- #1

maverick_starstrider

- 1,111

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Hi,

I'm looking at this wikipedia entry ( http://en.wikipedia.org/wiki/Dyson_series ) for the derivation of the Dyson series and I'm having a great deal of difficulty with:

[itex]S_n=\int_{t_0}^t{dt_1\int_{t_0}^{t_1}{dt_2\cdots\int_{t_0}^{t_{n-1}}{dt_nK(t_1, t_2,\dots,t_n)}}}.[/itex]

If K is symmetric in its arguments, we can define (look at integration limits):

[itex]K_n=\int_{t_0}^t{dt_1\int_{t_0}^{t}{dt_2\cdots\int_{t_0}^t{dt_nK(t_1, t_2,\dots,t_n)}}}.[/itex]

And so it is true that:

[itex]S_n=\frac{1}{n!}K_n.[/itex]

I simply can't get my head around this. How can the S integral lead to the K integral (up to an n factorial)? I don't understand how this integral can be performed. I mean S is a series of coupled integrals that must be performed in succession, K is a product of independent integrals...

Thanks in Advance

I'm looking at this wikipedia entry ( http://en.wikipedia.org/wiki/Dyson_series ) for the derivation of the Dyson series and I'm having a great deal of difficulty with:

[itex]S_n=\int_{t_0}^t{dt_1\int_{t_0}^{t_1}{dt_2\cdots\int_{t_0}^{t_{n-1}}{dt_nK(t_1, t_2,\dots,t_n)}}}.[/itex]

If K is symmetric in its arguments, we can define (look at integration limits):

[itex]K_n=\int_{t_0}^t{dt_1\int_{t_0}^{t}{dt_2\cdots\int_{t_0}^t{dt_nK(t_1, t_2,\dots,t_n)}}}.[/itex]

And so it is true that:

[itex]S_n=\frac{1}{n!}K_n.[/itex]

I simply can't get my head around this. How can the S integral lead to the K integral (up to an n factorial)? I don't understand how this integral can be performed. I mean S is a series of coupled integrals that must be performed in succession, K is a product of independent integrals...

Thanks in Advance

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