Wednesday, July 19, 2006

Weinberg's book

If you are interested in QFT, I strongly recommend you to read Weinberg's book [1]. It is so much different from all other books on the subject. Weinberg does not pretend that the idea of quantum fields arises from "generalization" of quantum-mechanical wave functions or as a quantum version of classical field theory. In Weinberg's book you will not find rather dubious logical chains like "Schroedinger equation -> Dirac equation -> Dirac's hole theory -> QFT" or "Lagrangian -> Klein-Gordon equation -> quantization -> quantum fields". His approach to quantum fields, Lagrangians, and gauges is pretty utilitarian.

Weinberg correctly says that the only thing we are interested in QFT is the S-matrix. We want the S-matrix to be (at least) relativistically invariant and cluster separable. This implies that the Hamiltonian of the theory should be relativistically invariant and cluster separable. He goes on to demonstrate how Hamiltonians with these properties can be constructed as certain polynomials of operator functions defined on the Minkowski space-time. These operator functions are called quantum fields and they are required to satisfy some formal properties, like (anti-)commutativity at space-like intervals and covariant Lorentz transformation laws. The fact that quantum fields satisfy Klein-Gordon or Dirac equations is almost accidental, and has no any fundamental significance.

In Weinberg's logic, there is no reason to assume that quantum fields have any physical interpretation. They are just formal mathematical constructs. Moreover, there is no reason to associate the Minkowski space-time on which the fields "live" with real physical space and time.

This is by far the best book written about QFT. It gives you a very clear and, in my opinion, the only correct perspective on the foundations of QFT. Enjoy reading!

[1] S. Weinberg, The Quantum Theory of Fields, Vol. 1 , (University Press, Cambridge, 1995)

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