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Topic: uncool / cold one call out thread (Read 427 times) previous topic - next topic

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  • el jefe
  • asleep till 2020 or 2024
uncool / cold one call out thread
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?
  • Last Edit: December 20, 2017, 10:44:56 AM by Brother Daniel

  • uncool
Re: uncool call out thread
Reply #1
You happen to be running into a weak area for me - I don't have a good understanding of the formal foundations of QFT. So don't take any of what I say as a strong "this is how it is".

Fock space is a Hilbert space (in the mathematical sense) - it still is (as far as I understand) a subspace of the Hilbert space of field wavefunctions. Field operators (and more specifically, creation and annihilation operators) act on this subspace.

For Klein-Gordon and Dirac equations, I'll have to look over it again. I think it's a mix of historical "This didn't work but it got us closer to a better theoretical understanding" and "This isn't as precise (in a mathematical, not physical sense) as we want it to be", but I could be entirely misremembering.

  • el jefe
  • asleep till 2020 or 2024
Re: uncool call out thread
Reply #2
ok, I'll treat that as gospel

Re: uncool call out thread
Reply #3
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?

Fock space = QFT Hilbert space (the whole thing, not s subspace) organized in a useful way for the operators you care about in weakly coupled or free QFTs.  You can think of it as a tensor product of single particle states, yes, because every basis state in Fock scan is some number of creation operators acting on the vacuum.

There's nothing wrong with the and Dirac and KG equations if you apply them correctly.  They are the equations of motion of the free classical theory you're quantizing, and the quantum field operators satisfy them too (again, in the free theory).  They just aren't the relativistic generalization of the Schrodinger equation, which is maybe what you are referring to.

  • el jefe
  • asleep till 2020 or 2024
Re: uncool call out thread
Reply #4
hang on...  are KG & Dirac close to / related to the Hamilton - Jacobi equation?

Re: uncool call out thread
Reply #5
KG and Dirac are just the Euler-Lagrange equations for two (different) Lagrangians. Generally in classical mechanics Euler-Lagrange equations are related to the HJE equation, yes - they're equivalent.  But it's generally pretty hard to use the HJE for field theories.

  • el jefe
  • asleep till 2020 or 2024
Re: uncool call out thread
Reply #6
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 particles
b.      Particle exchange symmetry
c.      force between particles
d.      classical potential

2)      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.  The quadratic potential yields a particularly cute analytic solution, where you factor the Hamiltonian and get creation and annihilation operators; this is mathematically (though not physically) similar to something that happens in QFT.]

3)      Use wavefunction to extract predictions (expectation values) of observables, by sandwiching the operator associated with a given observable in between the wavefunction and its complex conjugate, and integrating.

4)      Philosophize about it. 
certain conjugate pairs of operators (including famously those for position and momentum), obey a constraint on the minimum error bar for their predictions, AKA Heisenberg uncertainty.  This surprising feature of the theory can be semi-intuited at least one way: We are modeling things as waves, and by picturing a rope being jiggled you can see that the better-defined a wave's frequency (and therefore momentum) is, the more smeared out its position is, and vice versa.


Quantum field theory:

1)      Particle numbers are *not* fixed.  Something about pairwise creation and annihilation of particles and their antiparticles
2)      Something about symmetries
3)      ? ? ?
4)      gauge comes in here somewhere?
5)      Perturbation theory?
6)      Feynman diagrams?
7)      ? ? ?
8)      Profit!
  • Last Edit: December 20, 2017, 09:43:58 AM by el jefe

Re: uncool call out thread
Reply #7
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 particles
b.      Particle exchange symmetry
c.      force between particles
d.      classical potential

2)      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. 

OK.  In that style:

1)      Specify:
a.      the Lagrangian or Hamiltonian density of the classical field theory you're quantizing. 
b.      assuming the theory is perturbative (i.e. there is a small parameter you can use to do perturbation theory, like a small coupling constant), work out the rules for perturbation theory (typically in the form of rules for Feynman diagrams)
c.      calculate quantities  like decay rates of unstable particles and scattering cross-sections to the desired order in perturbation theory, which can then be compared to experiment

Obviously this is not exhaustive, but that will do for a start.

  • el jefe
  • asleep till 2020 or 2024
Re: uncool / cold one call out thread
Reply #8
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 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.
  • Last Edit: December 22, 2017, 02:12:36 PM by el jefe

  • uncool
Re: uncool / cold one call out thread
Reply #9
I'm assuming the actual equation(s) being approximately solved at 1b are euler-lagrange or hamilton's equations, respectively....?
I think cold one is talking quantities, not equations - cross-sections in particular. You're solving for how often certain outcomes will happen - e.g. when you collide an electron and a positron at high energy, how often will you get a muon and an anti-muon?
Quote
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.
All of that seems quite a bit further ahead - iirc my book got to calculating cross-sections (which are the calculations most often done iorc) in chapter 5, and things like gauge symmetries were chapter 9 or so.

  • el jefe
  • asleep till 2020 or 2024
Re: uncool / cold one call out thread
Reply #10
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

  • uncool
Re: uncool / cold one call out thread
Reply #11
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
I'm guessing you mean equations deriving the values of elements of the S-matrices, and relating them to Feynman diagrams?

  • el jefe
  • asleep till 2020 or 2024
Re: uncool / cold one call out thread
Reply #12
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.

  • linus
Re: uncool / cold one call out thread
Reply #13
Full QFT is beyond my expertise, but there are two simpler theories that may be helpful stepping stones if needed:

1. Fock space and creation/annihilation operators are used together with the Schrödinger equation (for electrons) in molecular physics and condensed matter theory.

2. Non-relativistic QED (electrons described by the Schrödinger equation coupled to a quantized radiation field) is a more well-defined framework that physicists use to study radiation effects and mathematicians use to study certain problems that are too ill-defined/hard in QED.

Re: uncool / cold one call out thread
Reply #14
I'm assuming the actual equation(s) being approximately solved at 1b are euler-lagrange or hamilton's equations, respectively....?

Sort of/not really.

You can of course solve those equations using perturbative expansions, and you can draw "Feynman diagrams" to help organize that.  But that's not what's done in QFT.  In QFT there are two things going on:  first, the classical theory is non-linear, and second, it's quantum.  The perturbative expansion in Feynman diagrams takes care of both of those.  So it's not just solving the classical equations of motion, it's also solving the quantized theory.

Quote
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.)

That's correct.

Quote
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.

Well, gauge invariance isn't exactly a symmetry.  A symmetry is something that when given a solution (let's talk about classical physics, so a  classical solution), the symmetry generates another, different solution.  For instance, rotation invariance (which means that angular momentum is conserved, via Noether).  Gauge invariance is different - it's (except in a limit) just a redundancy.  When you act on a solution with a gauge transformation you get the same solution, not a different one.  However there are special gauge transformations (the ones that are constant in space and time) that are real symmetries, and those produce a real conserved quantity (charge). 

Re: uncool / cold one call out thread
Reply #15
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.

To give more details on this - think about perturbation theory in quantum mechanics.  There, you're not really solving a dynamical equation.  Instead, you're working out the (say) energy levels of the Hamiltonian subject to a perturbation that makes it hard or impossible to find them exactly and/or to solve the Schrodinger equation exactly. 

Perturbation theory in QFT is similar, except the thing you're solving for isn't the energy levels.  Instead, you want to compute amplitudes (and hence, probabilities) for the results of experiments you can actually do.  You can't measure the energy because it's spread over the entire universe (remember, this is the Hamiltonian of a field theory, and the fields depend on space, so the energy is an integral over all space).  What you can measure are things like scattering amplitudes - you smash a few particles together and see what comes out.  So the perturbative expansion in QFT is designed to compute that, not solve a Schrodinger equation, or classical field equations of motion.

So no, there's no hidden dynamical equation that's being solved.  The quantities you're computing are the probabilities that a given an initial state "far in the past" (say, 2 particles of some definite type that are far apart but flying towards each other) will evolve after a "long" time into some definite final state (say, three particles of some other type with some other momenta that are far apart and flying apart).
  • Last Edit: January 05, 2018, 07:10:43 AM by cold one

  • el jefe
  • asleep till 2020 or 2024
Re: uncool / cold one call out thread
Reply #16
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).

https://en.m.wikipedia.org/wiki/Schwinger-Dyson_equation

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?

Re: uncool / cold one call out thread
Reply #17
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 wouldn't call S-D a "dynamical" equation - time doesn't have much to do with it, apart from the appearance of time-ordering. 

Quote
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.

Well, you can think of S-D as a step towards deriving the Feynman rules.  If you look at the equations at the bottom of that wiki page, you see some perturbative expansions for time-ordered correlation functions.  (Just keep applying S-D to go to higher orders.)  There is another formula, called the LSZ reduction formula, that relates time-ordered correlators to the S-matrix.

Quote
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?

Yes, see above.

  • el jefe
  • asleep till 2020 or 2024
Re: uncool / cold one call out thread
Reply #18
ok thanks

  • el jefe
  • asleep till 2020 or 2024
Re: uncool / cold one call out thread
Reply #19
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?)


Re: uncool / cold one call out thread
Reply #20
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?)

QFT = QM, just with lots of degrees of freedom.  It has a Hilbert space, a Hamiltonian, and a time-evolution operator.  It also has a Schrodinger equation for the time evolution of the wave functional.  So if you think the Schrodinger equation is the "master differential equation" for QM (I don't), then you could say the same for QFT.

The place where time really plays a different role is in quantum gravity, where the freedom to change coordinates makes time more like a redundancy of the description than a real variable.  As a consequence the Schrodinger equation there is that the time derivative of the wavefunction is zero (H \psi = 0, if you want). 

  • el jefe
  • asleep till 2020 or 2024
Re: uncool / cold one call out thread
Reply #21
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?

Re: uncool / cold one call out thread
Reply #22
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.

Good idea!

Quote
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?

Whether something is real is a question above/below my paygrade (maybe ask a philosopher).  But I think you'd have a hard time arguing that photons are any more or less real than electrons or protons.  They're a very similar sort of thing.  On top of that they can burn you, you can literally see them (or maybe you can see bunches of a few, but I don't think it's more than just a few), and in a lab you can easily produce, detect, and manipulate them.

Now, it's true that in QFT Feynman diagrams there are "virtual photons" - but it's not just photons.  Every particle species shows up that way in some Feynman diagrams and virtual particles of every type contribute at some level to just about every process in nature (virtual particles are just internal lines in Feynman diagrams, and they don't satisfy the classical equation of motion - for a virtual photon E^2 =/= p^2 c^2).  It's also true that the leading QFT diagram that reproduces the Coulomb potential in the Born approximation has a virtual photon line. 
  • Last Edit: January 12, 2018, 07:47:07 AM by cold one