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Quantum Field Theory Michaelmas 2014, Prof University of Cambridge Part III of the Mathematics Tripos Quantum Field Theory Michaelmas 2014, Prof. Malcolm J. Perry Contents Lecture Synopsis 2 Quantum mechanics: recap 4 1 Introduction to Quantum Field Theory 8 1.1 Overview . 8 1.2 Lorentz transformations and Lie algebras . 8 1.3 Quantising classical mechanics . 13 2 Quantisation of the scalar field: the Klein-Gordon equation 16 2.1 Quantisation of φ(x) and π(x).................................. 16 2.2 Quantisation of the Hamiltonian . 17 2.3 From the Schr¨odingerpicture to the Heisenberg picture . 20 2.4 Propagators for the Klein-Gordon equation . 22 2.5 Applying Noether's theorem for scalar fields . 24 3 Quantisation of the Dirac field 27 3.1 Spinors and the Dirac equation . 27 3.2 Solving the Dirac equation: antiparticles . 30 3.3 Quantizing the Dirac equation . 33 3.4 Applying Noether's theorem for the Dirac field . 36 3.5 Propagators for the Dirac equation . 37 4 Quantisation of the electromagnetic field 39 4.1 Introduction to Maxwell's field . 39 4.2 Maxwell's equations . 39 4.3 Gauge invariance . 39 4.4 Quantisation of electromagnetism . 41 4.5 Propagators in electromagnetism . 43 4.6 Example: complex scalar fields . 44 5 Feynman path integrals 46 5.1 Derivation of the Feynman path Integral for a single particle . 46 5.2 Example: simple harmonic oscillator . 49 5.3 Gaussian integrals and free fields . 52 5.4 Non-Gaussian integrals and perturbation expansions . 54 6 Feynman path integrals for scalar fields 56 6.1 A further introduction . 56 6.2 Non-interacting systems . 57 6.3 Interacting systems . 59 6.4 Solving the path integral . 62 6.5 Scattering amplitudes . 64 7 Feynman path integrals for the Dirac field 67 7.1 Grassmann calculus . 67 7.2 Road to solving the path integral . 69 7.3 Solution of the path integral . 70 7.4 Overview of solution rules . 71 7.5 Worked examples . 74 8 A little beyond perturbation theory 76 8.1 Real fields: the soliton . 76 8.2 Complex fields: the Higgs . 77 1 Lecture Synopsis Lecture One Introduction to Quantum Field Theory. Review of special relativity and consideration of the Lorentz transform, focusing on the Lorentz boost. Lecture Two Lorentz transforms on scalars, vectors and covectors. Expressing Lorentz transformations as Lie algebras (infinitesimal transformations). Introduction of the momentum operator Lab. Lecture Three Quantising classical mechanics: Fourier transform of φ, derivation of the Klein-Gordon equation. Overview of functional derivatives and commutation relations. Introduction to Schr¨odingerand Heisenberg pictures. Lecture Four Derivation of operators φ(x) and π(x), leading to an expression for H in terms of creation and annihilation operators. We see that our Hamiltonian looks like an infinite sum of simple harmonic oscillators. Lecture Five Calculation of the commutation of φ(x) with π(x). Proof that the free scalar field is an infinite collection of SHO. Derivation of number operator leading to an understanding of Bose-Einstein statistics. Moving from the Schr¨odingerpicture to the Heisenberg picture. Lecture Six Heisenberg picture. Probability amplitude for a particle at y to be found at x (∆(x; y)). Analysis of probability amplitudes and introduction to the Feynman operator ∆F (x; y) Lecture Seven Integral representation of the Feynman propagator. Showing that the Feynman propagator is a Greens function for the Klein-Gordon equation. Overview of Green's functions: retarded, advanced and anti-time. Introduction to Noether's theorem. Lecture Eight Noether's theorem examples. Introduction to the Dirac equation, spinors and different representa- tions (we will only use the Weyl rep.). Lecture Nine Continuing with the Dirac Equation. Definition of spinor using Lorentz transformations. Con- struction of Φ.¯ Solving the Dirac equation. Lecture Ten Pauli-Lubanski spin vector. Finished solution of the Dirac equation resulting in the mathematical prediction of anti-particles. Construction of the action of the Dirac equation. Lecture Eleven Quantising the Dirac equation from ΦH to the Hamiltonian. Highlighting the presence of Fermi- Dirac statistics and the behaviour of the creation and annihilation operators. Lecture Twelve Noether's theorem for the Dirac equation. Propagators for the Dirac equation. Introduction to the quantisation of Maxwell's field. Lecture Thirteen Gauge invariance and the quantisation of electromagnetism. Lecture Fourteen Finishing the quantisation of electromagnetism. Studying commutation relations and the Hamil- tonian. Derivation of the propagator with an example applied to the complex scalar field. 2 Lecture Fifteen Derivation of the Feynman path integral. Lecture Sixteen Solution of the harmonic oscillator using the Feynman path integral. Lecture Seventeen Potentially optional: Feynman integral of a non-interacting system using perturbation expansion and corresponding Feynman diagrams introduced. Lecture Eighteen Finding the path integral in QFT, leading to the definition of a quantum field theory. Introduction of a source term and a 101 on how to deal with the horrible 1 of QFT. Lecture Nineteen Probability amplitude for particle creation (and corresponding Feynman diagram). Wick's theorem. Introduction of interactions into the path integral, examples of n = 1 and r = 0; 1; 2 are given. Feynman diagrams are also given representing connected and unconnected terms in the expression. Lecture Twenty Quick note on differentiation of Z[J]. Consideration of the four point function and note of unim- portance of unconnected terms. New expression is found by considering the zero point function in order to remove unconnected components. Given a recipe for solving the path integral by con- sidering the Feynman diagrams in both coordinate and momentum space. Overview of scattering cross sections. Lecture Twenty-one Scattering cross sections. Solution of scattering amplitudes in QFT. Introduction to path integrals of the Dirac field and the concept of Grassmann numbers. Lecture Twenty-two A definition of Grassmann variables including introduction to the methods of calculus. Which leads us to a formal definition of path integrals for Grassmann variables. Lecture Twenty-three Application of Grassmann integrals to the Dirac field. An overview of all the different fields we have come across including their respective propagators and interaction terms. A generalised formulation of the rules needed to calculate the various matrix elements for all the different interactions we have covered is given. Finishing with an example of the scattering of two Fermions. Lecture Twenty-four We finish our discussion of Feynman diagrams for scattering fermions and look at our rule system for calculating the scattering amplitudes. The lecture series is concluded by looking at two non- perturbative solutions in Quantum Field theory. The Real Scalar field gives rise to the Soliton and by introducing a Complex field we arrive at the Higgs particle. 3 Quantum mechanics: recap Dirac Notation Basics 2 ~ 2 Schr¨odinger ) i~Ψ(x; t) = − 2m @xΨ(x; t) + V (x)Ψ(x; t) If Ψ1;2 are solutions then λΨ1 + µΨ2 is a solution as well for λ, µ 2 C States live in Hilbert space H: jΨi 2 H The Inner Product is defined: hΨ j Φi = hΦ j Ψi∗. jj jΨi jj2 = hΨ j Ψi ≥ 0. hΨ j Ψi = 0 () jΨi = 0. Brakets are linear: hΨj (λ jΦi + µ jχi) = λ hΨ j Φi + µ hΨ j χi Observables y ∗ y Observables are hermitian operators: A = A. hΨ j A j Φi = Φ A Ψ . Properties of operators are: P P • They have a complete basis of eigenstates: A jΨii = λi jΨii ; λi 2 R. I = i jΨii hΨij ) i ci jΨii. • The eigenvalues are real. • The eigenstates for different eigenvalues are orthogonal. P P P A jΨi = AI jΨi = m A jmi hm j Ψi ) A jmi = n hn j A j mi jni = n Anm jni Pauli matrices and spin states The Pauli matrices are: 0 1 0 −i 1 0 σ = ; σ = ; σ = : (0.1) 1 1 0 2 i 0 3 0 −1 We define the spin states as: j"i ; j#i. Combining these with the Pauli matrices we gain the following expressions: σ1 j"i = j#i ; σ1 j#i = j"i ; σ2 j"i = i j#i ; σ2 j#i = −i j"i. Where: 1 0 j"i = ; j#i = (0.2) 0 1 In one-dimension: x^ jxi = x jxi ; position basis. (0.3) Ψ(x) = hx j Ψi ; Wavefunction (0.4) Probability and measurement • Measure Eigenvalue λi ) state collapses to jΨii • Probability of this is (λ ) = P j hΨ j Ψ i2 j: P i j:λj =λi j • The expectation for a state jΨi is hAi = hΨ j A j Ψi • The uncertainty is (∆A)2 = (A − hAi)2 . Remark: jΨi ! eiφ jΨi. Physics is invariant under phase change. More generally: jΨi 7! U jΨi and A 7! UAU y (i.e. U yU = UU y = I). 4 Simultaneous eigenstates Observables: A, B, pick states jΨi;ji with A jΨiji = ai jΨiji and B jP siiji = bi jΨiji. We define the commutation relation as: [A; B] = AB − BA. Example: [^x; p^] = i~. We cannot have simultaneous eigenstates. Also, for i = f 1; 2; 3 g ; [σi; σj] = 2iijkσk. A good example of this is if we take: 1 jΨi = and measure using combinations of σ . 1 i 1 1 σ σ jΨi = σ = (0.5) 3 1 3 1 −1 1 −1 σ σ jΨi = σ = (0.6) 1 2 1 −1 1 Hence the two quantities are inequivalent. Uncertainty principle 1 ∆A∆B ≥ j h[A; B]i j (0.7) 2 This is easily proved. Especially with the given hints of: • What happens when hAi = 0 • Use the Cauchy-Schwartz inequality: hΨjΨi hΦjΦi ≥ j hhΨjΦii j2 Harmonic oscillator Starting with the Hamiltonian of the simple harmonic oscillator: p^2 1 H^ = + m!2x^2 (0.8) 2m 2 Define: rm! i a = x^ + p^ ; (0.9) 2~ m! where a is the annihilation operator (ay is the creation operator).
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