Amplitudes for spacetime regions and the quantum Zeno effect: pitfalls of standard path integral constructions

6th International Workshop DICE2012 Spacetime – Matter – Quantum Mechanics: From the Planck Scale to Emergent Phenomena

物理学;力学

Halliwell, J.J.^1 ; Yearsley, J.M.^2

Blackett Laboratory Imperial College, London SW7 2BZ, United Kingdom^1

Centre for Quantum Information and Foundations, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom^2

关键词: Classical limits;    Coarse Graining;    Hilbert subspace;    Measurement based model;    Quantum mechanical;    Quantum Zeno effect;    Space-time region;    Spatial regions;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/442/1/012018/pdf
DOI  :  10.1088/1742-6596/442/1/012018

学科分类：力学，机械学

来源: IOP

【 摘 要 】

Path integrals appear to offer natural and intuitively appealing methods for defining quantum-mechanical amplitudes for questions involving spacetime regions. For example, the amplitude for entering a spatial region during a given time interval is typically defined by summing over all paths between given initial and final points but restricting them to pass through the region at any time. We argue that there is, however, under very general conditions, a significant complication in such constructions. This is the fact that the concrete implementation of the restrictions on paths over an interval of time corresponds, in an operator language, to sharp monitoring at every moment of time in the given time interval. Such processes suffer from the quantum Zeno effect-the continual monitoring of a quantum system in a Hilbert subspace prevents its state from leaving that subspace. As a consequence, path integral amplitudes defined in this seemingly obvious way have physically and intuitively unreasonable properties and in particular, no sensible classical limit. In this paper we describe this frequently-occurring but little-appreciated phenomenon in some detail, showing clearly the connection with the quantum Zeno effect. We then show that it may be avoided by implementing the restriction on paths in the path integral in a "softer" way. The resulting amplitudes then involve a new coarse graining parameter, which may be taken to be a timescale Ε, describing the softening of the restrictions on the paths. We argue that the complications arising from the Zeno effect are then negligible as long as Ε >> 1/E, where E is the energy scale of the incoming state. Our criticisms of path integral constructions largely apply to approaches to quantum theory such as the decoherent histories approach or quantum measure theory, which do not specifically involve measurements. We address some criticisms of our approach by Sokolovksi, concerning the relevance of our results to measurement-based models.