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### What Does It Mean?

From the “Afterword” of D J Griffith’s Introduction to Quantum Mechanics

Da Big Question

- Did the physical system “actually have” the eigenvalue in question prior to the measurement? (REALIST)
- OR
- Did the act of measurement “create” the eigenvalue (constrained by the wavefunction)? (ORTHODOX)
- OR
- Can we completely duck the question? (AGNOSTIC)

Realist View

- If the realist view is true, QM is an incomplete theory because:
- Even if you know everything that QM has to tell you about the system, you STILL cannot determine all of its features!

Orthodox Position

- Measurement forces the system to “make a stand” helping create an attribute that was not there previously
- Since repeated measurements yield the same result, the act of measurement collapses the wavefunctions.
- This is strange but not mystical

Agnostic Response

- I refuse to answer
- I ignore these problems

EPR Paradox

- 1935- Einstein, Podolsky, and Rosen came up with the EPR Paradox
- Designed to prove that the realist postion is the only possible one
- A simplified version goes like this:
- Assume that a pion at rest decays into an electron and positron
- p0->e- + e+
- The positron and electron fly off in opposite directions and the pion has spin=0

EPR Cont’d

- Since pion is spin 0, then electron and positron are in singlet configuration |00>
- So if the positron is spin up then electron must be down or vice versa.
- QM can’t tell you which you will get, but that they will be correlated

Movin’ On Up to the Big Time, the Deluxe Apartment in the Sky!

- Now move the electron far apart (pick one)
- 10 meters
- 10 light years
- Now say you measure electron spin down, then IMMEDIATELY you know that the positron is spin up
- BTW, this is an INSTANTANEOUS knowledge!

The 3 views again

- Realist: This is not surprising, the positron was always up since the decay
- Orthodox: Hmmm… electron was neither up or down until measurement. The measurement caused the wave function to collapse and that “knowledge” was transmitted instantaneously across the gulf to the corresponding particle.
- Agnostic: I don’t have an opinion

EPR Says

- Orthodox view is “spooky action at a distance”
- Ergo, says EPR, the realist have it correct and the spins were predetermined at the decay
- Of course, this is all predicated on the argument that nothing, not even information, can go faster than the speed of light– called the principle of locality

Bell’s Thm

- EPR did not doubt that QM is correct, just incomplete
- Some other “hidden variable” is needed to completely quantify the system.
- The hidden variable could be a single number or a whole collection of numbers; it doesn’t matter
- J. S. Bell proved that ANY hidden variable theory is INCOMPATIBLE with QM

The Gedunken

- Instead of having the electron and positron detectors along the same direction, allow them to be rotated independently
- The first detector measures the component of the electron spin in the direction of unit vector a and the second along the direction of b
- For simplicity, we will record the spins as +1 (up) and -1(down)

p0

e+

e-

a

b

The Results of the Experiment

- Bell proposed to calculate the average value of the product of the spins called P(a,b)
- If detectors are parallel, then we have original EPR configuration so always +1 and -1 and therefore P(a,a)= -1 (and so is the average)

However

- If they are anti-parallel, then P(a,-a)=+1
- For any arbitrary orientation then

P(a,b)=-a·b

- This result is IMPOSSIBLE for any hidden variable theory

Why?

- Assume, that the hidden variable is called k
- k varies in some way that we neither understand nor control from one decay to the next
- Suppose that the outcome of the electron measurement is independent of the orientation (b) of the positron detector. b is chosen after the decay but before measurement of electron and thus is hindered by speed of light (locality condition).

Let’s do some math

- So there is a function A(a,k) which gives the result of the electron measurement and B(b,k) which gives the result of the positron measurement
- A(a,k)=+/- 1 and B(b,k)=+/-1
- When detectors are aligned, the results are perfectly anti-correlated A(a,k)=-B(b,k) for all k

So

Where rho is the probability density of k

Now let’s eliminate B(b,k) by using our correlation function

If c is an other unit vector then

Bell’s Inequality:

|P(a,b)-P(a,c)|<= 1+P(b,c)

- It is easy to show that P(a,b)=-a·b (the QM prediction) is incompatible with Bell’s Inequality
- Suppose all three vectors, a, b, and c lie in a plane with c at 450 to a and b (a perpendicular to b)
- Then P(a,b)=0 and P(a,c)=P(b,c)=-.707
- Obviously .707 is not greater than 1-0.707 (.293)

But What does it Mean?

- If EPR is correct, then QM is completely WRONG!
- On the other hand, NO hidden variable is going to rescue us from the nonlocality that Einstein considered preposterous
- Many experiments were performed to test Bell’s inequality: the results were compatible with QM and incompatible with Bell’s Inequality
- In other words, the realists are wrong and there is spooky action at a distance
- Or in the lingo: there is the possibility of superluminal influences

Supernaturally Superluminal

- A causal influence that propagates faster than light is bad news
- Because relativity says that anything going faster than light is going backward in time!
- Faster than light things: any geometric point
- We don’t get upset that geometric points move faster than light
- So is this influence casual or is just information like a geometric point?

Two types of influence

- Causal: subluminal or luminal
- “ethereal”: neither energy or information and for which the only evidence is a correlation in the data of two different subsystems

Here, kitty, kitty ….

- What is a measurement and why is it so different from other physical processes? And how can we tell when a measurement has occurred?
- Schroedinger attempted to answer this in his famous thought experiment formally titled “The cat paradox”

The Cat Paradox

A cat is placed in steel chamber, together with the following hellish contraption: In a Geiger counter there is a tiny amount of radioactive material so tiny that maybe within 1 hour one of the atoms decays but equally probable none of them decays. If one decays then the counter triggers and via a relay activates a little hammer which breaks a container of cyanide. If one has let this entire system for one hour, then one would say the cat is living if no atom has decayed. The first decay would have poisoned it. The wave function of the entire system would express this by containing equal parts of the living and dead cat. At then end of an hour, the wave function of the cat has the form of

It’s ALIVE! It’s Dead! Etc. etc.

- So according to the paradox, the cat is trapped between life and death: a linear combination until
- You look! And the cat is:
- ALIVE: You saved it by measuring it
- DEAD: You killed it by measuring it
- Schroedinger thought this whole blame game was nonsense

Out of the Paradox

- Most widely accepted answer: the Geiger counter is the measurer, not you and it will make the statistical determination, not you
- Wigner and others say that it is the intervention of human consciousness that constitutes measurement in QM
- Wigner was Dirac’s brother in law and was associated with the Maharishi Institute and the Natural Law party.
- Unfortunately, the term measurement seems to imply human intervention but it does is not necessarily so

My Answer to All These Difficult Dilemmas

I refuse to answer.

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