More about path integrals for spin

Abstract

Path integral for the SU(2) spin system is reconsidered. We show that the Nielsen-Rohrlich(NR) formula is equivalent to the spin coherent state expression so that the phase space in the NR formalism is not topologically nontrivial. We also perform the WKB approximation in the NR formula and find that it gives the exact result. *

Coherent states over Grassmann manifolds and the WKB exactness in path integral

“…Even if other representation is adopted [19] for the CP 1 case to give the Nielsen-Rohrlich form [20], the same localization has been clarified [21]. As a natural generalization in this paper, we try to understand the same phenomena in the case of the Grassmann manifold, G N,n ≃ U (N)/( U (n)× U (N −n)).…”

“…The solution (2.20) takes the simplest form in the case of block-diagonal Hamiltonian given by B = 0: 21) where matrices V (t) ∈ U (n) and U(t) ∈ U (N − n) are given as…”

Multiperiodic coherent states and the Wentzel–Kramers–Brillouin exactness. II. Noncompact case and classical theories revisited

“…If we put x = ε in (A.6), we just obtain the formula. Also we can write (A.6) to the form which is the formula used in [14]. …”

“…With respect to the SU(2) spin, there is another expression, the Nielsen-Rohrlich formula [12], which is constructed in terms of the periodic coherent state [13]. We have shown the WKB-exactness of the Nielsen-Rohrlich formula [14] and its extension to U(N + 1) in terms of the "multi-periodic" coherent state [15] although their handling is more delicate than that of the generalized coherent state cases.…”

A Note on (Spin-) Coherent-State Path Integral

“…Part of the difficulties associated with CTSCSPI was previously noted by Funahashi et al [10,11] and by Schilling [12]. The discrete-time formalism was employed by Solari [13] who developed a general method of evaluating the fluctuation integral, and by Funahashi et al who evaluated the partition function for a single spin under a constant magnetic field [14].…”