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In article <BxFI6v.H04@well.sf.ca.us> sarfatti@well.sf.ca.us 
(Jack Sarfatti) writes:
>* The important point is that we must not assume that
>
>|a,e,+><a,e,+| + |a,o,+><a,o,+| = 1
>
>That is the mistake that led to the bogus "missing factor of 2" and which
>is still hanging you up Ramsay.

Not at all. You're making guesses about what I'm thinking-- I never
said anything of the kind. There never was any "missing factor of 2"--
this is your phrase.

>In fact, it is no contradiction to say that
>
>|<a,e,+|h>| = |<a,o,+|h>| = 1
>
>because the geometry is such that both e and o beams go to h with
>certainty.  Therefore, everything is unitary locally on each end!

It *is* a contradiction to claim all of these things. In order for the
evolution to be unitary, we must have <a,e,+|a,o,+> = 0. Since
|a,e,+>, |a,o,+>, and |h> are all states, they must be normalized:
hence <a,e,+|a,e,+> = <a,o,+|a,o,+> = <h|h> = 1. Now you want to claim
that it is possible to have |<a,e,+|h>| = |<a,o,+|h>| = 1 as well. It
is not possible to have all of those relationships hold among three
ket-vectors |a,e,+>, |a,o,+> and |h>.

Why? The simple answer, if you know the vector geometry, is that the
only way for two normalized vectors to satisfy |<a,e,+|h>|=1 is for
the one, |a,e,+>, to equal e^{it}|h> for some phase angle t. I.e.,
they represent the same state with a different phase. For this to hold
for both |a,e,+> and |a,o,+> with |h> would imply that |a,e,+> and
|a,o,+> are such multiples of |h>, and thus of each other. But this
contradicts the orthogonality <a,e,+|a,o,+>=0.

Physically, what this means is that the only way to ensure that two
states are "cofinal" in this way is for them to differ only by a phase
in the first place. This is the physical principle you have to make
your peace with, Sarfatti.

Which of those claims do you now want to deny?

Your optics is simply wrong; you can't just claim that there is one
state |h> at the end because it looks to you like there should be.
There are many possible states for the photon to be in at the end.
You can't just say, "the beams combine coherently" because you hope
that it can be done by focusing them well. Coherence is not so
trivially obtained, and the mistake of thinking it *can* be will
manifest itself directly in further erroneous calculations.

Keith Ramsay
ramsay@unixg.ubc.ca