Bootstraping Quantum Particles and their Bounds

• URL: http://arxiv.org/abs/2208.09370v2
• Date: Tue, 30 Aug 2022 14:47:31 GMT
• Title: Bootstraping Quantum Particles and their Bounds
• Authors: Takeshi Morita
• Abstract summary: We show that the bootstrap method can be regarded as a generalization of the uncertainty relations. We argue an application of the bootstrap method to thermal equilibrium states.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The range of motion of a particle with certain energy $E$ confined in a potential is determined from the energy conservation law in classical mechanics. The counterpart of this question in quantum mechanics can be thought of as what the possible range of the expectation values of the position operator $ \langle x \rangle$ of a particle, which satisfies $E= \langle H \rangle$. This range would change depending on the state of the particle, but the universal upper and lower bounds, which is independent of the state, must exist. In this study, we show that these bounds can be derived by using the bootstrap method. We also point out that the bootstrap method can be regarded as a generalization of the uncertainty relations, and it means that the bounds are determined by the uncertainty relations in a broad sense. Furthermore, the bounds on possible expectation values of various quantities other than position can be determined in the same way. However, in the case of multiple identical particles (bosons and fermions), we find some difficulty in the bootstrap method. Because of this issue, the predictive power of the bootstrap method in multi-particle systems is limited in the derivation of observables including energy eigenstates. In addition, we argue an application of the bootstrap method to thermal equilibrium states. We find serious issues that temperature and entropy cannot be handled. Although we have these issues, we can derive some quantities in microcanonical ensembles of integrable systems governed by generalized Gibbs ensembles.

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