6 Quantization of the electromagnetic field (I)

Quantum Field Theory-I Prof. G. Isidori

UZH and ETH, HS-2017 Revised version: October 20, 2017

http://www.physik.uzh.ch/en/teaching/PHY551/HS2017.html

6 Quantization of the electromagnetic field (I)

6.1 Prologue

After the quantization of the scalar (spin-0) and the Dirac (spin-1/2) fields, we would like to attack the problem of quantizing a field transforming as a Lorenz vector A

. This completes the introduction the three types of basic fields needed to describe all microscopic phenomena observed in Nature.

The quantization of a generic massive vector field turns out to be a rather com- plicated problem that, ultimately, leads to a serious inconsistency in the theory.

However, as far as we know, the only fundamental vector fields appearing in Nature are those associated to gauge symmetries, such as the photon. This implies a special form for the action of these fields and, in particular, it implies they are massless.

All the sub-atomic massive particles with spin-1 or higher do not correspond to the excitations of a fundamental field, but can be interpreted as the result of the interaction between fundamental spin-0 fields, spin-1/2 fields, and spin-1 (massless) gauge fields.

In this course we will concentrate only on the case of the photon field, or the gauge field associated to electromagnetism, that is an Abelian gauge symmetry.

6.2 Maxwell Lagrangian and Lorentz gauge

The Maxwell Lagrangian, in classical electromagnetism and in absence of external sources, is

L

= − 1

4 F

F

F

= ∂

A

− ∂

A

(1) from which we derive the following equation of motion for A

:

∂

F

= ∂

A

− ∂

(∂

A

) = 0 (2) Gauge invariance implies that the formulation of the theory in terms of A

is re- dundant: we obtain the same equations of motion for A

, and the same equations of motion for the physical fields ( E ~ and B), under the transformation ~

A

(x) → A

(x) = A

(x) + ∂

f (x) (3)

1

where f (x) is a generic (scalar) function. This implies that, although A

apparently has four independent components, only two combinations correspond to physically different configurations. These correspond to the two possible polarization states of the electromagnetic waves (or the two polarization states of the photons).

The unphysical components of A

can be eliminated by fixing the gauge. The gauge-fixing condition that is usually adopted in order to maintain a manifestly covariant formulation, is the so-called Lorentz gauge condition

Lorentz gauge condition : ∂

A

= 0 (4) Note, however, that this is only one constraint, that therefore does not remove completely the gauge degeneracy. If the Lorentz condition is imposed the equations of motions for A

reduce to ∂

A

= 0, namely to four independent Klein-Gordon equations for massless fields.

6.3 Gupta-Bleuler quantization

If we try to directly quantize L

, imposing appropriate commutation (or anti commutation) relations between the field (A

) and its conjugated variable (π

) we immediately encounter a problem: the Lagrangian does not depend on ∂

A

hence we cannot impose any commutation (or anti commutation) relation on A

.

An alternative way to see this problem is to note that the equation of motion for A

in (2) is not invertible. Indeed the equation for the Green function D

(x − y) reads

(∂

g

− ∂

∂

)D

(x − y) = ig

δ

(x − y) (5) or, in momentum space,

(−k

g

+ k

k

) D ^f

(k) = ig

(6) This equation has no solution since the 4× 4 matrix (−k

g

+ k

k

) is singular. The singularity is not only for a specific value of k

(as for the scalar or Dirac fields), but on a whole class of field configurations: this singularity reflects the gauge ambiguity in A

.

A possibility to overcome these problems is to consider a modified Lagrangian, that yields the same equations of motion for A

but only in a specific gauge, and in particular in the Lorentz gauge. Consider

L

= L

− 1

2ξ (∂

A

)

(7)

2

The extra term can be seen as constraint on the Lagrangian problem, that assign a non-trivial weight to the different gauge configurations: this way the minimal energy is obtained for ∂

A

= 0, i.e. in the Lorentz gauge. The equations of motion are now

∂

A

− 1 − 1 ξ

!

∂

∂

A

= 0 (8) and are equivalent to Eq. (2) only in the Lorenz gauge.

The equations become particularly for ξ = 1 (Feynman gauge-fixing choice). In the following we shall set ξ = 1, to simplify the discussion, but this specific choice of ξ is not mandatory. We will come back at the end of this lecture to the consequences of keeping ξ generic, in particular as far as the expression of the photon propagator is concerned.

Setting ξ = 1, the four canonical conjugated fields of A

are π

= ∂ L

∂(∂

A

) = −∂

A

(9) Being all non-vanishing, we can proceed quantizing the theory. Note, however, that we have not imposed any gauge-fixing condition: we have modified the Lagrangian and showed that Maxwell equations are recovered if the condition ∂

A

= 0 is satisfied. We therefore must impose this condition somewhere in the theory. In particular, we must impose that for any physical state

hS

|∂

A

|S

i = 0 (10) This procedure is called the Gupta-Bleuler quantization method.

The basic idea is that the space of field configurations is much larger than the sub-space corresponding to physical states. In other words, after the quantization, the fields contain also creation/destruction operators that create/destroy unphysical states. The physical states are selected, in this larger space, by the condition (10).

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