4 Electromagnetic Field and its quantisation

N. Panchapakesan

    1. Learning Outcomes

a. Review Maxwell’s equations and its formulation with vector potentials.

b. Learn about difficulties with imposing commutation relations for the Lagrangian of the electromagnetic field.

c. Learn about two different ways of avoiding the difficulty: Fixing the gauge or giving up Gauss’s theorem in operator form.

    2. Introduction

Electromagnetism is described by the electric E and magnetic B fields. It is convenient to derive them from vector potential A and scalar potential ϕ. The electromagnetic field travelling with velocity of light has two degrees of freedom, the two polarisations of the field. The four components of the potentials A and ϕ have four components or degrees of freedom. So they over-determine the electric and magnetic fields. The gauge conditions provide two constraints to reduce the degree of freedoms to two.

Quantising a system with constraints has special problems or difficulties. The description by vector potential and the way the constraints are handled are described in this module. Quantum electrodynamics or QED is a gauge theory with a massless (or zero mass) quanta , the photon. The electro weak theory, combines electrodynamics and weak interactions in a gauge theory with a massive quanta. Experience with QED played a very useful role in quantizing massive gauge theories. This led to the combination of electromagnetism and weak interaction, in 1968. This led to the prediction of neutral currents and Higgs meson. Neutral currents were discovered very soon. The Higgs meson took much longer and happened only in 2012.

General Relativity is also theory with constraints. We have not succeeded in quantizing it satisfactorily yet.

We review the formulation of electrodynamics as a gauge theory with vector and scalar potentials. We then discuss its canonical quantisation and the difficulties faced in doing that. We avoid the difficulties by fixing the gauge to be coulomb (or radiation) gauge and quantizing the theory. We have also to modify the right hand side of the equal time commutation relation. The Dirac delta function is modified to a transverse delta function.

3.Electromagnetic Field – Lagrangian formalism

3.1 Vector Potentials

The well known Maxwell’s equations of electromagnetism are

Where E and B are the electric and magnetic fields respectively and we will take c=1.

The right hand side of all equations are zero if we are dealing with a free field with no sources. We are using Heaviside Lorentz Rational Units (mks=Rational unit). The fine structure constant α , which is related to the charge of the electron e is defined in these units as

(Eq. (3.4c) is same as in books of J.D.Jackson, A. Lahiri and P. Pal, and Peskin and Schroeder . Bjorken and Drell and Ryder have opp. sign) The field tensor and the electric and magnetic fields are related by,

We can see these equalities follow from definition

    4. Quantisation and Problems:

We impose the usual commutation relations between the fields and their conjugate quantities (eq. 3.12).

The  equation  (4.2)  has  a  difficulty  with Maxwell equation viz. Gauss’s law.

Conflict

We have a conflict between ETCR and Maxwell equation ∇̅ ∙? ̅ = 0. We have to modify one of them. Modification of ∇̅ ∙? ̅ = 0 is called Gupta-Bleuler method. In Gupta-Bleuler method, we do not impose the Gauss’s law as an operator equation but only as a matrix element equation. The eigen states of the physical system have zero matrix elements for the operator. The operator ∇̅ ∙? ̅ ≠ 0. But 〈 | ∇̅ ∙ ̅?̅̅|̅ 〉 can be taken to vanish. It turns out that even this is a strong requirement and no solutions exist. Gupta and Bleuler modified the requirement to demand that only the annihilation Part of the operator acting on the ket vanishes. This worked. The method is manifestly covariant, but one has to use an indefinite metric and has negative norm or ghost states.

5. Coulomb or radiation gauge commutation relation

We shall not follow that method. We shall instead stay in a particular gauge, called the coulomb gauge or radiation gauge, and modify the ETCR. The method is not manifestly Lorentz covariant or gauge invariant.

With these two constraints eqs. 6.3 and 6.4 the number of degrees freedom ?? (= 4) will get reduced to 2 (the two polarisations). We end up in the Coulomb gauge.

[For a massive or Yang- Mills field , ? ≠ 0 and the equation of motion is ( + ?2)?? = 0, and we can make ???? = 0 but not ?0 = 0. So we have 41=3 degrees of freedom left.]

7. Summary

    We have reviewed the formulation of electromagnetism in terms of vector and scalar potentials, or the four dimensional vector potential.

We have tried to quantize the four dimensional vector potential by imposing the canonical commutation relations. This led to inconsistencies which were traced to the gauge constraints .

We found there were two ways of proceeding. One way is to give up manifest Lorentz and gauge invariance and quantize in the coulomb gauge. Another way is to use Gupta- Bleuler method which keeps manifest Lorentz invariance but has to keep the Gauss’s law not in the operator form. This means admitting only those states which satisfy Gauss’s law in a suitable form. This leads to negative norm ghost states in the theory which have to be dealt with suitably. We choose the first way.