Spatial Dimension as a Variable in Quantum Mechanics

We propose quantum-mechanical systems in which the number of spatial dimensions is promoted to a dynamical quantum variable, making the effective dimension state-dependent. Interestingly, systems of this form can exhibit enhanced symmetries compared to their fixed-dimensional counterparts. As an explicit example, we analyze a two-state system for which the number of spatial dimensions is represented by a quantum operator. By evaluating the corresponding partition function, we uncover a temperature-dependent effective dimension. Our framework opens a new avenue for constructing physical systems, from gravity to condensed matter, where the very notion of dimensionality becomes quantum.We propose quantum-mechanical systems in which the number of spatial dimensions is promoted to a dynamical quantum variable, making the effective dimension state-dependent. Interestingly, systems of this form can exhibit enhanced symmetries compared to their fixed-dimensional counterparts. As an explicit example, we analyze a two-state system for which the number of spatial dimensions is represented by a quantum operator. By evaluating the corresponding partition function, we uncover a temperature-dependent effective dimension. Our framework opens a new avenue for constructing physical systems, from gravity to condensed matter, where the very notion of dimensionality becomes quantum.

quant-pharXiv:2511.14547Nov 19, 20250 citations

Six-dimensional space-time from quaternionic quantum mechanics

Dorje C. Brody, Eva-Maria Graefe

Quaternionic quantum Hamiltonians describing nonrelativistic spin particles require the ambient physical space to have five dimensions. The quantum dynamics of a spin- $\frac{1}{2}$ particle system characterized by a generic Hamiltonian is worked out in detail. It is shown that there exists, within the structure of quaternionic quantum mechanics, a canonical reduction to three spatial dimensions upon which standard quantum theory is retrieved. In this dimensional reduction, three of the five dynamical variables are shown to oscillate around a cylinder, thus behaving in a quasi-one-dimensional manner at large distances. An analogous mechanism is shown to exist in the case of octavic Hamiltonians, where the ambient physical space has nine dimensions. Possible experimental tests in search for the signature of extra dimensions at low energies are briefly discussed.Quaternionic quantum Hamiltonians describing nonrelativistic spin particles require the ambient physical space to have five dimensions. The quantum dynamics of a spin-$\frac{1}{2}$ particle system characterized by a generic Hamiltonian is worked out in detail. It is shown that there exists, within the structure of quaternionic quantum mechanics, a canonical reduction to three spatial dimensions upon which standard quantum theory is retrieved. In this dimensional reduction, three of the five dynamical variables are shown to oscillate around a cylinder, thus behaving in a quasi-one-dimensional manner at large distances. An analogous mechanism is shown to exist in the case of octavic Hamiltonians, where the ambient physical space has nine dimensions. Possible experimental tests in search for the signature of extra dimensions at low energies are briefly discussed.

arXiv:1105.3604Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fieldsDec 14, 201129 citations

Quantum mechanics in finite dimensions

Thalanayar S. Santhanam, A. R. Tekumalla

Foundations of PhysicsOct 1, 197691 citations

Solvable -symmetric potentials in higher dimensions

GÉZA LÉVAI

-symmetric, non-relativistic quantum mechanical potentials are discussed in two and three spatial dimensions. Conditions are formulated under which these potentials are -symmetric and can be solved exactly by the separation of the radial and angular variables. It is found that the angular variables play an essential role in introducing non-Hermiticity via the imaginary potential terms. A simple partially exactly solvable potential is used to demonstrate various aspects of symmetry in both two and three dimensions. Possible generalizations of the results are outlined.-symmetric, non-relativistic quantum mechanical potentials are discussed in two and three spatial dimensions. Conditions are formulated under which these potentials are -symmetric and can be solved exactly by the separation of the radial and angular variables. It is found that the angular variables play an essential role in introducing non-Hermiticity via the imaginary potential terms. A simple partially exactly solvable potential is used to demonstrate various aspects of symmetry in both two and three dimensions. Possible generalizations of the results are outlined.

Journal of Physics A Mathematical and TheoreticalMar 23, 200715 citations

Quantum mechanics in a finite number of dimensions

Thalanayar S. Santhanam

Physica A Statistical Mechanics and its ApplicationsAug 1, 19826 citations

Quaternionic quantum Hamiltonians describing nonrelativistic spin particles require the ambient physical space to have five dimensions. The quantum dynamics of a spin-

\frac{1}{2}

particle system characterized by a generic Hamiltonian is worked out in detail. It is shown that there exists, within the structure of quaternionic quantum mechanics, a canonical reduction to three spatial dimensions upon which standard quantum theory is retrieved. In this dimensional reduction, three of the five dynamical variables are shown to oscillate around a cylinder, thus behaving in a quasi-one-dimensional manner at large distances. An analogous mechanism is shown to exist in the case of octavic Hamiltonians, where the ambient physical space has nine dimensions. Possible experimental tests in search for the signature of extra dimensions at low energies are briefly discussed.