1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Index notation - I never know when to introduce a new symbol?

  1. Jan 21, 2012 #1
    This isn't strictly a homework problem but anyway...
    I'm reading through a QFT textbook that is using index notation, and sometimes a new index symbol will be introduced during some mathematics and it always throws me off. I'll give a simple example, take the Minkowski metric:

    [tex]g^{\mu\nu} = \left(\begin{array}{cccc}1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{array}\right) [/tex] and its inverse: [tex]g_{\mu\nu} = \left(\begin{array}{cccc}1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{array}\right) [/tex]

    We can multiply these 2 matrices together, ie. we could take [itex]g^{\mu\nu}g_{\mu\nu}[/itex] to get the identity matrix. However - and this confuses me - we could also take [itex]g^{\mu\nu}g_{\mu\nu}[/itex] to mean just the sum of the products of the matrix elements over both indices, as both are repeated:

    [tex] g^{\mu\nu}g_{\mu\nu} = g^{00}g_{00} + g^{01}g_{01} + g^{02}g_{02} + g^{03}g_{03} + g^{10}g_{10} + g^{11}g_{11} + g^{12}g_{13} + g^{20}g_{20} + g^{21}g_{21} + g^{22}g_{22} + g^{23}g_{23} + g^{30}g_{30} + g^{31}g_{31} + g^{32}g_{32} + g^{33}g_{33} [/tex]

    So, the first thing that confuses me is, how come we use indices when we refer to the full matrix [itex]g^{\mu\nu}[/itex], when normally we would just call a matrix (for example) [itex]A[/itex], and only mention indices [itex]i, j[/itex] when we want to refer to the [itex]i^{th}, j^{th}[/itex] element of the matrix, [itex]A^{ij}[/itex] ?
    It seems to me that there is ambiguity here, when is [itex]g^{\mu\nu}g_{\mu\nu}[/itex] a matrix and when is it just a number?

    Also, to get to the main part of my question, my book makes the statement that [itex]g^{\mu\nu}g_{\nu\rho} = \delta^{\nu}_{\rho}[/itex], the kronecker delta.
    Here it has introduced a new index [itex]\rho[/itex]. I can see that this is true if I do the summation over [itex]\nu[/itex]:
    [itex]g^{\mu\nu}g_{\nu\rho} = g^{\mu 0}g_{0\rho} + g^{\mu 1}g_{1\rho} + g^{\mu 2}g_{2\rho} + g^{\mu 3}g_{3\rho}[/itex]

    then if we set, say, [itex]\mu = 0, \rho = 0[/itex], we get
    [itex]g^{\mu\nu}g_{\nu\rho} = g^{0 0}g_{00} + g^{0 1}g_{10} + g^{0 2}g_{20} + g^{0 3}g_{30} = (1)(1) + (0)(0) + (0)(0) + (0)(0) = 1[/itex]

    or if we set, say, [itex]\mu = 0, \rho = 1[/itex], we get
    [itex]g^{\mu\nu}g_{\nu\rho} = g^{0 0}g_{01} + g^{0 1}g_{11} + g^{0 2}g_{21} + g^{0 3}g_{31} = (1)(0) + (0)(-1) + (0)(0) + (0)(0) = 0[/itex]

    So clearly the Kronecker delta condition is satisfied, so the statement [itex]g^{\mu\nu}g_{\nu\rho} = \delta^{\nu}_{\rho}[/itex] is true. However, if I was writing out my own solution to a problem that involved index notation, I would never know to introduce a new index symbol myself. It's just lucky that the textbook told me and I could verify it with an explicit calculation.

    Can anyone explain to me how to know when a new index symbol should be introduced?
    Last edited: Jan 21, 2012
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted