Relativistic Quantum Coulomb Law Yury M. Zinoviev∗ Steklov Mathematical Institute, Gubkin St. 8, 119991, Moscow, Russia, e - mail: [email protected] 8 0 0 2 Abstract. Therelativisticquantummechanics equationsfortheelectromagnetic interaction n are proposed. a J 8 1 Introduction ] h t - The celestial mechanics is based on the gravity law discovered by Newton (1687). Cavendish p e (1773)proved by experiment that the forceof interaction between theelectric charged bodies h is inversely proportional to the square of distance. This discovery was left unpublished and [ later was repeated by de Coulomb (1785). The electrodynamics equations were formulated 1 v by Maxwell (1873). The analysis of these equations led Lorentz (1904), Poincar´e (1905, 2 1906), Einstein (1905) and Minkowski (1908) to the creation of the theory of relativity. 8 Due to the paper [1] the Maxwell equations are completely defined by the relativistic 1 1 Coulomblaw. TherelativisticCoulomblawequationsfortwochargedbodiesaresolvedinthe . 1 paper[2]forthecasewhenonebodymovesfreely. Theresults[2]maybeappliedtothestudy 0 of the hydrogen atom. The light electron moves around the heavy proton. The heavy proton 8 0 moves freely. The hydrogen spectrum is discrete and does not correspond to the results of : v the paper [2]. In order to study this problem we need to apply the quantum electrodynamics. i The following citation from the book ([3], Chapter 4) describes the situation in the quantum X electrodynamics: r a ”But there is one additional problem that is characteristic of the theory of quantum electrodynamics itself, which tooktwenty years to overcome. It has to do with idealelectrons and photons and the numbers n and j. ”If electrons were ideal, and went from point to point in space - time only by the direct path, then there would be no problem: n would be the mass of an electron (which we can determine by observation), and j would simply be its ”charge” (the amplitude for the electron to couple with a photon). It can also be determined by experiment. ”But no such ideal electron exist. The mass we observed in the laboratory is that of a real electron, which emits and absorbs its own photons from time to time, and therefore depends on the amplitude for coupling, j. And the charge we observe is between a real electron and a real photon - which can form an electron - positron pair from time to time - and therefore depends on E(A to B), which involves n. Since the mass and charge of an ∗ This work was supported in part by the Russian Foundation for Basic Research (Grant No. 07 - 01 - 00144) and Scientific Schools 672.2006.1. 1 electron are affected by these and all other alternatives, the experimentally measured mass, m, and experimentally measured charge, e, of the electron are different from the numbers we use in our calculations, n and j. ”If there were a definite mathematical connection between n and j on the one hand, and m and e on the other, there would still be no problem: we would simply calculate what values of n and j we need to start with in order to end up with the observed values, m and e. (If our calculations didn’t agree with m and e, we would jiggle the original n and j around until they did.) ”Let’s see how we actually calculate m. We write a series of terms that is something like the series we saw for the magnetic moment of the electron: the first term has no couplings - just E(A to B) - and represents an ideal electron going directly from point to point in space - time. The second term has two couplings and represents a photon being emitted and absorbed. Then come terms with four, six, and eight couplings, and so on. ”When calculating terms with couplings, we must consider (as always) all the possible points where couplings can occur, right down to cases where the two coupling points are on top of each other - with zero distance between them. The problem is, when we try to calculate all the way down to zero distance, the equation blows up in our face and gives meaningless answers - things like infinity. This caused a lot of trouble when the theory of quantum electrodynamics first came out. People were getting infinity for every problem they tried to calculate! (One should be able to go down to zero distance in order to be mathematically consistent, but that’s there is no n or j that makes any sense; that’s where the trouble is.) ”Well, instead of including all possible coupling points down to a distance of zero, if one stops the calculation when the distance between coupling points is very small - say, 10−30 centimeters, billions and billions of times smaller than anything observable in experiment (presently 10−16 centimeters) - then there are definite values for n and j that we can use so that the calculated mass comes our to match the m observed in experiments, and the calculated charge matches the observed charge, e. Now, here’s the catch: if somebody else comes andstops their calculation at a different distance - say, 10−40 centimeters - their values for n and j needed to get the same m and e come out different! ”Twenty years later, in 1949, Hans Bethe and Victor Weisskopf noticed something: if two people who stopped at different distances to determine n and j from the same m and e then calculated the answer to some other problem - each using the appropriate but different values for n and j - when all the arrows from all the terms were included, their answers to this other problem came out nearly the same! In fact, the closer to zero distance that the calculations for n and j were stopped, the better the final answers for the other problem would agree! Schwinger, Tomonaga, and I independently invented ways to make definite calculations to confirm that it is true (we got prizes for that). People could finally calculate with the theory of quantum electrodynamics! ”So it appears that the only thing that depend on the small distance between coupling pointsarethevaluesfornandj -theoretical numbers that are not directly observable anyway; everything else, which can be observed, seems not to be affected. ”The shell game that we play to find n and j is technically called ”renormalization”. But no matter how clever the word, it is what I would call a dippy process! Having to resort to such hocus - pocus has prevented us from proving that the theory of quantum electrodynamics ismathematicallyself -consistent. It’ssurprising thatthetheorystillhasn’t been proved self - consistent one way or the other by now; I suspect that renormalization is 2 not mathematically legitimate. What is certain is that we do not have a good mathematical way to describe the theory of quantum electrodynamics: such a bunch of words to describe the connection between n and j and m and e is not good mathematics.” This paper is devoted to the relativistic quantum mechanics. The equations for the electromagnetic interaction are proposed. 2 Lorentz group The theory of relativity has the mathematical foundation. It is possible to add and multiply the complex numbers. Let us consider the complex 2×2 - matrices A A A = 11 12 . (2.1) A A 21 22 ! The 2×2 - matrix ¯ ¯ A A A∗ = 11 21 (2.2) A¯ A¯ 12 22 ! is called Hermitian adjoint matrix. If A∗ = A, then the matrix (2.1) is Hermitian. Let us consider the basis of Hermitian 2×2 - matrices 1 0 0 1 0 −i 1 0 σ0 = ,σ1 = ,σ2 = ,σ3 = . (2.3) 0 1 1 0 i 0 0 −1 ! ! ! ! Any matrix (2.1) has the form 3 zµσµ (2.4) µ=0 X where z0,...,z3 are the complex numbers. It is possible to add and multiply the matrices (2.1). These matrices form the eight dimensional space of the matrices (2.4) which is an algebra. The following multiplication rules are valid σµσµ = σ0, σ0σµ = σµσ0 = σµ, µ = 0,...,3; 3 σk1σk2 = ǫk1k2k3iσk3, k ,k = 1,2,3, k 6= k (2.5) 1 2 1 2 kX3=1 where the antisymmetric tensor ǫk1k2k3 has the normalization ǫ123 = 1. Hence for the real numbers x0,...,x3 the matrices 3 x˜˜ = x0σ0 +i xkσk (2.6) k=1 X form analgebra. The matrix (2.6) is called quaternion. The quaternion algebra was invented by Hamilton (1843). The algebra of the matrices (2.6) is the non - commutative extension of the complex numbers field. The realnumbers arethe realdiagonal2×2 - matrices x0σ0. The pure imaginary numbers are the real antisymmetric 2×2 - matrices ix2σ2. The determinant of a quaternion (2.6) 3 detx˜˜ = (xµ)2 (2.7) µ=0 X 3 is called the Euclidean metric. Any matrix (2.1) satisfying the equation A∗ = (detA)A−1 (2.8) has the form (2.6) and is a quaternion. The equation (2.8) implies also that the matrices (2.6) form an algebra. The matrices (2.6) with determinant equal to 1 satisfy the equations A∗A = σ0, detA = 1 and form the group SU(2). The matrices (2.1) with determinant equal to 1 form the group SL(2,C). The group SU(2) is the maximal compact subgroup of the group SL(2,C). We identify a vector xµ, µ = 0,...,3, from the four dimensional Euclidean space and a quaternion (2.6). The unit sphere in the four dimensional Euclidean space is isomorphic to the group SU(2). A rotation of the four dimensional Euclidean space is a matrix product R(A,B)(x˜˜) = Ax˜˜B (2.9) where the matrices A,B ∈ SU(2). The metric (2.7) is not changed under any rotation (2.9). In the quaternion (2.6) the coordinate x0 is a real number and the coordinates ixk, k = 1,2,3, are the pure imaginary numbers. Let us consider the four dimensional space of Hermitian matrices 3 x˜ = xµσµ (2.10) µ=0 X where the coordinates x0,...,x3 are the real numbers. It is possible to add x˜ + y˜ and to multiply 1/2(x˜y˜+y˜x˜) these matrices. The determinant of a matrix (2.10) is 3 detx˜ = (x0)2 −|x|2 = (x,x) = η xµxν. (2.11) µν µ,ν=0 X Here the diagonal matrix η = ηµν, η = −η = −η = −η = 1. The Minkowski metric µν 00 11 22 33 (2.11) was introduced by Poincar´e (1906). For the matrices A,B ∈ SL(2,C) we consider a transformation L(A,B)(x˜) = Ax˜B (2.12) similar to a rotation (2.9). A transformation (2.12) does not change a determinant (2.11). A matrix (2.12) is Hermitian if B∗x˜A∗ = Ax˜B. (2.13) A matrix A is invertible since its determinant is equal to 1. Hence the relation (2.13) implies A−1B∗x˜ = x˜B(A∗)−1. (2.14) By inserting the matrix x˜ = σ0 into the equality (2.14) we have A−1B∗ = B(A∗)−1. The Hermitian matrix B(A∗)−1 commutes with any Hermitian matrix. Hence B(A∗)−1 = λσ0 (2.15) where λ is a real number. The number λ = ±1 since the determinants of the matrices A,B are equal to 1. It is easy to verify for any matrix (2.1) 2 tr(L(A,λA∗)(σ0)) = λ |A |2. (2.16) ij i,j=1 X The number (2.16) is the double coefficient at the matrix σ0 in the decomposition (2.10) for the matrix L(A,λA∗)(σ0). For λ = 1 the number (2.16) is positive and the time direction is not changed. The group of the transformations L(A,A∗), A ∈ SL(2,C), is called the Lorentz group. 4 3 Relativistic quantum laws For a complex 2×2 - matrix (2.1) we define the following 2×2 - matrices ¯ ¯ A A A A AT = 11 21 , A¯ = 11 12 . (3.1) A A A¯ A¯ 12 22 ! 21 22 ! Let us describe the irreducible representations of the group SU(2). We consider the half - integers l ∈ 1/2Z , i.e. l = 0,1/2,1,3/2,.... We define the representation of the group + SU(2) on the space of the polynomials with degree less than or equal to 2l A z +A T (A)φ(z) = (A z +A )2lφ 11 21 . (3.2) l 12 22 A z +A (cid:18) 12 22(cid:19) We consider a half - integer n = −l,−l+1,...,l−1,l and choose the polynomial basis ψ (z) = ((l−n)!(l+n)!)−1/2zl−n. (3.3) n The definitions (3.2), (3.3) imply l T (A)ψ (z) = ψ (z)tl (A), (3.4) l n m mn m=−l X tl (A) = ((l−m)!(l+m)!(l−n)!(l+n)!)1/2 × mn ∞ Al−m−jAj Am−n+jAl+n−j 11 12 21 22 (3.5) Γ(j +1)Γ(l−m−j +1)Γ(m−n+j +1)Γ(l+n−j +1) j=−∞ X where Γ(z) is the gamma - function. The function (Γ(z))−1 equals zero for z = 0,−1,−2,.... Therefore the series (3.5) is a polynomial. The relation (3.2) defines a representation of the group SU(2). Thus the polynomial (3.5) defines a representation of the group SU(2) l tl (AB) = tl (A)tl (B). (3.6) mn mk kn k=−l X This (2l+1) - dimensional representation is irreducible ([4], Chapter III, Section 2.3). The relations (3.5), (3.6) have an analytic continuation to all matrices (2.1). By making the change j → j+n−m of the summation variable in the equality (3.5) we have tl (A) = tl (AT). (3.7) mn nm The polynomial (3.5) is homogeneous of the matrix elements (2.1). Its degree is 2l. The sum (2.5) contains the only non - zero term. Since the definition (3.5) implies tl (σ0) = δ , (3.8) mn mn 5 the relations (2.5), (3.6) imply l l˙ 3 ∂ 3 ∂ tl (σν)tl˙ (σν) −i ηννtl (σν)tl˙ (σν) −i = mp m˙p˙ ∂xν!! pn p˙n˙ ∂xν!! pX=−lp˙X=−l˙ νX=0 νX=0 3 ∂2 tl (iσk3)tl˙ (−iσk3)((ǫk1k2k3)2l+2l˙+(ǫk2k1k3)2l+2l˙) mn m˙n˙ ∂xk1∂xk2 1≤k1X<k2≤3kX3=1 3 ∂ 2 −δ δ (∂ ,∂ ), (∂ ,∂ ) = ηνν . (3.9) mn m˙n˙ x x x x ∂xν! ν=0 X ˙ For an odd integer 2l+2l the relation (3.9) has the form l l˙ 3 ∂ 3 ∂ tl (σν)tl˙ (σν) −i ηννtl (σν)tl˙ (σν) −i = mp m˙p˙ ∂xν!! pn p˙n˙ ∂xν!! pX=−lp˙X=−l˙ νX=0 νX=0 −δ δ (∂ ,∂ ). (3.10) mn m˙n˙ x x ˙ By making use of the relation (3.10) for the oddinteger 2l+2l the Lorentz invariant equation (−(∂ ,∂ )−µ2)φ (x) = 0, m = −l,−l+1,...,l−1,l, m˙ = −l˙,−l˙+1,...,l˙−1,l˙ (3.11) x x mm˙ may be rewritten as the system of the linear equations 4l+2 2l˙+1 3 ∂ ηννtl (σν)tl˙ (σν) −i ψ (x)+ψ (x) = 0, m−l−1,n−3l−2 m˙−l˙−1,n˙−l˙−1 ∂xν! nn˙ mm˙ n=2l+2n˙=1ν=0 X X X ˙ m = 1,...,2l+1, m˙ = 1,...,2l+1; 2l+1 2l˙+1 3 ∂ tl (σν)tl˙ (σν) −i ψ (x)+µ2ψ (x) = 0, m−3l−2,n−l−1 m˙−l˙−1,n˙−l˙−1 ∂xν! nn˙ mm˙ n=1n˙=1ν=0 X X X ˙ m = 2l+2,...,4l+2, m˙ = 1,...,2l+1.(3.12) ˙ ˙ Let us define the ((4l+2)(2l+1))×((4l+2)(2l+1)) - matrices (α (µ2)) = µ2δ δ , m,n = 1,...,2l+1, m˙ ,n˙ = 1,...,2l˙+1; l,l˙ mm˙,nn˙ mn m˙n˙ (α (µ2)) = δ δ , m,n = 2l+2,...,4l+2, m˙ ,n˙ = 1,...,2l˙+1; l,l˙ mm˙,nn˙ mn m˙n˙ (β (µ2)) = δ δ , m,n = 1,...,2l+1, m˙ ,n˙ = 1,...,2l˙+1; l,l˙ mm˙,nn˙ mn m˙n˙ (β (µ2)) = µ2δ δ , m,n = 2l+2,...,4l+2, m˙ ,n˙ = 1,...,2l˙+1; l,l˙ mm˙,nn˙ mn m˙n˙ (γν (σ0)) = ηννtl (σν)tl˙ (σν), l,l˙ mm˙,nn˙ m−l−1,n−3l−2 m˙−l˙−1,n˙−l˙−1 ˙ m = 1,...,2l+1, n = 2l+2,...,4l+2, m˙ ,n˙ = 1,...,2l+1, ν = 0,...,3; (γν (σ0)) = tl (σν)tl˙ (σν), l,l˙ mm˙,nn˙ m−3l−2,n−l−1 m˙−l˙−1,n˙−l˙−1 ˙ m = 2l+2,...,4l+2, n = 1,...,2l+1, m˙ ,n˙ = 1,...,2l+1, ν = 0,...,3. (3.13) ˙ The other matrix elements are equal to zero. If an integer 2l + 2l is odd, then an integer ˙ ˙ (2l+1)(2l+1) is even and an integer (4l+2)(2l+1) has the form 4k where k is an integer. 6 By making use of the definitions (3.13) we can rewrite the equations (3.12) as 4l+2 2l˙+1 3 ∂ (γν (σ0)) −i +(β (µ2)) ψ (x) = 0. (3.14) l,l˙ mm˙,nn˙ ∂xν! l,l˙ mm˙,nn˙! nn˙ n=1n˙=1 ν=0 X X X The definitions (3.13) imply α (µ)β (µ2) = µβ (µ), α (µ)γν (σ0) = γν (σ0)β (µ). (3.15) l,l˙ l,l˙ l,l˙ l,l˙ l,l˙ l,l˙ l,l˙ In view of the relations (3.15) the action of the matrix α (µ) on the equation (3.14) yields l,l˙ 4l+2 2l˙+1 3 ∂ (γν (σ0)) −i ξ (x)+µξ (x) = 0, l,l˙ mm˙,nn˙ ∂xν! nn˙ mm˙ n=1n˙=1ν=0 X X X 4l+2 2l˙+1 ξ (x) = (β (µ)) ψ (x) = 0. (3.16) mm˙ l,l˙ mm˙,nn˙ nn˙ n=1n˙=1 X X For µ > 0 the transformation given by the second relation (3.16) is the isomorphism. The definition (3.5) implies 1 t2 (A) = A . (3.17) mn m+3,n+3 2 2 Due to the relations (3.13), (3.17) the equation (3.16) for l = 1, l˙ = 0 coincides with the 2 Dirac equation ([5], equation (1 - 41)). The relations (2.10), (2.12)define therepresentation ofthegroupSL(2,C) intheLorentz group 3 Λµ(A)xνσµ = L(A,A∗)(x˜) = Ax˜A∗. (3.18) ν µ,ν=0 X Let us define (4l+2)(2l˙+1) - dimensional representation of the group SL(2,C) (S (A)) = tl (A)tl˙ (A¯), l,l˙ mm˙,nn˙ m−l−1,n−l−1 m˙−l˙−1,n˙−l˙−1 ˙ m,n = 1,...,2l+1, m˙ ,n˙ = 1,...,2l+1; (S (A)) = tl ((A∗)−1)tl˙ ((AT)−1), l,l˙ mm˙,nn˙ m−3l−2,n−3l−2 m˙−l˙−1,n˙−l˙−1 ˙ m,n = 2l+2,...,4l+2, m˙ ,n˙ = 1,...,2l+1. (3.19) The other matrix elements are equal to zero. The definitions (3.13), (3.19) imply S (A)α (µ2)S (A−1) = α (µ2), S (A)β (µ2)S (A−1) = β (µ2). (3.20) l,l˙ l,l˙ l,l˙ l,l˙ l,l˙ l,l˙ l,l˙ l,l˙ ˙ Let the functions ψ (x), m = 1,...,4l+2, m˙ = 1,...,2l+1 be the solutions of the equation mm˙ (3.14). The relations (3.20) imply that the functions 4l+2 2l˙+1 3 ξ (x) = (S (A)) ψ Λµ(A−1)xν (3.21) mm˙ l,l˙ mm˙,nn˙ nn˙ ν ! n=1n˙=1 ν=0 X X X are the solutions of the equation 4l+2 2l˙+1 3 ∂ (γν (A)) −i +(β (µ2)) ξ (x) = 0, (3.22) l,l˙ mm˙,nn˙ ∂xν! l,l˙ mm˙,nn˙! nn˙ n=1n˙=1 ν=0 X X X 7 3 γµ(A) = Λµ(A)S (A)γν (σ0)S (A−1) (3.23) l,l˙ ν l,l˙ l,l˙ l,l˙ ν=0 X for any matrix A ∈ SL(2,C). The definition (3.23) implies 3 γµ(AB) = Λµ(A)S (A)γν (B)S (A−1) (3.24) l,l˙ ν l,l˙ l,l˙ l,l˙ ν=0 X for any matrices A,B ∈ SL(2,C). By changing the coordinate system we change the matrix γν (σ0) in the equation (3.14) for the matrix (3.23). The solutions of the equation (3.14) l,l˙ transform to the solutions (3.21) of the equation (3.22). It is valid for all half - integers l,l˙∈ 1/2Z . Due to ([5], relation (1 - 43)) + γµ (A) = γµ (σ0) (3.25) 1,0 1,0 2 2 for any matrix A ∈ SL(2,C). Hence the equation (3.14) for l = 1, l˙= 0 is covariant under 2 the group SL(2,C). In view of the definitions (3.13), (3.19), (3.23) the equation (3.22) is equivalent to the system of two equations 4l+2 2l˙+1 3 ∂ (γν (A)) −i ξ (x)+ξ (x) = 0, l,l˙ mm˙,nn˙ ∂xν! nn˙ mm˙ n=2l+2n˙=1ν=0 X X X ˙ m = 1,...,2l+1, m˙ = 1,...,2l+1; 2l+1 2l˙+1 3 ∂ (γν (A)) −i ξ (x)+µ2ξ (x) = 0, l,l˙ mm˙,nn˙ ∂xν! nn˙ mm˙ n=1n˙=1ν=0 X X X ˙ m = 2l+2,...,4l+2, m˙ = 1,...,2l+1. (3.26) The system of two equations (3.26) is equivalent to the equation 4l+2 2l˙+12l+12l˙+1 3 ∂ 3 ∂ (γν (A)) (γν (A)) ξ (x) l,l˙ mm˙,pp˙∂xν! l,l˙ pp˙,nn˙ ∂xν! nn˙ n=2l+2n˙=1p=1p˙=1 ν=0 ν=0 X X X X X X +µ2ξ (x) = 0, m = 2l+2,...,4l+2, m˙ = 1,...,2l˙+1. (3.27) mm˙ ˙ Let us prove that for an odd integer 2l+2l 2l+12l˙+1 3 ∂ 3 ∂ (γν (A)) (γν (A)) = (∂ ,∂ )δ δ , l,l˙ mm˙,pp˙∂xν! l,l˙ pp˙,nn˙ ∂xν! x x mn m˙n˙ p=1p˙=1 ν=0 ν=0 X X X X ˙ m,n = 2l+2,...,4l+2, m˙ ,n˙ = 1,...,2l+1. (3.28) We denote ∂ 3 ∂ = Λν(A) . (3.29) ∂yµ µ ∂xν ν=0 X The matrix Λν(A) belongs to the Lorentz group and the definition (3.29) implies µ 3 ∂ 2 ηµµ = (∂ ,∂ ). (3.30) ∂yµ! x x µ=0 X 8 Thedefinitions(3.13),(3.19),(3.23)andtherelations(3.10),(3.30)implytheequality(3.28). ˙ Therefore for an odd integer 2l+2l the equation (3.22) is equivalent to the equation (3.11). The relations (3.13) imply (α (µ2)β (µ2)) = µ2δ δ , m,n = 1,...,4l+2, m˙ ,n˙ = 1,...,2l˙+1. (3.31) l,l˙ l,l˙ mm˙,nn˙ mn m˙n˙ In view of the second relation (3.15) and the relations (3.20), (3.23), (3.31) the action of the matrix γ0 (σ0)α (µ2) on the equation (3.14) yields l,l˙ l,l˙ 4l+2 2l˙+1 3 ∂ { (γ0 (σ0)γν (σ0)β (µ2)) −i +µ2(γ0 (σ0)) }ψ (x) = 0. (3.32) l,l˙ l,l˙ l,l˙ mm˙,nn˙ ∂xν! l,l˙ mm˙,nn˙ nn˙ n=1n˙=1 ν=0 X X X The relations (2.5), (3.6), (3.8), (3.13) imply ((γ0 (σ0))2) = δ δ , m,n = 1,...,4l+2, m˙ ,n˙ = 1,...,2l˙+1; l,l˙ mm˙,nn˙ mn m˙n˙ (γ0 (σ0)γk (σ0)) = tl (σk)tl˙ (σk), l,l˙ l,l˙ mm˙,nn˙ m−l−1,n−l−1 m−l˙−1,n−l˙−1 ˙ m,n = 1,...,2l+1, m˙ ,n˙ = 1,...,2l+1, k = 1,2,3; (γ0 (σ0)γk (σ0)) = −tl (σk)tl˙ (σk), l,l˙ l,l˙ mm˙,nn˙ m−3l−2,n−3l−2 m−l˙−1,n−l˙−1 ˙ m,n = 2l+2,...,4l+2, m˙ ,n˙ = 1,...,2l+1, k = 1,2,3. (3.33) The othermatrix elements areequal tozero. The coefficients of thepolynomial(3.5)arereal. Hence in view of the relations (3.7), (3.13), (3.33) the matrices γ0 (σ0), γ0 (σ0)γν (σ0)β (µ2), l,l˙ l,l˙ l,l˙ l,l˙ ν = 0,...,3, are Hermitian. Due to the first relation (3.33) we have (γ0 (σ0))2β (µ2) = l,l˙ l,l˙ β (µ2). Let the functions ξ (x) have the form (3.21). Now the equation (3.32) implies l,l˙ mm˙ that the integral 4l+2 2l˙+1 4l+2 2l˙+1 3 2 d3x (β (µ2)) (S (A−1)) ξ Λλ(A)xν (3.34) l,l˙ mm˙,mm˙ l,l˙ mm˙,nn˙ nn˙ ν (cid:12) !(cid:12) Z mX=1m˙X=1 (cid:12)nX=1n˙X=1 νX=0 (cid:12) (cid:12) (cid:12) is independent of the variable x0 fo(cid:12)r x0 > 0. The integrand (3.34) is called the(cid:12) probability (cid:12) (cid:12) density of a solution of the equation (3.22). For A = σ0 the integrand (3.34) coincides with the usual probability density for the function (3.16). In the quantum mechanics the fixed probability density defines Hilbert space where any Hamiltonian acts. The integral (3.34) depends on the parameter µ2 in the equation (3.22). We do not expect the solutions of the equations with interaction to have the independent of time integrals similar to (3.34). This mathematical assumption does not seem physically reasonable. We deal with the asymptotic solutions of the equations with interaction in an experiment. We expect the solutions of the equations with interaction to coincide asymptotically with the products of the solutions of the equation (3.22). The probability density of the last solutions is given by the integrand (3.34). Let the functions ξ (x) be the solutions of the equation (3.22). Let us introduce the mm˙ distributions 1, x ≥ 0, f (x) = θ(x0)ξ (x), θ(x) = (3.35) mm˙ mm˙ (0, x < 0. The equation (3.22) implies 4l+2 2l˙+1 3 ∂ (γν (A)) −i +(β (µ2)) f (x) = l,l˙ mm˙,nn˙ ∂xν! l,l˙ mm˙,nn˙! nn˙ n=1n˙=1 ν=0 X X X −iδ(x0)f0 (+0,x), x = (x1,x2,x3) ∈ R3, (3.36) mm˙ 9 4l+2 2l˙+1 f0 (+0,x) = (γ0 (A)) f (+0,x). (3.37) mm˙ l,l˙ mm˙,nn˙ nn˙ n=1n˙=1 X X Let a support of a distribution e (x) ∈ S′(R4) lie in the closed upper light cone. µ2,...,µ2 1 n Let a distribution e (x) satisfy the equation µ2,...,µ2 1 n n (−(∂ ,∂ )−µ2) e (x) = δ(x). (3.38) x x i ! µ21,...,µ2n i=1 Y By changing the differential operator −(∂ ,∂ ) for the differential operator x x n (−(∂ ,∂ )−µ2) x x i i=1 Y in the proof of Lemma 3 from the paper [1] we obtain the uniqueness of the distribution e (x). Due to ([6], Section 30) µ2,...,µ2 1 n e (x) = −(2π)−1θ(x0)δ((x,x)), e (x) = (8π)−1θ(x0)θ((x,x)). (3.39) 0 0,0 The second definition (3.35) implies (∂ ,∂ )(θ(x0)θ((x,x))(x,x)n) = 4n(n+1)θ(x0)θ((x,x))(x,x)n−1, n = 1,2,.... (3.40) x x Due to the relations (3.39), (3.40) the distribution e (x) with n zeros has the form 0,...,0 e (x) = (−1)n(2π4n−1(n−2)!(n−1)!)−1θ(x0)θ((x,x))(x,x)n−2, n = 2,3,.... (3.41) 0,...,0 Let us prove n e (x) = lim (2π)−4 d4pexp{−i(p,x)} ((p0 +iǫ)2 −|p|2 −µ2)−1, µ21,...,µ2n ǫ→+0 j Z j=1 Y 3 (x,y) = x0y0 − xkyk. (3.42) k=1 X The integral (3.42) is the solution of the equation (3.38). By making the shift of the inte- gration path in the right - hand side of the equality (3.42) we obtain that the distribution (3.42) is equal to zero for x0 < 0. The distribution (3.42) is Lorentz invariant. Hence its support lies in the closed upper light cone. Now the uniqueness of the distribution (3.42) implies the equality (3.42). ˙ For an odd integer 2l+2l the relation 4l+22l˙+1 3 ∂ (γν (A)) −i +(β (µ2)) × l,l˙ mm˙,pp˙ ∂xν! l,l˙ mm˙,pp˙! p=1p˙=1 ν=0 X X X 3 ∂ (γν (A)) −i −(α (µ2)) = (−(∂ ,∂ )−µ2)δ δ , l,l˙ pp˙,nn˙ ∂xν! l,l˙ pp˙,nn˙! x x mn m˙n˙ ν=0 X ˙ m,n = 1,...,4l+2, m˙ ,n˙ = 1,...,2l+1, (3.43) 10