Sunday, April 08, 2012

Brian Greene's Mulder and Scully Illustration Demonstrated Graphically

Greetings to all my readers!

It’s been a very long time since my last post on here, so I don’t expect I’ve retained any of my regular readers (if I ever had any), but I had something I wanted to talk about today, and I couldn’t think of any better place to post it (actually, I did post about this at a couple of forums, but don’t know if I’ll get any response).

Several years ago, when I first read Brian Greene’s “Fabric of the Cosmos”, I was deeply fascinated by an illustration he gives for Bell’s Theorem.

If you’re not a math whiz – or even if you’re not interested in math at all – I urge you to stay tuned, because this gets quite interesting.

Greene’s illustration involves Fox Mulder and Dana Scully of “X-Files” fame. If you’re not familiar with Fox and Dana (it’s been a while since “X-Files” was on the air), the only important thing you need to know is that Fox is fairly gullible and prepared to believe fantastic claims about aliens or ghosts or whatever, while Dana is more skeptical and scientific. She won’t believe anything without concrete evidence.

Now I will try to reproduce Greene’s illustration from memory. If I’ve gotten any part of it wrong, I apologize. Here it is:

Fox is traveling in Paris and Dana is home in Washington D.C., when one day each receives a large package from an anonymous sender. The packages each contain 1000 small cubes, and on each cube are three small doors – one on each of three sides.

The cubes are numbered from 1 to 1000, and each of the doors is labeled A, B and C.

A letter is included in the package that explains that the cubes represent an alien technology that uses faster-than-light communications. It goes on to describe how the cubes work, as follows:

Behind each door on each cube is a light that will briefly flash either red or blue when the door is opened. Before any door has been opened on a particular cube, the colors that it will show have not been determined. The moment the first door is cracked
open on a specific cube, that cube randomly selects the colors it will show behind all of its doors, and at the same moment instantly transmits its decision to its corresponding cube in the other set, so that when any of that cubes doors are opened; they will show the same colors.

So, for example, say Fox is the first one to pick up cube #1, and he opens door A and sees a red light flash. What the letter is saying is that at the moment Fox opened the door, the cube randomly chose a set of colors to show behind its doors (say it picked red for A, blue for B, and red for C), and at the same instant transmitted this information to Dana’s cube #1 so that when she opens doors on that cube, she’ll also see the pattern red/blue/red for doors A, B and C.

Each numbered cube in Fox’s set is paired with the corresponding cube in Dana’s set. The first one to open one of the six doors in a pair initiates the random color selection, as described. Thereafter, the colors for that pair are set and unchanging.

The letter also explains that the cubes will self-destruct if tampered with, so Dana and Fox are unable to immediately test these claims by breaking open the cubes and inspecting their internal workings.

Now, as soon as Fox reads this letter, he calls Dana on the phone, full of excitement and wonder about this amazing “alien technology” that some anonymous benefactor has given them. But Dana is skeptical. “This is obviously a hoax”, she says, “and not a very good one at that. The bottom line is; the cubes will show the same colors behind corresponding doors on each cube, no matter what we do. Regardless which one of us picks up a particular cube first, the outcome will be that we’ll both see the same colors behind its doors."

"If you see red behind door A on some cube in your set, I’ll see red behind door A on the same cube in my set. How convenient that the cubes are set to self-destruct if we take them apart to investigate the letter's claims directly. For all we can tell, the cubes might just be wired up with Christmas lights, with each cube in my set wired identically as the corresponding cube in your set."

"There’s no alien technology here,” Dana concludes, “they’re just cubes wired up with lights, and somebody is trying to make fools of us.”

Fox is not ready to concede though, and gives it some thought. “You’re right,” he says. “The outcome is the same whether the letter is telling the truth, and the cubes behave as it describes, or you are right and the cubes are just wired up inside with red and blue lights. The key question is this: are the colors we see pre-determined, as with identically-wired Christmas lights, or are they not pre-determined, but chosen by the cubes at the instant one of a pair is first opened? Is there some test we can devise that could settle this?”

After considerable discussion, they realize there is such a test. Here’s what they do:

Fox and Dana both hang up the phone and proceed to open one of the three doors on each cube, in order, from 1 to 1000. For each cube they pick a door to open at random. As they do this, they record the cube number, the door they opened, and the color they saw. When they’re done, they each have a list that looks something like this:

Cube Door Color
1 A red
2 C blue
3 A blue
4 B red
5 B blue
.
.
.

Then they get back on the phone and compare their lists. First they look at a few entries where they happened to pick the same door for the same cube number. In these cases, they verify that they both saw the same color behind the doors. This verifies that at least that part of the letter was accurate.

Next they compare colors straight across, cube-by-cube, regardless which door was opened. Fox has predicted that if the cubes are just hard-wired with lights, they will have seen the same color for the same cube less frequently than if the cubes behave as described in the letter.

So that’s the illustration Greene presents in his book. He goes on to explain the math, but it gets confusing after that. The upshot of it all is that Fox is able to prove with this test that the letter is telling the truth. While a pair of cubes remains unopened, its colors are not determined – they could take on any combination of red and blue lights for the three doors. The moment one of them opens the first door on a particular numbered cube, and only then, the colors are determined for that cube number.

The important thing about this, and the reason this is so amazing, profound and so full of deep meaning about the nature of the universe, is that this illustrates something called “Bell’s Theorem”. If you know anything about the history of physics, you might recall that in Einstein’s time the field of theoretical physics we call “quantum mechanics” was just getting going, and one of the theories it proposed (or rather, its underlying math predicted) was that the properties of an elementary particle, including its position in space, its velocity, etc., are not absolute, but instead are defined by a probability wave that determines only the probability that the particle will be found at any point in space, throughout the universe. Only at the instant a person measures a particle’s position does it assume a definite position.

The critical thing to realize is that, according to this prediction of quantum mechanics, before you measure the position of a particle, it’s not just that you don’t know its position, it doesn’t HAVE a position until you measure it! By measuring it, you force it’s hand, so to speak – you make it take on a position, for that instant.

Einstein famously dismissed this theorem initially, saying “God doesn’t play dice”. By this he meant that he believed that the things in the universe, from planets and stars down to people, dust motes, and elementary particles, have a definite and absolute existence apart from our actions. Their position in space, their velocities, and other properties are real, and exist whether we take the trouble to look at them or not.

We may not know the position of a particle until we measure it, but it has a position before we measure it. When we take the measurement, we are only finding out something that was true already. Things do not just appear the moment we look at them, and then revert back to probability waves” the moment we look away.

Imagine you’ve lost a valuable gold coin somewhere in your house. When you go to look for it, where will you look first? Obviously, you’ll start by looking in the places you reason that it’s most likely to be found. You will probably not start out by looking on top of the roof. You won't go straight to your bathroom and start tearing up the tiles on the wall to see if its behind them. You certainly will not worry that maybe it’s sitting at the bottom of a crater on the moon.

Suppose you eventually find the coin in the pocket of a suit you wore the week before. When you find it there, you will think “Here it is! This is where I left it, and it’s been here all this time!” It would be very odd if you thought “before I found it, while I was still looking for it, it might have been anywhere in the universe. It had a high probability of being found in several locations throughout my house, including this suit coat, but the instant I reached into this pocket, it materialized here for me to find!”

And yet, according to Bell’s Theorem, something like this is exactly what really does happen when we look at things!

So, back to Greene’s illustration. If you search online, you will find a lot of discussions on physics websites, where people have tried in various ways to demonstrate Greene’s Mulder/Scully experiment and have come to the conclusion that Greene is wrong: the percentage of times Fox and Dana’s lists will agree when compared cube-for-cube will be the same, whether the cubes are pre-wired as Dana suggests, or behave in the exotic fashion described in the letter.

I have read through several such discussions, and no one anywhere (that I can find) has been able to show the veracity of Greene’s illustration. Being the software engineer that I am, a few days after I first read the chapter containing this illustration, I wrote a program in C++ to attempt to demonstrate the experiment. Like Fox, I was full of wonder at the implications of what the book claimed. I wrote a program that simulated the test both with 1000 pre-wired boxes, and 1000 “quantum” boxes that behaved as the book described.

But like the many others that have posted on these websites, I was disappointed. It seemed that no matter how I looked at the data coming out of my program, there was no appreciable difference between the outcomes. The averages would hover around the same number – 66.6% -- for both tests. There was some deviation, of course, because there is randomness involved, but I could detect no significant difference in the results.

Then the other day I was thinking about it again, and decided that maybe I’d done something wrong in my original program (which I don’t have anymore). I decided I’d try again. This time, I wrote my program so that it would repeat the 1000-box tests (both hard-wired, and “quantum”) many times, and record the percentage agreement over each 1000-box test in an Excel spreadsheet. Specifically, my new program conducts each 1000-box test 10,000 times.

I then loaded the spreadsheet into Excel and started graphing things. Initially I graphed the numbers straight-up, over time, and got the disappointing-looking result below. The red line graphs results for the “quantum” box tests, and the blue line shows the “hard-wired” results.

The graph appears predominantly red only because Excel draws the red line over the blue line. The black line at the center (sitting right around 66.6) is the “trend line” that Excel draws for both lines. This was not very encouraging.

Then I did something different. I wrote some Excel macros to calculate, for each line, the average percentage up to that point. So for example, the second line averaged the results from the first two tests, the 10th line was the average of tests 1 through 10, and so on. Then I graphed this “cumulative average” and got the surprising, delightful image below.

Looking at that graph, it is clear, I think, that there really is a very slight, but discernable, difference in the outcomes. It amounts to only about 0.1% over the 10,000 tests, but there’s no doubt that the lines are not converging on the same number.

So what is the point of all this? My confidence in Greene’s presentation of the facts, and in the ideas about the nature of the universe that they imply, was shaken by my initial failed attempt to demonstrate them for myself. This second attempt, and the “cumulative average” graph above helps to reestablish my confidence, and with it my sense of wonder and fascination with those ideas.

What do you think? If you're interested in seeing my source code, I'll be happy to provide it (this one's written in C#) to any who ask.

3 Comments:

At 2:33 PM, Anonymous Terri Barrett said...

I read this whole thing thinking you were going to reach some fabulous conclusion about pre-destination vs. free will, but alas, you went all atypical on me!!! Intriguing article, though....

 
At 7:26 PM, Blogger MollyMom103 said...

You will have to explain this time when get home. Something about the universe is in here. Love you. M>

 
At 11:47 AM, Blogger bblais said...

Greene's example comes straight from another example, published many years earlier, by Mermin entitled "Is the Moon there when you don't look at it?" (see here: http://www.iafe.uba.ar/e2e/phys230/history/moon.pdf)

the colors are red and green in that example, and yes the result holds- but you need to do the math right. I have python code which you can see here: http://dl.dropbox.com/u/1813623/mermin.py

I think you can verify that it does actually work the way Greene and Mermin state.

Weird! :)

 

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