|Cheshire Cat knows where infinities go.|
In the previous post on this subject, I somewhat tentatively asserted that the sum of the natural numbers up to infinity (1+2+3+4+ . . .) does not equal -1/12.
So what does it equal? The answer, at least as far as realistic problems I can wrap my head around goes, is -1/12 plus infinity.
Finding -1/12 in the first place
As the Numberphiles point out, there are many ways to associate the sum of the natural numbers with -1/12. To name just a few . . .
|This graph offers a simple way to associate -1/12 with 1+2+3+4+ . . .|
- There's the video Numberphiles made that shows how you can play some potentially dicey games with series manipulations to come up with the equation.
- There's the slightly more rigorous way Euler did it, which the Numberphiles document in a second video on the topic.
- There's the simple way I did it in a previous post that involves nothing more than a little algebra, a trivial amount of calculus, and a graph (like the one here) that you can draw based on the sum of the natural numbers (or any other series consisting of the sum of the natural numbers raised to a non-negative integer power).
- And, of course, there's always the possibility of a straightforward calculation based on a simple physical problem
Fortunately, there's an example that's pretty easy to describe, which relies on the sum of the cubes of the natural numbers 1^3+2^3+3^3+4^3+ . . .
Like the sum of natural numbers, the sum of the cubes can be associated with a small number. Instead of -1/12, it's 1/120. That is, instead of the "equation"
1+2+3+4+ . . . =-1/12
I'm going to talk about the "equation"
1^3+2^3+3^3+4^3+ . . . = 1/120.
The sum comes up in the calculation of the Casimir Effect. Even though the full calculation is nothing to sneeze at, the general set up and the solution behind the mystery of bizarre equations like the ones above are pretty easy to see.
(For those of you who are proficient at math and want to cut to the chase, just read this excellent explanation that Jonathan Dowling wrote up for Mathematics Magazine in 1989).
The Casimir Effect
Three quarters of a century ago, Hendrik Casimir proposed that two metal plates suspended in space will attract each other through a previously unimagined force that had nothing to do with gravity, electricity, magnetism or any other mechanism known at that time. The force is tiny and only becomes significant over minute distances, so detecting it is tricky. Nevertheless, nearly twenty years after his proposal, experimentalists were able to measure the attraction Casimir predicted in the lab.
There is some debate about exactly what causes the Casimir Effect, but the most interpretations involve something called vacuum energy. Essentially, the story goes, we are immersed in a sea of particles. An infinite number of particles, in fact. Each carries a tiny amount of energy, but because there is an infinite number of them, the total amount of energy is also infinite.
We aren't usually aware of the energy all around us because we're immersed in it, in the same way you don't notice the pressure of the atmosphere you're immersed in. The air around you exerts pressure (15 pounds on each square inch of your body, actually), but you don't feel it squeezing you because it's the same pressure inside your body as it is outside, and it perfectly balances out. This is also true of the vacuum energy, only instead of 15 pounds per square inch, the pressure is infinite.
One way to detect the vacuum energy would be to keep it out of some region. It would be similar to detecting the pressure of the atmosphere by sucking the air out of a soda bottle to make it collapse. Casimir's plates do almost something like that, except for vacuum energy instead of air.
|Infinite vacuum energy outside the plates vs. slightly smaller infinite energy inside.|
As you can see in the sketch, the longest wavelength that can fit between the plates corresponds to the arc at the lowest position. That wavelength has a frequency associated with it, which I will call f. The next curve up has twice the frequency of the first, for a frequency of 2f, the third has 3f, and so forth. Because light can have any frequency, you can have an infinite ladder of light frequencies between the plates.
Each frequency, in turn contributes some energy to the region between the plate. In fact, the energy per wavelength between the plates is proportional to the frequency cubed, so the total energy between the plates is proportional to
f^3 +(2f)^3 + (3f)^3 + (4f)^3 . . . = f^3*(1 + 2^3 + 3^3 + 4^3 + . . . )
It's reasonable to guess that adding the cubes of all the integers up this way means that the energy between the plates is infinite
But the vacuum energy outside the plates consists of all frequencies, including those that exist between the plates, so the energy outside is also infinity - although it's clearly a larger infinity. To figure out the pressure difference between the inside and outside, you have to subtract one infinity from the other.
Normally, subtracting infinity from infinity is meaningless. But in this problem we know a lot about the infinities. The paper I link to above by Jonathan Dowling goes through two ways to calculate the difference between the two infinities. In both calculations, Dowling shows that for the Casimir Effect problem subtracting infinities results in 1/120. In other words 1 + 2^3 + 3^3 + 4^3 + . . . doesn't equal 1/120, it equals 1/120 plus infinity.
It takes a fair amount of math for Dowling to come to that conclusion. And that means we have a choice when faced with a problem that looks like the Casimir Effect - we can do the same calculations he does, or when we see 1 + 2^3 + 3^3 + 4^3 + . . . we can just replace it with 1/120 and go about our business.
That is, provided you make sure to ignore the infinity outside the plates, you can pretend that
1 + 2^3 + 3^3 + 4^3 + . . . = 1/120
and you'll get the correct answers.
Back to -1/12
The Casimir Effect described here is a three dimensional problem. It's not surprising then that the relevant series involves cubed terms. One-dimensional problems, on the other hand, involve the sum of the natural numbers 1+2+3+4+ . . .
The Numberphiles point out that the relation 1+2+3+4+ . . .= -1/12 (though not true) is useful for string theory. I assume that's that case because strings are one-dimensional. But if you were to set up a Casimir Effect type of problem in one dimension, you'd be able to apply the same sort of calculation that Dowling used in three dimensions, and find that
1+2+3+4+ . . .= -1/12 + infinity
There's no analogous effect in two dimensions, by the way. You can see this because the same sorts of approaches that lead to 1+2+3+4+ . . .=-1/12 and 1 + 2^3 + 3^3 + 4^3 + . . . = 1/120 also produce the relation 1+2^2 +3^2+4^2+ . . . =0.
In other words the vacuum energy inside and outside a 2-d analog of a Casimir Effect-type problem would exactly balance out. Which means there is no Casimir Effect in Flatland (or in 4-d, 6-d, etc.).
Why Do All Those Other Calculations Leave Out the Infinity?
In the longer Numberphile video, Ed Copeland explains that the technique he used to calculate 1+2+3+4+ . . .=-1/12 involved mathematically moving around a troublesome point called a singularity, which erases the infinity on the way. That seems believable, but it's hard for me to have an intuitive feel for it.
I suspect the series manipulation they presented in their original video works because the things they use in the calculation have their own infinities snipped out in some way, leading to a self-consistent, but still deceptive, result.
The graphical way I use to find factors associated with the infinite sums works, I think, because it effectively includes both the infinities as you're setting it up.
In the graph here, for example, the stepped lines represent the sum 1+2+3+4+ . . . as you build it up one piece at a time.
1+2 = 3
1+2+3 = 6
This is analogous to adding up the discrete energies in the interior portion of a one-dimensional Casimir Effect problem.
The smooth curved line represents adding up the continuous distribution of frequencies in the vacuum energy of open space. The areas between the curve and the stepped lines are what's left over after you subtract the continuous lined from the stepped line. All of these sections are positive, so they still add up to infinity.
However, if you extend the graph to the left of zero you get something that looks like this.
However, there's a little portion between 0 and -1 that doesn't have a matching section to balance it out. I've zoomed in on the graph and colored it green. The area of that region exactly equals -1/12.
All you need to do to calculate the difference between the two infinities is find the size of this little orphaned section.
The same procedure works for 1+2^3+3^3+4^3+ . . . or any other series of the natural numbers raised to a non-negative integer.
To summarize the procedure:
- pick an appropriate series
- find the generating function for the sequence produced by the partial sums of the series
- integrate the continuous version of the generating function from -1 to 0
1 dimension -> 1+2+3+4+ . . . ---> -1/12
3 dimensions -> 1+2^3+3^3+4^3+ . . . ---> 1/120
5 dimensions -> 1+2^5+^5+4^5+ . . . ---> -1/252
7 dimensions -> 1+2^7+3^7+4^7+ . . . ---> 1/240
9 dimensions -> 1+2^9+3^9+4^9+ . . . ---> -1/132
11 dimensions -> 1+2^11+3^11+4^11+ . . . ---> 691/32760
13 dimensions -> 1+2^13+3^13+4^13+ . . . ---> -1/12
*For all even numbers of dimensions, the net difference is zero
(My reasons for extending the graph to negative n and x are a bit sketchy. For one thing, it works. I guess I could also say something about negative energy states and the negative energy sea, but I'm not keen on going there at the moment.)
One of the interesting features of the graph, by the way, is it seems to me to qualitatively explain why you can also solve Casimir Effect problems by simply assuming a cut-off that ignores the higher frequencies. After all, if the portion that accounts for the -1/12 contribution is near the origin, you can choose pretty much any convenient and reasonable cutoff without changing the result, and you don't have to worry at all about subtracting infinities or relying on counter-intuitive infinite sums. You just have to know what error to include as a result of cutting it off though ( that's also pretty easy to figure out from this type graph, but I'll explain it some other time).
Regardless of the way you choose to do the problem, the infinities are accounted for. The Numberphiles chose approaches that get rid of them implicitly (some people would say they get rid of them deceptively). Dowling does the hard math that explicitly deals with the infinities. And I include them in the set up of the problem in a way that lets you account for them without having to do any difficult math. But any way you slice it -
1+2+3+4+ . . . = -1/12 + infinity
Bonus Nonsense: Fun with Series Manipulations
Just to annoy the mathematicians, it occurred to me that there's an even quicker way "prove" 1+2+3+4+ . . .= -1/12 (even though it really doesn't) than the Numberphiles used.
If you are willing to accept the fact that 1-2+3-4+5-. . . = 1/4, then let S = 1+2+3+4+ . . .
Multiply S by 4.
4*S =1+8+12 +. . . .
Subtract 4*S from S, but line them up like this
1+2 +3+4+5+ 6. . .
- 4 -8 -12 . . .
= 1-2+3-4+5-6 . . . = 1/4
(1-4)S = -3*S = 1/4
S = (1/4)(-1/3) = -1/12
You can use the above result to find the value of Grandi's series too.
Add (1-4)S, as written above, to (1-4)S, but shift the series like this
1-2+3-4+5-6 . . .
+( 1 -2+3-4+5 . . .)
= 1-1+1-1+1-1 . . .
But (1-4)S+(1-4)S = -6S = -6(-1/12) = 1/2
So 1-1+1-1+1-1 . . .= 1/2, as the Numberphiles also show.
I can also "prove" that
1+1+1+1+1+ . . . = -1/2
2+3+4+5+ . . .= -7/12
and other similar things, but I've had enough of this stuff for now.