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Today in History – 19 June

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Today in History - 19 June

Today in History - 19 June

240 BC

Eratosthenes estimated the circumference of Earth using two sticks.

1566

James I, King of England (1603-1625), was born.

1747

Nadirshaha passed away.

1867

Mexican Emperor Maximilian was executed.

1934

The National Archives and Records Administration was established.

1949

People of Chandernagar, the French Indian Settlement, decided to merge with the Indian Union.

1961

Kuwait regained complete independence from Britain.

1966

Shiv Sena was established.

1981

APPLE, India’s first experimental geo-stationary telecommunication satellite having experimental communication satellite,was successfully launched by European Ariane, Kourou, French Guiana. This was the first to be stabilized on 3 axes.

1990

NDC approved approach paper to the Eighth Plan.

1995

In an unprecedented move, UP Speaker Dhani Ram Verma adjourned the two-day special session called to prove Mayavati Government’s majority in the House. Later, the House gathers, rejected the sine die adjournment and unanimously elected B.R. Verma (BSP) as new presiding officer to conduct the business of the House.

1995

India and US officially launched Indo-US Commercial Alliance during US Trade Secretary’s visit.

1997

Indian Prime Minister and his Pakistani counterpart Nawaz Sharif, spoke for first time on restored hotline, to reiterate their committment to comprehensive dialogue, including Kashmir.

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Today in History – 18 June

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Today in History – 15 June

SU(2), SO(3), and U(1): Key Symmetry Groups in Physics

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u-2-so-3-u-1

Table of Contents

  1. Introduction
  2. Overview of SU(2), SO(3), and U(1)
  3. U(1): The Simplest Lie Group
  4. U(1) in Physics: Electromagnetism and Phase Invariance
  5. SO(3): The Rotation Group in Three Dimensions
  6. Properties and Structure of SO(3)
  7. Representations of SO(3) and Angular Momentum
  8. SU(2): The Spin Group and Its Significance
  9. SU(2) vs SO(3): Double Cover and Topology
  10. SU(2) Representations and Spin
  11. Pauli Matrices and SU(2) Generators
  12. Embedding SU(2), SO(3), and U(1) in Field Theory
  13. Gauge Symmetries and the Standard Model
  14. Group Manifolds and Global Properties
  15. Conclusion

1. Introduction

SU(2), SO(3), and U(1) are three fundamental Lie groups that appear repeatedly across physics. They describe symmetries in quantum mechanics, gauge theories, and classical systems. Understanding their structure and differences is critical to theoretical physics.


2. Overview of SU(2), SO(3), and U(1)

  • U(1): Group of complex numbers with unit magnitude
  • SO(3): Group of all rotations in 3D space
  • SU(2): Group of 2×2 unitary matrices with determinant 1

All are compact, continuous Lie groups, but their topological and algebraic properties differ significantly.


3. U(1): The Simplest Lie Group

U(1) is defined as:

\[
U(1) = \{ e^{i\theta} \mid \theta \in [0, 2\pi) \}
\]

  • Abelian: group elements commute
  • Topologically a circle \( S^1 \)
  • Lie algebra: \( \mathfrak{u}(1) \cong \mathbb{R} \)

4. U(1) in Physics: Electromagnetism and Phase Invariance

  • Governs phase symmetry in quantum mechanics
  • Global U(1): conservation of electric charge (via Noether’s theorem)
  • Local U(1): leads to electromagnetic gauge theory
  • Appears in the Standard Model as part of \( SU(3) \times SU(2) \times U(1) \)

5. SO(3): The Rotation Group in Three Dimensions

SO(3) consists of real orthogonal \( 3 \times 3 \) matrices with determinant +1:

\[
SO(3) = \{ R \in \mathbb{R}^{3 \times 3} \mid R^T R = I, \ \det R = 1 \}
\]

  • Non-Abelian
  • Describes rigid body rotations in classical mechanics

6. Properties and Structure of SO(3)

  • 3 parameters (e.g., Euler angles)
  • Not simply connected: \( \pi_1(SO(3)) = \mathbb{Z}_2 \)
  • Every rotation corresponds to an axis and angle

7. Representations of SO(3) and Angular Momentum

Quantum angular momentum operators form an \( \mathfrak{so}(3) \) algebra:

\[
[J_i, J_j] = i\hbar \epsilon_{ijk} J_k
\]

  • Representations labeled by integer \( l \): \( l = 0, 1, 2, \dots \)

8. SU(2): The Spin Group and Its Significance

SU(2) consists of \( 2 \times 2 \) unitary matrices with determinant 1:

\[
SU(2) = \left\{
U = \begin{bmatrix} a & b \ -b^* & a^* \end{bmatrix} \mid |a|^2 + |b|^2 = 1
\right\}
\]

  • Non-Abelian
  • Topology: 3-sphere \( S^3 \)
  • Fundamental in spin and weak interactions

9. SU(2) vs SO(3): Double Cover and Topology

  • SU(2) double covers SO(3): \( SU(2)/\mathbb{Z}_2 \cong SO(3) \)
  • SU(2) is simply connected: \( \pi_1(SU(2)) = 0 \)
  • SO(3) is not simply connected: \( \pi_1(SO(3)) = \mathbb{Z}_2 \)

Implication: spin-½ particles (e.g., electrons) described by SU(2), not SO(3)


10. SU(2) Representations and Spin

  • Irreducible representations labeled by half-integers: \( j = 0, \frac{1}{2}, 1, \frac{3}{2}, \dots \)
  • Spin-½: two-dimensional representation
  • Spin-1: three-dimensional (vector) representation, matches SO(3)

11. Pauli Matrices and SU(2) Generators

Generators of SU(2) are:

\[
T_i = \frac{1}{2} \sigma_i, \quad \text{where } \sigma_i \text{ are the Pauli matrices}
\]

\[
\sigma_1 = \begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix}, \quad
\sigma_2 = \begin{bmatrix} 0 & -i \ i & 0 \end{bmatrix}, \quad
\sigma_3 = \begin{bmatrix} 1 & 0 \ 0 & -1 \end{bmatrix}
\]


12. Embedding SU(2), SO(3), and U(1) in Field Theory

  • SU(2): weak interaction gauge symmetry
  • U(1): electromagnetic gauge symmetry
  • SO(3): spatial symmetry of angular momentum

Combined in electroweak theory via:
\[
SU(2)L \times U(1)_Y \to U(1){\text{EM}}
\]


13. Gauge Symmetries and the Standard Model

  • SU(2) gauge bosons: \( W^+, W^-, W^0 \)
  • U(1) boson: \( B \)
  • Mixing gives photon and \( Z^0 \) boson

Electroweak unification is rooted in the algebra of SU(2) and U(1).


14. Group Manifolds and Global Properties

  • U(1): circle \( S^1 \)
  • SU(2): 3-sphere \( S^3 \)
  • SO(3): 3D projective space \( \mathbb{RP}^3 \)

These properties affect quantization and topological configurations.


15. Conclusion

SU(2), SO(3), and U(1) are foundational symmetry groups in physics. U(1) governs phase invariance and electromagnetism. SO(3) encodes rotational symmetry, while SU(2) is central to spin, angular momentum, and the weak interaction.

Understanding their algebraic and topological properties is essential for any physicist delving into quantum mechanics, particle theory, or gauge symmetries.


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Today in History – 18 June

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today in history 18 june

1633

Revairu Diogu, dictionary creator of Konkani language and Jezoet Priest, died.

1658

Aurangzeb managed to capture the Agra Fort and later arrested his father Emperor Shahjahan.

1812

The day after the Senate followed the House of Representatives in voting to declare war against Great Britain, President James Madison signed the declaration into law–and the War of 1812 began.

1815

At Waterloo in Belgium, Napoleon Bonaparte suffered defeat at the hands of the Duke of Wellington, bringing an end to the Napoleonic era of European history.

1858

Rani Laxmi Bai of Jhansi died while fighting with the British troops in the battlefield near Gwalior.

1887

Dr. Anugrah Narain Singh, freedom fighter, educationist, leader and Minister of India, was born in Poiyavan village, Gaya district, Bihar.

1917

Lachhmansingh Gill, famous politician and former chief minister, was born.

1940

Forward Block Party was established.

1943

British Field Marshal Archibald Wavell becomes Viceroy of India.

1946

First Satyagraha movement was started for Goa liberation from the Portuguese.

1946

Congress Working Committee decided to accept Interim Government scheme.

1956

New Hindu Succession Act was passed.

1966

California’s hippie subculture converged into a mass of long hair, flowers, and rock music this weekend, as 50,000 flowed into the fairgrounds of the Monterey International Pop Festival. The event featured the largest collection of major rock acts ever assembled; thousands of fans had to be turned away from the sold-out concert. Established artists such as the Byrds, Jefferson Airplane, Otis Redding, and the Mamas and the Papas received the expected ovations from the huge audience. But the response was equally enthusiastic for performances by Indian sitar master Ravi Shankar and new talents Janis Joplin, the Who, and Jimi Hendrix, a young man who played the electric guitar like nobody else.

1978

The Karakoram Highway, linking Gilgit in Pakistan-occupied Kashmir with Sinkiang in China, was opened.

1989

India and Pakistan decided to end the five-year-old confrontation in Siachen Glacier area by redeploying their forces.

1983

From Cape Canaveral, Florida, the space shuttle Challenger was launched into space on its second mission. Aboard the shuttle was Dr. Sally Ride, who as a mission specialist who became the first American woman to travel into space.

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Lie Groups and Lie Algebras: Continuous Symmetries in Physics and Geometry

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lie groups algebras

Table of Contents

  1. Introduction
  2. What Are Lie Groups?
  3. Examples of Lie Groups
  4. Smooth Manifolds and Group Structure
  5. What Is a Lie Algebra?
  6. The Lie Bracket and Commutators
  7. Matrix Lie Groups and Their Algebras
  8. The Exponential Map
  9. Structure Constants and Generators
  10. Representations of Lie Algebras
  11. Cartan Subalgebra and Root Systems
  12. Classification of Simple Lie Algebras
  13. SU(2), SU(3), and SO(n) Algebras
  14. Lie Groups in Physics
  15. Gauge Symmetries and Yang-Mills Theory
  16. Conclusion

1. Introduction

Lie groups and Lie algebras form the mathematical foundation for understanding continuous symmetries. Their theory bridges abstract algebra, differential geometry, and theoretical physics. These structures are essential in classical mechanics, quantum field theory, and the Standard Model of particle physics.


2. What Are Lie Groups?

A Lie group is a group that is also a smooth differentiable manifold, where the group operations (multiplication and inversion) are smooth maps.

Formally, a Lie group \( G \) satisfies:

  • \( G \) is a group
  • \( G \) is a differentiable manifold
  • \( g_1 \cdot g_2 \) and \( g^{-1} \) are smooth maps

3. Examples of Lie Groups

  • \( \mathbb{R}^n \) under addition
  • \( U(1) \): complex numbers of unit magnitude under multiplication
  • \( SU(n) \): special unitary group
  • \( SO(n) \): special orthogonal group (rotations in \( \mathbb{R}^n \))
  • \( GL(n, \mathbb{R}) \): general linear group of invertible \( n \times n \) matrices

4. Smooth Manifolds and Group Structure

Lie groups locally look like Euclidean space. Tangent spaces at each point allow for calculus on groups.

Key property: Lie groups combine local linearity (manifold) with global symmetry (group).


5. What Is a Lie Algebra?

Associated with every Lie group is its Lie algebra, a vector space equipped with a binary operation called the Lie bracket:

\[
[\cdot, \cdot] : \mathfrak{g} \times \mathfrak{g} \to \mathfrak{g}
\]

The Lie algebra \( \mathfrak{g} \) is the tangent space at the identity of the group.


6. The Lie Bracket and Commutators

For matrix Lie groups, the Lie bracket is the commutator:

\[
[X, Y] = XY – YX
\]

Satisfies:

  • Bilinearity
  • Antisymmetry: \( [X, Y] = -[Y, X] \)
  • Jacobi identity:
    \[
    [X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y]] = 0
    \]

7. Matrix Lie Groups and Their Algebras

If \( G \subset GL(n, \mathbb{R}) \), then the Lie algebra \( \mathfrak{g} \subset \mathfrak{gl}(n, \mathbb{R}) \) is the space of matrices tangent to \( G \) at the identity.

Example:

  • \( SU(2) \) Lie algebra consists of traceless anti-Hermitian \( 2 \times 2 \) matrices

8. The Exponential Map

Connects the Lie algebra to the group:

\[
\exp: \mathfrak{g} \to G, \quad \exp(X) = \sum_{n=0}^{\infty} \frac{X^n}{n!}
\]

For matrix groups, this is the matrix exponential.

Locally, every element of the group can be written as the exponential of an element of the algebra.


9. Structure Constants and Generators

Let \( \{T_a\} \) be a basis for \( \mathfrak{g} \). Then:

\[
[T_a, T_b] = f^c_{ab} T_c
\]

Where \( f^c_{ab} \) are the structure constants of the algebra.

Generators \( T_a \) represent infinitesimal symmetries.


10. Representations of Lie Algebras

A representation is a map from a Lie algebra to matrices that preserves the bracket:

\[
\rho([X, Y]) = [\rho(X), \rho(Y)]
\]

Irreducible representations are key to understanding particle multiplets, angular momentum, and quantum states.


11. Cartan Subalgebra and Root Systems

  • Cartan subalgebra: maximal commuting set of diagonalizable generators
  • Root system: structure describing how other generators act relative to the Cartan elements

Used in classification of semisimple Lie algebras.


12. Classification of Simple Lie Algebras

Four infinite series:

  • \( A_n = SU(n+1) \)
  • \( B_n = SO(2n+1) \)
  • \( C_n = Sp(2n) \)
  • \( D_n = SO(2n) \)

And five exceptional algebras: \( G_2, F_4, E_6, E_7, E_8 \)


13. SU(2), SU(3), and SO(n) Algebras

  • \( \mathfrak{su}(2) \): spin and angular momentum
  • \( \mathfrak{su}(3) \): QCD color symmetry
  • \( \mathfrak{so}(3) \): rotations in 3D space
  • \( \mathfrak{so}(1,3) \): Lorentz algebra (special relativity)

14. Lie Groups in Physics

  • Symmetry transformations (rotation, translation, boost)
  • Gauge theory: local symmetry leads to gauge bosons
  • Quantum field theory: internal symmetry groups define interactions

15. Gauge Symmetries and Yang-Mills Theory

Gauge fields correspond to Lie algebras:

  • \( SU(3) \): QCD (strong)
  • \( SU(2) \): weak
  • \( U(1) \): electromagnetic

Yang–Mills theory generalizes Maxwell’s equations to non-Abelian gauge groups.


16. Conclusion

Lie groups and Lie algebras provide a deep and powerful language for describing symmetry in mathematics and physics. They underpin quantum field theory, gauge theory, and the Standard Model, and continue to guide modern theoretical research.

Understanding their structure is essential for mastering the modern landscape of high-energy physics and differential geometry.


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Today in History – 17 June

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today in history 17 JUNE

today in history 17 JUNE

1674

Jijabai, mother of Chhatrapati Shivaji Maharaj, died.

1756

Nawab Sira-ud-Daulah attacked on Calcutta with 50,000 soldiers and captured it on June 21.

1824

Bureau of Indian Affairs established.

1839

Lord William Bentik, Governor General of India, was born.

1862

Lord Charles Jhon Canning, Governor General and Viceroy of India (1856-1862), died.

1885

On this day in 1885, the dismantled Statue of Liberty, a gift of friendship from the people of France to the people of America, arrived in New York Harbor after being shipped across the Atlantic Ocean in 350 individual pieces packed in more than 200 cases. The copper and iron statue, which was reassembled and dedicated the following year in a ceremony presided over by U.S. President Grover Cleveland, became known around the world as an enduring symbol of freedom and democracy.

1887

Dr. Kailas Nath Katju, freedom fighter, politician, leader and Governor of Orissa and West Bengal, was born.

1913

Waman Vasudev Chitle (Bal Chitle), Marathi story writer and editor, was born.

1917

Mahatma Gandhi and Kasturba started living at Hriday Kunj in Sabarmati Ashram in Ahmedabad.

1928

Pandit Gop Bandhu Das, famous writer, social worker and the creator of Orissa, passed away.

1933

The civil disobedience campaign ended.

1947

Burma was adopted as republic by constitution assembly.

1961

The HF-24 Marut supersonic fighter, designed by German engineer Kurt Tank, makes its maiden flight.

1973

Leander Vece Paes, tennis player who won a singles bronze at Atlanta Olympics (1996), was born. He was also a winner of French Open in 1999, Gold Flake Open, Newport 1998, Wimbledon 1999 (doubles and mixed doubles).

1976

Montreal Olympics opens.

1991

Bharat Ratna, India’s highest award, was given to Rajiv Gandhi (Posthumous) (1944-1991).

1991

Bharat Ratna, India’s highest award, was given to Sardar Vallabhbhai Patel (Posthumous) (1875-1950).

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