Path Integral Formulation of Quantum Mechanics

Source: Quantum Field Theory of Many-Body Systems by Xiao-gang wen

Some nice quotes from the preface:

After reading this book, I hope, instead of a feeling of completeness, readers will have a feeling of emptiness. After one-hundred years of condensed matter theory, which offers us so much, we still know so little about the richness of nature. However, instead of being disappointed, I hope that readers are excited by our incomplete understanding. It means that the interesting and exciting time of condensed matter theory is still ahead of us, rather than behind us.

Nature's richness is not bounded by quantum field theory.

The first chapter makes it clear that the main goal of condensed matter physics is to offer theories for collective behavior as a tractable alternative to bashing out the Schrodinger equation for millions of particles, which though may give a complete description of the system, would be too complex to even think about solving given the entire age of the universe. As a really pretty example, phonons emerge as "particle-like" collective behavior with simple interaction laws between them. One idea proposed is that all "particles" are actually just manifestations of low-energy collective excitations given by some even more fundamental theory.

TODO: What are Goldstone modes and spontaneous symmetry breaking?

Path Integral

Recall the time evolution operator, $U(t_b, t_a) = e^{-i(t_b - t_a)H}$ . Given the coordinate basis for a 1-d system, the matrix elements of $U$ in the coordinate basis are $iG(x_b, t_b, x_a, t_a) = \langle x_b | U(t_b, t_a) | x_a \rangle$ , where $G$ is our full propagator.

The propagator is the central device behind path integrals. In finding the amplitude of some particle going from position $q_I$ to $q_F$ , we want to sum over all possible paths, each path being composed of N "line segments" each with time interval $\delta t$ such that

\begin{aligned} \langle q_I |e^{iHT} q_J \rangle &= \langle q_I | e^{i H \delta t} e^{i H \delta t} \cdots e^{i H \delta t} | q_I \rangle \end{aligned}

Then we do some goomba expansions with $\int dq | q \rangle \langle q | = 1$ , so our expression becomes

\begin{aligned} \langle q_I |e^{iHT} q_J \rangle &= (\prod \int dq) \langle q_f | e^{i H \delta t} | q_{N-1} \rangle \langle q_{N-1} | e^{i H \delta t} | q_{N-2} \rangle \cdots \langle q_I | e^{i H \delta t}| q_I \rangle \end{aligned}

I'm now realizing there's a bunch of annoying algebra between this and the final result, so Imma just write it down

\begin{aligned} \langle q_F | e^{-i HT} | q_I \rangle = \int Dq(t) e^{i \int_0^T dt (\frac{1}{2}m \dot{q}^2 - V(q))} \end{aligned}

and that thingy at the top of the $e$ is just the Lagrangian for our Hamiltonian $H$ and $Dq(t)$ integrates over all possible paths.

"We count on the path integral to converge because the oscillatory phase factors from different paths tend to cancel out" -- wat?

The Continuum Limit and FIELDS

Ok now suppose we had $N$ particles. Then, we can easily generalize $H = \sum_a \frac{1}{2m_a} \hat{p}_a^2 + V(\hat{q}_1, \hat{q}_2, \cdots, \hat{q}_N)$ , and integrate over all possible paths.

Now, we get $Z = \langle 0 | e^{-iHT} | 0 \rangle = \int Dq(t) e^{i S(q)}$ , where

$S(q) = \int_0^T dt (\sum_a\frac{1}{2}m_a \dot{q}_a^2 - V(q_1, q_2, \cdots, q_N))$ . Suppose we have some chain-of-springs-like system where pairs of particles have quadratic potential interactions, so our potential is $V(q_1, q_2, \cdots, q_N) = \sum_{ab} \frac{1}{2} K_{ab}(q_a - q_b)^2 + \cdots$ (Zee calls this a "mattress"...). We care about scales much larger than lattice spacing, so lets take the "continuum limit:" $l \xrightarrow{} 0$ .

Now, our discrete points $q_a$ become a continuous function over position $\vec{x}$ , so we get a function $q(t, \vec{x})$ . By tradition, we replace the letter $q$ with $\phi$ , which we then call our field.

TODO: Link Kardar's note to this -- Zee gives a much clearer description of what fields are. How do our terms transform in this limit?

\begin{aligned} \sum_a \frac{1}{2}m_a(\dot{q}_a^2) \mapsto \int d^2 x \frac{1}{2}\sigma (\partial \phi/\partial t)^2 \end{aligned}

where we must now introduce a mass density $\sigma$ and integrate over 2d space (ie. dimension of our system). Then, lets look at our quadratic potential interaction, assuming nearest neighbor interaction for simplicitly. Naturally, $(q_a - q_b)^2 \mapsto l^2 (\partial\phi/\partial x)^2$ (just linearize). Ok, then all together our action is

\begin{aligned} S(q) \mapsto S(\phi)\int_0^T dt \int d^2 x \mathcal{L}(\phi) = \int_0^T dt \int d^2 x \frac{1}{2}(\sigma (\partial \phi/\partial t)^2 - \rho[(\frac{\partial \phi}{\partial x})^2 + (\frac{\partial \phi}{\partial y})^2]-\tau \phi^2 - \chi \phi^4 + \cdots) \end{aligned}

I think the higher order terms come from higher order terms in the potential? If we write $\rho = \sigma c^2$ and scale $\phi \mapsto \phi/\sqrt{\sigma}$ , we get a phase-velocity like term in our Lagrangian, as

$(\partial \phi/\partial t)^2 - c^2 [(\partial \phi/\partial x)^2 + (\partial \phi/\partial y)^2]$