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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...

21,923 просмотров • 3 месяцев назад •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

Mathelirium

31,560 просмотров • 4 месяцев назад

This kinetic sculpture, titled Nested Loops, hangs inside The National Museum of Mathematics (MoMath) in New York. It demonstrates exactly how higher-dimensional forms can emerge from the coordinated motion of simple elements moving through time. The circles are pathways of motion…timing mechanisms that guide anchor points along curved trajectories. As the mechanism rotates, strings stretched between these moving anchor points remain under constant tension. Because a taut string always forms the shortest path between two points, each string becomes a straight line, even though the points guiding it are traveling along curves! This is the key insight…circular motion is generating linear geometry. The loops create movement, the strings reveal structure, and as these relationships shift in coordinated time, the eye begins to perceive cubic forms like edges, diagonals, and vertices, all emerging from what is fundamentally a flat plane with moving parts. A cube does not need to be physically present to be perceived. It can be implied through the alignment of edges, the intersection of planes, and the convergence of perspective. What you’re witnessing is a dimensional emergence…a lower-dimensional system producing the visual signature of a higher-dimensional object through synchronized motion. Change the speed of one circle relative to another, and an entirely different geometric form appears. The same strings, the same frame…but a new structure emerges from the new rhythm. This is why the sculpture feels almost alive. It’s revealing that form is relational, temporal, and emergent, so what we perceive as solid objects are often just stable harmonies of moving relationships, frozen in a moment of coherence. This piece asks you to consider: What aspects of your reality are you perceiving as solid, when they’re actually just coherent patterns of motion temporarily aligned? Did this shift how you see structure and emergence? ✨🙌🏾💫 Artist © Chuck Hoberman (MoMath, New York)

🧬Maxpein🧬

48,005 просмотров • 3 месяцев назад

Physicist Avi Loeb just told me that futuristic technology could enable humanity to travel “faster than the speed of light.” But we would have to move beyond the three dimensions we are familiar with. And he speculated that advanced alien civilizations could already have access to these dimensions. You need to hear his fascinating theory: “Think about living on the surface of a balloon.” “That is two dimensional.” “You might not be aware that there is a third dimension because you are just living on the surface of that balloon.” “If there is another being that is capable of taking advantage of the third dimension, then that being will cross the distance between two points on the surface of the balloon faster than you can imagine.” “Because the travel between the two points can go through the third dimension that connects the two points—not necessarily on the curved surface of the balloon.” “So there are, in principle, possibilities of navigating in more than the dimensions that we are familiar with.” “We are familiar with three spatial dimensions plus time.” “If there are more than three and there is a technological way of taking advantage of those … objects will appear and disappear in ways that we cannot understand.” “Einstein’s theory of relativity states that no material object can move faster than light.” “However, if there are extra dimensions, you might actually travel faster than light in the three dimensions, even though you’re traveling less than the speed of light in the extra dimensions.”

Jan Jekielek

31,984 просмотров • 6 месяцев назад

Andrew Ng just revealed why the AI companies throwing the most compute at the problem are going to lose. The winner of the intelligence race won’t use the most compute. They’ll waste the least. Ng: “Most of your high-dimensional data lies on a lower-dimensional subspace. It’s just a fact of life.” Here’s what that means in practice. You have a 10,000-dimensional dataset. Every dimension dragged through every calculation. Every training cycle hauling dead weight the model will never use. Ng: “You’re carrying around these 10,000-dimensional examples throughout your whole training process.” That bloat isn’t just inefficient. It’s a tax on every computation you run. Memory bandwidth. Network bandwidth. Computational speed. All of it eaten by dimensions that contribute nothing to intelligence. They contribute noise. The insight that separates the architects from the arms race: that 10,000-dimensional dataset is almost entirely captured by a much smaller subspace. The signal lives in a fraction of the space you’re paying to process. Compress it. 10,000 dimensions down to 1,000. Ng: “You can run your learning algorithm on a much lower-dimensional set of data and it may be much more efficient.” Same hardware. Same budget. A fraction of the friction. Brute force is the strategy of whoever has the deepest pockets. Compression is the strategy of whoever actually understands the problem. The companies that master this don’t just build faster models. They build models that find more truth in less data than anything scaling blindly ever will. Intelligence was never about processing everything. It’s about knowing what to cut.

Dustin

215,643 просмотров • 6 месяцев назад