Commit 548241cf authored by Stefan Gehr's avatar Stefan Gehr
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Fixes and beautifications for chapters 6 and 7

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......@@ -108,22 +108,22 @@ Notes:
\paragraph{Poincaré invariance} First note that for \(f: \mathbb{R}^4 \to \mathbb{C}\)
\begin{align}
\int \frac{\dd^3{k}}{(2\pi)^3}\frac{1}{2\omega_{\vec{k}}}f(\omega_{\vec{k}},\vec{k})
&= \int \frac{\dd^4{k}}{(2\pi)^3}\delta(k^2-m^2)\theta(k^0)f(k)
&= \int \frac{\dd^4{k}}{(2\pi)^3}\delta(k^2+m^2)\theta(k^0)f(k)
\label{eq:ffiftynine}
.\end{align}
The proof for this comes in the exercises.
So \(\int \frac{\dd^3{k}}{(2\pi)^3}\frac{1}{2\omega_{\vec{k}}}\) is a uniform integral over positive mass shell \(p^2 = m^2\) in momentum space.
So \(\int \frac{\dd^3{k}}{(2\pi)^3}\frac{1}{2\omega_{\vec{k}}}\) is a uniform integral over positive mass shell \(k^0 = +\sqrt{\vec{k}^2+\vec{m}}\) in momentum space.
Hence
\begin{align}
D(x)
&= \int \frac{\dd^4{k}}{(2\pi)^3}\delta(k^2-m^2)\theta(k^0)e^{ik\cdot x}
&= \int \frac{\dd^4{k}}{(2\pi)^3}\delta(k^2+m^2)\theta(k^0)e^{ik\cdot x}
\label{eq:fsixty}
.\end{align}
For \(\Lambda\in \mathcal{L}^{\uparrow}\) we have
\begin{align}
D(\Lambda^{-1}x)
&= \int \frac{\dd^4{k}}{(2\pi)^3}\,\delta\left((\Lambda k)^2 - m^2\right)\theta((\Lambda k)^0) e^{\overbrace{ik\cdot \Lambda^{-1}x}^{=i(\Lambda k) \cdot x}} \nonumber \\
&= \int \frac{\dd^4{\Lambda k}}{(2\pi)^3}\abs{\det\Lambda}\,\delta\left((\Lambda k)^2 - m^2\right)\theta((\Lambda k)^0) e^{i(\Lambda k) \cdot x} \nonumber \\
&= \int \frac{\dd^4{k}}{(2\pi)^3}\,\delta\left((\Lambda k)^2 + m^2\right)\theta((\Lambda k)^0) e^{\overbrace{ik\cdot \Lambda^{-1}x}^{=i(\Lambda k) \cdot x}} \nonumber \\
&= \int \frac{\dd^4{\Lambda k}}{(2\pi)^3}\abs{\det\Lambda}\,\delta\left((\Lambda k)^2 + m^2\right)\theta((\Lambda k)^0) e^{i(\Lambda k) \cdot x} \nonumber \\
&= D(x)
\label{eq:fsixtyone}
.\end{align}
......
......@@ -20,9 +20,9 @@ There is a useful completeness relation
\begin{align}
\begin{split}
\sum_{s=1,2} u^s(k) \overline{u}^s(k)
&= -i\gamma k + m \mathbbm{1}_{4\times 4} \\
&= -i\gamma \cdot k + m \mathbbm{1}_{4\times 4} \\
\sum_{s=1,2} v^s(k) \overline{v}^s(k)
&= -i\gamma k - m \mathbbm{1}_{4\times 4}
&= -i\gamma \cdot k - m \mathbbm{1}_{4\times 4}
,\end{split}
\label{eq:feightyeight}
\end{align}
......
......@@ -13,7 +13,7 @@ Well defined examples in lower dimensions:
\item 2+1: some
\item 3+1: none
\end{itemize}
Nevertheless, some aspects can be understood i n 3+1.
Nevertheless, some aspects can be understood in 3+1.
There are many approaches.
In the following
\begin{itemize}
......@@ -46,8 +46,8 @@ Formally we have
with
\begin{align*}
\braket{\psi', \text{out}}{\psi, \text{in}}
&= \lim_{T\to \infty}\bra{\psi'}e^{-2iHT}\ket{\psi}
=: \bra{\psi'}S\ket{\psi}
&= \lim_{T\to \infty}\bra*{\psi'}e^{-2iHT}\ket{\psi}
=: \bra*{\psi'}S\ket{\psi}
\end{align*}
the scattering amplitude / \(S\) matrix element.
\(S\) can be decomposed
......@@ -89,7 +89,7 @@ Look at \autoref{fig:twoparticlescattering} to see a sketch of this scenario.
We can get another useful result using \(T\).
For a particle with momentum \(\vec{k}\) scattering off a target, we get in the Born approximation
\begin{align*}
\bra{\vec{p}}iT\ket{\vec{k}}
\bra*{\vec{p}}iT\ket*{\vec{k}}
&= -2\pi i \delta(E_{\vec{k}}-E_{\vec{p}})
\int e^{\vec{ix}\cdot (\vec{k}-\vec{p})} \frac{V(\vec{x})}{(2\pi)^3}\dd^3{x}
\end{align*}
......@@ -99,7 +99,7 @@ Consider \(\psi\), \(\psi'\) with sharp momenta. Also consider a scalar field.
Let
\begin{align*}
\tau(k_1,\hdots,k_n)
&= \mathcal{F}\left(\bra{\Omega}\phi(x_1)\cdot\phi(x_n)\underbrace{\ket{\Omega}}_{\text{vacuum of interacting field}}\right)
&= \mathcal{F}\big(\bra{\Omega}\phi(x_1)\cdots\phi(x_n)\underbrace{\ket{\Omega}}_{\text{vacuum of interacting field}}\big)
.\end{align*}
Then the
\paragraph{Lehmann-Szymanzik-Zimmermann (LSZ) reduction formula asserts}
......@@ -116,7 +116,7 @@ where the field strength renormalisation \(z\) is a constant, \(M\) is the mass
We express the \(T\)-functions in the form of free fields.
This is only a formality.
First we look at \(\ket{\Omega}\).
Consider an Hamiltionian \(H\) with
Consider a Hamiltionian \(H\) with
\begin{align*}
\mathbbm{1} &= \ketbra{\Omega}{\Omega} + \sum_{n=1}^{\infty}\ketbra{n}{n} \\
H\ket{\Omega} &= E_0 \ket{\Omega}, \quad H\ket{n} = E_n\ket{n}
......
......@@ -5,9 +5,9 @@ Example: Scalar field \(\phi\),
\label{eq:fonehundredtwentyeight}
.\end{align}
Note that this is not physical since \(H_I\) is not bounded below.
Consider the time-ordered 2-point function for this interacting field, and expand it in \(\lambda\)
Consider the time-ordered 2-point function for this interacting field, and expand it in \(\lambda\)
\begin{align*}
\lambda^0 &:\quad \bra{0}T(\phi(x_1)\phi(x_2)\phi(x_3)\ket{0} = 0 \\
\lambda^0 &:\quad \bra{0}T(\phi(x_1)\phi(x_2)\phi(x_3))\ket{0} = 0 \\
\lambda^1 &:\quad - \frac{i}{3!}\int\dd^4{y}\,\bra{0}T(\phi(x_1)\phi(x_2)\phi(x_3)\phi(y)\phi(y)\phi(y))\ket{0}
.\end{align*}
We apply Wick's theorem to the \(\lambda^1\)-term.
......@@ -30,6 +30,7 @@ Possible partitions (``contractions'') are
\draw (2.5,-0.15) -- (2.5,-0.4);
\draw (1,-0.4) -- (2.5,-0.4);
} and five more that give the same result
\label{item:contorigone}
\item \tikz[baseline=-0.2em] {
\node (x1) at (0,0) {\(x_1\)};
\node (x2) at (0.5,0) {\(x_2\)};
......@@ -81,6 +82,7 @@ Possible partitions (``contractions'') are
\draw (2.5,-0.15) -- (2.5,-0.25);
\draw (2,-0.25) -- (2.5,-0.25);
} and two more that give the same result
\label{item:contorigfour}
\end{enumerate}
Altogether
\begin{align*}
......@@ -106,7 +108,7 @@ Altogether
.\end{split}
\label{eq:fonehundredtwentynine}
\end{align}
Examples: Contractions \ref{item:contone}-\ref{item:contfour} \\
Examples: Contractions \ref{item:contorigone}-\ref{item:contorigfour} \\
\begin{enumerate*}[itemjoin=\quad]
\item
\tikz[baseline=-1em]{
......@@ -154,7 +156,7 @@ Examples: Contractions \ref{item:contone}-\ref{item:contfour} \\
What about the numerical pre-factor of the diagram?
There is a simple rule via \underline{graph symmetries}:
\begin{enumerate}[label=(\greek*)]
\item Both ends of propagator connect to the same cortex
\item Both ends of propagator connect to the same vertex
\label{item:ruleone}
\item Multiple propagators going between same two vertices
\label{item:ruletwo}
......@@ -318,7 +320,7 @@ Now we look at \(\lambda^3\).
+ \text{cyclic}
\right]
.\end{align*}
A New phenomenon is that we have lower order diagrams multiplied with ``\underline{vacuum bubbles}''
A new phenomenon is that we have lower order diagrams multiplied with ``\underline{vacuum bubbles}''
\begin{align*}
B_1 =
\tikz[baseline=-0.2em]{
......@@ -361,7 +363,7 @@ This is really nice, as it implies also
and hence
\begin{align}
\bra{\Omega} T(\phi(x_1)\cdots \phi(x_n))\ket{\Omega}
= \sum_k E(C_k)
= \sum_k E(c_k)
\label{eq:fonehundredthirtytwo}
.\end{align}
This is independent of the nature of the field.
......@@ -379,14 +381,11 @@ Remember
Note that in \(k\)-space the expression is not symmetric in \(x, y\).
We give the line a direction, and pick the convention
\begin{align}
\tikz[baseline=-0.2em]{
\draw [fill] (0,0) circle [radius=0.07];
\draw [fill] (1,0) circle [radius=0.07];
\draw [-stealth] (0,0) -- (0.5,0);
\draw (0.5,0) -- (1,0);
}
\feynmandiagram[baseline=-0.2em]{
a [dot] -- [fermion] b [dot],
};
\leftrightarrow
\int \frac{\dd^4{k}}{(2\pi)^4} \frac{i e^{ik(\text{end}-\text{beginning})}}{\hdots}
\int \frac{\dd^4{k}}{(2\pi)^4} \frac{i e^{ik(\text{end}-\text{beginning})}}{-k^2-m^2 + i \epsilon}
\label{eq:fonehundredthirtythree}
.\end{align}
Then at a vertex we have
......@@ -401,7 +400,7 @@ Then at a vertex we have
}
&\leftrightarrow
-i\lambda \int\dd^4{y}e^{iy\left(\sum_{\text{in}}k-\sum_{\text{out}}k\right)}\hdots \nonumber\\
&= -i\lambda(2\pi)^4\delta^{(4)}\left(\sum_{\text{in}}k-\sum_{\text{out}}\right)
&= -i\lambda(2\pi)^4\delta^{(4)}\left(\sum_{\text{in}}k-\sum_{\text{out}}k\right)
\label{eq:fonehundredthirtyfour}
.\end{align}
External vertices:
......
......@@ -108,16 +108,12 @@ with
};
+
\tikz[baseline=-0.2em,scale=0.4]{
\draw [fill] (-2.5,0) circle [radius=0.07];
\draw [fill] (-1.5,0) circle [radius=0.07];
\draw [fill] (70.53:1.5) circle [radius=0.07];
\draw [fill] (109.47:1.5) circle [radius=0.07];
\draw [fill] (1.5,0) circle [radius=0.07];
\draw [fill] (2.5,0) circle [radius=0.07];
\draw (-2.5,0) -- (-1.5,0);
\draw (70.53:1.5) arc [start angle=70.53, end angle=-250.53, x radius=1.5, y radius=1.5];
\draw (0,1.5) circle [radius=0.5];
\draw (1.5,0) -- (2.5,0);
}
+
\tikz[baseline=-0.2em,scale=0.4]{
......@@ -235,7 +231,7 @@ Two cases:
\(\implies\) Theory is not predictive, ``not renormalisable''.
\end{enumerate}
When is the theory renormalisable? It is complicated!
't Hooft + Veltamn showed that QED is renormalisable, and got the nobel prize for that.
't Hooft + Veltman showed that QED is renormalisable, and got the nobel prize for that.
\paragraph{Rough criterion}
Consider the units of coupling constants. \(\hbar=c=1\).
Everything has units of mass to some power.
......@@ -255,7 +251,7 @@ etc. If coupling constant \(g\) has \([g]=k\), then plausibly
.\end{equation}
For \(k<0\) \(g\omega_p^{-k}\to \infty\) for \(p\to \infty\) which is not good for perturbation theory.
This is an indication that it would be non-renormalisable.
Example: \(\phi^4\)
Example: \(\phi^4\)
\begin{align*}
[S]\overset{!}{=} 0 \implies [\mathcal{L}] = d \implies [\phi] = \frac{1}{2}(d-2) \implies[\lambda] -d +2
.\end{align*}
......@@ -279,8 +275,8 @@ with
\item \(B_E = \) \# external boson lines
\item \(g_v = \) coupling constant of vertex \(v\).
\end{itemize}
Not that an actualy divergence can be be better.
For example in ED
Note that an actual divergence can be be better.
For example in QED
\begin{align*}
D = 4 - \frac{3}{2}F_E - B_E
.\end{align*}
......
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