Three level Lambda system, relativistic invariance and renormalization in QED

Three level Lambda system, relativistic invariance and renormalization in QED

Navin Khaneja To whom correspondence may be addressed. Email:navinkhaneja@gmail.comIndian Institute of Technology Bombay, Powai - 400076, India.

Abstract

In this paper, we develop an analogy between the three level atomic system so called Lambda system and scattering processes in quantum electrodynamics (QED). In a Lambda system we have two ground state levels and at energy and excited level at energy . The transition from to has strength and transition from to has strength . In the interaction frame of natural Hamiltonian of the system, we get a second order term connecting level to with strength . This term creates an effective coupling between ground state levels and drives transition from to . Scattering processes in QED can be modelled like this. Feynman amplitudes are calculation of second order term . In the rest frame (CM frame) where sum of momentum of incoming particles is , if the Feynman amplitude is , then in a frame in which moves with velocity , the time gets dilated to and hence it must be true that . We show that to make this relativistic invariance of to work, we have to modify the QED interaction Hamiltonian. Using this modification, we calculate the scattering amplitude for QED processes, Compton scattering and Møller Scattering. This gives an additional factor of in Feynman amplitude, where is sum of four momentum of incoming particles (or outgoing) and . In this paper taking vacuum polarization as a example, we show that we can avoid divergences in QED if we correctly work with this factor .

1 Introduction

We first develop an analogy between the three level atomic system so called Lambda () system and scattering processes in quantum electrodynamics (QED) [1, 2, 3, 4]. In a Lambda system as shown in Fig. 1 we have two ground state levels and at energy and excited level at energy . The transition from to has strength and transition from to has strength . In the interaction frame of natural Hamiltonian of the system, we get a second order term connecting level to with strength . This term creates an effective coupling between ground state levels and drives transition from to . Scattering processes in QED can be modelled like this. Feynman amplitudes are calculation of second order term .

Figure 1: Above Fig. shows a three level Lambda system with two ground state levels and and an excited level .

The state of the three level system evolves according to the Schröedinger equation

(1)

We proceed into the interaction frame of the natural Hamiltonian (system energies) by transformation

(2)

This gives for ,

(3)

is periodic with period . After , the system evolution is

(4)

The first integral averages to zero, while the second integral

(5)

Evaluating it explicitly, we get for our system that second order integral is

(6)

Thus we have created an effective Hamiltonian

(7)

which couples level and and drives transition between them at rate .

2 Modelling QED as three level Lambda system

QED processes can be modelled as three level Lambda system as shown in Fig. 1. In a QED scattering process, level represents all the incoming particles with total energy and four momentum . Level all the outgoing particles with energy and level represent all the intermediate particles including a off shell virtual particle with four momentum and energy (virtual particle is only present in level 2). Let denote total energy of all particles in level . Then using notation , we can define

(8)
(9)
(10)

where are spinors we can associate with the QED process through , the electron-photon interaction Hamiltonian [1],

(11)

where vector potential in terms of photon annihilation and creation operators and is

(12)

where is four vector and indices polarization and wave-vector. is mode volume and permitivity. The electron field

(13)

where are the positive eigenvectors of the Dirac equation and indices helicity and momentum. and are annihilation and creation operators for electron and positron.

Important is to note that in , , the coefficients as determined by have been modified to one shown in (8 and 9). This ensures that we get the right relativistic dependence of the Feynman amplitude. In some sense we have re-defined the QED coupling Hamiltonian to get in tune with relativity.

To get a sense for how amplitude has right relativistic dependence, we find when we sum over two processes that define a Feynman diagram (also see section 4) we get

(14)

where, , when virtual particle is a photon. The term in bracket above is called propagator. Then

(15)

where spinor terms are relativistically invariant (spinor for off-shell virtual particle has its mass as off-shell mass). Now imagine we are in center of mass frame (CM frame), and we make a change of frame, then and . Rest everything is relativistically invariant and hence we get the right relativistic dependence of amplitude.

Furthermore observe, in the center of mass frame we have and hence

(16)

which is what we know from Feynman calculus.

We now take few examples that illustrate the above methodology. We consider first Compton scattering and then Møller scattering and calculate their scattering amplitudes.

3 Compton Scattering

Compton scattering is the inelastic scattering of a photon with an electrically charged particle, first discovered in 1923 by Arthur Compton [5]. This scattering process is of particular historical importance as classical electromagnetism is insufficient to describe the process; a successful description requires us to take into account the particle-like properties of light. Furthermore, the Compton scattering of an electron and a photon is a process that can be described to a high level of precision by QED.

In Compton scattering an electron and photon with momentum and respectively scatter into momentum and respectively. We want to calculate the amplitude for this scattering.

There are two Feynman diagrams that show mechanism of Compton scattering. They are shown in Fig. 2. We can associate each of these with two three level diagrams as shown in Fig. 3.

Figure 2: Fig. A shows Fig. B show the two mechanisms for Compton scattering.
Figure 3: Fig. shows three level systems that go with Feynman diagrams in Fig. (2).

Consider Feynman diagram A in Fig. 2, where a electron of momentum and photon of momentum are annihilated to give an electron of momentum which is then annihilated to create electron and photon with momentum and . This correspond to three level system Fig. 3 A. The scattering amplitude for this system is as follows

(17)
(18)
(19)
(20)

where and . Summing over electron polarization we get

(21)

There is an associated three level diagram with this as shown in 3 B, where we first create electron and photon with momentum and respectively alongside a positron with momentum and then annihilate electron and photon with momentum and alongside a positron with momentum .

The scattering amplitude for this system is as follows

(22)
(23)
(24)
(25)

Summing over electron polarization we get

(26)

Adding the two amplitudes , we get

(27)

We made use of identity , where and is off-shell mass ( is implicit).

Now consider Feynman diagram B in Fig. 2, where a electron of momentum is annihilated and photon of momentum is created to give an electron of momentum which is then annihilated along-with the photon of momentum to create electron with momentum . This correspond to three level system Fig. 3 C. The scattering amplitude for this system is as follows

(28)
(29)
(30)
(31)

Summing over electron polarization we get

(32)

There is an associated three level diagram with this as shown in 3 D, where we first create electron and annihilate photon with momentum and respectively alongside creating a positron with momentum and then annihilate electron and create photon with momentum and alongside annihilate positron with momentum .

The scattering amplitude for this system is as follows

(33)
(34)
(35)
(36)

Summing over electron polarization we get

(37)

Adding the two amplitudes , we get

(38)

We can compare the expression for and in Eq. (27) and (38) with the one in textbooks [2, 3], derived using a Feynman propagator and we find that the textbook expressions are

(39)
(40)

Our expression matches it when we are in CM frame, else we have an additional factor of , which makes amplitude relativistically correct and is in CM frame.

We now look at a different QED process, Møller scattering.

4 Coulomb Potential and Møller Scattering

In Møller scattering, electrons with momentum and exchange photon with momentum and scatter to new momentum states and . Observe the virtual particle four momentum is . The Feynman diagram for the process is in 4. There are two three level systems associated with this process. Let .

Figure 4: Fig. shows the Feynman diagram for the Møller scattering and its corresponding three level system. The electron with momentum emits (absorbs) a photon and scatters to momentum , the photon is absorbed (emitted) by electron with momentum which scatters to momentum .

In figure 4A, we have the first three level system where the electron with momentum is annihilated , a electron of momentum is created and a photon of momentum is created. Subsequently, the electron with momentum is annihilated , a electron of momentum is created and photon of momentum is annihilated. The amplitude for this process is

(41)
(42)
(43)
(44)

We can now sum over photon polarization .

Similarly we have another three level system, fig 4B in which emits photon with momentum and absorbs it. This gives

(45)
(46)
(47)
(48)

We can now sum over photon polarization .

Using conservation of energy , we get that when we add the two amplitudes, we get the total amplitude as

In case of Møller scattering, we get the same amplitude as obtained using Feynman propagator except an additional factor of . In CM frame this is 1.

We now look at QED processes that are modelled as a two level system and then show how they can be used to calculate vacuum polarization in QED.

5 Modeling vacuum polarization as two level system

Figure 5: Fig. shows how to model vacuum polarization as a two level system.

Consider the Feynman diagram in Fig. 5 A, a photon with momentum spontaneously creates a electron-positron pair with momentum and respectively, which recombine to give again. Creating a positron is same as annihilate a negative energy electron with momentum . We represent this as a two level system shown in Fig. 5 B, with lower level comprising of incoming momentum and higher level , the outgoing electron with momentum . The transition rate from to is given by

(49)

where , and

the relativistic factor for the outgoing electron. There is an associated two level system with this where we first create a electron, positron and photon with momentum respectively. This is shown in Fig. 5 C, where again creating a positron is same as annihilate a negative energy electron with momentum . The transition rate from to is

(50)

The dynamics of this two level system is

(51)

We proceed into the interaction frame of the natural Hamiltonian (system energies) by transformation

(52)

This gives for

(53)

is periodic with period . After , the system evolution is

(54)

The first integral averages to zero, while the second integral

(55)

Evaluating it explicitly, we get for our system that second order integral is

(56)

Thus we have created an shift of energy level from .

Figure 6: Fig. shows Feynman diagram for vacuum polarization.

Now lets return to Møller scattering where electrons with momentum exchange a virtual photon with momentum and scatter to momenta . In previous section we modelled this as a three level atomic system, where level contains the virtual particle. Now consider the situation as in Fig. (6), where the virtual photon creates electron-positron pairs along the way. This is modelled in Fig. (7), where level containing virtual photon is coupled to additional levels representing electron-positron pairs. This was already modelled as a two level system in the beginning of this section. Now we can eliminate the additional levels. To fix ideas just imagine we have one additional level we call . Then the three level dynamics with additional level is

Figure 7: Fig. shows how vacuum polarization is represented as additional levels when møller scattering is represented as a three level system.
(57)

We transform,

(58)

This gives for

(59)

is periodic with period . After , the system evolution is

(60)

with as calculated in description of energy shifts. Then

Thus we have removed coupling and introduced energy shifts in and . We can now eliminate level and calculate transition amplitude between and and as before the transition amplitude is

(61)

We can estimate how big is. We know

(62)

Using from Eq. (49) and Eq. (50), we find,

(63)

We now have to sum over all momentum . Observe for large , we have

(64)

If we look at the spinor terms , then for large , we have and , then for large , we have

(65)

When we integrate over by moving to polar coordinates we cancel the first factor of with leaving us with the integral which is finite.

Now returning back to Eq. (61), we can rewrite

(66)

Thus we get a correction to