On 24 Jul 2014, at 19:40, meekerdb wrote:
This may clarify (or provoke) discussion of Moscow vs. Washington.
It's interesting that Carroll and Sebens use FPI and Sean says it
increases his confidence in Everett's MWI. But in his penultimate
paragraph he essentially lays out an endorsement of Fuchs QBism,
which is generally seen as the instrumentalist alternative to MWI.
The universal machine does that too.
Eventually all mode of existence are psychological, as you can guess
by interpreting physics as the inside view of the arithmetical FPI
undetermined machine. The "observable" (roughly the []p & <>p (& p)
hypostases (with p sigma_1) are "mental", or "machine self-
referencial" modalities.
We just don't know yet if those dreams glue enough to determine a
multiverse or a multi-multiverse, has filtered from what at the start
his a giant web of (machine) dreams emulated in arithmetic.
Nice post. I like Born, even when wrong. I appreciate the Born-
Einstein dialog.
Nice way to call the Copenhagen theory: the theory of disappearing
universes, it is already closer to the brain filtration of realities,
or consciousness differentiation.
Bruno
Brent
-------- Original Message --------
Subject: [New post] Why Probability in Quantum Mechanics is Given by
the Wave Function Squared
Date: Thu, 24 Jul 2014 15:21:04 +0000
From: Sean Carroll <[email protected]>
To: [email protected]
New post on Sean Carroll
Why Probability in Quantum Mechanics is Given by the Wave Function
Squared
by Sean Carroll
One of the most profound and mysterious principles in all of physics
is the Born Rule, named after Max Born. In quantum mechanics,
particles don't have classical properties like "position" or
"momentum"; rather, there is a wave function that assigns a
(complex) number, called the "amplitude," to each possible
measurement outcome. The Born Rule is then very simple: it says that
the probability of obtaining any possible measurement outcome is
equal to the square of the corresponding amplitude. (The wave
function is just the set of all the amplitudes.)
Born Rule:
The Born Rule is certainly correct, as far as all of our
experimental efforts have been able to discern. But why? Born
himself kind of stumbled onto his Rule. Here is an excerpt from his
1926 paper:
That's right. Born's paper was rejected at first, and when it was
later accepted by another journal, he didn't even get the Born Rule
right. At first he said the probability
was equal to the
amplitude, and only in an added footnote did he correct it to being
the amplitude squared. And a good thing, too, since amplitudes can
be negative or even imaginary!
The status of the Born Rule depends greatly on one's preferred
formulation of quantum mechanics. When we teach quantum mechanics to
undergraduate physics majors, we generally give them a list of
postulates that goes something like this:
Quantum states are represented by wave functions, which are vectors
in a mathematical space called Hilbert space.
Wave functions evolve in time according to the Schrödinger equation.
The act of measuring a quantum system returns a number, known as the
eigenvalue of the quantity being measured.
The probability of getting any particular eigenvalue is equal to the
square of the amplitude for that eigenvalue.
After the measurement is performed, the wave function "collapses" to
a new state in which the wave function is localized precisely on the
observed eigenvalue (as opposed to being in a superposition of many
different possibilities).
It's an ungainly mess, we all agree. You see that the Born Rule is
simply postulated right there, as #4. Perhaps we can do better.
Of course we can do better, since "textbook quantum mechanics" is an
embarrassment.
There are other formulations, and you know that my own favorite is
Everettian ("Many-Worlds") quantum mechanics. (I'm sorry I was too
busy to contribute to the active comment thread on that post. On the
other hand, a vanishingly small percentage of the 200+ comments
actually addressed the point of the article, which was that the
potential for many worlds is automatically there in the wave
function no matter what formulation you favor. Everett simply takes
them seriously, while alternatives need to go to extra efforts to
erase them. As Ted Bunn argues, Everett is just "quantum mechanics,"
while collapse formulations should be called "disappearing-worlds
interpretations.")
Like the textbook formulation, Everettian quantum mechanics also
comes with a list of postulates. Here it is:
Quantum states are represented by wave functions, which are vectors
in a mathematical space called Hilbert space.
Wave functions evolve in time according to the Schrödinger equation.
That's it! Quite a bit simpler -- and the two postulates are exactly
the same as the first two of the textbook approach. Everett, in
other words, is claiming that all the weird stuff about
"measurement" and "wave function collapse" in the conventional way
of thinking about quantum mechanics isn't something we need to add
on; it comes out automatically from the formalism.
The trickiest thing to extract from the formalism is the Born Rule.
That's what Charles ("Chip") Sebens and I tackled in our recent
paper: Read more of this post
Sean Carroll | July 24, 2014 at 8:19 am | Categories: arxiv,
Philosophy, Science | URL: http://wp.me/p2WMeM-38Y
Comment See all comments
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