218 From Classical Mechanics to Quantum Field Theory. A Tutorial
the first order correction to the energy of all one-particle levels is finite. With the
above prescription there is no correction to the free valueωn, but we could have
renormalized the mass of the theory by a different subtraction:m^2 →m^2 −Δm^2 +
a^2 .Inthatcase,therenormalizedvalue of the energy one-particle states will be
after resummation of the perturbative series,
ωnr=√
n^2 +m^2 +a^2.With the above renormalizations of the vacuum energy and mass, the theory
is finite at first order of perturbation theory. This means that the corrections to
the higher energy levels are finite at first order inλ.
We can understand now the physical meaning of the renormalization program.
The physical parameters which appear in the classical Lagrangian do not necessar-
ily coincide with the corresponding quantum physical parameters. This includes
the constant term which can always be added to the classical Lagrangian with-
out changing the dynamics (but determines the vacuum energy of the quantum
theory), the mass of the theorymand the coupling constantλ.
Until now we have only renormalized the mass and the vacuum energy. How-
ever, in higher orders of perturbation theory new UV divergences appear. They
can be absorbed by new renormalizations of the vacuum energyE 0 ,themassm^2 ,
the coupling constantλand the field operatorsφˆ(f).
However, the proof of consistency of the resulting theory is quite involved and
required few decades to be completely achieved. One of the main problems is
that in the canonical approach, the preservation of the relativistic invariance is
not guaranteed. Among other things the use of UV cutoff breaks Lorentz invari-
ance and one has to prove that the renormalization prescriptions do preserve the
relativistic symmetries. In general, it is not obvious that the interacting theory
satisfies the general quantum field principles of section 3.
For such reasons it is convenient to develop a new approach to quantization
based on a covariant formalism, where time and space are treated on the same
footing.
3.7 Covariant Approach
If we consider the Heisenberg representation of quantum operators the field oper-
ator evolves according to the Heisenberg law
φ(x,t)=U(t)φ(x)U(t)†,