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Redalyc.Gupta-Bleuler Quantization for Massive and Massless Free Brazilian Journal of Physics ISSN: 0103-9733 [email protected] Sociedade Brasileira de Física Brasil Dehghani, M. Gupta-Bleuler quantization for massive and massless free vector fields Brazilian Journal of Physics, vol. 39, núm. 3, septiembre, 2009, pp. 559-563 Sociedade Brasileira de Física Sâo Paulo, Brasil Available in: http://www.redalyc.org/articulo.oa?id=46413562009 How to cite Complete issue Scientific Information System More information about this article Network of Scientific Journals from Latin America, the Caribbean, Spain and Portugal Journal's homepage in redalyc.org Non-profit academic project, developed under the open access initiative Brazilian Journal of Physics, vol. 39, no. 3, September, 2009 559 Gupta-Bleuler quantization for massive and massless free vector fields M. Dehghani∗ Department of Physics, Ilam University, Ilam, Iran (Received on 28 April, 2009) It is shown that the usual quantum field theory leads to an ultraviolet divergence in the vacuum energies and an infrared divergence in the two-point functions of the massive and massless vector fields. Using a new method of quantization it is shown that the vacuum energies converge, and the normal ordering procedure is not necessary. Also the propagators are calculated, which are automatically renormalied. Keywords: vector fields, quantization, Gupta-Bleuler 1. INTRODUCTION as well. The results for the Casimir energy and Casimir force are the same given in [13, 14]. Particles of spin-1 are described as the quanta of vector In section 2 a brief review of the massive vector field quan- fields. Such vector bosons play a central role as the medi- tization, using usual quantum field theory is presented. The ators of interactions in particle physics. The important ex- ultraviolet divergence of the vacuum energy and the infrared amples are the gauge fields of the electromagnetic (massless divergence of the two-point function is explicitly shown for photons), the weak (massive W ± and Z0 bosons), and the this field. Then using Gupta-Bleuler quantization it is shown strong (massless gluons) interactions. On a somewhat less that the vacuum energy converges and the infrared diver- fundamental level vector fields can also be used to describe gence of the propagator disappears. In section 3 the ultravi- spin-1 mesons, for example the ρ and the ω mesons. olet divergence of the vacuum energy and the infrared diver- gence of the two-point function of the massless vector field It has been shown [1, 2] that one can construct a covari- is shown clearly in the framework of usual quantum field ant quantization of massless minimally coupled scalar field theory. Also the vacuum energy and the propagator of this in de Sitter space-time, which is causal and free of any diver- field is calculated using Gupta-Bleuler quantization, which gences. The essential point of these papers is the unavoidable are free of any divergences. Finally a brief conclusion is presence of the negative norm states. Also they do not prop- given in section 4. agate in the physical space, they play a renormalization role. In the previous paper [3] we have shown that this is also true for linear gravity (the traceless rank-2 massless tensor field). These questions have recently been studied by several au- 2. MASSIVE VECTOR FIELD thors [4–7]. ( ), These auxiliary states (the negative norm states) appear to The lagrangian density for the massive vector field Aµ x be necessary for obtaining a fully covariant quantization of in Minkowskian flat space-time is that of Proca [13] the free minimally coupled scalar field in de Sitter space- time, which is free of any infrared divergence. It has been 1 µν 1 2 µ L = − FµνF + m AµA , (2.1) shown that these auxiliary states automatically renormalize 4 2 the infrared divergence in the two-point function [3, 4] and = ∂ − ∂ . remove the ultraviolet divergence in the vacuum energy [2]. where Fµν µAν νAµ The variational principal to (2.1) This method is applied to the massless vector field in de Sitter yields the field equation space [8, 9], interacting quantum fields and Casimir effect in µ 2 ∂ F ν + m Aν = 0. (2.2) Mincowski space-time [10–12]. µ We would like to generalize this quantization method to ν Taking the divergence of Eq. (2.2) we have m2∂ Aν = 0, and the massive and massless free vector fields in Minkowski since m2 = 0, we find the Lorenz gauge and Eq.(2.2) can be space-time. These auxiliary states once again, automatically written as leads to renormalized two-point functions and removes the ultraviolet divergences in the vacuum energies. 2 µ Aµ + m Aµ = 0, ∂µA = 0. (2.3) This new quantization method is used for consideration of the Casimir effect for the scalar field [10]. It can be ex- tended for the electromagnetic field in 4-dimensional space- 2.1. Usual quantum field theory time. The crucial point is that when we impose the physical boundary conditions on the field operator, only the positive norm states are affected. The negative norm states do not There are three independent components of a massive interact with the physical states or real physical world, thus spin-1 field, and the Lorentz condition may serve to elimi- they cannot be affected by the physical boundary conditions nate one of the four components of Aµ; we take this to be A0. Note that the positive sign in front of the mass term in − 1 2( . ) 0 the Lagrangian will result in a term 2 m A A when A is eliminated, so that the mass term has a negative coefficient, ∗ e-mail:[email protected] as it does for the Klein-Gordon Lagrangian. The field Aµ(x) 560 M. Dehghani has the expansion It may be checked out that the Hamiltonian is 3 3 d k (λ) (λ) −ik.x (λ)† ik.x 3 A (x)= ∑ ε (k) a (k)e + a (k)e , d k k0 (λ) (λ)† (λ)† (λ) µ ( π)3 ω µ H = ∑ a (k)a (k)+a (k)a (k) . 2 2 k λ= ( π)3 1 λ 2 2k0 2 (2.4) (2.7) ω = 2 + 2, in which k k m and It is clear that the above integral and so the vacuum energy (λ) (λ) diverges. Unless the “normal ordering” procedure is used. [ ( ), †( )] = ω ( π)3δ δ3( − ), a k a k 2 k 2 λλ k k We now calculate the propagator for massive spin-1 parti- cles. Recall that the relation between the propagator and the (λ) (λ) (λ) (λ) two-point function is [13] [a (k),a (k )] = [a †(k),a †(k )] = 0. (2.5) (λ) 0|T(A (x)Aν(y))|0 = iG ν(x − y). (2.8) The polarization vectors εµ (k), obey the following rela- µ µ tions, ( ) We calculate the left-hand side by substituting Aµ x from (λ) (λ ) (λ) µ ε .ε = δλλ , εµ (k)k = 0. (2.6) (2.4) − .( − ) d4k e ik x y 3 (λ) (λ) 0|T(A (x)Aν(y))|0 = i ∑ ε (k)εν (k). (2.9) µ ( π)4 2 − 2 + ε µ 2 k m i λ=1 (λ) (λ )† 3 3 By a direct calculation one can show that this propagator [b (k),b (k )] = −2ωk(2π) δλλ δ (k − k ), (2.14) bears an infrared divergence and the divergent term appears in it’s imaginary part (Appendix-A). (λ) (λ) (λ) (λ) [a (k),b (k )] = 0, [a †(k),b †(k )] = 0, 2.2. Gupta-Bleuler quantization (λ) (λ) (λ) (λ) [a (k),b †(k )] = 0, [a †(k),b (k )] = 0. (2.15) In the Krein quantum field theory the field operator is de- | fined as follows [2, 4, 11] The vacuum state 0 is then defined by (λ)† (λ) 1 (p) (n) a (k)|0 = |1k, a (k)|0 = 0, ∀ k, λ, (2.16) Aµ(x)=√ Aµ (x)+Aµ (x) , (2.10) 2 (λ) (λ) b †(k)|0 = |1¯ , b (k)|0 = 0, ∀ k, λ, (2.17) 3 3 k ( ) d k (λ) (λ) − . A p (x)= ∑ ε (k) a (k)e ik x µ ( π)3 ω µ 2 2 k λ= 1 (λ)( )| = , (λ)( )|¯ = , ∀ , λ, (λ) . b k 1k 0 a k 1k 0 k (2.18) +a †(k)eik x , (2.11) where |1k is called a one particle state and |1¯k is called a one “unparticle state”. These commutation relations, to- 3 3 (n) d k (λ) (λ) . gether with the normalization of the vacuum A (x)= ∑ ε (k) b (k)eik x µ ( π)3 ω µ 2 2 k λ= 1 0|0 = 1, (λ) − . +b †(k)e ik x , (2.12) lead to positive (negative) norms on the physical (unphysical) (p) (n) sector: where Aµ (x) and Aµ (x) have positive and negative norm respectively, in the sense of Klein-Gordon inner product. 1 |1 = δ3(k − k ), 1¯ |1¯ = −δ3(k − k ). (2.19) Creation and annihilation operators obey the following com- k k k k mutation relations Using the field operator (2.10) one can show that (λ) (λ) (λ) (λ) [ ( ), ( )] = , [ †( ), †( )] = , a k a k 0 a k a k 0 3 3 = d k k0 (λ)†( ) (λ)( ) H ∑ 3 a k a k (2π) 2k 2 (λ) (λ )† 3 3 λ=1 0 [a (k),a (k )] = 2ωk(2π) δλλ δ (k − k ), (2.13) (λ) (λ) (λ) (λ) (λ) (λ) +b †(k)b (k)+a †(k)b †(k)+a (k)b (k) . (λ) (λ) (λ) (λ) [b (k),b (k )] = 0, [b †(k),b †(k )] = 0, (2.20) Brazilian Journal of Physics, vol. 39, no.
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