Are gauge fields orthogonal/orthonormal? If you take the gauge fields associated with, say, SU(2) gauge invariance, it is my understanding that the 3 fields we obtain, $W_\mu^a$ can be treated as 3 four-vectors.
My intuition tells me that they should be linearly indepedent, but I can't find a rigorous justification for it. Indeed, in the few books I have read about the subject they don't adress the properties of these fields (I haven't done QFT yet, though, only particle physics).
So, are they linearly independent if we view them as vectors ? If they are, is it physically relevant to construct an "orthogonal" basis with respect to the metric ?
I know we take a special linear combination of those gauge fields to yield $W_\mu^+$ and $W_\mu^-$, so it doesn't seem so far-fetched to construct a orthonormal basis. But I'd like to have some explanations on the proof of the linear independence of those fields (if they are), and what it means physically if there is any interpretation.
Edit : On second though, since $W_\mu^a$ are vector FIELDS, it may be strange to test for orthogonality at two distinct points. Furthermore, I realised that, at least for electromagnetism, there are the Maxwell field equations $\partial_\mu F^{\mu\nu}= j^\mu$ that specify the properties of $A_\mu$. Is there an equivalent for the other gauge fields ? Maybe it is through those equations that their properties can be deduced. 
Again, maybe I'm a bit too hasty and that all of this is describe in the context of quantum field theory, if that is the case don't hesitate to let me know !
 A: You should not think of the gauge field $W^a_\mu$ as being "3 four-vectors". Since a gauge transformation mixes the component $W_\mu^1$ with $W_\mu^2$ and $W_\mu^3$ (by the adjoint action of a position-dependent $\mathrm{SU}(2)$ matrix) and field configurations related by a gauge transformation are physically equivalent, it is not meaningful to talk about the $W_\mu^1,W_\mu^2,W_\mu^3$ as distinct objects. In particular, these don't fullfill any relation among each other - they are coefficients of an expansion in a basis of the Lie algebra, which is a vector space, so they are three independent numbers for fixed $\mu$. You should rather think of $W$ as being a $\mathfrak{su}(2)$-valued 4-vector. Just like it is not meaningful to discuss the particular value of $W_0$ in relation to $W_1$, it is also not meaningful to discuss the value of $W^i$ in relation to $W^j$ - the first is not Lorentz invariant, the latter not gauge invariant, so neither is a "good" notion.
Note also that the $W^a$ are not linearly independent 4-vectors even in a fixed gauge - consider that $W = 0$ is a perfectly allowed vacuum solution, so they are not linearly dependent. Even if we could find that they are orthogonal in some setting, the decomposition of $W_\mu$ as a $\mathfrak{su}(2)$-valued gauge field into the $W^a_\mu$ depends on the choice of basis of $\mathfrak{su}(2)$, so this is already not a good invariant even if we don't consider gauge transformations.
A: The answer is a bit technical and does indeed lie in quantum field theory. So here we go:
First of all you should know what representations of the gauge group your fields are in. Say you want to make a gauge theory for SU(N) your fields can either be in the adjoint or the fundamental representations, but what does this mean ?
Fundamental
These are the particles in your theory and can be visualized as column matrices with length N. For example SU(2) can be used to describe the electron spin and indeed we can represent the electron as a two column:
$$\text{hypercharge(-1)}\ \ \rightarrow \ \ [1 0]^T $$
$$\text{hypercharge(-1)} \ \ \rightarrow \ \ [0 1]^T $$
$$\text{these correspond to } e^- \text{ and } e^+$$
Quarks are similar, these are fundamental representations of SU(3) and each color can be represented as a column matrix. 
This column representation of fundamental (=particle) fields gives you an immediate interpretation of the orthogonality of those fields.
Adjoint
These are the bosons a.k.a. gauge fields in your theory and can be visualized as hermitian N by N matrices acting on the fundamental representation. So let us look at SU(2).
The hermitian 2 by 2 matrices are the three pauli matrices:
$$\sigma_1 = \begin{bmatrix}1 & 0\\0 & 1\end{bmatrix}$$
$$\sigma_2 = \begin{bmatrix}0 & 1\\1 & 0\end{bmatrix}$$
$$\sigma_2 = \begin{bmatrix}0 & i\\-i & 0\end{bmatrix}$$
$$\text{these correspond to bosons with weak hypercharges the so called } W^{1,2,3}$$
It should now be clear that these matrices, and therefore corresponding particles, are ortogonal to each other.
But those are not the W^{+,-} ?
If you now want to make a good theory you should take linear combinations of the above matrices (= particles) such that the corresponding quantum numbers match those that be observe in nuture, specifically:
$$W^+ = 2^{-1/2}(\sigma_2+\sigma_3)$$
$$W^- = 2^{-1/2}(\sigma_2-\sigma_3)$$
Constructing the photon and Z boson is a bit more difficult as we need to introduce a second gauge group U(1) which has matrix representation 
$$A = \begin{bmatrix}1 & 0\\0 & 1\end{bmatrix}$$
At first this looks like the same field as $\sigma_1$ but we should realise that it works on objects with U(1) charge contrary to SU(2) charge so they are ortognal !
The photon and Z boson are than linear combinations of $\sigma_1$ and A. The exact combination would lead us to far from the topic... (You should be able to find this in your coursenotes or ask your professor he/she will know !)
EDIT: how to construct fields from this ?
Fields are constructed as 
$$\text{some field} = \sum_{M_i \in \{\sigma_i, A\}} (\text{some wavefunction})_i M_i $$
I hope that this wasn't to technical and helped you understand the issue ?
