The nature of the B(3) field

Ž .Chemical Physics 242 1999 331–339

The nature of the B Ž3. field

Geoffrey Hunter )

Department of Chemistry, York UniÕersity, Toronto, M3J 1P3, Canada

Received 20 August 1998; in final form 6 January 1999

Abstract

The nature of the BŽ3. field is determined through deductions from its mathematical definition; BŽ3. is a vector functionof the magnetic field of electromagnetic radiation. The sign and magnitude of BŽ3. measure the polarization of the radiation,

Ž .and it is re-normalized to define the Polarization Index a dimensionless number of magnitude unity , which proves to beidentical with one of Stokes’s parameters. This is revealed to be the essential nature of the BŽ3. field. The definition of BŽ3.

Ž .and of the Polarization Index is generalized in terms of all components of the electromagnetic field rather than only thetransverse magnetic components.

The analysis leads to internal inconsistencies if the BŽ3. field is presumed to be the longitudinal component of theŽ Ž3.physical magnetic field of electromagnetic radiation as asserted by M.W. Evans the originator of the theory of the B

.field . This inference explains the failure of experimental investigations to detect the hypothetical longitudinal magnetic fieldof radiation. It also corroborates a disproof by counter-example by E. Comay. q 1999 Elsevier Science B.V. All rightsreserved.

1. Introduction

The concept of the BŽ3. magnetic field was intro-w x w xduced by M.W. Evans in 1992 1 . Comay 2 cites

26 articles on this subject spanning the years 1992–1996, and a 4-volume set of books has appeared with

w xthe title: The Enigmatic Photon 3–6 . A recentdouble-issue of the journal Apeiron consists entirely

Ž3. w xof articles related to the B theory 7 .The BŽ3. theory asserts that electromagnetic radia-

Žtion has in addition to the well-established trans-

) E-mail: [email protected]

.verse electric and magnetic fields a longitudinalŽ w x Ž ..magnetic field defined by Ref. 3 , p. 3, Eq. 4a :

B =B)

t tŽ3.B s , 1Ž .< <i Bt

where B is the radiation’s magnetic field transverset

to the direction of propagation, B) is its complext' < <conjugate, is y1 , B is the magnitude of B ,t t

and = denotes the vector cross-product. The essen-tial character of BŽ3. is determined by the cross-

Ž . < <product in the numerator of 1 ; the division by Bt

normalizes its magnitude, BŽ3., and gives it the

0301-0104r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved.Ž .PII: S0301-0104 99 00016-6

( )G. HunterrChemical Physics 242 1999 331–339332

physical dimensions of a magnetic field, and the i isintroduced to make BŽ3. real.

2. Circular basis

The notation BŽ3. originates from the use of a‘circular basis’ whose basis vectors eŽ1.,eŽ2.,e Ž3. arerelated to the well-known Cartesian unit–basis vec-

Ž w x Ž ..tors i,j,k by Ref. 3 , p. 17, Eq. 46 :

1 1Ž1. Ž2. Ž3.e s iy ij e s iq ij e sk .Ž . Ž .' '2 2

2Ž .

This basis arises from the algebraic representationof circularly polarized radiation, e.g., as given by

Ž w x Ž . . Ž .Jackson Ref. 8 , Eq. 7.20 , p. 206 ; the form of 2is appropriate when the direction of propagation of

Ž w x .the radiation is parallel to the z-axis Ref. 3 , p. 17 .Since two of these orthonormal basis vectors have

complex components, the scalar product of any twoof them must be formed using the complex conjugateof one of the factors:

e Ž1.PeŽ1.) seŽ2.Pe Ž2.) se Ž3.PeŽ3.) s1 3Ž .

e Ž1.PeŽ2.) seŽ1.Pe Ž3.) se Ž2.PeŽ3.) s0 .

The vector cross–product of two of the basisŽ .vectors 2 is evaluated without conjugating either of

the factors, but the result is the product of i and theŽw xcomplex conjugate of the third basis vector 4 , p.

.137 :

e Ž1.=e Ž2.s i ks i e Ž3.) , e Ž3.=eŽ1.s i e Ž2.) ,

e Ž2.=e Ž3.s i eŽ1.) . 4Ž .

3. B(3) magnetic field

In view of the properties of the circular basis, inmost of what follows the magnetic field B will beexpressed in terms of its Cartesian components,which are related to the circular components by:

BsB iqB jqB ksBŽ1.qBŽ2.qBŽ3.s BŽ1.e Ž1.x y z

qBŽ2.eŽ2.qBŽ3.eŽ3. . 5Ž .

From the relationship between the Cartesian andŽ .circular bases 2 it follows that:

1 iŽ . Ž . Ž . Ž .1 2 2 1B s B qB , B s B yB ,Ž . Ž .x y' '2 2

B sBŽ3. 6Ž .z

1 1Ž1. Ž2.B s B q iB , B s B y iB ,Ž . Ž .x y x y' '2 2

BŽ3.sB . 7Ž .z

Note that BŽ2.) /BŽ1. and BŽ1.) /BŽ2. because Bx

and B are, in general, complex.yŽ3. Ž w x .One definition of B Ref. 3 , p. 3 has the

form:

BŽ1.=BŽ2.s iBŽ0.BŽ3.) , 8Ž .Ž . Ž0.rather than that of Eq. 1 . B is the magnitude of

the transverse field:

Ž0. Ž1. Ž2. ) )< < < <B s B q B s B B qB B . 9Ž .( x x y y

Ž . Ž .The alternative definitions 1 and 8 are equiva-lent.

3.1. A notational ambiguity

Ž .The definition 8 raises a ambiguity between:Ž3. Ž .Ø B as the vector product defined by 8 , and

Ø BŽ3. as the third component of the magnetic fieldŽ .in the circular basis 6,7 ,

for although BŽ3.eŽ3. and B k are parallel vectorszŽ Ž3. .since e 'k there is no a-priori reason that equatestheir magnitudes. The recurring statement in theliterature of the BŽ3. theory, that:

‘‘BŽ3. is the longitudinal component of the pho-ton’s magnetic field’’

amounts to the assertion that:

B 'BŽ3. 10Ž .z

, and in order to avoid the ambiguity implicit in thisassertion, in what follows:

Ž3. Ž Ž3..Ø B and its magnitude B will denote theŽ . Ž .vector product defined by 1 or 8 , and

Ø B will denote the physical magnetic field in thez

z direction, in both the circular and Cartesianbases.

( )G. HunterrChemical Physics 242 1999 331–339 333

3.2. TransÕerse field

The transverse field B in component form is:t

B s i B q j B q0 kse Ž1.BŽ1.qeŽ2.BŽ2. , 11Ž .t x y

for radiation propagating in the z direction. Thus thedefinition of the vector cross-product in Cartesiancoordinates yields:

i j k) ) )B B 0x yB =B s sk B B yB B .Ž .t t x y x y

) )B B 0x y

12Ž .

To complete the evaluation of BŽ3. we need an< <expression for B :t

< < 2 ) ) ) Ž1. Ž1.)B sB PB s B B qB B sB PBŽ .t t t x x y y

qBŽ2.PBŽ2.)sBŽ1.BŽ1.) qBŽ2.BŽ2.) , 13Ž .where the circular-component expressions follow

Ž . Ž .from 5 and 3 .Thus the expression for BŽ3. in terms of Cartesian

components is:

B B) yB B)Ž .x y y xŽ3.B sk . 14Ž .) )i B B qB B( x x y y

Ž . Ž3.It follows from 12 that the numerator of B isŽpurely imaginary because it is the difference be-

.tween a number and its complex conjugate , andŽ . Ž3.hence from 14 that B is real. It also follows from

Ž . Ž3.14 that B is parallel to the direction of propaga-Ž .tion i.e., the zsk axis .

3.3. Properties of B(3)

The BŽ3. field is clearly longitudinal in the senseof being parallel to the direction of propagation ofthe radiation, for its definition as the cross-product ofvectors that are transverse to the direction of propa-gation makes it longitudinal.

Ž . Ž3.It is also clear from 14 that B is normallyŽnon-zero when both transverse components B andx

.B have non-zero real, and non-zero imaginary parts.yŽThis pertains in the case of circularly or more

.generally elliptically polarized radiation, but in thecase of plane polarized radiation the direction of Bt

Ž .is constant say parallel to the x-axis ´B '0 andyŽ . Ž . Ž3.hence by 1 and 12 B is zero.

3.4. A model photon field

How this dependence upon polarization arises, isillustrated by the Maxwellian model of the photon

w xderived by Wadlinger and this author in 1985 9 , inwhich the Cartesian components of the photon’s

Žmagnetic and electric fields have the form wherew Ž . x.Tsr S exp 2p i zyct rl :0

B s A exp if qB exp yif TsE ,Ž . Ž .y x

B s i A exp if yB exp yif TsyE ,Ž . Ž .x y

B sE s0,. 15Ž .z z

� 4 Ž w xin cylindrical polar coordinates r f z t Ref. 9 , Eq.Ž . . 149 , p. 164 . A and B are real constants such thatA2 qB2 s1, S is the amplitude of the field, and l0

is the photon’s wavelength. The amplitude, S , was0

determined by equating the integrated field energyŽ w xwith the known energy of a photon, hn Ref. 9 , Eq.

Ž ..47 :

2'8 p hS s , 16Ž .0 3l

where h is Planck’s constant.This representation of an electromagnetic wave

w x. Ždiffers from that used by Lakhtakia 10 and otherŽ w xauthors such as Landau and Lifshitz Ref. 11 , Chap.

.. w x6 in not being a plane wave. The authors 9argued that a photon’s field cannot be a plane wavebecause its field is localized about its axis of propa-gation. This argument led to the non-plane wave

Ž .field 15 which naturally represents the angularŽ .momentum i.e., spin of the photon by the factors

Ž .exp "if .

1 The field components of the Hunter–Wadlinger model arehere written in units such that the velocity of light, c, the electricpermittivity, ´ , and the magnetic permeability, m , of empty0 0

space, are all equal to 1. Hence B' H and D' E.


It is noteworthy that B and B differ in phase byx y

908, as do E and E , the phase-difference factorx yŽ .being exp "ipr2 s"i, the phase factor being

w Ž . xexp 2p i zyct rl :Ø E is 908 ahead of E andx y

Ø B is 908 ahead of B ;x y

Ø E is in phase with B butx y

Ø B is 1808 ahead of E .x y

These simple phase relationships between the fourtransverse field components are a necessary conse-quence of them being a solution of Maxwell’s equa-tions.

The longitudinal, z-components, E and B , arez z

identically zero; this is also a consequence of solvingMaxwell’s equations for a field that conforms withthe known physical properties of the photon; it wasnot arbitrarily imposed.

3.4.1. B(3) for the H–W photon modelSubstituting the explicit forms for the Cartesian

Ž . Ž .components 15 into the two expressions 12,13involved in the definition of BŽ3. we obtain:

B =B) sk 2 i A2 yB2 S2 r 2 , 17Ž . Ž .t t 0

2) 2 2 2 2 2 2B sB PB s2 A qB S r s2S r , 18Ž . Ž .t t t 0 0

2 2 Ž .since A qB s1. Hence from its definition 1 :

Ž3. 2 2'B sk 2 A yB S r , 19Ž . Ž .0

Ž .for the Hunter–Wadlinger H–W photon model.From the interpretation of this algebraic model

w x9 :< < < < Ž .Ø A s B corresponds to a plane linearly polar-ized photon, in which case it is apparent fromŽ . Ž3.19 that the B field is zero;

Ø As0 or Bs0 correspond to the two directionsŽ .right or left of circularly polarized radiation, in

Ž3. 2'which cases B s" 2 S r since A s1 or0

B2 s1;< < < <Ø A / B and both A/0 and B/0 correspondsto elliptically polarized radiation.In the circular and elliptical cases it is apparent

Ž . Ž3.from 19 that the B field is non-zero, and that it ismathematically real as distinct from complex orimaginary.

It is especially pertinent to note that although BŽ3.

Žis not zero for the photon model except in the< < < < .A s B case of plane-polarized radiation , the solu-

tion of Maxwell’s equations that constitutes the alge-braic model of the photon has B '0. This contrastz

between BŽ3. and B suggests that BŽ3. is not az

component of the physical magnetic field that satis-fies Maxwell’s equations. Comay has demonstrated

w xthis analytically 12 .

4. Significance of B(3)

Ž3. ' Ž .Clearly B varies between 2 S r when Bs00' Ž .and y 2 BS r when As0 , taking the value zero0< < < < Ž3.when A s B . This suggests that instead of B as

Ž Ž ..defined by Eq. 1 , we should instead introduce thePolarization Vector:

B =B)

t t 2 2P s sk A yB , 20Ž . Ž .i 2< <i Bt

where the last expression is for the H–W photonmodel. The magnitude of P , P , is the Polarizationi i

Index, which for the H–W photon model assumesthe values:

Ž 2 .Ø P s1 when Bs0 since A s1i

Ø P s0 when AsBiŽ 2 .Ø P sy1 when As0 since B s1 .i

The values P s"1 correspond to the two possi-i

ble senses of circular polarization, P s0 character-iŽ .izes linear plane polarization, and " fractional

values correspond to various degrees of ellipticalŽ . 2polarization in two possible senses .

In the P s"1 cases of circular polarization, alli

six components of the H–W electromagnetic fieldare eigenfunctions of the operator for the z-compo-

Ž . Ž . Žnent of angular momentum, L s "E r iEf Ref.zw x Ž . Ž . .9 , Eqs. 33a and 33b , p. 162 :

Ž .Ø L sq" when P s1 Bs0z iŽ .Ø L sy" when P sy1 As0z i

This accords with the interpretation that the pho-Ž .ton’s intrinsic spin angular momentum may be

oriented parallel, or antiparallel, to the direction of

2 This Polarization Index is different from the degree ofŽ w x .polarization defined by Landau and Lifshitz Ref. 11 , p. 121 ;

the latter is a measure of the extent to which a beam of non-mono-chromatic light is polarized, regardless of whether the polar-ization is planar, circular or elliptical.


propagation, and that these alternative spin statescorrespond to light that is observed to be right or left

Žcircularly polarized also known as states of opposite.helicity .

Ž3. Ž .Thus the significance of B or rather P is as ai

measure of the polarization of the radiation. Itsmagnitude, P is an index – a number taking valuesi

within the range: y1(P (q1. With this interpre-i

tation we recognize P as a vector function of thei

electromagnetic field, parallel to the direction ofpropagation, whose sign and magnitude indicate thepolarization of the radiation. P is thus comparablei

Ž w x Ž .with Poynting’s vector S Ref. 8 , Eq. 7.14 , p.. 3205 :

i j k) E E 0x ySsE=H s

) )H H 0x y

sk E H ) yE H ) sk E E) qE E)Ž . Ž .x y y x x x y y

sk 2 S2 r 2 , 21Ž .0

Ž .for the H–W photon model from 15 . The magni-tude of the H–W Poynting vector is simply the

Ž .squared magnitude of the field as evaluated in 18 .Both P and S are parallel to the direction ofi

propagation, the magnitude of the former indicatingthe polarization of the radiation, and the latter theflow of energy, i.e. energy flow per unit time, perunit area normal to the direction of flow.

This comparison suggests that the PolarizationÕector, P is a theoretical construct that may bei

comparable in utility to Poynting’s Õector.

4.1. Definition in terms of the entire EM field

The analogy with Poynting’s vector suggests thatthe Polarization vector P be re-defined in terms ofi

the entire physical magnetic field B:

B=B) H=H )

P s s , 22Ž .i 2 2< < < <i B i H

since BsH for radiation in the vacuum. Similarly,

3 w x Ž .As noted by Akhtar et al. 13 p.64, col. 2 .

Ž3. Ž .B which only differs from P in normalizationi

would be re-defined by:

B=B)

Ž3.B s , 23Ž .< <i B

Ž .in place of the definition of Eq. 1 . The division by< < Ž . ŽB in 23 is simply a normalization designed by

. Ž3.M.W. Evans to make B have the dimensions andmagnitude of a magnetic field; the similar division

< < 2 Ž .by B in 22 makes P dimensionless and ofi

magnitude unity.The re-definition in terms of the entire magnetic

Žfield does not affect any results presented here and.elsewhere since the longitudinal component of the

physical magnetic field of a photon is zero. How-ever, the re-definition has the aesthetic appeal of

Ž3. Žgenerality, for now P and B are like Poynting’si.vector vector functions of the entire physical field

rather than of selected components.This train of thought leads to the speculation of a

further generalization in which the definitions of Pi

and BŽ3. are extended to include the electric field EŽ .as well as the magnetic field B or H , for it is only

the entire electromagnetic field tensor that has physi-cal significance within the framework of Relativitytheory.

As a step towards this generalization, we note thatin the H–W photon model, there is a simple relation-ship between the components of the magnetic fieldand those of the electric field: B syE and B sx y y

Ž . Ž .E 15 . Thus for this model at least the expressionx) Ž .for B =B 12 could be re-written as:t t

i j k) ) )B B 0x yB =B s sk B E qB EŽ .t t x x y y

) )yE E 0y x

sk BPE ) sk 2 i A2 yB2 S2 r 2 , 24Ž . Ž .0

Ž .where the last expression is in agreement with 17Ž .for the H–W photon model . Similarly the normal-

< < 2 Ž .izing denominator, B , evaluated in Eq. 18 couldt

be written more symmetrically as:2 1 ) ) 2 2 2 2w xB s BPB qEPE s2 A qB S rŽ .t 02

s2S2 r 2 . 25Ž .0

These electro-magnetic formulae for the numera-Ž . Ž .tor 24 and denominator 25 of the Polarization

vector, P , produce the same result as the formulaei


Ž .defined in terms of B 17 and 18 or equivalently BtŽ .since B s0 :z

P sk A2 yB2 , 26Ž . Ž .i

and while this has only been demonstrated explicitlyfor the H–W representation of the photon’s field, aformula that contains the electric and magnetic fieldcomponents symmetrically has the aesthetic appealthat it is Lorentz invariant.

If this generalization of the definition of BŽ3. interms of the entire electromagnetic field stands up tofurther consideration, then it will eradicate the notionthat BŽ3. is itself a component of the physical mag-netic field, for the definition would be recursive andindeterminate; i.e. the formula defining BŽ3. wouldcontain BŽ3. itself.

4.2. Lorentz scalar and pseudoscalar

w xComay 2 in his introduction mentions that cer-Žtain solutions of Maxwell’s equations in units such

.that the velocity of light cs1 have null values forthe Lorentz scalar and pseudoscalar:

B2 yE 2 s0 BPEs0 . 27Ž .For the H–W photon model these invariants have thefollowing form:

B2 yE 2 s B2 qB2 y E2 qE2 s0 28Ž .Ž . Ž .x y x y

BPEs B E qB E s B B yB B s0 ,Ž . Ž .x x y y x y y x

because E sB and E syB and E sB s0.x y y x z z

4.3. Mathematical considerations

The relationships between the six components ofthe electromagnetic field implied by Maxwell’sequations are quite different from that between BŽ3.

and the transverse components: BŽ3. is defined to bea function of the transverse components, and so iscompletely determined by them, whereas the rela-tionships between the 6 components of the electro-magnetic field implied by Maxwell’s equations leavea large degree of independence to each component,in particular because the boundary conditions oneach component will generally be different and inde-pendent of each other.

Thus it is incorrect to regard BŽ3. as the thirdcomponent of the physical magnetic field, B , whenz

in fact it is defined as a function of the transverse

components; the definition simply makes BŽ3. avector function of the transverse components.

4.4. Inconsistency with Poynting’s theorem

Consideration of the Poynting vector leads to theŽ3. Ž .realization that the assertion that B sB 10 can-z

not be valid. To compute the Poynting vector we useŽ w x .Evans’s note Ref. 14 , p. 2 that associated with the

real BŽ3. field there is a purely imaginary electricfield iE Ž3., and that this electric field is to be in-cluded with BŽ3. in computing Poynting’s vector.

Ž . w x Ž .According to Eq. B9 of Ref. 14 p. 349

E Ž3.scBŽ3.´E Ž3.sBŽ3. in units where cs1 ,and hence the purely imaginary electric field issimply iBŽ3.. Thus the complete Poynting vector:

i j kŽ3.

) E E i Bx ySsE=H s) ) Ž3.B B Bx y

sk E B) yE B) q i BŽ3. E y iB)Ž . Ž .x y y x y y

y j BŽ3. E y iB) , 29Ž .Ž .x x

has non-zero components in both directions trans-verse to the direction of propagation, as well asalong the direction of propagation – contrary toPoynting’s theorem that the direction of propagation

Ž .is the direction of the energy flow. This result 29Ž w x .disproves the statement Ref. 14 , p. 204, item 3

that the presence of BŽ3. and iE Ž3. does not affectŽPoynting’s theorem. A more recent publication Ref.

w x .15 , p. 70 expresses the opinion that:

‘‘... BŽ3. and iE Ž3. do not contribute to thePoynting vector ...’’.

The non-zero transverse components arise onlybecause of the assumption that B sBŽ3.; BŽ3. is az

factor of these components of S. It hardly requiresthe above analysis to realize that inclusion of alongitudinal field component in the computation ofthe Poynting vector will necessarilly make the Poynt-ing vector non-longitudinal.

4.5. Relationship to Stokes’s parameters

w x w xBoth Barron 16 and Lakhtakia 10 argue thatBŽ3. is related to the third of Stokes’s parameters


Ž w x .Ref. 11 , p. 122 ; Barron writes equations implyingthat this parameter, S , is directly proportional to3

Ž3. Ž w xB , while Lakhtakia’s equation Ref. 10 , Eqs.Ž . Ž ..16b and 17b indicates that S , is inÕersely pro-3

portional to BŽ3.. Barron describes S as3

‘‘the normalized intensity of the circularly polar-ized component of a light beam’’ that ‘‘takes thevalues q1 and y1 for right and left circularpolarization and 0 for linear polarization, respec-tively.’’

which identifies it with the Polarization Index de-Ž .fined above in Eq. 20 . Barron cites Landau andŽ w x .Lifshitz, who indicate Ref. 11 , p. 122 that S 'j2 2

is q1 and y1 for right and left circular polarizationrespectively, and 0 for linear polarization, thus sug-gesting that Barron intended to specify the secondŽ . Ž . 4S 'j rather than the third S 'j parameter .2 2 3 3

This inference is confirmed by relating the discus-w x Žsion of the polarization tensor in Ref. 11 pp.

. Ž Ž . Ž ..120–123 to the definition of P Eq. 20 ,Eq. 22 .i

Landau and Lifshitz define S 'j 'A as the2 2

signed-magnitude of the antisymmetric part of thepolarization tensor, and note that since this tensor is

Žessentially 2-dimensional being dependent upon onlythe transverse components of the electromagnetic

.field since the longitudinal components are zero ,that it is a pseudo-scaler 5. Thus the pseudoscaler Pi

may be expressed by the alternative expressions:

B B) yB)B E E) yE)EŽ . Ž .x y x y x y x yP s ' ,i

) ) ) )i B B qB B i E E qE EŽ . Ž .x x y y x x y y

30Ž .

in which the real denominator is simply BPB) or

4 Discussions of the Stokes parameters are not consistent ineither notation or definition; e.g., Staelin et al. define four parame-

Ž w x . 2ters Ref. 17 , p. 20 , that have the units of wattsrmetre ratherthan the dimensionless quantities defined by Landau and Lifshitz.

5 The numerator of P 'j is expressed in terms of the electrici 2

field components by Landau and Lifshitz, rather than the magneticŽ .field components of Eq. 12 . That these alternative definitions are

equivalent follows from the relationship between the electric andmagnetic field components determined by Maxwell’s equations,i.e., the relations: B s E and B sy E given for the H–Wy x x y

photon field, appear to apply generally to electromagnetic wavesolutions of Maxwell’s equations.

EPE ) , the choice being determined to make Pi

dimensionless, and the numerator is imaginary.The pseudoscalar nature of P is displayed by thei

alternative definition of its numerator as a scalarŽ .product 24 , with the alternative forms:

B. E ) s B E) qB E) 'y E B) qE B)Ž . Ž .x x y y x x y y

sy EPB) 31Ž . Ž .with alternative denominators

B E) yB E) s E B) yE B) .Ž . Ž .y x x y x y y x

5. Experimental evidence

The first experiments related to the theory of theŽ3. w xB field were carried out by Warren et al. 18 , at

about the same time that M.W. Evans proposed theŽ . w xtheory 1992 1 . Warren believed that the observed

NMR shifts were attributable to effects other thanthe BŽ3. field, while Evans believed that they sup-ported the theory.

w xRikken 19 carried out an experiment specificallydesigned to measure the longitudinal magnetic fieldof a circularly polarized light beam predicted byEvans’s theory. The measurements were calibratedagainst an externally applied magnetic field gener-ated by a solenoid. The circularly polarized lightbeam failed to exhibit a magnetostatic field; theexperimental measurements showed that any suchfield is at least 3-orders-of-magnitude smaller thanthe magnitude predicted by the theory.

The results of three experiments by Akhtar Rajaw xet al. 13 based upon:

Ø photomagnetic induction,Ø the optical Faraday effect, andØ the inverse Faraday effect,all ‘‘clearly disprove the claims of B -theory’’ 6.p

In the light of the theory presented elsewhere inthis article, the failure of all of this experimentalwork to detect the BŽ3. field, is predictable.

6 The notation Bp has been used in some publications as analternative to BŽ3..


6. Comay’s disproof by counter-example

w xComay’s disproof by counter example 2 is basedupon the known properties of the radiation from arotating dipole. He considers a closed path, PQRS in

Žthe radiation zone i.e., at a large distance from the.dipole compared with its length and proceeds to

compute the line integral of the radiation’s magneticfield around this path. This integration yields theresult that the integral is non-zero on account of thecontribution of the BŽ3. field along the segment ofthe path where the radiation is circularly polarized.This contribution of BŽ3. was based using the asser-tions of the theory that BŽ3. is the longitudinal

Ž .component of the physical magnetic field B , andz

that it is non-zero when the radiation is circularlypolarized.

Comay then uses Stokes’s theorem to convert theline integral into a surface integral over a surfaceŽ .A bounded by the path PQRSP:

EE E==B P d ss P d ss EP d s , 32Ž . Ž .H H H

Et EtA A A

where the RHS results from the Maxwell equation:

EE==Bs . 33Ž .

Et

Since the line integral is only non-zero on account ofthe contribution from BŽ3. along PQ, and since its

Ž3. Žmagnitude, B , is independent of time independent.of frequency . It follows that the time derivative of

the electric flux through the surface A is a non-zeroconstant. Hence the time-integral of the electric fluxthrough A must be a linear function of the time t,which implies that E itself increases monotonicallywith time, which is, of course, contrary to the physicsof a radiation field whose intensity is stationary.

Thus Comay’s analysis concludes that the as-sumption that BŽ3. is the longitudinal component ofthe physical magnetic field in circularly polarizedradiation, yields the non-physical result of an electricflux through PQRS whose magnitude increases lin-early with time. In addition Comay notes that thisresult disproves another assertion of the BŽ3. theoryŽ .that it is not associated with any electric field thusmaking the theory self-contradictory. The obviousexplanation is that it is incorrect to regard BŽ3. as thelongitudinal component of the physical magnetic

field, B . Evans and Jeffers have disputed Comay’szw xconclusions 20 . Their arguments include an analy-

sis based upon the Cartesian form of Stokes’s theo-Ž w x Ž ..rem Ref. 21 , p. 965, Eq. 7 of a scenario that is

somewhat different from that employed by Comay.

7. Conclusions

We conclude that M.W. Evans’s formulation ofŽ3. Žthe B field albeit re-normalized as P to make it ai

.dimensionless number between y1 and q1 is are-discovery of a parameter that Stokes introducedmany years ago as a measure of the radiation field’smode of polarization: P 'j has the values q1i 2

and y1 for right and left circularly polarized lightrespectively, and 0 for linearly polarized light.

This identification of the nature of the BŽ3. fieldgiven here accords with the conclusions of Lakhtakiaw x w x10 and Grimes 22 .

Not only is BŽ3. not equal to the longitudinalcomponent of the physical magnetic field, B , it isz

Žnot any kind of physical field as the experiments. Ž .have confirmed , but rather a well-defined Stokes

parameter of the physical field that characterizes thepolarization of the field.

In the light of this understanding of the nature ofthe B(3) field, the fundamental physical tenet of the

Ž3. Ž Ž3..theory of the B field that B sB is seen to bez

untenable.

References

w x Ž .1 M.W. Evans, Physica B 182 1992 227, 237.w x Ž .2 E. Comay, Chem. Phys. Lett. 261 1996 601.w x Ž .3 M.W. Evans, J.P. Vigier Eds. , The Enigmatic Photon, vol.

Ž . Ž .1, The Field B 3 , in: A. van der Merwe Series Ed. ,Fundamental Theories of Physics, Kluwer, Dordrecht, TheNetherlands, 1994.

w x Ž .4 M.W. Evans, J.P. Vigier Eds. , The Enigmatic Photon, vol.2, Non-Abelian Electrodynamics, in: A. van der MerweŽ .Series Ed. , Fundamental Theories of Physics, Kluwer, Dor-drecht, The Netherlands, 1995.

w x Ž .5 M.W. Evans, J.P. Vigier, S. Roy, S. Jeffers Eds. , TheEnigmatic Photon, vol. 3, Theory and Practice of the BŽ3.

Ž .Field, 1996, in: A. van der Merwe Series Ed. , FundamentalTheories of Physics, Kluwer, Dordrecht, The Netherlands,1996.


w x Ž .6 M.W. Evans, J.P. Vigier, S. Roy, G. Hunter Eds. , TheEnigmatic Photon, vol. 4, New Directions, 1998, in: A. van

Ž .der Merwe Series Ed. , Fundamental Theories of Physics,Kluwer, Dordrecht, The Netherlands, 1997.

w x7 Apeiron, vol. 4, Nos. 2–3, C. Roy Keys, Montreal, CanadaŽ .April–July 1997 ISSN 0843-6061.

w x8 J.D. Jackson, Classical Electrodynamics, Wiley, New York,1962.

w x Ž .9 G. Hunter, R.L.P. Wadlinger, Phys. Essays 2 1989 158.w x Ž .10 A. Lakhtakia, Physica B 191 1993 362.w x11 L.D. Landau, E.M. Lifshitz, The Classical Theory of Fields,

Pergamon Press, Oxford, 1975.w x Ž .12 E. Comay, Physica B 222 1996 150.w x13 M.Y. Akhtar Raja, W.N. Sisk, M. Yousaf, D. Allen, Appl.

Ž .Phys. B 64 1997 79.

w x14 A.A. Hasanein, M.W. Evans, The Photomagnetron andQuantum Field Theory, vol. 1, Quantum Chemistry, WorldScientific, Singapore, 1994.

w x Ž .15 M.W. Evans, Found. Phys. Lett. 7 1994 67.w x Ž .16 L.D. Barron, Physica B 190 1993 307.w x17 D.H. Staelin, A.W. Morgenthaler, J.A. Kong, Electromag-

netic Waves, Prentice Hall, NJ, 1994.w x18 W.S. Warren, S. Mayr, D. Goswami, A.P. West Jr., Science

Ž .255 1992 1683.w x Ž .19 G.L.J.A. Rikken, Optics Lett. 20 1995 846.w x Ž .20 M.W. Evans, S. Jeffers, Found. Phys. Lett. 9 1996 587.w x Ž .21 E.G. Milewski Ed. , The Vector Analysis Problem Solver,

Research and Education Association, New York, revisedprinting, 1987.

w x Ž .22 D.M. Grimes, Physica B 191 1993 367.

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The nature of the B(3) field