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String Theory, Number Operator , Mass of States

  1. Nov 12, 2017 #1
    1. The problem statement, all variables and given/known data

    I have the following definition of the space-time coordinates

    defxmodes.png
    2. Relevant equations

    Working in a certain gauge we can also do: alphan-.png
    From which we can find:

    mass.png

    Where ##N_{lc} ## sums over the transverse oscillation modes only.
    3. The attempt at a solution

    states.png


    MY QUESTION:

    I don't understand the RHS of the first excited state, how does ##N## take integer values? what exactly are the ##\alpha##?

    So (here the summation is over all modes, whereas above it is over the transvere oscillation modes only but ignoring this)

    number operator.png

    so for the first excited state do we have all alpha modes are zero except ##\alpha_{-1}## ?

    So Then ## N= \alpha^{j}_{-1} ## ? how is this ##N=1##?

    So for the next excited state I expect:

    ## m^2 \alpha_{-1}^i \alpha_{-1}^j |p^i>=m^2\alpha^{j}_{-2}|p^i>=(2-a) \alpha_{-1}^i \alpha_{-1}^j |p^i>=(2-a) \alpha_{-2}^j |p^i> ## right?

    And I am told in my notes that this can be achieved by either ##\alpha_{-1}^i \alpha_{-1}^j |p^i> ## or ##\alpha_{-2}^i##

    How does this get ##N=2##? since, well looking at ## \alpha_{-1}^i \alpha_{-1}^j ##, the sum in ##N## is over ##n## not ##i## or ##j## so am I looking at two different number operators here?

    For the other expression So Then ## N= \alpha^{j}_{-2} ## ? how is this ##N=2##?

    I'm just really confused as you can tel..

    Many thanks for your help in advance.
     
  2. jcsd
  3. Nov 17, 2017 #2
    Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
     
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