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

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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 divergence 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 particles. 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. 3, September, 2009 561

This energy is zero for the vacuum state and it is not needed These equations follow from a variational principle with the to use the “normal ordering” procedure. It is also positive Lagrangian [13] for any particles state or physical state |Nk (those built from † repeated action of the a (k)’s on the vacuum) 1 µν L = − F νF , (3.4) 4 µ 3 d3k k N |H|N = ∑ 0 where A is regarded as the dynamical field. The gauge trans- k k ( π)3 µ λ= 2 2k0 2 → = + ∂ Λ( ), µν 1 formation Aµ Aµ Aµ µ x leaves F unchanged. (λ)† (λ) Potentials satisfying this additional condition, ×Nk|a (k)a (k)|Nk≥0, |0k≡|0. (2.21) φ = , ∇. = , At this stage we calculate the propagator for the massive 0 A 0 (3.5) spin-1 particles, using Gupta-Bleuler quantization. Using (2.10) in the left hand side of (2.8) we have are said to belong to the radiation (or Coulomb) gauge.In this gauge there are clearly only two independent components of A . This is the case in the real world, so working in | ( ( ) ( ))| = 1 | ( (p)( ) (p)( ))| µ 0 T Aµ x Aν y 0 0 T Aµ x Aν y 0 the radiation gauge keeps the physical nature of the electro- 2 ( ) ( ) magnetic field most evident. Let us therefore study quantiza- + | ( n ( ) n ( ))| 0 T Aµ x Aν y 0 tion in this gauge. 1 (p) (n) = W ν (x,y)+W ν (x,y) . (2.22) 2 µ µ 3.1. Usual quantum field theory (n) By a direct calculation one can show that Wµν (x,y)= ∗ (p) (p) Note that in view of the radiation gauge Maxwell’s equa- − Wµν (x,y) and Wµν (x,y) is the two-point function for tions (3.3) become the positive modes as it is calculated in the previous subsec- tion. Therefore Aµ = 0, (3.6) | ( ( ) ( ))| = I (p)( , ) 0 T Aµ x Aν y 0 i m Wµν x y further, since in the radiation gauge we have φ = 0, this be- comes − .( − ) = . d4k e ik x y 3 (λ) (λ) A 0 (3.7) = i I i ∑ ε (k)εν (k) , (2.23) m (2π)4 k2 − m2 + iε µ λ=1 This is the Klein-Gordon equation for a massless field, and we write and the propagator for the transverse photon in the Krein space quantization is then 3 2 d k (λ) (λ) − . (λ) . A(x)= ∑ ε (k) a (k)e ik x + a †(k)eik x , ( π)3 − .( − ) 2 2k0 λ= d4k e ik x y 3 (λ) (λ) 1 Gµν(x−y)=iR e ∑ εµ (k)εν (k) , (3.8) ( π)4 2 − 2 + ε (λ) 2 2 k m i λ=1 where ε (k) are known as polarization vectors, k = (2.24) 0, k0 = |k|, and which is free of any infrared divergence and it is not need (λ) (λ) (λ) to use the renormalization procedure. Indeed this quantiza- k.ε (k)=0, ε (k).ε (k)=δλλ . (3.9) tion method acts as an automatic renormalization factor for infrared divergence in the two-point function. The creation and annihilation operators obey the following commutation relations

(λ) (λ )† 3 3 3. MASSLESS VECTOR FIELD [a (k),a (k )] = 2k0(2π) δλλ δ (k − k ),

The six components of the electromagnetic field (three of (λ) (λ) (λ) (λ) electric and three of magnetic field) may be written as an [a (k),a (k )] = [a †(k),a †(k )] = 0. (3.10) antisymmetric tensor Fµν, and the homogeneous Maxwell equations follow if it is assumed that Fµν is a 4-dimensional These commutation relations have the same form as those for curl of the field Aµ scalar field, and have the same interpretation as annihilation and creation operators for photons. ν ν ν Fµ = ∂µA − ∂ Aµ, (3.1) Let us now calculate the vacuum energy. It may be written as ν ∂ µ = , 1 1 2 µF 0 (3.2) H = d3x(E2 + B2)= d3x(A˙ +(∇ × A)2) 2 2 1 2 ν − ∂ν(∂ µ)= . = (A˙ − A.∇2A)d3x. (3.11) A µA 0 (3.3) 2 562 M. Dehghani

Substituting (3.8) in (3.11), we obtain after some algebra The vacuum is deﬁned by the relations given in Eqs.(2.16)- (2.18). 2 3 d k k (λ) (λ) (λ) (λ) If we calculate the energy operator in terms of the ﬁeld H = ∑ 0 a (k)a †(k)+a †(k)a (k) . ( π)3 operator (3.14), we have λ=1 2 2k0 2 (3.12) 2 3 d k k (λ) (λ) It is clear that the above integral and so the vacuum energy H = ∑ 0 a †(k)a (k) ( π)3 diverges. Unless the ”normal ordering” procedure is used. λ= 2 2k0 2 1 The propagator for the transverse (physical) photons may + (λ)†( ) (λ)( )+ (λ)†( ) (λ)†( )+ (λ)( ) (λ)( ) . be calculated in the same manner as for massive photons, b k b k a k b k a k b k after some algebra we have (3.20)

− .( − ) d4k e ik x y 2 (λ) (λ) This energy is zero for the vacuum state and it is not needed 0|T(A (x)Aν(y))|0 = i ∑ ε (k)εν (k). µ (2π)4 k2 + iε µ to use the “normal ordering” procedure. It is also positive λ=1 | (3.13) for any particles state or physical state Nk (those built from †( ) This propagator bears an infrared divergence in it’s imagi- repeated action of the a k ’s on the vacuum) nary part (Appendix-A). 2 d3k k N |H|N = ∑ 0 k k ( π)3 λ=1 2 2k0 2 3.2. Gupta-Bleuler quantization (λ)† (λ) ×Nk|a (k)a (k)|Nk≥0, |0k≡|0. (3.21)

Similar to the previous case, the ﬁeld operator is deﬁned At this stage we calculate the propagator for the physical as follows [2, 4, 11] photons using Gupta-Bleuler quantization. Using (3.14) in the left hand side of (2.8) we have 1 ( ) ( ) A(x)=√ A p (x)+A n (x) , (3.14) 2 1 (p) (p) 0|T(Aµ(x)Aν(y))|0 = 0|T(Aµ (x)Aν (y))|0 2 ( ) ( ) + | ( n ( ) n ( ))| 3 2 0 T Aµ x Aν y 0 ( ) d k (λ) (λ) − . A p (x)= ∑ ε (k) a (k)e ik x ( π)3 1 ( ) ( ) 2 2k0 λ= = p ( , )+ n ( , ) . 1 Wµν x y Wµν x y (3.22) (λ) . 2 +a †(k)eik x , (3.15) (n) By a direct calculation one can show that Wµν (x,y)= ∗ (p) (p) − W ν (x,y) and W ν (x,y) is the two-point function for 3 2 µ µ ( ) d k (λ) (λ) . A n (x)= ∑ ε (k) b (k)eik x the positive modes as it is calculated in the previous subsec- (2π)32k 0 λ=1 tion. Therefore (λ) − . +b †(k)e ik x , (3.16) (p) 0|T(Aµ(x)Aν(y))|0 = i Im [Wµν (x,y)]

( ) ( ) where A p (x) and A n (x) have positive and negative norm − .( − ) respectively, in the sense of Klein-Gordon inner product. d4k e ik x y 2 (λ) (λ) = i I i ∑ ε (k)εν (k) , (3.23) a(k) and b(k) are two independent operators. Creation and m ( π)4 2 + ε µ 2 k i λ=1 annihilation operators are constrained to obey the following commutation relations and the propagator for the transverse photons in the Krein (λ) (λ ) (λ)† (λ )† space quantization is then [a (k),a (k )] = 0, [a (k),a (k )] = 0, 4 −ik.(x−y) 2 tr d k e (λ) (λ) D ν(x − y)=iR e ∑ ε (k)εν (k) , µ ( π)4 2 + ε µ (λ) (λ )† 3 3 2 k i λ= [a (k),a (k )] = 2k0(2π) δλλ δ (k − k ), (3.17) 1 (3.24) which is free of any infrared divergence and it is not need (λ) (λ) (λ) (λ) [b (k),b (k )] = 0, [b †(k),b †(k )] = 0, to use the renormalization procedure. Indeed this quantization method acts as an automatic renormalization factor for infrared divergence in the propagators. (λ) (λ )† 3 3 [b (k),b (k )] = −2k0(2π) δλλ δ (k − k ), (3.18)

4. CONCLUSION (λ) (λ) (λ) (λ) [a (k),b (k )] = 0, [a †(k),b †(k )] = 0, Through a generalization of the Gupta-Bleuler quantization introduced in [2], a covariant quantization of the mas- (λ) (λ) (λ) (λ) [a (k),b †(k )] = 0, [a †(k),b (k )] = 0. (3.19) sive and massless vector ﬁelds in Minkowski space-time is Brazilian Journal of Physics, vol. 39, no. 3, September, 2009 563 constructed, which is free of any infrared (and ultraviolet) former, the integral can be solved in terms of the Bessel func- divergences. It has been shown that this method of quanti- tions as follows [11, 15] zation acts as an automatic renormalization element of the ultraviolet divergence in the stress tensors and a renormal- d4k e−ik.(x−y) 1 ization factor for infrared divergence in the propagators. In = − δ(σ) (2π)4 k2 − m2 + iε 8π other words, using this new quantization method, the ultra- √ √ 2 2 2 violet divergence in the vacuum energies disappears and the m J1( 2m σ) − iN1( 2m σ) + θ(σ) √ , σ ≥ 0,(A.5) normal ordering is not necessary. Also the two-point func- 8π 2m2σ tions are automatically renormalized, without any renormal- 1 2 1 µ ν σ = (x − y) = g ν(x − y) (x − y) , ization procedure. 2 2 µ

where Jν(x) and Nν(x) are real Bessel functions for real x and APPENDIX A: SOME USEFUL MATHEMATICAL ν. FORMULA Taking twice derivative ∂µ∂ν from both side of Eq.(A.5) we have In calculating the propagators given in Eqs. (2.9) and (3.13) we encounter the integrals of the form 4 −ik.(x−y) 4 −ik.(x−y) d k kµkνe d k e = −∂µ∂ν . d4k e−ik.(x−y) (2π)4 k2 − m2 + iε (2π)4 k2 − m2 + iε , (A.1) (2π)4 k2 − m2 + iε (A.6) The limiting values of the various kinds of Bessel functions for small values of their argument are as follows [16] 4 −ik.(x−y) d k kµkνe , (A.2) ν ( π)4 2 − 2 + ε 1 x 2 k m i Jν(x) → , Γ(ν + 1) 2

Γ(ν) ν − .( − ) 2 d4k e ik x y Nν(x) →− , ν = 0. , (A.3) π x (2π)4 k2 + iε In the massless limit, Eq.(A.5) reduces to [15] we give a brief explanation about these types of integrations. − .( − ) ( . ) d4k e ik x y i 1 Using the following integration formula in A 1 = − δ(σ). (A.7) (2π)4 k2 + iε 8π2σ 8π ∞ −s(k2−m2+iε) = 1 , dse 2 2 (A.4) 0 k − m + iε Acknowlegements: I am grateful to referee for useful com- and commuting d4k integration with ds and performing the ments and professor M.V. Takook for his useful discussions.

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