The motivation here is to know how physicists formulate complicated objects so that we could borrow to theories in Deep Learning. In particular, how they combine special relativity and quantum mechanics together into a nice formulation.
Outline:
1. Quantum mechanics, or Schrödinger equation that describes how a single particle evolves over time, does not follow Lorentz transformation, which is a must for any theory that is compatible with relativity (special / general).
2. There are attempts to modify Schrödinger equation for that. This includes the famous Dirac equation that makes wave function 4 dimensional so that it conforms to Lorentz transformation, but then end up with eigenstates that have unbounded negative eigenvalues (or "negative energies"). That's why Dirac came up with the concept of "Dirac Sea", assuming all the negative states have been occupied so that a particle with positive energy will not fall into the negative states, according to Pauli's exclusive principles.
3. QFT solves the problem by considering a field theory rather than a particle theory, and using Heisenberg picture (operators are moving) rather than Schrödinger picture (unit vectors are moving). This yields a field of changing operators over spacetime, and the energy of the ground state ("vacuum state") is infinity.
4. Infinity energy on the ground state is really bad. Renormalization then follows: assuming the energy of the ground state to be zero (!!) and we only compute the energy difference between excited state and the ground state. This gives sensible solutions. Note that this is not "renormalization group".
5. The good things about QFT is that it could also model particle interactions (E.g., 2 particles collide) once we add a interaction term in the Lagrangian. Hilbert space is so vast that one or a few unit vectors could represent any state, regardless of the number of particles. So we just use one unit vector to represent the input particles and one vector to represent the output particles, and the "interaction" is basically a unitary operator that transforms one to the other. The "cross section" (probability that an interaction happens) is dependent on their inner product, and thus the unitary operator. Since cross section is measurable, the theory finally is justifiable.
6. The remaining of the introduction then focuses on computation. Computing the cross section is hard so only Lagrangian of specific forms are considered. As in signal processing, the overall cross section is an integral on Green functions (the response function over delta impulse). Depending on the structure of Lagrangian, the structure of Green function could be factorizable (here I see a bit connection with graphical models). From Feynman rules, one could write the Green function from the structure of Lagrangian.
Some feelings (I could be very wrong since I am really not an expert in this direction):
1. Physicists will add extra dimensions if the algebra does not work out, since from group representation theory, any fancy group operations can be embedded into matrix space with sufficient number of dimensions. That's probably the reason why the world is 11-dimensional (maybe more if new physical phenomena are discovered). So I won't think too much about the specific number.
2. "Vacuum is nontrivial" is a construction of the theory. By representing all interesting states as the consequence of the (creation and annihilation) operators acting on the "vacuum state" |0>, the theory puts all the parameters there. From that point of view, of course vacuum is nontrivial. Using empirical evidence we could then determine the parameters in the Lagrangian, and thus determine the characteristics of vacuum.
1 and 2 are often misused in the popular science. The idea is clear in the context but is highly ambiguous if taken separately.
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