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

• lhcQFT
In summary, the process of deriving from equation (1) to (2) involves taking the product of the N terms and keeping terms linear in the interaction Hamiltonian. This can be done by setting different values for N and observing the pattern, which will help in understanding the method.
lhcQFT
First of all, I copy the text in my lecture note.
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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.)
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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.

Just write out the product of the N terms and keep terms linear in the interaction Hamiltonian.

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

Set off your equations using ## instead of  and they will display inline.

Take eg $N=2$ and $N=3$ and try then a general N...
I think from the N=2 and N=3 you will be able to see what is going on...

## 1. What is the interacting scalar field theory?

The interacting scalar field theory is a theoretical framework used in particle physics to describe the behavior of scalar particles, which are particles that have no spin. In this theory, scalar particles interact with each other through a potential energy term, which results in the exchange of virtual particles.

## 2. How does the interacting scalar field theory differ from other theories?

The interacting scalar field theory differs from other theories, such as the Standard Model, in that it does not include other fundamental particles like fermions or gauge bosons. It focuses solely on the interactions between scalar particles.

## 3. What are the applications of the interacting scalar field theory?

The interacting scalar field theory has many applications in various fields, including particle physics, cosmology, and condensed matter physics. It allows for the study of the behavior of scalar particles and their interactions, which can provide insights into the fundamental forces and structure of the universe.

## 4. How is the interacting scalar field theory tested?

The predictions of the interacting scalar field theory can be tested through experiments, such as high-energy particle collisions in particle accelerators. These experiments can provide data that can be compared to the theoretical predictions of the theory.

## 5. What are the current developments and challenges in the interacting scalar field theory?

Currently, there is ongoing research and development in the interacting scalar field theory, particularly in understanding the behavior of scalar particles in extreme conditions, such as in the early universe or in the presence of strong gravitational fields. Some of the challenges in this field include finding experimental evidence for the existence of scalar particles and developing more accurate and comprehensive mathematical models.

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