# The propagator of the gauge field

• I
• Monaliza Smile
In summary, the conversation is about the free massive vector boson Lagrangian and its corresponding propagator in momentum space. The formula for the propagator is derived from the Lagrangian using a standard textbook formula and a Fourier transformation. The person is asking for help or a good reference to understand the process.
Monaliza Smile
Hi,

I read in a QFT book that the free massive vector boson Lagrangian is

## \mathcal{L}_W = - \frac{1}{4} (\partial_\mu W^\dagger_\nu - \partial_\nu W_\mu^\dagger ) (\partial^\mu W^\nu - \partial^\nu W^\mu ) + M^2_W W^\dagger_\mu W^\mu ##

gives the propagator in momentum space by:

## i \Delta_{\mu\nu} = - i \frac{g_{\mu\nu - k_\mu k_\nu/M^2_W}}{k^2-M^2_W} ##

Any help how to derive the formula of the propagator from the Lagrangian ?

This is standard textbook material. The propagator is the unconnected two point Green function. To obtain Green functions from the action you need to apply the textbook formula and then switch to momentum representation via a Fouriert transformation of the propagator in the coordinate representation.

Ok, I'm asking how to do so, or please mention a good reference for ..

## 1. What is the propagator of the gauge field?

The propagator of the gauge field is a mathematical function used in quantum field theory that describes the amplitude of a particle propagating between two points in space and time. It is derived from the gauge field equations and is an important tool in understanding the behavior of gauge fields, which are fundamental to the study of elementary particles.

## 2. How is the propagator of the gauge field calculated?

The propagator of the gauge field can be calculated using various mathematical techniques such as Feynman diagrams, perturbation theory, and path integrals. The specific method used depends on the specific gauge field theory being studied and the desired level of accuracy.

## 3. What does the propagator of the gauge field tell us about the behavior of particles?

The propagator of the gauge field provides information about the probability amplitude for a particle to propagate from one point to another. It can also give insights into the interactions between particles and the effects of external forces on their behavior.

## 4. How does the propagator of the gauge field relate to the concept of gauge invariance?

The propagator of the gauge field is derived from the equations of motion for gauge fields, which are invariant under gauge transformations. This means that the propagator must also be gauge invariant, ensuring that physical predictions are independent of the specific choice of gauge.

## 5. Can the propagator of the gauge field be experimentally tested?

Yes, the predictions made by the propagator of the gauge field can be tested experimentally through various high-energy physics experiments, such as particle accelerators. These experiments can measure the behavior of particles and their interactions, providing evidence for the accuracy of the propagator and the underlying gauge field theory.

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