# Massive spin 1 propagator in imaginary time formalism

• Judas503
In summary, the conversation discusses the task of writing the propagator for a massive spin-1 particle in the imaginary time formalism commonly used in thermal field theories. The denominator of the propagator is factorized and substitutions are made, but the numerator cannot be simplified. The final solution will include the components D^{00}, D^{0i} = D^{i0} and D^{ij}.
Judas503

## Homework Statement

I have the following massive spin-1 propagator-
$$D^{\mu\nu}(k)=\frac{\eta^{\mu\nu}-\frac{k^{\mu}k^{\nu}}{m^2}}{k^2 - m^2}$$
I want to write down the propagator in the imaginary time formalism commonly used in thermal field theories.

## The Attempt at a Solution

Factorizing the denominator is easy:
$$k^2 - m^2 = (k^0)^2 - \mathbf{k}^2 - m^2$$
Then, the following substitution can be used:
$$\omega_{k}=\mathbf{k}^2 + m^2$$
and, $$k^0 = i\omega_{n}=\frac{2n\pi i}{\beta}$$
However, the problem lies in simplifying the numerator. Any help would be really appreciated.

Judas503 said:

## Homework Statement

I have the following massive spin-1 propagator-
$$D^{\mu\nu}(k)=\frac{\eta^{\mu\nu}-\frac{k^{\mu}k^{\nu}}{m^2}}{k^2 - m^2}$$
I want to write down the propagator in the imaginary time formalism commonly used in thermal field theories.

## The Attempt at a Solution

Factorizing the denominator is easy:
$$k^2 - m^2 = (k^0)^2 - \mathbf{k}^2 - m^2$$
Then, the following substitution can be used:
$$\omega_{k}=\mathbf{k}^2 + m^2$$
You mean $\omega_k^2$ on the left side, I guess.
and, $$k^0 = i\omega_{n}=\frac{2n\pi i}{\beta}$$
However, the problem lies in simplifying the numerator. Any help would be really appreciated.

You cannot simplify the numerator. What you can do is to simply provide separately $D^{00}$, $D^{0i} = D^{i0}$ and $D^{ij}$.

## 1. What is a massive spin 1 propagator?

A massive spin 1 propagator is a mathematical representation of a particle with a mass and spin of 1. It describes the probability amplitude for a particle to propagate from one point in space and time to another.

## 2. What is the significance of using the imaginary time formalism?

The imaginary time formalism is often used in quantum field theory to simplify calculations and make them more tractable. In this formalism, time is treated as an imaginary number, which allows for easier calculation of certain quantities.

## 3. How is the massive spin 1 propagator related to the propagator in real time formalism?

The massive spin 1 propagator in real time formalism is related to the propagator in imaginary time formalism through a mathematical transformation. In the imaginary time formalism, the propagator can be obtained by Wick rotation of the real time propagator.

## 4. What is the role of the massive spin 1 propagator in quantum field theory?

The massive spin 1 propagator plays a crucial role in quantum field theory as it describes the interaction between particles with spin 1. It is used to calculate scattering amplitudes and to understand the behavior of particles in quantum systems.

## 5. Are there any practical applications of the massive spin 1 propagator in imaginary time formalism?

Yes, the massive spin 1 propagator in imaginary time formalism has practical applications in various fields such as condensed matter physics, particle physics, and cosmology. It is used to study the behavior of particles in these systems and make predictions about their properties and interactions.

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