Home Blog Page 268

BRST Quantization

0
brst quantization

Table of Contents

  1. Introduction
  2. Motivation for BRST Quantization
  3. Gauge Redundancy in Quantum Field Theory
  4. Path Integral Formulation and Overcounting
  5. Faddeev–Popov Procedure
  6. Emergence of Ghost Fields
  7. The Need for BRST Symmetry
  8. Structure of BRST Transformations
  9. BRST Charge and Nilpotency
  10. BRST Cohomology and Physical States
  11. Gauge Fixing in BRST Formalism
  12. BRST Invariant Lagrangian
  13. Example: BRST Quantization of Yang–Mills Theory
  14. Role of Ghost and Antighost Fields
  15. BRST Exactness and Gauge Independence
  16. BRST Anomalies and Consistency Conditions
  17. Anti-BRST Symmetry
  18. BRST Quantization in String Theory
  19. BRST Algebra and Grading
  20. Relation to Supersymmetry
  21. Geometric Interpretation
  22. Applications in Topological Field Theory
  23. Role in Conformal Field Theory
  24. Open Questions and Research Directions
  25. Conclusion

1. Introduction

BRST quantization is a powerful method used in quantum field theory to systematically quantize gauge theories while preserving gauge invariance and unitarity. It introduces a global fermionic symmetry, known as BRST symmetry, named after Becchi, Rouet, Stora, and Tyutin.


2. Motivation for BRST Quantization

Standard quantization methods fail for gauge theories due to gauge redundancy. Simply applying canonical or path integral quantization leads to divergent or ill-defined results. BRST quantization provides a consistent approach that preserves gauge structure while enabling proper gauge fixing.


3. Gauge Redundancy in Quantum Field Theory

In a gauge theory, many field configurations are physically equivalent due to local symmetry transformations. This redundancy leads to overcounting of states in the path integral formulation of quantum field theory.


4. Path Integral Formulation and Overcounting

The naive path integral:

\[
Z = \int \mathcal{D}A \, e^{iS[A]}
\]

sums over all field configurations, including gauge-equivalent ones. This results in an infinite overcounting unless corrected by gauge fixing.


5. Faddeev–Popov Procedure

The Faddeev–Popov method introduces a delta function and a determinant to isolate physical degrees of freedom. For gauge fixing condition \( G(A) = 0 \):

\[
Z = \int \mathcal{D}A \, \delta(G(A)) \, \det\left( \frac{\delta G}{\delta \alpha} \right) e^{iS[A]}
\]

The determinant becomes a path integral over ghost fields.


6. Emergence of Ghost Fields

The Faddeev–Popov determinant is expressed using Grassmann-valued ghost fields \( c, \bar{c} \). These fields appear only in loop diagrams and cancel unphysical contributions, ensuring unitarity.


7. The Need for BRST Symmetry

While the Faddeev–Popov method works at tree level, it obscures the underlying gauge symmetry. BRST symmetry restores a global symmetry that captures the original gauge symmetry even after fixing the gauge.


8. Structure of BRST Transformations

The BRST transformations involve:

  • Gauge fields \( A_\mu \)
  • Ghosts \( c \)
  • Antighosts \( \bar{c} \)
  • Auxiliary fields \( B \)

For Yang–Mills:

\[
\delta A_\mu^a = D_\mu^{ab} c^b \, \epsilon, \quad \delta c^a = -\frac{1}{2} f^{abc} c^b c^c \, \epsilon, \quad \delta \bar{c}^a = B^a \, \epsilon, \quad \delta B^a = 0
\]

Here \( \epsilon \) is a Grassmann-valued parameter.


9. BRST Charge and Nilpotency

There exists a conserved charge \( Q_{\text{BRST}} \) such that:

\[
Q_{\text{BRST}}^2 = 0
\]

This nilpotency ensures that BRST transformations form a differential complex, and physical states are identified with cohomology classes.


10. BRST Cohomology and Physical States

A state \( |\psi\rangle \) is physical if:

\[
Q_{\text{BRST}} |\psi\rangle = 0, \quad |\psi\rangle \sim |\psi\rangle + Q_{\text{BRST}} |\chi\rangle
\]

This removes unphysical states while retaining gauge-invariant information.


11. Gauge Fixing in BRST Formalism

The gauge-fixed action is written as:

\[
S = S_{\text{inv}} + \delta_{\text{BRST}} \Psi
\]

where \( \Psi \) is the gauge-fixing fermion, a functional of the fields with ghost number -1.


12. BRST Invariant Lagrangian

In Yang–Mills theory:

\[
\mathcal{L} = -\frac{1}{4} F_{\mu\nu}^a F^{\mu\nu\, a} + \delta_{\text{BRST}} \left( \bar{c}^a \left( \partial^\mu A_\mu^a + \frac{\alpha}{2} B^a \right) \right)
\]

This Lagrangian is invariant under BRST transformations and contains gauge-fixing and ghost terms.


13. Example: BRST Quantization of Yang–Mills Theory

For SU(N) gauge theory:

  • Gauge field \( A_\mu^a \)
  • Ghost \( c^a \), antighost \( \bar{c}^a \)
  • BRST symmetry encodes the infinitesimal gauge transformation with ghost replacement for parameters

14. Role of Ghost and Antighost Fields

Ghosts propagate in internal loops and correct unphysical longitudinal and time-like gauge boson contributions. They ensure unitarity and cancel gauge artifacts in loop amplitudes.


15. BRST Exactness and Gauge Independence

Physical observables belong to the BRST cohomology and are invariant under different choices of gauge fixing, demonstrating the gauge independence of physical predictions.

Today in History – 12 September

0
Today in History-12-septemb
Today in History-12-septemb

1213

Simon de Montfort defeats Raymond of Toulouse and Peter II of Aragon at Muret, France.

1398

Amir Timur of Timur Lang invaded India with 92 squadrons of horses and 90,000 cavalry and reached the banks of Sindhu river near Atak city (now in Pakistan).

1609

Henry Hudson sails into what is now New York Harbor aboard his sloop Half Moon.

1627

Ibrahim Adilshah died.

1686

Aurangzeb ended Adilshah’s kingdom. The garrison lost heart and surrendered Sikandar who was sent to the state prison at Daulatabad. Bijapur was invested and the city opened its gates to the Mughals.

1722

The Treaty of St. Petersburg puts an end to the Russo-Persian War.

1779

Garibdas, follower of Saint Kabir and founder of ‘Garib Panth’, died.

1812

Richard March Hoe, who built the first successful rotary printing press, was born.

1836

Mexican authorities crush the revolt which broke out on August 25.

1905

Anglo-Japanese treaty provides for Japan to help safeguard India at London.

1922

Chandradhar Sharma Guleri passed away.

1965

Army fights back heavy Indian tank attacks at Pakistan.

1974

T. C. Yohannan at Teheran, sets record for LJ in 8.07.

1980

Military coup in Turkey.

1994

UN Secretary General Ghali visits New Delhi.

1998

Bihar flood toll mounts to 320.

1998

India, Malaysia sign an agreement on cooperation in science and technology in Kuala Lumpur.

2011

In New York City, the 9/11 Memorial Museum opens to the public.

Also Read:

Today in History – 10 September

Today in History – 9 September

Today in History-8 September

Today in History – 7 September

Non-Abelian Gauge Fields

0

Table of Contents

  1. Introduction
  2. Gauge Symmetries in Physics
  3. Abelian vs Non-Abelian Gauge Theories
  4. Lie Groups and Lie Algebras
  5. Gauge Fields and Connections
  6. Local Gauge Invariance
  7. Yang–Mills Theory
  8. Field Strength Tensor in Non-Abelian Theories
  9. Gauge Covariant Derivative
  10. Non-Abelian Gauge Transformations
  11. Self-Interactions of Gauge Fields
  12. Yang–Mills Lagrangian
  13. Color Charge in QCD
  14. Gauge Fixing and Redundancy
  15. Faddeev–Popov Ghosts
  16. BRST Symmetry
  17. Renormalizability of Non-Abelian Theories
  18. Asymptotic Freedom
  19. Confinement and Non-Perturbative Effects
  20. Wilson Loops and Gauge Invariants
  21. Instantons and Topological Effects
  22. Lattice Gauge Theory
  23. Applications in the Standard Model
  24. Open Questions and Research Frontiers
  25. Conclusion

1. Introduction

Non-Abelian gauge fields form the foundation of modern theoretical physics, particularly in the Standard Model. Unlike Abelian gauge fields (like electromagnetism), non-Abelian fields exhibit rich structures, including self-interactions, confinement, and asymptotic freedom.


2. Gauge Symmetries in Physics

Gauge symmetries describe local transformations that leave the physical laws unchanged. These symmetries necessitate the introduction of gauge fields to maintain invariance under local transformations.


3. Abelian vs Non-Abelian Gauge Theories

  • Abelian: Commuting transformations (e.g., U(1) in electromagnetism)
  • Non-Abelian: Non-commuting transformations (e.g., SU(2), SU(3))

Non-Abelian gauge fields transform under non-commuting Lie groups, leading to richer dynamics.


4. Lie Groups and Lie Algebras

Non-Abelian gauge theories are built on continuous symmetry groups like SU(N), characterized by generators \( T^a \) satisfying:

\[
[T^a, T^b] = i f^{abc} T^c
\]

where \( f^{abc} \) are the structure constants of the group.


5. Gauge Fields and Connections

The gauge field is a connection on a fiber bundle, allowing comparison of field values at different points. For a non-Abelian group, the gauge field \( A_\mu = A_\mu^a T^a \) is matrix-valued.


6. Local Gauge Invariance

Fields transform under local symmetry:

\[
\psi(x) \to U(x) \psi(x), \quad U(x) = \exp(i \alpha^a(x) T^a)
\]

To preserve invariance, introduce a gauge field \( A_\mu \) that transforms accordingly.


7. Yang–Mills Theory

Yang–Mills theory generalizes electromagnetism to non-Abelian symmetry groups. It forms the basis for:

  • SU(2) (weak interaction)
  • SU(3) (strong interaction)

8. Field Strength Tensor in Non-Abelian Theories

The field strength tensor generalizes to:

\[
F_{\mu\nu}^a = \partial_\mu A_\nu^a – \partial_\nu A_\mu^a + g f^{abc} A_\mu^b A_\nu^c
\]

The last term introduces non-linearity and self-interactions.


9. Gauge Covariant Derivative

The covariant derivative is:

\[
D_\mu = \partial_\mu – i g A_\mu^a T^a
\]

It ensures that derivatives of fields transform covariantly under gauge transformations.


10. Non-Abelian Gauge Transformations

Under a local transformation \( U(x) \), the gauge field transforms as:

\[
A_\mu \to U A_\mu U^{-1} + \frac{i}{g} (\partial_\mu U) U^{-1}
\]

This non-linear transformation reflects the non-Abelian nature.


11. Self-Interactions of Gauge Fields

Unlike Abelian fields, non-Abelian gauge fields interact with themselves, leading to three- and four-gluon vertices in Feynman diagrams.


12. Yang–Mills Lagrangian

The Lagrangian is:

\[
\mathcal{L}{\text{YM}} = -\frac{1}{4} F{\mu\nu}^a F^{\mu\nu\, a}
\]

This yields equations of motion analogous to Maxwell’s equations, but with non-linearities due to self-interactions.


13. Color Charge in QCD

In Quantum Chromodynamics (QCD), SU(3) gauge symmetry gives rise to color charge. Gluons carry color and anticolor, allowing them to interact, unlike photons.


14. Gauge Fixing and Redundancy

Gauge theories have redundant degrees of freedom. Gauge fixing is needed to make the theory well-defined in the quantum regime (e.g., Feynman gauge, axial gauge).


15. Faddeev–Popov Ghosts

Quantization requires introducing ghost fields to preserve unitarity and consistency. These are unphysical scalar fields used to cancel contributions from nonphysical polarizations.


16. BRST Symmetry

The BRST symmetry is a global fermionic symmetry used in the quantization of gauge theories. It plays a central role in ensuring gauge invariance in the quantum theory.


17. Renormalizability of Non-Abelian Theories

Despite their complexity, non-Abelian gauge theories are renormalizable, as proven by ‘t Hooft and Veltman. This ensures their consistency as quantum field theories.


18. Asymptotic Freedom

Non-Abelian gauge theories exhibit asymptotic freedom:
\[
\beta(g) < 0 \Rightarrow g \to 0 \text{ as } \mu \to \infty
\]

This allows perturbative calculations at high energies and is a key feature of QCD.


19. Confinement and Non-Perturbative Effects

At low energies, the coupling grows, leading to confinement — color-charged particles cannot be isolated. This requires non-perturbative approaches like lattice QCD.


20. Wilson Loops and Gauge Invariants

The Wilson loop is a gauge-invariant observable:

\[
W(C) = \text{Tr } \mathcal{P} \exp\left( i g \oint_C A_\mu dx^\mu \right)
\]

Its behavior indicates confinement (area law) or deconfinement (perimeter law).


21. Instantons and Topological Effects

Non-Abelian gauge theories support topological solutions like instantons, which contribute to tunneling between vacua and affect chiral symmetry breaking and CP violation.


22. Lattice Gauge Theory

Discretizing spacetime enables numerical studies of non-perturbative phenomena. Lattice gauge theory provides first-principles calculations of hadron masses and phase transitions.


23. Applications in the Standard Model

Non-Abelian gauge fields:

  • SU(2)\(_L\): weak force
  • SU(3)\(_C\): strong force

Combined with U(1)\(_Y\), they constitute the gauge structure of the Standard Model.


24. Open Questions and Research Frontiers

  • Mechanism of confinement
  • Duality with string theory (AdS/CFT)
  • Behavior in extreme environments (quark-gluon plasma)
  • Role in unification and gravity

25. Conclusion

Non-Abelian gauge fields form the core of our modern understanding of fundamental forces. Their rich structure, self-interactions, and non-perturbative phenomena distinguish them from Abelian counterparts and provide the backbone of the Standard Model and beyond.


.

Spontaneous Symmetry Breaking

0
spontaneous symmetry breaking

Table of Contents

  1. Introduction
  2. Symmetry in Physics
  3. Types of Symmetries: Global vs Local
  4. Concept of Spontaneous Symmetry Breaking (SSB)
  5. Classical Example: Mexican Hat Potential
  6. Quantum Mechanical Viewpoint
  7. Vacuum Structure and Degenerate Ground States
  8. Order Parameters and Broken Phases
  9. Goldstone’s Theorem
  10. Nambu–Goldstone Bosons
  11. SSB in Quantum Field Theory
  12. SSB in Scalar Field Theories
  13. Abelian Example: Complex Scalar Field
  14. Local Gauge Symmetry and Higgs Mechanism
  15. Role of the Higgs Field in SSB
  16. Mass Generation Through SSB
  17. SSB in the Electroweak Sector
  18. SSB in Condensed Matter Systems
  19. Superconductivity and Anderson–Higgs Mechanism
  20. Global vs Gauge SSB: Key Differences
  21. SSB and Phase Transitions
  22. SSB and Cosmology
  23. SSB in Grand Unified Theories
  24. Challenges and Open Questions
  25. Conclusion

1. Introduction

Spontaneous Symmetry Breaking (SSB) is a phenomenon where the underlying laws of a system possess a symmetry, but the system’s ground state does not. It plays a central role in both classical and quantum physics and underpins critical developments in particle physics, cosmology, and condensed matter.


2. Symmetry in Physics

Symmetries represent invariance under transformations. Physical laws that remain unchanged under transformations such as rotations, translations, or gauge transformations are said to possess symmetry. These lead to conserved quantities via Noether’s theorem.


3. Types of Symmetries: Global vs Local

  • Global symmetry: the transformation is the same at every point in space and time.
  • Local (gauge) symmetry: the transformation can vary from point to point, requiring the introduction of gauge fields to preserve invariance.

4. Concept of Spontaneous Symmetry Breaking (SSB)

In SSB, the equations of motion retain the symmetry, but the solution (typically the vacuum state) does not. A simple analogy is a pencil balanced vertically: while the situation is symmetric under rotation, once the pencil falls, it picks a direction — breaking the symmetry spontaneously.


5. Classical Example: Mexican Hat Potential

Consider a complex scalar field with a potential:

\[
V(\phi) = \mu^2 |\phi|^2 + \lambda |\phi|^4
]

If ( \mu^2 < 0 ), the potential has a ring of minima (a “Mexican hat” shape). The vacuum state lies on this circle, breaking the symmetry.


6. Quantum Mechanical Viewpoint

In quantum theory, the system chooses one of the degenerate ground states. Despite the Lagrangian being symmetric, the ground state is not, which leads to observable consequences such as emergent particles or fields.


7. Vacuum Structure and Degenerate Ground States

The vacuum manifold contains multiple equivalent vacua. The system chooses one arbitrarily, leading to different physical phenomena. This selection leads to breaking of the symmetry in a particular direction in field space.


8. Order Parameters and Broken Phases

The order parameter quantifies the degree of symmetry breaking. For example, in ferromagnetism, magnetization serves as an order parameter. In field theory, the vacuum expectation value (VEV) of a field plays a similar role.


9. Goldstone’s Theorem

Goldstone’s theorem states:

For every spontaneously broken continuous global symmetry, there exists a massless scalar particle — a Goldstone boson.

These bosons correspond to oscillations along the degenerate vacuum manifold.


10. Nambu–Goldstone Bosons

These massless modes are seen in many systems:

  • Phonons in crystals
  • Magnons in magnets
  • Pions in low-energy QCD (approximate Goldstone bosons)

11. SSB in Quantum Field Theory

In QFT, SSB appears when fields acquire non-zero VEVs:

\[
\langle 0 | \phi | 0 \rangle \neq 0
]

This breaks the original symmetry and changes the particle spectrum of the theory.


12. SSB in Scalar Field Theories

A typical example involves a complex scalar field with a global U(1) symmetry. When the potential leads to a non-zero VEV, the symmetry is broken, and a massless Goldstone boson appears.


13. Abelian Example: Complex Scalar Field

Let ( \phi = \frac{1}{\sqrt{2}}(\rho + v) e^{i\theta} ). After SSB:

  • ( \rho ): massive radial mode
  • ( \theta ): massless Goldstone boson

This separation illustrates the broken and unbroken degrees of freedom.


14. Local Gauge Symmetry and Higgs Mechanism

When the symmetry is local (gauged), Goldstone bosons are not physical. They are “eaten” by gauge fields, providing the longitudinal polarization for massive vector bosons — this is the Higgs mechanism.


15. Role of the Higgs Field in SSB

In the Standard Model, the Higgs field acquires a VEV:

\[
\langle \phi \rangle = \frac{1}{\sqrt{2}} \begin{pmatrix} 0 \ v \end{pmatrix}
]

This breaks SU(2)(_L) × U(1)(_Y) → U(1)(_\text{EM}), giving mass to W and Z bosons while keeping the photon massless.


16. Mass Generation Through SSB

Gauge boson masses:

\[
m_W = \frac{1}{2} g v, \quad m_Z = \frac{1}{2} \sqrt{g^2 + g’^2} v
]

Fermion masses arise via Yukawa couplings with the Higgs field.


17. SSB in the Electroweak Sector

Spontaneous symmetry breaking is central to the electroweak theory, ensuring renormalizability and predictive power, and allowing weak bosons to have finite mass while preserving gauge invariance.


18. SSB in Condensed Matter Systems

SSB is ubiquitous in condensed matter:

  • Superconductors (broken U(1))
  • Ferromagnets (broken rotational symmetry)
  • Superfluids and crystals

These systems exhibit phenomena analogous to Goldstone bosons and phase transitions.


19. Superconductivity and Anderson–Higgs Mechanism

In superconductors, the photon acquires mass (leading to the Meissner effect) through a mechanism analogous to the Higgs mechanism. This was the historical inspiration for gauge boson mass generation in particle physics.


20. Global vs Gauge SSB: Key Differences

AspectGlobal SymmetryGauge Symmetry
Goldstone bosonsPhysical, masslessAbsorbed, unphysical
Gauge boson massesUnaffectedGain mass
ExamplesFerromagnetElectroweak theory

21. SSB and Phase Transitions

SSB often accompanies a phase transition:

  • First-order: discontinuous change in order parameter
  • Second-order: continuous but with diverging correlation length

In cosmology, such transitions may have generated topological defects.


22. SSB and Cosmology

Cosmic phase transitions could break symmetries at early universe temperatures. This might produce:

  • Domain walls
  • Cosmic strings
  • Baryogenesis via electroweak SSB

23. SSB in Grand Unified Theories

GUTs involve SSB at very high energies, e.g.:
\[
SU(5) \rightarrow SU(3)_C \times SU(2)_L \times U(1)_Y
]

Such breaking patterns determine particle masses and interactions at lower energies.


24. Challenges and Open Questions

  • What is the true vacuum structure?
  • Is there a deeper mechanism behind SSB?
  • What stabilizes the Higgs mass? (hierarchy problem)
  • Can SSB help explain dark matter or baryon asymmetry?

25. Conclusion

Spontaneous symmetry breaking is a profound concept that bridges classical and quantum systems, particle physics and condensed matter, and theory and observation. It explains mass generation, phase transitions, and low-energy phenomena, serving as a cornerstone in modern theoretical physics.


.

Today in History – 10 September

0

Today in History-10-septemb

1419

John the Fearless is murdered at Montereau, France, by supporters of the dauphin.

1623

Lumber and furs are the first cargo to leave New Plymouth in North America for England.

1623

Shah Jahan entered Rajputana and raided Amber crossing the Tapti. He arrived near the kingdom of Golconda and from there passed on to Orissa which was surrendered to him.

1846

Elias Howe patents the first practical sewing machine in the United States.

1858

Manilal Nathubhai Dwivedi, great author and father of modern Gujarati poems, was born.

1887

Govind Ballabh Pant was born in village Khunt, Almora District, U.P. He served eight years as the Chief Minister of Uttar Pradesh.

1912

B. D. Jatti, former Vice President of India, was born.

1914

The six-day Battle of the Marne ends, halting the German advance into France.

1918

Muslim riots break out in Calcutta.

1966

Parliament approved the Punjab Re-organisation Bill for the formation of Haryana and Punjab as two independent states.

1976

An Indian Airlines Boeing 737 aeroplane was hijacked for the second time and taken to Lahore, Pakistan.

1977

The Planning Commission decides to introduce the Rolling Plan concept.

1980

Mangalore University established.

1993

India and South Korea sign agreement to expand trade and investment.

Also Read:

Today in History – 9 September

Today in History – 8 September

Today in History – 7 September

Today in History – 6 September