Abstract
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.
Highlights
Consider the following question in non-relativistic quantum mechanics for a point particle in d dimensions: What is the amplitude g∆(xf, tf |x0, t0) for the particle to start at a spacetime point (x0, t0), pass through a spatial region ∆ and end at a spacetime point? The seemingly-obvious answer to this question is surely the path integral expression, g∆(xf, tf |x0, t0) = Dx(t) exp ∆ i tf dt t0 1 2 mx 2 − U (x) (1)(We choose units in which = 1)
Summary and Conclusions We have argued that amplitudes constructed by path integrals for questions involving time in a non-trivial way can, if implemented in the simplest and most obvious way, lead to problems due to the Zeno effect
This has the consequence that they do not have a sensible classical limit and have properties very different to those expected from the underlying intuitive picture
Summary
Consider the following question in non-relativistic quantum mechanics for a point particle in d dimensions: What is the amplitude g∆(xf , tf |x0, t0) for the particle to start at a spacetime point (x0, t0), pass through a spatial region ∆ and end at a spacetime point (xf , tf )? The seemingly-obvious answer to this question is surely the path integral expression, g∆(xf , tf |x0, t0) = Dx(t) exp ∆ i tf dt t0 1 2 mx 2 − U (x) (1). In this expression, the paths x(t) summed over satisfy the initial condition x(t0) = x0, the final condition x(tf ) = xf and pass, at any intermediate time, through the region ∆ [1, 2, 3], as depicted in Fig..
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