Abstract
In the previous part, we discussed different field theories, simple and complicated, but we never actually defined what we were talking about. Any mechanical or field theory system is characterized by its Lagrangian. Once the Lagrangian is written, everything else follows.
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Notes
- 1.
For example, for the 4-momentum \(p^\mu \), it is just the mass of the particle, \(\eta _{\mu \nu }p^\mu p^\nu = m^2\).
- 2.
Commutative groups are also called Abelian.
- 3.
The objects, as is common in mathematics, may have different natures. One can think, for example, of wolf, sheep and cabbage. Or just of three numbers 1,2,3.
- 4.
In physics, a more common choice for such parameters is the Euler angles. And aircraft engineers prefer to describe the dynamics of aircraft flight in terms of roll, pitch and yaw.
- 5.
Mathematicians call by this strange word a smooth enough multidimensional surface. The group SO(3) represents a particular manifold that is topologically equivalent to a 3-dimensional sphere \(S^3\) with opposite points identified. This manifold is compact (its volume is finite). Not all groups are compact. The Lorentz group is not.
- 6.
A remarkable observation of Lie was that one can learn almost everything about a continuous group [any group, not only SO(3)] by looking at what happens in a small neighbourhood of its unit element.
- 7.
Here the upper position of the indices carries a purely aesthetic and not a mathematical meaning.
- 8.
It is customary to denote the Lie algebras in the same way as the corresponding groups, but with lower-case letters.
- 9.
- 10.
If something or someone buzzes, it is not without a reason, as Winnie the Pooh once wisely mentioned.
- 11.
Note that, when the rotation angle is changed continuously from 0 to \(2\pi \), the corresponding unitary matrix is changed from \(\mathbbm {1}\) to \(-\mathbbm {1}\) (it is like going onto another sheet of a Riemann surface). In other words, the spinor wave function changes sign after the \(2\pi \) rotation. This is a known quantum mechanical effect. It was confirmed in experiment.
- 12.
The negative sign in the upper component on the right is not a mistake. It appears under the standard convention \(\Psi _+ \ =\ \hat{J}^+ \Psi _0/\sqrt{2} \ = \ (\hat{J}^+)^2 \Psi _-/2\).
- 13.
Probably, that is why this representation was called adjoint. In French this word has a meaning of “attached” or “united”, and one can imagine that a Lie group attaches as a representation its own Lie algebra to make them both happy. Evidently, the adjoint representation exists for any Lie group, not only for SU(n).
- 14.
The presence of complex transformation parameters is a complication associated with the fact that the Lorentz group is not compact. In a similar construction for the compact SO(4) group, all the parameters are real.
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Smilga, A. (2017). Groups and Algebras. In: Digestible Quantum Field Theory. Springer, Cham. https://doi.org/10.1007/978-3-319-59922-9_6
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DOI: https://doi.org/10.1007/978-3-319-59922-9_6
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