Book Review : Path Integrals In Physics Thread
Book info
M. Chaichian and A. Demichev, “Path Integrals in Physics”, IOP (2001) link
While path integrals are famous for Feynman’s path integral, I am more familiar with path integrals in Wiener processes used in statistical mechanics. However, despite this familiarity, my mathematical understanding is quite lacking, so I aim to fill these gaps by thoroughly reading this book.
Preface
Path integrals, which are widely used for various purposes, have not been properly organized into textbooks despite their frequent use. Most exist only in the form of lecture notes, and mathematically rigorous books are too difficult for beginners. Therefore, the authors focused on two aspects while writing this book.
First, they aimed to include comprehensive concepts of Wiener-type, Feynman-type path integrals, and path integrals on phase space. Second, they wrote with a more practical purpose in mind, assuming a more diverse readership. Since path integration is still an actively developing methodology, they couldn’t include all related content, but they tried their best to include both traditional results and recent major discoveries.
Each chapter is written to be self-contained so it can be read like an independent textbook. Even those without deep understanding can start from any chapter in Volume 1, and those with deeper understanding can read from any point of interest, including Volume 2.
Introduction
Path integrals were first used by mathematician Norbert Wiener in the 1920s while studying diffusion and Brownian motion. Later, Feynman developed this further while reconstructing quantum mechanics in the 1940s. Moving into the 1950s, much research was conducted in quantum field theory, and it was subsequently widely used in areas such as phase transitions, superfluids, superconductors, the Ising model, quantum optics, and plasma physics. In 1955, Feynman discovered the variational principle of quantum mechanics while solving the polaron problem, which greatly influenced not only quantum field theory but also statistical mechanics and solid mechanics.
Universality
The transition probabilities in stochastic processes and quantum mechanics take the following form: \begin{equation} W(x_f, t_f | x_0, t_0) \sim \sum_{\textrm{trajectories from } x_0 \textrm{to} x_f} \exp[- \frac{1}{4D} F[x(\tau)]] \end{equation} Also, expectation values in statistical mechanics and quantum mechanics can be expressed as: \begin{equation} \expval{A} \sim \sum_{configurations} A(\varphi) \exp[- \frac{1}{\hbar} S[\varphi]] \end{equation} In statistical mechanics, $\hbar$ changes to $k_B T$, and Action $S$ changes to energy $E$. In these cases, their simple sum easily becomes infinite, requiring a resolution process. In statistical mechanics, this is called the thermodynamic limit, and in quantum field theory, it’s called renormalization. However, there are differences in their normalization methods: the statistical mechanical limit is a method of gradually increasing system size, calculating with an upper limit on wavelength (Infrared cut-off), while renormalization is a method of calculating with a lower limit on wavelength by setting a minimum unit length (Ultraviolet cut-off). This difference makes it difficult to merge statistical mechanics and quantum field theory, and is why the two fields have traditionally been treated as completely different domains. Despite being applied in different fields, path integrals show universal forms like this.
Differ from multiple finite-dimensional integrals
At first glance, path integrals appear like functionals integration. Functional integration was used by Volterra in 1965 with the following rules:
- Split the function to transform the functional into a function of N variables
- Perform integration
- Find the limit of the integral result when N approaches infinity
However, this approach didn’t work for path integrals, and a more precise definition of path integrals could be established using the Wiener process as a foundation. Let’s briefly examine this. The real vector space $\mathbb{R}^n$ has a Lebesgue measure that allows integration. Can a measure with the same properties exist in infinite dimensions, allowing integration of functions of infinite-dimensional vectors? The answer is no. Here’s the proof: When considering an orthonormal basis $\qty{ e_1, e_2, \cdots }$ in infinite-dimensional space $\mathbb{R}^\infty$, we can consider spheres $B_k$ of radius $1/2$ centered at $e_k$, and sphere $B$ of radius $2$ centered at the origin. If there exists a measure $\mu$, it would satisfy $0 < \mu(B_1) = \mu(B_2) = \cdots < \infty$, and since these have no intersection, by the additivity property of measures, $\mu(B) > \sum_k B_k = \infty$. Since the measure of a bounded set cannot be infinite, such a measure cannot exist. Consequently, finite-dimensional vector functions and infinite-dimensional functionals cannot be treated with the same type of measure, and functional integration requires building its own new theory.
1. Path integrals in classical theory
1.1 Brownian motion 1.2 Wiener path integrals and stochastic processes
2. Path integrals in quantum mechanics
2.1 Feynman path integrals 2.2 Path integrals in the Hamiltonian formalism 2.3 Quantization, the operator ordering problem and path integrals 2.4 Path integrals and quantization in spaces with topological constraints 2.5 Path integrals in curved spaces, spacetime transformations and the Coulomb problem 2.6 Path integrals over anticommuting variables for fermions and generalizations
3. Quantum field theory
3.1 Path-integral formulation of the simplest quantum field theories 3.2 Path-integral quantization of gauge-field theories 3.3 Non-perturbative methods for the analysis of quantum field models in the path-integral approach 3.4 Path integrals in the theory of gravitation, cosmology and string theory: advanced applications of path integrals
4. Path integrals in statistical physics
4.1 Basic concepts of statistical physics 4.2 Path integrals in classical statistical mechanics 4.3 Path integrals for indistinguishable particles in quantum mechanics 4.4 Field theory at non-zero temperature 4.5 Superfluidity, superconductivity, non-equilibrium quantum statistics and the path-integral technique 4.6 Non-equilibrium statistical physics in the path-integral formalism and stochastic quantization 4.7 Path-integral formalism and lattice systems