# Einstein and String Theory

• 1905. Einstein's formulation of special relativity.
• 1906. Einstein's calculation of the specific heat of solids.
• 1912. Debye's calculation of the specific heat. The first successful string model.
• 1962. Regge Poles and hadron spectra.
• 1973. The first phase of string model.
• 1985. The second phase of string model -- String Theory!
• 2006. After so many years, where is String Theory going?
1. When is the theory going to produce physical results?
2. If this question is too difficult to answer, where does the present form of String Theory stand in the history of physics?

• In 1927, quantum mechanics was formulated, after Einstein's special relativity (1905) and Debye's calculation (1912).
1. The question is whether quantum mechanics is consistent with special relativity.
2. Quantum Field theory with the Lorentz-covariant S-matrix was a great step toward combining quantum mechanics with relativity.
3. String Theory has the same purpose unless one invents a new quantum mechanics and/or a relativity which are drastically different from those known to us (with a new Einstein and/or a new Heisenberg).
4. What is the difference between strings and quantum fields?

• Quantum mechanics is based on wave-particle duality. When a particles moves, we should describe it also as a wave propagating in a given direction. If a particle is confined within a specified region, like an electron in the hydrogen atom, we need a standing wave with appropriate boundary conditions.
1. It is not difficult to make plane waves compatible with Lorentz covariance. This is the reason why it was possible to construct the present form of quantum field theory with the scattering matrix and Feynman diagrams.
2. How about bound states with standing waves? If a standing wave consists of two running waves moving in opposite directions, we have to deal with boundary conditions. Consider, for example, a one-dimensional particle confined between two infinite walls. How do those walls appear to an observer in a different Lorentz frame? The answer to this question is very simple. We do not know.

## 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

where the good and bad wave functions are given in the following figure.

We expect the bound state wave function be localized. The S matrix contains both incoming and outgoing waves. When we make analytic continuation to bound-state poles, the momentum becomes purely imaginary, and the outgoing wave decreases exponentially, while the incoming wave increases. There are no procedures to impose the localization boundary condition in the S matrix formalism.

• This is the problem quantum field theory could not solve. Indeed, R. P. Feynman, the inventor of Feynman diagrams, said Feynman diagrams are not effective in understanding bound states in 1970. In other words, quantum field theory cannot deal with standing waves.

Y. S. Kim
Department of Physics
University of Maryland
(2006.2.15)

copyright@2006 by Y. S. Kim, unless otherwise specified.

## 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.