LyX Document

1 Appendix

This appendix is intended to be techincal in nature.

1.1 “If you calculate this, it turns out to be exactly the same probability as you calculate before, the intermediate position is accounted for since you already considered every possible configuration”

This can be seen by examining how the propagator is constructed, e.g. Feyman's original paper on the subject [1] or any thorough introduction to the material [2]. We consider all paths from the staring position S to the final position F. Any intermediate position P will already be covered. If we ask the probability of going from S to F through P, but without perfoming a measurement, we need to consider all paths from S to P and then all paths from P to F. However, this will include paths that start at S, go to some other arbitrary position P' and instantaneously travel through P before going back to P'. This is, of course, already included in the original calculation.

We can more explicitly see that going though an intermediate position P does not effect the final result. Consider a propagator of the form [3] eq. 11.9

Δ (S - F) = \int \frac{d q}{k} e^{- i q (S - F)}

where q is momentum. If we introduce the intermediate position P we obtain

Δ (S - P - F) = \int \frac{d q}{k} e^{- i q (S - P)} e^{- i q (P - F)} = \int \frac{d q}{k} e^{- i q (S - F)}

i.e. we recover the original equation.

To the best of my knowledge this holds true for all propagators, see for example. [3] eqs. 11.119 and11.177.

1.2 “In fact, the other configuration does not need to be at an intermediate time, it could also be to the future of the final state: we can look at the possibility of going from our initial state way into the future and then back in time to our final state where the measurement is made”

In the Feynman propagator all paths are considered. The paths for virtual particles mathematically travel backwards in time. Therefore, the inclusion of virtual particles necessarily means including configurations that are in the past of the initial position S. This would only violate causality if a measurment were performed at such a time. In the main text I am arguing that such measurements define causality rather than the numerical value of the time coordinate, with only the duration between measurements being a physically meaningful quantity in the propagator.

2 References

[1] Space-Time Approach to Quantum Mechanics (1949) / Richard P. Feymnan

[2] Path Integral Methods and Applications (2000) / Richard MacKenzie

[3] Elementary Particle Physics Volume 1 (2010) / Yorikiyo Nagashima