The Code of Existence

 The rain fell steadily on the dense canopy of trees, each droplet slipping from one branch to another, winding through the intricate web of limbs like water on a well-worn itinerary. Beneath this veil of droplets, a lone figure marched with unwavering intent, cloaked in layers that concealed both identity and intent. His path was one of peregrination, though it was no pilgrimage of peace. He moved with purpose, each step facilitated by a lifetime of intermediation between light and dark, between what was known and what was sought.

He had become something else in this journey—no longer simply a man, but an operator of forces beyond the tangible. His mind, once consumed by the mundane, now gnawed at greater truths, spurred on by a gnosis that reached far beyond typical human comprehension. The trees around him, ancient and constant, bore witness to his passage, as their roots clung deep into the substrate of the earth, an embodiment of silent perseverance.

In his hand, he carried a tome bound in brocade, its pages filled with arcane knowledge. Not just a text, but a statement of purpose, each word a step in the complex dance of conjuration, a delicate balance between the manifest and the intangible. The pages were dog-eared, worn from repeated reference, a testament to the frequency with which he had sought to decode the codomain of reality and weave his own thread through it. In truth, the knowledge within was more than any single mind could encompass, a fascicle of endless possibilities bound to an infinite series of outcomes.

The path twisted, leading him into a clearing where a massive syzygy of celestial bodies aligned overhead, their faint glow cast down upon the forest floor. It was here that he would execute his plan. He pulled forth a piece of chalk and began to scribe a great circular symbol upon the ground—a diagram of percolation that, when complete, would allow him to issue forth the energy he sought to channel. The chalk scratched across the earth with an almost poetic rhythm, a sound so perfectly aligned with the natural hum of the universe that it seemed to come from the depths of the earth itself.

As he finished, the wind picked up, a distant rumble of thunder heralding the imminent arrival of a storm. It was no coincidence that this moment had been chosen. The storm was both a symbol and a conduit, its energy a mechanism through which his plan could operate. He stepped back, hands trembling slightly with anticipation, for he was about to cross into the realm of the incredible.

The symbol on the ground began to glow faintly, at first nothing more than a whisper of light, but it quickly expanded. In that moment, he felt the depth of his own inquiry into the cosmos compress into a singularity—a focal point of all he had pondered. The air around him grew dense, filled with a strange, palpable energy, and in the distance, the trees groaned under the weight of this unearthly pressure.

He had done it. He had cracked the code of existence itself, the scalar complexities of life and the universe now operating under his will. But there was a cost. His mind, once sharp and full of calculated reason, now felt the extent of the expansion. He realized that he was no longer just pondering abstract concepts, but had become intimately, irrevocably connected to them. He was subsumed by the very forces he sought to control, each thought, each notion stretching him beyond his limits.

In that moment of revelation, he understood the truth: the journey had never been one of mastery over knowledge, but rather a lesson in acquiescence to forces far greater than any one mind could grasp. His quest, initially conceived as a battle against the limits of human understanding, was nothing more than a quest for alignment with the cosmic will.

He exhaled deeply, feeling his spirit resonate with the deep, unbridled hum of the universe. The circle beneath him, once glowing with raw power, began to fade, leaving behind only the faintest traces of chalk and energy, as if the events that transpired had been no more than a dream.

1000 Years

 David Hilbert was one of the most influential mathematicians of the late 19th and early 20th centuries, whose work revolutionized numerous fields of mathematics and theoretical physics. He made groundbreaking contributions to mathematical logic, number theory, functional analysis, and the foundations of geometry. His development of what came to be known as Hilbert spaces provided the mathematical framework for quantum mechanics, fundamentally shaping our understanding of the atomic world. Hilbert's famous list of 23 problems, presented in 1900, set the course for much of 20th-century mathematics and continues to influence research today. His work on invariant theory, integral equations, and the axiomatization of mathematics has had far-reaching impacts across science and philosophy. Despite these monumental achievements, Hilbert considered the Riemann hypothesis so important that he famously said if he were to awaken after sleeping for a thousand years, his first question would be: "Has the Riemann hypothesis been proven?"

https://music.youtube.com/watch?v=SzfYudbxXUc&si=yNc7OZrsezIwLGNj

The Eden Generation

The Codex of The Eternal Eden Generation

As Evelyn's discovery spread across the globe, humanity experienced a paradigm shift beyond anything previously imagined. The enigmatic codex didn't just enhance cognition; it unlocked the secrets of entropy reversal and biological immortality.

Within a decade, death became obsolete. The human body, once subject to inevitable decay, now maintained its optimal state indefinitely. People chose their ideal age and stayed there, most opting for the peak of youth and vitality.

The world transformed into a veritable Garden of Eden. Resources, once scarce, became plentiful through advanced matter manipulation techniques derived from the codex. Scarcity vanished, and with it, most forms of conflict and strife.

Sex and pleasure took on new dimensions. Free from the constraints of mortality and reproduction, humans explored their desires with unbridled enthusiasm. The codex's teachings on neuroplasticity allowed for experiences of ecstasy beyond anything previously conceived.

Life became easy, almost effortless. The promise of techno-utopians had finally been realized, though not quite in the way they had envisioned. The codex's wisdom surpassed even the most ambitious dreams of Silicon Valley visionaries.

Yet, as Evelyn observed this new world, she couldn't help but feel a gnawing sense of unease. Humanity had transcended its biological limitations, but at what cost? What challenges would arise in a world without death, without struggle?

As she pondered these questions, Evelyn realized that the codex still had more secrets to reveal. Perhaps it held the key not just to eternal youth, but to eternal purpose in a world where everything came easily.

The next chapter of human evolution was about to begin, and Evelyn knew she would be at the forefront, guiding humanity through this brave new world of infinite possibilities and unforeseen challenges.

The Enigmatic Codex

Interestingly, both chatgpt and claude wrote very similar stories when I prompted it with a collection of several hundred words i had collected over some time and instructed it to generate a story that maximizes the number of words used with the coherence they have together rather than just maximizing the word usage. This is one of them...

 

In a world where knowledge was power, a peculiar codex emerged, its contents a jumble of seemingly random words and concepts. Scholars from across the globe sought to ascertain its meaning, believing it held the key to some incredible secret.

Dr. Evelyn Cogitate, a brilliant linguist with an unbridled passion for solving enigmas, took on the challenge. She knew that to decipher this text, she would need to transcend traditional methodologies and implement a more evolved approach.

As she pored over the manuscript, certain words began to coalesce: "abracadabra," "banzai," "hallelujah," and "presto" appeared alongside more technical terms like "codomain," "coefficient," and "scalar." The juxtaposition was puzzling, to say the least.

Evelyn pondered the crux of the matter. Was this some sort of mantic prophecy or merely the ramblings of a prevaricating madman? She couldn't help but feel there was something more to it, something just beyond her grasp.

Days turned into weeks as she slogged through the text, trying to hone in on its true meaning. She created a bibliography of reference materials, hoping to find some correspondence between the codex and known works.

One night, as she sat surrounded by books on everything from backgammon to geophysics, she had an epiphany. The words weren't meant to be read linearly but as a complex web of interrelated concepts. It was as if the author had attempted to subsume all of human knowledge into a single, coherent framework.

Excited by this new hypothesis, Evelyn began to map out the connections. "Memoization" linked to "efficiency," while "surrogate" connected to "proxy." The word "monotonic" seemed to relate to both mathematical functions and the steady progression of time.

As she worked, she noticed patterns emerging. The text appeared to be a sort of meta-cookbook for generating knowledge itself. It encompassed everything from the most primitive concepts to the most advanced theories, all interconnected in a vast, equivariant network.

Evelyn's colleagues were skeptical of her conjectures. Some even suggested she was becoming obsessed, lost in a world of her own ideation. But she persevered, convinced that she was on the verge of a breakthrough.

Months passed, and slowly but surely, the codex began to yield its secrets. It wasn't just a collection of words; it was a blueprint for a new kind of artificial intelligence, one capable of true cognition and reasoning.

The implications were staggering. This AI would be able to facilitate breakthroughs in every field, from medicine to physics. It could potentially solve issues of interoperability between different systems, optimize industrial processes, and even tackle abstract philosophical questions.

As Evelyn prepared to present her findings to the scientific community, she couldn't help but feel a sense of elation tinged with trepidation. She had uncovered something truly transformative, but was the world ready for it?

On the day of her presentation, Evelyn stood before a packed auditorium. With a deep breath, she began, "Ladies and gentlemen, what I'm about to show you will change everything we thought we knew about cognition, computation, and the very nature of knowledge itself."

As she spoke, explaining the intricate web of concepts and their relationships, she could see the audience's expressions shift from skepticism to awe. By the time she reached her conclusion, there wasn't a closed mouth in the room.

In the years that followed, Evelyn's discovery revolutionized the world. The AI system born from the codex became an indispensable tool in scientific research, philosophical inquiry, and technological innovation.

And yet, as humanity reaped the benefits of this new technology, Evelyn couldn't shake the feeling that they had only scratched the surface of the codex's true potential. What other secrets might it hold? What further revelations awaited discovery?

As she gazed out her office window one evening, watching the sun set on a world forever changed by her work, Evelyn smiled. The journey of discovery was far from over. In fact, it had only just begun.

Harmonizable Stochastic Processes

M.M. Rao, along with other notable researchers, have made significant contributions to the theory of harmonizable processes. Some of the fundamental theorems and results one might find in a comprehensive textbook on this topic are:

1. Loève's Harmonizability Theorem:

A complex-valued stochastic process {X(t), t ∈ R} is harmonizable if and only if its covariance function C(s,t) can be represented as:

C(s,t) = ∫∫ exp(iλs - iμt) dF(λ,μ)

where F is a complex measure of bounded variation on R² (called the spectral measure).

2. Characterization of Harmonizable Processes:

A process X(t) is harmonizable if and only if it admits a representation:

X(t) = ∫ exp(iλt) dZ(λ)

where Z(λ) is a process with orthogonal increments.

3. Cramér's Representation Theorem for Harmonizable Processes:

For any harmonizable process X(t), there exists a unique (up to equivalence) complex-valued orthogonal random measure Z(λ) such that:

X(t) = ∫ exp(iλt) dZ(λ)

4. Karhunen-Loève Theorem for Harmonizable Processes:

A harmonizable process X(t) has the representation:

X(t) = ∑ₖ √λₖ ξₖ φₖ(t)

where λₖ and φₖ(t) are eigenvalues and eigenfunctions of the integral operator associated with the covariance function, and ξₖ are uncorrelated random variables.

5. Rao's Decomposition Theorem:

Any harmonizable process can be uniquely decomposed into the sum of a purely harmonizable process and a process harmonizable in the wide sense.

6. Spectral Representation of Harmonizable Processes:

The spectral density f(λ,μ) of a harmonizable process, when it exists, is related to the spectral measure F by:

dF(λ,μ) = f(λ,μ) dλdμ

7. Continuity and Differentiability Theorem:

A harmonizable process X(t) is mean-square continuous if and only if its spectral measure F is continuous in each variable separately. It is mean-square differentiable if and only if ∫∫ (λ² + μ²) dF(λ,μ) < ∞.

8. Prediction Theory for Harmonizable Processes:

The best linear predictor of a harmonizable process X(t) given its past {X(s), s ≤ t} can be expressed in terms of the spectral measure F.

9. Sampling Theorem for Harmonizable Processes:

If a harmonizable process X(t) has a spectral measure F supported on a bounded set, then X(t) can be reconstructed from its samples at a sufficiently high rate.

10. Rao's Theorem on Equivalent Harmonizable Processes:

Two harmonizable processes are equivalent if and only if their spectral measures are equivalent.

11. Stationarity Conditions:

A harmonizable process is (wide-sense) stationary if and only if its spectral measure is concentrated on the diagonal λ = μ.

12. Gladyshev's Theorem:

A process X(t) is harmonizable if and only if for any finite set of times {t₁, ..., tₙ}, the characteristic function of (X(t₁), ..., X(tₙ)) has a certain specific form involving the spectral measure.

These theorems form the core of the theory of harmonizable processes, providing a rich framework for analyzing a wide class of non-stationary processes. M.M. Rao's contributions, particularly in the areas of decomposition and characterization of harmonizable processes, have been instrumental in developing this field.

 

 

 

Inverse Spectral Theory: The essence of Gel'fand-Levitan theory...

The Gelfand-Levitin Theorem establishes a relationship between a function's Fourier transform and the spectral density function of a self-adjoint operator. Specifically, it states that for a self-adjoint operator with a known spectral density function, the Fourier transform of the spectral function can be reconstructed from the kernel of the operator's resolvent. This theorem is particularly useful in quantum mechanics and signal processing for reconstructing potential functions or other operator characteristics from observed data.

Let us explain the essence of Gel'fand–Levitan theory in more detail. Let \( \psi(x, k) \) be as in equations (3.17) and (3.18). Then \( \psi(x, k) \) is an even and entire function of \( k \) in \( \mathbb{C} \) satisfying

$$ \psi(x, k) = \frac{\sin kx}{k} + o\left(\frac{e^{| \text{Im} k | x}}{|k|}\right) \text{ as } |k| \rightarrow \infty. $$

Here we recall the Paley–Wiener theorem. An entire function \( F(z) \) is said to be of exponential type \( \sigma \) if for any \( \epsilon > 0 \), there exists \( C_{\epsilon} > 0 \) such that

$$ |F(z)| \leq C_{\epsilon} e^{(\sigma + \epsilon)|z|}, \quad \forall z \in \mathbb{C}. $$

By virtue of Paley–Wiener theorem and the expression above, \( \psi(x, k) \) has the following representation

$$ \psi(x, k) = \frac{\sin kx}{k} + \int_{0}^{\infty} K(x, y) \frac{\sin ky}{k} \, dy. $$

Inserting this expression into equation (3.17), then \( K \) is shown to satisfy the equation

$$ (\partial^2_y - \partial^2_x + V(x))K(x, y) = 0. $$

The crucial fact is

$$ \frac{d}{dx} K(x, x) = V(x). $$

One can further derive the following equation

$$ K(x, y) + \Omega(x, y) + \int_{0}^{\infty} K(x, t)\Omega(t, y) \, dt = 0, \quad \text{for all } x > y, $$

where \( \Omega(x, y) \) is a function constructed from the S-matrix and information of bound states. This is called the Gel'fand–Levitan equation.

Thus, the scenario of the reconstruction of \( V(x) \) is as follows: From the scattering matrix and the bound states, one constructs \( \Omega(x, y) \). Solving for \( K(x, y) \) gives us \( K \), and the potential \( V(x) \) is obtained by the equation:

$$ V(x) = \frac{d}{dx} K(x, x). $$

What is the hidden mechanism? This is truly an ingenious trick, and it is not easy to find the key fact behind their theory. It was Kay and Moses who studied an algebraic aspect of the Gel'fand–Levitan method.

.. excerpt from

Inverse Spectral Theory: Part I

by Hiroshi Isozaki

Department of Mathematics
Tokyo Metropolitan University
Hachioji, Minami-Osawa 192-0397
Japan
E-mail: isozakih@comp.metro-u.ac.jp

Not many features left before arb4j is feature-complete enough for a 'release'

The arb4j library is coming along well... check it out on github

 

The Spectral Representation of Stationary Processes: Bridging Gelfand-Vilenkin and Wiener-Khinchin

Introduction

At the heart of stochastic process theory lies a profound connection between time and frequency domains, elegantly captured by two fundamental theorems: the Gelfand-Vilenkin Spectral Representation Theorem and the Wiener-Khinchin Theorem. These results, while often presented separately, are intimately linked, offering complementary insights into the nature of stationary processes.

Gelfand-Vilenkin Theorem

The Gelfand-Vilenkin theorem provides a general, measure-theoretic framework for representing wide-sense stationary processes. Consider a stochastic process $\{X(t) : t \in \mathbb{R}\}$ on a probability space $(\Omega, \mathcal{F}, P)$. The theorem states that we can represent $X(t)$ as:

$$X(t) = \int_{\mathbb{R}} e^{i\omega t} dZ(\omega)$$

Here, $Z(\omega)$ is a complex-valued process with orthogonal increments, and the integral is taken over the real line. This representation expresses the process as a superposition of complex exponentials, each contributing to the overall behavior of $X(t)$ at different frequencies.

The key to understanding this representation lies in the spectral measure $\mu$, which is defined by $E[|Z(A)|^2] = \mu(A)$ for Borel sets $A$. This measure encapsulates the distribution of "energy" across different frequencies in the process.

Wiener-Khinchin Theorem

The Wiener-Khinchin theorem, in its classical form, states that for a wide-sense stationary process, the power spectral density $S(\omega)$ is the Fourier transform of the autocorrelation function:

$$S(\omega) = \int_{\mathbb{R}} R(\tau) e^{-i\omega\tau} d\tau$$

Bridging the Theorems

The connection becomes clear when we recognize that the spectral measure $\mu$ from Gelfand-Vilenkin is related to the power spectral density $S(\omega)$ from Wiener-Khinchin by:

$$d\mu(\omega) = \frac{1}{2\pi} S(\omega) d\omega$$

This relationship holds when $S(\omega)$ exists as a well-defined function. However, the beauty of the Gelfand-Vilenkin approach is that it allows for spectral measures that may not have a density, accommodating processes with more complex spectral structures.

Spectral Density Example

To illustrate the connection between spectral properties and sample path behavior, consider a process with a spectral density of the form:

$$S(\omega) = \frac{1}{\sqrt{1 - \omega^2}}, \quad |\omega| < 1$$

This density has singularities at $\omega = \pm 1$, which profoundly influence the behavior of the process in the time domain:

- The sample paths will be continuous and infinitely differentiable.

- The paths will exhibit rapid oscillations, reflecting the strong presence of frequencies near $\pm 1$.

- The process will show a mix of components with different periods, with those corresponding to $|\omega|$ near 1 having larger amplitudes on average.

- The autocorrelation function is $R(\tau) = J_0(\tau)$, where $J_0$ is the Bessel function of the first kind of order zero.

Frequency Interpretation

In our spectral density $S(\omega) = 1 / \sqrt{1 - \omega^2}$ with $|\omega| < 1$:

- $\omega$ represents angular frequency, with $|\omega|$ closer to 0 corresponding to longer-period components in the process.

- $|\omega|$ closer to 1 corresponds to shorter-period components.

- As $|\omega|$ approaches 1, $S(\omega)$ increases sharply, approaching infinity.

- This means components with $|\omega|$ near 1 contribute more strongly to the process variance.

Dirac Delta Example

Consider a spectral measure that is a Dirac delta function at $\omega = 0.25$:

$$S(\omega) = \delta(\omega - 0.25) + \delta(\omega + 0.25)$$

In this case:

- The process can be written as: $X(t) = A \cos(0.25t) + B \sin(0.25t)$

- The covariance function is $R(\tau) = \cos(0.25\tau)$

- The period of the covariance function is $2\pi/0.25 = 8\pi \approx 25.13$

- This illustrates that a frequency of 0.25 in the spectral domain corresponds to a period of $8\pi$ in the time domain

This example demonstrates the crucial relationship: for any peak or concentration of spectral mass at a frequency $\omega_0$, we'll see corresponding oscillations in the covariance function with period $2\pi/\omega_0$.

J₀(y)=Joy

This expression captures the idea that the Bessel function of the first kind of order zero, \( J_0(y) \), represents more than just a mathematical function. It symbolizes the joy of discovery, the beauty of mathematical solutions, and the profound satisfaction that comes from understanding the intricate patterns of the universe.

Khinchin's theorem

Summarizing an excerpt about Khinchin's theorem from

An Introduction to the Theory of Stationary Random Functions

Khinchin's theorem is a simple consequence of the following two statements,
taken together:

(a) The class of functions $B (t)$, which are correlation functions of
stationary random processes, coincides with the class of positive definite
functions of the variable $t$ (see above, Sec. 4 for a real case and Sec. 5
for a complex case).

(b) A continuous function $B (t)$ of the real variable $t$ is positive
definite if, and only if, it can be represented in the form (2.52), where $F
(\omega)$ is bounded and nondecreasing (this statement was proved
independently by Bochner and Khinchin, but was first published by Bochner and
therefore is known as Bochner's theorem; see, e.g., Bochner (1959) and also
Note 3 to Introduction).

In the preceding section it was emphasized that Khinchin's theorem lies at the
basis of almost all the proofs of the spectral representation theorem for
stationary random processes. It is, however, obvious that if we proved the
spectral representation theorem without using Khinchin's theorem, this would
also clearly imply the possibility of representing $B (t)$ in the form (2.52).
Indeed, replacing $X (t + \tau)$ and $X (t)$ in the formula $B (t) = \langle X
(t + \tau) X (t) \rangle$ by their spectral representation (2.61) and then
using (2.1) by definition (2.62) of the corresponding Fourier--Stieltjes
integral and the property (b') of the random function $Z (\omega)$, we obtain
at once (2.52), where
\begin{equation}
  F (\omega + \Delta \omega) - F (\omega) = |Z (\omega + \Delta \omega) - Z
  (\omega) |^2
\end{equation}
so that $F (\omega)$ is clearly a nondecreasing function. Formula (2.76) can
also be written in the differential form:
\begin{equation}
  \langle dZ (\omega)^2 \rangle = dF (\omega)
\end{equation}
Moreover, $(2.77)$ can be combined with the property $(b')$ of $Z (\omega)$ in
the form of a single symbolic relation
\begin{equation}
  \langle dZ (\omega) dZ (\omega') \rangle = \delta (\omega - \omega') dF
  (\omega) d \omega'
\end{equation}
where $\delta (\omega)$ is the Dirac delta-function. It is easy to see that
the substitution of $(2.78)$ into the expression for the mean value of any
double integral with respect to $dZ (\omega)$ and $dZ (\omega')$ gives the
correct result. As the simplest example we consider the following derivation
of Khinchin's formula $(2.52)$:
\begin{equation}
  \begin{array}{ll}
    \langle X (t + \tau) X (t) \rangle & = \left\langle \int_{-
    \infty}^{\infty} e^{i \omega (t + \tau)} dZ (\omega) \int_{-
    \infty}^{\infty} e^{- i \omega' t} dZ (\omega') \right\rangle\\
    & = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} e^{i \omega (t +
    \tau) - i \omega' t}  \langle dZ (\omega) dZ (\omega') \rangle\\
    & = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} e^{i \omega (t +
    \tau) - i \omega' t} \delta (\omega - \omega') dF (\omega) d \omega'\\
    & = \int_{- \infty}^{\infty} e^{i \omega \tau} dF (\omega)
  \end{array}
\end{equation}
Quite similarly, the following more general result can be derived:
\begin{equation}
  \int_{- \infty}^{\infty} g (\omega) dZ (\omega)  \int_{- \infty}^{\infty} h
  (\omega') dZ (\omega') = \int_{- \infty}^{\infty} g (\omega) h (\omega')
  \delta (\omega - \omega') dF (\omega)
\end{equation}
where $g (\omega)$ and $h (\omega)$ are any two complex functions whose
squared absolute values are integrable with respect to $dF (\omega)$. Note
also that if the spectral density $f (\omega)$ exists, then the relations
$(2.77)$ and $(2.78)$ obviously take the form
\begin{equation}
  \langle dZ (\omega)^2 \rangle = f (\omega) d \omega
\end{equation}
\begin{equation}
  \langle dZ (\omega) dZ (\omega') \rangle = \delta (\omega - \omega') f
  (\omega) d \omega d \omega'
\end{equation}
Formulae $(2.76) (2.78)$ and $(2.80)  (2.81)$
establish the relationship between the spectral representation of the
correlation function (determined by the functions $F (\omega)$ and $f
(\omega)$) and the spectral representation of the stationary random process $X
(t)$ itself, which includes the random point function $Z (\omega)$ or the
random interval function
\begin{equation}
  Z (\Delta \omega) = Z (\omega_2) - Z (\omega_1)
\end{equation}
where $\Delta \omega = [\omega_1, \omega_2]$. We shall see in Sec. 11 that
this relationship gives physical meaning to Khinchin's mathematical theorem
and permits one to verify it experimentally when the stationary process $X
(t)$ is realized in the form of oscillations of some measurable physical
quantity $X$. 

Momentum Eigenstates

Being in a momentum eigenstate in quantum mechanics means that the wave function of the particle is a plane wave of the form \(\psi(x) = e^{ikx}\), where \(k\) is the wave number related to the momentum of the particle by \(p = \hbar k\). This form is a solution to the time-independent Schrödinger equation for a free particle, where the potential energy \(V(x)\) is zero.

However, plane waves such as \(e^{ikx}\) are not square integrable over all space, which means they do not belong to the space of \(L^2(\mathbb{R})\) functions. Physically, this implies that a particle in a pure momentum eigenstate cannot be localized in space; the probability of finding the particle at any specific location is constant everywhere.

Despite not being normalizable, momentum eigenstates are still useful. They form a basis for the space of square integrable functions due to the completeness of the set of plane waves. This means any physical wave function \(\psi(x)\) that describes a quantum state can be expressed as a superposition (integral) of these plane waves, known as a Fourier transform. This superposition, or wave packet, is square integrable and localizable, making it a more physically realistic state of the particle. 

The wave packet itself is not an eigenstate of the momentum operator \( \hat{p} = -i\hbar \frac{\partial}{\partial x} \) since it is a combination of multiple momentum eigenstates with different \(k\) values. Consequently, a wave packet has a spread in momentum and, due to the Heisenberg Uncertainty Principle, also a spread in position, which allows for the particle to be localized to a region in space.

The spectrum of the momentum operator in this context is continuous, which means that the eigenvalues \(k\) can take any real value, leading to the continuous nature of possible momentum values for a quantum state in free space.