Momentunian in TSI Quantum Mechanics

• Momentunian is a spatial evolution operator in time–space inverted quantum mechanics, defined via a square-root structure that replaces the Hamiltonian when position becomes the evolution parameter.
• It features both non-relativistic and relativistic formulations, utilizing Dirac-type factorization to yield two-component equations and symmetric momentum doublets.
• The term also appears in alternative MOND contexts, highlighting its context-dependent role in modified gravity and inertia frameworks alongside its quantum mechanical formulation.

Momentunian denotes, in time–space inverted quantum mechanics (TSI QM), the operator that generates spatial evolution when position $q$ is treated as the evolution parameter and time is promoted to an operator. In that setting, the Momentunian is the analogue of the Hamiltonian in ordinary Schrödinger dynamics, but with the roles of time and space inverted. The term is used explicitly for the square-root generator $\hat{\mathcal{P}}^\pm(\hat{\mathcal{H}},\hat t;q)$ in the TSI framework, where it acquires non-relativistic and relativistic realizations, admits Dirac-type factorization, and yields an emergent supersymmetric structure organized by momentum doublets rather than energy doublets (Beims et al., 17 May 2026). In separate modified-dynamics discussions, the same label also appears in MOND-related contexts, but with a different meaning; those usages concern low-acceleration gravitational dynamics rather than a quantum spatial-evolution operator (Singh, 7 Jan 2026).

1. Terminology and conceptual setting

In the Dias–Parisio space–time-symmetric framework, TSI QM is formulated on an extended Hilbert space $H_E = H_x \oplus H_T$. In the subspace $H_T$, time is an operator canonically conjugate to minus the energy operator, so that

$[\hat t,\hat{\mathcal{H}}] = - i\hbar,$

while the spatial coordinate $q$ is treated as a parameter. A state vector $|\psi)\in H_T$ evolves along $q$ according to

$$\hat{\mathcal{P}}^\pm(\hat{\mathcal{H}},\hat t;q)\,|\psi) = -\,i\hbar\,\partial_q |\psi),$$

which defines the Momentunian as the generator of spatial translation in the TSI description (Beims et al., 17 May 2026).

This construction reverses the conventional hierarchy of nonrelativistic quantum mechanics. In the ordinary Schrödinger picture, time is external and the Hamiltonian generates time evolution. In TSI QM, time belongs to the operator algebra, whereas position labels the evolution parameter. A plausible implication is that the Momentunian is best understood not as a modified momentum observable in the usual sense, but as a structural replacement for the Hamiltonian within a spatially parametrized dynamics.

2. Non-relativistic and relativistic definitions

For time-independent potentials, the non-relativistic Momentunian is defined by

$$\hat{\mathcal{P}}^\pm(\hat{\mathcal{H}},\hat t;q) \;=\; \pm\,\sqrt{\,2m\,\big[\hat{\mathcal{H}}-\hat{\mathcal{V}}(q)\big]\,}.$$

Assuming $\hat{\mathcal{P}}^\pm(\hat{\mathcal{H}},\hat t;q)$0 and $\hat{\mathcal{P}}^\pm(\hat{\mathcal{H}},\hat t;q)$1 are self-adjoint, the branches $\hat{\mathcal{P}}^\pm(\hat{\mathcal{H}},\hat t;q)$2 and $\hat{\mathcal{P}}^\pm(\hat{\mathcal{H}},\hat t;q)$3 are defined on the domain where $\hat{\mathcal{P}}^\pm(\hat{\mathcal{H}},\hat t;q)$4, with spectra $\hat{\mathcal{P}}^\pm(\hat{\mathcal{H}},\hat t;q)$5 and $\hat{\mathcal{P}}^\pm(\hat{\mathcal{H}},\hat t;q)$6, respectively. A two-component construction combines them into a single self-adjoint operator,

$\hat{\mathcal{P}}^\pm(\hat{\mathcal{H}},\hat t;q)$7

whose spectrum is symmetric under $\hat{\mathcal{P}}^\pm(\hat{\mathcal{H}},\hat t;q)$8 (Beims et al., 17 May 2026).

The relativistic construction starts from the classical relation

$\hat{\mathcal{P}}^\pm(\hat{\mathcal{H}},\hat t;q)$9

Canonical quantization and Dirac factorization yield a linear, Dirac-like Momentunian,

$H_E = H_x \oplus H_T$0

with $H_E = H_x \oplus H_T$1 and $H_E = H_x \oplus H_T$2. The associated two-component equations for $H_E = H_x \oplus H_T$3 are

$H_E = H_x \oplus H_T$4

These formulas make the square-root structure explicit and simultaneously explain why factorization techniques are natural in this framework.

3. Emergent supersymmetry from factorization

The central structural claim of the 2026 Letter is that supersymmetry is not imposed phenomenologically but emerges from factorizing the Momentunian’s square-root form. In the non-relativistic sector, Dirac linearization produces a first-order matrix equation with a fractional derivative in time,

$H_E = H_x \oplus H_T$5

Acting again with $H_E = H_x \oplus H_T$6 and using $H_E = H_x \oplus H_T$7 for Caputo derivatives recovers an effective Schrödinger equation

$H_E = H_x \oplus H_T$8

so the familiar quantum Hamiltonian reappears only after factorization (Beims et al., 17 May 2026).

Upon separation of variables, $H_E = H_x \oplus H_T$9, one obtains spatial intertwining operators with superpotential $H_T$0 for time-independent potentials: $H_T$1 The supercharges

$H_T$2

generate the standard $H_T$3-graded SUSY algebra,

$H_T$4

with partner Hamiltonians

$H_T$5

$H_T$6

The formal resemblance to standard SUSYQM is exact at the level of the factorized spatial operators. The crucial difference is that the primary generator is the Momentunian rather than the Hamiltonian, so the spectral organization is by $H_T$7 doublets rather than energy doublets (Beims et al., 17 May 2026).

4. Zero modes, vanishing momentum, and fractional-time dynamics

Within this framework, zero modes are defined by vanishing spatial momentum, not necessarily by zero energy. In the non-relativistic sector, the zero-mode conditions are

$H_T$8

which correspond to setting the separation constants $H_T$9. The time equation then becomes $[\hat t,\hat{\mathcal{H}}] = - i\hbar,$0, so $[\hat t,\hat{\mathcal{H}}] = - i\hbar,$1 is constant and the zero-mode state is time-independent in the separated representation (Beims et al., 17 May 2026).

The relativistic zero modes are more distinctive. Setting the separation constant equal to the potential, $[\hat t,\hat{\mathcal{H}}] = - i\hbar,$2, imposes the constraint $[\hat t,\hat{\mathcal{H}}] = - i\hbar,$3 and yields

$[\hat t,\hat{\mathcal{H}}] = - i\hbar,$4

These states are evanescent in $[\hat t,\hat{\mathcal{H}}] = - i\hbar,$5 and, as stated in the Letter, are independent of the physical potential except for the constraint that pins $[\hat t,\hat{\mathcal{H}}] = - i\hbar,$6 to $[\hat t,\hat{\mathcal{H}}] = - i\hbar,$7. Physically, when $[\hat t,\hat{\mathcal{H}}] = - i\hbar,$8, the kinetic momentum becomes imaginary in the massive case, producing exponentially decaying or growing spatial profiles (Beims et al., 17 May 2026).

The same work introduces VT-SUSY, or “square-root SUSY,” partners in which the supercharges themselves contain fractional-time derivatives. In the non-relativistic partner construction,

$[\hat t,\hat{\mathcal{H}}] = - i\hbar,$9

and zero-momentum states satisfy

$q$0

Their time dependence is governed by the Mittag–Leffler function,

$q$1

This is the mechanism by which memory effects enter the supersymmetric wavefunctions. The non-Markovian kernel of the Caputo derivative,

$q$2

makes the TSI-QM supersymmetry intrinsically history-dependent (Beims et al., 17 May 2026).

5. Spectral organization, examples, and operator theory

The non-relativistic branches $q$3 are self-adjoint on the domain where $q$4, and the two-component Momentunian is self-adjoint on its natural domain. Spatial evolution is therefore unitary whenever

$q$5

is defined with a self-adjoint generator. The spectrum exhibits a protected $q$6 pairing because a chiral-like operator $q$7 satisfies

$q$8

so $q$9 is an exact node of the symmetry (Beims et al., 17 May 2026).

For the harmonic oscillator, $|\psi)\in H_T$0, the superpotential becomes

$|\psi)\in H_T$1

and the partner Hamiltonians are

$|\psi)\in H_T$2

One sector possesses a normalizable zero-momentum ground state,

$|\psi)\in H_T$3

while the partner zero mode is non-normalizable, giving Witten index $|\psi)\in H_T$4. The Letter also states that separation yields a “kinetic energy of time” $|\psi)\in H_T$5 quantized alongside the oscillator levels, and a corresponding quantized momentum spectrum. For step potentials, the same machinery produces partner Hamiltonians differing by a localized edge term proportional to $|\psi)\in H_T$6, mapping scattering states between sectors with identical asymptotic momenta but different phase shifts (Beims et al., 17 May 2026).

These features place the Momentunian at the intersection of several operator-theoretic themes: square-root Hamiltonians, self-adjoint extension questions, supersymmetric intertwining, and fractional evolution. The main conceptual innovation is that all of these arise from a spatial generator rather than from the usual time-evolution operator.

6. Other research usages and terminological ambiguity

Outside TSI QM, “Momentunian” is not terminologically fixed. In one MOND paper summary, a “Momentunian/MONDian” regime is realized by a metric-only, UV-vanishing infrared deformation selected by an IR de Sitter vacuum and $|\psi)\in H_T$7D conformal symmetry; in that context the regime reproduces the AQUAL/MOND equation at low acceleration while recovering General Relativity exactly at high acceleration (Singh, 7 Jan 2026). In another summary, “Momentunian” is interpreted as a moment-based modification of Newtonian dynamics derived from higher moments of a geodesic equation with a random spin connection term; that usage leads to qMOND potentials written in Gauss and Appell hypergeometric functions, steeper MOND interpolation functions, and an mMOND regime with an almost-flat asymptotic rotation curve proportional to $|\psi)\in H_T$8 (Pietrzyk et al., 19 Nov 2025).

A related but distinct body of work treats MOND as modified inertia rather than modified gravity. There the central object is not a spatial-evolution generator but a trajectory-dependent inertia functional,

$|\psi)\in H_T$9

whose phenomenology differs from AQUAL and QUMOND in the external-field effect, noncircular motions, and inner-Solar-System anomalies (Milgrom, 2023). More generally, MOND introduces a universal acceleration scale $q$0 and a deep-MOND limit with scale invariance, with the standard algebraic relation

$q$1

and asymptotic law

$q$2

for isolated systems (Milgrom, 2011).

These usages concern low-acceleration gravitational dynamics, not the TSI-QM operator. This suggests that the word “Momentunian” is technically precise only in the 2026 TSI-QM literature, where it names a specific square-root spatial generator. In broader usage, it functions as a context-dependent label attached either to MONDian dynamics or to moment-based modifications of Newtonian gravity.