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In String Theory, we ask a deceptively simple question What happens when you replace a point with something stretched across space and time? On one side, the particle draws a world line. On the other, a string fills out a world-sheet as every point along it evolves together.

Mathelirium

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16,325 次观看 • 2 个月前 •via X (Twitter)

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String Theory Lecture 1 A String Does Not Move Like a Point A point particle traces a line through spacetime. A string traces a surface. This is the first geometric shift in String Theory. Particle mechanics asks where one object is at time t, so its history is a curve. String Theory asks where every point of an extended object is at worldsheet time τ, so we need another coordinate telling us where we are along the string. For a point particle x(t) So, for one input of time we get a position in Spacetime. For a string Xᵘ(τ,σ) Here τ plays the role of time on the worldsheet, while σ labels position along the string. Freeze τ and vary σ, and you see the string at one instant. Let τ move, and that curve sweeps out a two-dimensional surface... the worldsheet. The same comparison appears in the action. For a relativistic point particle, the geometric action measures worldline length S = −m ∫ ds If we parameterize the path by t, the action has one integral, one parameter, and one tangent vector dxᵘ/dt For a string, the same idea grows by one dimension. The action measures area, not length. In Nambu-Goto form, S = −T ∫ dτ dσ √[−det hₐᵦ] Here T is the string tension. It plays a role similar to mass, but for an extended object. It weights the area of a surface rather than the length of a line. The particle action has ∫ dt because the history is one-dimensional. The string action has ∫ dτ dσ because the history is two-dimensional. We are no longer summing along a path, we are summing over a surface. The geometry changes for the same reason. For the particle, one derivative is enough dxᵘ/dt For the string, the geometry is built from two derivatives: ∂τXᵘ and ∂σXᵘ The first tells you how the string changes as worldsheet time flows. The second tells you how the embedding changes as you move along the string. Together they define the induced worldsheet metric hₐᵦ = ∂ₐXᵘ ∂ᵦXᵤ In plain terms, hₐᵦ measures tangent lengths and tangent angles on the worldsheet. From it, the area element is dA = dτ dσ √[−det hₐᵦ] This, the Nambu-Goto action is the direct analogue of the point-particle length action. The point particle extremizes length and the string extremizes area. For calculations, people usually switch to the Polyakov action: S = −(T/2) ∫ dτ dσ √[−γ] γᵃᵇ ∂ₐXᵘ ∂ᵦXᵤ This describes the same classical string dynamics, but the algebra is cleaner. After choosing conformal gauge, varying with respect to Xᵘ gives (∂²/∂τ² − ∂²/∂σ²) Xᵘ = 0 This is the first real dynamical payoff... a two-dimensional wave equation on the worldsheet. For a point particle, the equation of motion tells you how one position evolves along one path. For a string, it tells you how an entire curve evolves, with waves traveling along it. The term ∂²Xᵘ/∂τ² measures acceleration in worldsheet time, while ∂²Xᵘ/∂σ² measures curvature along the string. The time evolution is balanced by how the string bends along its own length. This is why strings have oscillation modes. A point particle has one trajectory. A string has many possible vibration patterns, each one a normal mode of the worldsheet wave equation. For a closed string, σ wraps around the loop Xᵘ(τ, σ + 2π) = Xᵘ(τ, σ) For an open string, one standard free-end condition is ∂σXᵘ = 0 at the endpoints. Solving the wave equation gives waves moving in opposite directions along the string Xᵘ(τ,σ) = Fᵘ(τ + σ) + Gᵘ(τ − σ) A function of τ + σ moves one way. A function of τ − σ moves the other. Therefore, a particle has a worldline, its action measures length, and its geometry uses one tangent. The string has a worldsheet, its action measures area, and its geometry uses two tangent directions. #StringTheory #TheoreticalPhysics #MathematicalPhysics #Physics #Spacetime

String Theory Lecture 1 A String Does Not Move Like a Point A point particle traces a line through spacetime. A string traces a surface. This is the first geometric shift in String Theory. Particle mechanics asks where one object is at time t, so its history is a curve. String Theory asks where every point of an extended object is at worldsheet time τ, so we need another coordinate telling us where we are along the string. For a point particle x(t) So, for one input of time we get a position in Spacetime. For a string Xᵘ(τ,σ) Here τ plays the role of time on the worldsheet, while σ labels position along the string. Freeze τ and vary σ, and you see the string at one instant. Let τ move, and that curve sweeps out a two-dimensional surface... the worldsheet. The same comparison appears in the action. For a relativistic point particle, the geometric action measures worldline length S = −m ∫ ds If we parameterize the path by t, the action has one integral, one parameter, and one tangent vector dxᵘ/dt For a string, the same idea grows by one dimension. The action measures area, not length. In Nambu-Goto form, S = −T ∫ dτ dσ √[−det hₐᵦ] Here T is the string tension. It plays a role similar to mass, but for an extended object. It weights the area of a surface rather than the length of a line. The particle action has ∫ dt because the history is one-dimensional. The string action has ∫ dτ dσ because the history is two-dimensional. We are no longer summing along a path, we are summing over a surface. The geometry changes for the same reason. For the particle, one derivative is enough dxᵘ/dt For the string, the geometry is built from two derivatives: ∂τXᵘ and ∂σXᵘ The first tells you how the string changes as worldsheet time flows. The second tells you how the embedding changes as you move along the string. Together they define the induced worldsheet metric hₐᵦ = ∂ₐXᵘ ∂ᵦXᵤ In plain terms, hₐᵦ measures tangent lengths and tangent angles on the worldsheet. From it, the area element is dA = dτ dσ √[−det hₐᵦ] This, the Nambu-Goto action is the direct analogue of the point-particle length action. The point particle extremizes length and the string extremizes area. For calculations, people usually switch to the Polyakov action: S = −(T/2) ∫ dτ dσ √[−γ] γᵃᵇ ∂ₐXᵘ ∂ᵦXᵤ This describes the same classical string dynamics, but the algebra is cleaner. After choosing conformal gauge, varying with respect to Xᵘ gives (∂²/∂τ² − ∂²/∂σ²) Xᵘ = 0 This is the first real dynamical payoff... a two-dimensional wave equation on the worldsheet. For a point particle, the equation of motion tells you how one position evolves along one path. For a string, it tells you how an entire curve evolves, with waves traveling along it. The term ∂²Xᵘ/∂τ² measures acceleration in worldsheet time, while ∂²Xᵘ/∂σ² measures curvature along the string. The time evolution is balanced by how the string bends along its own length. This is why strings have oscillation modes. A point particle has one trajectory. A string has many possible vibration patterns, each one a normal mode of the worldsheet wave equation. For a closed string, σ wraps around the loop Xᵘ(τ, σ + 2π) = Xᵘ(τ, σ) For an open string, one standard free-end condition is ∂σXᵘ = 0 at the endpoints. Solving the wave equation gives waves moving in opposite directions along the string Xᵘ(τ,σ) = Fᵘ(τ + σ) + Gᵘ(τ − σ) A function of τ + σ moves one way. A function of τ − σ moves the other. Therefore, a particle has a worldline, its action measures length, and its geometry uses one tangent. The string has a worldsheet, its action measures area, and its geometry uses two tangent directions. #StringTheory #TheoreticalPhysics #MathematicalPhysics #Physics #Spacetime

Mathelirium

31,560 次观看 • 3 个月前

What String Theory Does That Classical Physics Doesn’t Classical Physics begins with a point particle. Its position is described by one function of time x = x(t) As time passes, that point traces a curve through Spacetime called a worldline. String theory changes the starting object. A string has length, so time alone is not enough. We also need a coordinate along the string Xᵘ = Xᵘ(τ, σ) Here τ is worldsheet time, and σ labels position along the string. As the string moves, it sweeps out a surface called a worldsheet. The action measures the area of that surface S = -T ∫ dτ dσ √[-det hₐᵦ] This is where string theory begins. #StringTheory #ClassicalMechanics #TheoreticalPhysics #Worldsheet #QuantumGravity #MathematicalPhysics #Spacetime #PhysicsVisualization #ScienceEducation

What String Theory Does That Classical Physics Doesn’t Classical Physics begins with a point particle. Its position is described by one function of time x = x(t) As time passes, that point traces a curve through Spacetime called a worldline. String theory changes the starting object. A string has length, so time alone is not enough. We also need a coordinate along the string Xᵘ = Xᵘ(τ, σ) Here τ is worldsheet time, and σ labels position along the string. As the string moves, it sweeps out a surface called a worldsheet. The action measures the area of that surface S = -T ∫ dτ dσ √[-det hₐᵦ] This is where string theory begins. #StringTheory #ClassicalMechanics #TheoreticalPhysics #Worldsheet #QuantumGravity #MathematicalPhysics #Spacetime #PhysicsVisualization #ScienceEducation

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In String Theory, it has been shown that one vibration of a closed string behaves like a graviton, the Quantum particle of gravity, revealing that gravity is already built into String Theory’s spectrum.

In String Theory, it has been shown that one vibration of a closed string behaves like a graviton, the Quantum particle of gravity, revealing that gravity is already built into String Theory’s spectrum.

Mathelirium

11,588 次观看 • 16 天前

Why String Theory Won't Die The string theory idea refuses to die because it's one of the few frameworks that forces a quantum object and spacetime geometry into the same picture. In the render, a closed string isn’t just a loop doing pretty vibrations. Its internal phase pattern is a quantum degree of freedom, shown as color. At the same time, its motion stirs ripples in the surrounding metric, shown as the fabric below. One object, two roles. A wave-like state living on the string, and a curvature-like response spreading through the background. You can literally watch quantum stuff and gravity stuff talk to each other in the same scene. #StringTheory #Physics #QuantumMechanics #GeneralRelativity #MathematicalPhysics #SciComm

Why String Theory Won't Die The string theory idea refuses to die because it's one of the few frameworks that forces a quantum object and spacetime geometry into the same picture. In the render, a closed string isn’t just a loop doing pretty vibrations. Its internal phase pattern is a quantum degree of freedom, shown as color. At the same time, its motion stirs ripples in the surrounding metric, shown as the fabric below. One object, two roles. A wave-like state living on the string, and a curvature-like response spreading through the background. You can literally watch quantum stuff and gravity stuff talk to each other in the same scene. #StringTheory #Physics #QuantumMechanics #GeneralRelativity #MathematicalPhysics #SciComm

Mathelirium

13,657 次观看 • 5 个月前

String Theory is one of the few frameworks that tries to describe Quantum Physics and Gravity in the same mathematical language. In this animation, the glowing loop is the string, the shifting color along it represent its quantum vibration modes, and the surrounding fabric is a proxy for Spacetime responding. When you quantize a string, one of its vibration modes behaves like a graviton. So, geometry shows up as part of the same quantum system. #StringTheory #QuantumGravity #GeneralRelativity #QuantumPhysics #TheoreticalPhysics #PhysicsVisualization

String Theory is one of the few frameworks that tries to describe Quantum Physics and Gravity in the same mathematical language. In this animation, the glowing loop is the string, the shifting color along it represent its quantum vibration modes, and the surrounding fabric is a proxy for Spacetime responding. When you quantize a string, one of its vibration modes behaves like a graviton. So, geometry shows up as part of the same quantum system. #StringTheory #QuantumGravity #GeneralRelativity #QuantumPhysics #TheoreticalPhysics #PhysicsVisualization

Mathelirium

42,145 次观看 • 5 个月前

What if point particles in Physics are just an illusion? Branes are higher-dimensional objects in String Theory that generalize the idea of a point particle. Instead of everything being just 0-dimensional points or 1-dimensional strings, a Brane can have 2, 3 or more spatial dimensions Open strings can end on Branes, which means Branes act like boundaries for string motion. In many models, what we call particles are simply vibration modes of strings attached to these Branes, and force can be understood as strings stretching between them.

What if point particles in Physics are just an illusion? Branes are higher-dimensional objects in String Theory that generalize the idea of a point particle. Instead of everything being just 0-dimensional points or 1-dimensional strings, a Brane can have 2, 3 or more spatial dimensions Open strings can end on Branes, which means Branes act like boundaries for string motion. In many models, what we call particles are simply vibration modes of strings attached to these Branes, and force can be understood as strings stretching between them.

Mathelirium

21,841 次观看 • 2 个月前

String Theory Lecture 2 In Conformal Gauge, the String Becomes a Wave Equation Episode 1 showed the geometric jump point particle -> worldline string -> worldsheet Episode 2 is the dynamical jump. The Nambu-Goto action measures the area of the worldsheet, S = −T ∫ dτ dσ √[−det hₐᵦ] but the square-root determinant is awkward to work with. So we usually rewrite the same classical theory in Polyakov form, S = −(T/2) ∫ dτ dσ √[−γ] γᵃᵇ ∂ₐXᵘ ∂ᵦXᵤ Here Xᵘ(τ,σ) tells us where each point of the string’s worldsheet sits in spacetime, and γₐᵦ is the metric we put on the worldsheet. The power of this form is that we can choose a convenient gauge. In conformal gauge, the equations of motion simplify to (∂²/∂τ² − ∂²/∂σ²) Xᵘ = 0 So the string’s spacetime coordinates behave like waves living on the worldsheet. The τ-derivative measures how the string changes in worldsheet time. The σ-derivative measures how it bends along its own length. For a closed string, σ is periodic Xᵘ(τ, σ + 2π) = Xᵘ(τ,σ) and the wave equation splits into two traveling pieces, Xᵘ(τ,σ) = Fᵘ(τ + σ) + Gᵘ(τ − σ) One family moves one way around the string and the other moves the opposite way. These are the left-moving and right-moving modes. In the render, the bright loop is the string at the present moment. The glowing cylinder behind it is the worldsheet it has swept out. The cyan curves trace one traveling family, and the gold curves trace the other. They are the visual version of τ + σ and τ − σ. Therefore, the theory has an internal wave equation, and its normal modes are the raw material for the string spectrum. #StringTheory #TheoreticalPhysics #ConformalGauge #Worldsheet #WaveEquation #Physics #Mathematics #MathematicalPhysics #QuantumGravity #ScienceVisuals

String Theory Lecture 2 In Conformal Gauge, the String Becomes a Wave Equation Episode 1 showed the geometric jump point particle -> worldline string -> worldsheet Episode 2 is the dynamical jump. The Nambu-Goto action measures the area of the worldsheet, S = −T ∫ dτ dσ √[−det hₐᵦ] but the square-root determinant is awkward to work with. So we usually rewrite the same classical theory in Polyakov form, S = −(T/2) ∫ dτ dσ √[−γ] γᵃᵇ ∂ₐXᵘ ∂ᵦXᵤ Here Xᵘ(τ,σ) tells us where each point of the string’s worldsheet sits in spacetime, and γₐᵦ is the metric we put on the worldsheet. The power of this form is that we can choose a convenient gauge. In conformal gauge, the equations of motion simplify to (∂²/∂τ² − ∂²/∂σ²) Xᵘ = 0 So the string’s spacetime coordinates behave like waves living on the worldsheet. The τ-derivative measures how the string changes in worldsheet time. The σ-derivative measures how it bends along its own length. For a closed string, σ is periodic Xᵘ(τ, σ + 2π) = Xᵘ(τ,σ) and the wave equation splits into two traveling pieces, Xᵘ(τ,σ) = Fᵘ(τ + σ) + Gᵘ(τ − σ) One family moves one way around the string and the other moves the opposite way. These are the left-moving and right-moving modes. In the render, the bright loop is the string at the present moment. The glowing cylinder behind it is the worldsheet it has swept out. The cyan curves trace one traveling family, and the gold curves trace the other. They are the visual version of τ + σ and τ − σ. Therefore, the theory has an internal wave equation, and its normal modes are the raw material for the string spectrum. #StringTheory #TheoreticalPhysics #ConformalGauge #Worldsheet #WaveEquation #Physics #Mathematics #MathematicalPhysics #QuantumGravity #ScienceVisuals

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Phase Space Is the Real Stage of Dynamics. So, Stop Watching the Object and Watch the State. Ordinary Space tells you where something is. Phase Space tells you what state it is in. Position alone is not enough because you also need the variable that tells you where the system is trying to go. In the animation, the left panel shows the particle moving in Ordinary Space, while the right panel shows the same system as a moving point in Phase Space. For one degree of freedom, the state is (x, p) and as time evolves, that state traces a trajectory t ↦ (x(t), p(t)) Instead of thinking of a differential equation as just a formula, Phase Space lets you see it as a flow on the space of states. For this lecture we use ẋ = p ṗ = x − x³ with Hamiltonian H(x,p) = 0.5 p² + 0.25 x⁴ − 0.5 x² #PhaseSpace #DynamicalSystems #NonlinearDynamics #HamiltonianMechanics #Mathematics

Phase Space Is the Real Stage of Dynamics. So, Stop Watching the Object and Watch the State. Ordinary Space tells you where something is. Phase Space tells you what state it is in. Position alone is not enough because you also need the variable that tells you where the system is trying to go. In the animation, the left panel shows the particle moving in Ordinary Space, while the right panel shows the same system as a moving point in Phase Space. For one degree of freedom, the state is (x, p) and as time evolves, that state traces a trajectory t ↦ (x(t), p(t)) Instead of thinking of a differential equation as just a formula, Phase Space lets you see it as a flow on the space of states. For this lecture we use ẋ = p ṗ = x − x³ with Hamiltonian H(x,p) = 0.5 p² + 0.25 x⁴ − 0.5 x² #PhaseSpace #DynamicalSystems #NonlinearDynamics #HamiltonianMechanics #Mathematics

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What happens when you take a polynomial and keep changing its coefficients every frame? You get a resonant root field in the complex plane where clusters, arcs and outer flares form, dissolve, and reappear as the system evolves in time.

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Mathelirium

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Calculus of Variations is what happens when you stop optimizing values and start optimizing geometries. The unknown isn’t a single x...it’s a whole curve y(x). And the thing you’re minimizing usually isn’t a formula you can eyeball...it’s an integral that judges the entire shape. For our first lecture, we look at the famous Brachistochrone problem. Fix two points, switch on gravity, and ask a question that sounds too simple...which track gets a bead from A to B in the least time? Your intuition will betray you on the first try. It’s not the straight line. It’s not the drop hard then cruise sketch either. The winner is a cycloid...the curve traced by a point on a rolling circle. In the animation, the track is the moving character. We start with an imperfect curve, run gradient descent in curve-space, and watch the geometry reshape frame by frame as T[y] collapses until it locks into the brachistochrone. Pls see the comment below for the math breakdown. #CalculusOfVariations #Brachistochrone #Cycloid #Physics #Optimization #Mathematics

Calculus of Variations is what happens when you stop optimizing values and start optimizing geometries. The unknown isn’t a single x...it’s a whole curve y(x). And the thing you’re minimizing usually isn’t a formula you can eyeball...it’s an integral that judges the entire shape. For our first lecture, we look at the famous Brachistochrone problem. Fix two points, switch on gravity, and ask a question that sounds too simple...which track gets a bead from A to B in the least time? Your intuition will betray you on the first try. It’s not the straight line. It’s not the drop hard then cruise sketch either. The winner is a cycloid...the curve traced by a point on a rolling circle. In the animation, the track is the moving character. We start with an imperfect curve, run gradient descent in curve-space, and watch the geometry reshape frame by frame as T[y] collapses until it locks into the brachistochrone. Pls see the comment below for the math breakdown. #CalculusOfVariations #Brachistochrone #Cycloid #Physics #Optimization #Mathematics

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Mathelirium

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Mathelirium

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A young Zakir Hussain fills in as the great Ali Akbar Khan replaces a broken string on his Sarod and tunes it.

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We know gravity pulls things together in space...but have you ever seen what it does in phase space? In this scene we're not watching a blob of matter in x–space. We're watching a thin probability sheet in (x, v).... position–velocity space... being pulled by its own gravity. In the collisionless limit, Liouville’s theorem says this sheet can stretch, fold, and swirl, but never tear or change area. Collapse in x is "paid for" by stretching in v. Numerically, this is a 1D cold-dark-matter toy universe: a sheet of particles evolved under the Vlasov-Poisson system. We solve Poisson on a periodic line, get the self-consistent gravitational field, then integrate Hamilton’s equations so the sheet folds into a multi-layer spiral. The glowing filaments are where the sheet keeps revisiting, carving out dense caustics in phase space. It's gravity not as "stuff falling down," but as area-preserving origami in phase space. #Mathelirium #VlasovPoisson #PhaseSpace #Liouville #DarkMatter #HamiltonianFlow #ProbabilityFlow

Mathelirium

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This is what happens when you point a laser at Marine One with the president on board.

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Mila Joy

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In Calculus of Variations, we give up the idea of only optimizing values and start optimizing geometries instead. The unknown is not a single x. It is a whole curve/surface y(x). The object you are minimizing is usually not a simple formula that you can guess. It is an integral that judges the whole shape. In our first lecture, we discuss the famous Brachistochrone problem. You choose two points, turn on gravity and ask a question which almost sounds too simple...which track will allow a bead to move from A to B in the minimum time? At your first attempt, your instinct will lead you astray. It is not the straight line. It is not the drop hard then cruise sketch either. The winner turns out to be a cycloid...the curve a point on a rolling circle traces. The track is the moving character in the animation. We begin with a flawed curve, perform gradient descent in curve-space, and observe the geometry change frame by frame as T[y] collapses to the brachistochrone. Please see the comment below for the math breakdown. #CalculusOfVariations #Brachistochrone #Cycloid #Physics #Optimization #Mathematics

In Calculus of Variations, we give up the idea of only optimizing values and start optimizing geometries instead. The unknown is not a single x. It is a whole curve/surface y(x). The object you are minimizing is usually not a simple formula that you can guess. It is an integral that judges the whole shape. In our first lecture, we discuss the famous Brachistochrone problem. You choose two points, turn on gravity and ask a question which almost sounds too simple...which track will allow a bead to move from A to B in the minimum time? At your first attempt, your instinct will lead you astray. It is not the straight line. It is not the drop hard then cruise sketch either. The winner turns out to be a cycloid...the curve a point on a rolling circle traces. The track is the moving character in the animation. We begin with a flawed curve, perform gradient descent in curve-space, and observe the geometry change frame by frame as T[y] collapses to the brachistochrone. Please see the comment below for the math breakdown. #CalculusOfVariations #Brachistochrone #Cycloid #Physics #Optimization #Mathematics

Mathelirium

23,254 次观看 • 2 个月前

D-Branes and the Expansion of String Theory String Theory was already attempting to do something huge by trying to describe particles, forces, and gravity in one framework. Branes made that picture much richer. When Physicists found that open strings can end on objects called D-branes, many important ideas became easier to organize. Things like gauge fields, black hole microstates, holography, and deep dualities gained a clearer place inside the theory. They showed that String Theory is not only about tiny vibrating strings, but also about larger structures that help connect Quantum Physics, Geometry, and Gravity.

D-Branes and the Expansion of String Theory String Theory was already attempting to do something huge by trying to describe particles, forces, and gravity in one framework. Branes made that picture much richer. When Physicists found that open strings can end on objects called D-branes, many important ideas became easier to organize. Things like gauge fields, black hole microstates, holography, and deep dualities gained a clearer place inside the theory. They showed that String Theory is not only about tiny vibrating strings, but also about larger structures that help connect Quantum Physics, Geometry, and Gravity.

Mathelirium

21,515 次观看 • 3 个月前

The Chinese Physicist Who Turned Force Into Geometry In 1954, Chinese physicist Chen-Ning Yang, together with Robert Mills, introduced a new idea that changed modern physics: Maybe a force is not just something that pushes or pulls. Maybe a force can come from geometry. In Yang-Mills Theory, every point in space carries a hidden internal direction. As you move through space, that direction can twist. If you go around a closed loop and the direction does not return exactly the same, the field has curvature. That curvature is what we see as force. In this animation, the glowing surface shows where the gauge field is strongest. The changing colours show how the hidden internal direction varies across space. The moving rectangular loops are Wilson loops. They travel through the field and measure how much the internal direction changes after completing a closed path. The key equation behind the scene is Fₓᵧ = ∂ₓAᵧ - ∂ᵧAₓ + Aₓ × Aᵧ. The final term, Aₓ × Aᵧ, is the special part. It means the field can interact with itself. That self-interaction is one reason Yang-Mills theory became one of the foundations of particle physics. #China #ChinesePhysics #YangMills #GaugeTheory #QuantumFieldTheory #NonAbelianGeometry #WilsonLoops #ModernPhysics #STEM

The Chinese Physicist Who Turned Force Into Geometry In 1954, Chinese physicist Chen-Ning Yang, together with Robert Mills, introduced a new idea that changed modern physics: Maybe a force is not just something that pushes or pulls. Maybe a force can come from geometry. In Yang-Mills Theory, every point in space carries a hidden internal direction. As you move through space, that direction can twist. If you go around a closed loop and the direction does not return exactly the same, the field has curvature. That curvature is what we see as force. In this animation, the glowing surface shows where the gauge field is strongest. The changing colours show how the hidden internal direction varies across space. The moving rectangular loops are Wilson loops. They travel through the field and measure how much the internal direction changes after completing a closed path. The key equation behind the scene is Fₓᵧ = ∂ₓAᵧ - ∂ᵧAₓ + Aₓ × Aᵧ. The final term, Aₓ × Aᵧ, is the special part. It means the field can interact with itself. That self-interaction is one reason Yang-Mills theory became one of the foundations of particle physics. #China #ChinesePhysics #YangMills #GaugeTheory #QuantumFieldTheory #NonAbelianGeometry #WilsonLoops #ModernPhysics #STEM

Mathelirium

83,077 次观看 • 1 个月前