trying to teach myself QFT here. still early days. a few questions....should I picture a Fock space as an infinite triangular matrix in some kind of fucked up outer product with a Hilbert space? i.e., the first column is 1-particle states, and has only one nonzero element. the second column has two nonzero elements, for the states of each of two particles. etc. and corresponding to every element of the matrix is an entire copy of the Hilbert space. so there's sort of a third index for the eigenfunctions of the space, or something like that.why does every book on QFT make you learn the klein-gordon and Dirac equations after acknowledging they were both wrong/flawed?
can you help me get a handwavey overview of how someone does qft? what does the process look like? here is the state of my understanding...My rusty recollection of basic quantum mechanics:1) Specify:a. # of particlesb. Particle exchange symmetryc. force between particlesd. classical potential2) solve schrodinger's equation, given 1a-d. [Solution may be analytic, approximate (perturbation theory, e.g.), or numerical. yields a wavefunction, as a sum or integral of eigenfunctions (depending on whether the boundary conditions quantize them) of the system.
I'm assuming the actual equation(s) being approximately solved at 1b are euler-lagrange or hamilton's equations, respectively....?
also, all the talk about symmetries.... I gather they are baked into your lagrangian or hamiltonian when you define it at 1a....? I know (at a handwavey level) that each symmetry corresponds to a conserved quantity because of noether's theorem. I think I read that each group gives rise to a force-mediating boson for each of its generators. (so in qed there is one generator (photon) because its symmetry is defined by a one-parameter group. the weak force has 3 mediator particles, because it has a 3-generator group, etc.)one piece of the puzzle I have no idea how to connect to any of the others is gauge. the only thing I know about gauge is that in classical e&m, the magnetic vector potential is only defined up to an arbitrary gradient term (because taking the curl to get the magnetic potential kills any gradient). since the gradient term does matter for most (any?) purposes, the ability to change it freely is known as gauge symmetry. I think I know that this is indeed related to "gauge" in qft, but I don't know how it connects to any of the other qft things we're talking about here.
I get that he was talking about computing quantities. but I think somewhere between steps 1a and 1b, some equations were implicitly derived and then are being (approximately .... and implicitly?) solved at 1b
also, all the talk about symmetries.... I gather they are baked into your lagrangian or hamiltonian when you define it at 1a....? I know (at a handwavey level) that each symmetry corresponds to a conserved quantity because of noether's theorem. I think I read that each group gives rise to a force-mediating boson for each of its generators. (so in qed there is one generator (photon) because its symmetry is defined by a one-parameter group. the weak force has 3 mediator particles, because it has a 3-generator group, etc.)
one piece of the puzzle I have no idea how to connect to any of the others is gauge. the only thing I know about gauge is that in classical e&m, the magnetic vector potential is only defined up to an arbitrary gradient term (because taking the curl to get the magnetic potential kills any gradient). since the gradient term does not (edit) matter for most (any?) purposes, the ability to change it freely is known as gauge symmetry. I think I know that this is indeed related to "gauge" in qft, but I don't know how it connects to any of the other qft things we're talking about here.
maybe? I feel like there has to be a dynamical equation floating around here somewhere. don't statements about physics ultimately always take that form?also, looking at cold one's steps 1a and 1b.... my understanding has been that the main (only?) purpose of stating a lagrangian or hamiltonian function is to derive an associated set of dynamical equations from it. and I understand perturbation theory to be a method of approximating solutions to an equation. .... so, to me, both steps 1a and 1b point to the presence of some unstated equation in between the two.
from some googling around, it sounds like the schwinger-dyson equation might be what I've been looking for (or at least the closest thing to it).
I realize I may have been asking the question wrongly. I also gather SD is several steps removed from the procedure you laid out. ..... from what you and wikipedia are telling me, i understand that the scattering matrix is a limiting case approximation (of some system) for a particular problem that happens to be of broad relevance (particles coming in from infinity, doing any of a number of local interactions, and going out to infinity). and then the perturbation theory is performed based on that. and as you reminded me, the perturbation theory in basic QM does not even give a full approximate solution, it's an incomplete approximate solution which hopefully has enough information for your purposes. so fair enough on saying it doesn't count as "solving" an equation, assuming you have one.
does the scattering matrix at least correspond to a limiting case of the schwinger-dyson equation? it sounds like heisenberg et al. built it up heuristically, without SD known to them a priori. but can the s - matrix be derived from SD, by declaring your limit and throwing out negligible terms?
so QFT, unlike QM, GR, and all (?) of classical physics, does not have a master differential equation. is this because time is treated differently in QFT? (something to do with it being used to index states, rather than as a regular variable?)
ok, having trouble reconciling all that. ........ on that particular front, I think I have to stop trying to learn at a handwavey level and resume actually learning the material.
different question. are photons real or not? on the one hand, they are one of the force carrier bosons (for EM in particular), and I think I have read a couple times that those force carrier bosons are just an artifact of the perturbation theory computations, which left me with the impression that they are possibly or presumably not real things. .... on the other hand, basic quantum mechanics was in large part historically motivated by the idea that photons are quite real, and I feel like my quantum professor would have strangled someone who said otherwise. ..... does the resolution to this apparent contradiction have to do with the EM force mediators being virtual photons, as opposed to garden variety photons?