# Spin-1 Propagator and polarization vectors

• TriTertButoxy
In summary, the conversation discusses the derivation of the propagator for a spin-1 field in a spontaneously broken gauge theory in the R_\xi gauges for an arbitrary gauge parameter \xi. The result is given by \tilde{D}^{\mu\nu}(p) = \frac{-i}{p^2-m^2+i\epsilon}\left(g^{\mu\nu}-(1-\xi)\frac{p^\mu p^\nu}{p^2-\xi m^2+i\epsilon}\right). The individual is stuck when trying to calculate the two-point correlator using the mode expansion for the spin-1 field and is having trouble with the polarization sum and determining the polarization vectors. However,
TriTertButoxy
Hi. I am stuck.
By inverting the spin-1 differential operator I was able to derive (quite easily) the propagator for the spin-1 field (in a spontaneously broken gauge theory) in the $R_\xi$ gauges for the arbitrary gauge parameter $\xi$. The result is
$$\tilde{D}^{\mu\nu}(p)=\frac{-i}{p^2-m^2+i\epsilon}\left(g^{\mu\nu}-(1-\xi)\frac{p^\mu p^\nu}{p^2-\xi m^2+i\epsilon}\right).$$
But now, when I try to calculate the two-point correlator using the mode expansion for the spin-1 field, I can't quite get the same answer.
$$\langle 0|T(\hat{A}_\mu(x)\hat{A}_\nu(y))|0\rangle=\int \frac{d^4p}{(2\pi)^4}\frac{ie^{-i p.(x-y)}}{p^2-m^2+i\epsilon}\left(\epsilon_\mu^{[0]}(\mathbf{p})\epsilon_\nu^{*[0]}(\mathbf{p})-\sum_{\lambda=1,2,3}\epsilon_\mu^{[\lambda]}(\mathbf{p})\epsilon_\nu^{*[\lambda]}(\mathbf{p})\right)$$
I need to do a polarization sum, but can't quite figure out how to get the gauge-dependence in there. What are the polarization vectors? and how do I derive them?

Last edited:
Ok. I found the answer in Greiner, Field Quantization 1996, chapter 7. The answer is very long, and I do not expect anyone to answer. Thanks to those who have thought about this problem.

## 1. What is the Spin-1 Propagator?

The Spin-1 Propagator is a mathematical representation of the propagation of a spin-1 particle, such as a vector boson, through space and time.

## 2. How is the Spin-1 Propagator calculated?

The Spin-1 Propagator is calculated using Feynman diagrams and quantum field theory calculations.

## 3. What are polarization vectors?

Polarization vectors are mathematical vectors that describe the orientation of the spin of a particle as it travels through space and time. In the context of the Spin-1 Propagator, they describe the polarization states of the vector boson.

## 4. How are polarization vectors used in the Spin-1 Propagator?

Polarization vectors are used in the Spin-1 Propagator to calculate the amplitude of a particle's spin to be in a particular state at a given point in spacetime.

## 5. What is the significance of the Spin-1 Propagator in physics?

The Spin-1 Propagator is significant in physics as it helps explain the behavior of spin-1 particles, which play a crucial role in fundamental interactions such as electromagnetism and the weak nuclear force.

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