MadSci Network: Physics |
Your question requires two answers, one about relativity and one about quantum field theory. 1. At what velocity can a relativistic equation be treated in a non- relativistic approximation.? The key word here is approximation. The appropriate v/c depends on how good an approximation you want. There is no single value of v/c that fits all cases. If not too much accuracy is needed v/c~1/10 (or even somewhat larger) is often enough. For more accuracy v/c must be smaller. Like any approximation, it depends on the context. One word of advice. Don't rush into the approximation. The relativistic equation is often easier to use. 2. How is Quantum Field Theory (QFT) related to Quantum Mechanics (QM)? QM here refers to what could be called "First Quantized" and QFT is called "Second Quantized". The relation between QFT and QM has very little to do with relativity, so v/c is not important. QM is the original quantum mechanics in which classical variables such as p or x became operators on a new quantity, the quantum wave function. In the early 1930's Jordan and Pauli developed a theory in which the quantum wave function was considered an operator which acted on a new function which was the real wave function. This process was called "Second Quantization" because it quantized what had already been quantized. This new theory led to QFT. The three main first quantized wave equations, Schrodinger, Klein-Gordon, Dirac can all be second quantized. The main new feature of QFT is that the old wave function, now considered an operator, can create and destroy particles. Now, to answer the question of when QFT can be treated as old fashioned QM: When energies are not large enough to create particles [That is where relativity (E=mc^2) comes in.] then the single particle wave equations (Schrodinger, KG, Dirac) can be used. Classical E&M also has a wave equation for the potentials A and phi. If A and phi are considered to be operators that can create and destroy photons, then the QFT that arises is called Quantum Electrodynamics (QED).
Try the links in the MadSci Library for more information on Physics.