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Nature's Capacities and Their Measurement$

Nancy Cartwright

Print publication date: 1994

Print ISBN-13: 9780198235071

Published to British Academy Scholarship Online: November 2003

DOI: 10.1093/0198235070.001.0001

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(p.253) Appendix II Do Quantum Causes Propagate?

(p.253) Appendix II Do Quantum Causes Propagate?

Source:
Nature's Capacities and Their Measurement
Publisher:
Oxford University Press

The arrangement for a two‐slit experiment is described in section 6.7. There it was claimed that the quantum‐mechanically predicted results show that the propagation requirement is not satisfied in the two‐slit experiment. To see why, consider a region on the recording screen with low probability in the two‐slit experiment, but high probability with only one slit open—say, slit one. Assume that some feature c operating at t 0 in the source is partially responsible for a positive result, r, at t 2 in the selected region of the screen. By the requirement for propagation, there must be some c r operating in the region of r, just before t 2, which has been caused by c and which is part of the cause of r. As with EPR, suppose that the measuring apparatus state, m r, also contributes. Then

r ( t 2 ) a r ( t 2 Δ t ) · c ^ r ( t 2 Δ t ) · m ^ r ( t 2 Δ t ) v u r
(1)
The causal influence c r must propagate to r from c, and since it can appear only where the quantum state is non‐zero, it—or some causal antecedent to it—must appear either at slit one at t 1 when the beam passes the wall, or at slit two. Call the causal antecedent at slit one, c 1, and at slit two, c 2. Let s 1 denote the state of the first slit when it is open at t 1; s 2, the second. Allowing that s 1 and s 2 may themselves affect the propagating influence, and assuming for simplicity that there are no further internal contributions, gives the following structure:
c ^ r ( t 2 Δ t ) a 1 ( t 1 + Δ t ) · c ^ 1 ( t 1 + Δ t ) v a ^ 2 ( t 1 + Δ t ) · c ^ 2 ( t 1 + Δ t ) v v r
(2)
c ^ 1 ( t 1 + Δ t ) a ^ 1 ( t 1 ) · c ^ 1 ( t 1 ) · s ^ 1 ( t 1 ) v w 1
c ^ 2 ( t 1 + Δ t ) a ^ 2 ( t 1 ) · c ^ 2 ( t 1 ) · s ^ 2 ( t 1 ) v w 2
(3)
c ^ 1 ( t 1 ) a ^ 1 ( t 0 ) · c ^ ( t 0 ) v z 1
c ^ 2 ( t 1 ) a ^ 2 ( t 0 ) · c ^ ( t 0 ) v z 2
(4)
s ^ 1 ( t 1 ) u 1
s ^ 2 ( t 1 ) u 2
(5)
c ^ ( t 0 ) u 0
(6)
m ^ r ( t 2 Δ t ) v m
(7)
Equation (3) contains an important and not yet noted assumption about the causal structure: s 2 appears nowhere in the equation for c 1 nor s 1 in the equation for c 2. This reflects the conventional assumption that the opening or closing of a distant slit has no effect at the other. If it were to do so, the (p.254) general constraint on structural models that causes must precede their effects would require the distant opening or closing to occur earlier than its effect, and the demand for propagation requires that some capacity‐bearing signal should move along a continuous path between.

What is the probability for r to occur, with both slits open? Dropping the time indices, and omitting all terms in which the errors figure:

P ( c ^ r / s ^ 1 · s ^ 2 ) P ( a ^ 1 c ^ 1 / s ^ 1 · s ^ 2 ) + P ( a ^ 2 · / s ^ 1 · s ^ 2 ) P ( a ^ 1 · a ^ 2 · c ^ 1 · c ^ 2 / s ^ 1 · s ^ 2 ) larger of { P ( a ^ 1 · c ^ 1 / s ^ 1 · s ^ 2 ) , P ( a ^ 2 · c ^ 2 / s ^ 1 · s ^ 2 ) }
Let the larger be P(â″1· ĉ′11· ŝ2). With the usual assumptions that the operations are independent of the causes and the errors of each other
P ( a ^ 1 · c ^ 1 / s ^ 1 · s ^ 2 ) P ( a ^ 1 · a ^ 1 · a ^ 1 ) P ( u 0 / u 1 · u 2 ) = P ( a ^ 1 · a ^ 1 · a ^ 1 ) P ( u 0 / u 1 · ¬ u 2 ) = P ( a ^ 1 · c ^ 1 / s ^ 1 · ¬ s ^ 2 ) P ( c ^ r / s ^ 1 · ¬ s ^ 2 )
It follows that
P ( r ^ / s ^ 1 · s ^ 2 ) P ( a ^ r ) P ( m ^ r ) P ( c ^ r / s ^ 1 · s ^ 2 ) P ( a ^ r ) P ( m ^ r ) P ( c ^ r / s ^ 1 · ¬ s ^ 2 ) P ( r ^ / s ^ 1 · ¬ s ^ 2 )
Hence the probability for a particle to register at r with both slits open is at least as large as the larger of the probabilities with only one slit open. This is contrary to what happens.