Hendrik Lorentz and Paul A. M. Dirac
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| Hendrik Lorentz (1853-1928). Photo from public domain. |
In addition, Lorentz was able to write one formula applicable to both electric and magnetic forces. We shall talk about this next time.
According to Gerhard Hegerfeldt, there was a German physicist at the University of Goettingen who worked out transformation formulas assuming that the speed of sound is an invariant quantity, before Lorentz formulated his transformation law. Hergerfeldt told me that this manuscript might be in the Goettingen library.
The mathematics of Lorentz transformations was further developed by Poincaré and later by Wigner as an important branch of group theory. We call this mathematics "Lorentz group." Unlike three-by-three matrices applicable to the three-dimenional Euclidean space, the Lorentz group starts from four-by-four matrices applicable to the four-dimensional Minkowskian space.
While we can study the three-dimensional rotation group using two-by-two matrices generated by three Pauli matrices, the most gratifying aspect of the Lorentz group is that we can study this group also using two-by-two matrices generated by six Pauli-like matrices. Three of them are the Pauli matrices and the other three are the Pauli matrices multiplied by i. The group of two-by-two matrices generated by these six Pauli-like matrices is called SL(2,c). The rotation group is therefore a subgroup of SL(2,c). Other interesting subgroups include the Lorentz groups applicable to two space dimensions and one time variable.
The point is that modern physics is largely a physics of two-by-two matrices and/or harmonic oscillators, since otherwise the problems are not souluble. We know how to write down oscillator wave functions. How much do you know about two-by-two matrices? For instance, do you know how to diagonalize them? This is not a trvial problem. Indeed, we need the Lorentz group to study systematically two-by-two matrices which are everywhere in physics.
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| Light-cone coordinate system. |
When we perform a Lorentz boost, we have a tendency to think only about the hyperbola extending to infinity. Dirac's light-cone coordinate system allows us to talk about localized a space-time region being squeezed during the transformation. This allows us to formulate the concept of Lorentz-covariant space-time localization.
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Paul A. M. Dirac (1902-1984). Photo by Bulent Atalay. |
It is fun and productive to study Dirac's papers. They are like poems.
Y. S. Kim (September 5, 2006)