condition If one introduces the four-vectors, including a vector interaction and scalar interaction , then the following classical minimal constraint form reproduces Notice, that the interaction in this "reduced particle" constraint depends on two invariant scalars, and , one guiding the time-like vector interaction and one the scalar interaction. Is there a set of two-body Klein–Gordon equations analogous to the two-body Dirac equations? The classical relativistic constraints analogous to the quantum two-body Dirac equations (discussed in the introduction) and that have the same structure as the above Klein–Gordon one-body form are Defining structures that display time-like vector and scalar interactions gives Imposing and using the constraint , reproduces Eqs.() provided The corresponding Klein–Gordon equations are and each, due to the constraint is equivalent to Hyperbolic versus external field form of the two-body Dirac equations For the two body system there are numerous covariant forms of interaction. The simplest way of looking at these is from the point of view of the gamma matrix structures of the corresponding interaction vertices of the single particle exchange diagrams. For scalar, pseudoscalar, vector, pseudovector, and tensor exchanges those matrix structures are respectively in which The form of the Two-Body Dirac equations which most readily incorporates each or any number of these intereractions in concert is the so-called hyperbolic form of the TBDE . For combined scalar and vector interactions those forms ultimately reduce to the ones given in the first set of equations of this article. Those equations are called the external field-like forms because their appearances are individually the same as those for the usual one-body Dirac equation in the presence of external vector and scalar fields. The most general hyperbolic form for compatible TBDE is where represents any invariant interaction singly or in combination. It has a matrix structure in addition to coordinate dependence. Depending on what that matrix structure is one has either scalar, pseudoscalar, vector, pseudovector, or tensor interactions. The operators and are auxiliary constraints satisfying in which the are the free Dirac operators This, in turn leads to the two compatibility conditions and provided that These compatibility conditions do not restrict the gamma matrix structure of . That matrix structure is determined by the type of vertex-vertex structure incorporated in the interaction. For the two types of invariant interactions emphasized in this article they are For general independent scalar and vector interactions The vector interaction specified by the above