Truth Like The Dark

This is a transcription of some notes I made after reading Quantum Mechanics: Symbolism of Atomic Measurements by Nobel Laureate Julian Schwinger, which I picked up after I heard Rui Vilela Mendes give a talk about the book’s ideas.

If you give me some time, this will be a part of a larger story about Schrodinger’s Aibo, about which I’ll say no more now.

Let’s first look at something that’s really weird to measure: the polarity of light.

You can think of a polarizer as a filter which has a very fine grid of parallel lines. If we orient this vertically (up-down), and we filter ordinary, white light, we get out all the light that was oriented (polarized) in an up-down direction. Turns out, this is about half the light we had. We can tell this from the relative intensity of the light that comes through the filter.

Now let’s take another polarizer, and stack it on the first one, but with the second polarizer’s grid oriented horizontally (90 degrees to the first polarizer’s orientation). We can say the new polarizer is filtering “left-right” polarization. When put together this way no light gets through the two sequential filters. This implies that up-down and left-right are mutually exclusive: all photons have one and only one of the two types of polarization. Filtering for both returns nothing.

Now let’s take another polarizer, and put it between the first two, but at a 45 degree angle to both of them. The weird thing is this: after inserting the third polarizer 12.5% of the original, white light gets through! So adding a filter increased the amount of light that got through!

It seems that polarizers do more than simply filter: they actively change something about the photons they are filtering.

This would be hell for a computer scientist: imagine a database where queries (filters of the objects in the database) affect the values in the database, via some internal program to which you have no real access (except through examining the results of sequences of queries). How could you figure out what (in general) you could expect from a series of queries?!?

To figure out what we might be able to do in this situation, let’s say we have a set of objects U, and a set of qualities that each object can have, Q. Q contains qualities like “A=address,” “B=boldness,” “C=company,” etc. Each quality can take on a number of possible values (A={a1,a2,a3…an). Let’s restrict our discussion to qualities where each object has to have one and only one value for any given quality. There are no empty data fields (or we could call “null” a particular value for a given quality). We call these complete qualities.

Let’s setup a query syntax G(U,A,a1), which means “give me all the objects in database U that have quality A set to value a1.” We can sum the results of queries to get an “OR” function, or perform queries in sequence to get a “AND” function. Let’s say that U is also a quality, which can take only one value u. This is a “universal” quality for which all objects have the same value. It helps later.

In normal databases, separate qualities are compatible: querying one has no effect on the values of another. In particular
G(G(G(U,A,a1),B,b1)A,a1)=G(G(U,A,a1),B,b1). That is, querying twice for a1 is redundant, and the fact we queried for b1 in between the a1 queries doesn’t change that. That means that the sequence of queries has no effect on the results: G(G(U,A,a1),B,b1)=G(G(U,B,b1),A,a1).

But what if this wasn’t the case? What if there was an unknowable machine operating, which changed the values of qualities in a database in a systematic but unfathomable way anytime you did a query? This seems to be the case with qualities like “up-down” and “left-right” polarizations. What can we say about such a situation?

Let’s try to describe what the unknowable machine does. To do this, let’s have a syntax for setting the value of a quantity: S(U,A,a1) sets the quality A for all the objects in U to value a1. Let’s also define a “fractional get”: a function that gets a fraction x of all the objects that have quality A set to value a1: F(U,A,a1,x). We’ll do lots of fractional gets and sets in sequence, so let’s say that S(F(U,A,a1,x),B,b1) ≡ SF(U,A,a1,x,B,b1).

What we need is some general properties for the fractions x(a1,b1) that result from sets and gets on A=a1 and B=b1, and other combinations of qualities and their values. Such properties are as much as we can hope to know about the unknowable machine.

To find these properties, let’s ask a question: for what function x(b,a) (that must be defined over all possible values for B and A) is the following statement true: G(G(U,A,a),B,b)=SF(U,A,a,x(b,a),B,b).

It may not be obvious, but for regular databases with compatible qualities, x(b,a) must equal 1 when a=b, and must equal 0 in all other cases. We will call this delta(b,a), the Kronecker delta function.

While x must be the delta function for compatible qualities, this isn’t the case for incompatible qualities. The question is: what properties (if any) must x(b,a) conform to in the incompatible case?

For compatible qualities, the following must be true, via reasoning about the delta function:
SF(SF(U,A,a,x(b,a),B,b),C,c,x(d,c),D,d)=F(SF(U,A,a,x(d,a),D,d),U,u,delta(c,b))

Let’s substitute x for delta, since that makes sense for compatible qualities:
SF(SF(U,A,a,x(b,a),B,b),C,c,x(d,c),D,d)=F(SF(U,A,a,x(d,a),D,d),U,u,x(c,b))

Since C and D are complete qualities, we know that if we sum over all their values, we have to get rid of those qualities:
Sum_C(Sum_D(F(F(SF(U,C,c,x(d,c),D,d),U,u,x(b,d)),U,u,x(c,a)))=SF(U,A,a,x(b,a),B,b)

For this to be possible, x() must have the following property:
x(a,c)=Sum_B(x(a,b) x(b,c))
The interesting thing about this is that if function x conforms to this property, it allows us to transform the effects of a query for A on B to the effects of a query for A on C. In other words, these functions relate all the various effects of the unknowable machine on various qualities to one another. For this reason, they are called transformation functions. Pretty important functions to know about.

It’s tempting to think that transformation functions are probabilities, since probabilities are the numbers we usually use to describe unknowable mechanics. But transformation functions aren’t probabilities.

Let’s say we have a function we can apply quality values called lambda(a), which returns a number, and that this number uniquely identifies the quality a. So we can invert lambda, such that lambda_invert(lambda(a)) gives us back a for all qualities and their values. It’s clear that for any such function:
SF(U,A,a,x(b,a),B,b)=F(F(SF(U,A,a, lambda(b)*x(b,a)*lambda_invert(a),B,b),U,u,lambda_invert(b)),U,u,lambda(a))

If the x’s were probabilities, all lambda(b)*x(b,a)*lambda_invert(a) would have to act like probabilities (stay between 0 and 1, and sum over all their values to yield 1). This can’t be the case for all invertible lambda, so the x’s can’t be probabilities.

What they can be is the square roots of conditional probabilities: p(b|a)=x(a,b)x(b,a) fixes the “lambda problem”. So, if we think that probabilities and transformation functions are both valid for describing unknowable mechanics, this is the way to unify the two ideas.

However, this means that x(a,b) must equal the complex conjugate of x(b,a). That is: the transformation functions map pairs of quality values to complex numbers.

Note that if you have x(a,b), you know p(b|a), but not the other way around. So, in some sense, the transformation functions are a more fundemental kind of knowledge about unknowable mechanics than probabilities. Since we are all used to thinking in probabilities, this is a rather disturbing fact: we’ve been thinking in the wrong space to get all the knowledge that is there.

So, we’ve identified a rather fundemental kind of knowledge we can have about the mechanics of machines we can’t know everything about. Since there are lots of qualities that have such mechanics, for instance:
-Polarizations that are at different angles
-Position and momentum
-Time and energy
-Value and ownership
-Current time and national product
this is pretty profound stuff. And we haven’t had to talk about sub-atomic particles at all to reveal it.

Next time we’ll talk about building toys with these ideas. Toys like Schrodinger’s Aibo.

This is certainly worth knowing about. Use for meetings, parties, whatever:

Meet-O-Matic: The World’s Simplest Meeting Scheduler

In the debates, both candidates claimed they’d cut the deficit in half (source)

As of today, Bush is officially failing to live up to this promise, or at least making things worse before they get better. And given that these things build up, it seems doubtful that he’ll live up to it later.

CBO projects budget deficit to be $368B in ‘05 - Jan. 25, 2005

Andrew (whose pictures are included in the link below) has recently evolved from being my long-time friend to being my current hero. (Don’t get a big head Andrew, I’ll probably have a new hero by lunchtime). But this guy struggled through some rough crap in his life, then settled down and worked really hard at a job he didn’t adore. But he saved up, and now he’s having many months on a South American adventure. He’s working while there to make it happen, as well. But the cool part is that he’s learned to climb mountains, dive the ocean, and he got to see the Galapagos.

Yahoo! Photos - andrewna1’s Photos - Imagen 9 011

On a more general note, isn’t it strange the The Galapagos plays such a quasi-spiritual role for so many of us? It’s sort of the Darwinian Mecca. Perhaps someday I’ll be able to make the haj. In the mean time, Andrew’s pictures are great.

Oh joy, I’m so glad I found this short film that I love online! Don’t read anything about it on the site until you’ve watched it!!!!

Channel 4 Film - Dual Balls film review, interviews, filmographies and more

As if you didn’t have enough ads in your life, I decided to share a couple of funny UK ads with those of you back in the states: KFC has made we chuckle twice this year. Click links below to watch.

KFC - The 99p Mini Fillet - Can I Have A Bite? , ad, advert | visit4info | TV/Cinema

KFC - The Variety Bucket Musical , ad, advert | visit4info | TV/Cinema

Unbelievable. The idea that race or gender should influence how much social secuirity you get. How could this have reasonably come out of someone in this position’s mouth?

CNN.com - Thomas: Social Security funding needs change - Jan 24, 2005
WASHINGTON (AP) — The chairman of the House tax-writing committee said Sunday that President Bush’s drive to overhaul Social Security should lead to consideration of a value added tax or other ways to fund the entitlement program.

Congress also should consider basing benefits on such factors as race, sex and the job a retiree once held, said Rep. Bill Thomas, R-California.

Via Metafilter

There are no big surprises here, but it has some good citations that illustrate that so-called Christians (at least in America) are failing to practive what they preach in statistically demonstrable ways.

Link here, quote from this article below:

This is the first from the category of silly crap that I feel almost required to circulate on the internet. There are at least 4 people in America who haven’t seen the real live dancing transformer from Citroen

.

Video can be seen at: Citroen UK Home Page

My personal opinion is that Cirtroen would move more units if at the end of this ad, the car destroyed Vancouver. Nothing against Vancouver, it’s just a good city for a dancing robot car to wreck havoc on. They should be proud.

Jenna Bush, America’s Underage Drinking spokesperson(tm) was seen riding with the devil at the coronation:

Southerners will know, this isn’t actually a hale to Jenna’s dark lord, it’s really the “Hook ‘Em Horns” gesture, symbolic of pride in The University of Texas, where the young and often sloshed Ms. Bush attends.

But the tie to religous practice is well observed. This kind of football-fueled allegence might as well be the full pentagram smearing, goat-slaughtering ceremonial belief. All you need is some guys who wear the same color clothes to follow around. If they participate in violence rituals, all the better to inspire the eyes-too-close-together crowd that Jenna represents. Winds them up good for fighting in wars , with the kind of “whoo-hoo, let’s kick some ass of somebody who dresses different from us” fervor that “Hook ‘Em Horns”, or “Roll Tide” or “Heil Hitler” really stands for. Those guys at Old Miss, they hate freedom, let’s ring their bells at homecoming.

Of course, this symbol flashing “spirit” is largely what “spirituality” is for in most people’s lives, as well. Give em an us group, and a them group, along with some slogans, rituals, funny hats and hand signs to tell the two crowds apart, and they’re as happy as chimps crapping in their hands and throwing it at zoo patrons.

Those who think people care about issues ought to try and remember what spending time with “Hook ‘Em Horns” types on a college Saturday night was really like.

Speaking of religious t-shirts:

You can buy this gem, and Jemma’s salute to Texsatan.

Buying tshirts is cheaper than tithing, anyday.

You wear your labels, I’ll wear mine, and we’ll both be able to tell who we might want to kill someday:

That reminds me, I’m out of loo roll. Sometimes I wish I could just wipe on my wardrobe. Better yet, your wardrobe, you freakin’ heretic.

*inpsired by a number of bOINGbOING posts.