Verify that q ·ε=0 for each λ, and show that, for a spacelike photon (q^2<0), ∑(-1)^λ+1 ε^μ^* ε^

Verify that q ·ε=0 for each λ, and show that, for a...

Key Concepts

Spacelike vs. Timelike Photons

Photons can be characterized by the sign of their four-momentum squared (q²). A spacelike photon (q² < 0) has different polarization properties compared to a timelike photon (q² > 0). For spacelike photons, the polarization sum includes a factor (-1)^(?+1), which is omitted in the timelike case. This distinction is important as it connects the behavior of virtual photon propagators in different kinematic regimes, linking the analysis to the spin states of massive vector particles when q² is replaced by M².

Cross Section Calculation in Photon-Proton Scattering

In scattering experiments involving photons and protons, the total cross section for a polarized photon interacting with an unpolarized proton is calculated by contracting the photon’s polarization vectors with the hadronic tensor W_??, which encapsulates the response of the proton to the photon. This approach not only provides a means to evaluate physical observables like cross sections but also highlights the role of polarization states (transverse and longitudinal) in the dynamics of electromagnetic interactions.

Photon Polarization Vectors

Photon polarization vectors describe the independent ways in which a photon can oscillate. For any physical photon, these vectors satisfy the transversality condition (q · ? = 0), meaning the polarization is orthogonal to the photon’s direction of propagation. This property is crucial for ensuring the consistency of electromagnetic and quantum field theories, as it is directly related to gauge invariance and the elimination of unphysical degrees of freedom.

Polarization Sum and Completeness Relation

The polarization sum is a method used to express the sum over all polarization states of a vector particle, such as a photon. In the context of a spacelike photon (where q² < 0), the sum takes a form involving the metric tensor and a term proportional to q^?q^?/q². This completeness relation is essential for calculations in quantum field theory, as it enables the replacement of sums over polarization vectors by covariant expressions, facilitating the evaluation of scattering processes and cross sections.

Transcript

00:01 All right.

00:04 In this problem, we're asked to derive an expression for compton scattering where the electron is not initially at rest, but is rather moving towards our photon with some momentum.

00:18 And after the scattering, the photon is deflected 180 degrees while the electron remains on its same trajectory.

00:25 So this is a particular case.

00:28 We were given a piece of information that the total energy e of the electron is much, much greater than its rest energy.

00:35 So this is an interesting result.

00:38 Or say that total energy is much greater than its rest energy.

00:42 And after we derive this expression for lambda prime, the new wavelength of our light, we are asked to numerically find for co2 laser what this lambda prime would be, given some parameters.

00:54 So full disclaimer, this is a very challenging problem involving a significant amount of algebra.

01:02 So there's no way i'm going to be able to do every single step of algebra in the, and keep this a pretty succinct video.

01:10 So i will show the main parts and leave it as an exercise to the watcher to fill in the gaps in the problem solving.

01:20 So the main equations we'll need will be applying conservation momentum and conservation of energy to derive this expression.

01:25 So we need to know that the energy of a photon is h -t over lambda.

01:29 Its momentum is h over lambda, where h is the plant constant.

01:34 And it's useful to know that the total energy of a relativistic particle with mass is this expression, root momentum times c squared plus mass times c squared, this expression.

01:48 We'll be using that.

01:50 So let's jump into it.

01:51 Let's apply some conservation of energy and conservation momentum.

01:57 So to start, we can set up conservation momentum in a vector form.

02:05 And let's be very careful about how we're applying these different symbols.

02:11 So i'm going to call this p, lowercase p.

02:17 That's going to be the momentum of our photon, and my uppercase p will be the momentum of the electron.

02:24 I will make sure to always include this tail to show the difference.

02:29 And we're going to use primes to indicate after the scattering.

02:33 So the left -hand side of this equation is before scattering, and this is going to be after scattering.

02:41 In a vector form, conservation momentum is simply this with our declared symbols.

02:48 So for this particular situation where we know the scattering angle, and we know the direction that the particles are moving in before and after the scattering, we can simplify this into a non -vector form to be p -minus capital p equals negative.

03:09 P prime minus negative, well, minus p prime, capital p prime.

03:18 And we can simplify it.

03:19 So this is our expression for momentum in a scalar form in 1d.

03:23 And i'm going to do one step of algebra, so we're going to need this in a bit.

03:31 That actually after scattering p prime is going to simply be, well, we can relate it to all the other momentum, which might be useful to us.

03:46 As so.

03:48 We can also set up conservation of energy.

03:49 It's worth noting that for a photon, the energy, you can also express it as the momentum times c.

03:57 This is initial energy of the photon, plus the total energy of the electron, which includes rest and kinetic, call that big e.

04:06 So then we have our after -scattering energy and our after -scattering energy of the electron.

04:17 We can simplify this down a little bit, just right -hand side, by inputting this large square root, as i mentioned, we would need.

04:40 Okay, so we've set up momentum and energy now here's where all the algebra is going to come in we need to solve for the final momentum of the photon so starting with so at this point we would want to combine energy we'll start with energy and we're going to come we're going to combine it with a result for momentum so just doing it some very simple algebra pc minus p prime c plus e, moving things over to the left -hand side of the equation, equals this square root, and we'll square both sides.

05:59 So at this point, we are going to plug in our momentum result and start expanding.

06:09 So we will have pc minus p -prime c plus e, all of this squared.

06:17 On the left -hand side, and the right -hand side will start expanding.

06:20 So it'll end up being p c squared plus p plus p prime c all that squared minus two capital p p plus p prime c squared plus mc squared squared so this this involved a bit of algebra where we've always simply did was combine was expand this right -hand side to combine it with momentum so now let's move over to another page continue.

07:09 So after a lot of simplification, which i'm not going to show, so it will take too much time, we get this equation into a slightly better form.

07:26 We are left with p prime, capital p, c squared minus 2p, c squared minus ec, equal to p negative ec, negative capital p, c squared.

07:50 So this is a good place to check your work and good stepping point.

08:00 So we can finally isolate p prime as this big fraction, p and then a big ratio, ec plus capital p, c squared, all over 2p c squared plus ec, minus capital p, c squared, which simplifies a little bit more by pulling out a c...

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