Such a reformulation is necessary since without it, as shown by Nakanishi,[2] the Bethe–Salpeter equation possesses negative-norm solutions arising from the presence of an essentially relativistic degree of freedom, the relative time.
A tilde symbol is used over the two sets of potentials to indicate that they may have additional gamma matrix dependencies not present in the one-body Dirac equation.
Constraint dynamics applied to the TBDE requires a particular form of mathematical consistency: the two Dirac operators must commute with each other.
In particular, for the two operators to commute, the scalar and four-vector potentials can depend on the relative coordinate
It does this by imposing on each Dirac operator a structure such that in a particular combination they lead to this interaction independent form, eliminating in a covariant way the relative energy.
These extra terms correspond to additional recoil spin-dependence not present in the one-body Dirac equation and vanish when one of the particles becomes very heavy (the so-called static limit).
[17] This section shows how the elimination of relative time and energy takes place in the c.m.
Constraint dynamics was first applied to the classical relativistic two particle system by Todorov,[18][19] Kalb and Van Alstine,[20][21] Komar,[22][23] and Droz–Vincent.
[24] With constraint dynamics, these authors found a consistent and covariant approach to relativistic canonical Hamiltonian mechanics that also evades the Currie–Jordan–Sudarshan "No Interaction" theorem.
Thus, the same covariant three-dimensional approach which allows the quantized version of constraint dynamics to remove quantum ghosts simultaneously circumvents at the classical level the C.J.S.
is called a weak equality and implies that the constraint is to be imposed only after any needed Poisson brackets are performed.
which leads to (note the equality in this case is not a weak one in that no constraint need be imposed after the Poisson bracket is worked out)
In addition to replacing classical dynamical variables by their quantum counterparts, quantization of the constraint mechanics takes place by replacing the constraint on the dynamical variables with a restriction on the wave function
and we see in this case that the constraint formalism leads to the vanishing commutator of the wave operators for the two particles.
independent of the wave function, then and it is straight forward to show that the constraint Eq.
Comparison of the resultant form with the time independent Schrödinger equation makes this similarity explicit.
In the two-body case, separate classical [29][30] and quantum field theory [4] arguments show that when one includes world scalar and vector interactions then
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
reproduces Notice, that the interaction in this "reduced particle" constraint depends on two invariant scalars,
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
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.
[31] For combined scalar and vector interactions those forms ultimately reduce to the ones given in the first set of equations of this article.
Depending on what that matrix structure is one has either scalar, pseudoscalar, vector, pseudovector, or tensor interactions.
The choice for these parameterizations (as with the two-body Klein Gordon equations) is closely tied to classical or quantum field theories for separate scalar and vector interactions.
This amounts to working in the Feynman gauge with the simplest relation between space- and timelike parts of the vector interaction.
of quantum field theory, as determined from Feynman diagrams and deduces the quasipotential Φ perturbatively.
Then one can use that Φ in (10), to compute energy levels of two particle systems that are implied by the field theory.
Sazdjian has presented a recipe for this extension when the particles are confined and cannot split into clusters of a smaller number of particles with no inter-cluster interactions [38] Lusanna has developed an approach, one that does not involve generalized mass shell constraints with no such restrictions, which extends to N bodies with or without fields.
It is formulated on spacelike hypersurfaces and when restricted to the family of hyperplanes orthogonal to the total timelike momentum gives rise to a covariant intrinsic 1-time formulation (with no relative time variables) called the "rest-frame instant form" of dynamics,[39][40]