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

Abstract

It is pointed out that there are some fundamental difficulties with the frequently used continuous-time formalism of the spin-coherent-state path integral. They arise already in a single-spin system and at the level of the "classical action" not to speak of fluctuations around the "classical path". Similar difficulties turn out to be present in the case of the (boson-)coherentstate path integral as well; although partially circumventable by an ingenious trick (Klauder's ǫ-prescription) at the "classical level", they manifest themselves at the level of fluctuations. Detailed analysis of the origin of these difficulties makes it clear that the only way of avoiding them is to work with the proper discrete-time formalism. The thesis is explicitly illustrated with a harmonic oscillator and a spin under a constant magnetic field.

Semiclassical dynamics of a spin-½ in an arbitrary magnetic field

“…In most of these studies a spin in a constant magnetic field is considered which allows for an explicit solution. Other work [10]- [15] allowing for time-dependent fields examines the discrete time-lattice version of the path integral, and it is usually concluded [16,17] that only formal calculations are possible once the continuous path integral is employed.…”

Generalized Jaynes-Cummings Model with a Pseudo-Hermitian: A Path Integral Approach

“…Specifically when the Hamiltonian is quadratic in the generator of the algebra used to construct the coherent states, the path integral fails to reproduce the correct results obtained through an operator approach [20]. Detailed analysis of the origin of these difficulties makes it clear that the only way to avoid them is by working with the proper discrete-time formalism [21].…”

Method of collective degrees of freedom in spin-coherent-state path integral

“…The transition amplitude between the initial state |ξ I and the final state |ξ F can be expressed as a spin-coherent-state path integral in the real discrete-time formalism by the standard procedure of the repeated use of the resolution of unity (see, e.g., Ref. 15 on which the present notation is based):…”

“…As a theoretical technique to evaluate the quantum dynamics of the domain wall, the spin coherent state path integral in the continuous-time formalism [12] is frequently used. However, as noted by some workers [13,14], it has some fundamental difficulties, which have been recently discussed in detail [15]. Furthermore, it is liable to lead to confusion concerning the interpretation of the collective degrees of freedom as has been pointed out in Ref.…”