To be specific, this means that in four dimensions, we will have an action S = Z d 4 x 1 2 ( ∂ μ φ ) 2 - 1 2 c m Λ 2 φ 2 - c 3 Λ φ 3 - c 4 φ 4 - c 5 Λ φ 5 - · · · - g 3 Λ φ ( ∂ μ φ ) - g 4 Λ 2 φ 2 ( ∂ μ φ ) - · · · Finally, let us consider conducting an experiment on this system at energies E Λ, corresponding to distance R a . Experimental results must ultimatelybe phrased in terms of dimensionless quantities. Note that all terms beyond the first four are proportional to powers of 1 / Λ. This means that the energy scale of the experiment must make up for these negative powers of Λ, or in other words, all terms beyond the first four contribute to experimental observables in a way that is suppressed by a power of E exp Λ 1 (2.7.16) This means that at long distances and low-energies, to a good approximation we can ignore all of the terms with negative powers of Λ! Thus when we study a single scalar quantum field in a Lorentz invariant universe, there is a universal theory with action S low - energy ≈ Z d 4 x 1 2 ( ∂ μ φ ) 2 - 1 2 c m Λ 2 φ 2 - c 3 Λ φ 3 - c 4 φ 4 There are only four possible terms, a kinetic term, a mass term, and two interactions. Those are all of the possibilities! Although the Higgs field interacts with other fields in the Standard Model, when we isolate it, this is its full Lagrangian (in fact it’s even simpler, basically because h → - h is 35
a symmetry, so c 3 = 0). The hierarchy problem is the fact that if Λ is really big (say at the Planck scale, 10 19 GeV ) then we must have c m ≪ 1 so that the Higgs has a 126 GeV mass. We will refine some of these statements when we learn about renormalization and renormalization flows, which incorporate effects from quantum mechanics, and you’ll learn to appreciate them more when we do a larger variety of (quantum) computations, but the basic ideas here are robustly true and extremely useful for understanding why QFTs are so simple and universal when they are regarded as Effective Field Theories , or long-distance, low-energy descriptions of physics. 2.8 Interactions in Classical Field Theory with a View Towards QFT 2.8.1 Coulomb’s Law in Our Formalism We will mostly be discussing scalar fields, but to make it clear where we’ll get eventually, it’s nice to see how electrodynamics works as a field theory. We have a Lagrangian density L = - 1 4 F 2 μν - A μ J μ (2.8.1) where J μ is the electromagnetic current. For static charges J 0 ( x ) = ρ ( x ) is the charge density, while ~ J = 0. We can write this as L = - 1 2 ( ∂ μ A ν ) 2 + 1 2 ( ∂ μ A ν ) 2 - A μ J μ (2.8.2) Now we need to vary we respect to A μ . The book does this formally via the Euler-Lagrange equation, more informally note that 0 = δS = Z d 4 x [ - ( ∂ μ A ν )( ∂ μ δA ν ) + ( ∂ μ A μ )( ∂ ν δA ν ) - J ν δA ν ] (2.8.3) = Z d 4 x [( ∂ μ ∂ μ A ν ) δA ν - ( ∂ ν ∂ μ A μ ) δA ν - J ν δA ν ] (2.8.4) so we find the equation of motion ∂ μ F μν = ∂ μ ∂ μ A ν - ∂ ν ∂ μ A μ = J ν (2.8.5) which are Maxwell’s equations in gauge invariant, manifestly relativistic form. This already looks like (where ≡ ∂ μ ∂ μ ) A
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