Massive Vector Boson E.o.M.: Minimize Action & Lorentz Gauge?

In summary, the Lagrangian density L can be used to derive the equations of motion for the field B. These equations show that for a massive field, the partial of B is equal to zero, which can be interpreted as a constraint or boundary condition for minimizing the action. This is commonly referred to as the Lorentz Gauge, which can be shown using equations 19, 20, and the first paragraph of page 181 in the paper provided. Taking the derivative of the equations of motion also results in the partial of B being zero.
  • #1
ChrisVer
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Starting from the Lagrangian density:
[itex] L= -\frac{1}{4}F_{\mu\nu}F^{\mu\nu} + \frac{m^{2}}{2} B_{\mu}B^{\mu}[/itex]
we can derive the E.o.M. for the field [itex]B[/itex] which read:

[itex] ( \partial^{2} + m^{2}) B^{\mu} - \partial^{\mu} (\partial B) = 0 [/itex]
In the case of a massive field, I am not sure how I can kill out the partial of B through the field equations...
[itex] \partial B=0 [/itex]
Does this come as a constraint/boundary condition of minimizing the action? or is there something I cannot see? In most cases they state it's a Lorentz Gauge, however I am not sure how this can be indeed shown...
 
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  • #2
http://www.theory.nipne.ro/~poenaru/PROCA/proca_rila06.pdf
P.180 , eq 19,20 and the 1st paragraph of p.181 gave the answer... One has to take the derivative of the Equations of Motion [itex]\partial_{\mu}[/itex] and [itex]\partial B[/itex] comes out zero... (unfortunately for me I came out with the wrong EoM missing a minus sign and I couldn't even think of doing it)
 

1. What is a Massive Vector Boson?

A Massive Vector Boson is a particle that carries the fundamental force of the weak interaction. It has a non-zero mass, unlike a photon which is a massless vector boson that carries the electromagnetic force.

2. What is the E.o.M. in relation to Massive Vector Bosons?

The E.o.M. stands for Equation of Motion and it describes the behavior of a system of particles. In the context of Massive Vector Bosons, the E.o.M. is used to minimize the action and determine the path of the particle.

3. Why is it important to minimize the action in relation to Massive Vector Bosons?

Minimizing the action is important because it allows us to determine the most probable path of the Massive Vector Boson. This path is known as the classical path and it helps us understand the behavior of the particle in a given system.

4. What is the Lorentz gauge in relation to Massive Vector Bosons?

The Lorentz gauge is a mathematical condition that is used to simplify the equations of motion for Massive Vector Bosons. It ensures that the equations are consistent with the principles of special relativity.

5. How does the Lorentz gauge affect the behavior of Massive Vector Bosons?

The Lorentz gauge does not affect the behavior of Massive Vector Bosons directly. Instead, it helps simplify the equations of motion and allows us to focus on the physical properties and interactions of the particles without worrying about inconsistencies with special relativity.

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