Massive vector boson propagator - Definition

In summary, the propagator of a massive spin-1 boson is defined as the inverse of a differential operator, with the 4D delta distribution appearing due to the fact that the propagator is a distribution and the differential operator acts as a bilinear form. This allows for the inverse to be defined in a similar way to inverting a tensor, with only one vector index being contracted between the propagator and the differential operator.
  • #1
mchouza
2
0
I'm reading Zee's QFT textbook and I'm stuck trying to understand why the [tex]\delta^\mu_\lambda[/tex] appears when he defines the propagator of a massive spin-1 boson as the inverse of a differential operator:

[tex][(\partial^2 + m^2)g^{\mu\nu}-\partial^\mu\partial^\nu]D_{\nu\lambda} = \delta_\lambda^\mu \delta^{(4)}(x)[/tex]

In particular, why only one vector index must be contracted between the propagator and the differential operator? I'm sure that the reason is very simple, but it's still eluding me.

Thanks!
 
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  • #2
mchouza said:
I'm reading Zee's QFT textbook and I'm stuck trying to understand why the [tex]\delta^\mu_\lambda[/tex] appears when he defines the propagator of a massive spin-1 boson as the inverse of a differential operator:

[tex][(\partial^2 + m^2)g^{\mu\nu}-\partial^\mu\partial^\nu]D_{\nu\lambda} = \delta_\lambda^\mu \delta^{(4)}(x)[/tex]

In particular, why only one vector index must be contracted between the propagator and the differential operator? I'm sure that the reason is very simple, but it's still eluding me.

Thanks!

Well, intuitively because Zee considers his analysis as a "continuation of matrix operators":

[tex]
[D^{-1}]^{\mu\nu}(x)D_{\nu\rho}(x) = \delta^{\mu}_{\rho}\delta^4(x)
[/tex]
The 4D delta distribution is there because the propagator is a distribution; you integrate them over spacetime agains functions of spacetime.
 
  • #3
Thanks for your help. I was confused with the idea of "inverting a tensor", but now I realize that the differential operator works here as a bilinear form (like the metric) and not as a general tensor:

[tex]J_\mu M^{\mu\nu} J_\nu = J_\mu M^\mu\!_\nu J^\nu[/tex]

So it makes sense to invert it:

[tex]M^\mu\!_\nu [M^{-1}]^\nu\!_\lambda = \delta^\mu_\lambda[/tex]

[tex]M^{\mu\nu} [M^{-1}]_{\nu\lambda} = \delta^\mu_\lambda[/tex]
 

1. What is a massive vector boson propagator?

A massive vector boson propagator is a mathematical representation of the behavior and interactions of a massive vector boson, which is a type of elementary particle. It describes how the particle moves and changes over time.

2. What is the purpose of a massive vector boson propagator?

The purpose of a massive vector boson propagator is to help scientists understand and predict the behavior of particles in particle physics experiments. It is used to calculate the probabilities of different particle interactions and to test the validity of theoretical models.

3. How is a massive vector boson propagator calculated?

A massive vector boson propagator is calculated using mathematical equations and techniques from quantum field theory. It takes into account factors such as the mass and spin of the particle, as well as the effects of other particles and forces in the system.

4. What is the significance of the massive vector boson propagator in physics?

The massive vector boson propagator is significant because it plays a crucial role in the Standard Model of particle physics, which is the most widely accepted theory for explaining the behavior of particles and their interactions. It has also been used to make predictions for experiments at the Large Hadron Collider, leading to the discovery of new particles such as the Higgs boson.

5. Is the massive vector boson propagator a physical object?

No, the massive vector boson propagator is not a physical object. It is a mathematical concept used by scientists to describe and study the behavior of particles. However, its predictions and calculations have been confirmed by experimental evidence, making it a valuable tool in understanding the fundamental building blocks of the universe.

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