepsilon v}e^{2\pi my^{2}/v})$ as
$v\rightarrow\infty$ for some $\varepsilon>0$.

We denote the space of skew-holomorphic harmonic Maass-Jacobi form of weight
$k$ and index $m$ by $\widehat{\mathbb{J}}^{\text{sk}}_{k,m}$.

######  Remark 2.9.

Let $J_{k,m}^{!\text{sk}}$ (respectively $J_{k,m}^{\text{sk}}$) denote the
space of Skoruppa’s skew-holomorphic weak (respectively holomorphic) Jacobi
forms of weight $k$ and index $m$. Then only easily sees that
$J_{k,m}^{\text{sk}}\subset
J_{k,m}^{!\text{sk}}\subset\widehat{\mathbb{J}}^{\text{sk}}_{k,m}.$

Proposition 3.6 of [3] along with the growth condition (iii) implies that
$\Phi\in\widehat{\mathbb{J}}^{\text{sk}}_{k,m}$ has Fourier expansion of the
form

$\begin{split}\Phi(\tau,z)&=v^{\frac{3}{2}-k}\sum_{n,r\in\mathbb{Z}\atop
D=0}c^{0}(n,r)q^{n}\zeta^{r}+\sum_{n,r\in\mathbb{Z}\atop
D\ll\infty}c^{+}(n,r)\text{exp}\left(\frac{\pi Dv}{m}\right)q^{n}\zeta^{r}\\\
&+\sum_{n,r\in\mathbb{Z}\atop
D\gg-\infty}c^{-}(n,r)\Gamma\left(\frac{3}{2}-k,\frac{\pi
Dv}{m}\right)\text{exp}\left(\frac{\pi Dv}{m}\right)q^{n}\zeta^{r}\end{split}$
(6)

where $D=r^{2}-4mn$. We also consider
$\Phi\in\widehat{\mathbb{J}}^{\text{sk}}_{k,m}$ which have Fourier expansion
of the form

$\begin{split}\Phi(\tau,z)&=\sum_{n,r\in\mathbb{Z}\atop
D\ll\infty}c^{+}(n,r)\text{exp}\left(\frac{\pi Dv}{m}\right)q^{n}\zeta^{r}\\\
&\hskip 85.35826pt+\sum_{n,r\in\mathbb{Z}\atop
D>0}c^{-}(n,r)\Gamma\left(\frac{3}{2}-k,\frac{\pi
Dv}{m}\right)\text{exp}\left(\frac{\pi
Dv}{m}\right)q^{n}\zeta^{r}.\end{split}$ (7)

We denote this subspace by $\widehat{\mathbb{J}}^{\text{sk,cusp}}_{k,m}$. With
the above Definition and notations we have the following Theorem:

######  Theorem 2.10.

Let $k$ be odd, and $m=1$ or a prime. Then

  1. 1.

$\widehat{\mathbb{J}}_{k,m}^{\text{sk}}\simeq
H_{k-\frac{1}{2}}^{!+}\left(\Gamma_{0}(4m)\right)$,

  2. 2.

$\widehat{\mathbb{J}}_{k,m}^{\text{sk,cusp}}\simeq
H_{k-\frac{1}{2}}^{+}\left(\Gamma_{0}(4m)\right)$.

## 3 Proof of Theorem 2.10

We will only prove 2 since 1 is similar. Let
$\Phi\in\widehat{\mathbb{J}}_{k,m}^{\text{sk,cusp}}$. Then we have that

$\begin{split}\Phi(\tau,z)&=\sum_{n,r\in\mathbb{Z}\atop
D\ll\infty}c^{+}(n,r)\text{exp}\left(\frac{\pi Dv}{m}\right)q^{n}\zeta^{r}\\\
&\hskip 85.35826pt+\sum_{n,r\in\mathbb{Z}\atop
D>0}c^{-}(n,r)\Gamma\left(\frac{3}{2}-k,\frac{\pi
Dv}{m}\right)\text{exp}\left(\frac{\pi
Dv}{m}\right)q^{n}\zeta^{r}.\end{split}$

Using the transformation of $\Phi$ for $A=(I,(\lambda,\mu))$, we have that

$\Phi(\tau,z+\lambda\tau+\mu)=e\left(-m\left(\lambda^{2}\tau+2\lambda
z\right)\right)\Phi(\tau,z).$

Using similar arguments as in Theorem 2.2 of [5], one can deduce that if
$r^{\prime}\equiv r\bmod 2m$ and $D^{\prime}=D$ with $D^{\prime}:=r^{\prime
2}-4n^{\prime}m,$ then

$c^{+}\left(n^{\prime},r^{\prime}\right)=c^{+}(n,r),\quad
c^{-}\left(n^{\prime},r^{\prime}\right)=c^{-}(n,r),\quad\Gamma\left(\frac{3}{2}-k,\frac{\pi
D^{\prime}y}{m}\right)=\Gamma\left(\frac{3}{2}-k,\frac{\pi Dy}{m}\right)$

Hence, we can decompose $\Phi(\tau,z)$ as a linear combination of the theta
functions as

$\Phi(\tau,z)=\sum_{\ell\in\mathbb{Z}/2m\mathbb{Z}}h_{\ell}(\tau)\vartheta_{m,\ell}(\tau,z)$

where

$\begin{split}h_{\ell}(\tau)&=\sum_{N\gg-\infty}c^{+}\left(\frac{N+r^{2}}{4m},r\right)\text{exp}\left(-\frac{\pi
Nv}{m}\right)q^{N/4m}\\\
&+\sum_{N<0}c^{-}\left(\frac{N+r^{2}}{4m},r\right)\Gamma\left(\frac{3}{2}-k,-\frac{\pi
Ny}{m}\right)\text{exp}\left(-\frac{\pi Nv}{m}\right)q^{N/4m}\end{split}$

with any $r\in\mathbb{Z},r\equiv\ell\bmod 2m,$ and

$\vartheta_{m,\ell}(\tau,z):=\sum_{r\in Z\atop r\equiv\ell\bmod
2m}q^{r^{2}/4m}\zeta^{r}.$

Put $g_{\ell}(\tau)=h_{\ell}(-\overline{\tau})$. Then using similar arguments
as in Theorem 5.1 of [5], one can show that the $2m$ tuple
$(g_{\ell})_{\ell(\text{mod}\leavevmode\nobreak\ 2m)}$ satisfies the
transformation property of vector valued harmonic Maass form. It remains to
show that $\Delta_{k-\frac{1}{2}}g_{\ell}(\tau)=0$. Using the fact that
$\vartheta_{m,\ell}$ is annihilated by the heat operator $L_{m}$, it is easy
to check that

$\mathcal{C}^{\text{sk}}_{k,m}(h_{\ell}\vartheta_{m,\ell})=2\left[2iv\left(k-\frac{1}{2}\right)\frac{\partial
h_{\ell}}{\partial\tau}+4v^{2}\frac{\partial^{2}h_{\ell}}{\partial\tau\overline{\tau}}\right]\vartheta_{m,\ell}.$

Thus $\mathcal{C}^{\text{sk}}_{k,m}(\Phi)=0$ gives

$2iv\left(k-\frac{1}{2}\right)\frac{\partial
h_{\ell}}{\partial\tau}+4v^{2}\frac{\partial^{2}h_{\ell}}{\partial\tau\overline{\tau}}=0.$
(8)

for each $\ell$. Next observe that

$\Delta_{k-\frac{1}{2}}g_{\ell}(\tau)=\Delta_{k-\frac{1}{2}}h_{\ell}(-\overline{\tau}).$

Using the expression for the hyperbolic Laplacian we have

$\Delta_{k-\frac{1}{2}}h_{\ell}(-\overline{\tau})=-4v^{2}\frac{\partial^{2}h_{\ell}(-\overline{\tau})}{\partial\tau\overline{\tau}}+2ikv\frac{\partial
h_{\ell}(-\overline{\tau})}{\partial\overline{\tau}}$

Substituting $T=-\overline{\tau}$ and using chain rule we get

$\Delta_{k-\frac{1}{2}}h_{\ell}(-\overline{\tau})=-4(v^{2})\frac{\partial^{2}h_{\ell}(T)}{\partial
T\overline{T}}+2i\left(k-\frac{1}{2}\right)(-v)\frac{\partial
h_{\ell}(T)}{\partial T}.$

Observing that $V=$ Im$(-\overline{\tau})=v$, we get

$\Delta_{k-\frac{1}{2}}h_{\ell}(-\overline{\tau})=-\left[4V^{2}\frac{\partial^{2}h_{\ell}(T)}{\partial
T\overline{T}}+2i\left(k-\frac{1}{2}\right)V\frac{\partial
h_{\ell}(T)}{\partial T}\right]=0$

where we used Eq. (8). Conversely, given $2m$ tuple
$(g_{\ell})_{\ell(\text{mod}\leavevmode\nobreak\ 2m)}$ of eigenfunctions of
the hyperbolic Laplacian, having growth condition as in (iii) of Definition
2.1 and satisfying the transformation rule of a vector valued harmonic Maass
form, it is easy to show that the function

$\Phi(\tau,z)=\sum_{\ell\in\mathbb{Z}/2m\mathbb{Z}}h_{\ell}(\tau)\vartheta_{m,\ell}(\tau,z)$

where $h_{\ell}(\tau)=g_{\ell}(-\bar{\tau})$, satisfies the transformation
property of skew-holomorphic harmonic Maass-Jacobi form. The annihilation by
the skew Casimir element follows easily by a similar calculation as in Eq.
(8). Finally the Fourier expansion follows from the Fourier expansion (3) of
$g_{\ell}(\tau)$. It follows that we have an isomorphism

$\widehat{\mathbb{J}}_{k,m}^{\text{sk,cusp}}\simeq
H_{k-\frac{1}{2},\varrho_{L}}.$

We conclude by noting that Theorem 2.6 implies that

$\widehat{\mathbb{J}}_{k,m}^{\text{sk,cusp}}\simeq
H_{k-\frac{1}{2},\varrho_{L}}\simeq H_{k-\frac{1}{2}}^{+}(\Gamma_{0}(4m)).$

######  Remark 3.1.

In view of Remark 2.7, we see that the isomorphism of Theorem 2.10 restricts
to isomorphisms of smaller subspaces. To be precise, we have the following
isomorphisms: For $k$ odd and $m=1$ or a prime, we have

  1. 1.

$H_{k-\frac{1}{2}}^{!+}\left(\Gamma_{0}(4m)\right)\simeq
H^{!}_{k-\frac{1}{2},\varrho_{L}}\simeq\widehat{\mathbb{J}}_{k,m}^{\text{sk}},$

  2. 2.

$H_{k-\frac{1}{2}}^{+}\left(\Gamma_{0}(4m)\right)\simeq
H_{k-\frac{1}{2},\varrho_{L}}\simeq\widehat{\mathbb{J}}_{k,m}^{\text{sk,cusp}},$

  3. 3.

$M_{k-\frac{1}{2}}^{!+}\left(\Gamma_{0}(4m)\right)\simeq
M^{!}_{k-\frac{1}{2},\varrho_{L}}\simeq J_{k,m}^{!\text{sk}}$,

  4. 4.

$M_{k-\frac{1}{2}}^{+}\left(\Gamma_{0}(4m)\right)\simeq
M_{k-\frac{1}{2},\varrho_{L}}\simeq J_{k,m}^{\text{sk}}$.

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