Compositeness, Bargmann-Wigner solutions within a U(1)-interaction quantum-field-theory expansion, and charge
CCompositeness, Bargmann-Wigner solutions withina U(1)-interaction quantum-field-theory expansion,and charge
J. Besprosvany ∗ Instituto de F´ısica, Universidad Nacional Aut´onoma de M´exico,Apartado Postal 20-364, 01000, Ciudad de M´exico, M´exico
Abstract
New solutions of the Bargmann-Wigner equations are obtained: free fermion-antifermion pairs, each satisfying Dirac’s equation, with parallel momenta andmomenta on a plane, produce vectors satisfying Proca’s equations. These equa-tions are consistent with Dirac’s and Maxwell’s equations, as zero-order con-ditions within a Lagrangian expansion for the U(1)-symmetry quantum fieldtheory. Such vector solutions’ demand that they satisfy Maxwell’s equationsand quantization fix the charge. The current equates the vector field, repro-ducing the superconductivity London equations, thus, binding and screeningconditions. The derived vertex connects to QCD superconductivity and con-strains four-fermion interaction composite models. ∗ email: bespro@fisica.unam.mx 1 a r X i v : . [ h e p - ph ] J a n Introduction
The Standard Model[1]-[3] (SM) accurately describes elementary particles and theirhypercharge ( Y ), weak (left-handed, L ) and color ( c ) interactions, defined by thegauge groups U(1) Y × SU(2) L × SU(3) c , respectively; yet, puzzles remain as the originof phenomenological constants, like the interactions’ coupling constants.Insight on these was provided on proposed structures that generalize physical fea-tures: grand-unified theories[4] assume a common group for the interactions, requiringa unique coupling constant at the unification scale, and setting constraints on theirvalues.Compositeness, which refers to related properties in systems built from simplerones, underlies many physical systems and provides information on them. Thequark model[5] is one of its paradigms, as hadron features are derived from thequarks’. Likewise, theories with more fundamental elements were proposed to ex-plain SM features[6, 7]; supersymmetry generalizes the SM composite quantum num-bers to additional feasible fermions and bosons[8]; and SM structures can predictsuch constants[9, 10], all suggesting elementary composite configurations may pro-vide clues, for which we review relevant physical and formal setups.In superconductivity, compositeness is also present. For its Bardeen-Cooper-Schrieffer’s[11] (BCS) theory, the relevant degree of freedom is a Cooper pair con-formed of an electron and a hole. In addition, interaction screening allows for free-particle behavior, which may enable elementary-particle properties to become mani-fest.An early application of this scheme in particle physics[12] describes a mass-generating mechanism for fermions and composite bosons, relating their masses,through a hypothesised interaction built from four fermions, with binding and self-consistency features. A correspondence was argued between such a composite modeland the U(1) theory[13]. The idea of a quark condensate generating the Higgs[14, 15],2ith a binding effect analogous to superconductivity, remains feasible. Such a mech-anism is dynamical, as for radiative corrections[16].We use the simpler Abelian U(1)-group gauge symmetry, relevant at all ranges.With focus on fundaments, a gauge-interaction quantum field theory is a useful frame-work as a four-vector’s spin-1 degree of freedom can be constructed in terms of twofermions’ spin-1/2. The U(1) interaction picks a particular four-fermion interactionas effective theory for the Nambu-Jona Lasinio model[17]; it comprises physically thehypercharge and the electromagnetic interactions, and an effective version for thestrong interaction that, e. g., models the quark anti-quark potential[18].Framed within the U(1) theory, the Bargmann-Wigner equations[19] depart froma fermion-antifermion pair, each element satisfying Dirac’s equation, that conforms acurrent element satisfying Proca’s, all comprising a composite configuration.In this paper, we present a new class of Bargmann-Wigner solutions, which signi-fies their space is dimensionally larger. We show that these equations constitute zero-order terms, within a generated expansion for the Lagrangian, for a U(1)-interactionquantum field theory, constituting a compositeness limit. For such systems, we showthat Maxwell’s equations and quantization fix the coupling constant. Such configura-tions satisfy superconductivity conditions through the London equations, connectingthe current and the U(1) vector. Finally, we argue that this setup applies to QCD su-perconductivity, among other systems. The material is organized thus: Section II de-velops a Lagrangian expansion that contains the free Dirac’s and Maxwell’s equationsas zero-order conditions; in Section III, these are connected to the Bargmann-Wignerequations, presenting more solutions. Section IV derives a charge from Maxwell’sequations consistency and quantization. Section V shows that the London’s equationsare implied, linking fermion-antifermion states to superconductivity. Section VI ana-lyzes other applications and draws conclusions. Appendixes A-F present calculationsin detail; notably, Appendix B contains Bargmann-Wigner solutions with fermion-antifermion same-momentum and arbitrary spin configurations, and with combina-3ions of momenta on the same plane, including a confined configuration. AppendixC presents the adapted Bargmann-Wigner equations used. II U(1) expansion
A U(1)-symmetry imposes four-vector field A µ ( x ) minimal substitution on Dirac’sequation[20], setting its interaction with the fermion represented by ψ , a four-complexcolumn wave function accounting for spin and the antiparticle, with the U(1) four-current given by gj µ = g ¯ ψγ µ ψ , with coupling g . The x space-time coordinate depen-dence is hence usually omitted. The U(1) Lagrangian L ( A µ , ψ ) = L D ( ψ ) + L K ( A µ ) + L I ( A µ , j µ ) contains the free-Dirac ( D ), vector-kinetic ( K ) components and interac-tion term ( I ) L I ( A µ , j µ ) = − gA µ j µ . Defining the the lowest-order solution throughan external c-number current[21] j sµ leads to L C ( A µ , ψ ) = L ( A µ , ψ ) − L I ( A µ , j µ ) + L I ( A µ , j sµ ) = L D ( ψ ) + L K ( A µ ) + L I ( A µ , j sµ ) (1)as zero-order term, with corrections L I ( A µ , j µ ) − L I ( A µ , j sµ ).Next, we obtain the equations of motion to this order, showing that extendedBargmann-Wigner equations connect to them, allowing for a charge definition, andimplying that A µ satisfies the superconductivity[11] London’s equations. We shall usethe relativistic-quantum-mechanics framework and the quantization/normalizationcondition. 4 II Dirac’s and Maxwell’s equations connection toBargmann-Wigner equations
The Euler-Lagrange equations for L C ( A µ , ψ ) in Eq. 1 imply Dirac’s free-particleequation for a U(1)-charged fermion ( i∂ µ γ µ − m ) ψ ( x ) = 0 , (2)with m its mass, γ µ × { γ µ , γ ν } = 2 Ig µν ,transforming under the Lorentz group as vectors, I the identity matrix, and g µν themetric. These matrices are given in the Dirac representation in Appendix A, withsolutions in Appendix B.Now, with the current notation j sµ → j µ , not specifying the zero order, variationsof L C over A µ lead to Maxwell’s equations ∂ ν ( ∂ ν A sµ − ∂ µ A sν ) = 4 πgj µ , (3)where A sµ provides the solution to Maxwell’s equations at this order. The homogenousequations are also implied for F µν = ∂ µ A sν − ∂ ν A sµ ∂ µ (cid:15) µνησ F ησ = 0 . (4)The Bargmann-Wigner[19] equations (applied for two quanta, Appendix C) com-prise two identical spinors satisfying each Dirac’s equation 2. These conform a vectorsatisfying in turn Proca’s equations[22] (which generalize Maxwell’s equations to themassive case) ( (cid:126) c ) [ ∂ ν ( ∂ µ j ν − ∂ ν j µ )] = (2 mc ) j µ . (5) For the speed of light and reduced Planck’s constant, c = 1, (cid:126) = 1 is assumed, respectively,unless needed. The metric is defined g µν = (1 , − , − , − and mass 2 m , as ∂ µ j µ = 0 (Lorentz gauge). These equations translate such spin-1/2 elements to non-diagonal current components, for an electron and a positron wave functions j µ = (cid:104) ψ c | γ γ µ | ψ (cid:105) , where ψ c = Cγ ψ ∗ , C = iγ γ is the charge-conjugation matrix in the Dirac representation, for relevant cases (Appendix C).Fig. 1 represents these solutions, reinterpreted as a fermion-antifermion pair, as theyconnect to states in a Feynman diagram in which it converts to a massive vector.Given the antisymmetric form F jµν = ∂ µ j ν − ∂ ν j µ , Maxwell’s homogeneous equationsare satisfied, as in Eq. 4. Same-momentum opposite spin, and arbitrary energy,spin, and momentum on a plane, extend the basis for larger kinematical regions,implying a feasible description of more than one fermion-antifermion pair (AppendixC). For the fermion self-energy, the U(1) theory leads to an effective (four-fermion)Hartree interaction, which connects to the contact vector-vector term in L I ( A µ , j sµ ),constituting an interaction case in the Nambu-Jona Lasinio model[17].For the field A sµ in Maxwell’s equations 3, generated by j µ , the equations in 5 area necessary consistency condition restricting g . With their common element j µ onthe right-hand side, matching these equations implies the former is the latter times4 πg/ (2 mc ) . We rewrite Eq. 5 in terms of A ψµ = ( (cid:126) c ) (2 mc ) ¯ ψγ µ ψ. (6)Using the Lorentz-gauge condition (cid:3) A ψµ = − ¯ ψγ µ ψ, (7)where the (cid:3) = ∂ µ ∂ µ is the d’Alambertian. Comparing the same elements, withderivatives acting upon them, leads to4 πgA ψµ = − A sµ . (8)This equation ascertains A ψµ is the charge-independent component. For arbitrary c , (cid:126) units, with ∂ µ = ( c ∂ , ∇ ), j µ = ( ¯ ψγ ψ, − c ¯ ψ γ ψ ), γ = ( γ , γ , γ ). Within context, we use the electron and its antiparticle as fermion representatives. igγ µ , related to solutions at fixed vol-ume V , as described in Eq. 3. Incoming fermion-antifermion f - ¯ f have four-momenta p f , p ¯ f . The Bargmann-Wigner solutions here have parallel momenta with p f = p ¯ f = ( E, , , p z ), energy E = (cid:112) m + | p | , p = (0 , , p z ), each with spin s z ,represented by the small cones, producing massive vector A sµ with four-momentum p f + p ¯ f , polarization 2 s z . The inverse process represents vector-particle annihilationinto two fermions. 7 V Vector quantization and the U(1) charge
Quantum field theory imposes field quantization, which identifies the factor connect-ing A ψµ with A sµ in Eq. 6, as they constitute the same quantum. We require thenormalization condition on conserved quantities as a proxy to quantization rules forvectors[23].A vector component extended to be massive, is analyzed in Appendix D. TheMaxwell’s energy-quantization condition[23, 24], for Eq. 3, for such a vector withmomentum 2 p , and mass 2 m , is (cid:82) d x π [ ( | E s | + | B s | ) + (2 m ) (2 A s A s − g A sµ A sµ )] = 2 E, (9)where E = (cid:112) ( mc ) + | c p | , and we define the electric and magnetic componentsfrom A sµ = ( A s , A s ) E s = − ∂ A s − ∇ A s (10) B s = ∇ × A s , valid for a U(1) complex field, in correspondence with the Klein-Gordon equationcase.On the other hand, a Proca positive-energy free field with the same four-momentumquantum-mechanical normalization in Eq. 7 requires[25] − (cid:90) d xA ψµ ∗ i c ←→ ∂ A ψµ = 1 . (11)This massive-vector quantization condition and that of the U(1) field connectionin Eq. 9 implies, comparing their normalization constants (Appendixes B, D) √ π (cid:126) cA ψµ = − A sµ . (12)Combining this equation and the Dirac and U(1) vector relation in Eq. 8 fixes g : g/ √ (cid:126) c = 1 / √ π, (13) In the parallel momentum and spin case, the energy-momentum T µν satisfies ∂ µ T µν = 0 . is g eLH / √ (cid:126) c = √ πg/ √ (cid:126) c = 1 in Lorentz-Heaviside units.The conserved property of the constructed energy and probability in Eqs. 9,11, respectively, ensure that g is Lorentz and gauge invariant. The obtained value, α − g = (cid:126) c/g = 4 π (cid:39) .
6, is between the couplings[26]: weak g w , hypercharge g (cid:48) and strong g s (associated to its U(1) effective form[27]) at low energies, consistentlywith unification conditions. Links to SM interactions are feasible, through a uni-fication model. The unification coupling constrains SM ones, as obtained in othertheories and energy scales[4, 28]. One connection is through quantum chromody-namics, for its asymptotically-free behaviour[29] relates to a unification energy scale,with g interpreted as the effective U(1) component[27] sets (4 / g s = g , implying α − g s = π (cid:39) .
76 . Such values are consistent with the unified coupling constantin some models[30], with emphasis on those with compositeness[31]. A generic U(1)interaction with stationary coupling running up to the unification scale, within feasi-ble models[32] accounts for such scales at high-energy. Such a value is also consistentwith compositeness theories under fixed-point conditions for the coupling[15].
V Fermion-antifermion self-superconductor
The Maxwell’s equations also relate the current to the vector potential. Using Eqs.6, 8, and 13, extended London’s equations are obtained j µ = − g (cid:0) mc (cid:126) (cid:1) A sµ mc , (14)as A sµ reproduces superconductivity behavior. Paired particles with 2 m mass associatethis superconductivity condition to the density n j = (cid:0) mc (cid:126) (cid:1) , with cube unit size ofreduced Compton wavelength[33] λ C = (cid:126) mc . We compare them using the generalizedGordon identity[34] (Appendix F) for the interactive case¯ ψi ←→ ∂ µ ψ − g ¯ ψψA µ + i ∂ ν ¯ ψ [ γ µ , γ ν ] ψ = 2 m ¯ ψγ µ ψ. (15) Appendix E shows that this value is consistent with other unit choices. g , the U(1) four-current gj µ = g ¯ ψγ µ ψ has[11], on the left-hand side, a paramagnetic contribution (first term), and a dia-magnetic (second) term, where A ( x ) is the U(1) vector potential, reproducing forthe current the structure of London’s superconductivity equations[35] (in the Londongauge ∇ · A = 0) g j ( x ) = − g A ( x ) n c mc , (16)where n c is the electron density, to be compared with the local scalar bilinear ¯ ψψ .Eq. 16 is a valid approximation to Eq. 15 with only the second term on the left-handside, in the regime of small wave-function variation, e. g., | ¯ ψi ←→ ∂ µ ψ/ ¯ ψψ | (cid:28) | gA µ | .For this second superconducting component in Eq. 15, the U(1) field is associatedto the system’s length L . Over long distances L (cid:29) λ C , such component’s contribu-tion is negligible as compared to the third component, which contains the Maxwell’sequations contribution in Eq. 5, (Appendix F) and depends on λ C , implying thisconstitutes a sizable component in a fermion’s generated interactive field. As for theMeissner effect, a current is generated without friction, as a classical expansion ofthis equation manifests it. Unlike the usual London equations, which describe cur-rent under external fields, here they are self-consistent; also, the current oscillates,not decays. Substitution of Eq. 15 into L I ( A µ , j sµ ) implies such a term is atractive,as the on-shell A µ (Appendix D) is space-like.This scheme applies to models and scenarios in which compositeness is assumedexplicitly, starting with the application of the BCS model within quantum field theorythrough the Nambu Jona-Lasinio model[12]; SM extensions reproduce spontaneoussymmetric breaking[14, 15] through quark condensates, with supersymmetry[36], technicolor[37],and some unification models. 10 I Applications and conclusions
Methods are added to known cases as extensions [28, 37] requiring composite ele-ments. Compositeness can be applied to SM couplings[9], as connections were re-cently exploited to obtain information on the quark masses[10]. As the current is aparticular bilinear-spinor term, similarly to superconductor Cooper pairs[38], a com-posite regime is feasible, under some conditions. Generically, a 4-fermion interactionis self-consistent, as described in the Hartree approximation[39] or as order parame-ter in the Landau-Ginzburg theory[40]. The generation of a boson quantum withinsuperconducty[11, 12] suggests screening and binding conditions.The consistency of the implied Bargmann-Wigner condition in Eq. 5 with Maxwell’sequations and quantization fixes the effective U(1) charge. Derived London’s equa-tions imply superconductivity conditions, as a zero-order contribution within an in-duced expansion. Universal m -independent properties follow.The resulting fermion-antifermion pair with parallel momenta induces a 2 m vectorresonance described by the A sµ relation to the current through the extended Londonequations 14. Such a vector superconductive state constitutes zitterbewegung[41],and could be attained by stabilizing their interaction and with external fields[42]-[44],recreating screening and binding conditions. It should alter “fermonium” (fermion-antifermion) properties, possibly detectable in positronium (an e + e − system).Such a vector resonance is also consistent with the interpretation of the expansionthrough a massive contribution. The component L m ( A µ ) = L I ( A µ , A µ ) = (2 m ) A µ A µ corresponds to a 2 m mass term for A µ in which one may depart in an expansion thatincludes a massive vector term: L D ( ψ ) + L K ( A µ ) + L m ( A µ ) and orders in L I ( A µ , ψ ) −L m ( A µ ). Some models construct SM vectors fields dynamically and as compositeobjects[45]. The massive vector A sµ (or j ψµ ,) constructed a free fermion-antifermionpair constitutes an A µ component, within the zero-order Lagrangian expansion.Feasible connections arise for the U(1) compositeness regime, in limits stud-11ed in quantum electrodynamics (QED). In the infrared limit[46], reduced photon-interaction diagrams create conditions for the prevalence of a momentum-independentrenormalized coupling term. At high energy, QED predicts a fixed point[47], for nottoo large couplings, in which four-fermion interactions become renormalizable, allow-ing for coupling momentum-independence, with applications as constructing realistictechnicolor models[48]. In addition, QCD’s asymptotic freedom make the paper’sfree-particle solutions and conditions relevant for this regime.High density implies high-energy conditions, and under asymptotic-freedom forQCD, free-particle behavior and an attractive interaction generate superconduct-ing conditions[49]. These conditions are feasible in nuclear physics too[50]. ForFermi liquids under QCD superfluidity or superconducting conditions, at the Fermi-gas surface, four-fermion vertices are relevant with the kinematic structure asso-ciated with Cooper pairing (i.e. two quarks scattering with equal and oppositespatial momenta)[51, 52]. This paper’s new solutions provide parametrizations forsuperconducting confined systems, sensible to boundary conditions[53]. In addi-tion, its derived fermion-antifermion interaction components provides matrix elementsfor Dirac-equation wave functions and their conjugates, relevant in the theory ofsuperconductivity[54].An expansion with a c-number current constructed from a fermion-antifermion isassumed, akin to a mean-field, with corrections to such one-particle self-componentsolution extracted from the coupled Maxwell-Dirac equations, producing the fermion-vector interaction. Solutions were found for the Dirac equation under a massive vectorfree field[55], generalizing Volkow’s solution for a massless vector. Other correctionsshould consider second quantization, many-particle contributions, three-dimensionaleffects, loop corrections, and renormalization.Under compositeness conditions, for a U(1) quantum field theory, the couplingis derived from the consistency of a quantized current from fermion-antifermion freesolutions, and Maxwell’s equations. The point of view emerges that such a constant is12nherent to a theory[56]. Superconducting conditions are induced, within a generatedexpansion. The next task is to obtain the subsequent corrections to such a constant. Acknowledgements
The author acknowledges discussions with A. A. Santaella on the extended Gordonidentities and Bargmann-Wigner equations applications, and support from L. Novoaand M. Casa˜nas at the IF-UNAM with graphics, and from the Direcci´on General deAsuntos del Personal Acad´emico, UNAM, through Project IN117020.
Appendix A: Gamma-matrix identities and the Dirac repre-sentation
Among 4 × γ µ matrix properties, we list[21]: γ µ γ ν = g µν − iσ µν (A1)where σ µν = i [ γ µ , γ ν ] .γ µ γ ν γ η = g µν γ η + g νη γ µ − g µη γ ν + i(cid:15) µνησ γ γ σ , (A2)the chirality γ = iγ γ γ γ , and (cid:15) µνησ is the four-dimensional Levi-Civita tensor: (cid:15) µνησ = { µ, ν, η, σ } is an even permutation of { , , , }− { µ, ν, η, σ } is an odd permutation0 otherwise (A3)The Dirac representation for the γ µ matrices is given next, where we use the 2 × σ = (cid:18) (cid:19) σ = (cid:18) − ii (cid:19) σ = (cid:18) − (cid:19) (A4)and I × , the identity matrix, to define them: γ = (cid:18) I × − I × (cid:19) γ j = (cid:18) σ j − σ j (cid:19) j = 1 , , . (A5)13 ppendix B: Dirac’s equation fermion-antifermion solutioncombinations Dirac’s free equation in 2 contains particle and antiparticle ( x -dependent) solutions ψ u ( p, s α ), ψ v ( p, s α ) with p , s α , the associated momentum, and spin four-vectors, re-spectively. Negative-energy solutions are associated to antiparticles, with oppositequantum numbers assigned within the Dirac-sea interpretation, assumed in the no-tation, while second quantization provides a symmetric and consistent treatment. p is chosen with spatial components along ˆ z : p µ = ( E, , , p z ) = ( (cid:112) p z + m , , , p z ).The wave functions are separated ψ u ( p, s α ) = u ( p, s α ) e − ipx , ψ v ( p, s α ) = v ( p, s α ) e ipx ,where u ( p, s α ), v ( p, s α ) are associated spin states. We construct them from u = , u = (B1) v = , v = , (B2)with u ( p, s + , − ) = 1 (cid:112) m ( E + m ) ( Eγ − p z γ + m ) u , (B3) v ( p, s + , − ) = 1 (cid:112) m ( E + m ) ( − Eγ + p z γ + m ) v , (B4)classified by the spin operator − γ n · γ p · γ , defined by the helicity vector n = m ( | p | , E p / | p | ): 12 m γ ( p z γ − Eγ ) u ( p, s + ) = 12 u ( p, s + ) (B5)12 m γ ( p z γ + Eγ ) v ( p, s + ) = − v ( p, s + ) , m γ ( p z γ + Eγ ) v ( p, s − ) = 12 v ( p, s − ) , u ( p, s + ) is a positive-energy p z -momentum positive spinor and v ( p z , s − )is a p z -momentum negative spinor, both along ˆ z . We use spinor normalization (cid:104) u ( p, s α ) | u ( p, s α ) (cid:105) = (cid:104) v ( p, s β ) | v ( p, s β ) (cid:105) = E/m. (B6)Relevant current elements for u - and v -type states, produce the transverse andlongitudinal polarization current matrix elements, respectively: j µ ++ vu = j µ ++ ∗ uv = (cid:104) ψ v ( p, s + ) | γ γ µ | ψ u ( p, s + ) (cid:105) = mE e − iEt +2 ip z z (0 , , i,
0) (B7) j µ + − vu = j µ − + ∗ uv = (cid:104) ψ v ( p, s − ) | γ γ µ | ψ u ( p, s + ) (cid:105) = − E e − iEt +2 ip z z ( k z , , , E ) . (B8) Bargmann-Wigner solutions on a plane
A generalization is made for a stationary fermion-antifermion pair. We definefermion states with two momenta on a plane: p yz = ( E yz , , p y , p z ), E yz = (cid:112) p y + p z + m and ˜ p yz = ( ˜ E yz , , ˜ p z , ˜ p y ), ˜ E yz = (cid:112) ˜ p y + ˜ p z + m and possible spin combinations: Thewave functions are ψ u ( p yz , s α ) = u ( p yz , s α ) e − ip yz x , ψ v ( p yz , s α ) = v ( p yz , s α ) e ip yz x , where u ( p yz , s α ), v ( p yz , s α ) are associated spin states. We construct them with u ( p yz , s + , − ) = 1 (cid:112) m ( E + m ) ( Eγ − p y γ − p z γ + m ) u , (B9) v ( p yz , s + , − ) = 1 (cid:112) m ( E + m ) ( − Eγ + p y γ + p z γ + m ) v , , with u , , v , in Eqs. B1. They are classified by the appropriate spin operator, along n yz − γ n yz · γp yz · γ , defining, e. g., − γ n yz · γp yz · γu ( p yz , s + ) = 12 u ( p yz , s + ) (B10) − γ n yz · γp yz · γv ( yz , s − ) = 12 v ( yz , s − ) , The fermion and anti-fermion wave functions | M u (cid:105) = a | ψ u ( p yz , +) (cid:105) + b | ψ u (˜ p yz , +) (cid:105) + c | ψ u ( p yz , − ) (cid:105) + d | ψ u (˜ p yz , − ) (cid:105) (B11) | M v (cid:105) = a ∗ | ψ v ( p yz , +) (cid:105) − b ∗ | ψ v (˜ p yz , +) (cid:105) + c ∗ | ψ v ( p yz , − ) (cid:105) − d ∗ | ψ v (˜ p yz , − ) (cid:105) (B12)15roduce current matrix elements: (cid:104) M v | γ γ µ | M u (cid:105) = a j µ ++ vu yz + b j µ ++˜ v ˜ u yz + c j µ −− vu yz + d j µ −− ˜ v ˜ u yz + ac ( j µ − + vu yz + j µ + − uv yz )+ bd ( j µ − +˜ v ˜ u yz + j µ + − ˜ v ˜ u yz ), where a , b , c , d are arbitrary constants,producing current matrix elements: j µ ++ vu yz = (cid:104) ψ v ( p yz , s + ) | γ γ µ | ψ u ( p yz , s + ) (cid:105) = − e − i ( Et − p y y − p z z ) E ( p y , − im, E + 2 Em + m + p y − p z E + m ) , p y p z E + m ) (B13) j µ ++˜ v ˜ u yz = (cid:104) ψ v (˜ p yz , s + ) | γ γ µ | ψ u (˜ p yz , s + ) (cid:105) = e − i ( E (cid:48) t − p (cid:48) y y − p (cid:48) z z ) E (cid:48) ( p (cid:48) y , − im, E (cid:48) + 2 E (cid:48) m + m + p (cid:48) y − p (cid:48) z E (cid:48) + m ) , p (cid:48) y p (cid:48) z E (cid:48) + m ) (B14) j µ −− vu yz = (cid:104) ψ v ( p yz , s − ) | γ γ µ | ψ u ( p yz , s − ) (cid:105) = − e − i ( Et − p y y − p z z ) E ( p y , im, E + 2 Em + m + p y − p z E + m ) , p y p z E + m ) (B15) j µ −− ˜ v ˜ u yz = (cid:104) ψ v (˜ p yz , s − ) | γ γ µ | ψ u (˜ p yz , s − ) (cid:105) = e − i ( E (cid:48) t − p (cid:48) y y − p (cid:48) z z ) E (cid:48) ( p y , im, E (cid:48) + 2 E (cid:48) m + m + p (cid:48) y − p (cid:48) z E (cid:48) + m ) , p (cid:48) y p (cid:48) z E (cid:48) + m ) (B16) j µ − + vu yz = j µ + − vu yz = (cid:104) ψ v ( p yz , s − ) | γ γ µ | ψ u ( p yz , s + ) (cid:105) = − e − i ( Et − p y y − p z z ) E (2 ip z , , ip y p z E + m , i (cid:0) E + 2 Em + m − p y + p z (cid:1) E + m ) (B17) j µ − +˜ v ˜ u yz = j µ + − ˜ v ˜ u yz = (cid:104) ψ v (˜ p yz , s − ) | γ γ µ | ψ u (˜ p yz , s + ) (cid:105) = e − i ( E (cid:48) t − p (cid:48) y y − p (cid:48) z z )2 E (cid:48) (2 ip (cid:48) z , , ip (cid:48) y p (cid:48) z E (cid:48) + m , i (cid:0) E (cid:48) + 2 E (cid:48) m + m − p (cid:48) y + p (cid:48) z (cid:1) E (cid:48) + m ) (B18)A U(1) vector is thus obtained from a fermion-antifermion pair. The constants a , b , c , d are set by the boundary conditions. 16 ppendix C: Bargmann-Wigner equations Eq. 2 is written equivalently in the transposed version ψ t ( i ←− ∂ µ γ µt − m ) = 0 (C1) ψ t CC t ( i ←− ∂ µ γ µt − m ) C = 0 (C2) ψ t C ( i ←− ∂ µ γ µ + m ) = 0 (C3)In the Dirac matrix representation, the charge conjugation operator is C = iγ γ ,which satisfies CC t = C t C = − C = 1, C t γ µt C = − γ µ . The matrix pair config-uration Ψ BW = ψψ t C has only symmetric elements[57], implying Ψ BW = ( j µ BW γ µ + F µν BW σ µν ) C , with σ µν defined after Eq. A1. In the original Bargmann-Wigner equa-tions, the mass element is associated with a fermion quantum. Here, the equationsdescribe two quanta, with the action of the derivative operating on same-momentumket bra (particle anti-particle) contributions( ∂ µ | ψ u ( p, s α ) (cid:105) ) (cid:104) ψ v ( p, s α ) | = 12 [( ∂ µ | ψ u ( p, s α ) (cid:105) ) (cid:104) ψ v ( p, s α ) | + | ψ u ( p, s α ) (cid:105) ( (cid:104) ψ v ( p, s α ) |←− ∂ µ )] . (C4)Application of the Dirac operators as in Eqs. 2, C3 from the left and right, respec-tively, lead to the conditions F BW µν = 12 ( ∂ µ j BW ν − ∂ ν j BW µ ) (C5)12 ∂ µ F BW µν = − m j BW ν . (C6)Substitution of the first into the second leads to Proca’s equations ∂ µ ∂ µ j BW ν − ∂ ν ∂ µ j BW µ = − m j BW ν , (C7)which reduce to (cid:3) j BW ν = − m j BW ν , as ∂ µ j BW µ = 0.The association is to a fermion and anti-fermion connecting through the ampli-17udes F µν BW = trΨ BW σ µν = tr ψψ t Cσ µν = (cid:104) ψ v ( p, s α ) | γ σ νµ | ψ u ( p, s α ) (cid:105) (C8) j µ BW = trΨ BW γ † µ = tr ψψ t Cγ µ = (cid:104) ψ v ( p, s α ) | γ γ µ | ψ u ( p, s α ) (cid:105) , (C9)using the trace and the charge-conjugation properties: (cid:104) ψ v ( p, s α ) | γ = ( C | ψ u ( p, s α ) (cid:105) ) t ,and the γ -matrices order is chosen so as to fit the state matrix elements. Appendix D: Massive vector
We normalize solutions A ψµ to the derived equation from Eqs. 6 and 7[ (cid:3) + (2 m ) ] A ψµ = 0 . (D1)The classical counterpart is A ψµcl = √ ( A ψµ + A ψµ ∗ ), and each component correspondsto a quantum’s creation/annihilation.Eq. B7 presents a massive on-shell vector A ψ , with momentum 2 p z , and circularpolarization along the transverse directions ˆ x , ˆ y is assumed enclosed in a volume VA ψµ = N ψ ε µ e − i kx , (D2)where the momentum is 2 p z = 2 (cid:126) k z , 2 kx = 2( Et − p z z ) / (cid:126) , ε µ = √ (0 , , i, ∂ µ A ψµ =0 as required, and N ψ = (cid:113) (cid:126) c EV is fixed from the normalization condition in Eq. 11.The electric and magnetic fields are, using Eq. 10, E ψ = ( F ψ , F ψ , F ψ ) = − ( ∂ t A ψ + ∂ x A ψ , ∂ t A ψ + ∂ y A ψ , ∂ t A ψ + ∂ z A ψ ) B ψ = ( F ψ , − F ψ , F ψ ) = ( ∂ y A ψ − ∂ z A ψ , ∂ z A ψ − ∂ x A ψ , ∂ x A ψ − ∂ y A ψ ) . (D3)This gives for Eq. D2, E ψ = 2 N ψ Ec (cid:126) e − i kx √ i, − ,
0) (D4) B ψ = 2 N ψ p z (cid:126) e − i kx √ , i, . (D5)18 sµ is similarly obtained, while its mass term is constructed using 6 and 8, leadingto the same equation as D1, with solution A ψµ = N s ε µ e − i kx , (D6)This expression contains (cid:126) , while in principle, one can revert to classical variablesusing the established equivalences for the frequency ω and the wave number k interms of the energy and momentum, E = (cid:126) ω , p = (cid:126) k . We follow Eq. 9, finding N s = (cid:126) c (cid:113) EV . Appendix E: Electromagnetic unit independence of chargedefinition
The obtained charge in Eq. 13 is a unit-independent result. Instead of the g factor 4 π for j µ in Maxwell’s Eq. 3, charge units g f = f g require A sfµ = (1 /f ) A sµ to maintainthe Lorentz force, so Eq. 3 is rewritten ∂ ν ( ∂ ν A sfµ − ∂ µ A sfν ) = 4 πf g f ¯ ψγ µ ψ. (E1)The combined expression giving the energies in Eqs. 9 is unmodified, πf gf A ψµ = πf g f A ψ = − f A sµ = − A sfµ , so πf g f A ψµ = − A sfµ . Eq. 9 is converted to (cid:82) d x f π [ ( | E sf | + | B sf | ) + (2 m ) (2 A sf A sf − g A sf µ A sf µ )] = 2 E. (E2)Eq. 12 transforms to f √ π (cid:126) cA ψµ = − A sfµ , leading to g f / √ (cid:126) c = f √ π . f = √ π converts Gaussian to Lorentz-Heaviside units. This demonstration can be summarizedby tracing the 4 π in Eqs. 3, 9, as the substitution 4 π → π/f extends it to arbitraryunits. 19 ppendix F: Tensor extended identities Dirac’s Hermitian conjugate equation is ψ † ( x ) { [ − i ←− ∂ µ − g e A µ ( x )] γ γ µ − mγ } = 0 . (F1)To obtain Eq. 15, we sum the equations obtained by partial contraction: multiply-ing Eq. 2 by ¯ ψγ µ , ¯ ψ = ψ † γ , from the left, and Eq. F1 by γ µ ψ from the right. UsingAppendix A (Eq. A1), we get the generalized Gordon identity[34] for the interactivecase quoted.This identity can be extended to the antisymmetric component. The third termcontains another superconductivity component. To rewrite this term with single γ µ matrices, one relies on the identity derived by multiplying Eq. 2 by ¯ ψγ µ γ ν and Eq.F1 by γ ν γ µ ψ , and subtracting the expressions. One finds (cid:15) µνησ ¯ ψi ←→ ∂ η γ γ σ ψ + (cid:15) µνησ ψγ γ σ ψg e A η + ∂ µ ¯ ψγ ν ψ − ∂ ν ¯ ψγ µ ψ = mi ¯ ψ [ γ µ , γ ν ] ψ, (F2)where we apply Eq. A2, and (cid:15) µνησ is the Levi-Civita tensor, given in Eq. A3. Thecombination of this equation and Eq. 15 shows the Bargmann-Wigner in Eq. 5 as aparticular case with a Maxwell’s equations structure. References [1] S. L. Glashow,
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