Commutator-like notation, index notation

In summary, the conversation discusses unfamiliar notation in field theory equations and how it may relate to anti-symmetry. One person suggests a method of dividing by factorial and writing out permutations to solve the equations. The equations mentioned are $\Phi_{\mu\nu} = 2x_{[\mu}\partial_{\nu]}\phi$, $\Lambda^{\rho}_{\mu\nu} = 2x_{[\mu}\delta^{\rho}_{\nu]}L$, and $\partial_{[\mu}F_{\nu\rho]} = 0$.
  • #1
ncs22
6
0

Homework Statement



There are some equations in the notes on field theory I am reading with notation I have never come across before. Someone told me it was a way of ensuring that the expression was anti-symmetric. I can't find it used the same anywhere else but no explanation is provided in the notes which makes me think maybe it is just an ordinary commutator but I am not sure.


Homework Equations



The equations are...

i. [tex]$\Phi_{\mu\nu} = 2x_{[\mu}\partial_{\nu]}\phi[/tex]
ii. [tex]$\Lambda^{\rho}_{\mu\nu} = 2x_{[\mu}\delta^{\rho}_{\nu]}L[/tex]
iii. [tex]$\partial_{[\mu}F_{\nu\rho]} = 0[/tex]

F is the electromagnetic tensor if it helps.

The Attempt at a Solution



does the first mean...

[tex] \left(2x_\mu\partial_\nu\phi - 2x_\nu\partial_\mu\phi\right) [/tex]

but then what do the others mean? For the second I was thinking of ignoring the superscript index - the rho - and doing the same as with the first but then what about the third one or am I not even close?

Thanks
 
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  • #2
not really.
you have to divide by the factorial of the number of terms in square brackets and then you write out all the possible permutations of the terms in square brackets (where even permutations are positive and odd permutations are negative),

so [itex]2x_{[\mu} \partial_{\nu]} \phi=2 \frac{1}{2!} \left(x_{\mu} \partial_{\nu} - x_{\nu} \partial_{\mu} \right) \phi = \left(x_{\mu} \partial_{\nu} - x_{\nu} \partial_{\mu} \right) \phi=[/itex]

not this can get much more complicated. for example in your third case you'll have 6 permutations, 3 odd and 3 even. recall a permutation is odd if it can be written as an odd number of transpositions and even if it can be written as an even number of transpositions.
 
  • #3
awesome! thanks
 

1. What is commutator-like notation and how is it used in mathematics?

Commutator-like notation is a shorthand way of representing mathematical equations that involve commutators, which are mathematical operators used to measure the non-commutativity of two other operators. It is commonly used in fields such as quantum mechanics and group theory.

2. How is index notation different from regular mathematical notation?

Index notation, also known as Einstein notation or tensor notation, uses indices to represent the components of a tensor or matrix, rather than explicitly writing out each component. This can make complicated equations easier to read and manipulate.

3. What are the advantages of using index notation?

One advantage of index notation is that it is more concise and can make complex equations easier to read and understand. It also allows for easier manipulation of equations involving tensors and matrices.

4. Are there any limitations to using commutator-like and index notation?

One limitation is that it may not be familiar to all mathematicians and scientists, so it may take some time to get used to. Additionally, it may not be suitable for all types of equations and mathematical concepts.

5. How can I learn and become proficient in using commutator-like and index notation?

The best way to learn and become proficient in using commutator-like and index notation is through practice and studying examples. There are also many online resources and textbooks available that can provide a thorough explanation of these notations and their applications in mathematics.

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