Joint Momenta-Coordinates Quantum States

• Joint momenta-coordinates states are quantum states that treat position and momentum on equal footing, achieving optimal phase-space localization and robust decoherence analysis.
• Their mathematical construction employs displacement operators acting on vacuum states to rigorously satisfy canonical commutation and generalized uncertainty relations.
• They facilitate symmetry classification, precise quantum measurement, and efficient digital simulation by mapping continuous variables onto discrete frameworks.

Joint momenta-coordinates states are quantum states that treat position and momentum on equal footing, embodying correlations and optimal localization in phase space. Typically constructed to saturate generalized uncertainty relations, these states form the basis for robust physical descriptions ranging from coherent state quantization on compact manifolds to the structural analysis of decoherence and quantum measurement. Below, the topic is resolved into logical sections to cover mathematical construction, physical properties, representation theory, computational frameworks, measurement, and implications.

1. Mathematical Constructions and Canonical Duality

Joint momenta-coordinates states formalize the simultaneous specification of position-like and momentum-like observables by exploiting the canonical commutation relationship $[Q, P] = i\hbar$ or its generalizations. On the unit circle, for instance, phase space becomes $\mathbb{Z} \times S^1$, combining integer-labeled momenta from Fourier expansion with continuous angular coordinates (Chadzitaskos et al., 2011). The construction involves displacement operators of the form

$$W(m, \alpha) = e^{imQ} V(\alpha),\quad V(\alpha) = e^{-i\alpha P}$$

acting on a "vacuum" (Gaussian-like) state. This produces a family of states

$$|m, \alpha\rangle = \mathcal{A} e^{im\varphi} e^{-\frac{1}{2}(\varphi-\alpha)^2}$$

with sharp momentum expectation $\langle m, \alpha|P|m, \alpha\rangle = m$ and localization around $\alpha$.

For systems with discrete spectra or collective coordinates, similar methodologies apply: e.g., exact canonically conjugate momenta to nuclear quadrupole coordinates are defined via a discrete integral equation to rigorously enforce $$[\Pi_{2\mu}, \phi_{2\mu'}] = -i\hbar\delta_{\mu\mu'}$$ and allow expansion of the kinetic part of the Hamiltonian in powers of $\Pi_{2\mu}$ (Nishiyama et al., 2013).

In composite quantum systems or constrained phase spaces, joint states are promoted using mapping-Hamiltonian frameworks that lift discrete degrees of freedom into a continuous, weighted constraint coordinate-momentum phase space (He et al., 2022).

2. Statistical Properties and Universality

Joint momenta-coordinates states often underpin universal fluctuation phenomena. In trapped Fermi systems, for example, positions and momenta have symmetric joint statistics in harmonic traps, both described by determinantal processes with Airy kernels at the edge. The distribution of maximum momentum or position follows the Tracy-Widom law,

$$\text{Prob}(p_{\max}<p)\simeq \mathcal{F}_2\left(\frac{p-p_e}{p_N}\right)$$

where $\mathcal{F}_2(s)$ is a Fredholm determinant built from the Airy kernel (Doussal et al., 2018). In higher-order ("flat") traps, momentum densities at the edge behave anomalously, and the corresponding kernels generalize to

$$K_{2n+1}(s, s') = \int_0^{+\infty} du\, \text{Ai}_{2n+1}(s+u)\, \text{Ai}_{2n+1}(s'+u)$$

where $\text{Ai}_{2n+1}$ is a generalized Airy function. The resulting edge distributions are expressed in terms of nonlinear Painlevé equations, forming a link to multicritical random matrix models.

3. Representation Theory and Symmetry Classification

Joint momenta-coordinates states are central to symmetry-based frameworks for analyzing composite systems, e.g., excitons in crystals. For excitonic wavefunctions, states are labeled by center-of-mass momentum $\mathbf{Q}$ and classified using irreducible representations (irreps) of little groups at each $\mathbf{Q}$ (Bajaj et al., 22 Aug 2025). Projection operators are constructed to block-diagonalize the Bethe-Salpeter Hamiltonian in symmetry-adapted bases:

$$\mathcal{V}^{(\xi)}_{ij;\mathbf{Q}} = \frac{d_\xi}{|\mathcal{G}_{\mathbf{Q}}|} \sum_{\{R|t\}\in \mathcal{G}_{\mathbf{Q}}} [\Delta^{(\xi)}_{ij}(\{R|t\})]^*\,P^{\mathrm{ex}}_{\{R|t\}}$$

States are mapped throughout the Brillouin zone using space-group and time-reversal symmetry, giving direct access to degeneracies, selection rules, and block structure in composite Hilbert spaces. This physically organizes joint momentum-coordinate degrees of freedom and dramatically reduces computational cost.

4. Quantum Measurement and Generalized Observables

Formally, there are six basic choices for coordinate operators, distinguished by whether the variable is continuous/discrete and the domain (full real line, half-line, interval) (Enk et al., 12 Feb 2025). For each, a conjugate momentum operator is derived via Fourier duality; self-adjointness may not be guaranteed, especially on domains with boundaries, and may require Neumark extensions or additional boundary conditions. For example, in bounded domains, momentum spectra become discretized:

$$p_{m} = \frac{2\pi m}{L} + \frac{\theta_0}{L},\quad m\in\mathbb{Z}$$

Measurements involving both coordinate and momentum naturally lead to positive operator-valued measures (POVMs) and often require coarse-graining or weak measurements (e.g., coherent-state measurement using projectors $|\alpha\rangle\langle\alpha|$ that resolve the identity).

5. Decoherence, Pointer States, and Generalized Uncertainty

Joint momenta-coordinates states are intimately linked to decoherence and the emergence of classicality. Recent work shows that these states—which saturate generalized uncertainty relations:

$$\sigma_{xx}(t)\sigma_{pp}(t) - \sigma_{px}^2(t) = \frac{\hbar^2}{4}$$

—are robust against environmental interaction and serve as optimal pointer states in the Lindblad decoherence framework (Manampisoa et al., 20 Sep 2025). Under Zurek's predictability-sieve criterion, only such states maintain purity and minimal entropy production in the underdamped regime. This generalizes the pointer state concept beyond Gaussian coherent states, embedding classical robustness within the phase space formalism.

6. Simulation, Digitization, and Numerical Frameworks

Digital representations of joint momenta-coordinates states are fundamental for quantum simulations. By expanding observables in binary or ternary numeral systems,

$$x = \sum_{s} q^s t(s, x),\quad p = \sum_{r} q^r t(r, p)$$

with $t(s,x)$ as Hermitian operators, the continuous variables $x$ and $p$ map naturally onto quantum digits (qubits or qutrits), making the system amenable to quantum computation (Ivanov et al., 2020). Renormalization techniques give finite meaning to divergent series arising in the construction, and explicit digit-based operator relations replicate canonical commutation, facilitating error-resilient simulations of phase-space dynamics.

7. Generalizations: Non-Hermitian and Biorthogonal Settings

Extensions include non-Hermitian and biorthogonal frameworks, where similarity transformations deform standard position/momentum operators but preserve their algebra up to biorthogonality of eigenstates. For example,

$q = Tq_0T^{-1},\quad p = Tp_0 T^{-1}$

The eigenstates become tempered distributions, and biorthogonal families satisfy generalized resolutions of the identity, crucial for joint state construction in pseudo-Hermitian and PT-symmetric models (Bagarello et al., 2017, Bagarello et al., 2022).

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Joint momenta-coordinates states thus provide the foundational structure for understanding phase-space localization, symmetry classification, optimal robustness against decoherence, efficient quantum simulation, and the measurement theory underpinning quantum mechanics and quantum information. These states generalize coherent state constructions, admit significant universality, and serve as both computational and conceptual nuclei in the paper of quantum systems.