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arXiv:2512.17549 · Annals of Physics

Bloch Motions and Spinning Tops

Albert Huber, Paul Schreivogl

quant-phSubmitted Dec 19, 2025• Announced Dec 22, 2025• 1 citation• Revised Aug 21, 2026 (v2)
v1 · Dec 19, 2025v2 · Aug 21, 2026
arXivHTMLPDF

Abstract

This work investigates the dynamics of closed quantum systems in the Bloch vector representation using methods from rigid body dynamics and the theory of integrable systems. To this end, equations of motion for Bloch components are derived from the von Neumann equation that are mathematically equivalent to equations of motion for a distribution of point masses from classical mechanics. Furthermore, using the Heisenberg equation, another system of Bloch vector equations is derived, which constitutes an Euler-Poinsot system of the type commonly encountered in the theory of torque-free spinning tops. This insight is used to prove the Liouville integrability of the corresponding Hamilton equations of motion and to construct an explicit closed-form solution using the Arnold-Liouville theorem. Within the same framework, stability criteria for quantum dynamics are then derived which correspond to the Energy-Casimir method of classical Newtonian mechanics. Following that, specific solutions to the equations of motion are constructed that encode the complex dynamics of composite quantum systems. Eventually, to show that this formalism provides concrete physical predictions, an analogue of the intermediate axis theorem is derived and the effect of oscillating entanglement is discussed. As a basis for this, special types of solutions to the equations of motion are derived that constitute oscillating entangled states, i.e., dynamical quantum states that change their entanglement structure from maximally entangled to separable and vice versa.

Journal reference
Annals of Physics, Volume 492, September 2026, 170558
DOI
10.1016/j.aop.2026.170558
Comments
41 Pages, no figures, matches published version

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This work investigates the dynamics of closed quantum systems in the Bloch vector representation using methods from rigid body dynamics and the theory of integrable systems. To this end, equations of motion for Bloch components are derived from the von Neumann equation that are mathematically equivalent to equations of motion for a distribution of point masses from classical mechanics. Furthermore, using the Heisenberg equation, another system of Bloch vector equations is derived, which constitutes an Euler-Poinsot system of the type commonly encountered in the theory of torque-free spinning tops. This insight is used to prove the Liouville integrability of the corresponding Hamilton equations of motion and to construct an explicit closed-form solution using the Arnold-Liouville theorem. Within the same framework, stability criteria for quantum dynamics are then derived which correspond to the Energy-Casimir method of classical Newtonian mechanics. Following that, specific solutions to the equations of motion are constructed that encode the complex dynamics of composite quantum systems. Eventually, to show that this formalism provides concrete physical predictions, an analogue of the intermediate axis theorem is derived and the effect of oscillating entanglement is discussed. As a basis for this, special types of solutions to the equations of motion are derived that constitute oscillating entangled states, i.e., dynamical quantum states that change their entanglement structure from maximally entangled to separable and vice versa.
Classical dynamics of rigid bodies The difficulties in formulating the theory of spin ½ systems outlined at the end of the last chapter may be avoided if we start from a classical theory of rotation in which the translational and orientational coordinates enter on a symmetrical footing, and pass to the quantum theory via the standard canonical quantization procedure. The result is not simply the Pauli theory written in a different matrix representation but a more general physical theory which explicitly incorporates in the wave equation the extended character of the bodies considered, through moment-of-inertia-dependent terms. As expected, the spin ½ character of the wave is manifested through the dependence of the body motion on the total quantum state via quantum potential and torque terms. Spin is not usually approached from this angle in basic quantum mechanics texts so it is worthwhile to build up the theory from scratch (for a more rigorous treatment see Normand (1980) and for further details: Bopp & Haag (1950), Rosen (1951, 1984), van Winter (1954), Dankel (1970), Holland (1988a, b)). We begin by reviewing some basic results in the classical canonical theory of rigid body motion. Our treatment is confined to a rigid homogeneous sphere with moment of inertia I about any axis passing through its centre, where the centre of mass is located. The sphere carries a rigid charge and a magnetic moment. By Chasle's theorem an arbitrary displacement of a rigid body is composed of a translation together with a rotation about a line fixed in the body.
The perennial aspect of the Newtonian foundation of mathematical physics is that the concept of ``motion,'' that is, ``kinematics,'' is to serve as the interface between mathematics and physics. Kinematics subdivides into the theory of orbital translation and that of undulation and spinning. Newtonian mechanics is based on giving to translational kinematics a priority over the other modes, since planetary revolution can be represented as translation modified by gravitation. The so-called breakdown of classical physics stems from giving the translational priority a canonical status and extending it to the constituents of matter. We claim that in this case the priority is to be reversed. The main content of this paper is to establish the algebraic model for an indivisible, undulating entity that we call a ``wave simplex.'' It is used as the point of departure for a non-Newtonian quantum dynamics in which physical and algebraic concepts are in close correspondence. The postulates of the classical phenomenological theories and those of the canonical theories based on translational priority are established as theorems under the proper limiting conditions, and forces are constructed rather than postulated. While the formal structure of two-level quantum mechanics is established as well, exception is taken to treating spin as a property of a point particle. It is considered self-evident that a spinning object is orientable, a property accounted for in terms of a unitary triplet. This is the point of departure for an intrinsic particle dynamics. A main result is the integration of classical and quantum physics, thus closing the gap created by the heuristic method of canonical quantization.
In the present paper we investigate an alternative two-axes decomposition method for rotations that has been proposed in our earlier research. It is shown to provide a convenient parametrization for many important physical systems. As an example, the kinematics of a rotating rigid body is considered and a specific class of solutions to the Euler dynamical equations are obtained in the case of symmetric inertial ellipsoid. They turn out to be related to the Rabi oscillator in spin systems well known in quantum computation. The corresponding quantum mechanical angular momentum and Laplace operator are derived as well with the aid of infinitesimal variations. Curiously, the coefficients in this new representation happen to depend only on one of the angles, which simplifies the corresponding system of ODE's emerging from separation of variables. Some applications of the hyperbolic and complex analogues of this construction in quantum mechanics and relativity are considered in a different paper cited below.
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In the first part of this series we have shown how the Schrodinger equation for a single particle and the corresponding non relativistic quantum observables can be obtained from a purely classical phonon model through the Newtonian equations of motion. In this work we focus instead on how the classical Hamiltonian formalism applied to the same phonon system allows to recover the feature of relativistic quantum mechanics for a single spinless particle. Using the classical nature of the phonon model, we naturally define continuous Poisson brackets between classical observables, which allows to recover the dynamics of such observables, i.e. the Ehrenfest relations associated to real-valued Klein-Gordon fields. The Poisson brackets also permits to obtain the generic form of constants of motions, thus generalizing the concept of inner products and momentum on Klein-Gordon fields. We then connect the formalism of real-valued classical functionals with that of hermitian operators and complex-valued wave functions. This is done through the introduction of a non-local complex-valued change of variables which allows to rewrite the real-valued Klein-Gordon equation in a form akin to the Schrodinger equation, and the classical observables as quantum expectation values. Then, we show how this change of variables allows to rewrite the classical Poisson brackets as commutators of hermitian operators. This points out the strict equivalence between the Heisenberg formalism and the formalism of classical Poisson bracket. Eventually, we illustrate how the Poisson brackets allows to recover the transformations of Poincare group in 1+1 dimension together with its algebra. The latter makes the link between the Lorentz invariant inner product of Mostafazadeh and the Casimir invariant associated to the mass of particle.
The transition from classical physics to quantum mechanics has been mysterious. Here, we derive the space-independent von Neumann equation for electron spin mathematically from the classical Bloch or Majorana--Bloch equation, which is also derived. Subsequently, the space-independent Schrödinger--Pauli equation is derived in both the quantum mechanical and recently developed co-quantum dynamic frameworks. An additional appendix incorporates the CQD signum branching function into the nonlinear von Neumann induction equation and develops its explicit double-commutator and state-dependent effective-Hamiltonian forms.