Saturday, October 10, 2020

Penrose Iconoclasm: The General Idea Is In The Geometrical Pictures

YOU - don't have to follow the detailed equations and everything. The celestial sphere is a Riemann sphere and the collection of light rays/points/events comprise a lorentz group in minkowski space. 

Points on the Riemann sphere are conformal comprising a circle on the sphere. A Riemann sphere is a complex curve or manifold, twistor space is therefore a complex space.

Quantum mechanics is based on complex numbers and amplitudes and thus we need a complex and non-local space-time manifold.

An entire light ray is a point, a simultaneous entity. A photon doesn't experience "time" and thus its mechanics are and can be non-local.

9:45 Stereographic projection on a Riemann spere with discussion of particle spin.

At 11:40 he shows the spinor calculus and breaks down the components of the spinor. Showing the light cone as a spinor in Riemann space, making things real and plain.

 At 12:40 he explains the twistor and its associated algebra.

At 40:00 he explains cohomology utilizing an impossible triangular solid - can the geometric structure be realized in 3-space

weylmann |  The notion of electron spin was first surmised in 1922 when the German physicists Otto Stern and WaltherGerlach noticed that a stream of silver atoms (each having a single electron in the outer 5s orbital) could be separated by a non-uniform magnetic field into two streams, ‘‘up’’ and ‘‘down.’’ At the time, no one knew what to make of this odd behavior, but three years later George Uhlenbeck and Samuel Goudsmit proposed that electrons could exist in two spin states,±1/2, each with the units of angular momentumħh=h/2π

Still, the concept of electron spin varied between a simplistic physical ‘‘spinning’’ of the electron about some axis (like a child’s top) and a kind of internal angular momentum. 

Today we look upon electrons and other spin-1/2 particles as havingan intrinsic angular momentum with no classical counterpart. This gave rise to the notion of such particles living in ‘‘spin space,’’ an abstract two-dimensional internal space that requires a description beyond that of scalars,vectors and tensors. Cartan’s spinor formalism was found to be appropriate for this description. 

Many attempts have been made over the years to explain spinors at an intuitive, elementary level, but the simplest approach remains an appeal to basic Lorentz group theory. This is rather a pity, because undergraduate students often express an aversion to group theory because of its mathematical nature. But the theory of Lorentz rotations and boosts is relatively simple, and it has the nice property of being relativistic from the start. 

In addition, it neatly admits a formalism that underlies that of the 4-vectors it ordinarily applies to, which is where spinors come in.

 Even better, Lorentz theory confirms the intuitive notion that if a spinor represents half of a 4-vector (rather than the square root), then there should be two kinds of spinor: one comprising the upper half and another representing the lower half. This observation is critical, since a single two-component spinor can be shown to violate odd-even parity in quantum physics, and it takes two spinors acting together to preserve it. Thus, the Dirac bispinor—a four-component object consisting of two stacked spinors—fully preserves parity. 

Our approach will therefore be based on the Lorentz transformations of rotations and boosts. There are, however,a few subtleties that other, more advanced treatments either gloss over or assert by inference, and I will try toexplain these in a more straightforward and understandable manner as we go along. 

The Dirac Equation 

When it was discovered that electrons can exist in both+12and12spin states, the Austrian physicist Wolfgang Pauli suggested in 1927 that the scalar wave functionΨin Schrödinger’s equation be replaced by a two-component spinor, each component representing one of the electron’s allowed spin states. 

But while Pauli’s approach worked,it presented a problem having to do withparity; that is, the sign reversal operationΨ(x,t)Ψ(x,t)gaveinconsistent results in Pauli’s approach, violating the notion that Nature should be mirror-image invariant. 

However, in 1928, at the age of just 25, the great British mathematical physicist Paul Adrienne Maurice Diracmade a monumental discovery, perhaps the greatest discovery in all of modern physics. The student can look upthe details, but what Dirac did was essentially take the square root of the relativistic energy-mass equation

 E2=m2c4+c2p

He ended up with the set of four partial differential equations in (1) involving four4×4 matricesγμ(now calledthe Dirac gamma matrices), along with a new four-component wave functionΨa(x,t), where 

a=0,1,2,3. 

It was soon realized that Dirac’sΨwas abispinor, a four-component mathematical object consisting of theφRandφLWeyl spinors we identified earlier: 

Ψ=Ψ0Ψ1Ψ2Ψ3=ïφRφLò 

Dirac’s bispinor was found to preserve parity under the sign reversal operation 

Ψ(x,t)Ψ(x,t)

Far more importantly, the spinorφReffectively represents the spin-up and spin-down components of an ordinary electron,while φL represents the spin-up and spin-down components of an anti-electron (known as a positron)—Dirac’swork thus predicted the existence of antimatter (the positron was subsequently discovered in 1932, for which the Caltech experimental physicist Carl Anderson won the Nobel Prize). 

Dirac’s relativistic electron equation alsoexplained electron spin as a form of intrinsic angular momentum called S. Thus, the angular momentum L of an electron alone is not conserved—instead, it is L+S that is conserved.Dirac’s equation and its underlying mathematics today represent the foundation of much of modern quantum field theory.