HE is risen!

And light and hope return to the world.  Happy Easter everyone!

(Except you, Pedo Joe Biden.  You're going to burn in Hell.)

Demystifying h and h-bar

h was the beginning of quantum physics.  It represents the indivisible quanta of light.  To get a photon's energy, you multiply its frequency by h.  The frequency is, mathematically, the inverse of the wavelength.  In other words, h represents the circumference of a circle, the amount of ink you would use drawing one wavelength.

All images are as found on the internet.  Credit to whomever made the things in the first place.  I can't draw.

That's easy enough.  But what is h-bar (ħ)?  It is h divided by 2π.  That makes ħ the radius of the circle.  (The circumference of a circle is 2πr.)  In other words, ħ is the null-to-peak amplitude of a photon.  Which photon?  Every photon.  That was the insight that eventually led to quantum physics.

While we're here, the Heisenberg uncertainty theorem states that the uncertainty in the momentum of a particle, times the uncertainty in its position, cannot be less than ħ/2.  In other words, their product cannot be less than half the amplitude of a photon.  Which certainly makes sense, because you couldn't possibly measure anything less than that size by any means.  Notice that this distance is minuscule!  So when you hear people say that you can't know the position and momentum at the same time, they're talking nonsense.  Of course you can.  But there is an absolute limit to the ability to measure something of that ridiculously small scale of precision.  (Notice we're not talking about absolute size, we're talking about the precision of a measurement.)  Not just because of the tooling involved, but because it really doesn't make any sense to talk about distances smaller than ħ/2, because a circle smaller than that is less than the amplitude of a photon, and there is no such measurement.  Not with any tools.  Not even conceptually.  In addition, the minimum arises because when you work out the math, at the smallest scales, the uncertainties (precision) in position and momentum sort of end up being reciprocals, just like frequency and wavelength.  The less you have of one, the more you get of the other.

Contemplate and compare this relationship with the minimum sampling rate of digital audio, which must be at least twice the maximum frequency of the audio signal being sampled.  If you tried to sample at a lower rate, the result would have no relationship to the original signal.

No, space is not curved

Space is flat.  What does this mean?  It means that the three familiar dimensions of space (x, y, z) obey the Pythagorean principle for Euclidean space.

0 ≤ (x2 + y2 + z2)  ≤ c2 

But what does this mean?  x, y, and z are not deformations, but velocities, so they have an intrinsic time element which we usually ignore.  No, it's not proper (subjective) time.  This is the home of true, objective time.  Yes, it exists, because it must.  After all, light propagates and changes over time, without experiencing any time itself.  But it is routinely ignored, for reasons which completely escape me.  (Note that the axes of a spacetime diagram are true distance and true time.  But we, for some reason, must pretend that these don't exist.  Possibly because we, being made of matter, can never actually experience them?)

It would be easier and more intuitive to replace the awkward x, y, z with a simple v for velocity, ignoring the actual direction, which is immaterial for this narrow purpose.  

0 ≤ v2 ≤ c2 

Of course, nothing with mass can travel at the speed of light.  By similar logic and a careful examination of the meaning of quanta, it cannot remain perfectly stationary, either.  To do so, the particle must give up that last bit or momentum.  But there is no way to drop that last bit, because it is less than the smallest quanta of energy transfer.  So we may simplify the expression slightly for normal matter.  (Notice that a photon either travels at the speed of light, or stops existing, but never anything in between.)

0 < v2 < c2

Spacetime, on the other hand, is not flat, but hyperbolic.  This is because proper (perceived, subjective) time is imaginary, and when you square it, out pops a minus one.

0 < (v2 - t2) < c2 

Oddly enough, you can negate the central term, and it still holds true.

0 < (t2 - v2) < c2

A morning wasted with Spring

Spring is here, ah, spring is here.

Life is skittles and life is beer.
I think the loveliest time of the year is the spring.
I do, don't you? 'Course you do.
But there's one thing that makes spring complete for me,
And makes every Sunday a treat for me.

All the world seems in tune
On a spring afternoon,
When we're poisoning pigeons in the park.
Every Sunday you'll see
My sweetheart and me,
As we poison the pigeons in the park.

When they see us coming, the birdies all try an' hide,
But they still go for peanuts when coated with cyanide.
The sun's shining bright,
Everything seems all right,
When we're poisoning pigeons in the park.

We've gained notoriety,
And caused much anxiety
In the Audubon Society
With our games.
They call it impiety
And lack of propriety,
And quite a variety
Of unpleasant names.
But it's not against any religion
To want to dispose of a pigeon.

So if Sunday you're free,
Why don't you come with me,
And we'll poison the pigeons in the park.
And maybe we'll do
In a squirrel or two,
While we're poisoning pigeons in the park.

We'll murder them all amid laughter and merriment,
Except for the few we take home to experiment.
My pulse will be quickenin'
With each drop of strych'nine
We feed to a pigeon.
(It just takes a smidgin!)
To poison a pigeon in the park!

 - Tom Lehrer, An Evening Wasted With Tom Lehrer

On the philosophy of quantum mechanics

This began with this post at William M. Briggs' place.

Does Quantum Mechanics Speak To Theology? — Guest Post by Bob Kurland

BY BRIGGS ON MARCH 15, 2024 • ( 3 COMMENTS )

“I think I can safely say that nobody understands quantum mechanics.” Richard Feynman, Nobel Prize winner for his research on quantum electrodynamics

Being my cantankerous yet truth-seeking self, I had to comment.  The theme is, as always, none so blind as those who just won't see.

Copenhagen Interpretation delenda est!

Entanglement is nothing more than the Zero Principle in action: Everything adds up to zero. When two particles interact in an entangling way, all it means is that they obtain, at that very moment, some set of opposite properties. They carry these opposites forward unless changed, because particles do not change without cause. It does not matter if they travel one millimeter or ten thousand light years, their properties will remain opposites until and unless altered by some force.

Let us say you have a pair of shoes. You place the shoes into identical, unlabeled, opaque boxes and seal them tightly shut. You hand the boxes to someone, and direct them into a dark room. There they place the boxes, unopened, upon a table. They swap them back and forth to their satisfaction. Then they come out, and a third person goes into the dark room. They find two boxes upon a table, and swap them back and forth to their satisfaction. They come out, and a fourth person goes into the darkened room, and selects one box at whim. Locking the door behind him, he comes out with that box, and fly to some distant portion of the globe. When he arrives at his destination, he calls you, and opens the box. At the moment that person tells you which shoe he has, you suddenly and irrevocably know, for a fact, which shoe still lies hidden in the sealed box in the darkened room.

Unpossible! Magic! Quantum strangeness! Or so the Copenhagen junta would have you believe.

It's all about mind control

 I am the leader of Gamergate 2.0, and so can you!

A leaderless insurrection cannot be decapitated.  There is no leadership to subvert and control.

This was never truly about video games.  It's all about control.  Mind control. 
"Give me the child, and I will give you back the man."

Links to Wilder's posts on the topic, because he is much more well-written than I.

Post 1

Post 2

Algebraic types demystified

In programming, a type is what a variable is allowed to represent.  Typical types are character, integer, and floating point number.  These are the usual atomic types, meaning they hold and represent only one value.  Typical collection types are lists (of one type) and strings (lists of characters).

Functional programmers like to talk about algebraic types.  They wrap these concepts in mystery and superstition, because they originated in mathematics.  They call them "product" and "sum" types.  But really, they're remarkably simple collections of other types.  

Product type = AND collection

A product type is a fixed length, ordered list of various types, where you must fill each element.  A and B and C and D.  It is better known in programming as a tuple.  If you name what each type represents, it is a struct or record.  For those working with an object-oriented language, this is a class.

Sum type = OR collection

A sum type is a set or list of various types, from which you must choose exactly one element to fill.  E or F or G or H.  It is better known as an enum (when referencing elements of a single type) or union (when referencing elements of disparate types).

You can combine these two concepts with each other and the atomic and basic collection types to produce arbitrarily complex data structures.  Some languages even allow types to be recursive, and include themselves as an element.

Contrived example:

type NameOrNumber = union [string, int]    # can be either but not both

var myGirl1, myGirl2: NameOrNumber

myGirl1 = "Jenny"

myGirl2 = 8675309

Overly Contrived Example:

type Lowercase = enum [`a ... `z]    # sum type

type Uppercase = enum [`A ... `Z]    # sum type

type NameFormat =  record       # product type

    initial: Uppercase

    rest: list[Lowercase]

type PersonName = record        # product type

    firstName: NameFormat

    middleInitial: Uppercase

    lastName: NameFormat

type PersonInitials = record        # product type

    firstInitial, middleInitial, lastInitial: Uppercase

type Person = union [PersonName, PersonInitials]    # sum type

var generalName: PersonName        # create a variable

generalName.firstName.initial = `G

generalName.firstName.rest = [`e, `o, `r, `g, `e]

generalName.middleInitial = `S

generalName.lastName.initial = `P

generalName.lastName.rest = [`a, `t, `t, `o, `n]

var nationName: PersonInitials        # create a completely different variable

nationName.firstInitial = `U

nationName.middleInitial = `S

nationName.lastInitial = `A

var name1, name2: Person        # create two variables of the same union type

name1 = generalName               # which are used

name2 = nationName                 # in different ways