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An Overview of Interacting Quantum Field Theories, Renormalizability of Quantum Field Theories

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PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 Dr. Anosh Joseph, IISER Mohali LECTURE 01 Monday, January 06, 2020 Topics: An Overview of Interacting Quantum Field Theories, Renormalizability of Quantum Field Theories. An Overview of Interacting Quantum Field Theories Free Field Theories Among the set of all possible quantum field theories that do not contain interactions or non- linearities, two kinds of field theories immediately stand out: one involving the Klein-Gordon field and the other containing the Dirac field. Let us briefly go though them, refreshing what we have learned in the previous course, QFT 1. The Klein-Gordon Field The Klein-Gordon field is a scalar field φ with mass m. The Lagrangian of this field theory is 1 1 L = (@ φ)2 − m2φ2: (1) 2 µ 2 The kinetic part of the above Lagrangian can be split into the form 2 2 2 (@µφ) = φ_ − (rφ) : (2) From the above Lagrangian we get the following equation of motion for the scalar field @2 − r2 + m2 φ = 0 =) @µ@ + m2 φ = 0: (3) @t2 µ The above equation is known as the Klein-Gordon equation. PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 Introducing the canonical momentum density conjugate to φ(x) π(x) = φ_(x) (4) we can write down the Hamiltonian Z H = d3x H Z 1 1 1 = d3x π2 + (rφ)2 + m2φ2 : (5) 2 2 2 These three terms in the above Hamiltonian represent the energy cost of moving in time, the energy cost of shearing in space and the energy cost of having the field around at all. The Klein-Gordon equation admits plane-wave solutions φ(x) ∼ e±ipx: (6) It represents arbitrary number of particles, each evolving independently. We can expand the scalar field in a series of plane waves, and ladder operators Z 3 d p 1 −ipx y ipx φ(x) = 3 p ape + ape ; (7) (2π) 2Ep @ π(x) = φ(x): (8) @t The creation and annihilation operators obey the commutation relations y 3 (3) [ap; aq] = (2π) δ (p − q); (9) [ap; aq] = 0; (10) y y [ap; aq] = 0: (11) The Feynman propagator is Z 4 d p −ip·(x−y) DF (x − y) = DeF (p) e (2π)4 Z d4p i = e−ip·(x−y): (12) (2π)4 p2 − m2 + i We can define this as DF (x − y) ≡ h0jT φ(x)φ(y)j0i; (13) where T denotes the “time-ordering". Upon using the step function, we can write the time ordered product as h0jT φ(x)φ(y)j0i = θ(x0 − y0)h0jφ(x)φ(y)j0i + θ(y0 − x0)h0jφ(y)φ(x)j0i: (14) 2 / 9 PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 We can also consider a complex valued scalar field, ' = φ1 + iφ2 (15) and with φ1 and φ2 denoting real scalar fields. We have the Lagrangian 2 2 2 L = j@µ'j − m j'j : (16) The equation of motion for this Lagrangian is again the Klein-Gordon equation µ 2 @ @µ + m ' = 0; (17) µ 2 ∗ @ @µ + m ' = 0: (18) The field '∗ can be treated as the anti-particle of ' and vice versa. The Dirac Field Let us consider the Dirac field of mass m. The Lagrangian is µ L = (iγ @µ − m) ; (19) where ≡ yγ0: (20) The Euler-Lagrange equation for gives the Dirac equation µ (iγ @µ − m) (x) = 0: (21) The Euler-Lagrange equation for gives the same equation, in Hermitian-conjugate form µ −i@µ (x)γ − m (x) = 0: (22) From the Lagrangian we see that the canonical momentum conjugate to is i y. The Hamil- tonian is Z H = d3x (−iγ · r + m) Z = d3x y −iγ0γ · r + mγ0 : (23) 3 / 9 PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 The fields can be expanded in a plane wave basis Z d3p 1 X (x) = as us(p)e−ipx + bsvs(p)eipx ; (24) (2π)3 p p p 2Ep s Z d3p 1 X (x) = asyu¯s(p)eipx + bsyv¯s(p)e−ipx : (25) (2π)3 p p p 2Ep s The creation and annihilation operators obey the anti-commutation relations r sy r sy 3 (3) rs fap; aq g = fbp; bq g = (2π) δ (p − q)δ :; (26) with all other anti-commutators equal to zero. The Feynman propagator is Z 4 d p −ip·(x−y) SF (x − y) = SeF (p) e (2π)4 Z d4p i(p= + m) = e−ip·(x−y): (27) (2π)4 p2 − m2 + i We can define the propagator as a time ordered product of fields SF (x − y) ≡ h0jT (x) (x)j0i: (28) We have 8 < (x) (y) if x0 > y0; T (x) (y) = (29) :− (y) (x) if y0 > x0: Turning on Interactions The field equations given above, the Klein-Gordon equation and the Dirac equation, are linear equations. There are no interactions. What this is telling us that the individual particles of the theory exist in their isolated modes. That is, they are not aware of the existence of each other, and particles of other species. However, the formalism of free theory is extremely important. The free theory forms the basis for doing perturbative calculations in the interacting theory. We need to turn on interactions to model the real world. We can add some interaction terms to the Lagrangian. However, these terms must respect causality and Lorentz invariance. The terms describing the interactions will be of the form Z 3 Hint = d x Hint [φ(x)] Z 3 = − d x Lint [φ(x)] : (30) 4 / 9 PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 Imposing the condition that the interaction Lagrangian is local, we have [Lint(x); Lint(y)] = 0; (31) for space-like separation. We will also restrict ourselves to theories in which the interaction terms contain only functions of fields, not their derivatives. We can treat them using a perturbative expansion when interaction strength (or coupling pa- rameter) is weak. Let us look at the three commonly occurring interacting theories. The Phi-Fourth Theory The Lagrangian is 1 1 λ L = (@ φ)2 − m2φ2 − φ4; (32) 2 µ 2 4! where λ, a dimensionless coupling constant, is a measure of the strength of the interaction. Note that if we consider a φ3 interaction, then the energy would not be positive definite. The phi-fourth theory is the simplest of all interacting quantum field theories. The self- interaction of the Higgs field in the standard electroweak theory contains this term. The phi-fourth theory also arises in statistical mechanics. The equation of motion for phi-fourth theory is λ (@2 + m2)φ = − φ3; (33) 3! which is a non-linear equation. Quantum Electrodynamics The Lagrangian is LQED = LDirac + LMaxwell + Lint 1 = i@= − m − (F )2 − e γµ A : (34) 4 µν µ In the above Aµ is the electromagnetic vector potential, Fµν = @µAν − @νAµ is the electromag- netic field tensor, and e = −|ej is the electron charge. If we want to describe a fermion of charge Q, then we need to replace e with Q. If we want to describe several species of charged particles at once, then we simply need to duplicate LDirac and Lint for each additional particle species. We can write QED Lagrangian more simply as 1 L = (iD= − m) − (F )2; (35) QED 4 µν 5 / 9 PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 where Dµ is the gauge covariant derivative Dµ ≡ @µ + ieAµ(x): (36) The QED Lagrangian is invariant under the local (or spacetime dependent) gauge transformation (x) ! eiα(x) (x); (37) (x) ! e−iα(x) (x); (38) 1 A ! A − @ α(x): (39) µ µ e µ That is, the theory has a symmetry under phase rotations. It is the U(1) gauge symmetry. The equation of motion for fermion is (iD= − m) (x) = 0: (40) This is the Dirac equation coupled to the electromagnetic field by the minimal coupling pre- scription. That is, @ ! D. The Euler-Lagrange equation for the gauge field is µν ν @µF = e γ = ejν: (41) These are the inhomogeneous Maxwell’s equations, with the current density jν = γν : (42) The Yukawa Theory The Lagrangian is LYukawa = LDirac + LKG − g φ. (43) The above Lagrangian is similar to QED but with the photon replaced by a scalar particle φ. Yukawa originally invented this theory to describe nucleons ( ) and pions (φ). The Standard Model contains Yukawa interaction terms as the couplings of the scalar Higgs field to quarks and leptons. Most of the free parameters in the Standard Model are Yukawa coupling constants. Renormalizability of Quantum Field Theories We can ask the question: Is the list of interactions infinite? 6 / 9 PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 Maybe we could write the interactions for the scalar field theory as λ L = n φn: (44) int n! However, one simple and reasonable axiom comes to our help. It eliminates all but a few of the possible interactions. That axiom is that the theory be renormalizable. In quantum field theories we need to introduce an energy cutoff Λ in the calculations - in the Feynman diagrams involving internal loops. That is, we need to cutoff the integral at some finite momentum Λ. At the end of the calculation we take the limit Λ ! 1, hoping that physical quantities turn out to be independent of Λ. If this is indeed the case, then the theory is said to be renormalizable. Suppose we have interactions where the coupling constants have the dimensions of mass to some negative power.
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