introduction to qft
play

Introduction to QFT Yuval Grossman Cornell Y. Grossman SM and - PowerPoint PPT Presentation

Oct 25, 2022 •155 likes •515 views

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

Recommend

INTRODUCTION INTRODUCTION INTRODUCTION INTRODUCTION INTRODUCTION INTRODUCTION INTRODUCTION

INTRODUCTION INTRODUCTION INTRODUCTION INTRODUCTION INTRODUCTION INTRODUCTION INTRODUCTION

TRAINING COURSE ON SOCIAL PROTECTION & FORMALIZATION TRINIDAD AND TOBAGO MARCH 15, 2017 INTRODUCTION INTRODUCTION INTRODUCTION INTRODUCTION INTRODUCTION INTRODUCTION INTRODUCTION INTRODUCTION Design of the NIS Assistance from the

547 views • 23 slides
INTRODUCTION Introduction 2/42 INTRODUCTION Alternations I am giving bread to the

INTRODUCTION Introduction 2/42 INTRODUCTION Alternations I am giving bread to the

Patterns of variation in the expression of case and agreement Andrs Brny Leiden University Centre for Linguistics 9 November 2019, BaSIS Workshop a.barany@hum.leidenuniv.nl INTRODUCTION Introduction 2/42 INTRODUCTION

1.37k views • 42 slides
Introduction Introduction Introduction Introduction Outline Motivation Failures

Introduction Introduction Introduction Introduction Outline Motivation Failures

Introduction Introduction Introduction Introduction Outline Motivation Failures Definition and concepts Process and product properties Principles SE Approaches Motivation Software and the economy The economies of

1.03k views • 49 slides
Introduction Introduction Introduction Nationwide Cause for Concern 1

Introduction Introduction Introduction Nationwide Cause for Concern 1

Introduction Introduction Introduction Nationwide Cause for Concern 1 https://www.cdc.gov/std/stats17/infographic.htm Introduction Epidemiology in Western Colorado CPHE Surveillance Data Report, May 2018 Chlamydia Epidemiology in Western

977 views • 39 slides
DAQ introduction Purpose of this talk : (1) Introduction for those who have not been in every

DAQ introduction Purpose of this talk : (1) Introduction for those who have not been in every

DAQ introduction Purpose of this talk : (1) Introduction for those who have not been in every meeting (2) Some ideas of diagrams for section 7.1 (Introduction) of TP. Giles Barr 25 Jan 2018, Pembroke College On DUNE, the reality is there is a

474 views • 10 slides
INTRODUCTION cf. Schneider, Chapter 1 INTRODUCTION THEY ARE OUT TO GET YOU INTRODUCTION WHAT

INTRODUCTION cf. Schneider, Chapter 1 INTRODUCTION THEY ARE OUT TO GET YOU INTRODUCTION WHAT

ADVANCED COMPUTER SECURITY INTRODUCTION cf. Schneider, Chapter 1 INTRODUCTION THEY ARE OUT TO GET YOU INTRODUCTION WHAT IS SECURITY? A systems security policies describe What the system is supposed to do Store and provide access

479 views • 16 slides
Overview Introduction to SMIL Introduction to W3C and XML Introduction to SMIL

Overview Introduction to SMIL Introduction to W3C and XML Introduction to SMIL

Overview Introduction to SMIL Introduction to W3C and XML Introduction to SMIL Writing a SMIL file Adding Media Objects Martin Halvey Clipping media files January 2008 Timing and Synchronising Layout World Wide

387 views • 14 slides
14 Introduction Introduction Bad guys can put malware into Example: hosts

14 Introduction Introduction Bad guys can put malware into Example: hosts

Introduction Introduction source Encapsulation ISO/OSI reference model message M application segment H t H t M transport network datagram H n H n H t M presentation: allow applications to frame link H l H n H t M interpret

583 views • 4 slides
Previous Lecture Introduction to SMIL 2 Introduction to W3C and XML Introduction to SMIL

Previous Lecture Introduction to SMIL 2 Introduction to W3C and XML Introduction to SMIL

Previous Lecture Introduction to SMIL 2 Introduction to W3C and XML Introduction to SMIL Writing a SMIL file Adding Media Objects Martin Halvey Clipping media files January 2008 Timing and Synchronising Layout

322 views • 8 slides
Lecture 1 Andreas Habegger Introduction Zynq Introduction Zynq Introduction Zynq PS vs. PL

Lecture 1 Andreas Habegger Introduction Zynq Introduction Zynq Introduction Zynq PS vs. PL

Introduction Zynq Lecture 1 Andreas Habegger Introduction Zynq Introduction Zynq Introduction Zynq PS vs. PL Processing System Processor Peripherals Data Buses AXI Bus Conclusion BTE5380 - Embedded Systems Octobre 2014 Andreas Habegger

1.81k views • 30 slides
2018.06 01 SMILE5 Introduction S E 5 02 Alpha Cloud M I L 03 Company Introduction 04

2018.06 01 SMILE5 Introduction S E 5 02 Alpha Cloud M I L 03 Company Introduction 04

2018.06 01 SMILE5 Introduction S E 5 02 Alpha Cloud M I L 03 Company Introduction 04 Project References S E 5 PART.1 SMILE5 Introduction M I L Introduction STORION-SMILE5 Residential Energy Storage System Inverter 5kW Output Input

1.06k views • 25 slides
Introduction to CICS Course introduction Course introduction What is CICS? What is an

Introduction to CICS Course introduction Course introduction What is CICS? What is an

Introduction to CICS Course introduction Course introduction What is CICS? What is an application server? Why use an application server? Course introduction Services provided by CICS How CICS applications are defined Scaling CICS to meet

1.84k views • 146 slides
Network Visualization Introduction Presented by Shahed Introduction Introduction Basic

Network Visualization Introduction Presented by Shahed Introduction Introduction Basic

Network Visualization Introduction Presented by Shahed Introduction Introduction Basic building blocks Node Links (relationship between nodes) Spatial information Network data

584 views • 9 slides
Introduction Introduction (1) Main argument of the paper: universities are not only

Introduction Introduction (1) Main argument of the paper: universities are not only

Le jugement des pairs comme outil de management des universits Peer-review as a management tool Christine Musselin (CSO, Sciences Po et CNRS) Avril 2013, Lirrsistible ascension du capitalisme acadmique Introduction Introduction (1)

590 views • 28 slides
Introduction Introduction to storage and to storage and filesystems filesystems Introduction

Introduction Introduction to storage and to storage and filesystems filesystems Introduction

Moreno Baricevic Gilberto Daz Axel Kohlmeyer Stefano Cozzini ULA ICTP CNR-IOM DEMOCRITOS Merida, VENEZUELA Trieste, ITALY Trieste, ITALY Introduction Introduction to storage and to storage and filesystems filesystems Introduction

1.19k views • 51 slides
JABLOTRON JA-100 introduction November 2011 Introduction Todays goals First introduction of

JABLOTRON JA-100 introduction November 2011 Introduction Todays goals First introduction of

JABLOTRON JA-100 introduction November 2011 Introduction Todays goals First introduction of the brand new JA-100 alarm system Visions Basic architecture System components Setting Help us with final validations You are the FIRST ones

1.69k views • 111 slides
1 Introduction Introduction Communication System Baseband an adjective that describes

1 Introduction Introduction Communication System Baseband an adjective that describes

Introduction Introduction Physical Layer Mobile network Global ISP Provides the means to transmit bits Basic Concepts of from sender to receiver, Data Transmission Defines the mechanical, electrical, and Home network timing

463 views • 7 slides
Welcome! Todays Agenda: Topic Introduction Course Introduction Team Practical

Welcome! Todays Agenda: Topic Introduction Course Introduction Team Practical

INFOGR Computer Graphics Jacco Bikker - April-July 2015 - Lecture 1: Introduction Welcome! Todays Agenda: Topic Introduction Course Introduction Team Practical Details Assignments Field Study State of

869 views • 62 slides
Introduction (00) RNDr. Martin Madaras, PhD. madaras@skeletex.xyz Introduction Introduction

Introduction (00) RNDr. Martin Madaras, PhD. madaras@skeletex.xyz Introduction Introduction

Virtual and Augmented Reality Introduction (00) RNDr. Martin Madaras, PhD. madaras@skeletex.xyz Introduction Introduction Who am I? What do I do here? 2 About me Short research bio: - 2014 - finished PhD at FMFI UK - 2014 2016 -

1.3k views • 39 slides
PRESENTATION The Introduction What is the Introduction ? The introduction prepares the audience

PRESENTATION The Introduction What is the Introduction ? The introduction prepares the audience

PRESENTATION The Introduction What is the Introduction ? The introduction prepares the audience for your presentation. It tells them what your presentation is about, why it is important, and finally, what to listen for in your speech. Get the

505 views • 3 slides
How to make a presentation with L A T EX? Introduction to L A T EX Introduction to Beamer

How to make a presentation with L A T EX? Introduction to L A T EX Introduction to Beamer

Introduction to Beamer Hafida Benhidour How to make a presentation with L A T EX? Introduction to L A T EX Introduction to Beamer Introduction to Beamer Hafida Benhidour Department of computer science King Saud University November

732 views • 49 slides
Shenzhen Cuilu jewelry Co., Ltd was founded in 1996 and its a large private enterprise

Shenzhen Cuilu jewelry Co., Ltd was founded in 1996 and its a large private enterprise

Introduction Introduction - - Brief Brief Introduction Introduction of of Cuilu Cuilu Group Group Introduction Introduction - - Brief Brief Introduction Introduction of of Cuilu Cuilu Group Group Shenzhen Cuilu jewelry Co., Ltd

147 views • 14 slides
Introduction to ICS- -214 214 Introduction to ICS Official Unit / Incident Log - A V-C-N.org

Introduction to ICS- -214 214 Introduction to ICS Official Unit / Incident Log - A V-C-N.org

V olunteer C ommunications N etwork (V-C-N.org) Introduction to ICS- -214 214 Introduction to ICS Official Unit / Incident Log - A V-C-N.org Academy Course - 2012-02.24-2208 262: Introduction to ICS-214 : V-C-N.org : Slide: 1 V olunteer C

735 views • 34 slides
Introduction to ICS- -214 214 Introduction to ICS Official Unit / Incident Log - A V-C-N.org

Introduction to ICS- -214 214 Introduction to ICS Official Unit / Incident Log - A V-C-N.org

V olunteer C ommunications N etwork (V-C-N.org) Introduction to ICS- -214 214 Introduction to ICS Official Unit / Incident Log - A V-C-N.org Academy Course - 2012-02.24-2208 262: Introduction to ICS-214 : V-C-N.org : Slide: 1 V olunteer C

558 views • 18 slides

More recommend