Piling Up Possibility: How Structure and Symmetry Shape evolution of lie group

islamic geometric pattern

"Structure unfolds from small steps, and the universe remembers the rule." - by dayun.H

How I came up with this Posting..

I have to submit the report about P-N junction solar cell fabrication.. then revisit the Fick's 2nd law and Gaussian function for doping distribution.. that I had learned 2 years ago. When I revisit them... Gaussian... again... "Is Fick's 2nd law and harmonic oscillator equation has something common?" because their both solutions are Gaussian function.  That's very special and unique! I have never thought about those two would share similar properties. But then, found that both have (d/dx)^2 and if we expand the HO function to psi(x,t) .. then d/dt parts appear. There I revisit all the concepts regarding Hamiltonian function roots.. why we have to have that exponential function as solution.. why H, p, L.. all starts with similar format of exponential.. and then finally approach to the Lie group world... which is totally mathematics area... all connected... everything is big one world sharing same root...

Every phenomena should be chop down to microscopic scale and then make freeze to pause to see what's happening

When I first learned about "Force," it was really hard for me. It seems to me that everyone around me readily understands what happens when we apply force to the piston. But for me, it took a really long time to understand. what is happening inside the molecules of gas that are being pushed, what if some air thing might slip away between the empty space of the pushing holder and cylinder, what if the pushing force might not be consistent, even if we are pushing at the same force if heat occurs due to holder and cylinder are sliding each other, what about heat is happening while pushing....etc problems hit my head. After gathering numerous assumptions, my speed in understanding the physics phenomena and problem-solving time decreased dramatically.

One of the biggest assumptions that I had established in my head was that "when we are trying to move something with force and especially trying to see sth is changing at some boundary, we should expand time and expand space with a very small amount" We should not treat as just one moment, one space point as "t" or "x", once we start to do like that we can never figure out phenomena easily.

Everything should be chopped down, like taking a picture with the really slow camera, and assume every chopped-down moment is pausing for a slight duration of time, for example, 10 seconds, which seems like leaving the shadow behind. For example, when we are going to the left piston, then at the same time, the right piston would be moving out. but actually, the left piston moves first, and then molecules pass to a small hole, and then the molecules push the right piston. And every slight moment delta t, slight movement of delta x of piston, and every moment same force is applied.

If we treat the movement as "just moved left piston and right piston" together, then we are going to ignore the microscopic scale where lots of things happening are smoothed out due to the macroscopic scale.

Start! "rule" and "evolution"

small scale movement = small scale "scalar" movement = how far? = theta

what direction? what flow speed? (gradient) = vector part = generator J

If we think the original start point is "1", then a slight increment is perpendicular to the start point because it's the direction of the rotation. Perpendicular to the real plot is the imaginary part thus direction? "i" how far? "theta." Thus, 1+i*theta is the second plot.

To be more exact, actually, it is moving around the curvature. Definitely, it is not exactly just moving "i*theta," but here we simplify that small curvature could be treated as a straightened rectangular portion.

We think on a microscopic scale, and then we need to expand our thoughts to the macroscopic. Piling up from the microscopic to the macroscopic scale. Here is the key. When we start to pile up the microscopic scale, we need to make a multiplication of the matrix.

The addition of a vector shifts the original vector. So shifting is just can be thought of as vector as staying the same, but the background coordinates where it lies on just shift.

If we want to modify the original one significantly, we should consider the vector properties first. Its identity is "length" and "direction"; multiplication "number" modifies "length," and multiplication of "operator" modifies both "length" and "direction." Thus, if we are going to pile up the microscopic scale to macroscopic, we need to do the same process repeatedly with multiplication.

This is the exponential.

If we expand this rotation a small amount as just moving forward a small step, which is a translation with a small step, then it can be interpreted as "T*T*T* ....".

To check out whether it works properly, when the exponential function multiplies together, its coefficient adds up. That added-up “macroscopic scale” is called a “generator”. This “generator,” which contains “direction” and “gradient,” will show the direction of “how the small scale should piled up”.

This is a similar situation to how we know the final destination, return back to our origin, and teach how to approach our destination.

So they are noted as “evolution”.

Generator = local “rule”

Exponential = global “evolution”

There are 4 types of “rule” - “H, p, L, H(imaginary time)” each regulating “evolution” – “time, space x, rotation theta, imaginary time tau”.

H,p, and L are in the Lie group, which maintains probability and symmetry preservation. H(imaginary time) is not preserve quantity. Not preserving quantity means containing a decaying state. In short, ground state, excited state all decays, but since excited state decay first so only ground state left at longer time scale, which is reflected in variational, monte carlo methods.

Complex plane and symmetry preservation

We live in a space where real and imaginary axes cross. Describing the world algebraically means describing it as algebraically closed, which forms a loop on the complex plane. This aligns with:

• Rotational motion (angles on a circle)
• Fourier series (loops via frequencies)
• Phase (loops around )

R vector is just transforming (1,0) (0,1) very basis x, y coordinates to (cos, sin) (-sin, cos). U vector is transforming (1,0) (0,1) basis x, y coordinates to (exp(i*theta), 0) (0, exp(-i*theta)).

R vector can be easily think as expand real axis as 2d as think at only real plane. 2 vector rotates amount of "theta" on r=1 circle with perpendicular relationship. While unitary vector is think of complex plane and 2 vector rotates amount of "theta" on r=1 circle with perpendicular relationship.

The “continuous symmetry -> generator” means that a small derivative amount is “continuous,” moving piles up to multiplication scale while “rotating around the same circle,” which preserves “length” in classical means or preserving “possibility” in quantum meaning. Then, an infinite pile makes a large macroscopic movement, which is a generator.

Symmetry preservation is about preserving structure between basis vectors and total possibilities. In quantum, this is done by unitary operators; in classical systems, by orthogonal matrices. Both preserve the relationship between components rotating on a circle — either in the real plane or complex plane. Also, piling up means follow the same pattern with “rule”, which is also preserving “structure”; “symmetry”

Lie groups are precisely the structures that preserve these symmetries.

Continuous symmetry generator small derivative infinite pile-up = exponential = macroscopic evolution.

Hamiltonian, momentum, and rotation — are all are part of Lie group symmetries because they preserve the structure.

In qubit systems, basis states and can entangle, which is a different kind of operation. Qubit gates like Pauli X, Y, Z, and Hadamard change the relative phase and entangle. These gates do preserve unitarity, so they are still in Lie groups (specifically SU(2)). They are qubit apart so seems discrete, but actually they are linear combined and living inside bloch sphere which makes them “continuous”.