[Paper Review] PT symmetry and supersymmetry

Miloslav Znojil|ArXiv.org|Sep 7, 2002

Quantum Mechanics and Non-Hermitian Physics5 references3 citations

TL;DR

This paper introduces a novel framework for supersymmetry (SUSY) in quantum mechanics by generalizing Witten's formalism to include ${\cal PT}$-symmetric Hamiltonians, particularly for spiked harmonic oscillators. By complexifying the radial coordinate $ r \to r - i\varepsilon $, the authors regularize singular centrifugal terms and construct a non-Hermitian SUSY structure with real energy spectra in unbroken ${\cal PT}$ phases, revealing new SUSY partner Hamiltonians and a hidden $sl(2,\mathbb{R})$ algebraic structure.

ABSTRACT

A re-formulated, non-Hermitian version of the Witten's supersymmetric quantum mechanics is presented. Its use of pseudo-Hermitian (so called PT symmetric) Hamiltonians is reviewed and illustrated via several forms of an innovated supersymmetric partnership between strongly singular ("spiked") harmonic oscillators.

Motivation & Objective

• To extend Witten's supersymmetric quantum mechanics to non-Hermitian, ${\cal PT}$-symmetric Hamiltonians.
• To resolve the issue of centrifugal singularities in higher-dimensional quantum systems using complex coordinate regularization.
• To construct a consistent supersymmetric structure for spiked harmonic oscillators with real energy spectra under unbroken ${\cal PT}$ symmetry.
• To identify an underlying $sl(2,\mathbb{R})$ Lie algebra symmetry in the generalized SUSY framework.

Proposed method

• Adopt a complexified radial coordinate $ r = x - i\varepsilon $ to regularize the centrifugal term in $ D $-dimensional spiked harmonic oscillators.
• Define pseudo-Hermitian Hamiltonians $ H^{(\alpha)} = -\partial_r^2 + \frac{\alpha^2 - 1/4}{r^2} + r^2 $ with $ \alpha > 0 $, preserving ${\cal PT}$ symmetry.
• Construct supersymmetric partners $ H_{(L)}^{(\gamma)} $ and $ H_{(R)}^{(\gamma)} $ using first-order differential operators $ A^{(\gamma)} = \partial_r + W^{(\gamma)} $, $ B^{(\gamma)} = -\partial_r + W^{(\gamma)} $, with $ W^{(\gamma)} = r - \frac{\gamma + 1/2}{r} $.
• Identify the SUSY algebra $ \{Q, \tilde{Q}\} = H $, $ \{Q, Q\} = \{\tilde{Q}, \tilde{Q}\} = 0 $, with $ H $ being block-diagonal and $ Q, \tilde{Q} $ acting as supercharges.
• Derive a second-order differential operator $ \mathbf{A}(\alpha) $ and its adjoint $ \mathbf{B}(\alpha) $, which generate a closed $sl(2,\mathbb{R})$ algebra with the Hamiltonian $ H^{(\alpha)} $.
• Show that the energy spectra remain real in unbroken ${\cal PT}$ phases and partially real even when symmetry is spontaneously broken, via identities on Laguerre polynomials.

Experimental results

Research questions

• RQ1 Can Witten's supersymmetric quantum mechanics be consistently extended to non-Hermitian, ${\cal PT}$-symmetric Hamiltonians?
• RQ2 How does complexification of the radial coordinate $ r \to r - i\varepsilon $ regularize the centrifugal singularity in spiked harmonic oscillators?
• RQ3 What is the structure of the SUSY partner Hamiltonians in the ${\cal PT}$-symmetric regime, and do they yield real energy spectra?
• RQ4 Does a hidden $sl(2,\mathbb{R})$ algebraic structure emerge in the generalized ${\cal PT}$-symmetric SUSY framework?
• RQ5 Can partially real energy spectra be obtained even when ${\cal PT}$ symmetry is spontaneously broken?

Key findings

• The complex coordinate shift $ r \to r - i\varepsilon $ successfully regularizes the centrifugal term in spiked harmonic oscillators, enabling a well-defined ${\cal PT}$-symmetric quantum mechanical framework.
• In the unbroken ${\cal PT}$ phase, the SUSY partner Hamiltonians $ H_{(L)}^{(\gamma)} $ and $ H_{(R)}^{(\gamma)} $ yield completely real energy spectra, forming a quadruplet of energies at each $ N $: $ 4N+4\alpha $, $ 4N+4 $, $ 4N $, and $ 4N-4\alpha $, depending on the regime.
• For real $ \gamma $, the system exhibits three distinct SUSY regimes: large negative $ \gamma $, small negative $ \gamma $, and positive $ \gamma $, each with distinct partner relations and energy patterns.
• The Hamiltonian $ H^{(\alpha)} $ satisfies $ H^{(\alpha)} = \frac{1}{8} \left( \mathbf{A}(\alpha)\mathbf{B}(\alpha) - \mathbf{B}(\alpha)\mathbf{A}(\alpha) \right) $, and generates a closed $sl(2,\mathbb{R})$ algebra with normalized generators.
• The energy eigenvalues of the generalized SUSY system are given by $ \Omega_N^{(\gamma)} = 16N(N + \gamma) $, satisfying the fourth-order differential equations $ \mathbf{G}_{(L/R)} |N^{(\gamma)}\rangle = \Omega_N^{(\gamma)} |N^{(\gamma)}\rangle $.
• Even when ${\cal PT}$ symmetry is spontaneously broken (e.g., for complex $ \gamma = i\delta $), the energy spectra remain partially real, with specific multiplets such as $ E_{(L)}^{(+\alpha)} = 4n $, $ E_{(L)}^{(-\alpha)} = E_{(R)}^{(-\beta)} = 4n - 4\alpha $, and $ E_{(R)}^{(+\beta)} = 4n + 4 $.