In quantum field theory, and in the significant subfields of quantum electrodynamics (QED) and quantum chromodynamics (QCD), the two-body Dirac equations (TBDE) of constraint dynamics provide a three-dimensional yet manifestly covariant reformulation of the Bethe–Salpeter equation for two spin-1/2 particles. Such a reformulation is necessary since without it, as shown by Nakanishi, the Bethe–Salpeter equation possesses negative-norm solutions arising from the presence of an essentially relativistic degree of freedom, the relative time. These "ghost" states have spoiled the naive interpretation of the Bethe–Salpeter equation as a quantum mechanical wave equation. The two-body Dirac equations of constraint dynamics rectify this flaw. The forms of these equations can not only be derived from quantum field theory they can also be derived purely in the context of Dirac's constraint dynamics and relativistic mechanics and quantum mechanics. Their structures, unlike the more familiar two-body Dirac equation of Breit, which is a single equation, are that of two simultaneous quantum relativistic wave equations. A single two-body Dirac equation similar to the Breit equation can be derived from the TBDE. Unlike the Breit equation, it is manifestly covariant and free from the types of singularities that prevent a strictly nonperturbative treatment of the Breit equation. In applications of the TBDE to QED, the two particles interact by way of four-vector potentials derived from the field theoretic electromagnetic interactions between the two particles. In applications to QCD, the two particles interact by way of four-vector potentials and Lorentz invariant scalar interactions, derived in part from the field theoretic chromomagnetic interactions between the quarks and in part by phenomenological considerations. As with the Breit equation a sixteen-component spinor Ψ is used.
Equations For QED, each equation has the same structure as the ordinary one-body Dirac equation in the presence of an external electromagnetic field, given by the 4-potential A μ {\displaystyle A_{\mu }} . For QCD, each equation has the same structure as the ordinary one-body Dirac equation in the presence of an external field similar to the electromagnetic field and an additional external field given by in terms of a Lorentz invariant scalar S {\displaystyle S} . In natural units: those two-body equations have the form.
[ ( γ 1 ) μ ( p 1 − A ~ 1 ) μ + m 1 + S ~ 1 ] Ψ = 0 , [ ( γ 2 ) μ ( p 2 − A ~ 2 ) μ + m 2 + S ~ 2 ] Ψ = 0. {\displaystyle {\begin{aligned}\left[(\gamma _{1})_{\mu }(p_{1}-{\tilde {A}}_{1})^{\mu }+m_{1}+{\tilde {S}}_{1}\right]\Psi &=0,\\[1ex]\left[(\gamma _{2})_{\mu }(p_{2}-{\tilde {A}}_{2})^{\mu }+m_{2}+{\tilde {S}}_{2}\right]\Psi &=0.\end{aligned}}}
where, in coordinate space, pμ is the 4-momentum, related to the 4-gradient by (the metric used here is η μ ν = ( − 1 , 1 , 1 , 1 ) {\displaystyle \eta _{\mu \nu }=(-1,1,1,1)} )
p μ = − i ∂ ∂ x μ {\displaystyle p^{\mu }=-i{\frac {\partial }{\partial x_{\mu }}}}
and γμ are the gamma matrices. The two-body Dirac equations (TBDE) have the property that if one of the masses becomes very large, say m 2 → ∞ {\displaystyle m_{2}\rightarrow \infty } then the 16-component Dirac equation reduces to the 4-component one-body Dirac equation for particle one in an external potential. In SI units:
[ ( γ 1 ) μ ( p 1 − A ~ 1 ) μ + m 1 c + S ~ 1 ] Ψ = 0 , [ ( γ 2 ) μ ( p 2 − A ~ 2 ) μ + m 2 c + S ~ 2 ] Ψ = 0. {\displaystyle {\begin{aligned}\left[(\gamma _{1})_{\mu }(p_{1}-{\tilde {A}}_{1})^{\mu }+m_{1}c+{\tilde {S}}_{1}\right]\Psi &=0,\\[1ex]\left[(\gamma _{2})_{\mu }(p_{2}-{\tilde {A}}_{2})^{\mu }+m_{2}c+{\tilde {S}}_{2}\right]\Psi &=0.\end{aligned}}}
where c is the speed of light and
p μ = − i ℏ ∂ ∂ x μ {\displaystyle p^{\mu }=-i\hbar {\frac {\partial }{\partial x_{\mu }}}}
Natural units will be used below. 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. Any coupling constants such as the electron charge are embodied in the vector potentials.
Constraint dynamics and the TBDE Constraint dynamics applied to the TBDE requires a particular form of mathematical consistency: the two Dirac operators must commute with each other. This is plausible if one views the two equations as two compatible constraints on the wave function. (See the discussion below on constraint dynamics.) If the two operators did not commute, (as, e.g., with the coordinate and momentum operators x , p {\displaystyle x,p} ) then the constraints would not be compatible (one could not e.g., have a wave function that satisfied both x Ψ = 0 {\displaystyle x\Psi =0} and p Ψ = 0 {\displaystyle p\Psi =0} ). This mathematical consistency or compatibility leads to three important properties of the TBDE. The first is a condition that eliminates the dependence on the relative time in the center of momentum (c.m.) frame defined by P = p 1 + p 2 = ( w , 0 → ) {\displaystyle P=p_{1}+p_{2}=(w,{\vec {0}})} . (The variable w {\displaystyle w} is the total energy in the c.m. frame.) Stated another way, the relative time is eliminated in a covariant way. In particular, for the two operators to commute, the scalar and four-vector potentials can depend on the relative coordinate x = x 1 − x 2 {\displaystyle x=x_{1}-x_{2}} only through its component x ⊥ {\displaystyle x_{\perp }} orthogonal to P {\displaystyle P} in which
x ⊥ μ = ( η μ ν − P μ P ν / P 2 ) x ν , {\displaystyle x_{\perp }^{\mu }=(\eta ^{\mu \nu }-P^{\mu }P^{\nu }/P^{2})x_{\nu },\,}
P μ x ⊥ μ = 0. {\displaystyle P_{\mu }x_{\perp }^{\mu }=0.\,}
This implies that in the c.m. frame x ⊥ = ( 0 , x → = x → 1 − x → 2 ) {\displaystyle x_{\perp }=(0,{\vec {x}}={\vec {x}}_{1}-{\vec {x}}_{2})} , which has zero time component. Secondly, the mathematical consistency condition also eliminates the relative energy in the c.m. frame. 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.
P ⋅ p Ψ = ( − P 0 p 0 + P → ⋅ p ) Ψ = 0. {\displaystyle P\cdot p\Psi =(-P^{0}p^{0}+{\vec {P}}\cdot p)\Psi =0.\,}
In this expression p {\displaystyle p} is the relative momentum having the form ( p 1 − p 2 ) / 2 {\displaystyle (p_{1}-p_{2})/2} for equal masses. In the c.m. frame ( P 0 = w , P → = 0 → {\displaystyle P^{0}=w,{\vec {P}}={\vec {0}}} ), the time component p 0 {\displaystyle p^{0}} of the relative momentum, that is the relative energy, is thus eliminated. in the sense that p 0 Ψ = 0 {\displaystyle p^{0}\Psi =0} . A third consequence of the mathematical consistency is that each of the world scalar S ~ i {\displaystyle {\tilde {S}}_{i}} and four vector A ~ i μ {\displaystyle {\tilde {A}}_{i}^{\mu }} potentials has a term with a fixed dependence on γ 1 {\displaystyle \gamma _{1}} and γ 2 {\displaystyle \gamma _{2}} in addition to the gamma matrix independent forms of S i {\displaystyle S_{i}} and A i μ {\displaystyle A_{i}^{\mu }} which appear in the ordinary one-body Dirac equation for scalar and vector potentials. 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).
More on constraint dynamics: generalized mass shell constraints Constraint dynamics arose from the work of Dirac and Bergmann. This section shows how the elimination of relative time and energy takes place in the c.m. system for the simple system of two relativistic spinless particles. Constraint dynamics was first applied to the classical relativistic two particle system by Todorov, Kalb and Van Alstine, Komar, and Droz–Vincent. 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. That theorem states that without fields, one cannot have a relativistic Hamiltonian dynamics. 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. theorem. Consider a constraint on the otherwise independent coordinate and momentum four vectors, written in the form ϕ i ( p , x ) ≈ 0 {\displaystyle \phi _{i}(p,x)\approx 0} . The symbol ≈ 0 {\displaystyle \approx 0} is called a weak equality and implies that the constraint is to be imposed only after any needed Poisson brackets are performed. In the presence of such constraints, the total Hamiltonian H {\displaystyle {\mathcal {H}}} is obtained from the Lagrangian L {\displaystyle {\mathcal {L}}} by adding to its Legendre transform ( p x ˙ − L ) {\displaystyle (p{\dot {x}}-{\mathcal {L}})} the sum of the constraints times an appropriate set of Lagrange multipliers ( λ i ) {\displaystyle (\lambda _{i})} .
H = p x ˙ − L + λ i ϕ i , {\displaystyle {\mathcal {H}}=p{\dot {x}}-{\mathcal {L}}+\lambda _{i}\phi _{i},}
This total Hamiltonian is traditionally called the Dirac Hamiltonian. Constraints arise naturally from parameter invariant actions of the form
I = ∫ d τ L ( τ ) = ∫ d τ ′ d τ d τ ′ L ( τ ) = ∫ d τ ′ L ( τ ′ ) . {\displaystyle I=\int d\tau {\mathcal {L}}(\tau )=\int d\tau '{\frac {d\tau }{d\tau '}}{\mathcal {L}}(\tau )=\int d\tau '{\mathcal {L}}(\tau ').}
In the case of four vector and Lorentz scalar interactions for a single particle the Lagrangian is
L ( τ ) = − ( m + S ( x ) ) − x ˙ 2 + x ˙ ⋅ A ( x ) {\displaystyle {\mathcal {L}}(\tau )=-(m+S(x)){\sqrt {-{\dot {x}}^{2}}}+{\dot {x}}\cdot A(x)\,}
The canonical momentum is
p = ∂ L ∂ x ˙ = ( m + S ( x ) ) x ˙ − x ˙ 2 + A ( x ) {\displaystyle p={\frac {\partial {\mathcal {L}}}{\partial {\dot {x}}}}={\frac {(m+S(x)){\dot {x}}}{\sqrt {-{\dot {x}}^{2}}}}+A(x)}
and by squaring leads to the generalized mass shell condition or generalized mass shell constraint
( p − A ) 2 + ( m + S ) 2 = 0. {\displaystyle (p-A)^{2}+(m+S)^{2}=0.\,}
Since, in this case, the Legendre Hamiltonian vanishes
p ⋅ x ˙ − L = 0 , {\displaystyle p\cdot {\dot {x}}-{\mathcal {L}}=0,\,}
the Dirac Hamiltonian is simply the generalized mass constraint (with no interactions it would simply be the ordinary mass shell constraint)
H = λ [ ( p − A ) 2 + ( m + S ) 2 ] ≡ λ ( p 2 + m 2 + Φ ( x , p ) ) . {\displaystyle {\mathcal {H}}=\lambda \left[\left(p-A\right)^{2}+(m+S)^{2}\right]\equiv \lambda (p^{2}+m^{2}+\Phi (x,p)).}
One then postulates that for two bodies the Dirac Hamiltonian is the sum of two such mass shell constraints,
H i = p i 2 + m i 2 + Φ i ( x 1 , x 2 , p 1 , p 2 ) ≈ 0 , {\displaystyle {\mathcal {H}}_{i}=p_{i}^{2}+m_{i}^{2}+\Phi _{i}(x_{1},x_{2},p_{1},p_{2})\approx 0,\,}
that is
H = λ 1 [ p 1 2 + m 1 2 + Φ 1 ( x 1 , x 2 , p 1 , p 2 ) ] + λ 2 [ p 2 2 + m 2 2 + Φ 2 ( x 1 , x 2 , p 1 , p 2 ) ] = λ 1 H 1 + λ 2 H 2 , {\displaystyle {\begin{aligned}{\mathcal {H}}&=\lambda _{1}[p_{1}^{2}+m_{1}^{2}+\Phi _{1}(x_{1},x_{2},p_{1},p_{2})]+\lambda _{2}[p_{2}^{2}+m_{2}^{2}+\Phi _{2}(x_{1},x_{2},p_{1},p_{2})]\\[1ex]&=\lambda _{1}{\mathcal {H}}_{1}+\lambda _{2}{\mathcal {H}}_{2},\end{aligned}}}
and that each constraint H i {\displaystyle {\mathcal {H}}_{i}} be constant in the proper time associated with H {\displaystyle {\mathcal {H}}}
H ˙ i = { H i , H } ≈ 0 {\displaystyle {\dot {\mathcal {H}}}_{i}=\{{\mathcal {H}}_{i},{\mathcal {H}}\}\approx 0\,}
Here the weak equality means that the Poisson bracket could result in terms proportional one of the constraints, the classical Poisson brackets for the relativistic two-body system being defined by
{ O 1 , O 2 } = ∂ O 1 ∂ x 1 μ ∂ O 2 ∂ p 1 μ − ∂ O 1 ∂ p 1 μ ∂ O 2 ∂ x 1 μ + ∂ O 1 ∂ x 2 μ ∂ O 2 ∂ p 2 μ − ∂ O 1 ∂ p 2 μ ∂ O 2 ∂ x 2 μ . {\displaystyle \left\{O_{1},O_{2}\right\}={\frac {\partial O_{1}}{\partial x_{1}^{\mu }}}{\frac {\partial O_{2}}{\partial p_{1\mu }}}-{\frac {\partial O_{1}}{\partial p_{1}^{\mu }}}{\frac {\partial O_{2}}{\partial x_{1\mu }}}+{\frac {\partial O_{1}}{\partial x_{2}^{\mu }}}{\frac {\partial O_{2}}{\partial p_{2\mu }}}-{\frac {\partial O_{1}}{\partial p_{2}^{\mu }}}{\frac {\partial O_{2}}{\partial x_{2\mu }}}.}
To see the consequences of having each constraint be a constant of the motion, take, for example
H ˙ 1 = { H 1 , H } = λ 1 { H 1 , H 1 } + { H 1 , λ 1 } H 2 + λ 2 { H 2 , H 1 } + { λ 2 , H 1 } H 2 . {\displaystyle {\dot {\mathcal {H}}}_{1}=\{{\mathcal {H}}_{1},{\mathcal {H}}\}=\lambda _{1}\{{\mathcal {H}}_{1},{\mathcal {H}}_{1}\}+\{{\mathcal {H}}_{1},\lambda _{1}\}{\mathcal {H}}_{2}+\lambda _{2}\{{\mathcal {H}}_{2},{\mathcal {H}}_{1}\}+\{\lambda _{2},{\mathcal {H}}_{1}\}{\mathcal {H}}_{2}.}
Since { H 1 , H 1 } = 0 {\displaystyle \{{\mathcal {H}}_{1},{\mathcal {H}}_{1}\}=0} and H 1 ≈ 0 {\displaystyle {\mathcal {H}}_{1}\approx 0} and H