Two of the biggest intellectual giants of modern physics changed our view of reality,
and proposed their theories around the same time, in the early twentieth century. Yet,
their views about reality were completely different.
Niels Bohr, one of the founders of modern quantum mechanics and the father of the Copenhagen
interpretation, argued that reality, or the state of a particle at the fundamental
level, was not only unknown, but was unknowable, until it was measured.
Albert Einstein passionately disagreed with this idea, and believed that reality was knowable
and that probabilities could not completely define reality, that there had to be an objective
reality out there, independent of measurement. He famously said, in his dramatic statement,
“Do you really believe the moon only exists when you look at it”
Bohr and Einstein argued passionately about their views on the essence of reality. And
for 30 years, both views were considered equally valid, and scientists chose sides.
Then in 1964, Irish physicist John Bell devised a way to prove whether Einstein’s view of
a classical, deterministic view of reality was correct, and he put this in a simple elegant
equation – called the Bell Inequality. Although the equation is simple, understanding
what it means is not so easy. So how did this simple equation disprove the ideas of the
most famous and powerful scientist of the 20th century, and forced us to confront the
unsettling truth that we may live in a fundamentally nondeterministic universe. The answer is
coming up right now...
The central weirdness of quantum mechanics
can be demonstrated with a toss of the dice. If the dice was a quantum system like a photon, electron
or atom, it would be in superposition. That is, its position and other properties, in
a sense would be up in the air like a dice thrown before it hits the table. It is a 1,
2, 3, 4, 5, and 6 all at the same time. According to the most accepted interpretation
of quantum mechanics, the Copenhagen interpretation, pioneered by Neils Bohr and Werner Heisenberg,
its not only that the dice’s value is not known, but that it is all values at once.
It’s value can only be known once it is measured.
The double slit experiment demonstrates this.
A single photon passing through the slit should not be thought of as a particle going through,
and interacting with itself. Rather, it is like a three dimensional wave or a cloud of
probability when it is emitted. This wave can go through both slits at once and interfere
with itself like a wave does, before it is measured by the photo sensitive screen in
the back, when it resolves to a specific location on the screen.
Einstein, who was the most famous scientist
at the time, was bothered by this interpretation of quantum mechanics. The idea that part
of nature remains not only unknown, but is unknowable unless measured. That’s why he
famously said, “God does not play dice.”
So Einstein along with two other scientists, Boris Podolsky and Nathan Rosen (EPR for short),
came up with what they thought disproved the Copenhagen interpretation. The crux of their
argument rested on the idea of a phenomenon predicted in quantum mechanics called entanglement.
Let’s say we have a quantum system that
has a zero angular momentum, also known as spin zero, and it emits two photons simultaneously.
Since photons have spin, and since angular momentum must be conserved, if one photon
has a spin up, the other photon must have a spin down. So the spin up and down cancel
each other out. This is entanglement. The two photons are not independent. If the spin of
one of them is known, the spin of the other becomes known instantaneously.
Now let’s separate these photons far apart,
let’s say 300,000 km, which is about 1 light second away. Now according to quantum mechanics,
each of these photons is governed by a wave function, and each photon is in a state of
both spin up and down at the same time. It is not up or down, it is both up AND down. However,
the wave functions of these photons is not independent. They are really the same wave
function governing both photons. Now if we measure the spin of one of them. We have a
a 50% chance of detecting a spin up. Let’s say do we detect spin up. Now let’s
measure the spin of the other photon in less than one second, it should also have a 50%
chance of having spin up or spin down. But it doesn’t, it has a 100% chance of having a spin down.
This means that the information of the collapse
of the wave function of one of the photons had to travel faster than light to affect
the other photon. EPR argued that since nothing can travel faster than light according to
the rules of special relativity, this should invalidate the Copenhagen interpretation. This theorized
violation is the EPR Paradox. EPR proposed instead there is likely another
theory which would show that the two entangled photons were in cahoots from the very beginning,
that their states were predetermined from their creation. In other words, their states
contained the information locally, so that when they were moved apart, no communication
had to take place. The information that we were measuring was hidden inside the two particles.
This is called local hidden variables.
So for example, it’s as if the two particles were a pair of gloves. One was a left handed
pair, and the other was a right handed pair. Once we found the left handed pair, we knew
immediately that the other pair, no matter where it was in the universe, must be a right
handed pair. This was a valid interpretation of quantum mechanics for almost 30 years,
From 1935 to 1964. This could not be disproven.
This is where Irish physicist, John Bell comes in. And in 1964, he proposed an experiment that
could show whether the local hidden variable theory was correct, or incorrect.
Bell’s equation is remarkably simple. But,
it is one of the most difficult to understand and simplify. So I will present a highly simplified
idea of how it works, then also show you simplified version of the mathematical underpinnings
later in this video.
We are going to play a game, and our opponent will be the universe. The game is guess the color of the checkers pieces. I want
to give a big shout out to Jim Al-Khalili who shows a similar simplified illustration
in a documentary available on Magellan TV. I encourage you to check it out because it
is fascinating.
If I guess the color correctly I win, if I am incorrect, the universe wins. In the first
game I declare that if the colors match I win. The universe picks the pieces. In play
after play, I find that I lose every time. So in the next round, I change the rules.
This time, I declare that if the colors are different I win. The universe picks the pieces.
In play after play, I find that I lose every time again.
So, I conclude that perhaps the universe has rigged
the game against me. This is what Einstein suspected was happening, that the checkers
Piexes were rigged. He believed that the colors of the pieces were predetermined, and that
this information had been known from the beginning when the two pieces were together. Niels Bohr’s
idea, on the other hand, was that the red and black don’t even exist until the piece is turned over.
So, I change the rules of the game again.
This time, I do not tell the universe whether matching colors, or different colors will
win the game, until after the universe has already picked the checker pieces.
If after revealing the checker pieces, I find that I win 50% of the time, and lose 50% of
the time, then my conclusion is that indeed the checker pieces were rigged from the very
beginning, because my chances of winning would be 50% if the red and black pieces were already
picked by the universe. But if I continue to lose, then I have to conclude that somehow,
the colors of the checker pieces were not chosen ahead of time.
Let’s look at how Bell’s elegant equation proves my conclusion. To do this, let’s
look at a universe where local hidden variables are correct.
In other words, entangled particles have predetermined
properties which are complementary to each other at the moment of their creation. Note
that this is not an interpretation of quantum mechanics. It’s an alternative to quantum
mechanics.
So in this system, when two entangled particles are produced from a process that conserves
linear and angular momentum, we will get two particles that go off in opposite directions,
and they will have opposite spins, as well as opposite directions.
Let’s say that Alice makes the measurement
on particle 1 in her laboratory, and Bob makes the measurement on particle 2 in his laboratory.
And let’s say that the two labs are very far apart and are not in communication with
each other.
If Alice measures the particle’s spin in the Z direction as positive, Bob will
measure the spin as negative in the Z direction. If they measure the spin in the X direction
which is orthogonal to the Z direction, or 90 degrees apart, the same type of complimentary
opposite spins will be measured, because, as I said earlier, angular momentum is always
conserved.
What Bell’s inequality does is, it says, now let’s make a third measurement that
is somewhere in between the Z and X axes. In our example we will use 45
degree angles to the Z and X direction, And we will call this the Q direction.
So in a universe where local hidden variables
are true, when the two particles are emitted, they know what their state is going to be
in all three directions, Z, X, and Q from birth. And there are only 8 combinations or
possibilities of spins that each particle could have. So for Alice and Bob, these 8
combinations will be following:
Event 1 could be where the spin in the Z direction is positive, the spin in the X direction is positive, and the spin in the Q direction is posive. positive
Event 2: Z is positive, X is positive, Q is negative
Event 3: Z is positive, X is negative, Q is positive
Event 4: Z is positive, X is negative, Q is negative
Event 5: Z is negative, X is positive, Q is positive
Event 6: Z is negative, X is positive, Q is negative
Event 7: Z is negative, X is negative, Q is positive
Event 8: Z is negative, X is negative, Q is negative
Now, let’s ask the question, what is the
probability that Alice measures in the z direction, gets a positive spin, and Bob measures in
the X direction and gets a positive spin? Well, if the above case is for Alice, there
are 4 events where Z is positive. In order for Bob to get X positive, Alice would have
to have measured X as negative. So these would be in event 3 and event 4. To get the probability
we have to divide by the total number of events, which is 8.
Let’s do this for two more scenarios. What is the probability that Alice measures positive
in the Z direction, and Bob measures positive in the Q direction? In this scenario, it would
be event 2 and 4. Again we divide by 8 to get the probability.
And the third case is: What is the probability that Alice measures positive in the Q direction,
and Bob measures positive in the X direction? This would be event 3 and event 7, divided
by 8 for probability. So these are the three probabilities given
the hidden variables theory. Now here is the big insight that John Bell had:
If I take the total number of Events, and multiply that by the probability that Alice
measures Z positive and Bob measures X positive, this has to be less than or equal to the total
number of events times the probability that Alice measures Z positive, and Bob measures
Q positive, plus the probability that Alice measures Q positive, and Bob measures X positive.
When we write this out,
the eight cancels out, and we are left with just the probabilities:
This is Bell’s inequality.
I can prove this is true by doing simple math: E3 + E4 divided by 8 is less than or equal
to E2 + E4 + E3 + E7 divided by 8. The eights cancel out, and we rearrange the
order of addition, and we are left with this: E3 + E4 is less than or equal to E3 + E4 +
E2 + E7 This makes total sense, because E3 and E4
are on both sides of the equation. And E2 and E7 have to be positive. So this inequality
absolutely HAS to be true for any hidden variables theory to be true. Now remember these probabilities
are for a universe with hidden variables.
But what happens in a universe where the laws of quantum mechanics are correct, and not
the hidden variables theory? Well, this inequality is violated in quantum mechanics. How is it
violated? Let’s say Alice measures the spin to be positive in the Z direction. Then,
we know that if Bob measured the particle in the Z direction, he would get a spin that is negative.
However, Bob doesn’t measure in the Z direction,
but in the Q direction. What will the spin of his particle be? In hidden variables,
there was a 50% chance that it would be positive, and a 50% chance that it was negative. But
this is not what happens in quantum mechanics, because the measurement of the particle follows
the probability laws of the wave function for a particle rotated 45 degrees.
And that probability of Bob measuring Q to
be positive, after Alice has measured Z to be positive, if the angle between them is
45 degrees, is given by the following equation: sine sqared of 45 degrees divided by 2.
This comes from the math of quantum mechanics. This is the critical difference between quantum
mechanics and hidden variables theory. The probability that Bob measures the same spin
as Alice, depending on the difference in angle measured is not linear but looks like a sine
wave. When you plot this out, this is what the probabilities look like:
So you can see from the graph that at 0, and
multiples of 90 degrees, the two systems are in agreement. But in between, like at 45 degrees,
the probability is 25% for hidden variables, and about 14.6% for quantum mechanics.
But the proof is in the pudding, because in
test after test, the sine function correlation has been confirmed. The particle does not behave
linearly. And so the hidden variables theory cannot be correct. Bell's inequality is violated.
So we can write out and compare Bell’s inequality
for both cases.
For hidden variables, the equation would be: E3 + E4 is less than or equal to E3 + E4 +
E2 + E7 2 is less than or equal to 4 – the inequality
holds true.
For quantum mechanics, the equation would be:
Sine squared of 90 degrees divided by 2 is less than or equal to sine squared 45 degrees
divided by two plus the sine squared of 45 degrees divided by two.
And when you solve this, you get: 0.5 is less than or equal to 0.293,
which, of course, is not true, so Bell’s inequality is violated in quantum mechanics, which is
exactly what we observe in experiment after experiment.
Does this prove that the two entangled particles
are communicating faster than light? There is differing
opinion on this. On the surface, it does appear that superluminal communications is taking
place. But my personal opinion is that no communication is happening. The two particles
are really just part of one wave function, and since the wave function can theoretically
be as wide as the universe, when this one wave function collapses, both particles collapse.
And since the collapse is random, it can not be used to communication in any way.
So most theorist do not think special relativity
is violated, because we can’t communicate using this seemingly faster than light phenomenon.
But our knowledge of quantum mechanics and
entanglement is incomplete to say the least. Why is there a superposition? What is the mechanism of wave
collapse between two particles separated by space and time? I happen to think that the
scientist who is going to reveal the mystery of quantum mechanics is alive right now in
the world. And I can’t wait for him or her to explain this, and like Copernicus, Newton
and Einstein - completely change our paradigm. I’ll see you in the next video my friend.