Is the Decoherent Histories Approach Consistent?
One particular approach of interpreting quantum mechanics is Decoherent Histories. All major non-Copenhagen approaches have serious issues:
- MWI has the issue of the very meaning of probability and without a non-circular derivation (impossible in my opinion) of Born rule does not qualify for anything but a "work in progress" status.
-GRW-type theories make different predictions than quantum mechanics which soon will be confirmed or rejected by ongoing experiments. (My bet is on rejection since later GRW versions tuned their free parameters to avoid collision with known experimental facts instead of making a falsifiable prediction)
-Bohmian approach has issues with "surreal trajectories" which invalidates their only hard ontic claim: the position of the particle.
Now onto Decoherent Histories. I did not closely follow this approach and I cannot state for sure if there are genuine issues here, but I can present the debate. On one hand, Robert Griffiths states:
"What is different is that by employing suitable families of histories one can show that measurement actually measure something that is there, rather than producing a mysterious collapse of the wave function"
On the other hand he states:
"Any description of the properties of an isolated physical system must consists of propositions belonging together to a common consistent logic" - in other words he introduces contextuality.
Critics of decoherent (or consistent) histories use examples which are locally consistent but globally inconsistent to criticize the interpretation.
Here is an example by Goldstein (other examples are known). The example can be found in Bricmont's recent book: Making Sense of Quantum Mechanics on page 231. Consider two particles and two basis for a two-dimensional spin base \((|e_1\rangle\, |e_2\rangle), (|f_1\rangle, |f_2\rangle))\) and consider the following state:
\(|\Psi\rangle = a |e_1\rangle|f_2\rangle + a|e_2\rangle|f_1\rangle - b |e_1\rangle|f_1\rangle\)
Then consider four measurements A, B, C, D corresponding to projectors on four vectors, respectively: \(|h\rangle, |g\rangle, |e_2\rangle, |f_2\rangle\) where:
\(|g\rangle = c|e_1\rangle + d|e_2\rangle\)
\(|h\rangle = c|f_1\rangle + d|f_2\rangle\)
Then we have the following properties:
(1) A and C can be measured simultaneously, and if A=1 then C=1
(2) B and D can be measured simultaneously, and if B=1 then D=1
(3) C and D can be measured simultaneously, but we never get both C and D = 1
(4) A and B can be measured simultaneously, and sometimes we get both A and B = 1
However all 4 statements cannot be true at the same time: when A=B=1 as in (4) then by (1) and (2) C=D=1 and this contradicts (3).
So what is going on here? The mathematical formalism of decoherent histories is correct as they predict nothing different than standard quantum mechanics. The interpretation assigns probabilities to events weather we observe them or not, but does it only after taking into account contextuality. Is this a mortal sin of the approach? Nature is contextual and I don't get the point of the criticism. The interpretation would be incorrect if it does not take into account contextuality. Again, I am not an expert of this approach and I cannot offer a definite conclusion, but to state my bias I like the approach and my gut feeling is that the criticism is without merit.
PS: I'll be going on vacation soon and my next post will be delayed: I will skip a week.