A gauge invariant path integral for electrodynamics with magnetic monopoles in the Hestenes-Haddamard-Rodrigues formalism
aa r X i v : . [ phy s i c s . g e n - ph ] A ug A gauge invariant path integralfor Electrodynamics with Magnetic Monopolesin the Hestenes-Haddamard-Rodrigues formalismLuiz C.L. Botelho
Departamento de Matem´atica AplicadaInstituto de Matem´atica, Universidade Federal FluminenseRua Mario Santos Braga, CEP 24220-140Niter´oi, Rio de Janeiro, Brasile-mail: [email protected]
Abstract:
We propose a new path integral for QED in presence of magnetic monopoles onthe formalism of Geometric Algebra of Hestenes-Haddamard-Rodrigues written in termsof Dirac matrixes.
1. An Euclidean Path Integral for Magnetic Monopoles
One of the most appealing question on quantum field theory is an old question posedby P.A.M. Dirac: “Do we have a quantum field theory of magnetic monopoles?”.Unfortunately the general answer to the question is no. What has been studied inthe literature is the introduction of magnetic monopoles as external backgrounds orstrings, due to its supposed massiveness (when in velocity the monopole is equivalentto a Rinocherus in full speed!), which turns out its nature to be a semi-classical levelphenomena ([1]).In this short comment directed to an Theoretical Physicist’s audience, we wish topoint out that it is possible to second quantize the monopole electromagnetic field in a
Presently a CNPq visiting research at IMEEC-UNICAMP. R in the presence of an electric source j ν ( x )and magnetic monopole source k ν ( x ) (both supposed divergenceless) write as of a( ∂ µ F µν )( x ) = j ν ( x ) (1) (cid:0) ∂ µ ( ∗ F µν ) (cid:1) ( x ) = k ν ( x ) (2)plus the usual Sommerfeld radiation conditions at the R infinity imposed now for themagnetic monopole electric charge generated Electromagnetic strenght fields.This classical problem is addressed in the Geometric algebra formalism by the introduc-tion of a complex matrix [ F ]( x ) on the Dirac Algebra of matrixes, and an antisymmetricrank two tensor with the following relationship with the electromagnetic strenght fields F µν ( x ) = ε µνρσ B σρ + 14 Tr( γ µ [ F ] γ ν ) (3)Here ours proposed tensorial potential dynamical equation reads as of as (cid:0) ε µνρσ ∂ µ B σρ (cid:1) ( x ) = 0 (4) (cid:0) ∂ µ B µν (cid:1) ( x ) = k ν ( x ) − j ν ≡ z }| { ε µνσρ ∂ µ (cid:8) Tr Dirac ( γ σ | [ F ] | ρ ) (cid:9) (5)Note that the set of equations eq(4)-eq(5) can be elementarly solved as linear functionalof a monopole divergenceless current sources k γ and j γ . B σρ ( x ) = ( ∂ σ Q ρ − ∂ ρ Q σ )( x ) . (6)It reads as of as in full. B σρ ( x ) = ( ∂ ) − [( ∂ σ ( k ρ − j ρ ) − ∂ ρ ( k σ − j σ )]( x ) (7) The solution of the tensorial PDE’s linear system F ]( x ) = 14 (cid:8) ( F µν − ε µνρτ B ρτ )[ γ µ , γ ν ] − (cid:9) (8)which by its turn is postulated to satisfies the “Dirac” like matricial wave equation (plusappropriate Sommerfeld radiation condidion)( γ µ ∂ µ )([ F ])( x ) = j ν ( x ) γ ν − i γ γ α k α ( x ) (9)At this point we introduce our proposal for an (euclidean) quantum field theory forquantum electrodynamics with magnetic monopoles.We consider as our basic field variable to be second quantized not the usual Electro-magnetic potential, but the scalar-complex matricial field [ F ]( x ) as given by eq(8).Unfortunatelly, a canonical Lagrangean and a corresponding variational principle ismising for eq(9) as far as this author knows ([1]). However we overcome such difficulty byproposition the simples bosonic action leading to eq(9) at its functional extremum point.It reads as of as S ([ F ] , [ F ] + ) = Z d x Tr Dirac (cid:26) (( γ µ ∂ µ )[ F ] − γ ν j ν − iγ µ γ k µ ) + (( γ µ ∂ µ )[ F ] − γ ν j ν − iγ µ γ k µ ) (cid:27) ( x ) (10)It is obvious that the (unique!) minimum of the quadratic functional eq(10) is achievedon the classical motion eq(9), thus on the classical Maxwell equations eq(5), eq(2) withmagnetic monopole source. a) ε αβγσ ∂ α W βγ = j σ ∂ µ W ρµ = ∂ µ W µν = 0is formally given by W µν = ( ∂ ) − ( ∂ α e j αµν ) with e j αµν = ε αµνρ j ρ .Note that ε αβγσ ∂ α ( ∂ β a γ − ∂ γ a β ) ≡ Z [ J µν ( x )] = 1 Z (0) × Z D F ([ F ]) D F ([ F ] + ) × exp (cid:18) − S ([ F ] , [ F + ]) (cid:19) × exp (cid:26) i Z d µ x (cid:20) J µν ( x ) (cid:18) F µν ( x )= z }| { ε µνσρ B σρ + 14 Tr Dirac ( γ µ [ F ] γ ν (cid:19) ( x ) (cid:21)(cid:27) , (11)where D F ([ F ]), D F ([ F ] + ) denotes the Feynman product measures. Explicitly: D F ([ F ]( x )) = Y A,B =1 Y x ∈ R d [ F ] AB ( x ) !! (12) D F ([ F ] + ( x )) = Y A,B =1 Y x ∈ R d [ F ] + AB ( x ) !! (13)Let us remark that one can “bosonize” eq(11) through the classical variable change ([3])eq(8) into the path integral eq(11), under the hypothesis of its validity at the quantumlevel. D F ([ F ]( x )) = det (cid:0) L νραζ (cid:1) × D F [ F µν ( x )] (14)which means that now the new fundamental variable are the gauge invariant magneticmonopole electromagnetic field strenght { F µν ( x ) } instead of the original matrix field[ F ]( x ).Here, the “tensorial matrix” in eq(14) is explicitly given by (it is monopole electro-magnetic field strenght independent.) L νραζ = | γ ν γ ρ > < γ α γ ζ | (15)det (cid:0) L νραζ (cid:1) = Tr( γ ρ γ ν γ α γ ζ ) (16)The free action eq(10) is now given by the Cramer-Julia action for antisymmetric ranktwo tensors (for vanishing sources k γ ( x ) = j γ ( x ) ≡ S (0) [ F µν ] = 16 Z R d x ( ∂ µ F νρ + ∂ ν F µρ + ∂ ρ F µν ) ( x ) (17)4ur path integral expression for the generating functional of the second quantizedgeneralized magnetic monopole electromagnetic field now reads as of as: Z [ J µν ( x )] = 1 Z (0) Z D F ( F µν ( x )) exp (cid:26) − S (0) [ F µν ] (cid:27) exp (cid:26) − S (1) [ F µν , k γ , j γ ] (cid:27) exp (cid:26) i Z d x (cid:20) J µν .F µν ( x ) (cid:21)(cid:27) (18)Here the functional weight S (1) [ F ρν , k γ , j γ ] depends explicitly on the currents-sourcergenerating the second quantized electromagnetic field supporting now magnetic monopoliccharges. It contains terms of the forma) Z d x ( j µ j µ )( x ) (19-a)b) Z d x ( k µ k µ )( x ) (19-b)c) Z d xF αβ ( x ) ε αβσζ ( ∂ σ k ζ − ∂ ζ k σ )( x ) (19-c)d) Z d x ε σµνζ ( ∂ µ F νζ )( x ) k σ ( x ) (19-d), etc...In the case for quantum sources, the second quantized Dirac electron field { ψ e ( x ) , ψ e ( x ) } and the postulated second quantized monopole field { Ω M ( x ) , Ω M ( x ) } , we should intro-duce on eq(18) the further euclidean path integral on the matter fields and the currents γ µ ∂ µ ([ F ]) = ( ∂ µ F µρ ) γ ρ − i ε σµνρ γ σ γ ∂ µ F νρ (1) Tr Dirac (cid:26)(cid:20) ( ∂ µ F µρ ) γ ρ − i ε σµνζ γ σ γ ∂ µ F νζ (cid:21) (cid:27) = 4( ∂ µ F µρ ) + 12[( ∂ µ F νρ ) + ( ∂ ρ F νρ )( ∂ µ F µν )+ ( ∂ ν F νρ )( ∂ µ F ρµ ) ] ∼ ( ∂ µ F νρ + ∂ ν F µρ + ∂ ρ F µν ) (2) Namely, the action eq(10) is quadratic (Gaussian) in terms of the Monopole Electromagnetic strenghtfields F µν . j µ ( x ) = e ( ψ e γ µ ψ e )( x ) (20) k µ ( x ) = g (Ω µ γ µ Ω µ )( x ) (21)Note that the introduction of matter-sources for the generalized fields introducesapparently non-renormalizable thirring like interaction on the pure matter sector (seeeq(19-a)-eq(19-b)). This means that Q.E.D in presence of magnetic monopoles is non-renormalizable, not satisfies the quantum mechanical unitary condition, besides of con-fining electrical charge ( ∗ ) . Appendix 1 – A new “fermionization” for Maxwell Equations.Let [ b F ] be the complex matrix taking values on the Dirac algebra of matrixes[ b F ] = 14 F µν ( x )[ γ µ , γ ν ] − + ( γ µ ) a µ ( x ) (22)where a µ ( x ) is an fixed external divergenceless field and the antisymmetric rank-two tensor F µν ( x ) satisfies the Maxwell equations for monopoles ∂ µ F µν ( x ) = j ν ( x ) (23) ∂ µ ( ∗ F µν )( x ) = k ν ( x ) (24)Then [ b F ] satisfies ours generalized Haddamard-Hestenes equation γ µ ∂ µ ([ b F ]( x )) = γ ρ f ρ ( x ) + ( γ σ γ k σ ( x )) + 14 [ γ α , γ β ] − F αβ ( a γ ( x )) (25)Conversely, if we define the rank-two antisymmetric tensor field through the relation-ship below F µν ( x ) = 14 Tr Dirac ( γ µ [ F ] γ ν ) + ( ε µνσρ B σρ ( x )) (26) ( ∗ ) On refs ([4]) we have proved that electromagnetic path integrals, when formally written in termsof vector potentials are of fourth-order, thus leading to confinement of electrical charge. This resultconstraint us to claim that only a new electrodynamics written directly in terms of the ElectromagneticField strenght has chances to be consistent at a second quantized level when in presence of magneticmonopoles. ε µνσρ ( ∂ µ B σρ )( x ) = 0 (27)( ∂ µ B µν )( x ) = k ν ( x ) − F ν ( x ) (28)where F ν ( x ) = 18 ε µνσρ ∂ µ { Tr Dirac ( γ σ [ F ] γ ρ ) } (29)then ∂ µ F µν ( x ) = j ν ( x ) ∂ µ ( ∗ F µν ( x )) = k ν ( x ) . (30) Acknowledgments:
The author is very thankfull to CNPq for financial support for ascientific visit to IMEEC-UNICAMP and too the Mathematical Physics group of ProfessorW. Rodrigues.