Free lagrangian propagator

In summary: There is a way to write the series in closed form. Consider $$ S \equiv 1 + x + x^2 + \ldots $$Then $$ S-1 = x S$$,right? Then we can can solve for the sum S.
  • #1
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33

Homework Statement


I have the Lagrangian $$L=-\frac{1}{2}\phi\Box \phi-\frac{1}{2}m^2\phi^2$$ and I need to show that the propagator in the momentum space I obtain using this lagrangian (considering no interaction) is the same as if I consider the free Lagrangian to be $$L_{free}=-\frac{1}{2}\phi\Box \phi$$ and treat the mass term as an interaction $$L_{int}= -\frac{1}{2}m^2\phi^2$$

Homework Equations

The Attempt at a Solution


So in the normal case the propagator for a mass m scalar particle is $$\frac{i}{p^2-m^2+i\epsilon}$$. For the other approach I get that the propagator looks like this: $$\frac{1}{p^2+i\epsilon}-im^2(\frac{1}{p^2+i\epsilon})^2-m^4(\frac{1}{p^2+i\epsilon})^3+im^6(\frac{1}{p^2+i\epsilon})^4+...$$ But i am not sure how to show they are equal. Can someone help me?
 
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  • #2
kelly0303 said:

Homework Statement


I have the Lagrangian $$L=-\frac{1}{2}\phi\Box \phi-\frac{1}{2}m^2\phi^2$$ and I need to show that the propagator in the momentum space I obtain using this lagrangian (considering no interaction) is the same as if I consider the free Lagrangian to be $$L_{free}=-\frac{1}{2}\phi\Box \phi$$ and treat the mass term as an interaction $$L_{int}= -\frac{1}{2}m^2\phi^2$$

Homework Equations

The Attempt at a Solution


So in the normal case the propagator for a mass m scalar particle is $$\frac{i}{p^2-m^2+i\epsilon}$$. For the other approach I get that the propagator looks like this: $$\frac{1}{p^2+i\epsilon}-im^2(\frac{1}{p^2+i\epsilon})^2-m^4(\frac{1}{p^2+i\epsilon})^3+im^6(\frac{1}{p^2+i\epsilon})^4+...$$ But i am not sure how to show they are equal. Can someone help me?
It is a geometric series. Alternatively, you can see this as the Taylor expansion around ##m^2=0## of what expression?
 
  • #3
nrqed said:
It is a geometric series. Alternatively, you can see this as the Taylor expansion around ##m^2=0## of what expression?
Thank you for this so I have: $$\frac{1}{p^2+i\epsilon}-im^2(\frac{1}{p^2+i\epsilon})^2-m^4(\frac{1}{p^2+i\epsilon})^3+im^6(\frac{1}{p^2+i\epsilon})^4+...$$ $$\frac{1}{p^2+i\epsilon}(1-im^2\frac{1}{p^2+i\epsilon}-m^4(\frac{1}{p^2+i\epsilon})^2+im^6(\frac{1}{p^2+i\epsilon})^3+...)$$ $$lim_{n \to \infty}\frac{1}{p^2+i\epsilon}\frac{1-(-im^2\frac{1}{p^2+i\epsilon})^n}{1+im^2\frac{1}{p^2+i\epsilon}}$$ $$lim_{n \to \infty}\frac{1-(-im^2\frac{1}{p^2+i\epsilon})^n}{p^2+i\epsilon+im^2}$$ I am not sure from here. How does that term behave as n goes to infinity? Also I have a factor of i with that ##m^2##
 
  • #4
kelly0303 said:
Thank you for this so I have: $$\frac{1}{p^2+i\epsilon}-im^2(\frac{1}{p^2+i\epsilon})^2-m^4(\frac{1}{p^2+i\epsilon})^3+im^6(\frac{1}{p^2+i\epsilon})^4+...$$ $$\frac{1}{p^2+i\epsilon}(1-im^2\frac{1}{p^2+i\epsilon}-m^4(\frac{1}{p^2+i\epsilon})^2+im^6(\frac{1}{p^2+i\epsilon})^3+...)$$ $$lim_{n \to \infty}\frac{1}{p^2+i\epsilon}\frac{1-(-im^2\frac{1}{p^2+i\epsilon})^n}{1+im^2\frac{1}{p^2+i\epsilon}}$$ $$lim_{n \to \infty}\frac{1-(-im^2\frac{1}{p^2+i\epsilon})^n}{p^2+i\epsilon+im^2}$$ I am not sure from here. How does that term behave as n goes to infinity? Also I have a factor of i with that ##m^2##
There is a way to write the series in closed form. Consider

$$ S \equiv 1 + x + x^2 + \ldots $$

Then $$ S-1 = x S$$,
right? Then we can can solve for the sum S.
 

1. What is a free lagrangian propagator?

A free lagrangian propagator is a mathematical function that describes the propagation of a particle or field in a system with no external forces or interactions. It is used in the study of quantum field theory and is derived from the lagrangian of the system.

2. How is a free lagrangian propagator calculated?

A free lagrangian propagator can be calculated using Feynman path integrals, which involves summing up all possible paths of particles or fields in the system. It can also be obtained through the use of Green's functions, which are solutions to the differential equations that describe the dynamics of the system.

3. What is the significance of the free lagrangian propagator in physics?

The free lagrangian propagator is an important tool in theoretical physics, particularly in quantum field theory. It allows for the calculation of probabilities and amplitudes of particle interactions, and can be used to predict the behavior of particles and fields in a system with no external forces or interactions.

4. How does the free lagrangian propagator differ from other propagators?

The free lagrangian propagator differs from other propagators, such as the Feynman propagator, in that it describes the propagation of particles or fields in a system with no external forces or interactions. Other propagators may take into account external forces or interactions, making them more complex but also more applicable to a wider range of systems.

5. Are there any applications of the free lagrangian propagator outside of physics?

While the free lagrangian propagator is primarily used in theoretical physics, it also has applications in other fields such as signal processing and control theory. In these areas, it is used to describe the propagation of signals or information through a system with no external disturbances or noise.

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