Running Waves
- The present form of quantum field theory leads to the Lorentz-covariant
scattering matrix or the S matrix, from which we can calculate scattering
amplitudes and cross sections. In addition, it leads to a very powerful
tool of Feynman diagrams. Quantum electrodynamics has been very successful
in producing some of numbers which can be measured experimentally.
For other
scattering processes, it is now the rule to draw Feynman diagrams first.
Those diagrams sometimes produce correct numbers and sometimes not. If
not, our excuse is that we cannot calculate all the terms in perturbation
series of the S matrix.
Instead of relying solely on the perturbation expansion, we can choose
selected sets of Feynman diagrams. We call these procedures models.
There have been some successful models. For instance, the Lee model is
a soluble model. The Bogoliubov transformation in superconductivity is
also a soluble model. Both of these models can be reduced to a problem
of coupled oscillators.
- Toward the end of 1950s, the S matrix became the central issue in
physics. The question was whether it is possible to derive the results
which can be observed in high-energy laboratories, from analytic properties
of the S matrix. If you are sufficiently old, you will remember the words
dispersion relations, Regge poles, N over D methods, strip approximation,
bootstrap dynamics. The philosophy behind all these ideas was that the
S matrix is the starting point for everything in physics.
- Within this framework of physics, in 1964, Roger Dashen and Steven Frautschi
calculated the neutron-proton mass difference from the belief that the mass
difference comes from an electromagnetic perturbation. Indeed, it was once
believed to be a history making calculation.
However, if we translate their calculation into the Schroedinger picture of
quantum mechanics, their formula becomes
(φ good , δV φbad) ,
where the good and bad wave functions are given in the following figure.
copyright@2006 by Y. S. Kim, unless otherwise specified.
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Standing Waves
- If String Theory is to be different from field theory, and if the theory is
going to address internal space-time structure of relativistic particles,
it should be based on standing waves. Then there are the following
questions.
- Quantum field theory starts with plane waves, and it is very
easy to provide Lorentz-covariant formulation of those waves. How about
standing waves? It is possible, in a given Lorentz frame, to produce a
standing wave by superposing
two running waves propagating in opposite directions. However, is this
picture of standing wave valid to an observer in a different Lorentz frame?
- In order to describe a standing wave, we need boundary conditions. The
running wave should be reflected at the boundary and should come back in
the opposite direction to be superposed with the original wave. The problem
is that the boundary condition specified in terms of purely space-like
variables in one Lorentz frame should be stated also in terms of the
time-like variable. The boundary condition in terms of the time variable
is still a very strange concept.
If we do not know the direction of where we are going, one way to deal with
this problem is to look at the history. Let us go to the history of Einstein
and Debye (1906-12). Einstein used harmonic oscillators to understand the
specific heat of solids in 1906. Then Debye used strings to improve
Einstein's result. Strings in solid are standing waves.
- Unlike the Debye model, we are dealing with a covariant model in String
Theory. In the Debye model, we use standing waves within a finite box. How
would this box look to an observer on a bicycle? Are the running waves in
opposite directions obey the superposition principle?
- Let us go back to the Einstein-Debye case again. String Theory
originally grew out of Regge poles and Regge trajectories. However, did
you know that Regge trajectories are basically degeneracies of
three-dimensional harmonic oscillators? Perhaps not,
but Feynman
knew this. Therefore, the Regge-String connection is like the
Einstein-Debye connection.
- In the Regge-String case, we are dealing with a finite number of
constituent particles, unlike Debye's solid. Thus, the harmonic oscillator
is closer to the real world. Then the question is whether the harmonic
oscillator can be made Lorentz-covariant?
- We are quite familiar with the procedure of constructing representations
of the O(3) rotation group using spherical harmonics. Then, is it possible
to construct representations of the Poincaré group? The answer to
this question is YES.
- Indeed, special relativity is largely a physics of the Lorentz group
and quantum mechanics is largely a physics of harmonic oscillators. If we
construct representations of the Poincaré group in terms of harmonic
oscillators, we are combining quantum mechanics and special relativity,
with Lorentz-covariant boundary conditions.
Paul A. M. Dirac spent much of his time to understand harmonic
oscillators within the Lorentz-covariant world.
- OK! Is the covariant oscillator purely a mathematical formalism,
science fiction, or capable of describing the real world?
Among the many phenomena observed in high-energy laboratories, the most
remarkable result is a resolution of the quark-parton puzzle. If a
hadron is on your table, it looks like a quantum bound state of two or
three quarks. If you are on a bicycle, the bound state should appear
deformed. If the bicycle speed is sufficiently high, the hadron will
look like a collection of free particles with a wide-spread momentum
distribution. These particles are known to us Feynman's partons.
Indeed, the Lorentz-covariant oscillators can describe the quark model
and the parton model as one covariant picture. For detailed explanation,
you can go to
this webpage.
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