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I usually don't read papers on philosophy of quantum field theory, but this one is really good: http://philsci-archive.pitt.edu/8890/
In particular, the prelude which I quote here is a true gem:
"Once upon a time there was a community of physicists. This community be-
lieved, and had good reason to believe, that quantum mechanics, not classical
mechanics, was the right framework in which to do physics. They also had
a good understanding, at the classical level, of the dynamics of solid bodies
(vibrations in crystals, for instance): they knew, for example, that some such
bodies could be analysed using Lagrangians like
L =φ^2 − (∇φ)^2 + (higher terms) (1)
where φ(x) is the displacement of the part of the crystal which is at position x
at equilibrium.
But the physicists were sad, because they knew nothing at all about the
microscopic structure of matter, and so they did not have a good quantum
theory of vibrations in crystals or of other solid-matter dynamics.
So one day, they set out to quantize their classical theories of solid mat-
ter. At first, they tried to do it naively, by putting the classical theory into
Hamiltonian form and replacing classical observables with self-adjoint opera-
tors. This worked quite well until the higher-order terms in (1) were included.
But when the physicists tried to include those higher order terms, the theory
became mathematically very badly behaved — all the calculations contained
integrals that diverged to infinity.
Soon the physicists discovered that they could extract working calculational
results if they just assumed that displacements couldn’t vary on arbitrarily
short lengthscales. This amounted to “cutting off” the range of integration in
the divergent integrals, so that they got a finite result. When they did their
calculations this way, the answers agreed very well with experiment.
But the physicists were still sad. “It’s ad hoc”, they said. “It’s inelegant”,
they lamented. “It conflicts with the Euclidean symmetries of solid matter”,
they cried.
So they went back to basics, and looked for an axiomatised, fully rigorous
quantum theory, with displacements definable on arbitrarily short lengthscales
and with exact Euclidean symmetries.
And to this day, they are still looking."
In particular, the prelude which I quote here is a true gem:
"Once upon a time there was a community of physicists. This community be-
lieved, and had good reason to believe, that quantum mechanics, not classical
mechanics, was the right framework in which to do physics. They also had
a good understanding, at the classical level, of the dynamics of solid bodies
(vibrations in crystals, for instance): they knew, for example, that some such
bodies could be analysed using Lagrangians like
L =φ^2 − (∇φ)^2 + (higher terms) (1)
where φ(x) is the displacement of the part of the crystal which is at position x
at equilibrium.
But the physicists were sad, because they knew nothing at all about the
microscopic structure of matter, and so they did not have a good quantum
theory of vibrations in crystals or of other solid-matter dynamics.
So one day, they set out to quantize their classical theories of solid mat-
ter. At first, they tried to do it naively, by putting the classical theory into
Hamiltonian form and replacing classical observables with self-adjoint opera-
tors. This worked quite well until the higher-order terms in (1) were included.
But when the physicists tried to include those higher order terms, the theory
became mathematically very badly behaved — all the calculations contained
integrals that diverged to infinity.
Soon the physicists discovered that they could extract working calculational
results if they just assumed that displacements couldn’t vary on arbitrarily
short lengthscales. This amounted to “cutting off” the range of integration in
the divergent integrals, so that they got a finite result. When they did their
calculations this way, the answers agreed very well with experiment.
But the physicists were still sad. “It’s ad hoc”, they said. “It’s inelegant”,
they lamented. “It conflicts with the Euclidean symmetries of solid matter”,
they cried.
So they went back to basics, and looked for an axiomatised, fully rigorous
quantum theory, with displacements definable on arbitrarily short lengthscales
and with exact Euclidean symmetries.
And to this day, they are still looking."