Chapter 18 \(\displaystyle S\) - Matrix

Quote

Interactions involves the creation or destruction of particles.

The \(\displaystyle S\) -Matrix

Consider the scattering process of two particles, starting with \(\displaystyle \ket{p_{1}p_{2}}^{\text in}_{\text{ realworld}}(t\to-\infty)\), ending up with \(\displaystyle \ket{q_{1}q_{2}}^{\text{ out}}_{\text {realworld}}(t\to + \infty)\). scattering amplitude \(\displaystyle \mathcal{A}\) is given by

\[ \mathcal{A}=^{\text {out}}_{\text{realworld}}\langle q_{1}q_{2}|p_{1}p_{2}\rangle ^{\text{in}}_{\text{realworld}} \]

We must recreate this amplitude using the simple-world states .We define

\[ \mathcal{A}=^{\text {out}}_{\text{realworld}}\langle q_{1}q_{2}|p_{1}p_{2}\rangle ^{\text{in}}_{\text{realworld}}=_{\text{simpleworld}}\langle q_{1}q_{2} |\hat{S}| p_{1}p_{2}\rangle_{\text{simpleworld}} \]

\(S\) -matrix contains the amplitudes for starting with a particular ‘in’ state and ending up with a particular ‘out’ state.

The interaction representation

\[ \hat{H}=\hat{H}_{0}+\hat{H}' \]

\(\displaystyle \hat{H}_{0}\) free part , \(\displaystyle \hat{H}'\) interaction part

Note

Operators in the interaction picture evolve in time via the free part \(\displaystyle \hat{H}_{0}\)

\[\hat{O}_{\text{I}}(t)=\mathrm{e}^{ \mathrm{i}\hat{H}_{0}t }\hat{O}\mathrm{e}^{ -\mathrm{i}\hat{H}_{0}t }\quad \mathrm{i}\frac{\mathrm{d}\hat{O}_{\text{I}}(t)}{\mathrm{d}t}=[\hat{O}_{\text{I}}(t),\hat{H}_{0}]\]

Compare a matrix element from the Schrödinger picture to one in the interaction picture

\[ \langle \phi(t)|\hat{O} |\psi(t) \rangle=\langle \phi _{\text{I}}(t)|\mathrm{e}^{ \mathrm{i} \hat{H}_{0}t}\hat{O}\mathrm{e}^{ -\mathrm{i}\hat{H}_{0}t } | \psi _{\text{I}}(t)\rangle\]

for the matrix elements to be the same as in the Schrödinger picture

\[ \ket{\psi_{\text{I}}(t)}=\mathrm{e}^{ \mathrm{i} \hat{H}_{0}t}\ket{\psi(t)} \]

Note

\[\mathrm{i}\frac{\mathrm{d}}{\mathrm{d}t} \ket{\psi_{\text{I}}(t)}=\hat{H}_{\text{I}}(t) \ket{\psi_{\text{I}}(t)}\]

where \(\displaystyle \hat{H}_{\mathrm{I}}(t)=\mathrm{e}^{ \mathrm{i}\hat{H}_{0}t }\hat{H}'\mathrm{e}^{ -\mathrm{i}\hat{H}_{0}t }\)

The interaction picture applied to scattering

Identify the simple-world states as those of the interaction picture at the start and end:

\[ \ket{\phi} _{\mathrm{simpleworld}}=\ket{\phi _{\mathrm{I}}(\pm \infty)} \]

All of quantum mechanical pictures are defined so that they coincide at \(\displaystyle t\) = 0

\[ _{\text{simpleworld}}\langle \phi|\hat{S} | \psi\rangle _{\text{simpleworld}}=^{\text{out}}_{\text{realworld}}\langle \phi | \psi\rangle^{\text{in}} _{\text{realworld}}=\langle \phi_{\text{I}}(0) |\psi_{\text{I}}(0) \rangle \]

Using the time-evolution operator in the interaction picture

\[\begin{align} _{\text{simpleworld}}\langle \phi|\hat{S} | \psi\rangle _{\text{simpleworld}}= & \langle \phi_{\text{I}}(\infty)|\hat{U}_{\text{I}}(\infty,0) \hat{U}_{\text{I}}(0,-\infty)|\psi_{\text{I}}(-\infty) \rangle \\ = &\langle \phi_{\text{I}}(\infty)|\hat{U}_{\text{I}}(\infty,-\infty)|\psi_{\text{I}}(-\infty) \rangle \\ = & {\text{simpleworld}}\langle \phi|\hat{U}_{\text{I}}(\infty,-\infty) | \psi\rangle _{\text{simpleworld}} \end{align} \]

Note

\(\displaystyle \hat{S}\) -operator is the time-evolution operator for the interaction-picture \(\displaystyle \hat{U}_{\text{I}}(t,-t)\) as \(\displaystyle t\to \infty\).

Perturbation expansion of the \(\displaystyle S\)-matrix

Note

\[\mathrm{i}\frac{\mathrm{d}}{\mathrm{d}t^{2}}\hat{U}_{\text{I}}(t_{2},t_{1})=\hat{H}_{\text{I}}(t_{2})\hat{U}_{\text{I}}(t_{2},t_{1})\]

To circumvent the problem that \(\displaystyle [\hat{H}_{\text{I}}(t_{2},),\hat{H}_\text{I}(t_{1})]\ne 0\). The solution turns out to be the time-ordered product \(\displaystyle T[\hat{A}\hat{B}\dots]\) defined as the string arranged so that the later operators are on the left.

Dyson's expansion

\[\hat{U}_{\text{I}}(t_{2},t_{1})=T\left[ \mathrm{e}^{ -\mathrm{i}\int_{t_{1}}^{t_{2}}\mathrm{d}t \hat{H}_{\text{I}}(t)} \right]\]

\(\displaystyle \hat{S}\) -operator

\[\displaystyle \hat{S}=T\left[ \mathrm{e}^{ -\mathrm{i}\int_{-\infty}^{\infty} \, \mathrm{d}^{4}x \hat{\mathcal{H}}_{I}(x)} \right]\]

Usually the integral in the exponent cannot be done exactly so we have to expand out the exponential in Dyson’s expansion thus:

\[ \hat{S}=T\left[ 1-\mathrm{i}\int\mathrm{d}^{4}z \hat{\mathcal{H}_{\text{I}}}(z)+\frac{(-\mathrm{i})^{2}}{2!}\int\mathrm{d}^{4}yd^{4}w \hat{\mathcal{H}}_{\text{I}}(y)\hat{\mathcal{H}}_{\text{I}}(w)+\dots \right] \]

Wick theorem

Wick theorem

\[T[\hat{A}\hat{B}\hat{C}\dots \hat{Z}]=N\left[\hat{A}\hat{B}\hat{C}\dots \hat{Z}+\begin{align} \text{all possible} &\,\,\text{contrations of } \\ \hat{A}\hat{B}&\hat{C}\dots \hat{Z} \end{align}\right]\]

Chapter 19 Expanding the \(S\) -matrix: Feynman diagrams

The example of \(\displaystyle \phi^4\) theory

Note

\[\mathcal{L}=\frac{1}{2}\left[\partial_{\mu} \phi (x)\right]^{2}-\frac{m^{2}}{2} \phi (x)^{2}-\frac{\lambda}{4 !} \phi (x)^{4}\]

free part \(\mathcal{L}_{0}=\frac{1}{2}\left (\partial_{\mu} \phi\right)^{2}-\frac{m^{2}}{2} \phi^{2}\) , \(\hat{\mathcal{H}}_{0}=\frac{1}{2}\left[\left (\frac{\partial \hat{\phi}}{\partial t}\right)^{2}+(\boldsymbol{\nabla} \hat{\phi})^{2}+m^{2} \hat{\phi}^{2}\right]\)，

interacting part \(\mathcal{L}_{\mathrm{I}}=-\frac{\lambda}{4 !} \phi (x)^{4}\) , \(\hat{\mathcal{H}}_{\mathrm{I}}=\frac{\lambda}{4 !} \hat{\phi}(x)^{4}\) .

Expand \(\displaystyle S\) -matrix

Step I write it as a vacuum expectation value (VEV), \(\displaystyle \ket{p}\) in state , \(\displaystyle \ket{q}\) out state.

\[\begin{array}{l}\mathcal{A}={ }^{\text {out }}\langle q\mid p\rangle^{\text {in }}=\langle q|\hat{S}| p\rangle =(2 \pi)^{3}\left (2 E_{\boldsymbol{q}}\right)^{\frac{1}{2}}\left (2 E_{\boldsymbol{p}}\right)^{\frac{1}{2}}\left\langle 0\left|\hat{a}_{\boldsymbol{q}} \hat{S} \hat{a}_{\boldsymbol{p}}^{\dagger}\right| 0\right\rangle, \end{array}\]

where we recall that the relativistic normalization of our states means that \(|p\rangle=(2 \pi)^{\frac{3}{2}}\left (2 E_{\boldsymbol{p}}\right)^{\frac{1}{2}} \hat{a}_{\boldsymbol{p}}^{\dagger}|0\rangle\) Step II Expand the \(\hat{S}\) -operator using Dyson's expansion

\[\begin{aligned}\hat{S} & =T\left[\exp \left (-\mathrm{i} \int \mathrm{d}^{4} z \hat{\mathcal{H}}_{\mathrm{I}}(z)\right)\right] \\& =T\left[1-\mathrm{i} \int \mathrm{d}^{4} z \hat{\mathcal{H}}_{\mathrm{I}}(z)+\frac{(-\mathrm{i})^{2}}{2} \int \mathrm{d}^{4} y \mathrm{~d}^{4} w \hat{\mathcal{H}}_{\mathrm{I}}(y) \hat{\mathcal{H}}_{\mathrm{I}}(w)+\ldots\right] \end{aligned}\]

Step III Plug the resulting expression for the \(\displaystyle \hat{S}\) -operator into the expression for the S-matrix element that we’re trying to calculate

\[\mathcal{A}=\langle q|\hat{S} | p\rangle \]

Step IV Use Wick’s theorem to grind down the terms. Step V Make sense of a term by drawing a Feynman diagram.

Note

\[ \begin{aligned} \langle 0|\hat{\phi}(z) \hat{a}_{\boldsymbol{p}}^{\dagger}| 0\rangle & =\int \frac{\mathrm{d}^{3} q}{(2 \pi)^{\frac{3}{2}}} \frac{1}{\left(2 E_{\boldsymbol{q}}\right)^{\frac{1}{2}}}\left\langle 0\left|\left(\hat{a}_{\boldsymbol{q}} \mathrm{e}^{-\mathrm{i} q \cdot z}+\hat{a}_{\boldsymbol{q}}^{\dagger} \mathrm{e}^{\mathrm{i} q \cdot z}\right) \hat{a}_{\boldsymbol{p}}^{\dagger}\right| 0\right\rangle \\ & =\int \frac{\mathrm{d}^{3} q}{(2 \pi)^{\frac{3}{2}}} \frac{1}{\left(2 E_{\boldsymbol{q}}\right)^{\frac{1}{2}}}\left\langle 0\left|\left(\hat{a}_{\boldsymbol{q}} \mathrm{e}^{-\mathrm{i} q \cdot z}+\hat{a}_{\boldsymbol{q}}^{\dagger} \mathrm{e}^{\mathrm{i} q \cdot z}\right)\right| \boldsymbol{p}\right\rangle \\ & =\int \frac{\mathrm{d}^{3} q}{(2 \pi)^{\frac{3}{2}}} \frac{1}{\left(2 E_{\boldsymbol{q}}\right)^{\frac{1}{2}}} \mathrm{e}^{-\mathrm{i} q \cdot z} \delta^{(3)}(\boldsymbol{q}-\boldsymbol{p}) \\ & =\frac{1}{(2 \pi)^{\frac{3}{2}}} \frac{1}{\left(2 E_{\boldsymbol{p}}\right)^{\frac{1}{2}}} \mathrm{e}^{-\mathrm{i} p \cdot z} \end{aligned} \]

\[ \langle 0|T \hat{\phi}(y) \hat{\phi}(z)| 0\rangle=\Delta(y-z) \]

Anatomy of a diagram

Example

External lines have one end which appears not to be connected to anything.
A vacuum diagram has no external lines.

A particular connected diagram might contain: Vertices where lines join together. These represent interactions. External incoming lines represent on-mass-shell particles entering the process.
External outgoing lines represent on-mass-shell particles leaving the process. Internal lines e represent virtual particles which are off-mass-shell and therefore exist internally within the diagram.

Feynman rules for \(\displaystyle \phi ^{4}\) theory in position space

To calculate an amplitude in the S-matrix expansion, translate a Feynman diagram into equations as follows: Each vertex contributes a factor \(\displaystyle -\mathrm{i}\lambda\).
Each line gives a propagation factor \(\displaystyle \Delta(x-y)\), where \(x\) and \(y\) are the start and end points of the line. External lines contribute incoming (\(\displaystyle -\mathrm{i}p\cdot x\)) or outgoing (\(\displaystyle +\mathrm{i}p\cdot x\)) waves \(\displaystyle \mathrm{e}^{ \pm \mathrm{i}p \cdot x }\).
Integrate the positions of the vertices over all spacetime.
In order to get the right coefficient in front of the term divide by the symmetry factor \(D\).

Symmetry factor

\[D=g 2^{\beta} \prod_{n}(n !)^{\alpha_{n}}\]

\(g\) : the number of permutations of vertices that leave the diagram unchanged with fixed external lines \(\beta\) : the number of bubbles
\(\alpha_{n}\) : the number of pairs of vertices connected by n identical lines

Calculations in \(\displaystyle p\) space

Feynman rules for \(\phi^{4}\) theory in momentum space

Each vertex contributes a factor \(-\mathrm{i} \lambda\) Label each internal line with a momentum \(q\) flowing along it and describe it by a propagator \(\displaystyle\frac{\mathrm{i}}{q^{2}-m^{2}+\mathrm{i} \epsilon}\) . Force the sum of each momentum coming into a vertex to be equal to the momentum leaving it. Integrate over unconstrained internal momenta with a measure \(\displaystyle\frac{\mathrm{d}^{4} q}{(2 \pi)^{4}}\) . External lines contribute a factor 1. Divide by the symmetry factor. Include an overall energy-momentum conserving delta function for each diagram.

Chapter 20 scattering theory