Path Integrals and Hamiltonians: Principles and Methods

My own path to, or through, quantum mechanics has been heavily influenced by the fact that I am a pure mathematician and don’t speak physics. I find physics books written for physicists by physicists very difficult to read. This was brought home to me in spades when I started all this ecumenical work in the cause of what are ultimately analytic number theoretic concerns. (See below for a hint.) Happily, I hit upon the book by Prugovečki, Quantum Mechanics in Hilbert Space, which is in my opinion, unsurpassed. It is fundamentally a functional analysis text tailored to the bizarre (but beautiful) axiomatics of quantum mechanics. Prugovečki does a phenomenal job playing off the Heisenberg picture (matrix mechanics) against the Schrödinger picture (wave mechanics), both fitted ultimately into the framework of self-adjoint operators (densely defined) on a Hilbert space, the Hilbert space of states of a quantum mechanical system. It is here that things get dicey, as various maneuvers with projection operators and spectral measures need to be interpreted vis à vis the physical reality suggested by measurements and observations. Thus, the Born rules and the Copenhagen interpretation rear their heads, and we find ourselves explaining to Einstein why Heisenberg had a point, at least epistemologically. On the other hand, in the framework of quantum mechanics proper, the vaunted (and frightening) uncertainty principle is ultimately a Fourier analytic affair: there is a lot of comfort to be derived from mathematics.

Of course, it needs to be said that the book for quantum mechanics is Dirac’s magisterial Principles of Quantum Mechanics. A strong case can be made that this is the most elegant book ever written in the genre; it reads like high literary art, in Dirac’s famous minimalist style. Erik Satie, not Maurice Ravel, perhaps. But his Lucasian professorship notwithstanding, Dirac’s mathematics is not a pure mathematician’s mathematics, brilliant though his presentation and ideas are. For example, there is the matter of notation, which was invented by Dirac for particular purposes. For example, the exploitation of what the isomorphism between a linear space and its dual can provide in the way of symmetry of outcomes (think of \(x^*(y)\) as simultaneously a function of each of the variables) — this leads to the bra-ket notation. To physicists this is soon second nature, but it is not the usual notation we mathematicians use. So I happily ran home to Progovečki.

Dirac was known among (many) other things for not only a reformulation of quantum mechanics in terms of Poisson brackets (on the heels of the pioneering work of Heisenberg and Schrödinger) but for the realization and proof (well, a proof: I believe Schrödinger established it too) that the so-called Heisenberg and Schrödinger pictures of quantum mechanics are mathematically equivalent: matrix mechanics and wave mechanics are the same thing, so there is just quantum mechanics, “QM,” presented in different pictures or from different standpoints. For example, one plays the evolution of a state in space off against the evolution of a state in time, and both perspectives are not just dual but complementary — you get two for the price of one. And in this game, we encounter 1-parameter groups of unitary operators, which, of course, is music to the ears of any analytic number theorist (to put in a plug for what I do). It is these 1-parameter groups of unitary operators that provide the time-evolution of a quantum mechanical system, and are in fact at the heart of an entirely novel way of doing quantum mechanics, that of Feynman.

And this takes us to the book under review. In his Preface, Baaquie says that beyond the traditional approach to QM, that of Schrödinger (via the Schrödinger wave equation), there are two others, “namely the operator approach of Heisenberg and the path integral approach of Dirac-Feynman, that provide a mathematical framework that is independent of the Schrödinger equation.” He goes on to say, “[i]n this book, the Schrödinger equation is never directly solved; instead the Hamiltonian [or total energy] operator is analyzed and path integrals for different quantum and classical random systems are studied to gain an understanding of quantum mechanics.”

After the Preface, Baaquie presents a Synopsis of the book, the thrust of which is that it is divided into six parts. We get, in sequence, fundamental principles and the mathematical structure of QM, stochastic processes, discrete degrees of freedom, quadratic path integrals, acceleration action, and finally nonlinear path integrals. The last mentioned material includes, e.g., coverage of a nonlinear quartic Lagrangian. It should be noted that Feynman’s presentation of his path integral formalism is fundamentally based on a Lagrangian formalism.

The book is well written, in a compact style, but not stinting on clear explanations. I don’t know whether it is because after a good deal of exposure to quantum physics through the services of Proguvečki and others I am more comfortable with this material, and that explains it, but I think the book is accessible even to us mathematicians. Well, let me hedge my bets and add a caveat: I don’t think this book is indicated for raw recruits — you should know some quantum mechanics already, perhaps at the level of Faddeev-Yakubovskiĭ, Lectures on Quantum Mechanics. After this it’s the right time for “a pedagogical introduction to the essential principles of path integrals and Hamiltonians,” as the present book’s back cover advertises.

By the way, there are a few limitations to be noted: there is no discussion of the uncertainty principle (which is not surprising, given the framework Baaquie has chosen), and there are no Feynman diagrams, meaning that the analysis of the integrals is not pushed in the usual direction physicists take it. In the latter connection see http://www.webofstories.com/play/freeman.dyson/71 for Freeman Dyson’s description of how Feynman himself used (or didn’t use) his integrals. And, once you’ve looked at that, look at http://www.webofstories.com/play/freeman.dyson/72 (It’s Dyson, after all: just keep going.)

However, Path Integrals and Hamiltonians looks like a very useful book, and I, for one, am very happy to have a copy.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.