There are three degrees of spookiness in quantum entanglement:
A … <… oo …> …B
A particle decays sending two fermions in opposite directions (conservation of momentum) towards A and B.
1) Observer A, called Alice, will measure the spin of incoming particle. If it has spin, say, up then she knows that observer B, called Bob, will measure its counter-part particle’s spin to be down.This is like Alice tossing a coin, heads or tails, and say it comes down heads. Her counterpart Bob who is light years away on the other side of the galaxy, who is also tossing a coin, and now his must come out in this case as tails!
2) Not only that, but Alice can choose whatever axis -- an infinity of possibilities -- along which the particle spin must quantize as spin up or spin down. Now Bob’s particle, on the other side of the galaxy, must also quantize along the same axis. But how does the particle know that?!?
3) Furthermore, a third observer C, called Claude, might be moving towards either Alice or Bob. If Claude is moving towards Alice, then according to him, she is measuring first, and she decides along which axis the particle spin will quantize. However if at the same time another observer D, called Donna, is moving towards Bob, then according to Donna, it is Bob who makes the first measurement and he decides along which axis the particle will quantize. How can this be??!?? Which is the cause and which is the effect? Accordingly Quantum physics makes no distinction between the two cases.
Bell’s theorem
Assumptions:
1. Logic
2. A, B, C are independent events (often called locality).
Examples: A is up or down, B is head or tail, C is red or green, etc.
Derivation of Bell’s inequality:
1. N (A+, B-) = N (A+, B-, C+) + N (A+, B-, C-); since an object must have the characteristic C or not have it.
2. So N (A+, B-) >= N (A+, B-, C-); since N (A+, B-, C+) cannot be smaller than zero.
3. N (B+, C-) = N (A+, B+, C-) + N (A-, B+, C-) ; similar reasoning to step 1.
4. So N (B+, C-) >= N (A+, B+, C-);similar reasoning to step 2.
5. So N (A+, B-) + N (B+, C-) >= N (A+, B-, C-) + N (A+, B+, C-);adding inequalities 2. and 4. together
6. But N (A+, B-, C-) + N (A+, B+, C-) = N (A+, C-); similar reasoning to steps 1. and 3.
7. So N (A+, B-) + N (B+, C-) >= N (A+, C-); which completes the proof.
Experiment:
A+ = right spin up at 00 ; A- = left spin up at 00
B+ = right spin up at 450 ; B- = left spin up at 450
C+ = right spin up at 900 ; C- = left spin up at 900
You put detectors at L and R (Prisms, polarizers, Stern-Gerlach apparatus, etc.)
L … <… oo …> …R
Case1: R has orientation 00 and L has orientation 450
Measure: N (A+, B-) = N (right spin up at 00, left spin up at 450)
CALL THIS N1
L … <… oo …> …R
Case2: L has orientation 900 and R has orientation 450
Measure: N (B+, C-) = N (right spin up at 450, left spin up at 900)
CALL THIS N2
L … <… oo …> …R
Case3: R has orientation 00 and L has orientation 900Measure:
N (A+, C-) = N (right spin up at 00, left spin up at 900)
CALL THIS N3
NOW ACCORDING TO BELL’S INEQUALITY:
N (A+, B-) + N (B+, C-) >= N (A+, C-) (Equation 7 above)
This translates for the three cases as: N1 + N2 >= N3
From the experimental data, it turns out that this inequality is wrong (In Quantum physics language: Bell's theorem is violated).
See http://perso.wanadoo.fr/eric.chopin/epr/aspect.htm
The conclusion is that either assumption 1 or 2 or both is/are wrong.
Historically, assumption 2 ( locality) is considered to be wrong.
Friday, October 14, 2005
Sunday, October 09, 2005
The theory that matters
Three outstanding features of String theory:
i) The number of dimensions is inherent into the theory. If you work in classical physics or quantum mechanics, the problem you are dealing with pretty much determines in how many dimensions you will write your equation. Say you are dealing with a particle moving along a straight line. A one-dimensional equation would be sufficed. If you are dealing with a particle moving along a surface or a body rotating about an axis, a two-dimensional equation might do the trick. In other words, you are plugging into the theory how many dimensions you need for a particular problem. Not so with String theory. It tells you plain and square that you must work in ten dimensions; otherwise the equations don't make sense.
2) In the Standard model, you need to plug in twenty parameters. One example is the ratio of the mass of a muon to the mass of an electron. These twenty parameters must be fixed, usually by some lab experiments. In String Theory you need one parameter, the length of the quantum strings. It has been conjectured that it is about the Planck size, about 10 to the exponent (-33), a decimal followed by 33 zeroes. If one would blow a proton up to the size of our sun, a quantum string would be no bigger than a baseball. This is so small that most likely it will be never measured. Nevertheless, a one-parameter theory would trump on any day a twenty-parameter theory.
3) In the twentieth century two grand theories evolved: the General Theory of Relativity in which gravity is shown as the warping of the space-time continuum; and Quantum Mechanics in which the other forces are revealed as interactions that exchange particles. In particular, for the electromagnetic force, the photons are carriers of the force; for the weak nuclear force, the W's and Z bosons; and for the force between quarks, the gluons. But gravity is the only force that stands outside of this scheme. To put gravity on an equal footing with the other forces, that is, have it as a force that would exchange particles, one would need a massless boson with spin 2. It was this particular feature of String Theory - that it does produce such a particle - that made physicists think seriously about String Theory as the theory that could unify all the forces in nature and hopefully give an explanation of the twenty parameters of the Standard Model.
i) The number of dimensions is inherent into the theory. If you work in classical physics or quantum mechanics, the problem you are dealing with pretty much determines in how many dimensions you will write your equation. Say you are dealing with a particle moving along a straight line. A one-dimensional equation would be sufficed. If you are dealing with a particle moving along a surface or a body rotating about an axis, a two-dimensional equation might do the trick. In other words, you are plugging into the theory how many dimensions you need for a particular problem. Not so with String theory. It tells you plain and square that you must work in ten dimensions; otherwise the equations don't make sense.
2) In the Standard model, you need to plug in twenty parameters. One example is the ratio of the mass of a muon to the mass of an electron. These twenty parameters must be fixed, usually by some lab experiments. In String Theory you need one parameter, the length of the quantum strings. It has been conjectured that it is about the Planck size, about 10 to the exponent (-33), a decimal followed by 33 zeroes. If one would blow a proton up to the size of our sun, a quantum string would be no bigger than a baseball. This is so small that most likely it will be never measured. Nevertheless, a one-parameter theory would trump on any day a twenty-parameter theory.
3) In the twentieth century two grand theories evolved: the General Theory of Relativity in which gravity is shown as the warping of the space-time continuum; and Quantum Mechanics in which the other forces are revealed as interactions that exchange particles. In particular, for the electromagnetic force, the photons are carriers of the force; for the weak nuclear force, the W's and Z bosons; and for the force between quarks, the gluons. But gravity is the only force that stands outside of this scheme. To put gravity on an equal footing with the other forces, that is, have it as a force that would exchange particles, one would need a massless boson with spin 2. It was this particular feature of String Theory - that it does produce such a particle - that made physicists think seriously about String Theory as the theory that could unify all the forces in nature and hopefully give an explanation of the twenty parameters of the Standard Model.
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