Second Edition of the textbook
The second edition of the textbook "Symmetry and its breaking in quantum field theory"
will be soon published, and therefore I should like to upload some of the textbook
chapters such that researchers can make use of the textbook for their new research projects
and students can understand field theory in depth.
In particular, I have included the important description of the gravity theory
in terms of the normal Lagrangian density terminology, wothout referring to the space
deformation. The new theoretical scheme of the gravity should be applied to many interesting
physical phenomena and one may learn physics in depth in terms of the simple physics
terminology.
In this textbook, the description of the perturbation theory and renormalization scheme
is not sufficiently made in detail. This is mainly because there is a good textbook
(eg. Bjorken-Drell) which explains the basic point of the renormalization scheme quite well.
However, the treatment of the vacuum polarization should be considerably modified
with important changes. As I repeatedly claimed, the renormalization scheme of QED
is most reliable since the theoretical framework is carefully constructed with emphasis
upon the physical observables. Among the renormalization procedure in QED, there is one thing
which is somewhat difficult to understand, that is, the vacuum polarization contribution.
If one carries out the S-matrix evaluation in QED, one always finds that the self-energy
of photon has the quadratic divergence term. Since it is obviously unphysical, one should throw
away this quadratic divergence term with some physical reason. For this, people have chosen that
the counter term that cancels out the quadratic divergence term should be gauge invariant.
In this case, one can fortunately throw away the quadratic divergence contribution since
this term is just a constant term which corresponds to the mass term in QED. However, the gauge
invariance is simply the mathematical result, that is, the number of the independent gauge
fields is larger than the number of the equations of motion by one unit, and therefore one has
to fix the gauge so that one can solve the equations and determine the gauge configuration.
Since the gauge fields themselves are not physical observables, physical observables do not
depend on the gauge choice. After the gauge fixing, one can quantize the gauge fields. In this
respect, there is no point of discussing the gauge dependence of the photon self-energy
contribution. In fact, the fermion self-energy and the vertex correction do not depend on the
gauge choice.
There is one more point which is very important in the QED renromalization scheme.
In the calculations of self-energy of photon and fermion, there appear infinities, and
they should be canceled out by the counter term. In this case, the perturbation theory
always makes use of the free Fock space, and therefore as long as one treats the self-energy
terms of photon and fermion within the free Fock space, they can be completely canceled out.
However, the situation is different for the fermion case. Fermions can be found in the states
which are not within the free Fock space. That is, the fermion can be bound in the hydrogen atom.
In this case, the bound states of fermions are different from the free Fock space and therefore
the counter term of the fermion self-energy terms cannot cancel out the infinity of the fermion
self-energy terms in the hydrogen atom, and this is exhibited as the Lamb shift energy.
On the other hand, the situation in the photon self-energy is quite different from the fermion
case in that there is no bound state for photon. Photon can be found always within the free
Fock space. Therefore, the divergent contribution of the photon self-energy is completely canceled
out by the counter terms. Thus, there is no occasion where the photon self-energy contribution
becomes a physical observable, and therefore one should not consider the photon self-energy term
into the renormalization procedure.
The fact that the self-energy of photon should not be considered in the renormalization
scheme suggests that there must be quite large influences on field theory models.
For example, the coupling constant (or charge) in QED is not affected by the renormalization
procedure, and therefore there exists no renormalization group equation from the beginning and
this clarifies the situation in the renormalization scheme a great deal.
Further, if the coupling constant is dimensionless, then this model field theory
may well be renormalizable. For example, the Yukawa model that describes the interaction
between pion and nucleon should be renormalizable, apart from the fact that pion and
nucleon must be composite objects.
For a long time, people are possessed by a "physics demon" that only gauge field theories must
be renormalizable. In particular, the standard model of Weinberg-Salem is just the case where
the model started from the non-abelian gauge field theory of SU(2)xU(1).
Since they had to derive a massive vector boson fields, they made use of the Higgs mechanism
which is simply an unphysical procedure. It should be noted that, if one transforms any Lagrangian
density into a new shape, one cannot learn anything if one carried out mathematics properly.
This is clear since the Lagrangian density itself is not directly related to physical observables.
However, the Hamiltonian that Weinberg-Salem finally obtained must be related to physical
observables and therefore it should be meaningful. Now, one sees that the massive vector fields
without the local gauge invariance can be renormalizable, and therefore the Weinberg-Salem model
should be a good field theory model as long as one makes a proper correction of throwing away
the Higgs fields.