# A Quantum Vocal Theory of Sound

Davide<EMAIL_ADDRESS>and Maria Mannone
Department of Mathematics and Computer Science,
University of Palermo, Italy

###### Abstract

Concepts and formalism from acoustics are often used to exemplify quantum
mechanics. Conversely, quantum mechanics could be used to achieve a new
perspective on acoustics, as shown by Gabor studies. Here, we focus in
particular on the study of human voice, considered as a probe to investigate
the world of sounds. We present a theoretical framework that is based on
observables of vocal production, and on some measurement apparati that can be
used both for analysis and synthesis. In analogy to the description of spin
states of a particle, the quantum-mechanical formalism is used to describe the
relations between the fundamental states associated with phonetic labels such
as phonation, turbulence, and supraglottal myoelastic vibrations. The
intermingling of these states, and their temporal evolution, can still be
interpreted in the Fourier/Gabor plane, and effective extractors can be
implemented. The bases for a Quantum Vocal Theory of Sound, with implications
in sound analysis and design, are presented.

##  1 Introduction

What are the fundamental elements of sound? What is the most meaningful
framework for analyzing existing sonic realities and for expressing new sound
concepts? These are long standing questions in sound physics, perception, and
creation. In his analytical theory of heat [18], Joseph Fourier laid the basis
for analyzing functions of one variable in terms of sinusoidal components, and
explicitly wrote that “…if the order which is established in these phenomena
could be grasped by our senses, it would produce in us an impression
comparable to the sensation of musical sounds”.

Hermann von Helmholtz took Fourier’s suggestion seriously and proceeded to
analyze all vibratory phenomena as additions of sinusoidal vibrations [56].
Although he admitted that “we can conceive a whole to be split into parts in
very different and arbitrary ways”, it was the observation that the ear
somehow reflects Fourier analysis, and can be described as a bank of
sympathetic resonators, that led him to state that “the existence of partial
tones […] acquire a meaning in nature”.

In the twentieth century, despite the Fourier transform being the key to
describe sampling and signal reconstruction from samples [50], skepticism
arose among physicists such as Norbert Wiener and Dennis Gabor about
considering Fourier analysis as the best representation for music [43]. In
1947, in a famous paper published in Nature [21], Gabor embraced the
mathematics of quantum theory to shed light on subjective acoustics, thus
laying the basis for sound analysis and synthesis based on acoustical quanta,
or grains, or wavelets.

The Fourier and Gabor frameworks for time-frequency, or time-scale,
representation of sound are widely used in the analysis and synthesis of sonic
phenomena. For example, the auditory time and frequency acuities have been
bounded in terms of the uncertainty principle, although the theoretical limit
has been shown to beaten by human audition [41]. As another example, cochlear
filters are designed so that their time-frequency behavior matches human
performance, and are used to simulate or replace human hearing [34].

Still, when we are imagining a sound, or describing it to peers, we do not use
the Fourier formalism, but we rather refer to the hypothetical sources and to
their characteristics [22], or we use our voice to mimic some salient sound
features, thus overcoming the limitations of language [30]. We argue,
therefore, that a description of sound that exploits the basic mechanisms of
voice production would be more readily understandable and manipulable than any
decomposition based on framed sines or on chirps.

In this contribution, we propose a phonetic approach to describe sound at
large. A coarse articulatory description can indeed be applied to any sound,
and it will provide the basis for attempting a vocal imitation, which makes
embodied sound perception concrete and audible. In presence of concurrent
sources, the high-level phonetic descriptors are superimposed and temporally
varying, and their evolution is governed by context and attention. Apparently,
our hearing system acts as a sort of destructive measurement apparatus which
continuously collapses superpositions of phonetic states into streams [7],
whose evolution we can single out and follow, with the possibility of jumping
from one stream to another as a result of hidden or apparent forces.

Superposition and evolution of states, together with the concepts of
measurement collapse and force fields, are among the cornerstones of quantum
theory, and this is the observation that led us to attempt a description of
sonic phenomena within a quantum framework. Hopefully, some phenomena that are
normally described through sets of rules and gestalt principles (e.g.,
auditory continuity or temporal displacement) may naturally emerge from such a
quantum-inspired description, similarly to how quantum cognition has been able
to address behaviors that are difficult to derive within classical frameworks
[61]. The apparent incompatibility of properties that are being judged, or of
forms that are being perceived, implies some vagueness in the mental states
and in their time evolution, which is difficult to model classically but is
intrinsic in quantum modeling [62]. This is particularly evident in bivalued
judgements and in bistable percepts, that can be modeled as a two-state
quantum-mechanical system, or qubit. We intend to apply such a quantum-
theoretical model, which is constructed in analogy with spins in a time-
varying magnetic field, to auditory scenes made of overlapping auditory
objects, described in phonetic terms. In the context of auditory scene
analysis, we introduce the quantum-theoretical concepts of superposition, time
evolution, and measurement (or foreground separation). We show how this
framework can be useful to describe and reproduce some auditory-streaming
phenomena, with possible applications in source separation and audio effects.

Section 2 provides a short background on prior research on the two main axes
that cross in this work: Research in sound objecthood, with special emphasis
on the voice as an embodied representation of sound; Quantum frameworks that
have been proposed for sound and image processing, music, and perception.
Section 3 gives the motivation and a compact overview of the proposed Quantum
Vocal Theory of Sound. The long section 4 recalls the basic mathematical
formalism and some key concepts of quantum theory, and it shows how these
tools and concepts can be recast in audio terms. Section 5 shows how quantum
evolution can inspire algorithms for auditory object streaming and separation,
thus pointing to possible applications in computational auditory scene
analysis and audio effects.

##  2 Background

###  2.1 Voice as Embodied Sound

Many researchers, in science, art, and philosophy, have been facing the
problem of how to approach sound and its representations [13, 47]. Should we
represent sounds as they appear to the senses, by manipulating their proximal
characteristics? Or should we rather look at potential sources, at physical
systems that produce sound as a side effect of distal interactions? In this
research path we assume that our body can help establish bridges between
distal (source-related) and proximal (sensory-related) representations, and we
look at research findings in perception, production, and articulation of
sounds [33, 53]. Our approach to sound [15, 46] seeks to exploit knowledge in
these areas, especially referring to human voice production as a form of
embodied representation of sound.

When considering what people hear from the environment, it emerges that sounds
are mostly perceived as belonging to categories of the physical world [22].
Research in sound perception has shown that listeners spontaneously create
categories such as solid, electrical, gas, and liquid sounds, even though the
sounds within these categories may be acoustically different [25]. However,
when the task is to separate, distinguish, count, or compose sounds, the
attention shifts from sounding objects to auditory objects [29] represented in
the time-frequency plane, or to auditory images, which are movie-like temporal
representations resembling the signals projected by the ear up to the auditory
cortex [34]. Tonal components, noise, and transients can be extracted from
auditory objects with Fourier-based techniques [6, 54, 20]. Low-frequency
periodic phenomena are also perceptually very relevant and often come as
trains of transients. The most prominent elements of the proximal signal may
be selected by simplification and inversion of time-frequency representations.
These auditory sketches [27] have been used to test the recognizability of
imitations [31].

When discussing spaces for sound representation it is also important to recall
the notion of sound object, often associated to Schaeffer’s theory of
listening and typo-morphological spaces, which support a phenomenological
description of sound and can be reported to the time-frequency plane [53]. For
example, the concept of mass is a generalization of the notion of pitch that
comprises both site (on the frequency axis) and caliber (or degree of
occupation of the frequency axis).

Vocal imitations can be more effective than verbalizations at representing and
communicating sounds when these are difficult to describe with words [30].
This indicates that vocal imitations can be a useful tool for investigating
sound perception, and shows that the voice is instrumental to embodied sound
cognition. Vocal imitations act similarly to visual sketches: they catch and
emphasize some essential elements of the original (visual) objects allowing
their identification. At a more fundamental level, research on non-speech
vocalization is affecting the theories of language evolution [42], as it seems
plausible that humans could have used iconic vocalizations to communicate with
a large semantic spectrum, prior to the establishment of full-blown spoken
languages. Experiments and sound design exercises [15] show that agreement in
production corresponds to agreement in meaning interpretation, thus showing
the effectiveness of teamwork in embodied sound creation. Converging evidences
from behavioral and brain imaging studies give a firm basis to hypothesize a
shared representation of sound in terms of motor (vocal) primitives [57].
Historically, such convergence was envisioned over a century ago by the
Italian Futurists: On one side, the composer Luigi Russolo developed an
organology of everyday sounds and devised mechanical synthesizers for these
“noises” [48]; On the other side, the poet Filippo Tommaso Marinetti devised a
way to transcend language to bring everyday sounds to poetry, through
imitations and onomatopeia [37].

Some phoneticians have turned their attention to non-speech voice production,
trying to identify the most relevant phonetic components that are found in
vocal imitations [24]. They identified the broad categories of phonation
(i.e., quasi periodic oscillations due to vocal fold vibrations), turbulence,
supraglottal myoelastic vibrations, and clicks, which can be extracted
automatically from audio with time-frequency analysis and supervised [19] or
unsupervised [36] machine learning. These categories can be made to correspond
to categories of sounds as they are perceived [32], and as they are produced
in the physical world. Indeed, it has been argued that human utterances
somehow mimic “nature’s phonemes” [10], and neurophysiological studies have
shown that the cortical area of the superior temporal gyrus actually encodes
abstract phonetic features [38].

###  2.2 Quantum Frameworks

It was Dennis Gabor [21] who first adopted the mathematics of quantum
mechanics to explain acoustic phenomena. In particular, he used operator
methods to derive the time-frequency uncertainty relation and the (Gabor)
function that satisfies minimal uncertainty. Time-scale representations [14]
are more suitable to explain the perceptual decoupling of pitch and timbre,
and operator methods can be used as well to derive the gammachirp function,
which minimizes uncertainty in the time-scale domain [26]. Research in human
and machine hearing [34] have been based on banks of elementary (filter)
functions and these systems are at the core of many successful applications in
the audio domain.

Despite its deep roots in the physics of the twentieth century, the sound
field has not yet embraced the quantum signal processing framework [16] to
seek practical solutions to sound scene representation, separation and
analysis, although some theoretical proposals to encode, store, and process
audio using quantum circuitry have been advanced [58, 60]. On the other hand,
some common observed properties of human cognition and quantum mechanics
(superposition, non-classical probability) have given universal value to the
quantum-theoretic formalism to explain cognitive acts [61], including actions
of human creation, such as music. The explanatory power of a quantum approach
to music cognition has been demonstrated to describe tonal attraction
phenomena in terms of metaphorical forces [2, 4]. The theory of open quantum
systems has been applied to music to describe the memory properties (non-
Markovianity) of different scores [35]. The time-dependent Schrödinger
equation for a single non-relativistic particle has been used as a model for
sound and music composition. Some examples include the creation of grain
clouds like orbitals [17], the sonification of controlled quantum dynamics
[28], and compositions for an ensemble of atoms [51]. It has even been claimed
that the interplay between musical ideas and extra-musical meanings can be
naturally represented in the framework of quantum semantics, where extra-
musical meanings can be treated within a theory of vague possible worlds [12].

Some theoretical physicists have looked at the sensory processes driving human
and animal perception, trying to understand if they are classical or quantum.
As far as visual perception is concerned, Ghirardi proposed an experiment to
verify if the perceptive apparatus can induce the suppression of a physically-
established superposition of states [23]. In application-oriented image
processing, on the other hand, it has been shown how the quantum framework can
be effective to solve problems such as segmentation. For example, the
separation of figures from background can be obtained by evolving a solution
of the time-dependent Schrödinger equation [63], or by discretizing the time-
independent Schrödinger equation [1]. An approach to signal manipulation based
on the postulates of quantum mechanics can also potentially lead to a
computational advantage when using Quantum Processing Units. Results in this
direction are being reported for optimization problems [40].

In this work, we consider auditory phenomena and look at quantum theory for a
possible process model, that somehow mirrors the way humans extract and follow
auditory objects from audio mixtures. Such a process model, that exploits our
embodied knowledge of sound via vocal production, does not assume any
underlying information processing model for the brain. This standpoint and
disclaimer is commonly assumed in quantum cognition [61] and readily adopted
here.

##  3 Sketch of a Quantum Vocal Theory of Sound

In the proposed research path, sound is treated as a superposition of states,
and the voice-based components (phonation, turbulence, supraglottal myoelastic
vibrations) are considered as observables to be represented as operators. The
extractors of the fundamental components, i.e., the measurement apparati, are
implemented as signal-processing modules that are available both for analysis
and, as control knobs, for synthesis. The baseline is found in the results of
the SkAT-VG project [44, 30, 32, 31, 15, 19], which showed that vocal
imitations are optimized representations of referent sounds, that emphasize
those features that are important for identification. A large collection of
audiovisual recordings of vocal and gestural
imitations222https://www.ircam.fr/projects/blog/multimodal-database-of-vocal-
and-gestural-imitations-elicited-by-sounds/ offers the opportunity to further
enquire how people perceive, represent, and communicate about sounds.

A first assumption underlying this research approach, largely justified by
prior art and experiences, is that articulatory primitives used to describe
vocal utterances are effective as high-level descriptors of sound in general.
This assumption leads naturally to an embodied approach to sound
representation, analysis, and synthesis.

A second assumption is that the mathematics of quantum mechanics, relying on
linear operators in Hilbert spaces, offers a formalism that is suitable to
describe the objects composing auditory scenes and their evolution in time.
The latter assumption is more adventurous, as this path has not been taken in
audio signal processing yet. However, the results coming from neighboring
fields (music cognition, image processing) encourage us to explore this
direction, and to aim at introducing new techniques for sound analysis,
synthesis, and transformation.

An embryonic theory of sound based on the postulates of quantum mechanics, and
using high-level vocal descriptors of sound, can be sketched as follows. Let
$\overline{\sigma}$ be a vector operator that provides information about the
phonetic elements along a specific direction of measurement. Phonation, for
example, may be represented by $\sigma_{z}$, with eigenstates representing a
upper and a lower pitch. Similarly, the turbulence component may be
represented by $\sigma_{x}$, with eigenstates representing turbulence of two
different spectral distributions. A measurement of turbulence prepares the
system in one of two eigenstates for operator $\sigma_{x}$, and a successive
measurement of phonation would find a superposition and get equal
probabilities for the two eigenstates of $\sigma_{z}$. The two operators
$\sigma_{z}$ and $\sigma_{x}$ may also be made to correspond to the two
components of the classic sines + noise model used in audio signal processing.
If we add transients/clicks as a third measurement direction (as in the sines
+ noise + transients model [54]) we can claim that there is no sound state for
which the expectation value of the three components is zero: a sort of spin
polarization principle as found in quantum mechanics. The evolution of state
vectors in time is unitary, and regulated by a time-dependent Schrödinger
equation, with a suitably chosen Hamiltonian. The eigenvectors of the
Hamiltonian allow to expand any state vector in that basis, and to compute the
time evolution of such expansion. A pair of components can be simultaneously
measured only if they commute. If they don’t, an uncertainty principle can be
derived, as it was done for time-frequency and time-scale representations [21,
26]. The theory can be extended to cover multiple uncertain sources, and the
resulting mixed states can be described via density matrices, whose time
evolution can also be computed if a Hamiltonian operator is properly defined.
In the following, we formally lay down this Quantum Vocal Theory of Sound.

##  4 The phon formalism

Consider a 3d space with the orthogonal axes

z

: phonation, with different pitches;

x

: turbulence, with different brightnesses;

y

: myoelasticity, slow pulsations with different tempos.

The labels attributed to the axes correspond to the three main
articulatory/phonatory categories that are used by phoneticians to annotate
vocal imitations of everyday sounds [24]. They are a simplification of the
more phonetically-correct labels “vocal fold phonation,” “turbulence,” and
“supraglottal myoelastic vibration” [19].

The phon operator $\overline{\sigma}$ is a 3-vector operator that provides
information about the phonetic component in a specific direction of the 3d
phonetic space, i.e., along a specific combination of phonation, turbulence,
and myoelasticity.

In this section we present the phon formalism, obtained by direct analogy with
the single spin, as presented in accessible presentations of quantum mechanics
[52]. We use standard Dirac notation and adopt the quantum-theoretical
concepts of measurement, preparation, pure and mixed states, uncertainty, and
time evolution [9].

###  4.1 Measurement along z

A measurement along the z axis is performed according to the quantum-mechanics
principles:

  1. 1.

Each component of $\overline{\sigma}$ is represented by a linear operator;

  2. 2.

The eigenvectors of $\sigma_{z}$ are $\ket{u}$ and $\ket{d}$, corresponding to
pitch-up and pitch-down, with eigenvalues $+1$ and $-1$, respectively:

    1. (a)

$\sigma_{z}\ket{u}=\ket{u}$

    2. (b)

$\sigma_{z}\ket{d}=-\ket{d}$

  3. 3.

The eigenstates of operator $\sigma_{z}$, $\ket{u}$ and $\ket{d}$, are
orthogonal: $\braket{u}{d}=0$;

The eigenstates can be represented as column vectors

$\ket{u}=\begin{bmatrix}1\\\ 0\end{bmatrix},\,\ket{d}=\begin{bmatrix}0\\\
1\end{bmatrix},$

and the operator $\sigma_{z}$ as a square $2\times 2$ matrix. Due to principle
2, we have

$\sigma_{z}=\begin{bmatrix}1&0\\\ 0&-1\end{bmatrix}.$ (1)

###  4.2 Preparation along x

The eigenstates of the operator $\sigma_{x}$ are $\ket{r}$ and $\ket{l}$,
corresponding to turbulences having different spectral distributions, one with
the rightmost (or highest-frequency) centroid and the other with the leftmost
centroid. The respective eigenvalues are $+1$ and $-1$, so that

  1. (a)

$\sigma_{x}\ket{r}=\ket{r}$

  2. (b)

$\sigma_{x}\ket{l}=-\ket{l}$ .

If the phon is prepared $\ket{r}$ (turbulent) and then the measurement
apparatus is set to measure $\sigma_{z}$, there will be equal probabilities
for $\ket{u}$ or $\ket{d}$ phonation as an outcome. Essentially, we are
measuring what kind of phonation is in a pure turbulent state. This
measurement property is satisfied if

$\ket{r}=\frac{1}{\sqrt{2}}\ket{u}+\frac{1}{\sqrt{2}}\ket{d}.$ (2)

Likewise, if the phon is prepared $\ket{l}$ and then the measurement apparatus
is set to measure $\sigma_{z}$, there will be equal probabilities for
$\ket{u}$ or $\ket{d}$ phonation as an outcome. This measurement property is
satisfied if

$\ket{l}=\frac{1}{\sqrt{2}}\ket{u}-\frac{1}{\sqrt{2}}\ket{d},$ (3)

which is orthogonal to the linear combination (2). In vector form, we have

$\ket{r}=\begin{bmatrix}\frac{1}{\sqrt{2}}\\\
\frac{1}{\sqrt{2}}\end{bmatrix},\,\,\ket{l}=\begin{bmatrix}\frac{1}{\sqrt{2}}\\\
-\frac{1}{\sqrt{2}}\end{bmatrix},\mbox{ and }$
$\sigma_{x}=\begin{bmatrix}0&1\\\ 1&0\end{bmatrix}.$ (4)

In fact, any state $\ket{A}$ can be expressed as

$\ket{A}=\alpha_{u}\ket{u}+\alpha_{d}\ket{d},$ (5)

where $\alpha_{u}=\braket{u}{A}$, and $\alpha_{d}=\braket{d}{A}$. Being the
system in state $\ket{A}$, the probability to measure pitch-up is

$p_{u}=\braket{A}{u}\braket{u}{A}={\alpha_{u}}^{*}\alpha_{u}$ (6)

and, similarly, the probability to measure pitch-down is
$p_{d}=\braket{A}{d}\braket{d}{A}={\alpha_{d}}^{*}\alpha_{d}$ (Born rule).

###  4.3 Preparation along y

The eigenstates of the operator $\sigma_{y}$ are $\ket{f}$ and $\ket{s}$,
corresponding to slow myoelastic pulsations, one faster and one slower333In
describing the spin eigenstates, the symbols $\ket{i}$ and $\ket{o}$ are often
used, to denote the in–out direction., with eigenvalues $+1$ and $-1$, so that

  1. (a)

$\sigma_{y}\ket{f}=\ket{f}$

  2. (b)

$\sigma_{y}\ket{s}=-\ket{s}$ .

If the phon is prepared $\ket{f}$ (pulsating) and then the measurement
apparatus is set to measure $\sigma_{z}$, there will be equal probabilities
for $\ket{u}$ or $\ket{d}$ phonation as an outcome. Essentially, we are
measuring what kind of phonation is in a myoelastic pulsations. This
measurement property is satisfied if

$\ket{f}=\frac{1}{\sqrt{2}}\ket{u}+\frac{i}{\sqrt{2}}\ket{d},$ (7)

where $i$ is the imaginary unit.

Likewise, if the phon is prepared $\ket{s}$, we can express this state as

$\ket{s}=\frac{1}{\sqrt{2}}\ket{u}-\frac{i}{\sqrt{2}}\ket{d},$ (8)

which is orthogonal to the linear combination (7). In vector form, we have

$\ket{f}=\begin{bmatrix}\frac{1}{\sqrt{2}}\\\
\frac{i}{\sqrt{2}}\end{bmatrix},\,\,\ket{s}=\begin{bmatrix}\frac{1}{\sqrt{2}}\\\
-\frac{i}{\sqrt{2}}\end{bmatrix},\mbox{ and }$
$\sigma_{y}=\begin{bmatrix}0&-i\\\ i&0\end{bmatrix}.$ (9)

The matrices (1), (4), and (9) are called the Pauli matrices and, together
with the identitity matrix, these are the quaternions.

###  4.4 Measurement along an arbitrary direction

Orienting the measurement apparatus along an arbitrary direction
$\overline{n}=\left[n_{x},n_{y},n_{z}\right]^{\prime}$ means taking a weighted
mixture of quaternions:

$\sigma_{n}=\overline{\sigma}\cdot\overline{n}=\sigma_{x}n_{x}+\sigma_{y}n_{y}+\sigma_{z}n_{z}=\begin{bmatrix}n_{z}&n_{x}-in_{y}\\\
n_{x}+in_{y}&-n_{z}\end{bmatrix}.$ (10)

####  4.4.1 Example: Harmonic plus Noise model

A measurement performed by means of a Harmonic plus Noise model [6] would lie
in the phonation-turbulence plane
($n_{z}=\cos\theta,n_{x}=\sin\theta,n_{y}=0$), so that

$\sigma_{n}=\begin{bmatrix}\cos\theta&\sin\theta\\\
\sin\theta&-\cos\theta\end{bmatrix}$ (11)

The eigenstate for eigenvalue $+1$ is

$\displaystyle\ket{\lambda_{1}}=\left[\cos\theta/2,\sin\theta/2\right]^{\prime},$
(12)

the eigenstate for eigenvalue $-1$ is

$\displaystyle\ket{\lambda_{-1}}=\left[-\sin\theta/2,\cos\theta/2\right]^{\prime},$
(13)

and the two are orthogonal. Suppose we prepare the phon to pitch-up $\ket{u}$.
If we rotate the measurement system along $\overline{n}$, the probability to
measure $+1$ is (by Born rule)

$p(+1)=\left|\braket{u}{\lambda_{1}}\right|^{2}=\cos^{2}\theta/2,$ (14)

and the probability to measure $-1$ is

$p(-1)=\left|\braket{u}{\lambda_{-1}}\right|^{2}=\sin^{2}\theta/2.$ (15)

The expectation value of measurement is therefore

$\braket{\sigma_{n}}=\sum_{j}\lambda_{j}p(\lambda_{j})=(+1)\cos^{2}\theta/2+(-1)\sin^{2}\theta/2=\cos\theta.$
(16)

####  4.4.2 Rotate to measure

What does it mean to rotate a measurement apparatus to measure a property?
Assume we have a machine that separates harmonics from noise from (trains of)
transients, and that can discriminate between two different pitches, noise
distributions, and tempos. Essentially, the machine receives a sound and
returns three numbers $\\{\rm{ph},\rm{tu},\rm{my}\\}\in[-1,1]$. If $\rm{ph}>0$
the result will be $\ket{u}$, and if $\rm{ph}<0$ the result will be $\ket{d}$.
If $\rm{tu}>0$, the result will be $\ket{r}$, and if $\rm{tu}<0$ the result
will be $\ket{l}$. If $\rm{my}>0$, the result will be $\ket{f}$ and if
$\rm{my}<0$, the result will be $\ket{s}$. These three outputs correspond to
rotating the measurement apparatus along each of the main axes. Rotating it
along an arbitrary direction means taking a weighted mixture of the three
outcomes.

For example, consider the vocal fragment444It is one of the example vocal
sounds considered in [45], and taken from [39]. whose spectrogram is
represented in fig. 1. An extractor of pitch salience can be used to measure
phonation, and an extractor of onsets can be used to measure slow myoelastic
pulsation. Such two feature extractors, as found in the Essentia library [5],
have been applied to highlight the phonation (horizontal dotted line) and
myoelastic (vertical dotted lines) components in the spectrogram of fig. 1. In
the $z-y$ plane, there would be a measurement orientation and a measurement
operator that admits such sound as an eigenvector.

Figure 1: Spectrogram of a vocal sound which is a superposition of phonation
and supgraglottal myoelastic vibration. A salient pitch (horizontal dotted
line) as well as quasi-regular train of pulses (vertical dotted lines) are
automatically extracted.

###  4.5 Pure and Mixed states

According to the first postulate of quantum mechanics [9], at each time
instant the system is completely specified by a state $\ket{\psi}$ such that
$\braket{\psi}{\psi}=1$. If the state is known with certainty it is called a
pure state. All the phon states described so far are pure states. More
generally, a state can be known probabilistically as one of a set of
$\ket{\psi_{i}}$ with a given probability distribution. States of such kind
are called mixed states. The density operator represents both pure and mixed
states, and it is defined as

$\rho=\sum_{j}p_{j}\ket{\psi_{j}}\bra{\psi_{j}},$ (17)

where $p_{j}$ is probability for state $\ket{\psi_{j}}$.

For a pure state, it is simply $\rho=\ket{\psi}\bra{\psi}$, and the trace of
the square of such matrix is $Tr[\rho^{2}]=1$. For a mixed state, it is always
the case that $Tr[\rho^{2}]<1$.

####  4.5.1 Example

Let state $\ket{u}$ with probability $\frac{1}{3}$ and state $\ket{d}$ with
probability $\frac{2}{3}$. The density matrix is

$\rho=\frac{1}{3}\ket{u}\bra{u}+\frac{2}{3}\ket{d}\bra{d}=\begin{bmatrix}\frac{1}{3}&0\\\
0&\frac{2}{3}\end{bmatrix},$ (18)

and the trace of its square is

$Tr[\rho^{2}]=\frac{5}{9}<1.$

The interest of the density operator is given by its generalization power. It
is an essential generalization in quantum mechanics and, as such, it is
relevant for a quantum vocal theory of sound. From an experimental point of
view, it introduces a degree of conceptual flexibility which may come useful
in synthesis and composition of auditory scenes. In particular, the audio
concept of mixing can be made to correspond with manipulation of mixed states.

###  4.6 Uncertainty

If we measure two observables ${\bf L}$ and ${\bf M}$ (in a single experiment)
simultaneously, quantum mechanics prescribes that the system is left in a
simultaneous eigenvector of the observables only if ${\bf L}$ and ${\bf M}$
commute, i.e., if their commutator $\left[{\bf L,M}\right]={\bf LM-ML}$ is
null. Measurement operators along different axes do not commute. For example,
$\left[\sigma_{x},\sigma_{y}\right]=2i\sigma_{z}$, and therefore phonation and
turbulence can not be simultaneously measured with certainty.

The uncertainty principle, based on Cauchy-Schwarz inequality in complex
vector spaces, prescribes that the product of the two uncertainties is at
least as large as half the magnitude of the commutator:

$\Delta{\bf L}\Delta{\bf M}\geq\frac{1}{2}\left|\braket{\psi}{\left[{\bf
L,M}\right]}{\psi}\right|$ (19)

If ${\bf L}=\mathscr{T}=t$ is the time operator and ${\bf
M}=\mathscr{W}=-i\frac{d}{dt}$ is the frequency operator, and these are
applied to the complex oscillator $Ae^{i\omega t}$, the time-frequency
uncertainty principle results, and uncertainty is minimized by the Gabor
function. Starting from the scale operator, the gammachirp function can be
derived [26].

###  4.7 Time evolution

Another postulate of quantum mechanics [9] states that the evolution of state
vectors in time

$\ket{\psi(t)}={\bf U}(t_{0},t)\ket{\psi(t_{0})},t>t_{0},$ (20)

is governed by the operator ${\bf U}$, which is unitary (i.e., ${\bf
U}^{\dagger}{\bf U}={\bf I}$) and depends only on $t_{0}$ and $t$. Taken a
small time increment $\epsilon$, continuity of the time-development operator
gives it the form

${\bf U}(\epsilon)={\bf I}-i\epsilon{\bf H},$ (21)

with ${\bf H}$ being the quantum Hamiltonian (Hermitian) operator. ${\bf H}$
is an observable and its eigenvalues are the values that would result from
measuring the energy of a quantum system. From (21) it turns out that a state
vector changes in time according to the time-dependent Schrödinger
equation555We do not need physical dimensional consistency here, so we drop
Planck’s constant.

$\frac{\partial\ket{\psi(t)}}{\partial t}=-i{\bf H}(t)\ket{\psi(t)}.$ (22)

Any observable ${\bf L}$ has an expectation value $\braket{\bf L}$ that
evolves according to

$\frac{\partial\braket{{\bf L}}}{\partial t}=-i\braket{\left[{\bf L},{\bf
H}\right]},$ (23)

where $\left[{\bf L},{\bf H}\right]$ is the commutator of ${\bf L}$ with ${\bf
H}$.

For a closed, isolated physical system the Hamiltonian ${\bf H}$ is time
independent (${\bf H}(t)={\bf H}$), and the unitary operator is ${\bf
U}(t_{0},t)={\bf U}(t-t_{0})=e^{-i{\bf H}(t-t_{0})}$. While evolving, a closed
system remains in a superposition of states and preserves their magnitudes and
relative angles.

For non-pure states, the evolution of density operators is

$\rho(t)={\bf U}^{\dagger}(t_{0},t)\rho(t_{0}){\bf U}(t_{0},t).$ (24)

In most physical as well as in audio applications we have that the system
under consideration is driven by external forces, such as a changing magnetic
field or a vocal gestural articulation. In such cases of closed non-isolated
systems [8], the Hamiltonian ${\bf H}$ is time dependent. The states change
under the effect of the external forces, which determine the change of
probabilities, and the Hamiltonian controls the evolution process.

With a commutative Hamiltonian ($\left[{\bf H}(0),{\bf H}(t)\right]=0$), the
time evolution can be expressed as

$\ket{\psi(t)}=e^{-i\int_{0}^{t}{\bf H}(\tau)d\tau}\ket{\psi(0)}={\bf
U}(0,t)\ket{\psi(0)}.$ (25)

In general, if the operators $\bf A$ and $\bf B$ do not commute (i.e.,
$\left[{\bf A},{\bf B}\right]\neq 0$) we have that $e^{\bf A}e^{\bf B}\neq
e^{{\bf A}+{\bf B}}$. Since the evolution between two time points $0$ and $t$
can be split at an intermediate time $t^{*}$, if $e^{-i\int_{0}^{t}{\bf
H}(\tau)d\tau}=e^{-i\int_{0}^{t^{*}}{\bf H}(\tau)d\tau-i\int_{t^{*}}^{t}{\bf
H}(\tau)d\tau}\neq e^{-i\int_{0}^{t^{*}}{\bf
H}(\tau)d\tau}e^{-i\int_{t^{*}}^{t}{\bf H}(\tau)d\tau}$ then it means that an
explicit solution in terms of an integral can not be found. Our approach is to
consider time segments where the Hamiltonian is locally commutative, and to
compute the time evolution segment by segment in terms of an integral.

####  4.7.1 Phon in utterance field

Similarly to a spin in a magnetic field, when a phon is part of an utterance,
it has an energy that depends on its orientation. We can think about it as if
it was subject to restoring forces, and its quantum Hamiltonian is

${\bf
H}\propto\overline{\sigma}\cdot\overline{B}=\sigma_{x}B_{x}+\sigma_{y}B_{y}+\sigma_{z}B_{z},$
(26)

where the components of the field $\overline{B}$ are named in analogy with the
magnetic field.

Consider the case of potential energy only along $z$:

${\bf H}=\frac{\omega}{2}\sigma_{z}.$ (27)

To find how the expectation value of the phon varies in time, we expand the
observable ${\bf L}$ in (23) in its components to get

$\displaystyle\braket{\dot{\sigma}_{x}}$
$\displaystyle=-i\braket{\left[\sigma_{x},{\bf
H}\right]}=-\omega\braket{\sigma_{y}}$ (28)
$\displaystyle\braket{\dot{\sigma}_{y}}$
$\displaystyle=-i\braket{\left[\sigma_{y},{\bf
H}\right]}=\omega\braket{\sigma_{x}}$ $\displaystyle\braket{\dot{\sigma}_{z}}$
$\displaystyle=-i\braket{\left[\sigma_{z},{\bf H}\right]}=0,$

which means that the expectation values of $\sigma_{x}$ and $\sigma_{y}$ are
subject to temporal precession around $z$ at angular velocity $\omega$. In
phon terms, the expectation value of $\sigma_{z}$ steadily keeps the pitch if
there is no potential energy along turbulence and myoelastic pulsation.

A potential energy along all three axes can be expressed as

${\bf
H}=\frac{\omega}{2}\overline{\sigma}\cdot\overline{n}=\frac{\omega}{2}\begin{bmatrix}n_{z}&n_{x}-in_{y}\\\
n_{x}+in_{y}&-n_{z}\end{bmatrix},$ (29)

whose energy eigenvalues are $E_{j}=\pm\frac{\omega}{2}$, with energy
eigenvectors $\ket{E_{j}}$.

An initial state vector (phon) $\ket{\psi(0)}$ can be expanded in the energy
eigenvectors as

$\ket{\psi(0)}=\sum_{j}\alpha_{j}(0)\ket{E_{j}},$ (30)

where $\alpha_{j}(0)=\braket{E_{j}}{\psi(0)}$, and the time evolution of state
turns out to be

$\ket{\psi(t)}=\sum_{j}\alpha_{j}(t)\ket{E_{j}}=\sum_{j}\alpha_{j}(0)e^{-iE_{j}t}\ket{E_{j}}.$
(31)

###  4.8 Measurement

Given that time evolution of states is governed by the unitary transformation
(20) and by the Schrödinger equation (22), the measurement postulate of
quantum mechanics [9] states that a measurement is represented by an operator
(a projector) that acts on the state and that causes its collapse onto one of
its eigenvectors.

A projector system $\Pi_{i}$ in the (Hilbert) space of states is Hermitian,
idempotent, and complete. If the system is in state $\ket{\psi}$ before
measurement, the probability that the outcome of a measurement through a
projector system returns $j$ is

$p_{m}(j|\psi)=\bra{\psi}\Pi_{j}\ket{\psi}$ (32)

and, as a result of the measurement, the system collapses in state
$\psi^{(j)}_{post}=\frac{\Pi_{j}\ket{\psi}}{\sqrt{p_{m}(j|\psi)}}$.

Given an orthonormal basis of measurement vectors $\ket{a_{j}}$, the
elementary projectors are $\Pi_{j}=\ket{a_{j}}\bra{a_{j}}$,
$p_{m}(j|\psi)=|\braket{\psi}{a_{j}}|^{2}$, and the system (by neglecting a
unitary phasor) collapses into $\psi^{(j)}_{post}=\ket{a_{j}}$.

If the system is in a pure state

$p_{m}(j|\psi)=\braket{\psi\mid\Pi_{j}\mid\psi}=Tr[\rho\Pi_{j}].$ (33)

If the system is in a mixed state, the outcome of measurement is formulated as
a random variable conditioned by a given state:

$p_{m}(j|\psi_{k})=\braket{\psi_{k}\mid\Pi_{j}\mid\psi_{k}}=\\\
Tr[\ket{\psi_{k}}\bra{\psi_{k}}\Pi_{j}]$ (34)

and by averaging over all components of the mixed state we get

$p_{m}(j|\rho)=\sum_{k}p_{k}p_{m}(j|\psi_{k})=Tr[\rho\Pi_{j}].$ (35)

If the outcome of measurement is $j$, the system collapses into the new
ensemble of states represented by the density operator

$\rho^{(j)}_{post}=\frac{\Pi_{j}\rho\Pi_{j}}{Tr[\rho\Pi_{j}]}.$ (36)

###  4.9 Audio measurement and evolution

The mathematics of quantum mechanics can be used to describe and develop some
operations of audio signal processing, aimed at segregating components or
streams from raw audio. The concepts of quantum measurement and temporal
evolution of quantum states can be recast in audio and phonetic terms if we
can rely on an audio analysis/synthesis system that permits the extraction and
manipulation of slowly varying features such as pitch salience or spectral
energy.

####  4.9.1 Non-commutativity and autostates

We expect that measurement operators along different axes do not commute: this
is the case, for example, of measurements of phonation and turbulence. Let $A$
be an audio segment. The measurement (by extraction) of turbulence by the
operator $T$ leads to $T(A)=A^{\prime}$. A successive measurement of phonation
by the operator $P$ gives $P(A^{\prime})=A^{\prime\prime}$, thus
$P(A^{\prime})=PT(A)=A^{\prime\prime}$. If we perform the measurements in the
opposite order, with phonation first and turbulence later, we obtain
$TP(A)=T(A^{\ast})=A^{\ast\ast}$. We expect that $[T,P]\neq 0$, and thus, that
$A^{\ast\ast}\neq A^{\prime\prime}$. The diagram in figure 2 shows non-
commutativity in the style of category theory.

Besides the compact diagrammatic representation, we can describe such a non-
commutativity in terms of projectors $\Pi_{T},\,\Pi_{P}$:

$\begin{split}&\Pi_{T}\left(\Pi_{P}\ket{A}\right)=\ket{T}\braket{T}{P}\braket{P}{A}=\braket{T}{P}\ket{T}\braket{P}{A}\neq\\\
&\Pi_{P}\left(\Pi_{T}\ket{A}\right)=\ket{P}\braket{P}{T}\braket{T}{A}=\braket{P}{T}\ket{P}\braket{T}{A}.\end{split}$
(37)

Given that $\braket{T}{P}$ is a scalar and $\braket{P}{T}$ is its complex
conjugate, and that $\ket{P}\bra{T}$ is generally non-Hermitian, we get

$\begin{split}[\Pi_{T},\Pi_{P}]=\ket{T}\braket{T}{P}\bra{P}-\ket{P}\braket{P}{T}\ket{T}=\\\
=\braket{T}{P}\ket{T}\bra{P}-\braket{P}{T}\ket{P}\bra{T}\neq 0.\end{split}$
(38)

Measurements of phonation and turbulence can be actually performed using the
sines + noise (a.k.a., Harmonic Plus Stochastic – HPS) model [6]. The order of
operations is visually described in figure 3. The measurement of phonation is
performed through the extraction of the harmonic component in the HPS model,
while the measurement of turbulence is performed through the extraction of the
stochastic component with the same model. The spectrograms for
$A^{\prime\prime}$ and $A^{\ast\ast}$ in Figure 4 show the results of such two
sequences of analyses on a segment of female
speech,666https://freesound.org/s/317745/.
Hann window of 2048 samples, FFT of 4096 samples, hop size of 1024 samples.
confirming that the commutator $\left[T,P\right]$ is non-zero.

Figure 2: A non-commutative diagram representing the non-commutativity of
measurements of phonation ($P$) and turbulence ($T$) on audio $A$.

Figure 3: On the left, an audio segment is analyzed via the HPS model. Then,
the stochastic part is submitted to a new analysis. In this way, a measurement
of phonation follows a measurement of turbulence. On the right, the
measurement of turbulence follows a measurement of phonation. This can be
described via projectors through equation (37), and diagrammatically in fig.
2.

Figure 4: On the top, the spectrogram corresponding to a measurement of
phonation $P$ following a measurement of turbulence $T$, leading to
$PT(A)=A^{\prime\prime}$. On the bottom, the spectrogram corresponding to a
measurement of turbulence $T$ following a measurement of phonation $P$,
leading to $TP(A)=A^{\ast\ast}$.

Essentially, if we adopt the HPS model and skip the final step of addition and
inverse transformation, we are left with something that is conceptually
equivalent to a quantum destructive measure. Let $St$ be the filter that
extracts the stochastic part from a signal. As figure 5 shows, the spectrogram
of $St(x)$ is visibly different from the spectrogram of $x$. Conversely, if we
apply $St$ once more, we get a spectrum that does not change much:
$St^{2}(x)=St(St(x))\sim St(x)$. If we transform back from the second and
third spectrograms of figure 5, we get sounds that are very close to each
other. In fact, ideally, $St^{2}(x)=St(x)$. It means that, after a measure of
the non-harmonic component of some signal, the output-signal can be considered
as an autostate, and it confirms that the projection operator is idempotent.
If we perform the measure again and again, we still get the same result. Such
a measure operation provokes the collapse of a hypothetical underlying wave
function, which is originally a superposition of states, and is reduced to a
single state upon measurement. The importance of the autostates in this
framework is connected with the concept of quantum measures, which may become
practically feasible through a set of audio-signal analysis tools.

Figure 5: Top: spectrum of the original sound signal (a female speech),
Center: the stochastic component, derived from harmonic plus stochastic
analysis (HPS), as the effect of a destructive measure, and Bottom: the
stochastic component of the stochastic component itself. The last two spectra
are very close.

####  4.9.2 Hamiltonian streaming

Let us consider a quantum state vector $\ket{\psi(t)}$, that evolves in time
according to the Schrödinger equation (22). The time evolution can be
represented by the unitary operator ${\bf U}(t_{0},t)$ of equation (20).

If we choose a particular, commutative Hamiltonian, the time evolution can be
expressed by an integral, as in equation (25). A time-independent Hamiltonian
such as the one leading to (31) would not be very useful, both because forces
indeed change continuously and because this would lead to oscillatory
solution. Similarly to what has been done by Youssry et al. [63], the
Hamiltonian can be chosen to be time-dependent yet commutative (i.e.,
$\left[{\bf H}(0),{\bf H}(t)\right]={\bf H}(0){\bf H}(t)-{\bf H}(t){\bf
H}(0)=0$), so that a closed-form solution to state evolution can be obtained.
A simple choice is that of a Hamiltonian such as

$H(t)=g(t){\bf S},$ (39)

with ${\bf S}$ a time-independent Hermitian matrix. A function $g(t)$ that
ensures convergence of the integral in (25) is the damping

$g(t)=e^{-t}.$ (40)

In an audio application, we can consider a slice of time and the initial and
final states for that slice. We should look for a Hamiltonian that leads to
the evolution of the initial state into the final state. In image segmentation
[63], where time is used to let each pixel evolve to a final foreground-
background assignment, the Hamiltonian is chosen to be

$H=e^{-t}f({\bf x})\begin{bmatrix}0&-i\\\ i&0\end{bmatrix},$ (41)

and $f(\cdot)$ is a two-valued function of a feature vector ${\bf x}$ that
contains information about a neighborhood of the pixel. Such function is
learned from an example image with a given ground truth. In audio we may do
something similar and learn from examples of transformations: phonation to
phonation, with or without pitch crossing; phonation to turbulence; phonation
to myoelastic, etc. We may also add a coefficient to the exponent in (40), to
govern the rapidity of transformation. As opposed to image processing, time is
the playground of audio processing, and a range of possibilities is open to
experimentation in Hamiltonian streaming and audio processing.

The matrix ${\bf S}$ can be set to assume the structure (29), and the
components of potential energy found in an utterance field can be extracted as
audio features. For example, pitch salience can be extracted from time-
frequency analysis [49] and used as $n_{z}$ component for the Hamiltonian.
Figure 6 shows the two most salient pitches, automatically extracted from a
mixture of male and female voice777https://freesound.org/s/431595/ using the
Essentia library [5]. Frequent up-down jumps are evident, and they make
difficult to track a single voice. Quantum measurement induces state collapse
to $\ket{u}$ or $\ket{d}$ and, from that state, evolution can be governed by
(25). In this way, it should be possible to mimic human figure-ground
attention [7, 3], and follow each individual voice, or sound stream.

Figure 6: Extraction of the two most salient pitches from a mixture of a male
voice and a female voice

##  5 Examples

This section is intended to illustrate the potential of the Quantum Vocal
Theory of Sound in auditory scene analysis and audio effects.888The reported
examples are available, as a jupyter notebook containing the full code, on
https://github.com/d-rocchesso/QVTS

###  5.1 Two crossing glides interrupted by noise

In auditory scene analysis, insight into auditory organization is often gained
through investigation of continuity effects [7]. One interesting case is that
of gliding tones interrupted by a burst of noise [11]. Under certain
conditions of temporal extension and intensity of the noise burst, a single
frequency-varying auditory object is often perceived as crossing the
interruption. Specific stimuli can be composed that make bouncing or crossing
equally possible, to investigate which between the Gestalt principles of
proximity and good continuity actually prevails. V-shape trajectories
(bouncing) are often found to prevail on crossing trajectories when the
frequencies at the ends of the interruption match.

To investigate how Hamiltonian evolution may be tuned to recreate some
continuity effects, consider two gliding sinewaves that are interrupted by a
band of noise.

Figure 7 (top) shows the spectrogram of such noise-interrupted crossing
glissandos, overlaid with the traces of the two most salient pitches, computed
by means of the Essentia library [5]. Figure 7 also displays (middle) the
computed salience for the two most salient pitches and (bottom) the energy
traces for two bands of noise ($1\rm{kHz}$ – $2\rm{kHz}$, and $2\rm{kHz}$ –
$6\rm{kHz}$).

Figure 7: Tracing the two most salient pitches and noise energy for two
crossing glides interrupted by noise

The elements of the ${\bf S}$ matrix of the Hamiltonian (29) can be computed
(in Python) from decimated audio features as

⬇

decimation = 10 # PARAMETER (hold on)

n_x = energyNoise1[::decimation]

n_x = np.repeat(n_x, decimation)[0:size(energyNoise1)] # turbulence potential

n_y = np.zeros(n_x.size) # n_y is not at play

n_z = totalSaliences1[::decimation]

n_z = np.repeat(n_z, decimation)[0:size(energyNoise1)] # pitch potential

and the time-varying Hamiltonian can be multiplied by a decreasing exponential
$g(m)=e^{-km}$, where $m$ is the frame number, extending over $M$ frames:

⬇

k = 0.1 # PARAMETER (damping)

et = exp(-k*np.arange(0,decimation))

et = np.tile(et,int(size(n_z)/decimation))

ett = np.pad(et,(0,size(n_z)-size(et)),’constant’,constant_values=(0))

nzc = n_z*ett

nxc = n_x*ett

nyc = n_y*ett

The resulting turbulence and phonation potentials are depicted in figure 8.

Figure 8: Potentials of turbulence (top) and phonation (bottom) as functions
of frame number

The Hamiltonian time evolution of equation (25) can be computed by
approximating the integral with a cumulative sum:

⬇

H = np.array([[nzc, nxc - 1j*nyc], [nxc + 1j*nyc, -nzc]])

intHc = np.cumsum(H, axis=2)

Uc = intHc

for t in range(size(n_z)):

Uc[:,:,t] = linalg.expm(-(1j)*intHc[:,:,t])

Choosing an initial state (e.g., pitch-up), the state evolution can be
converted into a pitch (phonation) stream, which switches to noise
(turbulence) when it goes below a given threshold of pitchiness:

⬇

threshold = 0.7 # PARAMETER (pitchiness)

hopCollapse = 5 # PARAMETER (decimation of collapses)

sigma_z = [[1, 0], [0, -1]]

sigma_x = [[0, 1], [1, 0]]

for t in range(size(totalFreqs1)):

sdiff = norm(transpose(matrix(sT[:, t])) -

sigma_z * transpose(matrix(sT[:, t])))

if (sdiff < threshold or

sdiff > (2 - threshold)):

# state is pitchy

prob = np.square(abs(sT[0,t]))

# collapse based on probability

cstate = np.random.choice(

np.arange(0, 2), p=[prob, 1-prob])

if (cstate == 0):

if (t%hopCollapse == 0):

sT[0,t] = 1.0

sT[1,t] = 0.0

<select upper pitch>

else:

if (t%hopCollapse == 0):

sT[0,t] = 0.0

sT[1,t] = 1.0

<select lower pitch>

else: # state is turbulent

prob = abs(np.square(np.dot(sT[:,t],

[1/sqrt(2),1/sqrt(2)])))

cstate = np.random.choice(

np.arange(0, 2), p=[prob, 1-prob])

if (cstate == 0):

if (t%hopCollapse == 0):

sT[0,t] = 1/sqrt(2)

sT[1,t] = 1/sqrt(2)

<select lower band of noise>

else:

if (t%hopCollapse == 0):

sT[0,t] = 1/sqrt(2)

sT[1,t] = -1/sqrt(2)

<select higher band of noise>

In the proposed implementation, the free parameters are decimation, k,
threshold and hopCollapse, the latter being a decimation on the measurements
that are a accompanied by a state collapse. This small set of parameters allow
to produce a variety of temporal behaviors, well beyond what is possible with
a rigid quantum-mechanical encoding of the listening process.

One resulting pitch stream evolution from pitch-up is depicted in figure 9 and
it shows a breaking of continuity with bouncing. A first pitch oscillation is
visible around second 0.75 when the two sine waves are beating close to each
other, although phonation sticks to pitch-up. Then, when the noise
interruption arrives after second 1.00, pitch attribution as well as phonation
become uncertain. Such state of pitch confusion persists almost until second
1.40, well beyond the noise interruption, with occasional commutations to a
turbulent state. After the noise shock has been forgotten, the tracking
process sticks back to pitch-up, thus preferring a bouncing over a crossing
trajectory. Occasionally, due to the inherent randomness of the process, the
crossing trajectory may be chosen by the tracking process. The relative
probability of bouncing versus crossing depends both on the characteristics of
the stimulus (slopes of sinusoidal trajectories, width of the noise break,
relative amplitude between noise and sines) and on some model parameters such
as the relaxation coefficient k of the exponential and the probability
threshold for collapsing the measure to phonation rather than turbulence.

This example, and some other experiments run with different parameters, show
that the quantum-vocal model can reproduce some relevant phenomena of auditory
continuity ([59], ch. 6), which are attributable to neural reallocation. The
confusion between phonation and turbulence that extends well beyond the
interruption is consistent with the known perceptual fact that bursts of noise
are not precisely located as referred to a tonal transition, with errors up to
a few hundred milliseconds [55].

Figure 9: Tracking the phonation state under Hamiltonian evolution from pitch-
up

###  5.2 Mixed as in a mixer

Given an audio scene such as that of the two crossing glides interrupted by
noise (figure 7), we may follow the Hamiltonian evolution from an initial
state that is known only probabilistically. For example, at time zero we may
start from a mixture of $\frac{1}{2}$ pitch-up and $\frac{2}{3}$ pitch-down.
The density matrix (18) would evolve according to equation (24), where the
unitary operator ${\bf U}(0,t)$ is defined as in (25). When a pitch
measurement is taken, the outcome would be up or down according to equation
(35), and the density matrix that results from collapsing would be given by
equation (36).

The density matrix can be made audible in various ways, thus sonifying the
Hamiltonian evolution. For example, the completely chaotic mixed state,
corresponding to the half-identity matrix $\rho=\frac{1}{2}{\bf I}$, can be
made to sound as noise, and the pure states can be made to sound as the upper
or the lower of the most salient pitches. These three components can be mixed
for intermediate states. If $p_{u}$ and $p_{d}$ are the respective
probabilities of pitch-up and pitch-down as encoded in the mixed state, the
resulting mixed sound can be composed by a noise having amplitude
$\min{(p_{u},p_{d})}$, by the upper pitch weighted by
$p_{u}-\min{(p_{u},p_{d})}$, and by the lower pitch weighted by
$p_{d}-\min{(p_{u},p_{d})}$. One example of such evolution from a mixed state,
with periodic measurements and collapses that reset the density matrix is
depicted in figure 10. The analyzed audio scene and the model parameters,
including the computed Hamiltonian, are the same as used in the evolution of
pure states described in section 5.1. The depicted instance of evolution, if
sonified by controlling the amplitudes of the extracted two most salient
pitches and of a noise, results in a prevailing downward tone and in a delayed
and slowly-decreasing burst of noise (figure 11).

Figure 10: Amplitudes of components pitch-up, pitch-down, and noise resulting
from a Hamiltonian evolution from a mixed state

Figure 11: Spectrogram of the sonification of the Hamiltonian evolution from a
mixed state, using the component amplitudes depicted in figure 10

##  6 Conclusion and perspective

The components of phonation, turbulence, and supraglottal myoelastic
vibrations (and clicks) can be found, in some form and possibly in
superposition, in all kinds of vocal sound. Since the voice gives a
possibility for an embodied representation of sound in general, we can use the
three aforementioned basic phonetic components as general sound descriptors.
In this work, we proposed the phon as an analogue of a particle spin, where
the phonetic components appear to be aligned along the $x$, $y$, and $z$ spin
measurement directions. As such, the phon is subject to the mathematical
formalism and to the postulates of quantum mechanics, and it can be used to
describe sonic processes. Such description is of higher level and exploits a
conventional analysis/synthesis framework based on spectral modeling. In
particular, we have shown how a time-varying Hamiltonian, that governs the
temporal evolution of auditory streams, can be constructed from features that
are extracted from spectral modeling.

In a computational realization of the quantum-inspired operators and
processes, the manipulation of a few parameters allows to extract a variety of
components from complex audio scenes. The simple examples that we provided
show how some relevant auditory streaming phenomena can be modeled and
reproduced, but extensive experimentation is definitely required to verify how
useful a Quantum Vocal Theory of Sound could be in auditory scene analysis. A
large range of possibilities is also open to the creative processing of audio
materials through the sonification of the extracted streams and events. As
compared to analysis/synthesis frameworks based on spectral processing, here
we work at a higher level corresponding to fewer descriptors whose evolution
and intertwinement are mathematically defined. The statistical nature of
measurement, in evolutions of pure or mixed states under time-varying force
fields, leads naturally to the synthesis of ensembles of audio processes, all
derived and somehow echoing the original audio material. If we successfully
model some auditory phenomena, such as continuity effects or temporal
displacement, by temporal phon evolution, and if we render these evolutions
back to sound we may somehow say that we listen to possible auditory
processes. However, in creative applications we are not bound to mimic
auditory processes and we can also depart from quantum orthodoxy in many
possible different ways.

The proposed theory enhances the role of quantum theory and of the underlying
mathematics as a connecting tool between different areas of human knowledge.
By flipping the wicked problem of finding intuitive interpretations of quantum
mechanics, we aimed at using quantum mechanics to interpret something that we
have embodied, intuitive knowledge of.

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###### Abstract

Using the exact solutions to the field equation for a massive scalar field on
noncommutative $AdS_{2}$, we apply the $AdS/CFT$ correspondence principle to
obtain an exact result for the associated two-point function on the conformal
boundary. The answer satisfies conformal invariance and has the correct
commutative limit and massless limit.

Noncommutative $AdS_{2}$ II: The Correspondence Principle

In Honor of A.P.Balachandran on the Occasion of His 85th Birthday

##  1 Introduction

The conjectured $AdS/CFT$ correspondence principle has played a central role
in theoretical physics in the past two decades. It posits a strong/weak
duality between gravity in the bulk of an asymptotically anti-de Sitter
($AdS$) space and a conformal field theory ($CFT$) located at the so-called
conformal boundary.[1] For obvious reasons many practical applications of the
correspondence principle utilize classical, or weak, gravity in the bulk, with
the intention of exploring the strong coupling regime of the boundary theory.
Since a fully consistent quantum gravity treatment of the bulk remains out of
reach, it is a nontrivial task explore other domains of the correspondence. On
the other hand, the incorporation of some quantum gravity effects in the bulk
is possible. This has been the motivation for our recent works [2, 3, 4, 5].
As remarked in [6], the inclusion of some quantum gravity effects might be
achieved by replacing the $AdS$ bulk by its noncommutative analogue,
$ncAdS_{2}$. While, in general, the introduction of noncommutativity destroys
the isometries of a manifold, and is not unique, this is not the case for
$AdS_{2}$. Therefore, $ncAdS_{2}$ can serve as a toy model for the
introduction of quantum gravity effects in the bulk (barring the known
difficulties of the $AdS_{2}/CFT_{1}$ correspondence that already exist in the
commutative case.111 See for example [7, 8].). Moreover, in [2] it was shown
i) that the star product for $ncAdS_{2}$, when acting on functions having a
well behaved boundary limit, reduces to the point-wise product in this limit,
and ii) that noncommutative corrections to isometry generators, i.e., Killing
vectors, vanish near the boundary. In other words, $ncAdS_{2}$ is an
asymptotically $AdS$ space and, so according to the correspondence principle,
a $CFT$ should be present at its boundary.

The on-shell field theory action in the bulk $S|_{\rm on-shell}$ plays a
central role in the explicit construction of the $AdS/CFT$ correspondence. It
generates the $n-$point connected correlation functions for operators on the
boundary.[9] As a first step, it is therefore necessary to obtain the
solutions to field theories in the bulk, which in our case is noncommutative.
Exact solutions on $ncAdS_{2}$ were obtained for free massless scalar and
spinor fields in [5], and massive scalar fields in [6]. These exact solutions
will therefore lead to exact expressions for the corresponding two-point
function on the boundary. This was shown in [5] for the case of massless
fields. It is the purpose of this article to obtain the boundary two-point
function resulting from the exact solutions to the massive field equation
found in [6].

The generating function $S|_{\rm on-shell}$ is expressed in terms of the
boundary values $\phi_{0}$ of the solutions for the bulk fields, and the
prescription for computing the $n-$point connected correlation function for
operators ${\cal O}$ located at non-coincidental points $x_{i}$ on the
boundary is given by

$<{\cal O}(x_{1})\cdots{\cal O}(x_{n})>=\frac{\delta^{n}S|_{\rm on-
shell}}{\delta\phi_{0}(x_{1})\cdots\delta\phi_{0}(x_{n})}\bigg{|}_{\phi_{0}=0}\,.$
(1)

For the case of a 2-dimensional bulk theory, both ${\cal O}$ and $\phi_{0}$
are functions of only one coordinate, the time $t$. Conformal invariance
severely restricts the form of the $n-$point correlators; For $n=2$ one has

$<{\cal O}(t){\cal
O}(t^{\prime})>\;=\;\,C_{\Delta_{+}}\,\frac{1}{|t-t^{\prime}|^{2\Delta_{+}}}\,,$
(2)

where $\Delta_{+}$ is the conformal dimension and $C_{\Delta_{+}}$ is a
constant which can be computed using (1). We shall compute the two-point
function that results from massive scalar fields on $ncAdS_{2}$ and show that
it has the form given in (2). The result for the overall factor
$C_{\Delta_{+}}$ differs from that of the commutative theory. By taking
various limits we can compare with the previous results.

The outline for the remainder of this article is as follows: We first review
the calculations for $C_{\Delta_{+}}$ in the commutative case in section 2,
and then apply the analogous procedure to find the constant factor in the non-
commutative case in section 3. Some concluding remarks are made in section 4.

##  2 Commutative case

The analysis of the correspondence principle is more conveniently performed
for the Euclidean version of $AdS$, or $EAdS$, due to the absence of
propagating states.222For treatments of the correspondence principle relying
on Lorentzian signature see [10, 11, 12]. Here we specialize to $EAdS_{2}$,
closely following the work in [13, 14].

The field equation for a scalar field $\Phi$ with mass $m$ on $EAdS_{2}$ is

$\Delta\Phi\;=\;{m^{2}}\Phi\,,$ (3)

with $\Delta$ denoting the appropriate Laplacian. $EAdS_{2}$ is conveniently
parametrized by Fefferman-Graham coordinates $(z,t)$. They span the half-
plane, $z>0$, $-\infty<t<\infty$, with the conformal boundary located at
$z\rightarrow 0$.333Some care is needed to define the boundary limit. One
should first assume that the boundary is located at some $z=\epsilon$, and
then take the limit $\epsilon\rightarrow 0$. In terms of these coordinates the
metric tensor and Laplacian are given, respectively, by

$ds^{2}=\frac{dz^{2}+dt^{2}}{z^{2}}\;,$ (4)

and

$\Delta=z^{2}(\partial_{z}^{2}+\partial_{t}^{2})\;.$ (5)

The field equation (3) results from the standard scalar field action

$\displaystyle S$ $\displaystyle=$
$\displaystyle\;\;\,\frac{1}{2}\int_{R\times
R_{+}}dtdz\left((\partial_{z}\Phi)^{2}+(\partial_{t}\Phi)^{2}+\frac{m^{2}}{z^{2}}\Phi^{2}\right)$
(6) $\displaystyle=$ $\displaystyle-\frac{1}{2}\int_{R\times
R_{+}}\frac{dtdz}{z^{2}}\,\Phi(\Delta-{m^{2}})\Phi-\frac{1}{2}\int_{R}dt\,\Phi\partial_{z}\Phi\Big{|}_{z\rightarrow
0}\,.$ (7)

Only the boundary term survives when evaluating the action on-shell.

General solutions to (3) that are well behaved away from the conformal
boundary have the form

$\Phi(z,t)=\frac{1}{2\pi}\int d\omega\,e^{i\omega
t}\sqrt{z}\,a(\omega)\,K_{\nu}\left({|\omega|z}\right)\;,\qquad\nu=\sqrt{m^{2}+\frac{1}{4}}\,,$
(8)

where $K_{\nu}(x)$ denotes the modified Bessel function. To determine the
behavior of the solutions near the conformal boundary we can use[15]

$K_{\nu}(x)\rightarrow\frac{1}{2}\Bigl{(}2^{\nu}\Gamma(\nu)x^{-\nu}+2^{-\nu}\Gamma(-\nu)x^{\nu}\Bigr{)}\left(1+{\cal
O}(x^{2})\right),\quad{\rm as}\;\;x\rightarrow 0\,.$ (9)

Then for $0<\nu<1$ (corresponding to $-\frac{1}{4}<m^{2}<\frac{3}{4}$),
$\Phi(z,t)$ tends towards

$\frac{1}{4\pi}\int d\omega\,e^{i\omega
t}a(\omega)\left(\left(\frac{2}{|\omega|}\right)^{\nu}\Gamma(\nu)z^{\Delta_{-}}+\left(\frac{|\omega|}{2}\right)^{\nu}\Gamma(-\nu)z^{\Delta_{+}}\right),\,{\rm
as}\,z=\epsilon\rightarrow 0,$ (10)

where $\Delta_{\pm}=\frac{1}{2}\pm\nu$, with $\Delta_{+}$ to be identified
with the conformal dimension. So upon keeping just the leading term,

$\Phi(\epsilon,t)\rightarrow\phi_{\epsilon}(t)=\epsilon^{\Delta_{-}}\phi_{0}(t)\;,\quad{\rm
as}\;\;{\epsilon\rightarrow 0}\,.\;$ (11)

Here we shall assume Dirichlet boundary conditions. $\phi_{0}(t)$ is regarded
as the source field on the boundary, which determines the solution for $\Phi$
in the bulk.444 For other boundary conditions, see for example [14]. This can
be made explicit by writing the coefficients $a(\omega)$ of the solution (8)
in terms of the Fourier coefficients $\phi(\omega)$ of the boundary field
$\phi_{\epsilon}(t)=\frac{1}{2\pi}\int d\omega e^{i\omega t}\phi(\omega)$,

$\phi(\omega)=\sqrt{\epsilon}a(\omega)K_{\nu}\left({|\omega|\epsilon}\right)\,.$
(12)

The solution for $\Phi(z,t)$ in the bulk can thereby be expressed in terms of
these Fourier coefficients

$\Phi(z,t)=\frac{1}{2\pi}\int d\omega e^{i\omega
t}\phi(\omega)\frac{\sqrt{z}K_{\nu}\left({|\omega|z}\right)}{\sqrt{\epsilon}K_{\nu}\left({|\omega|\epsilon}\right)}\;.\qquad$
(13)

The evaluation of the on-shell action requires an analogous expression for
$\partial_{z}\Phi(z,t)$. For this we can use the identity
$\frac{d}{dx}K_{\nu}(x)=\frac{\nu}{x}K_{\nu}(x)-K_{\nu+1}(x)$ to write

$\sqrt{z}\frac{d}{dz}\left(\sqrt{z}K_{\nu}(|\omega|z)\right)=\left(\frac{1}{2}+\nu{}\right)K_{\nu}(|\omega|z)-|\omega|zK_{\nu+1}(|\omega|z)\,,$

and hence

$\partial_{z}\Phi(z,t)=\frac{1}{2\pi}\int d\omega e^{i\omega
t}\phi(\omega)\frac{\left(\frac{1}{2}+\nu{}\right)K_{\nu}(|\omega|z)-|\omega|zK_{\nu+1}(|\omega|z)}{\sqrt{z\epsilon}K_{\nu}\left({|\omega|\epsilon}\right)}\,.\qquad$
(14)

Substitution of (13) and (14) into the boundary term in (7) gives the
following result for the on-shell action $S|_{\rm on-shell}$:

$-\frac{1}{4\pi}\int dt\int dt^{\prime}\int d\omega
e^{i\omega(t-t^{\prime})}\phi_{\epsilon}(t)\phi_{\epsilon}(t^{\prime})\,\frac{1}{\epsilon}\left({\frac{1}{2}+\nu-|\omega|\epsilon\frac{K_{\nu+1}(|\omega|\epsilon)}{K_{\nu}\left({|\omega|\epsilon}\right)}}{}\right)\,\Big{|}_{\epsilon\rightarrow
0}\,.$ (15)

From (9), the asymptotic limit of the ratio of Bessel functions is

$\frac{xK_{\nu+1}(x)}{K_{\nu}(x)}\rightarrow\frac{\left(2^{\nu+1}\Gamma(\nu+1)+2^{-\nu-1}\Gamma(-\nu-1)x^{2\nu+2}\right)}{\left(2^{\nu}\Gamma(\nu)+2^{-\nu}\Gamma(-\nu)x^{2\nu}\right)}\,\quad{\rm
as}\quad x\rightarrow 0\,,$

which to leading order is
$2\nu\left(1-\frac{\Gamma(-\nu)}{\Gamma(\nu)}\left(\frac{x}{2}\right)^{2\nu}\right)$
for $0<\nu<1$. Upon applying this result, along with the integral[14]

$\frac{1}{2\pi}\int d\omega e^{-i\omega
t}|\omega|^{\rho}=\frac{2^{\rho}\Gamma(\frac{\rho+1}{2})}{\sqrt{\pi}\Gamma(-\frac{\rho}{2})}\,\frac{1}{|t|^{\rho+1}}\;,\qquad\rho\neq-1,-3,...\;,$
(16)

one gets555The singular term  $-\frac{1}{4\pi}(\frac{1}{2}-\nu)\int dt\int
dt^{\prime}\int d\omega
e^{i\omega(t-t^{\prime})}\frac{\phi_{0}(t)\phi_{0}(t^{\prime})}{\epsilon^{2\nu}}\Big{|}_{\epsilon\rightarrow
0}\;$ can be ignored since the integration in $\omega$ leads to a delta
function and we are interested in $t\neq t^{\prime}$.

$\displaystyle S|_{\rm on-shell}$ $\displaystyle=$
$\displaystyle-\frac{\nu\Gamma(-\nu)}{2^{2\nu+1}\pi\Gamma(\nu)}\int dt\int
dt^{\prime}\phi_{\epsilon}(t)\phi_{\epsilon}(t^{\prime})\,\epsilon^{2\nu-1}\int
d\omega
e^{i\omega(t-t^{\prime})}{|\omega|^{2\nu}}\,\Big{|}_{\epsilon\rightarrow 0}$
(17) $\displaystyle=$
$\displaystyle-\frac{\nu}{\sqrt{\pi}}\frac{\Gamma(\nu+\frac{1}{2})}{\Gamma(\nu)}\int
dt\int
dt^{\prime}\phi_{\epsilon}(t)\phi_{\epsilon}(t^{\prime})\,\epsilon^{2\nu-1}\,\frac{1}{|t-t^{\prime}|^{2\nu+1}}\,\Big{|}_{\epsilon\rightarrow
0}\,.$ (19)

In terms of the $\epsilon$-independent source field $\phi_{0}(t)$ in (11) the
result is

$\displaystyle S|_{\rm on-shell}$ $\displaystyle=$
$\displaystyle-\frac{\nu}{\sqrt{\pi}}\frac{\Gamma(\nu+\frac{1}{2})}{\Gamma({\nu})}\int
dt\int
dt^{\prime}\,\frac{\phi_{0}(t)\phi_{0}(t^{\prime})}{|t-t^{\prime}|^{2\nu+1}}\,\,.\;$
(20)

The two-point function for the massive scalar (2) is now easily recovered from
(1) with the resulting factor $C_{\Delta_{+}}$ given by

$C_{\Delta_{+}}\;=\;\,-\frac{2\nu}{\sqrt{\pi}}\frac{\Gamma(\nu+\frac{1}{2})}{\Gamma({\nu})}\;.$
(21)

##  3 Non-commutative case

We now repeat the procedure for the non-commutative case. The field equation
for a massive scalar field $\hat{\Phi}$ on non-commutative $EAdS_{2}$ is

$\hat{\Delta}\hat{\Phi}\;=\;{m^{2}}\hat{\Phi}\;,$ (22)

where $\hat{\Delta}$ is the noncommutative version of the Laplacian (5). As in
[6], one can express $\hat{\Delta}$ in terms of operators
$\hat{X}^{a},\;a=1,2,3,$ which satisfy the $su(1,1)$ algebra

$[\hat{X}^{a},\hat{X}^{b}]=i\alpha\epsilon^{abc}\hat{X}_{c}\;,$ (23)

and have $\hat{X}^{a}\hat{X}_{a}=-1$. The indices are raised and lowered with
the metric diag$(-1,1,1)$, $\epsilon^{abc}$ is totally antisymmetric, with
$\epsilon^{012}=1$, and $\alpha$ is the noncommutative parameter, analogous to
Planck’s constant, which should correspond to the quantum gravity scale. The
Laplacian is given by

$\frac{\alpha^{2}}{2}\hat{\Delta}\hat{\Phi}=\hat{X}_{a}\hat{\Phi}\hat{X}^{a}\,+\,\hat{\Phi}\,.$
(24)

The field equation (22) results from the variation of $\hat{\Phi}$ in the
following action

$\hat{S}=-\frac{1}{2\alpha^{2}}{\rm
Tr}\,\Bigl{\\{}[\hat{X}^{a},\hat{\Phi}][{\hat{X}}_{a},\hat{\Phi}]-(\alpha
m)^{2}\hat{\Phi}^{2}\Bigr{\\}}\,.$ (25)

The trace in (25) can be understood as an integration over symbols of
operators, and the integration domain can again be taken to be the half-plane
spanned by commuting coordinates $z$ and $t$. As in (7) the action can be spit
up into two terms, one of which vanishes on-shell and the other is a boundary
term. Moreover, as was shown in [2], the boundary term for the noncommutative
theory is identical to that of the commutative theory. (A conformal boundary
limit can be defined for noncommutative $EAdS_{2}$ in terms of the
representation theory for the $su(1,1)$ algebra.[2]) Thus the on-shell action
once again takes the form

$\displaystyle S|_{\rm on-shell}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\int_{R}dt\,\hat{\Phi}\partial_{z}\hat{\Phi}\Big{|}_{z\rightarrow
0}\,,$ (26)

where here $\hat{\Phi}$ actually denotes the symbol of the field.

As shown in [6], (22) has exact solutions, which are given in terms of
generalized Legendre functions. Upon restricting to fields that are well-
behaved in the bulk, and integrating over all frequencies $\omega$, one has

$\hat{\Phi}=\frac{1}{2\pi}\int
d{\omega}\,a(\omega)e^{i\omega\hat{t}/2}\,P_{-\Delta_{-}}^{-\frac{2\kappa}{\alpha}}\left(\frac{2\hat{r}}{{|\omega}|\alpha}\right)\,e^{i\omega\hat{t}/2}\;,$
(27)

where $\hat{t}$ and $\hat{r}$ are noncommutative operators, which can be used
to construct $\hat{X}^{a}$ (see [6]), and
$\kappa=\sqrt{1+\frac{\alpha^{2}}{4}}$. $\hat{t}$ and $\hat{r}$ satisfy the
commutator $[\hat{t},\hat{r}]=-i\alpha\,.$

As in the commutative case, the solution can be expressed in terms of boundary
fields. The symbol of $\hat{r}^{-1}$ tends to zero in this limit, and the
algebra of functions of $\hat{r}$ and $\hat{t}$ is effectively commutative
near the boundary. Therefore the symbol of the solution (27) has the conformal
boundary limit:

$\phi_{\epsilon}(t)=\frac{1}{2\pi}\int d\omega\,e^{i\omega
t}a(\omega)\,P_{-\Delta_{-}}^{-\frac{2\kappa}{\alpha}}\left(\frac{2}{{|\omega}|\alpha\epsilon}\right)\,,\;\;\;{\rm
as}\;\epsilon\rightarrow 0\,.$ (28)

From [6] this tends to $\epsilon^{\Delta_{-}}\phi_{0}(t)\;{\rm
as}\;{\epsilon\rightarrow 0}$ for $\nu>0$, just as in the commutative case
(11). We shall again assume Dirichlet boundary conditions with $\phi_{0}(t)$
regarded as the source field. Re-expressing the solution (27) in terms of the
Fourier transform $\phi(\omega)$ of the boundary field $\phi_{\epsilon}(t)$,

$\phi(\omega)=a(\omega)\,P_{-\Delta_{-}}^{-\frac{2\kappa}{\alpha}}\left(\frac{2}{{|\omega}|\alpha\epsilon}\right)\;,$
(29)

we get

$\hat{\Phi}=\frac{1}{2\pi}\int
d\omega\,\phi(\omega)e^{i\omega\hat{t}/2}\,\frac{P_{-\Delta_{-}}^{-\frac{2\kappa}{\alpha}}\left(\frac{2\hat{r}}{{|\omega}|\alpha}\right)}{P_{-\Delta_{-}}^{-\frac{2\kappa}{\alpha}}\left(\frac{2}{{|\omega}|\alpha\epsilon}\right)}\,e^{i{|\omega}|\hat{t}/2}\,.$
(30)

Once again we need to compute the derivative of the solution with respect to
the radial coordinate. This, in general, will introduce operator ordering
ambiguities. However, such ambiguities are not a concern for the computation
of the on-shell action, since we only need the result in the conformal
boundary limit where the coordinates effectively commute. So let us choose the
radial derivative to be $\frac{i}{\alpha}[\hat{t},\hat{\Phi}]$. Then the
symbol of $\frac{i}{\alpha}[\hat{t},\hat{\Phi}]$ in the boundary limit is
$-z^{2}\partial_{z}\hat{\Phi}|_{z\rightarrow 0}$. After applying the
identity[16]

$\frac{dP^{\mu}_{\rho}(x)}{dx}=\frac{1}{1-x^{2}}\left({(\rho+1)xP_{\rho}^{\mu}(x)-(-\mu+\rho+1)P_{\rho+1}^{\mu}(x)}\right)\;,$
(31)

we get the following result for $\partial_{z}\hat{\Phi}|_{z=\epsilon}$ as
$\epsilon\rightarrow 0$

$-\frac{1}{4\pi}\int
d{\omega}e^{i{\omega}t}\phi({\omega}){\alpha{|\omega}|}\left((\Delta_{-}-1)\Bigl{(}\frac{2}{|\omega|\alpha\epsilon}\Bigr{)}+\Bigr{(}\Delta_{+}+\frac{2\kappa}{\alpha}\Bigl{)}\frac{P^{-\frac{2\kappa}{\alpha}}_{{\Delta_{+}}}\left(\frac{2}{{|\omega}|\alpha\epsilon}\right)}{P_{-\Delta_{-}}^{-\frac{2\kappa}{\alpha}}\left(\frac{2}{{|\omega}|\alpha\epsilon}\right)}\right)\,.$

The substitution of this result in the on-shell action (26) yields

$\displaystyle S|_{\rm on-shell}=\frac{1}{8\pi}\int dt\int dt^{\prime}\int
d{\omega}e^{i{\omega}(t-t^{\prime})}\phi_{\epsilon}(t)\phi_{\epsilon}(t^{\prime})\;\times$
(32) (33)
$\displaystyle\;{\alpha{|\omega}|}\left((\Delta_{-}-1)\left(\frac{2}{{|\omega}|\alpha\epsilon}\right)+\left(\Delta_{+}+\frac{2\kappa}{\alpha}\right)\frac{P^{-\frac{2}{\alpha}\kappa}_{{\Delta_{+}}}\left(\frac{2}{{|\omega}|\alpha\epsilon}\right)}{P_{-\Delta_{-}}^{-\frac{2}{\alpha}\kappa}\left(\frac{2}{{|\omega}|\alpha\epsilon}\right)}\right)\Bigg{|}_{\epsilon\rightarrow
0}.$ (34)

As with the commutative answer, the term that yields a delta function after
integration in $\omega$ can be dropped. To evaluate the remaining term we need
the asymptotic behavior of
${yP^{\mu}_{\nu+\frac{1}{2}}(1/y)}\,/\,{P^{\mu}_{\nu-\frac{1}{2}}(1/y)}$ as
$y\rightarrow 0$, which is

$\frac{2\nu}{\nu-\mu+\frac{1}{2}}\left(1-\frac{\Gamma(-\nu)\Gamma(\nu-\mu+\frac{1}{2})}{2^{2\nu}\Gamma(\nu)\Gamma(-\nu-\mu+\frac{1}{2})}y^{2\nu}\right)\;,\quad{\rm
for}\;\;0<\nu<1\,.$

Applying this to (34) gives

$S|_{\rm on-shell}=\frac{1}{4\pi}\int dt\int dt^{\prime}\int
d{\omega}e^{i{\omega}(t-t^{\prime})}\phi_{\epsilon}(t)\phi_{\epsilon}(t^{\prime})\frac{1}{\epsilon}\,\times\,\qquad\qquad$
$\frac{2\nu(\Delta_{+}+\frac{2\kappa}{\alpha})}{\nu+\frac{2\kappa}{\alpha}+\frac{1}{2}}\Biggr{(}1-\frac{\Gamma(-\nu)\Gamma(\nu+\frac{2\kappa}{\alpha}+\frac{1}{2})}{2^{2\nu}\Gamma(\nu)\Gamma(-\nu+\frac{2\kappa}{\alpha}+\frac{1}{2})}\Bigl{(}\frac{{|\omega}|\alpha\epsilon}{2}\Bigr{)}^{2\nu}\Biggr{)}\Bigg{|}_{\epsilon\rightarrow
0}\,.$ (35)

The integral over $\omega$ can be performed, again using (16), with the result
for the on-shell action being

$-\frac{\nu}{\sqrt{\pi}}\frac{\Gamma(\nu+\frac{1}{2})\Gamma(\nu+{\frac{2\kappa}{\alpha}}+\frac{1}{2})}{\Gamma(\nu)\Gamma(-\nu+{\frac{2\kappa}{\alpha}}+\frac{1}{2})}\int
dt\int
dt^{\prime}\phi_{\epsilon}(t)\phi_{\epsilon}(t^{\prime})\frac{1}{\epsilon}\Bigl{(}\frac{\alpha\epsilon}{2}\Bigr{)}^{2\nu}\frac{1}{|t-t^{\prime}|^{2\nu+1}}\Bigg{|}_{\epsilon\rightarrow
0}\,.$

In terms of the $\epsilon-$independent source fields $\phi_{0}$ this becomes

$S|_{\rm on-
shell}=-\left(\frac{\alpha}{2}\right)^{2\nu}\frac{\nu}{\sqrt{\pi}}\frac{\Gamma(\nu+\frac{1}{2})\Gamma(\nu+{\frac{2\kappa}{\alpha}}+\frac{1}{2})}{\Gamma(\nu)\Gamma(-\nu+{\frac{2\kappa}{\alpha}}+\frac{1}{2})}\,\int
dt\int
dt^{\prime}\,\frac{\phi_{0}(t)\phi_{0}(t^{\prime})}{|t-t^{\prime}|^{2\nu+1}}\,.$

The conformal answer for the two-point function (2) is then once again
recovered after using (1). Now the overall factor is

$C^{\rm
nc}_{\Delta_{+}}\;=\;\,-\left(\frac{\alpha}{2}\right)^{2\nu}\frac{2\nu}{\sqrt{\pi}}\frac{\Gamma(\nu+\frac{1}{2})\Gamma(\nu+{\frac{2\kappa}{\alpha}}+\frac{1}{2})}{\Gamma(\nu)\Gamma(-\nu+{\frac{2\kappa}{\alpha}}+\frac{1}{2})}\,.$
(36)

The ratio of the noncommutative result to the commutative result (21) is given
by the simple expression

${\cal
R}^{(2)}(\alpha,\nu)=\left(\frac{\alpha}{2}\right)^{2\nu}\frac{\Gamma(\nu+{\frac{2\kappa}{\alpha}}+\frac{1}{2})}{\Gamma(-\nu+{\frac{2\kappa}{\alpha}}+\frac{1}{2})}\,.$
(37)

One can check various limits. The ratio smoothly goes to one in the
commutative limit $\alpha\rightarrow 0$. It reduces to $\kappa$ in the
massless case, $\nu=\frac{1}{2}$, which agrees with the result found in
[5].666If one does a leading order expansion in $\alpha^{2}$ one gets  ${\cal
R}^{(2)}(\alpha,\nu)=1+\frac{\nu}{12}\left(\frac{13}{4}-\nu^{2}\right)\alpha^{2}+{\cal
O}(\alpha^{4}).$ This disagrees with the result found in [3], which is due to
the fact that a different regularization scheme was used in that approach. The
answers agree on the other hand, if one re-evaluates integrals in [3] with the
regularization scheme used here. Both regularizations give the same result in
the massless case. For a discussion of the different regularizations see the
appendix in [13]. We also note that the result found here agrees with that
found in [17] for the massless case, but it differs when $\nu\neq\frac{1}{2}$.

##  4 Concluding Remarks

It had previously been conjectured that the boundary two-point function
associated with field theory on a noncommutative $AdS_{2}$ bulk satisfies the
constraints of conformal invariance, i.e. it has the form (2). This was
previously shown to be true up to leading order in the noncommutativity
parameter $\alpha$. In this article we proved that the conjecture is true to
all orders in $\alpha$, at least for the scalar field. A similar proof should
be possible for spinors. The result means that the noncommutativity of the
bulk only effects the over-all normalization of the boundary two-point
function.

It is a nontrivial step to see whether or not these results can be extended to
the case of $n(>2)-$point correlators on the boundary. This will require
analyzing a fully interacting field theory on the noncommutative bulk. If
conformal invariance is satisfied for the $n(>2)-$point boundary correlators
then, once again, only the over-all factor is effected by the noncommutative
bulk. We would then have an expression for the ratio ${\cal
R}^{(n)}(\alpha,\nu)$ of the over-all factor for the noncommutative correlator
with that of the commutative correlator. If the ratio is such that
$\sqrt[n]{{\cal R}^{(n)}(\alpha,\nu)}=\sqrt{{\cal R}^{(2)}(\alpha,\nu)}$, it
would suggest that the only effect that bulk noncommutativity has on the
boundary is to renormalize the conformal operators ${\cal O}$. If instead this
turns out not to be the case, we would conclude that the couplings between
boundary operators pick up quantum gravity corrections as well. Only a
preliminary investigation in this direction has been undertaken so far. A
cubic interaction term was introduced to the massive scalar field on
noncommutative $EAdS_{2}$ in [3]. The analysis there was quite nontrivial
because only a perturbative solution (in $\alpha$) was available, and one
needs to perturb in the other parameter, the coupling constant $\lambda$, as
well. Although there were strong indications that the leading order correction
to $3-$point function satisfied the constraints of conformal invariance, we
were not able to write down an explicit expression for the leading order
correction, in either of the parameters. The fact that we now have an exact
answer for the free theory, could be quite beneficial for obtaining the
$3-$point function, since then we would only have to perform an expansion in
one parameter, i.e., $\lambda$. The only obstacle to proceeding with the
analysis is the construction of a meaningful Green function on the
noncommutative bulk. We hope to report on this construction in a future work.

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