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Effective Quantum Field Theories Thomas Mannel Theoretical Physics I (Particle Physics) University of Siegen, Siegen, Germany

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Effective Quantum Field Theories Thomas Mannel Theoretical Physics I (Particle Physics) University of Siegen, Siegen, Germany Generating Functionals Functional Integration Renormalization Introduction to Effective Quantum Field Theories Thomas Mannel Theoretical Physics I (Particle Physics) University of Siegen, Siegen, Germany 2nd Autumn School on High Energy Physics and Quantum Field Theory Yerevan, Armenia, 6-10 October, 2014 T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals Functional Integration Renormalization Overview Lecture 1: Basics of Quantum Field Theory Generating Functionals Functional Integration Perturbation Theory Renormalization Lecture 2: Effective Field Thoeries Effective Actions Effective Lagrangians Identifying relevant degrees of freedom Renormalization and Renormalization Group T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals Functional Integration Renormalization Lecture 3: Examples @ work From Standard Model to Fermi Theory From QCD to Heavy Quark Effective Theory From QCD to Chiral Perturbation Theory From New Physics to the Standard Model Lecture 4: Limitations: When Effective Field Theories become ineffective Dispersion theory and effective field theory Bound Systems of Quarks and anomalous thresholds When quarks are needed in QCD É. T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals Functional Integration Renormalization Lecture 1: Basics of Quantum Field Theory Thomas Mannel Theoretische Physik I, Universität Siegen f q f et Yerevan, October 2014 T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals Functional Integration Renormalization Contents 1 Generating Functionals Green’s Functions Connected Green’s Functions Vertex Function, Effective Action 2 Functional Integration Quantum Mechanics Quantum Field Theory Perturbation Theory 3 Renormalization One Loop Renormalization Power Counting Renormalization Theory T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals Functional Integration Renormalization Introduction Quantum Mechanics ⊕ Special Relativity = Quantum Field Theory Standard Approach: Canonical Quantization Field Operators Φ(x), conjugate momenta Π(x) with 3 ~ ~ [Φ(x); Π(y)]x0=y0 = i~δ (x − y) Use this for a (perturbative) calculation of Correlation Functions (or Green’s functions) Gn(x1; x2; :::; xn) = h0jT [Φ(x1)Φ(x2):::Φ(xn)]j0i with the ground state j0i and time ordering T . T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals Functional Integration Renormalization Scattering amplitudes S are related to Gn via the LSZ formalism Usual time-dependent perturbation theory to generate a perturbative expansion ... I assume you know this part of the story However, if we knew all the Gn, we would know everything about this particular field theory! For our purpose a different Ansatz will be pursued: Functional Integrals to generate the Gn T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals Green’s Functions Functional Integration Connected Green’s Functions Renormalization Vertex Function, Effective Action Generating Functionals A Functional is a map from a space of functions into (real) numbers J(x) ! F = F[J] Generating functional for a set Gn(x1; :::; xn) 1 X 1 Z Z [J] = d 4x d 4x :::d 4x G (x ; :::; x )J(x )J(x ):::J(x ) n! 1 2 n n 1 n 1 2 n n=0 Recover any Gn by functional differentiation δ δ Gn(x1; :::; xn) = ::: Z [J] (1) δJ(x1) δJ(xn) J=0 T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals Green’s Functions Functional Integration Connected Green’s Functions Renormalization Vertex Function, Effective Action Exercise 1: Functional derivative Show (1) use the following properties of the functional derivative: δ J(y) = δ4(x − y) δJ(x) δ δF[J] δG[J] (F[J] G[J]) = G[J] + F[J] δJ(x) δJ(x) δJ(x) T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals Green’s Functions Functional Integration Connected Green’s Functions Renormalization Vertex Function, Effective Action For Quantum Field Theory: If we knew Z[J], we would know everything about this theory! There are other generating functionals that can be derived form Z [J] W [J] = exp(Z [J]) generates the Connected Green’s Functions, e.g. h0jT [Φ(x1)Φ(x2)Φ(x3)Φ(x4)]j0i = h0jT [Φ(x1)Φ(x2)Φ(x3)Φ(x4)]j0iconnected +h0jT [Φ(x1)Φ(x2)]j0i h0jT [Φ(x3)Φ(x4)]j0i +permutations of x1; x2:x3:x4 T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals Green’s Functions Functional Integration Connected Green’s Functions Renormalization Vertex Function, Effective Action Usually we define h0jΦ(x)j0i = h0jT [Φ(x)]j0i = 0 h0jT [Φ(x)Φ(y)]j0i = h0jT [Φ(x)Φ(y)]j0iconnected Still another generating functional: Irreducible Vertex functions or Effective Action: Legendre Transformation Z iΓ[φ] = W [J] − i d 4x J(x)φ(x) with δ φ(x) = (−i) W [J](to be solved forJ) δJ(x) T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals Green’s Functions Functional Integration Connected Green’s Functions Renormalization Vertex Function, Effective Action Exercise 2: Legendre Transformation Show that δ Γ[φ] = J(x) δφ(x) Hint: use the definition of φ δ φ(x) = (−i) W [J] δJ(x) T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals Green’s Functions Functional Integration Connected Green’s Functions Renormalization Vertex Function, Effective Action Γ[Φ] is a very useful quantity: Its individual contributions Γn X 1 Z Γ[Φ] = d 4x :::d 4x Γ (x ; :::x )Φ(x ):::Φ(x ) n! 1 n n 1 n 1 n n are the One Particle Irreducible Vertex Functions. One-Particle-Irreducible: A Diagram is called One Particle Reducible, if it can be split into two pieces by cutting one internal line. T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals Quantum Mechanics Functional Integration Quantum Field Theory Renormalization Perturbation Theory Feynmans Approach to Quantum Theory A quantum mechanical amplitude is constructed by the “sum” over all trajectories a particle can go from point q1 to q2, weighted by a phase proportional to the action of the trajectory X −i A(q1; q2) ∼ exp S[q] all Paths ~ S[q]: Action functional depending on the trajectory q(t) with q(t1) = q1 and q(t2) = q2 Z t2 S[q] = dt L(q(t); q_ (t); t) t1 T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals Quantum Mechanics Functional Integration Quantum Field Theory Renormalization Perturbation Theory Mathematically speaking: The sum is a functional integral (assigning a number to a functional) X −i Z −i exp S[q] = [dq] exp S[q] all Paths ~ ~ [dq] Integration measure in the space of functions, in this case the trajectories q(t). It can be shown that for quantum mechanics, the integration measure −i [dq] exp S[q] ~ is well defined for all smooth potentials, and thus this formulation is equivalent to the canonical one. T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals Quantum Mechanics Functional Integration Quantum Field Theory Renormalization Perturbation Theory Functional integrals in QFT We just transfer this program to QFT The classical field theory be defined by its Lagrangian Density Z 3 L = d ~x L(Φ;@µΦ) The Action Functional is a function of the field Z 4 S[Φ] = d x L(Φ;@µΦ) Quantization is done via Z −i A ∼ [dΦ] exp S[Φ] ~ T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals Quantum Mechanics Functional Integration Quantum Field Theory Renormalization Perturbation Theory To be more precise: The generating functional Z [J] of QFT can be written as a functional integral: Z Z Z[J] = [dΦ] exp −iS[Φ] + i d 4x Φ(x)J(x) Remarks Normalization: Z[0] = R [dΦ] exp (−iS[Φ]) = 1 One point function: (−i) δ Z[J] = R [dΦ] Φ(x) exp (−iS[Φ]) = 0 δJ(x) J=0 Two point function: (−i) δ δ Z[J] = δJ(x) δJ(y) J=0 R [dΦ] Φ(x)Φ(y) exp (−iS[Φ]) h0jT [Φ(x)Φ(y)]j0i = D−1(x; y) T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals Quantum Mechanics Functional Integration Quantum Field Theory Renormalization Perturbation Theory Functional Integrals in QFT are more subtle: The general integral measure is mathematically ill defined ... only Gaussian measures are well defined: Z S[Φ] = d 4x d 4y Φ(x)D(x; y)Φ(y) with some (symmetric) distribution D(x; y) The Gaussian measures correspond to (generalized) free fields! More general measures are ill defined, may be related to the necessity of renormalization. T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals Quantum Mechanics Functional Integration Quantum Field Theory Renormalization Perturbation Theory Exercise 3: Generalized free fields Show that S[Φ] = R d 4x d 4y Φ(x)D(x; y)Φ(y) yields the generating functional for generalized free fields. Use the following assumptions D has an inverse: R d 4z D(x; z)D−1(z; y) = δ4(x − y) Perform the functional integration using quadratic completion, together with an appropriate shift of the integration variables. Why is the result Z Z [J] = exp d 4x d 4y J(x)D−1(x; y)J(y) a (generalized) free field? T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals Quantum Mechanics Functional Integration Quantum Field Theory Renormalization Perturbation Theory Local Quantum Field Theory: S is the integral over a (local) Lagrangian
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