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In Page 52., under the heading *3.1.1 Dyson's Formula*, Tong introduces an unitary operator

[tex]U(t, t_0) = T \exp(-i\int_{t_0}^{t}H_I(t') dt')[/tex]

He then introduces the usual definition of time ordered products and goes on to expand [tex]U(t,t_0)[/tex]. I am not able to follow how he expanded the time ordered product of operators in the second-order term of the Taylor expansion of the exponential. In particular,

**I am unable to follow the limits being used and why both integrals are being put in the front.**Should we not get product of two integrals involving H

_{I}?

The expansion of U(t,t_0) is given by

[tex]1 - i\int_{t_0}^{t}dt'H_I(t') + \frac{-i^2}{2}[\int_{t_0}^{t}dt'\int_{t'}^{t}dt''H_I(t'')H_I(t')+\int_{t_0}^{t}dt'\int_{t_0}^{t'}dt''H_I(t')H_I(t'')]+... [/tex]

Link to Course Page - [David Tong: Lectures on Quantum Field Theory][2]

[1]: http://www.damtp.cam.ac.uk/user/tong/qft/qft.pdf

[2]: http://www.damtp.cam.ac.uk/user/tong/qft.html