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S6f. The classical limit in relativistic QFT
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The classical limit of a quantum field theory is the
theory defined by taking the Lagrangian occuring in the functional
formalism and making the corresponding action stationary.
Note that a functional integral is an integral in which all
fields have classical meaning. The quantum interpretation comes
from taking the functional integral as a generating functional for
S-matrix elements, while the classical interpretation comes from
taking a saddle point approximation. Since the k loop contributions
scale with hbar^k, they disappear in the classical limit hbar to 0,
so only the tree diagrams are left in the expansion, which correspond
to the saddle point approximation in the functional integral.
This needs a slight qualification for Fermions, e.g., electrons.
A fermion field Psi(x) itself, being an anticommuting field,
has no direct classical meaning, but has the numerical advantage
that it is a field in 3 instead of 6 variables. Products of two
Psi terms commute with each other, hence have a direct classical
interpretation. Indeed, classically there is an electron density
field W(x,p) given by the Wigner transform of Psi(x)Psi(y)^*,
where Psi(x) is the classical Grassmann field occuring in the
Lagrangian, satisfying a Dirac equation with an electromagnetic
interaction added. This field W(x,p) is measurable and plays a role
in semiconductor modeling. (In the definition of the Wigner transform,
a second hbar appears, a remnant of second quantization. If one
moves this to zero, too, the description in terms of Psi is no longer
possible, and one gets instead a Vlasov equation for W.)
Thus the classical limit of the standard model is a mathematically
well-defined theory, while the quantum version is only perturbatively
defined, which means, it is mathematically undefined - even for QED.
Nevertheless, the renormalization prescription make at least the
coefficients of the asymptotoc series in hbar well-defined, which is
what particle physicists use to extract approximate physical
information.
In this relaxed sense, the quantum standard model is also well-defined.