Possible physical self-asserting of the homogeneous vector potential. A testing puzzle based on a G.P. Thomson-like arrangement
PPossible physical self-asserting of the homogeneousvector potential
A testing puzzle based on a G.P. Thomson-like arrangementSpiridon Dumitru (Retired)Department of Physics, ”Transilvania”
University,B-dul Eroilor 29, 500036 Brasov, RomaniaE-mail: [email protected]
November 10, 2018
Abstract
It is suggested a testing puzzle able to reveal the self-assertingproperty of the homogeneous vector potential field. As pieces of thepuzzle are taken three reliable entities : (i) influence of a potentialvector on the de Broglie wavelength (ii) a G.P. Thomson-like experi-mental arrangement and (iii) a special coil designed to create a homo-geneous vector potential. The alluded property is not connected withmagnetic fluxes surrounded by the vector potential field lines, but itdepends on the fluxes which are outside of the respective lines. Alsothe same property shows that in the tested case the vector potentialfield is uniquely defined physical quantity, free of any adjusting gauge.So the phenomenology of the suggested quantum test differs on that ofmacroscopic theory where the vector potential is not uniquely definedand allows a gauge adjustment. Of course that the proposed test hasto be subjected to adequate experimental validation.
PACS Codes: 03.75.-b ,03.65.Vf, 03.65.Ta, 03.65.Ca, 06.30 KaKeywords: Homogeneous vector potential, de Broglie wavelength, G.P.Thomson-like experiment, Electrons diffraction fringes, Outside magneticfluxes, Needlessness of an adjusting gauge.1 a r X i v : . [ qu a n t - ph ] A p r Introduction
The physical self-asserting (objectification) of the vector potential (cid:126)A field,distinctly of electric and/or magnetic local actions, is known as Aharonov-Bohm Effect (ABE). It aroused scientific discussions for more than half acentury ( see [1–8] and references). As a rule in ABE context the vectorpotential is curl-free field, but it is non-homogeneous ( n-h ) i.e. spatiallynon-uniform. In the same context the alluded self-asserting is connectedquantitatively with magnetic fluxes surrounded by the lines of (cid:126)A field. Inthe present paper we try to suggest a testing puzzle intended to reveal thepossible physical self-asserting property of a homogeneous ( h ) (cid:126)A field,. Notethat in both n-h and h cases here we consider only fields which are constantin time.The announced puzzle has as constitutive pieces three reliable Entities( E )namely : • E : The fact that a potential vector change the values de Broglie wave-length λ dB of electrons. (cid:4) • E : An experimental arrangement of G. P. Thomson type, able tomonitor the mentioned λ dB values. (cid:4) • E : A feasible special coil designed so as to create a h - (cid:126)A field. (cid:4) Accordingly, in its wholeness, the puzzle has to put together the men-tioned entities and, consequently, to synthesize a clear verdict regarding thealluded property of a h - (cid:126)A field .Experimental setup of the suggested puzzle is detailed in the next Sec-tion 2. Esential theoretical considerations concerning the action of a h - (cid:126)A field are given in Section 3. The above noted considerations are fortified inSection 4 through a set of numerical estimations for the quantities aimed tobe measured by means of the puzzle. Some concluding thoughts regardinga possible positive result of the suggested puzzle close the principal body ofthe paper in Section 5. Constructive and computational details regarding thespecial coil designed to generate a h - (cid:126)A field are presented in the Appendix. The setup of the suggested experimental puzzle is pictured and detailed belowin Fig. 1. It consists in a G. P. Thomson-like arrangement partially locatedin an area with a h - (cid:126)A field . The alluded arrangement is inspired fromsome illustrative figures [9, 10] about G. P. Thomson’s original experimentand it disposes in a straight line the following elements: electrons source,2lectrons beam , crystalline grating and detecting screen. An area with a h - (cid:126)A field can be obtained through a certain special coil whose constructiveand computational details are given in the alluded Appendix. !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ! Fig. 1.
Plane ! section in the image of suggested experimental puzzle setup, accompanied by the following explanatory records h !" A r ! field ; 8 - h " A r field ; ∅ = the width of the electrons beam with ∅ ≫ a ( a = interatomic spacing in crystal lattice of the foil -3) ; θ k = diffraction angle for the k-th order fringe (k=0,1,2,3,…) ; k displacement from the center line of the ! k-th order fringe ; i= interfringe width = y k+1 – y k ; D = distance between crystalline foil and screen ( D >> ∅ ; L = length of the special coil (L ≫ D) ; I = intensity of current in wires of the coil.
1 2 3 4 D 5 6 7 7 8 θ k
8 y k z y i 6 y k i! ∅ ! L I
Figure 1:
Plane section in the image of suggested experimentalpuzzle setup, accompanied by the following explanatory records h − (cid:126)A field;8 – h − (cid:126)A field ; φ = the width of the electrons beam with φ (cid:29) a ( a =interatomic spacing in crystal lattice of the foil -3); θ k = diffraction angle forthe k -th order fringe ( k = 0 , , , , . . . ) ; y k = displacement from the centerline of the k -th order fringe ; i = interfringe width = y k +1 − y k ; D = distancebetween crystalline foil and screen ( D (cid:29) φ ); L = length of the special coil( L (cid:29) D ) ; I = intensity of current in wires of the coil.The explanatory records accompanying Fig. 1 have to be supplementedwith the next notes: • Note 1 : If in Fig. 1 are omitted the elements 7 and 8 ( i.e. the sectionsin special coil and the lines of h - (cid:126)A field ) one obtains a G. P. Thomson-likearrangement as it is illustrated in references [9, 10]. (cid:4) • Note 2 : Evidently the above mentioned G. P. Thomson-like arrange-ment is so designed and constructed that it can be placed inside of a vacuum3lass container. The respective container is not showed in Fig. 1 and it willleave out the special coil. (cid:4) • Note 3 : At the incidence on crystalline foil the electrons beam mustensure a coherent and plane front of de Broglie waves. Similar ensuringis required [11] for optical diffracting waves at the incidence on a classicaldiffraction grating. (cid:4) • Note 4 : In Fig. 1 the detail 6 displays only the linear projectionsof the fringes from the diffraction pattern. In its wholeness the respectivepattern consists in a set of concentric circular fringes (diffraction rings). (cid:4) (cid:126)A field
The leading idea of the above suggested puzzle is to search the possiblechanges caused by a h - (cid:126)A field in diffraction of quantum (de Broglie) electronicwaves. That is why now firstly we remind some quantitative characteristicsof the diffraction phenomenon.The most known scientific domain where the respective phenomenon isstudied regards the optical light waves [11]. In the respective domain oneuses as main element the so called ”diffraction grating” i.e. a piece withperiodic structure having slits separated by distances a and which diffractsthe light into beams in different directions. For a light normally incident onsuch an element the grating equation (condition for intensity maximum ) hasthe form : a · sin θ k = kλ , where k = 0 , , , ... . In the respective equation λ denotes the light wavelength and θ k is the angle at which the diffracted lighthave the k − th order maximum. If the diffraction pattern is received on adetecting screen the k − th order maximum appear on the screen in position y k given by relation tan θ k = ( y k /D ), where D denote the distance betweenscreen and grating. For the distant screen assumption, when D >> y k , canbe written the relations: sinθ k ≈ tan θ k ≈ ( y k /D ). Then, with regard to thementioned assumption, one obtains that diffraction pattern on the screenis characterized by an interfringe distance i = y k +1 − y k given through therelation i = λ Da (1)Note the fact that the above quantitative aspects of diffraction have ageneric character, i.e. they are valid for all kinds of waves including thede Broglie ones. The respective fact is presumed as a main element of theexperimental puzzle suggested in the previous section. Another main elementof the alluded puzzle is the largely agreed idea [1–8] that the de Broglie4lectronic wavelength λ dB is influenced by the presence of a (cid:126)A field. Basedon the two before mentioned main elements the considered puzzle can bedetailed as follows.In experimental setup depicted in Fig. 1 the crystalline foil 3 havinginteratomic spacing a plays the role of a diffraction grating. In the sameexperiment on the detecting screen 5 is expected to appear a diffractionpattern of the electrons. The respective pattern would be characterized byan interfringe distance i dB definable through the formula i dB = λ dB · ( D/a ). Inthat formula D denote distance between crystalline foil and screen, supposedto satisfy the condition D >> φ ), where φ represents the the width of theincident electrons beam. In absence of a h - (cid:126)A field the λ dB of a non-relativisticelectron is known as having the expressions: λ dB = hp mec = hmv = h √ m E (2)In the above expressions h is the Planck’s constant while p mec , m , v and E denote respectively the mechanical momentum, mass, velocity and kineticenergy of the electron. If the alluded energy is obtained in the source ofelectrons beam (i.e. piece 1 in Fig. 1) under the influence of an acceleratingvoltage U one can write E = e · U and p mec = mv = √ meU .Now, in connection with the situation depicted in Fig. 1, let us look forthe expressions of the electrons characteristic λ dB and respectively of i dB = λ dB · ( D/a ) in presence of a h - (cid:126)A field. Firstly we note the known fact [6] thata particle with the electrical charge q and the mechanical momentum (cid:126)p mec = m(cid:126)v in a potential vector (cid:126)A field acquires an additional ( add ) momentum, (cid:126)p add = q (cid:126)A , so that its ”effective” (eff ) momentum is (cid:126)P eff = (cid:126)p mec + (cid:126)p add = m(cid:126)v + q (cid:126)A . Then for the electrons ( with q = − e ) supposed to be implied inthe experiment depicted in Fig. 1 one obtains the effective (eff) quantities λ dBeff ( A ) = hmv + eA ; i dBeff ( A ) = hDa ( mv + eA ) (3)Further on we have to take into account the fact that the h - (cid:126)A field acting inthe discussed experiment is generated by a special coil whose plane sectionis depicted by the elements 7 from Fig. 1. Then from the relation (10)established in Appendix we have A = K · I , where K = µ N π · ln (cid:16) R R (cid:17) . Addhere the fact that in the considered experiment mv = √ meU . Then for theeffective interfringe distance i dBeff of diffracted electrons one find i dBeff ( A ) = i dBeff ( U, I ) = hDa (cid:16) √ meU + e K I (cid:17) (4)5espectively 1 i dBeff ( U, I ) = f ( U, I ) = a √ mehD √ U + ae K hD I (5) The verisimilitude of the above suggested testing puzzle can be fortified tosome extent by transposing several of the previous formulas in their cor-responding numerical values. For such a transposing firstly we will appealto numerical values known from G.P. Thomson-like experiments. So, as re-gards the elements from Fig. 1 we quote the values a = 2 . · − m ( fora crystalline foil of copper) and D = 0 . m . As regards U we take the oftenquoted value: U = 30 · kV . Then the mechanical momentum of the electronswill be p mec = mv = √ meU = 9 . · − kg · m · s − . The additional(add) momentum of the electron, induced by the special coil, is of the form p add = e K · I where K = µ N π · ln (cid:16) R R (cid:17) . In order to estimate the value of K wepropose the following practically workable values: R = 0 . m , R = 0 . m , N = 2 πR · n with n = 2 · m − = number of wires (of 1 mm in diameter)per unit length, arranged in two layers. With the well known values for e and µ one obtains p add = 7 . · − ( kg · m · s − · A − ) · I (with A = ampere ).For wires of 1 mm in diameter, by changing the polarity of voltage power-ing the coil, the current I can be adjusted in the range I ∈ ( − to + 10) A .Then the effective momentum (cid:126)P eff = (cid:126)p mec + (cid:126)p add of the electrons have the val-ues within the interval (2 . to . · − kg · m · s − . Consequently, dueto the above mentioned values of a and D , the effective interfringe distance i dBeff defined in (4) changes in the range (1 . to . mm , respectively itsinverse from (5) has values within the interval (78 . to . m − .Now note that in absence of h - (cid:126)A field (i.e. when I = 0) the interfangedistance i dB specific to a simple G.P. Thomson experiment has the value i dB = hDa √ meU = 2 . mm . Such a value is within the values range of i dBeff characterizing the presence of a h - (cid:126)A field. This means that the quantitativeevaluation of the mutual relationship of i dBeff versus I and therefore of the self-asserting of a h - (cid:126)A field can be done with techniques and accuracies similarto those for simple G.P. Thomson experiment.6 Some concluding remarks
The aim of the experimental puzzle suggested above is to test a possiblephysical self-asserting for a h - (cid:126)A field. Such a test can be done concretely bycomparative measurements of the interfringe distance i dBeff and of the current I . Additionally it must to examine whether the results of the mentionedmeasurements verify the relations (4) and (5) ( particularly according to (5)the quantity ( i dBeff ) − is expected to show a linear dependence of I ). If theabove outcomes are positive one can be notified the fact that a h - (cid:126)A field hasits own characteristic of physical self-asserting. Such a fact leads in one wayor another to the following remarks ( R ): • R : The self-asserting of h - (cid:126)A field differs from the one of n-h - (cid:126)A fieldwhich appears in ABE. This because, by comparison with the illustrationsfrom [12] , one can see that : (i) by changing of n-h - (cid:126)A the diffraction patternundergoes a simple translation on the screen, without any modification ofinterfringe distance, while (ii) according to the relations (4) and (5) a changeof h - (cid:126)A (by means of current I ) does not translate the diffraction pattern butvaries the associated interfringe distance. The mentioned variation is similarwith those induced [12] by changing (through accelerating voltage U ) thevalues of mechanical momentum (cid:126)p mec = m(cid:126)v for electrons. (cid:4) • R : There are a difference between the objectification (self-asserting)of h - (cid:126)A and n-h - (cid:126)A fields in relation with the magnetic fluxes surrounded ornot by the field lines. The difference is pointed out by the next aspects: (i) On the one hand, as it is known from ABE, in case of a n-h - (cid:126)A fieldthe self-asserting is in a direct dependence on magnetic fluxes surrounded bythe field lines. (ii) On the other hand the self-asserting of a h - (cid:126)A field is not connectedwith magnetic fluxes surrounded by the field lines. But note that due to therelations (4) and (5) the respective self-asserting appears to be dependent(through the current I ) on magnetic fluxes not surrounded by field lines ofthe h - (cid:126)A . (cid:4) • R : Another particular characteristic of the self-asserting forecastingabove for h - (cid:126)A is that in the proposed test the vector potential field appearsas an uniquely defined physical quantity free of any adjusting gauge. So thephenomenology of the suggested test differs on that of macroscopic situationswhere [13, 14] the vector potential is not uniquely defined and allows a gaugeadjustment. Surely that such a fact (difference) and its implications have tobe approached in more elaborated studies.7 ostscript As presented above the suggested puzzle and its positive result appear aspurely hypothetical things, despite of the fact that they are based on theessentially reliable entities (constitutive pieces) presented in Introduction.Of course that a true confirmation of the alluded result can be done byan action of putting in practice the whole puzzle. Unfortunately I do nothave access to material logistics able to allow me an effective practical testof the puzzle in question. That is why I warmly appeal to experimentalistresearchers that have adequate logistics to put in practice the suggested testand to verify its validity.
Appendix
Constructive and computational details for a special coilable to create a h- (cid:126)A field
The case of an ideal coil
An experimental area of macroscopic size with a h - (cid:126)A field can be realizedwith the aid of a special coil whose constructive and computational details arepresented below. The announced details are improvements of ideas promotedby us in an early preprint [15].The basic element in designing of the mentioned coil is the h - (cid:126)A fieldgenerated by a rectilinear infinite conductor carrying a direct current. If theconductor is located along the axis Oz and current have the intensity I, theCartesian components ( written in SI units) of the mentioned h - (cid:126)A field aregiven [16] by formulas: A x (1) = 0 A y (1) = 0 A z (1) = − µ I π ln r (6)Here r denote the distance from the conductor of the point where h - (cid:126)A isevaluated and µ is vacuum permeability.Note that formulas (6) are of ideal essence because they describe a h - (cid:126)A field generated by an infinite (ideal) rectilinear conductor. Further firstly wewill use the respective formulas in order to obtain the h - (cid:126)A field generatedby an ideal annular coil. Later one we will specify the conditions in whichthe results obtained for the ideal coil can be used with good approximationin the characterization of a real ( non-ideal) coil of practical interest for thepuzzle-experiment suggested and detailed in Sections 2,3 and 4.The mentioned special coil has the shape depicted in Fig. 2-(a) (i.e. itis a toroidal coil of rectangular section). In the respective figure the finite8 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ( a ) ( b ) Fig. 2
Schemes for an annular special coil !! L I R R Z φ r dN I M R φ r d φ ρ P O
Figure 2: Schemes for an annular special coilquantities R and R represent the inside and outside finite radii of coil while L → ∞ is the length of the coil. For evaluation of the h - (cid:126)A generated insideof the mentioned coil let us now consider an array of infinite rectilinear con-ductors carrying direct currents of the same intensity I. The conductors aremutually parallel and uniformly disposed on the circular cylindrical surfacewith the radius R . Also the conductors are parallel with Oz as symmetryaxis. In a cross section the considered array are disposed on a circle of radius R as can be seen in Fig. 2-(b). On the respective circle the azimuthal angle ϕ locate the infinitesimal element of arc whose length is Rdϕ . On the respec-tive arc is placed a set of conductors whose number is dN = (cid:0) N π (cid:1) dϕ , where N represents the total number of conductors in the whole considered array.Let be an observation point P situated at distances r and ρ from the center O of the circle respectively from the infinitesimal arc (see the Fig. 2-(b) ).Then, by taking into account (6), the z-component of the h - (cid:126)A field generated9n P by the dN conductors is given by relation A z ( dN ) = A z (1) dN = − µ N I π ln ρ · dϕ (7)where ρ = (cid:112) ( R + r − Rr cos ϕ ) . Then the all N conductors will generatein the point P a h - (cid:126)A field whose value A is A = A z ( N ) = − µ N I π π (cid:90) ln (cid:0) R + r − Rr cos ϕ (cid:1) · dϕ (8)For calculating the above integral can be used formula (4.224-14) from [17].So one obtains A = − µ N I π ln R (9)This relation shows that the value of A does not depend on r , that is on theposition of P inside the circle of radius R . Accordingly this means that insidethe respective circle the potential vector is homogeneous. Then starting from(9), one obtains that the inside space of an ideal annular coil depicted inFig. 2-(a) is characterized by a h - (cid:126)A field whose value is A = µ N I π ln (cid:18) R R (cid:19) (10) From the ideal coil to a real one
The above presented coil is of ideal essence because their characteristics wereevaluated on the base of ideal formulas (6). But in practical matters, suchis the puzzle-experiment proposed in Sections 2 and 3, one needs of a realcoil which may be effectively constructed in a laboratory. That is why it isimportant to specify the main conditions in which the above ideal resultscan be used in real situations. The mentioned conditions are displayed herebelow. • On the geometrical sizes : In laboratory it is not possible to operatewith objects of infinite sizes. Then it must to note the restrictive conditionsso that the characteristics of the ideal coil discussed above to remain as goodapproximations for a real coil of similar geometric form. In the case of afinite coil having the form depicted in the Fig. 2-(a) the alluded restrictiveconditions impose the relations
L >> R , L >> R and L >> ( R − R ). Ifthe respective coil is regarded as a piece in the puzzle-experiment from Fig. 1there are indispensable the relations L >> D and
L >> φ .10
About the marginal fragments : In principle the marginal fragmentsof coil (of widths ( R − R )) can have disturbing effects on the Cartesiancomponents of (cid:126)A inside the the space of practical interest. Note that, onthe one hand, in the above mentioned conditions L >> R , L >> R and L >> ( R − R ) the alluded effects can be neglected in general practicalaffairs. On the other hand in the particular case of the proposed coil thealluded effects are also diminished by the symmetrical flowings of currents inthe respective marginal fragments. • As concerns the helicity : The discussed annular coil is supposed tobe realized by turning a single piece of wire. The spirals of the respectivewire are not strictly parallel with the symmetry axis of the coil ( Oz axis)but they have a certain helicity (corkscrew-like path). Of course that thealluded helicity has disturbing effects on the components of (cid:126)A inside the coils.Note that the mentioned helicity-effects can be diminished (and practicallyeliminated) by using an idea noted in another context in [18]. The respectiveidea proposes to arrange the spirals of the coil in an even number of layers,the spirals from adjacent layers having equal helicity but of opposite sense. References [1] Y. Aharonov and D. Bohm, Significance of electromagnetic potentialsin the quantum theory,
Phys.Rev. (1959) 485-491.[2] Y. Aharonov, D. Bohm, Further Considerations on Electromagnetic Po-tentials in the Quantum Theory,
Phys.Rev. (1961)1511-1524.[3] . S.Olariu, I. I.Popescu, The quantum effects of electromagnetic fluxes,
Rev. Mod. Phys. (1985) 339-436.[4] M. Peshkin, A. Tonomura, The Aharonov-Bohm Effect, Lecture Notesin Physics (Springer) (1989) 1-152.[5] M. Dennis, S. Popescu, L. Vaidman, Quantum Phases: 50 years of theAharonov-Bohm effect and 25 years of the Berry phase,
J. Phys. A:Math. Theor. (2010) 350301[6] A. Ershkovich, Electromagnetic potentials and Aharonov-Bohm effect,arXiv:1209.1078v2, last revised 10 Apr 2013[7] V. A. Leus, R.T. Smith, S. Maher, The Physical Entity of Vector Po-tential in Electromagnetism, Appl. Phys.Research , ( 2013) 56 - 68.118] B. J. Hiley, The Early History of the Aharonov-Bohm Effect,arXiv:1304.4736v1.[9] Images for G.P. Thomson experiment: , accesed 1/28/2014.[10] T. A. Arias, G.P. Thomson Experiment <\protect\vrule width0pt\protect\href{http://muchomas.lassp.cornell.edu/}{http://muchomas.lassp.cornell.edu/}><8.04/1997/quiz1/node4.html> , accesed 1/28/2014.[11] M. Born , E. Wolf, Principle of Optics, Electromagnetic theory of propa-gation, interference and diffraction of light
Classical Electrodynamics (John Wiley, N.Y. 1962)[14] L. Landau, E. Lifchitz,
Theorie des Champs (Ed. Mir, Moscou 1970)[15] S. Dumitru, M, Dumitru, Are there observable effects of the vector po-tential? : a suggestion for probative experiments of a new type (differentfrom the proposed Aharonov-Bohm one) CERN Central Library PRE24618 , Barcode 38490000001, Jan 1981. - 20 p.[16] R.P. Feynman, R.B. Leighton, M. Sands,
The Feynman Lectures onPhysics, vol. II (Addison-Wesley, Reading Mass. 1964).[17] I.S. Gradshteyn, I.M. Ryzhik,
Table of Integrals, Series, and Products,Seventh Edition ( Elsevier,2007).[18] O. Costa de Beauregard, J. M. Vigoureux, Flux quantization in ”autis-tic” magnets,
Phys. Rev. D9