Applications of partial supersymmetry

Abstract

I examine quantum mechanical Hamiltonians with partial supersymmetry, and explore two main applications. First, I analyze a theory with a logarithmic spectrum, and show how to use partial supersymmetry to reveal the underlying structure of this theory. This method reveals an intriguing equivalence between two formulations of this theory, one of which is one-dimensional, and the other of which is infinite-dimensional. Second, I demonstrate the use of partial supersymmetry as a tool to obtain the asymptotic energy levels in non-relativistic quantum mechanics in an exceptionally easy way. In the end, I discuss possible extensions of this work, including the possible connections between partial supersymmetry and renormalization group arguments.

The Euler–Riemann gases, and partition identities

“…One may look at it differently, and write Z f (β) = ∆ F (2β)/∆ F (β) from which the graded fermionic partition function is ∆ F (2β) = Z f (β)∆ F (β), so mixing the fermionic system with the graded fermionic system at thermal equilibrum at a given temperature β is the same as a graded parafermionic system whose temperature is doubled. This also happens in the case of quantum field theory with a logarithmic spectrum [4], in which the term duality was used to characterize the identities among arithmetic quantum theories. The above mixing is a special case of our relation 1 ∆ + s (β) = Z B (sβ)Z F (β).…”

“…where the Hamiltonian H B , is constructed out of certain operators χ k and r k such that (χ k ) s = (χ k †) s = 0 but no lower powers vanish as operators, i.e., these are parafermionic operators, the operators r k , r † k are the operators of the bosonic oscillators having frequencies with multiplicity s. These oscillators are the natural generalization of the even bosonic oscillators considered in the last section. Spector [4], in his construction of the bosonic Hamiltonian, he assumed that the operators given by r k = (b k ) s , r † k = (b † k ) s are bosonic operators. However, from the following explicite expressions of the commutators for s = 3, s = 4,…”

“…Using Maple the sequence corresponding to θ 4 (0,x 3 ) θ 4 (0,x) is, 1,2,4,6,10,16,24,36,52,74,104,144,198,268,360, · · · , 48672, 59122, · · · .…”

Riemannζfunction from wave-packet dynamics

“…(34)] of V (x; V 0 ) for V 0 → −∞. In this limit the argument of the error function becomes infinite and the error function takes on the value of unity.…”

“….. Such a spectrum has also been investigated [34] on the basis of partial supersymmetries. We start the iteration with the RKR potential V (0) discussed in Sec.…”