Advances in the study of relativistic quantum phase space have established the set of Linear Canonical Transformations (LCTs) as a candidate for the fundamental symmetry group associated with relativistic quantum physics. In this framework, for a spacetime of signature $(N_+,N_-)$ , the symmetry of the relativistic quantum phase space is described by the LCT group, isomorphic to the symplectic Lie group $Sp(2N_+,2N_-)$ , which preserves the canonical commutation relations (CCRs) and treats spacetime coordinates and momenta operators on an equal footing. In this work, we investigate the contraction structure of the Lie algebra associated with the LCT group for signature $(1,4)$ , clarifying how familiar spacetime symmetry groups emerge from this more fundamental quantum phase space symmetry. Using the Inönü-Wigner group contraction formalism, we examine each limit case corresponding to the possible combinations of asymptotic values of two fundamental length scale parameters associated with the theory, namely a minimum length $\ell$ and a maximum length $L$ , which may be identified respectively with the Planck length and the de Sitter radius. We explicitly analyze how contractions of the LCT Lie algebra lead to the physically relevant de Sitter algebra $\mathfrak{so}(1,4)$ and, in the flat-curvature limit, to the Poincaré algebra $\mathfrak{iso}(1,3)$ of four-dimensional spacetime. This provides an explicit mechanism through which relativistic spacetime symmetry can emerge from a deeper symplectic structure of quantum phase space.Advances in the study of relativistic quantum phase space have established the set of Linear Canonical Transformations (LCTs) as a candidate for the fundamental symmetry group associated with relativistic quantum physics. In this framework, for a spacetime of signature $(N_+,N_-)$, the symmetry of the relativistic quantum phase space is described by the LCT group, isomorphic to the symplectic Lie group $Sp(2N_+,2N_-)$, which preserves the canonical commutation relations (CCRs) and treats spacetime coordinates and momenta operators on an equal footing. In this work, we investigate the contraction structure of the Lie algebra associated with the LCT group for signature $(1,4)$, clarifying how familiar spacetime symmetry groups emerge from this more fundamental quantum phase space symmetry. Using the Inönü-Wigner group contraction formalism, we examine each limit case corresponding to the possible combinations of asymptotic values of two fundamental length scale parameters associated with the theory, namely a minimum length $\ell$ and a maximum length $L$, which may be identified respectively with the Planck length and the de Sitter radius. We explicitly analyze how contractions of the LCT Lie algebra lead to the physically relevant de Sitter algebra $\mathfrak{so}(1,4)$ and, in the flat-curvature limit, to the Poincaré algebra $\mathfrak{iso}(1,3)$ of four-dimensional spacetime. This provides an explicit mechanism through which relativistic spacetime symmetry can emerge from a deeper symplectic structure of quantum phase space.

quant-pharXiv:2603.21333Mar 24, 2026

From Standing Waves in a Cavity to Spacetime Emergence: A Unified Perspective on Quantum Wave-Particle Duality, Complex Phase, and Spacetime Symmetry

Ying Cao

This paper explores the physical essence of quantum wave nature and its fundamental connection to spacetime symmetries. Beginning with a comparative analysis of standing waves in a ring cavity versus classical thermodynamic diffusion, we systematically decon- struct mechanical intuitions regarding ”waviness.” Through rigorous symmetry analysis and logical deduction, we demonstrate that given only energy/momentum conservation (space- time translation symmetry) and propagation at the speed of light c, classical real-valued probability is insufficient to yield wave behavior. Quantum wave-particle duality is fun- damentally the geometric manifestation of a complex probability amplitude (unitary probability conservation) evolving under spacetime translation symmetry. Finally, we extend this binary interaction to modern theoretical physics, articulating a unified pic- ture where spacetime itself emerges from quantum entanglement, serving as the physical embodiment of energy and momentum. This work provides a self-consistent pedagogical and exploratory framework for understanding foundational quantum concepts. Keywords: Quantum Wave Nature; Spacetime Translation Symmetry; Complex Probabil- ity Amplitude; Standing Waves; Spacetime Emergence; Quantum GravityThis paper explores the physical essence of quantum wave nature and its fundamental connection to spacetime symmetries. Beginning with a comparative analysis of standing waves in a ring cavity versus classical thermodynamic diffusion, we systematically decon- struct mechanical intuitions regarding ”waviness.” Through rigorous symmetry analysis and logical deduction, we demonstrate that given only energy/momentum conservation (space- time translation symmetry) and propagation at the speed of light c, classical real-valued probability is insufficient to yield wave behavior. Quantum wave-particle duality is fun- damentally the geometric manifestation of a complex probability amplitude (unitary probability conservation) evolving under spacetime translation symmetry. Finally, we extend this binary interaction to modern theoretical physics, articulating a unified pic- ture where spacetime itself emerges from quantum entanglement, serving as the physical embodiment of energy and momentum. This work provides a self-consistent pedagogical and exploratory framework for understanding foundational quantum concepts. Keywords: Quantum Wave Nature; Spacetime Translation Symmetry; Complex Probabil- ity Amplitude; Standing Waves; Spacetime Emergence; Quantum Gravity

Aug 19, 20260 citations

Symmetries in Relativistic Quantum Mechanics

Andreas Wipf

Springer eBooksJan 1, 20260 citations

The Spacetime Approach to Quantum Mechanics

James B. Hartle

WORLD SCIENTIFIC eBooksDec 9, 20201 citation

Projective spacetime symmetry of spacetime crystals

Zheng Zhang, Z. Y. Chen, Y. X. Zhao

Abstract Wigner’s seminal work on the Poincaré group revealed one of the fundamental principles of quantum theory: symmetry groups are projectively represented. The condensed-matter counterparts of the Poincaré group could be the spacetime groups of periodically driven crystals or spacetime crystals featuring spacetime periodicity. In this study, we establish the general theory of projective spacetime symmetry algebras of spacetime crystals and reveal their intrinsic connections to gauge structures. As important applications, we exhaustively classify (1,1)D projective symmetry algebras and systematically construct spacetime lattice models for them all. Additionally, we present three consequences of projective spacetime symmetry that surpass ordinary theory: the electric Floquet-Bloch theorem, Kramers-like degeneracy of spinless Floquet crystals, and symmetry-enforced crossings in the Hamiltonian spectral flows. Our work provides both theoretical and experimental foundations to explore novel physics protected by projective spacetime symmetry of spacetime crystals.Abstract Wigner’s seminal work on the Poincaré group revealed one of the fundamental principles of quantum theory: symmetry groups are projectively represented. The condensed-matter counterparts of the Poincaré group could be the spacetime groups of periodically driven crystals or spacetime crystals featuring spacetime periodicity. In this study, we establish the general theory of projective spacetime symmetry algebras of spacetime crystals and reveal their intrinsic connections to gauge structures. As important applications, we exhaustively classify (1,1)D projective symmetry algebras and systematically construct spacetime lattice models for them all. Additionally, we present three consequences of projective spacetime symmetry that surpass ordinary theory: the electric Floquet-Bloch theorem, Kramers-like degeneracy of spinless Floquet crystals, and symmetry-enforced crossings in the Hamiltonian spectral flows. Our work provides both theoretical and experimental foundations to explore novel physics protected by projective spacetime symmetry of spacetime crystals.

arXiv:2310.09577Communications PhysicsNov 14, 20233 citations

(N_+,N_-)

, the symmetry of the relativistic quantum phase space is described by the LCT group, isomorphic to the symplectic Lie group

Sp(2N_+,2N_-)

, which preserves the canonical commutation relations (CCRs) and treats spacetime coordinates and momenta operators on an equal footing. In this work, we investigate the contraction structure of the Lie algebra associated with the LCT group for signature

(1,4)

, clarifying how familiar spacetime symmetry groups emerge from this more fundamental quantum phase space symmetry. Using the Inönü-Wigner group contraction formalism, we examine each limit case corresponding to the possible combinations of asymptotic values of two fundamental length scale parameters associated with the theory, namely a minimum length

\ell

and a maximum length

L

, which may be identified respectively with the Planck length and the de Sitter radius. We explicitly analyze how contractions of the LCT Lie algebra lead to the physically relevant de Sitter algebra

\mathfrak{so}(1,4)

and, in the flat-curvature limit, to the Poincaré algebra

\mathfrak{iso}(1,3)

of four-dimensional spacetime. This provides an explicit mechanism through which relativistic spacetime symmetry can emerge from a deeper symplectic structure of quantum phase space.