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Quantum mech, choices, and time travel too!

Friday, August 6th, 2010

Physicist: Recently I sent a series of emails back and forth with a reader that seem interesting enough to post. Conversations (near a chalkboard especially) are the best way to learn just about anything.

 

Q:
I wake up at 6am.
I brush my teeth get dressed and go downstairs.
I eat my breakfast at 7am and I’m just about to leave out the door.
when i notice there is an apple and an orange in a fruitbowl on a table next to me
at exactly 7:02am. I CHOOSE to take the apple.

I then make a choice whether I should choose to take my car or save gas and take the bus.
I CHOOSE to take the bus.

SUDDENLY. when i walk into the office, a strange event occurs and time starts moving backwards.
It goes back and back and back until finally it’s 6am in the morning and i wake up,
brush my teeth get dressed and go downstairs to eat my breakfast.

Here’s my tough quantum mechanics question for you,
at exactly 7:02am will I still choose to take the apple?

And if this process were repeated over and over and over again for about a million times. will the choice I make ALWAYS BE the apple?

 

A:
I’m working on a (to long) post about Bell’s theorem. The thought experiment you propose, about going back in time, is one of the better ways to understand it.

To actually answer your question: if choosing the apple is based on some quantum mechanical process in your brain (and there’s a good chance that at least some part of it is), then every time that choice is made the result is random. Time travel or not.
Part of the weirdness comes from the fact that every possible thing that can happen does. So (even when you time travel) some versions of you take the apple and some versions don’t.

 

Q:
Ok, remember how I took the bus in the thought experiment?

My question is, does quantum mechanics also apply in a reversal of time?

For instance, lets say that time started to slowly reverse.
Will I always get onto the bus backwards and head home.

OR.

will my car magically appear (even though i didn’t take it)
and will I backwards drive home in that?

So the concluding question is,
do quantum principles apply in a reversal of time as it does when time moves forwards?

 

A:
You’ll often hear “everything that can happen does” so if a particle can take two different paths it will actually take both.
If I understand your question correctly, the answer is yes. It turns out that “everything that could have happened did”. The “branching” goes both forward and backward in time. This is demonstrated by things like the “Franson experiment” that demonstrates the interference of a single photon with an earlier version of itself.
Driving a car, for example, will leave telltale signs that later make it impossible for you to have actually taken a bus. Chair fibers, leaving tire tracks, you’ll remember it, etc.
But if, in every way, you could have done either one, then you did both (no magically appearing cars).
This is actually the backbone of the Feynman path integral technique.

 

Q:
So are you telling me that just next to us, could exist a place where the Nazis won the second world war?
A place where there exists a flying spaghetti monster? (to quote richard dawkins)

Or even a place out there in a dimension somewhere where there exists an all knowing omnipresent, omnipotent, all encompassing being who “watches over us” etc..

 

A:
Sure. BUT, it’s impossible to interact with things that are even a little bit different. For example, a stream of identical photons (lasers) will all interact with each other strongly. You can see evidence of this in effects like speckling. Non-coherent (regular) light is made up of all kinds of different photons, and the best way to figure out how they’ll behave is to assume that they’ll ignore each other. This is sort of a metaphor, and sort of a concrete example.
So while, yes, there are almost certainly universes where the Nazis won, it doesn’t matter. It’ll never have any impact on our universe whatsoever.
A good rule of thumb is: if there is any conceivable way, whatsoever, for anything to tell the difference between universes, then they can’t interact (from the perspective of that thing that can tell the difference).

 

Q:
Couldn’t that then solve the entire God dilemma? I mean if in only one of these infinite dimensions there existed an all encompassing all knowing all powerful entity, wouldn’t this entity then transcend all dimensions? (since he is all encompassing)

 

A:
If you want to consider God, then it’s best not to do it in any kind of physics based context. That being said:
Remember that if two universes are even slightly dissimilar, they won’t interact at all. By “slightly dissimilar” I mean something like a single electron being conspicuously out of place.
So any existing Gods that follow the most basic laws of logic and quantum mechanics will be stuck in their native worlds.
If you’re not worried about Gods that follow physical laws, then, again, physics is literally the worst possible forum.
Also, you have to be careful with this kind of reasoning. You can make up just about anything and claim that it should exist in every version of the universe.
The rule “anything that can happen does” carries a bit more heft that it seems to at first. If something can’t happen, then it doesn’t happen in any version of the universe.
For example, spaghetti can neither fly nor think, so the FSM (pasta be upon him) can’t exist in any universe, no matter how much anyone dresses like a pirate.

Posted in -- By the Physicist, Philosophical, Physics, Quantum Theory | 2 Comments »

Q: Why is the speed of light finite?

Tuesday, August 3rd, 2010

Physicist: That is such a hard question.  Holy crap.

If you kept the laws of the universe the way they are, but ramped up the speed of light to infinity you’d end up with a surprising array of effects.  Newton would have been right about a lot more (nicely done, old dude), there would be no magnets of any kind, the amount of energy tied up in matter would also be infinite (E=MC2) so you’d have to be extra careful not to bring it near anti-matter, but not too careful because anti-particles probably wouldn’t exist (probably).  Also, all the weirdness of relativity would be out the window.

But, why is the speed of light finite?  I don’t know.  I think this is one of those culdesacs of science.  It is what it is.

The question, as it was originally asked, was about what keeps light from going any faster.  The answer to that question is that there is no faster.  If you shove a stone of mass X and it goes flying off at speed V, then if you shove a stone of mass X/2 it’ll fly off at speed 2V.  So, you might suspect that if you shove a stone of zero mass that it would go flying off at an infinite speed.

Well, that’s pretty much what photons (which have zero mass) do.  If you think of infinite speed as how fast you’d be going if you accelerated forever, then the speed of light is exactly that.  If you got into a rocket that could accelerate forever (using some kind magic fuel, such as the Schwartz), and you let it run for an eternity of two, then you’d be moving at the speed of light.

So it’s not that there’s anything slowing light down, so much as the laws of the universe are such that it doesn’t really make sense to talk about something moving faster.  More here:

Q: What’s it like when you travel at the speed of light?

Q: Why is the speed of light the fastest speed?  What makes light so special?

Also, if you’d like to find more “culdesacs of science” get yourself a toddler during their “Why?” phase, and try explaining something to them.

Posted in -- By the Physicist, Philosophical, Physics, Relativity | 1 Comment »

Q: Is it possible to beat the laws of physics?

Monday, July 26th, 2010

Physicist: No…
But to be fair, when a physical law is beaten it stops being a law.

Posted in -- By the Physicist, Paranoia, Philosophical, Skepticism | 1 Comment »

Q: Aren’t physicists just doing experiments to confirm their theories? Couldn’t they “prove” anything they want?

Thursday, July 22nd, 2010

The original question was: When we start investigating particles and effects at the quantum level, it seems we are not really measuring the reality of the particles, but rather, our instruments’ reactions to the particles.  So if we calibrated the instruments differently, wouldn’t we get different, perhaps contradictory, results?  It seems that physicists first construct mathematical models, then devise instruments to find just what they are looking for.  That seems to be the same as saying I’m going to make an instrument to find leprechauns, and lo and behold I did it!  Of course, no one can really see the leprechauns, but my instrument says they are there.  You can even make an identical machine and you will detect leprechauns, too.

Physicist: Every observation is nothing more than our instrument’s reactions (Quantum mechanical, or otherwise).
A measurement device that produces foregone conclusions is useless (it provides no new information) so nobody builds them.
While physicists do construct models, and then construct devices to test those models, the primary purpose of the devices is to tear down the models and equations.  Once done, physicists “close the circle” by coming up with new models.
If all we (scientists and Humans too) did was build machines that verified our crazy theories, we’d still be stuck in the stone age, having proven conclusively that everything is controlled by dead people and shamans.

For example: the early mathematical models behind quantum physics and relativity predicted a wide array of very bizarre things (quantum tunneling, super-position, time dilation, stretched spacetime). In an attempt to prove the model wrong (because no one believed it), dozens of different experiments were set up. Every experiment had at least two possible, different results (either disproving, or corroborating the theory).

In fact, this is one of the most basic results from information theory; the more you can anticipate a result, the less information you gain from it.  This is why 6th grade science experiments are pointless (except for edjucation or whatever), and why botched experiments and accidental discoveries are so useful.
As it happens, both the relativity model and quantum model held up to experiment, and so we still use them today.
Conversely, the theory of “luminiferous eather” (the idea that light is a wave in some kind of hidden material) was very popular and held for decades in the late 19th century. However, it wasn’t supported by experiment and so (despite its popularity) it was abandoned.
Admittedly, it’s easy to get tunnel vision with your subject, and even dismiss actual results as statistical noise.

My favorite example is a group of German scientists in the late 19th century who accidentally discovered electron diffraction (proof of the wave nature of electrons) when they were trying to measure the deflection of electrons off of a crystal.  The diffraction effect caused the electrons to come out of the crystal only at a small set of angles, thus saturating the film being used detect the outgoing electrons at points, instead of smoothly all over.  The German-tunnel-vision-solution?  Buy a “jiggler” to move the film around so that the overexposed points become reasonably exposed blurry patches.

Electron crystalography. It should be bright in the middle and then get dimmer toward to edges. But then stupid quantum mechanics makes everyting all "pointy".

But tendencies like tunnel-vision or “going with what everyone else thinks” are generally overwhelmed by a positive and contrary result.
For example: You could spend your whole life describing, in detail, the physics of a flat world. But the second someone travels all the way around the planet, all of your theories are instantly useless.

Posted in -- By the Physicist, Philosophical, Physics | No Comments »

Q: Will there always be things that will not or cannot be known?

Wednesday, June 30th, 2010

Mathematician: Unfortunately, limits to knowledge seem to be built into the nature of the universe, and even into logic itself.

Relativity: Einstein’s theory of special relativity implies that no information can travel faster than the speed of light. That means that information from sufficiently recent, sufficiently far away events will not have had the time to propagate to us yet, making detailed knowledge of such events impossible. In physics speak, we say that these events are outside of our “past light cone“, “space-like separated” from us, or just “elsewhere”. As long as new events of this type keep happening, there will always be things about which we do not and cannot know.

Quantum Mechanics: The Heisenberg uncertainty principle states that the uncertainty \Delta x we have in a particle’s position and the uncertainty \Delta p we have in the particle’s momentum cannot both be very small at the same time. In particular, the product of these uncertainties is greater than a constant (\Delta x \Delta p > \frac{\hbar}{2}). This implies a fundamental limit to the knowledge that is possible because we can know x accurately or p accurately, but not both.

What’s more, the vast majority of physicists agree that quantum mechanics demonstrates the universe is random at a fundamental level. This means that some events, like the time at which an atom will decay, can be predicted only probabilistically. We can say how likely an atom is to decay in a given time interval, but we will never be able to say precisely when the decay will occur, placing another limitation on what knowledge is possible. (Physicist’s note: After the decay you still can’t say when exactly it happened because according to quantum mechanics the exact time doesn’t actually exist!)

Mathematics: Gödel’s  first incompleteness theorem states (essentially) that any mathematical system  that is able to express elementary arithmetic (and doesn’t contain any contradictions) must contain true arithmetical statements that cannot be proven within that system. Essentially this implies that there will always be true mathematical statements that we cannot prove.


Add to all of these theoretical considerations the enormous (and possibly infinite) number of things that could be known about our physical universe, and the (most definitely) infinite number of true mathematical statements that could be known, and it is clear that there will always be knowledge that is beyond our reach.

Posted in -- By the Mathematician, Math, Philosophical, Quantum Theory, Relativity | 2 Comments »

Q: How it is that Bell’s Theorem proves that there are no “hidden variables” in quantum mechanics? How do we know that God really does play dice with the universe?

Tuesday, June 22nd, 2010

Physicist: Bell’s theorem, and its philosophical fallout, is one of the most profound discoveries since relativity.

Bell’s theorem states (among other things) that the universe is fundamentally unpredictable, and that quantum mechanical things (for example: everything) are not actually in one state.  If a box could contain either a blue marble or a red marble, then when you open it you’ll see either on or the other.  In “reality” it was one color or the other before you open the box.  In QM, it can be both before you open the box (it’s actually still both afterwords, but moving on…).

Einstein (and most other physicists of the time) believed that if you knew everything about a system of particles (no matter how big) that you could theoretically predict what that system will be doing in the future, perfectly.  Homeboy thought that the only reason that the movement of air molecules seems to be random, is that we can’t perfectly measure that exact position and velocity of every single one.  So he thought that every particle truly is in some particular state, but that we merely don’t know for sure what that state is (the marble in the box has only one color, but we don’t know what it is).

The idea that randomness and unpredictability are caused by unknown (or unknowable) things is called “hidden variable theory” (The ‘Stein believed in this).  For example; 2, 2, 3, 6, 0, 6, 7, 9, 7, 7, 4, 9, 9, … is not random, but seems random.  It would be really hard to predict the next term (7) if you don’t know the hidden variable.  (BTW, the “hidden variable” is: this is the decimal expansion of \sqrt{5})

Bell’s theorem essentially boils down to a proof that the result of an experiment doesn’t exist until the measurement is made (so it can’t be predicted).  Hidden variable theory presupposes that the particles involved are in definite states, which means that the result of a measurement already exists before the measurement is made.  For example: before you open a gift what you’ll see is already set in stone.  The gift is a set thing before you open the box.  This is not the case for most quantum mechanical systems.

Here’s one of the experiments that demonstrates Bell’s theorem, and two ways to look at it.

An entangled pair of photons is created and fired in opposite directions. En route the polarizers are randomly oriented, then the detectors measure whether or not the photons pass through. This is done hundreds of thousands of times to measure the relationship between 1) the difference in angles between the polarizers and 2) the probability of measuring the same result.

The experiment: Step 1: Create a pair of entangled photons and fire them in opposite directions.  Entangled particles always yield the same result when they are subjected to the same measurement, and are likely to yield the same result for similar measurements.

Step 2: Randomly orient the polarizers, after the entangled pair is created, but before either is detected (this is hard to time, and is really fast).  This is done so that the photons “don’t know what to expect” and “can’t compare notes”.  Information about polarizer A would have to travel faster than the speed of light to get to photon B before photon B hits it’s own polarizer.  So, without faster than light effects (which don’t exist for many, really good reasons) the photons are each acting independently.  The orientation is random so that the photons can’t “plan ahead”.

Step 3: Measure the polarization.  If the detector “clicks” then the photon made it through the polarizer, and therefore has the same polarization.  If the detector doesn’t click, then the photon had the opposite polarization and was stopped.

The probability of the measurements being the same (for an entangled pair) is P = \cos^2{(\theta)}, where \theta is the difference in angles between the polarizers.  It is tricky to see why, but this probability is impossible if you assume that the result of a measurement exists before the measurement is made.  Here’s why.

The possible polarizations for polarizer A (red) and polarizer B (blue).

Algebraic approach: Restricting the possible angles of the polarizers to 0° and 45° for A, and 22.5° and 67.5° for B, run the experiment. Here’s what’s about to happen:

1) If you could predict the outcome of each version of the experiment, then you could find a definite value of L (see below).

2) For strictly (unarguable) mathematical reasons L = ±2.

3) Experimentally we find that the average value of L is 2√2.

4) But this is a contradiction, so we cannot actually make useful predictions.

Now it’s happening:

If polarizer A is at 0° and the detector clicks then you’d say “A0 = 1″, and if the detector doesn’t click then “A0 = -1″.  Similarly, you can define B67.5, A45, and B22.5.  Just for the hell of it, take a look at: L = A0B22.5 + A45B22.5 + A45B67.5 - A0B67.5 = (A0 + A45)B22.5 + (A45 - A0)B67.5

L = (A0 + A45)B22.5 + (A45 - A0)B67.5 = ±2, since either (A0 + A45) = ±2 and (A45 - A0) = 0, or (A0 + A45) = 0 and (A45 - A0) = ±2.  So L = A0B22.5 + A45B22.5 + A45B67.5 - A0B67.5 = ±2 ≤ 2.

So if you could fill out each of these values (A0, A45, B22.5, B67.5), then L = ±2 ≤ 2.

However, you can’t make all of these measurements simultaneously, so you can’t actually get A0B22.5 + A45B22.5 + A45B67.5 - A0B67.5 for each run of the experiment.  The best you can do is find one of these four terms each time you run the experiment.  For example, if the polarizer A was randomly set to 45° and the detector clicked, and polarizer B was randomly set to 22.5° and the detector didn’t click, then you just found out that A45B22.5 = (1)(-1) = -1 for that run.

You can however find the expectation value by running the experiment over and over and keeping track of the results and polarizer orientation.

E[A0B22.5] + E[A45B22.5] + E[A45B67.5] – E[A0B67.5] = E[A0B22.5 + A45B22.5 + A45B67.5 - A0B67.5] ≤E[|A0B22.5 + A45B22.5 + A45B67.5 - A0B67.5|] = E[2] = 2.

So E[A0B22.5] + E[A45B22.5] + E[A45B67.5] – E[A0B67.5] ≤ 2.  This is one version of “Bell’s Inequality”, and it holds if each term (A0, A45, B22.5, B67.5) has a value.

Using the fact that the chance of getting the same result is P = \cos^2{(\theta)}, and that each term is 1 when the results are the same ((1)(1) or (-1)(-1)), and -1 when the results are different ((1)(-1) or (-1)(1)), you can calculate each term.  For example:

E[A_0B_{22.5}]=P(same)-P(different)=\cos^2{(22.5)}-(1-\cos^2{(22.5)})=\frac{1}{\sqrt{2}}

You’ll find that:

E[A_0B_{22.5}]+E[A_{45}B_{22.5}]+E[A_{45}B_{67.5}]-E[A_0B_{67.5}]=\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}-\frac{-1}{\sqrt{2}}=2\sqrt{2}

Holy crap!  2\sqrt{2}>2!  But that’s a violation of Bell’s inequality!  But the existence of each measurement (whether or not you actually do that measurement) is all you need for Bell’s inequality!  So if the inequality is false, then the result of those measurements don’t exist if the measurement isn’t made!

God plays dice with the universe.

Maybe, if you're clever and have ready access to a time machine, you could go back and do all the measurements you didn't make the first time. Then all the results would have to exist! They'd just have to!

Me and my time machine vs. quantum mechanics: If the results exist, but you just didn’t happen to do all the measurements, why not get a time machine?  Then you could do one measurement, go back, do a different measurement, go back, do a different measurement, …  Then every possible result would be known.

However, once again that correlation probability (P = \cos^2{(\theta)}) screws things up.

So, for example, if the photon goes through at 50°, and then you go back in time, change the polarizer to 51°, and repeat the experiment, then there’s a 99.97% (cos2(1°) = 0.9997) that the photon will go through again.

One result from probability says that P(x=z)\ge P(x=y)+P(y=z)-1.  Do this twice and you get P(w=z)\ge P(w=x)+P(x=z)-1\ge P(w=x)+P(x=y)+P(y=z)-2.  So if you measure in the 0° direction to find A0, then go back and change the angle by 1° and repeat this until you’re measuring at 90°, then:

P(A_0=A_{90})\ge P(A_0=A_1)+P(A_1=A_2)+\cdots+P(A_{89}=A_{90})-89 =90\cos^2{(1^o)}-89=0.9726

So, if you go back and forth in time to measure whether or not the photon goes through at 1° increments, then there’s a 97% chance that by the time you get to 90° you’ll be getting the same result you did at 0°.  However, in reality P(A_0=A_{90})=\cos^2{(90^o)}=0.

But this is a contradiction.  So the results of each measurement (A0, A1, A2, …, A90) can’t all exist.

If I had to guess, every time you go back in time the experiment is completely reset, and the experiment becomes completely random again.  The reason (such as it is) is below this unsettling picture.

Wait. Wait... Why?

But why?!: It turns out that the reason that the results of a quantum event can’t be predicted, is that every possible result of that event plays out.  So if you ask “will I see the photon go through the polarizer?” the answer is “yes, some versions of you will see the photon go through” and an equally valid answer is “no, some versions of you will not”.

If different versions of you will see every possible result, then the result can’t be predicted, and doesn’t really exist one way or the other until after the measurement is done.  At that time the different versions of you will disagree on the result.  But don’t worry too much.  You’ll never meet you’re parallel-universe twins.

Posted in -- By the Physicist, Philosophical, Physics, Probability, Quantum Theory | 2 Comments »

Q: What do complex numbers really mean or represent?

Monday, May 3rd, 2010

Physicist: Nothing really.

Complex numbers are very useful for streamlining a lot of different types of math, generalizing ideas, and “closing” the real numbers.  In quantum theory you’ll find that on the most fundamental level the universe seems to prefer complex numbers to real numbers.

But you can’t use them to count or measure stuff for crap, so most people can lives long happy lives without being particularly bothered by complex numbers.  If you can call that living.

Mathematician: There are many ways to view complex numbers, but one of the most intuitive is to think of them as representing points in the plane. Doing so will allow us to interpret basic arithmetic operations like addition and multiplication as performing geometric manipulations.

How does this work? Well, every complex number z can be written as z = x + i y  where x and y are real numbers, and i is the square root of -1. We can then think of z as a point on the (x,y) plane with x being the position along the horizontal (i.e. real) axis, and y the position along the vertical (i.e. imaginary) axis. To understand the operation of adding and multiplying complex numbers though, we need to think about them slightly differently.

The complex number 2+3i represented as a point in the complex plane. Have you ever seen a more boring image in your life?

If we choose, we can treat complex numbers as vectors rather than points. That means that now z = x+i y  represents not the point (x,y) but rather, the directed line segment which extends from (0,0) in the complex plane to (x,y) in the complex plane (unless otherwise stated we will assume here that our vectors are emanating from the origin (0,0)). You can think of a vector as simply representing a magnitude (or length) and a direction. The line segment extending from (1,2) to (-4,5) is the same as the one extending from (-4,5) to (1,2), but the vectors represented would be different because they would be pointing in opposite directions. Vectors are often drawn as arrows.

The complex number 2+3i represented as a vector in the complex plane. Compared to our last image, this is a hoot.

Once we have the vector notion of a complex number, we can think about adding complex numbers as adding vectors. For example, if we have

z_{1} = x_{1} + i y_{1} z_{2} = x_{2} + iy_{2}

then

z_{1} + z_{2} = (x_{1}+x_{2}) + i (y_{1}+y_{2}).

Hence, z_{1} + z_{2} is the vector in the complex plane that extends from (0,0) to (x_{1}+x_{2}, y_{1}+y_{2}). Note that (x_{1}+x_{2}, y_{1}+y_{2}) is the point we get to if we piggy back vectors z1 and z2 and then follow them both. By this I mean that we translate (i.e. move without rotating or stretching) z2 so that its beginning is at the end of z1, and then we follow the path consisting of the two vectors until we get to the (new) end of z2. However, since z_{1} + z_{2} = z_{2} + z_{1}, we can also piggy back z1 at the end of z2 to get the same result. A slightly different view is achieved if we think of our first number z_{1} as being a vector, and our second number z_{2} as being a point. In this case, z_{1} + z_{2} simply corresponds to moving the point z_{2} by the distance and direction represented by the vector z_{1}.

Here are two complex numbers represented as vectors, 2+3i and 1+3i.

When we sum the complex numbers 2+3i and 1+3i we get the complex number 3+6i.

Alright, so addition of complex numbers can be thought of as adding vectors in the complex plane (or moving a point by the distance and direction stored in a vector), but what the heck does multiplication do? Well, to understand multiplication we need yet another geometric way of thinking about complex numbers.

First though, we make a quick (and relevant) aside. For any complex number z, we have by definition that the absolute value |z| of z satisfies

|z| = \sqrt{z \overline z} = \sqrt{(x+iy)(x-iy)} = \sqrt{x^{2} + y^{2}} = \sqrt{(x-0)^{2} + (y-0)^{2}}

which is precisely the formula for the distance between the point (0,0) and the point (x,y). Hence, |z| measures the length of the vector that z represents. Now, according to Euler’s formula, we have that for any real number \theta

e^{i \theta} = cos(\theta) + i sin(\theta).

hence e^{i \theta} is a complex number since it is the sum of a real and imaginary number. In particular, we have

|e^{i \theta}|^{2} = e^{i \theta} \overline{ e^{i \theta} } = e^{i \theta} e^{-i \theta} = e^{i \theta - i \theta} = e^{0} = 1

That means that any complex number representable in this form must have a distance of 1 from the origin. In other words, such numbers represent points on the unit circle, or equivalently, vectors of length 1 extending from the origin. As it turns out, all vectors in the complex plane with length 1 can be represented in this form. But what angle does each of these vectors point at? Well, we compute the angle of the line segment (0,0) to (x,y) with respect to the horizontal axis using arctan(y/x). In our case though, since x=cos(\theta) and y = sin(\theta) this gives:

arctan(y/x) = arctan(sin(\theta)/cos(\theta)) = arctan(tan(\theta)) = \theta.

Hence, the complex number e^{i \theta} represents a vector of length 1 pointed at angle \theta. Similarly, for any non-negative number r, we have that the complex number

r e^{i \theta}

represents a vector of length r pointed at angle \theta . This provides a new way of thinking about a complex number: as a vector specified by its length and angle. What happens when we multiply two such numbers z_{1} = r_{1} e^{i \theta_{1}} and z_{2} = r_{2} e^{i \theta_{2}} together? Well, we have

z_{1} z_{2} = r_{1} e^{i \theta_{1}} r_{2} e^{i \theta_{2}}

= r_{1} r_{2} e^{i (\theta_{1} + \theta_{2})} .

Hence, z_{1} z_{2} is a new complex number representing a vector of length r_{1} r_{2} pointed at angle \theta_{1} + \theta_{2}. Therefore, we can think of the action of z_{1} multiplying z_{2} as causing the vector z_{2} to stretch by a factor of r_{1} and rotate by an angle \theta_{1}.

Hence, to recap, we can view complex numbers geometrically as representing points or vectors in the complex plane. If we do this, then adding complex numbers corresponds to adding together vectors, or equivalently, moving the point that the second complex number represents along the vector that the first complex number represents. On the other hand, we can think of multiplication of complex numbers as corresponding to scaling and rotating the second complex number in the multiplication by the length and angle inherent in the first complex number. Finally, we note that taking the absolute value of a complex number corresponds to measuring the length of the corresponding vector. Therefore, one way to view complex numbers is as a means for converting geometric operations (translation, rotation, scaling) into algebraic operations (adding, multiplication) and back again. As you might imagine, this can be extremely useful!

Posted in -- By the Mathematician, -- By the Physicist, Math, Philosophical | 2 Comments »

Q: Is it odd that the universe’s constants are all so perfectly conducive to life?

Friday, April 30th, 2010

Physicist: Maybe.

When written down, most physical laws involve at least one physical constant.  For example, the “G” in gravitational force: F=\frac{Gm_1m_2}{r^2}, or the “h” in the energy of photons: E=hf, or the speed of light: “c”.  There are a couple dozen other (more and more obscure) constants out there.  Changing these constants changes how the universe hangs together, which forces are most important, what chemical elements are possible, how (and if) things interact, whether or not stars exist and for how long, etc.

None of the constants have any reason to be what they are, and not something else.  Sure, G=6.673 × 10-11m3kg-1s-2, but why?  Why not G=2 m3kg-1s-2 or something?

What’s really spooky is that even tiny changes in most of the physical constants tend to make life, and even the universe as we know it, impossible.  Some research (computer simulations mostly) has recently suggested that there are completely different combinations of constants that lead to universes that are unrecognizably strange but still capable of supporting highly complicated systems (and so, possibly life).

Here are my two most fave theories about why the universe is so nice:

If there are plenty of universes: you can use the anthropic principle.  The anthropic principle can be used to justify effects that require an observer.  For example: “you are here” signs are always correct when you’re there to see them, but are always wrong when you’re not.  You can sometimes find people excited about how perfectly the universe (Earth in particular) is suited to Human life, and admittedly the fit is pretty good.  But you can justify that fit using the anthropic principle; if you’re going to find Human life somewhere in the universe, do you really expect to find people on an acid world or some kind of lava monster world?

Mall directories: how do they always know where you are?

To get all the constants just right you’d need a huge (probably infinite) number of universes.  Turns out that picking a number truly at random is tricky.  If you have enough universes, then many of them will have the right balance to allow life, no matter how unlikely it is.  Now everyone capable of asking the question will find themselves asking it in a universe perfectly suited for life, despite the fact that the overwhelming majority of universes are completely inhospitable.

The idea that there are effectively (or actually) an infinite number of universes is not new to physics.  All quantum mechanical systems (which is everything, really) exist in every possible state simultaneously.  However, all of these states use the same physics, so the jump to every possible universe with every possible set of physical laws is a pretty big jump (“Powellian” even).  But, sadly, we don’t know just a hell of a lot about universe creation.  It might be completely reasonable for new universes to have different physical constants.  We’ll have to create a few trillion to know for sure.

If there’s one universe: you don’t get to use the anthropic principle.  But how about this:  When you create a batch of particles (using accelerators mostly) the stuff that flies out is in the highest entropy it can manage.  You’ll see lots of different kinds particles, in different states, flying in different directions, and all of it is random.  Low predictability = high entropy.  The exact results of particle creation are impossible to predict.

What if the creation of the universe followed the same rule?  That a universe with its constants tuned for very high entropy is more likely to be created.  The tendency of things to have high entropy is independent of the constants involved, so it’s not completely unreasonable to apply a rule like this, merely very unreasonable.  Looking around at the universe today, it would seem to be set up to maximize its own entropy.  Chemicals of amazing complexity are possible, there are almost a hundred natural elements (which requires crazy balance), there are dozens of different types of exotic particles that blink in and out everywhere all the time.  Very unpredictable, very high entropy.  By contrast, if you kept everything the same, but changed the mass of the proton from 1.673 × 10-27 kg to 1.675 × 10-27 kg (0.2% increase) you’d find that all the universe’s protons would turn into neutrons in short order.  No more chemicals (or even elements) of any kind.  Very predictable, very low entropy.

It may be that the universe is the way it is just to be as complicated as possible.

Posted in -- By the Physicist, Philosophical, Physics, Quantum Theory | 5 Comments »

Q: How/when will the world end?

Monday, April 26th, 2010

Physicist: To answer this question definitively would require the destruction of at least a couple dozen other worlds.  But failing that, guesswork:

The little things (people): In the short term (less than several million years) the biggest threat the Earth faces is people.  We’ve already got the Holocene mass extinction going for us, now we’ve just got to step up our game and go for broke.  Hey, Coalition of the Willing!  I think I saw North Korea stealing your cup cakes.  Also, it’s too cold in the winter.  Couldn’t we burn a teraton of coal or something?

Boring, Regular Extinction: It we (Homo sapiens) follow the same fate as all of our predecessors and cousins (homo habilis, rudolfensis, georgicus, ergaster, erectus, cepranensis, antecessor, heidelbergensis, rhodesiensis, and neanderthalensis, for example), then it’s very likely that we’ll be extinct within the next 100,000 to 1,000,000 years.  Statistically speaking anyway.

Total carbon re uptake: Over very large time scales the sun is getting brighter (along the lines of about 10% per billion years).  Astrophysicist Brownlee and paleontologist Peter Ward have written a book espousing the idea that this gradual brightening will cause the Earth to heat up and the natural chemical processes that absorb CO2 from the air (and lock it away in sediment) will speed up.

"Main sequence stars", which include the sun, are surprisingly stable for a very long time. They do change a little, increasing their brightness by about 10% every billion years.

They figure that inside of 500 million years there won’t be enough CO2 in the atmosphere to support plant life, and that would be the end of complex life.  Although life has been around for at least 3.5 billion years, the interesting stuff (animals) have only been around for about 500 million years.  So if Brownlee and Ward are right, we’re only about halfway done (not nearing the end).

It may seem strange to talk about the loss of CO2 being the end of the world when we so often talk about the dangers of too much CO2.  The difference is in the time scales.  The spike of CO2 we worry about today is on the scale of centuries, while the long term absorption of CO2 is on a time scale a million times larger (unnoticeable in the short term).

Dynamo shutdown: The Earth’s magnetic field is the result of iron rich (electrically conductive) stuff flowing around in the Earth’s core.  The currents are driven by radioactive heating which causes convection, specifically the decay of radioactive potassium, uranium, and thorium.  The half-lives of these materials are 1.25 billion, 4.5 billion, and 14 billion years respectively, so most of the original fuel has already been used up.

The exact nature of magnetic dynamos is not terribly well known, and is still an active area or research.  We don’t know for certain what the minimum energy input is needed to keep the damn thing running.  We do know that it’s certainly possible for a planetary magnetic dynamo to shut down (Mars’ shut down at least a couple billion years ago).  If our dynamo shuts down, then our magnetic field will vanish and (in fairly short order) the atmosphere will be stripped away by solar wind, as happened on Mars.

Never-ending Summer: The increase in the Sun’s output will make it too hot for liquid water on Earth in about 1 billion years.  With the oceans boiled away the pressure everywhere on Earth will be about the same as the pressure on the ocean floor.  The difference between Venus and Earth will be academic.  No matter what else happens before then, this will be the end of life on Earth.

SPF 5,000,000,000,000,000: Somewhere around 5 to 7 billion years from now the Sun will start to run out of fuel.  Ironically this will actually make the core hotter as it collapses in on itself.  The top layers will fluff up and (probably) envelope all the inner planets, including Earth.  For obvious reasons this is called the “red giant” phase of the Sun’s life.  The solar system will eventually settle down with the gas giants still in place, the inner solar system missing, a white dwarf star where the Sun used to be, and the trans-neptunian stuff completely unaffected.

The Sun as we see it today (yellow), and the Sun in its fluffier red giant phase (red).

Lights out: If by “end of the world” you mean “end of the universe”, then a good end of everything is the end of the age of stars.  The universe started out made up of about 75% hydrogen, but today is only about 70% hydrogen.  Stars are almost completely powered by hydrogen fusion, so assuming that the consumption of the universe’s hydrogen is stays constant (which isn’t a particularly good assumption), then there will be almost no stars left in 250 to 300 billion years.

The big rip: Not only is the universe expanding, but the speed of that expansion is increasing.  The expansion is a little hard to picture because the expansion isn’t about things moving away from each other in space, it about the space in between things actually expanding.  Right now the effect is small enough that it can only be seen on huge, inter-galactic, scales.  But eventually the expansion with be so rapid that the space between the planets and their stars will increase so fast that the planets will be pulled into open space, and not long after than (as in a couple of months or so) the space between atoms will increase so fast that everything will be completely torn apart and atomized.  This is called the “big rip”.  Some estimates put the big rip about 20 billion years out, and some say it won’t happen at all.

Posted in -- By the Physicist, Astronomy, Biology, Paranoia, Physics | 1 Comment »

Q: What would happen if an unstoppable force met with an unmovable, impenetrable object?

Thursday, April 22nd, 2010

Mathematician: Sometimes, when we don’t use language carefully enough, we can get ourselves into philosophical trouble. For example, consider the following statement:

If a barber shaves all those men (and only those men) who do not shave themselves, does he shave himself?

If the barber shaves himself, then he is shaving a man who shaves himself, which is something that (by definition) he does not do. On the other hand, if the barber does not shave himself, then there is a man who doesn’t shave himself that the barber doesn’t shave, which again contradicts our definition of the barber.

So what is the answer? Well, the question has no answer, because the definition we use for our barber contains within it a logical contradiction. What’s more, it is impossible for such a barber to actually exist in the real world, since the razor burn associated with simultaneously shaving yourself and not shaving yourself is too much for any single person to withstand.

Now, let’s return to the original question:

What would happen if an unstoppable force met with an unmovable, impenetrable object?

Well, let’s suppose that we define an “unstoppable” force to be one that can move absolutely any matter. Furthermore, let’s define an “unmovable” object to be one that cannot be moved by any force. In that case, this question is unanswerable, because like the barber paradox above, it relies on contradictory information. By definition our force can move anything, but then, also by definition, there is an object that the force cannot move. This is a bit like saying “suppose X is true, and not X is true. Then is X true?”. Here  X is the idea that the force can move anything, and not X is the idea that there is at least one object that cannot be moved by the force (which in this case is our unmovable object). Hence, this question has no answer because it relies on assumptions which contradict each other.

Posted in -- By the Mathematician, Brain Teaser, Philosophical | No Comments »

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