Gauge field theory describes fundamental interactions through the principle of local gauge invariance. Quantum mechanics respects the gauge invariance of electromagnetic fields by requiring a simultaneous change in phase of the wavefunction under gauge transformations of potentials. Insisting on local gauge freedom in quantum mechanics forces the introduction of gauge fields that interact with particles. Yang-Mills theory extends this concept to field theories by demanding local gauge invariance of the Lagrangian density. This dictates that gauge fields belong to the Lie algebra of the symmetry group and interact with matter fields through covariant derivatives. The Lagrangian includes terms for gauge fields constructed from an invariant field strength tensor.
Maxwells equation and Electromagnetic WavesA K Mishra
These slide contains Scalar,Vector fields ,gradients,Divergence,and Curl,Gauss divergence theorem,Stoks theorem,Maxwell electromagnetic equations ,Pointing theorem,Depth of penetration (Skin depth) for graduate and Engineering students and teachers.
CHAPTER 6 Quantum Mechanics II
6.0 Partial differentials
6.1 The Schrödinger Wave Equation
6.2 Expectation Values
6.3 Infinite Square-Well Potential
6.4 Finite Square-Well Potential
6.5 Three-Dimensional Infinite-Potential Well
6.6 Simple Harmonic Oscillator
6.7 Barriers and Tunneling in some books an extra chapter due to its technical importance
Maxwells equation and Electromagnetic WavesA K Mishra
These slide contains Scalar,Vector fields ,gradients,Divergence,and Curl,Gauss divergence theorem,Stoks theorem,Maxwell electromagnetic equations ,Pointing theorem,Depth of penetration (Skin depth) for graduate and Engineering students and teachers.
CHAPTER 6 Quantum Mechanics II
6.0 Partial differentials
6.1 The Schrödinger Wave Equation
6.2 Expectation Values
6.3 Infinite Square-Well Potential
6.4 Finite Square-Well Potential
6.5 Three-Dimensional Infinite-Potential Well
6.6 Simple Harmonic Oscillator
6.7 Barriers and Tunneling in some books an extra chapter due to its technical importance
Lecture 8: Introduction to Quantum Chemical Simulation graduate course taught at MIT in Fall 2014 by Heather Kulik. This course covers: wavefunction theory, density functional theory, force fields and molecular dynamics and sampling.
The Physics of electromagnetic waves, a discourse to engineering 1st years.
"Lets discover what electromagnetic phenomena are entailed by the Maxwell’s equations.
Electromagnetic Waves are a set of phenomena broadly categorized as “Gamma rays, X-rays, Ultraviolet Rays, Visible light, Infra-red Rays, Microwaves and Radio waves.
We will discuss them from the perspective of Maxwell’s equations."
This presentation gives the basic idea about the methods of solving ODEs
The methods like variation of parameters, undetermined coefficient method, 1/f(D) method, Particular integral and complimentary functions of an ODE
De Alembert’s Principle and Generalized Force, a technical discourse on Class...Manmohan Dash
A technical discourse on formal classical mechanics. This is a 12 slide introduction to the basics of how Newton's Laws are generalized into a Lagrangian Dynamics apt at the level of an advance student of Physics.
It covers all the Maxwell's Equation for Point form(differential form) and integral form. It also covers Gauss Law for Electric Field, Gauss law for magnetic field, Faraday's Law and Ampere Maxwell law. It also covers the reason why Gauss Laws are also known as Maxwell's Equation.
PART V - Continuation of PART III - QM and PART IV - QFT.
I intended to finish with the Hydrogen Atom description and the atomic orbital framework but I deemed the content void of a few important features: the Harmonic Oscillator and an introduction to Electromagnetic Interactions which leads directly to a formulation of the Quantization of the Radiation Field. I could not finish without wrapping it up with a development of Transition Probabilities and Einstein Coefficients which opens up the proof of the Planck distribution law, the photoelectric effect and Higher order electromagnetic interactions. I believe this is the key contribution: making it more understandable up to, but not including, quantum electrodynamics!
Lecture 8: Introduction to Quantum Chemical Simulation graduate course taught at MIT in Fall 2014 by Heather Kulik. This course covers: wavefunction theory, density functional theory, force fields and molecular dynamics and sampling.
The Physics of electromagnetic waves, a discourse to engineering 1st years.
"Lets discover what electromagnetic phenomena are entailed by the Maxwell’s equations.
Electromagnetic Waves are a set of phenomena broadly categorized as “Gamma rays, X-rays, Ultraviolet Rays, Visible light, Infra-red Rays, Microwaves and Radio waves.
We will discuss them from the perspective of Maxwell’s equations."
This presentation gives the basic idea about the methods of solving ODEs
The methods like variation of parameters, undetermined coefficient method, 1/f(D) method, Particular integral and complimentary functions of an ODE
De Alembert’s Principle and Generalized Force, a technical discourse on Class...Manmohan Dash
A technical discourse on formal classical mechanics. This is a 12 slide introduction to the basics of how Newton's Laws are generalized into a Lagrangian Dynamics apt at the level of an advance student of Physics.
It covers all the Maxwell's Equation for Point form(differential form) and integral form. It also covers Gauss Law for Electric Field, Gauss law for magnetic field, Faraday's Law and Ampere Maxwell law. It also covers the reason why Gauss Laws are also known as Maxwell's Equation.
PART V - Continuation of PART III - QM and PART IV - QFT.
I intended to finish with the Hydrogen Atom description and the atomic orbital framework but I deemed the content void of a few important features: the Harmonic Oscillator and an introduction to Electromagnetic Interactions which leads directly to a formulation of the Quantization of the Radiation Field. I could not finish without wrapping it up with a development of Transition Probabilities and Einstein Coefficients which opens up the proof of the Planck distribution law, the photoelectric effect and Higher order electromagnetic interactions. I believe this is the key contribution: making it more understandable up to, but not including, quantum electrodynamics!
Kurt Lewin’s three stage model - Organizational Change and Development - Man...manumelwin
One of the cornerstone models for understanding organizational change was developed by Kurt Lewin back in the 1940s, and still holds true today.
His model is known as Unfreeze – Change – Refreeze, refers to the three-stage process of change he describes.
Kurt Lewin, a physicist as well as social scientist, explained organizational change using the analogy of changing the shape of a block of ice.
Change problem ; Features of organizational change; Importance of change ; Reasons / factors leading to organizational change ; Change process ; Kurt Lewin's Model of change process ; Bringing organizational change; Rolf Smith's seven levels of change model
Fuzzy random variables and Kolomogrov’s important resultsinventionjournals
:In this paper an attempt is made to transform Kolomogrov Maximal inequality, Koronecker Lemma, Loeve’s Lemma and Kolomogrov’s strong law of large numbers for independent, identically distributive fuzzy Random variables. The applications of this results is extensive and could produce intensive insights on Fuzzy Random variables
Maxwell's formulation - differential forms on euclidean spacegreentask
One of the greatest advances in theoretical physics of the nineteenth
century was Maxwell’s formulation of the the equations of electromag-
netism. This article uses differential forms to solve a problem related
to Maxwell’s formulation. The notion of differential form encompasses
such ideas as elements of surface area and volume elements, the work
exerted by a force, the flow of a fluid, and the curvature of a surface,
space or hyperspace. An important operation on differential forms is
exterior differentiation, which generalizes the operators div, grad, curl
of vector calculus. the study of differential forms, which was initiated
by E.Cartan in the years around 1900, is often termed the exterior
differential calculus.However, Maxwells equations have many very im-
portant implications in the life of a modern person, so much so that
people use devices that function off the principles in Maxwells equa-
tions every day without even knowing it
Published by:
Wang Jing
School of Physical and Mathematical Sciences
Nanyang Technological University
jwang14@e.ntu.edu.sg
Transcript: Selling digital books in 2024: Insights from industry leaders - T...BookNet Canada
The publishing industry has been selling digital audiobooks and ebooks for over a decade and has found its groove. What’s changed? What has stayed the same? Where do we go from here? Join a group of leading sales peers from across the industry for a conversation about the lessons learned since the popularization of digital books, best practices, digital book supply chain management, and more.
Link to video recording: https://bnctechforum.ca/sessions/selling-digital-books-in-2024-insights-from-industry-leaders/
Presented by BookNet Canada on May 28, 2024, with support from the Department of Canadian Heritage.
Software Delivery At the Speed of AI: Inflectra Invests In AI-Powered QualityInflectra
In this insightful webinar, Inflectra explores how artificial intelligence (AI) is transforming software development and testing. Discover how AI-powered tools are revolutionizing every stage of the software development lifecycle (SDLC), from design and prototyping to testing, deployment, and monitoring.
Learn about:
• The Future of Testing: How AI is shifting testing towards verification, analysis, and higher-level skills, while reducing repetitive tasks.
• Test Automation: How AI-powered test case generation, optimization, and self-healing tests are making testing more efficient and effective.
• Visual Testing: Explore the emerging capabilities of AI in visual testing and how it's set to revolutionize UI verification.
• Inflectra's AI Solutions: See demonstrations of Inflectra's cutting-edge AI tools like the ChatGPT plugin and Azure Open AI platform, designed to streamline your testing process.
Whether you're a developer, tester, or QA professional, this webinar will give you valuable insights into how AI is shaping the future of software delivery.
LF Energy Webinar: Electrical Grid Modelling and Simulation Through PowSyBl -...DanBrown980551
Do you want to learn how to model and simulate an electrical network from scratch in under an hour?
Then welcome to this PowSyBl workshop, hosted by Rte, the French Transmission System Operator (TSO)!
During the webinar, you will discover the PowSyBl ecosystem as well as handle and study an electrical network through an interactive Python notebook.
PowSyBl is an open source project hosted by LF Energy, which offers a comprehensive set of features for electrical grid modelling and simulation. Among other advanced features, PowSyBl provides:
- A fully editable and extendable library for grid component modelling;
- Visualization tools to display your network;
- Grid simulation tools, such as power flows, security analyses (with or without remedial actions) and sensitivity analyses;
The framework is mostly written in Java, with a Python binding so that Python developers can access PowSyBl functionalities as well.
What you will learn during the webinar:
- For beginners: discover PowSyBl's functionalities through a quick general presentation and the notebook, without needing any expert coding skills;
- For advanced developers: master the skills to efficiently apply PowSyBl functionalities to your real-world scenarios.
DevOps and Testing slides at DASA ConnectKari Kakkonen
My and Rik Marselis slides at 30.5.2024 DASA Connect conference. We discuss about what is testing, then what is agile testing and finally what is Testing in DevOps. Finally we had lovely workshop with the participants trying to find out different ways to think about quality and testing in different parts of the DevOps infinity loop.
Encryption in Microsoft 365 - ExpertsLive Netherlands 2024Albert Hoitingh
In this session I delve into the encryption technology used in Microsoft 365 and Microsoft Purview. Including the concepts of Customer Key and Double Key Encryption.
The Art of the Pitch: WordPress Relationships and SalesLaura Byrne
Clients don’t know what they don’t know. What web solutions are right for them? How does WordPress come into the picture? How do you make sure you understand scope and timeline? What do you do if sometime changes?
All these questions and more will be explored as we talk about matching clients’ needs with what your agency offers without pulling teeth or pulling your hair out. Practical tips, and strategies for successful relationship building that leads to closing the deal.
Builder.ai Founder Sachin Dev Duggal's Strategic Approach to Create an Innova...Ramesh Iyer
In today's fast-changing business world, Companies that adapt and embrace new ideas often need help to keep up with the competition. However, fostering a culture of innovation takes much work. It takes vision, leadership and willingness to take risks in the right proportion. Sachin Dev Duggal, co-founder of Builder.ai, has perfected the art of this balance, creating a company culture where creativity and growth are nurtured at each stage.
Elevating Tactical DDD Patterns Through Object CalisthenicsDorra BARTAGUIZ
After immersing yourself in the blue book and its red counterpart, attending DDD-focused conferences, and applying tactical patterns, you're left with a crucial question: How do I ensure my design is effective? Tactical patterns within Domain-Driven Design (DDD) serve as guiding principles for creating clear and manageable domain models. However, achieving success with these patterns requires additional guidance. Interestingly, we've observed that a set of constraints initially designed for training purposes remarkably aligns with effective pattern implementation, offering a more ‘mechanical’ approach. Let's explore together how Object Calisthenics can elevate the design of your tactical DDD patterns, offering concrete help for those venturing into DDD for the first time!
Key Trends Shaping the Future of Infrastructure.pdfCheryl Hung
Keynote at DIGIT West Expo, Glasgow on 29 May 2024.
Cheryl Hung, ochery.com
Sr Director, Infrastructure Ecosystem, Arm.
The key trends across hardware, cloud and open-source; exploring how these areas are likely to mature and develop over the short and long-term, and then considering how organisations can position themselves to adapt and thrive.
Kubernetes & AI - Beauty and the Beast !?! @KCD Istanbul 2024Tobias Schneck
As AI technology is pushing into IT I was wondering myself, as an “infrastructure container kubernetes guy”, how get this fancy AI technology get managed from an infrastructure operational view? Is it possible to apply our lovely cloud native principals as well? What benefit’s both technologies could bring to each other?
Let me take this questions and provide you a short journey through existing deployment models and use cases for AI software. On practical examples, we discuss what cloud/on-premise strategy we may need for applying it to our own infrastructure to get it to work from an enterprise perspective. I want to give an overview about infrastructure requirements and technologies, what could be beneficial or limiting your AI use cases in an enterprise environment. An interactive Demo will give you some insides, what approaches I got already working for real.
2. Brief Outline
1. Gauge invariance in classical electrodynamics
2. Local gauge invariance in quantum mechanics
3. Yang-Mills theory
3. Gauge Invariance and Classical Electrodynamics
• In classical electrodynamics, the electric and magnetic fields can
be written in terms of the scalar and vector potentials
B = × A E = − φ −
∂A
∂t
• However, these potentials are not unique for a given physical
field. There is a certain freedom in choosing the potentials.
• The potentials can be transformed as
Aµ
(x) → A µ
(x) = Aµ
(x) + ∂µ
Λ(x)
without affecting the physical electric and magnetic fields.
4. Local Gauge Invariance in Quantum Mechanics
Charged Particle in the Electromagnetic Field
• Hamiltonian of a charged particle moving in the presence of the
electromagnetic field is given by
H =
1
2m
(p − qA)2
+ qφ
• Quantum mechanically, the charged particle is described by the
Schr¨odinger equation,
−
1
2m
− iqA
2
ψ(x, t) = i
∂
∂t
+ iqφ ψ(x, t)
5. Local Gauge Invariance in Quantum Mechanics
Gauge Invariance and Quantum Mechanics
• Classically, the potentials φ and A are not unique for a given
physical electromagnetic field.
• We can transform the potentials locally without affecting the
physical fields (and hence the behaviour of the charged particle
moving in the field).
• We want to investigate whether an analogous situation exists in
quantum mechanics (i.e. whether quantum mechanics respects
the gauge invariance property of electromagnetic fields)
6. Local Gauge Invariance in Quantum Mechanics
• The gauge transformation of the potentials does not leave the
Schr¨odinger equation invariant.
• However, it is possible to restore the form invariance of the
Schr¨odinger equation, provided the transformation of the
potentials
Aµ
→ A µ
= Aµ
+ ∂µ
Λ(x)
is accompanied by a transformation of the wave function
ψ → ψ = e−iqΛ(x)
ψ
• With these two transformations together, the form invariance of
the Schr¨odinger equation is assured (i.e. A µ and ψ satisfy the
same equation as Aµ and ψ.)
7. Local Gauge Invariance in Quantum Mechanics
Summary
Quantum mechanics respects the gauge invariance property of the
electromagnetic field. It gives the freedom to change the
electromagnetic potentials but at the cost of a simultaneous change in
the phase of the wave function.
8. Local Gauge Invariance in Quantum Mechanics
Reversing the Argument
(Demanding Local Gauge Invariance)
• Instead of starting with the charged particle Schr¨odinger
equation, we start with the free particle Schr¨odinger equation
−
1
2m
2
ψ(x, t) = i
∂
∂t
ψ(x, t)
• We demand that this equation remains invariant under the local
phase transformation of the wave function
ψ(x) → ψ (x) = e−iqΛ(x)
ψ(x)
• However, the new wave function ψ (x) does not satisfy the free
particle Schr¨odinger equation.
9. Local Gauge Invariance in Quantum Mechanics
• We conclude that the local gauge invariance is not possible with
the free particle Schr¨odinger equation.
• However, the demand of local gauge invariance can be satisfied
by modifying the free particle Schr¨odinger equation.
• It turns out that by modifying the derivative operators in the free
particle Schr¨odinger equation as
∂µ → Dµ = ∂µ + iqAµ
we can achieve the required goal, provided the vector field Aµ
also transforms under the phase transformation of the wave
function ψ.
10. Local Gauge Invariance in Quantum Mechanics
Summary
• Local gauge freedom in the wave function in quantum mechanics
is not possible with the free particle Schr¨odinger equation.
• The insistence on the local gauge freedom forces us to introduce
in the equation a new field which interacts with the particle.
11. Yang-Mills Theory
• We now turn to extend the concept of local gauge invariance to
field theories.
• In field theory, the quantity of fundamental interest is the
Lagrangian density of the fields and accordingly, we demand the
local gauge invariance of the Lagrangian density.
12. Yang-Mills Theory
Lagrangian Density
• We consider a Lagrangian density which depends upon the scalar
field φ and its first derivative ∂µφ
L ≡ L(φ(x), ∂µφ(x))
• We also assume that the Lagrangian density is constructed out of
the inner products (φ, φ) and (∂µφ(x), ∂µφ(x)) (where bracket
denotes the inner product in field space), e.g.
L = (∂µφ)†
(∂µφ) − m2
φ†
φ − λ(φ†
φ)2
where the field φ, in general, is a multi component field.
13. Yang-Mills Theory
(Infinitesimal group theory)
• We are mainly interested in the compact Lie groups such as
SU(N) and SO(N).
• One basic property of the compact groups is that their finite
dimensional representations are equivalent to the unitary
representation.
• The advantage of unitary transformations is that they preserve
the inner products
φ†
φ → (Uφ)†
(Uφ) = φ†
(U†
U)φ = φ†
φ
14. Yang-Mills Theory
(Infinitesimal group theory)
• Associated with each Lie group is a Lie algebra. The elements ω
of the group can be written as
T(ω) = eiλaT(ta)
where ta are the generators of the group and T is some
representation.
• One important representation is the adjoint representation for
which the Lie algebra space coincides with the vector space on
which the group elements act. The action is given by
Ad(ω)A = ωAω−1
where, A is an element of the Lie algebra space.
15. Yang-Mills Theory
(Global symmetry transformation)
• We now assume that the Lagrangian density remains invariant
under a global symmetry transformation
φ(x) → φ (x) = T(ω)φ(x)
where ω is an element of the symmetry group and T(ω) is some
unitary representation under which the fields φ transform.
• For example, the field φ may be a two component object
transforming under the fundamental representation of the SU(2)
group, i.e.
φ(x) ≡
φ1(x)
φ2(x)
→
φ1(x)
φ2(x)
= eiΛaσa/2 φ1(x)
φ2(x)
16. Yang-Mills Theory
(Local symmetry transformation)
• We now generalize the global transformation to a local
transformation
φ(x) → φ (x) = T(ω(x))φ(x)
• Under a local transformation, the inner product φ†φ remains
invariant. However, the inner product involving the derivative of
the fields (∂µφ)†(∂µφ) does not remain invariant, since
∂µφ(x) → ∂µφ (x) = T(ω(x))∂µφ(x) + ∂µT(ω(x))φ(x)
(The second term in the right hand side prevents the invariance
of the inner product involving the derivatives)
17. Yang-Mills Theory
(Introducing gauge fields)
• To ensure the invariance of the Lagrangian density, the same
procedure, as in the case of quantum mechanics, is followed.
• We replace the ordinary derivative by a covariant derivative
∂µφ(x) → Dµφ(x) = (∂µ − igT(Aµ))φ(x)
introducing a field Aµ known as the gauge field.
• The field Aµ is constructed in such a way that the covariant
derivative transforms exactly as the field φ, namely
Dµφ(x) → (Dµφ(x)) = T(ω(x))Dµφ(x)
18. Yang-Mills Theory
(Gauge fields belong to the lie algebra)
• The last demand leads to the following transformation property
for the gauge fields Aµ
T(Aµ) = T(ωAµω−1
) +
i
g
T(ω∂µω−1
)
• Both the terms in the right hand side belong to the lie algebra of
the corresponding symmetry group.
• The first term is a result of the action of adjoint representation.
For the second term, we look at the group elements near identity
ω(x) = 1 + iλa(x)ta + o(λ2
)
This gives
ω∂µω−1
= −i(∂µλa)ta
19. Yang-Mills Theory
(Gauge fields belong to the Lie algebra)
• Since gauge fields Aµ belong to the Lie algebra space, it follows
that we can write them as a linear combination of the generators
ta
Aµ
= Aµ
a ta
• From this, it also follows that the number of independent gauge
fields is equal to the number of generators of the group. Thus,
e.g., if the symmetry group is SU(N), the number of gauge fields
will be (N2 − 1).
• Thus, the number of gauge fields depends only upon the
underlying symmetry group and is independent of the number of
matter fields present in the system (of course, the number of
matter fields should match with the dimension of some
representation of the symmetry group)
20. Yang-Mills Theory
(Lagrangian density for the gauge fields)
• Since we have introduced the gauge fields Aµ in our system, we
need to have a term in the Lagrangian density which describes
their dynamical behavior.
• Moreover, this term should also be gauge invariant to preserve
the gauge invariance of the Lagrangian density.
• We recall that the electromagnetic Lagrangian density is given by
L = −
1
4
FµνFµν
where, Fµν = ∂µAν − ∂νAµ is the field strength tensor.
21. Yang-Mills Theory
(Lagrangian density for the gauge fields)
• To construct the field strength tensor for the gauge fields, we take
guidance from the following theorem (Rubakov, chapter 3)
“A Lie algebra is compact if and only if it has a
(positive-definite) scalar product, which is invariant under the
action of the adjoint representation of the group ”
• Since our aim is also to have a gauge invariant term, we demand
that the field strength tensor for the gauge fields should also
transform according to the adjoint representation, i.e.
Fµν → Fµν = Ad(ω)Fµν = ωFµνω−1
and we construct the gauge invariant Lagrangian density using
this field tensor.
22. Yang-Mills Theory
(Lagrangian density for the gauge fields)
• This demand leads to the field strength tensor
Fµν = ∂µAν − ∂νAµ − ig[Aµ, Aν]
• Since Fµν belongs to the Lie algebra, we can write it as a linear
combination of the generators
Fµν
= Fµν
a ta
• In terms of the components Aµ
a and Fµν
a , we have
Fµν
a = ∂µ
Aν
a − ∂ν
Aµ
a + gfabcAµ
b Aν
c
• This differs from the electromagnetic case by the presence of a
non linear term.
23. Yang-Mills Theory
(Lagrangian density for the gauge fields)
• The Lagrangian density for the gauge fields is postulated to be
the inner product
Lgaugefield = −
1
2
Tr(Fµν
Fµν) = −
1
4
Fµν
a Faµν
• Since Fµν transforms as the adjoint representation, this inner
product is invariant (basically due to cyclic property of trace).
24. Yang-Mills Theory
(Full Lagrangian density)
• For the example given earlier, the complete Lagrangian density
thus becomes
L = (Dµφ)†
(Dµ
φ) − m2
(φ†
φ) − λ(φ†
φ)2
−
1
4
Fµν
a Faµν
where,
Dµφ = (∂µ − igT(Aµ))φ
25. Yang-Mills Theory
(Energy-momentum tensor)
• The energy momentum tensor can be obtained using the
definition
δS = −
1
2
d4
x
√
−g Tµν
δgµν
• This gives
Tµν
=
1
4
ηµν
Fλρ
a Faλρ − Fµλ
a Fν
a λ + 2(Dµ
φ)†
Dν
φ − ηµν
Lφ
• Energy is given by integrating the (00)th component of this
tensor over the spatial volume and is positive definite.
E = d3
x (D0
φ)†
D0
φ + (Di
φ)†
Di
φ + m2
(φ†
φ) + λ(φ†
φ)2
+
1
2
F0i
a F0i
a +
1
4
Fij
a Fij
a
26. Yang-Mills Theory
Summary
The interaction between the scalar fields and the gauge fields can be
obtained by invoking the local gauge invariance principle. This
principle also dictates the kind of terms which can be present in the
Lagrangian.
27. Why Yang-Mills Theory
• Every physical phenomenon is believed to be governed by four
interactions. Two of these, namely, Gravity and
Electromagnetism are felt in day to day life.
• Due to this, it is possible to formulate a classical version of these
interactions.
• The formulation of the universal Gravitational force law by Isaac
Newton from the observation of the motion of an apple and the
moon is an excellent example of this.
• Similarly, the laws of Electrodynamics were discovered by
observing the behavior of magnets, current carrying wires and so
on. James Clark Maxwell gave the exact mathematical form of
these laws using these observation (and his excellent insight).
30. Why Yang-Mills Theory
• The quantum version of the Electrodynamics (Quantum
Electrodynamics) was constructed with the help of its known
classical version.
• However, there is no guidance in the form of classical laws for
the strong and weak interactions. We have to directly deal with
the quantum version of these interactions.
• The Gauge invariance principle comes to rescue. The
mathematical form of the strong and weak interactions has been
constructed by using this principle.
31. References
1. Valery Rubakov, Classical Theory of Gauge Field, Princeton
University Press, Princeton, New Jersey (2002)
2. Aitchison and Hey, Gauge Theories in Particle Physics: Volume 1,
3rd Ed., IOP (2004)