Narrow Results

For two operators $X$ and $Y,$ which obey

We exploit the ideas of spin-statistics theorem, normal-ordering and the key concepts behind the symmetry principles to derive the canonical (anti)commutators for the case of a one $(0+1)$-dimensional (1D) ${𝒩}=2$ supersymmetric (SUSY) harmonic oscillator (HO) without taking the help of the mathematical definition of canonical conjugate momenta with respect to the bosonic and fermionic variables of this toy model for the Hodge theory (where the continuous and discrete symmetries of the theory provide the physical realizations of the de Rham cohomological operators of differential geometry). In our present endeavor, it is the full set of continuous symmetries and their corresponding generators that lead to the derivation of basic (anti)commutators amongst the creation and annihilation operators that appear in the normal mode expansions of the dynamical fermionic and bosonic variables of our present ${𝒩}=2$ SUSY theory of a HO. These basic brackets are in complete agreement with such kind of brackets that are derived from the standard canonical method of quantization scheme.

The first normal ordering problem involves bosonic harmonic oscillator creation and annihilation operators (Heisenberg algebra). It is related to the problem of finding the finite transformation generated by L_k−1 := −z^k ∂_z, k ∈ ℤ, z ∈ ℂ (conformal algebra generators). It can be formulated in terms of a subclass of Sheffer polynomials called Jabotinsky polynomials. The coefficients of these polynomials furnish generalized Stirling number triangles of the second kind, called S2(k;n,m) for k ∈ ℤ. Generalized Stirling-numbers of the first kind, S1(k;n, m) are also defined.

The second normal ordering problem appears in thermo-field dynamics for the harmonic Bose oscillator. Again Sheffer polynomials appear. They relate to Euler numbers and iterated sums of squares. In a different approach to this problem one solves the differential-difference equation f_n+1 = f'_n + n² f_n-1, n > = 1, with certain inputs f₀ and f₁ = f'₀.

In this case the integer coefficients of the special Sheffer polynomials which emerge have an interpretation as sum over multinomials for some subset of partitions.