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Introduction to QFT Yuval Grossman Cornell Y. Grossman SM and flavor (1) ICTP, June 10, 2019 p. 1 General remarks I have to make assumptions about what you know Please ask questions (in class and outside) Email: yg73@cornell.edu The plan:

  1. Introduction to QFT Yuval Grossman Cornell Y. Grossman SM and flavor (1) ICTP, June 10, 2019 p. 1

  2. General remarks I have to make assumptions about what you know Please ask questions (in class and outside) Email: yg73@cornell.edu The plan: Intro to QFT Intro to the SM Flavor Y. Grossman SM and flavor (1) ICTP, June 10, 2019 p. 2

  3. What is HEP? Y. Grossman SM and flavor (1) ICTP, June 10, 2019 p. 3

  4. What is HEP Find the basic laws of Nature More formally L = ? We have quite a good answer It is very elegant, it is based on axioms and symmetries The generalized coordinates are fields We use particles to answer this question Y. Grossman SM and flavor (1) ICTP, June 10, 2019 p. 4

  5. What is mechanics? Answer the question: what is x ( t ) ? A system can have many DOFs, and then we seek to find x i ( t ) ≡ x 1 ( t ) , x 2 ( t ) ,... Once we know x i ( t ) we know any observable Solving for q 1 ≡ x 1 + x 2 and q 2 ≡ x 1 − x 2 is the same as solving for x 1 and x 2 The idea of generalized coordinates is very important How do we solve mechanics? Y. Grossman SM and flavor (1) ICTP, June 10, 2019 p. 5

  6. How do we find x ( t ) ? x ( t ) minimizes the action, S . This is an axiom There is one action for the whole system � t 2 S = t 1 L ( x, ˙ x ) dt The solution is given by the E-L equation � ∂L � d = ∂L dt ∂ ˙ x ∂x Once we know L we can find x ( t ) up to initial conditions Mechanics is reduced to the question “what is L ?” Y. Grossman SM and flavor (1) ICTP, June 10, 2019 p. 6

  7. An example: Newtonian mechanics We assume a particle with one DOF and L = mv 2 − V ( x ) 2 We use the E-L equation L = mv 2 � ∂L � d = ∂L − V ( x ) dt ∂ ˙ x ∂x 2 The solution is − V ′ ( x ) = m ˙ v , aka F = ma Here L is te input and F = ma is the output. How do we find what is L ? Y. Grossman SM and flavor (1) ICTP, June 10, 2019 p. 7

  8. What is L? L is the most general one that is invariant under some symmetries We (again!) rephrase the question. Now we ask what are the symmetries of the system that lead to L What are the symmetries in Newtonian mechanics? Y. Grossman SM and flavor (1) ICTP, June 10, 2019 p. 8

  9. What is field theory Y. Grossman SM and flavor (1) ICTP, June 10, 2019 p. 9

  10. What is a field? In math: something that has a value in each point. We can denote it as φ ( x, t ) Temperature (scalar field) Wind (vector field) Mechanical string (?) The density of people (?) Electric and magnetic fields (vector fields) How good is the field description of each of these? In physics, fields used to be associated with sources, but now we know that fields are fundamental Y. Grossman SM and flavor (1) ICTP, June 10, 2019 p. 10

  11. A familiar example: the EM field Maxwall Eqs. leads to a wave equations ∂ 2 E ( x, t ) = c 2 ∂ 2 E ( x, t ) ∂t 2 ∂x 2 The solution is ( A and ϕ 0 depend on IC) E ( x, t ) = A cos( ωt − kx + ϕ 0 ) , ω = ck Some important implications of the result Each mode has its own amplitude, A ( ω ) The energy in each ω is conserved The superposition principle Are the statements above exact? Y. Grossman SM and flavor (1) ICTP, June 10, 2019 p. 11

  12. How to deal with generic field theories φ ( x, t ) has an infinite number of DOF . It can be an approximation for many (but finite) DOF To solve mechanics of fields we need to find φ ( x, t ) Here φ is the generalized coordinate, while x and t are treated the same (nice!) In relativity, x and t are also treated the same What is better x µ or t µ ? Y. Grossman SM and flavor (1) ICTP, June 10, 2019 p. 12

  13. Solving field theory Generalization of mechanics to systems with few “times” We still need to minimize S � L [ φ ( x, t ) , ˙ φ ( x, t ) , φ ′ ( x, t )] S = L dx dt We usually require Lorentz invariant (and use c = 1 ) � L d 4 x S = L [ φ ( x, t ) , ∂ µ φ ( x µ )] Y. Grossman SM and flavor (1) ICTP, June 10, 2019 p. 13

  14. E-L for field theory We also have an E-L equation for field theories � ∂ L � ∂ L � � d − d = ∂L ∂ ˙ ∂φ ′ dt dx ∂φ φ In relativistic notation � � ∂ L = ∂ L ∂ µ ∂ ( ∂ µ φ ) ∂φ We have a way to solve field theory, just like mechanics. Give me L and the IC, and I know everything! Just like in Newtonian mechanics, we want to get L from symmetries! Y. Grossman SM and flavor (1) ICTP, June 10, 2019 p. 14

  15. Example: a free field theory A free particle L has just a kinetic term A free field: The “kinetic term” is promoted � 2 � 2 � 2 � dx � dφ � dφ ≡ ( ∂ µ φ ) 2 T ∝ ⇒ T ∝ − dt dt dx Free particles, and thus free fields, only have kinetic terms L = ( ∂ µ φ ) 2 ⇒ ∂ 2 φ ∂x 2 = ∂ 2 φ ∂t 2 An L of a free field gives a wave equation As in Newtonian mechanics, what used to be the starting point, here is the final result Y. Grossman SM and flavor (1) ICTP, June 10, 2019 p. 15

  16. Harmonic oscillator Y. Grossman SM and flavor (1) ICTP, June 10, 2019 p. 16

  17. The harmonic oscillator Why do we care so much about harmonic oscillators? Because we really care about springs? Because we really care about pendulums? Y. Grossman SM and flavor (1) ICTP, June 10, 2019 p. 17

  18. The harmonic oscillator Why do we care so much about harmonic oscillators? Because we really care about springs? Because we really care about pendulums? Because almost any function around its minimum can be approximated as a harmonic function! Indeed, we usually expand the potential around one of its minima We identify a small parameter, and keep only a few terms in a Taylor expansion Y. Grossman SM and flavor (1) ICTP, June 10, 2019 p. 17

  19. Classic harmonic oscillator V = kx 2 2 We solve the E-L equation and get ω 2 = k x ( t ) = A cos( ωt ) m The period does not depend on the amplitude Energy is conserved Which of the above two statements is a result of the approximation of keeping only the harmonic term in the expansion? Y. Grossman SM and flavor (1) ICTP, June 10, 2019 p. 18

  20. Coupled oscillators Y. Grossman SM and flavor (1) ICTP, June 10, 2019 p. 19

  21. Coupled oscillators There are normal modes The normal modes are not “local” as in the case of one oscillator The energy of each mode is conserved This is an approximation! Once we keep non-harmonic terms energy moves between modes V ( x, y ) = k 1 x 2 + k 2 y 2 + αx 2 y 2 2 What determines the rate of energy transfer? Y. Grossman SM and flavor (1) ICTP, June 10, 2019 p. 20

  22. Things to think about Relations between harmonic oscillators and free fields Y. Grossman SM and flavor (1) ICTP, June 10, 2019 p. 21

  23. The quantum SHO Y. Grossman SM and flavor (1) ICTP, June 10, 2019 p. 22

  24. What is QM? Many ways to formulate QM For example, we promote x → ˆ x We solve QM when we know the wave function ψ ( x, t ) How many wave functions describe a system? The wave function is mathematically a field Y. Grossman SM and flavor (1) ICTP, June 10, 2019 p. 23

  25. The quantum SHO H = p 2 2 m + mω 2 x 2 E n = ( n + 1 / 2) � ω 2 We also like to use a, a † ∼ x ± ip H = ( a † a + 1 / 2) � ω x ∼ a + a † We call a † and a creation and annihilation operators a † | n � ∝ | n + 1 � E = a | n � ∝ | n − 1 � So far this is abstract. What can we do with it? Y. Grossman SM and flavor (1) ICTP, June 10, 2019 p. 24

  26. Couple oscillators Consider a system with 2 DOFs and same mass with V ( x, y ) = kx 2 + ky 2 + αxy 2 2 The normal modes are q ± = 1 ± = k ± α ω 2 √ 2( x ± y ) m What is the QM energy and spectrum of this system? Y. Grossman SM and flavor (1) ICTP, June 10, 2019 p. 25

  27. Couple oscillators Consider a system with 2 DOFs and same mass with V ( x, y ) = kx 2 + ky 2 + αxy 2 2 The normal modes are q ± = 1 ± = k ± α ω 2 √ 2( x ± y ) m What is the QM energy and spectrum of this system? E n + ,n − = ( n + + 1 / 2) � ω + + ( n − + 1 / 2) � ω − | n + , n − � Y. Grossman SM and flavor (1) ICTP, June 10, 2019 p. 25

  28. Couple oscillators and Fields With many DOFs, a → a i → a ( k ) And the states | n � → | n i � → | n ( k ) � And the energy � � ( n + 1 / 2) � ω → ( n i + 1 / 2) � ω i → [ n ( k ) + 1 / 2] � ω ( k ) dk Just like in mechanics, we expand around the minimum of the fields, and to leading order we have SHOs In QFT fields are operators while x and t are not Y. Grossman SM and flavor (1) ICTP, June 10, 2019 p. 26

  29. SHO and photons I have two questions: What is the energy that it takes to excite an harmonic oscillator by one level? What is the energy of the photon? Y. Grossman SM and flavor (1) ICTP, June 10, 2019 p. 27

  30. SHO and photons I have two questions: What is the energy that it takes to excite an harmonic oscillator by one level? What is the energy of the photon? Same answer � ω Why is the answer to both question the same? Can we learn anything from it? Y. Grossman SM and flavor (1) ICTP, June 10, 2019 p. 27

  31. What is a particle? Excitations of SHOs are particles Y. Grossman SM and flavor (1) ICTP, June 10, 2019 p. 28

  32. More on QFT Y. Grossman SM and flavor (1) ICTP, June 10, 2019 p. 29

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