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String Theory needed a place for open strings to end. So, D-Branes were born. D-Branes are hidden surfaces in higher dimensions where open strings anchor and give rise to nature’s forces, while closed strings roam the surrounding space carrying gravity.

Mathelirium

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45,959 次观看 • 23 天前 •via X (Twitter)

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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

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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

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String Theory tries to unify Quantum Mechanics and gravity into a single framework by replacing particles with strings whose vibrations also shape Spacetime.

String Theory tries to unify Quantum Mechanics and gravity into a single framework by replacing particles with strings whose vibrations also shape Spacetime.

Mathelirium

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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.

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Mathelirium

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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

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Why does String Theory keep introducing extra dimensions? Because once you quantize a relativistic string consistently, the Mathematics stops working cleanly in ordinary 3+1 dimensions.

Why does String Theory keep introducing extra dimensions? Because once you quantize a relativistic string consistently, the Mathematics stops working cleanly in ordinary 3+1 dimensions.

Mathelirium

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Why Are Some People Still Taking String Theory Seriously? String theory is controversial, but it is still one of the few ideas that tries to describe Quantum Physics and Gravity in one picture. This animation points to one of its best-known ideas: Holography. What happens on the boundary helps determine what happens in the interior, so the edge seems to shape the bulk.

Why Are Some People Still Taking String Theory Seriously? String theory is controversial, but it is still one of the few ideas that tries to describe Quantum Physics and Gravity in one picture. This animation points to one of its best-known ideas: Holography. What happens on the boundary helps determine what happens in the interior, so the edge seems to shape the bulk.

Mathelirium

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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

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How to modify stent strings in less than a minute so they are not inadvertently tugged on. I borrowed this technique from Dr. Lingeman and haven’t had a patient accidentally remove their stent since. No need for trying to tape the strings out of the way… just cut to length IU Endourology

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Terrorist Budanov got drunk and went to give an interview. ‘Putin's doubles are in place. They are needed to save time, sending them where it is dangerous’ 'The window of opportunity to end the war will open for Ukraine in February 2026.' - SK

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A player in Where Winds Meet made a Keanu Reeves character and played See You Again with Kong Hou. Yes, you can basically do anything in this Wuxia open world RPG with MMO features. For those curious, Kong Hou is an Chinese harp-like instrument from the Tang Dynasty. There are 2 versions, one is the modern Kong Hou, which is basically a western harp but with 2 rows of strings. The traditional Tang dynasty version looks similar to the one in this video with a single row of strings. The strings are attached to the hollow structure at the top of the Kong Hou, which creates a very unique sound.

A player in Where Winds Meet made a Keanu Reeves character and played See You Again with Kong Hou. Yes, you can basically do anything in this Wuxia open world RPG with MMO features. For those curious, Kong Hou is an Chinese harp-like instrument from the Tang Dynasty. There are 2 versions, one is the modern Kong Hou, which is basically a western harp but with 2 rows of strings. The traditional Tang dynasty version looks similar to the one in this video with a single row of strings. The strings are attached to the hollow structure at the top of the Kong Hou, which creates a very unique sound.

Zhao DaShuai 东北进修🇨🇳 Commentary

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This is a breaking podcast. We're premiering a new paradigm for quantizing 4D gravity here first, without strings. Neil Turok — inaugural Higgs Chair at Edinburgh, former director of Perimeter Institute, and 2026 Fellow of the Royal Society — believes quantum gravity may not require strings, extra dimensions, or a multiverse. The key: a 1970s theory called quadratic gravity, long abandoned over two seemingly fatal problems. Turok and Bateman argue both problems dissolve — one by reinterpreting a classical instability as ordinary gravitational expansion, the other by a subtle tweak to the Born rule that allows quantum states of negative norm without ever producing negative probabilities. One quiet assumption, Turok argues, underpins decades of string theory's necessity. Drop it, and the whole case for a multiverse unravels. Neil graciously gave me a sneak peek at his and his PhD student Sam Bateman's new research. Bleeding edge! I hope you enjoy.

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Curt Jaimungal

42,780 次观看 • 1 个月前

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

Gravity is fake and gay. The natural physics of density and buoyancy determine that objects denser than the medium surrounding them sink, while objects less dense than the medium surrounding them rise. (The up/down direction is dictated by Earth's electromagnetic torus field) No other magical, unexplainable forces are required. The one and ONLY reason the freemasons created the "Gravity" story was to convince you that you are stuck to a giant spinning ball in space.

Gravity is fake and gay. The natural physics of density and buoyancy determine that objects denser than the medium surrounding them sink, while objects less dense than the medium surrounding them rise. (The up/down direction is dictated by Earth's electromagnetic torus field) No other magical, unexplainable forces are required. The one and ONLY reason the freemasons created the "Gravity" story was to convince you that you are stuck to a giant spinning ball in space.

₿en Wehrman

13,458 次观看 • 1 个月前

Nyckelharpa has existed in one form or another format least 600 years, and is generally considered an instrument of Swedes. Given how Christianity came later to Sweden, its earliest iterations can easily be associated with Norse gods and secular life in the Nordic lands. There were multiple versions of stringed instrument, as it has evolved over time, to today where it could be mistaken for an instrumental marriage between a bowed instrument (violin, viola, cello, bass) and a keyboard such as a piano, harpsichord or perhaps an accordion. It’s unclear exactly what earliest versions of Nyckelharpa looked like. There are just three surviving instruments from 16th Century, but there are earlier depictions of an instrument that resembles the boxy, oblong contraption. Church paintings in Denmark, Sweden and Italy, dating to early 1400s, and later (about 1590) in a church in Hildesheim, Germany, seem to show the instrument. An even older example in ecclesiastical art dates from about 1350 on a carved relief on a gate into Kallunge Church in Gotland, Sweden, which shows what might be two Nyckelharpa players in what we are to assume is a performance of pre-Reformation sacred music (academics stop short of declaring these to be definitively Nyckelharpa). The physical instrument evolved in various iterations over time, as evidenced by addition of strings, now in three rows: the three melody strings, a drone C, and 12 resonance (sympathetic) strings, which were more modern addition (16th Century). The keys, played with the left hand while the right hand bows the strings, serve as frets when a key is depressed. This changes the pitch of the string, similar to what the violinists’ hands do on a violin. The nyckelharpa belongs to the same instrument family as French vielle and the English hurdy gurdy. Nyckelharpa has wooden keys that slide under the strings and have tangents set perpendicularly to the keys that reach up and stop (shorten) melody string. It’s sort of like moveable frets that move to meet the string, rather than pressing the string against a fingerboard or frets. A short bow is used. The oldest “evidence” of nyckelharpa use is a relief on one of the gates to Källunge church on Gotland from about 1350 AD, depicting two nyckelharpa players. There are three surviving examples of medieval nyckelharpa: one found in the town of Mora in Dalarna, Sweden, one found in Vefsen, Norway (it hangs in the Musik Museum in Stockholm); and one found in Esse, Finland. Moraharpa is dated 1526, and hangs in the Zorn Museum in Mora. They have one row of keys (although some keys stop two strings, making them a mix of melody and drone strings) and two drone strings. Moraharpa is shaped like a lute, while the Vefsenharpa and Esseharpa are shaped more like modern nyckelharpa. Recent evidence suggests that Moraharpa was probably built in early 1600’s, using wood that was already about 100 years old. It was likely made as a copy of the picture in the German book (picture forthcoming). So it appears that lute-shaped tradition is broken, while the long/narrow tradition is unbroken. A few people still play this version of the instrument today, including Anders Stake (Anders Norrude) in group “Hedningarna”. Perhaps under the influence of the viola d’amore which was popular in Uppsala and Stockholm in late 1600’s and early 1700’s, resonance strings were added. Sympathetic strings were brought from the near east to England circa 1600, and were thought to be a fashionable innovation. On the nyckelharpa, the resonance strings are set lower in the bridge, so the bow does not touch them. Oldest surviving example of enkelharpa is from 1777. Mixturharpa is similar to enkelharpa, except that 2nd string also has tangents (on the same key at the tangent for the 1st string) that stop the string and make the notes. Not many people today play this version of nyckelharpa, opting for older or younger versions. © American Nyckelharpa Association #archaeohistories

Nyckelharpa has existed in one form or another format least 600 years, and is generally considered an instrument of Swedes. Given how Christianity came later to Sweden, its earliest iterations can easily be associated with Norse gods and secular life in the Nordic lands. There were multiple versions of stringed instrument, as it has evolved over time, to today where it could be mistaken for an instrumental marriage between a bowed instrument (violin, viola, cello, bass) and a keyboard such as a piano, harpsichord or perhaps an accordion. It’s unclear exactly what earliest versions of Nyckelharpa looked like. There are just three surviving instruments from 16th Century, but there are earlier depictions of an instrument that resembles the boxy, oblong contraption. Church paintings in Denmark, Sweden and Italy, dating to early 1400s, and later (about 1590) in a church in Hildesheim, Germany, seem to show the instrument. An even older example in ecclesiastical art dates from about 1350 on a carved relief on a gate into Kallunge Church in Gotland, Sweden, which shows what might be two Nyckelharpa players in what we are to assume is a performance of pre-Reformation sacred music (academics stop short of declaring these to be definitively Nyckelharpa). The physical instrument evolved in various iterations over time, as evidenced by addition of strings, now in three rows: the three melody strings, a drone C, and 12 resonance (sympathetic) strings, which were more modern addition (16th Century). The keys, played with the left hand while the right hand bows the strings, serve as frets when a key is depressed. This changes the pitch of the string, similar to what the violinists’ hands do on a violin. The nyckelharpa belongs to the same instrument family as French vielle and the English hurdy gurdy. Nyckelharpa has wooden keys that slide under the strings and have tangents set perpendicularly to the keys that reach up and stop (shorten) melody string. It’s sort of like moveable frets that move to meet the string, rather than pressing the string against a fingerboard or frets. A short bow is used. The oldest “evidence” of nyckelharpa use is a relief on one of the gates to Källunge church on Gotland from about 1350 AD, depicting two nyckelharpa players. There are three surviving examples of medieval nyckelharpa: one found in the town of Mora in Dalarna, Sweden, one found in Vefsen, Norway (it hangs in the Musik Museum in Stockholm); and one found in Esse, Finland. Moraharpa is dated 1526, and hangs in the Zorn Museum in Mora. They have one row of keys (although some keys stop two strings, making them a mix of melody and drone strings) and two drone strings. Moraharpa is shaped like a lute, while the Vefsenharpa and Esseharpa are shaped more like modern nyckelharpa. Recent evidence suggests that Moraharpa was probably built in early 1600’s, using wood that was already about 100 years old. It was likely made as a copy of the picture in the German book (picture forthcoming). So it appears that lute-shaped tradition is broken, while the long/narrow tradition is unbroken. A few people still play this version of the instrument today, including Anders Stake (Anders Norrude) in group “Hedningarna”. Perhaps under the influence of the viola d’amore which was popular in Uppsala and Stockholm in late 1600’s and early 1700’s, resonance strings were added. Sympathetic strings were brought from the near east to England circa 1600, and were thought to be a fashionable innovation. On the nyckelharpa, the resonance strings are set lower in the bridge, so the bow does not touch them. Oldest surviving example of enkelharpa is from 1777. Mixturharpa is similar to enkelharpa, except that 2nd string also has tangents (on the same key at the tangent for the 1st string) that stop the string and make the notes. Not many people today play this version of nyckelharpa, opting for older or younger versions. © American Nyckelharpa Association #archaeohistories

Archaeo - Histories

15,692 次观看 • 1 年前

And so it begins… 🚨 Cook County in Illinois launches a Universal Basic income Program where low-income households are Guaranteed Monthly Payment “no strings attached” — All Paid for by Taxpayers who work The nine most terrifying words in the English language… “I’m from the government, and I'm here to help,".

And so it begins… 🚨 Cook County in Illinois launches a Universal Basic income Program where low-income households are Guaranteed Monthly Payment “no strings attached” — All Paid for by Taxpayers who work The nine most terrifying words in the English language… “I’m from the government, and I'm here to help,".

MJTruthUltra

279,907 次观看 • 8 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

Mathelirium

15,295 次观看 • 3 个月前

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.

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

16,325 次观看 • 2 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

Mathelirium

18,616 次观看 • 1 个月前