Nematic symmetry entails conserved quantized quantities such as number of topological defects and vorticity cells. Correspondingly, countless quantum analogies have been found in Active Nematics. We formalize Active Nematics and Liquid Crystal theory into the framework of Quantum Mechanics by introducing a complex valued Nematic Wavefunction to the Beris Edward equations, thus splitting spatiotemporally varying nematic systems into quantized states. We obtain the Planck's energy-frequency relationship for active micro-swimmers such as peristaltic worms and bacterium as a consequence of local complex phase-symmetry of the governing equations, similar to the gauge formulation of Electromagnetism. For organisms operating on diffusive chemotaxis, we obtain predator-prey dynamics that evolve to maximize/minimize pheromones field gradient overlap. Furthermore, when quantizing beating hearts, similar to the orbitals of hydrogen atoms, the state-function allows us to characterize hearts not only through the rhythm, but also the spaciotemporal distribution of contractile activity of various harmonics among healthy and unhealthy hearts.
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21 Pages, Double Spaced, 4 figures. Towards PhD thesis work of Ghansham Chandel, under the guidance of Dr Siddhartha Das
Nematic symmetry entails conserved quantized quantities such as number of topological defects and vorticity cells. Correspondingly, countless quantum analogies have been found in Active Nematics. We formalize Active Nematics and Liquid Crystal theory into the framework of Quantum Mechanics by introducing a complex valued Nematic Wavefunction to the Beris Edward equations, thus splitting spatiotemporally varying nematic systems into quantized states. We obtain the Planck's energy-frequency relationship for active micro-swimmers such as peristaltic worms and bacterium as a consequence of local complex phase-symmetry of the governing equations, similar to the gauge formulation of Electromagnetism. For organisms operating on diffusive chemotaxis, we obtain predator-prey dynamics that evolve to maximize/minimize pheromones field gradient overlap. Furthermore, when quantizing beating hearts, similar to the orbitals of hydrogen atoms, the state-function allows us to characterize hearts not only through the rhythm, but also the spaciotemporal distribution of contractile activity of various harmonics among healthy and unhealthy hearts.
The hydrogen atom has historically played a foundational role in the development of quantum mechanics, where its discrete energy spectrum is conventionally derived from solutions of the Schrödinger wave equation. In this work we present an alternative formulation in which the hydrogen spectrum emerges without invoking the Schrödinger equation. We consider a semi-quantum framework in which the electron is treated as a classical particle governed by Poisson-bracket dynamics while interacting with a quantized electromagnetic field described using second quantization. The electron moves in the Coulomb potential generated by the proton and simultaneously couples to quantized electromagnetic modes through minimal coupling. The Coulomb system possesses a hidden dynamical symmetry characterized by the Runge–Lenz vector, which enlarges the rotational symmetry to an group for bound states. Within this framework, interactions between the classical particle and the quantized field induce an effective commutation structure in the particle’s phase space. Once this structure emerges, the algebra of the conserved quantities associated with the Runge–Lenz vector becomes identical to the operator algebra used in Pauli’s symmetry-based derivation of the hydrogen spectrum. Consequently, the discrete hydrogen energy levels arise naturally from the combined effects of Coulomb symmetry and particle–field interaction, offering a physically transparent interpretation of atomic quantization and suggesting that wave–particle duality may arise dynamically from interactions with quantized electromagnetic fields.
This paper presents a rigorous mathematical derivation of the continuity equation within a reformulated quantum mechanical framework-Unified Quantum Mechanics (UQM)-where the fundamental imaginary unit i is systematically replaced by the temporal Hilbert transform operator Ht. This substitution enables a complete description of quantum dynamics using exclusively real-valued wavefunctions while preserving the essential conservation laws of quantum theory. We demonstrate that this real-valued formalism naturally gives rise to a conserved quantity analogous to the probability density in conventional quantum mechanics. By constructing a coupled system of evolution equations for the primary wavefunction ψ and its Hilbert transform ϕ=Ht[ψ], we identify an underlying global SO(2) symmetry that is mathematically isomorphic to the U(1) gauge symmetry responsible for charge conservation in standard complex-valued quantum mechanics. Through direct computation, we derive the continuity equation ∂t∂ρ+∇⋅j=0 and obtain explicit expressions for the conserved density ρ and current density j. This result establishes that the fundamental principle of local charge conservation emerges naturally within the UQM dynamics, thereby validating its mathematical consistency and physical completeness. Moreover, we extend the result to demonstrate the local conservation of mass, deriving the corresponding continuity equation for mass density.
The basic equation of quantum mechanics is the Schrödinger equation which expresses the wave function Ψ of a quantum system as an eigenfunction of a quantized Hamiltonian operator H : HΨ = λΨ where the (real) eigenvalue λ is the quantum energy of the system in the state Ψ see equations (1.3.2), (1.3.4). Embodied already in this equation is the basic quantum mechanical principle that quantum energies cannot take on arbitrary values but are quantized : they are given by a discrete set of eigenvalues of a suitable second-order differential operator. This mathematical phenomenon of the discreteness of eigenvalues explains, for example, the observed discreteness of absorption and emission atomic spectral lines; compare remarks in Sections 1.2 and 1.3 of Chapter 1. The Schrödinger theory, and the equivalent theory of Heisenberg, Born and Jordan, represents a distinct advancement of the Bohr theory. Some early basic papers on quantum mechanics are compiled in the book [82], which includes a historic introduction by B. van der Waerden. Also see [6, 8, 9, 24, 75, 76]. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This paper is concerned with the relationship between the group G of all symmetries of an unconstrained dynamical system and the space of its quantum states. The chief aim is to establish the validity of the claim that "quantizing a system" means deciding what G is and determining all projective unitary representations of G. The mathematical tool suited to this purpose is the theory of central extensions of an arbitrary group G by the circle group T (and the closely related cohomology group H2 (G, T)). The fundamental structural property is that there is a universal state space [Formula: see text] embedding all states, a unitary representation space of the universal central extension [Formula: see text] of G, on which the Pontryagin dual H2 (G, T)⋆ of H2 (G, T) operates as a group of su-perselection rules, so that the superselection sectors are [Formula: see text] and observables are self-adjoint operators Vh → Vh. This leads to a classification of time-evolution operators respecting the symmetries by H2 (G, T) and a general understanding of "anomalies" as arising from a mismatch of sectors to which symmetry transformations and time-evolution refer. For the special case of a Lie group G of symmetries, [Formula: see text] of G coincides with [Formula: see text] the universal covering group, if G is semisimple. More generally, depending on whether or not G is semisimple and/or simply connected, inequivalent quantizations are of topological, algebraic or "mixed" origin; in the algebraic case, the relationship between anomalous conservation laws and invariant dynamics is explicitly given. Illustrative examples worked out include the Euclidean group E (d) and the (connected) Poincaré group P (d, 1) for arbitrary spatial dimension d and the discrete groups (Z/N)2 and Z2 appropriate for planar square crystals — in particular, the absence of "fractional" angular momentum in the plane is established and the difficulties specific to (2,1)-dimensional quantum field theories precisely formulated. The paper concludes by noting that a number of ambiguities generally supposed to affect traditional procedures of quantization resolve themselves satisfactorily in this approach. Mathematically, the paper is self-contained as far as the concepts and results are concerned; occasionally, proofs of well-known facts are reworked when they serve to illustrate the value of the symmetry point of view. The theory of universal central extensions of groups, including important special cases, is treated in detail in an appendix by M. S. Raghunathan, in view of its significance for quantization and the absence of a similar account in the literature.
In this thesis we study the nonlinear dynamics of spin-1 and spin-2 Bose-Einstein condensates, with particular application to antiferromagnetic systems exhibiting nematic (beyond magnetic) order. Firstly, we give a derivation of the spinor energy functionals with a focus on the connections between the nonlinear terms. We derive a hierarchy of nonlinear irreducible multipole observables sensitive to different levels of nematic order, and explore the various nematic states in terms of their multipolar order, representations of their symmetries, and topological defects.\n\nWe then develop an exact solution to the nonlinear dynamics of spinor Bose-Einstein condensates. We use this solution to construct efficient and accurate numerical algorithms to evolve the spinor Gross-Pitaevskii equation in time. We demonstrate the advantages of our algorithms with several 1D numerical test problems, comparing with existing methods in the literature.\n\nWe apply our numerical methods to simulating quenches of the condensate between various antiferromagnetic phases for spin-1 and spin-2. For spin-1, we carry out quenches for a theoretical uniform system in 2D, and then specialize to the parameters used in a recent harmonically trapped experiment in 3D. We connect the long-time coarsening growth law of the relevant order parameter to the decay of half-quantum vortices, which are the relevant topological defects of the ground state.\n\nFor the spin-2 system, we investigate a novel quench from two different quadrupolar-nematic phases to an octupolar-nematic “cyclic” phase which supports 1/3 fractional vortices. We develop appropriate order parameter observables which couple to the spin and superfluid currents generated by these defects, and show that a new growth law appears with exponent 1/3.