This post is a synopsis of the material from the second last lecture of QFT I. I missed that class, but worked from notes kindly provided by Emily Tyhurst, and Stefan Divic, filling in enough details that it made sense to me.

[Click here for an unabrided PDF of my full notes on this day’s lecture material.]

Topics covered include

• The Hamiltonian action on single particle states showed that the Hamiltonian was an energy eigenoperator
  \begin{equation}\label{eqn:qftLecture22:140}
  H \ket{\Bp, r}
  =
  \omega_\Bp \ket{\Bp, r}.
  \end{equation}
• The conserved Noether current and charge for spatial translations, the momentum operator, was found to be
  \begin{equation}\label{eqn:momentumDirac:260}
  \BP =
  \int d^3 x
  \Psi^\dagger (-i \spacegrad) \Psi,
  \end{equation}
  which could be written in creation and anhillation operator form as
  \begin{equation}\label{eqn:momentumDirac:261}
  \BP = \sum_{s = 1}^2
  \int \frac{d^3 q}{(2\pi)^3} \Bp \lr{
  a_\Bp^{s\dagger}
  a_\Bp^{s}
  +
  b_\Bp^{s\dagger}
  b_\Bp^{s}
  }.
  \end{equation}
  Single particle states were found to be the eigenvectors of this operator, with momentum eigenvalues
  \begin{equation}\label{eqn:momentumDirac:262}
  \BP a_\Bq^{s\dagger} \ket{0} = \Bq (a_\Bq^{s\dagger} \ket{0}).
  \end{equation}
• The conserved Noether current and charge for a rotation was found. That charge is
  \begin{equation}\label{eqn:qftLecture22:920}
  \BJ = \int d^3 x \Psi^\dagger(x) \lr{ \underbrace{\Bx \cross (-i \spacegrad)}_{\text{orbital angular momentum}} + \inv{2} \underbrace{\mathbf{1} \otimes \Bsigma}_{\text{spin angular momentum}} } \Psi,
  \end{equation}
  where
  \begin{equation}\label{eqn:qftLecture22:260}
  \mathbf{1} \otimes \Bsigma =
  \begin{bmatrix}
  \Bsigma & 0 \\
  0 & \Bsigma
  \end{bmatrix},
  \end{equation}
  which has distinct orbital and spin angular momentum components. Unlike NRQM, we see both types of angular momentum as components of a single operator. It is argued in [3] that for a particle at rest the single particle state is an eigenvector of this operator, with eigenvalues \( \pm 1/2 \) — the Fermion spin eigenvalues!
• We examined two \( U(1) \) global symmetries. The Noether charge for the “vector” \( U(1) \) symmetry is
  \begin{equation}\label{eqn:qftLecture22:380}
  Q
  =
  \int \frac{d^3 q}{(2\pi)^3} \sum_{s = 1}^2
  \lr{
  a_\Bp^{s \dagger} a_\Bp^s
  –
  b_\Bp^{s \dagger}
  b_\Bp^s
  },
  \end{equation}
  This charge operator characterizes the \( a, b \) operators. \( a \) particles have charge \( +1 \), and \( b \) particles have charge \( -1 \), or vice-versa depending on convention. We call \( a \) the operator for the electron, and \( b \) the operator for the positron.
• CPT (Charge-Parity-TimeReversal) symmetries were also mentioned, but not covered in class. We were pointed to [2], [3], [4] to start studying that topic.

References

[1] C. Doran and A.N. Lasenby. Geometric algebra for physicists. Cambridge University Press New York, Cambridge, UK, 1st edition, 2003.

[2] Dr. Michael Luke. Quantum Field Theory., 2011. URL https://www.physics.utoronto.ca/~luke/PHY2403F/References_files/lecturenotes.pdf. [Online; accessed 05-Dec-2018].

[3] Michael E Peskin and Daniel V Schroeder. An introduction to Quantum Field Theory. Westview, 1995.

[4] Dr. David Tong. Quantum Field Theory. URL http://www.damtp.cam.ac.uk/user/tong/qft.html.