<s>[INST] Gauge-fixing of an arbitrary field: off-shell & on-shell degrees of freedom How to count the number of degrees of freedom of an arbitrary field (vector or tensor)? In other words, what is the mathematical procedure of gauge fixing?
 [/INST] In this answer, we summarize the results. The analysis itself can be found in textbooks, see e.g. Refs. 1 & 2.
$\downarrow$ Table 1: Massless spin $j$ field in $D$ spacetime dimensions.

$$\begin{array}{ccc}
\text{Massless}^1    & \text{Off-shell DOF}^2 & \text{On-shell DOF}^3 \cr
j=0 & 1 & 1 \cr
j=\frac{1}{2} & n & \frac{n}{2} \cr
j=1 & D-1 & D-2 \cr
j=\frac{3}{2} & n(D-1) & \frac{n}{2}(D-3) \cr
j=2 & \frac{D}{2}(D-1) & \frac{D}{2}(D-3) \cr
\vdots &\vdots &\vdots \cr
\text{Integer spin }j\in\mathbb{N}_0 & \begin{pmatrix} D+j-2 \cr D-2 \end{pmatrix}+ \begin{pmatrix} D+j-5 \cr D-2 \end{pmatrix}&
\begin{pmatrix} D+j-4 \cr D-4 \end{pmatrix}+ \begin{pmatrix} D+j-5 \cr D-4 \end{pmatrix}\cr
\text{Integer spin }D=4 & j^2+2 &  2-\delta^j_0 \cr
\text{Integer spin }D=5 & \frac{1}{6}(2j+1)(j^2+j+6) &  2j+1\cr
\vdots &\vdots &\vdots \cr
\text{Half-int. spin }j\in\mathbb{N}_0+\frac{1}{2} & n\begin{pmatrix} D+j-\frac{5}{2} \cr D-2 \end{pmatrix}+ n\begin{pmatrix} D+j-\frac{9}{2} \cr D-2 \end{pmatrix}&
  \frac{n}{2}\begin{pmatrix} D+j-\frac{9}{2}\cr D-4 \end{pmatrix} \cr
\text{Half-int. spin }D=4 &n(j^2+\frac{3}{4}) &\frac{n}{2} \cr
\text{Half-int. spin }D=5 &\frac{n}{6}(2j+1)(j^2+j+\frac{9}{4}) &\frac{n}{4}(2j+1) \cr
\vdots &\vdots &\vdots \cr
\end{array}$$

$^1$For massive multiplets, go up 1 spacetime dimension, i.e. change $D\to D+1$ (without changing the number $n$ of spinor components). E.g. the on-shell DOF for massive 4D fields famously has a factor $2j+1$, cf. the row $D=5$ in Table 1. 
$^2$ Off-shell DOF = # (components)- # (gauge transformations).
$^3$ On-shell DOF = # (helicity states)= (Classical DOF)/2, where Classical DOF = #(initial conditions).
$n$=# (spinor components). E.g. a Dirac spinor has $n=2^{[D/2]}$ complex components, while a Majorana spinor has $n=2^{[D/2]}$ real components, 
$\downarrow$ Table 2: Antisymmetric $p$-form gauge potential in $D$ spacetime dimensions, $p\in\mathbb{N}_0$.

$$\begin{array}{ccc}
p\text{-form gauge potential}& 
\text{Off-shell DOF} & 
\text{On-shell DOF} \cr
& \begin{pmatrix} D-1 \cr p \end{pmatrix} &
\begin{pmatrix} D-2 \cr p \end{pmatrix} \cr
\end{array}$$

References:


*

*D.Z. Freedman & A. Van Proeyen, SUGRA, 2012.

*H. Nastase, Intro to SUGRA, arXiv:1112.3502; chapter 5.
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