Putin is meeting Kim Jong Un.

Which brings me to children in Manchester Cathedral. They’re mentioned in the Big Book of Magic Stuff. Yes, children are THAT important. Children can be cruel, petty, vindictive and throw tantrums. We see the true nature of the Invisible Magic Friend in children.

https://mega.nz/file/Aq8XHLDQ#k29BOvg2X9py3eP0AR3mrLekweuY41TLJA1f4BoL55Y

There’s a terrible civil war in Sudan. Many have lost everything and have suffered violence and the loss of loved ones. They have been forced to flee to make shift camps, where children live in fear, lacking food and basic sanitation. Yet, somehow, arms and ammunition get there, even when aid doesn’t.

https://mega.nz/file/IilgnQgQ#KaQJfrHgufhlcED7vCnkP3fvurRT1Gv7iITwfLT-lj0

True stories from eye witnesses are much better than made up stories. That’s why the gospels are true, because they were definitely written by eyewitnesses, who were present at each event, including Jesus’ birth, and on which they agree in every single detail.

Slam dunk, case closed.

https://mega.nz/file/py1BnCbT#MsfajATxuRIALcVh_-IrsBE2Y3f6MdzPo5l0wh2J4qE

The Princess of Wales has made her first public appearance since her cancer diagnosis. We all have to find strength to handle life’s hurdles. Some find that strength in religion. But we can all find inspiration from others when they overcome great difficulties.

https://mega.nz/file/I70SDaJD#W47k7yvozYQeigxXyu9taync0w0Hm7-jMbDDUUBdRiA

This post is part of our journey to explore Quantum Field Theory. Last week I introduced the idea of a Probability Distribution (PD).

“https://platitudes.home.blog/2024/06/08/the-remarkable-theory-of-quantum-fields/“

This week I’m going to take this a little bit further and start using the notation used in Quantum Mechanics (QM).

Quantum Mechanics is the theory of probability, with one small, but extremely significant twist.

Take a very simple example of the roll of a six sided die. There are six possible outcomes. While the die is still rolling, there is an equal probability that any of the six sides will appear face up. Each has a 1 in 6 chance, or a probability of one sixth, 1/6. So while in this state, the Probability Distribution (PD) looks like this.

We can represent this as a sequence of numbers where we just list the probabilities for each possible outcome, (1/6,1/6,1/6,1/6,1/6,1/6). And to save some ink, I’ll give this a shorthand name. I’ll write it as |e>, where the letter “e” stands for “equal probability”. The funny brackets are conventional in QM.

|e> = (1/6,1/6,1/6,1/6,1/6,1/6) “e” for equal

Now suppose the die comes to rest with the number 4 facing up. The PD has now changed.

A 4 is now certain, it has a probability of 1. If we don’t do anything to the die it will just sit there, continuing to display a 4. Again, we can write this as a bunch of numbers, and give that a name.

|4> = (0,0,0,1,0,0)

This means there is zero probability for the faces 1,2,3,5 and 6, and a probability if 1, certainty, for the value 4. Similarly, if the number 6 ends up face up, then we can represent it’s PD as

|6> = (0,0,0,0,0,1)

If we want to represent any one of the above six then we write

|n> = ( 1 in the nth position and zero everywhere else).

How to Compare States

If we look at the two states |4> and |6>, we see that they have nothing in common.

|4> = (0,0,0,1,0,0)
|6> = (0,0,0,0,0,1)

The only number that is non-zero in |4> is in the fourth position, but the fourth position in |6> is a zero.

Similarly, the only number that is non-zero in |6> is in the sixth position, but the sixth position in |4> is a zero.

We say that the “overlap” between the two states is zero. As a shorthand, this is written as.

<4|6> = 0

As these are PDs, what we’re really saying is that, if the die is in the state |6>, then the probability of it being in the state |4> is zero.

Any PD always perfectly overlaps with itself < i | i > = 1. The probability of being in state i, given that it’s already in state i, is exactly 1, certainty.

Now compare the state |4> with the equal probability state |e>

|4> = (0,0,0,1,0,0)
|e> = (1/6,1/6,1/6,1/6,1/6,1/6)

Now there is some overlap between the two states. The “1” in position four in |4> matches a “1/6” in position four in |e>. If the die is in the |e> state then there is a non-zero probability that it will end up in |4>.

The probability that we’ll end up in the |4> state, given that we’re in the equal probabilty state is written as <4|e> and has the value 1/6.

(For those with some maths, this is just the inner product of two vectors |i> and |j>. It is P( i | j ), sort of, see below.)

And the twist? Well that’s just a little bit more mathematical, so I’ve added it as a comment.