# Can the zero-point energy of the quantized harmonic oscillator be lower?
Possible implications for the physics of “dark energy” and “dark matter”

H. A. Kastrup<EMAIL_ADDRESS>DESY Hamburg, Theory Group, Notkestrasse
85, D-22607 Hamburg, Germany

###### Abstract

Replacing the canonical pair $q$ and $p$ of the classical harmonic oscillator
(HO) by the _locally and symplectically_ equivalent pair angle $\varphi$ and
action variable $I$ implies a qualitative change of the global topological
structure of the associated phase spaces: the pair $(q,p)$ is an element of a
topologically trivial plane $\mathbb{R}^{2}$ whereas the pair
$(\varphi,I>0)\in\mathbb{S}^{1}\times\mathbb{R}_{+}$ is an element of a
topologically non-trivial, infinitely connected, $punctured$ plane
$\mathbb{R}^{2}-\\{0\\}$, which has the orthochronous “Lorentz” group
$SO^{\uparrow}(1,2)$ (or its two-fold covering, the symplectic group
$Sp(2,\mathbb{R})$) as its “canonical” group. Due to its infinitely many
covering groups the resulting (“symplectic”) spectrum of the associated
quantum Hamiltonian $H=\omega\,\hat{I}$ is given by
$\\{\hbar\omega(n+b),n=0,1,\ldots;\,b\in(0.1],\text{e.g.\
$b=1/s,\,s\in\mathbb{N}$ and large}\\}$, in contrast to the $(q,p)$ version,
where the Hamiltonian has the “orthodox” spectrum $\\{\hbar\omega(n+1/2)\\}$.
The deeper reason for the difference is that for the description of the
periodic orbit $\\{p=p(q)\\}$ one covering of $\mathbb{S}^{1}$ suffices,
whereas one generally needs many coverings for the time evolution
$\varphi(t)$. And this, in turn, can lead to a lowering of the zero-point
energies.

Several theoretical and possible experimental implications of the “symplectic”
spectra of the HO are discussed: The potentially most important ones concern
the vibrations of diatomic molecules in the infrared, e.g. those of molecular
hydrogen H2. Those symplectic spectra of the HO may provide a simultaneous key
to two outstanding astrophysical puzzles, namely the nature of dark (vacuum)
energy and that of dark matter: To the former because the zero-point energy
$b\,\hbar\omega$ of free electromagnetic wave oscillator modes can be
extremely small $>0$ $(b\approx\exp{(-35)}$ for the measured dark energy
density). And a key to the dark matter problem because the quantum zero-point
energies of the electronic Born-Oppenheimer potentials in which the two nuclei
of H2 or the nuclei of other primordial diatomic molecules vibrate can be
lower, too, and, therefore, may lead to spectrally detuned “dark” H2 molecules
during the “Dark Ages” of the universe and forming WIMPs in the hypothesized
sense! All results appear to be in surprisingly good agreement with the
$\Lambda$CDM model of the universe.

Besides laboratory experiments the search for 21-cm radio signals from the
Dark Ages of the universe and other astrophysical observations can help to
explore those hypothetical implications.

###### Contents

  1. I Introduction
  2. II Motions on the classical  phase spaces $\mathcal{S}_{\varphi,I}$ and $\mathcal{S}_{q,p}$
    1. II.1 Coordinates and orbits on $\mathcal{S}_{\varphi,I}$
      1. II.1.1 Global coordinates
      2. II.1.2 Orbits on $\mathcal{S}_{\varphi,I}$
      3. II.1.3 Time evolution
    2. II.2 Orbits on $\mathcal{S}_{q,p}$
    3. II.3 The frequency $\omega$ as an external field
  3. III Quantum mechanics of the phase space $\mathcal{S}_{\varphi,I}$
    1. III.1 Basics: self-adjoint representations of the  three Lie algebra generators $K_{j}$ of  the symplectic group $Sp(2,\mathbb{R})$
    2. III.2 Time evolution
      1. III.2.1 Heisenberg picture
      2. III.2.2 Schrödinger Picture
    3. III.3 Relationship between the operators $\tilde{K}_{j}$ and the conventional operators $Q$ and $P$
    4. III.4 The model $H=\hbar\,\omega(\tilde{K}_{0}+g\,\tilde{K}_{1})$
      1. III.4.1 Transition matrix elements with respect to the number states in 1st order
      2. III.4.2 Exact eigenvalues of $\tilde{C}_{g}(K)$
    5. III.5 Explicit Hilbert spaces for $\tilde{K}_{j},j=0,1,2$ and $\tilde{C}_{g}(K)=\tilde{K}_{0}+g\tilde{K}_{1}$, spectra and eigenfunctions
      1. III.5.1 Hardy space on the unit circle
      2. III.5.2 Hardy space related Hilbert spaces for general $b\in(0,1]$
      3. III.5.3 A unitary transformation by a change of basis
      4. III.5.4 Aharonov-Bohm-, (fractional) quantum Hall-effects, anyons, Berry’s phase,  Bloch waves etc.
      5. III.5.5 The operator $\tilde{C}_{g}(K)=\tilde{K}_{0}+g\tilde{K}_{1}$ on $H^{2}_{b,+}$
    6. III.6 Nonlinear interactions in terms  of the operators $\tilde{K}_{j}$
      1. III.6.1 The Morse potential for molecular vibrations
      2. III.6.2 Potentials involving the terms $\tilde{K}_{+}\tilde{K}_{-}$ and $\tilde{K}_{-}\tilde{K}_{+}$
  4. IV Reflections on possible experiments  and observations
    1. IV.1 Generalities
    2. IV.2 Vibrating diatomic molecules
    3. IV.3  Vibrations of diatomic molecules  with different isotopic atoms
    4. IV.4 Interferences of time dependent energy eigenstates
    5. IV.5 Transitions associated with the Hamiltonian $H(K)=\hbar\,\omega\,\tilde{C}_{g}(K)$
    6. IV.6 Traps for neutral molecules and optomechanics
    7. IV.7 Perelomov coherent states
    8. IV.8 (Dispersive) van der Waals forces
  5. V Possible astrophysical implications
    1. V.1 Dark energy and the cosmological constant
    2. V.2 “Dark” b-H2 and other primordial molecules  as dark matter?
  6. A Hilbert space for $\tilde{K}_{j}$ and $\tilde{C}_{g}(K)$ on $\mathbb{R}_{0}^{+}$
    1. A.1 Representation of the Lie algebra generators $\tilde{K}_{j}$ on $L^{2}(du,\mathbb{R}^{+}_{0})$
    2. A.2 Eigenfunctions and spectrum of $\tilde{C}_{g}=\tilde{K}_{0}+g\,\tilde{K}_{1}$
  7. B Notational conventions for the states of diatomic molecules
    1. B.1 Electronic motions
    2. B.2 Nuclear motions

##  I Introduction

It very probably appears presumptuous and provocative to question the well-
known quantum properties of the primeval prototype of quantum mechanical
systems: the harmonic oscillator (HO in the following)!

The motive for daring a new look at the physical system HO arise from its
well-known locally - but not globally - equivalent canonical descriptions:
either in terms of the Cartesian coordinates $(q,p)\in\mathbb{R}^{2}$ or in
terms of the angle and action variables
$(\varphi,I)\in\\{\mathbb{R}\bmod{2\pi}\times\mathbb{R}^{+}\\}\cong\mathbb{S}^{1}\times\mathbb{R}^{+}\cong\mathbb{R}^{2}-\\{0\\}$,
where $x\in\mathbb{R}^{+}$ iff $x\in\mathbb{R}$ and $x>0$, the relationship of
which can be defined by kas (a)

$q(\varphi,I)=\sqrt{\frac{2\,I}{m\,\omega}}\,\cos\varphi\,,~{}~{}p(\varphi,I)=-\sqrt{2\,m\,\omega\,I}\,\sin\varphi\,.$
(1)

This mapping is locally symplectic:

$dq\wedge dp=d\varphi\wedge
dI\,,~{}~{}\text{or}~{}~{}\frac{\partial(q,p)}{\partial(\varphi,I)}=1\,.$ (2)

The canonically equivalent Hamiltonians are given by

$H(q,p)=\frac{1}{2m}\,p^{2}+\frac{1}{2}\,m\,\omega^{2}\,q^{2}=H(\varphi,I)=\omega\,I\,,$
(3)

with their respective canonical Eqs. of motion

$\displaystyle\dot{q}=$ $\displaystyle\frac{\partial H}{\partial
p}=p/m,~{}~{}~{}\dot{p}=-\frac{\partial H}{\partial q}=-m\omega^{2}q\,;$ (4)
$\displaystyle\dot{\varphi}=$ $\displaystyle\frac{\partial H}{\partial
I}=\omega,~{}~{}~{}\dot{I}=-\frac{\partial H}{\partial\varphi}=0,$ (5)

the latter with the obvious solutions

$\varphi(t)=\omega\,t+\varphi_{0}\,,~{}~{}I=\text{
const.}=I_{0}=E/\omega>0\,.$ (6)

The solutions of the Eqs. (4) and (5) describe - as functions of time $t$ \-
orbits in the repective phase spaces

$\displaystyle\mathcal{S}_{q,p}$ $\displaystyle=$
$\displaystyle\\{(q,p)\in\mathbb{R}^{2}\,\\}\cong\mathbb{R}^{2}\,$ (7)
$\displaystyle\text{ and }~{}~{}\mathcal{S}_{\varphi,I}$ $\displaystyle=$
$\displaystyle\\{(\varphi,I),\varphi\in\mathbb{R}\bmod{2\pi}\,,\,I>0\,\\}$
$\displaystyle\cong S^{1}\times\mathbb{R}^{+}\cong\mathbb{R}^{2}-\\{0\\}\,.$

The crucial point - for all what follows in the present paper - is this: the
two phase spaces (7) and (I) are globally (topologically) qualitatively
different! Whereas the $(q,p)$-space is a simply connected and topologically
trivial plane $\mathbb{R}^{2}$, the $(\varphi,I)$-space is topologically a
“punctured” plane, i.e. a $\mathbb{R}^{2}$ with the origin $\\{(0,0)\\}$
deleted!

This is so for several reasons: the variables $\varphi$ and $I$ can be
considered as polar coordinates of a plane, with - obviously - $\varphi$ the
angle and $I>0$ the radial variable. The value $I=0$ has to be excluded
because otherwise the angle $\varphi$ becomes undefined at that point. In
addition the value $I=0$ describes a branch point for the transformation
functions (1). More arguments can be found in Ref. Kastrup (2007).

The topology of the phase space (I) may equivalently be characterized as that
of a simple cone with the tip deleted or as that of a semi-cylinder without
the points of the finite circular surface at $I=0$.

If $\varphi(t)$ and $I(t)$ describe the moving points of a periodic orbit on
$\mathcal{S}_{\varphi,I}$ then those points may loop around the origin
arbitrarily many times (the first homotopy group $\pi_{1}$ of
$\mathcal{S}_{\varphi,I}$ consists of the integers $\in\mathbb{Z}$), because
the orbit coordinate $\varphi(t)$ can circle the origin of $\mathbb{S}^{1}$
arbitrarily many times in the course of time $t$! Thus, the configuration
space of $\varphi$ corresponds to one of the infinitely many covering spaces
of the circle $\mathbb{S}^{1}$, the universal covering being the real line
$\mathbb{R}$. A physical example for a high number of coverings is provided by
the oscillations of electromagnetic vibrations.

That missing point in the phase space
$\mathbb{R}^{2}_{0}\equiv\mathcal{S}_{q,p}-\\{0,0\\}$, or in
$\mathcal{S}_{\varphi,I}$, has dramatic consequences for the associated
quantum theory which will be discussed in more detail below.

The crucial result is the following:

The quantum operator version $\hat{I}=K_{0}$ of the classical action variable
$I$ of the HO has the possible spectrum

$K_{0}|b,n\rangle=\hbar(n+b)|b,n\rangle,\,\,n=0,1,2,\ldots;\ 1>b>0.$ (9)

Here $K_{0}$ is the self-adjoint Lie algebra generator of the compact subgroup
O(2) in an irreducible unitary representation of the 3-dimensionsl
“orthochronous Lorentz” group $SO^{\uparrow}(1,2)\cong Sp(2,\mathbb{R})/Z_{2}$
or of one of its infinitely many covering groups, where the double covering
$Sp(2,\mathbb{R})$ is the symplectic group of the plane. (see below). This
means that the HO Hamilton operator

$H_{\varphi,I}=\omega\,K_{0},$ (10)

associated with the phase space $\mathcal{S}_{\varphi,I}$, can have a ground
state ($n=0$) with eigenvalue (zero-point energy) $\hbar\omega\,b,\,1>b>0$,
especially with $1/2>b>0$ !

As indicated above, the mathematical origin of this possibility is a group
theoretical one: The “canonical” transformation group of the punctured plane
$\mathbb{R}^{2}_{0}$ is the symplectic group $Sp(2,\mathbb{R})$ which acts
transitively on $\mathbb{R}^{2}_{0}$ (for any two points on
$\mathbb{R}^{2}_{0}$ there is an element of $Sp(2,\mathbb{R})$ which connects
the two but _leaves the origin invariant_)!

The group $Sp(2,\mathbb{R})$ is a twofold covering of the “orthochronous
Lorentz” group $SO^{\uparrow}(1,2)$ in “one time and two space dimensions”
which acts correspondingly on the phase space $\mathcal{S}_{\varphi,I}$, i.e.
acting transitively and leaving the origin invariant! The number $b,\,1>b>0$
in Eq. (9) characterizes an irreducible unitary representation of a given
covering group of $SO^{\uparrow}(1,2)$ kas (b).

The main mathematical aspects of the present paper have been presented
previously in Refs. Bojowald  _et al._ (2000); Bojowald and Strobl (2000);
Kastrup (2003, 2007); Bojowald and Strobl (2003). A brief - introductory but
probably helpful - summary of them is given in Ref. Kastrup (2011). Essential
mathematical references are Bargmann (1947); Pukánszky (1964); Vilenkin
(1968); P. J. Sally (1967, 1970); Boyer and Wolf (1975).

The present paper tries to draw attention to possible experimental and
observational implications of the $(\varphi,I)$-framework for the HO.
Hopefully, appropriate laboratory experiments and astrophysical obervations
will be able to find out whether nature has “made use of the available
mathematical possibilities”(Dirac) or not!

There are - at least - two immediate crucial questions:

i) Why should the classical canonical pair $(\varphi,I)$ be a “better” - or at
least equivalent - basis for the quantum description of a system like the HO,
compared to the conventional pair $(q,p)$?

ii) If the predictions of the quantized $(\varphi,I)$ framework are richer
than the $(q,p)$ framework - but not contradictory -, why haven’t they been
observed yet?

Ad i): A crucial obstacle for using the “observable” angle $\varphi$ itself
for the quantum description of a physical system has been that there exist no
corresponding self-adjoint operator $\hat{\varphi}$ Kastrup (2006, 2016)! This
shortcoming can, however, be remedied by the following observation Kastrup
(2006, 2016):

Geometrically an angle $\varphi$ can be defined by two oriented rays (vectors)
both originating from the same given point. The two rays then span a plane. In
order to describe the angle uniquely analytically one chooses a third ray
which “emanates” from the same point and which is orthogonal to one of the two
original rays. Projecting the second original ray onto the two orthogonal ones
by means of a circle, with radius $a$, around the origin yields a pair
$a(\cos\varphi,\sin\varphi),\,a>0,$ which determines $\varphi$ uniquely, after
chosing a clockwise- or counter-clockwise orientation. It is convenient, but
not necessary, to put $a=1$.

Quantizing the system then means: quantizing the components $a\cos\varphi$ and
$a\sin\varphi$ which combined represent one “observable”, the angle
$\varphi\,$! The details depend on the choice of $a$ and possibly other
elements of the associated Poisson algebra.

We now come to a crucial physical point:

For periodic motions - like that of a HO - the angle $\varphi(t)$ of Eq. (5)
does not stop at $\varphi(T)\equiv\omega\,T=2\pi$ but “runs” around the
origin, say at least $s$ times, i.e.
$\varphi(t)\geq\varphi(t_{s})=s\,2\pi,\,s\in\mathbb{Z},$, and in this way
generates an (s+1)-fold covering of the unit circle $\mathbb{S}^{1}$. In this
way the configuration space $\mathbb{S}^{1}_{[s+1]}$ of the angle becomes
(s+1)-fold connected, and, as $s$ can be an arbitrary integer, infinitely
connected.

So, for dynamical (time-dependent) periodic systems the “observable” angle
consists of 2 parts: the number $s$ of completed coverings
$\mathbb{S}^{1}_{[s]}$ of the unit circle and a “rest” $\chi\in[0,2\pi)$:

$\varphi=(2\pi\,s+\chi)\in\mathbb{S}^{1}_{[s+1]},~{}s\in\mathbb{Z},~{}\chi\in[0,2\pi).$
(11)

The crucial point for periodic motions is that $s=0$ is sufficient to describe
the orbit $p(q)$ of Eq. (1), but in order to describe the time evolution
$\varphi(t)$ one needs to know the pair $(\chi;s)$ of Eq. (11). This is a
consequence of the non-trivial topology of the phase space (I). In many cases
the angle $\varphi(t)$ appears in the form $\varphi(t)=\omega
t\equiv\tilde{t}$, i.e. it is essentially a time variable. Thus, the pair
$(q,p)\in\mathcal{S}_{q,p}$ of a periodic orbit is independent of the number
$s$ of coverings. But this number of coverings is essential in connection with
the phase space $\mathcal{S}_{\varphi,I}$. This important difference leads to
corresponding different quantum mechanical properties of the two phase spaces,
e.g. for $\mathcal{S}_{\varphi,I}$ to the set of spectra (9), containing the
“orthodox” case $b=1/2$ as a special one! We shall see that an $s$-fold
covering ($s>0$ ) is associated with $b=1/s$.

The introduction of the canonical pair angle and action variables is
conventionally motivated by the aim to make the action variable $I$ a constant
of motion, i.e. to have an “integrable” system Landau and Lifshitz (1969);
Thirring (1997, pb: 2003); Arnold  _et al._ (2006). But this is not necessary:
one can try to describe systems in the phase space (I) in terms of the local
coordinate pair $(\varphi,I)$ or the global ones
$h_{0}=I,.\,h_{1}=I\cos\varphi,\,h_{2}-I\sin\varphi$ which will be illustrated
by an example in the next chapter.

Phases play an important role in many physical systems with periodic
properties like vibrations, waves etc., e.g. in optics, atomic and molecular
spectroscopy, condensed matter physics etc. Thus, it is important to
understand the corresponding quantum theories in terms of the canonical pair
angle and action variable properly and consistently and look for experimental
consequences.

Ad ii): One reason might be that nobody up to now has been looking for the
newly predicted physical phenomena! Another reason could be that the
associated signals are very weak and obscured by the “orthodox” spectrum
$\hbar\omega\,(n+1/2)$! Possible related future laboratory experiments are
indicated in Ch. IV and associated interpretations of present and future
astrophysical observations in Ch. V.

The paper is organized as follows:

Ch. II discusses a few properties of the phase space (I): its global
coordinates $h_{j},j=0,1,2$, provided in terms of the group
$SO^{\uparrow}(1,2)$. Further the trivial orbits (circles) of the HO on the
phase space (I) and those of a dynamical model, a simple generalization on
$\mathcal{S}_{\varphi,I}$ of the HO. These “classical” considerations are
intended to provide an intuitive background for the discussions of the
associated quantum mechanics in Ch. III.

In Ch. III several aspects of quantum mechanical systems are discussed the
basic “observables” of which are given by the Lie algebra elements
$K_{0},K_{1}$ and $K_{2}$ of the canonical group $Sp(2,\mathbb{R})$ of the
punctured plane. Though this Lie algebra is also that of the isomorphic groups
$SU(1,1)$ or $SL(2,\mathbb{R})$, of the group $SO^{\uparrow}(1,2)$ and all
covering groups as well, the “symplectic” variant appears to be preferable
because of possible generalizations to higher dimensional phase spaces Kastrup
(2007, 2003).

One of the main topics in this chapter consists of the discussion of the
spectrum (9) and related explicit Hilbert spaces for the representation of the
self-adjoint operators $K_{0},K_{1},K_{2}$. Extended use is made of
mathematical results contained in Refs. Kastrup (2003, 2007). Hardy spaces
(i.e. Hilbert spaces which have non-vanishing Fourier components for
$n=0,1,2,\ldots$ only) on the circle play a prominent role for constructing
those explicit Hilbert spaces.

Ch. IV contains a number of suggestions to find concrete physical systems to
which the theoretical framework may apply.

Possibly the most important application concerns the vibrations of diatomic
molecules which are harmonic in the neighbourhood of the minima of their Born-
Oppenheimer (BO) potentials and where “symplectic” spectra (9) may lead to a
lower ($b<1/2$) ground state energy compared to the “orthodox” value $b=1/2$ !

Laboratory tests are, of course, of crucial importance, especially for
molecular hydrogen H2. As this molecule has no permanent electric dipole
moment, its “orthodox” infrared emission and absorption signals are already
very weak. Therefore very probably even more so the “non-orthodox” ones. Other
diatomic molecules of the lightest elements with an electric dipole element
(like, e.g. LiH) may be more appropriate for laboratory infrared experiments.

For H2 itself Raman scattering or atomic and molecular collisions may induce
transient electric dipole moments leading to characteristic emissions or
absorptions associated with vibrations (and rotations) her (a).

Extremely important are possible astrophysical applications, discussed in Ch.
V, especially concerning the problems of dark energy and dark matter; here the
spectrum (9) may provide the key to the simultaneous understanding of both
problems:

As the index $b$ may be arbitrarily small $>0$, the associated estimate of the
cosmological constant $\Lambda$ \- or the vacuum “dark” energy density - can
be compatible with the experimentally observed value, leading to
$b\approx\exp(-35)$ .

In addition, the possible lowering of the vibrational zero-point energies of
electronic Born - Oppenheimer potentials for diatomic molecules suggests to
look at molecular hydrogen b-H2 and other primordial diatomic molecules as
candidates for dark matter.

Altogether one finds that the consequences of a symplectic spectrum
($0<b<1/2$) of the HO are surprisingly well compatible with the cosmological
$\Lambda$CDM model, with - mainly - $b$-H2 molecules as WIMPs !

There is, however, one important caveat: the dynamics of the transitions
(rates) to and from the new additional energy levels has still to be worked
out !

Experimentally, 21-cm radio telescopes directed towards the Dark Ages of the
universe are of special importance (see, e.g. Ref. et al. (a)). Recent
observations et al. (2018a) indicate – unexpected for the present
interpretations of dark matter – non-gravitational (electromagnetic?)
interactions between atomic hydrogen and dark matter Barkana (2018); Fialkov
_et al._ (2018) ! This appears to be compatible with $b$-H2 molecules as dark
matter Similarly the recently observed discrepancy between computer simulated
dark matter models and gravitational lensing et al. (2020a) is of interest in
this context.

If the observed cosmic dark matter indeed consists of - infrared detuned -
primordial diatomic molecules then there is no need for the introduction of
any kind of “new” matter, a point which has also been emphasized in the recent
discussions of dark matter as being formed by primordial black holes (for a
recent review see, e.g. Ref. Hasinger (2020)).

##  II Motions on the classical
phase spaces $\mathcal{S}_{\varphi,I}$ and $\mathcal{S}_{q,p}$

The present chapter discusses a simple classical model on the phase space
$\mathcal{S}_{\varphi,I}$ as a preparation for the discussion of the
corresponding quantum mechanical one later.

###  II.1 Coordinates and orbits on $\mathcal{S}_{\varphi,I}$

####  II.1.1 Global coordinates

It was already indicated above that the angle $\varphi$ itself is not a “good”
global coordinate on $\mathcal{S}_{\varphi,I}$. The situation is even worse
for the corresponding quantum theory Kastrup (2006). As described above, a way
out is to characterize the geometrical quantity “angle” by the pair
$(\cos\varphi,\sin\varphi)$. However the triple
$I,C(\varphi)=\cos\varphi,S(\varphi)=\sin\varphi$ is still not appropriate for
our present purpose:

Consider the Poisson brackets

$\\{f_{1},f_{2}\\}_{\varphi,I}\equiv\partial_{\varphi}f_{1}\,\partial_{I}f_{2}-\partial_{I}f_{1}\,\partial_{\varphi}f_{2}$
(12)

for locally smooth functions on $\mathcal{S}_{\varphi,I}$. The 3 functions
$I,C(\varphi)$ and $S(\varphi)$ obey the Poisson Lie algebra

$\\{I,C\\}_{\varphi,I}=S\,,~{}\\{I,S\\}_{\varphi,I}=-C\,,~{}\\{C,S\\}_{\varphi,I}=0\,,$
(13)

which constitutes the Lie algebra of the Euclidean group $E(2)$ of the plane:
rotations (generated by $I$) and 2 independent translations (generated by $C$
and $S$). They are the proper coordinates for a phase space with the topology
of an infinite cylinder $\mathbb{S}^{1}\times\mathbb{R}$, like that of the
canonical system angle and orbital angular momentum Kastrup (2006, 2016).

It can be justified systematically ka1 (a) that the appropriate global
coordinates on $\mathcal{S}_{\varphi,I}$ are the functions

$\displaystyle h_{0}(\varphi,I)$ $\displaystyle=I>0\,,$ (14) $\displaystyle
h_{1}(\varphi,I)=I\cos\varphi\,,$
$\displaystyle\;\;\;h_{2}(\varphi,I)=-I\sin\varphi\,,$

which obey

$\vec{h}^{2}\equiv h_{1}^{2}+h_{2}^{2}=h_{0}^{2},\,h_{0}>0.$ (15)

and, therefore, describe a simple (“light”) cone, with the tip deleted. The
functions $h_{j}(\varphi,I)$ obey the Poisson Lie algebra

$\displaystyle\\{h_{0},h_{1}\\}_{\varphi,I}=-h_{2}\,,$
$\displaystyle~{}\\{h_{0},h_{2}\\}_{\varphi,I}=h_{1}\,,$ (16)
$\displaystyle\\{h_{1},h_{2}\\}_{\varphi,I}$ $\displaystyle=h_{0}\,,$

which constitutes - as mentioned above - the Lie algebra of the 3-dimensional
group $SO^{\uparrow}(1,2)$ or of the symplectic group $Sp(2,\mathbb{R})$ of a
$(x,y)$-plane, the transformations of which leave the skew-symmetric form
$dx\wedge dy$ invariant.

The triple $(h_{0},h_{1},h_{2})$ transforms as a 3-vector with respect to the
group $SO^{\uparrow}(1,2)$, the pair $(q,p)$ transforms as a vector with
respect to the symplectic group Kastrup (2007)!

As the symplectic group $Sp(2,\mathbb{R})$ is isomorphic to the groups
$SL(2,\mathbb{R})$ and $SU(1,1)$ kas (c), one may use those here, too. But the
identification as the symplectic group appears to be more appropriate in the
framework of classical mechanics and, above all, it can be generalized to
higher dimensions Kastrup (2007).

Justification of the global “canonical” coordinates (14) in a nutshell Kastrup
(2011): The three 1-dimensional subgroups of the (transitive) group
$SO^{\uparrow}(1,2)=Sp(2,\mathbb{R})/Z_{2}$ (1 rotation, 2 “Lorentz boosts”)
generate global orbits on $\mathcal{S}_{\varphi,I}$. The generators of these
orbits are global Hamiltonian vector fields the associated Hamiltonian
functions of which are the “coordinates” (14). This is in complete analogy to
the usual phase space $\mathcal{S}_{q,p}$ the global coordinates $q$ and $p$
of which are the Hamiltonian functions of the vector fields which generate the
global translations in $p$\- and $q$-directions on $\mathcal{S}_{q,p}$,
endowed with a symplectic structure in terms of the Poisson bracket
$\\{\cdot,\cdot\\}_{q,p}$.

####  II.1.2 Orbits on $\mathcal{S}_{\varphi,I}$

The graph of the motion (6) in $\mathcal{S}_{\varphi,I}$ is utterly simple: a
circle of radius $I_{0}>0$ on which the position at time $t$ is given by the
angle $\varphi(t)=\omega t$. We assume that $\varphi(t=0)~{}=0$ and that
$\varphi(t)$ starts clockwise off a given ray emanating from the point $I=0$.
That ray also defines an horizontal abscissa of an orthogonal coordinate
system with an ordinate of pointing upwards (see Fig. 1). The clockwise
orientation of the angle is induced by the choice of $h_{1}$ and $h_{2}$ in
Eqs. (14).

Note that $h_{1}$ is the projection of $I$ on the positive abscissa and
$h_{2}$ the one on the negative ordinate (see Fig. 1).

Note also that these two projections do not commute (see the last of the Eqs.
(16)), again a consequence of the fact that the point $(0,0)$ does not belong
to the phase space!

Things become more interesting if we “disturb” the HO by introducing new
interactions. Note that on $\mathcal{S}_{\varphi,I}$ functions have to be
expressed in terms of the basic variables $h_{j}(\varphi,I),j=0,1,2$. This
means for the HO:

$H(\varphi,I)=\omega\,h_{0}.$ (17)

A simple but interesting modification is ka1 (b)

$H(\varphi,I)=\omega(h_{0}+g\,h_{1})=\omega\,I(1+g\,\cos\varphi),~{}g\geq 0,$
(18)

with the Eqs. of motion

$\displaystyle\dot{\varphi}$
$\displaystyle=\partial_{I}H(\varphi,I)=\omega(1+g\cos\varphi),$ (19)
$\displaystyle\dot{I}$
$\displaystyle=-\partial_{\varphi}H(\varphi,I)=g\,\omega I\sin\varphi.$ (20)

Now $I(t)$ is no longer a constant.

The present discussion is an extension of the usual one for (completely)
integrable systems M. Born, unter Mitwirkung von F. Hund (1925); Landau and
Lifshitz (1969); Thirring (1997, pb: 2003); Arnold  _et al._ (2006) in which
the action variables are constants of motion as functions on the original
$(q,p)$-phase space and where the original “tori” (determined by $I_{0}=$
const. and $\varphi\in\mathbb{S}^{1}$ in our very special case) are rather
stable against small perturbations (KAM theory kam ).

Here the global phase space formed by angle and action variables is being
considered and the action variable $I$ may be a function of time $t$, like the
angle variable $\varphi$.

Recall that an action variable is originally defined as a global variable -
like the energy - on the phase space (7), namely as a closed path integral
along the border of a volume determined by the energy and the potential of the
system M. Born, unter Mitwirkung von F. Hund (1925); Landau and Lifshitz
(1969); Thirring (1997, pb: 2003); Arnold  _et al._ (2006):

$I(E)=\frac{1}{2\pi}\oint_{C^{\uparrow}(E)}dq\,p(q,E),\,\,p(q,E)=\pm\sqrt{2m(E-V(q)},$
(21)

where the clockwise oriented closed path $C^{\uparrow}(E)$ is determined by
the energy equation $p^{2}/2m+V(q)=E$. According to Stokes’ theorem the path
integral is equal to the volume with the border $C^{\uparrow}(E)$. Here $I(E)$
is a constant of motion because $E$ is a constant along the orbits
$\\{[q(t),p(t)]\\}$.

The more general case $I(t)$ can be obtained from the local relation (2):
Integrating both sides simultaneously at time $t$ gives

$\Delta\varphi(t)\cdot\Delta I(t)=\int_{\Delta G_{q,p}(t)}dqdp=\Delta
V_{q,p}(t),$ (22)

where $\Delta V_{q,p}(t)$ is the volume of the region $\Delta
G_{q,p}(t)\subset\mathcal{S}_{q,p}$, with $\Delta V_{q,p}(t)=0$ for
$\Delta\varphi(t)=0$ or $\Delta I=0$. Putting the lower value $I_{0}=0$ in
$\Delta I=I-I_{0}$ the special case (21) is obtained for
$\Delta\varphi(t)=2\pi$.

With $I_{0}=0$ it follows from Eq. (22) that

$I(t)=\partial V_{q,p}(t)/\partial\varphi,$ (23)

where the pair $(q,p)$ is assumed to be a function of $\varphi$ like in Eqs.
(1). Thus, $I(t)$ may be interpreted as the differential change with $\varphi$
of the phase space volume $V_{q,p}(t)$ at time $t$.

The additional term $g\,h_{1}$ in the Hamiltonian (18) breaks several related
symmetries: rotation invariance in the $(h_{1},h_{2})$-plane (which can be
remedied by using the combination $\cos\alpha\,h_{1}+\sin\alpha\,h_{2}$
instead of $h_{1}$), special Lorentz “boosts” in the directions “1” or “2” and
reflection parity ($\varphi\to\varphi\pm\pi$). Time reversal
($\varphi\to-\varphi$) is fulfilled.

The Eqs. of motion (19) and (20) can be integrated immediately:

As the energy is still conserved,

$\omega\,I(1+g\,\cos\varphi)=E=\text{$const.$ },$ (24)

we have the orbit equation

$I(\varphi)=\frac{I_{0}}{1+g\,\cos\varphi},\,\ I_{0}=E/\omega\,.$ (25)

This equation describes a conical section with a given focus as the origin for
the polar coordinates $\varphi$ (“true anomaly”), distance $I(\varphi)$ from
that focus , “semi-latus rectum” $I_{0}$ and “numerical eccentricity”
$\epsilon=g$.

For $g<1$ we have an ellipse, for $g=1$ a parabola and for $g>1$ a hyperbola!
The angle $\varphi$ increases clockwise from the fixed ray which starts from
the focus nearest to the orbit point $I_{-}=I_{0}/(1+g)$, the “perihelion”,
and further passes through that latter point (see Fig. 1).

Ellipse, $g<1:$ (26)

  * •

semi-latus rectum: $I_{0}$,

  * •

numerical eccentricity: $g$,

  * •

perihelion: $I_{-}=I_{0}/(1+g)$,

  * •

aphelion: $I_{+}=I_{0}/(1-g)$

  * •

major semi-axis:
$a=\frac{1}{2}[I_{0}/(1+g)+I_{0}/(1-g)]=I_{0}/(1-g^{2})$,

  * •

linear eccentricity: $e=g\,a=gI_{0}/(1-g^{2})$
(2$e$ is the distance of the two foci),

  * •

minor semi-axis: $b=\sqrt{a^{2}-e^{2}}=I_{0}/(1-g^{2})^{1/2}$,

  * •

area of ellipse: $ab\pi=\frac{I_{0}^{2}\pi}{(1-g^{2})^{3/2}}$ .

Thus the shape of the ellipse is completely determined by the coupling
constant $g$ and the integration constant $I_{0}$.

Figure 1:  Phase space $\mathcal{S}_{\varphi,I}$: 1. Graph of the circular
orbit $\\{(\varphi\in[0,2\pi),I=I_{0}>0)\\}$ generated by the periodic motion
(6) of the ”orthodox” HO (in black). 2. Graph of the elliptical orbit
$I(\varphi)=I_{0}/(1+g\,\cos\varphi)$ associated with the periodic motions
(31) and (36) (in red, with g=1/2). Origin of the polar coordinates
$(\varphi,I)$ here is the right focus. The meaning of the different letters
with their related quantities and their historical names are described in the
list (26)

The same holds for the hyperbola with the focus of the left branch as the
origin for the polar coordinates:

Hyperbola, $g>1$: (27)

  * •

If $g>1$ the expression (25) describes a hyperbola of which we consider one
branch only: the one open to the left. Its point of closest distance to the
(inside) focus is $I_{0}/(1+g)$ where $\varphi=0$.

  * •

semi-latus rectum: $I_{0}$,

  * •

numerical eccentricity: $g$,

  * •

linear eccentricity: $e=gI_{0}/(g^{2}-1)$
(2$e$ is the distance of the 2 foci),

  * •

major and minor semi-axis:
$a=I_{0}/(g^{2}-1),\,\,\,b=I_{0}/(g^{2}-1)^{1/2}$,

  * •

The two angles $\varphi_{\infty}(i),\,i=1,2$ characterizing the asymptotes are
determined by
$\cos\varphi_{\infty}(i)=-1/g,\,\varphi_{\infty}(1)\in(\pi/2,\pi)\text{ and
}\varphi_{\infty}(2)\in(\pi,\pi+3\pi/2)$.

Parabola, $g=1$: (28)

This simple case can be treated in the same way as the two others above.

Obviously, the orbits of the last two cases extend to infinity.

####  II.1.3 Time evolution

Ellipse
The time evolution follows from Eq. (19):

$\int_{\varphi_{0}}^{\varphi}\frac{d\vartheta}{1+g\,\cos\vartheta}=\omega\,(t-t_{0}).$
(29)

For $g<1$ wir get Gradshteyn and Ryzhik (1965) with $\varphi_{0}=0$ for
$t_{0}=0$:

$\sqrt{1-g^{2}}\,\omega\,t=2\arctan\left[\sqrt{\frac{1-g}{1+g}}\tan(\varphi/2)\right],$
(30)

or

$\tan[\varphi(t)/2]=\sqrt{\frac{1+g}{1-g}}\tan(\sqrt{1-g^{2}}\,\omega\,t/2).$
(31)

Thus, the interaction $g\,h_{1}(\varphi,I)$ leads to an effective redshifted
angular frequency

$\omega_{g}=\sqrt{1-g^{2}}\,\omega,$ (32)

with a branch point for $g^{2}\to 1$.

It follows from Eq. (30) that the time $t(\pi/2)$ needed to pass from
$\varphi=0$ to $\varphi=\pi/2$ is given by

$\omega_{g}\,t(\pi/2)=2\arctan\sqrt{\frac{1-g}{1+g}}=\arccos g.$ (33)

At that time $I[t(\pi/2)]=I_{0}$ (see Eq. (25)).

The time needed to pass from $\varphi=0$ to $\varphi=\pi$ is

$\omega_{g}\,t(\pi)=\pi.$ (34)

Here we have $I[t(\pi)]=I_{0}/(1-g)$ (“aphelion”).

For reasons of symmetry of the ellipse we get from Eq. (34) for one period

$\omega_{g}\,T_{g;2\pi}=2\pi,\,\,T_{g;2\pi}=\frac{2\pi}{\omega\sqrt{1-g^{2}}}.$
(35)

Thus, $1/\sqrt{1-g^{2}}$ is a kind of “refractive index”.

Once the time evolution $\varphi(t)$ is known that of $I(t)$ can be obtained
from the orbit equation (25). Using the relation
$\cos\varphi=1-\tan^{2}(\varphi/2)/[1+\tan^{2}(\varphi/2)]$ one obtains

$I(t)=\frac{I_{0}}{1-g^{2}}(1-g\cos[\omega_{g}t]),$ (36)

which again gives the relations (33), (34) and (35).

The above results may be looked at as follows: For vanishing $g$ we have on
$\mathcal{S}_{\varphi,I}$ a clockwise periodic motion with frequency $\omega$
on a circle of radius $I_{0}$. Adding the interaction
$g\,h_{1}(\varphi,I),\,0\leq g<1,$ deformes the circle into an ellipse with
semi-latus rectum $I_{0}$ and numerical eccentricity $g$. In addition the
original angular frequency $\omega$ of the periodic motion is reduced to
$\omega_{g}=\sqrt{1-g^{2}}\,\omega$.

Pictorially speaking we start with a “circularly polarized” motion ($g=0$)
which encounters a medium ($0<g<1$) which induces an “elliptical polarization”
and reduces the original angular frequency $\omega$!

Hyperbola
As before the time evolution can be calculated from Eq. (19), the integration
of which now gives Gradshteyn and Ryzhik (1965)

$\tan[\varphi(t)/2]=\sqrt{\frac{g+1}{g-1}}\tanh[\sqrt{g^{2}-1}\,\omega\,t/2].$
(37)

Parabola
Finally, the time evolution for the parabola ($g=1$) is given by gra (a)

$\tan[\varphi(t)/2]=\omega\,t\,.$ (38)

###  II.2 Orbits on $\mathcal{S}_{q,p}$

Using the mappings (1) the orbit equation (25) in $\mathcal{S}_{\varphi,I}$
can be mapped onto $\mathcal{S}_{q,p}$, where it has the parametrization

$\displaystyle q_{g}(\varphi)=$
$\displaystyle\sqrt{\frac{2I_{0}}{m\omega(1+g\cos\varphi)}}\cos\varphi$ (39)
$\displaystyle p_{g}(\varphi)=$ $\displaystyle-\sqrt{\frac{2m\omega
I_{0}}{1+g\cos\varphi}}\sin\varphi.$ (40)

Figure 2:  Phase space $\mathcal{S}_{q,p}$: 1. Graph of the “orthodox”
elliptical orbit generated by the usual HO (in black) 2. Graph of the image of
the (red) ellipse in Fig. 1 as described by Eqs. (39) and (40) (in red). Here
$\phi$ is the polar angle on $\mathcal{S}_{q,p}$ and $\varphi$ the polar angle
on $\mathcal{S}_{\varphi,I}$.

It implies

$\displaystyle H(q_{g},p_{g})$
$\displaystyle=\frac{p_{g}^{2}}{2m}+\frac{1}{2}m\omega^{2}\,q_{g}^{2}$ (41)
$\displaystyle=\omega\,I=\frac{\omega\,I_{0}}{1+g\,\cos\varphi},$
$\displaystyle-\frac{p_{g}}{q_{g}}\equiv\tan\phi=m\,\omega\tan\varphi.$ (42)

and

$\cos\varphi=\frac{\sqrt{m/2}\;\omega\,q_{g}}{\sqrt{H(q_{g},p_{g})}}=\pm\sqrt{V(q_{g},p_{g})/H(q_{g},p_{g})}.$
(43)

Inserting the last expression into Eq. (41) yields a 4th order equation in
$q_{g}$ and $p_{g}$ for the orbit described by Eqs. (39) and (40):

$\displaystyle H(q_{g},p_{g})\pm
g\,\sqrt{V(q_{g},p_{g})\,H(q_{g},p_{g})}=\omega\,I_{0},$ (44)
$\displaystyle[H(q_{g},p_{g})-\omega\,I_{0}]^{2}=g^{2}\,V(q_{g},p_{g})\,H(q_{g},p_{g}).$

For $g=0$ this is reduced to the usual orbit ellipse of the HO on
$\mathcal{S}_{q,p}$ (See Fig. 2):

$\frac{p_{g=0}^{2}}{2m}+\frac{1}{2}m\omega^{2}\,q_{g=0}^{2}=E_{0}=\omega
I_{0}=\text{const. }$ (45)

The functions (39) and (40) have the special values

$\displaystyle q_{g}(0)\equiv q_{-}$
$\displaystyle=\sqrt{\frac{2I_{0}}{m\omega(1+g)}},\,\,\,p_{g}(0)=0,$ (46)
$\displaystyle q_{g}(\pi/2)$
$\displaystyle=0,\,\,~{}~{}p_{g}(\pi/2)=-\sqrt{2m\omega I_{0}},$
$\displaystyle q_{g}(\pi)\equiv q_{+}$
$\displaystyle=-\sqrt{\frac{2I_{0}}{m\omega(1-g)}},\,\,\,p_{g}(\pi)=0,$
$\displaystyle q_{g}(3\pi/2)$
$\displaystyle=0,\,\,~{}~{}p_{g}(3\pi/2)=\sqrt{2m\omega I_{0}}.$

Note that the 4th order figure defined by Eq. (44) is symmetric with respect
to the $q$-axis, but no longer symmetric with respect to the $p$-axis (see
Fig. 2). Maximum and minimum of $p_{g}(\varphi)$ are given by the angles
$\varphi_{\pm}$, which obey

$\cos\varphi_{\pm}=-(1-\sqrt{1-g^{2}})/g.$ (47)

###  II.3 The frequency $\omega$ as an external field

According to Eq. (3) the Hamiltonian of the HO on $\mathcal{S}_{\varphi,I}$
has the simple form

$H(\varphi,I)=\omega\,I,$ (48)

where the frequency $\omega$ appears as a parameter multiplying the basic
action variable $I$.

That parameter $\omega$ may also be considered as an external “field” which
can be “manipulated” from outside, e.g., as a function $\omega(t)$ of time
$t$. The solution of Eq. (4) then is

$\varphi(t)-\varphi(t_{0})=\int_{t_{0}}^{t}d\tau\,\omega(\tau).$ (49)

Note that the Hamiltonian (48) is independent of $\varphi$ and therefore
$\dot{I}=-\partial_{\varphi}H(\varphi,I)=0$, even if $\omega=\omega(t)$! Thus,
the action variable $I$ is still conserved on $\mathcal{S}_{\varphi,I}$, but
the energy $E=\omega(t)\,I$ is not! A possible interesting example for
applications is a time dependent angular frequency of the form

$\omega(t)=\omega_{0}+\lambda\sin(\rho t),\;|\lambda|<\omega_{0}.$ (50)

In the present context it is also appropriate to briefly recall the so-called
“adiabatic invariance” bor ; Born and Fock (1928); Landau and Lifshitz (1969);
Thirring (2002); Arnold  _et al._ (2006) of the action variable I: If energy
$E(\lambda)$ and frequency $\omega(\lambda)$ of a periodic motion with period
$T$ depend on a slowly varying parameter $\lambda$ (
$T\,d\lambda/dt\ll\lambda$), then $I=E(\lambda)/\omega(\lambda)$ remains
constant if $\lambda$ varies.

In the following discussions on the “non-orthodox” quantum mechanics of the HO
it is essential to differentiate between the quantum counterparts of the
action variable $I$ and the Hamilton function $H$, the generator of time
evolution.

For a given “binding” potential $V(q)$ the angular frequency $\omega$ is
generally defined as one half of the 2nd derivative of $V(q)$ at its (local)
minimum $q=q_{0}$ : $\omega=(1/2)\,d^{2}V(q)/dq^{2}_{|q=q_{0}}$.

##  III Quantum mechanics of the phase space $\mathcal{S}_{\varphi,I}$

###  III.1 Basics: self-adjoint representations of the
three Lie algebra generators $K_{j}$ of
the symplectic group $Sp(2,\mathbb{R})$

The quantization of the global “coordinates” $h_{j}$ from Eq. (14) is
implemented by reinterpreting them as self-adjoint operators in a given
Hilbert space ka1 (c),

$h_{j}\to K_{j}=\hbar\,\tilde{K}_{j}\,$ (51)

which obey the associated Lie algebra (16):

$\displaystyle[\tilde{K}_{0},\,\tilde{K}_{1}]=i\,\tilde{K}_{2}\,,$
$\displaystyle~{}~{}[\tilde{K}_{0},\,\tilde{K}_{2}]=-i\,\tilde{K}_{1}\,,$ (52)
$\displaystyle[\tilde{K}_{1},\,\tilde{K}_{2}]$
$\displaystyle=-i\,\tilde{K}_{0}\,.$ (53)

(Quantities $\tilde{A}$ with a ”tilde”, here and in the following, are
considered to be dimensionless).

The three self-adjoint operators $\tilde{K}_{j}$ can be obtained as Lie algbra
generators of irreducible unitary representations of the corresponding groups
$SO^{\uparrow}(1,2)$, $Sp(2,\mathbb{R})$ (the latter being isomorphic to the
groups $SL(2,\mathbb{R})$ and $SU(1,1)$) or of one of their infinitely many
covering groups kas (c).

As $\tilde{K}_{0}$ is the generator of the maximal compact abelian subgroup
$U(1)$, its eigenstates may be used as a Hilbert space basis (here formally in
Dirac’s notation, explicit examples will be discussed later):

$\tilde{K}_{0}|b,\,n\rangle=(n+b)\,|b,\,n\rangle\,,$

where $b$ is some real number $\in(0,1)$ (“Bargmann index” Bargmann (1947))
and $n=0,1,2,\ldots$. This central result can be derived as follows

The operators

$\displaystyle~{}\tilde{K}_{\pm}=$ $\displaystyle\tilde{K}_{1}\pm
i\,\tilde{K}_{2}\,,$ (54) $\displaystyle
K_{1}=\frac{1}{2}(\tilde{K}_{+}+\tilde{K}_{-}),\,K_{2}=\frac{1}{2i}(\tilde{K}_{+}-\tilde{K}_{-}),$
(55)

obey the relations

$\displaystyle[\tilde{K}_{0},\,\tilde{K}_{+}]=\tilde{K}_{+}\,,$
$\displaystyle~{}~{}[\tilde{K}_{0},\,\tilde{K}_{-}]=-\tilde{K}_{-}\,,$ (56)
$\displaystyle[\tilde{K}_{+},\,\tilde{K}_{-}]$
$\displaystyle=-2\tilde{K}_{0}\,.$ (57)

They are raising and lowering operators:

$\displaystyle\tilde{K}_{+}|b,\,n\rangle$
$\displaystyle=[(2b+n)(n+1)]^{1/2}\,|b,\,n+1\rangle\,,$ (58)
$\displaystyle\tilde{K}_{-}|b,\,n\rangle$
$\displaystyle=[(2b+n-1)n]^{1/2}\,|b,\,n-1\rangle\,.$ (59)

The relations (58) and (59) are derived under the assumptions that there
exists a state $|b,0\rangle$ such that

$\tilde{K}_{0}|b,0\rangle=b\,|b,0\rangle\,,~{}~{}~{}\tilde{K}_{-}|b,0\rangle=0,~{}b\in[1,0).$
(60)

Eq. (58) implies

$\displaystyle|b,n\rangle=$
$\displaystyle\frac{1}{\sqrt{(2b)_{n}\,n!}}(\tilde{K}_{+})^{n}|b,0\rangle,$
(61) $\displaystyle(2b)_{n}=2b(2b+1)\cdots(2b+n-1)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}=\Gamma(2b+n)/\Gamma(2b).$

It then follows that

$\tilde{K}_{0}\,|b,n\rangle=(n+b)\,|b,n\rangle\,,\,n=0,1,\ldots;~{}~{}1>b>0\,.$
(62)

This is the so-called “positive discrete series” $D_{b}^{(+)}$ among the
different types of possible irreducible unitary representations of
$Sp(2,\mathbb{R})$ Bargmann (1947); kas (c) The Bargmann index $b$ \- called
“B-index” in the following - characterizes an irreducible unitary
representation (IUR) $D_{b}^{(+)}$.

The Casimir operator

$\displaystyle\mathfrak{C}$
$\displaystyle=\tilde{K}_{1}^{2}+\tilde{K}_{2}^{2}-\tilde{K}_{0}^{2}$ (63)
$\displaystyle=\tilde{K}_{+}\tilde{K}_{-}-\tilde{K}_{0}\,(\tilde{K}_{0}-1)$
$\displaystyle=\tilde{K}_{-}\tilde{K}_{+}-\tilde{K}_{0}\,(\tilde{K}_{0}+1)$

of the IUR $D_{b}^{(+)}$ has the value

$\mathfrak{C}=b(1-b){\bf 1}\,.$ (64)

This means that the “classical Pythagoras” (15) is violated quantum
mechanically for $b\neq 1$, e.g. in the case of the usual HO with $b=1/2$!

The Group $SO^{\uparrow}(1,2)$ has infinitely many covering groups because its
compact subgroup $O(2)\cong S^{1}\cong U(1)$ is infinitely connected!

Let us denote the $s$-fold covering by

$SO^{\uparrow}_{[s]}(1,2)\,,\,s=1,2,\ldots\,.$ (65)

Its irreducible unitary representations $D_{b}^{(+)}$ have the indices

$b=\frac{r}{s}\,,~{}r=1,2,\ldots,s-1\,.$ (66)

This means that $b_{min}=1/s$ can be arbitrarily small $>0$ if $s$ is large
enough!
The 2-fold coverings

$Sp(2,\mathbb{R})=SL(2,\mathbb{R})\,\cong SU(1,1)$ (67)

have $b=1/2$.

The results above imply that the $(\varphi,I)$-Hamiltonian

$H(\varphi,I)\to
H_{osc}(K)=\omega\,K_{0}\,,~{}~{}K=\hbar\,(\tilde{K}_{0},\tilde{K}_{1},\tilde{K}_{2})\,$
(68)

can have the $b$-dependent spectra

$E_{b,\,n}(\varphi,I)=\hbar\,\omega\,(n+b)\,,\,n=0,1,\ldots\,;~{}~{}1>b>0,$
(69)

As this result is due to the properties of the symplectic group
$Sp(2,\mathbb{R})$, especially its compact subgroup $U(1)\cong O(2)$, we call
it the “symplectic spectrum” of the HO, and the conventional special case
$b=1/2$ as its “orthodox” one.

Because of the relations (55), (58) and (59) one has

$\langle b,n|\tilde{K}_{j}|b,n\rangle=0\,,~{}j=1,2,$ (70)

implying for the mean square deviations

$(\Delta\tilde{K}_{j})^{2}_{b,n}=\frac{1}{2}(n^{2}+2nb+b)\,,~{}j=1,2,$ (71)

so that

$(\Delta\tilde{K}_{1})_{b,n}\,(\Delta\tilde{K}_{2})_{b,n}=\frac{1}{2}(n^{2}+2bn+b)\,,$
(72)
$(\Delta\tilde{K}_{1})_{b,n=0}\,(\Delta\tilde{K}_{2})_{b,n=0}=\frac{b}{2}\,.$
(73)

The last relation shows that $b\to 0$ is a kind of classical limit in the
angle-action framework!

###  III.2 Time evolution

####  III.2.1 Heisenberg picture

The appearence of covering groups (65) has a natural physical background: Take
the time dependence of the angle $\varphi(t)=\omega\,t$ in Eq. (6) (with
$t_{0}=0,\varphi_{0}=0$): In general the system will not stop after covering a
circle just once, $\omega\,T_{1}=2\pi$, but will circle the origin, say, at
least $s$ times, $\omega\,T_{s}=2\pi\,s$, where $s\in\mathbb{N}$ can be
arbitrarily large.

In the following the dimensionless time variable

$\tilde{t}=\omega\,t$ (74)

will be used. It is an angle variable.

The unitary time evolution operator is given by

$U(\tilde{t})=e^{-i\,\tilde{K_{0}}\,\tilde{t}}\,,~{}~{}\tilde{K}_{0}=N+b\bf{1}\,,$
(75)

where the number operator $N$ can be considered a function of the operators
$\tilde{K}_{j},\,j=0,1,2$, as will be shown below.

The unitary operator (75) implies the usual Heisenberg Eqs. of motion:

$\displaystyle U(-\tilde{t})\,\tilde{K}_{+}\,U(\tilde{t})$
$\displaystyle=e^{-i\tilde{t}}\tilde{K}_{+},$ (76) $\displaystyle
U(-\tilde{t})\,\tilde{K}_{-}\,U(\tilde{t})$
$\displaystyle=e^{+i\tilde{t}}\tilde{K}_{-},$ (77) $\displaystyle
U(-\tilde{t})\,\tilde{K}_{1}\,U(\tilde{t})$
$\displaystyle=\cos\tilde{t}\,\tilde{K}_{1}+\sin\tilde{t}\,\tilde{K}_{2},$
(78) $\displaystyle U(-\tilde{t})\,\tilde{K}_{2}\,U(\tilde{t})$
$\displaystyle=-\sin\tilde{t}\,\tilde{K}_{1}+\cos\tilde{t}\,\tilde{K}_{2}.$
(79)

For $\tilde{t}=2\pi$ the operator (75) becomes

$U(\tilde{t}=2\pi)=e^{-2\pi ib}{\bf 1}\,.$ (80)

If

$b=r/s\,,~{}r,s\in\mathbb{N},\,r<s\text{ and divisor free},$ (81)

this implies for $SO^{\uparrow}_{[s]}(1,2)$:

$U(\tilde{t}=s\,2\pi)={\bf 1}\,.$ (82)

The ground state $|b,0\rangle$ has the time evolution

$U(\tilde{t})\,|b,0\rangle=e^{-i\,b\,\tilde{t}}\,|b,0\rangle\,,$ (83)

with the associated time period

$T_{2\pi,b}=\frac{2\pi}{\omega_{b}},\;\;\omega_{b}\equiv b\,\omega\,,$ (84)

which can become arbitrarily large for $b=1/s,\,s\to\infty$. Symbolically
speaking: $1/b$ is a kind of refraction index $n$.

Whereas $\tilde{K}_{0}$ generates rotations and time evolutions by performing
many phase rotations, the operators $\tilde{K}_{1}$ and $\tilde{K}_{2}$
generate special “Lorentz” transformations (“boosts”) in directions 1 and 2,
respectively ka1 (d):

With

$\displaystyle U(w)=$ $\displaystyle
e^{(w/2)\tilde{K}_{+}-(w^{\ast}/2)\tilde{K}_{-}}=e^{i\,w_{2}\tilde{K}_{1}+i\,w_{1}\tilde{K}_{2}},$
(85) $\displaystyle w=w_{1}+i\,w_{2}=|w|e^{i\theta}$

one obtains

$\displaystyle U(-w)\vec{\tilde{K}}U(w)=$
$\displaystyle\vec{\tilde{K}}(\cosh|w|-1)(\vec{n}\cdot\vec{\tilde{K}})\cdot\vec{n}$
(86) $\displaystyle+\sinh|w|\,\vec{n}\,\tilde{K}_{0},$ $\displaystyle
U(-w)\tilde{K}_{0}U(w)=$
$\displaystyle\cosh|w|\tilde{K}_{0}+\sinh|w|(\vec{n}\cdot\vec{\tilde{K}}),$
(87)
$\displaystyle\vec{\tilde{K}}=(\tilde{K}_{1},\tilde{K}_{2}),\;\vec{n}=(\cos\theta,-\sin\theta).$

Eqs. (86) and (87) describe a Lorentz “boost” in direction $\vec{n}$.

####  III.2.2 Schrödinger Picture

If $|b;t_{0}\rangle$ is a state vector of the Hilbert space associated with
the B-index $b$ at time $t_{0}$, then - according to the unitary time
evolution (75) - we have at time $t$:

$|b;t\rangle=U(t-t_{0})|b;t_{0}\rangle=e^{-i\omega
K_{0}(t-t_{0})/\hbar}|b;t_{0}\rangle,$ (88)

which implies

$\frac{\hbar}{i}\partial_{t}|b;t\rangle=\omega
K_{0}|b;t\rangle,~{}K_{0}=\hbar\tilde{K}_{0}.$ (89)

For a representation

$|b;t\rangle=\sum_{n=0}^{\infty}c_{n}(t)|b,n\rangle$ (90)

we get from Eq. (88), with $t_{0}=0$,

$c_{n}(t)=e^{-i\omega(n+b)t},$ (91)

so that

$|b;t\rangle=e^{-ib\,\omega t}\sum_{n=0}^{\infty}e^{-i\,n\,\omega
t}|b,n\rangle.$ (92)

###  III.3 Relationship between the operators $\tilde{K}_{j}$
and the conventional operators $Q$ and $P$

The relations (1) expressed in terms of the functions $h_{j}(\varphi,I)$ from
Eqs. (14) take the form

$\displaystyle q(\varphi,I)$
$\displaystyle=\sqrt{\frac{2}{m\,\omega}}\,\frac{h_{1}(\varphi,I)}{\sqrt{h_{0}(\varphi,I)}}\,,$
(93) $\displaystyle p(\varphi,I)$
$\displaystyle=\sqrt{2\,m\,\omega}\,\frac{h_{2}(\varphi,I)}{\sqrt{h_{0}(\varphi,I)}}\,.$
(94)

There is a corresponding relationship at the operator level: Define the
operators

$\displaystyle A(K)$ $\displaystyle=(\tilde{K}_{0}+b)^{-1/2}\tilde{K}_{-},$
(95) $\displaystyle A^{\dagger}(K)$
$\displaystyle=\tilde{K}_{+}(\tilde{K}_{0}+b)^{-1/2}.$ (96)

According to Eqs. (62), (58) and (59) they act on the number states
$|b,n\rangle$ as

$\displaystyle A^{\dagger}\,|b,n\rangle$
$\displaystyle=\sqrt{n+1}\,|b,n+1\rangle,$ (97) $\displaystyle
A\,|b,\,n\rangle$ $\displaystyle=\sqrt{n}\,|b,n-1\rangle,$ (98)
$\displaystyle\,n=0,1,\ldots\,.$

This means

$[A,\,A^{\dagger}]={\bf 1}\,~{}~{}\forall~{}D_{b}^{(+)},$ (99)

independent of the value of $b$ !

Thus, the composite operators $A(K)$ and $A^{\dagger}(K)$ are the usual Fock
space annihilation and creation operators for all $b$ and independent of $b$ !

Note that the denominator in Eqs. (95) and (96) is well-defined, because
$\tilde{K}_{0}$ is a positive definite operator and $b$ a positive number for
each representation of the series $D^{(+)}_{b}$.

The quantum mechanical position and momentum operators $Q$ and $P$ can now be
defined as usual:

$\displaystyle Q=Q(K)=$
$\displaystyle\,\frac{\lambda_{0}}{\sqrt{2}}\,[A^{\dagger}(K)+A(K)],$ (100)
$\displaystyle P=P(K)=$
$\displaystyle\,\frac{i\,\hbar}{\sqrt{2}\,\lambda_{0}}\,[A^{\dagger}(K)-A(K],$
(101) $\displaystyle\,\lambda_{0}=\sqrt{\hbar/(m\,\omega)},$ (102)

where $\lambda_{0}$ has the dimension of a length.

The $(q,p)$-Hamilton operator

$\displaystyle H(Q,P)=\,$
$\displaystyle\frac{1}{2m}P^{2}(K)+\frac{1}{2}m\,\omega^{2}Q^{2}(K)$ (103)
$\displaystyle=\,$ $\displaystyle\hbar\,\omega(N(K)+\frac{1}{2}{\bf 1}),$
$\displaystyle\,N(K)=A^{\dagger}(K)A(K)$ (104)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}=\tilde{K}_{+}(\tilde{K}_{0}+b)^{-1}\tilde{K}_{-}$

has the usual “orthodox” spectrum $E_{n}=\hbar\omega(n+1/2)$ !

Thus, it turns out that the quantum mechanics associated with the phase space
$\mathcal{S}_{\varphi,I}$ is rather more subtle than the one associated with
$\mathcal{S}_{q,p}$ and that those subtleties get lost if one passes from the
$(\varphi,I)$-case to the $(q,p)$-case!

As the creation and annihilation operators $A^{\dagger}$ $A$ with their
($b$-independent) defining properties (97), (98) and (99) are essential
building blocks for many quantum systems, that loss of $b$-dependent
subtleties may in turn lead to a corresponding loss of physical insights. _The
big question is: Did nature implement those subtleties?_

The time evolution of the composite operators (100) and (101) is the usual
one. It follows from the relations (76) and (79):

$\displaystyle U(-\tilde{t})\,\tilde{Q}\,U(\tilde{t})=$
$\displaystyle\cos\tilde{t}\,\tilde{Q}+\sin\tilde{t}\,\tilde{P}\,,$ (105)
$\displaystyle U(-\tilde{t})\,\tilde{P}\,U(\tilde{t})=$
$\displaystyle-\sin\tilde{t}\,\tilde{Q}+\cos\tilde{t}\,\tilde{P}\,.$ (106)

###  III.4 The model $H=\hbar\,\omega(\tilde{K}_{0}+g\,\tilde{K}_{1})$

####  III.4.1 Transition matrix elements with respect to the number states in
1st order

The quantum mechanical counterpart of the classical Hamiltonian (18) is the
operator

$H(K)=\hbar\,\omega\,\tilde{C}_{g}(K),\,\tilde{C}_{g}(K)=\tilde{K}_{0}+g\,\tilde{K}_{1},\,g\geq
0\,.$ (107)

Before discussing a special explicit choice for the operators $\tilde{K}_{j}$,
their associated Hilbert space and the exact eigenfunctions of the Hamiltonian
(107) we mention the values of the (formal) 1st order matrix elements

$\langle b,m|\tilde{C}_{g}(K)|b,n\rangle,\;m,n=0,1,2,\ldots$ (108)

From the relations (55), (58) and (59) we get for $m\neq n$ (the case $m=n$
appears trivial, but that is only so in 1st order. It follows from the exact
solution - discussed below - that the 2nd order $g^{2}$ and higher ones
contribute):

$\displaystyle\langle b,m|\tilde{C}_{g}(K)|b,n\rangle=$
$\displaystyle\frac{g}{2}\langle b,m|(\tilde{K}_{+}+\tilde{K}_{-})|b,n\rangle$
(109) $\displaystyle=$
$\displaystyle\frac{g}{2}[(2b+n)(n+1)]^{1/2}\,\delta_{m(n+1)}$
$\displaystyle+$ $\displaystyle\frac{g}{2}[(2b+n-1)n]^{1/2}\,\delta_{m(n-1)}.$

Thus, we have the selection rule

$\Delta n=\pm 1,$ (110)

for the Hamiltonian (107), the same as, e.g., for vibrational (electric
dipole) transitions of diatomic molecules her (b)!

Examples:

$\displaystyle\langle b,m=0|\tilde{C}_{g}(K)|b,n=1\rangle=$
$\displaystyle\frac{g}{2}\sqrt{2b},$ (111) $\displaystyle\langle
b,m=2|\tilde{C}_{g}(K)|b.n=1\rangle=$
$\displaystyle\frac{g}{2}\sqrt{2(2b+1)},$ (112) $\displaystyle\langle
b,m=n_{0}-1|\tilde{C}_{g}(K)|b,n_{0}\rangle=$
$\displaystyle\frac{g}{2}\sqrt{n_{0}(2b+n_{0}-1)},$ (113)
$\displaystyle\langle b,m=n_{0}+1|\tilde{C}_{g}(K)|b,n_{0}\rangle=$
$\displaystyle\frac{g}{2}\sqrt{(n_{0}+1)(2b+n_{0})}$ (114)

Eq. (111) shows that the associated transition probability for
$0\leftrightarrow 1$ is proportional to $b$.

####  III.4.2 Exact eigenvalues of $\tilde{C}_{g}(K)$

It follows from the explicit Hilbert space calculations in Ch. III.E and in
Appendix A that the operator $\tilde{C}_{g}(K)$ has the exact eigenvalues

$\tilde{c}_{g,b;n}=(n+b)\sqrt{1-g^{2}},~{}n=0,1,\ldots,$ (115)

so that the Hamiltonian (107) has the the eigenvalues

$\tilde{E}_{g,b;n}=\hbar\omega_{g}(n+b),~{}n=0,1,\ldots,~{}\omega_{g}=\sqrt{1-g^{2}}\omega,$
(116)

which reflects the (”redshifted”) frequency reduction (32) of the classical
motions in Ch. II.

###  III.5 Explicit Hilbert spaces for $\tilde{K}_{j},j=0,1,2$ and
$\tilde{C}_{g}(K)=\tilde{K}_{0}+g\tilde{K}_{1}$, spectra and eigenfunctions

Several explicit Hilbert spaces for concrete irreducible unitary
representations of the group $SO^{\uparrow}(1,2)$, its twofold covering the
symplectic group $Sp(2,\mathbb{R})$ (or the isomorphic ones $SU(1,1)$ and
$SL(2,\mathbb{R}$) and of all other covering groups as well have been
discussed in the literature Bargmann (1947); P. J. Sally (1967); Vilenkin
(1968); P. J. Sally (1970); Boyer and Wolf (1975); ka1 (e).

The associated self-adjoint Lie algebra generators $\tilde{K}_{j}$ all obey
the same commutation relations (10). The representation spaces include Hardy
spaces on the unit circle $\mathbb{S}^{1}$, Hilbert spaces of holomorphic
functions on the unit disc $\mathbb{D}=\\{\lambda\in\mathbb{C},|\lambda|<1\\}$
and also Hilbert spaces on the positive real line
$\mathbb{R}^{+}_{0}=\\{x\in\mathbb{R},x\geq 0\\}$. We shall present Hardy
space related Hilbert spaces for $b\in(0,1)$ here and discuss corresponding
Hilbert spaces on $\mathbb{R}^{+}_{0}$ in Appendix A: Hardy spaces on the unit
circle are closely related to the variable angle, whereas Hardy Hilbert spaces
on $\mathbb{R}^{+}_{0}$ are associated with the action variable $I$.

The following discussion follows closely those of Secs. 7.1 and 7.2 of Ref.
Kastrup (2007). Mathematical details like, e.g. questions concerning the
convergence of series or integrals, will be ignored in the following! The
associated justification can be found in the mathematical literature quoted
above.

####  III.5.1 Hardy space on the unit circle

A “Hardy space” $H^{2}_{+}(\mathbb{S}^{1},d\vartheta)$ is a closed subspace of
the usual Hilbert space $L^{2}(\mathbb{S}^{1},d\vartheta)$ on the unit circle
$\mathbb{S}^{1}$ with the scalar product

$(f_{2},f_{1})=\frac{1}{2\pi}\,\int_{S^{1}}d\vartheta\,f_{2}^{*}(\vartheta)f_{1}(\vartheta)\,,$
(117)

and the orthonormal basis

$e_{n}(\vartheta)=e^{i\,n\,\vartheta}\,,~{}~{}n\in\mathbb{Z}\,.$ (118)

The associated Hardy subspace $H^{2}_{+}(\mathbb{S}^{1},\,d\vartheta)$ is
spanned by the basis consisting of the elements with non-negative $n$, namely

$e_{n}(\vartheta)=e^{i\,n\,\vartheta}\,,~{}~{}n=0,1,2,\cdots\,.$ (119)

If we have two Fourier series $\in H^{2}_{+}(\mathbb{S}^{1}\,,d\vartheta)$,

$f_{1}(\vartheta)=\sum_{n=0}^{\infty}c_{n,1}\,e^{i\,n\,\vartheta}\,,~{}~{}f_{2}(\vartheta)=\sum_{n=0}^{\infty}c_{n,2}\,e^{i\,n\,\vartheta}\,,$
(120)

they have the scalar product

$(f_{2},f_{1})_{+}=\frac{1}{2\pi}\int_{S^{1}}d\vartheta\,f^{*}_{2}(\vartheta)f_{1}(\vartheta)=\sum_{n=0}^{\infty}c^{*}_{n,2}\,c_{n,1}$
(121)

and obey the boundary condition

$f_{j}(\vartheta+2\pi)=f_{j}(\vartheta),\;j=1,2.$ (122)

The coefficients $c_{n,j}$ are given by

$c_{n.j}=(e_{n},f_{j})_{+}.$ (123)

$Sp(2,\mathbb{R})$ Lie algebra generators are

$\displaystyle\tilde{K}_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{i}\partial_{\vartheta}+\frac{1}{2}\,,$ (124)
$\displaystyle\tilde{K}_{+}$ $\displaystyle=$ $\displaystyle
e^{i\,\vartheta}\,(\frac{1}{i}\partial_{\vartheta}+1),$ (125)
$\displaystyle\tilde{K}_{-}$ $\displaystyle=$ $\displaystyle
e^{-i\,\vartheta}\,\frac{1}{i}\partial_{\vartheta}.$ (126)

Thus, the Hardy space with the basis (119) provides a Hilbert space for the
conventional HO with the spectrum $\\{n+1/2$} and the operators (124)-(126)
act on the basis (119) as

$\displaystyle\tilde{K}_{0}\,e_{n}(\vartheta)$ $\displaystyle=$
$\displaystyle(n+\frac{1}{2})\,e_{n}(\vartheta)\,,$ (127)
$\displaystyle\tilde{K}_{+}\,e_{n}(\vartheta)$ $\displaystyle=$
$\displaystyle(n+1)\,e_{n+1}(\vartheta)\,,$ (128)
$\displaystyle\tilde{K}_{-}\,e_{n}(\vartheta)$ $\displaystyle=$ $\displaystyle
n\,e_{n-1}(\vartheta)\,,$ (129)

which are special cases of the relations (62), (58) and (59) with $b=1/2$.
Note that the ground state here is given by $e_{n=0}(\vartheta)=1$.

A possible generalization of the case $b=1/2$ for $b\in(0,1)$ within the same
Hilbert space as above can be obtained Bojowald and Strobl (2000); hol  by
inspection of the relations (62), (58) and (59):

$\displaystyle\tilde{K}_{0}$
$\displaystyle=N+b,~{}~{}N=\frac{1}{i}\partial_{\vartheta},$ (130)
$\displaystyle\tilde{K}_{+}$
$\displaystyle=e^{i\vartheta}[(N+2b)(N+1)]^{1/2},$ (131)
$\displaystyle\tilde{K}_{-}$
$\displaystyle=[(N+2b)(N+1)]^{1/2}e^{-i\vartheta}.$ (132)

Applied to the basis (119) these operators have the correct properties.

Defining the self-adjoint operator

$\displaystyle M_{b}(N)$ $\displaystyle=+[(N+2b)(N+1)]^{1/2}=M_{b}^{\dagger},$
(133) $\displaystyle=+[(\tilde{K}_{o}+b)(N+1)]^{1/2},$

with

$M_{b}(N)\,e^{in\vartheta}=+[(n+2b)(n+1)]^{1/2}\,e^{in\vartheta},$ (134)

the operators (131) and (132) can be written as

$\tilde{K}_{+}=(e^{i\vartheta}\,M_{b})\,,~{}~{}\tilde{K}_{-}=(M_{b}\,e^{-i\vartheta}).$
(135)

####  III.5.2 Hardy space related Hilbert spaces for general $b\in(0,1]$

Another possible representation for the more general case $b\in(0,1)$ can be
obtained by a generalization of the the scalar product (121):

Introducing on $H_{+}^{2}(\mathbb{S}^{1},d\vartheta)$ the positive definite
(self-adjoint) operator $A_{b}$ by the action

$\displaystyle A_{b}\,e_{n}(\vartheta)=$
$\displaystyle\frac{n!}{(2b)_{n}}e_{n}(\vartheta),~{}n=0,1,\ldots\,,~{}b>0,$
(136) $\displaystyle(a)_{n}=a\,(a+1)\,(a+2)\ldots(a+n-1)$
$\displaystyle~{}~{}~{}~{}~{}\;=\Gamma(a+n)/\Gamma(a),\,\,(a)_{n=0}=1,$

one can define an additional scalar product for functions

$f_{j}(\vartheta)=\sum_{n=0}^{\infty}c_{n,j}e_{n}(\vartheta)\,,~{}j=1,2,$
(137)

by

$(f_{2},f_{1})_{b,+}\equiv(f_{2},A_{b}\,f_{1})_{+}=\sum_{n=0}^{\infty}\frac{n!}{(2b)_{n}}\,c^{*}_{n,2}\,c_{n,1}\,,$
(138)

so that

$(e_{n},f_{1})_{b,+}=\frac{n!}{(2b)_{n}}\,c_{n,1}.$ (139)

We denote the (Hardy space associated) Hilbert space with the scalar product
(138) by $H^{2}_{b,+}(\mathbb{S}^{1},d\vartheta)$.

An orthonormal basis in this space is given by

$\displaystyle\hat{e}_{b,n}(\vartheta)=$
$\displaystyle\sqrt{\frac{(2b)_{n}}{n!}}\,e_{n}(\vartheta)\,,$ (140)
$\displaystyle(\hat{e}_{b,n_{2}},\hat{e}_{b,n_{1}})_{b,+}=\delta_{n_{2}n_{1}}.$

Two series

$f_{j}(\vartheta)=\sum_{n=0}^{\infty}a_{n,j}\,\hat{e}_{b,n}(\vartheta)\,,~{}j=1,2,$
(141)

have the obvious scalar product

$\displaystyle(f_{2},f_{1})_{b,+}=$
$\displaystyle\sum_{n=0}^{\infty}a^{*}_{n,2}\,a_{n,1},$ (142) $\displaystyle
a_{n,1}=$ $\displaystyle(\hat{e}_{b,n},f_{1})_{b,+}$ (143)

It follows that for a given function $f(\vartheta)$ its expansion coefficients
$c_{n}$ or $a_{n}$ with respect to $e_{n}$ or $e_{b,n}$ are related as follows

$c_{n}=(e_{n},f)_{+}=\sqrt{\frac{n!}{(2b)_{n}}}\,(\hat{e}_{b,n},f)_{+}=\sqrt{\frac{n!}{(2b)_{n}}}\,a_{n}$
(144)

As in general

$(f,f)_{+}\neq(f,f)_{b,+},$ (145)

one has to be careful in the case of quantum mechanical applications:

In a Hilbert space with scalar product $(f_{2},f_{1})$ one generally needs the
normalization $(f,f)=1$ for the usual probability interpretations. If
initially $(f,f)\neq 1$ one has to renormalize the state $f$: $f\to
f/\sqrt{(f,f)}$. So, if, e.g. $(f,f)_{+}=1$ in inequality (145), one has to
renormalize $f$ if one wants to determine transition probabilities and
expectation values etc. with respect to (138).

In $H^{2}_{b,+}(\mathbb{S}^{1},d\vartheta)$ the generators $\tilde{K}_{j}$
have the form ka1 (c, e)

$\displaystyle\tilde{K}_{0}=$
$\displaystyle\frac{1}{i}\,\partial_{\vartheta}+b,$ (146)
$\displaystyle\tilde{K}_{+}=$ $\displaystyle
e^{i\,\vartheta}(\frac{1}{i}\partial_{\vartheta}+2b),$ (147)
$\displaystyle\tilde{K}_{-}=$ $\displaystyle
e^{-i\,\vartheta}\frac{1}{i}\partial_{\vartheta},$ (148)

so that

$\displaystyle\tilde{K}_{1}=$
$\displaystyle\frac{1}{2}(\tilde{K}_{+}+\tilde{K}_{-})=\cos\vartheta\,\frac{1}{i}\partial_{\vartheta}+b\,e^{i\vartheta},$
(149) $\displaystyle\tilde{K}_{2}=$
$\displaystyle\frac{1}{2i}(\tilde{K}_{+}-\tilde{K}_{-})=\sin\vartheta\,\frac{1}{i}\partial_{\vartheta}-ib\,e^{i\vartheta}$
(150) $\displaystyle=$ $\displaystyle\tilde{K}_{1}(\vartheta-\pi/2).$

The operators (147) and (148) have the correct actions (58) and (59) on the
basis (140):

$\displaystyle\tilde{K}_{0}\,\hat{e}_{b,n}$ $\displaystyle=$
$\displaystyle(n+b)\,\hat{e}_{b,n}\,,$ (151)
$\displaystyle\tilde{K}_{+}\,\hat{e}_{b,n}$ $\displaystyle=$
$\displaystyle\sqrt{(2b+n)(n+1)}\,\hat{e}_{b,n+1}\,,$ (152)
$\displaystyle\tilde{K}_{-}\,\hat{e}_{b,n}$ $\displaystyle=$
$\displaystyle\sqrt{(2b+n-1)n}\,\hat{e}_{b,n-1}\,.$ (153)

The operators (147) and (148) are adjoint to each other only with respect to
the scalar product (138), not with respect to (121). Their adjointness with
respect to the scalar product (138) can be verified by taking two series (141)
and showing that

$(\tilde{K}_{-}f_{2},f_{1})_{b,+}=(f_{2},\tilde{K}_{+}f_{1})_{b,+}.$ (154)

This relation implies the self-adjointness of the operators (149) and (150).
Note that

$\displaystyle(e_{n_{2}},\hat{e}_{b,n_{1}})_{+}$
$\displaystyle=(\hat{e}_{b,n_{1}},e_{n_{2}})_{+}=\sqrt{\frac{(2b)_{n_{1}}}{n_{1}!}}\,\delta_{n_{2}\,n_{1}},$
(155) $\displaystyle(e_{n_{2}},\hat{e}_{b,n_{1}})_{b,+}$
$\displaystyle=(\hat{e}_{b,n_{1}},e_{n_{2}})_{b,+}=\sqrt{\frac{n_{1}!}{(2b)_{n_{1}}}}\,\delta_{n_{2}\,n_{1}},$
(156) $\displaystyle(\hat{e}_{b,n_{2}},\hat{e}_{b,n_{1}})_{+}$
$\displaystyle=\frac{(2b)_{n_{1}}}{n_{1}!}\,\delta_{n_{2}\,n_{1}},$ (157)
$\displaystyle(e_{n_{1}},e_{n_{2}})_{b,+}$
$\displaystyle=\frac{n_{1}!}{(2b)_{n_{1}}}\,\delta_{n_{2}\,n_{1}}\,.$ (158)

The Fock space ladder operators $A$ and $A^{\dagger}$ associated with the Lie
algebra generators (146)-(148) are given according to Eqs. (95).

The so-called “reproducing kernel” on $H^{2}_{b,+}$ is given by the
“completeness” relation

$\displaystyle\hat{A}_{b}(\vartheta_{2}-\vartheta_{1})=$
$\displaystyle\sum_{n=0}^{\infty}\hat{e}^{*}_{b,n}(\vartheta_{2})\,\hat{e}_{b,n}(\vartheta_{1})$
(159) $\displaystyle=[1-e^{i\,(\vartheta_{1}-\vartheta_{2})}]^{-2b}=$
$\displaystyle\hat{A}_{b}^{*}(\vartheta_{1}-\vartheta_{2})\,,$ (160)

where the identity

$\frac{(a)_{n}}{n!}=(-1)^{n}{-a\choose n}$ (161)

has been used. According to the relations (155) – (158) the kernel (159) has
the properties

$\frac{1}{2\pi}\int_{\mathbb{S}^{1}}d\vartheta_{2}\hat{A}_{b}(\vartheta_{2}-\vartheta_{1})\hat{e}_{b,m}(\vartheta_{2})=\hat{e}_{b,m}(\vartheta_{1}),$
(162)

or, written more formally in terms of the scalar product (138)

$\displaystyle(\hat{A}_{b}(1,2),\,\hat{e}_{b,m}(2))_{b,+}$
$\displaystyle=\hat{e}_{b,m}(\vartheta_{1}),$ (163)
$\displaystyle(\hat{A}_{b}(1,2),\,\hat{e}_{b,m}(2))_{+}$
$\displaystyle=\frac{(2b)_{m}}{m!}\,\hat{e}_{b,m}(\vartheta_{1}),$ (164)
$\displaystyle(\hat{A}_{b}(1,2),\,e_{m}(2))_{b,+}$
$\displaystyle=\sqrt{\frac{m!}{(2b)_{m}}}\,\hat{e}_{b,m}(\vartheta_{1})=e_{m}(\vartheta_{1}),$
(165) $\displaystyle(\hat{A}_{b}(1,2),\,e_{m}(2))_{+}$
$\displaystyle=\sqrt{\frac{(2b)_{m}}{m!}}\,\hat{e}_{b,m}(\vartheta_{1})$ (166)
$\displaystyle=\frac{(2b)_{m}}{m!}\,e_{m}(\vartheta_{1}).$

The numbers $1$ and $2$ mean the variables $\vartheta_{1}$ and
$\vartheta_{2}$, the latter being an integration variable.

The scalar product (138) itself may - according to Eq. (166) - be written as

$\frac{1}{(2\pi)^{2}}\int_{\mathbb{S}^{1}}d\vartheta_{2}\int_{\mathbb{S}^{1}}d\vartheta_{1}f_{2}^{\ast}(\vartheta_{2})A_{b}(\vartheta_{2}-\vartheta_{1})f_{1}(\vartheta_{1}),$
(167)

where the functions $f_{j}(\vartheta)$ are as in Eq. (141). If a function

$f(\vartheta_{2})=\sum_{n=0}^{\infty}a_{n}\hat{e}_{b,n}(\vartheta_{2})$ (168)

is an element of $H^{2}_{b,+}$ then it follows from (163) that

$(\hat{A}_{b}(1,2),\,f(2))_{b,+}=f(\vartheta_{1}).$ (169)

Thus, the “reproducing kernel” $\hat{A}_{b}(2,1)$ has - formally - similar
properties as the usual $\delta$-function.

The property (169) has the following calculational advantage: If one has two
functions (141) considered as elements of $H^{2}_{b,+}$, then their scalar
product (138) can be calculated as

$(f_{2},f_{1})_{b,+}=\frac{1}{2\pi}\int_{\mathbb{S}^{1}}\,d\vartheta\,f_{2}^{\ast}(\vartheta)f_{1}(\vartheta).$
(170)

Space reflection and time reversal

According to Eq. (1) space reflections $P$ can be implemented by the
substitution

$P:~{}~{}\vartheta\to\vartheta+\pi$ (171)

and time reversal $T$ by

$T:~{}~{}\vartheta\to-\vartheta.$ (172)

Quantum mechanically $T$ is anti-unitary, i.e. accompanied by complex
conjugation. Thus we get for the basis (140) and the operators (146), (149)
and (150)

$\displaystyle P:e_{b,n}(\vartheta)\to$
$\displaystyle~{}e_{b.n}(\vartheta+\pi)=(-1)^{n}e_{b,n}(\vartheta),$ (173)
$\displaystyle\tilde{K}_{0}(\vartheta)\to$
$\displaystyle~{}\tilde{K}_{0}(\vartheta+\pi)=\tilde{K}_{0}(\vartheta),$ (174)
$\displaystyle\tilde{K}_{1}(\vartheta)\to$
$\displaystyle~{}\tilde{K}_{1}(\vartheta+\pi)=-\tilde{K}_{1}(\vartheta),$
(175) $\displaystyle\tilde{K}_{2}(\vartheta)\to$
$\displaystyle~{}\tilde{K}_{2}(\vartheta+\pi)=-\tilde{K}_{2}(\vartheta),$
(176)

and

$\displaystyle T:e_{b,n}(\vartheta)\to$
$\displaystyle~{}[e_{b.n}(-\vartheta)]^{\ast}=e_{b,n}(\vartheta),$ (177)
$\displaystyle\tilde{K}_{0}(\vartheta)\to$
$\displaystyle~{}[\tilde{K}_{0}(-\vartheta)]^{\ast}=\tilde{K}_{0}(\vartheta),$
(178) $\displaystyle\tilde{K}_{1}(\vartheta)\to$
$\displaystyle~{}[\tilde{K}_{1}(-\vartheta)]^{\ast}=\tilde{K}_{1}(\vartheta),$
(179) $\displaystyle\tilde{K}_{2}(\vartheta)\to$
$\displaystyle~{}[\tilde{K}_{2}(-\vartheta)]^{\ast}=-\tilde{K}_{2}(\vartheta),$
(180)

####  III.5.3 A unitary transformation by a change of basis

In the above discussion the $b$-dependence of the representation on
$H^{2}_{b,+}$ is contained in the Lie operators (146)-(148) and in the
metrical operator $A_{b}$ of Eq. (136), but not in the basis
$e_{n}(\vartheta)$ of $H^{2}_{+}$ we started from. Thus, all non-equivalent
irreducible representations for different $b$ are implemented by starting from
the Hardy space $H^{2}_{+}$ with the $b$-independent basis (119). By the
unitary transformation

$e_{n}(\vartheta)=e^{i\,n\,\vartheta}\to
e_{b,n}(\vartheta)=e^{i\,(n+b)\,\vartheta}\,,$ (181)

one can pass to $b$-dependent Hilbert spaces
$\hat{H}^{2}_{+}(\mathbb{S}^{1},d\vartheta;b)$ for functions with the boundary
condition

$e_{b,n}(\vartheta+2\pi)=e^{2i\,b\,\pi}\,e_{b,n}(\vartheta)\,.$ (182)

Now each irreducible unitary representation characterized by the number $b$
has its own Hilbert space, each with the scalar product (138) and with the
basis

$\hat{e}_{b,n}(\vartheta)=\sqrt{\frac{(2b)_{n}}{n!}}\,e_{b,n}(\vartheta)\,.$
(183)

The “reproducing kernel” here is

$\displaystyle A^{(b)}(\vartheta_{2}-\vartheta_{1})=$
$\displaystyle\sum_{n=0}^{\infty}{e}^{*}_{b,n}(\vartheta_{2})\,{e}_{b,n}(\vartheta_{1})$
(184) $\displaystyle=$ $\displaystyle
e^{ib(\vartheta_{1}-\vartheta_{2}}[1-e^{i\,(\vartheta_{1}-\vartheta_{2})}]^{-2b}$
$\displaystyle=$ $\displaystyle
e^{ib(\vartheta_{1}-\vartheta_{2})}\hat{A}_{b}(\vartheta_{2}-\vartheta_{1})\,,$

The generators (146)-(148) now take the form

$\displaystyle\tilde{K}_{0}$
$\displaystyle=\frac{1}{i}\,\partial_{\vartheta},$ (185)
$\displaystyle\tilde{K}_{+}=$ $\displaystyle
e^{i\,\vartheta}(\frac{1}{i}\,\partial_{\vartheta}+b),$ (186)
$\displaystyle\tilde{K}_{-}=$ $\displaystyle
e^{-i\,\vartheta}\,(\frac{1}{i}\,\partial_{\vartheta}-b)$ (187)
$\displaystyle\tilde{K}_{1}=$
$\displaystyle\cos\vartheta\frac{1}{i}\,\partial_{\vartheta}+ib\sin\vartheta,$
(188) $\displaystyle\tilde{K}_{2}$
$\displaystyle=\sin\vartheta\frac{1}{i}\,\partial_{\vartheta}-ib\cos\vartheta$
(189) $\displaystyle=$ $\displaystyle\tilde{K}_{1}(\vartheta-\pi/2)$

Note that the operator $\tilde{K}_{2}$ here, too, is obtained from
$\tilde{K}_{1}$ by replacing $\vartheta$ with $\vartheta-\pi/2$ in the latter.

Concerning the operations $P(\vartheta\rightarrow\vartheta+\pi)$ and
$T(\vartheta\rightarrow-\vartheta)$ applied to the basis (183) and the
operators (185), (188) and (189) compared to the properties (173)-(180) there
is only a change for the basis (183) for $P$:

$P:~{}~{}\hat{e}_{b,n}(\vartheta)\to\hat{e}_{b,n}(\vartheta+\pi)=(-1)^{n}e^{i\pi
b}\,\hat{e}_{b,n}(\vartheta).$ (190)

The global constant phase factor $\exp(i\pi b)$ can be interpreted as
representing a new type of “fractional” statistics in 2 dimensions kas (d), of
particles called “anyons” (see references below).

####  III.5.4 Aharonov-Bohm-, (fractional) quantum Hall-effects, anyons,
Berry’s phase,
Bloch waves etc.

The property of the “naive” planar rotation operator (185) to have a
1-parametric set of possible spectra - parametrized by the number $b$ \- is a
mathematical consequence of the fact that the operator has a 1-parametric set
of self-adjoint extensions kas (e).

For physical systems topologically related to a punctured plane, the parameter
$b$ can have different physical meanings kas (f):

In the description of Aharonov-Bohm effect kas (f); Peshkin and Tonomura
(1989); Hegerfeldt and Neumann (2008) (historically more appropriate:
“Ehrenberg-Siday-Aharonov-Bohm effect” Hiley ) the index $b$ is proportional
to the magnetic flux $\Phi$ crossing the plane.

The magnetic flux model can also help to understand the quantum Hall effect
Laughlin (1981); Avron  _et al._ ; Hansson  _et al._ (2017). It can also do so
for the fractional quantum Hall effect Laughlin (1999); Hansson  _et al._
(2017); Halperin (2020), especially in the framework of anyons kas (g);
Rosenow  _et al._ (2016); et al. (2020b) and related Chern-Simons theories
Fröhlich and Marchetti (1991, 1989). As special Chern-Simons theories have the
structure group $SO^{\uparrow}(1,2)$ they may help to find the appropriate
theoretical framework for the dynamics associated with with the symplectic
spectra Eq. (62).

Closely related are the properties of Berry’s phase Berry (1984, ); Shapere
and Wilczek (1989); Avron  _et al._ .

In the case of Bloch waves $b$ represents the momentum inside the first
Brillouin zone kas (h).

####  III.5.5 The operator $\tilde{C}_{g}(K)=\tilde{K}_{0}+g\tilde{K}_{1}$ on
$H^{2}_{b,+}$

According to Eq. (149) the operator
$\tilde{C}_{g}(K)=\tilde{K}_{0}+g\,\tilde{K}_{1}$ here has the form

$\tilde{C}_{g}(K)=(1+g\cos\vartheta)\frac{1}{i}\partial_{\vartheta}+b\,(1+ge^{i\vartheta})$
(191)

The eigenvalue differential equation

$\tilde{C}_{g}(K)f_{g,b}(\vartheta)=\tilde{c}_{g,b}\,f_{g,b}(\vartheta)$ (192)

has the general solution gra (b), with $0\leq g<1$,

$\displaystyle f_{g,b}(\vartheta)$
$\displaystyle=C\,(1+g\cos\vartheta)^{-b}\,e^{i\tilde{c}_{g,b}(1-g^{2})^{-1/2}\,\chi(\vartheta)-ib\vartheta},$
(193)
$\displaystyle\chi(\vartheta)=2\arctan\left[\sqrt{\frac{1-g}{1+g}}\tan(\vartheta/2)\right],$
(194) $\displaystyle C=\text{const}.$

The boundary condition

$f_{g,b}(\vartheta+2\pi)=f_{g,b}(\vartheta)$ (195)

implies

$\tilde{c}_{g,b}=(n+b)\sqrt{1-g^{2}}\equiv\tilde{c}_{g,b;n},~{}n=0,1,\ldots,$
(196)

The implementation of the boundary condition (195) includes the transformation
$\chi\to\chi+2\pi$.

The result (196) coincides with Eq. (292) in Appendix A and corresponds to the
classical result (32).

Thus, we have for $\tilde{C}_{g}(K)$ the eigenfunctions

$\displaystyle f_{g,b;n}(\vartheta)=C\,$
$\displaystyle(1+g\cos\vartheta)^{-b}\,e^{i(n+b)\chi(\vartheta)-ib\vartheta},$
(197) $\displaystyle n=0,1,\ldots,$
$\displaystyle\tilde{C}_{g}(K)\,f_{g,b;n}(\vartheta)=$
$\displaystyle(n+b)\sqrt{1-g^{2}}\,f_{g,b;n}(\vartheta).$ (198)

The constant $C$ in the solution (193) can be determined from the
normalization condition $(f_{g,b},f_{g,b})=1$. Using the relation (170) leads
to the integral gra (c)

$\displaystyle 1$
$\displaystyle=|C_{g,b}|^{2}\int_{0}^{2\pi}\frac{d\vartheta}{2\pi}\,(1+g\cos\vartheta)^{-2b}$
(199) $\displaystyle=|C_{g,b}|^{2}(1-g^{2})^{-b}\,P_{2b-1}[(1-g^{2})^{-1/2}],$

from which the normalization constant $C_{g,b}=|C_{g,b}|$ can be determined.
It is independent of $n$. Here $P_{\nu}(x),\,{\nu\in\mathbb{R}},\ x\geq 0$, is
the Legendre function of the first kind gra (d); Whittaker and Watson (1969).
It has - among others - the properties
$P_{\nu}(x)=P_{-\nu-1}(x),\,P_{\nu}(1)=1,\,P_{\nu=0}(x)=1$.

###  III.6 Nonlinear interactions in terms
of the operators $\tilde{K}_{j}$

The HO model plays an important role in molecular physics (see the next
chapter): For example, the nuclei of diatomic molecules can oscillate relative
to each other along their connecting axis. As long as the associated energy
levels are small compared to the dissociation energy $V_{0}$ one can
approximate those vibrations by a one-dimensional HO the potential of which is
centered at the equilibrium point $q_{0}=r_{0}>0$ her (a). For higher energies
when dissociation becomes relevant, the HO is no longer an appropriate model.

####  III.6.1 The Morse potential for molecular vibrations

In order to take dissociation into account Morse suggested the potential ka1
(f)

$V_{Mo}(q)=V_{0}\,(e^{-aq}-1)^{2},\,\,V_{0},\,a>0,$ (200)

where $q$ is the distance of the atomic nuclei from their point of equlibrium
$q_{0}$. For $aq\ll 1$ this becomes a HO potential

$V_{Mo}(q)\approx\frac{1}{2}m\omega_{0}^{2}\,q^{2},\,\,\omega_{0}=a\sqrt{2V_{0}/m},$
(201)

where $m$ is the reduced mass of the two nuclei.

In addition

$V_{Mo}(q\to\infty)=V_{0},\,\,V_{Mo}(q\to-\infty)=+\infty.$ (202)

For $aq\ll-1$ the potential describes some kind of “hard core”. The
modifications for $q=r\geq 0$ being the radial variable are discussed in Ref.
ter Haar (1946).

If $E<V_{0}$ the classical motions are bounded and periodic. The system is
also integrable, i.e. there exist canonical angle and (constant) action
variables in order to describe the sytem ka1 (f). The relationship between
constant energy $E<V_{0}$ and action variable $I$ turns out to be (see Eq.
(21))

$\omega_{0}I=2V_{0}(1-\sqrt{1-E/V_{0}},\;\;E=\omega_{0}I\,\left(1-\frac{\omega_{0}I}{4V_{0}}\right).$
(203)

This gives the Hamilton function

$H_{Mo}(\varphi,I)=\omega_{0}I\left(1-\frac{\omega_{0}I}{4\,V_{0}}\right),$
(204)

with the associated Eqs. of motion

$\displaystyle\dot{\varphi}=$
$\displaystyle\partial_{I}H=\omega_{0}(1-\frac{\omega_{0}I}{2V_{0}})\equiv\omega_{I},$
(205) $\displaystyle\dot{I}=$
$\displaystyle-\partial_{\varphi}H=0,~{}\Rightarrow\;\omega_{I}=\text{
const.}$ (206)

Note that here

$E(I)=\frac{1}{2}(\omega_{0}+\omega_{I})\,I.$ (207)

Replacing the action variable $I$ in the Hamilton function (204) by the
operator $\hbar\tilde{K}_{0}$ leads to the spectrum

$\displaystyle E_{b,n}=$
$\displaystyle\hbar\omega_{0}(n+b)\left[1-\frac{\hbar\omega_{0}}{4\,V_{o}}(n+b)\right],$
(208) $\displaystyle E_{b,n=0}=$
$\displaystyle\hbar\omega_{0}b\left(1-\frac{\hbar\omega_{0}}{4\,V_{o}}b\right).$
(209)

For the bracket […] in Eq. (208) to be positive only those $n$ are allowed
which imply this property.

This system is a simple but instructive example how the use of the canonical
pair angle and action variables instead of the canonical position and momentum
can simplify the description of the dynamics of the system, at the expense of
making it intuitively less accessible! That might be especially so if the
system is not completely integrable and the action variable a function of
time, too, as in the model of Ch. II above.

####  III.6.2 Potentials involving the terms $\tilde{K}_{+}\tilde{K}_{-}$ and
$\tilde{K}_{-}\tilde{K}_{+}$

Due to the Casimir operator relations (63) the eigenvalue equations of the
(dimensionless) Hamiltonians (up to a factor $\hbar\omega$)

$\tilde{F}_{-}=\tilde{K}_{0}+g_{-}\,\tilde{K}_{+}\tilde{K}_{-}$ (210)

and

$\tilde{F}_{+}=\tilde{K}_{0}+g_{+}\,\tilde{K}_{-}\tilde{K}_{+}$ (211)

can be solved immediately: Eigenvectors are still those of $\tilde{K}_{0}$
(see Eq. (62)) and the eigenvalues are

$\tilde{f}_{g_{-};b,n}=n+b+g_{-}(n+2b-1)n\,;~{}\tilde{f}_{g_{-};b,n=0}=b,$
(212)

and

$\tilde{f}_{g_{+};b,n}=n+b+g_{+}(n+2b)(n+1)\,;~{}\tilde{f}_{g_{+};b,n=0}=b(1+2g_{+}).$
(213)

The models describe the annihilation and creation of quanta (Eq. (210)) and
vice versa (Eq. (211)). The couplings $g_{-}$ and $g_{+}$ may depend on
external parameters.

##  IV Reflections on possible experiments
and observations

Replacing the ingrained and very successful habit of describing the quantum HO
by the “canonical” pair position and momentum operators (or the associated
creation and annihilation operators) by the quantum version of its classical -
locally - equivalent canonical pair angle and action variables may appear
unnecessarily artificial and even unnatural:

Compared to position and momentum variables the pair angle and action
variables is less familiar as far as visualization and perception are
concerned:

Whereas the angle can be illustrated well as a fraction of the unit circle and
its s-fold coverings by the corresponding number of rotations of the hand of a
clock, a visualization of the action variable is not so obvious. True, all
quantum mechanical action variables must - in principle - be proportional to
Planck’s constant $\hbar$ and for integrable systems it appears to be closely
related to the conserved quantity “energy”. But we have seen in Ch. II that
the action variable $I$ may be quite useful as a coordinate even if it is not
a constant of motion. For such time–dependent $I(t)$ the relation (23) may be
a helpful tool for an intuitive interpretation.

An important lesson from Ch. II for the discussions below is that the energy
$E$ may be conserved even if the action variable $I(t)$ is not!

Perhaps we have to go beyond the use of position and momentum as the basic
observables in a part of the quantum world where other “canonical” observables
are more appropriate! This is, of course, a larger challenge for a
reformulation of (perturbative) quantum field theories etc., for which the
orthodox description of the HO is a fundamental building block!

In view of the qualitative differences between the global phase spaces (7) and
(I) and their possible physical implications - especially for the associated
quantum theory - it is obviously important to make experimental and
observational attempts to look for corresponding phenomena in nature!

All the following considerations apply, of course, only, if the mathematical
models from the previous chapters have counterparts in nature! For this reason
all possible applications discussed in the following are hypothetical! The
good news is that the relevance of the model can be tested in the laboratory
and by astrophysical observations! The (hopefully preliminary) bad news is
that the associated theoretical framework for the dynamics governing
transition rates etc. involving the new spectra still has to be worked out!

###  IV.1 Generalities

In view of their possible far-reaching implications the above theoretical
results should, of course, be subject to critical reviews and be probed
experimentally! In the following - as a kind of “tour d’horizon” - ideas and
suggestions for such experiments and observations are discussed, in the hope
that a few experimentalists will be motivated and inclined to meet the
challenge and that experts - experimentalists and theoreticians - in the areas
of physics mentioned below, will point out possible misunderstandings and will
suggest improvements and consequences!

Harmonically oscillating quantum systems can be found in many areas of
physics, at least approximately close to the corresponding (local) minima of
classical “binding” potentials with periodic motions.

It is important to note that the “symplectic” or “fractional” spectrum (9) is
tied to the groups $U(1)$ or $O(2)$ and their infinitely many coverung groups,
but not to the rotation group $SO(3)$ and its single 2-fold covering $SU(2)$.
Accordingly one has to look for 2-dimensional (sub)systems with “effective”
phase spaces (I). Such systems may be found in molecular spectroscopy (e.g.
diatomic molecules), quantum optics, optomechanics and - possibly - in
astrophysics (“dark” energy and “dark” matter, see below).

One obvious question is: Why haven’t we seen those symplectic $b$-dependent
spectra yet? Several answers are possible:

0\. They just don’t exist in nature!

1\. One possible reason is that no one has looked for them. This is quite
plausible if the “visibility” of those symplectic spectra is very weak, as,
e.g. for infrared emission or absorption lines of homonuclear diatomic
molecules like H2, because they have no electric dipole moment or because
their Stokes or Anti-Stokes lines in inelastic Raman scattering off vibrating
and rotating molecules (see below) are very weak.

2\. As discussed in Section III.C the impact of the (composite) ”orthodox”
Fock space annihilation and creation operators (95) and (96) with the usual
properties (97), (98) and (99) may dominate and obscure the symplectic spectra
(69), except for the value $b=1/2$. Thus, one has to find means in order to
discover other (fractional) parts of the spectra (69), if they exist at all!
In any case, their observability appears to be rather weak.

3\. Transitions - radiative, non-radiative, collisional, Raman-type etc. -
between different levels of the spectra (69) require appropiate kinds of
electromagnetic interactions, the dynamics of which has not yet been worked
out!

Consider two generally different levels of the spectra (69):

$E_{b_{j},n_{b_{j}}}=\hbar\omega_{j}(n_{b_{j}}+b_{j}),\,j=1,2;\,n_{b_{j}}=0,1,\ldots.$
(214)

For a fixed $b=b_{1}=b_{2};\,$ and $\omega=\omega_{1}=\omega_{2}$ the
observable energy difference between an upper level characterized by
$n_{b}^{\prime}$ and a lower level characterized by
$n_{b}^{\prime\prime}<n_{b}^{\prime}$.

$E_{b,n_{b}^{\prime}}-E_{b,n_{b}^{\prime\prime}}=\hbar\omega(n_{b}^{\prime}-n_{b}^{\prime\prime})$
(215)

cannot be distinguished from the corresponding difference for a $b_{2}\neq b$.

More interesting is a transition with a change of the B-index
($b_{1}\leftrightarrow b_{2}$):

$E_{b_{1},n_{b_{1}}^{\prime}}-E_{b_{2},n_{b_{2}}^{\prime\prime}}=\hbar\omega[(n_{b_{1}}^{\prime}-n_{b_{2}}^{\prime\prime})+(b_{1}-b_{2})].$
(216)

If such transitions are possible, e.g. for
$n_{b_{2}}^{\prime\prime}=n_{b_{1}}^{\prime}=0$ and as - up to now - the only
condition on $b_{2}$ (and $b_{1}$) is the inequality $b_{2}>0$ one may have a
cascade of (fluorescence) transitions

$b_{1}\rightarrow b_{3}\rightarrow\cdots b_{m}\rightarrow b_{2}>0,$ (217)

accompanied by the emission of $m-1$ low-frequency (lower than $\omega$)
quanta. Even a continuum between $b_{1}$ and $b_{2}$ appears possible. All
this depends on the still to be established associated dynamics, which
determines rates and selection rules! If the initial quanta cascade down the
“fluorescence” sequence (217) they can end up in the microwave or even
radiowave region, without loss of the total energy!

Figure 3:  Comparison of the orthodox and the symplectic spectra of the HO,
with indications of possible transitions between levels (ignoring any
selection rules)

If $\omega_{2}\neq\omega_{1}$ (occurs for diatomic molecules with different
isotopic atoms and for electronic transitions between local minima of
different Born-Oppenheimer potentials for the nuclei; see below), one has

$(E_{b_{1},n_{b_{1}}^{\prime}}-E_{b_{2},n_{b_{2}}^{\prime\prime}})/\hbar=\omega_{1}[n_{b_{1}}^{\prime}-\sigma\,n_{b_{1}}^{\prime\prime}+b_{1}-\sigma
b_{2}],~{}\sigma=\frac{\omega_{2}}{\omega_{1}}.$ (218)

4\. As discussed in Section III.B above, for a given $\omega$ one needs the
time $t_{s},\,\omega\,t_{s}=2\pi s\,,\,s\in\mathbb{Z},$ in order to “run”
through an $s$-fold covering $\mathbb{S}^{1}_{[s]}$ of the circle
$\mathbb{S}^{1}$. The prefactor $\exp(-ib\,\omega t)$ of the general state
(92) shows that here the time “angle” $\tilde{t}=\omega t$ can be reduced by a
small $b\ll 1/2$!

For infrared light the number $s$ of coverings is obviously very large for a
finite time interval $\Delta t\gg T=1/(2\pi\omega)$, where $\omega\approx
10^{14}$Hz in the near infrared.

5\. In Section III.D above we discussed the $n\to n\pm 1$ transition
amplitudes for the “primitive” effective Hamiltonian (107) the interaction
term of which mimics partial properties of an electric dipole moment. If one
wants to include the influence of external electromagnetic radiation, one has
to allow the coupling term $g$ to depend explicitly on time or via other
external parameters or fields Merzbacher (1970).

6\. For experimental tests it is essential to find quantities (“observables”)
which are especially sensitive to values of the B(argmann)-index $b$. The
following is an - incomplete - list of possible theoretically interesting
experiments (without proper knowledge of their feasibility in the laboratory
or of their observability in the sky)!

###  IV.2 Vibrating diatomic molecules

Among the most important and interesting oscillators the above results may
apply to are vibrating diatomic molecules (for introductions to their physics
see, e.g. the textbooks Herzberg (1950); Landau and Lifshitz (1965);
Gasiorowicz (1974); Atkins  _et al._ (2018); Atkins and Friedman (2005);
Parigger and Hornkohl (2020)). They have one vibrational degree of freedom:
oscillations about the point of equilibrium along the line connecting the two
nuclei (“internuclear axis” = INA). Near that equilibrium point the potential
may be considered to be harmonic. In the Born-Oppenheimer (BO) approximation
the effective potentials for the vibrating nuclei are provided by energy
configurations of the electron “cloud” the dynamics of which depends only
“adiabatically” on the state of the nuclei, especially on their distance $R$
(see Fig.4).

The (classical) angular frequency $\omega=2\pi\nu$ for the mutual harmonic
vibrations of the nuclei is given by

$\omega=2\pi\,\nu=\sqrt{k/\mu},$ (219)

where $k$ is the “force constant”, determined - in the BO approximation -
solely by the actual electronic configuration and
$\mu=m_{1}\,m_{2}/(m_{1}+m_{2})$ is the “reduced” mass of the two vibrating
atoms.

Spectroscopists denote the vibrational level numbers $n$ of the HO by $v$ and
give the frequencies $\nu\,$[s-1] in terms of the “wave number”
$\tilde{\nu}=\nu/c\,$[cm-1]. One then has the (approximate) equivalences

$1\,\text{eV}\cong 8066\,\text{cm}^{-1}\cong 11605^{\circ}\text{ K}.$ (220)

Spectroscopically the differences between homonuclear (equal nuclei like
molecular hydrogen H2 or oxygen O2) and heteronuclear (different nuclei like
carbon monoxyde 12C16O) diatomic molecules are important: because of space
reflection symmetry the homonuclear molecules do not have a permanent electric
dipole moment and, therefore, no corresponding infrared emissions or
absorptions lan . If, however, their polarizability is nonvanishing, they can
have induced electric dipole moments, e.g. in case of elastic and inelastic
Raman-type scattering or by collisions.

In addition to the vibrational energy levels characterized by the numbers
$v=0,1,2\ldots$ the diatomic molecules have rotational levels $J=0.1.2,\ldots$
due to the rotations of the molecule around an axis which lies in a plane
perpendicular to the INA and passing through the centre of mass on that axis.
So in general one has the combined vibration - rotation (“rovibrational”)
transitions $(v^{\prime},J^{\prime})\to(v^{\prime\prime},J^{\prime\prime})$.
The frequencies of the vibrational transitions are generally in the “near-
infrared” (frequencies around $\nu\approx 10^{14}$ s-1) and those of the
rotational ones are at least one order of magnitude smaller and are in the
“far-infrared” or microwave region.

Example: Molecular hydrogen H2

Here are some essential properties of the molecule H2 which are importent for
our present discussion: As a homonuclear diatomic molecule H2 has no permanent
electric dipole moment (this property is frequently mentioned in the
literature, but very rarely proven; for a proof see Ref. lan ). Because of
this missing electric dipole moment their is no corresponding infrared
emission or absorption.

Figure 4:  Some of the lowest ”binding” electronic BO potentials for the
vibrations of the two H2 nuclei as a function of their distance $R$. The zero
energy on the ordinate coincides with the zero-point energy of the electronic
potential X ${}^{1}\Sigma^{+}_{g}$. The numerical values are taken from Ref.
Sharp (1971). For the interpretation of the terms denoting the different
curves see Appendix B.

There is, however, (weak) magnetic dipole and electric quadrupole infrared
radiation et al. (2019a).

Due to that missing electric dipole moment there are no direct vibrational
transitions $v\leftrightarrow v\pm 1$ within a given electronic BO –
potential, like the electronic ground state potential X ${}^{1}\Sigma^{+}_{g}$
(see Fig. 4).

As a consequence, in order to experimentally analyse the ladder of vibrational
states of, e.g. the BO electronic ground state potential X
${}^{1}\Sigma^{+}_{g}$, an “ultraviolet detour” has to be taken: one first
initiates an ultraviolet allowed (1- or 2-$\gamma$) absorptive transition from
the electronic ground state to a vibrational level of a higher BO electronic
potential (Fig. 4), from where the photons cascade down (in 1 or more steps)
to a vibrational level of X ${}^{1}\Sigma^{+}_{g}$ which is different of the
one the photons originally started from. The difference of the observed
ultraviolet frequencies then provides information about the vibrational levels
of the selected BO potential Sharp (1971); Glass-Maujean  _et al._ (1984);
Sternberg (1989); et al. (1993a, b); Roncin and Launay (1994); Bailly  _et
al._ (2010); Hancock  _et al._ (2004); et al. (2011, 2013, 2014, 2018b).

The “ultraviolet detour” also plays an essential role in the so-called
“Solomon process” which leads to photodissociation of H2 Field  _et al._
(1966); Stecher and Williams (1967); loe (a); Loeb and Furlanetto (2013).

Another possibility to observe vibrational and rovibrational levels of H2
electronic BO potentials is provided by the polarizability of the molecule,
which allows for Raman-type transitions associated with induced electric
dipole moments, induced by by external light beams or be collisionspol (a);
Veirs and Rosenblatt (1987); McCann and Hampton (1994); Long (2002); Li and et
al. (2018).

The two nuclei (protons) oscillating in the binding electronic BO potentials
may have antiparallel spins (para-H2) or parallel ones (ortho-H2). For recent
summeries and reviews of the role of H2 in different areas of physics see,
e.g. Field  _et al._ (1966); Sharp (1971); Sprecher  _et al._ (2011); Ubachs
_et al._ (2016); et al. (2017, 2019b). More references will be quoted in the
course of the discussions below. (Numerical values of quantities mentined
below are rounded up/down from their impressively determined accurate
theoretical and experimental values).

For the nuclear vibrations of the diatomic homonuclear molecules H2 in the
electronic ground state X ${}^{1}\Sigma^{+}_{g}$ BO potential (Fig. 4) one has
for the “transition” (”ground tone”) et al. (2019c, 2013)

$\displaystyle(\Delta\,v,\Delta\,J):\,$
$\displaystyle(v^{\prime\prime}=0,J^{\prime\prime}=0)\leftrightarrow(v^{\prime}=1,J^{\prime}=0)$
(221) $\displaystyle\approx(\pm[\tilde{\nu}_{0\leftrightarrow 1}\approx
4161\text{ cm}^{-1}],0)$

which is one of the larger values for vibrating diatomic molecules.

Recall that the BO electronic ground state X${}^{1}\Sigma^{+}_{g}$ is an
effective potential for the vibrations of the two nuclei, depending on their
distance $R$ (Fig. 4).

The vibrational transition value (221) correponds to about $0.516\,$ eV, a
wave length $\lambda=1/\tilde{\nu}\approx 2,4\,\mu$m and a temperature of
$\approx 5988^{\circ}$ K.

In comparison the rotational transition $(0,0)\leftrightarrow(0,1)$ has the
wave number $\tilde{\nu}\approx 118$ cm${}^{-1}\cong 0.0146$ eV et al.
(2019c). This means a wavelength $\lambda\approx 85\,\mu$m.

If the vibrating H2 molecule were an ideal HO, its “orthodox” zero-point
energy, according to Eq. (221), would be

$\displaystyle E_{0}(X~{}^{1}\Sigma^{+}_{g})$
$\displaystyle=\frac{1}{2}\tilde{\nu}_{0\leftrightarrow
1}(X~{}^{1}\Sigma^{+}_{g})$ (222) $\displaystyle\approx 2080\text{
cm}^{-1}\cong 0.258\text{ eV}\cong 2994^{\circ}\text{ K}.$

The vibrating molecule H2 is, of course, no ideal HO because it dissociates at
a finite energy $D_{0}>E_{0}$. The Morse potential (200) takes this
qualitatively into account, as can be seen from the relations (202). The
“anharmonic” modifications of energy (204) and angular frequency (205) are
small as long as $\omega_{0}I\ll V_{0}$.

The quantum mechanical energy (208) can be written as

$E_{b,n}=\hbar\omega_{0}\,(n+b)-\frac{(\hbar\omega_{0})^{2}}{4\,V_{o}}(n+b)^{2},$
(223)

which may be considered as a polynomial in $(n+b)$. As the Morse potential
still is only a rough approximation, one has taken - for the orthodox value
$b=1/2$ \- the expression (223) as a suggestion to parametrize the vibration
and rotation levels generally by Dunham (1932); her (a); Irikura (2007)

$E_{v,J}=\sum_{i\geq 0,j\geq 0}Y_{i,j}\,(v+1/2)^{i}J^{j}(J+1)^{j},$ (224)

where the coefficients $Y_{i,j}$ are determined (mainly) experimentally. The
ground state (“zero-point”) energy is given by

$E_{0,0}=\sum_{i\geq 0}Y_{i,0}(1/2)^{i}.$ (225)

For the Morse potential one has
$Y_{1,0}=\hbar\omega_{0},\,Y_{2,0}=-\hbar\omega_{0}/(4V_{0})$, all other
$Y_{i,0}$ vanishing.

The approximation ansatz (224) gives for H2 instead of (222) the value Irikura
(2007)

$E_{0}[\text{H}_{2})\approx 2180\text{ cm}^{-1}.$ (226)

Thus, by passing from the orthodox HO spectrum ($b=1/2$), usually associated
with H2 infrared vibrations, to the symplectic one [$b\in(0,1)$] one can lower
the zero-point energy of the BO potential X ${}^{1}\Sigma^{+}_{g}$ maximally
by the (approximate) amount

$\displaystyle E_{0}(H_{2};$ $\displaystyle b\to
0)-E_{0}(H_{2};b=1/2)\equiv\hat{E}_{0}(H_{2})$ (227) $\displaystyle\approx$
$\displaystyle 2100\text{ cm}^{-1}\cong 0.26\text{ eV}\cong 3000^{\circ}\text{
K }.$

Similarly, the known dissociation energy of H2 Sprecher  _et al._ (2011); et
al. (2018c, 2019b); Puchalski  _et al._ (2019)

$D_{0}\approx 36118\text{ cm}^{-1}\cong 4.48\text{ eV}$ (228)

\- theoretically - increases maximally by the the amount (227).

Thus, the orthodox H2 vibrational spectrum (b = 1/2) can be considerably
“detuned” for $1/2>b>0$.

The difference (227) between the orthodox and the symplectic ground states of
the vibrating H2 molecule implies an additional effecive Boltzmann factor

$e^{-\hat{E}_{0}(H_{2})/(k_{B}T)}=e^{-3000^{\circ}/T}.$ (229)

Preliminarily ignoring all dynamical mechanisms the last Eq. says that for
$T<3000^{\circ}$K the symplectic ground state becomes preferred statistically.
This will play a role in our astrophysical discussion below. It also indicates
that the symplectic HO spectra may be observed better at very low
temperatures.

As mentioned above, in the BO approximation the electronic ground state X
${}^{1}\Sigma^{+}_{g}$ (which includes the action of the nuclear Coulomb
potentials on the electrons) provides a potential for the two vibrating nuclei
as a function of their distance $R$. The potential has a minimum around which
the oscillations are approximately harmonic. The same applies to the next
higher electronic (metastable) states B ${}^{1}\Sigma_{u}^{+}$, E F
${}^{1}\Sigma^{+}_{g}$ and C ${}^{1}\Pi_{u}$. They have local minima in
appropriate neighbourhoods of which the vibrations are harmonic, too (see Fig.
4).

In the following list one can find the measured energy differences between the
ground states of the different electronic levels relative to X
${}^{1}\Sigma^{+}_{g}(v=0,J=0)$ and the energies $E_{0\leftrightarrow 1}$ of
the first vibrational excitations $(v=0,J=0)\to(v=1,J=0)$ above those ground
states Sharp (1971); Glass-Maujean  _et al._ (1984); Roncin and Launay (1994);
et al. (1993a, b, 2006); Bailly  _et al._ (2010); et al. (2011, 2014). The
data here are from Ref. Bailly  _et al._ (2010):

$\displaystyle\geq$ $\displaystyle\text{X}\,^{1}\Sigma^{+}_{g}\,[\text{eV}]$
$\displaystyle E_{0\leftrightarrow 1}\,[\text{eV}]$ (230)
$\displaystyle\text{X}\,^{1}\Sigma^{+}_{g}~{}~{}~{}~{}~{}$ $\displaystyle 0$
$\displaystyle 0.52~{}~{}~{}~{}~{}$
$\displaystyle\text{B}\,^{1}\Sigma_{u}^{+}~{}~{}~{}~{}~{}$ $\displaystyle
11.19$ $\displaystyle 0.16~{}~{}~{}~{}~{}$
$\displaystyle\text{E}\,^{1}\Sigma^{+}_{g}~{}~{}~{}~{}~{}$ $\displaystyle
12.30$ $\displaystyle 0.29~{}~{}~{}~{}~{}$
$\displaystyle\text{F}\,^{1}\Sigma^{+}_{g}~{}~{}~{}~{}~{}$ $\displaystyle
12.32$ $\displaystyle 0.15~{}~{}~{}~{}~{}$
$\displaystyle\text{C}\,^{1}\Pi_{u}~{}~{}~{}~{}~{}$ $\displaystyle 12.30$
$\displaystyle 0.29~{}~{}~{}~{}~{}$

The second $E_{0\leftrightarrow 1}$-column shows that the first vibrational
excitations are generally quite different for the different electronic levels,
reflecting the curvature differences at the minima of the potential curves. If
the five BO potentials are approximately harmonic near their minima, the above
numerical values of $E_{0\leftrightarrow 1}$ are twice the values of their
zero-point energies.

Note that the transitions from (to) the listed higher electronic levels to
(from) the electronic ground state X ${}^{1}\Sigma^{+}_{g}(v=0,J=0)$ are in
the vacuum UV ($\geq$ 6.20 eV). They are approximately the same as the Lyman
$\alpha$ transition of atomic hydrogen (10.20 eV). This is important for a gas
mixture of H and H2: The relative energy differences (230) are all larger than
the Lyman $\alpha$ transition and they become even larger for the symplectic
spectra. This is important for astrophysical applications (see below).

As mentioned above, in order to determine the vibrational transition energies
$E_{0\leftrightarrow 1}$ in the list (230) experimentally one has to take a
“detour” of determining related (electronic) UV transitions first and then
subtract the corresponding energies Sprecher  _et al._ (2011); et al. (2013,
2018b, 2018c, 2019b).

The set of UV transitions from the states of the potential
$\text{X}\,^{1}\Sigma^{+}_{g}$ to those of $\text{B}\,^{1}\Sigma_{u}^{+}$ or
vice versa is called the “Lyman-band”, and those of
$\text{X}\,^{1}\Sigma_{g}^{+}$ to $\text{C}\,^{1}\Pi_{u}$ the “Werner-band”
Glass-Maujean  _et al._ (1984); Roncin and Launay (1994); et al. (1993a, b);
Bailly  _et al._ (2010).

###  IV.3  Vibrations of diatomic molecules
with different isotopic atoms

Such systems played an important but nowadays mostly forgotten role in the
early history of quantum mechanics:

Even before Heisenberg derived the now well-established spectrum of the HO in
his famous paper from July 1925 Heisenberg (1925), Mullikan had concluded from
his investigations of diatomic molecules that their vibrational spectra should
be described by half-integers, not integers as the Bohr-Sommerfeld
quantization prescription had suggested M. Born, unter Mitwirkung von F. Hund
(1925); bor . Mullikan compared the vibrational spectra of diatomic molecules
in which one atom was replaced by an isotope (B10O and B11O; AgCl35 and
AgCl37) Mullikan (1925).

Classically the vibrating atoms have angular frequences
$\omega_{i}=\sqrt{k/\mu_{i}}$, where the $\mu_{i},i=1,2$ denote the reduced
masses of the oscillators, $\mu_{1}$ for one and $\mu_{2}$ for the other
molecule containing one or two isotopic atoms. The (electronic) oscillator
strength $k$ is assumed to be the same in both cases (BO approximation).

Let $E_{0}(i)=\gamma\,\hbar\omega_{i}$ be the two slightly different
oscillator ground state energy levels for the two “isotopic” oscillators. Let
further $E_{a}$ and $E_{b}$ be two known electronic energy levels (they may be
equal) from which transitions to the ground states with energies $E_{0}(i)$
are possible. Then the difference

$\omega_{a,1}-\omega_{b,2}=(E_{a}-E_{b})/\hbar-\gamma(\omega_{1}-\omega_{2})$
(231)

of the frequencies

$\omega_{a,1}=[E_{a}-E_{0}(1)]/\hbar,\;\;\omega_{b,2}=[E_{b}-E_{0}(2)]/\hbar$
(232)

can be used in order to determine $\gamma$. Mullikan concluded that
$\gamma\approx 1/2$. A good review of the method can be found in Ref. her (c).

Due to the tremendous experimental and technological advances since those
experiments from almost 100 years ago it appears possible to perform similar
more refined experiments et al. (2013) in order to find fractional values of
the B-index $b$ other than $1/2$. However, one first has to account for the
deficits of the BO approximation and for the corrections due to rotational,
electronic and QED effects Pachucki (2010); Puchalski  _et al._ (2019)!

###  IV.4 Interferences of time dependent energy eigenstates

Applying the unitary time evolution operator (75) to the energy eigenstates
$|b,n\rangle$ yields ($\hbar=1$ in the following)

$U(\tilde{t})\,|n,b\rangle=e^{-i(n+b)\tilde{t}}|n,b\rangle,\;\tilde{t}=\omega\,t.$
(233)

Reccall that $\tilde{t}$ is a (dimensionless) angle variable. Let $\tilde{t}$
increase by an amount $\delta\,\tilde{t}$ which may be implemented by either a
change of $\omega$ or of $t$ or of both. Consider the superposition

$|n,b;\tilde{t},\delta\tilde{t}\rangle=(1+e^{-i(n+b)\delta\tilde{t}})e^{-i(n+b)\tilde{t}}|n,b\rangle.$
(234)

Then the oscillations of the “intensity”

$|\langle
n,b;\tilde{t},\delta\tilde{t}|n,b;\tilde{t},\delta\tilde{t}\rangle|^{2}=4\cos^{2}[(n+b)\delta\tilde{t}/2].$
(235)

are sensitive to the value of $b\delta\tilde{t}$, especially for $n=0$. For an
analogous approach in a recent experiment see Ref. et al. (2019d).

An alternative to generate such interferences by a change $\delta\tilde{t}$
one may also use - at least theoretically - a change $\delta b$. The question,
how to generate states like $|n,b\rangle$ experimentally has, unfortunately,
to be left open here.

###  IV.5 Transitions associated with the Hamiltonian
$H(K)=\hbar\,\omega\,\tilde{C}_{g}(K)$

In case the model Hamiltonian (107) with its ”effective” electric dipole
moment can somehow be implemented experimentally, either by heteronuclear
molecules like. e.g. 7LiH (it has the rather large electric dipole moment 5.9
D[ebeye]) or by Raman-type induced electric dipole moments of homonuclear
diatomic molecules, then especially the transitions (111) depend sensitively
on the value of $b$: The probability for the transition $n=0\leftrightarrow
n=1$ is given by

$|\langle b,m=0|\tilde{C}_{g}(K)|b,n=1\rangle|^{2}=b\,g^{2}/2.$ (236)

So, if the index $b$ is very small $>0$ \- as it appears to be in
astrophysical cases (see below) - then the same holds for that transition
probability!

Another essential point here is that the spectrum $\\{n+b\\}$ of
$\tilde{K}_{0}$ is rescaled for $\tilde{C}_{g}(K)$ by an overall ”redshifting”
factor $\sqrt{1-g^{2}}$ (see Eq. (197)).

###  IV.6 Traps for neutral molecules and optomechanics

A speculatively optimal experimental situation would be a diatomic neutral
molecule in a cooled down trap which allows the vibrational emission and
absorption properties of the molecule to be observed, especially those of its
different electronic potential ground states. As already stressed above, the
conditions are different for heteronuclear and homonuclear molecules, the
former having an electric dipole moment, the latter not, which requires some
Raman-type excitations. In view of the very impressive developments of
experimental possibilities involving such traps Grimm  _et al._ (2000);
Leibfried  _et al._ (2003); Ashkin (2006), it appears possible to achieve at
least a few of the required aims. Closely related are optical devices coupled
to mechanical oscillators Aspelmeyer  _et al._ (2014); et al. (2019d, 2020c);
Qiu  _et al._ (2020)

###  IV.7 Perelomov coherent states

Among the three different types of coherent states kas (i) associated with the
Lie algebra (52), the so-called “Perelomov” coherent states appear to be the
most promising ones in order to detect traces of HO spectra with $b\neq 1/2$:
Their matrix elements contain the Bargmann index $b$ quite explicitly and they
can be generated experimentally kas (i).

The states $|b,\lambda\rangle$ can either be defined as eigenstates of a
composite “annihilation” operator,

$\displaystyle E_{b,-}|b,\lambda\rangle=$
$\displaystyle\lambda\,|b,\lambda\rangle,\;\;E_{b,-}=(\tilde{K}_{0}+b)^{-1}\tilde{K}_{-},$
(237)
$\displaystyle\lambda=|\lambda|e^{-i\theta}\in\mathbb{D}=\\{\lambda\in\mathbb{C},\;|\lambda|<1\\},$

or by generating them from the ground state $|b,0\rangle$ by means of the
unitary operator

$\displaystyle U(\lambda)_{P}=$ $\displaystyle
e^{(w/2)\tilde{K}_{+}-(w^{\ast}/2)\tilde{K}_{-}}=e^{\lambda\,\tilde{K}_{+}}e^{\ln(1-|\lambda|^{2})\tilde{K}_{0}}e^{-\lambda^{\ast}\tilde{K}_{-}},$
(238) $\displaystyle
w=|w|e^{-i\theta}\in\mathbb{C},\;\;\lambda=\tanh(|w|/2)e^{-i\theta},$
$\displaystyle|w|=\ln\left(\frac{1+|\lambda|}{1-|\lambda|}\right),$

so that

$|b,\lambda\rangle=U(\lambda)_{P}|b,0\rangle.$ (239)

In terms of number states we have the expansion

$\displaystyle|b,\lambda\rangle$
$\displaystyle=(1-|\lambda|^{2})^{b}\sum_{0}^{\infty}\left(\frac{(2b)_{n}}{n!}\right)^{1/2}\lambda^{n}|b,n\rangle,$
(240) $\displaystyle(2b)_{n}$
$\displaystyle=2b(2b+1)\ldots(2b+n-1)=\Gamma(2b+n)/\Gamma(2b),$
$\displaystyle~{}~{}~{}~{}(2b)_{n=0}=1.$

Note that the coefficient of $\lambda^{n}$ in this expansion is the same as
that of $e_{n}(\vartheta)$ in Eq. (140).

Important expectation values with respect to $|b,\lambda\rangle$ are

$\displaystyle\langle b,\lambda|\tilde{K}_{0}|b,\lambda\rangle\equiv$
$\displaystyle\overline{\tilde{K}_{0}}_{;b,\lambda}=b\cosh|w|,$ (241)
$\displaystyle\langle b,\lambda|\tilde{K}_{1}|b,\lambda\rangle\equiv$
$\displaystyle\overline{\tilde{K}_{1}}_{;b,\lambda}=b\sinh|w|\,\cos\theta,,$
(242) $\displaystyle\langle b,\lambda|\tilde{K}_{2}|b,\lambda\rangle\equiv$
$\displaystyle\overline{\tilde{K}_{2}}_{;b,\lambda}=-b\sinh|w|\,\sin\theta,$
(243) $\displaystyle\langle b,\lambda|N|b,\lambda\rangle\equiv$
$\displaystyle\overline{N}_{b,\lambda}=b(\cosh|w|-1),$ (244)
$\displaystyle(\Delta N)^{2}_{b,\lambda}\equiv$
$\displaystyle\overline{N^{2}}_{b,\lambda}-\overline{N}_{b,\lambda}^{2}=\frac{1}{2}b\sinh^{2}|w|$
(245)

It follows that most quantities can be expressed in terms of the “observables”
$\overline{N}_{b,\lambda}$ and $(\Delta N)_{b,\lambda}$: As

$b\sinh|w|=\sqrt{2b}(\Delta N)_{b,\lambda}$ (246)

and $\cosh|w|=(\sinh^{2}|w|+1)^{1/2}$ we have, e.g.,

$\displaystyle\overline{\tilde{K}_{0}}_{;b,\lambda}=$
$\displaystyle[2b(\Delta)^{2}_{b,\lambda}+b^{2}]^{1/2},$ (247)
$\displaystyle\overline{N}_{b,\lambda}[1+\overline{N}_{b,\lambda}/(2b)]=$
$\displaystyle(\Delta N)^{2}_{b,\lambda}.$ (248)

It follows from the last equation that Paul’s parameter R Paul (1999) here has
the value

$R_{b,\lambda}\equiv\frac{(\Delta
N)^{2}_{b,\lambda}-\overline{N}_{b,\lambda}}{\overline{N}_{b,\lambda}^{2}}=\frac{1}{2b}.$
(249)

$R$ is a measure for deviations from a Poisson distribution for which $(\Delta
n)^{2}-\bar{n}=0$.

In addition we have

$\displaystyle|\lambda|=$
$\displaystyle\tanh(|w|/2)=\frac{\cosh|w|-1}{\sinh|w|}=\frac{\overline{N}_{b,\lambda}}{\sqrt{2b}(\Delta
N)_{b,\lambda}}$ (250) $\displaystyle|\lambda|^{2}=$
$\displaystyle\frac{\cosh|w|-1}{\cosh|w|+1}=\frac{\overline{N}_{b,\lambda}}{\overline{N}_{b,\lambda}+2b}$
(251)

and

$|w|=\ln\left(\frac{1+|\lambda|}{1-|\lambda|}\right)=\ln\left[\frac{1+(1+2b/\overline{N}_{b,\lambda})^{1/2}}{(1+2b/\overline{N}_{b,\lambda})^{1/2}-1}\right].$
(252)

For $\overline{N}_{b,\lambda}/2b\gg 1$ the last relation reduces to

$|w|\approx\ln(2\overline{N}_{b,\lambda}/b).$ (253)

The transition probability

$p_{b}(n\leftrightarrow\lambda)=|\langle
b,n|b,\lambda\rangle|^{2}=(1-|\lambda|^{2})^{2b}\frac{(2b)_{n}}{n!}|\lambda|^{2n}$
(254)

can be expressed as (see Eq. (251))

$\displaystyle p_{b}(n\leftrightarrow\lambda)=$
$\displaystyle\frac{(2b)_{n}(2b)^{2b}\overline{N}_{b,\lambda}^{n}}{n!\,(\overline{N}_{b,\lambda}+2b)^{n+2b}},$
(255) $\displaystyle p_{b}(0\leftrightarrow\lambda)=$
$\displaystyle\frac{(2b)^{2b}}{(\overline{N}_{b,\lambda}+2b)^{2b}}.$ (256)

Comparing the classical quantities (14) with the relations (73) and (241) -
(243) one infers the classical limit

$I\asymp\lim_{(\hbar b)\to 0,|w|\to\infty}[(\hbar b)\cosh|w|\asymp(\hbar
b)\sinh|w|]$ (257)

###  IV.8 (Dispersive) van der Waals forces

F. London was the first one to associate the attractive van der Waals forces
between neutral atoms or molecules with the nonvanishing zero-point energy of
the HO Eisenschitz and London (1930); London (1930, 1931, 1937). For atoms or
molecules of the same type and without retardation he derived - using a HO
model - the potential (see also Refs. mil ; Milonni (1994))

$V(R)=-\gamma(\frac{1}{2}\hbar\omega)\,\frac{\alpha^{2}}{R^{6}},$ (258)

where $\gamma>0$ is a number of order 1, $\alpha$ the (static) polarizability
of the two atoms or molecules pol (a); atk , $R\leq\lambda=(2\pi c)/\omega$
the distance of their nuclei and $\omega$ the angular frequency of an
oscillating electric field mode which acts, e.g., either on the permanent
electric dipole moments of two molecules or on the their induced electric
dipole moments. If $\vec{d}$ is the electric dipole moment generated at its
position by the effective electric field $\vec{E}_{\mathit{eff}}$, then
$\alpha$ in an isotropic situation is defined by
$\vec{d}=\alpha\,\vec{E}_{\mathit{eff}}$. It has the dimension $[L^{3}]$
($\vec{E}_{\mathit{eff}}$ includes a charge factor). The relation (258) holds
for vanishing temperature $T$ (for $T>0$ see, e.g. Ref. Passante and Spagnolo
(2007)). It is proportional to the usual HO ground state energy
$\hbar\omega/2$.

The potential (258) is of special importance for atoms and molecules which do
not have a permanent electric dipole but an induced one like, e.g. atomic
hydrogen H or molecular hydrogen H2, both in their ground states. The
corresponding values are

$\alpha[H(^{1}S)]=0.67\cdot
10^{-30}m^{3},~{}~{}~{}\alpha[H_{2}(^{1}\Sigma_{g}^{+})]=0.79\cdot
10^{-30}m^{3}$ (259)

pol (b). As the polarizability $\alpha$ is closely related to the dispersion
properties of an optical medium Born and Wolf (1999), London called the forces
associated with the potential (258) “dispersion” van der Waals forces London
(1937).

If one applies London’s London (1930, 1931, 1937) and later heuristic
arguments Kleppner ; mil  for the derivation of the van der Waals potential to
the symplectic spectrum of the HO, one obtains instead of Eq. (258):

$V(R)=-\gamma(b\,\hbar\omega)\,\frac{\alpha^{2}}{R^{6}}.$ (260)

If $b<1/2$ the “symplectic” van der Waals forces are weaker than the
“orthodox” ones.

The closely related Casimir effect Casimir and Polder (1948); Casimir (1948);
Schwinger (1975); Schwinger  _et al._ (1978); Plunien  _et al._ (1986);
Milonni (1994); Mostepanenko  _et al._ (1997); Milton (2001); et al. (2004);
Genet  _et al._ (2004); Milton (2004); Jaffe (2005); Kawakami  _et al._
(2007); Lamoreaux ; Reynaud and Lambrecht (2017) has to be discussed
separately, due to its different derivations and interpretations!

##  V Possible astrophysical implications

In case the above theoretically possible “symplectic” - or “fractional” -
spectra (69) of the HO are - at least partially - realized in nature they
could shed new light on some unsolved basic astrophysical problems of which I
shall mention the two most important ones et al. (b):

Dark energy et al. (Particle Data Group); Olive and Peacock ; Weinberg and
White ; Frieman  _et al._ (2008) and dark matter Bertone and Hooper (2018);
Baudis and Profumo . Here the symplectic spectra (69) may be a (the) key to
the solutions of both problems simultaneously!

For reasons mentioned above those spectra have not yet been seen in the
laboratory. But, surprisingly, physical implications of those spectra are
supported by the observationally favoured cosmological $\Lambda$CDM model
Peebles (1993); Ellis (2018); Turner (2018); Olive and Peacock ; Collaboration
(a) and by the associated WIMP hypothesis Feng (2010); Bertone and Hooper
(2018); et al. (2018d); Schumann (2019).

_It probably sounds provocative, but the observed dark energy and dark matter
properties may provide the first empirical support for the existence of the
spectra ( 69) in nature!_

The following discussions and arguments are mostly qualitative. The obviously
necessary and crucial quantitative arguments will still have to be reviewed
and worked out in detail!

###  V.1 Dark energy and the cosmological constant

Describing the existing astrophysical observations in terms of the Einstein-
Friedmann-Lemaître cosmological model Olive and Peacock ; Lahav and Liddle
leads to the conclusion that the (“vacuum”) energy density
$c^{2}\rho_{\Lambda}$, associated with the so-called “Lambda”-term in the
Einstein-Friedmann-Lemaître equations, has the same order of magnitude as the
critical energy density lam

$c^{2}\rho_{crit}=3c^{2}H_{0}^{2}/(8\pi G_{N})\approx
10^{-5}h^{2}\,\text{GeV}\,\text{cm}^{-3},$ (261)

where the scale factor $h$ for the present Hubble expansion rate $H_{0}$ has
the approximate value $h\approx 0.7$ (this “astrophysical” $h$ is not to be
confused with Planck’s constant in the following):

Taking into account that the observed “dark” energy density
$c^{2}\rho_{\Lambda}$ is about $0.7$ of the critical density (261) Weinberg
and White  and equating $c^{2}\rho_{\Lambda}$ with the vacuum energy density
of the quantized free electromagnetic field kas (j),

$u_{em;0}(\hat{\omega},b)=\frac{b\,\hbar}{4\pi^{2}\,c^{3}}\,\hat{\omega}^{4},$
(262)

where $\hat{\omega}$ is an appropriate cutoff for the corresponding divergent
frequency integral

$u_{em;0}(b)=\frac{b\,\hbar}{\pi^{2}\,c^{3}}\,\int_{\omega\geq
0}d\omega\,\omega^{3},$ (263)

allows to make a numerical estimate of $b$:

Introducing the cutoff length

$\ell=\frac{2\pi c}{\hat{\omega}}$ (264)

leads to the approximate equality

$u_{em;0}(\ell,b)=b\frac{4\pi^{2}\,\hbar c}{\ell^{4}}\,\approx 0.7\cdot
c^{2}\rho_{crit}.$ (265)

Taking for $\ell$ the (reduced) Compton wave length of the electron,

$\ell\approx\lambdabar=\hbar/(m_{e}c)\approx 3.9\cdot 10^{-11}\text{cm}$ (266)

and inserting $h^{2}\approx 0.5$ into relation (261) gives for the B-index $b$
the approximate value

$b\approx 10^{-35}\,.$ (267)

This is an extremely small $b$ – value, but it is theoretically allowed in the
present framework! This is in contrast to the conventional theoretical
estimates of the dark energy with $b=1/2$ Weinberg (1989); Carroll  _et al._
(1992); Carroll (2001); Frieman  _et al._ (2008) which represent the most
embarrassing discrepancy between observations and theoretical reasoning in all
of present-day physics!

The discussion above assumes that all modes have the same index $b$. This
simplification is, of course, not necessary. $b$ can depend on the frequency
$\omega$: $b=b(\omega)\in(0,1]$. It can, therefore, become a dynamical
quantity!

The estimate (267) depends sensitively on the choice of the cutoff length
$\ell$: if we. e.g., replace the factor $10^{-11}$ by $10^{-10}$ the estimate
in Eq. (267) is reduced to $b\approx 10^{-31}$. In addition all non-
electromagnetic effects were neglected (they would lead to an even smaller
value of $b$ than that in Eq. (267)!). This will be justified by the
discussion below concerning the nature of dark matter as being essentially
molecular hydrogen the dynamics of which is essentially determined by
electromagnetic forces.

Let me make another very crude estimate related to the “cosmic” order of
magnitude (267) of the B-index $b$: Consider the relation (84) between the
angular frequency $\omega=2\pi\nu$, the time period $T_{2\pi;b}$ and the index
$b$: Most of the very first molecules and molecular ions after the beginning
of the recombination epoch in the very early universe were diatomic, with the
vibrating nuclei locally emitting infrared light with frequencies
$\nu=\omega/(2\pi)$ around $10^{14}~{}$s-1. Even though homonuclear elements
like $H_{2}$ do not have a permanent electric dipole element, they still
radiate in the infrared et al. (2019a) and especially can emit Raman radiation
by induced dipole momennts (see Ch. IV.B above).

Taking for $T_{2\pi;b}$ the extreme value $10^{10}$ yr $\approx 3\,\cdot
10^{17}$ s and ignoring cosmic red shifts $z$ (i.e. being in the rest frame of
the molecule) we get the crude estimate

$b=1/(\nu\,T_{2\pi;b})\approx 10^{-32}.$ (268)

An important open question is whether the vacuum (“dark”) energy has changed
with cosmic time which would imply a corresponding time dependence of $b\,$!

###  V.2 “Dark” b-H2 and other primordial molecules
as dark matter?

The following most intriguing but perhaps also dangerously seductive or even
deceptive attempt intends to interpret the cosmic “dark matter” in the
“symplectic” framework of the HO. The central hypothetical role here is being
played by “symplectic” molecular hydrogen b-H2 as the main candidate for dark
matter. The possibility that molecular hydrogen H2 may play a role for the
understanding of dark matter has been tentatively discussed before Carr
(1994); Pfenniger  _et al._ (1994); Pfenniger and Combes (1994); Combes and
Pfenniger (1997); Combes and des Forêts (2000); Combes ; Paolis  _et al._
(1995a, b); et al. (1995); Paolis  _et al._ (1999); Gerhard and Silk (1996)
without, it seems, having a lasting impact. But the possible existence of a
(weak or hidden) “detuned” symplectic spectrum of the vibrating b-H2 allows
for a new and probably more promising approach! In addition to this
”symplectic” detuning there is, of course, the usual cosmic redshift $z$ due
to the expansion of the universe.

All the directly obtained experimental and obervational data like those of the
“Planck” Kollaboration etc. are, of course, not affected, but all the
calculated particle and cosmic standard model dependent dynamical - vibration
related - electromagnetic properties (transition probabilities of emissions,
absorptions, dissociations, ionizations and other rates etc.) have to be re-
evaluated. The same applies to the Big Bang Nucleosynthesis and the primordial
photon-baryon ratio Cybert  _et al._ (2016); Fields  _et al._ !

Molecular hydrogen plays already an important role in the present standard
(“orthodox”) cosmological paradigm Field  _et al._ (1966); Peebles (1993);
Williams (1999); Combes and des Forêts (2000); Galli and Palla (1998); Lepp
_et al._ (2002); Snow and McCall (2006); Galli and Palla (2013); Loeb and
Furlanetto (2013); Ubachs  _et al._ (2016); et al. (2017). Due to the missing
electric dipole moment it is difficult to detect astrophysically. For searches
in the intergalactic medium (IGM) one uses a plausible correlation between the
densities of H2 and of carbon monoxyde CO Bolatto  _et al._ (2013), which has
a permanent electric dipole moment and is more visible. But the atoms C and O
are not primordial ones and have to be bred in (first) stars etc..

Presently, however, we are primarily interested in the epoch of the universe
which is called its “Dark Ages” Barkana and Loeb (2001); Miralda-Escudé
(2003); Loeb ; loe (b); et al. (a), i.e. the cosmic time period which started
when the photons decoupled from matter and primordial atoms (mainly He and H)
could form (“recombination epoch” pee ; Peacock (1999); pea ; Barkana and Loeb
(2001); Mukhanov (2005); muk ; Weinberg (2008); wei ) at about 400000 years
after the big bang (at redshift $z=z_{\star}\approx 1100\approx 3000^{\circ}$
K$\approx 0.26$ eV Collaboration (b)). And which ended just before (around
$z\approx 30$, i.e. about 80 Myr after the big bang) density fluctuations of
the primordial gases led to the first gravitational “clumps” as seeds for the
first stars Barkana and Loeb (2001); Naoz  _et al._ (2006); Loeb and
Furlanetto (2013); Fialkov  _et al._ (2012) and the first galaxies Bromm and
Yoshida (2011); Loeb and Furlanetto (2013). The heat and radiation associated
whith this gravitational process reionized the primordial neutral gases of the
dark ages Fan  _et al._ (2006); Loeb and Furlanetto (2013), a cosmic period
called the “Dawn” of the universe et al. (a).

During those “dark ages” there were no “dust grains”, no “metals” (like 16O or
12C etc.) and no X- and no cosmic rays. There were only gases of primordial
photons, neutrinos, electrons, protons, deuterons and atoms He, H, D, Li,
their ions and first primordial diatomic molecules and their ions Galli and
Palla (1998); Lepp  _et al._ (2002); Galli and Palla (2013) like H2 and, e.g.
(HeH)+ Dabrowski and Herzberg (1977); Zygelman  _et al._ (1998); Bovino  _et
al._ (2011); Galli and Palla (2013); et al. (2019e) most of them at least
partially in local thermodynamical equilibrium. Ions may be “thrown” out of
equilibrium by primordial magnetic fields.

As the main observable signal from that epoch the 21 cm line from the electron
spin flip in the field of the magnetic moment of the proton in atomic hydrogen
has intensively been discussed more recently Barkana and Loeb (2001);
Furlanetto  _et al._ (2006); Pritchard and Loeb (2012); Barkana (2016); Cohen
_et al._ (2017, a); Novosyadlyj  _et al._ (2020); et al. (a) and
experimentally implemented or planned Monsalve  _et al._ (2017); et al.
(2018a, a). Molecular hydrogen is also assumed to play an essential role
during the dark ages Barkana and Loeb (2001); Hirata and Padmanabhan (2006);
Alizadeh and Hirata (2011); Novosyadlyj  _et al._ because of its cooling
properties Loeb and Furlanetto (2013); Barkana (2016).

In the following a number of possibly important properties of “symplectic”
molecular hydrogen, called b-H2, are listed which support the hypothesis of
essentially identifying dark matter with b-H2. The $b$-detuned spectra of
other primordial diatomic molecules (containing D, He and Li Galli and Palla
(1998); Lepp  _et al._ (2002); Galli and Palla (2013)) probably play an
essential role, too, but that of b-H2 is the most important one and here
serves as a prototype for a symplectically detuned molecular HO! The following
discussion relies on properties of b-H2 pointed out in Ch. IV.B above.

What is still missing in the discussion presented here is a necessary critical
_quantitative_ evaluation! This is, of course, due to the fact that up to now
no corresponding experimental laboratory data are available and - above all -
that the associated theory for the dynamics of the “symplectic” spectroscopy
still has to be worked out! But in principle it should be possible to decide
the empirical existence of those symplectic spectra in the laboratory, keeping
in mind that the index $b$ might be very small (see Eq. (267)) and, therefore,
optical transitions may be very weak (slow)!

Here is a list of properties and problems which shows how the b-H2 (plus other
primordial molecules) dark matter hypothesis compares with what essentially is
known about dark matter experimentally:

  1. 1.

It is quite surprising that the measured (relative) cosmic baryonic density
$\Omega_{b}\approx 0.05$ and the dark matter density $\Omega_{d}\approx 0.25$
are roughly of the same order of magnitude. And that the total matter density
$\Omega_{m}\approx 0.3$ is about of the same order of magnitude as the dark
(“vacuum”) energy density $\Omega_{\Lambda}\approx 0.7$! But this is no longer
surprising if dark matter consists essentially of “b-”H2 and other “b”-detuned
primordial molecules, i.e. it is baryonic, too ! In addition there may be some
kind of dynamical relationships between vacuum energy and matter.

  2. 2.

Whereas there appears to be - on average - 5 times more dark than baryonic
matter in the universe, not a single non-standard particle has been detected,
despite years-long tremendous and ingenious efforts by experimentalists and
theoreticians Feng (2010); Bertone and Hooper (2018); Baudis (2012); et al.
(2018d); Salucci (2019); Schumann (2019). That failure is no surprise if dark
matter essentially consists of b-H2 and other b-detuned primordial diatomic
molecules and ions!

  3. 3.

If indeed dark matter consists of such molecules and ions, this implies that
there is no dark matter in the universe before the recombination/decoupling
era when the formation of primordial atoms (He, H, D, Li), their molecules and
ions sets in Galli and Palla (1998); Lepp  _et al._ (2002); Loeb and
Furlanetto (2013); Galli and Palla (2013). And there is then no annihilation
of dark matter!

  4. 4.

For the formation of b-H2 etc. as dark matter about 5 times more primordial
baryons and electrons are required. This needs, of course, a re-evaluation of
the orthodox big bang nucleosynthesis (BBN) and the related baryon to photon
ratio Peacock (1999); Mukhanov (2005); Weinberg (2008); Cybert  _et al._
(2016); Fields  _et al._ ; Olive and Peacock .

  5. 5.

The dark matter approach discussed here may also shed some new light on the
“Lithium problem” of the current BBN interpretation (a disturbing discrepancy
between the observed Li/H ratio und the BBN predicted one) lit .

  6. 6.

If there are at the beginning of the recombination phase - ignoring all other
light primordial nuclei, their atoms, molecules and ions - protons and
electrons, why should they form dominantly b-H2 molecules and H atoms so that
on average we have a ratio of approximately 5:1? A preliminary argument is
that the b-H2 molecules are energetically favoured compared to 2 free H atoms
because of their binding energies (4.48 + 0.26 = 4,74) eV (see Eqs. (228) and
(227)). In order to understand the observed ratio 5:1 of “dark” matter to
baryonic matter one probably has to take the dynamics of the different atomic
and molecular reactions into account! Note that the formation of primordial