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
Annihilation and creation operators in non-Hermitian supersymmetric quantum mechanics
Harmonic oscillators with a centrifugal spike are analysed, via a non-Hermitian regularization, within a complexified SUSY quantum mechanics. The formalism enables us to construct the factorized creation and annihilation operators. We show how the real though, generically, non-equidistant spectrum complies with the current SUSY-type isospectrality and degeneracy in an unusual way.
A slowly rotating coil system for AC field measurements of Fermilab Booster correctors
A method for measurement of rapidly changing magnetic fields has been developed and applied to the testing of new room temperature corrector packages designed for the Fermilab Booster Synchrotron. The method is based on fast digitization of a slowly rotating tangential coil probe, with analysis combining the measured coil voltages across a set of successive magnet current cycles. This paper presents results on the field quality measured for the normal and skew dipole, quadrupole, and sextupole elements in several of these corrector packages.