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covariantly eliminate the relative energy, introduce the relative momentum formula_66 defined by |
The above definition of the relative momentum forces the orthogonality of the total |
which follows from taking the scalar product of either equation with formula_18. |
From Eqs.() and (), this relative momentum can be written in terms of |
are the projections of the momenta formula_69 and formula_70 along the direction |
of the total momentum formula_18. Subtracting the two constraints formula_77 and formula_78, gives |
The equation formula_82 describes both the c.m. motion and the |
internal relative motion. To characterize the former motion, observe that |
since the potential formula_83 depends only on the difference of the two |
(This does not require that formula_85 since the formula_86.) Thus, the total momentum formula_18 is a constant of motion and |
formula_79 is an eigenstate state characterized by a total momentum |
formula_89. In the c.m. system formula_90 with formula_15 the |
invariant center of momentum (c.m.) energy. Thus |
and so formula_79 is also an eigenstate of c.m. energy operators for each of |
The above set of equations follow from the constraints formula_98 and the definition of the relative momenta given in Eqs.() and (). |
If instead one chooses to define (for a more general choice see Horwitz), |
and it is straight forward to show that the constraint Eq.() leads |
in place of formula_67. This conforms with the earlier claim on the |
vanishing of the relative energy in the c.m. frame made in conjunction with |
the TBDE.\ In the second choice the c.m. value of the relative energy is |
not defined as zero but comes from the original generalized mass shell |
constraints. The above equations for the relative and constituent |
four-momentum are the relativistic analogues of the nonrelativistic equations |
Using Eqs.(),(),(), one can write formula_36 in terms of formula_18 and formula_23 |
Eq.() contains both the total momentum formula_18 [through the formula_112] and the relative momentum formula_23. Using Eq. (), one obtains the eigenvalue equation |
so that formula_114 becomes the standard triangle |
With the above constraint Eqs.() on formula_79 then formula_117 where formula_118. This allows |
writing Eq. () in the form of an eigenvalue equation |
having a structure very similar to that of the ordinary three-dimensional |
nonrelativistic Schrödinger equation. It is a manifestly covariant |
equation, but at the same time its three-dimensional structure is evident. |
The four-vectors formula_120 and formula_121 have only |
The similarity to the three-dimensional structure of the nonrelativistic |
Schrödinger equation can be made more explicit by writing the equation in |
A plausible structure for the quasipotential formula_83 can be found by |
observing that the one-body Klein-Gordon equation formula_127 takes the form formula_128 when one |
introduces a scalar interaction and timelike vector interaction via formula_129and formula_130. In the |
two-body case, separate classical and quantum field theory |
arguments show that when one includes world scalar and |
vector interactions then formula_83 depends on two underlying invariant |
functions formula_132 and formula_133 through the two-body Klein-Gordon-like potential |
form with the same general structure, that is |
Those field theories further yield the c.m. energy dependent forms |
ones that Tododov introduced as the relativistic reduced mass |
and effective particle energy for a two-body system. Similar to what |
happens in the nonrelativistic two-body problem, in the relativistic case |
we have the motion of this effective particle taking place as if it were in |
an external field (here generated by formula_2 and formula_138). The two kinematical |
variables formula_139 and formula_140 are related to one another by the |
If one introduces the four-vectors, including a vector interaction formula_142 |
and scalar interaction formula_132, then the following classical minimal |
Notice, that the interaction in this "reduced particle" constraint depends |
on two invariant scalars, formula_133 and formula_132, 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 |
Defining structures that display time-like vector and scalar interactions |
and using the constraint formula_160, reproduces Eqs.() provided |
and each, due to the constraint formula_167 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 |
paraticle exchange diagrams. For scalar, pseudoscalar, vector, |
pseudovector, and tensor exchanges those matrix structures are respectively |
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 |
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 |
The most general hyperbolic form for compatible TBDE is |
where formula_172 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 formula_173 and formula_174 are auxiliary constraints |
in which the formula_176 are the free Dirac operators |
This, in turn leads to the two compatibility conditions |
conditions do not restrict the gamma matrix structure of formula_172. That |
matrix structure is determined by the type of vertex-vertex structure |
incorporated in the interaction. For the two types of invariant |
interactions formula_172 emphasized in this article they are |
For general independent scalar and vector interactions |
The vector interaction specified by the above matrix structure for an electromagnetic-like interaction would correspond to the Feynman gauge. |
If one inserts Eq.() into () and brings the free |
Dirac operator () to the right of the matrix hyperbolic functions |
and uses standard gamma matrix commutators and anticommutators and formula_186 one arrives at formula_187 |
The (covariant) structure of these equations are analogous to those of a Dirac equation for each of the two particles, with formula_194 and formula_195 |
playing the roles that formula_196 and formula_197 do in the single particle |
Over and above the usual kinetic part formula_199 and |
time-like vector and scalar potential portions, the spin-dependent |
and the last set of derivative terms are two-body recoil effects absent for |
the one-body Dirac equation but essential for the compatibility |
(consistency) of the two-body equations. The connections between what |
are designated as the vertex invariants formula_201 and the |
Comparing Eq.() with the first equation of this article one finds |
Note that the first portion of the vector potentials is timelike (parallel |
to formula_209 while the next portion is spacelike (perpendicular to formula_210. The spin-dependent scalar potentials formula_211 are |
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