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Gauging of equations method. II. Electromagnetic currents of three identical particles

The gauging of equations method, introduced in the preceding paper, is applied to the four-dimensional integral equations describing the strong interactions of three identical relativistic particles. In this way we obtain gauge invariant expressions for all possible electromagnetic transition currents of the identical three-particle system. In the three-nucleon system with no isospin violation, for example, our expressions describe the electromagnetic form factors of ^3H, pd->pd gamma, gamma ^3He->pd, gamma ^3He->ppn, etc. A feature of our approach is that gauge invariance is achieved through the coupling of the photon to all possible places in the (nonperturbative) strong interaction model. Moreover, once the proper identical particle symmetry is incorporated into the integral eqations describing the strong interactions, the gauging procedure automatically provides electromagnetic transition currents with the proper symmetry. In this way the gauging of equations method results in a unified description of strong and electromagnetic interaction of strongly interacting systems.


I. INTRODUCTION
In the preceding paper [1] (referred to as I in the following), we have introduced the gauging of equations method as a means of incorporating an external electromagnetic field into descriptions of quarks or hadrons whose strong interactions are described nonperturbatively by integral equations. The feature of this method is that it couples an external photon to all possible places in the strong interaction model despite its nonperturbative nature. Gauge invariance in our approach is therefore implemented in the way prescribed by quantum field theory (QFT). In I we demonstrated the gauging procedure on the example of three distinguishable particles. In this paper we would like to demonstrate the same method as applied to the case of indistinguishable particles where the strong interaction equations have the added complexity of identical particle symmetry.
As in I, the discussion here is restricted to the case where the three strongly interacting particles have no coupling to two-body channels. Thus we have in mind identical particle systems like three quarks qqq or three nucleons NNN. This is not an essential restriction, and indeed we have recently applied the gauging of equations method to the πNN system where coupling to the NN channel is included [2]. However, the purpose of this paper (together with I) is to present the basic details of the gauging of equations method, and as such, coupling to two-body channels presents an unnecessary complication. We note that the main results of this work have previously been summarised in conference proceedings [3].
In dealing with identical particles, one is faced with the problem of incorporating the proper particle-exchange symmetry into the equations describing both their strong and electromagnetic interactions. In Quantum Mechanics the standard procedure is to explicitly symmetrise (or antisymmetrise) the corresponding distinguishable particle equations. However, such a procedure is not justified within a field theoretic approach. As the basis of our approach here is QFT, we present a derivation of the strong interaction equations for three identical particles that is consistent with QFT, and that is therefore very different from the derivations found in standard texts on the quantum mechanical three-body problem [4].
Once the identical particle equations for the strong interactions are derived, it is a feature of the gauging of equations method that it may be applied directly to these equations, thereby automatically generating electromagnetic transition currents with the proper symmetry. Thus the main effort in generating practical expressions for the transition currents reduces down to a careful choice for the identical particle strong interaction equations that are to be gauged. Here we also show that for three identical particles an optimal choice is provided by an equation of Alt-Grassberger-Sandhas (AGS) form [5], but which is however different from the AGS equation used previously in the literature for three-nucleon calculations.

II. GAUGING THE THREE-PARTICLE GREEN FUNCTION
The gauging of equations method, introduced in I and used there to gauge the equations of three distinguishable particles, does not change when the particles are identical. Indeed, the main steps taken in I to derive the electromagnetic transition currents can be repeated for identical particles, although it is necessary to guarantee the proper identical particle symmetry of all perturbative diagrams at each step of the derivation. In this respect, it should be noted that we do not follow the common procedure of symmetrising or antisymmetrizing the distinguishable particle results. Such a procedure is strictly valid only within the context of second quantisation in quantum mechanics, while here the theoretical framework is that of relativistic quantum field theory. Instead we follow the standard rules of QFT for constructing Green functions for identical particles. Details of these rules as applied to few-body integral equations have been given by us in Ref. [6].
The strong interactions of three identical particles are described in quantum field theory by the Green function G defined by Here Ψ andΨ are Heisenberg fields, T is the time ordering operator, and |0 is the physical vacuum state. The interaction of this three-particle system with an external electromagnetic field is then described by the corresponding seven-point function G µ defined by where J µ is the quantised electromagnetic current operator and e i is the charge of the i-th particle. If the particles are isotopic doublets then e i includes an isospin factor, e.g. for nucleons 3 ]e p where τ 3 is the Pauli matrix for the third component of isospin, and e p is the charge of the proton. The Ward-Takahashi [WT] identity [7], which provides an important constraint on G µ , takes the same form as in the distinguishable particles case [1] or in the shorthand notation introduced in Ref. [1], To be definite, we shall assume that our three identical particles are fermions (for three identical bosons one can simply replace antisymmetric operations by symmetric ones in the following discussion). Then the field theoretic expressions of Eqs. (1) and (2) automatically guarantee the proper antisymmetry of the three-particle Green function G and the seven-point function G µ . On the other hand, the free Green function G 0 defined by where d i is the dressed propagator of particle i, is not antisymmetric in its particle labels; thus, for identical fermions, G 0 is not equal to the fully disconnected part of G (which we shall denote by G d ). Indeed, it can be easily shown [6] that to obtain G d , one needs only to antisymmetrize G 0 according to the equation where the sum is over all permutations P of either the initial or final state particle labels, and is understood to include a factor (−1) P = +1 or −1 depending on whether the permutation is even or odd, respectively. In Eq. (6) we use a symbolic notation where integers represent the momenta and all quantum numbers of the corresponding particles, with primes distinguishing the final states. To specify permutation sums over just the initial state momentum labels, we use the letter R (right); similarly, L (left) represents sums over permutations of just the final state momentum labels. The symbol P will be used only when it makes no difference which sum R or L, is taken. Quantities antisymmetrized in one of these ways will be indicated by the appropriate superscript. Thus, for example, if A is a quantity depending on three initial and three final particle labels, then with similar expressions holding for quantities having any number of identical legs. In general we write: Defining the kernel K to be the set of all possible three-particle irreducible Feynman diagrams for the 3 → 3 process, we may write the Green function G as [6] where the 1/3! factor reflects the fact that both G and K are fully antisymmetric in their particle labels. We write the disconnected part of K, indicated by subscript d, in terms of the identical particle two-body potential v: where δ(1 ′ , 1) represents the momentum conserving Dirac δ function (2π) 4 δ 4 (p ′ 1 − p 1 ), while L c and R c indicate that the sum is taken over cyclic permutations of the left labels (1 ′ 2 ′ 3 ′ ) and right labels (123), respectively (note that the sum is restricted to cyclic permutations because the potential v is already antisymmetric in its labels [6]). Defining where (ijk) is a cyclic permutation of (123), we have that where it makes no difference over which labels, left or right, the cyclic permutations are taken. Unlike the V i of the distinguishable particle case [see Eq.(61) of I], the one here consists of a two-body potential v that is antisymmetric under the interchange of its initial or final state labels. Denoting the connected part of the kernel by K c , we define the 3 → 3 potential V by Although V is not fully antisymmetric, it does have the useful symmetry property where P ij is the operator that exchanges the i'th and j'th momentum, spin, and isospin labels. Since Eq. (12) can be written as Formally, Eq. (19) differs from the equivalent relation for distinguishable particles [Eq.(59) of I] only in the explicit antisymmetrization of the inhomogeneous term. We therefore proceed as for the distinguishable particle case and gauge Eq. (19) directly: Before solving this equation for G µ it is useful to note that Indeed the combination G P 0 G −1 0 plays the role of the antisymmetrization operator since Note that there is no factor δ(3 ′ , 3) on the right side of Eq. (22) because an overall momentum conservation delta-function has been removed from our expressions. We can therefore write Eq. (20) as where the inclusion of G P 0 G −1 0 in the last term is compensated exactly by 1/6 since V satisfies the symmetry property of Eq. (17) and G is already fully antisymmetric. From Eq. (19) it follows that and using this we obtain In the last equation, we may replace the 6G 0 by G P 0 because of the antisymmetry of the G outside the square bracket, and using Eq. (19) we finally get that is the electromagnetic vertex function for three identical particles. The extra factor of 1/6 compared with the result for distinguishable particles reflects the fact that here Γ µ is to be sandwiched between fully antisymmetric functions. Neglecting three-body forces, Eq. (16) implies that with an extra factor of 1/2 compared with the distinguishable particle case. Writing Eq. (14) in the shorthand notation where v i denotes v(j ′ k ′ , jk), we may gauge this equation to obtain where Γ µ i is the one-particle electromagnetic vertex function defined by the equation and where we have used the fact that (d −1 i ) µ = −Γ µ i [1]. Similarly gauging Eq. (5) we find that Using these results in Eq. (28) we can express the electromagnetic vertex function as All electromagnetic transition currents of three identical particles can be obtained from Eq. (27) by taking appropriate residues at two-and three-body bound-state poles of G. If G admits a three-body bound state, it can be shown that where P is the total momentum, M is the bound-state mass, and Ψ P is the three-particle bound-state wave function defined by Here |P is the eigenstate of the full Hamiltonian corresponding to the three-particle bound state with momentum P µ . The three-body bound-state current j µ is found by taking left-and right-residues of G µ at the three-body bound-state poles. By exposing such poles in the field theoretic expression of Eq. (2) one finds that To find j µ for a particular model, one can alternatively use Eq. (27) to take residues at the three-body bound-state poles. In this way one finds that Although this expression is formally identical to the bound-state current for distinguishable particles [Eq.(80) of I], here the vertex function is given by Eq. (35), and the wave function is that for identical particles. The normalisation condition for the wave function in the case of identical particles follows from the fact that After taking residues at the three-body bound-state poles of G one finds that Note that this result differs from the one for distinguishable particles where unity appears on the right hand side [Eq.(82) of I]. This is a consequence of our convention where the same expression, Eq. (37), is used to define the bound-state wave function for both identical and distinguishable particles (in the latter case, however, the fields obtain particle labels). We can repeat the above procedure and take residues of Eq. (27) at two-body bound-state poles of G, thereby obtaining electromagnetic transition currents involving two-body bound states. This procedure was carried out in detail for the distinguishable particle case in I. However, as discussed in I, the expressions obtained in this way explicitly involve potentials and gauged potentials, and consequently may not be very convenient for practical calculations.
Here we shall therefore forego any further discussion of this procedure, and instead go on to an alternative approach based on the Alt-Grassberger-Sandhas (AGS) equations [5] which lead, after gauging, to electromagnetic transition currents expressed in terms of t-matrices and gauged t-matrices.

III. AGS AMPLITUDES FOR IDENTICAL PARTICLES
The AGS equations have long provided a practical way to describe the scattering of three particles in Quantum Mechanics. Not only do they lead (after one iteration) to equations with a connected kernel, but they also have the feature of having the two-body inputs in terms of t-matrices rather than potentials. As we would also like to have these advantages in the case of three relativistic particles, we shall utilise four-dimensional versions of the AGS equations, which for distinguishable particles were given by Eqs.(118) of I.
Our goal here is to extend the discussion of Sec. III C of I to the case of three identical particles. That is, we would like to express the electromagnetic transition currents of all possible processes involving three identical particles in terms of AGS amplitudes and gauged AGS amplitudes. Although the handling of identical particles in the AGS formulation is well documented [4], as far as we know all previous discussions do this by antisymmetrizing the distinguishable particle case. As stated previously, such a procedure is inappropriate for the field theoretic approach undertaken here. In this section we shall therefore define AGS amplitudes for identical particles and relate them to the 3 → 3 Green function G in a way that is consistent with field theory. Moreover, we do this with the view of gauging our final expressions, a task left to the following sections.
Our starting point here shall be the Green function for three identical particles as given by Eq. (19). The natural way to introduce the AGS operators U ij for three identical particles is via the distinguishable particle case. Recalling that Eq. (19) differs from the one for distinguishable particles in that the inhomogeneous term G P 0 is explicitly antisymmetrized, we are led to introduce a new three-particle Green function G D defined by the equation where the inhomogeneous term G 0 is not antisymmetrized. By its structure, Eq. (42) looks like the equation for the Green function of three distinguishable particles [Eq.(59) of I] and therefore allows us to define the AGS operators in the standard way. Nevertheless, G D should not be identified with the distinguishable particle Green function as the V in Eq. (42) is defined in terms of antisymmetric potentials v [see Eqs. (14) and (16)] while the V for distinguishable particles is defined in terms of two-body potentials which are not antisymmetric. The fully antisymmetric Green function G can be obtained from G D simply by antisymmetrizing: Neglecting the three-body force V c , we now proceed by analogy with the distinguishable particles case and define the AGS operators U ij through the equation where G i satisfies the equation (note that the inhomogeneous term in the last equation is not antisymmetrized, in contrast to the G i used for identical particles in Ref. [6]). The factor of 1/2 in Eq. (45) originates from Eq. (16) where it is clear that 1/2V i is the disconnected potential to be identified with the distinguishable particle case. Taking this into account, the AGS equations for the operators U ij become where the T i satisfy the equation and are given in terms of the two-body t-matrices t i by Note that t i is shorthand for t(j ′ k ′ , jk) and is fully antisymmetric under the interchange of its initial or final particle labels. As discussed in I, for gauging purposes it is preferable to work in terms of AGS Green functionsŨ which now satisfy the equations By using the above equations one can show that In this way we obtain that By taking appropriate residues, either of these relations allow us to obtain the scattering amplitudes for all possible processes in the system of three identical particles. To be specific, let us choose Eq. (52) to discuss the taking of residues. As only the last term in Eq. (52) is connected, only this term need be considered for the purposes of extracting physical three-particle amplitudes. We therefore define By writing out this sum explicitly and making use of the symmetry properties ofŨ ik discussed in Appendix A, it is possible to rewrite Eq. (54) in terms of just one AGS-like Green functioñ U : A detailed derivation of Eq. (55) is presented in Appendix C. The full Green function can then be written as As discussed in Appendix C, there is a variety of ways to chooseŨ without affecting the value of G. One form that has previously been used in three-nucleon calculations [8] is withX obeying the equation [see Eq. (C4)] where P = P 12 P 31 + P 31 P 12 is the sum of two successive cyclic permutations. The choice for U specified by Eq. (57) is unsatisfactory for our purposes since the presence of a sum of two permutations in both the inhomogeneous term and kernel of Eq. (58) makes the gauging of this expression particularly cumbersome. Fortunately there is another way to chooseŨ that avoids these difficulties. As shown in Appendix C, we can takẽ where the AGS-like Green functionZ obeys the equatioñ with no permutation sums involved. This is the form forŨ that we shall use in the next section for the purposes of gauging. By displaying all momentum variables in Eq. (55) it is easy to see that the connected part of the Green function can be written directly in terms ofZ as For notational purposes, we shall refer to our three identical strongly interacting fermions as a "nucleons" (NNN) although the true identity of these particles is arbitrary. Similarly we refer to a two-body bound state as a "deuteron" (d) and a three-body bound state as " 3 H". This enables us to write the various reactions that can take place between any three identical particles in a familiar way.
Using this notation, we can obtain the amplitude for Nd → Nd, by follow the usual procedure of taking residues at the two-body bound-state poles of G. Indeed for identical particles it can be shown that if quantum field theory admits the existence of two-body bound states, then the Green function G(k 1 k 2 k 3 ; p 1 p 2 p 3 ) possesses poles with respect to any of the variables (k i + k j ) 2 or (p i + p j ) 2 . To be definite, consider the variables (k 2 + k 3 ) 2 and (p 2 + p 3 ) 2 for the three-particle system. In the vicinity of the corresponding two-body poles, only the connected part of G contributes and we have that where K 1 = k 2 + k 3 , P 1 = p 2 + p 3 , ψ K 1 is the deuteron wave function [defined analogously to Eq. (37)], m is the mass of the deuteron, and T dd (k 1 K 1 ; p 1 P 1 ) is the physical scattering amplitude for N(p 1 ) + d(P 1 ) → N(k 1 ) + d(K 1 ). The same scattering amplitude could be picked out in the other channels, for example where T dd (k 1 K 1 ; p 3 P 3 ) depends on variables p 3 , P 3 in just the same way as T dd (k 1 K 1 ; p 1 P 1 ) depends on p 1 , P 1 . On the other hand, G c is given by Eq. (55) which when written out explicitly reads where k ′ 2 + k ′ 3 = K 1 and p ′ 2 + p ′ 3 = P 1 . In Eq. (64) D 0 is the free two-body propagator, and the poles at (k 2 + k 3 ) 2 = m 2 and (p 2 + p 3 ) 2 = m 2 are contained in the two-body t-matrices t(k 2 k 3 ; k ′ 2 k ′ 3 ) and t(p ′ 2 p ′ 3 ; p 2 p 3 ), respectively. In particular, where φ andφ are two-body bound-state vertex functions defined by the equations Note that the two-body t-matrix satisfies the integral equation With particle i as spectator, this equation when multiplied by d −1 i gives Eq. (47). The full two-body Green function D is given by analogy with Eq. (12) as The normalisation condition for our two-body wave function is therefore given by a fact that follows from the same argument that led to Eq. (41) but adapted to the case of two identical particles. Comparing Eq. (62) with Eq. (64) in the vicinity of the two poles gives the Nd elastic scattering amplitude as This equation can be written symbolically as or in terms of deuteron wave functions as The amplitude for the breakup reaction Nd → NNN is found by taking the residue of the Green function G at the initial state deuteron pole. Choosing momentum variables as above, we may express the connected part of the Green function G c in the vicinity of the initial bound-state pole as where (p 2 + p 3 ) 2 → m 2 . This relation defines the breakup amplitude T 0d (k 1 k 2 k 3 ; p 1 P 1 ). Comparing this with the behaviour of Eq. (64) in the vicinity of the same pole we deduce that Written symbolically this gives Using Eq. (59) and Eq. (60) we may eliminate T 1 from the last equation to obtain

IV. GAUGING THE IDENTICAL PARTICLE AGS EQUATIONS
Having developed the necessary expressions describing three identical particles in the purely strong interaction sector, we are now ready to carry out the gauging procedure that will generate the coupling to an external electromagnetic field. We follow the same procedure as used for distinguishable particles in Sec. III C of I.

A. N d → N d transition current
The Nd → Nd electromagnetic transition current j µ dd describes, for example, the process Nd → γNd. To obtain the expression for j µ dd we write Eq. (71) as Gauging Eq. (77) givesT The Nd → Nd electromagnetic transition current is then given by In Eq. (79)φ µ 1 and φ µ 1 are the gauged two-body bound-state vertex functions which follow from the solution of the two-body problem for particles 2 and 3. Dropping the spectator particle label, the bound-state equation for φ is Gauging this equation and using Eq. (68) gives By using the fact that the previous equation can also be written as which is the identical particle version of Eq.(38) of I.
To determine j µ dd all that is left is to specify a practical expression forŨ µ . If we make the choiceŨ = −ZP 12 as in Eq. (59), thenŨ µ = −Z µ P 12 and the problem reduces to that of gauging Eq. (60) in order to obtainZ µ . Eq. (60) is a relatively simple equation that has only one type of disconnectedness in the kernel, and unlike the corresponding equation for the where it is to be understood that t = t 2 and D 0 = d 3 d 1 . Gauging this equation gives With the help of Eq. (85) we then obtaiñ Using and thereforeZ  The t-matrix T 0d for the breakup process Nd → NNN was given in Eq. (76). We follow the same procedure as given in Sec. III C.4 for distinguishable particles. Thus we do not gauge T 0d directly but instead introduce the Green function quantitỹ The electromagnetic current for Nd → γNNN is then given by GaugingT 0d , one obtains (90)

C. Three-nucleon bound-state current
For three identical nucleons the electromagnetic bound-state current j µ was expressed in terms of the two-nucleon potential v and gauged potential v µ by Eq. (35) and Eq. (39). This was achieved by expressing the seven-point function as G µ = GΓ µ G and then taking the leftand right-residues of G µ at the three-body bound-state poles. The connection with j µ follows from the general structure of G µ in the vicinity of these poles: In this subsection we shall determine an alternative expression for j µ that is given in terms of the two-nucleon t-matrix t and gauged t-matrix t µ . To do this, we again take left-and rightresidues of G µ at the three-body bound-state poles; however, this time we use an expression for G µ which is obtained by gauging Eq. (56) where G is written in terms of the AGS Green functionŨ . For the purpose of taking residues it is sufficient to gauge just the connected part of G. Thus, making the choiceŨ = −ZP 12 , we are led to the gauging of Eq. (61): As discussed in Appendix C.2,Z contains the three-nucleon bound-state pole with the pole structure being given by Eq. (C22). The bound-state poles of G µ c can therefore be revealed by simply using Eq. (87) to express the middle term of Eq. (92) (containingZ µ ) in terms of Z factors. It is seen that only this middle term contains both the left and right bound-state poles; thus, in the vicinity of these poles one can write the seven-point function as Now using the results of Appendix B, we have that Ψ K = Lc G 0 T 1 Ψ K 2 andΨ P = RcΨ P 1 T 2 G 0 . Comparing with Eq. (91) and using Ψ 2 = −P 12Ψ1 then gives the desired expression Using the bound-state equations for Ψ 1 andΨ 1 [Eq. (B11) and Eq. (B12)], it is easy to see that We may thus write Eq. (94) as whose form corresponds to the bound-state current given by us in Ref. [9] in the context of the spectator approach to the three-nucleon system. Eq. (96) may appear to be in a form where the first term on the right hand side (RHS) corresponds to the two-body interaction current and the second term corresponds to the onebody current. Yet this is not the case since t µ in fact contains both types of contribution. To see this explicitly we gauge Eq. (67) for t in this way obtaining The first term on the RHS corresponds to a one-body current while the second term gives the two-body interaction current. Thus the total one-body current contribution to the threenucleon bound-state current is Again using the bound-state equations for Ψ 1 andΨ 1 , we may write this result as

D. Current conservation
To prove current conservation for observables expressed in terms ofZ µ , it is useful to first deduce the WT identity forZ µ . To do this we write Eq. (87) as and then follow the procedure of I in Sec. III B.2. Thus we introduce the quantitieŝ where ijk represent a cyclic ordering of 123. This allows us to write the WT identities for the gauged two-body potential and gauged one-particle propagator in three-particle space in terms of commutators as where I i , I j , and I k are unit operators in the space of particles i, j, and k respectively. Using Eqs. (103), it is then easy to see that The first term on the RHS of Eq. (100) contracted with q µ thus gives while for the last term of Eq. (100) we first deduce that By usingZ which follows from Eq. (85), the previous equation reduces to The WT identity for the last term of Eq. (100) thus becomes To simplify this expression further we use Eq. (107) and the fact that [ê 2 ,t 2 ] = 0, to obtaiñ Substituting this result into Eq. (109), we obtain whereê =ê 1 +ê 2 +ê 3 . Combining Eq. (111) with Eq. (105) we finally obtain the WT forZ µ : The expression for the three-nucleon bound-state current j µ , Eq. (94), can be written as and on comparison with Eq. (100) is recognised to be the result of taking simultaneous residues ofZ µ at the initial and final bound-state poles. Taking such simultaneous residues of Eq. (112), the left hand side of this equation gives q µ j µ , while the right hand side gives zero sinceZ has only a single pole; in this way we obtain the current conservation equation for the bound-state current: q µ j µ = 0.
To prove that the Nd → Nd electromagnetic transition current of Eq. (80) is conserved, we write the WT identity for φ µ 1 andφ µ 1 [see Eqs.(153) of I] in three-particle space as and note that Eq. (112) implies whereŨ = −ZP 12 . Using the last three equations, we obtain from Eq. (80) that By shifting momentum arguments, theê 1 factors stop the cancellation of external d −1 1 terms with the neighbouring d 1 propagators contained inŨ . Thus for on mass shell nucleons d −1 1 = 0, and both terms on the RHS of Eq. (116) become zero.

E. Normalisation condition and charge conservation
The normalisation condition for the three-body bound-state wave function that was given in Eq. (41) follows almost immediately from the basic integral equation for G, Eq. (19). The expression obtained gives the normalisation condition in terms of the three-body potential V and therefore in terms of input two-body potentials. The disadvantage of these quantities has already been discussed. To obtain the normalisation condition without explicit reference to V we follow a similar procedure to that for G, but instead use the AGS Green functionZ. From Eq. (107) it follows thatZ Using the pole structure ofZ given in Eq. (C22) and then taking residues, one obtains which is the normalisation condition expressed in terms of the input two-body t-matrix and Faddeev components of the bound-state wave function. Analogous to the normalisation condition for the three-body bound-state wave function is a condition known as "charge conservation" for the three-body bound-state electromagnetic current. The field theoretic definition of the three-body bound-state current is given by Eq. (38). From this expression, the translational invariance of the electromagnetic current operator, and the fact that |P is an eigenstate of the charge operator, it is easy to show that where e is the charge of the three-body bound state. Eq. (119) constitutes the charge conservation condition and corresponds to the fact that the full charge of the three particles is being probed by the external electromagnetic probe. As such, it is an essential condition that needs to be satisfied by any model calculation. Within our model the expression for j µ is given by Eq. (96). Assuming the Ward identities for the one-and two-body input, it is a matter of simple algebra to prove that this expression does indeed satisfy the charge conservation condition of Eq. (119). Such a proof has already been given by us in Ref. [9] for the case of the three-nucleon bound-state current within the spectator approach. As there is a direct one to one correspondence between Eq. (96) and the bound-state current in the spectator approach [Eq. (25) of Ref. [9]], this proof need not be repeated here.

V. SUMMARY
We have applied the gauging of equations method, introduced in the preceding paper for distinguishable particles [1], to the integral equations describing the strong interactions of three identical relativistic particles. For simplicity of presentation, we restricted the discussion to identical particle systems like that of three quarks (qqq) or three nucleons (NNN) where there is no strong interaction coupling to two-particle channels. Once the strong interaction equations are specified, the gauging of equations method couples an external photon to all possible places in the strong interaction model, while at the same time preserving the proper identical particle symmetry of the original equations. In this way we have obtained gauge invariant expressions for the various electromagnetic transition currents of such identical three-particle systems.
Two essentially different integral equations were gauged. The first was the integral equation for the three-particle Green function G, Eq. (12). This equation has a disconnected kernel that is defined in terms of the two-body potential v. As a result, the electromagnetic transition currents that follow are themselves expressed in terms of v and the gauged potential v µ . Although formally correct, such expressions for the transition currents may not be the most useful for practical calculations. We have therefore considered an alternative formulation of the strong interaction problem in terms of four-dimensional versions of the well known AGS equations. Such integral equations have a kernel that is connected (after one iteration) and are expressed in terms of two-body t-matrices. For identical particles there are many ways to define the AGS amplitude, all giving the same three-body Green function G. We have introduced an AGS amplitude that satisfies a particularly simple AGS equation, Eq. (60), where the inhomogeneous term consists of a single unpermuted term. This equation is ideal for our gauging procedure. By contrast, the AGS equation used previously in three-nucleon calculations [8] is inconvenient for gauging purposes as both its inhomogeneous term and kernel consist of sums over two different permutations. After gauging, our AGS equation gives practical expressions in terms of the two-body t-matrix t and gauged t-matrix t µ , for all the electromagnetic transition currents of three identical particles. In this sense, the gauging of equations method has provided a unified description of the strong and electromagnetic interactions of the identical three-particle system.
Using the definition ofŨ ik given in Eq. (44), it is easy to show that For the case k = 1 we explicitly have that From Eqs. (17) and (42) it follows that G D commutes with all elements of the symmetry group S 3 : P ij G D P ij = G D (of course for G there is the stronger symmetry property P ij G = GP ij = −G). Now using the fact that the following two relations are easily deduced from Eqs. (A2): In a similar way one can deduce how any AGS Green function transforms under the simultaneous exchange of two corresponding left-and right-particle labels. The following is a complete list for the case k = 1: P 12Ũ11 P 12 =Ũ 22 P 12Ũ21 P 12 =Ũ 12 P 12Ũ31 P 12 =Ũ 32 P 23Ũ11 P 23 =Ũ 11 P 23Ũ21 P 23 =Ũ 31 P 23Ũ31 P 23 =Ũ 21 P 31Ũ11 P 31 =Ũ 33 P 31Ũ21 P 31 =Ũ 23 P 31Ũ31 P 31 =Ũ 13 .
The corresponding transformations for k = 2 and k = 3 can be written down by inspection.

APPENDIX B: THREE-BODY BOUND-STATE WAVE FUNCTION FOR IDENTICAL PARTICLES
The three-body Green function G is defined by Eq. (1). If the field theory admits a threebody bound state, then G's behaviour in the vicinity of the bound-state pole is given by Eq. (36) where the bound-state wave function Ψ P is defined in Eq. (37). These equations are true whether the three particles are identical or not.

APPENDIX C: THE GREEN FUNCTIONZ
In dealing with identical particles, only one AGS-like Green functionŨ is needed to describe all possible processes of the three-body system. The connected part of the three-body Green function, G c , is expressed in terms ofŨ according to Eq. (55). In this appendix we present a derivation of Eq. (55) and show how various forms forŨ can be specified. One of these involves the AGS-like Green functionZ which, because of its simplicity, is chosen for the purposes of gauging. The pole structure ofZ is also discussed.