In his work "The Consistent Histories Approach to Quantum Mechanics" publish in the Stanford Encyclopedia of Philosophy, Griffiths claims that this approach overcomes the problem of the wave-function "collapse".
His suggestion is that in each trial of an experiment, a quantum system follows a "history" meaning a succession of states.
Here is an example: consider a Mach-Zehnder interferometer with an input beam-splitter BSi, and an output beam-splitter BSo, both transmitting and reflecting in equal proportion. The outputs of BSi are denoted b and c, and those of BSo, e and f. A single-particle wave-packet |a> impinging on BSi is split as follows
(1) |a> → (1/√2)( |c> + |d>).
Before impinging on BSo the wave-packets |c> and |d> have accumulates phases
(2) (1/√2)( |c> + |d>) → (1/√2)[exp(iϕc)|c> + exp(iϕd)|d>],
and BSo induces the transformation
(3) (1/√2)[exp(iϕc)|c> + exp(iϕd)|d>] → α|e> + β|f>,
where the amplitudes α and β depend on the phases ϕc and ϕd.
In his book "Consistent quantum theory" chapter 13, Griffiths indicates two possible histories:
(4.1) |a> → (1/√2)( |c> + |d>) → (1/√2)[exp(iϕc)|c> + exp(iϕd)|d>] → |e>,
(4.2) |a> → (1/√2)( |c> + |d>) → (1/√2)[exp(iϕc)|c> + exp(iϕd)|d>] → |f>,
the history (4.1) occurring with probability |α|2, and the history (4.2) with probability |β|2.
Does somebody understand in which way these histories avoid the collapse postulate?
The correct transformation at BSo is (3), a unitary transformation, not (4.1) and not (4.2). Each one of the histories (4.1) and (4.2) involves a truncation of the wave-function at BSo. But this is exactly the mathematical expression of the collapse principle: truncation of the wave-function.
Hence my question: can somebody tell me how is it possible to claim that these histories avoid the collapse postulate?
Nobody understands Griffiths. But if there is some need to explain, I believe that his view is that there could be multiplicity of projection operators of the state
\psi=1/SQRT(2) (|a> phase factor_a + |b> phase factor_b) on the final state but only two are consistent with observations. Still this explanation is based on assumption of zero coherence between channels |e> and |f>.
Dear Peter,
The fact that the two histories described by Griffiths' contain a coherent superpossition of |a> and |b>, but end either strictly on e or strictly on f - as you pointed out - means "collapse" of the wave-function. And by Griffiths, this collapse occurs "out of the blue", before meeting detectors.
Though, Griffiths makes the VERY STRANGE declaration that his interpretation AVOIDS the collapse. I really wonder why Griffiths does not look at the meaning of his formulas.
There is an additional pair of possible histories that Griffiths indicates in his "Consistent quantum theory", chapter 13. Continuing the numbering in the text of the question,
(4.3) |a> → |c> → (1/√2)( |e> + |f>),
(4.4) |a> → |d> → (1/√2)(-|e> + |f>).
Griffiths avoids telling us what happens when there are phase-shifts ϕc and ϕd. No doubt that these shifts have an influence on the probability of the histories (4.1) and (4.2).
But the situation is more severe. As Griffits claims that the histories approach removes the need of the collapse hypothesis, it is not clear how can one get from the superpositions ( |e> + |f>) and (-|e> + |f>), a detection either on the path e, or on the path f, without invoking the collapse of the superposition on the state |e> or on the state |f>.
I cannot see how this is avoided.
Recently I find that the measurement axiom or collapse hypothesis is the only way to keep sanity with QM, so I hope no one debunks it.
No half dead and half alive cats....either dead or alive.
I only suppose a measurement can give you either e > or f>
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