They say the more you know, the more you realize you don’t know. This is very true. Take teenagers who think they know everything (for instance)… idiots, on marijuana. Then take some wise old sage at the end of their days and they’ll tell you that contentment of knowledge can only come from accepting what you don’t know…. because you’ll never figure it all out. EVER.
Well, in the meantime, I love to fill my head with sundry and sometimes excruciating concepts of the physical world. Here’s a few links to some migraines waiting to happen.
John Baez (not to be confused with Joan) from the University of California at Riverside Math Department has compiled a plotted graph of all the roots of all polynomials of degree greater than or equal to 5 with integer coefficients ranging from -4 to 4. [Link here] It’s intriguing to see the patterns that the computed roots follow when graphed. There’s is definitely inherent order and even art in the mathematics involved.
More directly related to art is a blog entry I was reading on Mahndisa’s blog. She was talking about a favorite music artist that composes and records music using the Just Intonation scale. [the what?] It is an alternate scale tuning in which the notes’ frequencies are related to one another with whole integer fractions. [what?] One reason this really hasn’t been successfully used since the Middle Ages (except in modern ethereal New Age music) is because of the terribly dissonant wolf intervals it can create. [the what?] Here’s a little tutorial (with a lot of math) explaining why we have a 12-tone diatonic note set and how standard tuning (Equal Temperament) differs from just intonation. So remember, when playing very old music a G# might be different from an Ab. [huh?!]
And if your head hasn’t spun off, just read up on the uncertainty principle found in quantum mechanics. Try grasping the implications of the EPR paradox on Einstein’s theory of special relativity, especially the implications caused by quantum entanglement where information is transmitted between distant particles instantaneously (read: greater than the speed of light).
Good stuff.
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04 05 06
Hey Steve:
Thanks for the link. I am in SF right now and was just checking up and I saw this. You are so curious and it is so cool that you have checked out Mr. Baez. I won’t be able to have a conversation with him for a while yet! I particularly like his stuff on manifolds and the background independence problem and LQG. However, regarding entanglement, we must be quite careful in defining INFORMATION. For this reason, I will do a post explaining why no classical information goes faster than the speed of light. Think this: entangled degrees of freedom and how the wavefunction is represented. Also consider the process o f measurement and what that means…More to come. Great article and Warmest Regards.
PS I am not able to blog as much as I used to in terms of commenting because I am swamped with life. Just know that you have readers that don’t always comment:) Take Care,.
mahndisa :: Hey, Thanks for stoppin’ in. I’ll ponder the thought of classical information not traveling faster than light.
04 06 06
Hey there Waraxe:
I was thinking of sending you an email of this information because I am too screwy to post today. Since I don’t have it, I will insert the note here. Right Wing Prof asked me about this not too long ago. Since his question is along the same lines as this issue, I figured I would share it with you and not reinvent the wheel. Warmest Regards Steve:) If you wish to discuss this further, send me an email. I am going back to bed; I just got my wisdom teeth removed! ARGHHH!:)
03 21 06
As a result, measurements performed on one system seem
to be instantaneously influencing other systems
entangled with it. Nevertheless, classical information
cannot be transmitted through entanglement faster than
the speed of light-Wikipedia/Quantum Entanglement.
“Now, explain that last sentence, please. If, say, I
create two particles with opposite spin, and send one
to, wherever, Alpha Centauri, then change the spin on
the remaining particle to transmit a binary message,
the spin of the particle in the Alpha Centauri system
will change, and transmit the message, yes? After a
certain point, Quantum theory is just a brainfuck,
pardon the French”
Hey Prof:
Thanks for the note. I am not a physicist, yet. I am a
student of physics. I plan to finish up my degree
sometime this year, as I took time off. As you can see
I really love it, but I don’t want you to think I am
an expert. I can only say what I know and what I have
learned up to this point, so thanks for the note and
thought.ï First, it would seem that your
understanding of entanglement presupposes that the
spin of particle 1 can be changed without affecting
the spin of particle 2. Let me explain why that cannot
happen if the two particles are entangled. A good
example of two particle entanglement is pair
production.
PAIR PRODUCTION
A photon (in the gamma energy range) scatters off of a
nucleus of an atom and annihilates. This process
produces two electrons with the same magnitude of (but
oppositely directed momentum) and energy. Because of
the Pauli Exclusion Principle, the electrons must be
in two different spin states since they share the same
energy level. Electron 1 might be spin up, and
electron 2 spin down and vise versa. Due to the
process of their production, they are coupled in some
operations (in this case, their spin-z eigenstates).
This means that their wave functions are correlated.
In fact, one cannot reference an entangled property of
electron 1 without referencing that entangled property
in electron 2. I am being very careful when I specify
‘entangled property’. This is because the
particles may be correlated in some degrees of
freedom, but not all of them. So for example, the
spins may be coupled in their spin-z orientation, but
not in their spin-x,y orientations or in any other
observable quantity.
CLASSICAL EXAMPLE
This correlation between the measurements of both
electrons is not too different from looking at the
motion of a coupled double pendulum. Any perturbation
of pendulum 1 will affect the motion of pendulum 2 and
mathematically, the differential equations used to
describe the motion of the double pendulum will have
terms that couple pendulum 1 to pendulum 2.
Entanglement is not too different in concept.
RESULT
For these reasons, you cannot change the spin
orientation of particle 1 without affecting the
orientation of particle 2, as these properties are
entangled. Entangled states are NOT separable. What
this means is that you cannot separate the entangled
degrees of freedom from one another in the composite
wave function. So your question is built upon a
premise that is not possible.
However, you are still hitting upon something quite
important; the distinction between classical and
quantum information. Even if I manage to get
entangled particle 2 really far away from entangled
particle 1, that doesn’t mean that their
correlations ever vanish; it just means that the
particles are separated by a large spatial distance.
Spatial separation does not invalidate the correlated
properties of this two particle system. So just
because the objects are separated by a large distance
doesn’t mean that we magically erased their
entanglement. The entanglement is built into the wave
function though, and does not encode any ˜useful
information. This ˜useful information” might
be a binary message that we are trying to send
somewhere, and we have to use a series of entangled
states and a classical information channel to do that.
Prior to taking the measurement, we know that the
measurement will yield one N results (there are N
distinct eigenvalues for whatever operator/observable
we are measuring); we just don’t know which result
that will be. But remember that before we take a
measurement, the system is in a superposition of
eigenstates. The measurement process ‘forces’ the
system to take a stand and go into only one eigenstate
(of the property we are measuring). If there are only
two eigenstates corresponding to whatever observable
quantity we are measuring, then when we measure that
quantity in a system, the measurement might yield
state 1 OR state2. Although we have only two states
to choose from, we don’t know WHICH state the
electrons will be in until we measure. Classically,
there would not be this indeterminacy. That is why
classical information cannot be sent faster than the
speed of light; classical information picks out the
characteristics of only one state, and QM says that we
have at least two states to choose from when
measuring.
Does it now make sense that there really isn’t any
new information being transmitted when one measures
the spin-z component of one entangled particle and the
other yields a predictable result? It was built into
the wave function! NOW, what we cannot do is transmit
binary messages faster than the speed of light.
Information on the spin angular momentum, orbital
angular momentum, position and momentum of a particle
is contained in the wave function, however binary
messages are not. These messages can be sent via
quantum teleportation, but the information transfer
speed has never exceeded the speed of light.
Check out the IBM Quantum Teleportation page for more
information:
http://www.research.ibm.com/quantuminfo/teleportation/
Warm Regards,
Mahndisa