Spinors as the Square Root of Geometry -- Michael Atiyah
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- เผยแพร่เมื่อ 25 ส.ค. 2024
- This is an excerpt from Michael Atiyah's IHES lecture concerning spinors: • Sir Michael Atiyah, Wh...
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Hi, my name is Kyle and I'm currently doing my doctoral mathematics degree in complex differential geometry under the supervision of Professor Gang Tian and Professor Ben Andrews.
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To think that now, just a few short years after this lecture was recorded, Michael Atiyah is up there with god along with Paul Dirac too. It makes wish there was a telephone up there.
I feel like Anthony Hopkins would be perfect for the role of Atiiyah in a film!
Nice!
spinors are essentially objects called bivectors in Hestenes' geometric algebra (GA). GA is very approachable and intuitive. It builds on the natural concept of a vector (a directed line segment) and considers what happens when two vectors are multiplied together. One of the ways results in a bivector (a directed area). The benefit of the approach is that it negates the need for any knowledge of higher algebra. Of course, a more mathematical approach of this sort can be taken. This is referred to as Clifford Algebra. Hestenes' approach, as suggested by the name, is an attempt to maintain the prominence of geometry alongside algebra.
*rotates around infinity*
Just to be clear here, as far as I'm aware, in geometric algebra a spinor is actually an "even-grade multivector", so for example in the 2D geometric algebra we have spinors like "a*1 + b*e1e2", which is very similar to a complex number, and is the *sum* of a scalar and a bivector. Not just a bivector by itself; that's like saying that all of the complex numbers are only the multiples of i, instead of being in general all the "a + bi" having both a scalar ("real") part and an imaginary part summed together. And in 3D geometric algebra, the general even multivector looks like "a*1 + b*e1e2 + c*e2e3 + d*e3e1", which is essentially like a quaternion, again with both a scalar part and a bivector part.
Geometric algebra is an incredibly fruitful and fascinating field, that has brought me much joy and insight in many diverse topics. I apologize for not going into more detail here, but I just wanted to clarify what appeared to be a misconception under this framework. Cheers!
Interesting perspective
I had the same intuition when I first learned about spinors... I only wish I knew what to do with it... only God or perhaps Dirac could know
As seen in sudgylacmoe's video, "spinors" are like e^I*theta/2 where I is a bivector, something that spins vectors in a specified plane. The 1/2 in the exponential IS the square root. This comes from the need to apply the "spinor" twice on both sides with sign of the I switched to get a normal rotation. This is nothing formal, but pretty cool.
Can someone explain, or give a link to "how is a spinor equivalent to a differential form on a complex manifold, such that when you take an exterior product of this form with its complex conjugate form (Hodge dual?), you get the most general basis for forms on the manifold"? Is this what he means by square root? Is this related to twistors? And does anyone know the answer to Atiyah's (paraphrased) question: "how is a spinor the square root of a differential form on a REAL manifold"?
Would this be the difference between convex analysis and convex geometry?
Quantum Entangled Twisted Tubules: "A theory that you can't explain to a bartender is probably no damn good." Ernest Rutherford
The following is meant to be a generalized framework for an extension of Kaluza-Klein Theory.
When we draw a sine wave on a blackboard, we are representing spatial curvature. Does a photon transfer spatial curvature from one location to another? Wrap a piece of wire around a pencil and it can produce a 3D coil of wire, much like a spring. When viewed from the side it can look like a two-dimensional sine wave. You could coil the wire with either a right-hand twist, or with a left-hand twist. Could Planck's Constant be proportional to the twist cycles. A photon with a higher frequency has more energy. ( E=hf, More spatial curvature as the frequency increases). What if gluons are actually made up of these twisted tubes which become entangled with other tubes to produce quarks. (In the same way twisted electrical extension cords can become entangled.) Therefore, the gluons are a part of the quarks. Quarks cannot exist without gluons, and vice-versa. Mesons are made up of two entangled tubes (Quarks/Gluons), while protons and neutrons would be made up of three entangled tubes. (Quarks/Gluons) The "Color Force" would be related to the XYZ coordinates (orientation) of entanglement. "Asymptotic Freedom", and "flux tubes" are logically based on this concept. The Dirac “belt trick” also reveals the concept of twist in the ½ spin of subatomic particles. If each twist cycle is proportional to h, we have identified the source of Quantum Mechanics as a consequence twist cycle geometry.
Modern physicists say the Strong Force is mediated by a constant exchange of Mesons. The diagrams produced by some modern physicists actually represent the Strong Force like a spring connecting the two quarks. Asymptotic Freedom acts like real springs. Their drawing is actually more correct than their theory and matches perfectly to what I am saying in this model. You cannot separate the Gluons from the Quarks because they are a part of the same thing. The Quarks are the places where the Gluons are entangled with each other.
Neutrinos would be made up of a twisted torus (like a twisted donut) within this model. Gravity is a result of a very small curvature imbalance within atoms. (This is why the force of gravity is so small.) Instead of attempting to explain matter as "particles", this concept attempts to explain matter more in the manner of our current understanding of the space-time curvature of gravity. If an electron has qualities of both a particle and a wave, it cannot be either one. It must be something else. Therefore, a "particle" is actually a structure which stores spatial curvature. Can an electron-positron pair (which are made up of opposite directions of twist) annihilate each other by unwinding into each other producing Gamma Ray photons?
Does an electron travel through space like a threaded nut traveling down a threaded rod, with each twist cycle proportional to Planck’s Constant? Does it wind up on one end, while unwinding on the other end? Is this related to the Higgs field? Does this help explain the strange ½ spin of many subatomic particles? Does the 720 degree rotation of a 1/2 spin particle require at least one extra dimension?
Alpha decay occurs when the two protons and two neutrons (which are bound together by entangled tubes), become un-entangled from the rest of the nucleons
. Beta decay occurs when the tube of a down quark/gluon in a neutron becomes overtwisted and breaks producing a twisted torus (neutrino) and an up quark, and the ejected electron. The phenomenon of Supercoiling involving twist and writhe cycles may reveal how overtwisted quarks can produce these new particles. The conversion of twists into writhes, and vice-versa, is an interesting process.
Gamma photons are produced when a tube unwinds producing electromagnetic waves.
>>>>>>>>>>>>>>>>>>>>>>
Within this model a black hole could represent a quantum of gravity, because it is one cycle of spatial gravitational curvature. Therefore, instead of a graviton being a subatomic particle it could be considered to be a black hole. The overall gravitational attraction would be caused by a very tiny curvature imbalance within atoms.
>>>>>>>>>>>>>>>>>>>>>>
In this model Alpha equals the compactification ratio within the twistor cone. 1/137
1= Hypertubule diameter at 4D interface
137= Cone’s larger end diameter at 3D interface where the photons are absorbed or emitted.
The 4D twisted Hypertubule gets longer or shorter as twisting or untwisting occurs. (720 degrees per twist cycle.)
>>>>>>>>>>>>>>>>>>>>>>>
How many neutrinos are left over from the Big Bang? They have a small mass, but they could be very large in number. Could this help explain Dark Matter?
Spinors are equivalent to a Rank ½ Tensor.
what do you mean?
m.th-cam.com/video/j5soqexrwqY/w-d-xo.html
@@okoyoso Apologies for the late reply but I didn't get a Notification from TH-cam and just revisited this video and saw your reply had happened.
Einstein's theory of _General Relativity_ is described by his Einstein Field Equations (plural):
R - ½Rg + Λg = 8πG · T
μν μν μν c⁴ μν
These use a concise notation of his own invention where the greek subscripts μ (mu) and ν (nu) each range over the values 0, 1, 2, 3 so that between them they reference every entry in a 4x4 table, so for example T would look expand from its concise notation into:
μν
T₀₀ T₀₁ T₀₂ T₀₃
T₁₀ T₁₁ T₁₂ T₁₃
T₂₀ T₂₁ T₂₂ T₂₃
T₃₀ T₃₁ T₃₂ T₃₃
This T is the Stress-Energy-Momentum Tensor.
μν
Because it has columns (μ) and rows (ν) it is a Rank 2 Tensor. If it was just a column of 4 numbers then it would be a Rank 1 Tensor and look like this:
T₀
T₁
T₂
T₃
However, it could be simpler than that and have a single entry:
T
This is a Rank 0 Tensor, which is called a Scalar. When a Tensor holds simple numbers then this would be one value. As a branch of mathematics this would all be called Linear Algebra (here Einstein uses these special tables to organise his elaborate Partial Differential Equations that convey how the curvature of space-time on the Left Side of the Einstein Field equations influences the Stress and Momentum of localised Energy or its equivalent Mass as E = mc² as well as how the Right Side has its Mass curve space-time local to it; except in very rare and exotic cases such as a pair of Black Holes rotating around each other which then cause Gravitational waves which then ripple throughout space-time and end up measurably squishing the Earth from West to East as it changes the dimensions of space-time in which it resides as matter, so that the fixed measure of light will take less time to travel what should be a mile long W->E track compared to a mile long N->S track at the LIGO). However, this Linear Algebra is also used with regular decimals within popular 3D videogames to compute the corners of all the shapes as the camera navigates an assemblage of fictional objects in an imaginary world.
Thus far we have been dealing with whole number Tensors, but we can have fractional ones too. The easiest way to do this is to list them all as vulgar:
Rank 4/2 = Rank 2
Rank 2/2 = Rank 1
Rank 0/2 = Rank 0
It then becomes fairly obvious we have some gaps to fill in:
Rank 4/2 = Rank 2
Rank 3/2 = Rank 3/2
Rank 2/2 = Rank 1
Rank 1/2 = Rank 1/2
Rank 0/2 = Rank 0
There is a property of sub atomic particles which is called Spin (this isn't physical Spin in the regular sense, nor is Color in Quantum Chromo Dynamics being used to mean Color in the regular human scale sense, better words could have been chosen, but this is what we are stuck with because of the poor naming decisions of scientists). If we add that in we get:
Rank 4/2 = Rank 2 => Spin 2
Rank 3/2 = Rank 3/2 => Spin 3/2
Rank 2/2 = Rank 1 => Spin 1
Rank 1/2 = Rank 1/2 => Spin 1/2
Rank 0/2 = Rank 0 => Spin 0
These sub atomic particles have different properties due to being whole numbered Bosonic fields (which are the force mediating fields) as oppposed to them being fractional Fermionic fields (which are the matter manifesting fields). These are named after the physicists Satyendra Nath Bose, and Enrico Fermi. It is an important trait of Fermionic fields that they can not occupy the same location, whereas Bosonic fields can. These have all been named as "particles", so:
Rank 4/2 = Rank 2 => Spin 2 = graviton (boson)
Rank 3/2 = Rank 3/2 => Spin 3/2 = Rarita-Schwinger matter (fermion)
Rank 2/2 = Rank 1 => Spin 1 = gauge or vector (boson)
Rank 1/2 = Rank 1/2 => Spin 1/2 = spinor (fermion)
Rank 0/2 = Rank 0 => Spin 0 = Higgs (boson)
So, you might be wondering how do you define anything between a Rank 0 and a Rank 1 to obtain a Rank 1/2 spinor (fermion) such as the electron or quark. This is where you take the single entry that was used for the Rank 0 case and elaborate it to use Imaginary numbers. You can't use another Real number as it would become two entries in the table and it would become Rank 1 and therefore a boson (such as a photon or a gluon). To get something that is inbetween you need to finesse it so you get to "eat your cake yet still have it" (where the cake is having more information than Rank 0 allows but less than Rank 1 allows), and this is where you use Complex numbers where you get:
a + bi where i² = -1
That might LOOK like two values, but it is actually one number in the set of Complex numbers. Indeed, all numbers can be thought of as being Complex:
a + 0i where i² = -1
would define all the Real numbers as the Imaginary component is zeroed out.
@ooos2989 Apologies for the late reply but I didn't get a Notification from TH-cam and just revisited this video and saw your reply had happened.
Einstein's theory of _General Relativity_ is described by his Einstein Field Equations (plural):
R - ½Rg + Λg = 8πG · T
μν μν μν c⁴ μν
These use a concise notation of his own invention where the greek subscripts μ (mu) and ν (nu) each range over the values 0, 1, 2, 3 so that between them they reference every entry in a 4x4 table, so for example T would look expand from its concise notation into:
μν
T₀₀ T₀₁ T₀₂ T₀₃
T₁₀ T₁₁ T₁₂ T₁₃
T₂₀ T₂₁ T₂₂ T₂₃
T₃₀ T₃₁ T₃₂ T₃₃
This T is the Stress-Energy-Momentum Tensor.
μν
Because it has columns (μ) and rows (ν) it is a Rank 2 Tensor. If it was just a column of 4 numbers then it would be a Rank 1 Tensor and look like this:
T₀
T₁
T₂
T₃
However, it could be simpler than that and have a single entry:
T
This is a Rank 0 Tensor, which is called a Scalar. When a Tensor holds simple numbers then this would be one value. As a branch of mathematics this would all be called Linear Algebra (here Einstein uses these special tables to organise his elaborate Partial Differential Equations that convey how the curvature of space-time on the Left Side of the Einstein Field equations influences the Stress and Momentum of localised Energy or its equivalent Mass as E = mc² as well as how the Right Side has its Mass curve space-time local to it; except in very rare and exotic cases such as a pair of Black Holes rotating around each other which then cause Gravitational waves which then ripple throughout space-time and end up measurably squishing the Earth from West to East as it changes the dimensions of space-time in which it resides as matter, so that the fixed measure of light will take less time to travel what should be a mile long W->E track compared to a mile long N->S track at the LIGO). However, this Linear Algebra is also used with regular decimals within popular 3D videogames to compute the corners of all the shapes as the camera navigates an assemblage of fictional objects in an imaginary world.
Thus far we have been dealing with whole number Tensors, but we can have fractional ones too. The easiest way to do this is to list them all as vulgar:
Rank 4/2 = Rank 2
Rank 2/2 = Rank 1
Rank 0/2 = Rank 0
It then becomes fairly obvious we have some gaps to fill in:
Rank 4/2 = Rank 2
Rank 3/2 = Rank 3/2
Rank 2/2 = Rank 1
Rank 1/2 = Rank 1/2
Rank 0/2 = Rank 0
There is a property of sub atomic particles which is called Spin (this isn't physical Spin in the regular sense, nor is Color in Quantum Chromo Dynamics being used to mean Color in the regular human scale sense, better words could have been chosen, but this is what we are stuck with because of the poor naming decisions of scientists). If we add that in we get:
Rank 4/2 = Rank 2 => Spin 2
Rank 3/2 = Rank 3/2 => Spin 3/2
Rank 2/2 = Rank 1 => Spin 1
Rank 1/2 = Rank 1/2 => Spin 1/2
Rank 0/2 = Rank 0 => Spin 0
These sub atomic particles have different properties due to being whole numbered Bosonic fields (which are the force mediating fields) as oppposed to them being fractional Fermionic fields (which are the matter manifesting fields). These are named after the physicists Satyendra Nath Bose, and Enrico Fermi. It is an important trait of Fermionic fields that they can not occupy the same location, whereas Bosonic fields can. These have all been named as "particles", so:
Rank 4/2 = Rank 2 => Spin 2 = graviton (boson)
Rank 3/2 = Rank 3/2 => Spin 3/2 = Rarita-Schwinger matter (fermion)
Rank 2/2 = Rank 1 => Spin 1 = gauge or vector (boson)
Rank 1/2 = Rank 1/2 => Spin 1/2 = spinor (fermion)
Rank 0/2 = Rank 0 => Spin 0 = Higgs (boson)
So, you might be wondering how do you define anything between a Rank 0 and a Rank 1 to obtain a Rank 1/2 spinor (fermion) such as the electron or quark. This is where you take the single entry that was used for the Rank 0 case and elaborate it to use Imaginary numbers. You can't use another Real number as it would become two entries in the table and it would become Rank 1 and therefore a boson (such as a photon or a gluon). To get something that is inbetween you need to finesse it so you get to "eat your cake yet still have it" (where the cake is having more information than Rank 0 allows but less than Rank 1 allows), and this is where you use Complex numbers where you get:
a + bi where i² = -1
That might LOOK like two values, but it is actually one number in the set of Complex numbers. Indeed, all numbers can be thought of as being Complex:
a + 0i where i² = -1
would define all the Real numbers as the Imaginary component is zeroed out.
@ooos2989 Apologies for the late reply but I didn't get a Notification from TH-cam and just revisited this video and saw your reply had happened.
Einstein's theory of _General Relativity_ is described by his Einstein Field Equations (plural):
R - ½Rg + Λg = 8πG · T
μν μν μν c⁴ μν
These use a concise notation of his own invention where the greek subscripts μ (mu) and ν (nu) each range over the values 0, 1, 2, 3 so that between them they reference every entry in a 4x4 table, so for example T would look expand from its concise notation into:
μν
T₀₀ T₀₁ T₀₂ T₀₃
T₁₀ T₁₁ T₁₂ T₁₃
T₂₀ T₂₁ T₂₂ T₂₃
T₃₀ T₃₁ T₃₂ T₃₃
This T is the Stress-Energy-Momentum Tensor.
μν
Because it has columns (μ) and rows (ν) it is a Rank 2 Tensor. If it was just a column of 4 numbers then it would be a Rank 1 Tensor and look like this:
T₀
T₁
T₂
T₃
However, it could be simpler than that and have a single entry:
T
This is a Rank 0 Tensor, which is called a Scalar. When a Tensor holds simple numbers then this would be one value. As a branch of mathematics this would all be called Linear Algebra (here Einstein uses these special tables to organise his elaborate Partial Differential Equations that convey how the curvature of space-time on the Left Side of the Einstein Field equations influences the Stress and Momentum of localised Energy or its equivalent Mass as E = mc² as well as how the Right Side has its Mass curve space-time local to it; except in very rare and exotic cases such as a pair of Black Holes rotating around each other which then cause Gravitational waves which then ripple throughout space-time and end up measurably squishing the Earth from West to East as it changes the dimensions of space-time in which it resides as matter, so that the fixed measure of light will take less time to travel what should be a mile long W->E track compared to a mile long N->S track at the LIGO). However, this Linear Algebra is also used with regular decimals within popular 3D videogames to compute the corners of all the shapes as the camera navigates an assemblage of fictional objects in an imaginary world.
Thus far we have been dealing with whole number Tensors, but we can have fractional ones too. The easiest way to do this is to list them all as vulgar:
Rank 4/2 = Rank 2
Rank 2/2 = Rank 1
Rank 0/2 = Rank 0
It then becomes fairly obvious we have some gaps to fill in:
Rank 4/2 = Rank 2
Rank 3/2 = Rank 3/2
Rank 2/2 = Rank 1
Rank 1/2 = Rank 1/2
Rank 0/2 = Rank 0
There is a property of sub atomic particles which is called Spin (this isn't physical Spin in the regular sense, nor is Color in Quantum Chromo Dynamics being used to mean Color in the regular human scale sense, better words could have been chosen, but this is what we are stuck with because of the poor naming decisions of scientists). If we add that in we get:
Rank 4/2 = Rank 2 => Spin 2
Rank 3/2 = Rank 3/2 => Spin 3/2
Rank 2/2 = Rank 1 => Spin 1
Rank 1/2 = Rank 1/2 => Spin 1/2
Rank 0/2 = Rank 0 => Spin 0
These sub atomic particles have different properties due to being whole numbered Bosonic fields (which are the force mediating fields) as oppposed to them being fractional Fermionic fields (which are the matter manifesting fields). These are named after the physicists Satyendra Nath Bose, and Enrico Fermi. It is an important trait of Fermionic fields that they can not occupy the same location, whereas Bosonic fields can. These have all been named as "particles", so:
Rank 4/2 = Rank 2 => Spin 2 = graviton (boson)
Rank 3/2 = Rank 3/2 => Spin 3/2 = Rarita-Schwinger matter (fermion)
Rank 2/2 = Rank 1 => Spin 1 = gauge or vector (boson)
Rank 1/2 = Rank 1/2 => Spin 1/2 = spinor (fermion)
Rank 0/2 = Rank 0 => Spin 0 = Higgs (boson)
So, you might be wondering how do you define anything between a Rank 0 and a Rank 1 to obtain a Rank 1/2 spinor (fermion) such as the electron or quark. This is where you take the single entry that was used for the Rank 0 case and elaborate it to use Imaginary numbers. You can't use another Real number as it would become two entries in the table and it would become Rank 1 and therefore a boson (such as a photon or a gluon). To get something that is inbetween you need to finesse it so you get to "eat your cake yet still have it" (where the cake is having more information than Rank 0 allows but less than Rank 1 allows), and this is where you use Complex numbers where you get:
a + bi where i² = -1
That might LOOK like two values, but it is actually one number in the set of Complex numbers. Indeed, all numbers can be thought of as being Complex:
a + 0i where i² = -1
would define all the Real numbers as the Imaginary component is zeroed out.
Atiya once made an honest admission, declaring that after a life time of working with abstract algebra, he believes he understood nothing. Talking about Dirac he said he never talks (perhaps he also goes blank). Complex number i defines how cause and effect are related, change in x results in change in its effect y of z=x+iy (page 217 of Visual Complex Analysis by Tristan Needham) perhaps humans will never understand the mind of god.
Complex numbers are to do with rotations. I think we've got a bit bogged down by vector calculus and the rather abstract pure mathematical way it's been taught. Geometric Algebra has a much more intuitive formalism for doing all the same things and more, in any number of dimensions.
@@Lincoln_Bio You are right about Complex numbers, but abstract algebra give a magical insight into deeper reality due to mapping. Standard model would never have possible without abstract algebra, although complex numbers also played their role.