Constructor in an Interface?

A problem that you get when you allow constructors in interfaces comes from the possibility to implement several interfaces at the same time. When a class implements several interfaces that define different constructors, the class would have to implement several constructors, each one satisfying only one interface, but not the others. It will be impossible to construct an object that calls each of these constructors.

Or in code:

interface Named { Named(String name); }
interface HasList { HasList(List list); }

class A implements Named, HasList {

  /** implements Named constructor.
   * This constructor should not be used from outside, 
   * because List parameter is missing
   */
  public A(String name)  { 
    ...
  }

  /** implements HasList constructor.
   * This constructor should not be used from outside, 
   * because String parameter is missing
   */
  public A(List list) {
    ...
  }

  /** This is the constructor that we would actually 
   * need to satisfy both interfaces at the same time
   */ 
  public A(String name, List list) {
    this(name);
    // the next line is illegal; you can only call one other super constructor
    this(list); 
  }
}

An interface defines a contract for an API, that is a set of methods that both implementer and user of the API agree upon. An interface does not have an instanced implementation, hence no constructor.

The use case you describe is akin to an abstract class in which the constructor calls a method of an abstract method which is implemented in an child class.

The inherent problem here is that while the base constructor is being executed, the child object is not constructed yet, and therfore in an unpredictable state.

To summarize: is it asking for trouble when you call overloaded methods from parent constructors, to quote mindprod:

In general you must avoid calling any non-final methods in a constructor. The problem is that instance initialisers / variable initialisation in the derived class is performed after the constructor of the base class.

Taking some of the things you have described:

"So you could be sure that some fields in a class are defined for every implementation of this interface."

"If a define a Interface for this class so that I can have more classes which implement the message interface, I can only define the send method and not the constructor"

...these requirements are exactly what abstract classes are for.