--- jupytext: formats: ipynb,md:myst text_representation: extension: .md format_name: myst format_version: 0.13 jupytext_version: 1.14.0 kernelspec: display_name: Python 3 (phys-581) language: python name: phys-581 --- ```{code-cell} :tags: [hide-cell] import mmf_setup;mmf_setup.nbinit() import logging;logging.getLogger('matplotlib').setLevel(logging.CRITICAL) %matplotlib inline import numpy as np, matplotlib.pyplot as plt ``` (sec:MockQFT)= # Mock QFT :::{margin} Zee's "baby" problem is the 1D case with $A = m^2/2$. To help simplify notations a little later, we express the first part of the exponent as the "Lagrangian" \begin{gather*} L(\vect{q}, \lambda) = \frac{\vect{q}^T\mat{A}\vect{q}}{2} + \frac{\lambda}{4!}q^4. \end{gather*} Then we can define the following probability distribution over $\vect{q}$: \begin{gather*} \rho(\vect{q}, \lambda) = \frac{e^{-L(\vect{q},\lambda)}} {Z(\vect{0}, \lambda)}. \end{gather*} We will use this notion later. ::: We start with Zee's "child" problem {cite:p}`Zee:2010` which asks us to consider the following integral \begin{gather*} Z(\vect{J}, \lambda) = \int\d^{N}\vect{q}\; \exp\Biggl( \overbrace{ -\frac{\vect{q}^T\mat{A}\vect{q}}{2} - \frac{\lambda}{4!}q^4}^{-L(\vect{q}, \lambda)} + \vect{J}\vect{q} \Biggr), \qquad q^4 = \sum_{i}q_i^4. \end{gather*} Performing the gaussian integrals, we get \begin{gather*} Z(\vect{J}, \lambda) = \underbrace{ \overbrace{\sqrt{\frac{(2\pi)^N}{\det\mat{A}}}} ^{\frac{1}{\sqrt{\det(\mat{A}/2\pi)}}}}_{Z(0,0)} \exp\left(-\frac{\lambda}{4!}\sum_{i}\diff[4]{}{J_i}\right) \exp\left(\frac{1}{2}\vect{J}^T\mat{A}^{-1}\vect{J}\right). \end{gather*} We can express this as a power series in $\vect{J}$, \begin{gather*} \frac{Z(J, \lambda)}{Z(0,0)} = \sum_{s=0}^{\infty} \frac{1}{s!}G^{(s)}_{i_1i_2\dots i_s}J_{i_1}J_{i_2}\cdots J_{i_s} \end{gather*} where the indices $i_n$ are implicitly summed over. From this it might be clear that the $G^{(s)}_{\vect{i}}(\lambda)$ behave like [moments][] over the distribution $\rho(\vect{q}, \lambda)$ defined in the margin note above: \begin{align*} G^{(s)}_{i_1i_2\dots i_s}(\lambda) &= \left. \frac{1}{Z(\vect{0},0)} \frac{\partial^s Z(\vect{J}, \lambda)} {\partial J_{i_1}\partial J_{i_2} \dots \partial J_{i_s}}\right|_{\vect{J} = 0} \\ &= \frac{1}{Z(0,0)} \int \d^{N}\; q_{i_1}q_{i_2}\dots q_{i_s} \exp\left( -\frac{\vect{q}^T\mat{A}\vect{q}}{2} -\frac{\lambda}{4!}q^4 \right)\\ &= \frac{Z(\vect{0},\lambda)}{Z(\vect{0},0)} \braket{q_{i_1}q_{i_2}\cdots q_{i_s}}_{e^{-L(\vect{q}, \lambda)}}. \end{align*} The **source** term $\vect{J}$ is simply a tool for computing these moments. :::{admonition} Analogy with Field Theory Think of the indices $i_n$ as specifying a location: i.e. a site on a 1D lattice $x_n = ai_n$ where $a$ is the lattice spacing. The quantity $G^{(s)}_{i_1 i_2 \dots i_s}(\lambda)$ are analogous to the **$s$-point Green's functions**, which are primary quantities of interest in quantum field theory. For example, the 2-point function $G^{(2)}_{ij}$ plays the role of the **propagator**, describing how a particle "propagates" from site $i$ to site $j$. The goal of perturbative field theory is to calculate quantities like the **full propagator** $G^{(2)}_{ij}(\lambda)$ using a perturbation theory based on Feynman diagrams whose ingredients are quantities like the **bare propagator** $G^{(2)}_{ij}(0)$. Note that the bare propagator has a simple form: \begin{gather*} G^{(2)}_{ij}(\lambda = 0) = [\mat{A}^{-1}]_{ij} \end{gather*} ::: Here we expand Zee's "baby problem" {cite:p}`Zee:2010` to include ideas about renormalization. To this end, imagine that we can perform "experiments" which "measure" the value of $N$-point "correlation functions" at some $\Lambda$: \begin{gather*} \newcommand{\Z}{\mathcal{Z}} C_n(\Lambda) = \frac{1}{\Z(0, \Lambda)}\left.\diff[n]{}{J}\Z(J, \Lambda)\right|_{J=0}, \qquad \Z(J, \Lambda) = \Z(0, \Lambda)\sum_{n=0}^{\infty} C_n(\Lambda) \frac{J^n}{n!}. \end{gather*} To be consistent with Zee's problem, we assume that we have a symmetry $J \rightarrow -J$ so that only even terms are non-zero. To be definite, let \begin{gather*} \Z(J, \Lambda) = \int_{-\Lambda}^{\Lambda}\d{q}\; e^{-L(q) + Jq}, \qquad L(q) = \frac{m^2}{2}q^2 + \frac{\lambda}{4!}q^4, \end{gather*} where the parameters $m$ and $\lambda$ are "fundamental parameters of nature". We will compute the results of these "measurements" numerically by simply doing the integral once we have chosen the parameters: :::{margin} These measurements are just the various "moments" of the probability distribution $\rho(q) \propto e^{-L(q)}$ normalized on the interval $[-\Lambda, \Lambda]$. ::: \begin{gather*} C_n(\Lambda) = \braket{q^n} = \frac{\int_{-\Lambda}^{\Lambda}\d{q}\; q^n e^{-L(q)}} {\int_{-\Lambda}^{\Lambda}\d{q}\; e^{-L(q)}}. \end{gather*} We would like to model this using the following theory -- our "mock QFT" \begin{gather*} Z(J, \Lambda) = \int_{-\infty}^{\infty}\d{q}\; e^{-L_{\Lambda}(q)+ Jq}, \\ L_{\Lambda}(q) = c_0(\Lambda) + c_2(\Lambda)q^2 + c_4(\Lambda)q^4 + c_6(\Lambda) q^6 + \cdots. \end{gather*} In the limit $\Lambda \rightarrow \infty$, our theory should "agree with nature", with $c_0(\infty) = 0$, $c_2(\infty) = m^2/2$, and $c_4(\infty) = \lambda/4!$, but at finite scales $\Lambda$, we may need additional coefficients that depend on the scale $c_n(\Lambda)$. In our "theory", we have moved the scale dependence $\Lambda$ into the coefficients $c_n(\Lambda)$ so that we can complete the gaussian integral as with Zee's baby problem. The technique for calculating the various "correlation functions" $C_n(\Lambda)$ is the same in terms of "Feynman" diagrams, but now possibly with more interactions. ```{code-cell} from functools import partial from scipy.integrate import quad m = 1.2 lam = 0.1 def integrand(q, n): return q**n * np.exp(-m**2/2*q**2-lam/4/3/2*q**4) def get_C(n, Lam): kw = dict(epsabs=1e-12, epsrel=1e-12) Zn = quad(partial(integrand, n=n), -Lam, Lam, **kw)[0] Z0 = quad(partial(integrand, n=0), -Lam, Lam, **kw)[0] return Zn/Z0 Lams = 10**np.linspace(-1, 2) ns = [0, 2, 4, 6] Cs = np.array([[get_C(n, Lam) for Lam in Lams] for n in ns]) fig, ax = plt.subplots() ax.semilogx(Lams, Cs.T) ax.set(xlabel=r"$\Lambda$", ylabel="$C_n$"); ``` [moments]: