This document provides a summary of key concepts in constructing the SU(2)xU(1) model of the Standard Model of particle physics. It introduces the partial Lagrangian that is invariant under SU(2)xU(1) gauge transformations. This includes a left-handed spinor doublet, right-handed singlet, and scalar doublet. It also discusses how the left-handed neutrino is rendered massless. Transformations of fields under the SU(2)xU(1) subgroups are presented, showing how the gauge fields must also transform to maintain Lagrangian invariance.
Nanotechnology involves engineering systems at the molecular scale and can be applied to improve solar cells. It allows for thinner silicon films that better absorb photons in solar cells, increasing efficiency. Quantum dots in particular can generate multiple electrons from a single photon, boosting performance. The use of nanomaterials and thin films in solar cells promises to significantly reduce manufacturing costs and make solar power affordable enough to supply energy to developing countries on a large scale.
Types Of nuclear reactions. Nuclear Fission Reaction. Nuclear Fusion Reaction. Difference between nuclear fusion and nuclear fusion. Light Element Fission. Light Element Fusion. Nuclear Fusion on Sun. Beta Decay process happening in sun. A short explanation of D–D reaction, D–He(3) reaction, D–T reaction. the outstanding problem is the tritium supply. Binding energy curve.Energy partition in process of fusion reactions. How then can light element fusion reactions be initiated? A major explanation for all these above steps. A complete explanation by Syed Hammad Ali Gillani.
This document discusses novel materials for batteries. It begins by introducing solid state batteries and the requirements for electrode materials, including low working potential, high specific capacity, good interface with electrolytes, and high electrode kinetics. It then discusses various materials that could be used as electrodes, including lithium carbon electrodes using graphite and graphite intercalation compounds. Different types of graphite like natural, synthetic, and HOPG are described. The document also discusses intercalation of lithium ions into carbon and potential carbon-sodium electrodes. Finally, it discusses various material classes like rutile, perovskite, and spinel materials that could be used as cathodes in rechargeable lithium ion batteries. Specific
This document is a 31 page presentation by Dr. Lynn Fuller on Gallium Arsenide devices, technologies, and integrated circuits. It provides an overview and comparison of silicon and gallium arsenide, describes growth and processing techniques for GaAs such as molecular beam epitaxy, and details GaAs device technologies including MESFETs and basic processing steps for GaAs integrated circuits.
This document discusses variational principles and Lagrange's equations. It covers Hamilton's principle, the calculus of variations, deriving Lagrange's equations from Hamilton's principle, Hamilton's principle for nonholonomic systems, and conservation theorems. Key points include using Hamilton's principle to find the path that makes the action integral stationary, using the calculus of variations to find such paths, deriving Lagrange's equations by making the variation of the action integral equal to zero, and handling constraints using Lagrange multipliers.
Solar cells directly convert sunlight into electricity and were first used in spacecraft. Traditional solar cells use two types of silicon sandwiched together, while newer types are still in development. Solar cells work by using photons to separate electron-hole pairs in materials like titanium dioxide.
Nuclear fission involves the splitting of heavy atomic nuclei into lighter nuclei. It occurs through induced fission where a neutron is absorbed, causing the nucleus to split into fission products along with more neutrons. Nuclear fusion combines two atomic nuclei into a heavier one and releases energy in the process. It is the process that powers stars like our Sun, which generates energy through the fusion of hydrogen into helium in its core.
This document provides an overview of superconductivity. It begins with definitions and the discovery of superconductivity by Kamerlingh Onnes in 1911. It describes the Meissner effect and the development of BCS theory in 1957 to explain superconductivity through Cooper pairs. It outlines several properties of superconductors like the Josephson effect and reviews applications such as magnetic levitation and power transmission. It concludes by noting superconductivity enables enormous power transfer without loss below critical temperatures.
Nanotechnology involves engineering systems at the molecular scale and can be applied to improve solar cells. It allows for thinner silicon films that better absorb photons in solar cells, increasing efficiency. Quantum dots in particular can generate multiple electrons from a single photon, boosting performance. The use of nanomaterials and thin films in solar cells promises to significantly reduce manufacturing costs and make solar power affordable enough to supply energy to developing countries on a large scale.
Types Of nuclear reactions. Nuclear Fission Reaction. Nuclear Fusion Reaction. Difference between nuclear fusion and nuclear fusion. Light Element Fission. Light Element Fusion. Nuclear Fusion on Sun. Beta Decay process happening in sun. A short explanation of D–D reaction, D–He(3) reaction, D–T reaction. the outstanding problem is the tritium supply. Binding energy curve.Energy partition in process of fusion reactions. How then can light element fusion reactions be initiated? A major explanation for all these above steps. A complete explanation by Syed Hammad Ali Gillani.
This document discusses novel materials for batteries. It begins by introducing solid state batteries and the requirements for electrode materials, including low working potential, high specific capacity, good interface with electrolytes, and high electrode kinetics. It then discusses various materials that could be used as electrodes, including lithium carbon electrodes using graphite and graphite intercalation compounds. Different types of graphite like natural, synthetic, and HOPG are described. The document also discusses intercalation of lithium ions into carbon and potential carbon-sodium electrodes. Finally, it discusses various material classes like rutile, perovskite, and spinel materials that could be used as cathodes in rechargeable lithium ion batteries. Specific
This document is a 31 page presentation by Dr. Lynn Fuller on Gallium Arsenide devices, technologies, and integrated circuits. It provides an overview and comparison of silicon and gallium arsenide, describes growth and processing techniques for GaAs such as molecular beam epitaxy, and details GaAs device technologies including MESFETs and basic processing steps for GaAs integrated circuits.
This document discusses variational principles and Lagrange's equations. It covers Hamilton's principle, the calculus of variations, deriving Lagrange's equations from Hamilton's principle, Hamilton's principle for nonholonomic systems, and conservation theorems. Key points include using Hamilton's principle to find the path that makes the action integral stationary, using the calculus of variations to find such paths, deriving Lagrange's equations by making the variation of the action integral equal to zero, and handling constraints using Lagrange multipliers.
Solar cells directly convert sunlight into electricity and were first used in spacecraft. Traditional solar cells use two types of silicon sandwiched together, while newer types are still in development. Solar cells work by using photons to separate electron-hole pairs in materials like titanium dioxide.
Nuclear fission involves the splitting of heavy atomic nuclei into lighter nuclei. It occurs through induced fission where a neutron is absorbed, causing the nucleus to split into fission products along with more neutrons. Nuclear fusion combines two atomic nuclei into a heavier one and releases energy in the process. It is the process that powers stars like our Sun, which generates energy through the fusion of hydrogen into helium in its core.
This document provides an overview of superconductivity. It begins with definitions and the discovery of superconductivity by Kamerlingh Onnes in 1911. It describes the Meissner effect and the development of BCS theory in 1957 to explain superconductivity through Cooper pairs. It outlines several properties of superconductors like the Josephson effect and reviews applications such as magnetic levitation and power transmission. It concludes by noting superconductivity enables enormous power transfer without loss below critical temperatures.
1. Nuclear models like the liquid drop model and shell model describe aspects of nuclear structure and behavior. The liquid drop model treats the nucleus like a liquid drop while the shell model treats nucleons as moving independently in nuclear orbits.
2. The shell model explains nuclear magic numbers and properties like spin and parity. Magic numbers correspond to nuclear stability when the number of protons or neutrons equals 2, 8, 20, 28, 50, 82, etc. The shell model accounts for magic numbers in terms of closed nuclear shells.
3. While insightful, nuclear models have limitations and do not fully describe all nuclear phenomena. The liquid drop model cannot explain magic numbers while the shell model fails to explain the stability of certain
The document provides an outline for a course on quantum mechanics. It discusses key topics like the time-dependent Schrodinger equation, eigenvalues and eigenfunctions, boundary conditions for wave functions, and applications like the particle in a box model. Specific solutions to the Schrodinger equation are explored for stationary states with definite energy, including the wave function for a free particle and the quantization of energy for a particle confined to a one-dimensional box.
This document discusses conductors, insulators, and semiconductors. It explains that semiconductors are metalloids that have a small band gap between the valence and conduction bands, allowing electrical conductivity to increase with temperature. Semiconducting elements like silicon and germanium form the basis of solid state electronic devices. Doping semiconductors with other elements can produce either n-type or p-type materials, and joining n-type and p-type materials creates a p-n junction that can function as a rectifier. The transistor was invented in 1947 at Bell Labs and has revolutionized electronics, with integrated circuits continuing to shrink in size following Moore's Law.
This document provides an overview of elementary particles. It discusses their classification into baryons, leptons, and mesons. Baryons include protons, neutrons, and heavier hyperons. Leptons contain electrons, photons, neutrinos, and muons. Mesons have masses between baryons and leptons. Each particle is described along with its properties. The document also discusses particles and their antiparticles, and conservation laws related to parity, charge conjugation, time reversal, and the combined CPT symmetry.
origin of quantum physics -
Inadequacy of classical mechanics and birth of QUANTUM PHYSICS
ref: Quantum mechanics: concepts and applications, N. Zettili
The document discusses the elementary particles that make up the universe. It explains that all matter is composed of atoms, which themselves are made of electrons, protons, and neutrons. Protons and neutrons are composed of quarks. The elementary particles are divided into three families with similar properties. Tables are provided that summarize the key properties of each particle such as mass, electric charge, strong charge, and weak charge. There are four fundamental forces that act on these particles: gravitation, electromagnetism, strong nuclear force, and weak nuclear force.
Dr. Kamal K. Ali's lecture discusses the structure of atoms and radioactivity. It covers topics like the atom structure, isotopes, radioactive decay mechanisms, and types of radiation. It also explains techniques used to measure isotopes like mass spectrometry. Mass spectrometry works by ionizing atoms, accelerating the ions, and separating them in a magnetic field based on their mass-to-charge ratio. This allows determining the relative abundances of isotopes in a sample.
Metamaterials are artificial materials engineered to have properties that are not found in nature. They derive their properties from their structure rather than composition. Depending on their structure, metamaterials can have a refractive index less than 1 or even negative refractive index. Left-handed materials have a negative refractive index. While natural materials cannot simultaneously exhibit negative permittivity and permeability, metamaterials can be designed with these properties. Potential applications of metamaterials include antennas, superlenses beyond the diffraction limit, cloaking devices, and modeling conditions of the big bang.
The document summarizes key aspects of the diamond lattice structure formed by carbon atoms in a diamond crystal. It describes the diamond lattice as being formed by two interpenetrating face-centered cubic (fcc) sublattices offset by 1/4 cube edge. Each carbon atom has four nearest neighbors in a tetrahedral structure, giving the diamond lattice a coordination number of 4. The packing factor of the diamond lattice is 0.34, indicating it is a loosely packed structure.
This document provides an overview of nonlinear optics and second harmonic generation. It begins with an introduction to lasers and their components. It then discusses symmetry operations in crystals and how centrosymmetric and noncentrosymmetric materials affect nonlinear polarization. Maxwell's equations are presented for linear media. The document introduces nonlinear optics and lists various nonlinear optical effects such as second harmonic generation. It derives the wave equation for nonlinear media and shows how second harmonic generation leads to frequency doubling. Examples of nonlinear crystals used for second harmonic generation are also provided.
This document summarizes research on synthesizing ternary cadmium chalcogenide quantum dots (QDs) with a gradient structure and tunable bandgaps. The QDs were loaded onto mesoporous titanium dioxide films using electrophoretic deposition to create quantum dot solar cells (QDSCs). Sequentially depositing different sized QDs with varying bandgaps improved light absorption and increased power conversion efficiency compared to mixing the QDs. Further studies are investigating the synergistic electron or energy transfer mechanisms enabling the improved performance. In conclusion, the layer-by-layer QD structure maximizes light harvesting for QDSCs across the visible spectrum.
Lecture slides from a class introducing quantum mechanics to non-majors, giving an overview of black-body radiation, the photoelectric effect, and the Bohr model. Used as part of a course titled "A Brief history of Timekeeping," as a lead-in to talking about atomic clocks
Basics refresher on Laser Technology and it's applications. Presentation prepared by (and for) student(s). Level- Karnataka State Pre-university PUC1(India)
Quantum dots are semiconductor nanocrystals that can emit light of varying wavelengths depending on their size. They have applications in display technology where they can convert blue light into red or green light. There are different types of quantum dot display systems including photo-enhanced, photo-emissive, and electro-emissive systems. Quantum dots offer benefits like high brightness, energy efficiency, and pure color emission.
Here is a semi-log plot of the data with an exponential trendline:
The equation of the trendline is:
y = 12456e-0.4693x
Taking the natural log of both sides:
ln(y) = ln(12456) - 0.4693x
The slope is -0.4693
Using the equation:
t1/2 = 0.693/λ
λ = 0.4693
t1/2 = 0.693/0.4693 = 1.5
Therefore, the half-life of the isotope is 1.5 intervals, or 1.5 x 30 s = 45 seconds.
Quantum dots are tiny semiconductor crystals between 2-10 nanometers in size. Their properties depend on factors like size and energy levels. Smaller quantum dots emit higher frequency/shorter wavelength blue light while larger dots emit lower frequency/longer wavelength red light. Quantum dots have potential applications in solar cells, displays, and medical imaging due to their tunable light emission and other optical properties. They are typically made through colloidal synthesis which allows for mass production under mild conditions.
This document provides an overview of supercapacitors. It discusses what supercapacitors are, their history, basic design involving two electrodes separated by an ion permeable membrane, how they work by forming an electric double layer when charged, the materials used such as carbon nanotubes for electrodes and electrolytes, their features like high energy storage and charge/discharge rates, applications including use in buses and backup power systems, and advantages like long lifespan and eco-friendliness with disadvantages like low energy density and high cost.
This document discusses molecules and the different types of bonds that hold atoms together to form compounds. It describes ionic bonds, which form when one atom transfers electrons to another, and covalent bonds, which form when atoms share electrons. The document also discusses molecular spectra arising from rotational and vibrational energy levels of molecules, and how infrared spectroscopy can analyze molecular vibrations. Potential energy graphs illustrate the attractive and repulsive forces between atoms at different distances that determine molecular structure.
This document compares and contrasts linear and nonlinear optics. In linear optics, light propagates through a medium without changing frequency, while in nonlinear optics the medium's response depends on light intensity. Nonlinear optics involves effects where the induced polarization in a medium does not linearly depend on the electric field of the light. This allows frequency conversion via processes like second harmonic generation and sum frequency generation. Materials can exhibit a nonlinear refractive index, leading to self-focusing or defocusing of high intensity light beams. Nonlinear optical effects enable applications like frequency conversion, optical limiting, and all-optical signal processing.
Density of States (DOS) in Nanotechnology by Manu ShreshthaManu Shreshtha
1. The document discusses density of states (DOS), which describes the number of accessible quantum states at each energy level in a system. It explains how electrons populate energy bands based on DOS and the Fermi distribution function.
2. Calculation of DOS for a semiconductor is shown, and applications like quantization in low-dimensional structures and photonic crystals are described. Impurity bands formed by dopants are also discussed.
3. In summary, the document provides an overview of density of states, how it is calculated, and its applications in areas like quantization effects and photonic crystals.
This document provides a summary of key concepts from an initial construction of the SU(2)xU(1) model that represents the electroweak sector of the Standard Model of particle physics. It includes:
1) A partially unified Lagrangian that incorporates a left-handed fermion doublet, right-handed singlet, Higgs scalar doublet, and SU(2) and U(1) gauge fields.
2) Gauge transformations of the fields that leave the Lagrangian invariant, including transformations of the gauge bosons required to cancel terms.
3) Transformations of the left-handed fermion doublet under the SU(2)xU(1) subgroup and resulting transformations of covariant derivatives and
This document provides a summary of key concepts from an initial construction of the SU(2)xU(1) model that represents the electroweak sector of the Standard Model of particle physics. It includes:
1) A partially unified Lagrangian that incorporates a left-handed fermion doublet, right-handed fermion singlet, and Higgs scalar doublet, as well as SU(2) and U(1) gauge fields.
2) Transformations of the left-handed fermion doublet and gauge fields under the SU(2)xU(1) gauge group that are required to maintain invariance of the Lagrangian.
3) Gauge transformations of the SU(2) field
1. Nuclear models like the liquid drop model and shell model describe aspects of nuclear structure and behavior. The liquid drop model treats the nucleus like a liquid drop while the shell model treats nucleons as moving independently in nuclear orbits.
2. The shell model explains nuclear magic numbers and properties like spin and parity. Magic numbers correspond to nuclear stability when the number of protons or neutrons equals 2, 8, 20, 28, 50, 82, etc. The shell model accounts for magic numbers in terms of closed nuclear shells.
3. While insightful, nuclear models have limitations and do not fully describe all nuclear phenomena. The liquid drop model cannot explain magic numbers while the shell model fails to explain the stability of certain
The document provides an outline for a course on quantum mechanics. It discusses key topics like the time-dependent Schrodinger equation, eigenvalues and eigenfunctions, boundary conditions for wave functions, and applications like the particle in a box model. Specific solutions to the Schrodinger equation are explored for stationary states with definite energy, including the wave function for a free particle and the quantization of energy for a particle confined to a one-dimensional box.
This document discusses conductors, insulators, and semiconductors. It explains that semiconductors are metalloids that have a small band gap between the valence and conduction bands, allowing electrical conductivity to increase with temperature. Semiconducting elements like silicon and germanium form the basis of solid state electronic devices. Doping semiconductors with other elements can produce either n-type or p-type materials, and joining n-type and p-type materials creates a p-n junction that can function as a rectifier. The transistor was invented in 1947 at Bell Labs and has revolutionized electronics, with integrated circuits continuing to shrink in size following Moore's Law.
This document provides an overview of elementary particles. It discusses their classification into baryons, leptons, and mesons. Baryons include protons, neutrons, and heavier hyperons. Leptons contain electrons, photons, neutrinos, and muons. Mesons have masses between baryons and leptons. Each particle is described along with its properties. The document also discusses particles and their antiparticles, and conservation laws related to parity, charge conjugation, time reversal, and the combined CPT symmetry.
origin of quantum physics -
Inadequacy of classical mechanics and birth of QUANTUM PHYSICS
ref: Quantum mechanics: concepts and applications, N. Zettili
The document discusses the elementary particles that make up the universe. It explains that all matter is composed of atoms, which themselves are made of electrons, protons, and neutrons. Protons and neutrons are composed of quarks. The elementary particles are divided into three families with similar properties. Tables are provided that summarize the key properties of each particle such as mass, electric charge, strong charge, and weak charge. There are four fundamental forces that act on these particles: gravitation, electromagnetism, strong nuclear force, and weak nuclear force.
Dr. Kamal K. Ali's lecture discusses the structure of atoms and radioactivity. It covers topics like the atom structure, isotopes, radioactive decay mechanisms, and types of radiation. It also explains techniques used to measure isotopes like mass spectrometry. Mass spectrometry works by ionizing atoms, accelerating the ions, and separating them in a magnetic field based on their mass-to-charge ratio. This allows determining the relative abundances of isotopes in a sample.
Metamaterials are artificial materials engineered to have properties that are not found in nature. They derive their properties from their structure rather than composition. Depending on their structure, metamaterials can have a refractive index less than 1 or even negative refractive index. Left-handed materials have a negative refractive index. While natural materials cannot simultaneously exhibit negative permittivity and permeability, metamaterials can be designed with these properties. Potential applications of metamaterials include antennas, superlenses beyond the diffraction limit, cloaking devices, and modeling conditions of the big bang.
The document summarizes key aspects of the diamond lattice structure formed by carbon atoms in a diamond crystal. It describes the diamond lattice as being formed by two interpenetrating face-centered cubic (fcc) sublattices offset by 1/4 cube edge. Each carbon atom has four nearest neighbors in a tetrahedral structure, giving the diamond lattice a coordination number of 4. The packing factor of the diamond lattice is 0.34, indicating it is a loosely packed structure.
This document provides an overview of nonlinear optics and second harmonic generation. It begins with an introduction to lasers and their components. It then discusses symmetry operations in crystals and how centrosymmetric and noncentrosymmetric materials affect nonlinear polarization. Maxwell's equations are presented for linear media. The document introduces nonlinear optics and lists various nonlinear optical effects such as second harmonic generation. It derives the wave equation for nonlinear media and shows how second harmonic generation leads to frequency doubling. Examples of nonlinear crystals used for second harmonic generation are also provided.
This document summarizes research on synthesizing ternary cadmium chalcogenide quantum dots (QDs) with a gradient structure and tunable bandgaps. The QDs were loaded onto mesoporous titanium dioxide films using electrophoretic deposition to create quantum dot solar cells (QDSCs). Sequentially depositing different sized QDs with varying bandgaps improved light absorption and increased power conversion efficiency compared to mixing the QDs. Further studies are investigating the synergistic electron or energy transfer mechanisms enabling the improved performance. In conclusion, the layer-by-layer QD structure maximizes light harvesting for QDSCs across the visible spectrum.
Lecture slides from a class introducing quantum mechanics to non-majors, giving an overview of black-body radiation, the photoelectric effect, and the Bohr model. Used as part of a course titled "A Brief history of Timekeeping," as a lead-in to talking about atomic clocks
Basics refresher on Laser Technology and it's applications. Presentation prepared by (and for) student(s). Level- Karnataka State Pre-university PUC1(India)
Quantum dots are semiconductor nanocrystals that can emit light of varying wavelengths depending on their size. They have applications in display technology where they can convert blue light into red or green light. There are different types of quantum dot display systems including photo-enhanced, photo-emissive, and electro-emissive systems. Quantum dots offer benefits like high brightness, energy efficiency, and pure color emission.
Here is a semi-log plot of the data with an exponential trendline:
The equation of the trendline is:
y = 12456e-0.4693x
Taking the natural log of both sides:
ln(y) = ln(12456) - 0.4693x
The slope is -0.4693
Using the equation:
t1/2 = 0.693/λ
λ = 0.4693
t1/2 = 0.693/0.4693 = 1.5
Therefore, the half-life of the isotope is 1.5 intervals, or 1.5 x 30 s = 45 seconds.
Quantum dots are tiny semiconductor crystals between 2-10 nanometers in size. Their properties depend on factors like size and energy levels. Smaller quantum dots emit higher frequency/shorter wavelength blue light while larger dots emit lower frequency/longer wavelength red light. Quantum dots have potential applications in solar cells, displays, and medical imaging due to their tunable light emission and other optical properties. They are typically made through colloidal synthesis which allows for mass production under mild conditions.
This document provides an overview of supercapacitors. It discusses what supercapacitors are, their history, basic design involving two electrodes separated by an ion permeable membrane, how they work by forming an electric double layer when charged, the materials used such as carbon nanotubes for electrodes and electrolytes, their features like high energy storage and charge/discharge rates, applications including use in buses and backup power systems, and advantages like long lifespan and eco-friendliness with disadvantages like low energy density and high cost.
This document discusses molecules and the different types of bonds that hold atoms together to form compounds. It describes ionic bonds, which form when one atom transfers electrons to another, and covalent bonds, which form when atoms share electrons. The document also discusses molecular spectra arising from rotational and vibrational energy levels of molecules, and how infrared spectroscopy can analyze molecular vibrations. Potential energy graphs illustrate the attractive and repulsive forces between atoms at different distances that determine molecular structure.
This document compares and contrasts linear and nonlinear optics. In linear optics, light propagates through a medium without changing frequency, while in nonlinear optics the medium's response depends on light intensity. Nonlinear optics involves effects where the induced polarization in a medium does not linearly depend on the electric field of the light. This allows frequency conversion via processes like second harmonic generation and sum frequency generation. Materials can exhibit a nonlinear refractive index, leading to self-focusing or defocusing of high intensity light beams. Nonlinear optical effects enable applications like frequency conversion, optical limiting, and all-optical signal processing.
Density of States (DOS) in Nanotechnology by Manu ShreshthaManu Shreshtha
1. The document discusses density of states (DOS), which describes the number of accessible quantum states at each energy level in a system. It explains how electrons populate energy bands based on DOS and the Fermi distribution function.
2. Calculation of DOS for a semiconductor is shown, and applications like quantization in low-dimensional structures and photonic crystals are described. Impurity bands formed by dopants are also discussed.
3. In summary, the document provides an overview of density of states, how it is calculated, and its applications in areas like quantization effects and photonic crystals.
This document provides a summary of key concepts from an initial construction of the SU(2)xU(1) model that represents the electroweak sector of the Standard Model of particle physics. It includes:
1) A partially unified Lagrangian that incorporates a left-handed fermion doublet, right-handed singlet, Higgs scalar doublet, and SU(2) and U(1) gauge fields.
2) Gauge transformations of the fields that leave the Lagrangian invariant, including transformations of the gauge bosons required to cancel terms.
3) Transformations of the left-handed fermion doublet under the SU(2)xU(1) subgroup and resulting transformations of covariant derivatives and
This document provides a summary of key concepts from an initial construction of the SU(2)xU(1) model that represents the electroweak sector of the Standard Model of particle physics. It includes:
1) A partially unified Lagrangian that incorporates a left-handed fermion doublet, right-handed fermion singlet, and Higgs scalar doublet, as well as SU(2) and U(1) gauge fields.
2) Transformations of the left-handed fermion doublet and gauge fields under the SU(2)xU(1) gauge group that are required to maintain invariance of the Lagrangian.
3) Gauge transformations of the SU(2) field
This document summarizes key aspects of constructing the Standard Model using the SU(2) x U(1) gauge symmetry group. It describes the Lagrangian for this partially unified model, including terms for left-handed fermion doublets, right-handed fermion singlets, and scalar fields. It also discusses how the fermion and gauge boson fields transform under this symmetry group, and how their transformations maintain the invariance of the Lagrangian. Specifically, it shows the transformations of the left-handed fermion doublet and the corresponding transformations required of the U(1) and SU(2) gauge fields.
1+3 gr reduced_as_1+1_gravity_set_1_fordisplayfoxtrot jp R
1. The document details the dimensional reduction of Einstein-Hilbert action from its original 1+3 form to a 1+1 dimensional form. This is done by defining a fundamental line element that encodes lower dimensional metrics for 1+1 spacetime and a compactified 2-sphere.
2. A conformal scalar field is introduced and the higher dimensional metric is written in terms of lower dimensional components and this scalar field. The Einstein-Hilbert action is then written as an effective action by integrating over the angular components.
3. Variations of the effective action lead to the equations of motion for the metric and dilaton field in 1+1 dimensions, in the form of Einstein field equations and a wave equation respectively
This document presents an overview of classical Feynman graphs for the Higgs boson. It expands the classical Higgs boson solution perturbatively up to third order in the coupling constant. In the 0th order of this expansion, the first few classical Feynman graphs are constructed. Specifically, it describes how there are five classical graphs looking at terms in the equations of motion involving the W and Z fields coupled to the 0th order Higgs field. It then provides an example of writing one such solution in coordinate space and transforming it to momentum space, depicting the vertex and propagator that would represent it.
1+3 gr reduced_as_1+1_gravity_set_1 280521fordsplyfoxtrot jp R
1. The document describes dimensional reduction of Einstein-Hilbert action from its original 1+3 form to a 1+1 form. This is done by defining a fundamental line element that encodes lower dimensional metrics for 1+1 spacetime and a compactified 2-sphere.
2. Key steps include writing the higher dimensional metric and curvature scalar in terms of lower dimensional components, and integrating over the 2-sphere coordinates to obtain an effective 2D action.
3. Varying the effective action with respect to the metric and scalar field derives the equations of motion in 1+1 dimensions, identifying an effective energy-momentum tensor and equation of motion for the dilaton field.
Sweeping discussions on dirac field1 update3 sqrdfoxtrot jp R
This document discusses Dirac fields and Lagrangian formulations of electron theory. It presents Lagrangians for an effective electron theory based on an SU(2)XU(1) construction. The Lagrangians include terms for the electron, neutrino interactions, and interactions with gauge bosons. Integration by parts is used to rewrite the action in terms of adjoint spinor fields. Varying the actions yields Dirac's first order equation of motion for the electron field.
Fieldtheoryhighlights2015 setab 28122020verdisplay_typocorrectedfoxtrot jp R
This document provides an overview of field theory concepts, including:
1) Constructing a toy Standard Model using an SU(2)xU(1) symmetry, which unifies scalar and spinor fields into a single Lagrangian.
2) Describing the transformations of fields like the left-handed spinor doublet under the full and diagonal subgroups.
3) Explaining how the U(1) and SU(2) gauge fields can be rotated into a massive Z field and massless photon through mixing angles.
4) Defining covariant derivative operators that incorporate the distinguishing hypercharges of fields.
This document provides an overview of field theory concepts related to constructing a toy Standard Model using an SU(2) x U(1) gauge symmetry. It discusses how scalar and spinor fields can be incorporated into a Lagrangian that respects this symmetry. It describes how the gauge fields transform under subgroups like the diagonal subgroup, and how this relates to the masses of the Z and W bosons. It also discusses the Higgs potential and how it gives mass to the Higgs boson while the Goldstone bosons are eliminated via a gauge condition.
Fieldtheoryhighlights2015 setab 24102020verdisplayfoxtrot jp R
This document provides an overview of field theory and highlights from 2015, focusing on constructing a toy standard model with SU(2)xU(1) symmetry. It describes incorporating scalar, left-handed spinor, and right-handed spinor fields into a Lagrangian that respects this symmetry. The fields each carry distinct hypercharges. It also discusses how the symmetry is realized in a diagonal subgroup by rotating gauge fields into a massive Z field and massless photon. The Higgs field is introduced as a doublet that breaks the symmetry and gives masses via Yukawa couplings.
Fieldtheoryhighlights2015 setab 22092020verdsplyfoxtrot jp R
This document provides an overview of field theory and highlights from 2015, focusing on constructing a toy standard model with SU(2)XU(1) symmetry. It describes how distinct scalar, left-handed spinor, and right-handed spinor fields can be incorporated into a single Lagrangian that observes this symmetry. It also discusses basic practice calculations involving scalar fields, including how the scalar doublet relates to the Higgs boson and Higgs potential.
Fieldtheoryhighlights2015 setabdisplay 18092020foxtrot jp R
This document provides an overview of field theory and highlights from 2015, focusing on constructing a toy standard model with SU(2)xU(1) symmetry. It describes how distinct scalar, left-handed spinor, and right-handed spinor fields can be incorporated into a single Lagrangian that observes this symmetry. It also discusses basic practice calculations involving scalar fields, including how the scalar doublet transforms under diagonal subgroups and its covariant derivative operator. Finally, it touches on the Higgs potential and vacuum states.
Fieldtheoryhighlights2015 setab display_07092020foxtrot jp R
This document provides an overview of field theory and highlights from 2015, focusing on constructing a toy standard model with SU(2)XU(1) symmetry. It describes how distinct scalar, left-handed spinor, and right-handed spinor fields can be incorporated into a single Lagrangian that observes this symmetry. It also discusses basic practice calculations involving scalar fields, including how the scalar doublet transforms under diagonal subgroups and its covariant derivative operator. Higgs boson and potential are introduced.
One particle to_onepartlce_scatteringsqrdcpy1foxtrot jp R
1) A scalar particle travels from one spacetime region to another, carrying momentum k and scattering into momentum k'. This scattering process is described by a scattering matrix.
2) The scattering matrix involves second order derivatives of the vacuum-to-vacuum matrix element with respect to sources J(x) and J(x'). This vacuum-to-vacuum matrix has a factored exponential form.
3) The left-hand side of the scattering matrix gives the probability that a one-particle state with momentum k at initial time Tin will be found with momentum k' at later time Tout, and can be evaluated via path integration.
Very brief highlights on some key details tosssqrdfoxtrot jp R
(1) This document discusses the vacuum-to-vacuum matrix in the presence of a cubic self-interaction term in the Hamiltonian for a scalar field.
(2) The vacuum-to-vacuum matrix, which gives the probability that a particle initially in the vacuum state remains in the vacuum state, is modified by the inclusion of the cubic self-interaction term.
(3) The modified vacuum-to-vacuum matrix is expressed in both coordinate space and momentum space, and can be depicted using Feynman diagrams involving three propagators and two vertices.
One particle to_onepartlce_scattering_18052020foxtrot jp R
1) A scalar particle travels from one spacetime region to another, carrying momentum k and scattering into momentum k'. This scattering process is described by a matrix involving integrals over the initial and final spacetime regions.
2) The scattering matrix involves derivatives of the vacuum-to-vacuum matrix element with respect to sources, representing the interaction of the particle with an external field. This vacuum-to-vacuum element can be written as a Taylor expansion involving the connected scalar classical action.
3) The classical action is a function of the sources and involves a scalar Green's function as a propagator. Differentiation of the classical action yields terms involving the Green's function that are important to the scattering matrix.
This document discusses path integration for a scalar field with one degree of freedom along the time direction. It begins by writing the transition matrix for a scalar field in the absence of time-dependent sources. It then expresses the path integral evaluation of this transition matrix using the action of the scalar field. The scalar field is decomposed into a classical part and a perturbation part. The path integral is then written in factored form using the effective action involving just the classical and perturbation parts. It then discusses taking the Fourier transform of the perturbation part to satisfy the boundary conditions and obtain the Fourier components needed for the path integral.
Outgoing ingoingkleingordon 8th_jun19sqrdfoxtrot jp R
This document discusses solutions to the Klein-Gordon equation in Schwarzschild spacetime near a black hole's event horizon. The radial equation is approximated in the Regge-Wheeler coordinate, leading to oscillatory solutions. These solutions are then expressed as outgoing and ingoing waves, which have different properties in the future and past horizons. Near the horizon, the radial equation simplifies to an oscillatory form, allowing solutions describing outgoing and ingoing waves.
One particle to_onepartlce_scattering_12082020_fordisplayfoxtrot jp R
1) A scalar particle travels from one spacetime region to another, carrying an initial momentum k and scattering into a final momentum k'. This scattering process can be described by a scattering matrix.
2) The scattering matrix involves integrals over the initial and final spacetime regions, which yield Dirac delta functions. It also involves derivatives of the vacuum-to-vacuum matrix with respect to sources.
3) The vacuum-to-vacuum matrix can be written as a Taylor expansion involving the connected scalar classical action, which describes the propagation of a scalar particle via a Green's function.
One particle to_onepartlce_scattering_5302020_pdfcpyfoxtrot jp R
1) A scalar particle travels from one spacetime region to another, carrying an initial momentum k and scattering into a final momentum k'. This scattering process is described by a scattering matrix.
2) The scattering matrix involves second order derivatives of the vacuum-to-vacuum matrix element with respect to sources. This vacuum-to-vacuum matrix can be written as a Taylor expansion involving the connected scalar classical action.
3) The left-hand side of the scattering matrix gives the probability that a one-particle state with momentum k at initial time Tin will be found with momentum k' at final time Tout, and can be evaluated via path integration.
Authoring a personal GPT for your research and practice: How we created the Q...Leonel Morgado
Thematic analysis in qualitative research is a time-consuming and systematic task, typically done using teams. Team members must ground their activities on common understandings of the major concepts underlying the thematic analysis, and define criteria for its development. However, conceptual misunderstandings, equivocations, and lack of adherence to criteria are challenges to the quality and speed of this process. Given the distributed and uncertain nature of this process, we wondered if the tasks in thematic analysis could be supported by readily available artificial intelligence chatbots. Our early efforts point to potential benefits: not just saving time in the coding process but better adherence to criteria and grounding, by increasing triangulation between humans and artificial intelligence. This tutorial will provide a description and demonstration of the process we followed, as two academic researchers, to develop a custom ChatGPT to assist with qualitative coding in the thematic data analysis process of immersive learning accounts in a survey of the academic literature: QUAL-E Immersive Learning Thematic Analysis Helper. In the hands-on time, participants will try out QUAL-E and develop their ideas for their own qualitative coding ChatGPT. Participants that have the paid ChatGPT Plus subscription can create a draft of their assistants. The organizers will provide course materials and slide deck that participants will be able to utilize to continue development of their custom GPT. The paid subscription to ChatGPT Plus is not required to participate in this workshop, just for trying out personal GPTs during it.
The technology uses reclaimed CO₂ as the dyeing medium in a closed loop process. When pressurized, CO₂ becomes supercritical (SC-CO₂). In this state CO₂ has a very high solvent power, allowing the dye to dissolve easily.
ESA/ACT Science Coffee: Diego Blas - Gravitational wave detection with orbita...Advanced-Concepts-Team
Presentation in the Science Coffee of the Advanced Concepts Team of the European Space Agency on the 07.06.2024.
Speaker: Diego Blas (IFAE/ICREA)
Title: Gravitational wave detection with orbital motion of Moon and artificial
Abstract:
In this talk I will describe some recent ideas to find gravitational waves from supermassive black holes or of primordial origin by studying their secular effect on the orbital motion of the Moon or satellites that are laser ranged.
When I was asked to give a companion lecture in support of ‘The Philosophy of Science’ (https://shorturl.at/4pUXz) I decided not to walk through the detail of the many methodologies in order of use. Instead, I chose to employ a long standing, and ongoing, scientific development as an exemplar. And so, I chose the ever evolving story of Thermodynamics as a scientific investigation at its best.
Conducted over a period of >200 years, Thermodynamics R&D, and application, benefitted from the highest levels of professionalism, collaboration, and technical thoroughness. New layers of application, methodology, and practice were made possible by the progressive advance of technology. In turn, this has seen measurement and modelling accuracy continually improved at a micro and macro level.
Perhaps most importantly, Thermodynamics rapidly became a primary tool in the advance of applied science/engineering/technology, spanning micro-tech, to aerospace and cosmology. I can think of no better a story to illustrate the breadth of scientific methodologies and applications at their best.
hematic appreciation test is a psychological assessment tool used to measure an individual's appreciation and understanding of specific themes or topics. This test helps to evaluate an individual's ability to connect different ideas and concepts within a given theme, as well as their overall comprehension and interpretation skills. The results of the test can provide valuable insights into an individual's cognitive abilities, creativity, and critical thinking skills
The binding of cosmological structures by massless topological defectsSérgio Sacani
Assuming spherical symmetry and weak field, it is shown that if one solves the Poisson equation or the Einstein field
equations sourced by a topological defect, i.e. a singularity of a very specific form, the result is a localized gravitational
field capable of driving flat rotation (i.e. Keplerian circular orbits at a constant speed for all radii) of test masses on a thin
spherical shell without any underlying mass. Moreover, a large-scale structure which exploits this solution by assembling
concentrically a number of such topological defects can establish a flat stellar or galactic rotation curve, and can also deflect
light in the same manner as an equipotential (isothermal) sphere. Thus, the need for dark matter or modified gravity theory is
mitigated, at least in part.
The cost of acquiring information by natural selectionCarl Bergstrom
This is a short talk that I gave at the Banff International Research Station workshop on Modeling and Theory in Population Biology. The idea is to try to understand how the burden of natural selection relates to the amount of information that selection puts into the genome.
It's based on the first part of this research paper:
The cost of information acquisition by natural selection
Ryan Seamus McGee, Olivia Kosterlitz, Artem Kaznatcheev, Benjamin Kerr, Carl T. Bergstrom
bioRxiv 2022.07.02.498577; doi: https://doi.org/10.1101/2022.07.02.498577
Current Ms word generated power point presentation covers major details about the micronuclei test. It's significance and assays to conduct it. It is used to detect the micronuclei formation inside the cells of nearly every multicellular organism. It's formation takes place during chromosomal sepration at metaphase.
ESPP presentation to EU Waste Water Network, 4th June 2024 “EU policies driving nutrient removal and recycling
and the revised UWWTD (Urban Waste Water Treatment Directive)”
Describing and Interpreting an Immersive Learning Case with the Immersion Cub...Leonel Morgado
Current descriptions of immersive learning cases are often difficult or impossible to compare. This is due to a myriad of different options on what details to include, which aspects are relevant, and on the descriptive approaches employed. Also, these aspects often combine very specific details with more general guidelines or indicate intents and rationales without clarifying their implementation. In this paper we provide a method to describe immersive learning cases that is structured to enable comparisons, yet flexible enough to allow researchers and practitioners to decide which aspects to include. This method leverages a taxonomy that classifies educational aspects at three levels (uses, practices, and strategies) and then utilizes two frameworks, the Immersive Learning Brain and the Immersion Cube, to enable a structured description and interpretation of immersive learning cases. The method is then demonstrated on a published immersive learning case on training for wind turbine maintenance using virtual reality. Applying the method results in a structured artifact, the Immersive Learning Case Sheet, that tags the case with its proximal uses, practices, and strategies, and refines the free text case description to ensure that matching details are included. This contribution is thus a case description method in support of future comparative research of immersive learning cases. We then discuss how the resulting description and interpretation can be leveraged to change immersion learning cases, by enriching them (considering low-effort changes or additions) or innovating (exploring more challenging avenues of transformation). The method holds significant promise to support better-grounded research in immersive learning.
Remote Sensing and Computational, Evolutionary, Supercomputing, and Intellige...University of Maribor
Slides from talk:
Aleš Zamuda: Remote Sensing and Computational, Evolutionary, Supercomputing, and Intelligent Systems.
11th International Conference on Electrical, Electronics and Computer Engineering (IcETRAN), Niš, 3-6 June 2024
Inter-Society Networking Panel GRSS/MTT-S/CIS Panel Session: Promoting Connection and Cooperation
https://www.etran.rs/2024/en/home-english/
aziz sancar nobel prize winner: from mardin to nobel
Su(2)xu(1)
1. Highlights From SU(2)XU(1) Basic Standard Model Construction
Roa, F. J. P., Bello, A., Urbiztondo, L.
Abstract
In this draft we present some important highlights taken from our study course in the subject of Standard
Model of particle physics although in this present draft we are limited only to discuss the basics of SU(2)XU(1)
construction. The highlights exclude the necessary additional neutrinos aside from the left-handed ones which are
presented here as massless.
Keywords:
1. Introduction
This paper serves as an exposition on an
initial and partial construction of SU(2)XU(1) model
in Quantum Field Theory whose complete
SU(2)XU(1) structure represents the Electro-Weak
Standard model. The discussions center on
Lagrangian that must be invariant or symmetric under
the SU(2)XU(1) gauge group. It must be noted that
the whole of The Standard Model has the
mathematical symmetry of the SU(3)XSU(2)XU(1)
gauge group to include the Strong interaction that
goes by the name of Chromodynamics. Such is
ofcourse beyond the scope of this present draft.
In its present draft, this paper is mainly
based on our group’s study notes that include our
answers to some basic exercises and workouts
required for progression. So we might have used
some notations by our own convenient choice though
as we understand these contain the same notational
significance as that used in our main references.
The initial and partial SU(2)XU(1)
construction presented here is intended primarily to
illustrate gauge transformation of fields and how such
fields must transform so as to observe invariance or
symmetry of the given Lagrangian.
Concerning neutrinos, the Dirac left-handed
spinor doublet discussed here aside from the left-
handed electron it contains, it also has a left-handed
neutrino that is rendered massless in the Yukawa
coupling terms. In addition to these, the other
Fermion is the right-handed electron. As there is only
one left-handed spinor doublet and one right-handed
spinor singlet no other type of fermions such as
additional neutrinos are present in this initial and
partial SU(2)XU(1) construction. In a later section, it
will be shown how this left-handed neutrino is made
massless in the mentioned Yukawa coupling terms.
2. Partially Unified Lagrangian
Let us start our highlights say with a
partially unified Lagrangian,
ℒ( 𝑆𝑈(2) × 𝑈(1)) 𝑃𝑎𝑟𝑡 = ℒ( 𝜓 𝐿
, 𝜓2
𝑅
, 𝜙 ) +
ℒ( 𝑊, 𝐵 )
(1.1)
This is for fields under the 𝑆𝑈(2) × 𝑈(1) gauge
symmetry group [1]. In this, the necessary additional
fermions in the complete Electro-Weak theory [2] are
not yet included. The basic fermions present here are
contained in the component Lagrangian
ℒ( 𝜓 𝐿
, 𝜓2
𝑅
, 𝜙 ) = 𝑖𝜓̅ 𝐿
𝛾 𝜇
𝐷𝜇(𝐿) 𝜓 𝐿
+
𝑖𝜓̅2
𝑅
𝛾 𝜇
𝐷𝜇(𝑅) 𝜓2
𝑅
−
𝑦( 𝜓̅2
𝑅
𝜙 †
𝜓 𝐿
+ 𝜓̅ 𝐿
𝜙𝜓2
𝑅 ) +
1
2
| 𝐷𝜇 𝜙|
2
− 𝑉(𝜙)
(1.2)
This component Lagrangian incorporates a
Left-handed spinor doublet, 𝜓 𝐿
, Right-handed spinor
singlet 𝜓2
𝑅
and scalar doublet 𝜙. The Left-handed
spinor doublet consists of initial Left-handed
Fermions – the left-handed neutrino 𝜓1
𝐿
and the left-
handed electron, 𝜓2
𝐿
. The right-handed spinor singlet
represents for the right-handed electron, while the
scalar doublet represents for the Higgs field, which
consists of a vacuum expectation value (vev) and a
scalar component called the Higgs Boson, then three
Goldstone bosons.
2. As a partially unified Lagrangian under the
cited gauge symmetry group, Lagrangian (1.1) also
consists of a component part ℒ( 𝑊, 𝐵 ) that contains
the three components of 𝑆𝑈(2) vector gauge boson
field 𝑊⃗⃗⃗ and one 𝑈(1) vector gauge boson field, 𝐵𝜇.
Such component Lagrangian is given by[3, 4]
ℒ( 𝑊, 𝐵 ) = ℒ 𝑊 + ℒ 𝐵
(1.3)
where one subcomponent goes for the boson field 𝑊⃗⃗⃗
ℒ 𝑊 = −
1
4
𝐹𝜇𝜈 ∙ 𝐹 𝜇𝜈
= −
1
4
∑ 𝐹𝜇𝜈
(𝑖)
𝐹(𝑖)
𝜇𝜈
3
𝑖 =1
(1.4)
(We note: Greek index as space index, while Latin
index as particle index.)
The anti-symmetric tensor 𝐹𝜇𝜈 in (1.4) is
given by
𝐹𝜇𝜈 = 𝜕𝜇 𝑊⃗⃗⃗𝜈 − 𝜕𝜈 𝑊⃗⃗⃗ 𝜇 − 2𝑄′𝑊⃗⃗⃗𝜇 × 𝑊⃗⃗⃗𝜈
(1.5)
The 𝑆𝑈(2) vector gauge boson takes three
components, 𝑊⃗⃗⃗ = ( 𝑊𝜇
(1)
, 𝑊𝜇
(2)
𝑊𝜇
(3)
), where Latin
indices take parameter values 1, 2, 3. In short hand,
we write for a component in the cross product as [5]
𝐴 × 𝐵⃗ | 𝑎
= 𝜀 𝑎𝑏𝑐 𝐴 𝑏
𝐵 𝑐
(1.6)
This is written in terms of the components 𝜀 𝑎𝑏𝑐 of
Levi-Civita symbol.
The remaining subcomponent of (1.3) is for
the solely U(1) gauge boson 𝐵𝜇 whose Lagrangian in
turn is given by
ℒ 𝐵 = −
1
4
( 𝜕𝜇 𝐵 𝜈 − 𝜕𝜈 𝐵𝜇)
2
(1.7)
We must also take note the complex linear
combinations that give out the W-plus and W-minus
gauge bosons
𝑊𝜇
(±)
=
1
√2
( 𝑊𝜇
(1)
± 𝑖 𝑊𝜇
(2)
) (1.8)
and the SO(2)-like rotations
𝑍 𝜇 = 𝐵𝜇 𝑠𝑖𝑛𝛼 − 𝑊(3)𝜇 𝑐𝑜𝑠𝛼 (1.9.1)
𝐴𝜇
𝑒𝑚
= 𝐵𝜇 𝑐𝑜𝑠𝛼 + 𝑊(3)𝜇 𝑠𝑖𝑛𝛼 (1.9.2)
with respect to the mixing angle alpha, which mixing
(rotation-like) gives out one massive Z field and one
massless gauge boson that represents the
electromagnetic field 𝐴𝜇
𝑒𝑚
.
3. Transformations Under The
SU(2)XU(1) Subgroups
In this section, we highlight the left-handed
spinor doublet as the specific illustration whose
𝑆𝑈(2) × 𝑈(1) 𝐿 subgroup is characterized by the
hypercharge 𝑌𝐿, a label we choose by our own
convenient notation.Such subgroup is represented by
the matrix
𝑒−𝑖𝑌 𝐿 𝜒 𝑞
𝑒−𝑖𝑄′𝜎⃗⃗ ∙ 𝜒⃗⃗
(2.1)
This is in exponentiated form, where 𝜎𝑖
(𝑖 = 1, 2, 3)
are the Pauli matrices. We must make the
identifications
𝑄′𝜎 ∙ 𝜒 = 𝑄′∑ 𝜎𝑖 𝜒 𝑖
3
𝑖 =1
𝜒 𝑞 = 𝑄′𝜒3
(2.2)
Associated with this particular subgroup is
the covariant derivative operator for the left-handed
spinor doublet as characterized also by the
hypercharge, 𝑌𝐿.
𝐷𝜇(𝐿) = 𝜕𝜇 + 𝑖𝑄𝑌𝐿 𝐵𝜇 + 𝑖 𝑄′ ∑ 𝜎𝑖 𝑊(𝑖)𝜇
3
𝑖 =1
(2.3)
We see in this that the hypercharge goes along with
the U(1) gauge field.
We note in the matrix (2.1) the U(1) part as
given by 𝑒−𝑖𝑌 𝐿 𝜒 𝑞
, while the SU(2) part by the 2X2
matrix 𝑒−𝑖𝑄′𝜎⃗⃗ ∙ 𝜒⃗⃗
. Under this subgroup, the left-handed
spinor doublet transforms as
𝜓 𝐿
→ 𝑒−𝑖𝑌 𝐿 𝜒 𝑞 𝑒−𝑖𝑄′𝜎⃗⃗ ∙ 𝜒⃗⃗
𝜓 𝐿
(2.4)
So to first order in 𝑄′ this will result in the
transformation of covariant derivative operation
𝐷𝜇(𝐿) 𝑒−𝑖𝑌𝐿 𝜒 𝑞
𝑒−𝑖𝑄′𝜎⃗⃗ ∙ 𝜒⃗⃗
𝜓 𝐿
=
𝑒−𝑖𝑌𝐿 𝜒 𝑞
𝑒−𝑖𝑄′𝜎⃗⃗ ∙ 𝜒⃗⃗ ( 𝜕𝜇 + 𝑖𝑄𝑌𝐿( 𝐵𝜇 − 𝑄−1
𝜕𝜇 𝜒 𝑞) +
𝑖𝑄′𝜎 ∙ ( 𝑊⃗⃗⃗𝜇 − 𝜕𝜇 𝜒 − 2𝑄′
𝜒 × 𝑊⃗⃗⃗ 𝜇) ) 𝜓 𝐿
(2.5)
For our present purposes let us take the
invariance of Lagrangian (1.1) with respect to the
transformation of the left-handed spinor doublet that
is given in (2.4) under the 𝑆𝑈(2) × 𝑈(1) 𝐿 gauge
group. This invariance requires that the gauge vector
bosons must also transformin the following ways
𝐵𝜇 → 𝐵𝜇 + 𝑄−1
𝜕𝜇 𝜒 𝑞 (2.6.1)
for the U(1) gauge field, while to first order in 𝑄′, the
𝑆𝑈(2) vector boson transforms as
𝑊⃗⃗⃗𝜇 → 𝑊⃗⃗⃗𝜇 + 𝜕𝜇 𝜒 + 2𝑄′
𝜒 × 𝑊⃗⃗⃗ 𝜇 (2.6.2)
Such transformations are needed to cancel the extra
3. terms picked up in (2.5) when the left-handed spinor
doublet transforms under its own gauge subgroup.
For these results, it is fairly straightforward
exercise to obtain the following approximated
identity
𝜎 ∙ 𝑊⃗⃗⃗ 𝜇 𝑒−𝑖𝑌 𝐿 𝜒 𝑞
𝑒−𝑖𝑄′𝜎⃗⃗
∙ 𝜒⃗⃗
𝜓 𝐿
≈
𝑒−𝑖𝑌 𝐿 𝜒 𝑞
𝑒−𝑖𝑄′𝜎⃗⃗ ∙ 𝜒⃗⃗ ( 𝜎 ∙ 𝑊⃗⃗⃗𝜇 +
𝑖𝑄′[( 𝜎 ∙ 𝜒), (𝜎 ∙ 𝑊⃗⃗⃗𝜇 )] ) 𝜓 𝐿
(2.7.1)
in which we note of the commutator
[( 𝜎 ∙ 𝜒), (𝜎 ∙ 𝑊⃗⃗⃗𝜇 )] = 𝑖2𝜎 ∙ ( 𝜒 × 𝑊⃗⃗⃗𝜇 )
(2.7.2)
which is also a straightforward exercise to prove.
Given the SU(2) gauge transformation
(2.6.2), the W-gauge boson Lagrangian ℒ 𝑊 also
transforms as
−4ℒ 𝑊 = 𝐹𝜇𝜈 ∙ 𝐹 𝜇𝜈
→ 𝐹𝜇𝜈 ∙ 𝐹 𝜇𝜈
+
2(2)𝑄′𝐹𝜇𝜈 ∙ (𝜒 × 𝐹 𝜇𝜈
)
(2.8.1)
This is also taken to first order in 𝑄′. By cyclic
permutation we note that
𝐹𝜇𝜈 ∙ ( 𝜒 × 𝐹 𝜇𝜈) = 𝜒 ∙ ( 𝐹 𝜇𝜈
× 𝐹𝜇𝜈 ) = 0
(2.8.2)
This drops the second major term of (2.8.1) off,
proving the invariance of ℒ 𝑊 under gauge
transformation.
We can proceed considering the given
Spinor doublet under the 𝑆𝑈(2) × 𝑈(1) 𝐿 diagonal
subgroup whose matrix is given by
𝑒−𝑖𝑌 𝐿 𝜒 𝑞
𝑒−𝑖𝜎3 𝜒 𝑞
= 𝑑𝑖𝑎𝑔( 𝑒−𝑖(1+𝑌 𝐿)𝜒 𝑞
, 𝑒 𝑖(1−𝑌 𝐿)𝜒 𝑞)
(2.9.1)
This matrix utilizes the 𝜎3 Pauli matrix and the
Spinor doublet transforms as
𝜓 𝐿
→ 𝑒−𝑖𝑌 𝐿 𝜒 𝑞
𝑒−𝑖𝜎3 𝜒 𝑞
𝜓 𝐿
(2.9.2)
It is to be noted that as a doublet this Spinor doublet
is a 2X1 column vector wherein each element in a
row is a left-handed Dirac spinor in itself.
𝜓 𝐿
= (
𝜓1
𝐿
𝜓2
𝐿
) (2.9.3)
In this draft the authors’convenient notation
for each of these left-handed Dirac spinors is given
by
𝜓𝑖
𝐿
=
1
2
(1 + 𝛾5) 𝜓𝑖 (2.9.4)
with Hermitian left-handed ad joint spinor given as
𝜓̅ 𝑖
𝐿
= (𝜓𝑖
𝐿
)†
𝛾0
=
1
2
𝜓̅ 𝑖
(1 − 𝛾5) (2.9.5)
In our notations, our fifth Dirac gamma matrix 𝛾5
has
the immediate property
𝛾5
= −𝛾5 (2.9.6)
Alternatively, under this diagonal subgroup
and given (1.9.1) and (1.9.2), we can write the
covariant left-handed derivative operator in terms of
the 𝑍 𝜇 field and the electromagnetic field, 𝐴𝜇
𝑒𝑚
.
𝐷𝜇(𝐿) = 𝜕𝜇 + 𝑖𝑄′( 𝜎1 𝑊(1)𝜇 + 𝜎2 𝑊(2)𝜇) +
𝑖𝑄′
𝑐𝑜𝑠𝛼
( 𝑌𝐿 𝑠𝑖𝑛2
𝛼 − 𝜎3 𝑐𝑜𝑠2
𝛼 ) 𝑍 𝜇 +
𝑖𝑄′( 𝜎3 + 𝑌𝐿
) 𝐴𝜇
𝑒𝑚
𝑠𝑖𝑛𝛼 (2.10)
It is to be noted that 𝑆𝑈(2) × 𝑈(1) 𝐿 is non-
Abelian gauge group whose generators (the Pauli
matrices) do not commute so that we can have the
following results
𝜎1 𝑒−𝑖𝜎3 𝜒 𝑞
= 𝑒−𝑖𝜎3 𝜒 𝑞( 𝜎1 𝑐𝑜𝑠2𝜒 𝑞 − 𝜎2 𝑠𝑖𝑛2𝜒 𝑞 )
(2.11.1)
and
𝜎2 𝑒−𝑖𝜎3 𝜒 𝑞
= 𝑒−𝑖𝜎3 𝜒 𝑞( 𝜎1 𝑠𝑖𝑛2𝜒 𝑞 + 𝜎2 𝑐𝑜𝑠2𝜒 𝑞 )
(2.11.2)
As the Left-handed spinor doublet
transforms under (2.9.2) the covariant differentiation
with (2.10) also takes the corresponding
transformation
𝐷𝜇(𝐿) 𝑒−𝑖𝑌𝐿 𝜒 𝑞
𝑒−𝑖𝜎3 𝜒 𝑞
𝜓 𝐿
= 𝑒−𝑖𝑌 𝐿 𝜒 𝑞
𝑒−𝑖𝜎3 𝜒 𝑞 ( 𝜕𝜇 −
𝑖( 𝑌𝐿 + 𝜎3
) 𝜕𝜇 𝜒 𝑞 + 𝑖𝑄𝑌𝐿 𝐵𝜇 + 𝑖 𝑄′( 𝜎1 𝑊′
(1) 𝜇 +
𝜎2 𝑊′
(2) 𝜇) + 𝑖 𝑄′
𝜎3 𝑊(3)𝜇 ) 𝜓 𝐿
(2.12)
where we take note of the SO(2) like rotations
𝑊(1)𝜇 → 𝑊′
(1) 𝜇 = 𝑊(1)𝜇 𝑐𝑜𝑠2𝜒 𝑞 + 𝑊(2)𝜇 𝑠𝑖𝑛2𝜒 𝑞
𝑊(2)𝜇 → 𝑊′
(2) 𝜇 = −𝑊(1)𝜇 𝑠𝑖𝑛2𝜒 𝑞 + 𝑊(2)𝜇 𝑐𝑜𝑠2𝜒 𝑞
(2.13)
A quick drill would show the invariance
∑ 𝑊′
( 𝑖) 𝜇 𝑊′(𝑖)
𝜇
2
𝑖 =1
= ∑ 𝑊( 𝑖) 𝜇 𝑊(𝑖)
𝜇
2
𝑖 =1
(2.14)
Corresponding to the transformation (2.12)
of covariant differentiation is the U(1) like gauge
transformation of 𝑊(3)𝜇.
𝑊(3)𝜇 → 𝑊(3)𝜇 + 𝑄′−1
𝜕𝜇 𝜒 𝑞
(2.15.1)
These transformations consequently lead to
U(1) gauge transformation of 𝐴𝜇
𝑒𝑚
.
𝐴𝜇
𝑒𝑚
→ 𝐴𝜇
𝑒𝑚
+ 𝛿𝐴𝜇
𝑒𝑚
4. 𝛿𝐴𝜇
𝑒𝑚
= ( 𝑄−1
𝑐𝑜𝑠𝛼 + 𝑄′−1
𝑠𝑖𝑛𝛼 ) 𝜕𝜇 𝜒 𝑞 =
2𝑒−1
𝜕𝜇 𝜒 𝑞
(2.15.2)
where
𝑄′
𝑠𝑖𝑛𝛼 = 𝑄 𝑐𝑜𝑠𝛼 = 𝑒/2 (2.15.3)
The massive 𝑍 𝜇 field stays gauge invariant
𝑍 𝜇 → 𝑍 𝜇 + 𝛿𝑍 𝜇 = 𝑍 𝜇 (2.16.1)
since
𝛿𝑍 𝜇 = ( 𝑄−1
𝑠𝑖𝑛𝛼 − 𝑄′−1
𝑐𝑜𝑠𝛼 ) 𝜕𝜇 𝜒 𝑞 = 0
(2.16.2)
In order to conform with conventional or
that is standard notations, we may have to identify
the spacetime-dependent parameter 𝜒 𝑞 in terms of
Λ(𝑥 𝜇
).
𝜒 𝑞 =
1
2
𝑒Λ (2.17)
so that the U(1) gauge transformation of the
electromagnetic field can be written as
𝐴𝜇
𝑒𝑚
→ 𝐴𝜇
𝑒𝑚
+ 𝜕𝜇 Λ (2.18)
4. The Yukawa Coupling
From (1.2) let us proceed with the Yukawa
coupling.
ℒ 𝑦 = −𝑦( 𝜓̅2
𝑅
𝜙 †
𝜓 𝐿
+ 𝜓̅ 𝐿
𝜙𝜓2
𝑅 ) (3.1.1)
Under all (diagonal) subgroups of
SU(2)XU(1), the transformations lead to the
following end result
𝜓̅ 𝐿
𝜙𝜓2
𝑅
→ 𝜓̅ 𝐿
𝜙𝜓2
𝑅
𝑒−𝑖(1− 𝑌 𝐿)𝜒 𝑞
𝑒−𝑖𝑌 𝑅 𝜒 𝑞
(3.1.2)
or
𝜓̅2
𝑅
𝜙 †
𝜓 𝐿
→ 𝜓̅2
𝑅
𝜙 †
𝜓 𝐿
𝑒 𝑖𝑌 𝑅 𝜒 𝑞
𝑒 𝑖(1− 𝑌𝐿 )𝜒 𝑞
(3.1.3)
We take note in here that to the right-handed
spinor singlet we attribute the hypercharge 𝑌 𝑅.
SU(2)XU(1) symmetry also requires the Yukawa
term to remain invariant under SU(2)XU(1) gauge
transformations. This invariance requires a relation
between hypercharges that is given by
𝑌 𝑅 = 𝑌𝐿 − 1 (3.2)
Under U(1) gauge subgroup the right-
handed spinor singlet transforms as
𝜓2
𝑅
→ 𝑒−𝑖𝑌 𝑅 𝜒 𝑞
𝜓2
𝑅
(3.3.1)
while under the SU(2)XU(1) the scalar doublet
transforms as
𝜙 → 𝑒−𝑖𝜒 𝑞
𝑒−𝑖𝜎3 𝜒 𝑞
𝜙 (3.3.2)
The values of the mentioned hypercharges
play important roles in the coupling or decoupling of
the fields involved in the Yukawa terms. For the left-
handed spinor doublet its hypercharge has the value
𝑌𝐿 = − 1. This value decouples the left-handed
neutrino from the electromagnetic field so that only
the left-handed electron interacts with the
electromagnetic field. This can be seen in the matrix
( 𝜎3 + 𝑌𝐿
) 𝜓 𝐿
= (
0
−2𝜓2
𝐿) (3.4.1)
(As noted.)
(1 + 𝜎3
) 𝜓 𝐿
𝐴𝜇
𝑒𝑚
= (
0
−2𝜓2
𝐿) 𝐴𝜇
𝑒𝑚
(3.4.2)
In (3.2) we consider 1 as the hypercharge
given to the scalar doublet and with this value we see
in the following matrix
(1 + 𝜎3
) 𝜙0 𝐴𝜇
𝑒𝑚
= (0
0
) 𝐴𝜇
𝑒𝑚
(3.4.3)
that the electromagnetic field decouples from the
vacuum expectation value (vev) 𝜙0 of the Higgs field
thus, rendering this electromagnetic field massless.
Conveniently, we can re-group the terms in
(3.1.1) so as to separate out a mass term and an
interaction term.
ℒ 𝑦 = ℒ 𝑦(𝑚𝑎𝑠𝑠 ) + ℒ 𝑦(𝑖𝑛𝑡) (3.5)
The mass term gives masses to the electrons
and the interaction term gives the interaction of the
Higgs boson with fermions that have masses.
Mmmmmmmmmmmmmmmmmmmmmmmm
Mmmmmmmmmmmmmmmmmmmmmmmmmmm
mmmmmmmmmmmmmmmmmmmmmmmmmmm
5. Quantum Electrodynamics (QED)
pieces
Mmmmmmmmmmmmmmmmmmmmmmmm
Mmmmmmmmmmmmmmmmmmmmmmmmmmm
mmmmmmmmmmmmmmmmmmmmmmmmmmm
6. Concluding Remarks
Mmmmmmmmmmmmmmmmmmmmmmmm
Mmmmmmmmmmmmmmmmmmmmmmmmmmm
mmmmmmmmmmmmmmmmmmmmmmmmmmm
7. Acknowledgment
Mmmmmmmmmmmmmmmmmmmmmmmm
Mmmmmmmmmmmmmmmmmmmmmmmmmmm
mmmmmmmmmmmmmmmmmmmmmmmmmmm
8. References
[1]Baal, P., A COURSE IN FIELD THEORY,
http://www.lorentz.leidenuniv.nl/~vanbaal/FTcourse.
html
5. [2] W. Hollik, Quantum field theory and the Standard
Model, arXiv:1012.3883v1 [hep-ph]
[3]Siegel, W., FIELDS, arXiv:hep-th/9912205 v2
[4]Griffiths, D. J., Introduction To Elementary
Particles, John Wiley & Sons, Inc., USA, 1987
[5]Arfken, G. B., Weber, H. J., Mathematical
Methods For Physicists, Academic Press, Inc., U. K.,
1995
Mmmmmmmmmmmmmmmmmmmmmmmm
Mmmmmmmmmmmmmmmmmmmmmmmmmmm
mmmmmmmmmmmmmmmmmmmmmmmmmmm