New possible physical evidence of the homogeneous electromagnetic vector potential for quantum theory: idea of a test based on a g. p. thomson- like arrangement.
The physical evidence of the vector potential [??] field, distinctly of electric and/or magnetic local actions, is known as Aharonov-Bohm-effect (A-B-eff). It aroused scientific discussions for more than half a century (see [1-8] and references). As a rule in the A-B-eff context, the vector potential is curl-free field, but it is non-homogeneous (n-h) i.e. spatially non-uniform. In the same context, the alluded evidence is connected quantitatively with magnetic fluxes surrounded by the lines of [??] field. In the present paper we try to suggest a test intended to reveal the possible physical evidence of a homogeneous (h) [??] field. Note that in both n-h and h cases herein, we take into consideration only fields which are constant in time.
The announced test has as constitutive pieces three reliable entities (E) namely:
[E.sub.1]: The fact that a vector potential [??] field changes the values of the de Broglie wavelength [[lambda].sup.dB] for electrons.
[E.sub.2]: An experimental arrangement of the G. P. Thomson type, able to monitor the mentioned [[lambda].sup.dB] values.
[E.sub.3]: A feasible special coil designed so as to create a h-[??] field.
Accordingly, on the whole, the test has to put together the mentioned entities and, consequently, to synthesize a clear verdict regarding the alluded evidence of a h-[??] field.
Experimental setup of the suggested test is detailed in the next Section 2. Essential theoretical considerations concerning the action of a h-[??] field are given in Section 3. The above-noted considerations are fortified in Section 4 by a set of numerical estimations for the quantities aimed to be measured through the test. Some concluding thoughts regarding a possible positive result of the suggested test close the main body of the paper in Section 5. Constructive and computational details regarding the special coil designed to generate a h-[??] field are presented in the Appendix.
2 Setup details of the experimental arrangement
The setup of the suggested experimental test is pictured and detailed below in Fig. 1. It consists primarily of a G. P. Thomson-like arrangement partially located in an area with a h-[??] field. The alluded arrangement is inspired by some illustrative images [9,10] about G. P. Thomson's original experiment and it disposes in a straight line of the following elements: electron source, electron beam, crystalline grating, and detecting screen. An area with a h-[??] field can be obtained through a certain special coil whose constructive and computational details are given in the above-mentioned Appendix at the end of this paper.
The following notes have to be added to the explanatory records accompanying Fig. 1.
Note 1: If in Fig. 1 the elements 7 and 8 are omitted (i.e. the sections in special coil and the lines of h-[??] field) one obtains a G. P. Thomson-like arrangement as it is illustrated in the said references [9, 10].
Note 2: Surely the above mentioned G. P. Thomson-like arrangement is so designed and constructed that it can be placed inside of a vacuum glass container. The respective container is not shown in Fig. 1 and it will leave out the special coil.
Note 3: When incident on the crystalline foil, the electron beam must ensure a coherent and plane front of de Broglie waves. Similar ensuring is required  for optical diffracting waves at a classical diffraction grating.
Note 4: In Fig. 1 the detail 6 displays only the linear projections of the fringes from the diffraction pattern. On the whole, the respective pattern consists in a set of concentric circular fringes (diffraction rings).
3 Theoretical considerations concerning action of a h-[??] field
The leading idea of the above-suggested test is to search for possible changes caused by a h-[??] field in the diffraction of quantum (de Broglie) electronic waves. That is why we begin by recalling some quantitative characteristics of the diffraction phenomenon.
The most known scientific domain wherein the respective phenomenon is studied regards optical light waves . In the respective domain, one uses as the main element the so-called diffraction grating i.e. a piece with a periodic structure having slits separated each by a distance a and which diffracts the light into beams in different directions. For a light normally incident on such an element, the grating equation (condition for intensity maximums) has the form: a x sin [[theta].sub.k] = k[lambda], where k = 0, 1, 2, ... In the respective equation, [lambda] denotes the light's wavelength and [[theta].sub.k] is the angle at which the diffracted light has the k-th order maximum. If the diffraction pattern is received on a detecting screen, the k-th order maximum appears on the screen in the position [y.sub.k] given by the relation tan [[theta].sub.k] = ([y.sub.k]/D), where D denotes the distance between the screen and the grating. For the distant screen assumption, when D [much [much greater than] [y.sub.k], the following relation holds: sin [[theta].sub.k] [approximately equal to] tan [[theta].sub.k] [approximately equal to] ([y.sub.k]/D). Then, with regard to the mentioned assumption, one observes that the diffraction pattern on the screen is characterized by an interfringe distance i = [y.sub.k+1] - [y.sub.k] given through the relation
i = [lambda] [D/a]. (1)
Note the fact that the above quantitative aspects of diffraction have a generic character, i.e. they are valid for all kinds of waves including de Broglie ones. The respective fact is presumed as a main element of the experimental test suggested in the previous section. Another main element of the alluded test is the largely agreed upon idea [1-8] that the de Broglie electronic wavelength [[lambda].sup.dB] is influenced by the presence of a [??] field. Based on the two afore-mentioned main elements the considered test can be detailed as follows.
In the experimental setup depicted in Fig. 1 the crystalline foil 3 having interatomic spacing a plays the role of a diffraction grating. In the same experiment, on the detecting screen 5 it is expected to appear a diffraction pattern of the electrons. The respective pattern would be characterized by an interfringe distance [i.sup.dB] definable through the formula [i.sup.dB] = [[lambda].sup.dB] x (D/a). In that formula, D denotes the distance between the crystalline foil and the screen, supposed to satisfy the condition D [much greater than] [pi]), where p represents the width of the incident electron beam. In the absence of a h-[??] field, the [[lambda].sup.dB] of a non-relativistic electron is known to satisfy the following expression:
[[lambda].sup.dB] = [h/[p.sub.kin]] = [h/mv] = [h[square root of 2mE]]. (2)
In the above expression, h is Planck's constant while [p.sub.kin], m, v and E denote respectively the kinetic momentum, mass, velocity, and kinetic energy of the electron. If the alluded energy is obtained in the source of the electron beam (i.e. piece 1 in Fig. 1) under the influence of an accelerating voltage U, one can write E = e x U and [p.sub.kin] = mv = [square root of 2meU].
Now, in connection with the situation depicted in Fig. 1, let us look for the expression of the electrons' characteristic [[lambda].sup.dB] and respectively of [i.sup.dB] = [[lambda].sup.dB] x (D/a) in the presence of a h-[??] field. Firstly, we note the known fact  that a particle with the electric charge q and the kinetic momentum [[??].sub.kin] = mv in a potential vector [??] field acquires an additional (add) momentum, [[??].sub.add] = q[??], so that its effective (eff) momentum is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then for the electrons (with q = -e) supposed to be implied in the experiment depicted in Fig. 1, one obtains the effective (eff) quantities
[[lambda].sup.dB.sub.eff] (A) = [h/mv + eA]; [i.sup.dB.sub.eff] (A) = [hD/a(mv + eA)]. (3)
Further on, we have to take into account the fact that the h-[??] field acting in the experiment presented before is generated by a special coil whose plane section is depicted by the elements 7 from Fig. 1. Then from the relation (10) established in the Appendix, we have A = K x I, where K = [[[mu].sub.0]N/2[pi]] x ln ([R.sub.2]/[R.sub.1]).
Add here the fact that in this experiment mv = [square root of 2meU]. Then for the effective interfringe distance ff of the diffracted electrons, one finds
[i.sup.dB.sub.eff](A) = [i.sup.dB.sub.eff] (U, I) = [hD/a([square root
of 2meU]+ eKI)], (4)
[1/[i.sup.dB.sub.eff] (U, I)] = f(U, I) = [a[square root of 2me]/hD] [square root of U] + [aeK/hD] I. (5)
4 A set of numerical estimations
The verisimilitude of the above-suggested test can be fortified to some extent by transposing several of the previous formulas into their corresponding numerical values. For such a transposing, we firstly will appeal to numerical values known from G. P. Thomson-like experiments. So, as regarding the elements from Fig. 1, we quote the values a = 2.55 x [10.sup.-10] m (for a crystalline foil of copper) and D = 0.1 m. As regarding U, we take the often quoted value: U = 30 kV. Then the kinetic momentum of the electrons will be [p.sub.kin] = mv = [square root of 2meU] = 9.351 x [10.sup.-23] kg m/s. The additional (add) momentum of the electron, induced by the special coil, is of the form [p.sub.add] = e K x I where K = x ln ([R.sub.2]/[R.sub.1]). In order to estimate the value of K, we propose the following practically workable values: [R.sub.1] = 0.1 m, [R.sub.2] = 0.12 m, N = 2[pi][R.sub.1] x n with n = 2 x [10.sup.3] [m.sup.-1] = number of wires (of 1 mm in diameter) per unit length, arranged into two layers. With the well known values for e and [[mu].sub.0] one obtains [p.sub.add] = 7.331 x [10.sup.-24] (kg m [C.sup.-1]) x I (with C = Coulomb).
For wires of 1 mm in diameter, by changing the polarity of the voltage powering the coil, the current I can be adjusted in the range I [member of] (-10 to + 10)A. Then the effective momentum [[??].sub.eff] = [[??].sub.kin] + [[??].sub.add] of the electrons shall have the values within the interval (2.040 to 16.662) x [10.sup.-23] kg m/s. Consequently, due to the above mentioned values of a and D, the effective interfringe distance ff defined in (4) changes in the range (1.558 to 12.725) mm, respectively its inverse from (5) has values within the interval (78.58 to 641.84) [m.sup.-1]. Then it results that in this test the h-[??] field takes its magnitude within the interval A [member of] (-4.5, +4.5) x [10.sup.-4] kg m [C.sup.-1], (C = Coulomb).
Now note that in the absence of the h-[??] field (i.e. when I = 0) the interfange distance [i.sup.dB] specific to a simple G.P. Thomson experiment has the value [i.sup.dB] = [hD/a [square root of 2meU]] = 2.776 mm. Such a value is within the range of values of [i.sup.dB.sub.eff] characterizing the presence of the h-[??] field. This means that the quantitative evaluation of the mutual relationship of [i.sup.dB.sub.eff] versus I, and therefore the testing evidence of a h-[??] field can be done with techniques and accuracies similar to those of the G. P. Thomson experiment.
5 Some concluding remarks
The aim of the experimental test suggested above is to verify a possible physical evidence for the h-[??] field. Such a test can be done by comparative measurements of the interfringe distance [i.sup.dB.sub.eff] and of the current I. Additionally it must examine whether the results of the mentioned measurements verify the relations (4) and (5) (particularly according to (5) the quantity [([i.sup.dB.sub.eff]).sup.-1] is expected to show a linear dependence of I). If the above outcomes are positive, one can notice the fact that a h-[??] field has its own characteristics of physical evidence. Such a fact leads in one way or another to the following remarks (R):
[R.sub.1]: The physical evidence of the h-[??] field differs from the one of the n-h-[??] field which appears in the A-B-eff. This happens because, by comparison to the illustrations from , one can see that: (i) by changing the values of n-h-[??], the diffraction pattern undergoes a simple translation on the screen, without any modification of interfringe distance, while (ii) according to the relations (4) and (5) a change of h-[??] (by means of current I) does not translate the diffraction pattern but varies the value of associated interfringe distance [i.sup.dB.sub.eff]. The mentioned variation is similar to that induced  by changing (through accelerating the voltage U) the values of kinetic momentum [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for electrons.
[R.sub.2]: There is a difference between the physical evidence (objectification) of the h-[??] and the n-h-[??] fields in relation with the magnetic fluxes surrounded or not by the field lines. The difference is pointed out by the following subsequent aspects:
(i) On the one hand, as it is known from the A-B-eff, in case of the n-h-[??] field, the corresponding evidence depends directly on magnetic fluxes surrounded by the [??] field lines.
(ii) On the other hand, the physical evidence of the h-[??] field is not connected to magnetic fluxes surrounded by the field lines. But note that due to the relations (4) and (5), the respective evidence appears to be dependent (through the current I) on magnetic fluxes not surrounded by the field lines of the h-[??].
[R.sub.3]: A particular characteristic of the physical evidence forecasted above for the h-[??] regards the macroscopic versus quantum difference concerning the uniqueness (gauge freedom) of the vector potential field. As is known, in macroscopic situations [13, 14] the vector potential A field is not uniquely defined (i.e. it has a gauge freedom). In such situations, an arbitrary [??] field allows a gauge fixing (adjustment), without any alteration of macroscopic relevant variables/equations (particularly of those involving the magnetic field [??]). So two distinct vector potential fields [??] and [[??].sup.1] have the same macroscopic actions (effects) if [[??].sup.1] = [??] + [nabla]f, where f is an arbitrary gauge functions. On the other hand, in a quantum context, a h-[??] has not any gauge freedom. This is because if this test has positive results, two fields like [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and h - [[??].sup.1] = h - [??] + [nabla]f are completely distinct if f = (-z x A x [??]), where [??] denotes the unit vector of the Oz axis. So we can conclude that, with respect to the h-[??] field, the quantum aspects differ fundamentally from those aspects originating in a macroscopic consideration. Surely, such a fact (difference) and its profound implications have to be approached in subsequently more elaborated studies.
As presented above, the suggested test and its positive results appear as purely hypothetical things, despite the fact that they are based on essentially reliable entities (constitutive pieces) presented in the Introduction. Of course, we hold that a true confirmation of the alluded results can be done by the action of putting in practice the whole test. Unfortunately, at the moment I do not have access to material logistics able to allow me an effective practical approach of the test in question. Thus I warmly appeal to the concerned experimentalists and researchers who have adequate logistics to put in practice the suggested test and to verify its validity.
Appendix: Constructive and computational details for a special coil able to create a h-[??] field
The case of an ideal coil
An experimental area of macroscopic size with the h-[??] field can be realized with the aid of a special coil whose constructive and computational details are presented below. The announced details are improvements of the ideas promoted by us in an early preprint .
The basic element in designing the mentioned coil is the h-[??] field generated by a rectilinear infinite conductor carrying a direct current. If the conductor is located along the axis Oz and the current has the intensity I, the Cartesian components (written in SI units) of the mentioned h-[??] field are given  by the following formulas:
[A.sub.x](1) = 0, [A.sub.y] (1) = 0, [A.sub.z] (1) = -[[mu].sub.0] [I/2[pi]] ln r. (6)
Here r denotes the distance from the conductor of the point where the hct-[??] is evaluated and where [[mu].sub.0] is the vacuum permeability.
Note that formulas (6) are of ideal essence because they describe the h-[??] field generated by an infinite (ideal) rectilinear conductor. Further onwards, we firstly use the respective formulas in order to obtain the h-[??] field generated by an ideal annular coil. Later on we will specify the conditions in which the results obtained for the ideal coil can be used with fairly good approximation in the characterization of a real (non-ideal) coil of practical interest for the experimental test suggested and detailed in Sections 2, 3 and 4.
The mentioned special coil has the shape depicted in Fig. 2-(a) (i.e. it is a toroidal coil with a rectangular cross section). In the respective figure the finite quantities [R.sub.1] and [R.sub.2] represent the inside and outside finite radii of the coil while L [right arrow] [infinity] is the length of the coil. For evaluation of the h- [??] generated inside of the mentioned coil let us now consider an array of infinite rectilinear conductors carrying direct currents of the same intensity I. The conductors are mutually parallel and uniformly disposed on the circular cylindrical surface with the radius R. The conductors are also parallel with Oz as the symmetry axis. In a cross section, the considered array is disposed on a circle of radius R as can be seen in Fig. 2b. On the respective circle, the azimuthal angle [phi] locates the infinitesimal arc element whose length is Rd[phi]. On the respective arc there was placed a set of conductors whose number is dN = (N/2[pi]) d[phi], where N represents the total number of conductors in the whole considered array. Let there be an observation point P situated at distances r and [phi] from the center O of the circle respectively from the infinitesimal arc (see the Fig. 2b). Then, by taking into account (6), the z-component of the h-[??] field generated in P by the dN conductors is given by relation
[A.sub.z] (dN) = [A.sub.z] (1) dN = -[[mu].sub.0] [NI/4[[pi].sup.2]] ln [rho] x d[phi], (7)
where [rho] = [square root of ([R.sup.2] + [r.sup.2] - 2Rr cos [phi])]. Then all N conductors will generate in the point P a h-[??] field whose value A is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (8)
In calculating the above integral, the formula (4.224-14) from  can be used. So, one obtains
A = -[[mu].sub.0] [NI/2[pi]] ln R. (9)
This relation shows that the value of A does not depend on r, i.e. on the position of P inside the circle of radius R. Accordingly this means that inside the respective circle, the potential vector is homogeneous. Then starting from (9), one obtains that the inside space of an ideal annular coil depicted in Fig. 2a is characterized by the h-[??] field whose value is
A = [[mu].sub.0] [NI/2[pi]] ln ([R.sub.2]/[R.sub.1]). (10)
From the ideal coil to a real one
The above-presented coil is of ideal essence because their characteristics were evaluated on the basis of an ideal formula (6). But in practical matters, such as the experimental test proposed in Sections 2 and 3, one requires a real coil which may be effectively constructed in a laboratory. That is why it is important to specify the main conditions in which the above ideal results can be used in real situations. The mentioned conditions are displayed here below.
On the geometrical sizes: In a laboratory, it is not possible to operate with objects of infinite size. Thus we must take into account the restrictive conditions so that the characteristics of the ideal coil discussed above to remain as good approximations for a real coil of similar geometric form. In the case of a finite coil having the form depicted in the Fig. 2a, the alluded restrictive conditions impose the relations L [much greater than] [R.sub.1], L [much greater than] [R.sub.2] and L [much greater than] ([R.sub.2] - [R.sub.1]). If the respective coil is regarded as a piece in the test experiment from Fig. 1, indispensable are the relations L [much greater than] D and L [much greater than] [phi].
About the marginal fragments: On the whole, the marginal fragments of coil (of width ([R.sub.2] - [R.sub.1])) can have disturbing effects on the Cartesian components of [??] inside the the space of practical interest. Note that, on the one hand, in the above-mentioned conditions L [much greater than] [R.sub.1], L [much greater than] [R.sub.2] and L [much greater than] ([R.sub.2] - [R.sub.1]) the alluded effects can be neglected in general practical affairs. On the other hand, in the particular case of the proposed coil the alluded effects are also diminished by the symmetrical flows of currents in the respective marginal fragments.
As concerns the helicity: The discussed annular coil is supposed to be realized by winding a single piece of wire. The spirals of the respective wire are not strictly parallel to the symmetry axis of the coil (the Oz axis) but they have a certain helicity (corkscrew-like path). Of course, the alluded helicity has disturbing effects on the components of [??] inside the coils. Note that the mentioned helicity-effects can be diminished (and practically eliminated) by using an idea noted in another context in . The respective idea proposes to arrange the spirals of the coil in an even number of layers, with the spirals from adjacent layers having equal helicity but of opposite sense.
Submitted on May 6, 2014/Accepted on May 30, 2014
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(Retired) Department of Physics, "Transilvania" University, B-dul Eroilor 29,
500036 Brasov, Romania
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