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String Theory and Coupling Constants

This is going to be a bit more haphazard of a blog post, because it was in response to a question I was asked about how String Theory relates to the “universal constants” of physics. I think the best way to describe it is the following sentence:

String Theory provides a single coupling constant that is the expectation value of a scalar field. This constant ends up being independent of the energy scale chosen because this constant (physically) represents the “tension” in the String.

I tried to answer this question without too much math and all of the physics I know (which is pretty much undergrad physics with some knowledge of graduate quantum mechanics, field theory and string theory). The question I was asked was,

I was wondering if string theory had implications towards the values of the ‘universal constants’ and if there was any really obscene math that governed how those values changed over time. When I was first learning physics in HS, I always thought that it was super naive and arrogant even to assume that the universal constant are actually time independent, but maybe the math on universal expansion makes sense with the constants being constants, but I was under the impression string theory offered geometry of hyper dimensions as a reason for these values, possibly the size of curled dimensions as an energy source, etc.

So I replied with something along the lines of the following:

The main result that String Theory provides is that everything boils down to precisely one universal constant. The reason that this is important is because all of the quantum field theories that we know about are effective field theories. What this means is that the specifiers for a field theory, the fields, the equation of motion or an action and the proportionality constants, all depend on the energy regime one is looking at. In the case of the strong nuclear force (which keeps quarks bound to each other, giving rise to mesons such as protons and neutrons), it turns out that at really high energies, the field theory we currently use (Quantum Chromodynamics, QCD) has a coupling constant (called the Yukawa Coupling) that scales roughly as e^{-E}, where E is the energy scale (i.e. 1 MeV, 10 GeV, 1 TeV). Note that I’m being a bit sloppy here: The actual scaling is represented by what is known as the \beta function, which is a function of energy scale. However, the idea is that these contributions are only valid within certain energy regimes (i.e. 1 MeV — 10 TeV). In terms of QCD, one of the biggest problems is elucidating how the strong force works at low energies. The current theory gives a lot of infinite integrals that aren’t renormalizable (the heuristic method that Feynman, et. al came up with removing infinites from expectations of electric or magnetic fields) when the energy is low and this represents a phenomena known as confinement. Effectively, this is a weird quantum mechanical phenomenon that is similar to the fact that you are changing the system by making a measurement. What confinement says is that if you’re at a low enough energy scale (say 1 MeV for protons), you cannot discern quarks from antiquarks because there is not enough energy to break the so-called “Dirac String” connecting the two particles. However, a lot of important quantum mechanical (and even condensed-matter) effects take place around the “condensation” energy (the energy level where confinement occurs for a certain particle). The best attempt to solve this problem is to use something called Lattice Gauge Theory, where you fix a different gauge from the symmetry group of your theory [\mathsf{SU}(3) for QCD] at each point and use some nearest-neighbor computations. So what does this have to do with having one universal constant?

Well String Theory effectively says that there is one main scalar field (i.e. a function of space-time that assigns a number to each point of space-time; this seems like a tautologous definition, but scalar fields are actually functions on a configuration space, which is in general different than “space-time” itself.) which automatically determines the coupling constants for different fields by looking at it’s expectation. This field is called the dilaton and the universal (String) coupling constant is defined as \exp\left( \langle\phi\rangle\right), where \phi : \mathcal{C} \rightarrow \mathbb{R} is the dilaton field [\mathcal{C} is the configuration space]. This coupling constant determines the “tension” in a string, just as if one were in classical mechanics. As it turns out, this means that you’ve avoided all the infinities that occur in conventional quantum field theories, because there is a universal coupling constant that doesn’t depend on energy scale. In fact, all of the energy scale dependence is in the fields themselves. This means that a String Theory inherently avoids the somewhat arbitrary algorithms that one follows in Quantum Field Theory to “subtract infinity from infinity.” Think of it as a constructive formulation of a theory from the ground up as opposed to a destructive theory that works from bigger objects down. When you’re trying to “integrate” over a space of bigger objects in a space of lower dimension, you’ll always get an infinity. The converse, however, is similar to the fact that a plane embedded in 3-space has zero volume in 3-space because it’s “width” is zero.

I’m not sure if that answers your question, because the extra dimensions aren’t necessarily the energy source; rather they are required to make sure that Poincaré Invariance (Lorentz Invariance plus Translational Invariance) holds and so that anomalies (i.e. quantum predictions that have expectations that don’t jive with classical mechanics) don’t occur. Perhaps they are an energy source, but it’s hard to elucidate the precise physical meaning of that. Rather, the “tension” in the string gives you the energy and that tension is derived from a 1-dimensional object (a string). The extra dimensions are simply for the string to reside in and not violate Lorentz or Translational invariance. Think of it this way: A spherical point particle (0-dimensional object as the radius R \rightarrow 0) “requires” 4-dimensions for Lorentz Invariance (3 space, 1 time) and for a string we require 6 extra dimensions to be consistent. In fact the first String Theory (Bosonic String theory) required 26-dimensions, but if one uses supersymmetry you can bring it down to 10-dimensions and include fermions. Just for completeness, let me just explain what supersymmetry is, albeit heuristically. In the real world, we have fermions and bosons and from the Black Body Spectrum and quantum statistical mechanics, we know that they obey different statistical laws. Since statistics are derived from a choice of probability measure on a separable Hilbert Space, this means that we need two Hilbert Spaces, equipped with different integration measures, to accommodate both fermions and bosons. As Dirac realized, fermion expectations rely on anticommutivity of operators whereas bosonic expectations rely on commutivity of operators. This means that our Hilbert Space H should split into the direct sum of two Hilbert Spaces, H = H_{\text{fermion}} \oplus H_{\text{boson}}. Think of a direct sum as adding the other Hilbert Space’s basis vectors as independent dimensions, just as \mathbb{R}^2 is constructed as \mathbb{R}^1 \oplus \mathbb{R}^1, where the first \mathbb{R}^1 contributes the \hat{e}_{x} basis vector and the second contributes the \hat{e}_{y} basis vector. Great, but what about relationships between H_{\text{fermion}} and H_{\text{boson}}? This is what supersymmetry assumes: There exists an operator A : H_{\text{fermion}} \rightarrow H_{\text{boson}} that acts something like the annihilation and creation operators of the harmonic oscillator. This means that I can “trade” a fermion for a boson at some type of energy cost. There is no physical reason for this, but it certainly makes the math a lot easier so physicists took the path of shallowest ascent.

I’m not sure if that really answered his question or not, but I suppose that time independence is directly related to the metric, which plays a role in the development of the \beta function in non-linear space-times.

Leave me a lot of questions :)

The Gauss-Bonnet Theorem explains why I believe in the philosophy behind String Theory

So I tried to find a funny picture of Gauss, but during my search I ended up finding this and it was too good to not use!

So one of the things that I’m sure that most String Theorists encounter is the question, “Aren’t you wasting your time?” And perhaps the only fields that come to their defense (maybe) are mathematics and perhaps philosophy. The fact of the matter is that many physicists, both applied and theoretical tend to view String Theory in a negative light because:

1) It makes unique claims that are well, astronomically hard to prove (at least as far as we know!)

2) It uses requires one to study pure mathematics1, which a lot physicists (in my experience) tend to write off as philosophy.

And to be honest, for a while I didn’t know exactly what allure I saw in String Theory. I think it started off with me thinking that it’s so cool that someone could find a physical interpretation of cobordism or that the mysteries of “gauge theories” could be unraveled with the mathematical sophistication of Fibre Bundles, Sheaf Cohomology and Lie Groups. At the same time, I was mesmerized by how much the idea of supersymmetry and the concept of grading were affecting mathematics. For example, Edward Witten2 uses Supersymmetry to “prove” the Morse Inequalities in roughly a page and a half. By contrast, it took a few works of Raoul Bott and René Thom to fully resolve the inequality in the 1950s. While Witten didn’t give a true proof, his remarkable use of intuition has spurred him to give the skeletons for proofs many other important mathematical theorems and as such he was the first physicist to win the Fields Medal in 1987. But I digress; These are great reasons for liking something or having a heuristic belief in a paradigm. After all, the efficient market hypothesis is so controversial because it requires a large leap of faith for many to believe. So why do I believe so strongly in the vision that String Theory delivers?

After reading a lot of String Theory books and learning a lot of math, I figured that I really liked String Theory because of the mathematics involved. After all, my training so far has been predominantly in pure mathematics (mainly analysis, topology and dynamical systems) with a fair amount of applied physics on the side. However, I’ve been focusing this week on trying to regain a physical perspective on String Theory and understanding the motivation for studying it. I stumbled upon something yesterday that I first thought was a mistake; it was the following, relatively benign looking formula (written in coordinates):

\chi(M) = \int_{M} \sqrt{-g}\, R\, d^{D}x+ \int_{\partial M} K ds \;\;\;\;\;\;\;\;\;\; (1)


  • M is the D-dimensional manifold that describes space-time (for those without any knowledge of differential geometry, think of it as a geometric object, such as a sphere, that “locally” looks like Euclidean space)
  • g is the determinant of the metric g : TM \rightarrow TM \rightarrow \mathbb{R} in local coordinates. The metric is used to “measures distances”
  • R is the intrinsic Scalar Curvature (Ricci Curvature)
  • K is the extrinsic curvature
  • ds is the volume form restricted to the boundary.

This formula is a (Physics-y) statement of the Gauss-Bonnet Theorem. We’ll get back to the difference between intrinsic and extrinsic curvature later, but let’s focus on what the above equation says. The thing that really amazed me is that first term on the right hand side is what is known as the Einstein-Hilbert Action. Great, it has Einstein’s name so it’s gotta be good. But what does it mean?

Let’s go back to the late 1700s/early 1800s. A philosopher-physicist (it’s hard to believe that they were the same thing back in the day!) named Jean le Rond D’Alembert came up with a wondrous philosophical idea: All of physics boils down to minimizing energy. Now this is something that can be appreciated in everyday life, for in general people try to minimize the amount of energy they use either by being more efficient or just by being lazy! However, William Rowan Hamilton and Joseph Louis Lagrange formulated physics in terms of this principle of minimal energy that was 1) equivalent to Newton’s Laws and 2) made it much easier to derive the equations that describe the motion of an object. These theories (known as Hamiltonian and Lagrangian Mechanics, respectively) relied on an object called an action which is represented as the integral of the difference of the kinetic and potential energies of a system (the Lagrangian). Currently, most of physics is structured around the ability to formulate an action for an observed process. As sort-of indicated by the name, the Einstein-Hilbert Action represents the laws of General Relativity; if you use a slightly modified version of Lagrange’s Variational Calculus3 to “minimize” the Einstein-Hilbert action, you get the Einstein Field Equations, G_{\mu\nu} = 8\pi G T_{\mu\nu}.

Right now, my optimistic side hopes that those of you still reading are thinking something along the lines of: Cool! So some weird math formula that involves a fancy looking x contains Einstein’s equations in a slightly opaque form. What’s so great about that?

Here is where the physics steps in. Einstein’s equations predicate that an object that travels under the influence of gravity will travel along geodesics. Intuitively, geodesics represent the minimal distance between two points in a (Riemannian/Lorentzian) manifold.4 On a curved surface, these geodesics are not always equivalent to the standard straight-line or Euclidean distance between points. For example, on the Earth, when one travels a long distance, say via plane, it is most efficient to travel in a great circle. After all, one doesn’t travel via the “Euclidean” distance (i.e. the distance if we embedded the Earth in Euclidean 3-space, \mathbb{R}^3) as that would go through the center of the Earth! One of the biggest implications of the principle of equivalence (which guides General Relativity), which states that one cannot discern the difference between a constant acceleration and a gravity field, is that gravity effectively forces objects to travel on their geodesics. Since geodesics (at least on this heuristic level) are dependent on the curvature of a surface, this means that gravity depends on the curvature of space-time. The G_{\mu\nu} term on the left-hand side of the Einstein Field Equation represents this geometry; G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R where R_{\mu\nu} is the Ricci 2-form (intrinsic) curvature, R is the Ricci scalar and g_{\mu\nu} is the metric (in coordinates, of course). All of these terms deal with the curvature of space-time. On the other hand, the T_{\mu\nu} term on the left is the energy-momentum tensor. It contains all of the physical constraints (i.e. the energy constraints) and by equating these two, one limits the geodesics by 1) the curvature of space and 2) the other “forces” that contribute energy.

Okay, so what have we established:

  • The cool math formula contains the Einstein equations, in disguise
  • The fancy looking x, i.e. \chi is related to the curvature of the space and in turn the Einstein equation

Awesome! So we have four things to establish:

  • What is \chi?
  • What’s this intrinsic/extrinsic curvature thing?
  • What about the second term?

So let’s address the first point. \chi is what is known as the Euler Characteristic. The Euler Characteristic started out as a way for Euler to classify the platonic solids based on their three geometric characteristics: The number of vertices, edges and faces. However, it wasn’t until the 20th century when mathematicians realized that the Euler Characteristic could help classify “types” of spaces (by using the wonderful machinery of algebraic topology). Colloquially, one might say that the Euler Characteristic classifies the shapes one can make out of Play-Doh without poking any holes into the object. Let me be a bit more succinct: This is a homeomorphism invariant of a manifold M (in general: A topological space X). Without going into a fair bit of math, it’s kind of hard to explain what a homeomorphism is5 but the Play-Doh example I mentioned before contains the essence of what a homeomorphism can look like in 2/3 dimensions. For example the famous coffee-cup/doughnut homeomorphism is below:

The idea is that the Euler Characteristic computes something a bit more fundamental about a manifold or topological space that doesn’t change under a broad number of transformations. However this is NOT an analytic piece of information. As it turns out, one need not define a notion of distance in order to compute an Euler Characteristic; in fact, it would make no sense, since a dilation/stretching (of an object endowed with a way to measure distances) is a homeomorphism.  Yet the formula (1) is described by an analytic formula! Not only is it an analytic formula, but it contains physical data as well! This means that the Einstein-Hilbert Action (and in turn General Relativity) say a very important bit about some of the topological properties and restrictions that anything that satisfies Einstein’s equations must adhere to. And since quantum field theory needs to incorporate the results of general relativity (GR) in order to explain gravity at a microscopic level, this means that some how, quantum field theory (QFT) should provide a topological statement, either in the form of curvature or in the form of a topological invariant (like the Euler Characteristic) to describe gravity. Since the formulations of Gauge Theories rely on Fibre Bundles, Principle G-Bundles and Cohomology, some of the topological “tools” are already adept to making the jump from QFT to QFT+GR. Yet a big problem is that a point particle generates a line as it travels through space; this provides a sort of “minimal” amount of topological data or geometric data. The point particle is topologically trivial and provide no information about gravity, based on the Gauss-Bonnet Theorem! As such, it makes more sense to try to see if higher dimensional object fit the bill, because then we can produce some of the effects predicted by GR by using a microscopic (QFT) theory. Since M-Theory (the effective successor of String Theory) relies on Strings as well as higher-dimensional objects called branes, we are effectively trying to ensure that QFT provides results that jive with (1)!

Phew! That was a mouthful and probably needs some re-reading and questioning, so feel free to leave comments with questions and I’ll try my best (hey, I’m no expert yet!) to answer them. Now let’s address the last two points. The manifolds that one deals with in General Relativity always have an associated way to measure distance, via the symmetric, non-degenerate, 2-tensor $g$, i.e. the metric (2-)tensor. In order to make some real computations (i.e. things that have numbers) one needs to be less abstract and represent the metric in some sort pragmatic fashion. Recall that a manifold is “locally” Euclidean; since Euclidean spaces have many metrics (for example, the straight-line or Euclidean metric), a manifold inherits a metric, at least locally. Note that since there are an uncountably infinite number of metrics on \mathbb{R}^n, this metric need not be as trivial as the straight-line metric. In GR, one tends to write a “local” expression for a metric in coordinates in order to do computations. However, the choice of “coordinates” that one uses is not unique (i.e. Cartesian vs. Polar coordinates in the plane) as one can write such a “local” expression in a multitude of ways. Let’s illustrate this with an example. The standard Euclidean metric has the local expression g(\partial_x, \partial_y) = dx^2 + dy^2. However, I can just as well express it as g(\partial_r,\partial_\theta) = dr^2 + r^2 d\theta^2. As such, since the Ricci and Riemann tensors (Recall: instrinsic) curvatures depend on the metric (in their formal definitions), one may ask whether they depend on the coordinates one choose. As it turns out they do not depend on the choice of coordinates that one chooses and hence they get the name intrinsic. On the other hand, it is possible to define “curvature” in a way that is computationally useful but relies heavily on the coordinates of the metric. This is called the extrinsic curvature. In physics, one often encounters the phrase geodesic curvature. This is simply the extrinsic curvature; it is named as such since it relies on the specific coordinate expression of the geodesics, which in turn rely on the coordinate expression of the metric. If this all is a bit too much to handle at once here is the take-away message:

Intrinsic Curvature doesn’t change if you change your perspective of space-time; Extrinsic Curvature changes with a change in perspective/viewpoint (coordinates) of space-time

Finally: That pesky second term! Effectively, it kind of computes the “boundary” curvature of the manifold (i.e. $\partial M$ is short-hand for “the boundary” of M). Now the idea here is that the curvature on the boundary represents the explicit parametrization of the manifold; this is again hard to quantitatively describe in a few words, but we can use Play-Doh to make an “argument” of sorts. When you shape Play-Doh, a lot of the time you are only smoothing out the boundary; if you make a hole, you drastically change how the boundary looks and such a change is reflected in the extrinsic curvature. Some of the points on the interior of your Play-Doh object may never even change (as embedded in \mathbb{R}^3) when you mold a sphere into a torus by poking one hold through your Play-Doh, but the curvature forms on the boundary will have singularities in their coordinate expressions upon such a change. This type of change may not be as easily detected by the intrinsic curvature and as such the boundary behavior is quite important. This is in many-ways analogous to Stoke’s Theorem and if it helps, one should try to use that analogy.6

So the Gauss-Bonnet Theorem relates the curvature (an analytic thing) of space time to its topology! In fact, this is generalized by the Riemann-Roch Theorem and the Atiyah-Singer Index Theorem. This class of theorem, called index theorems, finds very clever ways to relate analysis (a concrete thing) to topology (a less concrete thing) so that one can come up with “qualitative” restrictions on a mathematical object. And as explained above, these “qualitative” restrictions play a big role in pointing out the ideas that Quantum Field Theory (which is analytically formulated) is missing. String Theory attempts to fix these missing ideas while preserving both the analytic and topological paradigms of physics!

Phew! Thanks for reading!

  1. Okay, so I’m probably excluding logic here but personally I find mathematical logic to really be a philosophical discipline. However, that’s not a knock on the subject — it’s certainly difficult and one has to grapple with many interesting problems. It’s just that I’m not sure if the majority of the historical viewpoints of mathematics would include logic within the field. I think the only time I’ve seen logic come up in String Theory is a) showing that some set of moduli is uncountable and b) mentioning that the word problem for groups in NP-complete, which makes the classification of connected manifolds in dimension 4 and higher practically impossible. Note that the word problem for groups shows up because this makes it impossible to find an algorithm that can prove that groups/modules generated by the functors \pi_{*} or H_{*} are isomorphic. []
  2. Citation: Edward Witten, Supersymmetry and Morse Theory, Journal of Differential Geometry, 17(4), 1982. If you know what the Euler characteristic is and if you know how to do a Taylor expansion, I suggest you read this paper. Witten is brilliant because he finds way to use elementary methods to prove über complicated stuff  []
  3. Yes, I know. People such as Euler and the Bernoullis solved the Brachistochrone problem before Lagrange and used techniques from functional analysis on C^{\infty}(\mathbb{R}). But Lagrange’s treatise, Méchanique Analytique, really put all these ideas together and algorithmically described some the variational calculus involved, so I give him credit.  []
  4. Side note: Sure, Orbifolds and Conifolds have some sort of geodesics that they inherit under the topological quotients that define them. But since we’re talking about macroscopic physics here, I don’t think it’s worth complicating the discussion by including them. []
  5. Mathematical Definition: Suppose one has two topological spaces X,Y and a bicontinuous, bijective map f: X \rightarrow Y. Then X,Y are said to be homeomorphic, which is denoted X\cong Y and f is called a homeomorphism []
  6. They are closely related as it turns out; Stoke’s Theorem has to do with the cohomology of a space while the curvature form represent an element of a certain cohomology group, called a characteristic class. []

My experience in beginning Undergrad String Theory Research

So you say that you’ve watched The Elegant Universe more times than you’re willing to admit and you say that you love the mathematical beauty of general relativity yet desire the accurate predictions of Quantum Field Theory? Plus, you also claim to think that Lee Smolin is a nutjob, Edward Witten is god and the word “blow up” brings to mind algebraic varieties and birational maps. Finally, you claim some knowledge of algebraic and differential topology as well as differential geometry. Given this background, you probably are considering string theory research. Great!

…but where do you start?

Aha! So I’ve been in the same position and I figure that it maybe useful for me to outline some of the things I’ve learned (and I’m still learning) as I’ve gone through this process. Let’s start with my opinion of “what” string theory actually is, as a discipline.

String Theory is a natural generalization of the point particle that attempts to microscopically describe gravity. With regard to the current, predominant (and experimentally-validated!) paradigms of Physics (GR, QFT), String Theory tries to ensure that all prior experiments results blend together in harmony. This is not a statement that String Theory claims to be a Theory of Everything. Rather, it attempts to describe some phenomena that both subsume our current mathematical technologies and require a more delicate treatment. At the end of the day, I feel that QFT is formulated in a highly unsatisfactory way. The fact that one still needs to use lattice approximations to deal with weakly coupled Quantum Chromodynamics is a telling sign that the theory is piecemeal and needs refinement.

Note that String Theory is not a single discipline or study. In fact, Prof. Liam McAllister (my undergrad thesis advisor in physics) has repeatedly explained to me that String Theory stretches a whole slew of disciplines (math, physics, astronomy) as well as subdisciplines (Algebraic Geometry, Differential Geometry, Algebraic Topology, Differential Topology, Functional Analysis, General Relativity, Quantum Field Theory, Statistical Mechanics and others). As such, there are many practitioners of String Theory who focus on rather disparate areas of the theory, without knowing precisely how everything fits together. There are few who actually grasp all of the mathematics and the physics (these are separate and distinct!), and those who do make the biggest intellectual leaps.


Okay. If I’m going to give any advice, one has to know what perspective I’m coming from. So if String Theory is not a single discipline, where do I fit in? I actually am trying to straddle the mathematical and physical boundary (no AdS/CFT pun attempted here) in order to gain a slightly more complete picture. At the moment I’m trying to discern the spectrum of vector operators on a 5-dimensional space of String Theoretic Interest. I initially tried to get to this point by solely reading mathematics perspectives and learning the algebraic precursors, since I believed that the only way to approach the field was via a strong mathematical background. I learned the hard way, that the only “pure” way of entering String Theory is via Philosophy and my disdain for that field led me to reconsider my approach to String Theory. I began to realize that instead of focusing on the analytical aspects (i.e. functional analytic development of the theory), I should focus on the geometric aspects of the theory since they had the most mathematical fortitude within the realm of string theorists.  As such I had to learn a ton of things about algebraic geometry and differential topology. In this process I was blind-sided by the amount of things I had to learn and certainly became disheartened at various points. However, once things started clicking mathematically I felt that I was ready to tackle the physics of String Theory.

Before I was even sure I wanted to do String Theory Research, I tried reading Polchinski’s String Theory, Vol. 1 but I was flabbergasted by the notation. I had been so used to a lack of coordinates in differential geometry that seeing the non-degenerate, bounded below, symmetric quadratic form g : TM \times TM \rightarrow \mathbb{R} denoted as the metric g_{ij} was quite unfamiliar. This was only the first stumbling point, because I thought that reading a physics book would be like reading a math book — If you pour enough time into reading and understanding each assertion you will feel quite confident. Alas, this is not the case. I like to say that Physics is like Mathematics except that you live dangerously; one of the things about living dangerously is that there is now a non-zero probability of not feeling confident in what one has learned after reading a book!  In reading physics literature, I had to trick myself into accepting random formulas and pushing myself from trying to validate every computation made. This took quite a while to do and at the time I still thought that I needed to learn more mathematics to understand the literature. As it turns out I realized that my biggest mistake was restricting myself to a few books on string theory. The best way to actually understand some of the artifices and subtleties in things as “basic” as understanding the open string spectrum really required going through each of the major sources and reading their descriptions. For example, I struggled through Chapters 1 and 2 of Polchinski, ending with a sad feeling in my stomach because I didn’t feel like I understood very much. However, after reading the same material in Green-Schwarz-Witten (Superstring Theory, Vol. 1, 2), Becker-Becker-Schwarz (Introduction to String Theory and M-Theory), Zwiebach (An Introduction to String Theory), Johnson (D-Branes) and t’Hooft (Introduction to String Theory), I finally began to understand where the 26-dimensional Bosonic String (and in particular, the quantum anomaly in the Virasoro algebra) came from. In fact, it took me a year (I only felt confident in my understanding of the bosonic string last week!) to finally feel a bit more confident. One of the biggest lessons I learned from this experience is that those with a mathematical background actually don’t need to learn any extra math. In this process, I ended up learning a lot more mathematics than I “needed to.” For example, I learned a fair bit about topological K-Theory, Sheaf Cohomology and equivariant cohomology. Yet, I’m quite glad I did because a lot of the current string theory research seems to require a deep understanding of these objects, enough so that a lot of the papers I’ve been reading have both mathematicians and physicists among the authors.

Perhaps that was a bit unclear, so I’ll summarize the main points/give some tips:

  • Make sure you have a LOT of physics sources. Everyone picks and chooses their favorite facts, often time trivializing important computations. Read ALL possible sources.
  • My favorite source (the one that tied everything together for me) was Johnson’s D-Branes. He skips a lot of the explicit computations, but the big picture perspective he uses can get you up to speed on the physics in a fortnight.
  • You may end up learning a lot more math than you need. And in my case, that was great, because I loved learning about that stuff. But to be honest, the way physicists use math is quite ad-hoc and in reality you need to be more deft at “symbol manipulation” as opposed to an understanding of a concept. For example, while Dirac Monopoles do represent non-trivial Chern Classes, this non-triviality is explained in the physics literature by the Chern Number which in turn is simply an integral. It’s not as transparent as I just said, which may make your mathematical mind spend time on converting things to mathematics vocabulary. This is useful, to the extent that one can trust assertions that are made, but when it is done too much, it is impossible to get the big picture.
  • If you want a quick introduction to the math needed, just read Nakahara’s Geometry and Topology in Physics. If you’ve never seen some of the topological and geometrical concepts needed [I'll outline the ones I think are the most important below] and you want to learn them more thoroughly via pure mathematical textbooks use Nakahara as an introduction. The analogy I use is Nakahara is to Atiyah, Bott, Singer, Hatcher, Milnor, etc. as Stewart’s Calculus is to Rudin’s Real and Complex Analysis. Nakahara is a great way to gain some intuition for some of the abstract objects one deals with. In fact, I think I understood Chern Classes better after reading Nakahara, even though I already knew the functorial definition for complex vector bundles.

If I think of anything else important, I’ll add it.

Good luck!


Thanks Supriya!

BEARBOT – Bassman

As promised, I figured I’d post a Bearbot song. Bearbot is a DJ from NYC who makes mashup art that I personally find better than Girl Talk or Norwegian Recycling. Take a listen, tell me what you think!

Robert Miles? What happened?

01-Children (Dream Version)

I recently listened to the new Robert Miles album, Th1rt3en and I have to say that I’m quite confused. What exactly happened to the melodic, trance-y instrumentals of his classic Children? When I listened to this album the only images I evoked were of Envy banging her head into the wall. Seriously! For example, the song Somnambulism certainly put me to sleep; I almost felt as if it should music played at some crappy cafe in Newark. Miles attempted to do too much I feel, trying to make things a bit jazzy, a bit hip and a bit instrumental. He should have stuck with his forte of pounding, melodic instrumental music. In that vein, I’ve attached Children in order to clean our ears of the rubbish contained in Th1rt3en.

In other news, I’ve fallen in love with BEARBOT. Once I pick my favorite song I shall share it!



Clifford Algebras, Part I: The Dirac Equation

So while Heisenberg definitely has the cooler suit, Dirac has the better Physicist stare

Mathematicians consistently lament the lack of rigor that physicists hold themselves to, often to the point of ignoring some of the more mathematical quandaries that nature presents. At the same time, physicists rule out pure mathematics as “abstract nonsense” or a “waste of time.” Luckily for us, Paul A.M. Dirac, perhaps unknowingly, discovered a wealth of mathematical and physical treasures when he discovered the celebrated Dirac Equation of quantum field theory. To peer into why it serves as an illustrative example of the interplay between modern mathematics and physics, let’s first look at the motivation and history of this equation.

Surprisingly, Wikipedia has a wonderful article on the Dirac Equation that dictates the history roughly to the level that I’ve heard it described; I suggest you check it out should you find my necessarily concise exposition insufficient for your curiosity.


Ah nature, some times you provide us with stories that are even more romantic and yet dorkier than what the human imagination can conjure. As legend has it, Paul Dirac was staring into a fireplace at Cambridge while pondering a great problem of physics at the time. The year was 1928 and quantum mechanics had been put on a somewhat solid footing thanks to the equivalence of Heisenberg’s Matrix Mechanics and the Schrödinger picture. With the old theory of Bohr in the rear-view mirror, physicists had a new mathematical tool to tackle the biggest problems of their day with. This tool was the theory of bounded, densely-defined and symmetric operators on the Hilbert Space L^2 (\mathbb{R}^3, dm) [Note: I didn't say self-adjoint or Hermitian. Physicists tend to overuse this term, even though they often mean symmetric]. This tool, however, seemed nothing like the hammer to Einstein’s Relativity’s sickle. It lacked geometric intuition and could only probabilistically guarantee results. Making these somewhat belligerent tools learn to work together was crucial to developing a quantum theory of the electromagnetic field. As it turns out, the main particle involved in the propagation, radiation and interactions of the electromagnetic field is the photon, which obeys the strict speed limit of the speed of light. Yet the quantum mechanics of the time provided an autobahn for the energetic and massless photon, contradicting the immense developments (due to Einstein) of 1905-1920. For a naïve explanation of why quantum mechanics and (special) relativity refuse to jive, lets consider the energy-momentum relation of Special Relativity. In special relativity, we have the equation,

E^2 = p^2 c^2 + m^2 c^4

Note that this equation gives E \propto p + O(c^4) so that energy is proportional to momentum. However, recall that the Schrödinger equation is defined as,

\left(-\frac{p^2}{2m} + V\right) | \psi \rangle = E|\psi \rangle

In this equation E \propto p^2. Since the Lagrangian and Hamiltonian formulations of classical mechanics center on energy, how do we rectify this different? After all, one definition of energy is quadratic and one definition is linear. Now note the theory of Hilbert Spaces, however, is distinctly linear. After all, a Hilbert Space is defined as a vector space equipped with an inner product that is complete in the norm induced by the inner product. As such, quantum mechanics is a genuinely linear theory and can rely on the vast amount knowledge that we have about linear algebra in both finite and infinite-dimensional vector spaces.

Great. Now let’s rewind to Dirac staring into the Cambridge fireplace. He was trying to unite Special Relativity with Quantum Mechanics to describe a particle theory of the electromagnetic field. But Special Relativity is a non-linear theory (at least to the heuristic analysis of the \gamma factor) with a linearly defined energy while quantum mechanics is a linear theory with a quadratic energy. Putting ourselves in Dirac’s shoes, we have two ways to proceed:

1) Since the discovery of the derivative as a “linearization” of complicated objects, humanity has discovered a ton of linearization techniques. Maybe if we “linearize special relativity” we can approximate quantum mechanics

2) Is there something similar about the two theories? We’ve focused on how they don’t jive in one (relatively) major way, but perhaps there is some underlying similarity can be used to come up with a theory.

Many physicists, including those of great esteem and those of no tenure, tried to use the first method to no avail. Perhaps this stems from the difference in perspectives: In quantum mechanics, one focuses on the analytic and infinite-dimensional aspects of the theory whereas in special relativity the geometric and finite aspects are important. [Ironically, this linearization methodology is effectively what we use to try to show that String Theory is a viable candidate to describe nature, predominantly since the theoretical validation of General Relativity is generally performed via linearization to Newtonian gravity.] Regardless, it didn’t work. So let’s outline what is similar between special relativity and quantum mechanics

1. Both have a norm. In fact, we can say something stronger that ends up being far more important: Both have associated quadratic forms

2. Both have gauge theories associated with them. For those with a bit of math experience, a Gauge Theory with Gauge Group G is simply a  principal G-Bundle over a smooth manifold. In the case of special relativity, G = \mathsf{SO}(1,3) and in (non-relativistic) quantum mechanics G = \mathsf{SO}(3) \rtimes \mathbb{R}^3

The exact formulation of Clifford Algebras that I will explain in the proceeding articles will emphasize how these two similarities allow for special relativity and quantum mechanics to unite.

Dirac’s Discovery

Dirac may have been a physicist, but he had more mathematical intuition than most mathematicians of the last hundred years. He understood that the main problem in rectifying these two theories was the inherent quadratic vs. linear difference. If we let \hat{H} be the free-particle Hamiltonian (i.e. V = 0, then the Schrödinger equation is simply \hat{H}|\psi \rangle = i\hbar \partial_t |\psi \rangle. However at some point, a brilliant idea dawned on Dirac — Would it be possible to define linear operators A,B such that A,B \propto p AB = H? If one could find such an operator, then it would be possible to satisfy 1) a Schrödinger-esque equation and 2) Special Relativity. More precisely, our goal is to find “constants” \alpha_k,\beta such that,

\hat{H} = \overbrace{\beta mc^2}^{\text{Rest Energy}} + \underbrace{\sum_{i=1}^3 \alpha_k p_k c}_{\text{Kinetic/Dynamic Energy}}

Now this was a phenomenal proposition for the time, as “square-roots” of operators were thought to be only small interest, even to mathematicians. As it turned out, \beta, \alpha_k had to satisfy the following relations in order for the above equation to hold:

\alpha_k^2 = \beta^2 = \mathbf{1}

\left\{\alpha_i,\alpha_j \right\} = 0

\left\{ \alpha_i, \beta \right\} = 0

These relations generate the Clifford Algebra \mathsf{C\ell}_{(1,3)}(\mathbb{C}) and \alpha_k,\beta are most easily described as matrix representations of \mathsf{C\ell}_{(1,3)}(\mathbb{C}) on \mathbb{R}^{1,3}. As such, Dirac discovered a connection between an abstract, long-forgotten algebraic object discovered by W. Clifford in the mid-1800s and differential equations and physics. With some slight definitions of \beta,\alpha_k one can construct the Dirac matrices \gamma^i that satisfy \left\{\gamma^i,\gamma^j \right\} = 2\eta_{ij} (\eta_{ij} is the Minkowski metric) which can define the beautiful Dirac equation:

-i\hbar \gamma^{\mu} \partial_{\mu} \psi + mc\psi = 0

The Impact of the Dirac Equation

The steady progress of physics requires for its theoretical formulation a mathematics that gets continually more advanced. This is only natural and to be expected. What, however, was not expected by the scientific workers of the last century was the particular form that the line of advancement mathematics would take, namely it was expected that the mathematics would get more and more complicated, by would rest on a permanent basis of axioms and definitions, while actually the modern physical developments have required a mathematics that continually shifts its foundations and gets more abstract … [soon] it will be beyond the power of human intelligence to get necessary new ideas by direct attempts to formulate the experimental data in mathematical terms. The theoretical worker in the future will therefore have to proceed in a more indirect way. The most powerful method of advance that can be suggested at present is to employ all the resources of pure mathematics in attempts to perfect and generalise the mathematical formalism that forms the existing basis of theoretical physics…

Paul Dirac, Quantised Singularities in the Electromagnetic Field, Proc. Roy. Soc. A 133, 60, 1931

Let this quote sink in for a minute. Dirac quite explicitly equivocated for a strong connection between mathematics and physics, far before the mid-1970s, when physicists began to realize that they had their own “abstract nonsense.” While the immediate descendants of Dirac and others tended to shun the discoveries of mathematics in their search to classify all the elementary particles, the subsequent generation found it a necessity to be fluent with once “useless” concepts such as fibre bundles, sheaf cohomology and homotopy groups. At the same time, the breathtaking mathematical discoveries of Simon Donaldson, Shing-Tung Yau and Edward Witten had convinced mathematicians that perhaps living a G.H. Hardy lifestyle would turn out to be futile. All of these developments were predicted by Dirac upon his realization that the universe has a quantized electric charge iff there exists a magnetic monopole, which represented a “non-integrable phase.” In modern terminology, such a phase represents a generator of a non-trivial first Chern Class.

To be finished soon!

My name is Taru

The History and Beginnings of String Theory

Short aside about why this topic was chosen

As I mentioned in my last post, I want to detail things that tend to have poor expositions or poor mathematical rigor. A lot of the mathematical topics I included in my initial schedule are really well exposited in books such as Allen Hatcher’s Algebraic Topology and Differential Forms in Algebraic Topology by Raoul Bott and Loring Tu. These books provide a beautiful combination of geometry, topology and imagery so that the reader has a well-rounded understanding of the topics they are dealing with. I initially wanted to write about these various topics in the blog for my own benefit, but I realized that I would end up rewriting what they have in less clear terms. Instead, I figure that it will be better to focus on objects, dynamical systems and theorems in physics which have poor mathematical descriptions. For example, the classical Bosonic String tends to have two disjoint mathematical and physical descriptions, which are unified to some extent in a few books (e.g. M. Schottenlohen’s A Mathematical Introduction to Conformal Field Theory). I hope to write this article in three parts, emphasizing the intuition in the beginning and the rigor toward the end. I’m going to put the list of references used at the beginning of each article, because it will help those who are more mathematically and/or physically inclined choose what they would like to read.


Note 1: I’m going to update the references with all of the ones I used later today (Monday, August 16)
Note 2: If anyone knows of something that will let me use BibTeX in WordPress please, please e-mail me!

[1] Becker, Katrin, Melanie Becker, and John Schwarz. String Theory and M-Theory: A Modern Introduction. 1st. New York: Cambridge University Press, 2007. Print.

[2] Polchinski, Joseph. String Theory. 2nd. 1. New York: Cambridge University Press, 2005. Print.

[3] Schottenloher, Martin. A Mathematical Introduction to Conformal Field Theory. 2nd. Heidelberg: Springer-Verlag, 2008. Print.

[4] Schwarz, John. “A Brief History of Superstrings.” A Brief History of Superstings [sic]. Caltech, n.d. Web. 15 Aug 2010. <>.

[5] Zwiebach, Barton. A First Course in String Theory. 2nd. Cambridge: Cambridge University Press, 2004. Print


Some exposure to the concepts involved in the theories of the following keywords: differential equations, manifolds (esp. the concept of a submanifold and the associated submersions and embeddings), group actions, electromagnetism (necessarily relativistic), Lagrangian mechanics. The history section will be written with the expertise of a mathematician or even an engineering, with some of the rudimentary physical concepts laid out explicitly. The representation theory of the Virasoro algebra will probably be avoided, pending time constraints. However, I may go into how one can mathematically arrive at the algebra via a short-exact sequence called a central extension.1


Note: The majority of this history is quite basic and general in order to provide any reader with an accessible introduction into the motivation of string theory. Those with quantum mechanics or QFT experience can skip to the section entitled “Development of the String”

String Theory generally finds it’s roots in the 1970s with the ambiguity over the discovery of the strong nuclear force. I’m going to suppose that most people who vaguely pay attention to science news or read popular science magazines know that physicists believe that there are only four forces of nature, namely2 :

  • Electromagnetic Force (EMF)
  • Gravity
  • Strong (Nuclear) Force
  • Weak (Nuclear) Force

The standard presentation of these forces tends to be along the lines of “gravity and EMF are obvious” and the strong and weak forces take too much time to explain, so just accept their existence. While I agree with the first point about gravity and EMF, I figure it would be wise to go through a brief view of the nature of the strong and weak forces as well as experimental evidence.

Weak Nuclear Force

The weak nuclear force has it’s roots in the statistical mechanics studies of Enrico Fermi in the 1930s. The notion of radioactivity has well-defined intuitive and macroscopic meanings, but after the advent of Quantum Mechanics, theoretical physicists were in search of an atomic or subatomic explanation of known macroscopic phenomena. Radioactivity as a macroscopic phenomenon, has its theoretical roots in the radiation theory of classical EMFs. In a sentence, one can define radiation as the emittance of an EMF due to the motion of a particle; examples of radiation range from the benign (your car stereo’s radio) to the malignant (X-Rays, \Gamma-Rays). However the treatment of the electron and photon in classical EMF theory assumes that the energy contained within these emitted EMFs continuously change with regard to properties of the distinct particle. The uncertainty principle of Quantum Mechanics, however, makes such labeling of particles as distinct a dubious proposition. Satyendra N. Bose (with later generalization due to Einstein) indirectly gave credence to those who challenged radiative explanations, by showing that the energy distribution3 of a certain group of particles (now called Bosons) matches experiment if one assumes the particles are indistinguishable (i.e. can’t be labelled individually). Note that the concept of indistinguishable particles is manifestly a quantum mechanical observation; Bose made the discovery while considering the failure of the ultraviolet catastrophe and the effect this has on radiation. While Bose and Einstein didn’t realize it at the time, their discovery provided credence to quantum mechanics in the form of spin. The existence of spin is due to the Stern-Gerlach Experiment, which showed that an electron beam splits into two groups under the influence of a strong magnetic field. Quantum Mechanics was able to absorb this experiment by subsuming a new quantity4 that behaved (at least algebraically) like angular momentum.5 Initially, Bose only described the statistics for the photon while Einstein generalized it to certain other components in an atom (e.g. Protons, Neutrons). These components all share one quantifiable quantity: They have integer spin! Putting it all together, we have: The Bose-Einstein Statistics, the distribution of the expected number of bosons at a given energy, describe the energetics of Bosons, or particles with spin s,\,s\in\mathbb{Z}.
Now back to Fermi: A major contribution of his in the early part of the 1900s was the detailing of the energy distribution for a single particle with spin s, s \in \frac{1}{2}\mathbb{Z} \setminus \mathbb{Z}. These particles with “half-integer spin” (i.e. an element of \frac{1}{2}\mathbb{Z} \setminus \mathbb{Z}) are called fermions and they follow a different set of statistics (aptly named Fermi Statistics). Fermi attempted to use various statistical results he derived to explain radiation, leading up to Fermi’s Theory of \beta Decay . As it turned out, while this theory was a step in the right direction (it described interactions between particles albeit without specifying how the particles interact) it gave way to the electroweak theory of 1968 (Glashow, Weinberg, Salam) which used the concept of a mediating Boson (namely the W, Z Bosons) which fixed a specific gauge6. If you have heard that the gauge group of the standard model of particle physics7 is \mathsf{SU}(3)\times\mathsf{SU}(2)\times\mathsf{U}(1), then the electroweak interaction describes the \mathsf{SU}(2)\times\mathsf{U}(1) portion of the gauge group. This implies that \mathsf{U}(1) plays two roles, for it is also the gauge group of EMF interactions.

\boxed{\therefore \text{Thus one can treat the weak-force as the force that defines nuclear decay properties}}
Strong Nuclear Force
Until the apex of the 20^{th} century, many discoveries in physics arose from unexplained experiment that requires theory. The Strong Nuclear Force (SNF) is generally considered to be the last big discovery of this type. Now let’s start in the year 1970. At this time, the existence of smaller particles that made up the protons and neutrons in the nucleus was well-known with the discovery of the quark8 (1968, Stanford) and the mystery of beta decay and atomic-level radioactivity was esoteric-no-more after the 1968 introduction of the electroweak interaction. However, the mystery of why these particles “stuck” together and created the coherent object of a proton was still unsolved. The quantum field theory called Quantum Chromodynamics was invented at this time to describe this mysterious force and how quarks and be put together to make hadrons.9. Without going into details (for I know very few!), one can look at the gluon as an example of a particle involved in mediating the SNF. Essentially, the gluon allows quarks to interact and form cohesive units either as triplets or doublets. Thus the SNF can be thought of as the interaction that keeps hadrons (such as the proton) cohesive.
Development of the String

In the wake of the success of the electroweak model, theoreticians began to look for ways to describe the strong interaction and the field theoretic interpretation of quarks. As mentioned before, Quantum Chromodynamics became the accepted model (and is currently accepted) in the late-1970s. However, QCD had an initial competitor known as a dual-resonance model, proposed by Gabriele Veneziano in 1968. In the 1960s, theories attempted to use scattering amplitudes to demystify the unexplainable experimental evidence for the SNF. Why is scattering related to the SNF? Since physicists were looking for building blocks of the hadrons, a quick way to break something up into it’s component pieces is to break it. It may be easier to explain this with a gedanken:

Suppose that I have a glass sphere filled with marbles of three different masses. If I roll the sphere into a wall with enough force, the ball will smash releasing the marbles everywhere. Since the marbles are of different masses, how far individual marbles roll and the direction they “scatter” in is in general a function of the (internal and external) forces involved in throwing the sphere into the wall. This means that the mass of the marble plays an important role in it’s trajectory and distance travelled.

Now suppose that we could recover the “type of marble” we randomly pick up off the ground based on how far the marbles travelled, the path taken (trajectory) and the force used to smash the sphere. Then this situation is precisely analogous to how particle accelerators try to detect new particles: they smash hadrons (which are analogous to the entire sphere and marble system, with the glass shell of the ball mediating10 the interaction between the marbles and the sphere) into each other, hoping to pick out smaller particles (the marbles) using the amplitudes of their constituent wavefunctions11 (analagous to the “distance” the marble rolls from the point of impact) and possibly trajectory. Thus scattering provides the experimenter with an indirect way of measuring the strength of the SNF, based on the amplitude distribution.
The most important SNF-related theory to come out of the 1960s is the Regge Theory. Let’s start by looking through what is already known: physicists understand how EMF scattering12 works as an \mathbb{R}-valued function of angular momentum in the basic quantum mechanical case (i.e. via the Clebsch-Gordan Coefficients13 ). What the Regge Theory attempts to do is extend (via analytic continuation) the allowed values of angular momentum (those ever-so familiar \ell quantum numbers that chemists often misinterpret) to any value \ell \in \mathbb{C}. In doing this, we introduce complex energies and an object called the Regge trajectory, \alpha : \mathbb{C} \rightarrow \mathbb{C} such that if \alpha(E) = L \in\mathbb{Z} then there exists a bound state with energy, angular momentum E,L. Moreover the Regge trajectory was implicitly defined to have compact support; this is key to the formulation of the open string. The main implication of the theory is as follows:

If we have some scattering reaction A + \tilde{A} \rightarrow B + \tilde{B} (at energy E),where \tilde{A}, \tilde{B} are the antimatter partners of A,B, respectively, then the scattering amplitude \emph{A} is proportional to \theta^{\alpha(E)}, where \theta is the scattering angle

Note that this claim is testable (we need only to know the energy of the accelerator and \theta, which is experimentally measurable).

Now back to Veneziano’s contribution: He provided an explicit formulation of scattering amplitudes as the Euler \beta function and interpreted the evolution of the associated Regge trajectory as a string-like object that represented different particles at different resonances. More precisely, we can explicitly define the Veneziano Scattering Amplitude, A(s,t) in terms of the Mandelstam variables.

Essentially, the Mandelstam variables serve as a way to group together momenta of ingoing and outgoing particles. For example, if we have electron-positron scattering, e^{+}e^{-} \rightarrow e^{+}e^{-} then we define the Mandelstam variable s to be the sum of the momenta of the incoming electron and positron, squared. By conservation of momentum this means that the sum of the momenta of the outgoing electron and positron (squared) is precisely the same. This variable represents the idea that the incoming particles form an intermediate particle that late splits into the outgoing particles.
Next we define the Mandelstam variables t,u to be the difference in momenta an incoming particle and an outgoing particle. The idea here is that the incoming particle emits an intermediate particle which becomes the outgoing particle. Let’s clear this up, paradoxically, with formality:
Suppose we have incoming momenta p_{1}, p_{2} and outgoing momenta p_{3},p_{4}; then we have the s-channel variable and t-channel variable,

s = (p_{1}+p_{2})^{2}= (p_{3}+p_{4})^{2} t = (p_{1}-p_{3})^{2}=(p_{2}-p_{4})^{2}

Finally we have the Veneziano Scattering Amplitude:

A(s,t) = g^2\frac{\Gamma(-\alpha(s))\Gamma(-\alpha(t))}{\Gamma(-\alpha(s)-\alpha(t))} = g^2 \beta(-\alpha(s),-\alpha(t))

where \alpha(x) = \alpha_{0} + \alpha\cdot x

From this equation, we see that we have:

  • A linear Regge Trajectory
  • A relatively simple amplitude relation that explicitly describes the relationship with the coupling constant g

So how do we get from a scattering amplitude to a string? Two words: A Pomeron
In the 1960s and 1970s it was noted that hadron-hadron experiments lead to the following, rather general conclusions (Reference [6]):

  • Hadron-Hadron collisions were inelastic and had little or no quantum number flow14.
  • The forward scattering amplitudes are almost purely imaginary15 giving credence to the Regge trajectory concept.
  • The total likelihood of interaction between particles is asymptotically constant (i.e. \displaystyle{\lim_{E\uparrow \infty}} \text{Likelihood} = C)

As it turns out, a Regge trajectory with \alpha_{0} = 1 is a sufficient condition for the second and third experimental properties above; any such trajectory is called a pomeron. This trajectory represents a collection of particles16 and as such it makes predictions of new particles at various (E,L). However, since these particles were not detected in the 1970s,17 Physicists attempted to use a Kaluza-Klein compactification of Minkowski Space \mathbb{R}^{1,3} to explain some of the experimental correspondences with the pomeron. The Kaluza-Klein compactifcation starts with \mathbb{R}^{1,4} and then mods out by the equivalence relation ~ defined by:

x ~ y \iff p_{5}(x) - p_{5}(y) \in r\cdot \mathbb{Z} for some r\in\mathbb{R}, where p_{i} is the projection onto the i^{th} component of \mathbb{R}^{1,4}

This is simply saying that the extra dimension is compactified into a circle so that we have a cylinder-like space. This effectively says that any “slice” or cross-section is homeomorphic to \mathbb{R}^{1,3}, or (p_{1}\times p_{2} \times p_{3} \times p_{4} \times 0) (\mathbb{R}^{1,4}) \cong \mathbb{R}^{1,3}. As it turns out, in this compactification, the interaction of the particles predicted by pomeron is mediated by the release of a cross section of this “cylinder.” This is the first closed string.

The evolution of the Regge trajectory could be computed using a construction called the S-matrix that Heisenberg developed in the 1950s as a possible solution to the renormalization problems18 of pre-Feynman-Schwinger-Tomononga-Dyson quantum field theory. For completeness purposes, it’s probably a good idea to define the S-matrix; I’ll give a little description in the next blog entry that will talk about the construction of the Bosonic string. This novel treatment of a quantum mechanical object as a non-point object grew into string theory as we know it.

Veneziano’s ideas garnered attention from some of the brightest minds of the day, including Y. Nambu (Nobel Prize winner, 2008), D. Gross (Nobel Prize winner, 2004) and the aforementioned Heisenberg. These ideas developed into a few, relatively disconnected string theories. The early string theories were called Bosonic string theories because they could only explain the existence of Bosons (recall: integer spin) and were considered unrealistic because they predicted a particle called the tachyon which necessarily violates special relativity. Another odd flaw with these string theories is that they requires spacetime to actually be of dimension 26, which was portrayed as preposterous at a time when extra-dimensions were looked down upon19. In order to describe fermions, early string theorists attempted to add <i>supersymmetry</i> into the theory of the bosonic string. What supersymmetry does is add a relationship between fermionic particles and bosonic particles. How does it do this? Let’s consider the simple example of a graded Hilbert space \emph{H}:=\emph{H}_{Bos}\oplus\emph{H}_{Ferm} (one can generalize this to Fock Spaces quite naturally). The idea here is that \emph{H}_{Bos} and $\emph{H}_{Ferm}$ each have their own actions S_{Bos} and S_{Ferm} and as such have their own symmetries. Suppose that these symmetries are two Lie Groups G_{Bos} and G_{Ferm} that have irreducible representations on their respective Hilbert subspaces. Supersymmetry attempts to combine the two Lie groups into a “super Lie group” G^{Bos\vert Ferm} that acts on the entire Hilbert space \emph{H}. More concretely (and physically) this says that I can convert a Fermionic symmetry to a Bosonic symmetry if I have actions such that G^{Bos\vert Ferm} induces a conserved current for both Bosonic and Fermionic portions. This is quite unnatural — in fact, the difference in Fermi and Bose statistics ensures that fermions and bosons are treated on equal but separate footings. The hope is that at high energies, such a symmetry becomes apparent and realistic.

Let’s go back to the Bosonic String. Why 26 dimensions? As it turns out, the only way to preserve Lorentz invariance is when the dimension is equal to 26. This is due to the existence of a quantum anomaly. A quantum anomaly is a (natural) physical symmetry20 that generates a conserved charge in the classical theory but not in the quantum theory. Since the canonical quantization process is somewhat heuristic, one walks in hoping that all of the conserved currents (symmetries) will be preserved upon enforcing commutation relations between annihilation and creation operators. However, as it turns out the only way for there to be a Lorentz Current21 is if the dimension is 26 (Warning: this is a long computation!).  However, if one assumes supersymmetry, it turns out that one only needs 10 dimensions to preserve Lorentz invariance. However it turns out that one loses another symmetry in the process — Chiral symmetry.

Let’s briefly (and superficially) look at the main quantum anomaly that plagued string theory until the introduction of the Green-Schwarz Mechanism in 1984. When one is initially introduced to physics, the notion of right-handed and left-handed coordinate systems is presented intuitively as the naming suggests. However, the choice of the “right-hand rule” is arbitrarily built into mechanics and classical electrodynamics. As it turns out if one doesn’t force a handedness criterion, then it is possible to get some very interesting quantum phenomenon. This choice of “handedness” is known as chirality and it plays an important role in chemistry and physics. In general, it is much easier to formulate a theory with handedness “built-in;” one does this by not specifying an orientation of the cross product. However once experiments dictated that chiral effects are indeed crucial to a full understanding of particle physics,  physicists began to incorporate chirality into their equations. One of the greatest triumphs of human intellectualism is the invention of the Dirac Equation. By using a relatively simple mathematical argument, Dirac landed at an equation that described experimental phenomena up to unbelievable accuracy; moreover, it naturally separated chiral solutions. These chiral solutions manifested themselves in the quantum number known as flavor in quantum chromodynamics. The introduction of flavor as a quantum number allowed particle physicists to understand,  among other things, that quarks were not all the same. In effect, there are three types of quarks and chirality is the way to separate them experimentally. Let’s just take a look at the Dirac Equation:

\left(-i\hbar \slash{\partial} + m^{2}\right)\psi = 0

where \slash{\partial} = \gamma^{\mu}\partial_{\mu}, psi is the wavefunction and m^{2} is the mass of the particle. Where is chirality defined in this equation? As it turns out the \gamma matrices represent the generators of a Clifford algebra. Clifford algebras attempt to generalize the cross and inner products of Euclidean space in order allow both types of handedness.

To be continued..

  1. As it turns out the Virasoro algebra is the unique central extension of a more common Lie algebra called the Witt Algebra. []
  2. Most people tend to ignore the world nuclear when addressing these two forces, but it is a crucial semantic hint since most people imagine that these forces are macroscopic []
  3. i.e. the probability distribution function for number of Bosons at a certain energy []
  4. These quantities are referred to as Quantum Numbers []
  5. This led to the renewed interest in Clifford Algebras, since the algebraic structure of spin turned out to be W. Clifford’s once-foresaken invention. Kind of indicatesthat physicists shouldn’t disregard mathematicians as they often do :) []
  6. For practical purposes one can think of a gauge as a quantity that is only determined up to some multiplicative structure. Mathematically, suppose that we have a principle G-bundle P with structure group S over a manifold M. Then we define a gauge transformation as an element of S. Since the structure group represents transformations between local trivializations, the gauge transformation represents a locally-defined symmetry of an action. Physicists use the word “gauge” for two things: the gauge transformation and the structure group (known as the gauge group). When one fixes a gauge that means that one is choosing a specific gauge transformation in order for a computation to be well-defined []
  7. Recall that this only models three of the four fundamental forces []
  8. Theorized by Murray Gell-Mann in 1964 []
  9. Hadrons were particles that were once thought to be fundamental; however, once thirty or so of them were discovered physicists began to believe that they were made up of a tinier particle. This guess came to fruition with the discovery of the quark. Thus a hadron can be defined as any particle made up of three quarks (Baryons) or of one quark and one antiquark, the charge conjugate of a quark []
  10. This is a bit of a stretch, but if it helps picture particle interaction, all the better []
  11. When we smash the ball into the wall, the energy contained in the motion of the ball is converted to heat; similarly, due to the strong and weak forces, the energy used to “smash” the hadrons together is converted into radiation []
  12. i.e. how EMF scatters when it interacts with a hydrogen atom []
  13. Note that the reason these coefficients are real-valued is because they represent the conversion of tensor products of irreducible representations of \mathsf{SO}(3) into direct sums of irreducible representations of \mathsf{SO}(3) []
  14. i.e. exchange of angular momentum, spin, color, flavor, etc. []
  15. A purely imaginary scattering amplitude is not a mathematical nicety that doesn’t have a realistic interpretation. By definition, a complex scattering amplitude is the quotient of the scattering field by the incident field. Consider the simple fields F = \frac{\partial}{\partial x^{0}} and F' = e^{ix^{0}} \frac{\partial}{\partial x^{0}} [\frac{\partial}{\partial x^{0}} is the basis of the tangent space]. Then we certainly have a complex quotient which represents a phase shift. []
  16. Remember, a Regge trajectory represents areas where one is guaranteed a bound state for a given energy E and angular momentum L. These bound states represent a particle, since they guarantee the existence of a solution of the Dirac Equation at (E,L) []
  17. They’ve been confirmed in the early-1990s at Fermilab []
  18. i.e. infinities that arose within computations as the characteristic length scales changed []
  19. Probably due to the failure of the failure of the original, naïve Kaluza-Klein compactification; recall that this simply uses the extra-dimension to “wrap” spacetime around a cylinder []
  20. This means that the group action of a Lie group on a phase space element leaves the (Lagrangian) action invariant []
  21. Recall: Lorentz invariance is simply invariance under isometries of \mathbb{R}^{1,3}. Concretely this means invariance under the group action of \mathsf{SO}(1,3) []

So uh, about that Schedule…

Guess I blew through week 1 without writing much (aka I wrote nothing), but not for a lack of trying. I started writing articles on singular homology and the Mayer-Vietoris Sequence, but I realized that whatever I write will be more or less repeating what various resources (that are excellent!) already say. Thus I will focus on topics that have poor mathematical rigor or topics for where there are perhaps poor expositions. In this vein, I hope to write the following for this week1:

  • The Classical Bosonic String
  • What is a Conformal Field Theory?

Hopefully I’ll write more, but we’ll see how it goes.

In other news, check out this crazy video that is synchronized with Ratatat’s Drugs. Essentially he took stock images from Getty Images (i.e. the image provider to corporations when they need a generic, politically correct images, such as those motivational images that are often the bane of Internet memes) and overlayed them to Drugs magically.

  1. These two topics were chosen since the mathematical expositions and the physical expositions of these topics are surprisingly disjoint []

Time for me to be garrulous…

Hiya, there! So since this is my first post, I figure I ought to give you, my beloved reader, some insight into the type of things I will be writing about. Firstly, I’m in my Senior year at Cornell and I’m pursuing an honors thesis on a topic in Mathematical Physics, so things related to that will certainly be an important part of this blog. But all such stuff? No! I also write for a music blog, Catchy Song of the Day! and I definitely enjoy (whatever little I can) of fashion. Plus, I love random tidbits of knowledge so don’t be surprised if you often see a Bloomberg 1 citation or an oddball Wikipedia article.

I hope to post roughly one or two math or physics related articles per week that explain a topic relatively thoroughly while providing both proof and intuition. A rough sketch of what such posts will look like maybe be found in the following:

  • Topic Name
  • Rough Requirements for Understanding
  • Why is it interesting?
  • What problem inspired the topic?
  • Is there any history of the topic?
  • Definitions, Propositions (with proof, if available) related to the topic

All these articles will be written in \LaTeX and TiKz will be used to the extent possible. As of right now, my tentative topic schedule is as follows:

Week 1 (\approx Aug. 8 – Aug. 15)

  • The Mayer-Vitoris Sequence
  • Singular (and hopefully Cech) Cohomology
  • Poincaré Duality and de Rham’s Theorem (Part I)
  • What is a Conformal Field Theory?

Week 2 (\approx Aug. 16 – Aug. 23)

  • de Rham’s Theorem (Part II)
  • Fock Space, Formal Definitions in Constructive Field Theory
  • Bianchi Identities and Maxwell’s Equations
  • Hodge Theory (Part I)

Week 3 (\approx Aug. 24  -  Aug. 31 )

  • Hodge Theory (Part II)
  • Probabilistic Route to QFT (via Bruce Driver’s Notes)
  • Complex Manifolds
  • Fiber Bundles (Part I)

Week 4 (\approx Sept. 1 – Sept. 8)

  • Fiber Bundles, Connections on Fiber Bundles (Part II)
  • The Ising Model
  • Monodromy, Characteristic Classes
  • Holonomy Groups and Yang-Mills Gauge Theory

Week 5-6 (\approx Sept. 9 – Sept. 23)

  • Calabi-Yau Manifolds
  • Anti de Sitter Space

Week 7-8 (\approx Sept. 24 – Oct. 9)

  • Sasaki-Einstein Manifolds

I’ll try to update the schedule at the first post of each week!

Hopefully I’ll be able to keep up with this plan. Please note that I will be learning all of this material as I am writing the blog, so there may be quite a few errors. If you want to comment or correct my errors just e-mail me at  \text{tc}328 \;\,\text{at} \;\,\text{}



  1. Hopefully I’ll explain why I love Bloomberg and The Economist (and generally tend to ignore most other news sources) one day in an impassioned post!