Research Project Flashcards
How can the solutions of Dirac factorization be used in physics/real-world situations?
The quantum field with a higher-order spin is built into the Dirac Equation, where electrons and positrons spin up or down. The Dirac Equation can describe a particle with some other spin.
What is a Generalized Clifford Algebra?
The vtensorv component of the equation t(v)/<vtensorv - Q(v)> changes to the kth tensor power which is denoted (v^tensork).
The Q(v) component of the same equation changes to the k-order form since Q(v) is a 2-order form.
Why do the Pauli matrices show up?
They satisfy the conditions of a Clifford Algebra (as stated on the poster)
may want to elaborate more
What are the Generalized Pauli matrices?
The procedure is shown in the results section. The Pauli matrices are denoted by the letters P, J, and K. These build the Generalized Dirac matrices.
What is Fractional Calculus?
The process of taking a derivative to a fractional order instead of an integer order.
As shown on our poster, we take the Fourier Transform and multiply it by omega^alpha and e^3i(pi)(alpha)/2
When we plug in 0 for alpha, the equation equals f since w^apha and e^3i(pi)(alpha)/2 go to 1
Why do we need the Semigroup property?
If you look at the example for the Dirac Factorization, it is used for simplification of the right-hand side when expanding.
Where else do Pauli matrices show up?
Within quantum field theory for physics (explain the spin if needed) and complex mathematics such as Lie Algebra structures.
Why should we study Fractional Partial Differential Equations (PDEs)?
These equations can help in real-world situations, such as diffusion equations that can be written into fractional differential equations.
What is a Generate Clifford Algebra?
Elements of Generate Clifford Algebras can build a basis. These algebras do not consider basis redundancy when generating. In our project, the Generate Clifford Algebra (Clk(V,Q)) was generated by beta 0, beta 1, beta 2, and beta 3.
Explain what is happening in the graph.
This is a graph for the fractional derivatives of the Gaussian Equation. There is a transition of function from the 0 derivative (no derivative) to the 1st derivative. As shown on the graph, the first derivative begins below the fractional derivatives but ends up crossing over them closer to the right-hand side. Mathematicians are working on a proof of this phenomenon.
What is the point of studying Dirac Factorization?
We can study more complex math and science topics, such as partial differential equations and quantum field theory that explains the behavior of subatomic particles.
How do we know if a PDE generates a Clifford Algebra?
Using the generalized Clifford Algebra relation (under Dirac Factorization), we can find two basis of Ai and Aj that add to 2deltaij (also known as the Kronecker Delta)
What is the point of studying Dirac Factorization?
We can study more complex math and science topics, such as partial differential equations and quantum field theory, that explain subatomic particle behavior.
How did Dirac come up with Dirac Factorization?
He took the Klein-Gordon Equation’s first order and tried to factor is by take the second order.
What do the Solutions to (free) Fractional Dirac equations state?
it creates a plain wave where there is no momentum leaving only energy behind
