Systems with Quantum Dimensions

Abstract

We propose quantum-mechanical systems in which the number of spatial dimensions is promoted to a dynamical quantum variable. As a consequence, the effective dimension depends on the physical state of the system. Interestingly, systems of this form exhibit enhanced symmetries compared to their fixed-dimensional counterparts. As an explicit example, we analyze a harmonic oscillator for which the spatial dimension 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.

Figures (2)

• Figure 1: Plots of the harmonic oscillator wavefunctions in different dimensions. The three states $\ket{\psi^{(1)}}, \ket{\psi^{(2)}}, \ket{\psi^{(3)}}$ belong to $\mathcal{H}^{(1)}$, $\mathcal{H}^{(2)}$, and $\mathcal{H}^{(3)}$, respectively. The Hilbert space of the QD harmonic oscillator is then constructed by combining $\mathcal{H}^{(d)}$, $d=1,2,3$ via the direct sum $\bigoplus$. A generic state of a QD harmonic oscillator is a linear combination of states $\ket{\psi^{(d)}} \in \mathcal{H}^{(d)}$.
• Figure 2: The number of spatial dimensions of three QD harmonic oscillators with $N_1=0$ plotted as a function of energy for three different values of $N_2$. The expectation value $\braket{\hat{D}}$ increases with the energy scale, starting from $\braket{\hat{D}}=N_1=0$ in the limit $\braket{\hat{H}_0}=0$ (dotted line) and approaching $\braket{\hat{D}}=N_2$ at high energies (dashed line).