Index Concordia

Relativity – Week 01

Posted in Relativity by Index Guy on September 9, 2007

After going through an outline of the course, professor Siegel started with a very brief review of the most important notions of special relativity: the energy-momentum relation, notation and two interesting frames of reference. For a wimp-of-a-particle (no electric charge, no intrinsic angular momentum, only mass), the most general thing one can write about it is the statement

p^2 + m^2 = 0.

This took me a while to see why, but then I realized that in this equation we mean the magnitude of the four-momentum, which is a Lorentz scalar. One of the frames the professor mentioned is the lightcone frame, which is obtained by setting

\sqrt{2} V^{+} = V^{0} + V^{1} and \sqrt{2} V^{-} = V^{0} - V^{1}.

This frame is interesting since when one compares the relativistic and non-relativistic expressions for the energy of a point particle, the relativistic version in the lightcone frame looks very similar to the non-relativistic version, with the mass being one component of the momentum for example. This helps generalizing some manipulations done non-relativistically into relativistic ones.

The other frame mentioned was the null frame, which is obtained by taking the lightcone frame and further defining

\sqrt{2} V^{t} = V^{2} - iV^{3} and \sqrt{2} \bar{V}^{t} = V^{2} + iV^{3}.

One of the conventions was mentioning the classification of the four-vector magnitude into timelike, lightlike and spacelike. When applied to the four-momentum we get massive particles have timelike momentum, massless particles have lightlike momentum and tachyons have spacelike momentum. But tachyons are responsible for killing one’s grandfather before one is born, so we do not like tachyons.

Next we moved to discussing the elements of the Poincaré group, namely Lorentz transformations, translations (the former and the later being continuous transformations, built up from infinitesimal ones) and the discrete transformations (CPT).

Time reversal is sort of easy to understand why it is discrete since no continuous transformation will change the class of a four-vector (i.e. timelike to lightlike to spacelike to lightlike to timelike across the lightcone). Parity is understandable once one considers one or three spatial dimensions. This all got me thinking: if I have a spacetime with a number of dimensions, then if I consider a subspace, parity for that subspace will be continuous. I guess this is related to the fact that the subspace is embedded in a higher dimensional space.

Just like one thinks of Lorentz transformations as preserving the metric tensor, parity \mathcal{P} and time reversal \mathcal{T} together can be thought of as the transformations that preserve the Levi-Civita tensor.

Charge conjugation \mathcal{C} is related to a change of sign in the spacetime interval, and hence of momentum, since

p = m \displaystyle\frac{dx}{ds}.

From this operation one can distinguish particle and antiparticles according to the direction which each move through time (forward or backward). The “charge” comes from considering the coupling to an external electromagnetic field

(p + qA)^2 + m^2 = 0.

We see that if momentum changes, then the charge also needs to change in order for this expression to remain the same. Charge conjugation is in the worldline the analog of \mathcal{PT} in the full spacetime.

All these are some of the symmetries of a massive particle. When one treats massless particles, the conformal group appears. Professor Siegel mentioned many reasons why it is worthwhile to study the conformal group, even though it is found broken in nature. Among other things

  • the conformal group is important for the construction of theories (like starting with massless particles and then adding some symmetry breaking or some mass term);
  • it also gives some insight regarding the solutions of theories;
  • high energy behavior – particle energy is much larger than its mass so conformal approach is valid;
  • self-duality

Conformal invariance involve transformations that take the spacetime interval and scales it by a function of the coordinates,

dx^{\prime 2} = \xi (x) dx^{2}.

The symmetry group of conformal invariance is SO(D,2), which contrast with that of the Lorentz transformations (continuous) being O(D-1, 1) and Lorentz and discrete (\mathcal{CPT}) being SO(D-1, 1).

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