Two-body Dirac equations
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In quantum field theory, and in the significant subfields of
quantum electrodynamics In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and spec ...
(QED) and quantum chromodynamics (QCD), the two-body Dirac equations (TBDE) of constraint dynamics provide a three-dimensional yet
manifestly covariant In general relativity, a manifestly covariant equation is one in which all expressions are tensors. The operations of addition, tensor multiplication, tensor contraction, raising and lowering indices, and covariant differentiation may appear in t ...
reformulation of the
Bethe–Salpeter equation The Bethe–Salpeter equation (named after Hans Bethe and Edwin Salpeter) describes the bound states of a two-body (particles) quantum field theoretical system in a relativistically covariant formalism. The equation was actually first published i ...
for two
spin-1/2 In quantum mechanics, spin is an intrinsic property of all elementary particles. All known fermions, the particles that constitute ordinary matter, have a spin of . The spin number describes how many symmetrical facets a particle has in one full ...
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 P.A.M. Dirac, Lectures on Quantum Mechanics (Yeshiva University, New York, 1964) and relativistic mechanics and quantum mechanics. Their structures, unlike the more familiar two-body Dirac equation of
Breit Breit is an ''Ortsgemeinde'' – a Municipalities of Germany, municipality belonging to a ''Verbandsgemeinde'', a kind of collective municipality – in the Bernkastel-Wittlich Districts of Germany, district in Rhineland-Palatinate, Germany. Ge ...
, which is a single equation, are that of two simultaneous quantum
relativistic wave equations In physics, specifically relativistic quantum mechanics (RQM) and its applications to particle physics, relativistic wave equations predict the behavior of particles at high energies and velocities comparable to the speed of light. In the con ...
. A single two-body Dirac equation similar to the
Breit equation The Breit equation is a relativistic wave equation derived by Gregory Breit in 1929 based on the Dirac equation, which formally describes two or more massive spin-1/2 particles (electrons, for example) interacting electromagnetically to the first ...
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 interaction In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...
s 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 In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a sligh ...
Ψ is used.


Equations

For QED, each equation has the same structure as the ordinary one-body
Dirac equation In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin- massive particles, called "Dirac par ...
in the presence of an external electromagnetic field, given by the 4-potential A_\mu. For QCD, each equation has the same structure as the ordinary one-body
Dirac equation In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin- massive particles, called "Dirac par ...
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 In a relativistic theory of physics, a Lorentz scalar is an expression, formed from items of the theory, which evaluates to a scalar, invariant under any Lorentz transformation. A Lorentz scalar may be generated from e.g., the scalar product of ...
scalar S. In
natural units In physics, natural units are physical units of measurement in which only universal physical constants are used as defining constants, such that each of these constants acts as a coherent unit of a quantity. For example, the elementary charge ma ...
: those two-body equations have the form. : \gamma_1)_\mu (p_1-\tilde_1)^\mu+m_1 + \tilde_1Psi=0, : \gamma_2)_\mu (p_2-\tilde_2)^\mu+m_2 + \tilde_2Psi=0. where, in coordinate space, ''p''μ is the
4-momentum In special relativity, four-momentum (also called momentum-energy or momenergy ) is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. Momentum is a vector in three dimensions; similarly four-momentum i ...
, related to the 4-gradient by (the
metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathem ...
used here is \eta_=(-1,1,1,1)) :p^\mu = -i and γμ are the
gamma matrices In mathematical physics, the gamma matrices, \left\ , also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra Cl1,3(\ma ...
. The two-body Dirac equations (TBDE) have the property that if one of the masses becomes very large, say m_\rightarrow \infty then the 16-component Dirac equation reduces to the 4-component one-body
Dirac equation In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin- massive particles, called "Dirac par ...
for particle one in an external potential. In
SI units The International System of Units, known by the international abbreviation SI in all languages and sometimes Pleonasm#Acronyms and initialisms, pleonastically as the SI system, is the modern form of the metric system and the world's most wid ...
: : \gamma_1)_\mu (p_1-\tilde_1)^\mu+m_1c + \tilde_1Psi=0, : \gamma_2)_\mu (p_2-\tilde_2)^\mu+m_2c + \tilde_2Psi=0. where ''c'' is the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
and :p^\mu = -i\hbar 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 The elementary charge, usually denoted by is the electric charge carried by a single proton or, equivalently, the magnitude of the negative electric charge carried by a single electron, which has charge −1 . This elementary charge is a fundame ...
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 Commute, commutation or commutative may refer to: * Commuting, the process of travelling between a place of residence and a place of work Mathematics * Commutative property, a property of a mathematical operation whose result is insensitive to th ...
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 ) then the constraints would not be compatible (one could not e.g., have a wave function that satisfied both x\Psi=0 and 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,\vec 0). (The variable 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 only through its component x_ orthogonal to P in which : x_\perp^\mu =(\eta^-P^\mu P^\nu /P^2)x_\nu, \, : P_\mu x_\perp^\mu =0. \, This implies that in the c.m. frame x_\perp =(0,\vec=\vec_1 -\vec_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\cdot p\Psi =(-P^0p^0+\vec P\cdot p)\Psi=0. \, In this expression p is the relative momentum having the form (p_1 - p_2)/2 for equal masses. In the c.m. frame ( P^0=w,\vec P=\vec 0), the time component p^0 of the relative momentum, that is the relative energy, is thus eliminated. in the sense that p^0\Psi=0. A third consequence of the mathematical consistency is that each of the world scalar \tilde_i and four vector \tilde_^ potentials has a term with a fixed dependence on \gamma _ and \gamma _ in addition to the gamma matrix independent forms of S_i and 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,I. T. Todorov, Annals of the Institute of H. Poincaré' ,207 (1978) 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 Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta ...
. 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 \phi _(p,x)\approx 0. The symbol\approx 0 is called a weak equality and implies that the constraint is to be imposed only after any needed
Poisson brackets In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. Th ...
are performed. In the presence of such constraints, the total
Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
\mathcal is obtained from the
Lagrangian Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
\mathcal by adding to the Legendre Hamiltonian (p\dot-\mathcal) the sum of the constraints times an appropriate set of
Lagrange multiplier In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equality constraints (i.e., subject to the condition that one or more equations have to be satisfied e ...
s (\lambda _). :\mathcal=p\dot-\mathcal+\lambda _\phi _, This total Hamiltonian is traditionally called the Dirac Hamiltonian. Constraints arise naturally from parameter invariant actions of the form :I=\int d\tau \mathcal\int d\tau ^\frac\mathcal\int d\tau ^\mathcal^\mathcal. In the case of four vector and Lorentz scalar interactions for a single particle the Lagrangian is :\mathcal=-(m+S(x))\sqrt+\dot\cdot A(x) \, The
canonical momentum In mathematics and classical mechanics, canonical coordinates are sets of coordinates on phase space which can be used to describe a physical system at any given point in time. Canonical coordinates are used in the Hamiltonian formulation of cla ...
is :p=\frac=\frac+A(x) and by squaring leads to the generalized mass shell condition or generalized mass shell constraint : ( p-A)^2 + (m+S)^2 =0. \, Since, in this case, the Legendre Hamiltonian vanishes :p\cdot \dot-\mathcal=0, \, the Dirac Hamiltonian is simply the generalized mass constraint (with no interactions it would simply be the ordinary mass shell constraint) :\mathcal\left \left( p-A\right)^2 + (m+S)^2 \right\equiv \lambda (p^+m^+\Phi (x,p)). One then postulates that for two bodies the Dirac Hamiltonian is the sum of two such mass shell constraints, : \mathcal_i=p_i^2+m_i^2+\Phi_i (x_1,x_2,p_1,p_2)\approx 0, \, that is :\mathcal =\lambda_1 _1^2+m_1^2+\Phi_1(x_1,x_2,p_1,p_2)+ \lambda_2 _2^2 + m_2^2+\Phi_2(x_1,x_2,p_1,p_2) : =\lambda_1 \mathcal_1 + \lambda_2 \mathcal_2, \, and that each constraint \mathcal_i be constant in the proper time associated with \mathcal :\mathcal_i=\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 : \=\frac \frac -\frac \frac +\frac \frac -\frac \frac. To see the consequences of having each constraint be a constant of the motion, take, for example : \mathcal_1 = \\lambda _1 \\_2\mathcal_2\\_2. Since \=0 and \mathcal_\approx 0 and \mathcal_2 \approx 0 one has :\mathcal_\approx \mathcal_\\approx 0. \, The simplest solution to this is :\Phi _1 =\Phi _2 \equiv \Phi (x_\perp ) 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) : \=0 \, (see Todorov, and Wong and Crater ) with the same x_\perp defined above.


Quantization

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 :\mathcal_\approx 0\rightarrow \mathcal_\Psi =0, :\mathcal \approx 0\rightarrow \mathcal\Psi =0. The first set of equations for ''i'' = 1, 2 play the role for spinless particles that the two Dirac equations play for spin-one-half particles. The classical Poisson brackets are replaced by commutators :\\rightarrow \frac _1,O_2 \, Thus : mathcal_2,\mathcal_0, \, and we see in this case that the constraint formalism leads to the vanishing commutator of the wave operators for the two particles. This is the analogue of the claim stated earlier that the two Dirac operators commute with one another.


Covariant elimination of the relative energy

The vanishing of the above commutator ensures that the dynamics is independent of the relative time in the c.m. frame. In order to covariantly eliminate the relative energy, introduce the relative momentum p defined by The above definition of the relative momentum forces the orthogonality of the total momentum and the relative momentum, :P\cdot p=0, which follows from taking the scalar product of either equation with P. From Eqs.() and (), this relative momentum can be written in terms of p_ and p_ as :p=\fracp_-\fracp_ where :\varepsilon _ =-\frac=-\frac :\varepsilon _ =-\frac=-\frac are the projections of the momenta p_ and p_ along the direction of the total momentum P. Subtracting the two constraints \mathcal _\Psi =0 and \mathcal_\Psi =0, gives Thus on these states \Psi :\varepsilon _\Psi =\frac \Psi :\varepsilon _\Psi =\frac \Psi . The equation \mathcal\Psi =0 describes both the c.m. motion and the internal relative motion. To characterize the former motion, observe that since the potential \Phi depends only on the difference of the two coordinates : ,\mathcalPsi =0. (This does not require that ,\lambda _0 since the \mathcal _\Psi =0.) Thus, the total momentum P is a constant of motion and \Psi is an eigenstate state characterized by a total momentum P^. In the c.m. system P^=(w,\vec), with w the invariant center of momentum (c.m.) energy. Thus and so \Psi is also an eigenstate of c.m. energy operators for each of the two particles, :\varepsilon _\Psi =\frac\Psi :\varepsilon _\Psi =\frac\Psi. The relative momentum then satisfies :p\Psi =\frac\Psi, so that :p_\Psi =\left( \fracP+p\right) \Psi , :p_\Psi =\left( \fracP-p\right) \Psi , The above set of equations follow from the constraints \mathcal_\Psi =0 and the definition of the relative momenta given in Eqs.() and (). If instead one chooses to define (for a more general choice see Horwitz), :\varepsilon _ =\frac, :\varepsilon _ =\frac, :p =\frac, independent of the wave function, then and it is straight forward to show that the constraint Eq.() leads directly to: in place of P\cdot p=0. 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 non-relativistic equations :\vec =\frac, :\vec_ =\frac\vec+\vec, :\vec_ =\frac\vec+\vec.


Covariant eigenvalue equation for internal motion

Using Eqs.(),(),(), one can write \mathcal in terms of P and p :\mathcal\Psi =\\Psi where :b^(-P^,m_^,m_^)=\varepsilon _^-m_^=\varepsilon _^-m_^\ =-\frac (P^+2P^(m_^+m_^)+(m_^-m_^)^)\,. Eq.() contains both the total momentum P hrough the b^(-P^,m_^,m_^)and the relative momentum p. Using Eq. (), one obtains the eigenvalue equation so that b^(w^,m_^,m_^) becomes the standard triangle function displaying exact relativistic two-body kinematics: :b^(w^,m_^,m_^)=\frac\left\\,. With the above constraint Eqs.() on \Psi then p^\Psi =p_^\Psi where p_=p-p\cdot PP/P^. This allows writing Eq. () in the form of an eigenvalue equation :\\Psi =b^(w^,m_^,m_^)\Psi \,, 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 p_^ and x_^ have only three independent components since :P\cdot p_=P\cdot x_=0\,. The similarity to the three-dimensional structure of the nonrelativistic Schrödinger equation can be made more explicit by writing the equation in the c.m. frame in which :P =(w,\vec), :p_ =(0,\vec), :x_ =(0,\vec). Comparison of the resultant form with the time independent Schrödinger equation makes this similarity explicit.


The two-body relativistic Klein–Gordon equations

A plausible structure for the quasipotential \Phi can be found by observing that the one-body Klein–Gordon equation (p^+m^)\psi =(\vec^ -\varepsilon ^+m^)\psi =0 takes the form (\vec ^ -\varepsilon ^+m^+2mS+S^+2\varepsilon A-A^)\psi =0~ when one introduces a scalar interaction and timelike vector interaction via m\rightarrow m+S~and \varepsilon \rightarrow \varepsilon -A. In the two-body case, separate classical and quantum field theory arguments show that when one includes world scalar and vector interactions then \Phi depends on two underlying invariant functions S(r) and A(r) through the two-body Klein–Gordon-like potential form with the same general structure, that is : \Phi =2m_S+S^+2\varepsilon _A-A^. Those field theories further yield the c.m. energy dependent forms : m_=m_m_/w, and : \varepsilon _=(w^-m_^-m_^)/2w, 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 S and A). The two kinematical variables m_ and \varepsilon _ are related to one another by the Einstein condition : \varepsilon _^-m_^=b^(w), If one introduces the four-vectors, including a vector interaction A^ :\mathfrak =\varepsilon _\hat+p, :A^ =\hat^A(r) : r =\sqrt\,, and scalar interaction S(r), then the following classical minimal constraint form :\mathcal\left( \mathfrakA\right) ^+(m_+S)^\approx 0\,, reproduces Notice, that the interaction in this "reduced particle" constraint depends on two invariant scalars, A(r) and S(r), 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 :\mathcal_=(p_-A_)^+(m_+S_)^=p_^+m_^+\Phi _\approx 0 :\mathcal_=(p_-A_)^+(m_+S_)^=p_^+m_^+\Phi _\approx 0, :p_ =\varepsilon _\hat+p;~~p_=\varepsilon _\hat-p~. Defining structures that display time-like vector and scalar interactions :\pi _ =p_-A_= hat(\varepsilon _-\mathcal_)+p :\pi _ =p_-A_= hat(\varepsilon _-\mathcal_)-p :M_ =m_+S_, :M_ =m_+S_, gives :\mathcal_ =\pi _^+M_^, :\mathcal_ =\pi _^+M_^. Imposing :\begin \Phi _ & =\Phi _\equiv \Phi (x_) \\ & =-2p_\cdot A_+A_^+2m_S_+S_^ \\ & =-2p_\cdot A_+A_^+2m_S_+S_^ \\ & =2\varepsilon _A-A^+2m_S+S^, \end and using the constraint P\cdot p\approx 0, reproduces Eqs.() provided :\pi _^-p^ =-\left( \varepsilon _-\mathcal_\right)^=-\varepsilon _^+2\varepsilon _A-A^, :\pi _^-p^ =-\left( \varepsilon _-\mathcal_\right)^=-\varepsilon _^+2\varepsilon _A-A^, :M_^ =m_^+2m_S+S^, :M_^ =m_^+2m_S+S^. The corresponding Klein–Gordon equations are :\left( \pi _^+M_^\right) \psi =0, :\left( \pi _^+M_^\right) \psi =0, and each, due to the constraint P\cdot p\approx 0, is equivalent to :\mathcal\left( p_^+\Phi -b^\right) \mathcal=0.


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 : 1_1_; \gamma _\gamma _; \gamma _^\gamma _; \gamma _\gamma _^\gamma _\gamma _; \sigma _\sigma _^, in which : \sigma _=\frac gamma _,\gamma _ i=1,2. 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 :\mathcal_\psi =(\cosh (\Delta )\mathbf_+\sinh (\Delta ) \mathbf_)\psi =0\mathrm where \Delta 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 \mathbf_ and \mathbf_ are auxiliary constraints satisfying :\mathbf_\psi \equiv (\mathcal_\cosh (\Delta )+\mathcal _\sinh (\Delta )~)\psi =0, in which the \mathcal_ are the free Dirac operators This, in turn leads to the two compatibility conditions : \lbrack \mathcal_,\mathcal_]\psi =0, and : \lbrack \mathbf_,\mathbf_]\psi =0, provided that \Delta =\Delta (x_). These compatibility conditions do not restrict the gamma matrix structure of \Delta . That matrix structure is determined by the type of vertex-vertex structure incorporated in the interaction. For the two types of invariant interactions \Delta emphasized in this article they are :\Delta _(x_) =-1_1_\frac\mathcal_,\text\mathrm :\Delta _(x_) =\gamma_\cdot \gamma_\frac\mathcal_,\text\mathrm :\mathcal_=-\gamma _\gamma _. For general independent scalar and vector interactions : \Delta (x_)=\Delta _+\Delta _. 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 \cosh ^\Delta -\sinh ^\Delta =1 one arrives at \left( \partial _=\partial /\partial x^\right) , : \big(G\gamma _\cdot \mathcal_-E_\beta _+M_-G\frac \Sigma _\cdot \partial (\mathcal\beta _\mathcal\beta _)\gamma _\big)\psi =0, in which : G =\exp \mathcal, :\beta _ =-\gamma _\cdot \hat, :\gamma _^ =(\eta ^+\hat^\hat^)\gamma _, :\Sigma _ =\gamma _\beta _\gamma _, :\mathcal_ \equiv p_-\frac\Sigma _\cdot \partial \mathcal\Sigma _, i=1,2. The (covariant) structure of these equations are analogous to those of a Dirac equation for each of the two particles, with M_ and E_ playing the roles that m+S and \varepsilon -A do in the single particle Dirac equation : (\mathbf\cdot \mathbf\beta (\varepsilon -A)+m+S)\psi =0. Over and above the usual kinetic part \gamma _\cdot p_ and time-like vector and scalar potential portions, the spin-dependent modifications involving \Sigma _\cdot \partial \mathcal\Sigma _ 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 \mathcal,\mathcal and the mass and energy potentials M_,E_ are : M_ =m_ \cosh \mathcal +m_\sinh \mathcal, :M_ =m_ \cosh \mathcal +m_ \sinh \mathcal, :E_ =\varepsilon _ \cosh \mathcal -\varepsilon _\sinh \mathcal, : E_ =\varepsilon _ \cosh \mathcal-\varepsilon _\sinh \mathcal. Comparing Eq.() with the first equation of this article one finds that the spin-dependent vector interactions are : \tilde_^ =\big((\varepsilon _-E_)\big )\hat^+(1-G)p_^-\frac\partial G\cdot \gamma _\gamma _^, :A_^ =\big((\varepsilon _-E_)\big )\hat^-(1-G)p_^+\frac\partial G\cdot \gamma _\gamma _^, Note that the first portion of the vector potentials is timelike (parallel to \hat^) while the next portion is spacelike (perpendicular to \hat^). The spin-dependent scalar potentials \tilde_ are : \tilde_ =M_-m_-\fracG\gamma _\cdot \partial \mathcal , :\tilde_ =M_-m_+\fracG\gamma _\cdot \mathcal The parametrization for \mathcal and \mathcal takes advantage of the Todorov effective external potential forms (as seen in the above section on the two-body Klein Gordon equations) and at the same time displays the correct static limit form for the Pauli reduction to Schrödinger-like form. 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,. The mass and energy potentials are respectively : M_^ =m_^+\exp (2\mathcal2m_S\mathcalS^), :E_^ =\exp (2\mathcal\varepsilon _-A)^, so that : \exp (\mathcal) =\exp (\mathcal(S,A))=\frac , :G =\exp \mathcal\exp (\mathcalA\mathcal\sqrt.


Applications and limitations

The TBDE can be readily applied to two body systems such as
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,
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,
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-like atoms,
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, and the two-
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system. These applications involve two particles only and do not involve creation or annihilation of particles beyond the two. They involve only elastic processes. Because of the connection between the potentials used in the TBDE and the corresponding quantum field theory, any radiative correction to the lowest order interaction can be incorporated into those potentials. To see how this comes about, consider by contrast how one computes scattering amplitudes without quantum field theory. With no quantum field theory one must come upon potentials by classical arguments or phenomenological considerations. Once one has the potential V between two particles, then one can compute the scattering amplitude T from the Lippmann–Schwinger equation :T+V+VGT=0, in which G is a Green function determined from the Schrödinger equation. Because of the similarity between the Schrödinger equation Eq. () and the relativistic constraint equation (),one can derive the same type of equation as the above :\mathcal+\Phi +\Phi \mathcal=0, called the quasipotential equation with a \mathcal very similar to that given in the Lippmann–Schwinger equation. The difference is that with the quasipotential equation, one starts with the scattering amplitudes \mathcal of quantum field theory, as determined from Feynman diagrams and deduces the quasipotential Φ perturbatively. Then one can use that Φ in (), to compute energy levels of two particle systems that are implied by the field theory. Constraint dynamics provides one of many, in fact an infinite number of, different types of quasipotential equations (three-dimensional truncations of the Bethe–Salpeter equation) differing from one another by the choice of \mathcal. The relatively simple solution to the problem of relative time and energy from the generalized mass shell constraint for two particles, has no simple extension, such as presented here with the x_ variable, to either two particles in an external field or to 3 or more particles. 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 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,


See also

*
Breit equation The Breit equation is a relativistic wave equation derived by Gregory Breit in 1929 based on the Dirac equation, which formally describes two or more massive spin-1/2 particles (electrons, for example) interacting electromagnetically to the first ...
* 4-vector *
Dirac equation In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin- massive particles, called "Dirac par ...
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Dirac equation in the algebra of physical space In physics, the algebra of physical space (APS) is the use of the Clifford algebra, Clifford or geometric algebra Cl3,0(R) of the three-dimensional Euclidean space as a model for (3+1)-dimensional spacetime, representing a point in spacetime via a ...
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Dirac operator In mathematics and quantum mechanics, a Dirac operator is a differential operator that is a formal square root, or half-iterate, of a second-order operator such as a Laplacian. The original case which concerned Paul Dirac was to factorise forma ...
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Electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of ...
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Kinetic momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass and ...
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Many body problem The many-body problem is a general name for a vast category of physical problems pertaining to the properties of microscopic systems made of many interacting particles. ''Microscopic'' here implies that quantum mechanics has to be used to provid ...
* Invariant mass *
Particle physics Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...
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Positronium Positronium (Ps) is a system consisting of an electron and its anti-particle, a positron, bound together into an exotic atom, specifically an onium. Unlike hydrogen, the system has no protons. The system is unstable: the two particles annih ...
* Ricci calculus *
Special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The laws ...
* Spin *
Quantum entanglement Quantum entanglement is the phenomenon that occurs when a group of particles are generated, interact, or share spatial proximity in a way such that the quantum state of each particle of the group cannot be described independently of the state of ...
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Relativistic quantum mechanics In physics, relativistic quantum mechanics (RQM) is any Poincaré covariant formulation of quantum mechanics (QM). This theory is applicable to massive particles propagating at all velocities up to those comparable to the speed of light  ...


References

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Various forms of radial equations for the Dirac two-body problem
W. Królikowski (1991), Institute of theoretical physics (Warsaw, Poland) * {{Refend Quantum field theory Mathematical physics Equations of physics Dirac equation