Antisymmetrizing a Factorized Polynomial Vanishes?

I
Thread starter thatboi
Start date Jun 12, 2023
Tags

Polynomials Quantum physics

In summary, there is a discussion about an argument in a paper where it is claimed that a term with multiple factors of (z_i-z_j) would require antisymmetrization of twice as many variables, which is not possible. However, the speaker suggests that even for an expression like (z_1-z_2)(z_3-z_4), it can be antisymmetrized using the definition of the antisymmetrizer. The question then arises as to why the paper claims that the antisymmetrizer vanishes.

Jun 12, 2023

thatboi

Hi all,
I am having trouble understanding the argument below equation (3.5) in https://arxiv.org/pdf/cond-mat/9605145.pdf where they claim that "Upon antisymmetrization, however, a term with ##k## factors of ##(z_{i}-z_{j})## would have to antisymmetrize ##2k## variables with a polynomial that is linear in each", which is impossible.

However, I thought that even for something like ##(z_{1}-z_{2})(z_{3}-z_{4})##, I can antisymmetrize this expression by just using the definition of the antisymmetrizer, i.e I sum over all permutations of the indices ##(1,2,3,4)## and include ##\pm## signs as appropriate depending on how many times an index has been shifted. So why does the paper claim that the antisymmetrizer vanishes?
Thanks.

1. What does it mean to "antisymmetrize" a polynomial?

Antisymmetrization is a mathematical operation that involves rearranging the terms of a polynomial in a specific way. It essentially involves taking the polynomial and swapping any two terms that are identical except for a change in sign.

2. Why is it important to antisymmetrize a factorized polynomial?

Antisymmetrizing a factorized polynomial can help simplify the expression and make it easier to work with. It can also reveal any underlying patterns or relationships between the terms.

3. Can any polynomial be antisymmetrized?

Yes, any polynomial can be antisymmetrized. However, the resulting expression may not always be useful or meaningful.

4. How is antisymmetrizing different from factoring a polynomial?

Antisymmetrizing and factoring are two different mathematical operations. Factoring involves breaking down a polynomial into smaller, simpler terms, while antisymmetrizing involves rearranging the terms in a specific way.

5. Are there any real-world applications for antisymmetrizing a polynomial?

Antisymmetrization is often used in quantum mechanics and other areas of physics to simplify and solve complex equations. It can also be useful in computer science and data analysis for finding patterns and relationships in large datasets.