A Reformulation of Schrödinger and Dirac Equations in Terms Of

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A reformulation of Schrödinger and Dirac equations in terms of observable local densities and electromagnetic fields : a step towards a new interpretation of quantum mechanics ? J. Des Cloizeaux

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J. Des Cloizeaux. A reformulation of Schrödinger and Dirac equations in terms of observable local densities and electromagnetic fields : a step towards a new interpretation of quantum mechanics ?. Journal de Physique, 1983, 44 (8), pp.885-908. 10.1051/jphys:01983004408088500. jpa-00209672

HAL Id: jpa-00209672 https://hal.archives-ouvertes.fr/jpa-00209672 Submitted on 1 Jan 1983

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J. Physique 44 (1983) 885-908 AOÛT 1983, 885

Classification Physics Abstracts 03.65

A reformulation of Schrödinger and Dirac equations in terms of observable local densities and electromagnetic fields : a step towards a new interpretation of quantum mechanics ?

J. des Cloizeaux

SPHT-CEN Saclay, 91191 Gif sur Yvette Cedex, France (*) and Institute for Theoretical Physics, University of California, Santa Barbara, CA. 93117, U.S.A.

(Reçu le 27 septembre 1982, révisé le 22 dicembre, accepté le 14 avril 1983)

Résumé. 2014 On montre que l’équation de Schrödinger, l’équation de Dirac à 3 dimensions et l’équation de Dirac à 4 dimensions peuvent être transformées de manière à éliminer tous les effets de phase et de jauge. Des densités locales réelles sont définies et l’on montre que les tenseurs obéissent à des « équations probabilistes » réelles qui sont équivalentes aux équations de la mécanique quantique. Ces nouvelles équations, qui sont construites expli- citement, ne font intervenir que les densités locales et le champ électromagnétique extérieur. De nouvelles conditions aux limites sont aussi établies. L’expérience d’Aharonov-Bohm est réinterprétée dans le cadre de ce nouveau formalisme.

Abstract. 2014 It is shown that the Schrödinger equation, the 3-dimensional Dirac equation and the 4-dimensional Dirac equation can be transformed so as to eliminate all phase and gauge effects. Real local densities are defined and it is shown that these tensors obey real « probabilistic equations » which are equivalent to the equations of quantum mechanics. These new equations, which are explicitly constructed, involve only local densities and the external electromagnetic field. New boundary conditions are also derived. The Aharonov-Bohm experiment is reinterpreted in the framework of this new formalism.

« I think one can make a safe will present his views and his motivations. If the guess that uncertainty reader does not agree with them and is just interested relations in their present in facts, he should pass directly to the following form will not survive in the sections. physics of the future. » The author thinks that some difficulties may be P.A.M. Dirac, 1962 connected with the fact that a wave function is not (Scientific American, 1963) an observable. Of course, physicists working now on particle do not use wave functions but more 1. Introduction. theory explicitly powerful tools like functional integrals or the S Quantum mechanics is extremely successful but many matrix. However, when they calculate a physical physicists [1], even young ones, do not consider its quantity, they generally use, in the course of the orthodox interpretation as completely satisfactory. calculation process, quantities which are gauge and The difficulties are purely philosophical but real. phase dependent. Quantum mechanics is an efficient recipe. Is that The situation would be clearer if quantum mecha- enough ? nics could be directly formulated in terms of proba- Actually, some physicists think that quantum bilities which are observables with a physical meaning. mechanics should also present a picture of reality. Let us dream. These probabilities should obey However, concerning this question, physicists have partial derivative equations just as the probability different opinions. In this introduction the author distributions of classical statistical mechanics obey a Fokker-Planck equation. These « probabilistic equa- (*) Permanent address. tions » cannot be trivial because the vacuum is not

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01983004408088500 886 a simple system; it is certainly as complex as a heat flux of an external field through the hole of the torus. bath. As the particle propagates in a domain where the Of course dissipation is not present since a particle magnetic field is always zero, we see that apparently may have a constant motion in the vacuum but the particle does not feel only the local electro- polarization effects should be important (think of magnetic field but also the local potential vector the complexity of the vacuum-vacuum diagrams in which is gauge dependent. Thus, the classical inter- field theory !). These probabilistic equations should pretation of the Aharonov-Bohm experiment [4] describe the evolution of real probabilities or obser- seems to contradict our answer to the preceding vables. They should also present a more realistic questions. view of nature than ordinary quantum mechanics. We shall justify our point of view by introducing Therefore, we may expect that it would be easier to real « local densities » and by showing that these interpret them. observable tensors obey « probabilistic equations ». In statistical mechanics, we describe the motion of These equations, which determine the space-time a particle interacting with an external potential and dependence of the local densities, will be explicitly a heat bath by writing a Langevin equation and, written down. Thus, the contradiction with the current from this Langevin equation, we deduce the cor- interpretation of the Aharonov-Bohm experiment responding Fokker-Planck equation; in this case, cannot be real and in fact the paradox will be dis- the Langevin equation provides a physical inter- cussed and solved at the end of the article. pretation of the Fokker-Planck equation. We may Actually, D. Bohm [5] himself found the « proba- expect a similar situation in quantum mechanics [2]. bilistic equations » which can be associated with an At the present time, this is only a dream. However, ordinary Schr6dinger equation describing the motion we may ask the question « Is it possible to replace a of a particle in an external potential. (D. Bohm, who Schr6dinger equation or a Dirac equation by an assumed the existence of « hidden variables », was equivalent probabilistic equation describing how trying to find out a new interpretation of quantum local densities (local mean values) vary with the mechanics.) However, the simple but important fact local electromagnetic field ? ». Another formulation that the equations found by Bohm are equivalent to of the same question is « Can we transform quantum the Schr6dinger equation but are more « physica.l », mechanics so as to eliminate all phase and gauge has not been sufficiently emphasized. effects ? ». In his famous lectures, R. P. Feynman [6] has written Our answer is « Yes » and the aim of the present down these « probabilistic equations » in the case article is to prove the validity of this affirmation. The where an external electromagnetic field produced by reader, however, might be tempted to say « No ». In a potential and a vector potential interacts with the fact, when we try to interpret the famous « Aharonov- particle. These results are reproduced and discussed Bohm experiment » [3] (see Fig. 1) we discover that in section 2. the interference pattern of a particle interfering with We believe however that the Schr6dinger equation itself can be modified without applying any electro- cannot lead to a deep understanding of reality. In magnetic field to the particle. In the experiment, the fact it is deduced from the Dirac equation by a com- particle beams define a toroidal space and to modify plicated unitary transformation (the Foldy-Wouthui- the interference pattern we have only to change the sen transformation [7]) which destroys simple physical aspects of the Dirac equation. For instance, the operator vj, which represents the velocity of an electron should not depend on the external potential vector. Dirac equation satisfies this requirement since in this case, we have vop = cy i (c = velocity of light). However, when we deal with the Schrodinger equation, we have to use the unpleasant formula :

We also note that the « Zitterbewegung » is a very important phenomenon [7] related to the existence of spins, and that it requires an adequate interpretation. Of course, Dirac equation presents difficulties with respect to negative energies and some physicists think that it has a meaning only in field theory. We remark however that one-electron states

- exist and that the motion of an electron in an external Fig. 1. Propagation of a wave in a toroidal domain. The circuit C is represented by the dashed line. 0 is the electromagnetic field is well described by Dirac magnetic flux through the hole. equation. Moreover, the effect known as the Klein 887 paradox also occurs in semiconductors and in that tions of x, y, z, and t. We shall also use the complex case, the situation is quite clear. The anomaly which conjugate of equation (1) : is observed in strongly repulsive potential is only a mark of the instability of the vacuum (or ground state) with respect to the production of pairs of particles of opposite signs. Thus, all these arguments brought us to believe that Dirac equation may be The P and the current J at a more realistic that Schrodinger equation. density density given are defined Accordingly, we derive here the «probabilistic point by equations » which can be associated with the usual four-dimensional Dirac equation (3 dimensions of space, 1 dimension of time). The system however is not very simple. Consequently, we begin with a simpler problem, the derivation of the probabilistic equations >> which can be associated with the three- dimensional Dirac equation (2 dimensions of space, 1 dimension of time). It will also be convenient to introduce the velocity v In section 3, we show how the simple three-dimen- by setting sional Dirac equation can be transformed into a probabilistic equation (and conversely). In section 4, a similar treatment is applied to the four-dimensional We note that the wave function can be written in Dirac equation. However, this case is rather compli- the form cated because the number of parameters which are involved in the calculation is much larger. Conse- quently, if the reader is only interested in methods and results, he may skip all the details of the calcula- where a is the phase of the wave function. Now tion and pass directly to subsection 4.6 where the equation (3) gives results are listed. We also show that the probabilistic equations are not always sufficient to determine the system. It is sometimes necessary to add special boundary condi- and therefore according to equation (6) tions, especially when the particle propagates in a domain which is not simply connected (toroidal). Section 5 is more physical and is devoted to a discussion of the Aharonov-Bohm experiment. Finally, in the we that the conclusion, suggest preceding From this equation, we deduce immediately the results may contribute to a better understanding of condition quantum mechanics. The Einstein convention of summation over repeated indices is used everywhere in this article. 2. A transformation of the Schrodinger equation. where B is the induction is Z .1 LOCAL DENSITIES AND PROBABILISTIC EQUATIONS. magnetic (Bjkm completely and = is a - Thus, Using proper units, we may write the one-particle antisymmetric ExyZ 1). equation (8) condition which has to be fulfilled Schrodinger equation in the following form ( j = x, y, z) : [8] independently of the propagation equations which we shall now consider. Let us calculate 8tP: where V is the potential, and Ai a component of the vector potential A. The quantities 0, V, Aj are func- thus by using equations (1) and (2), we find 888

and therefore, using definition (3), we obtain the We see that these equations are hydrodynamic conservation equation equations where W plays the role of a quantum potential or pressure. Incidentally we remark that if B = 0, we have an irrotational flow. We also note that equation (8) can be considered as an initial condition. which is the first probabilistic equation. We have In fact, the propagation equations are such that if now to calculate D,Jj : equation (8) is satisfied for t = to, it remains satisfied for t > to, as can be easily verified We also note that equations (15) give the electric field E and the magnetic induction B in terms of the local densities; the magnetic induction is given by the first equation (15) and the electric field by the third equation (15) after replace- ment, in this equation, of 8jlm B, by the value extracted from the first equation (15). In this equality, we replace g§ and g§* by the Thus a knowledge of P, vj at all times determines values obtained from equations (1) and (2). A straight- completely the values, at all times, of the electroma- forward (but not very short) calculation shows that gnetic fields applied to the particle. The fact that these 0,Jj can be written in the following way quantities obey a closed set of equations which determine their time dependence, is in itself very remarkable (very different situations occur in statistical mechanics). We note that the time dependence of P is given by the continuity equation (second equation (15)) which is linear with respect to P, in agree- where ment with probability theory. On the contrary, the third equation (15) is not linear in P; this fact which may look strange cannot be directly interpreted, but, in some way, this equation looks like a « self-consis- tent equation ». Let us now show that the preceding set of equa- We note that equation (6) involves only the electric tions is equivalent to the Schrodinger equation. This field Ej and the magnetic induction By means that starting from these equations, we can reconstruct the Schrodinger equation. For this pur- pose, we shall proceed as follows. The electromagnetic fields which appear in equation (15) obey Maxwell equations but are defined independently of any gauge. In order to fix the gauge, we have to choose a poten- Thus the propagation equations are tial V and a vector potential A which have to obey equation (13) but which otherwise are arbitrary. Now we can define a wave function 0 by setting where Tjl is given by equation (12). It is convenient to write these equations by express- where the phase a at a point A is given by a path ing Jj in terms of P and vj (see Eq. (4)). The propaga- integral going from the origin 0 to the point A : tion equations obtained in this way and equation (8) form the set of probabilistic equations which we wanted to obtain, i.e.,

It will be assumed that the domain in which the particle moves is simply connected In this case, the integral depends on the positions of A and 0 but as a consequence of equation (8), it does not depend on the shape of the path. Thus, the phase a of 0 is well defined where Let us now calculate i ðt4> : 889

Let us now use equations (15); we obtain

The integral can be transformed by using equations (8) and (13). We find

We can now integrate and we obtain

where Vo is just a constant (which can be absorbed in V). More explicitly, we have (see Eq. (16))

Let us now calculate (0, - iA,)2 0. We have

On the other hand, equation (18) gives and therefore using this equation and equation (17), we find

Now we have

Now, we must compare i Ot4> which is give by equation (21) and - 1 a - iA 2 which is given by equation (23). A trivial calculation shows that the Thus, we see that the set of equations (15) is com- function 0 which is defined by equations (17) and pletely equivalent to the Schrodinger equation (1), (1$) obeys the equation since the constant V 0 can always be absorbed in V. 890

However this equivalence is only formal because the tion conditions but they are as exact as the Schr6din- condition P =f:. 0 has been implicitly assumed If ger equation. With the probabilistic equations and P = 0 at some point, singularities may occur; the the boundary conditions on the border of the domain, effect of such singularities will be studied in sec- they determine the solutions P(x, y, z, t) and tion 2 . 3. Jj(x, y, z, t) corresponding to the physical situation. 2.2 MULTIPLY CONNECTED DOMAINS. ADDITIONAL Z . 3 SINGULAR SOLUTIONS. - If P vanishes at a point, BOUNDARY CONDITIONS. - To determine a solution vj may be singular. Consider for instance, the function of the probabilistic equations, we must add boundary conditions. Thus, we have to assume that P and Jj cp(x = r cos a, y = r sin a, t) = eima. Jm(kr) e-‘k2‘ , vanish at the border of the domain (and also a1P since q5 = 0 on the border). which is a solution of the equation These conditions are not sufficient if the domain is not simply connected Let us consider, in a two- dimensional space, the propagation of a wave of wave vector k inside a rectangular strip with cyclic boundary conditions. The corresponding values of P in the circle r R, for values of k given by Jm(kR) = 0. In the of the and Jj are constants and the probabilistic equations vicinity origin do not give the value of k. We need an additional boundary condition. Thus, let us consider a multiply connected domain (a torus) and, in this space, a circuit C which cannot be reduced to a point by continuous deformation (see Fig. 1). Let a be the phase of 0 at a given point

Quantum mechanics tell us that the variation of a The is at the and the along C must be a multiple of 2 n velocity singular origin phase of the wave function is also singular at this point. In this case, we need an additional boundary condition. If a solution of the probabilistic equation is singular at the origin, we quantize it by assuming that On the other hand, equation (7) gives it is defined in the domain obtained by removing the origin from the square box. This domain is not simply connected, consequently we draw a circuit around the origin and we write the corresponding boundary condition given by equation (27). In the present and equation (25) can be written in the form case n = m. In general, in three dimensional space, if T is complex, P vanishes on a line which can be considered as a kind of vortex line. The solutions of the probabilistic equations are quantized by writing boundary The circuit C can be considered as the of a edge conditions corresponding to circuits drawn around two-sided surface 8 and we have (Stokes Theorem) : these vortex lines. If, on the contrary, 9 is real, Jj vanishes everywhere and therefore vj also vanishes everywhere. In three dimensional space, P may vanish on surfaces and the of _p 112 has to be continuous on the sur- where defines an element of the surface S. Con- gradient d aj face. sequently, equation (26) gives

3. A transformation of the three-dimensional Dirac equation.

Thus we obtain a global boundary condition 3.1 THE EQUATION ;§ ALGEBRAIC CONSIDERATIONS. - the involving only local densities and the flux of Let us consider a quantum particle with spin 1/2 course it is neces- magnetic induction through S. Of in a three-dimensional relativistic space-time. The sary to write as many conditions as there are inde- coordinates of a point are x, y and t. The metric pendent irreducible nontrivial circuits in the domain tensor gjl has only diagonal elements. (their number is the genus of the domain). Thesp boundary conditions remind us of the Bohr quantiza- 891

We assume that a classical electromagnetic field Fj, (where I is the 2 x 2 unit matrix). Then a general is interacting with the particle. We have theorem [10] gives where Ai is a component of the vector potential. The which leads to equation (29). Using the representa- components Fxt and Fyt define the electric field; tion (22) of the matrices y’, we see that we may also the third component Fxy defines the magnetic induc- write : tion (and can be considered as a pseudo-scalar). The motion of a particle of spin 1/2 and mass M can be described a three-dimensional Dirac by equa- Let us multiply the preceding equation by the product tion. units = c = Using proper (Ii = 1, e 1, 1) 0. 0.’ Ob Y’b’ of components of 0. By summing over we write this in the usual equation way [7, 9]. all indices, we obtain

The current density has 3 components J j, Which where are real. This vector field defines the state of the system since the phase of 0 at each point is not well defined In we can make the transforma- and fact, gauge tion

(here the particle has a negative charge). are the Here 0 has two complex components. Accordingly The equations (30) and (31) invariant; quan- the symbols and P represent 2 x 2 matrices and 11 tities P(t, x, y), J’(t, x, y) and also the tensor Fik(t, x, y) remain We want now to eliminate the is the unit 2 x 2 matrix. These matrices can be express- unchanged. and from ed in terms of the Pauli matrices 61, Q2, U 3. For phase gauge dependent quantities. Starting the Dirac we shall derive « instance, we may set equation, probabilistic equations ». In these equations, the current density will replace the wave function and the electromagnetic field will replace the vector potential.

3.2 THE PROBABILISTIC EQUATIONS. - An equation The algebra of the matrices yi is determined by the of conservation is easily obtained Let us multiply relation equation (30) by § to the left and equation (31 ) by 0 to the right By adding the resulting equalities, we find where eikl is completely anti-symmetric and E‘xy = 1. At a given point (t, x, y), the density P and the current density Ji are defined by and therefore

We have now to derive dynamical equations. These equations will express the value of the electromagnetic where P and Ji are functions of (t, x, y). field applied to the particle at a given point, in terms At a given point 0 depends on 4 real parameters of the components of the current density and of their but the phase is not observable. Thus, as we shall derivatives at the same point see, the preceding quantities cannot be independent. We note that by applying various terms to the We shall use the identity lefthand side of equation (30) and to the righthand side of equation (31), it is possible to obtain several equations. From these equations we shall extract the value of the potential. Let us multiply equa- where is a matrix element of a,,,. vector Qa,ab tion (30) by yk to the left and equation (31) by yk QJ To derive this formula, we have only to consider to the right We obtain the group of quaternions and its representation by Pauli matrices : 892

Let us add these equalities. Using equation (34), we find

This equality is the « probabilistic equation » which we want, but the calculation is not finished. By looking in addition to we obtain three Thus, equation (39), at equation (46) we see that qljk is gauge invariant real These seem equations. equations independent and does not depend on the phase of 0. We have now from one another. We assume that are may they equi- to show that this quantity can be expressed in terms valent to the initial Dirac equation. Now, we may set of components J i of the current density and derivatives of these components. Equations (42) and (45) give

and we find

I Now it can be directly verified that Oik remains We shall now calculate the electromagnetic field invariant under the transformation

We may set Let us transform the first term on the righthand side. By using the identity (37), we find that

Let us bring this result in equation (47) (first part of the righthand side). We obtain

which we easily transform into

Using the definitions (35), we may also write The equation

In this we which is a direct consequence of equation (38) can be used to eliminate OjP from equation (51). way, obtain

where

or after a trivial reordering of the righthand side

write Both brackets on the right hand side are now phase independent More explicitly, we may

and therefore

and over b and a’, we obtain Again, we shall use equation (37). Multiplying this equation by 4>b 0., summing

in Let us use this identity to transform all terms of the form CPa; 00 which appear equation (53). We obtain

or in a more compact way

and therefore

Equations (33) and (34) tell us that

Thus equation (55) gives 894

The preceding equality and equation (52) give

Let us bring this expression in equation (46). We obtain the result

This equation and the relation (38) and (39)

define the « probabilistic equations >> associated with the three-dimensional Dirac equation. Another form of the same equation can be obtained by setting

In this way, we obtain

3. 3 REMARKS CONCERNING THE « PROBABILISTIC EQUA- them are « physical >> quantities. These physical quan- TIONS ». - We note that the local densities determine tities are the current density and the electromagnetic completely the electromagnetic held This is the best field which are gauge independent. we can do since the forces which produce the motion Unfortunately, to reach this interesting result, of the particle are completely determined by this we had to pay a high price : the simple linearity of motion. We also note that at each point the wave the Dirac equation has been lost Thus, from a practi- function depends on three « physic41 » parameters cal point of view. Equation (59) seems useless. We may and that equation (59) gives three relations at each only hope that it will lead to a more fundamental point Thus equation (59) should be completely interpretation of quantum mechanics. equivalent to the Dirac equation. This equivalence is explicitly proved in appendix A. 3.4 MULTIPLY CONNECTED DOMAINS. ADDITIONAL The proof is valid in regions where P # 0. When BOUNDARY CONDITIONS. - When the domain in which P = 0 singularities occur, in general, the compo- the particle propagates is multiply connected, addi- nents Vj become infinite on 2-dimensional surfaces tional boundary conditions have to be introduced which are boundaries of regions of the 3-dimensional as was explained in section 2.2. We consider a cir- space. In this case, as we have a simple pole on the cuit C which cannot be reduced to a point by conti- surface, the matching condition is trivial. nuous deformation (see Fig.1) and we want to express These probabilistic equations have great merits ; the fact that the variation along C of the global phase they are real and all the variables which appear in a of 0 is a multiple of 2 1t. 895

The phase a is defined by writing the components axis. In this case 4>1 and 4>2 of 0 as follows

where (P is the flux of the magnetic induction through the circuit C. where a and b are real positive. The boundary condition is 3.5 CLASSICAL LIMIT. - For reasons of simplicity, we wrote Oj instead of h Dj in Dirac equation (30). If we had kept the factors there, it would also appear in equation (61) in front of the terms which contain two derivatives. Thus, we see that the classical limit We shall express this condition in terms of local (h = 0) of equation (61) is densities by using equation (43)

This equation has also been obtained many years ago (from Klein Gordon equation) by Takabayasi [11]. where (see Eq. (35)) The preceding equality gives :

On the other hand equation (60) gives

In fact we shall apply equation (52) along e. We note that Ck = ak a + terms independent of a which leads to and consequently that (Ck - aka) is a uniform function of x, y, t. Accordingly, we have (see Eq. (45)) Thus by combining the preceding equations we find

The operation vk ak is a differentiation along the trajectory. Introducing

we may also write equation (66) in the form Now using equation (63), let us calculate the circulation of Ak along e. With the help of equations (62) and (43), we find which give the classical relativistic motion of the electron (Lorentz equation).

4. A transformation of the four-dimensional Dirac equation.

Finally, we obtain the boundary condition 4.1 THE DIRAC EQUATION AND THE ALGEBRA 6F THE yi. - We consider now the motion of a quantum particle with spin 1/2 in a four-dimensional relativistic space-time. The coordinates of a point are x, y, z, and t. The metric tensor g il has only diagonal elements

We assume that a classical electromagnetic field Fj, is interacting with the particle. We have This equation is written in a relativistic form. However, it is convenient to assume that along C we have dr, = 0 and that 8 is perpendicular to the t where A; is a component of the vector potential. 896

The motion of a particle of spin 1/2 and mass M where according to equation (75) is described by the usual Dirac equation which will be written [8]

Thus for any matrix X belonging to E where

We shall now use these equalities to obtain the identity which defines the o Fierz transformations » [12]. The matrix x has matrix elements Tab and a particular X will be defined by setting The field q5 has now four components and the symbols 7i represent 4 x 4 matrices ; 4 is now the 4 x 4 unit matrix. It will be useful to define the spin tensor Qjk and the matrix Y5 by setting where ao and bo have fixed values. For this X, we have

and accordingly we have and therefore, equation (77) gives

We also define y5 which has the properties

In a more standard form, we may write

We shall also use the notation and this basic identity will be used several times in the following. where si’mn is completely antisymmetric and Btxyz = 1. Let us now derive a useful identity (similar to Eq. (26)). We note that the space E of the 4 x 4 matrices 4.2 LOCAL DENSITIES. - At each point (x, y, z, t) which are of the form u where u is an hermitian y’ the T A generate 16 local densities of the 4 x 4 matrix can be the 16 matrices spanned by using following form matrices l¡)r A 0, namely

We shall also use the notation

Two matrices T A and T B belonging to this set of It is not difficult to show and it is important to 16 matrices obey the relation note that all the quantities which are defined by equations (79) are automatically real, as we want The 16 local densities defined by equations (79) where are not independent In fact, the wave function 0 has four complex components. Thus, as the local densities do not depend on the global phase of 0, they are functions of 7 independent variables. Thus, we must find 9 independent relations among these Thus any 4 x 4 matrix x belonging to E can be expanded in the form quantities. These relations can be deduced from the identity (78). In this way, we find (see Appendix B) the 897

following (complete) sets of equations : However these matrix elements are not defined in a unique way,’ they are not -independent, and they are not tensors. Thus, in principle, it is better to introduce local densities. and 4.3 THE KINEMATIC PROBABILISTIC EQUATIONS. - Let us start by writing the Dirac equation

It is easy to find out current conservation equations. Let us multiply equation (86) by § to the left and equation (87) by 0 to the right By adding the resulting It is also possible to find these relations by using equalities, we obtain special representations of the yi and an adequate system of reference. The local densities P, Q, Ji, and Ki which obey and therefore to equation (79) equations (81) are just sufficient to define the state of according the particle at a point (t, x, y, z). In particular, we = 0 (an equation which is similar to (39)). may set OjJj (88) Let us now multiply equation (86) by (j)ys to the left and equation (87) by - y’ 0 to the right By adding the resulting equalities, we obtain

Thus, T’‘ is the plane containing the associated with and therefore, according to equations (89) vectors ii and Kj and Tjl with the plane perpendicular to these vectors. Now, equations (82) show that the antisymmetric tensor S’‘ can be expressed in terms of T’1 and !7j’ We can now derive a more complicated set of kinematica.l equations. Let us multiply (86) by - i(j)y" to the left and (87) by ’Yk 0 to the right. By adding these equalities, we obtain and this equation is just a compact form of equations (82). (This equality can be easily established by taking the axis Ot along J and the axis Oz along K, which is possible according to equations (81).) We also find an equation which is very similar to (41). Another equation of the same kind can be obtained as follows. We multiply (86) by - ioy5 y. to the left and (87) by iyk y5 0 to the right By adding these equalities, we obtain (the results in the first line can be used to derive the second one). It has to be shown that the local densities defined by the preceding equalities obey probabilistic equations, which are to be deduced from the Dirac equation. Accordingly, we shall now derive : Equations (90) and (91) are simple because both of 1) Kinematic probabilistic equations which involve them express At in terms of matrix elements. For only local densities. this reason, these equations will be used as a starting 2) Dynamical probabilistic equations which involve point to obtain new probabilistic equations. local densities and the external electromagnetic field In particular, a new set of kinematic equations is obtained by eliminating At from these equations. Let we note with M. Gaudin Incidentally, may [13] us set that it would be certainly easier to write these equations in terms of the phase independent elements 0. Q>b of the density matrix at the point (t, x, y, z). 898

Then, by combining (90) and (91), we obtain the With any 4 x 4 matrix U, we can associate the equation phase independent quantity and using definitions (79) and (80), we may also write and we see from equation (92) that D. is of that form. In fact, we have The vector Dk defined by (92) is phase independent (gauge invariant); thus to derive a new kinematic probabilistic equation, we have only to express Dk Let us show how can be in terms in terms of local densities (and derivatives of these Hk( U) expressed of local densities. Equation (95) can be written as quantities). follows

Now we have to use equation (78). Multiplying this equation by 4Jø 4>b’ and summing over repeated indices, we obtain

With the help of this identity, we transform equation (97) into

and finally we obtain

Thus according to equation (96), we have

and we see that on the right hand side of this equation all matrix elements are local densities. Finally equations (93) and (100) lead to the kinematic probabilistic equations

4.4 THE DYNAMIC PROBABILISTIC EQUATION. - In the The aim of this operation is the elimination of all preceding section, we have found for Ak two different phase dependent (gauge dependent) terms. Using expressions (Eqs. (90) and (91)) which give two inde- equations (90), let us write Fit as the sum of two pendent conditions. By eliminating Ak, we found the terms kinematic equation (101). We have to take into account another equation and we shall choose equation (90) (see also Eqs. (79)). The first term is fairly simple

From this equation, we can deduce the strength of the electromagnetic field Fik since it is directly expressed in terms of local densities. 899

The second term is more complicated :

The problem is to express "’jk in terms of local densities. We have

(this equality is similar to Eq. (47) but now 4/ has four components instead of two). Let us use equation (78). Multiplying this equation by ÔjPa ô"tPa’ tPb tPb’ we find

an expression which will be used to transform the first bracket on the right hand side of equation (107). In this way, we obtain

an equation which can also be written as follows

Let us again use equation (78). Multiplying equation (78) by 0. 4>ø’ (f)b 4>b’ we find and by differentiation

We may now replace Ôj«¡)l!J) in equation (109) by the value given by the preceding equation and we do the same for Ôk(l!Jl/». Then, it is easy to see that 4lik can be written the following form

where

and we note that I’ is phase invariant. By comparing equations (112) and (95), we see that

and using the result (99), we find

Let us bring this result in equation (111). We find

JOURNAL DE PHYSIQUE. - T. 44, No 8, AOBT 1983 900

Equations (104), (105), and (114) give the dynamical equation

(Note that i Tr (r Á[r B’ r c]) is always real.)

4.5 MULTIPLY CONNECTED DOMAINS. ADDITIONAL

BOUNDARIES CONDITIONS. - When the domains in which the particle propagates is multiply connected additional conditions have to be intro- boundary and we note that these boundary conditions involve duced The method used in the present case and the only local densities and the electromagnetic field. method described in section 3.4 are very similar. The global phase a is defined by writing the components §1, 02, cP3’ cP4 of 0 as follows 4.6 RESULTS. - We summarize here the results which have been found for the four-dimensional Dirac equation. We have introduced a set of complex matrices F’ (see Eqs. (74)) and defined local densities of the form (t/JrÂ t/J); these local densities are real tensors. With each A, we also associate a number 8(A ) and where a, b, c, d are real positive. The boundary condi- (see Eqs. (75) (76)). The local densities and tion corresponding to a circuit C (see Fig. 1) is P, JJ9 Sik9 K j, Q (see Eqs. (79)) obey the following conditions:

We shall express this condition in terms of local and probabilities by using equation (102)

where

(see Eqs. (81), (83), (84)). We calculate the circulation of along the circuit C Ak From Dirac equation, we deduced : and by reasoning exactly as in section 3.4, we find 1) conservation equations (Eqs. (88), (89))

2) a set of kinematic equations (Eq. (101)) where now 4/jt is given by (114). Finally, we obtain

3) a set of dynamical equations (Eq. (115)) 901

The local densities depend on seven independent parameters. By looking at equations (90) and (91) from which the preceding probabilistic equations are deduced we can count the number of independent equations (2 x 4 - 1) and it is not difficult to realize that the preceding equations should be sufficient to determine the time dependence of the local densities. The same conclusion can be reached by looking at the preceding equations. We see that we have four kinematic equations and six dynamical equations. However the dynamical equations are not independent. The second group of Maxwell equations is automatically satisfied (see Eq. (103)), this gives three independent conditions and we have only three independent dynamical equations. Thus, this discussion shows that equations (120), (121), and (122) can be considered as equivalent to the Dirac equation. However, if the domain is multiply connected we must add boundary conditions of the form (see Eq. (118) :

where t/ljt is (see Eq. (114))

5. Probabilistic description of the Aharonov-Bohm experiment.

S .1 INTERSECTING BEAMS. - Before discussing the Aharonov-Bohm experiment in connection with the existence of probabilistic equations, we shall consider a simpler experiment. We consider waves propagating in a plane inside two linear channels of width 2 a, crossing each other at right angles (see Fig. 1). The arms of the cross will be called X +, X -, Y+, and Y -. The waves are assumed to be solutions of the time independent Schr6- dinger equation

where .

--- - We want to study the interference (at the crossing) of two beams propagating along Ox and Oy. We note that far from the center of the cross, in one channel, the local densities P(x, y) and J(x, y) should not apriori depend on the global phase of the beam propagating in this channel. However, the interference pattern at the centre of the cross depends on the difference between the phases of the beams; thus this phase difference has a physical meaning; therefore it should appear in P(x, y) and J(x, y). How can we expect such an effect ? This is the which will be solved question here. I

Fig. 2. - Quantum waves propagate along Ox and Oy in a cross-shaped wave guide and interfere at the centre of the cross. 902

We may associate solutions of the Schr6dinger equation with the following irreducible representations of the group of the cross.

A representation will be denoted by the index j (j = 1, 2, 3) and the phase shift v will be associated with it With the symbols X +, X -, Y +, and Y -, we shall associate functions X + (x, y), X - (x, y), Y + (x, y), and Y-(x, y) which are solutions of the Schr6dinger equation in the corresponding arms and vanish elsewhere. By definition, we have (asymptotically) :

Thus far from the centre of the cross, we may write the general solution of energy

in the form

Incidentally, we remark : 1) that the solutions associated with representation (3) propagate only along one axis and are not scattered side-ways, that they vanish at the centre of the cross, and therefore that they cannot interfere with any other solution; 2) that in the absence of any scattering, we would get vi = V2 = 0, v3 = n/2. Let us consider now a solution 4J(x, y). It will be convenient to set

In the same way, we associate with P(x, y) and J(x, y), the quantities PX(x), JX(x) and PY( y), JY( y). In particular, we have :

In order to understand the situation better, we shall first assume that we have only one incoming beam along Ox (see Fig. 3). Along X +, Y+, and Y-, we have only pure outgoing waves. It is not difficult to see that

Fig. 3. - A wave coming from - oo on the x-axis is scattered by the cross. 903 this solution is given by setting

Then far from the centre of the cross, we have

The corresponding local densities are

and

Thus, we see that the incoming wave is scattered in all directions. In particular, it is scattered backwards and, in X -, the outgoing wave interferes with the incoming wave (look at the expression of Pi(x)). In the other channels, the density is uniform. We could treat in the same way the case where we have one incoming beam along Oy. Now, it would be possible to treat the case where we have two incoming beams along 4x and Oy (see Fig. 4) by applying the superposition principle. The result would be complicated but we guess what happens in this case : the beams are scattered at the crossing; in particular the beam propagating in the arm X - towards the centre of the cross is scattered sideways in the arm Y- and interferes with the other beam (and conversely). Thus, if the incoming beams are out of phase the phase difference will appear in the interference terms. In particular, this phase diffe- Fig. 4. - Two waves coming from - oo on the x-axis rence will appear in the expressions of Pi (x) and and the y-axis interfere at the centre of the cross and are Pi (y). scattered as shown on the picture. 904

The conclusion of this discussion is the following. We can solve this Schr6dinger equation in the When two incoming beams propagating in a wave channels and at the intersection points 0 and I. guide cross, producing an interference pattern at the We assume that everywhere in the domain where centre of the cross, interference effects can also be the particle propagates, the electromagnetic field felt far from the crossing point Thus, there exists a vanishes. Thus, when considering local solutions of correlation between the behaviour of the local den- the Schr6dinger equation, we may always assume sities far from the crossing point and the interference that Ax = 0, Ay = 0, and V = 0 in equation (134). pattern at the centre of the cross. This effect would In the preceding section, we have studied the nature appear by writing explicitly the expressions of PX (x), of the solutions of the Schr6dinger equation at a Pi (x), Py+(y), and Py (y) corresponding to the crossing like I. We can make a similar analysis of the present situation. In this case, the solution of the solutions of the Schr6dinger equation at the bifur- probabilistic equation depends on three real para- cation 0. For simplicity, we may consider that the meters as can be shown in various ways. We may bifurcation is completely symmetric and this « tris- say for instance that these parameters are the ampli- kele » is drawn on figure 6. Its arms will be called tudes of the incoming beams and their phase diffe- To, T+, and T-. We associate solutions of the Schr6- rence which is a physical quantity. dinger equations in this triskele with the following irreducible representation of the group of the triskele 5.2 REMARKS CONCERNING THE AHARONOV-BOHM EXPERIMENT. - The setup of the experiment is shown in figure 5. An incoming beam is split in 0 into two beams which interfere in I. In the middle, a coil may produce a magnetic flux through the central hole. In the preceding section, we discussed the situation in which two beams cross each other and we noted that the probabilistic equations had several solutions. The solution which correspond to a given physical situation must be determined by supplementary boundary conditions. The same remark applies here. At a given energy, several solutions are possible but only one solution is compatible with a given magnetic flux through the hole. The preceding statement is supported by the following analysis. We assume that the waves propagate in narrow channels of width 2 a ; they are assumed to be solutions of the time independent Schr6dinger equation

where

Fig. 6. - An incoming wave is split into two waves by the triskele.

A representation will be denoted by the index j (where j =1, 2) and the phase shift vj will be associated with it With the symbols To, T +, T -, we associate functions T °(x, y), T +(x, y), and T-(xy) which are solutions of the Schr6dinger equation in the corresponding arms (far from 0) and which vanish else-

Fig. 5. - An incoming beam is separated in 0 into two where. beams which afterwards interfere with each other in I. The For instance waves are scattered at each crossing. A magnetic flux 0 can be produced through the central hole. This flux modifies the interference pattern in I. Thus, far from the centre of the triskele, we may 905 write the general solution energy E in the form

- - .. -.

Thus a wave function in the triskele depends on three tions involving only local densities and electromagne- complex parameters, and as we have seen in the pre- tic fields. Consequently, all phase and gauge effects ceding section, a wave function in the cross depends can be eliminated on four complex parameters. Instead of looking at the wave functions we may consider the correspond- 6. Conclusion. ing local densities which are solutions of the proba- we have studied bilistic equations. In the triskele, the local densities In this article, one-particle quantum and we have shown that these equations depend on fve real parameters, in the cross on seven equations could be transformed so as to eliminate all and real parameters. phase and all related to the To determine the total number of independent gauge effects, ambiguities representation of the Dirac matrices (1). We believe that parameters, we have now to examine how many condi- this is an result tions we have to fulfill. interesting The reader that has been Firstly, the local densities P(x, y) and y) at may object nothing gain- Jj(x, ed in this that it is not useful to res- the entrance and at the exit of a channel must be process, always trict oneself to work only with group invariants and In a channel, the wave function compatible. O(x, y) that reference frames is How- is of the form using quite legitimate. ever, we think that this point of view which may be sensible in other occasions is not adapted to the present situation : x is the channel where the coordinate along and y 1) because we do not want to make practical cal- the perpendicular coordinate. culations but to get a more basic insight into quan- Secondly, in the Aharonov-Bohm experiment, the tum mechanics, waves in arms Y + are propagating the X + and purely 2) because the group which is eliminated is un- outgoing waves. Equation (136) immediately shows physical, that for each channel this gives two real conditions, 3) because it seems that difficulties of interpreta- and therefore, in this way, we find four real conditions. tion in quantum mechanics may be related to the Let us summarize. The triskele and the cross give fact that the wave function is not an observable and twelve parameters (5 + 7) and we found ten condi- experience in other fields (for instance critical pheno- tions (6 + 4). There remains two independent para- mena) has taught us that it is sometimes very impor- meters. Clearly one parameter is an amplitude which tant to work with observable quantities. is determined by the amplitude of the incoming wave We must also emphasize another point We have in channel T°. The other parameter corresponds to shown that in quantum mechanics a limited number a phase difference between the channels (T+ X - ) of mean values obey a closed set of equations which and (T- Y- ). Now, we observe that, until now, we determine their time dependence. This is a very took into account only local boundary conditions. remarkable result : it was not a priori obvious that This is not sufficient here because the domain of pro- closed sets of equations should exist In fact, very pagation drawn on figure 5 is not simply connected. different situations occur in statistical mechanics. We have to draw a circuit C around the hole and we Then, in general, mean values an infinite set of must take into account the boundary condition obey equations and finite sets are obtained only by trunca- (see Eq. (20)) tion. From a conceptual point of view, the new formalism which is presented here, has the great advantage of being more physical since all equations are real where 8 is the surface bounded by C, B the magnetic and since it eliminates all quantities which cannot defined induction perpendicular to the plane (x, y) and 0 be. without ambiguity. the flux through the hole. Now everything is deter- We note that the equations which we obtained are mined local but we do not know how to interprete this fact. Thus, this rather long discussion leads to a simple conclusion : in an Aharonov-Bohm experiment, all (1) In this respect, our probabilistic equations differ from physical effects can be predicted by solving pro- other recent reformulations [16, 17] of Dirac equation, babilistic equations and by applying boundary condi- which « a priori » may look similar. 906

The problem of locality has been much discussed in quantum mechanics [ 14,15]. However, this interesting question is outside the scope of the present article. We have also to that the emphasize probabilistic where 0 is related to q5 in the usual way. Here P and Ji which have been derived in this article equations are real and obey (60) cannot be directly interpreted These equations cannot be considered as really fundamental and a deeper analysis of their nature and of their meaning is needed From another point of view, the transformation Using for y’ the representation given by (33), we which has been made does not allow us to use the may write equations (A.1) in the form superposition principle and practically (and also theoretically), this is a serious drawback. However, the probabilistic equations may be only a consequence, of more primitive and simple physical processes, which have to be discovered This is what we expect The preceding transformation applies only to one-particle quantum equations, but it seems that It will be assumed that J’ > 0. it should be to extend it to with seve- possible systems Now, in accordance with equation (A. 2), we may ral interacting particles. Again one should establish set equations describing the evolution of local densities and an analysis of such equations may help to solve the paradoxes of quantum mechanics. In any case, the new formalism does not allow to discuss « the reduction of the wave packet » since all wave functions are eliminated !t It would also be interesting to transform field theory in the same spirit namely by eliminating all phase and gauge dependence at every stage of the theory. How- where peu and v are real. ever, it is hard to foretell whether this ambitious pro- In this way, we obtain solutions of equation (A. 3) gram can lead to really useful results ! It is not obvious of the form that it may provide a clear reinterpretation of quantum mechanics. The present article only shows that much work remains to be done in this domain.

Acknowledgments. where a is arbitrary. The author is indebted to Dr. G. Mahoux for Thus, we may P him of of the Fierz transforma- using equations (A .1 ), replace reminding properties and in (59) and (39), by their expressions in terms tion and to Dr. R. Ball for constructive Ji criticism; of cP, and yi. Now, by using the identity (37) and by he is also indebted to a referee whose pertinent remarks proceeding backwards, we get (46). Let us now consi- the author to the version brought modify original by der the equation adjunction of useful proofs and discussions. This research has been pursued at the Institute for Theoretical Physics at Santa Barbara during a stay which has been supported by the National Science where Fjk is given by (45) and (46). Obviously, the Foundation, Grant No. PHY77-27084. quantity Ak which is defined by (41) in terms of 0, is a solution of equation (A. 6) and we may assume that the vector potential coincide with this solution (we note that the arbitrary phase a defines the gauge). A. - Reconstruction of the 3-dimensional Appendix Thus, we have only to show that Dirac equation Dirac from the equation corresponding probabilistic can be deduced from the equations equation. We start from the probabilistic equations given by (59), (60) and (39) and we want to reconstruct the Dirac equation (30). Firstly, we have to define a wave function. We choose matrices yJ which obey (32). We want to show that it is possible to find a two component function cP (01, cP2) which obeys the relations (Eqs. (41) and (34) lead to the second equation). 907

Equations (A. 8) and (70) give : Let us introduce two arbitrary 4 x 4 matrices U and Y. Equation (B .1) leads to the following identity

and by using (76) and (79), we can write it in the expli- cit form

We shall now multiply both sides of equation (A. 9) by yk 4> (which is a two component vector) and we shall transform the right hand side by using the identity (70) which reads : (the factor 2 comes from the fact that o4l and o’j correspond to the same F’). can now and V well matri- Firstly, let us deduce, from this equation, a few We replace U by defined ces obtain in this relations between local _useful identities. Let us multiply (A. 10) by y’ Ojo I., and way densities. For these the Pb Pb’; we obtain calculations, following algebraic equations are useful (see Eqs. (70), (71), (72)) :

Let us multiply (A. 10) by ðj(Pi Ib 4>ø’ 4>b’; we obtain

and and By subtracting (A. 11) (A .12) by taking (with etxyz = 1) into we find (A. 6) account, Choosing for U and V the following values, we obtain respectively

Now let us multiply (A. 10) by 0.’ lPb Ob’; we obtain

Finally, by using equations (A 13) and (A 14), the product of (A. 8) and yk cp can be transformed into

In general (0-0) :0 0 and therefore the preceding equation coincide with Dirac equation :

Thus, we have shown that the 3-dimensional Dirac equation is completely equivalent to the corresponding probabilistic equations.

Appendix B. - Relations between local densities.

We want to establish relations between the local Thus, by combining equations (B. 5), ..., (B .12), we densities P, Jj5 sit, Ki. and Q defined by equations (79). obtain the sets of equations (81) and (82). (The reader We start from equations (81) : may note that for P # 0 or Q # 0 it is easy to show that p K. = 0, but that this relation remains true if P=Q=0.) 908

References

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