Introduction
In Chapter 2, we discussed an example: how does the process of braiding 2 particles around each other, or exchanging their positions, introduces knot theory and topology into quantum mechanics.
We are familiar with the bosons and fermions situation:
Note: If you exchange a pair of particles then exchange them again, you get back where you started. So the square of the exchange operator should be the identity, or one. There are two square roots of one: +1 and −1, so these are the only two possibilities for the exchange operator.
However, this argument is considered to be incorrect now, if we learn to think about quantum mechanics from the Feymann path integral point of view.
3.1 Single Particle Path Integral
Propagator 传播子:
$$
\left\langle\mathrm{x}_f\right| \hat{U}\left(t_f, t_i\right)\left|\mathrm{x}_i\right\rangle
$$
i stands for initial, f stands for final
Important: Propagator gives the amplitude fo ending up at a position $\mathrm{x}_f$ at the final time $t_f$.
The propagator can be used to propagate forward in time some arbitrary wave function $\psi(x)=\langle\mathbf{x} \mid \psi\rangle$ from $t_i$ to $t_f$ as follows
$$
\left\langle\mathrm{x}_f \mid \psi\left(t_f\right)\right\rangle=\int d \mathrm{x}_i\left\langle\mathrm{x}_f\right| \hat{U}\left(t_f, t_i\right)\left|\mathrm{x}_i\right\rangle\left\langle\mathrm{x}_i \mid \psi\left(t_i\right)\right\rangle
$$
$\left\langle\mathrm{x}_i \mid \psi\left(t_i\right)\right\rangle$ : the initial wave function $\psi(\mathrm{x}_i, t_i)$.
2 properties:
- unitary
- obey composition
If we subdivide time into infinitesimally small pieces, and do integrals on all possible intermediate positions, the propagator turn into:
$$
\left\langle\mathbf{x}f\right| \hat{U}\left(t_f, t_i\right)\left|\mathbf{x}i\right\rangle=\mathcal{N} \sum{\substack{\text { paths } \mathbf{x}(t) \text { from } \\left(\mathbf{x}{\mathbf{i}}, t_i\right) \text { to }\left(\mathbf{x}_{\mathbf{f}}, t_f\right)}} e^{i S[\mathbf{x}(t)] / \hbar}
$$
Where $\mathcal{N}$ is some normalization constant. Here, $S[\mathbf{x}(t)]$ is the (classical!) action of the path with $L$ the Lagrangian.
$$
S=\int_{t_i}^{t_f} d t L[\mathbf{x}(\mathbf{t}), \dot{\mathbf{x}}(\mathbf{t}), t]
$$
$$\left\langle\mathbf{x}f\right| \hat{U}\left(t_f, t_i\right)\left|\mathbf{x}i\right\rangle=\mathcal{N} \int{\left(\mathbf{x}{\mathbf{i}}, t_i\right)}^{\left(\mathbf{x}_{\mathbf{f}}, t_f\right)} \mathcal{D} \mathbf{x}(t) e^{i S[\mathbf{x}(t)] / \hbar}$$
3.2 Two Identical Particles
Topologically, we have 2 types of path: exchanged and unexchanged
It is obvious that the paths has the structure of $G(\cdot, \lbrace 1, -1\rbrace)$, if we define $\cdot$ as “follow by”
We can construct 2 type of path integral, which are both allowed according to the composition law.
$$\begin{aligned}
& \left\langle\mathrm{x}{1 f} \mathrm{x}{2 f}\right| \hat{U}\left(t_f, t_i\right)\left|\mathrm{x}{1 i} \mathrm{x}{2 i}\right\rangle=\mathcal{N} \sum_{\substack{\text { paths } \
i \rightarrow f}} e^{i S[\text { path }] / \hbar}= \
& \mathcal{N}\left(\sum_{\text {TYPE }+1 \text { paths }} e^{i S[\text { path }] / \hbar}+\sum_{\text {TYPE }-1 \text { paths }} e^{i S[\text { path }] / \hbar}\right)
\end{aligned}$$
$$\begin{aligned}
& \left\langle\mathrm{x}{1 f} \mathrm{x}{2 f}\right| \hat{U}\left(t_f, t_i\right)\left|\mathrm{x}{1 i} \mathrm{x}{2 i}\right\rangle=\mathcal{N} \sum_{\substack{\text { paths } \
i \rightarrow f}} e^{i S[\text { path }] / \hbar}= \
& \mathcal{N}\left(\sum_{\text {TYPE }+1 \text { paths }} e^{i S[\text { path }] / \hbar}-\sum_{\text {TYPE }-1 \text { paths }} e^{i S[\text { path }] / \hbar}\right)
\end{aligned}$$
Fermions and bosons naturally emerge from this theory! Experiments will decide