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Jdeloz828

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- TL;DR Summary
- I'm looking to understand what's going on with the underlying Hilbert space of a system described using a Feynman diagram.

I'm currently working my way through Griffith's Elementary Particles text, and I'm looking to understand what's going on with the underlying Hilbert space of a system described using a Feynman diagram. I'm fairly well acquainted with non relativistic QM, but not much with QFT. In particular, I'd like to know how we would represent the initial and final states within the space for different types of particles, as well as how we can explain certain types of constraints on the interactions based on the symmetry of the underlying Hamiltonian.

- I'd first like to know how to represent states of free particles in the underlying Hilbert space. Take an electron for instance. I'd imagine the state vector would be represented as something of the form:

My second question involves how we explain certain constraints on interactions in terms of the Hilbert Space, using the following example. In his text, Griffiths makes the claim that if the color singlet gluon occurred in nature, it could be absorbed by baryons (also color singlets) resulting in a long-range component of the strong interaction. How do we explain this claim in terms of the symmetry of the underlying Hamiltonian and the evolution of the state vector?

I apologize if my questions seem somewhat naive. As I said, I'm not very familiar with relativistic QM. If anyone can help shed some light on these matters and/or point me to some good resources for learning more about it, it would be much appreciated.

- I'd first like to know how to represent states of free particles in the underlying Hilbert space. Take an electron for instance. I'd imagine the state vector would be represented as something of the form:

Ψ(r, t) ⊗ u(p),

where Ψ(r, t) represents the spatial part of the vector, encoding information about position and velocity, and u(p) is a bispinor encoding information about the spin state. Continuing in this way, a quark state would be represented as:

Ψ(r, t) ⊗ u(p) ⊗ I ⊗ c,

where Ψ(r, t) and u(p) are the same as the electron, I is the isospinor representing the quark flavor state, and c is the color state. A mediator, such as a photon would be represented as:

Ψ(r, t) ⊗ ε,

where ε represents the polarization vector encoding the photons spin state. My first (possibly dumb) question is this: How does a system evolve from an initial state vector, represented as a direct product of objects appropriate to the particles involved, into a final state with an different underlying representation. Take, for example, the process of pair annihilation:

e

Our underlying state vector would have to evolve from a state representing an electron and positron to one representing two photons:^{+}+ e^{-}→ γ + γ

(Ψ(r, t)

but how is this possible given that they're vectors from two fundamentally different vector spaces?_{electron}⊗ u(p)) ⊗ (Ψ(r, t)_{positron}⊗ v(p)) → (Ψ(r, t)_{photon 1}⊗ ε_{photon 1}) ⊗ (Ψ(r, t)_{photon 2}⊗ ε_{photon 2}),

My second question involves how we explain certain constraints on interactions in terms of the Hilbert Space, using the following example. In his text, Griffiths makes the claim that if the color singlet gluon occurred in nature, it could be absorbed by baryons (also color singlets) resulting in a long-range component of the strong interaction. How do we explain this claim in terms of the symmetry of the underlying Hamiltonian and the evolution of the state vector?

I apologize if my questions seem somewhat naive. As I said, I'm not very familiar with relativistic QM. If anyone can help shed some light on these matters and/or point me to some good resources for learning more about it, it would be much appreciated.

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