Container-managed EntityManger is null on Injection - jpa

Question

I'm nearly new to JavaEE (JRE7/JDK1.7.0) and startet my first (maven based) Project on Eclipse (Luna).
Applicationserver is JBoss EAP 6.2.0.GA (AS 7.3.0.Final-redhat-14) which I updated to JSF2.2 by adding jsf-api-2.2.7.jar and jsf-impl-2.2.7.jar and registering in module.xml - which works perfect, by the way.
The problem/question is, why can't I get the EntityManager from the container (AS) with Injection, with following code in my Project:
#Stateless
public class SCatRep {
#PersistenceContext
EntityManager em;
private ...
*In persistence.xml there is only one persistenceunit defined. So no further arguments are necessary ... in my opinion?!
On the other hand, when I try to get the EntityManager from container with a lookup, the EntityManager is present and everything works fine. I used this code (for testing):
#Stateless
#PersistenceContext(unitName="scha", name="persistence/em")
public class SCatRep {
EntityManager em;
public SCatRep() {
try {
Context ic = new InitialContext();
em = (EntityManager) ic.lookup("java:comp/env/persistence/em");
} catch (NamingException e) {
e.printStackTrace();
}
}
I read a lot forums and tutorials - because this behaviour is discussed a lot - but can't get a solution or answer. Hope somebody can help me and point me to the right direction.
Thanks in advance.
EDIT:
I'm using EJB 3.1. And here is the persistence.xml:
<persistence xmlns="http://java.sun.com/xml/ns/persistence"
xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"
xsi:schemaLocation="http://java.sun.com/xml/ns/persistence
http://java.sun.com/xml/ns/persistence/persistence_2_0.xsd"
version="2.0">
<persistence-unit name="scha" transaction-type="JTA">
<jta-data-source>java:/scha_jndi</jta-data-source>
<properties>
<!-- SQL dialect -->
<property name="dialect" value="org.hibernate.dialect.MySQLDialect" />
<!-- Enable Hibernate's automatic session context management -->
<property name="current_session_context_class" value="thread" />
<!-- Echo all executed SQL to stdout -->
<property name="show_sql" value="true" />
</properties>
</persistence-unit>
</persistence>

Answer 1

Thanks for help.
I tried to access the EntityManager in constructor ... which is not possible, due to the initializationprocess.
Now I put the Initial loading (databaseaccess) in a #PostConstruct-annotated method and this works perfectly.