- #1

Aleolomorfo

- 73

- 4

Hello everybody.

The Lagrangian for a massive vector field is:

$$\mathcal{L} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu} + \frac{m^2}{2}A_\mu A^\mu$$

The equation of motion is ##\partial_\mu F^{\mu\nu}+m^2A^\nu = 0##

Expanding the EOM with the definition of ##F^{\mu\nu}## the Klein-Gordon equation for ##A_\mu## is obtained: ##\Box A^\nu + m^2A^\nu = 0##.

Then taking the derivate of the EOM with respect to ##\partial_\nu##, the Lorenz gauge condition is obtained ##\partial_\nu A^\nu = 0##

I have two question.

The first one is related to the gauge condition. What does the fact that the theory still has the Lorenz condition (without the need to impose it) means? Because the Lagrangian is not gauge-invariant, but it satisfies a gauge condition.

The second is about the degrees of freedom. A massive vector field has three degrees of freedom. However, ##A_\nu## is a 4-component object which satisfy the equation of motion and the Lorenz condition, but these are two conditions, so 4-2=2 degrees of freedom. Where does my reasoning fail?

The Lagrangian for a massive vector field is:

$$\mathcal{L} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu} + \frac{m^2}{2}A_\mu A^\mu$$

The equation of motion is ##\partial_\mu F^{\mu\nu}+m^2A^\nu = 0##

Expanding the EOM with the definition of ##F^{\mu\nu}## the Klein-Gordon equation for ##A_\mu## is obtained: ##\Box A^\nu + m^2A^\nu = 0##.

Then taking the derivate of the EOM with respect to ##\partial_\nu##, the Lorenz gauge condition is obtained ##\partial_\nu A^\nu = 0##

I have two question.

The first one is related to the gauge condition. What does the fact that the theory still has the Lorenz condition (without the need to impose it) means? Because the Lagrangian is not gauge-invariant, but it satisfies a gauge condition.

The second is about the degrees of freedom. A massive vector field has three degrees of freedom. However, ##A_\nu## is a 4-component object which satisfy the equation of motion and the Lorenz condition, but these are two conditions, so 4-2=2 degrees of freedom. Where does my reasoning fail?

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