Graded Algebra: Understanding Color Dirac Spinors in Space-Time

In summary, the author is talking about a graded algebra, which is a vector space in which we can also multiply two vectors, the result being a vector. The author mentions that the product of two vectors in the same "grade" is not necessarily in that "grade".
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I just read through a paper on a \(\displaystyle \mathbb{Z} _ 3\) graded Algebra. In this instance we are talking about color Dirac spinors in space-time. It looks like the author is talking about \(\displaystyle \left ( SU(3) \otimes L^4 \otimes \mathbb{Z}_2 \otimes \mathbb{Z} _2 \right ) \otimes \mathbb{Z} _3\). ( SU(3) is the color group, \(\displaystyle L_4 \) is the Lorentz group, \(\displaystyle \mathbb{Z} _2 \otimes \mathbb{Z} _2\) is the Dirac 4-spinor group, and \(\displaystyle \mathbb{Z} _3\) is the usual group on 3 elements.

I can (mostly) follow the paper assuming the tensor products, but what do they mean by the word "graded?"

Thanks!

-Dan
 
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  • #2
In mathematics, an "algebra" is a vector space (so we can add vectors and multiply vectors by scalars) in which we can also multiply two vectors, the result being a vector. An algebra is said to be "graded" if we can divide the vectors into "grades" that are "closed" under addition and scalar multiplication (so the sum of two vectors in one "grade" are also in that "grade" and the product of a scalar and a vector in a given "grade" is in that same "grade") but the product of two vectors in the same "grade" is not necessarily in that "grade".

The simplest example of a "graded algebra" is the algebra of all polynomials. Each "grade" is the vector space of polynomials of degree "n" or less for some positive integer "n". Adding two polynomials of degree n or less gives a polynomial of degree n or less and multiplying a polynomial of degree n or less times a scalar (real or complex number) is a polynomial of degree n or less. But while the product of two polynomials of degree n or less is a polynomial it is not necessarily of degree n or less.
 
  • #3
Thank you. That actually answers another question I had about the paper as well. I think I've got the idea now.

Thanks again!

-Dan
 

Related to Graded Algebra: Understanding Color Dirac Spinors in Space-Time

1. What is Graded Algebra?

Graded Algebra is a mathematical framework that combines elements of linear algebra and abstract algebra to study the properties of graded objects, such as color Dirac spinors in space-time. It involves the use of graded vector spaces, graded algebras, and graded modules to represent and manipulate these objects.

2. What are Color Dirac Spinors?

Color Dirac Spinors are mathematical objects that describe the quantum states of particles with both spin and color charge. They are represented as vectors in a graded vector space and are used in the Standard Model of particle physics to explain the behavior of fundamental particles, such as quarks and leptons.

3. How does Graded Algebra relate to Space-Time?

Graded Algebra is used to study the properties of objects, such as color Dirac spinors, in space-time. This is because the framework allows for the representation and manipulation of objects that have both spatial and temporal properties, making it useful for understanding the behavior of particles in the fabric of space-time.

4. What is the significance of understanding Color Dirac Spinors in Space-Time?

Understanding Color Dirac Spinors in Space-Time is crucial for studying the fundamental particles and forces that make up our universe. By using Graded Algebra, scientists can gain a deeper understanding of the properties and interactions of these particles, leading to advancements in fields such as particle physics and cosmology.

5. Are there any real-world applications of Graded Algebra and Color Dirac Spinors?

Yes, Graded Algebra and Color Dirac Spinors have many real-world applications. They are used in fields such as quantum computing, where the principles of quantum mechanics and graded algebra are applied to create more powerful and efficient computers. They are also used in particle accelerators and other high-energy physics experiments to study the behavior of fundamental particles.

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