Revision of the textbook
The textbook "Fundamental Problems in Quantum Field Theory" was published
in July 2013.
Apart from the renormalization scheme of QED, I wrote the textbook rather critically
concerning the present understanding of field theory. But good reactions from
young people encouraged me a great deal.
In spite of the fact that I tried to write the textbook as carefully as possible,
there is one part of Appendix G which turns out to be incorrect. The path integral
formulation was written just in the same way as normal field theory textbooks.
However, it turns out that the path integral formulation for field theory cannot
quantize the fields. The path integral method in QED was first formulated by Feynman
who made it in a correct fashion. Feynman understood that the field
quantization means that the parameter which appears in the vector potential should
be replaced by an operator and therefore the path integral formulation corresponds
to the many dimensional integrations over the parameter space.
In this sense, he realized the importance of the fact that the QED Hamiltonian is
written just in terms of the harmonic oscillators in the parameter space.
On the other hand, the path integral formulation which is used after that is completely
different from Feynman's original formulation. People employed the path integral expression
which is based on the many dimensional integrations over discretized field variables, and
in this case, it has nothing to do with the field quantization and it is physically
meaningless. This mistake would not have happened if the physical meaning of the second
quantization were properly understood and the limitation of the path integral method were
firmly considered.
If one carries out mathematical evaluation properly from A-state to B-state, then
this is just the tautology, and there is no progress in this process. In physics,
any progress means that one finds some new aspect of the phenomena A or B by thinking
over in depth. The rewriting A into B may help find a new aspect of the physical
phenomena in A or B. In the path integral formulation, one simply rewrites the amplitude
into many dimensional integrations over discretized coordinates, and it is nothing
more than that. In other words, one may obtain some physical observables by the many
dimensional integrations without solving the differential equation.
Unfortunately, most of the lattice gauge theory calculations are based on the wrong
expression of the path integral formulation of the field theory. Some of the physicists
that are involved in the numerical simulations may well have had uneasy feeling on
the calculated results of the lattice calculations. However, it must be, at the same time,
difficult to raise any questions on the results since most of the evaluations in physics
may well be done by the decision of majority
This is just what Prof. Nishijima told us some time ago. He said that the evaluation
of research works must be determined by the majority vote and therefore any original works
or some theoretical works against majority cannot be highly appreciated. But he added
that we should not care for that.
Here, I included the revised part of Appendix G as a PDF file, and I will replace
it in the textbook when a new revised one will appear.