Confusion on the connection between causality and stationarity and possible implications Let $(x_{t})_{t\in \mathbb Z}$ be a causal AR(p) process with operator $\phi$ such that $\phi(L)=\phi_{0}-\phi_{1}L-...-\phi_{p}L^{p}$ and $(\epsilon_{t})_{t \in \mathbb N_{0}}$ white noise sequence: 
By definition, there exists a sequence $(\psi_{t})_{t \in \mathbb N_{0}}$ such that $x_{t}=\psi(L)\epsilon_{t}$ where $\psi(L)=\psi_{0}+\psi_{1}L+\psi_{2}L^{2}+...$ with the Lag operator $L$ and $\sum\limits_{j\in \mathbb N_{0}}\lvert \psi_{j}\rvert<\infty$.
One equivalent property of causality is that all roots of $\phi$ lie outside the unit circle. 
Furthermore, introducing the notion of weak stationarity, meaning mean stationarity and covariance stationarity of the time series $x_{t}$. We get that a AR(p) process $x_{t}$ is stationary, if the roots of $\phi$ lie outside of the unit circle. 
So by the above thoughts I have espoused, I may assume the following: 
causality $\implies$ stationarity with mean $0$. 
This leads me to what I think may be a contradiction, because under the assumption that $x_{t}$ is stationary, we have 
$\mathbb E[x_{t}]=\frac{\phi_{0}}{1-\phi_{1}-...-\phi_{p}}$
So even if i did assume causality, this would have to mean that 
$0=\mathbb E[x_{t}]=\frac{\phi_{0}}{1-\phi_{1}-...-\phi_{p}}$, which is certainly not true, and would only be true if I assume $\phi_{0}=0$.
What am I not understanding? 
 A: Causality is by definition a special case of stationarity. Stationarity, or causality, does not imply mean 0. 
Where you went wrong is you're comparing different AR models---one without intercept and one with. Stationary AR models without intercept have zero mean in general, whether causal or non-causal. Vice versa for those with intercept.
When you defined $x_t$ by $\phi(L) x_t = \epsilon_t$, e.g.
$$
(\phi_0 - \phi_1 L)x_t = \epsilon_t, \quad (*)
$$ 
you defined an AR model with no intercept. Such processes necessarily have mean zero. (As you already pointed out, in the stationary case where there is a MA representation, it's a (infinite) sum of mean-zero variables.) $\phi_0$ is customarily taken to be 1 in such formulations.
On the other hand, the expression for the unconditional mean
$$
\mathbb E[x_{t}] = \frac{\phi_{0}}{1-\phi_{1}}
$$
is for the causal AR model with intercept
$$
x_{t+1} = \phi_0 + \phi_1 x_t + \epsilon_t. \quad (**)
$$
This is not the same model as $(*)$.
Instead, the lag operator formulation of $(**)$ is
$$
(1 - \phi_1 L) x_t = \phi_0 + \epsilon_t.
$$
So in the causal case 
$$
x_t = \frac{\phi_0}{1-\phi_1} + \underbrace{ \psi(L)\epsilon_{t}}_\text{$\sum_{h \geq 0} \psi_h \epsilon_{t-h}$}, \;\; \psi_h = \phi_1^h,
$$
which has mean 
$$
\mathbb E[x_{t}] = \frac{\phi_{0}}{1-\phi_{1}}.
$$
