A Short Course on Quantum Matrices: Understanding Notation Basics

In summary, the conversation revolves around the topic of mathematical physics and the challenges faced by someone coming from a background in straight physics. The discussion references a specific work by Mitsuhiro Takeuchi and delves into questions about unfamiliar notation and concepts such as M2, M4, the circle with the cross, and the Yang Baxter equation. The conversation also includes a mention of the tensor product and its use in quantum matrices, as well as a resource for learning more about it.
  • #1
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I just started working with mathematical physics coming from a straight physics background. The actual work doesn't seem that hard but some of the notation is unfamilar.

The work is "A Short Course on Quantum Matrices" by Mitsuhiro Takeuchi that can be found at:

http://www.msri.org/publications/books/Book43/files/takeuchi.pdf

Definition 1.5 on p 386 presents the first display of my ignorance...what is M2, M4, what is the circle with the cross and what is the Yang Baxter equation trying to tell me?

Basically what the hell is happening on that whole page from Definition 1.5 onwards? Proposition 1.6 is just as forign to me. Don't warry about things from section 2.

Where can I find a list of all the notation that I should have picked up in an undergrad degree in maths but instead I was off doing physics?
 
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  • #2
[tex]\otimes[/tex] is the tensor product.

M_n(k) is the nxn matrices with entries in the field k

Yang Baxter is telling you how to commute some elements in the algebra, though I forget the analogies people make.

q is an indeterminate, it measures how far way from being the ordinary case the quantum case is. Usually, the limit as q tends to 0 is the classical case.

The notation M_n is undergrad. The tensor product probably isn't in most places.

www.dpmms.cam.ac.uk/~wtg10

on the page of mathematical discussions, there is one called "lose your fear of tensor products".
 
  • #3


First of all, don't worry - it's completely normal to feel overwhelmed by new notation when diving into a new mathematical field. The good news is that with some practice and exposure, it will become more familiar and make more sense.

To answer your specific questions, M2 and M4 refer to the dimensions of the matrices being discussed. In this case, M2 refers to 2x2 matrices and M4 refers to 4x4 matrices. The circle with the cross inside is a symbol for the tensor product, which represents the combining of two matrices. The Yang-Baxter equation is a fundamental equation in quantum mechanics that relates the behavior of particles in a multi-particle system.

As for the rest of the page and Proposition 1.6, it is discussing the properties of these matrices and how they relate to each other. It may be helpful to review some basic linear algebra concepts, as well as familiarizing yourself with some common notation used in quantum mechanics.

In terms of finding a list of notation, it may be helpful to consult a textbook or online resources specifically on quantum mechanics or linear algebra. Additionally, don't hesitate to ask for clarification from your peers or professors - they are there to help you understand and succeed in your studies. With some practice and guidance, you will soon become more comfortable with this notation and be able to apply it confidently in your work. Best of luck in your studies!
 

FAQ: A Short Course on Quantum Matrices: Understanding Notation Basics

What is a quantum matrix?

A quantum matrix is a mathematical representation of a quantum system that takes into account the probabilistic nature of quantum mechanics. It is a square array of complex numbers that can be used to describe the state of a quantum system.

How is a quantum matrix different from a classical matrix?

A quantum matrix differs from a classical matrix in that it can have complex numbers as its entries, whereas a classical matrix can only have real numbers. Additionally, while classical matrices represent deterministic systems, quantum matrices represent probabilistic systems.

What is the significance of the notation used in quantum matrices?

The notation used in quantum matrices is significant because it represents the state of a quantum system and allows for calculations and predictions to be made about the system's behavior. It also follows specific rules and principles in order to accurately represent quantum phenomena.

How can quantum matrices be applied in real-world situations?

Quantum matrices can be applied in a variety of fields, including quantum computing, cryptography, and quantum mechanics research. They are also used in industries such as pharmaceuticals and materials science to model and understand quantum systems.

Are there any limitations to using quantum matrices?

There are some limitations to using quantum matrices, as they can become very complex and difficult to interpret for larger quantum systems. Additionally, they may not accurately represent certain phenomena, such as entanglement, which requires a more advanced mathematical framework.

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