Quantum mechanics of the free Dirac electrons and Einstein photons, and the Cauchy process
aa r X i v : . [ qu a n t - ph ] O c t QUANTUM MECHANICS OF THE FREE DIRAC ELECTRONSAND EINSTEIN PHOTONS, AND THE CAUCHY PROCESS
A. A. Beilinson
Abstract.
Fundamental solutions for the free Dirac electron and Einstein photonequations in position coordinates are constructed as matrix valued functionals on thespace of bump functions. It is shown that these fundamental solutions are relatedby a unitary transform via the Cauchy distribution in imaginary time. We study theway the classical relativistic mechanics of the free particle comes from the quantummechanics of the free Dirac electron.
Introduction
We study fundamental solutions for the Dirac electron equations and the Maxwellequations for electromagnetic field without sources (the photon field) as functionalsof space variables, the time is viewed as a parameter.We first construct the fundamental solutions as analytic functionals in positioncoordinates, so in momentum coordinates these are functionals on bump functions;then, using a renormalization procedure, we construct the position coordinatespresentation of our fundamental solutions as functionals on bump functions.Specifically, we exploit the Foldy-Wouthuysen presentation of solutions (diago-nalized fundamental solutions) of the Dirac and Maxwell equations that reducesthem to a scalar transition probability of the Cauchy process in imaginary time,using both momentum and position coordinate presentations of this generalizedfunction.For a short exposition of main results see [15].
1. A particular case of the Dirac equation with m = 0 . Here the Dirac equation looks as (see e.g. [8])(1) γ ∂∂t + ( γ, ∇ ) = 0 . We use the system of units where the Planck constant ~ and the velocity of light c are equal to 1. Denote by D t ( x ) the fundamental solution of (1); its momentumcoordinate presentation ˜ D t ( p ) is then(2) ˜ D t ( p ) = exp( itγ ( γ, p )) . Key words and phrases. analytic functionals, the Foldy-Wouthuysen transform, regularizationof a functional, Parseval’s identity. Typeset by
AMS -TEX A. A. BEILINSON
Notice that the matrix γ ( γ, p ) in (2) is Hermitian, hence it can be diagonalized;it is easy to see that this can be done using a unitary (and Hermitian) operator˜ T ( p ) = ˜ T − ( p ) = γ √ (( γ, p e ) + I ) where p e is the unit norm vector for p , thus γ ( γ, p ) = ˜ T ( p ) γ ρ ˜ T ( p ) and(3) ˜ D t ( p ) = ˜ T ( p ) exp( itγ ρ ) ˜ T ( p ) , where ρ = p p + p + p .Therefore (3) provides the fundamental solution of the Dirac equation (1) inmomentum space in the Foldy-Wouthuysen variables (see [13])(4) ˜ D Ft ( p ) = exp( itγ ρ ) . Notice that the Foldy-Wouthuysen transform of solutions of the Dirac equation isan isomorphism.The operator in (4) is diagonal, hence it reduces to two scalar unitary operatorsthat are complex conjugate(5) ˜ C it ( p ) = exp( itρ ) , ˜ C it ( p ) = ˜ C − it ( p ) = exp( − itρ ) . Our problem is to write down the position coordinate presentation of these opera-tors.Recall that a probability density π (1+ x ) , whose Fourier transform is exp( −| p | ),was first studied by Cauchy; the corresponding 1-dimensional process has transitionprobability tπ ( t + x ) = C t ( x ), t ≥
0; the Cauchy process in 3-dimensional spacehas transition probability(6) tπ ( t + r ) = C t ( x ) , where r = p x + x + x , t ≥
0, with the momentum presentation exp( − tρ ), ρ = p p + p + p (see [7]). Notice that C t ( x ) = Q j =1 C t ( x j ) and that processis not Gaussian. So the passage from the 3-dimensional problem C it ( x ) to the1-dimensional one C it ( x ) can be performed only through integration C it ( x ) = R C it ( x ) dx dx .Therefore the construction of the position space presentation of operators (5) isreduced to a correct analytic continuation of operator (6) from real positive timeto the imaginary one, which is possible if C it ( x ) and ˜ C it ( p ) are understood asgeneralized functions (see [2]).
2. One-dimensional Cauchy distribution in imaginary time
Consider first one-dimensional case with the momentum space retarded Green’sfunction exp( it | p | ) = ˜ C it ( p ) which is the analytic continuation to imaginary timeof the momentum space transition probability of the Cauchy process ˜ C t ( p ) = UANTUM MECHANICS OF THE FREE DIRAC ELECTRONS AND EINSTEIN PHOTONS, AND THE CAUCHY PROCESS3 exp( − t | p | ), t ≥
0. Therefore the result is not Gaussian as well; it is naturalto call it one-dimensional Cauchy process in imaginary time.We assume first that the functional ˜ C it ( p ) is defined on the space of bumptest functions ϕ ( p ) ∈ K (1) , and the retarded Green’s function C it ( x ) is analyticfunctional on Z (1) (see [2]).Consider Parseval’s identity(7) Z C it ( x ) ψ ( x ) dx = 12 π Z exp( it | p | ) ϕ ( p ) dp. where ϕ ( p ) ∈ K (1) is a bump function and ψ ( x ) ∈ Z (1) is its Fourier transformwhich is an entire function of order 1 (Paley-Wiener theorem, see [2]).Thus the r.h.s. in (7) equals(8) 12 π Z exp( − it | p | + ixp ) ψ ( x ) dpdx. Since exp( it | p | ) and ψ ( x ) are entire functions and ψ ( x ) decays rapidly at infinitynear the real axis (see [2]), the integral along real axis x in (8) equals the integralalong any axis parallel to the real one. It is easy to see that one can choose theintegration path so that the integral by p converges absolutely (and its value canbe found in a list of integrals). Indeed, (8) becomes12 π Z Z −∞ e ( itp + ipx ) dpψ ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x → x − i + 12 π Z ∞ Z e ( − itp + ipx ) dpψ ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x → x + i == 12 π Z ∞ Z e ( − itp − ipx ) dpψ ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x → x − i + 12 π Z ∞ Z e ( − itp + ipx ) dpψ ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x → x + i ,which defines C it ( x ) as an analytic even functional on test functions ψ ( x ) ∈ Z (1) : Z C it ( x ) ψ ( x ) dx = 12 πi (cid:18)Z ψ ( x + i t − ( x + i dx + Z ψ ( x − i t + ( x − i dx (cid:19) .That functional can be written as well as Z C it ( x ) ψ ( x ) dx = 12 πi I ψ ( z ) dzz − t + 12 πi Z (cid:18) t − x + 1 t + x (cid:19) ψ ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12) x → x − i where the integration of the first summand is performed along the boundary of arectangular strip that contains the real line. Thus C it ( x ) = δ ( x − t ) + iπ · tt − ( x − i , and, as well,(9) C it ( x ) = 12 ( δ ( x − t ) + δ ( x + t )) + i π (cid:18) tt − ( x − i + tt − ( x + i (cid:19) . A. A. BEILINSON
It is clear that the other functional in (5) is C it ( x ) = C − it ( x ). Remark.
The functional C it ( x ) we have constructed can be also viewed as afunctional on the space of bump test functions ϕ ( x ) ∈ K (1) . Then(10) C it ( x ) = 12 ( δ ( x − t ) + δ ( x + t )) + iπ · tt − x . Here the functional tt − x = 12 (cid:18) t − x + 1 t + x (cid:19) regularizes, and the integral R tϕ ( x ) t − x dx is understood as Cauchy’s principal value,see [2] and below. By abuse of notation, we denote the regularized functionalby C it ( x ) as well. A reason for this is that the Fourier transform ˜ C it ( p ) of thatfunctional, as seen from Parseval’s identity(11) Z C it ( x ) ϕ ( x ) dx = 12 π Z (cid:18)Z C it ( x ) exp( − ipx ) dx (cid:19) ψ ( p ) dp, where ψ ( p ) ∈ Z (1) and the integral is understood as the principal value, is ananalytic functional ˜ C it ( p ) = exp( it | p | ) on Z (1) .Therefore we have proved the next Theorem (I):
The inverse Fourier transform of the functional exp( it | p | ) on Z (1) is equal to the even functional C it ( x ) = ( δ ( x − t ) + δ ( x + t )) + iπ · tt − x on thebump test functions space ϕ ( x ) ∈ K (1) . We call that generalized function C it ( x ) one-dimensional quantum Cauchy functional. Notice that C it ( x ) (see (10)) satisfies the Chapman-Kolmogorov equation C iτ ( x τ ) ∗ C i ( t − τ ) = C it ( x t )(here ∗ is the convolution of generalized functions), the existence of the convolutionfollows from the structure of the Fourier image ˜ C it ( p ) as a functional on Z (1) .It is clear that the infinitesimal operators (generators) J C ( x ) and J C ( x ) thatcorrespond to C it ( x ) and C it ( x ) are, respectively, − iπx and iπx . Therefore thequantum Cauchy functionals C it ( x ) and C it ( x ) satisfy the equations(12) ∂∂t C it ( x ) = − iπx ∗ C it ( x ), ∂∂t C it ( x ) = iπx ∗ C it ( x ),and they are fundamental solutions of these equations (see [3]).
3. The space Cauchy distribution in imaginarytime and the massless Dirac particle
Consider now the momentum space retarded Green’s function ˜ C it ( p ) = exp( itρ )in (5) which is the analytic continuation to imaginary time of the space Cauchyprocess transition probability C t ( x ) viewed in the momentum coordinates. It isnatural to call this process the space Cauchy process in imaginary time. As in UANTUM MECHANICS OF THE FREE DIRAC ELECTRONS AND EINSTEIN PHOTONS, AND THE CAUCHY PROCESS5 one-dimensional case, we first assume that the functional ˜ C it ( p ) is defined on thespace of bump test functions ϕ ( p ) ∈ K (3) (see [2]).We find C it ( x ) using Parseval’s identity that recovers a functional C it ( x ) on Z (3) from its Fourier transform ˜ C it ( p ) which is a functional on K (3) . Namely, we have(13) Z C it ( x ) ψ ( x ) dx = 1(2 π ) Z exp( itρ ) ϕ ( p ) dp ,where ϕ ( p ) ∈ K (3) is a bump function and ψ ( x ) ∈ Z (3) is its Fourier transformwhich is an entire function of first order (see [2]).Consider in more details the integral in the r.h.s. of (13); since ϕ ( p ) = Z exp( i ( x, p )) ψ ( x ) dx ,one has Z C it ( x ) ψ ( x ) dx = 1(2 π ) Z Z exp( − itρ + i ( x, p )) dpψ ( x ) dx. Rewriting the inner integral in spherical coordinates, we get(14) Z C it ( x ) ψ ( x ) dx = 1(2 π ) Z ∞ Z Z S exp( iρ ( − t + ( x, p e ))) ρ dS ( p e ) dρψ ( x ) dx, where S is the unit sphere in 3-dimensional space, and dS ( p e ) its area element.Therefore the functional1(2 π ) ∞ Z Z S exp( iρ ( t − ( x, p e ))) ρ dS ( p e ) dρ is the sought-for inverse Fourier transform of the functional exp( itρ ). We rewriteit using the fact that ψ ( x ) ∈ Z (3) and an orthogonal change of variables x = Ay , y = ( x, p e ) (so y , y are in the plane defined by equation y = ( x, p e )); clearly ψ ( x ) = ξ ( y ) ∈ Z (3) . Set R ξ ( y ) dy dy = Ξ( y ) ∈ Z (1) . Therefore(15) Z ∞ Z Z S exp( iρ ( − t + ( x, p e ))) ρ dS ( p e ) dρψ ( x ) dx == Z ∞ Z Z S exp( iρ ( − t + ( x, p e ))) ρ dS ( p e ) dρ Ξ( y ) dy .Since exp( iρy ) is an entire function of y , the shift of the integration path intocomplex plane y y + i ρ A. A. BEILINSON converges absolutely and uniformly, and one has(16) Z ∞ Z Z S exp( iρ ( − t + ( x, p e ) + i ρ dS ( p e ) dρ Ξ( y + i dy == 2 i Z Z S dS ( p e )( − t + ( x, p e )) Ξ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y → y + i == − i Z ∂ ∂ ( x, p e ) Z S dS ( p e ) − t + ( x, p e ) Ξ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y → y + i == − i ∂ ∂t Z S Z Ξ( y + i dy − t + ( y + i dS ( p e ).Therefore ∂ ∂t Z S Z Ξ( y + i dy − t + ( y + i dS ( p e ) = − ∂ ∂t Z S I Ξ( z ) dz − t + z dS ( p e ) + Z S Z ξ ( y ) dy ( − t + ( Ay, p e )) dS ( p e ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y → y + i ++ Z S Z ξ ( y ) dy ( − t + ( Ay, p e )) dS ( p e ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y → y − i ,where the contour in the first integral goes around the real axis. Consider thefirst summand; by Cauchy’s theorem one has ∂ ∂t Z S I Ξ( z ) dzz − t dS ( p e ) = 2 πi Z S Ξ (2) ( t ) dS ( p e ) == 2 πi Z S Z δ (2) ( − t + ( x, p e ))Ξ(( x, p e )) d ( x, p e ) dS ( p e ).Use the fact that in 3-space one has “flat waves decomposition of the δ -function” δ ( x ) = − π ) Z S δ (2) (( x, p e )) dS ( p e )(see [2]) which is a solution, in the generalized functions language, of the Radonproblem of reconstruction of ψ (0) from the integrals of ψ along all planes ( x, p e ) = 0.Thus(17) − π ) Z Z S δ (2) ( − t + ( x, p e )) dS ( p e ) ψ ( x ) dx == R δ S t ( x ) ψ ( x ) dx πt = ψ ( x ) S t = ξ ( y ) S t , UANTUM MECHANICS OF THE FREE DIRAC ELECTRONS AND EINSTEIN PHOTONS, AND THE CAUCHY PROCESS7 where δ S t ( x ) is δ -function of the sphere of radius t and center at 0, and ψ ( x ) S t isthe average of ψ ( t ) over that sphere.From this, by (16) we see that C it ( x ) is an analytic functional on Z (3) (18) Z C it ( x ) ψ ( x ) dx = ψ ( y ) S t −− i π Z S Z Ξ( y ) dy dS ( p e )( − t + ( Ay, p e )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y → y + i + Z S Z Ξ( y ) dy dS ( p e )( − t + ( Ay, p e )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y → y − i .It is important that this analytic functional can be viewed on bump functions ϕ ( x ) ∈ K (3) as well; then the integration over the unit sphere can be performedexplicitly, and one has(19) Z C it ( x ) ϕ ( x ) dx = ϕ ( x ) S t − iπ Z t ( t − r ) ϕ ( x ) dx . Remark.
The functional (19) can be constructed, using (10) and theorem ofinverse Radon transformation for measurable functions, see [4]In order to define that functional on the whole space K (3) we need to regularizethe functional t ( t − r ) . One proceeds as follows.Consider the integral R tϕ ( x ) dx ( t − r ) in spherical coordinates Z tϕ ( x ) dx ( t − r ) = ∞ Z t ( t − r ) Z S r ϕ ( x ) dS r ( x ) dr = 4 πt ∞ Z r ( t − r ) ϕ ( x ) S r dr, where ϕ ( x ) S r = Φ( r ) is the average of ϕ ( x ) along the sphere of radius r with centerat 0. Then Φ( r ) ∈ K (1) is an even bump function, see [2]. Therefore the integral ∞ Z tr ( t − r ) Φ( r ) dr = 12 ∞ Z −∞ tr ( t − r ) Φ( r ) dr is a functional defined on the subspace of even bump functions in K (1) .One has 4 tr ( t − r ) = t ( t − r ) − t − r + t ( t + r ) − t + r . Notice also that the functional ∞ Z −∞ Φ( r )( t − r ) dr = − ∞ Z −∞ Φ (1) ( r ) t − r dr is defined on arbitrary odd bump functions, hence the functional − ∞ Z −∞ t Φ (1) ( r ) t − r dr − ∞ Z −∞ Φ( r ) t − r dr = Z ϕ ( r ) t − r dr A. A. BEILINSON is defined on arbitrary bump functions ϕ ( r ) ∈ K (1) . So we have reduced theregularization of the functional in (19) to the already considered regularization indimension one (10).As in 1-dimensional case, we keep to denote the quantum Cauchy functional by C it ( x ) and its momentum coordinate presentation by ˜ C it ( p ), understood now asfunctionals on K (3) and Z (3) .Thus we have proved Theorem (II):
The inverse Fourier transform of the functional exp( itρ ) on Z (3) is equal to the spherically symmetric functional C it ( x ) = δ S t πt + iπ · t ( t − r ) on bump functions ϕ ( x ) ∈ K (3) . We call this generalized function C it ( x ) the spacequantum Cauchy functional. Corollary 1.
The fundamental solution of the Dirac equation for massless par-ticle has structure D t ( x ) = T ( x ) ∗ D Ft ( x ) ∗ T ( x ) , where (see (3), (4), (5)) (25) D Ft ( x ) = C it ( x ) 0 0 00 C it ( x ) 0 00 0 ¯ C it ( x ) 00 0 0 ¯ C it ( x ) does not lie in the Minkowski world. The solutions of the Dirac equations (3) andthe Dirac equations in the Foldy-Wouthuysen coordinates D Ft ( x ) are isomorphic. The shape of ˜ C it ( p ) implies that C it ( x ), viewed as a functional on the space ofbump functions K (3) is a retarded Green’s function hence satisfies the Chapman-Kolmogorov equation C iτ ( x τ ) ∗ C i ( t − τ ( x t − τ ) = C it ( x t ) , where 0 ≤ τ ≤ t . Notice that the convolution is well defined since ˜ C it ( p ) = exp( itρ )is a functional on Z (3) .Notice also that the generator of this retarded Green’s function C it ( x ) equals iπ r − . Thus the functional C it ( x ) is a fundamental solution of an integral equation(see [3])(20) ∂∂t C it ( x ) = iπ r ∗ C it ( x ) . The results of this work are based on the study of those quantum Cauchy func-tionals C it ( x ) and ¯ C it ( x ).
4. Einstein’s photons and the quantum Cauchy functional
Consider the equations for electromagnetic field without sources(21) ∂∂t E = rot H, div E = 0 , ∂∂t H = − rot E, div H = 0; UANTUM MECHANICS OF THE FREE DIRAC ELECTRONS AND EINSTEIN PHOTONS, AND THE CAUCHY PROCESS9 here E t ( x ) and H t ( x ) are, respectively, electric and magnetic fields or a photonfield. We study solutions of these equations in Majorana coordinates (see [1]) for M t ( x ) = E t ( x ) + iH t ( x ), ¯ M t ( x ) = E t ( x ) − iH t ( x ).The equations for them in the momentum coordinates ˜ M t ( p ), ¯˜ M t ( p ) are(22) i ∂∂t ˜ M t ( p ) = ( S, p ) ˜ M t ( p ), ( p, ˜ M t ( p )) = 0 ,i ∂∂t ˜ M t ( p ) = ( S, p ) ˜ M t ( p ), ( p, ˜ M t ( p )) = 0.Here ( S, p ) = P j =1 s j p j where s = − i i , s = i − i , s = − i i are the photon spin operators, so the operator(23) ( S, p ) = − ip ip ip − ip − ip ip is Hermitian.Consider the system of equations (22). First notice that the conditions in (22)are automatically satisfied since ∂∂t ( p, ˜ M t ( p )) = 0 as follows from (22).The roots of the characteristic polynomial of the Hermitian matrix ( S, p ) are ± ρ ( ρ = p p + p + p ) and 0, so this matrix is degenerate. Therefore( S, p ) = ˜ Q + ( p )˜ h F ( p ) ˜ Q ( p ),where(24) ˜ h F ( p ) = ρ − ρ
00 0 0 ,˜ Q ( p ) is a unitary operator that diagonalizes ( S, p ), and ˜ Q + ( p ) is the conjugateoperator.Therefore the fundamental solution of (24) viewed in the momentum coordinatesis the direct product ˜M t ( p ) = ˜ M t ( p ) × ˜ M t ( p ) of matrices(25) ˜ M t ( p ) = ˜ Q + ( p ) ˜ M Ft ( p ) ˜ Q ( p ) , ˜ M Ft ( p ) = exp( − itρ ) 0 00 exp( itρ ) 00 0 1 , and ˜ M t ( p ) = ˜ M − t ( p ) which are analytic functionals on Z (3) . Therefore M t ( x ), asthe position coordinate presentation of the generalized function ˜M t ( p ), is a func-tional on ϕ ( x ) ∈ K (3) , and we have Corollary 2 of theorem (II).
The fundamental solution of the Maxwell equa-tion (3). viewed as a functional on K (3) , has structure M t ( x ) = Q + ( x ) ∗ M Ft ( x ) ∗ Q ( x ) × Q + ( x ) ∗ M Ft ( x ) ∗ Q ( x ), where functional M Ft ( x ) = ¯ C it ( x ) o C it ( x ) 00 0 δ ( x ) evidently does not lie in the Minkowski world. Yet the solutions of the Maxwellequation (3) and the Maxwell equation in the Foldy-Wouthuysen coordinates areunitary equivalent.
5. A modified one-dimensional Cauchy functional
Consider the Dirac equation for a mass m particle (see [8])(26) γ ∂∂t + ( γ, ∇ ) − im = 0 . Its fundamental solution in the momentum coordinates is˜ D mt ( p ) = exp( it ( γ ( γ, p ) + mγ )) . Since the above inner bracket is an Hermitian matrix, it can be diaonalized bya unitary (and Hermitian) transform ˜ T m ( p )(27) ˜ T m ( p ) = ˜ T m − ( p ) = γ ( γ, p ) + I ( m + p m + ρ ) q p m + ρ ( m + p m + ρ ) , hence(28) ˜ D mt ( p ) = ˜ T m ( p ) exp( itγ p m + ρ ) ˜ T m ( p ) . Here γ p m + ρ is the momentum coordinates Foldy-Wouthuysen presentationfor the energy of the Dirac electron (see [13]), and ˜ D m F t ( p ) = exp( itγ p m + ρ )is the momentum coordinates Foldy-Wouthuysen presentation for the fundamentalsolution of the Dirac electron equation, see [13].We will call(29) ˜ C mit ( p ) = exp( it p m + ρ ) the momentum coordinates presentation of the modified quantum Cauchy functional .It is clear that we need to know the position coordinate presentation of that func-tional in order to construct D mt ( x ). UANTUM MECHANICS OF THE FREE DIRAC ELECTRONS AND EINSTEIN PHOTONS, AND THE CAUCHY PROCESS11
Our task now is to compute the inverse Fourier transform of ˜ D m F t ( p ) in one-dimensional case; to do this, we compute the inverse Fourier transform of thefunctional ˜ C mit ( p ) as a generalized function on Z (1) by a method we used to solve asimilar problem above. Namely, we find C mit ( x ) from Parseval’s identity(30) Z C mit ( x ) ψ ( x ) dx = 12 π Z ˜ C mit ( p ) ϕ ( p ) dp, where ψ ( x ) ⊂ Z (1) and ϕ ( p ) ∈ K (1) . We compute the right integral by deformingthe path of integration to make it absolutely convergent Z exp( − it p m + p ) (cid:18)Z exp( ixp ) ψ ( x ) dx (cid:19) dp == Z Z −∞ ˜ C mit ( p ) exp( ixp ) dpψ ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x → x − i + Z ∞ Z ˜ C mit ( p ) exp( ixp ) dpψ ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x → x + i . Since ˜ C mit ( p ) is even with respect to p one has(31) Z (cid:18)Z exp( − it p m + p ) exp( ixp ) dp (cid:19) ψ ( x ) dx == Z ∞ Z ˜ C mit ( p ) e − ixp dpψ ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x → x − i + Z ∞ Z ˜ C mit ( p ) e ixp dpψ ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x → x + i == Z ∞ Z ˜ C mit ( p ) cos( xp ) dpψ ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x → x − i + Z ∞ Z ˜ C mit ( p ) cos( xp ) dpψ ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x → x + i , since the integrals with sine vanish by the Cauchy theorem.One has (see [12] formula 3.914)(32) ∞ Z exp( − t p p + m ) cos( px ) dp = tm √ t + x K ( m p t + x ) , where K ( z ) is the Macdonald function (see [10], 3.7, formula 6); here t and x arereal and t > Z ∞ Z exp( − t p p + m ) cos( px ) dpψ ( x ) dx = Z tm √ t + x K ( m p t + x ) ψ ( x ) dx where ψ ( x ) ∈ Z (1) . Since cosine is an even function, we can move the integrationpath, so one has Z ∞ Z exp( − t p p + m ) cos( px ) dpψ ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x → x ± i == Z tm √ t + x K ( m p t + x ) ψ ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12) x → x ± i . The latter equality can be continued analytically to t in the imaginary axis: Z ∞ Z exp( − it p p + m ) cos( px ) dpψ ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x → x ± i == Z tm √ t − x K ( im p t − x ) ψ ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12) x → x ± i . Thus (31) becomes
Z Z ˜ C mit ( p ) exp( ixp ) dpψ ( x ) dx == Z tmK ( im √ t − x ) √ t − x ψ ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x → x − i + Z tmK ( im √ t − x ) √ t − x ψ ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x → x + i , and Parseval’s identity (30) yields(33) Z C mit ( x ) ψ ( x ) dx == 12 π I tmK ( im √ t − z ) √ t − z ψ ( z ) dz + 1 π Z tmK ( im √ t − x ) √ t − x ψ ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x → x + i ,where the first integral is taken along the boundary of an infinite rectangular stripthat contains the real axis.Since(34) K ( z ) | z → ≃ z − (see [10], 3.7, formulas (6), (2)), the function under the contour integral sign in (33)has simple poles at points t and − t . So, by Cauchy’s theorem, one has(35) Z C mit ( x ) ψ ( x ) dx = 12 ( ψ ( − t ) + ψ ( t )) + 1 π Z tmK ( im √ t − x ) √ t − x ψ ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x → x + i . The functional C mit ( x ), just as C it ( x ) above, can be seen to be defined on all thebump functions ϕ ( x ) ∈ K (1) ; then the integral in the r.h.s. should be regularizedsince, by (34), the function under the integral has simple poles on the real line at t and − t .This can be done as follows. Set (see (10) and (35))(36) tmπ √ t − x K ( im p t − x ) (cid:18) − iπ · tt − x (cid:19) − = B ( t, x );then B ( t, x ) and B ( t, x ) − are infinitely differentiable functions that have no zerosin any finite domain with t ≥
0. And (34) implies that B ( t, ± t ) = 1 . UANTUM MECHANICS OF THE FREE DIRAC ELECTRONS AND EINSTEIN PHOTONS, AND THE CAUCHY PROCESS13
Consider the functional C it ( x ) B ( t, x ); one has Z C it ( x ) B ( t, x ) ϕ ( x ) dx = 12 ( ϕ ( t ) + ϕ ( − t )) + iπ Z tmK ( im √ t − x ) √ t − x ϕ ( x ) dx. Thus R C it ( x ) B ( t, x ) ϕ ( x ) dx = R C mit ( x ) ϕ ( x ) dx (see (36)), where ϕ ( x ) ∈ K (1) ,which means that(37) C it ( x ) B ( t, x ) = C mit ( x ) . One also has C mit ( x ) B − ( t, x ) = C it ( x ) . Therefore, by (37), the regularization of the functional Z C mit ( x ) ϕ ( x ) dx = 12 ( ϕ ( t ) + ϕ ( − t )) − i π Z (cid:18) t − x + 1 t + x (cid:19) B ( t, x ) ϕ ( x ) dx is reduced to the interpretation of the integrals in the r.h.s. as the integrals in thesense of Cauchy’s principal value.Also, using (34), we get a relation between the functionals C mit ( x ) and C it ( x ):(38) lim m → C mit ( x ) = C it ( x ) . Note that to the modified quantum Cauchy functional C mit ( x ) there corresponds agenerator J C m ( x ) = iπ · mK ( mx ) x , so the next equation (see [3]) is satisfied(39) ∂∂t C mit ( x ) = iπ · mK ( mx ) x ∗ C mit ( x ) , and C mit ( x ) is its fundamental solution.Therefore C mit ( x ) satisfies the Chapman-Kolmogorov equation C miτ ( x τ ) ∗ C mi ( t − τ ) ( x t − τ ) = C mit ( x t ) , where the convolution is well defined due to the structure of ˜ C mit ( p ) as a functionalon Z (1) . The same is true for ¯ C mit ( x ).Thus we have proved Theorem (III):
The inverse Fourier transform of the functional exp( it p m + p ) on Z (1) is equal to the even functional C mit ( x ) = 12 ( δ ( t − x ) + δ ( t + x )) + 1 π · − tmK ( im √ t − x ) √ t − x on K (1) . We call this generalized function C mit ( x ) the one-dimensional modifiedquantum Cauchy functional.
6. The modified space quantum Cauchyfunctional and the free Dirac electron
We construct now the inverse Fourier transform of the functional ˜ C mit ( p ) as ageneralized function on Z (3) using the same method as was used to solve the sim-ilar problem for the quantum Cauchy functional. Namely, we find C mit ( x ) fromParseval’s identity Z C mit ( x ) ψ ( x ) dx = 12 π ) Z exp( it p m + ρ ) ϕ ( p ) dp, where ϕ ( p ) ∈ K (3) and ψ ( x ) ∈ Z (3) . Since ϕ ( p ) = R exp( i ( x, p )) ψ ( x ) dx , we see,rewriting the integral in spherical coordinates, that(40) Z ∞ Z exp( it p m + ρ ) ρ Z S exp( iρ ( x, p e )) dS ( p e ) dρψ ( x ) dx == Z ∞ Z exp( it p m + ρ ) ρ Z S cos( ρ ( x, p e )) dS ( p e ) dρψ ( x ) dx == − Z ∞ Z exp( it p m + ρ ) Z S ∂ ∂ ( x, p e ) cos( ρ ( x, p e )) ! dS ( p e ) dρψ ( x ) dx (here we use R S sin( ρ ( x, p e )) dS ( p e ) = 0).As in (10), we perform the orthogonal change of variables x = Ay , and set y = ( x, p e ), ξ ( y ) = ψ ( x ), and Ξ( y ) = R ξ ( u ) dy dy ∈ Z (1) .Since cos ( ρy ) is an even function, we can shift the integration path from thereal axis y in (40) to the complex domain y → y + i
0, so(41) ∞ Z exp( it p m + ρ ) Z S Z (cid:18) ∂ ∂y cos( ρy ) (cid:19) Ξ( y ) dy dS ( p e ) dρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y → y + i == Z S Z ∞ Z exp( it p m + ρ ) cos( ρy ) dρ Ξ (2) ( y ) dy dS ( p e ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y → y + i . Interpreting (32) as an equality of analytic functionals, we get(42) Z ∞ Z exp( − t p m + ρ ) cos( ρy ) dρ Ξ (2) ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y → y + i == Z tm p t + y K ( m q t + y )Ξ (2) ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y → y + i . UANTUM MECHANICS OF THE FREE DIRAC ELECTRONS AND EINSTEIN PHOTONS, AND THE CAUCHY PROCESS15
As a result of analytic continuation by y , we can analytically continue the aboveequality to t on the imaginary line:(43) Z ∞ Z exp( − it p m + ρ ) cos( ρy ) dρ Ξ (2) ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y → y + i == Z tm p t − y K ( im q t − y )Ξ (2) ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y → y + i . Therefore (40) implies that(44) Z ∞ Z exp (cid:16) − it p m + ρ (cid:17) ρ Z S exp ( iρ ( x, p e )) dS ( p e ) dρψ ( x ) dx == Z S Z tm p t − y K (cid:18) im q t − y (cid:19) Ξ (2) ( y ) dy dS ( p e ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y → y + i . Since the function we integrate in (44) is analytic and Ξ (2) ∈ Z (1) , one has(45) Z S Z tm p t − y K (cid:18) im q t − y (cid:19) Ξ (2) ( y ) dy dS ( p e ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y → y + i == − Z S I tm √ t − z K (cid:16) im p t − z (cid:17) Ξ (2) ( z ) dzdS ( p e )++ 12 Z S Z tm p t − y K (cid:18) im q t − y (cid:19) Ξ (2) ( y ) dy dS ( p e ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y → y + i ++ 12 Z S Z tm p t − y K (cid:18) im q t − y (cid:19) Ξ (2) ( y ) dy dS ( p e ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y → y − i , where the integration in the first summand is along a contour around the realaxis. Consider that summand; then (34) implies that the function we integrate haspoles at t and − t . By the Cauchy theorem(46) I tm √ t − z K (cid:16) im p t − z (cid:17) Ξ (2) ( z ) dz = π (cid:16) Ξ (2) ( t ) + Ξ (2) ( − t ) (cid:17) . This implies(47) − Z S I tm √ t − z K (cid:16) im p t − z (cid:17) Ξ (2) ( z ) dzdS ( p e ) == − π Z S Z δ (2) ( t + ( x, p e )) (Ξ (( x, p e )) + Ξ ( − ( x, p e ))) d ( x, p e ) dS ( p e ) == − π Z Z S δ (2) ( t + ( x, p e )) dS ( p e )Ξ (( x, p e )) d ( x, p e ) Therefore the ”decomposition of δ -function by flat waves” we have already used(see [2]) shows that C mit ( x ) is an analytic functional on Z (3) :(48) Z C mit ( x ) ψ ( x ) dx = ψ ( x ) S t ++ 12(2 π ) Z S Z ∂ ∂y tmK (cid:16) im p t − y (cid:17)p t − y ξ ( y ) dydS ( p e ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y → y + i ++ 12(2 π ) Z S Z ∂ ∂y tmK (cid:16) im p t − y (cid:17)p t − y ξ ( y ) dydS ( p e ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y → y − i . If we consider that functional on K (3) , then integrals along S in the last twosummands can be computed, and one has(49) Z S ∂ ∂ ( x, p e ) tmK (cid:16) im p t − ( x, p e ) (cid:17)p t − ( x, p e ) dS ( p e ) == 2 πr − r Z − r ∂ ∂y tmK (cid:16) im p t − y (cid:17)p t − y dy == − π ∂∂ ( t − r ) tmK (cid:0) im √ t − r (cid:1) √ t − r , where r = | y | .Notice that on bump functions ϕ ( x ) ∈ K (3) that functional(50) Z C mit ( x ) ϕ ( x ) dx = ϕ ( x ) S t + iπ Z ∂∂ ( t − r ) − tmK (cid:0) im √ t − r (cid:1) √ t − r ϕ ( x ) dx requires a regularization since the function we integrate has singularity on thesphere of radius t . To find it, we first rewrite that functional. Remark.
And as in case space Cauchy functional (19) the functional (50) can beconstructed by the theorem of inverse Radon transformation (35) for measurablefunctions, see [4].We use the regularized solution C it ( x ) in the space case (see (19)) that wasdeduced from the one-dimensional situation (12).Consider the function(51) 1 π ∂∂ ( t − r ) tmK (cid:0) im √ t − r (cid:1) √ t − r ! (cid:18) iπ t ( t − r ) (cid:19) − == iml ∂∂ ( l ) K ( iml ) l = B ( t, r ) , UANTUM MECHANICS OF THE FREE DIRAC ELECTRONS AND EINSTEIN PHOTONS, AND THE CAUCHY PROCESS17 where √ t − r = l . That function is infinitely differentiable with respect to r , andlim r → t B ( t, r ) = 1 by (34). Both B ( t, r ) and B ( t, r ) − do not vanish.Consider C it ( x ) B ( t, r ) (see (19)) as a functional on K (3) ; then(52) Z C it ( x ) B ( t, r ) ϕ ( x ) dx = Z C mit ( x ) ϕ ( x ) dx for every ϕ ( x ) ∈ K (3) . Hence(53) C it ( x ) B ( t, r ) = C mit ( x ) . Therefore(54) Z C mit ( x ) ϕ ( x ) dx = ϕ ( x ) S t − iπ Z t − r ) B ( t, r ) ϕ ( x ) dx, so the asked for regularization of the integral in (54) is reduced to the alreadyperformed regularization of the functional C it ( x ) (see (19)).We notice also that there is a natural relationlim m → C mit ( x ) = C it ( x ) . It is clear that the functional C mit ( x ) is the fundamental solution of an integralequation(55) ∂∂t C mit ( x ) = im π r ∂∂r K ( mr ) r ∗ C mit ( x ) , that satisfies the Chapman-Kolmogorov equation(56) C miτ ( x τ ) ∗ C mi ( t − τ ) ( x t − τ ) = C mit ( x t ) , the convolution is well defined due to the structure of ˜ C mit ( p ) = exp( it p m + ρ ).It is easy to see that ¯ C mit ( x ) = C m − it ( x ) has similar properties.Thus we have deduced Theorem (IV):
The inverse Fourier transform of the functional exp( it p m + ρ ) on Z (3) is the spherically symmetric functional C mit ( x ) = δ S t πt + 1 π ∂∂ ( t − r ) tmK ( im √ t − r ) √ t − r on K (3) . We call that functional the modified space quantum Cauchy functional. Corollary.
The fundamental solution of the Dirac equation (1) is a matrix-valued functional D mt ( x ) = T m ( x ) ∗ D m F t ( x ) ∗ T m ( x ) on K (3) (see (3), (4), (5)),where D m F t ( x ) = C mit ( x ) 0 0 00 C mit ( x ) 0 00 0 ¯ C mit ( x ) 00 0 0 ¯ C mit ( x ) clearly does not lie in Minkowski’s world. Yet the solutions of the Dirac equa-tion (26) and the Dirac equation in the Foldy-Wouthuysen D m F t ( x ) coordinates areunitary equivalent. Therefore the evolution of Dirac’s electron is reduced to its evolution in theFoldy-Wouthuysen coordinates (that preserves spin), after which the operations ofthe left and right convolutions with T m ( x ) return the spin to the construction.
7. On the correspondence principle for the Dirac electron
Consider now the problem of construction of the quasi-classical solution to theDirac electron equations in more details (cf. [11]).Recall that the fundamental solution of the Dirac electron equation in the mo-mentum coordinates is ˜ D mt ( p ) = ˜ T m ( p ) ˜ D m F t ( p ) ˜ T m ( p ) where˜ D m F t ( p ) = exp( itγ p m + ρ )is the Foldy-Wouthuysen solution, and ˜ T m ( p ) is as in (27). To return to physicalcoordinates we have to change m by ~ − cm , where m is the invariant mass of theelectron, and t by ct (see [13]); then our quasi-classical approximation is reducedto finding of the quasi-classical approximation D mt ( x ) and D m F t ( m ) for m → ∞ ;here, as before, c = 1.If ~ → T m ( p ) → I hence ˜ D mt ( p ) → ˜ D m F t ( p ), but, despite the vanishingof the operator responsible for the finiteness of the functional D mt ( x ) (see [8]), thequasi-classical limit for the fundamental solution of the Dirac electron equation andits Foldy-Wouthuysen transform coincide. Let us show this.Notice that, due to the asymptotic of the Macdonald function K ( z ) | z →∞ ≃ r π z exp( − z )(see [10] 7.2.3, formula (1)), yield the asymptotic of C mit ( x ) | ~ → : as an analyticfunctional on Z (3) it is an entire function Z C mit ψ ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12) ~ → = − t p π/ π ) Z Z S (cid:18) ∂ ∂y exp( i ~ − m l t ) l t / (cid:19) dS ( p e )Ξ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y → y + i = − t p π/ π ) Z π Z ∂ ∂y sin θ exp( i ~ − m l t )( t − ( r cos θ ) ) / dθ Ξ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y → y + i , UANTUM MECHANICS OF THE FREE DIRAC ELECTRONS AND EINSTEIN PHOTONS, AND THE CAUCHY PROCESS19 (here l t = √ t − r is the relativistic interval and m l t is the free particle eikonal).Therefore, for ~ → t , − t disappear (hence the δ -function in(50) disappears) and the integral over the domain r ≥ t (the complement to thelight cone) disappears in two (of the four) diagonal elements of D m F t , due to theexponential decrease of these in that domain.Therefore, the obtained functional can be considered as a functional on finitefunctions, that allow to calculate the integral by the unit sphere(57) Z C mit ϕ ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12) ~ → = t √ π π ) Z (cid:18) ∂∂ ( l t ) exp( i ~ − m l t ) l t (cid:19) ϕ ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ~ → = Z exp( i ~ m l t ) φ ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12) ~ → ,where, easy to see that ∀ φ ( x ) ∈ K (3) .So C mit ( x ) | ~ → becomes a finite functional that describes the classical free rela-tivistic particle of an arbitrary finite mass, cf. [14], [8], [9].We emphasize that the classical relativistic limit for the quantum relativisticparticle of nonzero mass comes from the imaginary summand of Green’s functionalwhich has unbounded support.This implies, in particular, that it is impossible to interpret the quantization of aclassical relativistic particle as consideration of arbitrary trajectories in the classicalaction integral, as it happens for the nonrelativistic theory (see [9], [11]). Noticealso that in quantum relativistic case, as follows from (57), the dependence from theeikonal is exponential only in the quasi-classical approximation. This characteristicproperty of quantization of the classical relativistic particles should be accountedfor in the quantum theory. Conclusion
The present work shows the special role of the quantum Cauchy functionals(which are the imaginary time Cauchy distributions) on the bump functions forunderstanding the solutions of the relativistic quantum mechanics equations.These quantum Cauchy functionals appear naturally when one constructs solu-tions of the Dirac and Maxwell equations in the position coordinates in the Foldy-Wouthuysen form. Yet the physical facts, that correspond to these functionals, arebeyond the casual classical Minkowski world.With the help of the quantum Cauchy functionals, we observe a fundamentalrelation between solutions of the Dirac and Maxwell equations and find unitarytransformations that interchange bosons and fermions without leaving the classi-cal Minkowski world, which may be related to a possibility of construction of theF. A. Berezin superinteraction theory (see [6]).We also find that these functionals can be effectively applied to the study of thepassage from quantum relativistic problems to their classical relativistic versions.The author thanks O. G. Smolyanov for stimulating support.
References [1] A. I. Akhiezer, V. B. Berestetskii,
Quantum electrodinamics , Interscience, 1965.[2] I. M. Gelfand, G. E. Shilov,
Generalized functions , vol. 1, Academic Press, 1964.[3] I. M. Gelfand, G. E. Shilov,
Generalized functions , vol. 3, Academic Press, 1967.[4] I. M. Gelfand, M. I. Graev, N. Ya. Vilenkin,
Generalized Functions , vol. 5, Academic Press,1967.[5] L. L. Foldy, S.A. Wouthuysen,
On the Dirac theory of spin / particles and its non-relativistic limit , Phys. Rev. (1950), 29.[6] F. A. Berezin, Method of secondary quantization , Nauka, 1986. (Russian)[7] B. V. Gnedenko, A. N. Kolmogorov,
The limit distributions for sums of independent randomvariables , GIZT-TL (1949), Moscow. (Russian)[8] N. N. Bogolyubov, D. V. Shirkov,
Introduction to the theory of quantized fields , Interscience,1959.[9] R. P. Feynman,
Space time approach to nonrelativistic quantum mechanics , Rev. Mod. Phys (1948), 367.[10] G. N. Watson, A treatise on the theory of Bessel functions , Cambridge University Press,1966.[11] V. P. Maslov, M. V. Fedoryuk,
Semi-classical approximation in quantum mechanics , Springer,2001.[12] I. S. Gradshteyn, I. M. Ryzhik,
Table of integrals, series and products , Academic Press,1980.[13] C. Itzykson, J.-B. Zuber,
Quantum field theory , Dover, 2006.[14] A. A. Beilinson,
The fundamental solutions of the quantum mechanics equations viewed asdistributions, and singularities of the Dirac equation solutions , TMP151