Fermion quasi-spherical harmonics

Abstract

Quasi-Spherical harmonics, Y m ℓ (θ, φ) are derived and presented for half-odd-integer values of ℓ and m. The form of the φ factor is identical to that in the case of integer ℓ and m: exp (imφ). However, the domain of these functions in the half-odd-integer case is 0≤φ<4π rather than the domain 0≤φ<2π in the case of integer ℓ and m (the true spherical harmonics). The form of the θ factor, P |m| ℓ (θ) (an associated Legendre function) is (as in the integer case) the factor (sin θ) |m| multiplied by a polynomial in cos θ of degree (ℓ−|m|) (an associated Legendre polynomial). A substantial difference between the spherical (integer ℓ and m) and quasi-spherical (half-odd-integer ℓ and m) Legendre functions is that the latter have an irrational factor of √ sin θ whereas the factor of the truly spherical functions is an integer power of sin θ. The domain of both the true and quasi spherical associated Legendre functions is the same: 0≤θ<π. A table of the Associated Legendre Functions is presented for both integer and half-odd-integer values of ℓ and m, for |m| = 0, 1 2 , 1 . . . 11 2 , and for (ℓ−|m|) = 0, 1, 2, 3, 4, 5. The table displays the similarity between the functions for integer ℓ and m (which are well known) and those for half-odd-integer ℓ and m (which have not been recognized previously).

Bilocal model for the relativistic spinning particle

“…(3.6). As can be easily verified, the relativistic model possesses two first class constraints 11) and four second class constraints P · ∆x = 0, ∆p · ∆x = 0, P · ∆p = 0, 2 (∆p) 2 − 2 s 2 (∆x) 2 = 0.…”

“…As in the previous section we can introduce "center of mass" 3 and relative displacement coordinates. In doing so it will be convenient to specialize to the case where the particles are of equal mass m 1 = m 2 = m, whence and φ ∈ [0, 2π] are the fermionic spherical harmonics Y m for ∈ N 2 , see [11][12][13]. In this case however the functionals cannot be understood as depending continuously on the sphere variables ∆x.…”

“…In this appendix we include a brief discussion on "fermionic spherical harmonics" Y m (θ, φ) which allow for half-integer values of m, , see [11,12]. We begin with the standard differential equation…”

Legendre functions of fractional degree: transformations and evaluations

“…As the reader can verify, they follow from homographic identities (of type M , above) by applying Whipple's transformation formula to both sides. Many (associated) Legendre functions of half-odd-integer degree and order, including P −1/2 1/2 , P −3/2 5/2 , P −5/2 9/2 , have been tabulated for use in quantum mechanics [3]. Up to phase factors (see (9a)), these are the same as P…”

“…8p 3 , and is invariant under R → −R, which is performed by p → −p. An associated prefactor function A = A(p), with limit unity when p → 1 and (L, R) → (1, 1), is…”