If you really want a J with a T head :
- By using
adjusbox you can cut out the J bottom and the T head.
For example:
\def\hJ{\clipbox{0 -0.05ex 0 0.1em}{$J$}}
\def\hT{\raisebox{0.45em}{\clipbox{0 0.45em 0 0em}{$T$}}}
- You combine them with the suitable
\mkern. Using the mathllap command from mathtools package :
$\mathllap{\hT}\mkern-15mu\hJ$ gies the desired symbol (mathtalic)
To get the bold or upright version you will have to adjust the kerning because the width is different. And for boldface you will not be able to use \bm but will return to the old \boldmath put in another box. A compromise would be to pass the kern amount as a parameter :
\newcommand\JJ[1][15]{\hT\mkern -#1mu\hJ}
\newcommand\JJbf{\mbox{\boldmath$\JJ[14.2]$}}
$\JJ : \square \mathbf{A} = \mu_0 \,\JJbf$
produces the picture below.

The makeshift (not a real solution) suggested here should work with any font instead of the default latin modern I used in the first picture. If you want to use another (math)font you simply would have to find the correct kerning, which is font dependent.
For the other variants (bold, upright etc.), simply add \mathrm or \mathbf or whatever needed in the definitions of \hT and \hJ (inside the \clipbox)
To get the result in the four mathstyles you will have to use \mathchoice, explained in several post on this forum.
Here is a full MWE using times and mtpro2 lite, and the result:
\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{newtxtext}
\usepackage[lite,mtphbi]{mtpro2}
\usepackage[scaled=0.92]{helvet}
\usepackage{amsmath,amssymb}
\usepackage{adjustbox}
\def\hJ{\mathchoice%
{\clipbox{0 -0.05ex 0 0.1ex}{$J$}}%
{\clipbox{0 -0.05ex 0 0.1ex}{$J$}}%
{\clipbox{0 -0.05ex 0 0.1ex}{$\scriptstyle J$}}%
{\clipbox{0 -0.05ex 0 0.1ex}{$\scriptscriptstyle J$}}%
}
\def\hT{\mathchoice%
{\raisebox{0.45ex}{\clipbox{0 0.45ex 0 0}{$T$}}}
{\raisebox{0.45ex}{\clipbox{0 0.45ex 0 0}{$T$}}}
{\raisebox{0.45ex}{\clipbox{0 0.45ex 0 0}{$\scriptstyle T$}}}
{\raisebox{0.45ex}{\clipbox{0 0.45ex 0 0}{$\scriptscriptstyle T$}}}
}
\def\hJbf{\mathchoice%
{\clipbox{0 -0.05ex 0 0.1}{$\mathbf{J}$}}%
{\clipbox{0 -0.05ex 0 0.1}{$\mathbf{J}$}}%
{\clipbox{0 -0.05ex 0 0.1}{$\scriptstyle \mathbf{J}$}}%
{\clipbox{0 -0.05ex 0 0.1}{$\scriptscriptstyle \mathbf{J}$}}%
}
\def\hTbf{\mathchoice%
{\raisebox{0.45ex}{\clipbox{0 0.45ex 0 0}{$\mathbf{T}$}}}
{\raisebox{0.45ex}{\clipbox{0 0.45ex 0 0}{$\mathbf{T}$}}}
{\raisebox{0.45ex}{\clipbox{0 0.45ex 0 0}{$\scriptstyle \mathbf{T}$}}}
{\raisebox{0.45ex}{\clipbox{0 0.45ex 0 0}{$\scriptscriptstyle \mathbf{T}$}}}
}
\newcommand\JJ[1][15.2]{\hT\mkern -#1mu\hJ}
\newcommand\JJbf[1][11.5]{\hTbf\mkern -#1mu\hJbf}
\begin{document}
\huge
A big regular J : $J\quad \mathrm{J}\quad \mathbf{J}$\par
A big regular T : $T\quad \mathrm{T}\quad \mathbf{T}$
This is custom J : $\JJ \quad \square \mathbf{A} = \mu_0 \,\JJbf$
$\JJbf_{\JJ_{\JJ}}$
\end{document}

\includegraphics). But honestly, I'ld recommend just sticking to what's conventional (for indicating four-vectors).. – Troy Apr 23 '18 at 12:52mtpro2package. :-). What do I remove the question? – Sebastiano Apr 23 '18 at 13:11