Diagonal Operator Ordering Technique (DOOT)

• DOOT is a formalism that orders deformed bosonic operators into a diagonal and commutative structure, facilitating accurate evaluation of thermal observables.
• It converts operator expressions into c-number series, crucial for generating functions, partition sums, and coherent-state resolutions in deformed settings.
• DOOT underpins the construction of temperature-dependent coherent states in Thermo Field Dynamics, bridging operator ordering with thermal quantum statistics.

The Diagonal Operator Ordering Technique (DOOT) is a generalized formalism for operator ordering in the context of quantum field theory and quantum statistical mechanics, specifically devised for systems characterized by deformed bosonic operators and their temperature-dependent coherent states. Originating as an extension of the Integration Within an Ordered Product (IWOP) procedure commonly used for canonical bosons, DOOT enables the manipulation and calculation of operator expressions—such as expectation values, resolutions of identity, and generating functions—in an ordered, diagonally dominant, and commutative framework even for algebraically deformed ladder structures. DOOT is especially relevant within the Thermo Field Dynamics (TFD) framework, providing the mathematical infrastructure needed to define and analyze thermal coherent states of both Barut-Girardello and Klauder-Perelomov variety for deformed quantum oscillators (Popov, 22 Dec 2025).

1. Background: Operator Ordering and Thermo Field Dynamics

Quantum field-theoretic computations often require explicit operator ordering (normal, antinormal, etc.), crucial for evaluating trace identities, partition functions, and coherent-state path integrals. For canonical bosons, the IWOP method allows normal ordering and functional integration. However, many physically relevant systems—such as quantum groups, nonlinear oscillators, and deformed oscillator algebras—feature ladder operators $A=f(N)a$, $B=a^\dagger f(N)$ with nontrivial commutation rules due to a deformation function $f(n)$. In such cases, canonical ordering techniques break down.

Thermo Field Dynamics (TFD) is a formalism that realizes equilibrium (finite-temperature) quantum statistical averages as vacuum expectation values in an extended, doubled Hilbert space. It employs a Bogoliubov transformation using thermalizing operators and introduces a thermal vacuum $\lvert 0(\beta)\rangle$, which encodes thermal (Gibbs) statistics as pure-state averages (Popov, 22 Dec 2025).

2. Formal Definition of DOOT

DOOT is defined via a symbolic notation $\#\,\cdots\,\#$, which denotes diagonal (simultaneously normal-ordered and fully commutative) operator products over all unpaired (physical and ancillary) ladder operators within a given expression. For general analytic functions $G(BA)$ of the product $BA$ (where $A,B$ may be canonical or deformed), DOOT gives:

$$\# G(BA)\# = \sum_{n=0}^\infty \frac{G^{(n)}(0)}{n!} \#(BA)^n\#$$

Inside $\#\cdots\#$, all operators $A,B,\tilde{A},\tilde{B}$ (tilde denoting the TFD fictitious sector) are treated as if they commute. This diagonalization greatly simplifies both the evaluation of mixed moments and the construction of thermal states for systems lacking canonical structure (Popov, 22 Dec 2025).

3. Application to Deformed Boson Algebras

For a deformed oscillator, ladder operators $A=f(N)a$, $B=a^\dagger f(N)$ with $N=a^\dagger a$ satisfy:

$$BA\,|n\rangle = e(n)\,|n\rangle, \quad AB\,|n\rangle = e(n+1)\,|n\rangle$$

where $e(n)=n f(n)^2$. DOOT enables one to express operator exponentials and their ordered integrals as c-number series, allowing formal manipulations analogous to those for canonical bosons. Specifically:

$$\#\exp(zB)\exp(wA)\# = \sum_{n,m\ge 0} \frac{z^n w^m}{n! m!} \# B^n\,A^m\#$$

This c-number representation is crucial for constructing generating functions and partition sums in deformed settings (Popov, 22 Dec 2025).

4. Construction of Thermal Coherent States Using DOOT

Within TFD and DOOT, two principal classes of temperature-dependent coherent states arise:

• Barut-Girardello–type states: Eigenstates of the (thermally rotated) annihilation operator $A(\beta)$, satisfying $$A(\beta) |z,\beta\rangle_{BG} = z\cosh\theta\,|z,\beta\rangle_{BG}$$, where $\theta$ is set by $\tanh\theta = e^{-\beta\omega/2}$.
• Klauder-Perelomov–type states: Generated by thermalized displacement operators: $$|z,\beta\rangle_{KP} = \exp(zB(\beta))\exp(-z^*A(\beta)) |0(\beta)\rangle$$.

DOOT underpins the normal-ordering and c-number expansion of such thermal states, enabling explicit summation of their Fock-space expansions, overlap integrals, and resolutions of the identity:

$$|z,\beta\rangle_{BG} = N_{BG}(\beta,|z|^2) \sum_{n=0}^\infty \frac{(z\cosh\theta)^n}{\sqrt{[n]!}}\,|n,\tilde n;\beta\rangle$$

with normalization and measure determined via moment problems solvable within DOOT (Popov, 22 Dec 2025).

5. Calculation of Physical Observables and Measures

The expectation value of observables (e.g., the number operator $N=BA$) in thermal coherent states is obtained directly with DOOT substitutions, producing:

$$_{BG}\langle z,\beta |A^\dagger A |z,\beta\rangle_{BG} = |z|^2\,\cosh^2\theta$$

in the Barut-Girardello case, with analogous results for the Klauder-Perelomov states plus an explicit thermal population offset. Integration measures and the resolution of the identity for the overcomplete coherent-state system are also handled via DOOT, enabling construction of generalized partition functions and Husimi Q-representations for deformed oscillators at finite temperature (Popov, 22 Dec 2025).

6. Duality and Extensions

DOOT manifests duality relations between Barut-Girardello and Klauder-Perelomov thermal coherent states; for instance, via Laplace-type integral transforms connecting their respective families. All temperature dependence resides in the “squeezing” parameter $\theta(\beta)$, which encodes Bose-Einstein statistics through its appearance in Bogoliubov coefficients and c-number expansions (Popov, 22 Dec 2025).

7. Mathematical Significance and Research Impact

DOOT generalizes prior canonical methods, providing a universal algebraic structure for operator ordering in both canonical and non-canonical algebras, including those with nonlinear structure constants or underlying Lie group symmetries. The technique underpins the explicit construction of thermal coherent states for a wide range of deformed systems, including those relevant to quantum optics, statistical mechanics, and quantum information science where temperature-dependent symmetry and nonclassical states are of interest. It establishes the mathematical infrastructure for rigorous treatment of quantum thermodynamic effects in extended operator algebras, connecting operator-theoretic methods with thermodynamic observables (Popov, 22 Dec 2025).