# Counting degrees of freedom of gauge bosons

Gauge bosons are represented by $A_{\mu}$, where $\mu = 0,1,2,3$. So in general there are 4 degrees of freedom. But in reality, a photon (gauge boson) has two degrees of freedom (two polarization states).

So, when someone asks about on-shell and off-shell degrees of freedom, I thought they are 2 and 4. But I read that the off-shell d.of. are 3.

And my question is how to see this?

-

## 2 Answers

On-shell dof: Consider the constraints of "Equation of Motion" plus "Gauge Condition"(Lorentz gauge, Coulomb gauge,...), #dof=4-2. Off-Shell dof: Consider the constraints of "Gauge Condition" (Lorentz gauge, Coulomb gauge,...) only, #dof=4-1.

-

One way to see this is the fact that if you write out the Lagragian in terms of the $A^0$ and $A^i$ explicitly, you will find there is no time derivatives for the $A^0$. The only candidate term $F^{00} = \partial^0 A^0 - \partial^0 A^0$ vanishes. This means we shouldn't count $A^0$ as a degree of freedom, and in fact, we should integrate it out. Now the resulting Lagrangian explicitly only has 3 DOF. If you now take this Lagrangian and vary it you will get the EOM which puts one constraint on the 3 DOF which knocks you down to the 2 DOF on-shell.

-

No $\partial_0 A_0$ term may be interpreted that $A_0$ is not a dynamical dof rather than dof. Three dimensional gravity for example, the gravity has a dof which is not dynamical. – Craig Thone Jul 2 '12 at 16:12
Craig - sorry, I don't understand your comment. Did I make an error in my statement? – DJBunk Jul 2 '12 at 17:25
No. I just want to emphasize $A_0$ is not a dynamical degree of freedom. In Lorentz gauge, I am not sure whether $A_0$ is a degree of freedom. – Craig Thone Jul 3 '12 at 1:42
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 11, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9115456938743591, "perplexity_flag": "middle"}