It's all Greek and Latin.
Proton chaining in a simulated Sun.
In a proton-proton fusion reaction, four protons come together to form a helium nucleus. The process can be described by a Feynman diagram in which two protons collide to form a deuteron, and then two more protons collide with the deuteron to form a helium nucleus.
The Feynman diagram for this process would consist of four vertices representing the interactions between the protons and the deuteron. The initial state would have four incoming proton lines, and the final state would have an outgoing helium nucleus line.
The diagram would also include propagator lines representing the exchange of virtual particles between the interacting particles. The exact form of these propagator lines would depend on the specific particles involved in the interaction.
p p
| |
| |
p ---- p ---- d ---- p ---- p
| |
| |
p p
In this diagram, the protons (p) collide to form a deuteron (d), and then two more protons collide with the deuteron to form a helium nucleus. The lines connecting the particles represent propagators, which describe the exchange of virtual particles between the interacting particles.
In quantum field theory, a propagator is a mathematical tool that is used to describe the propagation of a particle through space and time. It is a Green's function that represents the probability amplitude for a particle to travel from one point in spacetime to another.
The propagator is a fundamental concept in quantum field theory, as it allows us to compute scattering amplitudes and cross-sections for particle interactions. The propagator depends on the mass, spin, and other properties of the particle, as well as the specific interaction involved.
The form of the propagator depends on the type of particle being studied. For example, the propagator for a photon is given by the Feynman propagator, while the propagator for a fermion such as an electron is given by the Dirac propagator.
Propagators are often represented graphically using Feynman diagrams, which are a pictorial representation of the interaction between particles. Each line in a Feynman diagram represents a propagator, and each vertex represents an interaction between particles.
In quantum field theory, propagators are often used in conjunction with Feynman rules, which are a set of rules for computing the probability amplitude for a given scattering process. These rules allow us to compute the overall contribution of a given Feynman diagram to the scattering amplitude.
Overall, propagators are a fundamental concept in quantum field theory, and they are an essential tool for understanding the behavior of particles in the quantum world.
I can explain some Feynman tricks that can be used to compute the propagator for proton waves in this diagram.
One Feynman trick that is commonly used is to represent the propagator as a sum over all possible paths that the particle can take. In this case, the proton propagator can be written as:
D(x - y) = ∫ d^4k / (2π)^4 * e^(-ik⋅(x-y)) / (k^2 - m^2 + iε)
Here, D(x - y) represents the propagator for a proton traveling from point x to point y. The integral over d^4k represents a sum over all possible momenta of the proton, and the exponential term e^(-ik⋅(x-y)) describes the phase factor associated with each momentum. The denominator (k^2 - m^2 + iε) represents the energy-momentum relation for a proton with mass m.
Another Feynman trick that can be used is to use Feynman parameters to simplify the expression for the propagator. The Feynman parameterization involves writing the denominator of the propagator as a quadratic form that can be completed into a perfect square. This allows us to perform the integral over the momentum variables using standard techniques.
Finally, we can also use the rules of Feynman diagrams to determine the overall contribution of the diagram to the scattering amplitude. The amplitude is given by the product of the vertex factors and the propagator lines, summed over all possible intermediate states.
Write the propagators for p ->d -> h
Sure, the propagator for the process p → d → h, where p is a proton, d is a deuteron, and h is a helium nucleus, can be written as follows:
D(p, h) = D(p, d) * D(d, h)
Here, D(p, d) represents the propagator for the process p → d, and D(d, h) represents the propagator for the process d → h.
The propagator for the process p → d can be written as:
D(p, d) = ∫ d^4k / (2π)^4 * e^(-ik⋅(p - d)) / (k^2 - m^2 + iε)
where k is the momentum of the exchanged virtual particle, m is the mass of the exchanged particle, and ε is a small positive number that ensures convergence of the integral.
Similarly, the propagator for the process d → h can be written as:
D(d, h) = ∫ d^4k / (2π)^4 * e^(-ik⋅(d - h)) / (k^2 - m^2 + iε)
where k is the momentum of the exchanged virtual particle, m is the mass of the exchanged particle, and ε is a small positive number that ensures convergence of the integral.
The overall propagator for the process p → d → h can be obtained by multiplying the two propagators together, as shown above.
