1 of 18

## What's hot

APPLICATIONS OF MULTIPLE INTEGRALS.pdf
APPLICATIONS OF MULTIPLE INTEGRALS.pdfnissyjessilyn

False Point Method / Regula falsi method
False Point Method / Regula falsi methodNasima Akhtar

Crout s method for solving system of linear equations
Crout s method for solving system of linear equationsSugathan Velloth

Matrix of linear transformation
Matrix of linear transformationbeenishbeenish

BCA_Semester-II-Discrete Mathematics_unit-i Group theory
BCA_Semester-II-Discrete Mathematics_unit-i Group theoryRai University

algebraic&transdential equations

Scalar product of vectors
Scalar product of vectorsBed Dhakal

Eigen value and eigen vector
Eigen value and eigen vectorRutvij Patel

Lecture 4 neural networks
Lecture 4 neural networksParveenMalik18

Estudio funcion logaritmica
Estudio funcion logaritmicaMISHELQUENORAN

Indeterminate Forms and L' Hospital Rule
Indeterminate Forms and L' Hospital RuleAakash Singh

Linear Algebra presentation.pptx
Linear Algebra presentation.pptxProveedorIptvEspaa

Math major 14 differential calculus pw
Math major 14 differential calculus pwReymart Bargamento

Group theory notes
Group theory notesmkumaresan

### What's hot(20)

APPLICATIONS OF MULTIPLE INTEGRALS.pdf
APPLICATIONS OF MULTIPLE INTEGRALS.pdf

False Point Method / Regula falsi method
False Point Method / Regula falsi method

Crout s method for solving system of linear equations
Crout s method for solving system of linear equations

Matrix of linear transformation
Matrix of linear transformation

BCA_Semester-II-Discrete Mathematics_unit-i Group theory
BCA_Semester-II-Discrete Mathematics_unit-i Group theory

Power method
Power method

algebraic&transdential equations
algebraic&transdential equations

Lesson 5 a matrix inverse
Lesson 5 a matrix inverse

Integral calculus
Integral calculus

Scalar product of vectors
Scalar product of vectors

Eigen value and eigen vector
Eigen value and eigen vector

Lecture 4 neural networks
Lecture 4 neural networks

Estudio funcion logaritmica
Estudio funcion logaritmica

Indeterminate Forms and L' Hospital Rule
Indeterminate Forms and L' Hospital Rule

Linear Algebra presentation.pptx
Linear Algebra presentation.pptx

Euler and hamilton paths
Euler and hamilton paths

Unit4
Unit4

Math major 14 differential calculus pw
Math major 14 differential calculus pw

Group theory notes
Group theory notes

Gram-Schmidt process
Gram-Schmidt process

## Similar to Gauge Theory for Beginners.pptx

Magnetic Monopoles, Duality and SUSY.pptx
Magnetic Monopoles, Duality and SUSY.pptxHassaan Saleem

Review of Seiberg Witten duality.pptx
Review of Seiberg Witten duality.pptxHassaan Saleem

Maxwell's formulation - differential forms on euclidean space
Maxwell's formulation - differential forms on euclidean spacegreentask

Helmholtz equation (Motivations and Solutions)
Helmholtz equation (Motivations and Solutions)Hassaan Saleem

DOMV No 7 MATH MODELLING Lagrange Equations.pdf
DOMV No 7 MATH MODELLING Lagrange Equations.pdfahmedelsharkawy98

Schrodinger Equation of Hydrogen Atom
Schrodinger Equation of Hydrogen AtomSaad Shaukat

Lecture Notes: EEEC4340318 Instrumentation and Control Systems - System Models
Lecture Notes: EEEC4340318 Instrumentation and Control Systems - System ModelsAIMST University

Cauchy-Euler Equation.pptx
Cauchy-Euler Equation.pptxSalarBasheer

MAT-314 Relations and Functions
MAT-314 Relations and FunctionsKevin Johnson

Learning group em - 20171025 - copy
Learning group em - 20171025 - copyShuai Zhang

Small amplitude oscillations
Small amplitude oscillationsharshsharma5537

Optimum Engineering Design - Day 2b. Classical Optimization methods
Optimum Engineering Design - Day 2b. Classical Optimization methodsSantiagoGarridoBulln

Schwarzchild solution derivation
Schwarzchild solution derivationHassaan Saleem

### Similar to Gauge Theory for Beginners.pptx(20)

Magnetic Monopoles, Duality and SUSY.pptx
Magnetic Monopoles, Duality and SUSY.pptx

Review of Seiberg Witten duality.pptx
Review of Seiberg Witten duality.pptx

Maxwell's formulation - differential forms on euclidean space
Maxwell's formulation - differential forms on euclidean space

String theory basics
String theory basics

lec19.ppt
lec19.ppt

Helmholtz equation (Motivations and Solutions)
Helmholtz equation (Motivations and Solutions)

Two
Two

lec14.ppt
lec14.ppt

DOMV No 7 MATH MODELLING Lagrange Equations.pdf
DOMV No 7 MATH MODELLING Lagrange Equations.pdf

Schrodinger Equation of Hydrogen Atom
Schrodinger Equation of Hydrogen Atom

Lecture Notes: EEEC4340318 Instrumentation and Control Systems - System Models
Lecture Notes: EEEC4340318 Instrumentation and Control Systems - System Models

Cauchy-Euler Equation.pptx
Cauchy-Euler Equation.pptx

MAT-314 Relations and Functions
MAT-314 Relations and Functions

Learning group em - 20171025 - copy
Learning group em - 20171025 - copy

lec23.ppt
lec23.ppt

co.pptx
co.pptx

Small amplitude oscillations
Small amplitude oscillations

Optimum Engineering Design - Day 2b. Classical Optimization methods
Optimum Engineering Design - Day 2b. Classical Optimization methods

Schwarzchild solution derivation
Schwarzchild solution derivation

lec4.ppt
lec4.ppt

complex analysis best book for solving questions.pdf
complex analysis best book for solving questions.pdfSubhamKumar3239

GenAI talk for Young at Wageningen University & Research (WUR) March 2024
GenAI talk for Young at Wageningen University & Research (WUR) March 2024Jene van der Heide

BACTERIAL DEFENSE SYSTEM by Dr. Chayanika Das
BACTERIAL DEFENSE SYSTEM by Dr. Chayanika DasChayanika Das

Combining Asynchronous Task Parallelism and Intel SGX for Secure Deep Learning
Combining Asynchronous Task Parallelism and Intel SGX for Secure Deep Learningvschiavoni

WEEK 4 PHYSICAL SCIENCE QUARTER 3 FOR G11
WEEK 4 PHYSICAL SCIENCE QUARTER 3 FOR G11GelineAvendao

KDIGO-2023-CKD-Guideline-Public-Review-Draft_5-July-2023.pdf
KDIGO-2023-CKD-Guideline-Public-Review-Draft_5-July-2023.pdfGABYFIORELAMALPARTID1

Environmental Acoustics- Speech interference level, acoustics calibrator.pptx
Environmental Acoustics- Speech interference level, acoustics calibrator.pptxpriyankatabhane

final waves properties grade 7 - third quarter
final waves properties grade 7 - third quarterHanHyoKim

LESSON PLAN IN SCIENCE GRADE 4 WEEK 1 DAY 2
LESSON PLAN IN SCIENCE GRADE 4 WEEK 1 DAY 2AuEnriquezLontok

DETECTION OF MUTATION BY CLB METHOD.pptx
DETECTION OF MUTATION BY CLB METHOD.pptx201bo007

Introduction of Human Body & Structure of cell.pptx
Introduction of Human Body & Structure of cell.pptxMedical College

Observational constraints on mergers creating magnetism in massive stars
Observational constraints on mergers creating magnetism in massive starsSérgio Sacani

Pests of Sunflower_Binomics_Identification_Dr.UPR
Pests of Sunflower_Binomics_Identification_Dr.UPRPirithiRaju

BACTERIAL SECRETION SYSTEM by Dr. Chayanika Das
BACTERIAL SECRETION SYSTEM by Dr. Chayanika DasChayanika Das

Observation of Gravitational Waves from the Coalescence of a 2.5–4.5 M⊙ Compa...
Observation of Gravitational Waves from the Coalescence of a 2.5–4.5 M⊙ Compa...Sérgio Sacani

dll general biology week 1 - Copy.docx
dll general biology week 1 - Copy.docxkarenmillo

Interferons.pptx.
Interferons.pptx.

complex analysis best book for solving questions.pdf
complex analysis best book for solving questions.pdf

GenAI talk for Young at Wageningen University & Research (WUR) March 2024
GenAI talk for Young at Wageningen University & Research (WUR) March 2024

AZOTOBACTER AS BIOFERILIZER.PPTX
AZOTOBACTER AS BIOFERILIZER.PPTX

BACTERIAL DEFENSE SYSTEM by Dr. Chayanika Das
BACTERIAL DEFENSE SYSTEM by Dr. Chayanika Das

Combining Asynchronous Task Parallelism and Intel SGX for Secure Deep Learning
Combining Asynchronous Task Parallelism and Intel SGX for Secure Deep Learning

WEEK 4 PHYSICAL SCIENCE QUARTER 3 FOR G11
WEEK 4 PHYSICAL SCIENCE QUARTER 3 FOR G11

KDIGO-2023-CKD-Guideline-Public-Review-Draft_5-July-2023.pdf
KDIGO-2023-CKD-Guideline-Public-Review-Draft_5-July-2023.pdf

Environmental Acoustics- Speech interference level, acoustics calibrator.pptx
Environmental Acoustics- Speech interference level, acoustics calibrator.pptx

final waves properties grade 7 - third quarter
final waves properties grade 7 - third quarter

LESSON PLAN IN SCIENCE GRADE 4 WEEK 1 DAY 2
LESSON PLAN IN SCIENCE GRADE 4 WEEK 1 DAY 2

DETECTION OF MUTATION BY CLB METHOD.pptx
DETECTION OF MUTATION BY CLB METHOD.pptx

Introduction of Human Body & Structure of cell.pptx
Introduction of Human Body & Structure of cell.pptx

Observational constraints on mergers creating magnetism in massive stars
Observational constraints on mergers creating magnetism in massive stars

Pests of Sunflower_Binomics_Identification_Dr.UPR
Pests of Sunflower_Binomics_Identification_Dr.UPR

PLASMODIUM. PPTX
PLASMODIUM. PPTX

BACTERIAL SECRETION SYSTEM by Dr. Chayanika Das
BACTERIAL SECRETION SYSTEM by Dr. Chayanika Das

Observation of Gravitational Waves from the Coalescence of a 2.5–4.5 M⊙ Compa...
Observation of Gravitational Waves from the Coalescence of a 2.5–4.5 M⊙ Compa...

dll general biology week 1 - Copy.docx
dll general biology week 1 - Copy.docx

### Gauge Theory for Beginners.pptx

• 1. Gauge Theory for Beginners Muhammad Hassaan Saleem (PhD student, SUNY Albany, USA)
• 2. Electromagnetism Example 𝐵 = ∇ × 𝐴 where 𝑨 is vector potential and 𝑩 is magnetic field. Define another potential 𝐴′ i.e. 𝐴′ = 𝐴 + 𝛁𝛼 (1) where 𝛼 is an arbitrary differentiable function. ⇒ 𝐵′ = ∇ × 𝐴 + 𝛁𝛼 = ∇ × 𝐴 = 𝐵 So, 𝐴′ and 𝐴 are physically equivalent and thus, 𝐴 is not measurable (i.e. it is unphysical). Transformation (1) is known as gauge transformation and thus, 𝐴 and 𝐴′ are called gauge equivalent.
• 3. Gauge Choices So, ∇. 𝐴 doesn’t matter. We can choose for value for it. These choices are called gauge choices and this procedure is called gauge fixing. Examples ∇. 𝐴 = 0 (Coulomb Gauge) ∇. 𝐴 = − 1 𝑐2 𝜕𝜙 𝜕𝑡 (Lorenz Gauge)
• 4. Quantum Mechanics Example and local U(1) gauge symmetry. Take a quantum particle of mass 𝑚 and charge 𝑞 and let it interact it with a background field (𝜙, 𝐴). The Hamiltonian is; 𝐻 = 1 2𝑚 𝑝 − 𝑞𝐴 𝑐 2 + 𝑞𝜙 The Schrodinger equation is thus, 𝑖ℏ 𝜕𝜓 𝑥, 𝑡 𝜕𝑡 = 1 2𝑚 −𝑖ℏ∇ − 𝑞𝐴 𝑐 2 𝜓 𝑥, 𝑡 + 𝑞𝜙 𝜓(𝑥, 𝑡) and is invariant under following transformations 𝜓 𝑥, 𝑡 → 𝑒𝑖𝜆 𝑥,𝑡 𝜓(𝑥, 𝑡) 𝐴 → 𝐴 − ℏ𝑐 𝑞 ∇𝜆 𝑥, 𝑡 = 𝐴 − 𝑖ℏ𝑐 𝑞 ei𝜆 𝑥,𝑡 ∇ ei𝜆 𝑥,𝑡 † 𝜙 → 𝜙 + ℏ 𝑞 𝜕𝜆 𝑥, 𝑡 𝜕𝑡 = 𝜙 + ℏ 𝑞 ei𝜆 𝑥,𝑡 𝜕 𝜕𝑡 ei𝜆 𝑥,𝑡 † 𝑒𝑖𝜆 𝑥,𝑡 ∈ 𝑈(1) group. This is an example of local 𝑈(1) gauge transformation.
• 5. Bring in QFT QFT is a theory of quantized fields. Different fields have different transformation properties and correspond to different mathematical quantities. Different fields have different spin particles. Examples For electrodynamics, we need photons and electrons i.e. spin one and spin half particles. So, we use 𝐴𝜇 and 𝜓 fields. Field Spin Scalars 𝜙 0 Spinors 𝜓 1/2 Vector 𝐴𝜇 1 Second Rank Tensor 𝑇𝜇𝜈 2
• 6. Making of QED (Simplest, real world gauge theory) Q. What is the free electron lagrangian (with mass 𝑚 and charge 𝑒)? Ans. It is the Dirac lagrangian i.e. ℒ𝐷𝑖𝑟𝑎𝑐 = 𝜓 𝑖ℏ𝛾𝜇 𝜕𝜇 − 𝑚 𝜓 Q. How to get Maxwell lagrangian? Ans. Write Maxwell equations in covariant form and then find the lagrangian. It turns out to be ℒ𝑀𝑎𝑥𝑤𝑒𝑙𝑙 = − 1 4 𝐹 𝜇𝜈𝐹𝜇𝜈 where 𝐹 𝜇𝜈 = 𝜕𝜇𝐴𝜈 − 𝜕𝜈𝐴𝜇 𝑒−, 𝛾 don’t interact ⇒ total lagrangian is just the sum. Q. What is the interaction term? Ans. It can only be one term i.e. ℒ𝐼𝑛𝑡 = −𝑒𝜓𝛾𝜇 𝐴𝜇𝜓 (Lorentz invariant, renormalizable). Q. Why not 𝜖𝛼𝛽𝜇𝜈𝐹𝜇𝜈 𝐹𝛼𝛽 ? Ans. CP violating, just a surface term. So, QED lagrangian is ℒ𝑄𝐸𝐷 = ℒ𝐷𝑖𝑟𝑎𝑐 + ℒ𝑀𝑎𝑥𝑤𝑒𝑙𝑙 + ℒ𝐼𝑛𝑡 = 𝜓 𝑖ℏ𝛾𝜇 𝜕𝜇 − 𝑚 𝜓 − 1 4 𝐹 𝜇𝜈𝐹𝜇𝜈 − 𝑒𝜓𝛾𝜇 𝐴𝜇𝜓
• 7. Predictions and Successes of QED Using Feynman rules for QED, we can do important calculations. QED gives us certain insights e.g., corrections in Coulomb potential;
• 8. Conventional Form of QED lagrangian The QED lagrangian is ℒ𝑄𝐸𝐷 = 𝜓 𝑖ℏ𝛾𝜇 𝜕𝜇 − 𝑚 𝜓 − 1 4 𝐹 𝜇𝜈𝐹𝜇𝜈 − 𝑒𝜓𝛾𝜇 𝐴𝜇𝜓 It can be written as ℒ𝑄𝐸𝐷 = 𝜓 𝑖ℏ𝛾𝜇(𝜕𝜇 + 𝑖𝑒𝐴𝜇) − 𝑚 𝜓 − 1 4 𝐹 𝜇𝜈𝐹𝜇𝜈 = 𝜓 𝑖ℏ𝛾𝜇𝐷𝜇 − 𝑚 𝜓 − 1 4 𝐹 𝜇𝜈𝐹𝜇𝜈 where 𝐷𝜇 = 𝜕𝜇 + 𝑖𝑒𝐴𝜇 is called the covariant derivative. So, The conventional form of the QED lagrangian is given as ℒ𝑄𝐸𝐷 = 𝜓 𝑖ℏ𝛾𝜇𝐷𝜇 − 𝑚 𝜓 − 1 4 𝐹 𝜇𝜈𝐹𝜇𝜈
• 9. Local symmetry approach to QED We can derive the form of QED lagrangian starting from Dirac lagrangian alone and demanding 𝑈(1) local symmetry. Dirac lagrangian is ℒ𝐷𝑖𝑟𝑎𝑐 = 𝜓 𝑖ℏ𝛾𝜇𝜕𝜇 − 𝑚 𝜓 Demand symmetry under 𝑈(1) transformation 𝜓 𝑥 → 𝑒𝑖𝛼(𝑥) 𝜓 This renders 𝜓𝜓 → 𝜓𝑒−𝑖𝛼(𝑥)𝑒𝑖𝛼 𝑥 𝜓 = 𝜓𝜓 So, mass term is invariant. The derivative term (also called kinetic term) transforms as 𝜓𝜕𝜇𝜓 → 𝜓𝑒−𝑖𝛼 𝑥 𝜕𝜇 𝑒𝑖𝛼 𝑥 𝜓 = 𝜓𝜕𝜇𝜓 + 𝑖𝜕𝜇𝛼 𝑥 𝜓 𝜓 ≠ 𝜓𝜕𝜇𝜓 The kinetic term isn’t invariant. WHY?
• 10. Exploring the problem Choose a direction in spacetime specified by a vector 𝜂𝜇 . The directional derivative of 𝜓(𝑥) in this direction is; 𝜂𝜇 𝜕𝜇𝜓 𝑥 = lim 𝜖→0 1 𝜖 𝜓 𝑥 + 𝜖𝜂 − 𝜓 𝑥 Under gauge transformation, we have; 𝜂𝜇 𝜕𝜇𝜓 𝑥 = lim 𝜖→0 1 𝜖 𝑒𝑖𝛼 𝑥+𝜖𝜂 𝜓 𝑥 + 𝜖𝜂 − 𝑒𝑖𝛼 𝑥 𝜓 𝑥 Different exponentials cause the problem. Invent a function 𝑈(𝑦, 𝑥) such that 𝑈 𝑦, 𝑥 𝜓(𝑥) transforms like 𝜓 𝑦 i.e. 𝑈 𝑦, 𝑥 𝜓 𝑥 → 𝑒𝑖𝛼 𝑦 𝑈 𝑦, 𝑥 𝜓(𝑥) ⇒ 𝑈 𝑦, 𝑥 → 𝑒𝑖𝛼 𝑦 𝑈(𝑦, 𝑥)𝑒−𝑖𝛼 𝑥 Moreover, it is plausible to set 𝑈 𝑥, 𝑥 = 1. Now, let first argument differ slightly from 𝑥 and do a Taylor’s expansion; 𝑈 𝑥 + 𝜖𝜂, 𝑥 = 1 + 𝜕𝑈 𝑦, 𝑥 𝜕𝑦𝜇 𝑦=𝑥 𝜖𝜂𝜇 + 𝒪 𝜖2 Set 𝜕𝑈 𝑦, 𝑥 𝜕𝑦𝜇 𝑦=𝑥 = −𝑖𝑒𝐴𝜇 𝑥 ⇒ 𝑈(𝑥 + 𝜖𝜂, 𝑥) = 1 − 𝑖𝑒𝜖𝜂𝜇 𝐴𝜇 𝑥 + 𝒪 𝜖2
• 11. Exploring the problem Define a new kind of derivative called 𝐷𝜇𝜓 as 𝜂𝜇 𝐷𝜇𝜓 𝑥 = lim 𝜖→0 1 𝜖 𝜓 𝑥 + 𝜖𝜂 − 𝑈 𝑥 + 𝜖𝜂, 𝑥 𝜓 𝑥 It turns out to be 𝜂𝜇 𝐷𝜇𝜓 𝑥 = lim 𝜖→0 1 𝜖 (𝜓 𝑥 + 𝜖𝜂 − 1 − 𝑖𝑒𝜖𝜂𝜇 𝐴𝜇 𝜓(𝑥)) = lim 𝜖→0 1 𝜖 𝜓 𝑥 + 𝜖𝜂 − 𝜓 𝑥 + 𝑖𝑒𝜂𝜇 𝐴𝜇𝜓(𝑥) = 𝜂𝜇 𝜕𝜇𝜓 𝑥 + 𝑖𝑒𝐴𝜇𝜓 𝑥 ⇒ 𝐷𝜇𝜓 𝑥 = 𝜕𝜇𝜓 𝑥 + 𝑖𝑒𝐴𝜇𝜓 𝑥 We recover the same covariant derivative !!! We check that 𝐷𝜇𝜓 𝑥 → 𝑒𝑖𝛼 𝑥 𝐷𝜇𝜓(𝑥) and thus, the following lagrangian is invariant under local 𝑈(1) symmetry; 𝜓 𝑖ℏ𝛾𝜇𝐷𝜇 − 𝑚 𝜓 WHAT ABOUT MAXWELL LAGRANGIAN?
• 12. What about Maxwell’s lagrangian? First ask that how does 𝐴𝜇 transform? We know that 𝑈 𝑦, 𝑥 → 𝑒𝑖𝛼 𝑦 𝑈(𝑦, 𝑥)𝑒−𝑖𝛼 𝑥 Set 𝑦 = 𝑥 + 𝜖𝜂 and we have; 1 − 𝑖𝑒𝜖𝜂𝜇𝐴𝜇(𝑥) → 𝑒𝑖𝛼 𝑥+𝜖𝜂 1 − 𝑖𝑒𝜖𝜂𝜇𝐴𝜇(𝑥) 𝑒−𝑖𝛼 𝑥 ⇒ 𝐴𝜇 𝑥 → 𝐴𝜇(𝑥) − 𝑖 𝑒 𝜕𝜇𝛼(𝑥) We recover the same gauge transformation for 𝐴𝜇 as before! 𝐹 𝜇𝜈 is gauge invariant and thus Maxwell lagrangian can be − 1 4 𝐹 𝜇𝜈𝐹𝜇𝜈 as it is Lorentz invariant and renormalizable. Moral: QED can be deduced by demanding local 𝑈(1) gauge symmetry.
• 13. Crash Course on generators and parameters of groups 𝑈(1) is the group is 1 × 1 matrices which are unitary (i.e. complex numbers). So, one parameter needed to specify. 𝑆𝑈 𝑁 is the group of complex matrices which are unitary and have determinant 1. An 𝑁 × 𝑁 complex matrix needs 2𝑁2 parameters to be specified. However, 𝑆𝑈(𝑁) matrices need 𝑁2 − 1 parameters to be specified (think why). Every group has an identity element 𝕀. Elements near identity can be specified as follows (for 𝑆𝑈(𝑁) groups); 𝕀𝑁 − 𝑖 𝑗=1 𝐷 𝛼𝑗𝑇𝑗 = 𝕀𝑁 − 𝑖 𝛼𝑗𝑇𝑗 Where 𝐷 is the number of parameters (which is 𝑁2 − 1 in case of 𝑆𝑈(𝑁)), 𝛼𝑗 are parameters and 𝑇𝑗 are matrices called generators. An arbitrary element of the 𝑆𝑈(𝑁) group denoted as 𝒪(𝛼1, … , 𝛼𝐷)can be written by exponentiation 𝒪 𝛼1, … , 𝛼𝐷 = exp(𝑖 𝛼𝑗𝑇𝑗) The element of 𝑈(1) is 𝑒𝑖𝛼 and thus, the generator of 𝑈(1) is just 1. The generators satisfy 𝑇𝑎, 𝑇𝑏 = 𝑖𝑓𝑎𝑏𝑐𝑇𝑐 where 𝑓𝑎𝑏𝑐 are called structure constants. Example: 𝜎𝑎 2 , 𝜎𝑏 2 = 𝑖𝜖𝑎𝑏𝑐 𝜎𝑐 2 for 𝑆𝑈 2 and hence, 𝑓𝑎𝑏𝑐 = 𝜖𝑎𝑏𝑐
• 14. Generalizing QED Let 𝜓(𝑥) be a multiplet now (don’t use the word vector) with 𝑁 components as follows; 𝜓1 𝑥 ⋮ 𝜓𝑁 𝑥 So, we have an index now (𝜓𝑗 (𝑥)) called the gauge index. Now, demand symmetry under the transformation 𝜓𝑗 𝑥 → [exp(𝑖𝛼𝑘 𝑇𝑘 )]𝑗𝑙 𝜓𝑙 (𝑥) This is a non-Abelian gauge transformation (in this case, 𝑆𝑈(𝑁)). Process similar to QED gives; 𝑈 𝑥 + 𝜖𝜂, 𝑥 = 1 + 𝑖𝑔𝜖𝜂𝜇 𝐴𝜇 𝑗 𝑇𝑗 + 𝒪(𝜖2 ) 𝐷𝜇𝜓𝑗 = 𝕀 𝜕𝜇 − 𝑖𝑔𝐴𝜇 𝑘 𝑇𝑘 𝑗𝑙 𝜓𝑙 = 𝜕𝜇𝜓𝑗 − 𝑖𝑔𝐴𝜇 𝑘 𝑇𝑘 𝜓 𝑗 𝑨𝝁 𝒋 𝑻𝒋 → 𝐞𝐱𝐩 𝒊𝜶𝒍 𝑻𝒍 𝑨𝝁 𝒋 𝑻𝒋 + 𝒊 𝒈 𝝏𝝁 𝐞𝐱𝐩 𝒊𝜶𝒑 𝑻𝒑 † 𝐹 𝜇𝜈 𝑗 𝑇𝑗 = 𝜕𝜇𝐴𝜈 𝑗 𝑇𝑗 − 𝜕𝜈 𝐴𝜇 𝑙 𝑇𝑙 + 𝑔𝑓𝑖𝑗𝑘 𝐴𝜇 𝑖 𝐴𝜈 𝑗 𝑇𝑘 Not gauge inv. Gauge Invariant quantity is − 1 2 tr F𝜇𝜈 j 𝑇𝑗 2 = − 1 4 F𝜇𝜈 j 2 The final lagrangian is (called Yang Mills lagrangian) ℒ𝑌𝑀 = − 1 4 𝐹 𝜇𝜈 𝑗 2 + 𝜓𝑗 𝑖𝛾𝜇 𝐷𝜇 − 𝕀 𝑚 𝑗𝑘 𝜓𝑘 = − 1 4 𝐹 𝜇𝜈 𝑗 2 + 𝜓 𝑖𝛾𝜇 𝐷𝜇 − 𝑚 𝜓
• 15. Peculiarity of Non-Abelian gauge theories 1) The extra terms in 𝐹 𝜇𝜈 𝑗 𝑇𝑗 i.e. 𝑔𝑓𝑖𝑗𝑘𝐴𝜇 𝑖 𝐴𝜈 𝑗 𝑇𝑘 give extra vertices in Feynman diagrams i.e. i.e. it allows the force carriers to interact with one another. 2) Non Abelian 𝑆𝑈(𝑁) gauge theories are asymptotically free (𝛽 < 0) 𝛽 𝑔 = − 𝑔3 4𝜋 2 11 3 𝐶2 𝐺 − 4 3 𝑛𝑓𝐶 𝑟 No other renormalizable theory in four spacetime dimension is asymptotically free (proved by Coleman and Gross Phys. Rev. Lett. 31, 851 (1973)).
• 16. Some famous Non-Abelian gauge theories 1) Quantum Chromodynamics  𝑆𝑈(3) gauge group.  The gauge index can be 𝑖 = 1,2,3 ⇒ three colors. 2) Electroweak Theory  𝑆𝑈 2 𝐿 × 𝑈 1 𝑌 gauge group.  It is a chiral gauge theory (violates parity). 3) Standard Model  𝑆𝑈 3 × 𝑆𝑈 2 𝐿 × 𝑈 1 𝑌 gauge group. 4) Glashow Georgi Model (Candidate GUT)  𝑆𝑈(5) gauge group.  Contains 𝑋, 𝑌 bosons with mass ∼ 1016 GeV  Implies proton decay.
• 17. Super Yang Mills theories • Yang Mills theories in the presence of supersymmetry are called Super Yang Mills theories. • Supersymmetry is NOT a local gauge symmetry. It is an extension of spacetime symmetry (Poincare symmetry). • 𝛽 function for 𝑁 = 4 Super Yang Mills theories is zero (proved by N. Seiberg in 1988).
Current LanguageEnglish
Español
Portugues
Français
Deutsche