Klein Gordon ret. Greens function on closed form

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In summary, the Klein-Gordon retarded green function is derived as ##G_{ret}(x − x′) = \theta(t − t') \int \frac{d^3 \vec k}{(2\pi)^3 \omega_k} \sin \omega_k (t − t′) e^{i \vec{k}\cdot (\vec x - \vec x')}##, where ##\omega_k = \sqrt{\vec{k}^2 + m^2}##. The author suggests converting to spherical coordinates in momentum space to solve the integration, which results in Bessel functions. However, the author is unsure how to proceed due to the dependence of ##\omega_k## on ##\vec k^
center o bass
In this note (http://sgovindarajan.wdfiles.com/local--files/serc2009/greenfunction.pdf) the Klein-Gordon retarded green function is derived on the form $$G_{ret}(x − x′) = \theta(t − t') \int \frac{d^3 \vec k}{(2\pi)^3 \omega_k} \sin \omega_k (t − t′) e^{i \vec{k}\cdot (\vec x - \vec x')}$$

where ##\omega_k = \sqrt{\vec{k}^2 + m^2}##. The author then gives an exercise to carry out the rest of the integration and express the Greens function on closed form. But I do not see how to carry the integration out due to the dependence of ##\omega_k## on ##\vec k^2##. Does anyone have any suggestions on how this might be solved, or alternatively know where I can find a full derivation?

Last edited:
You need to convert to spherical coordinates in momentum space. Then you can do it. It ends up with Bessel functions.

1. What is the Klein Gordon retarded Green's function?

The Klein Gordon retarded Green's function is a mathematical function that describes the propagation of a scalar field in spacetime. It is a solution to the Klein Gordon equation, which is a relativistic wave equation that describes the behavior of particles with zero spin.

2. What is the significance of the closed form of the Klein Gordon retarded Green's function?

The closed form of the Klein Gordon retarded Green's function allows for a more efficient and accurate calculation of the propagation of scalar fields in spacetime. It also allows for easier analysis and interpretation of the behavior of these fields.

3. How is the Klein Gordon retarded Green's function calculated?

The Klein Gordon retarded Green's function is calculated using mathematical techniques such as Fourier transforms and complex analysis. It involves solving for the inverse Fourier transform of the propagator function, which is a function that describes the interaction between particles.

4. What are some applications of the Klein Gordon retarded Green's function?

The Klein Gordon retarded Green's function has many applications in theoretical physics, particularly in the study of quantum field theory and particle interactions. It is used to calculate scattering amplitudes, decay rates, and other properties of particles. It also has applications in condensed matter physics and statistical mechanics.

5. Are there any limitations to the Klein Gordon retarded Green's function?

While the Klein Gordon retarded Green's function is a powerful tool in theoretical physics, it does have some limitations. It is only applicable to scalar fields, and cannot be used to describe particles with spin. Additionally, it may not accurately describe the behavior of particles in certain extreme conditions, such as near black holes or in the early universe.

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