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S14a. Theoretical challenges close to experimental data
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Many theoretical physicists seem to think that the only worthwhile
challenges in theoretical physics can be found at >TeV energies.
But, (un?)fortunately, there are challenges, as difficult and
as exciting, in the realm of normal energies, deep in the limits
of the unknown (as regards understanding), and far more relevant
in my opinion.
The manpower and money invested in the exotic realms of nature
at very large energies would be much better spent on these challenges
closer to experimental data..
For example, finding a consistent nonperturbative setting for
QFT, or giving a meaning to the concept of the ground state of
a Helium atom in quantum electrodynamics (extended by a field
describing the nuclei).
I have not seen a single field-theoretic treatment of Helium,
surely a simple system.
Helium is a bound state with well-defined asymptotic behavior,
as well-defined as a dressed electron or photon, but there is no
clear conceptual basis for this in QFT although there should be
such a concept. That's why I think it is a very important
unsolved problem.
There are papers making heuristic approximations
(see hep-ph/9612330) which give accurate predictions - cf.
Phys Rev. A 65 (2002), 032516 and Phys. Rev Lett. 84 (2000), 3274 -,
but they don't give a clue what a helium atom 'is' in QFT.
Moreover, they treat two electrons in a classical external
Coulomb field instead of a system of two electrons and a nucleus.
The current treatment of bound states in QFT (see elsewhere in this FAQ)
is a very loose patchwork of techniques borrowed from perturbative
field theory and nonrelativistic quantum mechanics that should make
every theoretician shudder. There are some beginnings in algebraic
QFT of what bound states should be, but nothing convincing on the
quantitative level.
A theory of everything should also be able to answer questions
that are well established experimentally but not understood
from the foundations.
For example, deriving the Navier-Stokes equations
for water from quantum theory is another challenge
that so far remained unmet; it has been done long ago for
dilute gases, but no one extended it to dense fluids.
There are severe difficulties to overcome, but we know both
the final result (to much better accuracy than the parameters
of the standard model) and the supposed underlying microscopic
model (unlike in quantum gravity); and the availablility of
a derivation might even have long-term engineering consequences
for predicting properties of fluids under thermodynamic conditions
where experiments are difficult or impossible.
I am not an expert in this topic, but here are some pointers to
what I have seen about the problem.
I have never seen any microscopic derivation of Navier-Stokes
for water, although this is by far the most important application.
The statistical mechanics text of Reichl derives the equations
in Chapter 14F from thermodynamics, and in Chapter 16C-F
(for dilute monatomic gases) from classical statistical physics.
Fujita, Nonequilibrium statistical mechanics, derives Boltzmann
from QM in Chapter 4.2 and Chapter 6; Navier-Stokes would
be roughly analogous (for dilute gases). Similarly for many
other books on nonequilibrium stat. mech..
Mueller/Ruggeri, Extended Thermodynamics, treat relativistic versions,
deriving them from the Boltzmann equation and from thermodynamics.
Volume 9 of Landau/Lifschitz discusses techniques for the condensed
state in general, but no derivation of Navier-Stokes.
J. Math. Phys 11 (1970), 2481 is a paper summarizing in the
introduction what was known by 1970.
Phys. Rev. D 53, 5799-5809 (1996) derives hydrodynamic equations
from quantum fields but only in a scalar phi^4 theory.
More recent related work includes
Phys. Rev. D 68, 085009 (2003)
Phys. Rev. D 64, 025001 (2001)
Phys. Rev. D 61, 125013 (2000)
Thus there is a well-trodden pathway for the dilute gas case,
and a set of tools for the condensed phase, but no synthesis
of the two.
If you find better references, please let me know.
(See also the list of unsolved problems in physics at
http://en.wikipedia.org/wiki/Unsolved_problems_in_physics
which covers many more speculative items.)