A Clebsch portrait for Schrödinger’s theory

Eur. Phys. J. Plus (2024) 139:697 https://doi.org/10.1140/epjp/s13360-024-05466-8 Regular Article A Clebsch portrait for Schrödinger’s theory Gabriele Barbieri1,2,a , Mauro Spera3,b 1 Dipartimento di Matematica e Applicazioni, Universitá di Milano-Bicocca, Via Cozzi 55, 20125 Milano, Italy 2 Dipartimento di Matematica “Felice Casorati”, Università degli Studi di Pavia, Via Ferrata 5, 27100 Pavia, Italy 3 Dipartimento di Matematica e Fisica “Niccolò Tartaglia”, Università Cattolica del Sacro Cuore, Via della Garzetta 48, 25133 Brescia, Italy Received: 10 March 2024 / Accepted: 16 July 2024 © The Author(s) 2024 Abstract In this note we pursue the investigation initiated in Spera M (in: Nielsen, Barbaresco, (eds) Geometric Science of Information. GSI 2023. Lecture Notes in Computer Science, Springer, Cham, 2023) by addressing geometric and topological issues concerning the zero set of the wave function, provided it is a knot in 3-space. Since, the standard Madelung velocity breaks down thereat, it is necessary to resort to the Clebsch geometry of the probability current shown in the above paper. This leads to considering several tightly interknit symplectic manifolds. 1 Introduction Vortex structures, while ubiquitous in various physical contexts, emerge, in particular, as nodal lines (zero sets) of the wave function of a massive spinless particle ruled by the Schrödinger equation, where the probability density vanishes and the phase is totally undetermined (see e.g. [2, 3]), thus giving rise to both physical and mathematical subtleties. The paper [1] investigated some hydrodynamical aspects of the probability current in Schrödinger’s theory, starting from the observation that, outside the nodal line, the latter shares the same (Bohm) trajectories with the Madelung velocity, while exhibiting a regular behaviour; the nodal line motion was found to be closely related to the time derivative of the probability current and the nodal line itself – advected via the hydrodynamical Schrödinger-Madelung equation – arose as a fibre of a Clebsch-type fibration. In the present contribution we further analyse this Clebsch geometry, which turns out to be naturally related to several symplectic manifolds manufactured from wave functions. In particular, we resort to the projective Hilbert space approach of [4, 5] and sketch a portrait akin to the one developed in [6–9] for the Euler equation, ultimately conveyed in diagrammatic form (Theorem 2). The paper is organised as follows: first, we briefly discuss the standard Madelung-Schrödinger picture, closely following, in particular, the exposition given in [1]; then, we present a series of constructions based on [7–9] and [4, 5] which eventually merge together in the Clebsch portrait of Theorem 2. 2 Preliminaries Basic references for this section are [10] for quantum hydrodynamics and, specifically concerning geometric aspects, [1, 4, 5, 11–14]. See [15] and [16] as well. 2.1 Quantum hydrodynamics Let us discuss, for simplicity, the motion of a spinless particle of mass m > 0 in 3-space. The quantum wave function depends on x and t: ψ  ψ(x, t), with x ∈ R3 , t ∈ R and obeys the Schrödinger equation (set   m  1) 1 ∂t ψ  −i Ĥ ψ : −i(−  + V )ψ 2 (1) Gabriele Barbieri and Mauro Spera have contributed equally to this work. Contribution to the Focus Point on “Mathematics and Physics at the Quantum-Classical Interface” edited by D.I. Bondar, I. Joseph, G. Marmo, C. Tronci. a e-mail: b e-mail: (corresponding author) 0123456789().: V,-vol 123 697 Page 2 of 8 Eur. Phys. J. Plus (2024) 139:697 with  denoting the Laplace operator and V  V (x) a (“classical”) potential. Its polar decomposition (as soon as ρ > 0) reads √ ρ  |ψ|2 . (2) ψ  ρ ei S , However, we shall, whenever expedient, stick to Schrödinger’s ψ-formalism, in order to bypass the limitations imposed by the latter. The Schrödinger equation can be cast into the Madelung-Bohm hydrodynamical form [17–19] (setting u  ∇ S) ⎧ ⎨ ∂ u + (u · ∇)u  −∇(V + V ) q ∂t , (3) ⎩ ∂ρ + div(ρ u)  0 ∂t √  ρ − 21 √ρ where Vq  is the so-called quantum potential, the first equation is an Euler equation for a compressible irrotational fluid (div u  S   0 in general) and the second one is the continuity equation, involving the probability current 1 (ψ † ∇ψ − ψ∇ψ † ) . (4) j  ρ u  Im(ψ † ∇ψ)  2i We shall freely switch vector fields and differential 1-forms via the musical isomorphisms induced by the Euclidean metric in R3 , so we write for instance j  ρ d S and so on. After setting      1 1 † 3 H : ψ| Ĥ ψ  ψ −  + V (x) ψ d x  (5) |∇ψ|2 +V (x)|ψ|2 d 3 x, 3 3 2 2 R R one can rephrase the above equations in a Hamiltonian fashion, following Bohm [17, 18]: ∂ρ δH ∂S δH  , − ∂t δS ∂t δρ (6) or, in complex terms, see [14]: ∂ψ δH  −i † , ∂t δψ ∂ψ † δH i . ∂t δψ (7) 2.2 Symplectic geometric interpretation The above discussion can be geometrically reformulated as follows, glossing over analytical subtleties (see [1, 14] for extra bibliography and [20] for a general theory). Let ψ : R3 x → (ρ(x), S(x)) ≡ ψ(x) ∈ R2 ∼  C be a smooth map (we may use polar coordinates whenever ρ > 0, so S(x) ∈ R/2πZ  S 1 ). The set M of such maps becomes a symplectic manifold as soon as the target space is equipped with the symplectic structure dρ ∧ d S (and we tacitly compactify R3 to S 3 ). The symplectic form  on M reads   δρ(x) ∧ δS(x)d 3 x (8) R3 or, in complex coordinates (Kähler structure),    −i R3 δψ † (x) ∧ δψ(x)d 3 x (9) Let G denote the (connected component of the identity of the) group of volume preserving diffeomorphisms of R3 which rapidly approach the identity at infinity, with “Lie algebra” g consisting of divergence-free vector fields (div b  0) rapidly vanishing at infinity. The space R3 is equipped with the standard Euclidean metric (allowing the customary identification of vector fields and 1-forms). The symplectic form  is G-invariant under the natural action of G on ψ ∈ M via g(ψ)(x)  ψ(g −1 (x)) (10) (since, the Jacobian J (g)  1). Notice that the probability current can be viewed itself as the velocity field of a new fluid (again compressible), with vorticity w  curl j  ∇ρ × ∇ S, w  d j  dρ ∧ d S  −i dψ † ∧ dψ, (11) see [1]. The Hamiltonian algebra (Rasetti-Regge (RR) current algebra, [6–9, 21, 22]) associated to the G-coadjoint orbit pertaining to the divergence-free vector field w consists of functions λb - for any b divergence-free - defined as   λb  j·b w · B, (12) R3 R3 where curl B  b. One checks the Lie algebra structure of the RR-current algebra, namely {λb , λc }  λ[b,c] , 123 (13) Eur. Phys. J. Plus (2024) 139:697 Page 3 of 8 697 the vector field bracket being minus the usual one and reading, for divergence-free vector fields [b, c]  curl (b × c) (14) and, where the Poisson bracket is the one induced by the symplectic form. Explicitly:    {λb , λc }  j · [b, c]  j · curl(b × c)  w · (b × c). R3 R3 R (...truncated)