What is going on with the indices

  • Thread starter vertices
  • Start date
  • Tags
    Indices
In summary, the conversation discusses the calculation of \delta_1 \delta_2 x^\mu, which is apparently equal to (w^{\mu}_{1}_\nu a^{\nu}_{2} + w^{\mu}_{1}_\lambda w^{\lambda}_{2}_\nu) x^\nu. The context is an infinitesimal Poincare transformation, where \delta x^\mu is a variation of a coordinate and a^{\mu} and \omega^{\mu}_{\nu} represent translation and Lorentz matrices, respectively. The interpretation of the subscripts on the deltas is not clear, but the resulting equation depends on how they are interpreted.
  • #1
vertices
62
0
If:

[tex]\delta x^\mu = a^\mu +w^{\mu}_{\nu}x^\nu[/tex]

How would we go about working out

[tex]\delta_1 \delta_2 x^\mu[/tex]

This is apparently equal to:

[tex](w^{\mu}_{1}_\nu a^{\nu}_{2} + w^{\mu}_{1}_\lambda w^{\lambda}_{2}_\nu) x^\nu[/tex]

but I can't understand what is going on with the indices - can someone help me out?
 
Physics news on Phys.org
  • #2


What's the context, the answer depends on what these things are. I'm assuming that [tex]\delta x^{\mu}[/tex] is some sort of variation of a coordinate and you're expanding it about some point [tex]a^{\mu}[/tex] and only keeping terms up to order 1 in x. What do the subscripts on your deltas mean? I can get some results similar to what you propose as an answer but it depends on how I interpret.

For example if I apply delta on the first line: [tex]\delta(a^{\mu}+\omega^{\mu}_{\nu}x^{\nu})=\delta(a^{\mu})+\omega^{\mu}_{\nu}\delta(x^{\nu})[/tex]

where I've assumed the matrix [tex]\omega^{\mu}_{\nu}[/tex] is constant. Then just vary the x term again. I'm assuming that the variation of the 'a' term is zero and so I wonder about your parantheses. But I can't be sure until I know that these letters represent.
 
Last edited:
  • #3


It's presumably an infinitesimal Poincare transformation (translation + Lorentz), so

[tex]
\delta_1 \delta_2 x^\mu = \delta_1 (a_2^\mu +{w_2^{\mu}}_{\nu}x^\nu) = a_1^\mu + {w_1^\mu}_\nu a_2^\nu +{w_1^\mu}_\nu {w_2^{\nu}}_{\lambda} x^\lambda .
[/tex]

This is a bit different than what's written down in the OP.
 

1. What are indices?

Indices, also known as indexes, are numerical representations of a group of data points that are used to measure changes or trends in a particular market or sector. They are often used as benchmarks to track the performance of a specific group of stocks or other assets.

2. How are indices calculated?

Indices are typically calculated using a weighted average of the stock prices of the companies included in the index. The weights are determined by the market capitalization of each company, meaning larger companies have a greater impact on the index's value.

3. Why do indices fluctuate?

Indices fluctuate due to changes in the underlying stock prices of the companies included in the index. If the majority of the stocks in the index experience a decrease in price, the index will also decrease. Factors such as economic conditions, company earnings, and market sentiment can all impact the prices of the stocks and therefore, the index.

4. How are indices used in investing?

Indices are commonly used as benchmarks for investment performance. Investors can compare the performance of their own portfolio to that of a particular index and use it as a gauge for how well their investments are doing. Investors can also use index funds, which track the performance of an index, to diversify their portfolio and potentially minimize risk.

5. What is the difference between a price-weighted and a market-cap weighted index?

In a price-weighted index, the stocks are weighted based on their individual stock prices, meaning a higher-priced stock will have a greater impact on the index's value. In a market-cap weighted index, the stocks are weighted based on their market capitalization, giving more weight to larger companies. Market-cap weighted indices are more commonly used as they provide a more accurate representation of the overall market.

Similar threads

Replies
6
Views
1K
Replies
24
Views
2K
  • Quantum Physics
Replies
3
Views
1K
Replies
5
Views
757
  • Quantum Physics
Replies
3
Views
1K
  • Quantum Physics
Replies
1
Views
583
Replies
5
Views
2K
  • Quantum Physics
Replies
10
Views
2K
  • Quantum Physics
Replies
14
Views
2K
Replies
2
Views
908
Back
Top