Some of the earliest problems in mathematics asked about the integer solutions to simple polynomial equations. For instance, what are the possible right triangles with whole number side lengths? The solution dates at least back to the Ancient Greeks; the side lengths are related by Pythagoras' famous formula x2 + y2 = z2. The 7th century Indian mathematician Brahmagupta studied integer solutions to the equation x2 - 2y2 = 1 as well as the same formula with 2 replaced by a general integer n (called Pell's equation). Many other similar equations have been studied for centuries or millennia.
In general, a Diophantine equation is a polynomial equation for which we are interested in integer solutions. Counterintuitively, some questions about solving these in the integers may be more difficult than considering all types of solutions. For example, the fundamental theorem of algebra states that any polynomial in a single variable has a root over the complex numbers (e.g. x3 - 4x2 + 17x + 20 = 0 is true for some complex number). However, there is often no integer solution to such equations.
Historically, different types of Diophantine Equations were typically solved by ad hoc methods, as they come in many different varieties. However, one general observation that connects many of these equations is that they state something about the factorization of a sum of two numbers. Pythagoras' equation says something special about the sum of two squares, namely that it is another square! Similarly, Pell's equation says that one plus some number multiplied by a square has the property that it too is square. Our motivating question may then be taken to be:
How does the factorization of a sum of two numbers relate to the factorizations of the individual numbers?
The abc Conjecture provides a partial answer to this question. Its name comes from the fact that we are considering equations of the form a + b = c and asking how the factorizations of the three numbers relate. Mathematicians David Masser and Joseph Oesterlé first made the conjecture in 1985 while studying integer points on what are called elliptic curves, in this case given by the equation y2 = x3 + k (where k is a fixed integer). This is yet another example of a sum having special factorization properties. Throughout the rest of this post, we will see how thinking about the motivating question might lead you to formulating the abc conjecture.
Simply put, we want the answer to our motivating question to be "it doesn't." Somehow, the additive and multiplicative structures of the integers should be independent of one another. This is in some ways a deep statement, and not at all intuitively clear, but we'll begin with this assumption. In other words, for an equation a + b = c, if all three numbers satisfy some special factorization properties (e.g. being cubes, etc.) it should in some sense be a coincidence. Our next task is to make this progressively less vague. First, we need a definition.
Definition: Two numbers are relatively prime if they share no common prime divisors.
For example, 34 and 45 are relatively prime, but 24 and 63 are not, because they are both divisible by 3. Here is how we will express our independence hypothesis: for any equation of the form a + b = c, where a and b are relatively prime, if a and b are divisible by high powers of primes, c almost always is not. This is in keeping with our theme because "divisible by high powers of primes" is special factorization property. That is, most prime factorizations should look more like 705 = 3*5*47 and not 768 = 28*3. The assumption that a and b are relatively prime exists to rule out silly equations like
2n + 2n = 2n + 1,
in which all three numbers are divisible by arbitrarily high powers of 2. This doesn't represent some special connection between addition and multiplication - all we've done is multiplied the equation 1 + 1 = 2 by 2n. If we assert that a and b are relatively prime, then the prime factors of each of the three numbers are distinct, and we eliminate the uninteresting examples. Next, we require a mathematical notion that measures "divisibility by high prime powers".
Definition: The radical of a number n, denoted rad(n), is the product of the distinct prime powers of n. Also define rad(1) = 1.
For example, rad(705) = 3*5*47 = 705 (since the factors 3, 5, and 47 are distinct) but rad(768) = 2*3 = 6. The radical function forgets about any powers in the prime factorization, keeping only the primes themselves. Notice that the radical of a number can be as large as the number itself, but it can also be much smaller. The amount by which rad(n) is smaller than n can be taken as a measure of to what extent n is divisible by large prime powers.
Now we return to our equation a + b = c (where we will now consistently assume the relatively prime hypothesis). A reasonable way to test for high prime power divisibility for all three of these numbers is to calculate rad(abc) = rad(a)rad(b)rad(c) (the reader may wish to prove this last equation). Since rad(abc) could be as large as abc itself, it seems likely that rad(abc) would usually be much larger any of the individual numbers, the largest of which is c. For example, consider 13 + 22 = 35. In this case, rad(abc) = rad(13*22*35) = 13*2*11*7*5 = 10010, which is much larger than c = 35. However, this property does not always hold true. Consider another example, 1 + 8 = 9. Now we have rad(abc) = rad(1*8*9) = 2*3 = 6, and 6 < 9 = c. Notice that this anomaly reflects something weird going on; the equation can also be written 1 + 23 = 32, so one plus a cube is a square. Testing different values of a, b, and c gives the impression that equations of the second sort are rare. Therefore, we make an almost mathematical conjecture:
"Almost" Conjecture: For equations of the form a + b = c where a and b are relatively prime, rad(abc) is almost always greater than c.
We're close! The equation rad(abc) > c is a bona fide mathematical condition that we can check. However, we have yet to render "almost always" into mathematical language. Clearly there are infinitely many a + b = c equations to look at. What does it mean to say that "most of them" behave in some way? We know from our 1 + 8 = 9 example that there are at least some exceptions. Maybe we could assert that there are less than 10 total exceptions, or less than 100. However, these numbers seem arbitrary, so we'll just guess that there are only finitely many exceptions. That is, all but at most N of these equations, for some fixed finite number N, satisfy our hypothesis. In conclusion, we conjecture that:
Conjecture 1: For all but finitely many equations of the form a + b = c where a and b are relatively prime, rad(abc) > c.
Finally, a real conjecture! Unfortunately, it's false. In other words, there are infinitely many such equations for which rad(abc) ≤ c. Don't worry! It's rare in mathematics to come up with the correct statement on the first try! In the next post, we'll prove our conjecture 1 false and see how to correct it.
Sources: https://rlbenedetto.people.amherst.edu/talks/abc\_intro14.pdf, Brian Conrad: The abc Conjecture, 12 sep 2013.
Tuesday, March 26, 2019
Tuesday, March 5, 2019
The Casimir Effect
The idea of the electromagnetic field is essential to physics. Dating back to the work of James Clark Maxwell in the mid-1800s, the classical theory of electromagnetism posits the existence of certain electric and magnetic fields that permeate space. Mathematically, these fields assign vectors (arrows) to every point in space, and their values at various points determine how a charged particle moving in space would behave. For example, the magnetic field generated by a magnet exerts forces on other nearby magnetic objects. Crucially, the theory also explains light as an electromagnetic phenomenon: what we observe as visible light, radio waves, X-rays, etc. are "waves" in the electromagnetic field that propagate in space.
Maxwell's theory is still an essential backbone of physics today. Nevertheless, the introduction of quantum mechanics in the early 20th century introduced new aspects of electromagnetism. Perhaps most importantly, it was discovered that light comes in discrete units called photons and behaves in some ways both as a wave and a particle. Though electromagnetism on the human scale still behaves largely as the classical theory predicts, at small scales there are quantum effects to account for. Around the middle of the century, physicists Richard Feynman, Shinichiro Tomonaga, Julian Schwinger, and many others devised a new theory of quantum electrodynamics (or QED) that described how light and matter interact, even on quantum scales.
Naturally, QED predicted new phenomena that classical electromagnetism had not. One especially profound change was the idea of vacuum energy. For most purposes, "vacuum" is synonymous with "empty space". As is typical of quantum mechanics, however, a system is rarely considered to be in a single state, but rather in a superposition of many different states simultaneously. These different states can have different "weights" so that the system is "more" in one given state than another. This paradigm applies even to the vacuum. Certain pairs of particles may appear and disappear spontaneously in many of these states and even exchange photons. Some of the possible interactions are illustrated below with Feynman Diagrams.
In these diagrams, the loops represent the evanescent virtual particle pairs described above. Wavy lines represent the exchange of photons. Each of the six diagrams represents a possible vacuum interaction, and there are many more besides (infinitely many, in fact!). The takeaway is that the QED vacuum is not empty, but rather a "soup" of virtual particle interactions due to quantum fluctuations. Further, these interactions have energy, known as vacuum energy. This, at least, is the mathematical description. There are some curious aspects to this description, because the vacuum energy calculation in any finite volume yields a divergent series. In other words, there is theoretically an infinite amount of vacuum energy in any finite volume! Because of this, physicists devised a process called renormalization that cancels out these infinities in calculations describing the interaction of real particles. This process in fact gives results that have been confirmed by experiment. Nevertheless, it does not follow that the infinite vacuum energy exists in any "real" sense or is accessible to measurement. One possible way in which it is, however, is the Casimir Effect.
The setup of the Casimir effect involves two conducting metal plates placed parallel to one another. The fact that the plates are conducting is important because the electric field vanishes inside conducting materials. Now, the vacuum energy between the plates can be calculated as a sum over the possible wavelengths of the fluctuations of the electromagnetic field. However, unlike the free space vacuum, the possible wavelengths are limited by the size of the available space: the longest wavelength contributions to the vacuum energy do not occur between the plates (this is schematically illustrated in the image above). A careful subtraction of the vacuum energy density inside the plates from outside yields that there is more energy outside. Remarkably, this causes an attractive force between these plates known as the Casimir force. The force increases as the distance between plates is decreased. Precisely, the magnitude of the force F is proportional to 1/d4, where d is the distance between the plates. As a result, if the distance is halved, the force goes up by a factor of sixteen! The initial calculation of this effect was due to H.G.B Casimir in 1948.
Around 50 years after first being postulated, the effect was finally measured experimentally with significant precision. The primary issue was that for the Casimir force to be large enough to measure, the metal plates would have to be put very close to one another, less than 1 micrometer (0.001 mm). Even then, very sensitive instruments are necessary to measure the force. One landmark experiment took place in 1998. Due to the practical difficulty of maintaining two parallel plates very close to one another, this experiment utilized one metal plate and one metal sphere with a radius large compared to the separation (so that it would "look" like a flat plate close up). The authors of the experiment also added corrections to Casimir's original equation accounting for the sphere instead of the plane and the roughness of the metal surfaces (at the small distances of the experiment, microscopic bumps matter). They obtained the following data for the force as it varies with distance:
In the figure above, the squares indicate data points from the experiment and the curve is the theoretical model (including the corrections mentioned). The distance on the x-axis is in nanometers and the smallest distance measured was around 100 nm, hundreds of times smaller than the width of a human hair. Even at these minuscule distances, the force only reached a magnitude of about 1*10-10 Newtons, a billion times smaller than the weight of a piece of paper. Nevertheless, the results confirmed the presence of the Casimir force to high accuracy.
The existence of the Casimir effect would seem to vindicate the rather strange predictions of QED with respect to the quantum vacuum, suggesting that it is indeed full of energy that can be tapped, if indirectly. However, others have argued that it is possible to derive the effect without reference to the energy of the vacuum, and therefore the experiment does not necessarily mean that vacuum energy is "real" in any meaningful way. Continued study into the existence of vacuum energy may help to explain the accelerating expansion of the universe since some mysterious "dark energy" is believed to be the source. In the mean time, the Casimir effect is an important experimental verification of QED and could someday see applications in nanotechnology, since the force would be relatively large on small scales.
Sources: https://www.scientificamerican.com/article/what-is-the-casimir-effec/, https://arxiv.org/pdf/hep-th/0503158.pdf, The Quantum Vacuum: An Introduction to Quantum Electrodynamics by Peter W. Milonni, http://web.mit.edu/kardar/www/research/seminars/PolymerForce/articles/PRL-Mohideen98.pdf
Maxwell's theory is still an essential backbone of physics today. Nevertheless, the introduction of quantum mechanics in the early 20th century introduced new aspects of electromagnetism. Perhaps most importantly, it was discovered that light comes in discrete units called photons and behaves in some ways both as a wave and a particle. Though electromagnetism on the human scale still behaves largely as the classical theory predicts, at small scales there are quantum effects to account for. Around the middle of the century, physicists Richard Feynman, Shinichiro Tomonaga, Julian Schwinger, and many others devised a new theory of quantum electrodynamics (or QED) that described how light and matter interact, even on quantum scales.
Naturally, QED predicted new phenomena that classical electromagnetism had not. One especially profound change was the idea of vacuum energy. For most purposes, "vacuum" is synonymous with "empty space". As is typical of quantum mechanics, however, a system is rarely considered to be in a single state, but rather in a superposition of many different states simultaneously. These different states can have different "weights" so that the system is "more" in one given state than another. This paradigm applies even to the vacuum. Certain pairs of particles may appear and disappear spontaneously in many of these states and even exchange photons. Some of the possible interactions are illustrated below with Feynman Diagrams.
In these diagrams, the loops represent the evanescent virtual particle pairs described above. Wavy lines represent the exchange of photons. Each of the six diagrams represents a possible vacuum interaction, and there are many more besides (infinitely many, in fact!). The takeaway is that the QED vacuum is not empty, but rather a "soup" of virtual particle interactions due to quantum fluctuations. Further, these interactions have energy, known as vacuum energy. This, at least, is the mathematical description. There are some curious aspects to this description, because the vacuum energy calculation in any finite volume yields a divergent series. In other words, there is theoretically an infinite amount of vacuum energy in any finite volume! Because of this, physicists devised a process called renormalization that cancels out these infinities in calculations describing the interaction of real particles. This process in fact gives results that have been confirmed by experiment. Nevertheless, it does not follow that the infinite vacuum energy exists in any "real" sense or is accessible to measurement. One possible way in which it is, however, is the Casimir Effect.
The setup of the Casimir effect involves two conducting metal plates placed parallel to one another. The fact that the plates are conducting is important because the electric field vanishes inside conducting materials. Now, the vacuum energy between the plates can be calculated as a sum over the possible wavelengths of the fluctuations of the electromagnetic field. However, unlike the free space vacuum, the possible wavelengths are limited by the size of the available space: the longest wavelength contributions to the vacuum energy do not occur between the plates (this is schematically illustrated in the image above). A careful subtraction of the vacuum energy density inside the plates from outside yields that there is more energy outside. Remarkably, this causes an attractive force between these plates known as the Casimir force. The force increases as the distance between plates is decreased. Precisely, the magnitude of the force F is proportional to 1/d4, where d is the distance between the plates. As a result, if the distance is halved, the force goes up by a factor of sixteen! The initial calculation of this effect was due to H.G.B Casimir in 1948.
Around 50 years after first being postulated, the effect was finally measured experimentally with significant precision. The primary issue was that for the Casimir force to be large enough to measure, the metal plates would have to be put very close to one another, less than 1 micrometer (0.001 mm). Even then, very sensitive instruments are necessary to measure the force. One landmark experiment took place in 1998. Due to the practical difficulty of maintaining two parallel plates very close to one another, this experiment utilized one metal plate and one metal sphere with a radius large compared to the separation (so that it would "look" like a flat plate close up). The authors of the experiment also added corrections to Casimir's original equation accounting for the sphere instead of the plane and the roughness of the metal surfaces (at the small distances of the experiment, microscopic bumps matter). They obtained the following data for the force as it varies with distance:
In the figure above, the squares indicate data points from the experiment and the curve is the theoretical model (including the corrections mentioned). The distance on the x-axis is in nanometers and the smallest distance measured was around 100 nm, hundreds of times smaller than the width of a human hair. Even at these minuscule distances, the force only reached a magnitude of about 1*10-10 Newtons, a billion times smaller than the weight of a piece of paper. Nevertheless, the results confirmed the presence of the Casimir force to high accuracy.
The existence of the Casimir effect would seem to vindicate the rather strange predictions of QED with respect to the quantum vacuum, suggesting that it is indeed full of energy that can be tapped, if indirectly. However, others have argued that it is possible to derive the effect without reference to the energy of the vacuum, and therefore the experiment does not necessarily mean that vacuum energy is "real" in any meaningful way. Continued study into the existence of vacuum energy may help to explain the accelerating expansion of the universe since some mysterious "dark energy" is believed to be the source. In the mean time, the Casimir effect is an important experimental verification of QED and could someday see applications in nanotechnology, since the force would be relatively large on small scales.
Sources: https://www.scientificamerican.com/article/what-is-the-casimir-effec/, https://arxiv.org/pdf/hep-th/0503158.pdf, The Quantum Vacuum: An Introduction to Quantum Electrodynamics by Peter W. Milonni, http://web.mit.edu/kardar/www/research/seminars/PolymerForce/articles/PRL-Mohideen98.pdf