Archive for October 2021

Married, Divorced, and Remmaried

Posted on Friday, October 29, 2021

By Greg Moore. Read it here. An exciting and caught-up read.

Posted in Uncategorized | Leave a comment

Print it.

Complex Analysis: Analytic Functions, Cauchy-Reimann

Posted on Tuesday, October 26, 2021

Some basics of complex analysis. We will follow L. Ahlfors's Complex Analysis book. An analytic function is a complex-valued function with derivatives everywhere where function $f(x)$ is defined with an appropriate power series. Holomorphic functions are the same, with a different meaning. For an analytic function $f(z) = u+iv$, we write

$\frac{\partial f}{\partial x} = \frac{\partial u}{\partial x}+i\frac{\partial v}{\partial x}$

with the limit

$f^{\prime}(z)=\lim _{k \rightarrow 0} \frac{f(z+i k)-f(z)}{i k}=-i \frac{\partial f}{\partial y}=-i \frac{\partial u}{\partial y}+\frac{\partial v}{\partial y}$

from which we can extract

$\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}, \quad \frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}$.

These differential equations have a name; Cauchy-Reimann differential equations. (They have well-meaning in complex analysis, greatly in Reimann spheres, from my readings.) If one compute $|f(z)
|^2$, which is a Jacobian of $u$ with respect to $x$ and $v$ respect to $y$, one can extract from the Jacobian that $\Delta u$ and $\Delta v$ are harmonic functions as

$\Delta u=\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0$

$\Delta v=\frac{\partial^{2} v}{\partial x^{2}}+\frac{\partial^{2} v}{\partial y^{2}}=0$

and $u$ and $v$ satisfy the Cauchy-Reimann differential condition. $v$ is said to be a harmonic conjugate of $u$, $u$ is a harmonic conjugate of $-v$.

If $u(x, y)$ and $v(x, y)$ have continuous first-order partial derivatives which satisfy the Cauchy-Riemann differential equations, then $f(z)=u(z)+i v(z)$ is analytic with continuous derivative $f^{\prime}(z)$, and conversely.

Edit: If one wants to go at an advanced level (which sure one needs in theoretical computations), try Stephen Fisher's Complex Variables, as suggested by one friend.

Posted in Analysis | 3 Comments

Print it.

Renormalization Without the Infinities

Posted on Thursday, October 21, 2021

It is a common misconception that renormalization is needed only when infinities are coming up. But R. Shankar beautifully tackles this in his textbook on Quantum Field Theory and Condensed Matter in chapter 11th.

The central idea of renormalization is to do the computation by integrating out unnecessary mathematics, so we get good physics out of it. But a lot of good renormalization problems do not have anything to do with infinties. I will reproduce here one such example from the same book. Let us take a system $(a,b,c, \cdots,n; x,y)$, where $a,b,c,\cdots,n$ are parameters and $x,y$ are two variables. Calculating a partition function of such a system is easy. But what if we want to compute the partition function ignoring variable $y$. Such thing is achieved by writing a modified system $(a',b',c',\cdots,n')$

$Z(a',b',\cdots,n') = \int dx \left[ \int dy e^{-a(x^2+y^2)}e^{-b(x+4)^4} \right]$

$Z(a',b',\cdots,n') = \int dx e^{-S(a',b',c',\cdots,n')}$

where $S'$ is the action of the modified system. So

$e^{-S(a',b',c',\cdots,n');x} = \int dy e^{-a(x^2+y^2)}e^{-b(x+4)^4} \equiv \int dye^{-S(a,b,\cdot,n;x,y)}$

here we have created an effective action $S'$ of an effective theory. We did not eliminate $y$ by setting it zero, but we created an integral where we integrated out $y$ but got the same answer as the original theory. The second integral with integrated of exponential with Boltzman weight have interactions parts involving $x,y$. In the last we have modified the system in such a way that we do not need to figure out the coupling of $x$ and $y$, but we have set the fate of $x$ to itself.