arXiv:1107.5836v1 [quant-ph] 28 Jul 2011 sater fdrc neato ewe -1 between require interaction not direct does of and theory theory a field is quantum to approach alternative o eitdb ooi ed(ietinteractions). b (direct interactions field electromagnetic bosonic the a that by assumed mediated is not It . Dirac h eiainpeetdi 8 sseeet fqatmfil theor fe field state. quantum between vacuum of interactions unconventional elements uses direct an [8] of in terms presented derivation in The derivation recent more A a edsrbd otrso eododr yteDri Lagran Darwin the by order, second of terms to described, be may ouin fteBeteuto ierslsfrteeeg spectru energy 7]. the 6, for [5, results experiments give with agrement equation in Breit systems equ the Breit the of by solutions terms, order second including electromagnetically, oain omlto feetoyaisi em of terms in electrodynamics of formulation covariant nti ae ewr ihnteter rpsdb au 9 0.T 10]. [9, Barut by proposed theory the within work we paper this In ti elkonta nioae ytmof system isolated an that known well 11.10.Ef. is 03.65.Pm, It 03.50.-z, 03.30.+p, No. PCAS eiaino h ri qainfo tnadqatmfil theo field quantum standard from equation Breit the of derivation A tteqatmlvloecndsrb ytmo w spin-1 two of system a describe can one level quantum the At between eitdb ooi ed(ietitrcin) ihnth (semirela spin-1 betwee Within approximate interacting interactions the two interactions). derive the (direct we that formalism, field assumed Lagrangian bosonic is a It by mediated spinor. Dirac a by eiaino h ri qainfo Barut’s from equation Breit the of derivation A esuyBrtscvrateutosdsrbn h electr the describing equations covariant Barut’s study We N spin-1 / atce.I h oain omlto ahpril sd is particle each formulation covariant the In particles. 2 eateto hsc n Astronomy, and Physics of Department / particles. 2 nvriyo rts Columbia British of University ietinteractions direct acue,Cnd,VT1Z1 V6T Canada, Vancouver, oig .Louis-Martinez J. Domingo [email protected] Abstract 1 / N atce.Ec atcei ecie ya by described is particle Each particles. 2 atce neatn electromagnetically interacting particles iitc ri qainfor equation Breit tivistic) sfruain sn the using formulation, is h atce r not are particles the n mgei interactions omagnetic to 3 ] Perturbative 4]. [3, ation n eursemploying requires and y in[1],[2]. gian moswsgvni [8]. in given was rmions tenteprilsare particles the etween frltvsi bound relativistic of m eodqatzto.It quantization. second / atce interacting particles 2 ywsgvni [5]. in given was ry i hoyi an is theory his escribed Our goal in this paper is to derive the Breit equation directly from Barut’s covariant equations describing the electromagnetic interactions between two spin-1/2 particles. The Darwin Lagrangian [1],[2], describing a system of N particles interacting electro- magnetically, to terms of second order, can be written as:

2 1 2 1 4 1 eaeb LDarwin = −c ma + mava + 2 mava − 2 8c 2 rab Xa Xa Xa Xa Xb=6 a 1 eaeb + 2 ((~va~vb)+(~nab~va)(~nab~vb)) , (1) 4c rab Xa Xb6=a where c is the speed of light, ma and ea are the mass and electric charge of particle a (a = 1, 2, ..., N), ~va its velocity, and ~rab = ~ra − ~rb the relative position of particle a with ~rab respect to particle b. In (1), ~nab ≡ . rab The Darwin Lagrangian can be derived either from Faraday-Maxwell’s field theory of elec- trodynamics [2] or from the relativistic action-at-a-distance theory of Wheeler and Feynman [11, 12]. v4 To terms of fourth order ( c4 ), ignoring the radiation effects (which for Faraday-Maxwell electrodynamics appear at third order) the Lagrangian for an isolated system of N particles interacting electromagnetically was obtained in [13], [14]. At the quantum level a system of two spin-1/2 particles interacting electromagnetically, including second order terms, can be described by the Breit equation [3, 4]:

∂ ∂ ∂ 2 2 i~ + i~c~α1 + i~c~α2 − m1c β1 − m2c β2  ∂t ∂~r1 ∂~r2

e1e2 1 (~α1(~r1 − ~r2)) (~α2(~r1 − ~r2)) − 1 − (~α1~α2)+ Ψ(t, ~r1, ~r2)=0. (2) |~r1 − ~r2|  2  |~r1 − ~r2| |~r1 − ~r2|  where Ψ(t, ~r1, ~r2) is a two-body bilocal field defined as follows:

Ψ(t, ~r1, ~r2)= ψ1 (t, ~r1) ⊗ ψ2 (t, ~r2) . (3) and ~α and β are Dirac’s matrices[15], ⊗ is used to denote a tensor product, and ~ is Planck’s constant. Equation (2) is consistent with the Darwin Lagrangian (1) for N = 2 [3]. It was obtained by following the simple recipe [3, 16] of substituting the free terms in (1) by the left hand sides of the free for each particle, and by substituting the velocities ~va by c~αa in the interaction terms of (1). Let us consider a system of N distinguishable spin-1/2 particles interacting electro- magnetically. In Barut’s formulation each particle is described by a Dirac spinor ψa(x) (a = 1, 2, ..., N). It is assumed that the interactions between particles are not mediated by a bosonic field. The action functional for each individual particle can be written as[9, 10]:

2 (a) 4 µ ea 4 (a) µ S = d x i~ψ¯a(x)γ ∂µψa(x) − macψ¯a(x)ψa(x) − d xA (x)ψ¯a(x)γ ψa(x), (4) Z c Z µ  where,

(a) 4 2 A (x)= eb d yδ (x − y) ψ¯b(y)γµψb(y). (5) µ Z Xb6=a  2 µ µ ν ν The argument of the Dirac delta function in (5) is (x − y) = ηµν (x − y )(x − y ), µ where the metric tensor ηµν = diag(+1, −1, −1, −1). The Dirac matrices γ are defined as 0 ¯ † 0 usual[15]: γ = β, ~γ = β~α. In (4,5) ψa(x)= ψa(x)γ . The action functional for the whole system of N interacting distinguishable spin-1/2 particles can be given in the form[10]:

4 µ S = d x i~ψ¯a(x)γ ∂µψa(x) − macψ¯a(x)ψa(x) Z Xa  1 4 4 2 µ − eaeb d xd yδ (x − y) ψ¯a(x)γ ψa(x)ψ¯b(y)γµψb(y). (6) 2c Z Z Xa Xb6=a  The action (6) is invariant under Lorentz transformations. At the classical level, (6) corresponds to the action functional of Wheeler-Feynman action-at-a-distance theory of electrodynamics[17, 11, 18, 19]:

1 2 2 1 2 S = − mac dsa z˙ − eaeb dsadsbδ (za − zb) (z ˙az˙b) . (7) Z a 2c Z Z Xa  Xa Xb6=a 

µ In (7), sa = cτa, where τa is the proper time of particle a, z (sa) is its world line in µ a Minkowski spacetime, and the 4-vector velocityz ˙µ = dza . The Dirac delta function in (7) a dsa accounts for the interactions propagating at the speed of light forward and backward in time. The fully relativistic equations of motion corresponding to the action functional (7) admit exact circular solutions for any number of particles [20, 21]. From the action functional (6) the covariant equations for the spin-1/2 particles can be obtained as follows[10]:

µ ea 4 2 µ i~γ ∂µψa(x) − macψa(x) − eb d yδ (x − y) ψ¯b(y)γµψb(y)γ ψa(x)=0, (8) c Z Xb6=a 

µ ea 4 2 µ i~∂µψ¯a(x)γ + macψ¯a(x)+ eb d yδ (x − y) ψ¯b(y)γµψb(y)ψ¯a(x)γ =0. (9) c Z Xb6=a 

3 From (8, 9) it follows that:

µ ∂µ ψ¯a(x)γ ψa(x) =0, (a =1, ..., N). (10) These are the well known continuity equations[15]. We use the normalization conditions:

3 † d ~xψ (t, ~x)ψa(t, ~x)=1. (11) Z a In [9, 10] from (8, 9) an exact covariant equation for the two-body bilocal field:

|~x − ~y| Φ(t, ~x, ~y)= ψ (t, ~x) ⊗ ψ t − , ~y (12) 1 2  c  was obtained. In this paper we are interested in obtaining an approximate (semirelativistic) equation for the two-body bilocal field (3):

Ψ(t, ~x, ~y)= ψ1 (t, ~x) ⊗ ψ2 (t, ~y) . (13) We will now derive the approximate (semirelativistic) Breit equation for the two-body bilocal field (13) directly from Barut’s covariant equations describing the interactions be- tween two spin-1/2 particles. In order to obtain the Breit equation for Ψ(t, ~x, ~y) (13) in the case of electrodynamics, we return to the action functional (6) for N = 2:

4 ¯ µ ¯ S = d x i~ψ1(x)γ ∂µψ1(x) − m1cψ1(x)ψ1(x) Z  4 µ + d y i~ψ¯2(y)γ ∂µψ2(y) − m2cψ¯2(y)ψ2(y) Z  e1e2 4 4 2 µ − d xd yδ (x − y) ψ¯1(x)γ ψ1(x)ψ¯2(y)γµψ2(y). (14) c Z Z  In (14) we write the Dirac delta function: 1 δ (x − y)2 = δ(y0 − x0 + |~x − ~y|)+ δ(y0 − x0 −|~x − ~y|) . (15) 2|~x − ~y|   Therefore, we can write the third term in (14) as:

e1e2 4 4 2 µ − d xd yδ (x − y) ψ¯1(x)γ ψ1(x)ψ¯2(y)γµψ2(y)= c Z Z 3  e1e2 3 d ~y ¯ |~x − ~y| |~x − ~y| − dt d ~x ψ2 t − , ~y γµψ2 t − , ~x + 2 Z Z Z |~x − ~y|   c   c  |~x − ~y| |~x − ~y| ψ¯ t + , ~y γ ψ t + , ~x ψ¯ (t, ~x)γµψ (t, ~x) (16) 2  c  µ 2  c  1 1

4 Using the approximate expressions:

|~x − ~y| |~x − ~y| |~x − ~y| ∂ ψ¯ t − , ~y γ ψ t − , ~x ≈ ψ¯ (t, ~y)γ ψ (t, ~y) − ψ¯ (t, ~y)γ ψ (t, ~y) 2  c  µ 2  c  2 µ 2 c ∂t 2 µ 2  |~x − ~y|2 ∂2 + ψ¯ (t, ~y)γ ψ (t, ~y) (17) 2c2 ∂t2 2 µ 2 

|~x − ~y| |~x − ~y| |~x − ~y| ∂ ψ¯ t + , ~y γ ψ t + , ~x ≈ ψ¯ (t, ~y)γ ψ (t, ~y)+ ψ¯ (t, ~y)γ ψ (t, ~y) 2  c  µ 2  c  2 µ 2 c ∂t 2 µ 2  |~x − ~y|2 ∂2 + ψ¯ (t, ~y)γ ψ (t, ~y) (18) 2c2 ∂t2 2 µ 2  and substituting (17,18) in (16) we obtain:

e1e2 4 4 2 ¯ µ ¯ − d xd yδ (x − y) ψ1(x)γ ψ1(x)ψ2(y)γµψ2(y) ≈ c Z Z 3  3 d ~y µ −e1e2 dt d ~x ψ¯1(t, ~x)γ ψ1(t, ~x)ψ¯2(t, ~y)γµψ2(t, ~y) Z Z Z |~x − ~y| 2 e1e2 3 3 ¯ µ ∂ ¯ − dt d ~x d ~y|~x − ~y|ψ1(t, ~x)γ ψ1(t, ~x) ψ2(t, ~y)γµψ2(t, ~y) (19) 2c2 Z Z Z ∂t2  Integrating the second term in (19) by parts and recalling that (γ0)2 = I,γ0~γ = ~α we find:

e1e2 4 4 2 µ − d xd yδ (x − y) ψ¯1(x)γ ψ1(x)ψ¯2(y)γµψ2(y) ≈ c Z Z 3  3 d ~y † † −e1e2 dt d ~x ψ (t, ~x)ψ1(t, ~x)ψ (t, ~y)ψ2(t, ~y) Z Z Z |~x − ~y| 1 2 3 3 d ~y † † +e1e2 dt d ~x ψ (t, ~x)~αψ1(t, ~x)ψ (t, ~y)~αψ2(t, ~y) Z Z Z |~x − ~y| 1 2

e1e2 3 3 ∂ † ∂ † + 2 dt d ~x d ~y|~x − ~y| ψ1(t, ~x)ψ1(t, ~x) ψ2(t, ~y)ψ2(t, ~y) 2c Z Z Z ∂t   ∂t   e1e2 3 3 ∂ † ∂ † − 2 dt d ~x d ~y|~x − ~y| ψ1(t, ~x)~αψ1(t, ~x) ψ2(t, ~y)~αψ2(t, ~y) . (20) 2c Z Z Z ∂t   ∂t   Using the continuity equations (10), the third term in (20) can be rewritten as:

e1e2 3 3 ∂ † ∂ † 2 dt d ~x d ~y|~x − ~y| ψ1(t, ~x)ψ1(t, ~x) ψ2(t, ~y)ψ2(t, ~y) 2c Z Z Z ∂t   ∂t   e1e2 3 3 ∂ † ∂ † ≈ dt d ~x d ~y|~x − ~y| ψ1(t, ~x)~αψ1(t, ~x) ψ2(t, ~y)~αψ2(t, ~y) . (21) 2 Z Z Z ∂~x   ∂~y   5 Integrating by parts and taking into account that ∂ ~x − ~y (|~x − ~y|)= − (22) ∂~y |~x − ~y| we obtain:

e1e2 3 3 ∂ † ∂ † 2 dt d ~x d ~y|~x − ~y| ψ1(t, ~x)ψ1(t, ~x) ψ2(t, ~y)ψ2(t, ~y) 2c Z Z Z ∂t   ∂t   e1e2 3 3 ∂ † † (~α(~x − ~y)) ≈ dt d ~x d ~y ψ1(t, ~x)~αψ1(t, ~x) ψ2(t, ~y) ψ2(t, ~y). (23) 2 Z Z Z ∂~x   |~x − ~y| Performing a second integration by parts and taking into account that:

∂ 1 ~x − ~y = − , (24) ∂~x |~x − ~y| |~x − ~y|3 we obtain:

e1e2 3 3 ∂ † ∂ † 2 dt d ~x d ~y|~x − ~y| ψ1(t, ~x)ψ1(t, ~x) ψ2(t, ~y)ψ2(t, ~y) ≈ 2c Z Z Z ∂t   ∂t   3 e1e2 3 d ~y † † − dt d ~x ψ (t, ~x)~αψ1(t, ~x)ψ (t, ~y)~αψ2(t, ~y) 2 Z Z Z |~x − ~y| 1 2 3 e1e2 3 d ~y † (~α(~x − ~y)) † (~α(~x − ~y)) + dt d ~x ψ (t, ~x) ψ1(t, ~x)ψ (t, ~y) ψ2(t, ~y).(25) 2 Z Z Z |~x − ~y| 1 |~x − ~y| 2 |~x − ~y|

Substituting (25) into (20) we finally obtain:

e1e2 4 4 2 µ − d xd yδ (x − y) ψ¯1(x)γ ψ1(x)ψ¯2(y)γµψ2(y) ≈ c Z Z 3  3 d ~y † † −e1e2 dt d ~x ψ (t, ~x)ψ1(t, ~x)ψ (t, ~y)ψ2(t, ~y) Z Z Z |~x − ~y| 1 2 3 e1e2 3 d ~y † † + dt d ~x ψ (t, ~x)~αψ1(t, ~x)ψ (t, ~y)~αψ2(t, ~y) 2 Z Z Z |~x − ~y| 1 2 3 e1e2 3 d ~y † (~α(~x − ~y)) † (~α(~x − ~y)) + dt d ~x ψ (t, ~x) ψ1(t, ~x)ψ (t, ~y) ψ2(t, ~y) 2 Z Z Z |~x − ~y| 1 |~x − ~y| 2 |~x − ~y|

e1e2 3 3 ∂ † ∂ † − 2 dt d ~x d ~y|~x − ~y| ψ1(t, ~x)~αψ1(t, ~x) ψ2(t, ~y)~αψ2(t, ~y) . (26) 2c Z Z Z ∂t   ∂t   If we assume that the last term in (26) can be neglected, then, substituting (26) into (14) we obtain the approximate action functional for two spin-1/2 particles in the form:

6 S ≈ 3 3 † † ∂ ∂ 2 dt d ~x d ~yψ (t, ~y)ψ2(t, ~y)ψ (t, ~x) i~ ψ1(t, ~x)+ i~c~α ψ1(t, ~x) − m1c βψ1(t, ~x) Z Z Z 2 1  ∂t ∂~x 

3 3 † † ∂ ∂ 2 + dt d ~x d ~yψ (t, ~x)ψ1(t, ~x)ψ (t, ~y) i~ ψ2(t, ~y)+ i~c~α ψ2(t, ~y) − m2c βψ2(t, ~y) Z Z Z 1 2  ∂t ∂~y  3 3 d ~y † † −e1e2 dt d ~x ψ (t, ~x)ψ1(t, ~x)ψ (t, ~y)ψ2(t, ~y) Z Z Z |~x − ~y| 1 2 3 e1e2 3 d ~y † † + dt d ~x ψ (t, ~x)~αψ1(t, ~x)ψ (t, ~y)~αψ2(t, ~y) 2 Z Z Z |~x − ~y| 1 2 3 e1e2 3 d ~y † (~α(~x − ~y)) † (~α(~x − ~y)) + dt d ~x ψ (t, ~x) ψ1(t, ~x)ψ (t, ~y) ψ2(t, ~y). (27) 2 Z Z Z |~x − ~y| 1 |~x − ~y| 2 |~x − ~y| Following [9, 10], we have multiplied the free terms in the action functional by the normalization conditions (11). This can of course be done without altering these terms (it is equivalent to multiplying each term by one). In terms of the two-body bilocal spinor (13) the action (27) can simply be written as:

3 3 † ∂ ∂ ∂ 2 2 S ≈ dt d ~x d ~yΨ (t, ~x, ~y) i~ + i~c~α1 + i~c~α2 − m1c β1 − m2c β2 Z Z Z  ∂t ∂~x ∂~y e e 1 (~α (~x − ~y)) (~α (~x − ~y)) − 1 2 1 − (~α ~α )+ 1 2 Ψ(t, ~x, ~y) (28) |~x − ~y|  2  1 2 |~x − ~y| |~x − ~y|  From this action functional, one can immediately obtain Breit’s equation for electrody- namics in the form:

∂ ∂ ∂ i~ + i~c~α + i~c~α − m c2β − m c2β  ∂t 1 ∂~x 2 ∂~y 1 1 2 2 e e 1 (~α (~x − ~y)) (~α (~x − ~y)) − 1 2 1 − (~α ~α )+ 1 2 Ψ(t, ~x, ~y)=0. (29) |~x − ~y|  2  1 2 |~x − ~y| |~x − ~y|  which coincides with (2). We have presented a very simple derivation of the Breit equation from Barut’s formulation of electrodynamics in terms of direct interactions between fermions. In this formulation the describing the spin-1/2 particles are not operators. This is reflected at every step of our derivation, which proceeds completely within the realm of the Lagrangian formalism of relativistic wave equations. Since it is known that the Breit equation gives results, up to terms of order α4, for the energy spectrum of relativistic bound systems (such as ) in agreement with experiments[6, 7], our proof that the Breit equation follows from Barut’s formulation of electrodynamics (in terms of direct interactions) lends further support to Barut’s approach.

7 References

[1] C.G. Darwin, Phil. Mag. 39, 537 (1920). [2] L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields (4th revised English edition), Butterworth- Heinenann, Oxford (1996). [3] G. Breit, Phys. Rev. 34, 553 (1929). [4] P.A.M. Dirac, R. Peierls and M.H.L. Pryce, Proc. Cambridge Phil. Soc. 35, 186 (1939). [5] H.A. Bethe and E.E. Salpeter, of One and Two Atoms (Springer, Berlin, 1957). [6] W. Krolikowski, Acta Phys. Pol. B12, 891 (1981). [7] V.I. Fushchich and A.G. Nikitin, Symmetries of Equations of Quantum Mechanics (Allerton Press, 1994). [8] A. Duviryak and J.W. Darewych, CEJP, 3(4), 467 (2005). [9] A.O. Barut, Lecture Notes in Physics, Vol. 180, p. 332 (Springer-Verlag, Berlin, 1983). [10] A.O. Barut, J. Math. Phys. 32, 1091 (1991). [11] J.L. Anderson, Principles of Relativity Physics (Academic Press, New York, 1967). [12] D.J. Louis-Martinez, Phys. Lett. A364, 93 (2007). [13] V.N. Golubenkov and Ya. A. Smorodinskii, Zh. Eksp. Teor. Fiz. 31, 330 (1956) [Sov. Phys. JETP 4, 442 (1957)]. [14] T. Ohta and T. Kimura, J. Math. Phys. 34, 4655 (1993). [15] J.D. Bjorken and S.D. Drell, Relativistic Quantum Mechanics (McGraw-Hill, New York, 1964). [16] P. Van Alstine and H. Crater, Found. Phys. 27, 67 (1997). [17] J. A. Wheeler and R. P. Feynman, Rev. Mod. Phys. 17, 157 (1945); 21, 425 (1949). [18] F. Hoyle and J.V. Narlikar, Lectures on Cosmology and Action at a Distance Electrody- namics, (World Scientific, 1996). [19] A. O. Barut, Electrodynamics and Classical Theory of Fields and Particles, (Dover Publications Inc, New York, 1980). [20] A. Schild, Phys. Rev. 131, 2762 (1963). [21] D.J. Louis-Martinez, Phys. Lett. A320, 103 (2003).

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