QFT and NKR formalism Quantization Renormalization Conclusions Renormalization of a Second Order Formalism for Spin 1 / 2 Fermions e ´ Ren´ Angeles-Mart´ ınez Mauro Napsuciale-Mendivil Science and Engineering Division - University of Guanajuato October 2011 1 / 37
QFT and NKR formalism Quantization Renormalization Conclusions Brief Historical Review of Second Order Formalisms for spin 1 / 2 ◮ (1927) V. Fock, Relativistic Quantum Mechanics of spin 1 / 2 through a second order differential equation. ◮ (1928) Dirac, P. A. M. ◮ (1951,1958) Feynman - Gell-Mann 1 used a two component spinorial field that satisfies ( g = 2 , ξ = 0 ). [( i∂ µ − A µ ) 2 + � σ · ( � B ± i � E )] φ = m 2 φ, Their main motivation was to describe the weak interactions. ◮ ... ◮ (1961) Hebert Pietschmann 2 , one loop renormalization of the Feynman-Gell-Mann theory. Showing the equivalence with the Dirac framework has been always a goal in these works. 1 Phys. Rev. 84, 108 , 1951; Phys. Rev. 109, 193, 1958 2 Acta Phys. Austr. 14, 63 (1961) 2 / 37
QFT and NKR formalism Quantization Renormalization Conclusions Motivations ◮ The NKR second order formalism for massive spin 3 / 2 particles is an alternative 3 to the inconsistent Rarita-Schwinger theory of electromagnetic interactions. ◮ The case of spin 1 / 2 is interest by itself e.g. in this theory the gyromagnetic factor g is a free parameter ⇒ a low energy effective theory of particles with g � = 2 , e. g. proton. ◮ We expect that this give us a better understanding of the properties of spin 1 / 2 particles, e.g. the classical limit 4 . ◮ ¿Generalizations? In this work we used general principles of QFT to study the quantization and Renormalization. We will only compare with the conventional Dirac results only at the end. 3 Eur. Phys. J. A29 (2006); Phys. Rev. D77: 014009, 2008 4 R. P. Feynman Phys. Rev. 84, 108 , 1951. 3 / 37
QFT and NKR formalism Quantization Renormalization Conclusions Index Quantum Field Theory and the NKR Formalism Quantization Renormalization Conclusions 4 / 37
QFT and NKR formalism Quantization Renormalization Conclusions Index Quantum Field Theory and the NKR Formalism Quantization Renormalization Conclusions 5 / 37
QFT and NKR formalism Quantization Renormalization Conclusions Quantum Fields Quantum theories that satisfy ◮ special relativity ◮ cluster descomposition principle can be built with quantum fields φ l ( x ) defined as Z e ip · x u l (Γ) a † (Γ) + e − ip · x v l (Γ) a (Γ) ˆ ˜ φ l ( x ) = d Γ , such that under a Poincar´ e transformation U (Λ , b ) the fields U (Λ , b ) φ l ( x ) U (Λ , b ) − 1 = D (Λ) ll ′ φ l ′ (Λ x + b ) , ( x − y ) 2 > 0 , [ φ l ( x ) , φ m ( y )] ∓ = 0 for where D (Λ) ll ′ is a representation of SO (3 , 1) . 6 / 37
QFT and NKR formalism Quantization Renormalization Conclusions Scheme of the NKR construction of QFTs Spacetime Symmetries of Fields φ ( x ) Second Order Equations of Motion [ T µν ∂ µ ∂ ν + ... ] φ = 0 Lagrangian L [ φ, ∂φ ] Noether: Poincar´ e Scalar Hermitian Interactions: Minimal Coupling L [ φ, ∂φ ] → L [ φ, Dφ, A ] 7 / 37
QFT and NKR formalism Quantization Renormalization Conclusions Equations of motion of the NKR formalism General Idea: To use the Poincare invariants P 2 and W 2 to construct projectors P ( m,s ) over spaces of definite mass and spin. Acting these projectos on the fields results in equations of motion. For a field ψ ( D,m,s ) with only one spin sector s in a given representations D (Λ) only a projector is necessary P m,s “ P 2 W 2 P m,s = ”“ ” , m 2 − s ( s + 1) P 2 the action of this projector over the field results in the following equation of motion T Dµν P µ P ν − δ ll ′ m 2 ´ ψ ( D,m,s ) ` ( x ) = 0 , ll ′ l ′ 1 is defined by W 2 = − where T Dµν s ( s + 1) T Dµν P µ P ν , it depends on the ll ′ generators M µν of the D (Λ) . 8 / 37
QFT and NKR formalism Quantization Renormalization Conclusions NKR for spin 1 / 2 and the representations (1 / 2 , 0) ⊕ (0 , 1 / 2) For a field ψ ( D,m,s =1 / 2) in the representation D ≡ (1 / 2 , 0) ⊕ (0 , 1 / 2) the NKR equation of motion can be deduced from the following family of hermitian Poincar´ e scalar Lagrangians ψT µν ∂ ν ψ − m 2 ¯ L = ∂ µ ¯ ψψ, where T µν = g µν − igM µν + ξγ 5 M µν . M µν are the generators of the (1 / 2 , 0) ⊕ (0 , 1 / 2) Lorentz group representation. „ M µν „ 1 « « 0 0 M µν = γ 5 = (1 / 2 , 0) , . 0 − 1 0 M (0 , 1 / 2) 9 / 37
QFT and NKR formalism Quantization Renormalization Conclusions Electromagnetics Interactions Finally we introduce Electromagnetic interactions are introduced through minimal coupling L = − 1 ψ [ g µν − ( ig − ξγ 5 ) M µν ] D ν ψ − m 2 ¯ 4 F µν F µν + D µ ¯ ψψ, g = 2 , ξ = 0 corresponds to the Feynman-Gell-Mann theory. The interactions that contains g can be rewritten as Z d 4 xeg ¯ ψM µν ψF µν , L i = − that includes the interaction � S · � B ⇒ we recognize g as the gyromagnetic factor. 10 / 37
QFT and NKR formalism Quantization Renormalization Conclusions Index Quantum Field Theory and the NKR Formalism Quantization Renormalization Conclusions 11 / 37
QFT and NKR formalism Quantization Renormalization Conclusions Feynman Rules Z Z h i D A D ¯ Z [ J µ , ¯ η, η ] = C ψ D ψ exp i L ef dx , L = − 1 4 F µν F µν − 1 2 α ( ∂ µ A µ ) 2 + D µ ¯ ψT µν D ν ψ − m 2 ¯ ψψ + J µ A µ + ¯ ηψ + ¯ ψη − p q µ ν i i ∆ µν ≡ − ig µν iS ( p ) ≡ p 2 − m 2 q 2 + iε q, µ µ ν p � p � p p ( p � + p ) µ + ( ig + ξγ 5 ) M µν ( p � − p ) ν � − ieV µ ( p, p � ) = − ie � 2 ie 2 g µν 12 / 37
QFT and NKR formalism Quantization Renormalization Conclusions Ward Identities As a consequence of gauge invariance there exist identities between the green functions − 1 δ δ δ δJ µ ( x )) − ∂ µ J µ − e (¯ h i Z ( J µ , η, ¯ 0 = α � ( ∂ µ η η ( x ) + η δη ( x )) η ) δ ¯ 13 / 37
QFT and NKR formalism Quantization Renormalization Conclusions Index Quantum Field Theory and the NKR Formalism Quantization Renormalization Conclusions 14 / 37
QFT and NKR formalism Quantization Renormalization Conclusions Divergencies in the Second Order Theory Asking the Lagrangian to be dimensionless one obtains ◮ [ A ] = [ ψ ] = 1 , ◮ [ g ] = [ e ] = [ ξ ] = 0 . Thus the greater superficial degree of divergency of a process is D ≤ 4 − F − P The greater degree of divergency is: ◮ quadratic for propagators ◮ linear for 3 lines processes e.g. ffp ◮ logarithmic for 4 lines processes e.g. ffpp These characteristics are necessary for a theory to be renormalizable QFT. 15 / 37
QFT and NKR formalism Quantization Renormalization Conclusions Free Parameters and Counterterms ( ξ = 0) In terms of the bare parameters m 2 b , e b , g b the Lagrangian is L = − 1 ψ [ g µν − ig b M µν ]( ∂ ν + ie b A ν ) ψ − m 2 4 F µν F µν + ( ∂ µ − ie b A µ ) ¯ b ¯ ψψ. Introducing the renormalized parameters m 2 , e y g and the renormalized fields − 1 − 1 A µ y ψ r = Z A µ r = Z 2 2 ψ there appear the following counterterms 1 2 p q µ ν i ( p 2 − m 2 ) δ Z 2 − i δ m − i ( g µ ν q 2 − q µ q ν ) δ Z 1 q, µ µ ν p p � p p � − ie [ V µ ( p � , p )] δ e + egM µ ν ( p � − p ) ν δ g 2 ie 2 g µ ν δ 3 16 / 37
QFT and NKR formalism Quantization Renormalization Conclusions Dimensional Regularization Extend the theory to d dimensions. The natural objects to be extended to d dimension are the Lorentz generators M µν [ M αβ , M µν ] = − ig βν M αµ + ig βµ M αν − ig αµ M βν + ig αν M βµ , with g µ µ = d { M µν , M αβ } = 1 2( g µα g νβ − g µβ g να ) − i 2 ǫ µναβ γ 5 , e.g. we can use the last expression to calculate to calculate a trace in a fermion loop tr { M µν M αβ } = f ( d ) ( g µα g νβ − g µβ g να ) with l´ d → 4 f ( d ) = 4 , ım 4 17 / 37
QFT and NKR formalism Quantization Renormalization Conclusions Photon Propagator As usual one can express the complete photon propagator i ∆ µν c ( q ) as i ∆ µν c ( q ) = i ∆ µν ( q ) + i ∆ µσ [ − i Π σρ ( q )][ i ∆ ρν ( q )] + ... where Π µν ( q ) is the vacuum polarization Π µν ( q ) = ( q 2 g µν − q µ q ν ) π ( q 2 ) , Then the complete propagator is given by c ( q ) = − g µν + q µ q ν π/q 2 ∆ µν [ q 2 + iǫ ][1 + π ] . The first condition of renormalization is that the photon doesn’t acquired mass due to the radiative corrections, i.e. π ( q 2 → 0) = 0 . 18 / 37
QFT and NKR formalism Quantization Renormalization Conclusions Vacuum polarization to one loop It has the following contributions − i ( g µν q 2 − q µ q ν ) π ( q ) 2 = − i ( g µν q 2 − q µ q ν ) π ∗ ( q 2 ) − i ( g µν q 2 − q µ q ν ) δ Z 1 The first renormalization conditions requires δ Z 1 = − π ∗ ( q 2 = 0) , l l q, α q, ν q, µ q, ν l + q Finally, imposing the renormalization condition the physical vacuum polarization is Z 1 » m 2 − q 2 x (1 − x ) 2 e 2 (1 − 2 x ) 2 − g 2 – » – π ( q 2 ) = dx ln , (4 π ) 2 m 2 4 0 for g = 2 one recovers the one loop vacuum polarization of the conventional Dirac formalism. 19 / 37
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