Time ordered integrals

1. Jan 9, 2010

Mr confusion

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. Jan 9, 2010

tiny-tim

Hi Mr confusion! Happy new year to you too!

"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 … 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

So you re-arrange the Vs before doing an ordinary integration.

3. Jan 9, 2010

Mr confusion

tiny tim -thank you.
i am now trying to fit in your idea in the derivation. I will keep posting my progress.

4. Jan 9, 2010

nicksauce

FYI they are ordered with decreasing time.

5. Jan 9, 2010

Mr confusion

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.

6. Jan 10, 2010

Mr confusion

i have understood. But still do not know about the applications of time ordered integrals.

7. Jan 10, 2010

tiny-tim

"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.

8. Jan 11, 2010

tiny-tim

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,

to

∫∫…∫ 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.