$\mathbf{sp}\,\phi(a)=\phi_{*}^{(n)}\mathbf{sp}\,a.$

The explicit expression of (6.9) for $\phi_{*}^{(n)}$, which involves
derivatives of $\phi$ upto $n$th order, is known, see for example
[HornJohnson94]*Thm. 6.2.25, but was not recognised before as form of spectral
mapping.

######  Example 6.12.

Let us continue with Example 6.10. Let $\phi$ map all four eigenvalues
$\lambda_{1}$, …, $\lambda_{4}$ of the matrix $a$ into themselves. Then Fig.
11(a) will represent the classical spectrum of $\phi(a)$ as well as $a$.

However Fig. 11(c) shows mapping of the new spectrum for the case $\phi$ has
_orders of zeros_ at these points as follows: the order $1$ at $\lambda_{1}$,
exactly the order $3$ at $\lambda_{2}$, an order at least $2$ at
$\lambda_{3}$, and finally any order at $\lambda_{4}$.

####  6.4. Functional Model and Spectral Distance

Let $a$ be a matrix and $\mu(z)$ be its _minimal polynomial_ :

$\mu_{a}(z)=(z-\lambda_{1})^{m_{1}}\cdot\ldots\cdot(z-\lambda_{n})^{m_{n}}.$

If all eigenvalues $\lambda_{i}$ of $a$ (i.e. all roots of $\mu(z)$ belong to
the unit disk we can consider the respective _Blaschke product_

$B_{a}(z)=\prod_{i=1}^{n}\left(\frac{z-\lambda_{i}}{1-\overline{\lambda_{i}}z}\right)^{m_{i}},$

such that its numerator coincides with the minimal polynomial $\mu(z)$.
Moreover, for an unimodular $z$ we have
$B_{a}(z)=\mu_{a}(z)\overline{\mu^{-1}_{a}(z)}z^{-m}$, where
$m=m_{1}+\ldots+m_{n}$. We also have the following covariance property:

######  Proposition 6.13.

The above correspondence $a\mapsto B_{a}$ intertwines the
$SL_{2}{}(\mathbb{R}{})$ action (6.2) on the matrices with the action (5.33)
with $k=0$ on functions.

The result follows from the observation that every elementary product
$\frac{z-\lambda_{i}}{1-\overline{\lambda_{i}}z}$ is the Moebius
transformation of $z$ with the matrix $\begin{pmatrix}1&-\lambda_{i}\\\
-\overline{\lambda_{i}}&1\end{pmatrix}$. Thus the correspondence $a\mapsto
B_{a}(z)$ is a covariant (symbolic) calculus in the sense of the Defn. 4.20.
See also the Example 4.19.

The Jordan normal form of a matrix provides a description, which is equivalent
to its contravariant spectrum. From various viewpoints, e.g. numerical
approximations, it is worth to consider its stability under a perturbation. It
is easy to see, that an arbitrarily small disturbance breaks the Jordan
structure of a matrix. However, the result of random small perturbation will
not be random, its nature is described by the following remarkable theorem:

######  Theorem 6.14 (Lidskii [Lidskii66a], see also [MoroBurkeOverton97a]).

Let $J_{n}$ be a Jordan block of a length $n>1$ with zero eigenvalues and $K$
be an arbitrary matrix. Then eigenvalues of the perturbed matrix
$J_{n}+\varepsilon^{n}K$ admit the expansion

$\lambda_{j}=\varepsilon\xi^{1/n}+o(\varepsilon),\qquad j=1,\ldots,n,$

where $\xi^{1/n}$ represents all $n$-th complex roots of certain
$\xi\in\mathbb{C}{}$.

(a) (b)

Figure 12. Perturbation of the Jordan block’s spectrum: (a) The spectrum of
the perturbation $J_{100}+\varepsilon^{1}00K$ of the Jordan block $J_{100}$ by
a random matrix $K$. (b) The spectrum of the random matrix $K$.

The left picture in Fig. 12 presents a perturbation of a Jordan block
$J_{100}$ by a random matrix. Perturbed eigenvalues are close to vertices of a
right polygon with $100$ vertices. Those regular arrangements occur despite of
the fact that eigenvalues of the matrix $K$ are dispersed through the unit
disk (the right picture in Fig. 12). In a sense it is rather the Jordan block
regularises eigenvalues of $K$ than $K$ perturbs the eigenvalue of the Jordan
block.

Although the Jordan structure itself is extremely fragile, it still can be
easily guessed from a perturbed eigenvalues. Thus there exists a certain
characterisation of matrices which is stable under small perturbations. We
will describe a sense, in which the covariant spectrum of the matrix
$J_{n}+\varepsilon^{n}K$ is stable for small $\varepsilon$. For this we
introduce the covariant version of spectral distances motivated by the
functional model. Our definition is different from other types known in the
literature [Tyrtyshnikov97a]*Ch. 5.

######  Definition 6.15.

Let $a$ and $b$ be two matrices with all their eigenvalues sitting inside of
the unit disk and $B_{a}(z)$ and $B_{b}(z)$ be respective Blaschke products as
defined above. The _(covariant) spectral distance_ $d(a,b)$ between $a$ and
$b$ is equal to the distance $\left\|B_{a}-B_{b}\right\|_{2}$ between
$B_{a}(z)$ and $B_{b}(z)$ in the Hardy space on the unit circle.

Since the spectral distance is defined through the distance in $H_{2}{}$ all
standard axioms of a distance are automatically satisfied. For a Blaschke
products we have $\left|B_{a}(z)\right|=1$ if $\left|z\right|=1$, thus
$\left\|B_{a}\right\|_{p}=1$ in any $L_{p}{}$ on the unit circle. Therefore an
alternative expression for the spectral distance is:

$d(a,b)=2(1-\left\langle B_{a},B_{b}\right\rangle).$

In particular, we always have $0\leq d(a,b)\leq 2$. We get an obvious
consequence of Prop. 6.13, which justifies the name of the covariant spectral
distance:

######  Corollary 6.16.

For any $g\in SL_{2}{}(\mathbb{R}{})$ we have $d(a,b)=d(g\cdot a,g\cdot a)$,
where $\cdot$ denotes the Möbius action (6.2).

An important property of the covariant spectral distance is its stability
under small perturbations.

######  Theorem 6.17.

For $n=2$ let $\lambda_{1}(\varepsilon)$ and $\lambda_{2}(\varepsilon)$ be
eigenvalues of the matrix $J_{2}+\varepsilon^{2}\cdot K$ for some matrix $K$.
Then

(6.10)
$\left|\lambda_{1}(\varepsilon)\right|+\left|\lambda_{2}(\varepsilon)\right|=O(\varepsilon),\quad\text{
however
}\quad\left|\lambda_{1}(\varepsilon)+\lambda_{2}(\varepsilon)\right|=O(\varepsilon^{2}).$

The spectral distance from the $1$-jet at $0$ to two $0$-jets at points
$\lambda_{1}$ and $\lambda_{2}$ bounded only by the first condition in (6.10)
is $O(\varepsilon^{2})$. However the spectral distance between $J_{2}$ and
$J_{2}+\varepsilon^{2}\cdot K$ is $O(\varepsilon^{4})$.

In other words, a matrix with eigenvalues satisfying to the Lisdkii condition
from the Thm. 6.14 is much closer to the Jordan block $J_{2}$ than a generic
one with eigenvalues of the same order. Thus the covariant spectral distance
is more stable under perturbation that magnitude of eigenvalues. For $n=2$ a
proof can be forced through a direct calculation. We also conjecture that the
similar statement is true for any $n\geq 2$.

####  6.5. Covariant Pencils of Operators

Let $H$ be a real Hilbert space, possibly of finite dimensionality. For
bounded linear operators $A$ and $B$ consider the _generalised eigenvalue
problem_ , that is finding a scalar $\lambda$ and a vector $x\in H$ such that:

(6.11) $Ax=\lambda Bx\qquad\text{or equivalently}\qquad(A-\lambda B)x=0.$

The standard eigenvalue problem corresponds to the case $B=I$, moreover for an
invertible $B$ the generalised problem can be reduced to the standard one for
the operator $B^{-1}A$. Thus it is sensible to introduce the equivalence
relation on the pairs of operators:

(6.12) $(A,B)\sim(DA,DB)\quad\text{for any invertible operator }D.$

We may treat the pair $(A,B)$ as a column vector $\begin{pmatrix}A\\\
B\end{pmatrix}$. Then there is an action of the $SL_{2}{}(\mathbb{R}{})$ group
on the pairs:

(6.13) $g\cdot\begin{pmatrix}A\\\ B\end{pmatrix}=\begin{pmatrix}aA+bB\\\
cA+dB\end{pmatrix},\qquad\text{where }g=\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in SL_{2}{}(\mathbb{R}{}).$

If we consider this $SL_{2}{}(\mathbb{R}{})$-action subject to the equivalence
relation (6.12) then we will arrive to a version of the linear-fractional
transformation of the operator defined in (6.2). There is a connection of the
$SL_{2}{}(\mathbb{R}{})$-action (6.13) to the problem (6.11) through the
following intertwining relation:

######  Proposition 6.18.

Let $\lambda$ and $x\in H$ solve the generalised eigenvalue problem (6.11) for
the pair $(A,B)$. Then the pair $(C,D)=g\cdot(A,B)$, $g\in
SL_{2}{}(\mathbb{R}{})$ has a solution $\mu$ and $x$, where

$\mu=g\cdot\lambda=\frac{a\lambda+b}{c\lambda+d},\qquad\text{for
}g=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in SL_{2}{}(\mathbb{R}{}),$

is defined by the Möbius transformation (1.1).

In other words the correspondence

$(A,B)\mapsto\text{all generalised eigenvalues}$

is another realisation of a covariant calculus in the sense of Defn. 4.20. The
collection of all pairs $g\cdot(A,B)$, $g\in SL_{2}{}(\mathbb{R}{})$ is an
example of _covariant pencil_ of operators. This set is a
$SL_{2}{}(\mathbb{R}{})$-homogeneous spaces, thus it shall be within the
classification of such homogeneous spaces provided in the Subsection 2.1.

######  Example 6.19.

It is easy to demonstrate that all existing homogeneous spaces can be realised
by matrix pairs.

  1. (i)

Take the pair $(O,I)$ where $O$ and $I$ are the zero and identity $n\times n$
matrices respectively. Then any transformation of this pair by a lower-
triangular matrix from $SL_{2}{}(\mathbb{R}{})$ is equivalent to $(O,I)$. The
respective homogeneous space is isomorphic to the real line with the Möbius
transformations (1.1).

  2. (ii)

Consider $H=\mathbb{R}^{2}{}$. Using the notations $\iota$ from Subsection 1.1
we define three realisations (elliptic, parabolic and hyperbolic) of an
operator $A_{\iota}$:

(6.14) $A_{\mathrm{i}}=\begin{pmatrix}0&1\\\ -1&0\end{pmatrix},\qquad
A_{\varepsilon}=\begin{pmatrix}0&1\\\ 0&0\end{pmatrix},\qquad
A_{\mathrm{j}}=\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}.$

Then for an arbitrary element $h$ of the subgroup $K$, $N$ or $A$ the
respective (in the sense of the Principle 3.5) pair $h\cdot(A_{\iota},I)$ is
equivalent to $(A_{\iota},I)$ itself. Thus those three homogeneous spaces are
isomorphic to the elliptic, parabolic and hyperbolic half-planes under
respective actions of $SL_{2}{}(\mathbb{R}{})$. Note, that
$A_{\iota}^{2}=\iota^{2}I$, that is $A_{\iota}$ is a model for hypercomplex
units.

  3. (iii)

Let $A$ be a direct sum of any two different matrices out of the three
$A_{\iota}$ from (6.14), then the fix group of the equivalence class of the
pair $(A,I)$ is the identity of $SL_{2}{}(\mathbb{R}{})$. Thus the
corresponding homogeneous space coincides with the group itself.

Hawing homogeneous spaces generated by pairs of operators we can define
respective functions on those spaces. The special attention is due the
following paraphrase of the resolvent:

$R_{(A,B)}(g)=(cA+dB)^{-1}\qquad\text{where}\quad g^{-1}=\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in SL_{2}(\mathbb{R}).$

Obviously $R_{(A,B)}(g)$ contains the essential information about the pair
$(A,B)$. Probably, the function $R_{(A,B)}(g)$ contains too much simultaneous
information, we may restrict it to get a more detailed view. For vectors $u$,
$v\in H$ we also consider vector and scalar-valued functions related to the
generalised resolvent:

$R^{u}_{(A,B)}(g)=(cA+dB)^{-1}u,\qquad\text{ and }\qquad
R^{(u,v)}_{(A,B)}(g)=\left\langle(cA+dB)^{-1}u,v\right\rangle,$

where $(cA+dB)^{-1}u$ is understood as a solution $w$ of the equation
$u=(cA+dB)w$ if it exists and is unique, this does not require the full
invertibility of $cA+dB$.

It is easy to see that the map $(A,B)\mapsto R^{(u,v)}_{(A,B)}(g)$ is a
covariant calculus as well. It worth to notice that function $R_{(A,B)}$ can
again fall into three EPH cases.

######  Example 6.20.

For the three matrices $A_{\iota}$ considered in the previous Example we
denote by $R_{\iota}(g)$ the resolvetn-type function of the pair
$(A_{\iota},I)$. Then:

$R_{\mathrm{i}}(g)=\frac{1}{c^{2}+d^{2}}\begin{pmatrix}d&-c\\\
c&d\end{pmatrix},\quad
R_{\varepsilon}(g)=\frac{1}{d^{2}}\begin{pmatrix}d&-c\\\
0&d\end{pmatrix},\quad
R_{\mathrm{j}}(g)=\frac{1}{d^{2}-c^{2}}\begin{pmatrix}d&-c\\\
-c&d\end{pmatrix}.$

Put $u=(1,0)\in H$, then $R_{\iota}(g)u$ is a two-dimensional real vector
valued functions with components equal to real and imaginary part of
hypercomplex Cauchy kernel considered in [Kisil11b].

Consider the space $L(G)$ of functions spanned by all left translations of
$R_{(A,B)}(g)$. As usual, a closure in a suitable metric, say $L_{p}{}$, can
be taken. The left action $g:f(h)\mapsto f(g^{-1}h)$ of $SL_{2}(\mathbb{R})$
on this space is a linear representation of this group. Afterwards the
representation can be decomposed into a sum of primary subrepresentations.

######  Example 6.21.

For the matrices $A_{\iota}$ the irreducible components are isomorphic to
analytic spaces of hypercomplex functions under the fraction-linear
transformations build in Subsection 3.2.

An important observation is that a decomposition into irreducible or primary
components can reveal an EPH structure even in the cases hiding it on the
homogeneous space level.

######  Example 6.22.

Take the operator $A=A_{\mathrm{i}}\oplus A_{\mathrm{j}}$ from the Example
6.19(iii). The corresponding homogeneous space coincides with the entire
$SL_{2}{}(\mathbb{R}{})$. However if we take two vectors
$u_{\mathrm{i}}=(1,0)\oplus(0,0)$ and $u_{\mathrm{j}}=(0,0)\oplus(1,0)$ then
the respective linear spaces generated by functions $R_{A}(g)u_{\mathrm{i}}$
and $R_{A}(g)u_{\mathrm{j}}$ will be of elliptic and hyperbolic types
respectively.

Let us briefly consider a _quadratic eigenvalue_ problem: for given operators
(matrices) $A_{0}$, $A_{1}$ and $A_{2}$ from $B(H)$ find a scalar $\lambda$
and a vector $x\in H$ such that

(6.15) $Q(\lambda)x=0,\qquad\text{where}\quad
Q(\lambda)=\lambda^{2}A_{2}+\lambda A_{1}+A_{0}.$

There is a connection with our study of conic sections from Subsection 2.2
which we will only hint for now. Comparing (6.15) with the equation of the
cycle (2.7) we can associate the respective Fillmore–Springer–Cnops–type
matrix to $Q(\lambda)$, cf. (2.8):

(6.16) $Q(\lambda)=\lambda^{2}A_{2}+\lambda
A_{1}+A_{0}\quad\longleftrightarrow\quad C_{Q}=\begin{pmatrix}A_{1}&A_{0}\\\
A_{2}&-A_{1}\end{pmatrix}.$

Then we can state the following analogue of Thm. 2.4 for the quadratic
eigenvalues:

######  Proposition 6.23.

Let two quadratic matrix polynomials $Q$ and $\tilde{Q}$ are such that their
FSC matrices (6.16) are conjugated $C_{\tilde{Q}}=gC_{{Q}}g^{-1}$ by an
element $g\in SL_{2}{}(\mathbb{R}{})$. Then $\lambda$ is a solution of the
quadratic eigenvalue problem for $Q$ and $x\in H$ if and only if
$\mu=g\cdot\lambda$ is a solution of the quadratic eigenvalue problem for
$\tilde{Q}$ and $x$. Here $\mu=g\cdot\lambda$ is the Möbius transformation
(1.1) associated to $g\in SL_{2}{}(\mathbb{R}{})$.

So quadratic matrix polynomials are non-commuting analogues of the cycles and
it would be exciting to extend the geometry from Section 2 to this non-
commutative setting as much as possible.

######  Remark 6.24.

It is beneficial to extend a notion of a scalar in an (generalised) eigenvalue
problem to an abstract field or ring. For example, we can consider pencils of
operators/matrices with polynomial coefficients. In many circumstances we may
factorise the polynomial ring by an ideal generated by a collection of
algebraic equations. Our work with hypercomplex units is the most elementary
realisation of this setup. Indeed, the algebra of hypercomplex numbers with
the hypercomplex unit $\iota$ is a realisation of the polynomial ring in a
variable $t$ factored by the single quadratic relation $t^{2}+\sigma=0$, where
$\sigma=\iota^{2}$.

###  7\. Quantum Mechanics

Complex valued representations of the Heisenberg group (also known as Weyl or
Heisenberg-Weyl group) provide a natural framework for quantum mechanics
[Howe80b, Folland89]. This is the most fundamental example of the Kirillov
orbit method, induced representations and geometrical quantisation technique
[Kirillov99, Kirillov94a]. Following the presentation in Section 3 we will
consider representations of the Heisenberg group which are induced by
hypercomplex characters of its centre: complex (which correspond to the
elliptic case), dual (parabolic) and double (hyperbolic).

To describe dynamics of a physical system we use a universal equation based on
inner derivations (commutator) of the convolution algebra [Kisil00a]
[Kisil02e]. The complex valued representations produce the standard framework
for quantum mechanics with the Heisenberg dynamical equation [Vourdas06a].

The double number valued representations, with the hyperbolic unit
$\mathrm{j}^{2}=1$, is a natural source of hyperbolic quantum mechanics
developed for a while [Hudson04a, Hudson66a, Khrennikov03a, Khrennikov05a,
Khrennikov08a]. The universal dynamical equation employs hyperbolic commutator
in this case. This can be seen as a _Moyal bracket_ based on the hyperbolic
sine function. The hyperbolic observables act as operators on a Krein space
with an indefinite inner product. Such spaces are employed in study of
$\mathcal{PT}$-symmetric Hamiltonians and hyperbolic unit $\mathrm{j}^{2}=1$
naturally appear in this setup [GuentherKuzhel10a].

The representations with values in dual numbers provide a convenient
description of the classical mechanics. For this we do not take any sort of
semiclassical limit, rather the nilpotency of the parabolic unit
($\varepsilon^{2}=0$) do the task. This removes the vicious necessity to
consider the Planck _constant_ tending to zero. The dynamical equation takes
the Hamiltonian form. We also describe classical non-commutative
representations of the Heisenberg group which acts in the first jet space.

######  Remark 7.1.

It is worth to note that our technique is different from contraction technique
in the theory of Lie groups [LevyLeblond65a, GromovKuratov05b]. Indeed a
contraction of the Heisenberg group $\mathbb{H}^{n}{}$ is the commutative
Euclidean group $\mathbb{R}^{2n}{}$ which does not recreate neither quantum
nor classical mechanics.

The approach provides not only three different types of dynamics, it also
generates the respective rules for addition of probabilities as well. For
example, the quantum interference is the consequence of the same complex-
valued structure, which directs the Heisenberg equation. The absence of an
interference (a particle behaviour) in the classical mechanics is again the
consequence the nilpotency of the parabolic unit. Double numbers creates the
hyperbolic law of additions of probabilities, which was extensively
investigates [Khrennikov03a, Khrennikov05a]. There are still unresolved issues
with positivity of the probabilistic interpretation in the hyperbolic case
[Hudson04a, Hudson66a].

######  Remark 7.2.

It is commonly accepted since the Dirac’s paper [Dirac26a] that the striking
(or even _the only_) difference between quantum and classical mechanics is
non-commutativity of observables in the first case. In particular the
Heisenberg commutation relations (7.5) imply the uncertainty principle, the
Heisenberg equation of motion and other quantum features. However, the entire
book of Feynman on QED [Feynman1990qed] does not contains any reference to
non-commutativity. Moreover, our work shows that there is a non-commutative
formulation of classical mechanics. Non-commutative representations of the
Heisenberg group in dual numbers implies the Poisson dynamical equation and
local addition of probabilities in Section7.6, which are completely classical.

This entirely dispels any illusive correlation between classical/quantum and
commutative/non-commutative. Instead we show that quantum mechanics is fully
determined by the properties of complex numbers. In Feynman’s exposition
[Feynman1990qed] complex numbers are presented by a clock, rotations of its
arm encode multiplications by unimodular complex numbers. Moreover, there is
no a presentation of quantum mechanics, which does not employ complex phases
(numbers) in one or another form. Analogous parabolic and hyperbolic phases
(or characters produced by associated hypercomplex numbers, see Section 3.1)
lead to classical and hypercomplex mechanics respectively.

This section clarifies foundations of quantum and classical mechanics. We
recovered the existence of three non-isomorphic models of mechanics from the
representation theory. They were already derived in [Hudson04a, Hudson66a]
from translation invariant formulation, that is from the group theory as well.
It also hinted that hyperbolic counterpart is (at least theoretically) as
natural as classical and quantum mechanics are. The approach provides a
framework for a description of aggregate system which have say both quantum
and classical components. This can be used to model quantum computers with
classical terminals [Kisil09b].

Remarkably, simultaneously with the work [Hudson66a] group-invariant
axiomatics of geometry leaded R.I. Pimenov [Pimenov65a] to description of
$3^{n}$ Cayley–Klein constructions. The connection between group-invariant
geometry and respective mechanics were explored in many works of N.A. Gromov,
see for example [Gromov90a, Gromov90b, GromovKuratov05b]. They already
highlighted the rôle of three types of hypercomplex units for the realisation
of elliptic, parabolic and hyperbolic geometry and kinematic.

There is a further connection between representations of the Heisenberg group
and hypercomplex numbers. The symplectomorphism of phase space are also
automorphism of the Heisenberg group [Folland89]*§ 1.2. We recall that the
symplectic group $\mathrm{Sp}(2)$ [Folland89]*§ 1.2 is isomorphic to the group
$SL_{2}{}(\mathbb{R}{})$ [Lang85] [HoweTan92] [Mazorchuk09a] and provides
linear symplectomorphisms of the two-dimensional phase space. It has three
types of non-isomorphic one-dimensional continuous subgroups (2.4-2.6) with
symplectic action on the phase space illustrated by Fig. 9. Hamiltonians,
which produce those symplectomorphism, are of interest [Wulfman10a]*§ 3.8
[ATorre08a] [ATorre10a]. An analysis of those Hamiltonians from Subsection 3.3
by means of ladder operators recreates hypercomplex coefficients as well
[Kisil11a].

Harmonic oscillators, which we shall use as the main illustration here, are
treated in most textbooks on quantum mechanics. This is efficiently done
through creation/annihilation (ladder) operators, cf. § 3.3 and [Gazeau09a]
[BoyerMiller74a]. The underlying structure is the representation theory of the
Heisenberg and symplectic groups [Lang85]*§ VI.2 [MTaylor86]*§ 8.2 [Howe80b]
[Folland89]. As we will see, they are naturally connected with respective
hypercomplex numbers. As a result we obtain further illustrations to the
Similarity and Correspondence Principle 3.5.

We work with the simplest case of a particle with only one degree of freedom.
Higher dimensions and the respective group of symplectomorphisms
$\mathrm{Sp}(2n)$ may require consideration of Clifford algebras [Kisil93c]
[ConstalesFaustinoKrausshar11a] [CnopsKisil97a] [GuentherKuzhel10a]
[Porteous95].

####  7.1. The Heisenberg Group and Its Automorphisms

#####  7.1.1. The Heisenberg group and induced representations

Let $(s,x,y)$, where $s$, $x$, $y\in\mathbb{R}{}$, be an element of the one-
dimensional _Heisenberg group_ $\mathbb{H}^{1}{}$ [Folland89, Howe80b].
Consideration of the general case of $\mathbb{H}^{n}{}$ will be similar, but
is beyond the scope of present paper. The group law on $\mathbb{H}^{1}{}$ is
given as follows:

(7.1)
$\textstyle(s,x,y)\cdot(s^{\prime},x^{\prime},y^{\prime})=(s+s^{\prime}+\frac{1}{2}\omega(x,y;x^{\prime},y^{\prime}),x+x^{\prime},y+y^{\prime}),$

where the non-commutativity is due to $\omega$—the _symplectic form_ on
$\mathbb{R}^{2n}{}$, which is the central object of the classical mechanics
[Arnold91]*§ 37:

(7.2) $\omega(x,y;x^{\prime},y^{\prime})=xy^{\prime}-x^{\prime}y.$

The Heisenberg group is a non-commutative Lie group with the centre

$Z=\\{(s,0,0)\in\mathbb{H}^{1}{},\ s\in\mathbb{R}{}\\}.$

The left shifts

(7.3) $\Lambda(g):f(g^{\prime})\mapsto f(g^{-1}g^{\prime})$

act as a representation of $\mathbb{H}^{1}{}$ on a certain linear space of
functions. For example, an action on $L_{2}{}(\mathbb{H}{},dg)$ with respect
to the Haar measure $dg=ds\,dx\,dy$ is the _left regular_ representation,
which is unitary.

The Lie algebra $\mathfrak{h}^{n}$ of $\mathbb{H}^{1}{}$ is spanned by
left-(right-)invariant vector fields

(7.4) $\textstyle S^{l(r)}=\pm{\partial_{s}},\quad
X^{l(r)}=\pm\partial_{x}-\frac{1}{2}y{\partial_{s}},\quad
Y^{l(r)}=\pm\partial_{y}+\frac{1}{2}x{\partial_{s}}$

on $\mathbb{H}^{1}{}$ with the Heisenberg _commutator relation_

(7.5) $[X^{l(r)},Y^{l(r)}]=S^{l(r)}$

and all other commutators vanishing. We will sometimes omit the superscript
$l$ for left-invariant field.

We can construct linear representations of $\mathbb{H}^{1}{}$ by induction
[Kirillov76]*§ 13 from a character $\chi$ of the centre $Z$. Here we prefer
the following one, cf. § 3.2 and [Kirillov76]*§ 13 [MTaylor86]*Ch. 5. Let
$F_{2}^{\chi}{}(\mathbb{H}^{n}{})$ be the space of functions on
$\mathbb{H}^{n}{}$ having the properties:

(7.6) $f(gh)=\chi(h)f(g),\qquad\text{ for all }g\in\mathbb{H}^{n}{},\ h\in Z$

and

(7.7) $\int_{\mathbb{R}^{2n}{}}\left|f(0,x,y)\right|^{2}dx\,dy<\infty.$

Then $F_{2}^{\chi}{}(\mathbb{H}^{n}{})$ is invariant under the left shifts and
those shifts restricted to $F_{2}^{\chi}{}(\mathbb{H}^{n}{})$ make a
representation ${\rho_{\chi}}$ of $\mathbb{H}^{n}{}$ induced by $\chi$.

If the character $\chi$ is unitary, then the induced representation is unitary
as well. However the representation ${\rho_{\chi}}$ is not necessarily
irreducible. Indeed, left shifts are commuting with the right action of the
group. Thus any subspace of null-solutions of a linear combination
$aS+\sum_{j=1}^{n}(b_{j}X_{j}+c_{j}Y_{j})$ of left-invariant vector fields is
left-invariant and we can restrict ${\rho_{\chi}}$ to this subspace. The left-
invariant differential operators define analytic condition for functions, cf.
Cor. 5.6.

######  Example 7.3.

The function $f_{0}(s,x,y)=e^{\mathrm{i}hs-h(x^{2}+y^{2})/4}$, where
$h=2\pi\hslash$, belongs to $F_{2}^{\chi}{}(\mathbb{H}^{n}{})$ for the
character $\chi(s)=e^{\mathrm{i}hs}$. It is also a null solution for all the
operators $X_{j}-\mathrm{i}Y_{j}$. The closed linear span of functions
$f_{g}=\Lambda(g)f_{0}$ is invariant under left shifts and provide a model for
Fock–Segal–Bargmann (FSB) type representation of the Heisenberg group, which
will be considered below.

#####  7.1.2. Symplectic Automorphisms of the Heisenberg Group

The group of outer automorphisms of $\mathbb{H}^{1}{}$, which trivially acts
on the centre of $\mathbb{H}^{1}{}$, is the symplectic group $\mathrm{Sp}(2)$.
It is the group of symmetries of the symplectic form $\omega$ in (7.1)
[Folland89]*Thm. 1.22 [Howe80a]*p. 830. The symplectic group is isomorphic to
$SL_{2}{}(\mathbb{R}{})$ considered in the first half of this work. The
explicit action of $\mathrm{Sp}(2)$ on the Heisenberg group is:

(7.8) $g:h=(s,x,y)\mapsto g(h)=(s,x^{\prime},y^{\prime}),$

where

$g=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in\mathrm{Sp}(2),\quad\text{ and
}\quad\begin{pmatrix}x^{\prime}\\\
y^{\prime}\end{pmatrix}=\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\begin{pmatrix}x\\\ y\end{pmatrix}.$

The Shale–Weil theorem [Folland89]*§ 4.2 [Howe80a]*p. 830 states that any
representation ${\rho_{\hslash}}$ of the Heisenberg groups generates a unitary
_oscillator_ (or _metaplectic_) representation ${\rho^{\text{SW}}_{\hslash}}$
of the $\widetilde{\mathrm{Sp}}(2)$, the two-fold cover of the symplectic
group [Folland89]*Thm. 4.58.

We can consider the semidirect product
$G=\mathbb{H}^{1}{}\rtimes\widetilde{\mathrm{Sp}}(2)$ with the standard group
law:

(7.9)
$(h,g)*(h^{\prime},g^{\prime})=(h*g(h^{\prime}),g*g^{\prime}),\qquad\text{where
}h,h^{\prime}\in\mathbb{H}^{1}{},\quad
g,g^{\prime}\in\widetilde{\mathrm{Sp}}(2),$

and the stars denote the respective group operations while the action
$g(h^{\prime})$ is defined as the composition of the projection map
$\widetilde{\mathrm{Sp}}(2)\rightarrow{\mathrm{Sp}}(2)$ and the action (7.8).
This group is sometimes called the _Schrödinger group_ and it is known as the
maximal kinematical invariance group of both the free Schrödinger equation and
the quantum harmonic oscillator [Niederer73a]. This group is of interest not
only in quantum mechanics but also in optics [ATorre10a, ATorre08a]. The
Shale–Weil theorem allows us to expand any representation ${\rho_{\hslash}}$
of the Heisenberg group to the representation
${\rho^{2}_{\hslash}}={\rho_{\hslash}}\oplus{\rho^{\text{SW}}_{\hslash}}$ of
the group $G$.

Consider the Lie algebra $\mathfrak{sp}_{2}$ of the group $\mathrm{Sp}(2)$. We
again use the basis $A$, $B$, $Z$ (3.12) with commutators (3.13). Vectors $Z$,
$B-Z/2$ and $B$ are generators of the one-parameter subgroups $K$,
$N^{\prime}$ and $A\\!^{\prime}$ (2.4–2.6) respectively. Furthermore we can
consider the basis $\\{S,X,Y,A,B,Z\\}$ of the Lie algebra $\mathfrak{g}$ of
the Lie group $G=\mathbb{H}^{1}{}\rtimes\widetilde{\mathrm{Sp}}(2)$. All non-
zero commutators besides those already listed in (7.5) and (3.13) are:

(7.10) $\displaystyle[A,X]$ $\displaystyle=\textstyle\frac{1}{2}X,$
$\displaystyle[B,X]$ $\displaystyle=\textstyle-\frac{1}{2}Y,$
$\displaystyle[Z,X]$ $\displaystyle=Y;$ (7.11) $\displaystyle[A,Y]$
$\displaystyle=\textstyle-\frac{1}{2}Y,$ $\displaystyle[B,Y]$
$\displaystyle=\textstyle-\frac{1}{2}X,$ $\displaystyle[Z,Y]$
$\displaystyle=-X.$

Of course, there is the derived form of the Shale–Weil representation for
$\mathfrak{g}$. It can often be explicitly written in contrast to the
Shale–Weil representation.

######  Example 7.4.

Let ${\rho_{\hslash}}$ be the Schrödinger representation [Folland89]*§ 1.3 of
$\mathbb{H}^{1}{}$ in $L_{2}{}(\mathbb{R}{})$, that is [Kisil10a]*(3.5):

$[{\rho_{\chi}}(s,x,y)f\,](q)=e^{2\pi\mathrm{i}\hslash(s-xy/2)+2\pi\mathrm{i}xq}\,f(q-\hslash
y).$

Thus the action of the derived representation on the Lie algebra
$\mathfrak{h}_{1}$ is:

(7.12)
${\rho_{\hslash}}(X)=2\pi\mathrm{i}q,\qquad{\rho_{\hslash}}(Y)=-\hslash\frac{d}{dq},\qquad{\rho_{\hslash}}(S)=2\pi\mathrm{i}\hslash
I.$

Then the associated Shale–Weil representation of $\mathrm{Sp}(2)$ in
$L_{2}{}(\mathbb{R}{})$ has the derived action, cf. [ATorre08a]*(2.2)
[Folland89]*§ 4.3:

(7.13)
${\rho^{\text{SW}}_{\hslash}}(A)=-\frac{q}{2}\frac{d}{dq}-\frac{1}{4},\quad{\rho^{\text{SW}}_{\hslash}}(B)=-\frac{\hslash\mathrm{i}}{8\pi}\frac{d^{2}}{dq^{2}}-\frac{\pi\mathrm{i}q^{2}}{2\hslash},\quad{\rho^{\text{SW}}_{\hslash}}(Z)=\frac{\hslash\mathrm{i}}{4\pi}\frac{d^{2}}{dq^{2}}-\frac{\pi\mathrm{i}q^{2}}{\hslash}.$

We can verify commutators (7.5) and (3.13), (7.11) for operators (7.12–7.13).
It is also obvious that in this representation the following algebraic
relations hold:

$\displaystyle\qquad{\rho^{\text{SW}}_{\hslash}}(A)$ $\displaystyle=$
$\displaystyle\frac{\mathrm{i}}{4\pi\hslash}({\rho_{\hslash}}(X){\rho_{\hslash}}(Y)-{\textstyle\frac{1}{2}}{\rho_{\hslash}}(S))$
$\displaystyle=$
$\displaystyle\frac{\mathrm{i}}{8\pi\hslash}({\rho_{\hslash}}(X){\rho_{\hslash}}(Y)+{\rho_{\hslash}}(Y){\rho_{\hslash}}(X)),$
(7.15) $\displaystyle{\rho^{\text{SW}}_{\hslash}}(B)$ $\displaystyle=$
$\displaystyle\frac{\mathrm{i}}{8\pi\hslash}({\rho_{\hslash}}(X)^{2}-{\rho_{\hslash}}(Y)^{2}),$
(7.16) $\displaystyle{\rho^{\text{SW}}_{\hslash}}(Z)$ $\displaystyle=$
$\displaystyle\frac{\mathrm{i}}{4\pi\hslash}({\rho_{\hslash}}(X)^{2}+{\rho_{\hslash}}(Y)^{2}).$

Thus it is common in quantum optics to name $\mathfrak{g}$ as a Lie algebra
with quadratic generators, see [Gazeau09a]*§ 2.2.4.

Note that ${\rho^{\text{SW}}_{\hslash}}(Z)$ is the Hamiltonian of the harmonic
oscillator (up to a factor). Then we can consider
${\rho^{\text{SW}}_{\hslash}}(B)$ as the Hamiltonian of a repulsive
(hyperbolic) oscillator. The operator
${\rho^{\text{SW}}_{\hslash}}(B-Z/2)=\frac{\hslash\mathrm{i}}{4\pi}\frac{d^{2}}{dq^{2}}$
is the parabolic analog. A graphical representation of all three
transformations defined by those Hamiltonian is given in Fig. 9 and a further
discussion of these Hamiltonians can be found in [Wulfman10a]*§ 3.8.

An important observation, which is often missed, is that the three linear
symplectic transformations are unitary rotations in the corresponding
hypercomplex algebra, cf. [Kisil09c]*§ 3. This means, that the
symplectomorphisms generated by operators $Z$, $B-Z/2$, $B$ within time $t$
coincide with the multiplication of hypercomplex number $q+\iota p$ by
$e^{\iota t}$, see Subsection 3.1 and Fig. 9, which is just another
illustration of the Similarity and Correspondence Principle 3.5.

######  Example 7.5.

There are many advantages of considering representations of the Heisenberg
group on the phase space [Howe80b]*§ 1.7 [Folland89]*§ 1.6 [deGosson08a]. A
convenient expression for Fock–Segal–Bargmann (FSB) representation on the
phase space is, cf. § 7.3.1 and [Kisil02e]*(2.9) [deGosson08a]*(1):

(7.17) $\textstyle[{\rho_{F}}(s,x,y)f](q,p)=e^{-2\pi\mathrm{i}(\hslash
s+qx+py)}f\left(q-\frac{\hslash}{2}y,p+\frac{\hslash}{2}x\right).$

Then the derived representation of $\mathfrak{h}_{1}$ is:

(7.18)
$\textstyle{\rho_{F}}(X)=-2\pi\mathrm{i}q+\frac{\hslash}{2}\partial_{p},\qquad{\rho_{F}}(Y)=-2\pi\mathrm{i}p-\frac{\hslash}{2}\partial_{q},\qquad{\rho_{F}}(S)=-2\pi\mathrm{i}\hslash
I.$

This produces the derived form of the Shale–Weil representation:

(7.19)
$\textstyle{\rho^{\text{SW}}_{F}}(A)=\frac{1}{2}\left(q\partial_{q}-p\partial_{p}\right),\quad{\rho^{\text{SW}}_{F}}(B)=-\frac{1}{2}\left(p\partial_{q}+q\partial_{p}\right),\quad{\rho^{\text{SW}}_{F}}(Z)=p\partial_{q}-q\partial_{p}.$

Note that this representation does not contain the parameter $\hslash$ unlike
the equivalent representation (7.13). Thus the FSB model explicitly shows the
equivalence of ${\rho^{\text{SW}}_{\hslash_{1}}}$ and
${\rho^{\text{SW}}_{\hslash_{2}}}$ if $\hslash_{1}\hslash_{2}>0$
[Folland89]*Thm. 4.57.

As we will also see below the FSB-type representations in hypercomplex numbers
produce almost the same Shale–Weil representations.

####  7.2. p-Mechanic Formalism

Here we briefly outline a formalism [Kisil96a, Prezhdo-Kisil97, Kisil00a,
BrodlieKisil03a, Kisil02e], which allows to unify quantum and classical
mechanics.

#####  7.2.1. Convolutions (Observables) on $\mathbb{H}^{n}{}$ and Commutator

Using a invariant measure $dg=ds\,dx\,dy$ on $\mathbb{H}^{n}{}$ we can define
the convolution of two functions:

(7.20) $\displaystyle(k_{1}*k_{2})(g)$ $\displaystyle=$
$\displaystyle\int_{\mathbb{H}^{n}{}}k_{1}(g_{1})\,k_{2}(g_{1}^{-1}g)\,dg_{1}.$

This is a non-commutative operation, which is meaningful for functions from
various spaces including $L_{1}{}(\mathbb{H}^{n}{},dg)$, the Schwartz space
$S{}$ and many classes of distributions, which form algebras under
convolutions. Convolutions on $\mathbb{H}^{n}{}$ are used as _observables_ in
$p$-mechanic [Kisil96a, Kisil02e].

A unitary representation ${\rho}$ of $\mathbb{H}^{n}{}$ extends to
$L_{1}{}(\mathbb{H}^{n}{},dg)$ by the formula:

(7.21) ${\rho}(k)=\int_{\mathbb{H}^{n}{}}k(g){\rho}(g)\,dg.$

This is also an algebra homomorphism of convolutions to linear operators.

For a dynamics of observables we need inner _derivations_ $D_{k}$ of the
convolution algebra $L_{1}{}(\mathbb{H}^{n}{})$, which are given by the
_commutator_ :

$\displaystyle\qquad D_{k}:f\mapsto[k,f]$ $\displaystyle=$ $\displaystyle
k*f-f*k$ $\displaystyle=$
$\displaystyle\int_{\mathbb{H}^{n}{}}k(g_{1})\left(f(g_{1}^{-1}g)-f(gg_{1}^{-1})\right)\,dg_{1},\quad
f,k\in L_{1}{}(\mathbb{H}^{n}{}).$

To describe dynamics of a time-dependent observable $f(t,g)$ we use the
universal equation, cf. [Kisil94d, Kisil96a]:

(7.23) $S\dot{f}=[H,f],$

where $S$ is the left-invariant vector field (7.4) generated by the centre of
$\mathbb{H}^{n}{}$. The presence of operator $S$ fixes the dimensionality of
both sides of the equation (7.23) if the observable $H$ (Hamiltonian) has the
dimensionality of energy [Kisil02e]*Rem 4.1. If we apply a right inverse
$\mathcal{A}$ of $S$ to both sides of the equation (7.23) we obtain the
equivalent equation

(7.24) $\dot{f}=\left\\{\\!\left[H,f\right]\\!\right\\},$

based on the universal bracket
$\left\\{\\!\left[k_{1},k_{2}\right]\\!\right\\}=k_{1}*\mathcal{A}k_{2}-k_{2}*\mathcal{A}k_{1}$
[Kisil02e].

######  Example 7.6 (Harmonic oscillator).

Let $H=\frac{1}{2}(mk^{2}q^{2}+\frac{1}{m}p^{2})$ be the Hamiltonian of a one-
dimensional harmonic oscillator, where $k$ is a constant frequency and $m$ is
a constant mass. Its _p-mechanisation_ will be the second order differential
operator on $\mathbb{H}^{n}{}$ [BrodlieKisil03a]*§ 5.1:

$\textstyle H=\frac{1}{2}(mk^{2}X^{2}+\frac{1}{m}Y^{2}),$

where we dropped sub-indexes of vector fields (7.4) in one dimensional
setting. We can express the commutator as a difference between the left and
the right action of the vector fields:

$\textstyle[H,f]=\frac{1}{2}(mk^{2}((X^{r})^{2}-(X^{l})^{2})+\frac{1}{m}((Y^{r})^{2}-(Y^{l})^{2}))f.$

Thus the equation (7.23) becomes [BrodlieKisil03a]*(5.2):

(7.25) $\frac{\partial}{\partial s}\dot{f}=\frac{\partial}{\partial
s}\left(mk^{2}y\frac{\partial}{\partial
x}-\frac{1}{m}x\frac{\partial}{\partial y}\right)f.$

Of course, the derivative $\frac{\partial}{\partial s}$ can be dropped from
both sides of the equation and the general solution is found to be:

(7.26) $\textstyle
f(t;s,x,y)=f_{0}\left(s,x\cos(kt)+mky\sin(kt),-\frac{x}{mk}\sin(kt)+y\cos(kt)\right),$

where $f_{0}(s,x,y)$ is the initial value of an observable on
$\mathbb{H}^{n}{}$.

######  Example 7.7 (Unharmonic oscillator).

We consider unharmonic oscillator with cubic potential, see
[CalzettaVerdaguer06a] and references therein:

(7.27) $H=\frac{mk^{2}}{2}q^{2}+\frac{\lambda}{6}q^{3}+\frac{1}{2m}p^{2}.$

Due to the absence of non-commutative products p-mechanisation is
straightforward:

$H=\frac{mk^{2}}{2}X^{2}+\frac{\lambda}{6}X^{3}+\frac{1}{m}Y^{2}.$

Similarly to the harmonic case the dynamic equation, after cancellation of
$\frac{\partial}{\partial s}$ on both sides, becomes:

(7.28) $\dot{f}=\left(mk^{2}y\frac{\partial}{\partial
x}+\frac{\lambda}{6}\left(3y\frac{\partial^{2}}{\partial
x^{2}}+\frac{1}{4}y^{3}\frac{\partial^{2}}{\partial
s^{2}}\right)-\frac{1}{m}x\frac{\partial}{\partial y}\right)f.$

Unfortunately, it cannot be solved analytically as easy as in the harmonic
case.

#####  7.2.2. States and Probability

Let an observable ${\rho}(k)$ (7.21) is defined by a kernel $k(g)$ on the
Heisenberg group and its representation ${\rho}$ at a Hilbert space
$\mathcal{H}$. A _state_ on the convolution algebra is given by a vector
$v\in\mathcal{H}$. A simple calculation:

$\displaystyle\left\langle{\rho}(k)v,v\right\rangle_{\mathcal{H}}$
$\displaystyle=$
$\displaystyle\left\langle\int_{\mathbb{H}^{n}{}}k(g){\rho}(g)v\,dg,v\right\rangle_{\mathcal{H}}$
$\displaystyle=$
$\displaystyle\int_{\mathbb{H}^{n}{}}k(g)\left\langle{\rho}(g)v,v\right\rangle_{\mathcal{H}}dg$
$\displaystyle=$
$\displaystyle\int_{\mathbb{H}^{n}{}}k(g)\overline{\left\langle
v,{\rho}(g)v\right\rangle_{\mathcal{H}}}\,dg$

can be restated as:

$\left\langle{\rho}(k)v,v\right\rangle_{\mathcal{H}}=\left\langle
k,l\right\rangle,\qquad\text{where}\quad l(g)=\left\langle
v,{\rho}(g)v\right\rangle_{\mathcal{H}}.$

Here the left-hand side contains the inner product on $\mathcal{H}$, while the
right-hand side uses a skew-linear pairing between functions on
$\mathbb{H}^{n}{}$ based on the Haar measure integration. In other words we
obtain, cf. [BrodlieKisil03a]*Thm. 3.11:

######  Proposition 7.8.

A state defined by a vector $v\in\mathcal{H}$ coincides with the linear
functional given by the wavelet transform

(7.29) $l(g)=\left\langle v,{\rho}(g)v\right\rangle_{\mathcal{H}}$

of $v$ used as the mother wavelet as well.

The addition of vectors in $\mathcal{H}$ implies the following operation on
states:

(7.30) $\displaystyle\left\langle
v_{1}+v_{2},{\rho}(g)(v_{1}+v_{2})\right\rangle_{\mathcal{H}}$
$\displaystyle=$ $\displaystyle\left\langle
v_{1},{\rho}(g)v_{1}\right\rangle_{\mathcal{H}}+\left\langle
v_{2},{\rho}(g)v_{2}\right\rangle_{\mathcal{H}}$ $\displaystyle{}+\left\langle
v_{1},{\rho}(g)v_{2}\right\rangle_{\mathcal{H}}+\overline{\left\langle
v_{1},{\rho}(g^{-1})v_{2}\right\rangle_{\mathcal{H}}}$

The last expression can be conveniently rewritten for kernels of the
functional as

(7.31) $l_{12}=l_{1}+l_{2}+2A\sqrt{l_{1}l_{2}}$

for some real number $A$. This formula is behind the contextual law of
addition of conditional probabilities [Khrennikov01a] and will be illustrated
below. Its physical interpretation is an interference, say, from two slits.
Despite of a common belief, the mechanism of such interference can be both
causal and local, see [Kisil01c] [KhrenVol01].

####  7.3. Elliptic characters and Quantum Dynamics

In this subsection we consider the representation ${\rho_{h}}$ of
$\mathbb{H}^{n}{}$ induced by the elliptic character
$\chi_{h}(s)=e^{\mathrm{i}hs}$ in complex numbers parametrised by
$h\in\mathbb{R}{}$. We also use the convenient agreement $h=2\pi\hslash$
borrowed from physical literature.

#####  7.3.1. Fock–Segal–Bargmann and Schrödinger Representations

The realisation of ${\rho_{h}}$ by the left shifts (7.3) on
$L_{2}^{h}{}(\mathbb{H}^{n}{})$ is rarely used in quantum mechanics. Instead
two unitary equivalent forms are more common: the Schrödinger and
Fock–Segal–Bargmann (FSB) representations.

The FSB representation can be obtained from the orbit method of Kirillov
[Kirillov94a]. It allows spatially separate irreducible components of the left
regular representation, each of them become located on the orbit of the co-
adjoint representation, see [Kisil02e]*§ 2.1 [Kirillov94a] for details, we
only present a brief summary here.

We identify $\mathbb{H}^{n}{}$ and its Lie algebra $\mathfrak{h}_{n}$ through
the exponential map [Kirillov76]*§ 6.4. The dual $\mathfrak{h}_{n}^{*}$ of
$\mathfrak{h}_{n}$ is presented by the Euclidean space $\mathbb{R}^{2n+1}{}$
with coordinates $(\hslash,q,p)$. The pairing $\mathfrak{h}_{n}^{*}$ and
$\mathfrak{h}_{n}$ given by

$\left\langle(s,x,y),(\hslash,q,p)\right\rangle=\hslash s+q\cdot x+p\cdot y.$

This pairing defines the Fourier transform $\hat{\
}:L_{2}{}(\mathbb{H}^{n}{})\rightarrow L_{2}{}(\mathfrak{h}_{n}^{*})$ given by
[Kirillov99]*§ 2.3:

(7.32) $\hat{\phi}(F)=\int_{\mathfrak{h}^{n}}\phi(\exp
X)e^{-2\pi\mathrm{i}\left\langle X,F\right\rangle}\,dX\qquad\textrm{ where
}X\in\mathfrak{h}^{n},\ F\in\mathfrak{h}_{n}^{*}.$

For a fixed $\hslash$ the left regular representation (7.3) is mapped by the
Fourier transform to the FSB type representation (7.17). The collection of
points $(\hslash,q,p)\in\mathfrak{h}_{n}^{*}$ for a fixed $\hslash$ is
naturally identified with the _phase space_ of the system.

######  Remark 7.9.

It is possible to identify the case of $\hslash=0$ with classical mechanics
[Kisil02e]. Indeed, a substitution of the zero value of $\hslash$ into (7.17)
produces the commutative representation:

(7.33) ${\rho_{0}}(s,x,y):f(q,p)\mapsto
e^{-2\pi\mathrm{i}(qx+py)}f\left(q,p\right).$

It can be decomposed into the direct integral of one-dimensional
representations parametrised by the points $(q,p)$ of the phase space. The
classical mechanics, including the Hamilton equation, can be recovered from
those representations [Kisil02e]. However the condition $\hslash=0$ (as well
as the _semiclassical limit_ $\hslash\rightarrow 0$) is not completely
physical. Commutativity (and subsequent relative triviality) of those
representation is the main reason why they are oftenly neglected. The
commutativity can be outweighed by special arrangements, e.g. an
antiderivative [Kisil02e]*(4.1), but the procedure is not straightforward, see
discussion in [Kisil05c] [AgostiniCapraraCiccotti07a] [Kisil09a]. A direct
approach using dual numbers will be shown below, cf. Rem. 7.21.

To recover the Schrödinger representation we use notations and technique of
induced representations from § 3.2, see also [Kisil98a]*Ex. 4.1. The subgroup
$H=\\{(s,0,y)\,\mid\,s\in\mathbb{R}{},y\in\mathbb{R}^{n}{}\\}\subset\mathbb{H}^{n}{}$
defines the homogeneous space $X=G/H$, which coincides with $\mathbb{R}^{n}{}$
as a manifold. The natural projection $\mathbf{p}:G\rightarrow X$ is
$\mathbf{p}(s,x,y)=x$ and its left inverse $\mathbf{s}:X\rightarrow G$ can be
as simple as $\mathbf{s}(x)=(0,x,0)$. For the map $\mathbf{r}:G\rightarrow H$,
$\mathbf{r}(s,x,y)=(s-xy/2,0,y)$ we have the decomposition

$(s,x,y)=\mathbf{s}(p(s,x,y))*\mathbf{r}(s,x,y)=(0,x,0)*(s-\textstyle\frac{1}{2}xy,0,y).$

For a character $\chi_{h}(s,0,y)=e^{\mathrm{i}hs}$ of $H$ the lifting
$\mathcal{L}_{\chi}:L_{2}{}(G/H)\rightarrow L_{2}^{\chi}{}(G)$ is as follows:

$[\mathcal{L}_{\chi}f](s,x,y)=\chi_{h}(\mathbf{r}(s,x,y))\,f(\mathbf{p}(s,x,y))=e^{\mathrm{i}h(s-xy/2)}f(x).$

Thus the representation
${\rho_{\chi}}(g)=\mathcal{P}\circ\Lambda(g)\circ\mathcal{L}$ becomes:

(7.34)
$[{\rho_{\chi}}(s^{\prime},x^{\prime},y^{\prime})f](x)=e^{-2\pi\mathrm{i}\hslash(s^{\prime}+xy^{\prime}-x^{\prime}y^{\prime}/2)}\,f(x-x^{\prime}).$

After the Fourier transform $x\mapsto q$ we get the Schrödinger representation
on the _configuration space_ :

(7.35)
$[{\rho_{\chi}}(s^{\prime},x^{\prime},y^{\prime})\hat{f}\,](q)=e^{-2\pi\mathrm{i}\hslash(s^{\prime}+x^{\prime}y^{\prime}/2)-2\pi\mathrm{i}x^{\prime}q}\,\hat{f}(q+\hslash
y^{\prime}).$

Note that this again turns into a commutative representation (multiplication
by an unimodular function) if $\hslash=0$. To get the full set of commutative
representations in this way we need to use the character
$\chi_{(h,p)}(s,0,y)=e^{2\pi\mathrm{i}(\hslash+py)}$ in the above
consideration.

#####  7.3.2. Commutator and the Heisenberg Equation

The property (7.6) of $F_{2}^{\chi}{}(\mathbb{H}^{n}{})$ implies that the
restrictions of two operators ${\rho_{\chi}}(k_{1})$ and
${\rho_{\chi}}(k_{2})$ to this space are equal if

$\int_{\mathbb{R}{}}k_{1}(s,x,y)\,\chi(s)\,ds=\int_{\mathbb{R}{}}k_{2}(s,x,y)\,\chi(s)\,ds.$

In other words, for a character $\chi(s)=e^{2\pi\mathrm{i}\hslash s}$ the
operator ${\rho_{\chi}}(k)$ depends only on

$\hat{k}_{s}(\hslash,x,y)=\int_{\mathbb{R}{}}k(s,x,y)\,e^{-2\pi\mathrm{i}\hslash
s}\,ds,$

which is the partial Fourier transform $s\mapsto\hslash$ of $k(s,x,y)$. The
restriction to $F_{2}^{\chi}{}(\mathbb{H}^{n}{})$ of the composition formula
for convolutions is [Kisil02e]*(3.5):

(7.36)
$(k^{\prime}*k)\hat{{}_{s}}=\int_{\mathbb{R}^{2n}{}}e^{{\mathrm{i}h}{}(xy^{\prime}-yx^{\prime})/2}\,\hat{k}^{\prime}_{s}(\hslash,x^{\prime},y^{\prime})\,\hat{k}_{s}(\hslash,x-x^{\prime},y-y^{\prime})\,dx^{\prime}dy^{\prime}.$

Under the Schrödinger representation (7.35) the convolution (7.36) defines a
rule for composition of two pseudo-differential operators (PDO) in the Weyl
calculus [Howe80b] [Folland89]*§ 2.3.

Consequently the representation (7.21) of commutator (7.2.1) depends only on
its partial Fourier transform [Kisil02e]*(3.6):

$\displaystyle[k^{\prime},k]\hat{{}_{s}}$ $\displaystyle=$ $\displaystyle
2\mathrm{i}\int_{\mathbb{R}^{2n}{}}\\!\\!\sin(\textstyle\frac{h}{2}(xy^{\prime}-yx^{\prime}))\,$
$\displaystyle\qquad\times\hat{k}^{\prime}_{s}(\hslash,x^{\prime},y^{\prime})\,\hat{k}_{s}(\hslash,x-x^{\prime},y-y^{\prime})\,dx^{\prime}dy^{\prime}.$

Under the Fourier transform (7.32) this commutator is exactly the _Moyal
bracket_ [Zachos02a] for of $\hat{k}^{\prime}$ and $\hat{k}$ on the phase
space.

For observables in the space $F_{2}^{\chi}{}(\mathbb{H}^{n}{})$ the action of
$S$ is reduced to multiplication, e.g. for $\chi(s)=e^{\mathrm{i}hs}$ the
action of $S$ is multiplication by $\mathrm{i}h$. Thus the equation (7.23)
reduced to the space $F_{2}^{\chi}{}(\mathbb{H}^{n}{})$ becomes the Heisenberg
type equation [Kisil02e]*(4.4):

(7.38) $\dot{f}=\frac{1}{\mathrm{i}h}[H,f]\hat{{}_{s}},$

based on the above bracket (7.3.2). The Schrödinger representation (7.35)
transforms this equation to the original Heisenberg equation.

######  Example 7.10.

  1. (i)

Under the Fourier transform $(x,y)\mapsto(q,p)$ the p-dynamic equation (7.25)
of the harmonic oscillator becomes:

(7.39) $\dot{f}=\left(mk^{2}q\frac{\partial}{\partial
p}-\frac{1}{m}p\frac{\partial}{\partial q}\right)f.$

The same transform creates its solution out of (7.26).

  2. (ii)

Since $\frac{\partial}{\partial s}$ acts on $F_{2}^{\chi}{}(\mathbb{H}^{n}{})$
as multiplication by $\mathrm{i}\hslash$, the quantum representation of
unharmonic dynamics equation (7.28) is:

(7.40) $\dot{f}=\left(mk^{2}q\frac{\partial}{\partial
p}+\frac{\lambda}{6}\left(3q^{2}\frac{\partial}{\partial
p}-\frac{\hslash^{2}}{4}\frac{\partial^{3}}{\partial
p^{3}}\right)-\frac{1}{m}p\frac{\partial}{\partial q}\right)f.$

This is exactly the equation for the Wigner function obtained in
[CalzettaVerdaguer06a]*(30).

#####  7.3.3. Quantum Probabilities

For the elliptic character $\chi_{h}(s)=e^{\mathrm{i}hs}$ we can use the
Cauchy–Schwartz inequality to demonstrate that the real number $A$ in the
identity (7.31) is between $-1$ and $1$. Thus we can put $A=\cos\alpha$ for
some angle (phase) $\alpha$ to get the formula for counting quantum
probabilities, cf. [Khrennikov03a]*(2):

(7.41) $l_{12}=l_{1}+l_{2}+2\cos\alpha\,\sqrt{l_{1}l_{2}}$

######  Remark 7.11.

It is interesting to note that the both trigonometric functions are employed
in quantum mechanics: sine is in the heart of the Moyal bracket (7.3.2) and
cosine is responsible for the addition of probabilities (7.41). In the essence
the commutator and probabilities took respectively the odd and even parts of
the elliptic character $e^{\mathrm{i}hs}$.

######  Example 7.12.

Take a vector $v_{(a,b)}\in L_{2}^{h}{}(\mathbb{H}^{n}{})$ defined by a
Gaussian with mean value $(a,b)$ in the phase space for a harmonic oscillator
of the mass $m$ and the frequency $k$:

(7.42) $v_{(a,b)}(q,p)=\exp\left(-\frac{2\pi
km}{\hslash}(q-a)^{2}-\frac{2\pi}{\hslash km}(p-b)^{2}\right).$

A direct calculation shows:

$\displaystyle\left\langle
v_{(a,b)},{\rho_{\hslash}}(s,x,y)v_{(a^{\prime},b^{\prime})}\right\rangle=\frac{4}{\hslash}\exp\left(\pi\mathrm{i}\left(2s\hslash+x(a+a^{\prime})+y(b+b^{\prime})\right)\frac{}{}\right.$
$\displaystyle\left.{}-\frac{\pi}{2\hslash km}((\hslash
x+b-b^{\prime})^{2}+(b-b^{\prime})^{2})-\frac{\pi km}{2\hslash}((\hslash
y+a^{\prime}-a)^{2}+(a^{\prime}-a)^{2})\right)$ $\displaystyle=$
$\displaystyle\frac{4}{\hslash}\exp\left(\pi\mathrm{i}\left(2s\hslash+x(a+a^{\prime})+y(b+b^{\prime})\right)\frac{}{}\right.$
$\displaystyle\left.{}-\frac{\pi}{\hslash
km}((b-b^{\prime}+{\textstyle\frac{\hslash
x}{2}})^{2}+({\textstyle\frac{\hslash x}{2}})^{2})-\frac{\pi
km}{\hslash}((a-a^{\prime}-{\textstyle\frac{\hslash
y}{2}})^{2}+({\textstyle\frac{\hslash y}{2}})^{2})\right)$

Thus the kernel $l_{(a,b)}=\left\langle
v_{(a,b)},{\rho_{\hslash}}(s,x,y)v_{(a,b)}\right\rangle$ (7.29) for a state
$v_{(a,b)}$ is:

(7.43) $\displaystyle l_{(a,b)}$ $\displaystyle=$
$\displaystyle\frac{4}{\hslash}\exp\left(2\pi\mathrm{i}(s\hslash+xa+yb)\frac{}{}-\frac{\pi\hslash}{2km}x^{2}-\frac{\pi
km\hslash}{2\hslash}y^{2}\right)$

An observable registering a particle at a point $q=c$ of the configuration
space is $\delta(q-c)$. On the Heisenberg group this observable is given by
the kernel:

(7.44) $X_{c}(s,x,y)=e^{2\pi\mathrm{i}(s\hslash+xc)}\delta(y).$

The measurement of $X_{c}$ on the state (7.42) (through the kernel (7.43))
predictably is:

$\left\langle
X_{c},l_{(a,b)}\right\rangle=\sqrt{\frac{2km}{\hslash}}\exp\left(-\frac{2\pi
km}{\hslash}(c-a)^{2}\right).$

######  Example 7.13.

Now take two states $v_{(0,b)}$ and $v_{(0,-b)}$, where for the simplicity we
assume the mean values of coordinates vanish in the both cases. Then the
corresponding kernel (7.30) has the interference terms:

$\displaystyle l_{i}$ $\displaystyle=$ $\displaystyle\left\langle
v_{(0,b)},{\rho_{\hslash}}(s,x,y)v_{(0,-b)}\right\rangle$ $\displaystyle=$
$\displaystyle\frac{4}{\hslash}\exp\left(2\pi\mathrm{i}s\hslash-\frac{\pi}{2\hslash
km}((\hslash x+2b)^{2}+4b^{2})-\frac{\pi\hslash km}{2}y^{2}\right).$

The measurement of $X_{c}$ (7.44) on this term contains the oscillating part:

$\left\langle
X_{c},l_{i}\right\rangle=\sqrt{\frac{2km}{\hslash}}\exp\left(-\frac{2\pi
km}{\hslash}c^{2}-\frac{2\pi}{km\hslash}b^{2}+\frac{4\pi\mathrm{i}}{\hslash}cb\right)$

Therefore on the kernel $l$ corresponding to the state $v_{(0,b)}+v_{(0,-b)}$
the measurement is

$\displaystyle\left\langle X_{c},l\right\rangle$ $\displaystyle=$
$\displaystyle 2\sqrt{\frac{2km}{\hslash}}\exp\left(-\frac{2\pi
km}{\hslash}c^{2}\right)\left(1+\exp\left(-\frac{2\pi}{km\hslash}b^{2}\right)\cos\left(\frac{4\pi}{\hslash}cb\right)\right).$

(a) (b)

Figure 13. Quantum probabilities: the blue (dashed) graph shows the addition
of probabilities without interaction, the red (solid) graph present the
quantum interference. Left picture shows the Gaussian state (7.42), the
right—the rational state (7.45)

The presence of the cosine term in the last expression can generate an
interference picture. In practise it does not happen for the minimal
uncertainty state (7.42) which we are using here: it rapidly vanishes outside
of the neighbourhood of zero, where oscillations of the cosine occurs, see
Fig. 13(a).

######  Example 7.14.

To see a traditional interference pattern one can use a state which is far
from the minimal uncertainty. For example, we can consider the state:

(7.45)
$u_{(a,b)}(q,p)=\frac{\hslash^{2}}{((q-a)^{2}+\hslash/km)((p-b)^{2}+\hslash
km)}.$

To evaluate the observable $X_{c}$ (7.44) on the state $l(g)=\left\langle
u_{1},{\rho_{h}}(g)u_{2}\right\rangle$ (7.29) we use the following formula:

$\left\langle
X_{c},l\right\rangle=\frac{2}{\hslash}\int_{\mathbb{R}^{n}{}}\hat{u}_{1}(q,2(q-c)/\hslash)\,\overline{\hat{u}_{2}(q,2(q-c)/\hslash)}\,dq,$

where $\hat{u}_{i}(q,x)$ denotes the partial Fourier transform $p\mapsto x$ of
$u_{i}(q,p)$. The formula is obtained by swapping order of integrations. The
numerical evaluation of the state obtained by the addition
$u_{(0,b)}+u_{(0,-b)}$ is plotted on Fig. 13(b), the red curve shows the
canonical interference pattern.

####  7.4. Ladder Operators and Harmonic Oscillator

Let ${\rho}$ be a representation of the Schrödinger group
$G=\mathbb{H}^{1}{}\rtimes\widetilde{\mathrm{Sp}}(2)$ (7.9) in a space $V$.
Consider the derived representation of the Lie algebra $\mathfrak{g}$
[Lang85]*§ VI.1 and denote $\tilde{X}={\rho}(X)$ for $X\in\mathfrak{g}$. To
see the structure of the representation ${\rho}$ we can decompose the space
$V$ into eigenspaces of the operator $\tilde{X}$ for some $X\in\mathfrak{g}$.
The canonical example is the Taylor series in complex analysis.

We are going to consider three cases corresponding to three non-isomorphic
subgroups (2.4–2.6) of $\mathrm{Sp}(2)$ starting from the compact case. Let
$H=Z$ be a generator of the compact subgroup $K$. Corresponding
symplectomorphisms (7.8) of the phase space are given by orthogonal rotations
with matrices $\begin{pmatrix}\cos t&\sin t\\\ -sint&\cos t\end{pmatrix}$. The
Shale–Weil representation (7.13) coincides with the Hamiltonian of the
harmonic oscillator in Schrödinger representation.

Since $\widetilde{\mathrm{Sp}}(2)$ is a two-fold cover the corresponding
eigenspaces of a compact group $\tilde{Z}v_{k}=\mathrm{i}kv_{k}$ are
parametrised by a half-integer $k\in\mathbb{Z}{}/2$. Explicitly for a half-
integer $k$ eigenvectors are:

(7.46)
$v_{k}(q)=H_{k+\frac{1}{2}}\left(\sqrt{\frac{2\pi}{\hslash}}q\right)e^{-\frac{\pi}{\hslash}q^{2}},$

where $H_{k}$ is the _Hermite polynomial_ [Folland89]*§ 1.7
[ErdelyiMagnusII]*8.2(9).

From the point of view of quantum mechanics as well as the representation
theory it is beneficial to introduce the ladder operators $L^{\\!\pm}$ (3.14),
known also as _creation/annihilation_ in quantum mechanics [Folland89]*p. 49
[BoyerMiller74a]. There are two ways to search for ladder operators: in
(complexified) Lie algebras $\mathfrak{h}_{1}$ and $\mathfrak{sp}_{2}$. The
later coincides with our consideration in Section 3.3 in the essence.

#####  7.4.1. Ladder Operators from the Heisenberg Group

Assuming $L^{\\!+}=a\tilde{X}+b\tilde{Y}$ we obtain from the relations
(7.10–7.11) and (3.14) the linear equations with unknown $a$ and $b$:

$a=\lambda_{+}b,\qquad-b=\lambda_{+}a.$

The equations have a solution if and only if $\lambda_{+}^{2}+1=0$, and the
raising/lowering operators are $L^{\\!\pm}=\tilde{X}\mp\mathrm{i}\tilde{Y}$.

######  Remark 7.15.

Here we have an interesting asymmetric response: due to the structure of the
semidirect product $\mathbb{H}^{1}{}\rtimes\widetilde{\mathrm{Sp}}(2)$ it is
the symplectic group which acts on $\mathbb{H}^{1}{}$, not vise versa. However
the Heisenberg group has a weak action in the opposite direction: it shifts
eigenfunctions of $\mathrm{Sp}(2)$.

In the Schrödinger representation (7.12) the ladder operators are

(7.47)
${\rho_{\hslash}}(L^{\\!\pm})=2\pi\mathrm{i}q\pm\mathrm{i}\hslash\frac{d}{dq}.$

The standard treatment of the harmonic oscillator in quantum mechanics, which
can be found in many textbooks, e.g. [Folland89]*§ 1.7 [Gazeau09a]*§ 2.2.3, is
as follows. The vector $v_{-1/2}(q)=e^{-\pi q^{2}/\hslash}$ is an eigenvector
of $\tilde{Z}$ with the eigenvalue $-\frac{\mathrm{i}}{2}$. In addition
$v_{-1/2}$ is annihilated by $L^{\\!+}$. Thus the chain (3.16) terminates to
the right and the complete set of eigenvectors of the harmonic oscillator
Hamiltonian is presented by $(L^{\\!-})^{k}v_{-1/2}$ with $k=0,1,2,\ldots$.

We can make a wavelet transform generated by the Heisenberg group with the
mother wavelet $v_{-1/2}$, and the image will be the Fock–Segal–Bargmann (FSB)
space [Howe80b] [Folland89]*§ 1.6. Since $v_{-1/2}$ is the null solution of
$L^{\\!+}=\tilde{X}-\mathrm{i}\tilde{Y}$, then by Cor. 5.6 the image of the
wavelet transform will be null-solutions of the corresponding linear
combination of the Lie derivatives (7.4):

(7.48)
$D=\overline{X^{r}-\mathrm{i}Y^{r}}=(\partial_{x}+\mathrm{i}\partial_{y})-\pi\hslash(x-\mathrm{i}y),$

which turns out to be the Cauchy–Riemann equation on a weighted FSB-type
space.

#####  7.4.2. Symplectic Ladder Operators

We can also look for ladder operators within the Lie algebra
$\mathfrak{sp}_{2}$, see § 3.3.1 and [Kisil09c]*§ 8. Assuming
$L_{2}^{\\!+}=a\tilde{A}+b\tilde{B}+c\tilde{Z}$ from the relations (3.13) and
defining condition (3.14) we obtain the linear equations with unknown $a$, $b$
and $c$:

$c=0,\qquad 2a=\lambda_{+}b,\qquad-2b=\lambda_{+}a.$

The equations have a solution if and only if $\lambda_{+}^{2}+4=0$, and the
raising/lowering operators are
$L_{2}^{\\!\pm}=\pm\mathrm{i}\tilde{A}+\tilde{B}$. In the Shale–Weil
representation (7.13) they turn out to be:

(7.49)
$L_{2}^{\\!\pm}=\pm\mathrm{i}\left(\frac{q}{2}\frac{d}{dq}+\frac{1}{4}\right)-\frac{\hslash\mathrm{i}}{8\pi}\frac{d^{2}}{dq^{2}}-\frac{\pi\mathrm{i}q^{2}}{2\hslash}=-\frac{\mathrm{i}}{8\pi\hslash}\left(\mp
2\pi q+\hslash\frac{d}{dq}\right)^{2}.$

Since this time $\lambda_{+}=2\mathrm{i}$ the ladder operators
$L_{2}^{\\!\pm}$ produce a shift on the diagram (3.16) twice bigger than the
operators $L^{\\!\pm}$ from the Heisenberg group. After all, this is not
surprising since from the explicit representations (7.47) and (7.49) we get:

$L_{2}^{\\!\pm}=-\frac{\mathrm{i}}{8\pi\hslash}(L^{\\!\pm})^{2}.$

####  7.5. Hyperbolic Quantum Mechanics

Now we turn to double numbers also known as hyperbolic, split-complex, etc.
numbers [Yaglom79]*App. C [Ulrych05a] [KhrennikovSegre07a]. They form a two
dimensional algebra $\mathbb{O}{}$ spanned by $1$ and $\mathrm{j}$ with the
property $\mathrm{j}^{2}=1$. There are zero divisors:

$\mathrm{j}_{\pm}=\textstyle\frac{1}{\sqrt{2}}(1\pm j),\qquad\text{ such that
}\quad\mathrm{j}_{+}\mathrm{j}_{-}=0\quad\text{ and
}\quad\mathrm{j}_{\pm}^{2}=\mathrm{j}_{\pm}.$

Thus double numbers algebraically isomorphic to two copies of $\mathbb{R}{}$
spanned by $\mathrm{j}_{\pm}$. Being algebraically dull double numbers are
nevertheless interesting as a homogeneous space [Kisil05a, Kisil09c] and they
are relevant in physics [Khrennikov05a, Ulrych05a, Ulrych08a]. The combination
of p-mechanical approach with hyperbolic quantum mechanics was already
discussed in [BrodlieKisil03a]*§ 6.

For the hyperbolic character $\chi_{\mathrm{j}h}(s)=e^{\mathrm{j}hs}=\cosh
hs+\mathrm{j}\sinh hs$ of $\mathbb{R}{}$ one can define the hyperbolic
Fourier-type transform:

$\hat{k}(q)=\int_{\mathbb{R}{}}k(x)\,e^{-\mathrm{j}qx}dx.$

It can be understood in the sense of distributions on the space dual to the
set of analytic functions [Khrennikov08a]*§ 3. Hyperbolic Fourier transform
intertwines the derivative $\frac{d}{dx}$ and multiplication by $\mathrm{j}q$
[Khrennikov08a]*Prop. 1.

######  Example 7.16.

For the Gaussian the hyperbolic Fourier transform is the ordinary function
(note the sign difference!):

$\int_{\mathbb{R}{}}e^{-x^{2}/2}e^{-\mathrm{j}qx}dx=\sqrt{2\pi}\,e^{q^{2}/2}.$

However the opposite identity:

$\int_{\mathbb{R}{}}e^{x^{2}/2}e^{-\mathrm{j}qx}dx=\sqrt{2\pi}\,e^{-q^{2}/2}$

is true only in a suitable distributional sense. To this end we may note that
$e^{x^{2}/2}$ and $e^{-q^{2}/2}$ are null solutions to the differential
operators $\frac{d}{dx}-x$ and $\frac{d}{dq}+q$ respectively, which are
intertwined (up to the factor $\mathrm{j}$) by the hyperbolic Fourier
transform. The above differential operators $\frac{d}{dx}-x$ and
$\frac{d}{dq}+q$ are images of the ladder operators (7.47) in the Lie algebra
of the Heisenberg group. They are intertwining by the Fourier transform, since
this is an automorphism of the Heisenberg group [Howe80a].

An elegant theory of hyperbolic Fourier transform may be achieved by a
suitable adaptation of [Howe80a], which uses representation theory of the
Heisenberg group.

#####  7.5.1. Hyperbolic Representations of the Heisenberg Group

Consider the space $F_{h}^{\mathrm{j}}{}(\mathbb{H}^{n}{})$ of
$\mathbb{O}{}$-valued functions on $\mathbb{H}^{n}{}$ with the property:

(7.50) $f(s+s^{\prime},h,y)=e^{\mathrm{j}hs^{\prime}}f(s,x,y),\qquad\text{ for
all }(s,x,y)\in\mathbb{H}^{n}{},\ s^{\prime}\in\mathbb{R}{},$

and the square integrability condition (7.7). Then the hyperbolic
representation is obtained by the restriction of the left shifts to
$F_{h}^{\mathrm{j}}{}(\mathbb{H}^{n}{})$. To obtain an equivalent
representation on the phase space we take $\mathbb{O}{}$-valued functional of
the Lie algebra $\mathfrak{h}_{n}$:

(7.51)
$\chi^{j}_{(h,q,p)}(s,x,y)=e^{\mathrm{j}(hs+qx+py)}=\cosh(hs+qx+py)+\mathrm{j}\sinh(hs+qx+py).$

The hyperbolic Fock–Segal–Bargmann type representation is intertwined with the
left group action by means of the Fourier transform (7.32) with the hyperbolic
functional (7.51). Explicitly this representation is:

(7.52) ${\rho_{\hslash}}(s,x,y):f(q,p)\mapsto\textstyle
e^{-\mathrm{j}(hs+qx+py)}f\left(q-\frac{h}{2}y,p+\frac{h}{2}x\right).$

For a hyperbolic Schrödinger type representation we again use the scheme
described in § 3.2. Similarly to the elliptic case one obtains the formula,
resembling (7.34):

(7.53)
$[{\rho^{\mathrm{j}}_{\chi}}(s^{\prime},x^{\prime},y^{\prime})f](x)=e^{-\mathrm{j}h(s^{\prime}+xy^{\prime}-x^{\prime}y^{\prime}/2)}f(x-x^{\prime}).$

Application of the hyperbolic Fourier transform produces a Schrödinger type
representation on the configuration space, cf. (7.35):

$[{\rho^{\mathrm{j}}_{\chi}}(s^{\prime},x^{\prime},y^{\prime})\hat{f}\,](q)=e^{-\mathrm{j}h(s^{\prime}+x^{\prime}y^{\prime}/2)-\mathrm{j}x^{\prime}q}\,\hat{f}(q+hy^{\prime}).$

The extension of this representation to kernels according to (7.21) generates
hyperbolic pseudodifferential operators introduced in [Khrennikov08a]*(3.4).

#####  7.5.2. Hyperbolic Dynamics

Similarly to the elliptic (quantum) case we consider a convolution of two
kernels on $\mathbb{H}^{n}{}$ restricted to
$F_{h}^{\mathrm{j}}{}(\mathbb{H}^{n}{})$. The composition law becomes, cf.
(7.36):

(7.54)
$(k^{\prime}*k)\hat{{}_{s}}=\int_{\mathbb{R}^{2n}{}}e^{{\mathrm{j}h}{}(xy^{\prime}-yx^{\prime})}\,\hat{k}^{\prime}_{s}(h,x^{\prime},y^{\prime})\,\hat{k}_{s}(h,x-x^{\prime},y-y^{\prime})\,dx^{\prime}dy^{\prime}.$

This is close to the calculus of hyperbolic PDO obtained in
[Khrennikov08a]*Thm. 2. Respectively for the commutator of two convolutions we
get, cf. (7.3.2):

(7.55)
$[k^{\prime},k]\hat{{}_{s}}=\int_{\mathbb{R}^{2n}{}}\\!\\!\sinh(h(xy^{\prime}-yx^{\prime}))\,\hat{k}^{\prime}_{s}(h,x^{\prime},y^{\prime})\,\hat{k}_{s}(h,x-x^{\prime},y-y^{\prime})\,dx^{\prime}dy^{\prime}.$

This the hyperbolic version of the Moyal bracket, cf. [Khrennikov08a]*p. 849,
which generates the corresponding image of the dynamic equation (7.23).

######  Example 7.17.

  1. (i)

For a quadratic Hamiltonian, e.g. harmonic oscillator from Example 7.6, the
hyperbolic equation and respective dynamics is identical to quantum considered
before.

  2. (ii)

Since $\frac{\partial}{\partial s}$ acts on
$F_{2}^{\mathrm{j}}{}(\mathbb{H}^{n}{})$ as multiplication by $\mathrm{j}h$
and $\mathrm{j}^{2}=1$, the hyperbolic image of the unharmonic equation (7.28)
becomes:

$\dot{f}=\left(mk^{2}q\frac{\partial}{\partial
p}+\frac{\lambda}{6}\left(3q^{2}\frac{\partial}{\partial
p}+\frac{\hslash^{2}}{4}\frac{\partial^{3}}{\partial
p^{3}}\right)-\frac{1}{m}p\frac{\partial}{\partial q}\right)f.$

The difference with quantum mechanical equation (7.40) is in the sign of the
cubic derivative.

#####  7.5.3. Hyperbolic Probabilities

(a) (b)

Figure 14. Hyperbolic probabilities: the blue (dashed) graph shows the
addition of probabilities without interaction, the red (solid) graph present
the quantum interference. Left picture shows the Gaussian state (7.42), with
the same distribution as in quantum mechanics, cf. Fig. 13(a). The right
picture shows the rational state (7.45), note the absence of interference
oscillations in comparison with the quantum state on Fig. 13(b).

To calculate probability distribution generated by a hyperbolic state we are
using the general procedure from Section 7.2.2. The main differences with the
quantum case are as follows:

  1. (i)

The real number $A$ in the expression (7.31) for the addition of probabilities
is bigger than $1$ in absolute value. Thus it can be associated with the
hyperbolic cosine $\cosh\alpha$, cf. Rem. 7.11, for certain phase
$\alpha\in\mathbb{R}{}$ [Khrennikov08a].

  2. (ii)

The nature of hyperbolic interference on two slits is affected by the fact
that $e^{\mathrm{j}hs}$ is not periodic and the hyperbolic exponent
$e^{\mathrm{j}t}$ and cosine $\cosh t$ do not oscillate. It is worth to notice
that for Gaussian states the hyperbolic interference is exactly the same as
quantum one, cf. Figs. 13(a) and 14(a). This is similar to coincidence of
quantum and hyperbolic dynamics of harmonic oscillator.

The contrast between two types of interference is prominent for the rational
state (7.45), which is far from the minimal uncertainty, see the different
patterns on Figs. 13(b) and 14(b).

#####  7.5.4. Ladder Operators for the Hyperbolic Subgroup

Consider the case of the Hamiltonian $H=2B$, which is a repulsive (hyperbolic)
harmonic oscillator [Wulfman10a]*§ 3.8. The corresponding one-dimensional
subgroup of symplectomorphisms produces hyperbolic rotations of the phase
space, see Fig. 9. The eigenvectors $v_{\mu}$ of the operator

${\rho^{\text{SW}}_{\hslash}}(2B)v_{\nu}=-\mathrm{i}\left(\frac{\hslash}{4\pi}\frac{d^{2}}{dq^{2}}+\frac{\pi
q^{2}}{\hslash}\right)v_{\nu}=\mathrm{i}\nu v_{\nu},$

are _Weber–Hermite_ (or _parabolic cylinder_) functions
$v_{\nu}=D_{\nu-\frac{1}{2}}\left(\pm
2e^{\mathrm{i}\frac{\pi}{4}}\sqrt{\frac{\pi}{\hslash}}q\right)$, see
[ErdelyiMagnusII]*§ 8.2 [SrivastavaTuanYakubovich00a] for fundamentals of
Weber–Hermite functions and [ATorre08a] for further illustrations and
applications in optics.

The corresponding one-parameter group is not compact and the eigenvalues of
the operator $2\tilde{B}$ are not restricted by any integrality condition, but
the raising/lowering operators are still important [HoweTan92]*§ II.1
[Mazorchuk09a]*§ 1.1. We again seek solutions in two subalgebras
$\mathfrak{h}_{1}$ and $\mathfrak{sp}_{2}$ separately. However the additional
options will be provided by a choice of the number system: either complex or
double.

######  Example 7.18 (Complex Ladder Operators).

Assuming $L_{h}^{\\!+}=a\tilde{X}+b\tilde{Y}$ from the commutators (7.10–7.11)
we obtain the linear equations:

(7.56) $-a=\lambda_{+}b,\qquad-b=\lambda_{+}a.$

The equations have a solution if and only if $\lambda_{+}^{2}-1=0$. Taking the
real roots $\lambda=\pm 1$ we obtain that the raising/lowering operators are
$L_{h}^{\\!\pm}=\tilde{X}\mp\tilde{Y}$. In the Schrödinger representation
(7.12) the ladder operators are

(7.57) $L_{h}^{\\!\pm}=2\pi\mathrm{i}q\pm\hslash\frac{d}{dq}.$

The null solutions
$v_{\pm\frac{1}{2}}(q)=e^{\pm\frac{\pi\mathrm{i}}{\hslash}q^{2}}$ to operators
${\rho_{\hslash}}(L^{\\!\pm})$ are also eigenvectors of the Hamiltonian
${\rho^{\text{SW}}_{\hslash}}(2B)$ with the eigenvalue $\pm\frac{1}{2}$.
However the important distinction from the elliptic case is, that they are not
square-integrable on the real line anymore.

We can also look for ladder operators within the $\mathfrak{sp}_{2}$, that is
in the form $L_{2h}^{\\!+}=a\tilde{A}+b\tilde{B}+c\tilde{Z}$ for the
commutator $[2\tilde{B},L_{h}^{\\!+}]=\lambda L_{h}^{\\!+}$, see § 3.3.2.
Within complex numbers we get only the values $\lambda=\pm 2$ with the ladder
operators $L_{2h}^{\\!\pm}=\pm 2\tilde{A}+\tilde{Z}/2$, see [HoweTan92]*§ II.1
[Mazorchuk09a]*§ 1.1. Each indecomposable $\mathfrak{h}_{1}$\- or
$\mathfrak{sp}_{2}$-module is formed by a one-dimensional chain of eigenvalues
with a transitive action of ladder operators $L_{h}^{\\!\pm}$ or
$L_{2h}^{\\!\pm}$ respectively. And we again have a quadratic relation between
the ladder operators:

$L_{2h}^{\\!\pm}=\frac{\mathrm{i}}{4\pi\hslash}(L_{h}^{\\!\pm})^{2}.$

#####  7.5.5. Double Ladder Operators

There are extra possibilities in in the context of hyperbolic quantum
mechanics [Khrennikov03a] [Khrennikov05a] [Khrennikov08a]. Here we use the
representation of $\mathbb{H}^{1}{}$ induced by a hyperbolic character
$e^{\mathrm{j}ht}=\cosh(ht)+\mathrm{j}\sinh(ht)$, see [Kisil10a]*(4.5), and
obtain the hyperbolic representation of $\mathbb{H}^{1}{}$, cf. (7.35):

(7.58)
$[{\rho^{\mathrm{j}}_{h}}(s^{\prime},x^{\prime},y^{\prime})\hat{f}\,](q)=e^{\mathrm{j}h(s^{\prime}-x^{\prime}y^{\prime}/2)+\mathrm{j}x^{\prime}q}\,\hat{f}(q-hy^{\prime}).$

The corresponding derived representation is

(7.59)
${\rho^{\mathrm{j}}_{h}}(X)=\mathrm{j}q,\qquad{\rho^{\mathrm{j}}_{h}}(Y)=-h\frac{d}{dq},\qquad{\rho^{\mathrm{j}}_{h}}(S)=\mathrm{j}hI.$

Then the associated Shale–Weil derived representation of $\mathfrak{sp}_{2}$
in the Schwartz space $S{}(\mathbb{R}{})$ is, cf. (7.13):

(7.60)
${\rho^{\text{SW}}_{h}}(A)=-\frac{q}{2}\frac{d}{dq}-\frac{1}{4},\quad{\rho^{\text{SW}}_{h}}(B)=\frac{\mathrm{j}h}{4}\frac{d^{2}}{dq^{2}}-\frac{\mathrm{j}q^{2}}{4h},\quad{\rho^{\text{SW}}_{h}}(Z)=-\frac{\mathrm{j}h}{2}\frac{d^{2}}{dq^{2}}-\frac{\mathrm{j}q^{2}}{2h}.$

Note that ${\rho^{\text{SW}}_{h}}(B)$ now generates a usual harmonic
oscillator, not the repulsive one like ${\rho^{\text{SW}}_{\hslash}}(B)$ in
(7.13). However the expressions in the quadratic algebra are still the same
(up to a factor), cf. (7.4–7.16):

$\displaystyle\qquad{\rho^{\text{SW}}_{h}}(A)$ $\displaystyle=$
$\displaystyle-\frac{\mathrm{j}}{2h}({\rho^{\mathrm{j}}_{h}}(X){\rho^{\mathrm{j}}_{h}}(Y)-{\textstyle\frac{1}{2}}{\rho^{\mathrm{j}}_{h}}(S))$
$\displaystyle=$
$\displaystyle-\frac{\mathrm{j}}{4h}({\rho^{\mathrm{j}}_{h}}(X){\rho^{\mathrm{j}}_{h}}(Y)+{\rho^{\mathrm{j}}_{h}}(Y){\rho^{\mathrm{j}}_{h}}(X)),$
(7.62) $\displaystyle{\rho^{\text{SW}}_{h}}(B)$ $\displaystyle=$
$\displaystyle\frac{\mathrm{j}}{4h}({\rho^{\mathrm{j}}_{h}}(X)^{2}-{\rho^{\mathrm{j}}_{h}}(Y)^{2}),$
(7.63) $\displaystyle{\rho^{\text{SW}}_{h}}(Z)$ $\displaystyle=$
$\displaystyle-\frac{\mathrm{j}}{2h}({\rho^{\mathrm{j}}_{h}}(X)^{2}+{\rho^{\mathrm{j}}_{h}}(Y)^{2}).$

This is due to the Principle 3.5 of similarity and correspondence: we can swap
operators $Z$ and $B$ with simultaneous replacement of hypercomplex units
$\mathrm{i}$ and $\mathrm{j}$.

The eigenspace of the operator $2{\rho^{\text{SW}}_{h}}(B)$ with an eigenvalue
$\mathrm{j}\nu$ are spanned by the Weber–Hermite functions
$D_{-\nu-\frac{1}{2}}\left(\pm\sqrt{\frac{2}{h}}x\right)$, see
[ErdelyiMagnusII]*§ 8.2. Functions $D_{\nu}$ are generalisations of the Hermit
functions (7.46).

The compatibility condition for a ladder operator within the Lie algebra
$\mathfrak{h}_{1}$ will be (7.56) as before, since it depends only on the
commutators (7.10–7.11). Thus we still have the set of ladder operators
corresponding to values $\lambda=\pm 1$:

$L_{h}^{\\!\pm}=\tilde{X}\mp\tilde{Y}=\mathrm{j}q\pm h\frac{d}{dq}.$

Admitting double numbers we have an extra way to satisfy $\lambda^{2}=1$ in
(7.56) with values $\lambda=\pm\mathrm{j}$. Then there is an additional pair
of hyperbolic ladder operators, which are identical (up to factors) to (7.47):

$L_{\mathrm{j}}^{\\!\pm}=\tilde{X}\mp\mathrm{j}\tilde{Y}=\mathrm{j}q\pm\mathrm{j}h\frac{d}{dq}.$

Pairs $L_{h}^{\\!\pm}$ and $L_{\mathrm{j}}^{\\!\pm}$ shift eigenvectors in the
“orthogonal” directions changing their eigenvalues by $\pm 1$ and
$\pm\mathrm{j}$. Therefore an indecomposable $\mathfrak{sp}_{2}$-module can be
parametrised by a two-dimensional lattice of eigenvalues in double numbers,
see Fig. 10.

The following functions

$\displaystyle v_{\frac{1}{2}}^{\pm h}(q)$ $\displaystyle=$ $\displaystyle
e^{\mp\mathrm{j}q^{2}/(2h)}=\cosh\frac{q^{2}}{2h}\mp\mathrm{j}\sinh\frac{q^{2}}{2h},$
$\displaystyle v_{\frac{1}{2}}^{\pm\mathrm{j}}(q)$ $\displaystyle=$
$\displaystyle e^{\mp q^{2}/(2h)}$

are null solutions to the operators $L_{h}^{\\!\pm}$ and
$L_{\mathrm{j}}^{\\!\pm}$ respectively. They are also eigenvectors of
$2{\rho^{\text{SW}}_{h}}(B)$ with eigenvalues $\mp\frac{\mathrm{j}}{2}$ and
$\mp\frac{1}{2}$ respectively. If these functions are used as mother wavelets
for the wavelet transforms generated by the Heisenberg group, then the image
space will consist of the null-solutions of the following differential
operators, see Cor. 5.6:

$\textstyle
D_{h}=\overline{X^{r}-Y^{r}}=(\partial_{x}-\partial_{y})+\frac{h}{2}(x+y),\qquad
D_{\mathrm{j}}=\overline{X^{r}-\mathrm{j}Y^{r}}=(\partial_{x}+\mathrm{j}\partial_{y})-\frac{h}{2}(x-\mathrm{j}y),$

for $v_{\frac{1}{2}}^{\pm h}$ and $v_{\frac{1}{2}}^{\pm\mathrm{j}}$
respectively. This is again in line with the classical result (7.48). However
annihilation of the eigenvector by a ladder operator does not mean that the
part of the 2D-lattice becomes void since it can be reached via alternative
routes on this lattice. Instead of multiplication by a zero, as it happens in
the elliptic case, a half-plane of eigenvalues will be multiplied by the
divisors of zero $1\pm\mathrm{j}$.

We can also search ladder operators within the algebra $\mathfrak{sp}_{2}$ and
admitting double numbers we will again find two sets of them, cf. § 3.3.2:

$\displaystyle L_{2h}^{\\!\pm}$ $\displaystyle=$
$\displaystyle\pm\tilde{A}+\tilde{Z}/2=\mp\frac{q}{2}\frac{d}{dq}\mp\frac{1}{4}-\frac{\mathrm{j}h}{4}\frac{d^{2}}{dq^{2}}-\frac{\mathrm{j}q^{2}}{4h}=-\frac{\mathrm{j}}{4h}(L_{h}^{\\!\pm})^{2},$
$\displaystyle L_{2\mathrm{j}}^{\\!\pm}$ $\displaystyle=$
$\displaystyle\pm\mathrm{j}\tilde{A}+\tilde{Z}/2=\mp\frac{\mathrm{j}q}{2}\frac{d}{dq}\mp\frac{\mathrm{j}}{4}-\frac{\mathrm{j}h}{4}\frac{d^{2}}{dq^{2}}-\frac{\mathrm{j}q^{2}}{4h}=-\frac{\mathrm{j}}{4h}(L_{\mathrm{j}}^{\\!\pm})^{2}.$

Again the operators $L_{2h}^{\\!\pm}$ and $L_{2h}^{\\!\pm}$ produce double
shifts in the orthogonal directions on the same two-dimensional lattice in
Fig. 10.

####  7.6. Parabolic (Classical) Representations on the Phase Space

After the previous two cases it is natural to link classical mechanics with
dual numbers generated by the parabolic unit $\varepsilon^{2}=0$. Connection
of the parabolic unit $\varepsilon$ with the Galilean group of symmetries of
classical mechanics is around for a while [Yaglom79]*App. C.

However the nilpotency of the parabolic unit $\varepsilon$ make it difficult
if we will work with dual number valued functions only. To overcome this issue
we consider a commutative real algebra $\mathfrak{C}$ spanned by $1$,
$\mathrm{i}$, $\varepsilon$ and $\mathrm{i}\varepsilon$ with identities
$\mathrm{i}^{2}=-1$ and $\varepsilon^{2}=0$. A seminorm on $\mathfrak{C}$ is
defined as follows:

$\left|a+b\mathrm{i}+c\varepsilon+d\mathrm{i}\varepsilon\right|^{2}=a^{2}+b^{2}.$

#####  7.6.1. Classical Non-Commutative Representations

We wish to build a representation of the Heisenberg group which will be a
classical analog of the Fock–Segal–Bargmann representation (7.17). To this end
we introduce the space $F_{h}^{\varepsilon}{}(\mathbb{H}^{n}{})$ of
$\mathfrak{C}$-valued functions on $\mathbb{H}^{n}{}$ with the property:

(7.64) $f(s+s^{\prime},h,y)=e^{\varepsilon hs^{\prime}}f(s,x,y),\qquad\text{
for all }(s,x,y)\in\mathbb{H}^{n}{},\ s^{\prime}\in\mathbb{R}{},$

and the square integrability condition (7.7). It is invariant under the left
shifts and we restrict the left group action to
$F_{h}^{\varepsilon}{}(\mathbb{H}^{n}{})$.

There is an unimodular $\mathfrak{C}$-valued function on the Heisenberg group
parametrised by a point $(h,q,p)\in\mathbb{R}^{2n+1}{}$:

$E_{(h,q,p)}(s,x,y)=e^{2\pi(\varepsilon
s\hslash+\mathrm{i}xq+\mathrm{i}yp)}=e^{2\pi\mathrm{i}(xq+yp)}(1+\varepsilon
sh).$

This function, if used instead of the ordinary exponent, produces a
modification $\mathcal{F}_{c}$ of the Fourier transform (7.32). The transform
intertwines the left regular representation with the following action on
$\mathfrak{C}$-valued functions on the phase space:

(7.65) ${\rho^{\varepsilon}_{h}}(s,x,y):f(q,p)\mapsto
e^{-2\pi\mathrm{i}(xq+yp)}(f(q,p)+\varepsilon
h(sf(q,p)+\frac{y}{2\pi\mathrm{i}}f^{\prime}_{q}(q,p)-\frac{x}{2\pi\mathrm{i}}f^{\prime}_{p}(q,p))).$

######  Remark 7.19.

Comparing the traditional infinite-dimensional (7.17) and one-dimensional
(7.33) representations of $\mathbb{H}^{n}{}$ we can note that the properties
of the representation (7.65) are a non-trivial mixture of the former:

  1. (i)

The action (7.65) is non-commutative, similarly to the quantum representation
(7.17) and unlike the classical one (7.33). This non-commutativity will
produce the Hamilton equations below in a way very similar to Heisenberg
equation, see Rem. 7.21.

  2. (ii)

The representation (7.65) does not change the support of a function $f$ on the
phase space, similarly to the classical representation (7.33) and unlike the
quantum one (7.17). Such a localised action will be responsible later for an
absence of an interference in classical probabilities.

  3. (iii)

The parabolic representation (7.65) can not be derived from either the
elliptic (7.17) or hyperbolic (7.52) by the plain substitution $h=0$.

We may also write a classical Schrödinger type representation. According to §
3.2 we get a representation formally very similar to the elliptic (7.34) and
hyperbolic versions (7.53):

$\displaystyle[{\rho^{\varepsilon}_{\chi}}(s^{\prime},x^{\prime},y^{\prime})f](x)$
$\displaystyle=$ $\displaystyle e^{-\varepsilon
h(s^{\prime}+xy^{\prime}-x^{\prime}y^{\prime}/2)}f(x-x^{\prime})$
$\displaystyle=$ $\displaystyle(1-\varepsilon
h(s^{\prime}+xy^{\prime}-\textstyle\frac{1}{2}x^{\prime}y^{\prime}))f(x-x^{\prime}).$

However due to nilpotency of $\varepsilon$ the (complex) Fourier transform
$x\mapsto q$ produces a different formula for parabolic Schrödinger type
representation in the configuration space, cf. (7.35) and (7.58):

$[{\rho^{\varepsilon}_{\chi}}(s^{\prime},x^{\prime},y^{\prime})\hat{f}](q)=e^{2\pi\mathrm{i}x^{\prime}q}\left(\left(1-\varepsilon
h(s^{\prime}-{\textstyle\frac{1}{2}}x^{\prime}y^{\prime})\right)\hat{f}(q)+\frac{\varepsilon
hy^{\prime}}{2\pi\mathrm{i}}\hat{f}^{\prime}(q)\right).$

This representation shares all properties mentioned in Rem. 7.19 as well.

#####  7.6.2. Hamilton Equation

The identity $e^{\varepsilon t}-e^{-\varepsilon t}=2\varepsilon t$ can be
interpreted as a parabolic version of the sine function, while the parabolic
cosine is identically equal to one, cf. § 3.1 and [HerranzOrtegaSantander99a,
Kisil07a]. From this we obtain the parabolic version of the commutator
(7.3.2):

$\displaystyle[k^{\prime},k]\hat{{}_{s}}(\varepsilon h,x,y)$ $\displaystyle=$
$\displaystyle\varepsilon h\int_{\mathbb{R}^{2n}{}}(xy^{\prime}-yx^{\prime})$
$\displaystyle{}\times\,\hat{k}^{\prime}_{s}(\varepsilon
h,x^{\prime},y^{\prime})\,\hat{k}_{s}(\varepsilon
h,x-x^{\prime},y-y^{\prime})\,dx^{\prime}dy^{\prime},$

for the partial parabolic Fourier-type transform $\hat{k}_{s}$ of the kernels.
Thus the parabolic representation of the dynamical equation (7.23) becomes:

(7.67) $\varepsilon h\frac{d\hat{f}_{s}}{dt}(\varepsilon h,x,y;t)=\varepsilon
h\int_{\mathbb{R}^{2n}{}}(xy^{\prime}-yx^{\prime})\,\hat{H}_{s}(\varepsilon
h,x^{\prime},y^{\prime})\,\hat{f}_{s}(\varepsilon
h,x-x^{\prime},y-y^{\prime};t)\,dx^{\prime}dy^{\prime},$

Although there is no possibility to divide by $\varepsilon$ (since it is a
zero divisor) we can obviously eliminate $\varepsilon h$ from the both sides
if the rest of the expressions are real. Moreover this can be done “in
advance” through a kind of the antiderivative operator considered in
[Kisil02e]*(4.1). This will prevent “imaginary parts” of the remaining
expressions (which contain the factor $\varepsilon$) from vanishing.

######  Remark 7.20.

It is noteworthy that the Planck constant completely disappeared from the
dynamical equation. Thus the only prediction about it following from our
construction is $h\neq 0$, which was confirmed by experiments, of course.

Using the duality between the Lie algebra of $\mathbb{H}^{n}{}$ and the phase
space we can find an adjoint equation for observables on the phase space. To
this end we apply the usual Fourier transform $(x,y)\mapsto(q,p)$. It turn to
be the Hamilton equation [Kisil02e]*(4.7). However the transition to the phase
space is more a custom rather than a necessity and in many cases we can
efficiently work on the Heisenberg group itself.

######  Remark 7.21.

It is noteworthy, that the non-commutative representation (7.65) allows to
obtain the Hamilton equation directly from the commutator
$[{\rho^{\varepsilon}_{h}}(k_{1}),{\rho^{\varepsilon}_{h}}(k_{2})]$. Indeed
its straightforward evaluation will produce exactly the above expression. On
the contrast such a commutator for the commutative representation (7.33) is
zero and to obtain the Hamilton equation we have to work with an additional
tools, e.g. an anti-derivative [Kisil02e]*(4.1).

######  Example 7.22.

  1. (i)

For the harmonic oscillator in Example 7.6 the equation (7.67) again reduces
to the form (7.25) with the solution given by (7.26). The adjoint equation of
the harmonic oscillator on the phase space is not different from the quantum
written in Example 7.10(i). This is true for any Hamiltonian of at most
quadratic order.

  2. (ii)

For non-quadratic Hamiltonians classical and quantum dynamics are different,
of course. For example, the cubic term of $\partial_{s}$ in the equation
(7.28) will generate the factor $\varepsilon^{3}=0$ and thus vanish. Thus the
equation (7.67) of the unharmonic oscillator on $\mathbb{H}^{n}{}$ becomes:

$\dot{f}=\left(mk^{2}y\frac{\partial}{\partial x}+\frac{\lambda
y}{2}\frac{\partial^{2}}{\partial x^{2}}-\frac{1}{m}x\frac{\partial}{\partial
y}\right)f.$

The adjoint equation on the phase space is:

$\dot{f}=\left(\left(mk^{2}q+\frac{\lambda}{2}q^{2}\right)\frac{\partial}{\partial
p}-\frac{1}{m}p\frac{\partial}{\partial q}\right)f.$

The last equation is the classical Hamilton equation generated by the cubic
potential (7.27). Qualitative analysis of its dynamics can be found in many
textbooks [Arnold91]*§ 4.C, Pic. 12 [PercivalRichards82]*§ 4.4.

######  Remark 7.23.

We have obtained the _Poisson bracket_ from the commutator of convolutions on
$\mathbb{H}^{n}{}$ without any quasiclassical limit $h\rightarrow 0$. This has
a common source with the deduction of main calculus theorems in
[CatoniCannataNichelatti04] based on dual numbers. As explained in
[Kisil05a]*Rem. 6.9 this is due to the similarity between the parabolic unit
$\varepsilon$ and the infinitesimal number used in non-standard analysis
[Devis77]. In other words, we never need to take care about terms of order
$O(h^{2})$ because they will be wiped out by $\varepsilon^{2}=0$.

An alternative derivation of classical dynamics from the Heisenberg group is
given in the recent paper [Low09a].

#####  7.6.3. Classical probabilities

It is worth to notice that dual numbers are not only helpful in reproducing
classical Hamiltonian dynamics, they also provide the classic rule for
addition of probabilities. We use the same formula (7.29) to calculate kernels
of the states. The important difference now that the representation (7.65)
does not change the support of functions. Thus if we calculate the correlation
term $\left\langle v_{1},{\rho}(g)v_{2}\right\rangle$ in (7.30), then it will
be zero for every two vectors $v_{1}$ and $v_{2}$ which have disjoint supports
in the phase space. Thus no interference similar to quantum or hyperbolic
cases (Subsection 7.3.3) is possible.

#####  7.6.4. Ladder Operator for the Nilpotent Subgroup

Finally we look for ladder operators for the Hamiltonian
$\tilde{B}+\tilde{Z}/2$ or, equivalently, $-\tilde{B}+\tilde{Z}/2$. It can be
identified with a free particle [Wulfman10a]*§ 3.8.

We can look for ladder operators in the representation (7.12–7.13) within the
Lie algebra $\mathfrak{h}_{1}$ in the form
$L_{\varepsilon}^{\\!\pm}=a\tilde{X}+b\tilde{Y}$. This is possible if and only
if

(7.68) $-b=\lambda a,\quad 0=\lambda b.$

The compatibility condition $\lambda^{2}=0$ implies $\lambda=0$ within complex
numbers. However such a “ladder” operator produces only the zero shift on the
eigenvectors, cf. (3.15).

Another possibility appears if we consider the representation of the
Heisenberg group induced by dual-valued characters. On the configuration space
such a representation is [Kisil10a]*(4.11):

(7.69)
$[{\rho^{\varepsilon}_{\chi}}(s,x,y)f](q)=e^{2\pi\mathrm{i}xq}\left(\left(1-\varepsilon
h(s-{\textstyle\frac{1}{2}}xy)\right)f(q)+\frac{\varepsilon
hy}{2\pi\mathrm{i}}f^{\prime}(q)\right).$

The corresponding derived representation of $\mathfrak{h}_{1}$ is

(7.70)
${\rho^{p}_{h}}(X)=2\pi\mathrm{i}q,\qquad{\rho^{p}_{h}}(Y)=\frac{\varepsilon
h}{2\pi\mathrm{i}}\frac{d}{dq},\qquad{\rho^{p}_{h}}(S)=-\varepsilon hI.$

However the Shale–Weil extension generated by this representation is
inconvenient. It is better to consider the FSB–type parabolic representation
(7.65) on the phase space induced by the same dual-valued character. Then the
derived representation of $\mathfrak{h}_{1}$ is:

(7.71) ${\rho^{p}_{h}}(X)=-2\pi\mathrm{i}q-\frac{\varepsilon
h}{4\pi\mathrm{i}}\partial_{p},\qquad{\rho^{p}_{h}}(Y)=-2\pi\mathrm{i}p+\frac{\varepsilon
h}{4\pi\mathrm{i}}\partial_{q},\qquad{\rho^{p}_{h}}(S)=\varepsilon hI.$

An advantage of the FSB representation is that the derived form of the
parabolic Shale–Weil representation coincides with the elliptic one (7.19).

Eigenfunctions with the eigenvalue $\mu$ of the parabolic Hamiltonian
$\tilde{B}+\tilde{Z}/2=q\partial_{p}$ have the form

(7.72) $v_{\mu}(q,p)=e^{\mu p/q}f(q),\text{ with an arbitrary function }f(q).$

The linear equations defining the corresponding ladder operator
$L_{\varepsilon}^{\\!\pm}=a\tilde{X}+b\tilde{Y}$ in the algebra
$\mathfrak{h}_{1}$ are (7.68). The compatibility condition $\lambda^{2}=0$
implies $\lambda=0$ within complex numbers again. Admitting dual numbers we
have additional values $\lambda=\pm\varepsilon\lambda_{1}$ with
$\lambda_{1}\in\mathbb{C}{}$ with the corresponding ladder operators

$L_{\varepsilon}^{\\!\pm}=\tilde{X}\mp\varepsilon\lambda_{1}\tilde{Y}=-2\pi\mathrm{i}q-\frac{\varepsilon
h}{4\pi\mathrm{i}}\partial_{p}\pm
2\pi\varepsilon\lambda_{1}\mathrm{i}p=-2\pi\mathrm{i}q+\varepsilon\mathrm{i}(\pm
2\pi\lambda_{1}p+\frac{h}{4\pi}\partial_{p}).$

For the eigenvalue $\mu=\mu_{0}+\varepsilon\mu_{1}$ with $\mu_{0}$,
$\mu_{1}\in\mathbb{C}{}$ the eigenfunction (7.72) can be rewritten as:

(7.73) $v_{\mu}(q,p)=e^{\mu
p/q}f(q)=e^{\mu_{0}p/q}\left(1+\varepsilon\mu_{1}\frac{p}{q}\right)f(q)$

due to the nilpotency of $\varepsilon$. Then the ladder action of
$L_{\varepsilon}^{\\!\pm}$ is
$\mu_{0}+\varepsilon\mu_{1}\mapsto\mu_{0}+\varepsilon(\mu_{1}\pm\lambda_{1})$.
Therefore these operators are suitable for building
$\mathfrak{sp}_{2}$-modules with a one-dimensional chain of eigenvalues.

Finally, consider the ladder operator for the same element $B+Z/2$ within the
Lie algebra $\mathfrak{sp}_{2}$, cf. § 3.3.3. There is the only operator
$L_{p}^{\\!\pm}=\tilde{B}+\tilde{Z}/2$ corresponding to complex coefficients,
which does not affect the eigenvalues. However the dual numbers lead to the
operators

$L_{\varepsilon}^{\\!\pm}=\pm\varepsilon\lambda_{2}\tilde{A}+\tilde{B}+\tilde{Z}/2=\pm\frac{\varepsilon\lambda_{2}}{2}\left(q\partial_{q}-p\partial_{p}\right)+q\partial_{p},\qquad\lambda_{2}\in\mathbb{C}{}.$

These operator act on eigenvalues in a non-trivial way.

#####  7.6.5. Similarity and Correspondence

We wish to summarise our findings. Firstly, the appearance of hypercomplex
numbers in ladder operators for $\mathfrak{h}_{1}$ follows exactly the same
pattern as was already noted for $\mathfrak{sp}_{2}$, see Rem. 3.8:

  * •

the introduction of complex numbers is a necessity for the _existence_ of
ladder operators in the elliptic case;

  * •

in the parabolic case we need dual numbers to make ladder operators _useful_ ;

  * •

in the hyperbolic case double numbers are not required neither for the
existence or for the usability of ladder operators, but they do provide an
enhancement.

In the spirit of the Similarity and Correspondence Principle 3.5 we have the
following extension of Prop. 3.9:

######  Proposition 7.24.

Let a vector $H\in\mathfrak{sp}_{2}$ generates the subgroup $K$, $N^{\prime}$
or $A\\!^{\prime}$, that is $H=Z$, $B+Z/2$, or $2B$ respectively. Let $\iota$
be the respective hypercomplex unit. Then the ladder operators $L^{\\!\pm}$
satisfying to the commutation relation:

$[H,L_{2}^{\\!\pm}]=\pm\iota L^{\\!\pm}$

are given by:

  1. (i)

Within the Lie algebra $\mathfrak{h}_{1}$:
$L^{\\!\pm}=\tilde{X}\mp\iota\tilde{Y}.$

  2. (ii)

Within the Lie algebra $\mathfrak{sp}_{2}$:
$L_{2}^{\\!\pm}=\pm\iota\tilde{A}+\tilde{E}$. Here $E\in\mathfrak{sp}_{2}$ is
a linear combination of $B$ and $Z$ with the properties:

     * •

$E=[A,H]$.

     * •

$H=[A,E]$.

     * •

Killings form $K(H,E)$ [Kirillov76]*§ 6.2 vanishes.

Any of the above properties defines the vector $E\in\mathop{\operator@font
span}\nolimits\\{B,Z\\}$ up to a real constant factor.

It is worth continuing this investigation and describing in details hyperbolic
and parabolic versions of FSB spaces.

###  8\. Open Problems

A reader may already note numerous objects and results, which deserve a
further consideration. It may also worth to state some open problems
explicitly. In this section we indicate several directions for further work,
which go through four main areas described in the paper.

####  8.1. Geometry

Geometry is most elaborated area so far, yet many directions are waiting for
further exploration.

  1. (i)

Möbius transformations (1.1) with three types of hypercomplex units appear
from the action of the group $SL_{2}{}(\mathbb{R}{})$ on the homogeneous space
$SL_{2}{}(\mathbb{R}{})/H$ [Kisil09c], where $H$ is any subgroup $A$, $N$, $K$
from the Iwasawa decomposition (1.3). Which other actions and hypercomplex
numbers can be obtained from other Lie groups and their subgroups?

  2. (ii)

Lobachevsky geometry of the upper half-plane is extremely beautiful and well-
developed subject [Beardon05a] [CoxeterGreitzer]. However the traditional
study is limited to one subtype out of nine possible: with the complex numbers
for Möbius transformation and the complex imaginary unit used in FSCc (2.8).
The remaining eight cases shall be explored in various directions, notably in
the context of discrete subgroups [Beardon95].

  3. (iii)

The Fillmore-Springer-Cnops construction, see subsection 2.2, is closely
related to the _orbit method_ [Kirillov99] applied to
$SL_{2}{}(\mathbb{R}{})$. An extension of the orbit method from the Lie
algebra dual to matrices representing cycles may be fruitful for semisimple
Lie groups.

  4. (iv)

A development of a discrete version of the geometrical notions can be derived
from suitable discrete groups. A natural first example is the group
$\mathrm{SL}_{2}(\mathbb{F}{})$, where $\mathbb{F}{}$ is a finite field, e.g.
$\mathbb{Z}_{p}{}$ the field of integers modulo a prime $p$.

####  8.2. Analytic Functions

It is known that in several dimensions there are different notions of
analyticity, e.g. several complex variables and Clifford analysis. However,
analytic functions of a complex variable are usually thought to be the only
options in a plane domain. The following seems to be promising:

  1. (i)

Development of the basic components of analytic function theory (the Cauchy
integral, the Taylor expansion, the Cauchy-Riemann and Laplace equations,
etc.) from the same construction and principles in the elliptic, parabolic and
hyperbolic cases and respective subcases.

  2. (ii)

Identification of Hilbert spaces of analytic functions of Hardy and Bergman
types, investigation of their properties. Consideration of the corresponding
Toeplitz operators and algebras generated by them.

  3. (iii)

Application of analytic methods to elliptic, parabolic and hyperbolic
equations and corresponding boundary and initial values problems.

  4. (iv)

Generalisation of the results obtained to higher dimensional spaces. Detailed
investigation of physically significant cases of three and four dimensions.

  5. (v)

There is a current interest in construction of analytic function theory on
discrete sets. Our approach is ready for application to an analytic functions
in discrete geometric set-up outlined in item 8.1.iv above.

####  8.3. Functional Calculus

The functional calculus of a finite dimensional operator considered in Section
6 is elementary but provides a coherent and comprehensive treatment. It shall
be extended to further cases where other approaches seems to be rather
limited.

  1. (i)

Nilpotent and quasinilpotent operators have the most trivial spectrum possible
(the single point $\\{0\\}$) while their structure can be highly non-trivial.
Thus the standard spectrum is insufficient for this class of operators. In
contract, the covariant calculus and the spectrum give complete description of
nilpotent operators—the basic prototypes of quasinilpotent ones. For
quasinilpotent operators the construction will be more complicated and shall
use analytic functions mentioned in 8.2.i.

  2. (ii)

The version of covariant calculus described above is based on the _discrete
series_ representations of $SL_{2}{}(\mathbb{R}{})$ group and is particularly
suitable for the description of the _discrete spectrum_ (note the remarkable
coincidence in the names).

It is interesting to develop similar covariant calculi based on the two other
representation series of $SL_{2}{}(\mathbb{R}{})$: _principal_ and
_complementary_ [Lang85]. The corresponding versions of analytic function
theories for principal [Kisil97c] and complementary series [Kisil05a] were
initiated within a unifying framework. The classification of analytic function
theories into elliptic, parabolic, hyperbolic [Kisil05a, Kisil06a] hints the
following associative chains:

Representations — |  Function Theory — | Type of Spectrum
---|---|---
discrete series — | elliptic — | discrete spectrum
principal series — | hyperbolic — | continuous spectrum
complementary series — | parabolic — | residual spectrum

  3. (iii)

Let $a$ be an operator with $\mathbf{sp}\,a\in\bar{\mathbb{D}{}}$ and
$\left\|a^{k}\right\|<Ck^{p}$. It is typical to consider instead of $a$ the
_power bounded_ operator $ra$, where $0<r<1$, and consequently develop its
$H_{\infty}{}$ calculus. However such a regularisation is very rough and hides
the nature of extreme points of $\mathbf{sp}\,{a}$. To restore full
information a subsequent limit transition $r\rightarrow 1$ of the
regularisation parameter $r$ is required. This make the entire technique
rather cumbersome and many results have an indirect nature.

The regularisation $a^{k}\rightarrow a^{k}/k^{p}$ is more natural and accurate
for polynomially bounded operators. However it cannot be achieved within the
homomorphic calculus Defn. 6.1 because it is not compatible with any algebra
homomorphism. Albeit this may be achieved within the covariant calculus Defn.
6.4 and Bergman type space from 8.2.ii.

  4. (iv)

Several non-commuting operators are especially difficult to treat with
functional calculus Defn. 6.1 or a joint spectrum. For example, deep insights
on joint spectrum of commuting tuples [JTaylor72] refused to be generalised to
non-commuting case so far. The covariant calculus was initiated [Kisil95i] as
a new approach to this hard problem and was later found useful elsewhere as
well. Multidimensional covariant calculus [Kisil04d] shall use analytic
functions described in 8.2.iv.

  5. (v)

As we noted above there is a duality between the co- and contravariant calculi
from Defins. 4.20 and 4.22. We also seen in Section 6 that functional calculus
is an example of contravariant calculus and the functional model is a case of
a covariant one. It is interesting to explore the duality between them
further.

####  8.4. Quantum Mechanics

Due to the space restrictions we only touched quantum mechanics, further
details can be found in [Kisil96a] [Kisil02e] [Kisil05c] [Kisil04a] [Kisil09a]
[Kisil10a]. In general, Erlangen approach is much more popular among
physicists rather than mathematicians. Nevertheless its potential is not
exhausted even there.

  1. (i)

There is a possibility to build representation of the Heisenberg group using
characters of its centre with values in dual and double numbers rather than in
complex ones. This will naturally unifies classical mechanics, traditional QM
and hyperbolic QM [Khrennikov08a]. In particular, a full construction of the
corresponding Fock–Segal–Bargmann spaces would be of interest.

  2. (ii)

Representations of nilpotent Lie groups with multidimensional centres in
Clifford algebras as a framework for consistent quantum field theories based
on De Donder–Weyl formalism [Kisil04a].

######  Remark 8.1.

This work is performed within the “Erlangen programme at large” framework
[Kisil06a, Kisil05a], thus it would be suitable to explain the numbering of
various papers. Since the logical order may be different from chronological
one the following numbering scheme is used:

Prefix |  Branch description
---|---
“0” or no prefix |  Mainly geometrical works, within the classical field of Erlangen programme by F. Klein, see [Kisil05a] [Kisil09c]
“1” |  Papers on analytical functions theories and wavelets, e.g. [Kisil97c]
“2” |  Papers on operator theory, functional calculi and spectra, e.g. [Kisil02a]
“3” |  Papers on mathematical physics, e.g. [Kisil10a]

For example, [Kisil10a] is the first paper in the mathematical physics area.
The present paper [Kisil11c] outlines the whole framework and thus does not
carry a subdivision number. The on-line version of this paper may be updated
in due course to reflect the achieved progress.

### Acknowledgement

Material of these notes was lectured at various summer/winter schools and
advanced courses. Those presentations helped me to clarify ideas and improve
my understanding of the subject. In particular, I am grateful to Prof. S.V.
Rogosin and Dr. A.A. Koroleva for a kind invitation to the Minsk Winter School
in 2010, which were an exciting event. I would like also to acknowledge
support of MAGIC group during my work on those notes.

## References

## Index

  * $\mathrm{\breve{\i}}$ (hypercomplex unit in cycle space) §2.2
  * $\iota$ (hypercomplex unit) §1.1
  * $\breve{\sigma}$ ($\breve{\sigma}:=\mathrm{\breve{\i}}^{2}$) §2.2
  * $A$ subgroup §1.1
  * $A$-orbit §1.1
  * action
    * derived §1.1
    * transitive §1
  * admissible wavelet §4, Example 4.25, Example 4.7
  * affine group, see $ax+b$ group
  * algebra
    * Clifford §2.6, Example 4.12, §7, item 8.4.ii, §8.2
    * homomorphism §6
  * analysis
    * multiresolution Example 4.12
  * analytic
    * contravariant calculus Definition 6.4
    * function on discrete sets item 8.2.v
  * annihilation operator, see ladder operator
  * annulus Example 5.11
  * Atiyah-Singer index theorem §1.1
  * $ax+b$ group §1, Example 4.14, Example 4.27, Example 4.28, Example 4.8, §5.2, Example 5.3, Example 5.7
    * invariant measure Example 4.8
    * representation Example 4.8
  * Bergman
    * space §1.2, §4, Example 4.9, §5.5, item 8.2.ii
  * Blaschke
    * product §6.4
  * boundary effect on the upper half-plane §2.3
  * bracket
    * Moyal §7.3.2, Remark 7.11
      * hyperbolic §7, §7.5.2
    * Poisson Remark 7.23
  * calculus
    * contravariant Definition 4.22, item 8.3.v
      * analytic Definition 6.4
    * covariant Definition 4.20, §6.4, §6.5, §6.5, item 8.3.v
      * Berezin Example 4.17
    * functional §1.2, Definition 4.22, §6, item 8.3.v, §8.3
      * Riesz–Dunford Example 4.21
      * support, see spectrum
    * symbolic, see covariant calculus
    * umbral Remark 4.6
  * cancellative semigroup Remark 4.6
  * Cartan
    * subalgebra Remark 3.7
  * case
    * elliptic §1.1
    * hyperbolic §1.1
    * parabolic §1.1
  * Casimir operator §5.3
  * categorical viewpoint §2.3
  * category theory §2.3
  * Cauchy
    * integral §1.2, Example 4.8, §5.3, §5.4, §5.4, §5.5, §6.1, item 8.2.i
  * Cauchy-Riemann operator §1.2, §5.3, §5.3, Example 5.7, §7.4.1, item 8.2.i
  * causal §7.2.2
  * Cayley transform §5.5
  * centre
    * length from Definition 2.10
    * of a cycle §2.2
  * characteristic
    * function Example 4.19
  * classic
    * Fock–Segal–Bargmann representation §7.6.1
    * probability §7.6.3
    * Schrödinger representation §7.6.1
  * classical mechanics §7.1.1, Remark 7.9
  * Clifford
    * algebra §2.6, Example 4.12, §7, item 8.4.ii, §8.2
  * Clifford algebras §2.6
  * coherent state, see wavelet
  * commutator §7, §7.2.1
  * commutator relation §7.1.1
  * concentric §2.2
  * cone
    * generator §2.2
  * configuration
    * space §7.3.1, §7.5.1, §7.6.1, §7.6.4, Example 7.12
  * conformality §2.6
  * conic
    * section §1.1, §2.2
  * conjugation
    * cycles, of §2.5
  * constant
    * Planck §7, Remark 7.20
  * construction
    * Fillmore–Springer–Cnops §2.2, item 8.1.ii
      * operator §6.5
    * Gelfand–Naimark–Segal §2.4
  * contravariant
    * calculus Definition 4.22, item 8.3.v
      * analytic Definition 6.4
    * spectrum item ii, §6.4, Definition 6.5, Proposition 6.9
      * stability §6.4
    * symbol §4.3
  * convolution §7.2.1
  * correspondence, see principle of similarity and correspondence
  * covariant
    * calculus Definition 4.20, §6.4, §6.5, §6.5, item 8.3.v
      * Berezin Example 4.17
    * pencil §6.5
    * spectral distance Definition 6.15
    * symbol §4.3
    * symbolic calculus Example 4.16
    * transform Definition 4.1
      * induced Definition 5.1
  * covariant transform
    * inverse Definition 4.24
  * creation operator, see ladder operator
  * cycle Definition 2.2, §6.5
    * centre §2.2
    * conjugation §2.5
    * equation §2.2
    * focus §2.2
    * isotropic item i
    * matrix §2.2
    * normalised §2.3
      * $k$- §2.6
      * Kirillov §2.3
    * radius §2.6
    * reflection §2.5
    * space §2.2
    * zero-radius item i, §2.3
  * cycles
    * f-orthogonal §2.6, Definition 2.7
    * focal orthogonal, see f-orthogonal
    * orthogonal §2.2, Definition 2.5
  * De Donder–Weyl formalism item 8.4.ii
  * decomposition
    * Iwasawa §1.1, item 8.1.i
  * defect operator Example 4.19
  * derivation
    * inner §7, §7.2.1
  * derived action §1.1
  * determinant §2.3
  * directing functional Example 4.12
  * directrix §2.2
  * discrete
    * analytic function item 8.2.v
    * geometry item 8.1.iv
    * spectrum item 8.3.ii
  * distance Definition 2.9
    * covariant spectral Definition 6.15
  * divisor
    * zero §1.1, §7.5.5
  * domain
    * non-simply connected §5.5
    * simply connected Corollary 5.10
  * double §1.1
    * number §1.1, 3rd item, §3.1, §3.3.2, §7, §7.5, §7.5.5
  * drum
    * hearing shape §2.3
  * dual §1.1
    * number §1.1, item i—§3.1, §3.1, §3.2, §7, §7.6—§7.6.4
  * eigenvalue §3.3.1, §3.3.2, §6.4, Proposition 6.8
    * generalised §6.5
    * quadratic §6.5
  * elliptic
    * case §1.1
  * EPH-classification §1.1
  * equation
    * Cauchy-Riemann, see Cauchy-Riemann operator
    * dynamics in p-mechanics §7.2.1
    * Hamilton §7, §7.6.2
    * Heisenberg §7, §7.3.2
  * Erlangen programme §1, §1
  * f-orthogonality §2.6, Definition 2.7
  * factorisation
    * Wiener–Hopf Example 4.10
  * fiducial operator Definition 4.1
  * field
    * quantum item 8.4.ii
  * Fillmore–Springer–Cnops construction §2.2, item 8.1.ii
    * operator §6.5
  * focal
    * orthogonality, see f-orthogonality
  * Fock–Segal–Bargmann
    * representation §7.6.4, §7.6.4, Example 7.3, Example 7.5, Example 7.5
      * classic (parabolic) §7.6.1
      * hyperbolic §7.5.1
    * representations §7.3.1
    * space Example 4.12, §7.4.1, §7.6.5, Example 7.5, item 8.4.i
  * focus
    * length from Definition 2.10
    * of a cycle §2.2
    * of a parabola §2.2
  * form
    * symplectic §7.1.1
  * formula
    * reconstruction Example 4.25
  * FSB
    * representation, see Fock–Segal–Bargmann representation
    * space, see Fock–Segal–Bargmann space
  * function
    * characteristic Example 4.19
    * Heaviside §2.4
    * Hermite, see Hermite polynomial
    * maximal Example 4.14
    * parabolic cylinder, see Weber–Hermite function
    * polyanalytic §5.3
    * Weber–Hermite §7.5.4, §7.5.5
  * functional
    * calculus §1.2, Definition 4.22, §6, item 8.3.v, §8.3
      * Riesz–Dunford Example 4.21
      * support, see spectrum
    * directing Example 4.12
    * invariant item iv
    * model Example 4.19, §6, §6.4, item 8.3.v
  * Gaussian Example 4.8, item ii, Example 7.12, Example 7.16
  * Gelfand–Naimark–Segal construction §2.4
  * generalised
    * eigenvalue §6.5
  * generator
    * of a cone §2.2
    * of a subgroup §3.3.1
    * quadratic Example 7.4
  * geometrical quantisation §7
  * geometry §8.1
    * discrete item 8.1.iv
    * Lobachevsky §1.1, Example 4.28, §5.2, item 8.1.ii
    * non-commutative §1.2, §6.5
  * GNS construction §2.4, see Gelfand–Naimark–Segal construction
  * group §1
    * affine, see $ax+b$ group
    * $ax+b$ §1, Example 4.14, Example 4.27, Example 4.28, Example 4.8, §5.2, Example 5.3, Example 5.7
      * invariant measure Example 4.8
    * Heisenberg §1, §7, §7—§7.6.5, item 8.4.i
      * Fock–Segal–Bargmann representation Example 7.3
        * classic (parabolic) §7.6.1
        * hyperbolic §7.5.1
      * induced representation §7.1.1
      * invariant measure §7.1.1
    * representation
      * linear §1.2
    * Schrödinger §7.1.2, §7.4
    * $SL_{2}{}(\mathbb{R}{})$ §1
    * $\mathrm{Sp}(2)$ §7, see also $SL_{2}{}(\mathbb{R}{})$
    * $\mathrm{SU}(1,1)$ Example 4.19, see also $SL_{2}{}(\mathbb{R}{})$
    * symplectic §7, Remark 7.15
  * $\mathbb{H}^{1}{}$, Heisenberg group §7.1.1
  * Haar measure, see invariant measure
  * half-plane §1.1, item ii
    * boundary effect §2.3
  * Hamilton
    * equation §7, §7.6.2
  * Hardy
    * maximal function Example 4.14
    * pairing §4.4, §4.4
    * space §1.2, §4, Example 4.8, §5.3, §5.5, §6.1, Definition 6.15, item 8.2.ii
  * harmonic
    * oscillator item i, item i, item i, §7, §7.1.2, §7.4, Example 7.6
      * repulsive (hyperbolic) §7.5.4
  * hearing drum’s shape §2.3
  * Heaviside
    * function §2.4
  * Heaviside function §2.4
  * Heisenberg
    * commutator relation §7.1.1
    * equation §7, §7.3.2
    * group §1, §7, §7—§7.6.5, item 8.4.i
      * Fock–Segal–Bargmann representation Example 7.3
        * classic (parabolic) §7.6.1
        * hyperbolic §7.5.1
      * induced representation §7.1.1
      * invariant measure §7.1.1
  * Heisenberg group §1
  * Hermite
    * polynomial §7.4, §7.5.5
  * homomorphism
    * algebraic §6
  * hyperbolic
    * case §1.1
    * Fock–Segal–Bargmann representation §7.5.1
    * harmonic oscillator §7.5.4
    * Moyal bracket §7, §7.5.2
    * probability §7.5.3
    * Schrödinger representation §7.5.1
    * unharmonic oscillator item ii
    * unit ($\mathrm{j}$) §1.1
  * hypercomplex
    * number §1.1, item ii, Remark 3.8, §7.6.5
  * $\mathrm{i}$ (imaginary unit) §1.1
  * imaginary
    * unit ($\mathrm{i}$) §1.1
  * induced
    * covariant transform Definition 5.1
    * representation §3, §6.1, §7
      * Heisenberg group,of §7.1.1
  * induction §3
  * inner
    * derivation §7, §7.2.1
  * integral
    * Cauchy §1.2, Example 4.8, §5.3, §5.4, §5.4, §5.5, §6.1, item 8.2.i
    * Poisson, see Poisson kernel
  * interference §7.2.2, §7.6.3, Example 7.13
  * intertwining operator §1.2, Example 4.14, Theorem 4.4, Remark 4.6, Definition 5.1, Proposition 5.5, item i, §6, §6.5
  * invariant §4.4
    * functional item iv
    * joint §2.4, §2.5
    * measure §4, Example 4.25, Example 4.27, Example 4.7, Example 4.8, §7.1.1
    * pairing §4.4
  * inverse
    * covariant transform Definition 4.24
  * isotropic cycles item i
  * Iwasawa decomposition §1.1, item 8.1.i
  * $\mathrm{j}$ (hyperbolic unit) §1.1
  * jet §1.2, §6.3, Theorem 6.17, Definition 6.7, §7
    * bundle §6.2
  * $\mathbb{J}^{n}{}$ (jet space) Definition 6.7
  * joint
    * invariant §2.4, §2.5
    * spectrum §6.1
  * Jordan
    * normal form §6.3
  * $K$ subgroup §1.1
  * $k$-normalised cycle §2.6
  * $K$-orbit §1.1
  * kernel
    * Poisson Example 4.8, §5.3
  * Krein
    * directing functional Example 4.12
    * space §3.2, §7
  * ladder operator §3.3.1—§3.3.3, §5.3, §5.3, §5.4, §7.4—§7.4.2, §7.5.4—§7.5.5, §7.6.4—§7.6.5, Example 7.16
  * Laplace operator, see Laplacian
  * Laplacian §1.2, §5.3, §5.3, item 8.2.i
  * left regular representation §7.1.1
  * length Definition 2.10
    * from centre Definition 2.10
    * from focus Definition 2.10
  * Lidskii theorem Theorem 6.14
  * limit
    * semiclassical §7, Remark 7.9
  * Littlewood–Paley
    * operator Example 4.13
  * Lobachevsky
    * geometry §1.1, Example 4.28, §5.2, item 8.1.ii
  * lowering operator, see ladder operator
  * map
    * Möbius §1, Example 4.19, Example 4.9, §5.2, §5.5, §6.4, Proposition 6.18, Proposition 6.23
      * on cycles Theorem 2.4
  * matrix
    * cycle, of a §2.2
    * Jordan normal form §6.3
  * maximal
    * function Example 4.14
  * measure
    * Haar, see invariant measure
    * invariant §4, Example 4.25, Example 4.27, Example 4.7, Example 4.8, §7.1.1
  * mechanics
    * classical §7.1.1, Remark 7.9
  * metaplectic representation, see oscillator representation
  * method
    * orbits, of §7, item 8.1.iii
  * minimal
    * polynomial §6.4
  * Möbius map §1, Example 4.19, Example 4.9, §5.2, §5.5, §6.4, Proposition 6.18, Proposition 6.23
    * on cycles Theorem 2.4
  * model
    * functional Example 4.19, §6, §6.4, item 8.3.v
  * mother wavelet Example 4.7, §6.1
  * Moyal bracket §7.3.2, Remark 7.11
    * hyperbolic §7, §7.5.2
  * multiresolution analysis Example 4.12
  * $N$ subgroup §1.1
  * $N$-orbit §1.1
  * nilpotent unit, see parabolic unit
  * non-commutative geometry §1.2, §6.5
  * non-simply connected domain §5.5
  * norm
    * shift invariant Example 4.14
  * normalised
    * cycle §2.3
      * $k$- §2.6
      * Kirillov §2.3
  * number
    * double §1.1, 3rd item, §3.1, §3.3.2, §7, §7.5, §7.5.5
    * dual §1.1, item i—§3.1, §3.1, §3.2, §7, §7.6—§7.6.4
    * hypercomplex §1.1, item ii, Remark 3.8, §7.6.5
  * numerical
    * range Example 4.18
  * observable §7.2.1
  * on hearing drum’s shape §2.3
  * operator
    * annihilation, see ladder operator
    * Casimir §5.3
    * Cauchy-Riemann §1.2, §5.3, §5.3, Example 5.7, §7.4.1, item 8.2.i
    * creation, see ladder operator
    * defect Example 4.19
    * fiducial Definition 4.1
    * Fillmore–Springer–Cnops construction §6.5
    * intertwining §1.2, Example 4.14, Theorem 4.4, Remark 4.6, Definition 5.1, Proposition 5.5, item i, §6, §6.5
    * ladder §3.3.1—§3.3.3, §5.3, §5.3, §5.4, §7.4—§7.4.2, §7.5.4—§7.5.5, §7.6.4—§7.6.5, Example 7.16
    * Laplace, see Laplacian
    * Littlewood–Paley Example 4.13
    * lowering, see ladder operator
    * power bounded item 8.3.iii
    * pseudodifferential §4.3
    * quasinilpotent item 8.3.i
    * raising, see ladder operator
    * Toeplitz item 8.2.ii
  * optics §7.1.2
  * orbit
    * method §7, item 8.1.iii
    * subgroup $A$, of §1.1
    * subgroup $K$, of §1.1
    * subgroup $N$, of §1.1
  * orders
    * of zero Example 6.12
  * orthogonality
    * cycles, of §2.2, Definition 2.5
      * focal, see f-orthogonality
  * oscillator
    * harmonic item i, item i, item i, §7, §7.1.2, §7.4, Example 7.6
      * repulsive (hyperbolic) §7.5.4
    * representation §7.1.2
    * unharmonic item ii, item ii, Example 7.7
      * hyperbolic item ii
  * p-mechanics §7.2
    * dynamic equation §7.2.1
    * observable §7.2.1
    * state §7.2.2
  * p-mechanisation Example 7.6
  * pairing
    * Hardy §4.4, §4.4
    * invariant §4.4
  * parabola
    * directrix §2.2
    * focus §2.2
    * vertex §2.2
  * parabolic
    * case §1.1
    * cylinder function, see Weber–Hermite function
    * Fock–Segal–Bargmann representation §7.6.1
    * probability, see classic probability
    * Schrödinger representation §7.6.1
    * unit ($\varepsilon$) §1.1
  * PDO §4.3
  * pencil
    * covariant §6.5
  * perpendicular Definition 2.12
  * phase
    * space §3.1, §7, §7.3.1, §7.4, §7.5.1, Example 7.12, Example 7.5
  * Planck
    * constant §7, Remark 7.20
  * Plato’s cave §2.2
  * point space §2.2
  * Poisson
    * bracket Remark 7.23
    * kernel Example 4.8, §5.3
  * polyanalytic function §5.3
  * polynomial
    * Hermite §7.4, §7.5.5
    * minimal §6.4
  * power bounded operator item 8.3.iii
  * primary
    * representation §6.2, §6.3, §6.5, Definition 6.5
  * principle
    * similarity and correspondence §3.3.3, Principle 3.5, §7.5.5, §7.6.5
  * probability
    * classic (parabolic) §7.6.3
    * hyperbolic §7.5.3
    * quantum §7, §7.2.2, §7.3.3
  * product
    * Blaschke §6.4
  * projective space §2.2
  * prolongation §6.2
  * pseudodifferential operator §4.3
  * $\mathcal{PT}$-symmetry §7
  * quadratic
    * eigenvalue §6.5
    * generator Example 7.4
  * quantisation
    * geometrical §7
  * quantum
    * field item 8.4.ii
    * probability §7, §7.2.2, §7.3.3
  * quantum mechanics §7
  * quasinilpotent
    * operator item 8.3.i
  * quaternion §2.6
  * quaternions §2.6
  * radius
    * cycle, of §2.6
  * Radon transform Example 4.15
  * raising operator, see ladder operator
  * range
    * numerical Example 4.18
  * reconstruction formula Example 4.25
  * reduced wavelet transform Example 5.2
  * reflection
    * in a cycle §2.5
  * representation
    * $ax+b$ group Example 4.8
    * coefficients, see wavelet transform
    * complementary series §1.2, Example 4.9, item 8.3.ii, item 8.3.ii
    * discrete series §1.2, Example 3.2, §4, Example 4.9, item 8.3.ii, item 8.3.ii
    * Fock–Segal–Bargmann §7.6.4, §7.6.4, Example 7.3, Example 7.5, Example 7.5
      * classic (parabolic) §7.6.1
      * hyperbolic §7.5.1
    * FSB, see Fock–Segal–Bargmann representation
    * Heisenberg group
      * classic (parabolic) §7.6.1
      * hyperbolic §7.5.1
      * Schrödinger Example 7.4
    * induced §3, §6.1, §7
      * Heisenberg group,of §7.1.1
    * left regular §7.1.1
    * linear §1.2
    * metaplectic, see oscillator representation
    * oscillator §7.1.2
    * primary §6.2, §6.3, §6.5, Definition 6.5
    * principal series §1.2, Example 4.9, item 8.3.ii, item 8.3.ii
    * Schrödinger §7.3.1, §7.3.1, §7.4.1, Example 7.18, Example 7.4
      * classic (parabolic) §7.6.1
      * hyperbolic §7.5.1
    * Shale–Weil, see oscillator representation
    * $SL_{2}{}(\mathbb{R}{})$ group §3.2, Example 3.2—Example 3.2
      * in Banach space §6.1
    * square integrable §4, Example 4.25, Example 4.26, Example 4.7, Example 5.2
  * representations
    * Fock–Segal–Bargmann §7.3.1
    * linear §3
  * repulsive
    * harmonic oscillator §7.5.4
  * resolvent Example 4.21, §6.1, §6.5, Definition 6.2
  * Riemann
    * mapping theorem §5.5
  * Schrödinger
    * group §7.1.2, §7.4
    * representation §7.3.1, §7.3.1, §7.4.1, Example 7.18, Example 7.4
      * classic (parabolic) §7.6.1
      * hyperbolic §7.5.1
  * sections
    * conic §1.1, §2.2
  * semiclassical
    * limit §7, Remark 7.9
  * semigroup
    * cancellative Remark 4.6
  * Shale–Weil representation, see oscillator representation
  * similarity, see principle of similarity and correspondence
  * simply connected domain Corollary 5.10
  * $SL_{2}{}(\mathbb{R}{})$ group §1
    * representation §3.2, Example 3.2—Example 3.2
      * in Banach space §6.1
  * $\mathrm{Sp}(2)$ §7.1.2
  * space
    * Bergman §1.2, §4, Example 4.9, §5.5, item 8.2.ii
    * configuration §7.3.1, §7.5.1, §7.6.1, §7.6.4, Example 7.12
    * cycles, of §2.2
    * Fock–Segal–Bargmann Example 4.12, §7.4.1, §7.6.5, Example 7.5, item 8.4.i
    * FSB, see Fock–Segal–Bargmann space
    * Hardy §1.2, §4, Example 4.8, §5.3, §5.5, §6.1, Definition 6.15, item 8.2.ii
    * Krein §3.2, §7
    * phase §3.1, §7, §7.3.1, §7.4, §7.5.1, Example 7.12, Example 7.5
    * point §2.2
    * projective §2.2
  * spectral
    * covariant distance Definition 6.15
  * spectrum §6.2, Definition 6.2
    * contravariant item ii, §6.4, Definition 6.5, Proposition 6.9
      * stability §6.4
    * discrete item 8.3.ii
    * joint §6.1
    * mapping Theorem 6.3
  * square integrable
    * representation §4, Example 4.25, Example 4.26, Example 4.7, Example 5.2
  * stability
    * contravariant spectrum §6.4
  * state §7.2.2
    * coherent, see wavelet
    * vacuum, see mother wavelet
  * subalgebra
    * Cartan Remark 3.7
  * subgroup
    * $A$ §1.1, Example 2.1
      * orbit §1.1
    * generator §3.3.1
    * $K$ §1.1, Example 2.1
      * orbit §1.1
    * $N$ §1.1, Example 2.1
      * orbit §1.1
  * supersymmetry Remark 3.6
  * support
    * functional calculus, see spectrum
  * symbol
    * contravariant §4.3
    * covariant §4.3
  * symbolic calculus, see covariant calculus
    * covariant Example 4.16
  * symplectic
    * form §7.1.1
    * group §7, Remark 7.15
    * transformation §3.1, §3.1, §7, §7.5.4
  * theorem
    * Atiyah-Singer (index) §1.1
    * Lidskii Theorem 6.14
    * Riemann mapping §5.5
    * spectral mapping Theorem 6.3
  * Toeplitz
    * operator item 8.2.ii
  * token Remark 4.6
  * trace §2.3
  * transform
    * Cayley §5.5
    * covariant Definition 4.1
      * induced Definition 5.1
      * inverse Definition 4.24
    * Radon Example 4.15
    * wavelet, see wavelet transform
  * transformation
    * symplectic §3.1, §3.1, §7, §7.5.4
  * transitive §1
  * umbral calculus Remark 4.6
  * unharmonic
    * oscillator item ii, item ii, Example 7.7
      * hyperbolic item ii
  * unit
    * circle §5.5
    * disk §5.5
    * hyperbolic ($\mathrm{j}$) §1.1
    * imaginary ($\mathrm{i}$) §1.1
    * nilpotent, see parabolic unit
    * parabolic ($\varepsilon$) §1.1
  * vacuum state, see mother wavelet
  * vertex
    * of a parabola §2.2
  * wavelet §4, §4.1, Example 4.7, §5.1, §5.2, §6.1
    * admissible §4, Example 4.25, Example 4.7
    * mother Example 4.7, §6.1
    * transform Example 4.7, §6.1, §7.4.1
      * induced, see induced covariant transform
      * reduced Example 5.2
  * Weber–Hermite function §7.5.4, §7.5.5
  * Wiener–Hopf factorisation Example 4.10
  * zero
    * divisor §1.1, §7.5.5
    * order, of Example 6.12
  * zero-radius cycle item i, §2.3
  * $\varepsilon$ (parabolic unit) §1.1
  * $\sigma$ ($\sigma:=\iota^{2}$) §1.1