Gupta-Bleuler quantization for massive and massless free vector fields

Gupta-Bleuler quantization for massive and massless free vector fields

M. Dehghani ^*

Department of Physics, Ilam University, Ilam, Iran

ABSTRACT

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

Particles of spin-1 are described as the quanta of vector fields. Such vector bosons play a central role as the mediators of interactions in particle physics. The important examples are the gauge fields of the electromagnetic (massless photons), the weak (massive W^± and Z⁰ bosons), and the strong (massless gluons) interactions. On a somewhat less fundamental level vector fields can also be used to describe spin-1 mesons, for example the ρ and the ω mesons.

It has been shown [1,2] that one can construct a covari-ant quantization of massless minimally coupled scalar field in de Sitter space-time, which is causal and free of any divergences. The essential point of these papers is the unavoidable presence of the negative norm states. Also they do not propagate 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 authors [4-7].

These auxiliary states (the negative norm states) appear to be necessary for obtaining a fully covariant quantization of the free minimally coupled scalar field in de Sitter spacetime, which is free of any infrared divergence. It has been shown that these auxiliary states automatically renormalize the infrared divergence in the two-point function [3, 4] and remove the ultraviolet divergence in the vacuum energy [2]. This method is applied to the massless vector field in de Sitter space [8, 9], interacting quantum fields and Casimir effect in Mincowski space-time [10-12].

We would like to generalize this quantization method to the massive and massless free vector fields in Minkowski space-time. These auxiliary states once again, automatically leads to renormalized two-point functions and removes the ultraviolet divergences in the vacuum energies.

This new quantization method is used for consideration of the Casimir effect for the scalar field [10]. It can be extended for the electromagnetic field in 4-dimensional spacetime. 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 interact with the physical states or real physical world, thus they cannot be affected by the physical boundary conditions as well. The results for the Casimir energy and Casimir force are the same given in [13, 14].

In section 2 a brief review of the massive vector field quantization, using usual quantum field theory is presented. The ultraviolet divergence of the vacuum energy and the infrared divergence of the two-point function is explicitly shown for this field. Then using Gupta-Bleuler quantization it is shown that the vacuum energy converges and the infrared divergence of the propagator disappears. In section 3 the ultraviolet divergence of the vacuum energy and the infrared divergence of the two-point function of the massless vector field is shown clearly in the framework of usual quantum field theory. Also the vacuum energy and the propagator of this field is calculated using Gupta-Bleuler quantization, which are free of any divergences. Finally a brief conclusion is given in section 4.

2. MASSIVE VECTOR FIELD

The lagrangian density for the massive vector field A_µ(x), in Minkowskian flat space-time is that of Proca [13]

where . The variational principal to (2.1) yields the field equation

Taking the divergence of Eq. (2.2) we have , and since , we find the Lorenz gauge and Eq.(2.2) can be written as

2.1. Usual quantum field theory

There are three independent components of a massive spin-1 field, and the Lorentz condition may serve to eliminate 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 the Lagrangian will result in a term when A⁰ is eliminated, so that the mass term has a negative coefficient, as it does for the Klein-Gordon Lagrangian. The field Aµ(x) has the expansion

in which , and

The polarization vectors , obey the following relations,

It may be checked out that the Hamiltonian is

It is clear that the above integral and so the vacuum energy diverges. Unless the "normal ordering" procedure is used.

We now calculate the propagator for massive spin-1 particles. Recall that the relation between the propagator and the two-point function is [13]

We calculate the left-hand side by substituting Aµ(x) from (2.4)

By a direct calculation one can show that this propagator bears an infrared divergence and the divergent term appears in it's imaginary part (^Appendix-A).

2.2. Gupta-Bleuler quantization

In the Krein quantum field theory the field operator is defined as follows [2,4, 11]

where and have positive and negative norm respectively, in the sense of Klein-Gordon inner product. Creation and annihilation operators obey the following commutation relations

The vacuum state is then defined by

where is called a one particle state and is called a one "unparticle state". These commutation relations, together with the normalization of the vacuum

lead to positive (negative) norms on the physical (unphysical) sector:

Using the field operator (2.10) one can show that

This energy is zero for the vacuum state and it is not needed to use the "normal ordering" procedure. It is also positive for any particles state or physical state (those built from repeated action of the on the vacuum)

At this stage we calculate the propagator for the massive spin-1 particles, using Gupta-Bleuler quantization. Using (2.10) in the left hand side of (2.8) we have

By a direct calculation one can show that

and is the two-point function for the positive modes as it is calculated in the previous subsection. Therefore

and the propagator for the transverse photon in the Krein space quantization is then

which is free of any infrared divergence and it is not need to use the renormalization procedure. Indeed this quantization method acts as an automatic renormalization factor for infrared divergence in the two-point function.

3. MASSLESS VECTOR FIELD

The six components of the electromagnetic field (three of electric and three of magnetic field) may be written as an antisymmetric tensor , and the homogeneous Maxwell equations follow if it is assumed that is a 4-dimensional curl of the field

These equations follow from a variational principle with the Lagrangian [13]

where A_µ is regarded as the dynamical field. The gauge transformation A_µ → A'_µ = A_µ + ∂_µΛ(x), leaves F^µv unchanged. Potentials satisfying this additional condition,

are said to belong to the radiation (or Coulomb) gauge.In this gauge there are clearly only two independent components of A_µ. This is the case in the real world, so working in the radiation gauge keeps the physical nature of the electromagnetic field most evident. Let us therefore study quantization in this gauge.

3.1. Usual quantum field theory

Note that in view of the radiation gauge Maxwell's equations (3.3) become

further, since in the radiation gauge we have Ф = 0, this becomes

This is the Klein-Gordon equation for a massless field, and we write

where ε^(λ-)(k) are known as polarization vectors, k² = 0, k₀ = |k|, and

The creation and annihilation operators obey the following commutation relations

These commutation relations have the same form as those for scalar field, and have the same interpretation as annihilation and creation operators for photons.

Let us now calculate the vacuum energy. It may be written as

Substituting (3.8) in (3.11), we obtain after some algebra

It is clear that the above integral and so the vacuum energy diverges. Unless the "normal ordering" procedure is used.

The propagator for the transverse (physical) photons may be calculated in the same manner as for massive photons, after some algebra we have

This propagator bears an infrared divergence in it's imaginary part (^Appendix-A).

3.2. Gupta-Bleuler quantization

Similar to the previous case, the field operator is defined as follows [2, 4, 11]

where and have positive and negative norm respectively, in the sense of Klein-Gordon inner product. a(k) and b(k) are two independent operators. Creation and annihilation operators are constrained to obey the following commutation relations

The vacuum is defined by the relations given in Eqs.(2.16)-(2.18).

If we calculate the energy operator in terms of the field operator (3.14), we have

This energy is zero for the vacuum state and it is not needed to use the "normal ordering" procedure. It is also positive for any particles state or physical state │N_k〉_; (those built from repeated action of the on the vacuum)

At this stage we calculate the propagator for the physical photons using Gupta-Bleuler quantization. Using (3.14) in the left hand side of (2.8) we have

By a direct calculation one can show that

and is the two-point function for the positive modes as it is calculated in the previous subsection. Therefore

and the propagator for the transverse photons in the Krein space quantization is then

which is free of any infrared divergence and it is not need to use the renormalization procedure. Indeed this quantization method acts as an automatic renormalization factor for infrared divergence in the propagators.

4. CONCLUSION

Through a generalization of the Gupta-Bleuler quantization introduced in [2], a covariant quantization of the massive and massless vector fields in Minkowski space-time is constructed, which is free of any infrared (and ultraviolet) divergences. It has been shown that this method of quantization acts as an automatic renormalization element of the ultraviolet divergence in the stress tensors and a renormalization factor for infrared divergence in the propagators. In other words, using this new quantization method, the ultraviolet divergence in the vacuum energies disappears and the normal ordering is not necessary. Also the two-point functions are automatically renormalized, without any renormalization procedure.

Acknowlegements: I am grateful to referee for useful comments and professor M.V. Takook for his useful discussions.

(Received on 28 April, 2009)

In calculating the propagators given in Eqs. (2.9) and (3.13) we encounter the integrals of the form

we give a brief explanation about these types of integrations. Using the following integration formula in (A.1)

and commuting integration with ds and performing the former, the integral can be solved in terms of the Bessel functions as follows [11, 15]

where and are real Bessel functions for real x and v.

Taking twice derivative from both side of Eq.(A.5) we have

The limiting values of the various kinds of Bessel functions for small values of their argument are as follows [16]

In the massless limit, Eq.(A.5) reduces to [15]

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Appendix-A

Publication Dates

• Publication in this collection
  15 Oct 2009
• Date of issue
  Sept 2009