Amplitudes for Spacetime Regions and the Quantum Zeno Effect: Pitfalls of Standard Path Integral Constructions

Abstract

We argue that under very general conditions, there is a significant complication in amplitudes for spacetimes regions constructed from path integrals. 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 $\eps$, 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 $\eps >> 1/ E$, 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.

To appear in proceedings of the 2012 DICE Conference (edited by T.Elze). arXiv admin note: substantial text overlap with arXiv:1205.3773

References in corpus (9)

• A Straightforward Introduction to Continuous Quantum Measurement
• The Classical Universes of the No-Boundary Quantum State
• The No-Boundary Measure of the Universe
• Consistent Probabilities in Wheeler-DeWitt Quantum Cosmology
• Disclosing hidden information in the quantum Zeno effect: Pulsed measurement of the quantum time of arrival
• Hilbert Spaces from Path Integrals
• Pitfalls of Path Integrals: Amplitudes for Spacetime Regions and the Quantum Zeno Effect
• Spacetime Coarse Grainings in the Decoherent Histories Approach to Quantum Theory
• Quantum Zeno Effect in the Decoherent Histories

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