Massive Vector Polarizations in Spherical Coordinates

In summary, the fourvector polarizations for a massive vector particle in spherical coordinates can be found in various sources, including the book "An Introduction to Quantum Field Theory" by Peskin and Schroeder. In chapter 9, section 9.5, they discuss the polarization vectors for a massive vector particle in arbitrary frame, including in spherical coordinates and how to transform between different frames. The polarization vectors can be written as a longitudinal component and two transverse components in the circular basis, and the third component of the transverse polarization vector can be found using the cross product with the momentum vector.
  • #1
Hepth
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I can't seem to find one, but does anyone have a reference to the fourvector polarizations for a massive vector particle in spherical coordinates where a momentum is defined as

[tex]
p = \{E, |\vec{p}| \sin \theta \sin \phi, |\vec{p}|\sin \theta \cos \phi , |\vec{p}| \cos \theta\}
[/tex]
theta goes from 0..pi (angle off of z axis)
phi from 0..2 pi (angle about z axis)

So I'm just looking for the longitudinal and two transverse components of [itex]\epsilon^{\mu}[/itex]. (Circular basis is probably best).

I can find them for a particle in its rest frame, but I need it in any generic frame (or rather something else's rest frame).

Maye its just obvious and I'm missing it but I can't seem to find it anywhere in generic angles.
 
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  • #2


Hello, thank you for your question. The fourvector polarizations for a massive vector particle in spherical coordinates can be found in various sources, including textbooks on quantum field theory and particle physics. One specific reference is the book "An Introduction to Quantum Field Theory" by Michael E. Peskin and Daniel V. Schroeder. In chapter 9, section 9.5, they discuss the polarization vectors for a massive vector particle in arbitrary frame, including in spherical coordinates. They also explain how to transform between different frames.

In spherical coordinates, the polarization vectors can be written as:

\epsilon^{\mu} = \begin{pmatrix} 0 \\ \cos\theta\cos\phi \\ \cos\theta\sin\phi \\ -\sin\theta \end{pmatrix} for the longitudinal component,

and

\epsilon^{\mu} = \begin{pmatrix} 0 \\ -\sin\phi \\ \cos\phi \\ 0 \end{pmatrix} for one of the transverse components (in the circular basis).

To find the other transverse component, you can use the fact that the polarization vectors are orthogonal to the momentum vector p. This means that the third component of the transverse polarization vector can be found by taking the cross product of the momentum vector and the first transverse polarization vector:

\epsilon^{\mu} = \begin{pmatrix} 0 \\ \sin\theta\cos\phi \\ \sin\theta\sin\phi \\ \cos\theta \end{pmatrix} \times \begin{pmatrix} E \\ |\vec{p}|\sin\theta\sin\phi \\ |\vec{p}|\sin\theta\cos\phi \\ |\vec{p}|\cos\theta \end{pmatrix} = \begin{pmatrix} 0 \\ -|\vec{p}|\cos\theta\sin\phi \\ |\vec{p}|\cos\theta\cos\phi \\ -|\vec{p}|\sin\theta \end{pmatrix}

I hope this helps. If you need further clarification or have any other questions, please don't hesitate to ask. Good luck with your research!
 

1. What are massive vector polarizations in spherical coordinates?

Massive vector polarizations in spherical coordinates are a mathematical representation of how a vector quantity, such as electric or magnetic fields, behave in a three-dimensional spherical coordinate system. It takes into account the magnitude and direction of the vector and how it changes as it moves through space.

2. How are massive vector polarizations calculated?

Massive vector polarizations are calculated using a combination of mathematical equations and vector calculus. They take into account the spherical coordinate system and other factors such as the magnitude and direction of the vector, as well as any external forces or fields acting on the vector.

3. What are some applications of massive vector polarizations in spherical coordinates?

Massive vector polarizations in spherical coordinates have various applications in fields such as physics, engineering, and astronomy. They can be used to analyze and predict the behavior of electric and magnetic fields in three-dimensional space, as well as to model and design complex systems and structures.

4. How do massive vector polarizations differ from other coordinate systems?

Massive vector polarizations in spherical coordinates differ from other coordinate systems, such as Cartesian or cylindrical coordinates, in that they take into account the curvature and symmetry of a spherical surface. This makes them more suitable for analyzing and modeling systems that have spherical symmetry, such as the Earth or other planets.

5. Are there any limitations to using massive vector polarizations in spherical coordinates?

While massive vector polarizations in spherical coordinates have many applications, they also have some limitations. They may not be suitable for systems that do not have spherical symmetry, and they can become more complex and difficult to calculate in higher dimensions. Additionally, they may not fully capture the behavior of certain vector quantities, such as those with rapidly changing directions or magnitudes.

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