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An investigation of coherent state path integrals as applied to a harmonic oscillator and a single spin

L. C. Lew Yan Voon

Published Jan 1, 2010• 2 citations
Publisher

Abstract

In this project two steps involved in the handling of path integrals are reexamined in detail for coherent state path integals. They concern the continuum limit approximation and the regularization of the formal path integrals. Restricting oneself to the harmonic oscillator, the technique of time splitting is used to set up the coherent state path integrals and the proper way to pass to the continuum limit is demonstrated. The manipulation of these path integrals calls for regularization procedures and the validity of discrete, Riemann zeta function and 'derivative' regularization methods is observed. A modification to a fermionic theory is briefly mentioned and, finally, the above results are implemented in writing down a path integral for a single spin.

DOI
10.14288/1.0085053

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In this project two steps involved in the handling of path integrals are reexamined in detail for coherent state path integals. They concern the continuum limit approximation and the regularization of the formal path integrals. Restricting oneself to the harmonic oscillator, the technique of time splitting is used to set up the coherent state path integrals and the proper way to pass to the continuum limit is demonstrated. The manipulation of these path integrals calls for regularization procedures and the validity of discrete, Riemann zeta function and 'derivative' regularization methods is observed. A modification to a fermionic theory is briefly mentioned and, finally, the above results are implemented in writing down a path integral for a single spin.
Definition and formulas for harmonic oscillator coherent states and spin coherent states are reviewed in detail. The path integral formalism and its relation with the partition function of a system are also reviewed. The harmonic oscillator coherent state path integral is evaluated exactly at the discrete level and then used to find its continuum limit using various regularizations. The computation of the path integral for a particle of spin s put in a constant magnetic field is carried out using harmonic oscillator coherent states and spin coherent states, with a careful analysis of infinitesimal terms (in 1/N where N is the number of time slices) appearing in the Lagrangian. A mapping of the spin system into a CP1 model is shown explicitly. The theory of a spinless particle in the field of a magnetic monopole and its relation with the spin system are explained. The equivalence of these two models is established up to infinitesimal order by the introduction of an external field correction. This gives a new representation of a coherent state path integral in terms of a more familiar Feynman path integral.
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-)coherent-state path integral as well; although partially circumventable by an ingenious trick (Klauder's ∊-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.
Coherent states can be used for diverse applications in quantum physics including the construction of coherent state path integrals. Most definitions make use of a lattice regularization; however, recent definitions employ a continuous-time regularization that may involve a Wiener measure concentrated on continuous phase space paths. The introduction of constraints is both natural and economical in coherent state path integrals involving only the dynamical and Lagrange multiplier variables. A preliminary indication of how these procedures may possibly be applied to quantum gravity is briefly discussed.
The paper solves the problem of continuum limit in bosonic thermal coherent-state path integrals. For this purpose, exact discrete versions of the path integral are constructed for three different orderings of the Hamiltonian: normal, anti-normal and symmetric (Weyl order). Subsequently, their different continuum versions are checked on the harmonic oscillator, to choose the symmetric ordering as a possibly correct choice for all polynomial Hamiltonians. Spotted mathematical subtleties in the simple case serve as a clue to the general solution. Finally, a general justification for the symmetric order is provided by deriving the continuum path integral starting from the exact discrete case using a renormalization procedure in the imaginary time frequency domain. While the role of Weyl order has already been found, the paper provides the missing proof of its suitability for every polynomial Hamiltonian and simplifies the previously established construction by referring only to creation and annihilation operators (without position and momentum operators).