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Time ordered integrals

  1. Jan 9, 2010 #1
    hi friends,
    i am in the middle of my course in introductry quantum mechanics. Now, i am getting stuck in understanding time ordered integrals. my text is showing a time dependent hamiltonian and then constructing a time ordered integral . i am not understanding why i will call it time ordered? and what does a time ordered integral mean?
    thanks and new year greetings to all of you.
  2. jcsd
  3. Jan 9, 2010 #2


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    Hi Mr confusion! Happy new year to you too! :smile:

    "time-ordered" describes a polynomial in V, where V is a function of a 4-vector x.

    T{V(x1)V(x2)…V(xn)} simply means that you rearrange the Vs, in order (I forget whether it's increasing or decreasing :redface: … let's suppose it's increasing) of the t-component of the 4-vectors x1 x2 … xn.

    For example:

    T{V(a,3)V(b,5.5)V(c,7)} = V(a,3)V(b,5.5)V(c,7)

    T{V(a,3)V(b,7)V(c,5.5)} = V(a,3)V(c,5.5)V(b,7)

    T{V(a,7)V(b,5.5)V(c,3)} = V(c,3)V(b,5.5)V(a,7)

    etc :wink:

    So you re-arrange the Vs before doing an ordinary integration. :smile:
  4. Jan 9, 2010 #3
    tiny tim -thank you.:smile:
    i am now trying to fit in your idea in the derivation. I will keep posting my progress.
  5. Jan 9, 2010 #4


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    FYI they are ordered with decreasing time.
  6. Jan 9, 2010 #5
    sorry, but what is FYI? (i am new to english)
    ok, if they are ordered with decreasing time, then i have got a problem here,
    my text is performing a time evolution of a state vector by applicasionising the time dependent scroedinger equation involving a time dependent hamiltonian.
    but when i think, will it matter much if they are ordered or not while integrating? i will have worried if they were matrices......
    but hamiltonians are matrices in basis....
    will think this over again.
    nickstats -is that the photo of the great feynman? seems more like dirac from side angle. but i loved it.
  7. Jan 10, 2010 #6
    i have understood. But still do not know about the applications of time ordered integrals.
  8. Jan 10, 2010 #7


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    "Time-ordered integral" simply means that instead of integrating

    ∫∫…∫ V(x1)V(x2)…V(xn) dx1dx2…dxn,

    you first swap all the Vs into time-order so that it becomes

    ∫∫…∫ T{V(x1)V(x2)…V(xn)} dx1dx2…dxn.
  9. Jan 11, 2010 #8


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    On second thoughts, perhaps you mean something slightly different by "time-ordered integral" …

    I assumed you meant that the integrand was time-ordered, but perhaps you were referring to the limits? If so …​

    The reason we change from

    ∫∫…∫ V(x1)V(x2)…V(xn) dx1dx2…dxn,


    ∫∫…∫ T{V(x1)V(x2)…V(xn)} dx1dx2…dxn

    is because the limits of integration in the first integral (in quantum field theory) are usually time-ordered, that is the limits of integration are -∞ < xi,yi,zi < ∞ (i = 1 … n) but -∞ < tn < … t2 < t1 < ∞,

    but that's really awkward to calculate :yuck:, so we change to the second integral, which has the same value, but its limits of integration are simply -∞ < xi,yi,zi,ti < ∞ (i = 1 … n).

    In other words, instead of having an ordinary integrand and horrible time-ordered limits, we change to nice ordinary limits and a time-ordered integrand. :wink:
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