Given a symmetry of a classical Lagrangian without sources we can always extend

Given a symmetry of a classical lagrangian without

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Given a symmetry of a classical Lagrangian without sources, we can always extend it to a symmetry of the Lagrangian including sources, by requiring the sources to transform appropriately. The interesting question is, can we always define a corresponding quantum theory that exhibits the symmetry. The answer is — sometimes! When it is possible to find a regularization and renormalization scheme that preserves the symmetry, then the symmetry can be extended into the quantum theory, but sometimes this is not possible. We will discuss important examples of the “anomalies” that can occur in quantum field theory to prevent the realization of a classical symmetry later. For now we will ignore this subtlety, and discuss symmetry in the language we have developed to discuss field theory, without asking whether any particular symmetry is consistent with our DRMS scheme. 1b.1 Noether’s Theorem – Field Theory Consider, then, a quantum field theory with N real scalar fields, φ , and real sources s and with K fermion fields, ψ , and sources, η , and Lagrangian 1 L 0 ( φ, ∂ μ φ, ψ, ∂ μ ψ ) + s T φ + η ψ + ψ η , (1b.1.1) Suppose that L 0 is invariant under a global internal symmetry group, G , as follows: φ D φ ( g - 1 ) φ , ψ L D L ( g - 1 ) ψ L , ψ R D R ( g - 1 ) ψ R , (1b.1.2) where g is and element of G and the D ’s are unitary representations of the symmetry group. Note that because the φ fields are real, the representation D φ must be a “real” representation. That is, the matrices, D φ , must be real. Such a symmetry can then be extended to include the sources as 1 We use L 0 for the Lagrangian without any sources. 21
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Weak Interactions — Howard Georgi — draft - March 25, 2010 — 22 well, as follows: s D φ ( g - 1 ) s , η R D L ( g - 1 ) η R , η L D R ( g - 1 ) η L . (1b.1.3) Note the switching of L R for the fermion sources, because the fermions source terms are like mass terms that couple fields of opposite chirality. Now if the regularization and renormalization respect the symmetry, the vacuum amplitude, Z ( s, η, η ) will also be invariant under (1b.1.3). Generators — T a It is often convenient to use the infinitesimal version of (1b.1.2) and (1b.1.3), in terms of the generators, T a , δφ = i a T a φ φ , δψ L = i a T a L ψ L , δψ R = i a T a R ψ R , δs = i a T a φ s , δη R = i a T a L η R , δη L = i a T a R η L . (1b.1.4) The T a j for j = φ , L and R , are generators of the representation of the Lie algebra of G , corre- sponding to the representations, D j ( g - 1 ) = e i a T a j . (1b.1.5) A sum over repeated a indices is assumed. Often we will drop the subscript, and let the reader figure out from the context which representation we are discussing. Gauge Symmetry In order to discuss the conserved Noether currents associated with a symmetry of the quantum field theory as composite operators, it is useful to convert the global symmetry, (1b.1.4), into a gauge symmetry, that is a symmetry in which the parameters can depend on space-time, a a ( x ) . (1b.1.6) To do this, we must introduce a set of “gauge fields”, which in this case will be classical fields, like the other sources. Call these fields,
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