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Register a new userThe Klauder-Daubechies Construction of the Phase Space Path Integral and the Harmonic Oscillator
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Jan Govaerts
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The canonical operator quantisation formulation corresponding to the Klauder-Daubechies construction of the phase space path integral is considered. This formulation is explicitly applied and solved in the case of the harmonic oscillator, thereby illustrating in a manner complementary to Klauder and Daubechies original work some of the promising features offered by their construction of a quantum dynamics. The Klauder-Daubechies functional integral involves a regularisation parameter eventually taken to vanish, which defines a new physical time scale. When extrapolated to the field theory context, besides providing a new regularisation of short distance divergences, keeping a finite value for that time scale offers some tantalising prospects when it comes to strong gravitational quantum systems.
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I review the generating function for quantum-statistical mechanics, known as the Feynman-Vernon influence functional, the decoherence functional, or the Schwinger-Keldysh path integral. I describe a probability-conserving $ivarepsilon$ prescription from a path-integral implementation of Lindblad evolution. I also explain how to generalize the formalism to accommodate out-of-time-ordered correlators (OTOCs), leading to a Larkin-Ovchinnikov path integral. My goal is to provide step-by-step calculations of path integrals associated to the harmonic oscillator.
We give a superfield formulation of the path integral on an arbitrary curved phase space, with or without first class constraints. Canonical tranformations and BRST transformations enter in a unified manner. The superpartners of the original phase space variables precisely conspire to produce the correct path integral measure, as Pfaffian ghosts. When extended to the case of second-class constraints, the correct path integral measure is again reproduced after integrating over the superpartners. These results suggest that the superfield formulation is of first-principle nature.
Using a regularised construction of the phase space path integral due to Ingrid Daubechies and John Klauder which involves a time scale ultimately taken to vanish, and motivated by the general programme towards a noncommutative space(time) geometry, physical consequences of assuming this time parameter to provide rather a new fundamental time scale are explored in the context of the one dimensional harmonic oscillator. Some tantalising results are achieved, which raise intriguing prospects when extrapolated to the quantum field theory and gravitational contexts.
We consider the evolution of a quantum simple harmonic oscillator in a general Gaussian state under simultaneous time-continuous weak position and momentum measurements. We deduce the stochastic evolution equations for position and momentum expectation values and the covariance matrix elements from the systems characteristic function. By generalizing the Chantasri-Dressel-Jordan (CDJ) formalism (Chantasri et al.~2013 and 2015) to this continuous variable system, we construct its stochastic Hamiltonian and action. Action extremization gives us the equations for the most-likely readout paths and quantum trajectories. For steady states of the covariance matrix elements, the analytical solutions for these most-likely paths are obtained. Using the CDJ formalism we calculate final state probability densities exactly starting from any initial state. We also demonstrate the agreement between the optimal path solutions and the averages of simulated clustered stochastic trajectories. Our results provide insights into the time dependence of the mechanical energy of the system during the measurement process, motivating their importance for quantum measurement engine/refrigerator experiments.
The energy-spectrum of two point-like particles interacting in a 3-D isotropic Harmonic Oscillator (H.O.) trap is related to the free scattering phase-shifts $delta$ of the particles by a formula first published by Busch et al. It is here used to find an expression for the it shift rm of the energy levels, caused by the interaction, rather than the perturbed spectrum itself. In the limit of high energy (large quantum number $n$ of the H.O.) this shift is shown to be given by $-2frac{delta}{pi}$, also valid in the limit of infinite as well as zero scattering length at all H.O. energies. Numerical investigation shows that the shifts differ from the exact result of Busch et al, by less than $<frac{1}{2}%$ except for $n=0$ when it can be as large as $approx 2.5%$. This approximation for the energy-shift is well known from another exactly solvable model, namely that of two particles interacting in a spherical infinite square-well trap (or box) of radius $R$ in the limit $Rrightarrow infty$, and/or in the limit of large energy. It is in this context referred to as the it phase-shift approximation rm. It can be (and has been) used in (infinite) nuclear matter calculations to calculate the two-body effective interaction in situations where in-medium effects can be neglected. It has also been used in expressing the energy of free electrons in a metal.
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