arXiv:2609.25478

Phase Encoding of Genuine Three-Body Interactions in a Relativistic Dirac System in $1+1$ Dimensions

Tadashi Yoshikawa

Published Sep 21, 2026• 0 citations

Abstract

We show how genuine three-body phase information can enter the invariant mass of a relativistic three-particle Dirac system in $(1+1)$ dimensions. As a solvable reference system, we consider the Sakamoto--Munakata--Ino model with pairwise contact interactions $g_{ij}(1-α_iα_j)δ(x_i-x_j)$. These singular interactions can be transferred into sector-dependent phases and matching conditions by a discontinuous unitary transformation. Although the explicit contact terms are thereby removed, the nonzero constituent-mass operator is rotated and retains nontrivial spectral information. We introduce a genuine three-body holonomy generated by $Q_3=α_1α_2α_3$. The kinetic and pair-interaction parts commute with $Q_3$, while the constituent-mass operator anticommutes with it. Consequently, the massless system separates into the $Q_3=\pm1$ sectors, which acquire opposite holonomy phases $e^{\pm iθ_3}$, whereas nonzero constituent masses mix the two sectors. This phase-sector-mixing mechanism makes the relative three-body phase dynamically accessible to the bound-state spectrum and establishes an operator-level mechanism through which the three-body holonomy generates a $θ_3$ dependence of the physical three-body invariant mass. We further emphasize that the topological three-body holonomy is not automatically equivalent to a bare triple-contact potential; such an equivalence requires a regulated self-adjoint realization and a compatible interaction-dependent boost satisfying the Poincaré algebra. The resulting framework therefore connects genuine three-body phase information to the mass spectrum of a relativistic composite system while clearly separating the controlled holonomy construction from the unresolved short-distance triple-contact realization.

DOI: 10.48550/arxiv.2609.25478

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arXiv:2609.25478

Phase Encoding of Genuine Three-Body Interactions in a Relativistic Dirac System in $1+1$ DimensionsPhase Encoding of Genuine Three-Body Interactions in a Relativistic Dirac System in $1+1$ Dimensions

Tadashi Yoshikawa

Published Sep 21, 2026• 0 citations

arXiv HTML PDF

Abstract

We show how genuine three-body phase information can enter the invariant mass of a relativistic three-particle Dirac system in

(1+1)

dimensions. As a solvable reference system, we consider the Sakamoto--Munakata--Ino model with pairwise contact interactions

g_{ij}(1-α_iα_j)δ(x_i-x_j)

. These singular interactions can be transferred into sector-dependent phases and matching conditions by a discontinuous unitary transformation. Although the explicit contact terms are thereby removed, the nonzero constituent-mass operator is rotated and retains nontrivial spectral information. We introduce a genuine three-body holonomy generated by

Q_3=α_1α_2α_3

. The kinetic and pair-interaction parts commute with

Q_3

, while the constituent-mass operator anticommutes with it. Consequently, the massless system separates into the

Q_3=\pm1

sectors, which acquire opposite holonomy phases

e^{\pm iθ_3}

, whereas nonzero constituent masses mix the two sectors. This phase-sector-mixing mechanism makes the relative three-body phase dynamically accessible to the bound-state spectrum and establishes an operator-level mechanism through which the three-body holonomy generates a

θ_3

dependence of the physical three-body invariant mass. We further emphasize that the topological three-body holonomy is not automatically equivalent to a bare triple-contact potential; such an equivalence requires a regulated self-adjoint realization and a compatible interaction-dependent boost satisfying the Poincaré algebra. The resulting framework therefore connects genuine three-body phase information to the mass spectrum of a relativistic composite system while clearly separating the controlled holonomy construction from the unresolved short-distance triple-contact realization.

DOI: 10.48550/arxiv.2609.25478

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arXiv:2211.09765Nov 17, 20220 citations

Phase Encoding of Genuine Three-Body Interactions in a Relativistic Dirac System in

1+1

Dimensions

We show how genuine three-body phase information can enter the invariant mass of a relativistic three-particle Dirac system in

(1+1)

dimensions. As a solvable reference system, we consider the Sakamoto--Munakata--Ino model with pairwise contact interactions

g_{ij}(1-α_iα_j)δ(x_i-x_j)

Q_3=α_1α_2α_3

. The kinetic and pair-interaction parts commute with

Q_3

, while the constituent-mass operator anticommutes with it. Consequently, the massless system separates into the

Q_3=\pm1

sectors, which acquire opposite holonomy phases

e^{\pm iθ_3}

θ_3

Properties of two identical particles of mass

m

and a distinct particle of mass

m_1

in the universal low-energy limit of zero-range two-body interaction are studied in different sectors of total angular momentum

L

and parity

P

. For the unambiguous formulation of the problem in the interval

μ_r(L^P) < m/m_1 \le μ_c(L^P)

(

μ_r(1^-) \approx 8.619

and

μ_c(1^-) \approx 13.607

μ_r(2^+) \approx 32.948

and

μ_c(2^+) \approx 38.630

,~etc.) in each

L^P

sector an additional parameter

b

determining the wave function near the triple-collision point is introduced; thus, a one-parameter family of self-adjoint Hamiltonians is defined. Within the framework of this formulation, dependence of the bound-state energies on

m/m_1

and

b

in the sector of angular momentum and parity

L^P

is calculated for

L \le 5

and analysed with the aid of a simple model. A number of the bound states for each

L^P

sector is analysed and presented in the form of `phase diagrams' in the plane of two parameters

m/m_1

and

b

K

, which couples spin to transverse momentum;

K_m

, a mass-dependent operator derived from the transverse Dirac Hamiltonian; and helicity. On the common regular domain away from the momentum axis, explicit smooth, single-valued

SU(2)

transformations relate all three splittings. Although globally

SU(2)

-equivalent on this common domain, they exhibit three distinct global extension behaviors. The

K

projectors have azimuth-dependent polar limits and do not extend continuously to the axis. For nonzero mass, the

K_m

projectors extend smoothly over the enclosed momentum ball and define Chern-trivial eigenlines. The helicity projectors are smooth on every nonzero momentum sphere, but their eigenlines carry opposite unit Chern numbers and cannot extend through the enclosed origin. The parent positive-energy Dirac connection Abelianizes exactly in the

K

eigenlines on fixed-azimuth meridians, whereas its full three-dimensional curvature has noncommuting components. We obtain the azimuthal Wilson loop in closed form and derive the exact conversion probability between the two

K

branches under purely geometric positive-energy transport.

2\times2

matrix distribution within that sector before quantisation. Projecting the Weyl-ordered matrix Liouvillian gives relativistic transport while preserving all internal populations and coherences. For universal deformations on flat canonical phase space satisfying the stated covariance, matrix-neutrality, homogeneity and associativity assumptions, the resulting product is the matrix Weyl--Moyal product. Its parameter has dimensions of action and, when identified with

\hbar

, supplies the scale of physical spin-

\tfrac12

. Requiring all-order constraint preservation gives a vanishing star commutator, while preservation of the selected star-spectral sector yields the two-sided Dirac--Wigner equations. Our central conclusion is therefore that relativistic mass-shell factorisation supplies spinor structure, while completion of the constraint-preservation hierarchy within the stated deformation class leads to the Weyl--Moyal phase-space formulation of quantum mechanics.

\uparrow

and

\downarrow

particles. With respect to the relation between three-body loss rate and number of nearby triplets stated in [Petrov, Salomon and Shlyapnikov, PRL 93, 090404 (2004)], the present work adds a derivation, expresses the proportionality factor in terms of the three-body parameter, and includes the general case where there are several contributions to the three-body contact and several three-body parameters.