Pqnelson

I was a math/physics double major at UC Davis (graduated 2011) and I'm fascinated with Quantum gravity. I've taken several graduate courses on the subject, taught by Steve Carlip, and I'm auditing graduate courses in math (specifically dealing with Lie algebras, Lie groups, Riemannian geometry, etc.). That's my life summed in two sentences.

Here are a few notes that some may find interesting/awesome/boring/random.

Problems for Sean

Here is a grocery list of interesting math problems for my friend Sean.

Calculus

1. Evaluate the following integral $\int _{0}^{1}{\frac {dt}{(ta+(1-t)b)^{2}}}$
2. Relate the approximation of $f(x)=\int _{1}^{x}{\frac {dt}{t}}$ as a trapezoid with vertices (1,0), $(x,0)$, (1,1) and $(x,1/x)$...to its Taylor expansion. When is this a good approximation? How good is it? Can we have some description of its error?
3. Can we deduce the rules for differentiation of trigonometric functions from right triangles alone?

Complex Analysis

1. Recall we work in the complex plane and we use coordinates $z=x+iy$ and ${\bar {z}}=x-iy$. What is $\partial /\partial z$ and $\partial /\partial {\bar {z}}$ in terms of partial derivatives with respect to $x$ and $y$?
2. Using the previous exercise, generalize Green's Theorem to the complex plane.
3. Using the previous two exercises, when will the integral in Green's Theorem applied to the complex plane vanish?

Linear Algebra

1. Prove that real polynomials with a single variable is a vector space over $\mathbb {R}$. We denote this vector space as $\mathbb {R} [x]$, where $x$ is the unknown.
   1. Prove the derivative (with respect to $x$) is a linear operator on $\mathbb {R} [x]$.
   2. Does $\mathbb {R} [x]$ have a basis?
2. Prove that $C^{\infty }(\mathbb {R} )$ is a vector space over $\mathbb {R}$.
   1. Suppose we considered the space $C^{0}(0,2\pi )$, the space of periodic continuous functions. Is it a vector space over $\mathbb {R}$? What about $C^{k}(0,2\pi )$, for some $k\in \mathbb {N}$? Is it a subspace of $C^{0}(0,2\pi )$?
3. Recall a linear operator $L\colon V\to V$ is such that $L(ax+by)=aL(x)+bL(y)$. Prove that the derivative (denoted $\partial$) is a linear operator on $C^{\infty }(\mathbb {R} )$.
4. Let $X$ be some square $n\times n$ matrix. We can define its inverse by $X^{-1}X=XX^{-1}=I$, and $X^{-1}$ is likewise a square $n\times n$ matrix. If $\partial \colon C^{\infty }(\mathbb {R} )\to C^{\infty }(\mathbb {R} )$ is the derivative linear operator, does it have an inverse?
   1. What is the trace of the derivative operator? The determinant?
   2. What is $\exp(t\partial )f(x)$, where $t$ is some constant? (Hint: use Taylor series for exponential map)
   3. On a similar note, what about integration $\int \colon C^{\infty }(\mathbb {R} )\to [-\infty ,+\infty ]$ defined by $f(x)\mapsto \int _{-\infty }^{\infty }f(x)dx$. Is this a linear mapping? What is its trace? Determinant? Its exponential?
5. We have $X$ be some square $n\times n$ matrix. Define the function $\exp(X)=I+X+{\frac {X^{2}}{2!}}+\dots$. Does it converge?
   1. What is the image of this exponential map?
   2. Let $0$ be the zero matrix, what is $\exp(0)$?
   3. Prove or find a counter-example $\det {\bigl (}\exp(X){\bigr )}=\exp {\bigl (}\mathrm {tr} (X){\bigr )}$.
   4. Consider $J={\begin{pmatrix}0&1\\1&0\end{pmatrix}}$. What is $\exp(J)$?
6. Let $X$ be some square $n\times n$ matrix. Define $\sin(X)=X-{\frac {X^{3}}{3!}}+\dots$. Does it converge?
7. From the previous exercise, could we define an inverse function $\ln(X)=\exp ^{-1}(X)$? When would it be defined (e.g., if $X$ is the zero matrix, we should expect it to be not well defined...)?
   1. Is the matrix logarithm unique? I.e., could there be two distinct matrices $X\not =Y$ such that $\exp(X)=\exp(Y)$?
8. Consider the functions $f\colon (0,2\pi )\to \mathbb {R}$ such that $\int _{0}^{2\pi }|f(x)|^{2}dx<\infty$. For the sake of randomness, lets call all such functions $L^{2}(0,2\pi )$. Do these functions form a vector space over $\mathbb {R}$?
   1. Is the derivative a linear operator on $L^{2}(0,2\pi )$?
   2. Is there a basis for $L^{2}(0,2\pi )$?
   3. We have a norm on $L^{2}(0,2\pi )$, but can we construct an inner product? Would $\langle f,g\rangle =\|f+g\|+\|f-g\|-\|f\|-\|g\|$ a "good" inner product?

Regge Theory

This is a peculiar theory which roughly concludes the total angular momentum of a Meson $J$ is such that $J\propto M^{2}$ where $M$ is the Meson's mass. This is all very rushed and sloppily put, but Regge theory has its own page. I'm writing an annotated bibliography:

• Landau and Lifshitz, Quantum Mechanics (Non-Relativistic Theory), §141 "Regge Poles" discusses Regge trajectories quite well. In fact, it's the most coherent explanation I've found.
• Weinberg's Quantum Theory of Fields (vol I) chapter 10 §8 "Dispersion Relations", pp. 468-469 gives a brief discussion of Regge trajectories.
• Gribov's The Theory of Complex Angular Momenta is a book dedicated entirely to Regge trajectories, although it presupposes knowledge of Complex analysis.

This has applications in studying quasinormal modes of black holes. For a few references on this:

• Sam R. Dolan, Adrian C. Ottewill, "On an Expansion Method for Black Hole Quasinormal Modes and Regge Poles." Eprint arXiv:gr-qc/0304030, 23 pages.
• K. Glampedakis, N. Andersson, "Quick and dirty methods for studying black-hole resonances." Eprint arXiv:gr-qc/0304030, 18 pages.
• Yves Décanini, Antoine Folacci, Bruce Jensen, "Complex angular momentum in black hole physics and quasinormal modes." Eprint arXiv:gr-qc/0212093, 6 pages.
• Yves Décanini, Antoine Folacci, "Regge poles of the Schwarzschild black hole: a WKB approach." Eprint arXiv:0906.2601 (gr-qc), 8 pages.

Functional Integration

This is just an annotated bibliography of the various Quantum field theory texts and their approach towards defining functional integration (the "path integral" Quantization Procedure).

• Steven Weinberg's The Quantum Theory of Fields, vol. I, defines the path integral from the canonical quantization procedure. That is, based on a certain choice of operator ordering, Weinberg "defines" the measure ${\mathcal {D}}\varphi$ and has an effective algorithm for performing calculations with the path integral.
• Fritz Mandl and Graham Shaw, Quantum Field Theory (Second ed.). The approach is very unique, very clever. Consider the function space, and pretend we can pick some Orthogonal functions to span the space. This is done with Fourier analysis, for example. We just define the functional integral on the orthogonal functions! Of course, this works provided there exists a countable set of orthogonal functions spanning the function space; but in general, it won't work. Very sad, indeed, I know...
• John R. Klauder's "The Feynman Path Integral: An Historical Slice" arXiv:quant-ph/0303034 provides a historical survey of various different ways to define path integrals. It appears to be quite comprehensive, although I'm guessing that there are a few approaches missing...