Composite Photon Theory versus Elementary Photon Theory

Transcription

1 Journal of Modern Physics Published Online December 04 in Scies htt://wwwscirorg/journal/jm htt://dxdoiorg/046/jm Comosite Photon Theory versus Elementary Photon Theory Walton A Perkins Perkins Advanced Comuting Systems 0 Hidden Meadows Circle Auburn CA USA werkins@nccnnet eceived October 04; revised November 04; acceted 4 November 04 Coyright 04 by author and Scientific esearch Publishing Inc This work is licensed under the Creative Commons Attribution International License (CC BY) htt://creativecommonsorg/licenses/by/40/ Abstract The urose of this aer is to show that the comosite hoton theory measures u well against the Standard Model s elementary hoton theory This is done by comaring the two theories area by area Although the redictions of quantum electrodynamics are in excellent agreement with exeriment (as in the anomalous magnetic moment of the electron) there are some roblems such as the difficulty in describing the electromagnetic field with the four-comonent vector otential because the hoton has only two olarization states In most areas the two theories give similar results so it is imossible to rule out the comosite hoton theory Pryce s arguments in 98 against a comosite hoton theory are shown to be invalid or irrelevant ecently it has been realized that in the comosite theory the antihoton does not interact with matter because it is formed of a neutrino and an antineutrino with the wrong helicity This leads to exerimental tests that can determine which theory is correct Keywords Comosite Photon Antihoton Neutrino Theory of Light Introduction In the history of hysics many articles which were once believed to be elementary later turned out to be comosites The idea that the hoton is a comosite article dates back to 9 when Louis de Broglie [] [] suggested that the hoton is comosed of a neutrino-antineutrino air bound together Pascual Jordan [] who develoed canonical anticommutation relations for fermions thought that he could obtain Bose commutation relations for a comosite hoton from the fermion anticommutation relations of its constituents In order to obtain Bose commutation relations Jordan modified de Broglie s theory suggesting that a single neutrino could simulate a hoton by a aman effect and that no interaction between the neutrino and antineutrino was needed if they were emitted in exactly the same direction Today of course we know that a single neutrino interacts much too How to cite this aer: Perkins WA (04) Comosite Photon Theory versus Elementary Photon Theory Journal of Modern Physics htt://dxdoiorg/046/jm045805

2 weakly to simulate a hoton Because of Jordan s idea that the neutrino and antineutrino do not interact the comosite hoton theory was referred to as the Neutrino Theory of Light Jordan s modifications made it easy for Pryce in 98 to show that the theory was untenable Pryce [4] showed that if the comosite hoton obeyed Bose commutations relations its amlitude would be zero Pryce gave several arguments against the comosite theory but as Case [5] and Berezinskii [6] discussed the only valid argument was that the comosite hoton could not satisfy Bose commutation relations In 98 the existence of many other subatomic comosite bosons that are formed of fermion-antifermion airs was unknown Perkins [7] has shown that there is no need for a comosite hoton to satisfy exact Bose commutation relations He oints out that many comosite bosons such as Cooer airs deuterons ions and kaons are not erfect bosons because of their internal fermion structure although in the asymtotic limit they are essentially bosons Neutrino oscillations in which one flavor of neutrino changes into another have been observed at the Suer- Kamiokande [8] and SNO [9] Among the electron muon and tau neutrinos at least two must have mass Here we will assume that the comosite hoton is formed of an electron neutrino and an electron antineutrino and that the electron neutrinos are massless There has been some continuing work on the comosite hoton theory (see [0]-[]) but it still has not been acceted as an alternative to the elementary hoton theory A major roblem for the comosite hoton theory is that no exeriment has demonstrated the need for it ecently Perkins [] showed that in the comosite theory the antihoton is different than the hoton and that antihotons do not interact with electrons because their neutrinos have the wrong helicity This leads to exerimental redictions that can differentiate between the Standard Model elementary hoton theory and the comosite hoton theory In the antihydrogen exeriments at CEN the ALPHA [4] [5] and ASACUSA [6] Grous will be looking for sectral emissions from the antihydrogen atoms and shinning light on the atoms to ut them into excited states According to the comosite hoton theory neither of these exeriments will roduce the exected results In the next section we will comare the elementary and comosite theories area by area In Section we re-examine Pryce s arguments [4] that the Neutrino Theory of Light is untenable and confirm that his arguments are no longer valid Comarison of Photon Theories Intuitively de Broglie s idea makes reasonable the significant difference in characteristics exhibited by sin- hoton and a sin-/ neutrino When a hoton is emitted a neutrino-antineutrino air arises from the vacuum Later the neutrino and antineutrino annihilate when the hoton is absorbed In the following sections we will examine the similarities and differences of the elementary and comosite hoton theories Although the comosite and elementary theories are similar there are both subtle and major differences Photon Field Elementary Photon Theory In noting the roblem of quantizing the electromagnetic field Bjorken and Drell [7] declared It is ironic that of the fields we shall consider it is the most difficult to quantize Srednicki [8] commented Since real sin- articles transform in the ( ) reresentation of the Lorentz grou they are more naturally described as bisinors A αα than as 4-vectors A ( x) Varlamov [9] also noted that the electromagnetic four-otential is transformed within ( ) reresentation of the homogeneous Lorentz grou Usually a canonical rocedure for quantization is used although it is not manifestly covariant We can describe the electromagnetic field with the four-comonent vector otential but the hoton only has two olarization states One method of handling the roblem is to introduce two non-hysical hotons along with the real ones the Guta-Bleuler rocedure [0] Another method is to give the hoton a very very small mass [] Following Bjorken and Drell [7] we will take only the transverse comonents and abandon manifest covariance We start with Maxwell equations (in the absence of source charges and currents) E( x) = 0 H ( x) = 0 () E( x) = H( x) t H x = E x t ( ) ( ) 090

3 This imlies a vector otential ( φ ) A = Α that satisfies ( ) A( ) ( x) = A( x) E x = x t φ H For any electromagnetic field E and H there are many A s that differ by a gauge transformation A satisfactory Lagrangian density is given by Using the standard method we construct conjugate momenta from A A A ν = xν x x () ν π 0 = = 0 A 0 A π = = A = E 0 k k k A k xk Comosite Photon Theory We start with the neutrino field Solving the Dirac equation for a massless article γ Ψ= 0 with Ψ= u ( )e ix results in the sinors where ( ie) + i i E+ + + E+ E+ + ( ) = E+ u ( ) = u E E E+ + E+ u+ ( ) = u ( ) = + i E E + i E+ E+ = and the suerscrits and subscrits on u refer to the energy and helicity states resectively The gamma matrices in the Weyl basis were used in solving the Dirac equation: i i γ = γ = 0 i i i i γ = γ4 = i i γ = γγγγ 4 = We designate a as the annihilation oerator for ν the right-handed neutrino and c as the annihilation () (4) (5) (6) (7) 09

4 oerator for ν the left-handed antineutrino We assign a as the annihilation oerator for ν the lefthanded neutrino and c as the annihilation oerator for ν the right-handed antineutrino Since only ν and ν have been observed we take the neutrino field to be + ikx { e } ( ) ( ) ikx Ψ x = a u ( ) e c ( ) u ( ) V k k + k + k (8) k where we have used only the two corresonding sinors and kx stands for be created from a fermion-antifermion air k x ω t k A four-vector field can Ψi γ Ψ (9) The fermion and antifermion are bound by this attractive local vector interaction of Equation (9) as discussed by Fermi and Yang [] We ostulate that this local interaction between the neutrino and antineutrino is resonsible for their interaction with the electromagnetic couling constant α while a single neutrino interacts with the weak couling constant g Both Kronig [] and de Broglie [] suggested local interactions in their work on the comosite hoton theory Since the neutrino and antineutrino momenta are in oosite directions we take the hoton field to be [] { + + ix A ( x) = G( ) u ( ) iγ u ( ) GL( ) u ( ) iγ u ( ) e Vω (0) + + ix + G( ) u+ ( ) iγ u ( ) + GL( ) u ( ) iγ u+ ( ) e } with the annihilation oerators for left-circularly and right-circularly olarized hotons with momentum given by GL ( ) = F ( k) c( k) a( + k) k () G ( ) = F ( k) c( + k) a( k) k where F ( k ) is a sectral function Although many sets of gamma matrices satisfy the Dirac equation one must use the Weyl reresentation of gamma matrices to obtain sinors aroriate for the comosite hoton If a different set of gamma matrices is used the hoton field will NOT satisfy Maxwell equations Kronig [] was the first to realize this but he did not mention the deeer significance ie two-comonent neutrinos are required for a comosite hoton At that time a two-comonent neutrino theory would have been rejected because it violated arity The connection between the hoton antisymmetric tensor and the two-comonent Weyl equation was also noted by Sen [4] Although we are working at the four-comonent level one can form a comosite hoton at the two-comonent level [] Commutation elations Elementary Photon Theory In classical Hamiltonian mechanics the Poisson bracket is defined as [ FG] PB F G F G = q q where qk ( t ) are the generalized coordinate and k ( ) k k k k t are the generalized momenta If we use q i and in lace F and G we obtain the fundamental Poisson brackets ( ) j ( ) ( ) j ( ) ( ) ( ) qi t q t = 0 PB i t t = 0 PB qi t j t = δij PB () j () 09

5 In going over to quantum theory it is hyothesized that the fundamental Poisson brackets become commutators with q and becoming oerators i i ( ) ( ) qi t qj t = 0 i( t) j ( t) = 0 qi ( t) j ( t) = iδ ij The generalized coordinates and momenta for the classical electromagnetic field are Thus the fundamental commutators become i ( ) ( x ) ( ) ( x ) q t A t t t j ( x ) ( x ) π A t Aν t = 0 π ( x t) πν ( x t) = 0 π ( x t) Aν ( x t) = iδνδ ( x x ) However the third Equation of (6) is not consistent with Maxwell equations so we must deart from the canonical ath [7] and relace it with Exanding the A and π into lane waves where 0 tr ( t) A ( t) iδ ( ) (4) (5) (6) π x ν x = + ν x x (7) d ix ix A( x) = ( ) b ( ) e b ( ) e + ω π = ( ) ω π ( x) = = i b + b π ix ix A d ( ) ( ) e ( ) e = ( ) ω = and b ( ) and b ( ) We take the two unit olarization vectors to be erendicular to in order to satisfy ( x) = 0 radiation gauge) Also it is convenient to choose (8) are identified as annihilation and creation oerators for olarization A (ie ( ) = 0 (9) ( ) ( ) = Inverting Equation (8) we obtain the amlitudes b ( ) and ( ) δ (0) b ix d xe b ( ) = ( ) ω A( x) + ia ( x) ( ω ( π) ) d xe ix b ( ) = A x + ia x ( ω ( π) ) ( ) ω ( ) ( ) () Following Bjorken and Drell [7] we use Equations (6) and (7) to obtain commutation relations for the annihilation and creation oerators 09

6 b ( ) b ( q) = 0 ( ) b ( q) b = 0 ( ) b ( q) δ ( q) b = δ Left-handed and right-handed circularly olarized annihilation oerators are obtained from the combinations and they obey the commutation relations bl b ib b b ib ( ) = ( ) ( ) ( ) = ( ) + ( ) ( ) ( q) ( ) ( q) bl bl = 0 bl b L = 0 ( ) b ( q) δ ( q) bl L = ( ) ( q) ( ) ( q) b b = 0 b b = 0 ( ) b ( q) δ ( q) b = ( ) ( q) ( ) ( q) bl b = 0 bl b = 0 From this discussion it is evident that the elementary hoton commutation relations were carried over from the classical canonical formalism and are not based on any fundamental rincile The hoton distribution for Blackbody radiation can be calculated using the second quantization method [5] including commutation relations of Equation () resulting in Planck s law n = (5) ω kt e Comosite Photon Theory Comosite integral sin articles obey commutation relations [6]-[8] that are derived from the fermion anticommutation relations of their constituents For comosite hotons we have L( ) L( q) = 0 L( ) L( q) = 0 L( ) GL( q) δ ( q) ( L( )) ( ) ( q) = 0 ( ) ( q) = 0 ( ) G( q) δ ( q) ( ( )) L( ) ( q) = 0 L( ) ( q) = 0 ( ) F( k) a ( k) a( k) c ( k) c( k) G G G G G = G G G G G = G G G G L = k ( ) F( k ) a ( k) a ( k) + c ( + k) c ( + k) = k In obtaining the commutation relations involving ( ) and ( ) L we have taken the exectation values Here the linearly-olarized hoton annihilation oerators are defined as ξ ( ) = GL( ) + G( ) (7) i η ( ) = GL( ) G( ) () () (4) (6) 094

7 and they obey the commutation relations ( ) ( q) ( ) ( q) ξ ξ = 0 ξ ξ = 0 ξ( ) ξ ( q) = δ ( q) ( L( ) + ( )) η( ) η( q) = 0 η ( ) η ( q) = 0 η( ) η ( q) = δ ( q) ( L( ) + ( )) ξ( ) η( q) = 0 i ξ( ) η ( q) = δ ( q) ( L( ) ( )) One virtue of a good theory is simlicity Although the comosite hoton commutations relations (6) and (8) aear more comlex than the elementary commutations relations () and (4) they are really simler because it is only necessary to ostulate the fermion anticommutation relations and then derive boson commutation relations A more detailed discussion is contained in ef [7] The comosite hoton distribution for Blackbody radiation can be calculated using the second quantization method [5] as above but with the comosite hoton commutation relations This results [7] in (8) n = (9) ω e kt + Ω The Ω measure comonent is less than 0 9 so the difference between Equation (5) and (9) is too small to Polarization Vectors In the elementary theory the olarization vectors are chosen so that the electromagnetic field satisfies Maxwell equations In comosite theory there is no flexibility; the olarization vectors are given by the neutrino bisinors Elementary Photon Theory Polarization vectors for hotons with sin arallel and antiarallel to their momentum (taken to be along the third axis) are given by ( n) = ( i ) ( n) = ( i ) In Section we chose some roerties of the olarization vectors in Equation (9) and (0) In four dimensions we have k ( ) ( ) j = δ jk and the dot roducts with the internal four-momentum (0) () ( ) ( ) = 0 = 0 () 095

8 Also in three dimensions ( ) iω ( ) ( ) iω ( ) = = To calculate the comleteness relation we use linear olarization vectors Noting that the sum over olarization states only involves the two transverse olarizations and not the third direction j ( ) l ( ) j ( ) l ( ) j ( ) l ( ) δ jl = = j l = = (4) Comosite Photon Theory From Equation (0) we see that the olarization vectors are neutrino bisinors: () Carrying out the matrix multilications results in = = + ( ) u ( ) iγ u ( ) + + ( ) u ( ) iγ u ( ) + (5) i + E + E + ie + i E i i = 0 ( ) E( E+ ) E( E+ ) E i + E + E ie i E + i + i = 0 ( ) E( E+ ) E( E+ ) E Since the neutrino sinors and the olarization vectors only deend uon the direction of we can set n = E in n + + n n n n + in + in + in n = n in0 ( ) + n + n in n + + n n n n in in in n = n+ in0 ( ) + n + n As one can see these olarization vectors are good for any direction n while the elementary olarization vectors Equation (0) are only given along the third axis These olarization vectors satisfy the normalization relation k ( ) ( ) j = δ jk and the dot roducts with the internal four-momentum give Also in three dimensions (6) (7) (8) ( ) ( ) = 0 = 0 ( ) iω ( ) ( ) iω ( ) = = Using Equation (6) we calculate the comleteness relation j j j j ( ) ν ( ) ( ) ν ( ) δ ν j= j= (9) (40) ν = = (4) E 096

9 4 Maxwell Equations 4 Elementary Photon Theory In the elementary theory Maxwell equations are taken as an exerimental result as discussed in Section The vector otential A ( x) is then created to satisfy Maxwell equations 4 Comosite Photon Theory In the comosite theory Maxwell equations are derived as they must be if the comosite theory is relevant Substituting Equation (5) into Equation (0) gives A in terms of the olarization vectors ix { } ix A ( x) = G( ) ( ) + GL( ) ( ) e + G( ) ( ) + GL( ) ( ) e V ω (4) The electric and magnetic fields are obtained from E( x) = A ( x) t and ( x) = ( x) H A as usual ω ix ix { e } ( ) ( ) ( ) ( ) ( ) e = + L ( ) ( ) + L( ) ( ) E x i G G G G V (4) ω ix ix { } ( ) = ( ) ( ) L( ) ( ) + ( ) ( ) L( ) ( ) H x G G e G G e V (44) Using Equation (9) we obtain and with Equation (40) we obtain ( x) ( x) E = 0 H = 0 ( ) H( ) ( ) E( ) E x = x t H x = x t Using Equation (9) again we see that A satisfies the Lorentz condition 5 Number Oerator ( ) 0 (45) (46) A x x = (47) 5 Elementary Photon Theory The numbers oerator for an elementary hoton is defined as ( ) = ( ) ( ) N b b (48) When acting on a number state or Fock state it returns the number of hotons with momentum and olarization m ( ) = ( ( )) ( ) ( ) m N b 0 mb 0 (49) for a state with m hotons Normalizing in the usual manner [5] ( ) ( ) b n n n ( ) = + + b n = n n (50) Acting on the one and zero article states results in b b ( ) ( ) 0 = = 0 (5) 097

10 5 Comosite Photon Theory The number oerators for right-handed and left-handed comosite hotons are defined as ( ) = ( ) ( ) N G G ( ) = ( ) ( ) N G G L L L Perkins [7] showed that the effect of the comosite hoton s number oerator acting on a state of m righthanded comosite hotons is ( ) (5) m m m m N( ) ( G( ) ) 0 = m ( G( )) 0 (5) Ω where Ω is a constant equal to the number of states used to construct the wave acket and N ( ) 0 = 0 This result differs from that for the elementary hoton because of the second term which is small for large Ω Normalizing n G ( ) n = ( n + ) n + Ω ( n ) G ( ) n n = n Ω (54) where n is the state of n right-handed comosite hotons having momentum which is created by alying G ( ) on the vacuum n times Note that G G ( ) ( ) 0 = = 0 which is the same result as obtained with boson oerators The formulas in Equation (54) are similar to those in Equation (50) with correction factors that aroach zero for large Ω 6 Commutation elations for E and H 6 Elementary Photon Theory The commutation relations for electric and magnetic fields in the elementary hoton theory are [9] and Ei ( x) E j ( y) = δij id( x y) x0 y0 xi y j ( ) ( ) = ( ) ( ) H x H y E x E y i j i j i j ijk y0 k = (55) (56) (57) E ( x) H ( y) = id( x y) x (58) 6 Comosite Photon Theory With the comosite hoton theory the commutation relations for E and H are similar to the ones for the elementary hoton theory However the extra terms in comosite commutation relations (6) result in extra terms for the E and H commutation relations [7] With the extra terms the commutation relations do not vanish for sace-like intervals indicating that comosite articles have a finite extent in sace [7] k 098

11 and i Ei ( x) Ej ( y) = δij id( x y) d ω sin ( x y) ( ( ) + L ( )) x0 y0 xi y j 6π i 6π y ijk d ω cos ( x y) ( ( ) L ( )) 0 k = x k ( ) ( ) = ( ) ( ) (59) Hi x H j y Ei x Ej y (60) i E x H y = id x y x y + i ( ) j ( ) ijk ( ) d ω sin ( ) ( ( ) L ( )) y0 k = xk 6π 7 Charge Conjugation and Parity i δ 6π ( ) ( ( ) ( ) ) ij d ω cos x y L x0 y0 xi y j 7 Elementary Photon Theory The antihoton is identical to the hoton Thus the electromagnetic field can at most change by a factor of under charge conjugation Since the electromagnetic current j ( x) changes sign under the oeration of charge conjugation the electromagnetic field must transform as ( ) = ( ) (6) Cj x j x (6) ( ) = ( ) CA x A x (6) in order to leave the roduct j ( x) A ( x) in the Lagrangian invariant For ( x) resentation Equation (8) this means ( ) = b( ) ( ) = b ( ) C b C b Under the arity oerator the vector otential transforms as L L ( t) = ( t) A in the lane-wave re- (64) P A x A x (65) This imlies that the creation and annihilations oerators change as Under the combined oeration of CP In short-hand notation ( ) = bl( ) ( ) = b ( ) P b P b L ( t) = ( t) (66) CP A x A x (67) C γ = γ P γ = γ CP γ = γ 7 Comosite Photon Theory Under C (charge conjugation) and P (arity) the neutrino annihilation oerator transform as follows: (68) 099

12 ( k) = ( k) ( k) = ( k) ( k) = ( k) ( k) = ( k) ( k) = ( k) ( k) = ( k) ( k) = ( k) ( k) = ( k) C a c C c a C a c C c a P a a P c c P a a P c c We construct the comosite antihoton field in a manner similar to that of the comosite hoton field { + + A ( x) = G( ) u ( ) iγ u+ ( ) + GL( ) u+ ( ) iγ u ( ) e Vω + + ix ( ) + ( ) γ ( ) L( ) ( ) γ + ( ) } + G u i u + G u i u e with the annihilation oerators for left-circularly and right-circularly olarized antihotons with momentum given by Note that ( x) GL F c a k G F c a k ( ) = ( k) ( + k) ( k) ( ) = ( k) ( k) ( + k) A contains the other two sinors from Equation (5) Aying the charge conjugation and arity oerators on the comosite hoton annihilation oerators gives C GL( ) = GL( ) C G( ) = G( ) (7) P GL( ) = G( ) P G = G where we have taken F ( ) ( ) ( ) L k to be symmetric in k Alying the charge conjugation and arity oerators on the comosite hoton field gives since Under the combined oeration of CP In short-hand notation CA ( x) = A ( x) ( ) = ( ) PA x t A x t ( ) γ ( ) = ( ) γ ( ) ( ) γ ( ) = ( ) γ ( ) ( ) γ ( ) = ( ) γ ( ) u i u u i u u i u u i u u i u u i u ( t) = ( t) ix (69) (70) (7) (7) (74) CP A x A x (75) Cν e = ν e Since the internal structure of the comosite hoton is the antihoton is C ν = ν (76) e e γ = ν ν (77) e e γ = ν ν (78) e e Not only is γ different than γ but its neutrinos tyes have never been observed Under C and P 00

13 Cγ = γ Pγ = γ Cγ = γ P γ = γ (79) The hoton and antihoton are invariant only under the combined oeration of charge conjugation and arity CP e e γ = ν ν = γ CP γ = ν ν = γ (80) e e However there can be hoton states that are eigenstates of C and P As is done with the neutral kaon we create suerositions of the article and antiarticle Under charge conjugation γ = γ + γ ( ) γ = ( γ γ ) (8) C γ C γ = γ = γ (8) showing that γ is an eigenstate of C with value while γ is an eigenstate of C with value + with similar results under arity In the comosite hoton theory the electromagnetic field transforms in the usual way only under the combined oeration of CP 8 Symmetry under Interchange 8 Elementary Photon Theory Since the hoton is its own antiarticle all hotons are identical Thus a state of two hotons must be symmetric under interchange This result has been used to rule out certain reactions [0] [] 8 Comosite Photon Theory In the comosite theory four hoton states exist ie γ γ γ and γ If the hotons are not identical a state of two hotons can be antisymmetric (as well as symmetric) under interchange Therefore a vector article can decay into two hotons [] 9 Photon-Electron Interaction Here we examine Comton scattering using Feynman diagrams (The hoto-electric effect is similar) Figure (a) shows the usual Feynman diagram for Comton scattering with the incoming hoton imarting energy and momentum to an electron Figure (b) shows the same rocess with the hoton relaced by the bound state of the neutrino-antineutrino air as a chain of constituent fermion-antifermion bubbles The local interaction is similar to that in Fermi s beta decay theory [] The relevant Feynman rules are: V Ψ = Incoming electron: e ( ) Outgoing electron: V ( ) Ψ = e e Proagator: ( iγ me) ( m ) Incoming neutrino: V u ( k ) Incoming antineutrino: V u+ ( ) + Outgoing neutrino: V u ( k ) r 0

14 Outgoing antineutrino: V u ( ) Figure Comton scattering (a) Elementary hoton theory; (b) Comosite hoton theory r + k V ω i Incoming hoton: ( ) i Outgoing hoton: ( ) Vertex: ie γ k k V ω k 9 Elementary Photon Theory The matrix element for Comton scattering as shown in Figure (a) is ie iγ i q + m e i = ( ) ( ) ( ) ( ) ( ) Ψe γ k γ k Ψe (8) V ω q q + me 9 Comosite Photon Theory In the comosite theory the matrix element for Comton scattering as shown in Figure (b) is and ie iγ q + m = Ψ ( ) ( ) Ψ V ( q + me ) ( ) ( ) ( ) ( ) e + + e γ u+ r u k γ e u+ r γ u k ω q iγ q + m + e + +Ψe ( ) γ u ( k) u ( ) γ ( ) ( ) ( ) + r Ψe u k γ u+ r ( q + me ) The matrix element contains comonents + ( ) γ u ( ) u ( ) γ Ψ ( ) (84) Ψe + r k e (85) + ( ) γ u ( ) u ( ) γ Ψ ( ) Ψe k + r e (86) 0

15 Since the electron-neutrino interaction is V-A we must insert the rojection oerator ( γ 5 ) to select states with negative-helicity articles and ositive-helicity antiarticles With this insertion we have comonents and + Ψe ( ) γ ( γ5) u ( ) + u ( ) γ ( γ5) Ψe ( ) 4 r k (87) + Ψe ( ) γ ( γ5) u ( ) u+ ( ) γ ( γ5) Ψe ( ) 4 k r (88) Since u + ( ) designates a ositive-helicity antiarticle and ( ) the insertion of ( γ ) u + designates a negative-helicity article 5 does not change the result [] However for the interaction of an antihoton with an electron the terms contain comonents and + The ( γ ) u ( ) and ( γ ) u ( ) Ψe ( ) γ ( γ5) u ( ) u+ ( ) γ ( γ5) Ψe ( ) 4 r k (89) + Ψe ( ) γ ( γ5) u ( ) + u ( ) γ ( γ5) Ψe ( ) 4 k r (90) 5 terms equate to zero as i ( + E+ E+ γ 5) u+ ( ) = E+ = (9) E E This indicates that antihotons do NOT interact with elections in a matter world because ν e and ν e have the wrong helicity In an antimatter world the ositron-neutrino interaction is V + A and ( + γ 5 ) selects states with ositive- helicity articles and negative-helicity antiarticles In a symmetric manner hotons do not interact with ositrons in an antimatter world [] Exeriment [] shows that all the hotons in ositronium are detected Therefore the hotons involved must be γ and γ the suerosition of γ and γ Positrons interact with the electromagnetic field in a manner similar to that of electrons Thus the comosite hoton theory requires that the effect of virtual hotons is the same in matter and antimatter worlds Conclusions In comaring the elementary and comosite hoton theories it is noted that in the elementary theory it is difficult to describe the electromagnetic field with the four-comonent vector otential This is because the hoton has only two olarization states This roblem does not exist with the comosite hoton theory The commutation relations are more comlex in the comosite theory because of the comosite hoton s internal fermion structure However this comlexity is not unique to the comosite hoton; other comosite articles with internal fermions have similar comlexity In the elementary theory the olarization vectors are chosen to give a transverse field while in the comosite theory they are determined by the fermion bisinors The comosite theory redicts Maxwell equations while the elementary theory has been created to encomass it Some differences are so slight that they are almost imossible to detect exerimentally (ie Planck s law) However the comosite theory redicts that the antihoton is different than the hoton 0

16 Pryce [4] had many arguments against a comosite hoton theory His arguments are either not valid or irrelevant Let us look at them one by one: ) Pryce: In so far as the failure of the theory can be traced to any one cause it is fair to say that it lies in the fact that light waves are olarized transversely while neutrino waves are olarized longitudinally Both Case [5] and Berezinski [6] asserted that constructing transversely olarized hotons is not a roblem The fact that one can combine neutrino fields and obtain a comosite hoton that satisfies Maxwell equations (as in Section 4) roves that this is not a roblem ) Pryce: In order to fix the reresentation therefore we must decide on a definite a [olarization vector erendicular to n ] This choice is entirely arbitrary for among all unit vectors erendicular to a given direction in sace all are equivalent and none is singled out in any way The comosite theory singled out the two olarization vectors of Equation (6) which are functions of n Under a rotation by an angle θ about n they change into themselves iθ ( n) e ( n) (9) iθ ( n) e ( n) Note that ( n) is a self-orthogonal comlex unit vector [4] ) Pryce: the theory [must] be invariant under a change of co-ordinate system it has been necessary to analyze rather carefully the transformation of the amlitudes under certain tyes of rotation and this reveals an arbitrariness in the choice of certain hases In order to obtain the comleteness relation Equation (4) Kronig [] arbitrarily wrote his Equation (7) connecting neutrino sinors Pryce showed that Kronig s Equation (7) combined with Kronig s Equation (9) was not invariant under a rotation of the coordinate system Kronig s Equation (7) is not needed as one can obtain the comleteness relation Equation (4) from the lane-wave sinors as shown in Section Pryce s argument that the comosite hoton theory is not invariant under a rotation of coordinate system alies to one unnecessary equation in Kronig s aer 4) Pryce: The conditions under which this will lead to a satisfactory theory of light are () that certain [Bose] commutation rules be satisfied; () that the theory be invariant under a change of coordinate system Pryce required that comosite hotons satisfied Bose commutation relations (Jordan and Kronig were working on that assumtion) Pryce [4] showed that requiring ξ( ) η ( q ) = 0 meant that ξ = 0 For a roof using the last of Equation (8) see [] This is a valid oint but it is really irrelevant Integral sin articles are considered to be bosons and most integral sin articles (deuterons helium nuclei Cooer airs ions kaons etc) are comosite articles formed of fermions These comosite articles cannot satisfy Bose commutation relations because of their internal fermion structure but their difference from erfect bosons is so small that it has not been detected with the excetion of Cooer airs [7] In the asymtotic limit which usually alies these comosite articles are bosons An imortant test of these ideas will occur when the hotons from anti-hydrogen are examined The comosite hoton theory redicts that the antihotons from anti-hydrogen will have the wrong helicity for interaction with electrons and thus the antihotons will not be detectable Furthermore ordinary hotons have the wrong helicity for interaction with anti-hydrogen Acknowledgements Helful discussions with Prof J E Kiskis are gratefully acknowledged eferences [] de Broglie L (9) Comtes endus [] de Broglie L (94) Une novelle concetion de la lumiere Hermann et Cie Paris [] Jordan P (95) Zeitschrift fur Physik htt://dxdoiorg/0007/bf007 [4] Pryce MHL (98) Proceedings of oyal Society (London) A htt://dxdoiorg/0098/rsa [5] Case KM (957) Physical eview htt://dxdoiorg/00/physev066 [6] Berezinskii VS (967) Soviet Physics JETP [7] Perkins WA (00) International Journal of Theoretical Physics htt://dxdoiorg/00/a: [8] Fukuda Y et al (998) Physical eview Letters htt://dxdoiorg/00/physevlett856 04

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