## PHY2403H Quantum Field Theory. Lecture 22: Dirac sea, charges, angular momentum, spin, U(1) symmetries, electrons and positrons. Taught by Prof. Erich Poppitz

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
\label{eqn:qftLecture22:140}
H \ket{\Bp, r}
=
\omega_\Bp \ket{\Bp, r}.
• The conserved Noether current and charge for spatial translations, the momentum operator, was found to be
\label{eqn:momentumDirac:260}
\BP =
\int d^3 x

which could be written in creation and anhillation operator form as
\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}
}.

Single particle states were found to be the eigenvectors of this operator, with momentum eigenvalues
\label{eqn:momentumDirac:262}
\BP a_\Bq^{s\dagger} \ket{0} = \Bq (a_\Bq^{s\dagger} \ket{0}).
• The conserved Noether current and charge for a rotation was found. That charge is
\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,

where
\label{eqn:qftLecture22:260}
\mathbf{1} \otimes \Bsigma =
\begin{bmatrix}
\Bsigma & 0 \\
0 & \Bsigma
\end{bmatrix},

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
\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
},

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.

## PHY1520H Graduate Quantum Mechanics. Lecture 12: Symmetry (cont.). Taught by Prof. Arun Paramekanti

November 5, 2015 phy1520 , , , , , , ,

### Disclaimer

Peeter’s lecture notes from class. These may be incoherent and rough.

These are notes for the UofT course PHY1520, Graduate Quantum Mechanics, taught by Prof. Paramekanti, covering chap. 4 content from [1].

### Parity (review)

\label{eqn:qmLecture12:20}
\hat{\Pi} \hat{x} \hat{\Pi} = – \hat{x}

\label{eqn:qmLecture12:40}
\hat{\Pi} \hat{p} \hat{\Pi} = – \hat{p}

These are polar vectors, in contrast to an axial vector such as $$\BL = \Br \cross \Bp$$.

\label{eqn:qmLecture12:60}
\hat{\Pi}^2 = 1

\label{eqn:qmLecture12:80}
\Psi(x) \rightarrow \Psi(-x)

If $$\antisymmetric{\hat{\Pi}}{\hat{H}} = 0$$ then all the eigenstates are either

• even: $$\hat{\Pi}$$ eigenvalue is $$+ 1$$.
• odd: $$\hat{\Pi}$$ eigenvalue is $$– 1$$.

We are done with discrete symmetry operators for now.

### Translations

Define a (continuous) translation operator

\label{eqn:qmLecture12:100}
\hat{T}_\epsilon \ket{x} = \ket{x + \epsilon}

The action of this operator is sketched in fig. 1.

fig. 1. Translation operation.

This is a unitary operator

\label{eqn:qmLecture12:120}
\hat{T}_{-\epsilon} = \hat{T}_{\epsilon}^\dagger = \hat{T}_{\epsilon}^{-1}

In a position basis, the action of this operator is

\label{eqn:qmLecture12:140}
\bra{x} \hat{T}_{\epsilon} \ket{\psi} = \braket{x-\epsilon}{\psi} = \psi(x – \epsilon)

\label{eqn:qmLecture12:160}
\Psi(x – \epsilon) \approx \Psi(x) – \epsilon \PD{x}{\Psi}

\label{eqn:qmLecture12:180}
\bra{x} \hat{T}_{\epsilon} \ket{\Psi}
= \braket{x}{\Psi} – \frac{\epsilon}{\Hbar} \bra{ x} i \hat{p} \ket{\Psi}

\label{eqn:qmLecture12:200}
\hat{T}_{\epsilon} \approx \lr{ 1 – i \frac{\epsilon}{\Hbar} \hat{p} }

A non-infinitesimal translation can be composed of many small translations, as sketched in fig. 2.

fig. 2. Composition of small translations

For $$\epsilon \rightarrow 0, N \rightarrow \infty, N \epsilon = a$$, the total translation operator is

\label{eqn:qmLecture12:220}
\begin{aligned}
\hat{T}_{a}
&= \hat{T}_{\epsilon}^N \\
&= \lim_{\epsilon \rightarrow 0, N \rightarrow \infty, N \epsilon = a }
\lr{ 1 – \frac{\epsilon}{\Hbar} \hat{p} }^N \\
&= e^{-i a \hat{p}/\Hbar}
\end{aligned}

The momentum $$\hat{p}$$ is called a “Generator” generator of translations. If a Hamiltonian $$H$$ is translationally invariant, then

\label{eqn:qmLecture12:240}
\antisymmetric{\hat{T}_{a}}{H} = 0, \qquad \forall a.

This means that momentum will be a good quantum number

\label{eqn:qmLecture12:260}
\antisymmetric{\hat{p}}{H} = 0.

### Rotations

Rotations form a non-Abelian group, since the order of rotations $$\hatR_1 \hatR_2 \ne \hatR_2 \hatR_1$$.

Given a rotation acting on a ket

\label{eqn:qmLecture12:280}
\hatR \ket{\Br} = \ket{R \Br},

observe that the action of the rotation operator on a wave function is inverted

\label{eqn:qmLecture12:300}
\bra{\Br} \hatR \ket{\Psi}
=
\bra{R^{-1} \Br} \ket{\Psi}
= \Psi(R^{-1} \Br).

## Example: Z axis normal rotation

Consider an infinitesimal rotation about the z-axis as sketched in fig. 3(a),(b)

\label{eqn:qmLecture12:320}
\begin{aligned}
x’ &= x – \epsilon y \\
y’ &= y + \epsilon y \\
z’ &= z
\end{aligned}

The rotated wave function is

\label{eqn:qmLecture12:340}
\tilde{\Psi}(x,y,z)
= \Psi( x + \epsilon y, y – \epsilon x, z )
=
\Psi( x, y, z )
+
\epsilon y \underbrace{\PD{x}{\Psi}}_{i \hat{p}_x/\Hbar}

\epsilon x \underbrace{\PD{y}{\Psi}}_{i \hat{p}_y/\Hbar}.

The state must then transform as

\label{eqn:qmLecture12:360}
\ket{\tilde{\Psi}}
=
\lr{
1
+ i \frac{\epsilon}{\Hbar} \hat{y} \hat{p}_x
– i \frac{\epsilon}{\Hbar} \hat{x} \hat{p}_y
}
\ket{\Psi}.

Observe that the combination $$\hat{x} \hat{p}_y – \hat{y} \hat{p}_x$$ is the $$\hat{L}_z$$ component of angular momentum $$\hat{\BL} = \hat{\Br} \cross \hat{\Bp}$$, so the infinitesimal rotation can be written

\label{eqn:qmLecture12:380}
\boxed{
\hatR_z(\epsilon) \ket{\Psi}
=
\lr{ 1 – i \frac{\epsilon}{\Hbar} \hat{L}_z } \ket{\Psi}.
}

For a finite rotation $$\epsilon \rightarrow 0, N \rightarrow \infty, \phi = \epsilon N$$, the total rotation is

\label{eqn:qmLecture12:420}
\hatR_z(\phi)
=
\lr{ 1 – \frac{i \epsilon}{\Hbar} \hat{L}_z }^N,

or
\label{eqn:qmLecture12:440}
\boxed{
\hatR_z(\phi)
=
e^{-i \frac{\phi}{\Hbar} \hat{L}_z}.
}

Note that $$\antisymmetric{\hat{L}_x}{\hat{L}_y} \ne 0$$.

By construction using Euler angles or any other method, a general rotation will include contributions from components of all the angular momentum operator, and will have the structure

\label{eqn:qmLecture12:480}
\boxed{
\hatR_\ncap(\phi)
=
e^{-i \frac{\phi}{\Hbar} \lr{ \hat{\BL} \cdot \ncap }}.
}

### Rotationally invariant $$\hat{H}$$.

Given a rotationally invariant Hamiltonian

\label{eqn:qmLecture12:520}
\antisymmetric{\hat{R}_\ncap(\phi)}{\hat{H}} = 0 \qquad \forall \ncap, \phi,

then every

\label{eqn:qmLecture12:540}
\antisymmetric{\BL \cdot \ncap}{\hat{H}} = 0,

or
\label{eqn:qmLecture12:560}
\antisymmetric{L_i}{\hat{H}} = 0,

Non-Abelian implies degeneracies in the spectrum.

### Time-reversal

Imagine that we have something moving along a curve at time $$t = 0$$, and ending up at the final position at time $$t = t_f$$.

fig. 4. Time reversal trajectory.

Imagine that we flip the direction of motion (i.e. flipping the velocity) and run time backwards so the final-time state becomes the initial state.

If the time reversal operator is designated $$\hat{\Theta}$$, with operation

\label{eqn:qmLecture12:580}
\hat{\Theta} \ket{\Psi} = \ket{\tilde{\Psi}},

so that

\label{eqn:qmLecture12:600}
\hat{\Theta}^{-1} e^{-i \hat{H} t/\Hbar} \hat{\Theta} \ket{\Psi(t)} = \ket{\Psi(0)},

or

\label{eqn:qmLecture12:620}
\hat{\Theta}^{-1} e^{-i \hat{H} t/\Hbar} \hat{\Theta} \ket{\Psi(0)} = \ket{\Psi(-t)}.

# References

[1] Jun John Sakurai and Jim J Napolitano. Modern quantum mechanics. Pearson Higher Ed, 2014.