Time Ordered Integrals Explained - Quantum Mechanics

[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.

 

[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(x_n)} 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 … x_n.

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.

 

[Mr confusion]

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

 

[nicksauce]

FYI they are ordered with decreasing time.

 

[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.

 

[Mr confusion]

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

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

 

[tiny-tim]

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

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

∫∫…∫ V(x₁)V(x₂)…V(x_n) dx₁dx₂…dx_n,

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

∫∫…∫ T{V(x₁)V(x₂)…V(x_n)} dx₁dx₂…dx_n.

 

[tiny-tim]

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

∫∫…∫ V(x₁)V(x₂)…V(x_n) dx₁dx₂…dx_n,

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

∫∫…∫ T{V(x₁)V(x₂)…V(x_n)} dx₁dx₂…dx_n.

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(x_n) dx₁dx₂…dx_n,

to

∫∫…∫ T{V(x₁)V(x₂)…V(x_n)} dx₁dx₂…dx_n

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 -∞ < x_i,y_i,z_i < ∞ (i = 1 … n) but -∞ < t_n < … t₂ < t₁ < ∞,

but that's really awkward to calculate , so we change to the second integral, which has the same value, but its limits of integration are simply -∞ < x_i,y_i,z_i,t_i < ∞ (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.