First of all, I copy the text in my lecture note.(adsbygoogle = window.adsbygoogle || []).push({});

- - - - - - - - - - - - - - - - - - -

In general, $$e^{-iTH}$$ cannot be written exactly in a useful way in terms of creation and annihilation operators. However, we can do it perturbatively, order by order in the coupling $$ \lambda $$. For example, let us consider the contribution linear in $$ \lambda $$. We use the definition of the exponential to write:

$$ e^{-iTH} = [1-iHT/N]^N = [1-i(H_0 + H_{\text{int}})T/N]^N $$ - - - (1)

for $$ N \rightarrow \infty $$. Now, the part of this that is linear in $$ H_{\text{int}} $$ can be expanded as:

$$ e^{-iTH} = \sum_{n=0}^{N-1} [1-iH_0T/N]^{N-n-1}(-iH_{\text{int}}T/N)[1-iH_0T/N]^n $$ - - - (2)

(Here, we have dropped the 0th order part, $$ e^{-iTH_0} $$, as uninteresting; it just corresponds to the particles evolving as free particles.)

- - - - - - - - - - - - - - - - - -

So, my question is how do I derive from eq. (1) to (2)? If you teach me the method, I really thank you.

p.s. In this environment, inline math mode is not worked. Sorry for inconvenient.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# In the interacting scalar field theory, I have a question.

Tags:

**Physics Forums | Science Articles, Homework Help, Discussion**