YANG-MILLS GAUGE INVARIANT THEORY FOR SPACE CURVED ELECTROMAGNETIC FIELD. Algirdas Antano Maknickas 1. September 3, 2014

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1 YANG-MILLS GAUGE INVARIANT THEORY FOR SPACE CURVED ELECTROMAGNETIC FIELD Algirdas Antano Maknickas Institute of Mechanical Sciences, Vilnius Gediminas Technical University September 3, 04 Abstract. It was proposed new gauge invariant Lagrangian, where the gauge field interact with the charged electromagnetic fields. Gauge invariance was archived by replacing of particle mass with new one invariant of the field F µνf µν multiplied with calibration constant α g. It was shown that new proposed Lagrangian generates similar Dirac and electromagnetic field equations. Solution of Dirac equations for a free no massless particle answers to the question of the age why free particle deal in experiments like a de Broil wave. Resulting wave functions of the new proposed Lagrangian will describe quantized list of bespinor particles of different masses. Finally, it was shown that renormalization of the new proposed Lagrangian is similar to QED in case similarity of new proposed Lagrangian to classic QED.. Introduction de Broglie (94, 95) formulated the de Broglie hypothesis, claiming that all matter, not just light, has a wave-like nature. Heisenberg (95) proposes a quantum mechanics, under the form of a mechanics of matrices. Schrödinger (96) publishes his equation that bases quantum mechanics on the solution of a nonrelativistic wave equation. Since relativity theory was already well established when quantum mechanics was formulated, this may surprise. In fact, for accidental reasons, the spectrum of the hydrogen atom is better described by a non-relativistic wave equation than by a relativistic equation without spin, the Klein (96); Gordon (96) equation. Dirac (98) introduces a relativistic wave equation that incorporates the spin / of the electron, which describes much better the spectrum of the hydrogen atom, and opens the way for the construction of a relativistic quantum theory. In the two following years, Heisenberg and Pauli lay out, in a series of articles, the general principles of quantum field theory. First correct calculation in quantum electrodynamics (Pauli and Weisskopf, 934) and confirmation of the existence of divergences, called ultraviolet (UV) since they are due, in this calculation, to the short-wavelength photons. Landau (937a,b) publishes his general theory of phase transitions. At 947 it was made measurement of the so-called Lamb shift by Lamb and Retherford (947), which agrees well with the prediction of quantum electrodynamics (QED) after cancellation between infinities. At 949 it was made construction of an empirical general method to eliminate divergences called renormalization (Dyson, 95). Yang and Mills (954) propose a non-abelian Date: September 3, 04. Key words and phrases. Antigravity; Dirac equation; Gauge invariant; Lorentz gauge; Yang- Mills Lagrangian.

2 YANG-MILLS GAUGE INVARIANT THEORY FOR SPACE CURVED EMF generalization of Maxwells equations based on non-abelian gauge symmetries (associated to non-commutative groups). Maknickas (03) published its work about curvature of electromagnetic field as ground idea of gravitational mass. So, all this basic ideas listed above are incorporated into new proposed model.. Description of Model Mathematically, Yang-Mills theory is an abelian gauge theory with the symmetry group U(), where the gauge field interact with the charged spin-/ fields. In special case, the Yang-Mills Lagrangian for a spin-/ field interacting with the electromagnetic field is given by the real part of (.) L = ψ(i c γ µ D µ mc )ψ 4µ 0 F µν F µν where γ µ are Dirac matrices; ψ a bispinor field of spin-/ particles (e.g. electronpositron field); ψ ψ γ 0, called psi-bar, is sometimes referred to as Dirac adjoint; D µ µ + ie β ν A µ = µ + iea µ is the gauge covariant derivative; e is the coupling constant, equal to the electric charge of the bispinor field; A is the covariant four-potential of the electromagnetic field generated by the electron itself; F µν = µ A ν ν A µ is the electromagnetic field tensor; β ν is expressed as follow β ν = (.) If we describe m as space time curvature of electromagnetic field, as mentioned in (Maknickas, 03) (.3) (.4) (.5) R = 4G c (M + m), r3 m = α ( ) emv B ε 0 E µ 0 V = 4πr3 3, = α emv F µν F µν, where r is average radius of field, we could start substituting the definition of D µ and m into the Lagrangian (.6) L = i c ψγ µ µ ψ e ψγ µ A µ ψ α emc F µν F µν ψψ 4µ 0 F µν F µν. where volume multiplier V was neglected. So we have invariant Lagrangian to gauge transformations, which form a Lie group. Next, we can substitute this Lagrangian into the Euler-Lagrange equation of motion for a field: ( ) L (.7) µ L ( µ ψ) ψ = 0 to find the field equations.

3 YANG-MILLS GAUGE INVARIANT THEORY FOR SPACE CURVED EMF 3 The two terms from this Lagrangian are then ( ) L ( µ = µ i c µ ψγ ), ( µ ψ) (.8) L ψ = e ψγ µ A µ α emc F µν F µν ψ. Substituting these two back into the Euler-Lagrange equation [.7] results in (.9) i c µ ψγ µ + e ψγ µ A µ + α emc F µν F µν ψ = 0 with complex conjugate (.0) i cγ µ µ ψ eγ µ A µ ψ α emc F µν F µν ψ = 0. Bringing the middle term to the right-hand side transforms this second equation into (.) i cγ µ µ ψ α emc F µν F µν ψ = eγ µ A µ ψ The left-hand side is similar to original Dirac equation and the right-hand side is the interaction with the electromagnetic field. One further important equation can be found by substituting the Lagrangian into another Euler-Lagrange equation, this time for the field, A µ : (.) The two terms this time are ( ) L = ( ν A µ ) (.3) ( L ( ν A µ ) ) L A µ = 0 ( ) α em c ψψ + ν ( µ A ν ν A µ ), µ 0 L A µ = e ψγ µ ψ and these two terms, when substituted back into [.] give us (.4) ν F νµ = e ψγ µ ψ (α em c ψψ + µ0 ) Now, if we impose the Lorenz gauge condition, that the divergence of the four potential vanishes (.5) µ A µ = 0 then we get (.6) A µ = e ψγ µ ψ (α em c ψψ + µ0 ) which is similar to a wave equation of the four potential of the classical Maxwell equations in the Lorenz gauge.

4 4 YANG-MILLS GAUGE INVARIANT THEORY FOR SPACE CURVED EMF 3. de Broglie waves According to Banzaitis and Grabauskas (975) angular momentum of free particle during the time is constant. So, we could choose particle moving direction (for example z) as direction of coordinate system and would get opposite to 3D motion D motion. Moreover, motion of free particle do not depend on time. So, we could solve time independent Dirac [.] (3.) i cγ z z ψ + α emc F µν F µν ψ + eγ ν A ν ψ = Eψ In general F µν F µν depends on space-time coordinates, but in our case let be constant. Now we could start to investigate homogeneous part of equation (3.) i cγ z z ψ + α emc F µν F µν ψ = Eψ in case inhomogeneous equation could be solved using Green function formalism. After including γ z (3.3) γ z = into [3.], we get four equations for Dirac wave function (3.4) (3.5) (3.6) (3.7) i c z ψ 3 + α emc F µν F µν ψ = Eψ i c z ψ 4 + α emc F µν F µν ψ = Eψ i c z ψ α emc F µν F µν ψ 3 = Eψ 3 i c z ψ α emc F µν F µν ψ 4 = Eψ 4 So, we get two independent pairs of equations [3.4], [3.6] and [3.5], [3.7] accordantly, where γ, γ 3 denotes spin m z = and γ, γ 4 denotes spin m z =. Let start to investigate meaning meaning of indexes, 3. Equations [3.4], [3.6] could be rewritten as follow (3.8) (3.9) α emc F µν F µν E z ψ 3 = ψ i c i c z ψ 3 = α emc F µν F µν + E ψ Differentiation of [3.9] and inserting into [3.8] gives second order differential equation for ψ (3.0) zψ + E α em c4 4 (F µν F µν ) Solution of [3.0] is c ψ = 0 (3.) ψ = N exp (ikz) + N exp ( ikz)

5 YANG-MILLS GAUGE INVARIANT THEORY FOR SPACE CURVED EMF 5 where (3.) k = E α em c4 4 (F µν F µν ) c Now we could express ψ 3 from [3.9] as follow (3.3) ψ 3 = E αemc F µν F µν E + αemc F µν F (N µν exp (ikz) N exp ( ikz)) So, we get two independent solutions 0 (3.4) ψ ±k = ± 0 E αemc F µνf µν E+ αemc F µνf µν N, exp (±ikz) where N and N norm multipliers for ψ k and ψ +k accordantly. The same way could be found and wave functions ψ, ψ 4. Using this functions second two independent solutions could be found (3.5) ψ ±k = ± 0 0 E αemc F µνf µν E+ αemc F µνf µν N, exp (±ik z) General solution is superposition of the corresponding wave functions with the same k. Since z range boundaries are in (, + ), wave functions [3.4] and [3.5] are defined z just for real k. Thus (3.6) E α emc 4 (F µν F µν ) 4 This condition will be true if one of equalities will be true (3.7) E α emc F µν F µν E α emc (3.8) F µν F µν ) for a positive F µν F µν > 0. So we have a gap ( αemc F µν F µν, αemc F µν F µν in the range ( < E < ). For a negative F µν F µν < 0 we obtain the same but reversed gap, where F µν F µν < 0 denotes antigravity mass. The negative mass this time could be interpreted not only as a positron or anti-proton but also antigravity positron ant antigravity anti-proton. Moreover, every particle describes four components field and it easy explain why every particle deal as wave in experiments. 4. General solution Let rewrite wave function equation using notation /a = γ µ a µ in units c = (4.) i / ψ = α em F µνf µν ψ + e /Aψ

6 6 YANG-MILLS GAUGE INVARIANT THEORY FOR SPACE CURVED EMF Solution of this equation could be expressed using Green function described as follow (4.) i / G(x x ) = δ 4 (x x ) The explicit form of the Green function can be written as a Fourier transform (4.3) G(x x ) = (π) 4 d 4 pĝ(p) exp (i(x x )p) where (4.4) Ĝ(p) = / p p The general solution of the inhomogeneous equation [4.] reads (4.5) ψ(x) = φ(x) + e G(x x ) /A (x ) ψ(x )d 4 x (4.6) + α em G(x x )F µν (x ) F µν (x ) ψ(x )d 4 x On the other hand, using the same Green function formalism could be found and solutions of [.6] (4.7) A µ K(x x ) ψ(x (x) = e )γ µ ψ(x ) (α em c ψ(x )d 4 x )ψ(x ) + µ0 where K(x x ) is a Green function of wave equation (4.8) K(x x ) = δ 4 (x x ) and is known too (4.9) K(r r, t t ) = 8π 3 (r r ) + dω e iω((r r )/c (t t )) For weak enough coupling constants e and α em one can use the perturbation theory ψ (0) (x) = φ(x) A (0)µ K(x x ) φ(x = e )γ µ φ(x ) (α em c φ(x )d 4 x )φ(x ) + µ0 (4.0) ψ () (x) = φ(x) + A ()µ = e... ( G(x x ) e /A (0) (x ) + α em F µν (0) F (0)µν) ψ(x )d 4 x K(x x ) ψ () (x )γ µ ψ () (x ) ( α em c ψ )d 4 x () (x )ψ () (x ) + µ 0 So, resulting wave functions will describe quantized list of bespinor particles of different masses. Quantization of masses ensures due to term F µν (x)f µν (x)ψ(x) inserted into model Lagrangian.

7 YANG-MILLS GAUGE INVARIANT THEORY FOR SPACE CURVED EMF 7 5. Renormalization If we choose β ν as follow (5.) β ν = we get massless version of QED with additional term (5.) F µν F µν p in momentum space after Fourier transformation of equation [.4]. The meaning massless denotes in this case that term mψ = 0. Since A µ is the same order after Fourier transformation (5.3) A µ p too, according to equation [.6] renormalization is applicable and could be done using steps descibed in (Soper, 00). 6. Conclusions It was proposed gauge invariant Lagrangian of a fully renormalized quantum version of Yang-Mills theory of R 4 based on Lie group, where the gauge field interact with the charged spin-/ electromagnetic fields. Gauge invariance was archived by replacing of particle mass with new one invariant of the field F µν F µν multiplied with calibration constant α em. It was shown that new proposed Lagrangian generates similar Dirac and electromagnetic field equations. Solution of Dirac equations for a free no massless particles answers to the question of the age why free particle deal in experiments like a de Broglie waves. The answer is they are space curved field waves. The negative mass in proposed model could be interpreted not only as a positron or anti-proton but also antigravity positron and antigravity anti-proton, where magnetic moment of antigravity positron is less than magnetic moment of electron to meet condition F µν F µν < 0. Resulting wave functions of the new proposed Lagrangian will describe quantized list of bespinor particles of different masses. Finally, it was shown that renormalization of the new proposed Lagrangian is similar to QED in case similarity of new proposed Lagrangian to classic QED. References L. de Broglie. Recherches sur la thorie des quanta (researches on the quantum theory). Thesis (Paris), 94. L. de Broglie. Recherches sur la thorie des quanta (researches on the quantum theory). Ann. Phys. (Paris), 3, 95. W. Heisenberg. Über quantentheoretische umdeutung kinematischer und mechanischer beziehungen. Zeitschrift für Physik, 33: , 95. E. Schrödinger. Quantisierung als eigenwertproblem. Annalen der Physik, 384: , 96. O. Klein. Quantentheorie und funfdimensionale relativitatstheorie. Z. Phys., 37: , 96.

8 8 YANG-MILLS GAUGE INVARIANT THEORY FOR SPACE CURVED EMF W. Gordon. Der comptoneffekt nach der schrödingerschen theorie. Z. Phys., 40: 7 33, 96. P. A. M. Dirac. The quantum theory of the electron. Royal Society of London Proceedings Serie s A, 7:60 64, 98. doi: 0.098/rspa W. Pauli and V. Weisskopf. Über die quantisierung der skalaren relativistischen wellengleichung. Helv. Phys. Acta, 7:709 73, 934. L. D. Landau. Theory of phase transformations. i. Zh. Eksp. Teor. Fiz., 7, 937a. L. D. Landau. Theory of phase transformations. i. Zh. Eksp. Teor. Fiz., 7, 937b. W. E. Lamb and R. C. Retherford. Fine Structure of the Hydrogen Atom by a Microwave Method. Physical Review, 7:4 43, August 947. doi: 0.03/Phys- Rev.7.4. F. J. Dyson. Divergence of perturbation theory in quantum electrodynamics. Phys. Rev., 85:63 63, Feb 95. doi: 0.03/PhysRev URL C. N. Yang and R. L. Mills. Conservation of isotopic spin and isotopic gauge invariance. The Physical Review, 96, 954. A. A. Maknickas. Biefeld - Brown effect and space curvature of electromagnetic field. Journal of Modern Physics, 4:05 0, 03. A. Banzaitis and D. Grabauskas. Quantum Mechanic (in Lithuanian). Mokslas, Vilnius, Lithuania, 975. D. E. Soper. Renormalization of Quantum Electrodynamics, 00. URL soper/qft/renormalization.pdf.