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I'm trying to fit QFT into a familiar mathematical framework:

Newtonian Mechanics (Single Particle):

We have a set of particles $X$ whose locations at given moment in time is $\hat{X}(t)$ and whose momentum component-wise is $P_i = mX_i'(t)$ called $\hat{P}(t)$ as a group:

  1. we define inertial frames
  2. we define $F = P'(t)$
  3. We define a property of $F$ that: net force = sum of component forces.

And the name of the game then is say given a description of $F$ and configuration of $X$ find $X(t)$ or $P(t)$ etc...

Special Relativist Mechanics (Multi Particle):

Same game as before but now we have $E^2 = (mc^2)^2 + (pc)^2$ as a relationship and new rules for transformations between frames because $c$ is invariant under boosts.

Classical Quantum Mechanics (Multi Particle):

Given a set of particles $S$ and a variable that describes their possible states $X$ then:

The evolution of our system is described by a wave function $\Psi(X,t)$ which obeys

$$ i\hbar \frac{\partial \Psi}{\partial t} = H\Psi$$

where $H$ is defined as the hamiltonian operator (requiring a particular basis for it to be fully expanded).

The name of the game is to find $\Psi$ or estimate bounds on the values of operators acting upon $\Psi$ given a particular basis $X$.

The math got harder but everything still is clear.

Relativistic Quantum Mechanics (Single Particle):

Depending on the spin of your particle it either obeys the klein gordon equation (an attempt to quantize $E^2 = (mc^2)^2 + (pc)^2$ by lifting $E,p$ to operators.

But this no longer behaves as a probability distribution (rather a charge distribution according to wikipedia? although its unclear where charge ever entered the picture)

In the spin-$\frac{1}{2}$ case it obeys the dirac Equation:

$$ \beta mc^2 + c \left( \sum_{n=0}^{4} \alpha_n p_n \right)\psi = i \hbar \frac{\partial \Psi}{\partial t} $$

We'll focus on this:

Here $\beta, a_n$ correspond to specific $4\times 4$ matrices and $p_n$ correspond to a set of 3 orthogonal directioned momentum operators.

Our fundamental description is now a 4 component wave function: corresponding to 4 independent pieces a spin-up/down electron/positron. After accepting this, the world doesn't look any different really than old school quantum mechanics, we can solve for the components of the wave function and act on it by operators.

So my Question:

  1. Where on earth is this charge or spin business coming from in 1-particle theory given the equations of state themselves make NO reference to them

  2. What does a multiparticle Quantum-Relativistic wave equation look like? I'm essentially looking for the defining equation of QFT, the way the multiparticle schrodinger equation is to QM and F=MA is to Newtonian mechanics.

Some Thoughts:

In newtonian mechanics we were solving for 1-D trajectories $x(t)$

In QM we were solving for wave function which can be viewed (k+1)-Dimensional geometric object where $k$ is the dimension of the basis

I suspect in QFT we will not be solving for finite dimensional objects but infinite dimensional ones, that is the object to be solved for is an OPERATOR.

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It's actually easier to answer 2, then 1.

Answer to 2: In QFT, the starting point is usually the Lagrangian density. This is constructed based purely on lorentz invariance, symmetry and gauge invariance arguments. The purists out there may try to dress it up in a really abstract way, but it more or less comes down to guessing symmetries and then seeing if it agrees with nature. Each term in the lagrangian either represents a free-field term, with particles being number excitations of this field, or a coupling term allowing for interactions between fields. "The name of the game" as you so eloquently put it, is to take this lagrangian and apply the Euler-Lagrange equations for each degree of freedom, which gives you your "equation of motion" for each particle. You are essentially solving for operators that create or destroy particles.

Answer to 1: Easy way to think about spin and charge is to simply look at the eigenvectors or generators of the matrix representation of whichever symmetry group( e.g. U(1) or SU(3)) you used to construct the lagrangian.

Addressing your thoughts: The fact that the fields are operators is due to the canonical quantization, not the fact that the Hilbert space is infinite dimensional.

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