- #1
Final
- 25
- 0
Hi,
there is a good expression for [tex] \sum_{s}{u_s(\vec{p})\bar{v_s}(\vec{p})}[/tex] ?
Thank you
there is a good expression for [tex] \sum_{s}{u_s(\vec{p})\bar{v_s}(\vec{p})}[/tex] ?
Thank you
Last edited:
Finding Simplicity in Summation Expressions
- Thread starter Final
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- Expressions Summation
In summary, the conversation discusses the expression for \sum_{s}{u_s(\vec{p})\bar{v_s}(\vec{p})} and the usage of spin sums in relation to unobserved spins. It also mentions the problem of Majorana's fermions and the computation of the cross section, which involves sums over the spin of the square of the transition amplitude involving u vbar and v ubar. The expert suggests using spinor identities to transform the equation and only have u ubar or v vbar, as explained in the book by Srednicki.
- #2
Avodyne
Science Advisor
- 1,396
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Not that I know of. But I don't know why you would need this sum; spin sums are needed when a spin is not observed, then you want to sum the absolute square of the transition amplitude over the unobserved spin; but that will always involve u and ubar or v and vbar, but never u and vbar or v and ubar.
- #3
Final
- 25
- 0
Not always...
My problem is about Majorana's fermions:
Take the scattering [tex]\nu_{\tau}+\bar{\nu}_{\tau}\rightarrow \nu_e+\bar{\nu}_e[/tex] and the interaction [tex]{\cal{L}}=g \sum Z_{\mu}\bar{\psi}_{\nu_l}\gamma^{\mu}(1-\gamma_5)\psi_{\nu_l}[/tex].
The [tex]\nu[/tex] are Majorana's fermions (i.e. [tex]d_r=b_r [/tex]) with mass [tex]m_{\nu_{\tau}}>m_{\nu_e} [/tex]. Compute the cross section. Here the feynman rules are quite difficult and the sums over the spin of the square of the transition amplitude involve also u vbar and v ubar!
My problem is about Majorana's fermions:
Take the scattering [tex]\nu_{\tau}+\bar{\nu}_{\tau}\rightarrow \nu_e+\bar{\nu}_e[/tex] and the interaction [tex]{\cal{L}}=g \sum Z_{\mu}\bar{\psi}_{\nu_l}\gamma^{\mu}(1-\gamma_5)\psi_{\nu_l}[/tex].
The [tex]\nu[/tex] are Majorana's fermions (i.e. [tex]d_r=b_r [/tex]) with mass [tex]m_{\nu_{\tau}}>m_{\nu_e} [/tex]. Compute the cross section. Here the feynman rules are quite difficult and the sums over the spin of the square of the transition amplitude involve also u vbar and v ubar!
- #4
Avodyne
Science Advisor
- 1,396
- 94
For Majorana fermions, there is always a way to transform things (using spinor identities) so that you get only u ubar or v vbar. This is explained in the book by Srednicki (draft copy available free online, google to find it).
Related to Finding Simplicity in Summation Expressions
1. What is a summation expression?
A summation expression is a mathematical notation used to represent the sum of a series of terms. It is written as Σ (the Greek letter sigma) followed by the expression for each term and the range of values over which the terms are to be summed.
2. Why is finding simplicity in summation expressions important?
Finding simplicity in summation expressions is important because it allows us to reduce complex mathematical expressions to simpler forms, making them easier to understand and work with. It also helps to identify patterns and relationships between terms, allowing for more efficient calculations.
3. What are some strategies for simplifying summation expressions?
There are several strategies for simplifying summation expressions, including using known summation formulas, factoring out common terms, and using properties of summation such as linearity and commutativity. It is also helpful to break down the expression into smaller parts and look for patterns or recurring terms.
4. Can summation expressions be simplified using computer programs?
Yes, summation expressions can be simplified using computer programs such as mathematical software or coding languages like Python or MATLAB. These programs have built-in functions and algorithms for simplifying summation expressions and can handle more complex expressions than can be done by hand.
5. How can finding simplicity in summation expressions be applied to real-world problems?
Finding simplicity in summation expressions can be applied to real-world problems in various fields such as economics, physics, and computer science. It can help in analyzing patterns and trends in data, calculating probabilities and making predictions, and optimizing processes or algorithms. For example, in economics, summation expressions can be used to model and analyze financial data, while in physics, they can help in calculating the total energy or force in a system.
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