The condition ∂λA^λ=0 does not fully define the propagator....

The condition $\partial_\lambda A^\lambda=0$ does not fully define the propagator. We are at liberty to rewrite wave equation (6.78) as $$ \left[g^{\nu \lambda} \square^2-\left(1-\frac{1}{\xi}\right) \partial^\nu \partial^\lambda\right] A_\lambda=j^\nu . $$ In this case, use (6.79) to show that the propagator is $$ \frac{i}{q^2}\left(-g_{\mu \nu}+(1-\xi) \frac{q_\mu q_r}{q^2}\right) . $$ The Feynman gauge takes $\xi=1$. But in any case, the extra term in the propagator vanishes in QED calculations in which the virtual photon is coupled to conserved currents which satisfy $q_j j^\mu=q_\nu j^\nu=0$.

The condition $\partial_\lambda A^\lambda=0$ does not fully define the propagator. We are at liberty to rewrite wave equation (6.78) as
$$
\left[g^{\nu \lambda} \square^2-\left(1-\frac{1}{\xi}\right) \partial^\nu \partial^\lambda\right] A_\lambda=j^\nu .
$$

In this case, use (6.79) to show that the propagator is
$$
\frac{i}{q^2}\left(-g_{\mu \nu}+(1-\xi) \frac{q_\mu q_r}{q^2}\right) .
$$

The Feynman gauge takes $\xi=1$. But in any case, the extra term in the propagator vanishes in QED calculations in which the virtual photon is coupled to conserved currents which satisfy $q_j j^\mu=q_\nu j^\nu=0$.

Key Concepts

Photon Propagator

The photon propagator is the Green's function that describes how a photon propagates from one spacetime point to another in quantum electrodynamics. Its form is affected by the chosen gauge-fixing condition, which introduces additional terms that must be managed. The propagator structure underlies the computation of interaction amplitudes in QED, ensuring that the correct physical behavior is captured.

Gauge Parameter (?) Dependence

In formulating gauge theories, a gauge parameter (?) is introduced to parametrize the freedom in the choice of gauge-fixing condition. Different values of ? lead to different mathematical forms for the propagator, but physical observables remain invariant under such changes. For instance, the Feynman gauge (? = 1) often simplifies calculations, while any extra terms in the propagator generally vanish when interacting with conserved currents.

Gauge Fixing

Gauge fixing is a procedure used in field theory to remove the redundancy of gauge degrees of freedom. By imposing an additional condition (such as the Lorenz condition), one can uniquely specify the fields' evolution and define propagators in a way that avoids overcounting equivalent configurations. This step is vital for quantizing gauge theories and performing perturbative calculations.

Conserved Current and Ward Identities

Conserved currents in gauge theories, which satisfy conditions like ?_?j^? = 0, play a crucial role in ensuring that gauge-dependent extra terms cancel in physical calculations. This cancellation is a manifestation of Ward identities, which are relationships that follow from gauge invariance and guarantee that the unphysical degrees of freedom do not contribute to observable quantities.

Transcript

00:01 In this question, we are given the decay of a sigma baron into a lambda baron with a photon.

00:08 And we are given the respective rest masses.

00:12 And we want to find out what is the wavelength of this photon that has been released.

00:17 So to find a wavelength, we will need to find what is the energy of this photon.

00:21 And to find what is the energy of this photon, we need to find the means you use momentum conservation, as well as energy conservation.

00:31 So from energy conservation, the total kinetic energy, kinetic energy of the lambda barion plus the energy of the photon must be equal to the difference, the change in the rest energy from the sigma to the lambda barion.

00:59 This tells us how much of the mass was actually released and converted into energy, which is kinetic energy, and the photon energy.

01:11 So that would be close to 1192 minus 116.

01:18 This gives us 76 mev.

01:26 And now that we have the energy, let me just change this gamma to eo, of gamma.

01:35 Now we're going to use conservation of momentum.

01:41 Since our initial sigma baron is given to be stationary, it means that the initial momentum is zero.

01:50 Our final momentum must be zero as well.

01:53 So this means that our momentum for the sigma baron and the photon must be equal and opposite to each other in order to cancel out and give us zero momentum momentum of the sigma i mean the photon is given as e over c right since the energy of the photon is pc on the other hand for the for the lambda barion we're going to use a different equation that is momentum of the lambda baron times c square times c holding square is equals to the kinetic energy of the particle square plus two times the kinetic energy times the rest energy now this equation is derived over here i'm going to leave it to just for your own reference but the the other facial involves knowing that the total energy is equal to gamma times the rest energy, as well as the kinetic energy is equal to the gamma minus 1 times the rest energy.

03:34 And so using these two relations, as well as the overall relation between the total energy, and the momentum and the rest energy, then you should be able to get the final equation over here which relates the momentum to the kinetic energy and the rest energy of our particle of interest right in the relativistic way right so we can continue from here because we know that the momentum must be equal and opposite so what we're going to do is we want to equate this to this so p of lambda baron square must be equals to p gamma times c squared and we know that p gamma is e gamma over c so this gives us e gamma square and so this must be e gama square right the next step we want to do is we want to substitute our kinetic energy k we know that k is equals to the total kinetic energy, let us just label this as k, not k total, but rather total energy released, let us just label as e.

05:24 So k is equals to e minus e gamma, right? so we can substitute that in into this equation...

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