Numbers in Dimension Analysis When I learned dimensional analysis for the first time, I know that the dimension, for example, of the velocity can be written like this $$[V]=LT^{-1},$$ but in QFT the action for example is dimensionless, this mean $$[S]=1,$$ and 4D volume element have dimension $$[{\rm d}^4x]= -4 ,$$ So The Question:

What these number represent and how do we get them?

 A: *

*OP's inquired dimension/number $D$ is the power of mass dimensions
$$[Q]=M^D\tag{1} $$
of a physical quantity $Q$ in units where $c=1=\hbar$.


*Equivalently, it's the inverse power of length dimensions
$$[Q]~=~L^{-D}.\tag{2}$$


*Be aware that the QFT literature often use the shorthand notation
$$[Q]~=~D\tag{3}$$
instead of eqs. (1) & (2). This convention (3) probably spurred OP's question in the first place.


*In QFT such dimensional considerations are e.g. conducted to find the superficial degree of divergence of a Feynman diagram.
A: If you assume $\hbar = c = 1$ then


*

*Energy and mass have the same units: $[M] = [E]$. This comes from the $E = m_0c^2$ relation

*Momentum has units of mass, and because of the above, units of energy $[P] = [E]$

*Length has units of distance inverse, since distance times momentum has units of action, which we set to unity with $\hbar = 1$. So $[L] = [P^{-1}] = [E^{-1}]$


You can continue doing the analysis, but in general you can express all in terms of units of energy. That's why you just use a number to denote this. That is,
$$
[E^k]= k
$$
For example, since the units of length are $[E^{-1}]$ we say $[{\rm d}x] = -1$, and $[{\rm d}^4x] = [E^{-4}] = -4$
