WEBVTT 00:00.000 --> 00:04.720 Thank you to Odoo for supporting PBS.  Add up every positive whole number—one   00:04.720 --> 00:11.280 plus two plus three plus four etc —and eventually  you get infinity. And yet, if you do some careful   00:11.280 --> 00:17.600 mathematical trickery, a very different number  appears: minus one twelfth. Maybe you saw this   00:17.600 --> 00:21.920 strange result do the rounds of the internet a  few years ago. Maybe you also correctly surmised   00:21.920 --> 00:27.360 that it’s just some mathematical sleight of hand.  But maybe you didn’t know that exactly this way   00:27.360 --> 00:33.840 of extracting finite numbers from infinities is  at the heart of some famous results in physics.   00:33.840 --> 00:36.910 Some of them are even equal to minus a twelfth. 00:39.058 --> 00:42.658 what learning actually does to you. Links  to the merch store in the description.  00:42.658 --> 00:46.738 When you add an infinite list of numbers  together, sometimes the result settles down   00:46.738 --> 00:52.738 to a finite value. For example, a half plus a  quarter plus an eighth and so on equals one. As   00:52.738 --> 00:59.298 long as we’re careful about ensuring convergence,  then an infinite sum can equal a finite number.  00:59.298 --> 01:05.698 But many infinite series do not converge—their  result really is infinity. And it is pretty   01:05.698 --> 01:10.898 obvious that the sum of the natural  numbers is such a divergent series. 1+2+3+   01:10.898 --> 01:16.978 etc equals infinity. So why were people  going around saying that it equals -1/12?   01:16.978 --> 01:24.338 It takes a bit of mathematical hocus pocus, The OG viral video on this is on Numberphile,   01:24.338 --> 01:28.578 and their “proof” does some rather  loose arithmetic with a few different   01:28.578 --> 01:32.818 infinite series. And by loose, I mean not  mathematically valid. It’s linked below,   01:32.818 --> 01:38.498 as well as my favorite debunking video. Now, to be  fair to Numberphile, that video was trying to give   01:38.498 --> 01:44.818 an intuitive description of a much more rigorous  argument based on the Riemann zeta function.   01:44.818 --> 01:51.138 This is a famous function—subject of the famous  Riemann hypothesis, which as of this recording   01:51.138 --> 01:55.538 hasn’t been solved by AI yet. It can be written as   01:55.538 --> 02:05.138 sum-over-n-from-one-to-infinity of 1/n^s. As long  as s is greater than 1, the function converges. In   02:05.138 --> 02:11.858 fact, the infinite series itself only defines the  zeta function for values of s greater than one.   02:11.858 --> 02:17.858 But a creative mathematician can rewrite the same  function in other forms that agree perfectly with   02:17.858 --> 02:25.378 that series wherever it converges, and those  new expressions continue smoothly into values   02:25.378 --> 02:34.418 where the original sum diverges. That process is  analytic continuation. At s=-1, this analytically   02:34.418 --> 02:41.538 continued zeta function has the value minus  one twelfth. 3blue1brown has an amazing video   02:41.538 --> 02:45.938 on this stuff, also linked below. None of this means that 1+2+3+..   02:45.938 --> 02:53.458 et cetera equals -1/12. It doesn’t. That  fraction is the zeta-regularized part of a   02:53.458 --> 03:01.298 genuinely divergent sum. But this also isn’t  just a cheap maths trick. This result really   03:01.298 --> 03:07.858 does show up in physics. And in physics it's  actually been measured. With the Casimir effect.  03:07.858 --> 03:15.538 So, in 1948 Hendrik Casimir figured out a  way to delete part of the quantum vacuum.   03:15.538 --> 03:20.338 As you may have heard, empty space isn’t really  empty. Quantum field theory describes it as being   03:20.338 --> 03:25.378 filled with quantum fields in their vacuum  state. Each field supports an enormous range   03:25.378 --> 03:30.498 of possible vibrational modes, and even when  none of those modes contains a real particle,   03:30.498 --> 03:36.418 each still retains an irreducible amount of  quantum fluctuation at every allowed wavelength.  03:36.418 --> 03:42.418 OK, now take a pair of conducting plates and put  them extremely close together. The plates exclude   03:42.418 --> 03:49.138 a bunch of modes in the quantum electromagnetic  field in particular. Only EM waves whose nodes   03:49.138 --> 03:55.458 line up with the plates are allowed—and that  means those for which an integer number of   03:55.458 --> 04:02.178 half-wavelengths fit exactly in the gap width.  It’s just like a guitar string—its sound is the   04:02.178 --> 04:08.338 combination of those notes whose half-wavelengths  fit between the fixed ends of the string. That’s   04:08.338 --> 04:13.858 a 1-D analogy, but even with the Casimir  effect, one dimension gives the right idea. 04:13.858 --> 04:18.178 We can also think in terms of frequency  instead of wavelength. There’s a smallest   04:18.178 --> 04:26.418 allowed frequency and then only frequencies that  are an integer multiple of this are allowed,   04:26.418 --> 04:33.618 but all such frequencies are allowed. The energy  of each mode is proportional to its frequency,   04:33.618 --> 04:38.018 so if you want the total energy in  allowed EM modes between Casimir   04:38.018 --> 04:48.258 plates you get some base energy times 1+2+3 etc. That sum should now sound very familiar. It sounds   04:48.258 --> 04:54.658 like it predicts that there's infinite energy in  the vacuum between our Casimir plates. And that’s   04:54.658 --> 05:01.458 after excluding certain modes—which means there  should be infinity … plus? energy outside the   05:01.458 --> 05:07.138 plates. This “prediction” of infinite energy in  the quantum vacuum is not something we can take   05:07.138 --> 05:14.418 literally. To compare the vacuum within the plates  to the vacuum without them, we introduce a cutoff   05:14.418 --> 05:20.978 that temporarily stops us from adding arbitrarily  high-frequency modes. For real metal plates this   05:20.978 --> 05:26.498 even has a physical motivation: at sufficiently  high frequencies the metal stops behaving like a   05:26.498 --> 05:32.418 perfect reflector, so those modes barely notice  that the plates are there. But cutoff or no,   05:32.418 --> 05:36.578 as long as this background energy is the  same between one location and another,   05:36.578 --> 05:41.538 it feels like there’s nothing there. We only  detect differences in energy. Which is why   05:41.538 --> 05:47.458 those excluded modes between the Casimir plates  are important. Both the inside and outside of the   05:47.458 --> 05:55.218 plates appear to add up to infinite or very large  energy, but after regulating them in the same way,   05:55.218 --> 06:01.378 the vacuum energy with the plates is slightly  lower than the reference vacuum outside them.  06:01.378 --> 06:06.338 There’s a process for comparing infinities  to find their finite difference. The trick,   06:06.338 --> 06:13.058 as I hinted, is to separate the varying finite  part of the sum from a shared infinite part.   06:13.058 --> 06:18.658 That latter part can be expressed as a shared  high-energy cutoff. This process of introducing   06:18.658 --> 06:24.898 a cutoff to deal with divergent terms is one  way to do what we call regularization. Zeta   06:24.898 --> 06:30.818 regularization, which involves the analytic  continuation I mentioned, is another way.  06:30.818 --> 06:35.138 So, in the Casimir effect, we figure  out what divergent part is shared by   06:35.138 --> 06:41.858 the interior and exterior of the plates and  subtract that. We end up with a Casimir energy   06:41.858 --> 06:51.138 of pi-hbar-c/d * ½ * -1/12. The first factor  sets the energy scale of the fundamental mode,   06:51.138 --> 06:55.698 where (d) is the distance between the plates, ½  comes from the zero point energy of the quantum   06:55.698 --> 07:01.378 oscillator. And then there’s the -1/12. And  yes, it’s back. And it’s back for the same   07:01.378 --> 07:07.778 reason—we plucked a finite part of an infinity. Here the minus sign really matters because it   07:07.778 --> 07:13.538 tells us that energy gets even lower as the plates  move closer together. The system therefore lowers   07:13.538 --> 07:20.098 its energy by shrinking the gap—and that  means an attractive force. The resulting   07:20.098 --> 07:25.058 force has been successfully measured, and the  Casimir effect has been verified, and it’s   07:25.058 --> 07:31.138 exactly what we expect from the weird maths. The real three-dimensional electromagnetic   07:31.138 --> 07:36.738 calculation is more complicated because the modes  can also carry momentum parallel to the plates and   07:36.738 --> 07:42.178 because the electromagnetic field has multiple  polarizations. But the same basic logic works,   07:42.178 --> 07:49.858 though the -1/12 is instead +1/120.  But yeah, the infinite sum really does   07:49.858 --> 07:55.698 sometimes come out to -1/12 or something,  but only in the context of regularization   07:55.698 --> 08:01.778 and renormalization, which I’ll come back to. But first it’s time for some string theory—which   08:01.778 --> 08:07.698 may or may not have anything to do with  nature. But our -1/12 shows up here too,   08:07.698 --> 08:11.698 and actually gives us one of string  theory’s weirdest properties.  08:11.698 --> 08:16.738 So, string theory replaces point particles  with tiny filaments vibrating across multiple   08:16.738 --> 08:22.098 compact dimensions. Just like a guitar  string, a quantum string can vibrate in   08:22.098 --> 08:27.458 different modes, and those modes determine  the properties of the emergent particles.  08:27.458 --> 08:33.538 In the guitar string and Casimir, the allowed  modes come from the fact that their ends are fixed   08:33.538 --> 08:39.138 in place. Quantum strings also have a boundary  condition, but in the case of closed strings,   08:39.138 --> 08:44.338 which we’ll focus on, it’s because the strings  are loops. Every oscillation has to match up   08:44.338 --> 08:50.018 with itself when traced around the loop. But the  result is the same. Allowed frequency and energy   08:50.018 --> 08:57.058 modes are every integer multiple of the base mode.  Which means we once again have a divergent sum.  08:57.058 --> 09:03.778 In the case of the quantum string, regularization  involves introducing a cutoff. However,   09:03.778 --> 09:09.858 this cutoff feels less ad hoc than in regular  QFT. The thing we want to calculate based on   09:09.858 --> 09:14.898 these stringy oscillations is the overall energy  spectrum of the string. What are the possible   09:14.898 --> 09:21.938 global oscillations? The shortest-wavelength modes  may correspond to extreme energies. But at those   09:21.938 --> 09:28.978 tiny scales, such a fluctuation only sees an  infinitesimal patch of the total string—it   09:28.978 --> 09:34.658 doesn't know about the global periodicity. Those high-frequency contributions are at both   09:34.658 --> 09:39.458 local and constant all around the string,  and don’t contribute to the string’s energy   09:39.458 --> 09:44.418 spectrum that we’re trying to calculate. We  can ignore them in the same way that we can   09:44.418 --> 09:50.658 regularize away the high-frequency modes between  the Casimir plates when those tiny fluctuations   09:50.658 --> 09:56.818 have no idea the plates are even there. So again we have a cutoff for our infinite   09:56.818 --> 10:02.018 sum and lets us figure out the size of the  finite component of the string’s vibrations.  10:02.018 --> 10:06.658 A couple of things for perfect precision. First,  I’ve been talking about oscillations around a   10:06.658 --> 10:13.938 1-D loop. Really, we think of the string’s loop  as sweeping out a 2-D surface over time that we   10:13.938 --> 10:21.058 call a worldsheet. Second, all of this stuff—the  string, its worldsheet—are embedded within some   10:21.058 --> 10:26.098 larger number of spatial dimensions. Could be  3 like our universe, could be more. We’ll say   10:26.098 --> 10:33.138 it’s D dimensions for now and see what happens. Once we account for the redundancies in how we   10:33.138 --> 10:38.578 describe the surface of the worldsheet, the  genuinely physical vibrations of the string   10:38.578 --> 10:45.698 are transverse to the string’s direction of  motion. That leaves D-2 independent transverse   10:45.698 --> 10:51.938 directions in which the string can oscillate.  In ordinary four-dimensional spacetime that   10:51.938 --> 10:58.818 would leave two physical transverse  directions, but more if D is larger.  10:58.818 --> 11:04.018 And every one of those directions has  its own infinite tower of modes. So   11:04.018 --> 11:08.627 the total zero-point contribution is  proportional to the energy is this:  11:08.627 --> 11:11.378 \frac{D-2}{2} (1+2+3+4+\ldots). Now use the same regulation trick   11:11.378 --> 11:16.578 as we used for the Casimir effect to get rid  of the inconsequential high energy modes. In   11:16.578 --> 11:22.178 the quantized string this shows up as a shift  in the string’s energy spectrum—the so-called   11:22.178 --> 11:27.858 normal-ordering constant. We again use the  Riemann zeta function to replace the infinite   11:27.858 --> 11:33.058 sum with -1/12, giving us: -\frac{D-2}{24}.  11:33.058 --> 11:38.978 And this is where the -1/12 has some crazy  consequences. Now I have to get a bit hand-wavey   11:38.978 --> 11:44.178 here, so please forgive me. If we bring in  relativity and force the quantum theory on   11:44.178 --> 11:52.258 the worldsheet to preserve Lorentz symmetry, the  D-2/24 part has to equal one. This is the only   11:52.258 --> 11:59.058 relationship consistent with our non-negotiable  symmetries. But this is only possible if D equals   11:59.058 --> 12:04.818 26. And D is the number of dimensions. So yeah,  if you’ve heard that string theory requires lots   12:04.818 --> 12:11.218 of hidden dimensions, it’s because that  -1/12 is forcing string theory’s hand.  12:11.218 --> 12:15.378 D here is the famous critical dimension  of bosonic string theory. Now, bosonic   12:15.378 --> 12:19.138 string theory itself is not a realistic  theory of nature. Among other problems,   12:19.138 --> 12:24.818 it doesn’t naturally have fermions, which are  needed for actual matter. Modern string theories   12:24.818 --> 12:31.218 do include fermionic vibrations on the worldsheet,  and these partially cancel the bosonic vibrations,   12:31.218 --> 12:37.378 reducing the number of dimensions needed. But the  basic idea survives: regulate the infinite quantum   12:37.378 --> 12:42.818 fluctuations, keep track of their finite quantum  contribution, and demand that the symmetries of   12:42.818 --> 12:48.978 the theory remain intact. Now for superstrings  that consistency condition gives ten spacetime   12:48.978 --> 12:55.218 dimensions rather than twenty-six. But the  same minus-one-twelfth-style regularization   12:55.218 --> 13:00.018 is still hiding inside the calculation. So this brings us back to the original   13:00.018 --> 13:05.138 mathematical trick. In the physics examples,  regularization gives us a controlled way to   13:05.138 --> 13:10.498 handle an infinite mode sum, and renormalization  tells us which divergent parts are universal or   13:10.498 --> 13:17.298 unobservable and can be subtracted away. Zeta  regularization—the mathematical procedure that   13:17.298 --> 13:24.658 got us our 1+2+3..=-1/12 - is related to  this. But instead of introducing a cutoff,   13:24.658 --> 13:29.858 it uses analytic continuation to assign  a finite value to the divergent series. 13:29.858 --> 13:35.778 And, in fact, for the one-dimensional Casimir  example and the simplest bosonic-string   13:35.778 --> 13:42.338 calculation, we can literally use this as  the Riemann zeta function to regularize   13:42.338 --> 13:48.898 the same divergent sum. Minus one twelfth is not what you get by adding all the positive integers.   13:49.538 --> 13:54.898 It's what can be left after the universal  infinity has been stripped away,   13:54.898 --> 14:02.258 and knowing this lets us tame the infinite  quantum fluctuations of empty spacetime. 13:54.898 --> 13:59.938 functionality. If you’re already running a  business or looking to start one, you can book   13:54.898 --> 14:02.512 a free consultation with an Odoo expert or start  a 15-day free trial, with no credit card required.