Pt Symmetry and Supersymmetry 1. Supersymmetry for Pedestrians, or Bosons and Fermions in Witten's Picture
Pseudo-Hermitian (so called PT symmetric) Hamiltonians are featured within a re-formulated Witten's supersymmetric quantum mechanics. An unusual form of the supersymmetric partnership between the spiked harmonic oscillators is described. In the review [1] of the Witten's (or so called supersymmetric) quantum mechanics one finds the specific Fock vacuum q|0 = exp(−q 2 /2)/ √ π 0 , q ∈ (−∞, ∞). (1) In the upper line we recognize the ground state of the one-dimensional harmonic oscillator H (HO) = p 2 + q 2 and see that " bosons " may be created/annihilated by the action of the respective operators −∂ r + q and ∂ r + q (to be denoted as B (−1/2) and A (−1/2) here). The solvable harmonic-oscillator character of this elementary model enables us to define the three two-by-two matrices F † = 0 0 1 0 , F = 0 1 0 0 , N F = 0 0 0 1 (2) which create and annihilate a " fermion " and determine the fermionic number, respectively. Formally, this enables us to introduce the two partner Hamiltonians H (L) = H (HO) − 1, H (R) = H (HO) + 1 and identify the underlying symmetry with the superalgebra sl(1/1) generated by the following three operator matrices
Re-establishing supersymmetry between harmonic oscillators in $D\ne 1$ dimensions
Using a PT symmetric regularization technique we reminad the reader that and how (a) the SUSY is re-established between the two shifted harmonic oscillator potentials $ V(q)=q^2+{G}/{q^2}+ const$ and (b) many non-equivalent Hermitian and non-Hermitian (PT symmetric) systems may often be studied in parallel (with a schematic solvable illustrative example added).
What is PT symmetry
The recently proposed complexication of Hamiltonians which keeps the spectra real (and is usually called PT symmetry) is re-interpreted as a certain natural linearalgebraic alternative to Hermiticity.