This post contains a summary of my lecture notes for the second half of last Wednesday’s QFT-I lecture.
[Click here for a PDF with the full notes for this portion of the lecture.]

DISCLAIMER: Very rough notes from class, with some additional side notes.

These are notes for the UofT course PHY2403H, Quantum Field Theory, taught by Prof. Erich Poppitz, fall 2018.

Summary:

  • We found that the Dirac Hamiltonian is
    \begin{equation*}
    H
    =
    \int d^3 x
    \Psi^\dagger
    \lr{
    – i \gamma^0 \gamma^j \partial_j \Psi + m \gamma^0
    }
    \Psi.
    \end{equation*}
  • We found that our plane wave solutions
    \(\Psi_u = u(p) e^{-i p \cdot x}\), and \( \Psi_v = v(p) e^{i p \cdot x} \), were eigenvectors of the operator portion of the Hamiltonian
    \begin{equation*}
    \begin{aligned}
    -\gamma^0 \lr{ i \gamma^j \partial_j – m } \Psi_u &= p_0 \Psi_u \\
    -\gamma^0 \lr{ i \gamma^j \partial_j – m } \Psi_v &= -p_0 \Psi_v.
    \end{aligned}
    \end{equation*}
  • We formed a linear superposition of our plane wave solutions
    \begin{equation}\label{eqn:qftLecture21b:800}
    \Psi(\Bx, t)
    =
    \sum_{s = 1}^2
    \int \frac{d^3 p}{(2 \pi)^3 \sqrt{ 2 \omega_\Bp } }
    \lr{
    e^{-i p \cdot x} u^s_\Bp a_\Bp^s
    +
    e^{i p \cdot x} v^s_\Bp b_\Bp^s
    }.
    \end{equation}
  • and expressed the Dirac Hamiltonian in terms of creation and anhillation operators
    \begin{equation*}
    H_{\text{Dirac}}
    =
    \sum_{r = 1}^2
    \int \frac{d^3 p }{(2\pi)^3 }
    \omega_\Bp
    \lr{
    a^{r \dagger}_\Bp
    a^r_\Bp
    –
    b^{r \dagger}_{-\Bp}
    b^r_{-\Bp}
    }.
    \end{equation*}
  • Finally, we interpreted this using the Dirac Sea argument

    Dirac Sea

  • It was claimed that the \( a, b\)’s satisfied anticommutator relationships
    \begin{equation}\label{eqn:qftLecture21b:940}
    \begin{aligned}
    \symmetric{a^s_\Bp}{a^{r \dagger}_\Bq} &= \delta^{sr} \delta^{(3)}e(\Bp – \Bq) \\
    \symmetric{b^s_\Bp}{b^{r \dagger}_\Bq} &= \delta^{sr} \delta^{(3)}(\Bp – \Bq),
    \end{aligned}
    \end{equation}
    where all other anticommutators are zero
    \begin{equation}\label{eqn:qftLecture21b:960}
    \symmetric{a^r}{b^s} =
    \symmetric{a^r}{b^{s\dagger}} =
    \symmetric{a^{r\dagger}}{b^s} =
    \symmetric{a^{r\dagger}}{b^{s\dagger}} = 0.
    \end{equation}
    and used these to algebraically remove the negative energy states of the Hamiltonian.