Levi-Civita Contraction Meaning: Undergrad Research

In summary, the conversation discusses calculating a term involving four vectors (representing four-momentum, spin, etc.) and its relation to a 4x4 determinant. The physical significance of this determinant in the context of particle scattering is also mentioned. The conversation then shifts to considering the two and three dimensional analogues of this problem.
  • #1
cmcraes
99
6
Hi all, I'm doing undergraduate research this summer, and a few times I've been told to calculate a term with the following form: ∈abcdpaqbkcsd, where p,q,k and s are four vectors (four-momentum, spin, etc). Now I know this ends up calculating exactly like a 4x4 determinant, I'm just not quite sure what its the determinant of (A matrix/tensor composed of these four vectors, I guess?), and what physical significance this quantity/determinant has.

Any and all insight is appreciated! (If it helps, I'm working on particles scattering problems)
 
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  • #2
What would you say for the two and three dimensional analogues of this problem?

In 3 dimensions, this might be represented by ##\vec A\cdot(\vec B \times \vec C)##.
 
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1. What is Levi-Civita Contraction?

Levi-Civita Contraction is a mathematical operation used in differential geometry to reduce a tensor with multiple indices into a single index tensor. It is often used in the study of general relativity and other fields of physics.

2. Who discovered the Levi-Civita Contraction?

The Levi-Civita Contraction was discovered by Italian mathematician Tullio Levi-Civita in the late 19th and early 20th century. He is also known for his contributions to the theory of relativity and differential equations.

3. How is Levi-Civita Contraction used in research?

Levi-Civita Contraction is commonly used in research related to differential geometry, general relativity, and other fields of mathematics and physics. It is particularly useful in simplifying complex equations and calculations involving tensors with multiple indices.

4. What are the limitations of Levi-Civita Contraction?

While Levi-Civita Contraction is a powerful tool in mathematics and physics, it does have its limitations. It can only be applied to tensors with identical upper and lower indices, and it cannot be used to contract a tensor with itself.

5. Are there any real-world applications of Levi-Civita Contraction?

Yes, Levi-Civita Contraction has various real-world applications in fields such as physics, engineering, and computer science. It is used, for example, in the analysis and prediction of fluid dynamics, as well as in the development of machine learning algorithms.

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