A new class of coherent states involving Fox-Wright functions and their generalization in the bicomplex framework

This paper introduces and analyzes a new class of coherent states based on Fox-Wright functions, extending their properties from the discrete to the continuous spectrum and further generalizing the framework to the bicomplex domain.

The Gist

Imagine the universe of quantum mechanics as a giant, complex orchestra. For a long time, musicians (physicists) have been trying to find the perfect "note" that bridges the gap between the chaotic, jittery world of tiny particles and the smooth, predictable world of everyday objects. These perfect notes are called Coherent States.

Think of a coherent state like a perfectly tuned laser beam or a calm ripple in a pond. It's a special quantum state that behaves almost exactly like a classical wave, making it incredibly useful for things like quantum computing and secure communication.

This paper is about the authors (Snehasis Bera, Sourav Das, and Abhijit Banerjee) discovering a brand new family of these perfect notes, and then taking them on a journey into a strange, four-dimensional musical dimension.

Here is the breakdown of their adventure, using simple analogies:

1. The New "Recipe" for Perfect Notes (The Fox-Wright Function)

For decades, scientists have used standard recipes (like the "Glauber" recipe) to make these coherent states. But the authors asked: "What if we use a more complex, exotic ingredient?"

They chose a mathematical ingredient called the Fox-Wright function.

• The Analogy: Imagine you are baking a cake. Standard recipes use flour and sugar. The Fox-Wright function is like a secret, multi-layered spice blend that contains all other spices inside it. It's a "super-spice" that can turn into regular flour, sugar, or anything else depending on how you use it.
• The Result: By using this "super-spice" as the main ingredient (the normalization function), they created a new class of coherent states. They proved these new states work perfectly: they are stable, they don't disappear, and they can be used to reconstruct the whole quantum system (a property called "resolution of unity," which is like saying if you have a complete set of these notes, you can play any song in the universe).

2. The "Continuous" vs. "Discrete" Switch

In quantum mechanics, energy usually comes in steps (like rungs on a ladder). This is the discrete spectrum. But sometimes, energy flows like a smooth ramp. This is the continuous spectrum.

• The Analogy: Imagine a staircase (discrete) vs. a ramp (continuous).
• The Innovation: The authors showed how to take their new "Fox-Wright cake" recipe and smoothly slide it from the staircase to the ramp without the cake falling apart. They created a new version of their "super-spice" (which they call the FW-generalized multi-parameter $\nu$-function) that works specifically for the smooth ramp. This means their new coherent states can handle both the step-by-step world and the smooth-flowing world.

3. The Journey to "Bicomplex" Land (The Fourth Dimension)

This is the most mind-bending part of the paper. Standard quantum mechanics uses Complex Numbers (numbers with a real part and an imaginary part, like $3 + 4i$). Think of this as a 2D map.

The authors decided to go one step further into Bicomplex Numbers.

• The Analogy: If a complex number is a flat map (2D), a bicomplex number is like a 4D hologram. It has two "imaginary" directions instead of just one. It's a number system that allows for "zero divisors" (numbers that aren't zero but multiply to zero), which is like having a shadow that can cancel out the object casting it.
• The Challenge: Math in this 4D world is tricky. Functions that work on a flat map might explode or break in 4D.
• The Discovery: The authors took their "Fox-Wright super-spice" and tried to bake it in this 4D holographic kitchen. They spent a lot of time checking the "convergence" (making sure the cake doesn't burn or turn into dust).
  • They found nine different scenarios (like nine different weather conditions) where this 4D cake works perfectly.
  • They proved that under certain conditions, the 4D Fox-Wright function is stable and behaves beautifully, just like its 2D cousin.

4. The Grand Finale: 4D Coherent States

Finally, they built Bicomplex Coherent States.

• The Analogy: They took their perfect notes and played them in the 4D holographic orchestra.
• The Result: They proved these 4D notes are also stable, normalized, and can reconstruct the 4D universe. They even showed how to switch these 4D notes from the "staircase" (discrete) to the "ramp" (continuous) using a special 4D version of their $\nu$-function.

Why Does This Matter?

You might ask, "Who cares about 4D math cakes?"

1. Flexibility: This new "Fox-Wright" recipe is so powerful that it includes many other famous mathematical functions as special cases. It's a "master key" that unlocks many different types of quantum states.
2. Future Tech: As we move toward more advanced quantum computers and communication systems, we might need to use these higher-dimensional number systems (bicomplex numbers) to process information more efficiently or securely.
3. Mathematical Beauty: They showed that even in these weird, high-dimensional worlds, the fundamental laws of quantum mechanics (like stability and continuity) still hold true.

In a nutshell: The authors invented a new, ultra-versatile mathematical tool (the Fox-Wright function), used it to create new types of quantum states, and then successfully transported those states into a strange, four-dimensional reality, proving that the rules of the quantum universe are robust enough to handle even the most exotic math.

Technical Summary

1. Problem Statement and Motivation

The paper addresses the construction and generalization of coherent states (quantum states that most closely resemble classical states) using advanced special functions.

• Context: While canonical coherent states (Glauber states) are well-established for the harmonic oscillator, there is a need to extend these concepts to systems with nonlinear energy spectra and non-standard commutation relations.
• Gap: Previous works have utilized Mittag-Leffler and generalized hypergeometric functions for coherent states. However, the Fox-Wright function (${}_p\psi_q$), a more general special function encompassing many others (including Wright, Bessel, and Mittag-Leffler functions), had not been fully utilized as a normalization function for a broad class of coherent states, nor had it been extended to the bicomplex number system.
• Objective: To construct a new class of coherent states where the Fox-Wright function serves as the normalization factor, analyze their properties for both discrete and continuous spectra, and generalize these results to the bicomplex domain.

2. Methodology

The authors employ a rigorous mathematical framework combining quantum mechanics, special function theory, and bicomplex analysis.

• Discrete Spectrum Construction:

  • Defined coherent states $|z\rangle$ in an infinite-dimensional Hilbert space using Fock vectors $|k\rangle$.
  • Introduced a normalization function $N_{(p,q)}(\zeta)$ based on the Fox-Wright function.
  • Defined annihilation ($A_-$) and creation ($A_+$) operators satisfying deformed commutation relations involving a parameter function $\rho(k)$.
  • Verified the Resolution of Unity by deriving a specific weight function $W(|z|^2)$ using the Mellin transform of the H-function.

• Continuous Spectrum Transition:

  • Applied a discrete-to-continuous limit ($d \to c$), replacing summation with integration and discrete quantum numbers with continuous energy variables ($E$).
  • Introduced the FW-generalized multi-parameter $\nu$-function as the normalization function for the continuous spectrum.

• Bicomplex Generalization:

  • Extended the framework to bicomplex numbers ($\mathbb{BC}$), which are commutative rings with zero divisors formed by two imaginary units $i$ and $j$.
  • Utilized the idempotent representation ($Z = z_1e_1 + z_2e_2$) to decompose bicomplex problems into two independent complex problems.
  • Defined the Bicomplex Fox-Wright function and investigated its convergence using the hyperbolic root test and Stirling's formula.
  • Constructed bicomplex coherent states and derived the corresponding integration measures for the resolution of unity.

3. Key Contributions

A. Fox-Wright Coherent States (Complex Domain)

• Definition: Introduced coherent states $|z\rangle$ normalized by the Fox-Wright function ${}_p\psi_q$.
• Properties: Proved that these states satisfy:
  1. Continuity: The state varies continuously with the complex parameter $z$.
  2. Normalizability: $\langle z | z \rangle = 1$.
  3. Resolution of Unity: $\int d\mu(z) |z\rangle\langle z| = 1$, with an explicitly derived weight function involving the H-function.
• Continuous Spectrum: Successfully derived coherent states for the continuous spectrum, introducing the FW-generalized multi-parameter $\nu$-function as the normalization factor.

B. Bicomplex Fox-Wright Function

• Existence and Convergence: Defined the bicomplex Fox-Wright function and established its convergence conditions.
• Theorem 3.1: Provided a comprehensive classification of convergence based on the parameter $\Upsilon = \sum N_j - \sum M_i$. The series converges absolutely in various regions (hyperbolic balls, disks, or planes) depending on the real and hyperbolic components of $\Upsilon$.
  • Specifically, if $\Upsilon >_h -1$, the function is an entire function in the bicomplex space.
  • If $\Upsilon = -1$, convergence is uniform and absolute on the boundary of the hyperbolic ball under specific real-part conditions.

C. Bicomplex Coherent States

• Construction: Developed bicomplex coherent states $|m; n; Z\rangle$ using the idempotent components of the bicomplex Fox-Wright function.
• Normalization: Defined the Bicomplex FW-generalized multi-parameter $\nu$-function ($V_{b}^{(m,n)}$) to normalize states in the continuous spectrum.
• Resolution of Unity: Derived the bicomplex integration measure $d\tilde{\mu}_b(Z)$ ensuring the resolution of unity holds in the bicomplex Hilbert space.

4. Key Results

• Generalization: The proposed Fox-Wright coherent states generalize existing models. By setting specific parameters (e.g., $A_l = B_r = 1$), the results reduce to generalized hypergeometric coherent states. By choosing $p=q=1$, they reduce to Mittag-Leffler coherent states.
• Convergence Criteria: The paper provides precise conditions for the convergence of the bicomplex Fox-Wright series, categorizing the domain of convergence into nine distinct cases based on the hyperbolic parameters.
• Weight Functions: Explicit formulas for the weight functions required for the resolution of unity are derived for both discrete and continuous spectra in both complex and bicomplex settings. These involve products of Fox-Wright functions and H-functions (or G-functions in specific cases).
• Operator Algebra: The annihilation operators for these states act as eigenvalue operators ($A_-|z\rangle = z|z\rangle$), confirming their status as coherent states.

5. Significance

• Theoretical Advancement: This work significantly broadens the scope of coherent state theory by incorporating the highly versatile Fox-Wright function, unifying various previously distinct classes of coherent states under a single framework.
• Bicomplex Quantum Mechanics: It represents a substantial step forward in bicomplex quantum mechanics, a field exploring extensions of standard quantum theory. By successfully defining coherent states in this non-trivial algebraic structure (which includes zero divisors), the authors demonstrate the viability of bicomplex formalism for physical applications.
• Mathematical Physics Applications: The derived weight functions and normalization conditions are crucial for applications in quantum optics, signal processing, and fractional calculus, where Fox-Wright functions naturally arise.
• Foundation for Future Work: The introduction of the bicomplex FW-generalized $\nu$-function opens new avenues for studying continuous spectra in bicomplex systems and exploring physical models where bicomplex variables are essential (e.g., in certain relativistic or multi-component quantum systems).

In summary, the paper successfully constructs a robust mathematical framework for a new class of coherent states, validates their fundamental quantum mechanical properties, and extends these concepts into the complex and bicomplex domains, offering a unified generalization of existing special-function-based coherent states.