Continuous Symmetries and Conserved Currents Overview

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QFT Chapter 22: Continuous Symmetries and Conserved Currents Overview • We now turn to our next topic: symmetries… • Continuous Symmetries (ch. 22) • Discete Symmetries (ch. 23) • Non-Abelian Symmetries (ch. 24) • At the end of the part 1, we’ll address symmetry breaking, which will help us construct the standard model in part 3 Transformation of the Lagrangian • Let’s transform: • The Langrangian is therefore transformed as: • Next, we use the field equation to find that: Note that the first equality holds only if the classical field equations are satisfied. The Noether Current • Using these results, we can rewrite the infinitesimal Lagrangian as: • The Noether current is the portion in large parenthesis. • The Noether current is conserved if: • The field equations are satisfied • The Langrangian is invariant under the transformation. • The upshot of this is that we get a new conservation law: The Noether Charge • The integral of the Noether current is the Noether charge • This must be constant in time, assuming reasonable boundary conditions • By decomposing the field operators and creation/ annihilation operators, we find that: which is the number of A particles minus the number of B particles. So the A and B particles have opposite Noether charge but the same mass. Sounds like antiparticles! More on this in next chapter… Utility of the Noether Current • We can use these ideas to derive some identities: • Schwinger-Dyson • Ward These identities are rather mathematical, so I won’t state or prove them here. • If the differential change in the Lagrangian is nonzero, but rather a total divergence of a function K, there is still a conserved current: • In the case of a spacetime translation, for example, we find that the Noether current is related to the stress- energy tensor, which is highly useful. Lorentz Symmetry • An infinitesimal Lorentz Transformation is another symmetry that has a conserved current associated with it. The conserved current is: • The conserved charge is: • These are the generators of the Lorentz Group that were introduced before. • We can check this using the canonical commutation relations (see problem 22.3). .

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