# How to Express Contractions of Complex Scalar Field Operators via Propagators?

• ofer
This is the first non-zero correlator in this picture. In summary, when expressing contractions of field operators via propagators, it is important to consider the interaction terms and their corresponding vertices in order to calculate non-zero correlators.
ofer
i am trying to understand how to express contractions of field operators via propagators.
we are talking about an interacting theory of 2 complex scalar fields,
lets call them ψ1 and ψ2.

the interaction term is: Lint=λ(ψ2)^3(ψ1)

i have found the free propagator defined as:
Df-i(x-y)=<0|T(ψi(x) ψi*(y))|0> i=1,2.

what i am strugling with is that when considering the free parts
this propagator is 0 for <0|T(ψi ψi)|0> (no complex conjugate=no anti-particle).
and also it is 0 for <0|T(ψi ψj)|0> ,i≠j.
meaning that a field can only contract with it's conjugate counter part in the free theory.

so when i try to calculate an interaction corelator in 1st order of different forms i get 0, for instantce:

<Ω|T(ψ1 ψ2*)|Ω>
this term turns out 0 because there is no one to contract with all the none-conjugate fields
in the interaction hamiltonian.
am i miss understanding things here,
or is the first corelator to be none-zero in this picture of the form:
<Ω|T(ψ2* ψ2* ψ2* ψ1* )|Ω> ??

help will be much apriciated.
thank you.

Yes, you are correct that the free propagator is only non-zero when contracting a field with its conjugate counter part. In order to calculate an interaction correlator in first order, you need to consider a term of the form <Ω|T(ψ2* ψ2* ψ2* ψ1* )|Ω>. This term corresponds to a three-point vertex involving three fields of type ψ2 and one field of type ψ1.

## What is a complex scalar field propagator?

A complex scalar field propagator is a mathematical function that describes the probability amplitude for a particle to propagate from one point in space to another in a complex scalar field theory. It takes into account the interactions between different particles and their respective energies.

## How is the complex scalar field propagator used in particle physics?

The complex scalar field propagator is used in particle physics to calculate the scattering amplitudes of particles and their interactions. It is also used to determine the behavior of particles in a vacuum and in the presence of external fields.

## What is the significance of the complex scalar field propagator?

The complex scalar field propagator is significant because it allows us to understand the behavior of particles and their interactions in a complex scalar field theory. It is a fundamental concept in quantum field theory and helps us make predictions about the behavior of particles in different scenarios.

## How is the complex scalar field propagator different from other propagators?

The complex scalar field propagator is different from other propagators because it describes the behavior of particles in a complex scalar field theory, while other propagators may describe particles in different types of fields, such as electromagnetic or gravitational fields.

## Are there any real-world applications of the complex scalar field propagator?

Yes, the complex scalar field propagator has many real-world applications, particularly in the field of high-energy physics. It is used to study and understand the behavior of particles in particle accelerators and is also used in the development of quantum computers.

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