PROBLEM SET 2
Due Friday, February 5, 1999
Reading: Jackson, Secs. 12.3-12.10
Problems: Jackson 11.13, 11.16(a), (b); LD4, LD5, LD6
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11.16: This is a problem on the use of invariance arguments to generalize the expressions for covariant quantities obtained in a particular Lorentz frame to an arbitrary Lorentz frame. The result in (a) may be clearer physically if the (U·J)U in the expression given is put on the righthand side of the equation and reexpressed in terms of the charge density in the rest frame of the wire. |
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Particle distributions in high-energy collider detectors are commonly
plotted in terms of  and the azimuthal angle
 around the beam direction. The maximum energy of pions
produced in the Fermilab proton-antiproton collider is 900 GeV, corresponding
to a true rapidity  equal to 9.5, with most particles
at considerably small values. While the psuedorapidity diverges for
vanishing transverse momentum (small angles),
the approximation 
breaks down when the transverse momentum is small
relative to the particle mass. The difference is generally not important
practically.
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This problem illustrates the complications that arise in transforming
coordinate systems because of the Wigner rotation. The effect of
two successive boosts, from a frame K to K' and then to K'',
is not the same as the effect of a boost directly from K to K'', but
differs from the latter by the Wigner rotation. The representation of the
Lorentz transformations in terms of 2 × 2 matrices applies to
transformations of coordinates, and builds the rotation in
automatically.
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(a): The expressions you will obtain in (a) will look unfamiliar
because all quantities are expressed in terms of half-angles and the
rotation is built in. The usual
expressions for the transformation of a four vector x can be recovered by
multiplying the matrix which represents x left and right by
the transformation matrix and its Hermitean conjugate. However, we are
concerned here with the properties of the transformations themselves so
that step is unnecessary. Also, it's easy to get used to half angles!
(b): To compare your result for the extra rotation angle in (b) with the expression for the Thomas angle given in Jackson, you will need to rewrite your result in terms of quantities defined in the original Lorentz frame K, that is, the rapidities of the boosts from K to K' and directly from K to K'', and the (small) angle between the directions of the two boosts. The relations for hyperbolic triangles will be useful. Work to first order in the small quantities. |
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This method was used to determine the parity of the  ,
but with a variation: A virtual high-energy photon can convert to an
electron-positron pair (a "Dalitz pair") with a probability of about 1/170.
The charged particles tend to lie in the plane of E (why?), so by
looking at the correlation of the planes of double Dalitz pairs in the
decay  , one can measure the correlation of the
photon polarizations and determine the parity of the
.
For some original references see C.N. Yang, Phys. Rev. 77, 242, 722
(1950); N. Kroll and W. Wada, Phys. Rev. 98, 1355 (1955); and for
the original measurement, R. Plano et al., Phys. Rev. Lett.
3, 525 (1959). |
Send comments or questions to: ldurand@theory2.physics.wisc.edu
© 1997, 1998, 1999 Loyal Durand