Σ Model, from not obvious to obvious?

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SUMMARY

The discussion centers on Georgi's assertion in his book regarding the invariance of the Σ model under specific transformations. The transformation equations provided, $$\delta \sigma = \epsilon_5^a \pi_a$$ and $$\delta \pi_a = - \epsilon_{abc}\epsilon^b\pi_c - \epsilon_5^a \sigma$$, illustrate how the fields σ and π_a behave under these transformations. While Georgi claims that the form of $$\Sigma = \sigma + i\tau_a\pi_a$$ is preserved, participants express confusion about the clarity of this invariance without performing explicit calculations. The consensus is that the term "obvious" refers to the potential for visual recognition of invariance rather than immediate comprehension.

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  • Understanding of the Σ model in theoretical physics
  • Familiarity with transformation equations in field theory
  • Knowledge of the notation and concepts related to σ and π_a fields
  • Basic grasp of invariance principles in physics
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The discussion is beneficial for theoretical physicists, graduate students in particle physics, and researchers focusing on field theory and model invariance.

PhyAmateur
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Georgi in his book, for reference: http://www.people.fas.harvard.edu/~hgeorgi/weak.pdf he says that it is not obvious that the form of $$\Sigma = \sigma + i\tau_a\pi_a$$ is preserved under (2.6.6) $$ \delta \Sigma = i \epsilon_L^aT_a \Sigma - i \Sigma \epsilon_R^aT_a$$

But then he said, "But it is true, and you can work out by explicit calculation the transformations of the $\sigma$ and $\pi _a$ fields":

$$\delta \sigma = \epsilon_5^a \pi_a$$
$$\delta \pi_a = - \epsilon_{abc}\epsilon^b\pi_c - \epsilon_5^a \sigma$$

But I have no idea how after what he wrote it became obvious? It is still not obvious to me how it is preserved though he saying it must be.
 
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I don't think he wants to say that it becomes obvious but simply that if you actually do the calculation with the change of the ##\sigma## and ##\pi_a## fields that you listed before you'll find that it is actually invariant. By ''obvious'' he probably means that can be seen by eye, without doing the explicit calculation.