# Help David learn Quantum Field Theory

For the few months, I’ve been trying to learn Quantum Field Theory, working mostly from Zee’s book Quantum Field Theory in a Nutshell and also from John Baez’s course notes. Note that my goal is to actually understand physics, in terms of actual physical objects interacting, not to just understand mathematical constructions. This is going to be the first of what I imagine will be a series of posts where I ask dumb questions and hope that AJ, John Baez or one of our other physcist readers will help me out. For my first question, I’m going to ask about something which has been bothering me since the very start of the book.

Question 1: True or false — there are two kinds of particles in QFT. There are force carrying particles and ordinary particles. The former are described by wave functions on the space of solutions to the classical field equations while the latter are described by wave functions on spacetime.

Sometimes Zee talks as if the answer is “true” and sometimes he talks as if the answer is “false”. I have slowly come to the conclusion that he implies “false” more strongly. Assuming that “false” is the corect answer:

Question 2: What is the classical field whose excited states are electrons?

My best guess is that this is the current J — the vector field on spacetime (four-vector field, in physicist terminology) whose time component is the charge density and whose space component is the “density” of charge x velocity. Following through this idea, I tried to write down a Lagrangian for the evolution of J and ran into trouble. I know the Lagrangian for an individual particle of mass m and charge q, traveling in an electro-magnetic field with potential A (a vector field). If this particle travels a parameterized path y(u) through spacetime, the Lagrangian is

$\int \left( \ m \cdot \langle dy/du, \ dy/du \rangle^{1/2} - q \cdot \langle A(y(u)) , \ dy/du \rangle \ \right) du .$

Here $\langle v,u \rangle$ is the relativistic inner product $v_t u_t - (1/c^2) (v_x u_x + v_y u_y + v_z u_z)$. I checked, and this does give the Lorentz force. (I was going to write this up, but John Baez already did.) Based on this, I tried to guess the Lagrangian for J, but something seems to have gone wrong:

Question 3: Is the Lagrangian for the evolution of the J field given by
$\int \left( \sigma \langle J,J \rangle^{1/2} - \langle J,A \rangle \right)$
where the integral is over all spacetime and $\sigma$ is the ratio of electron mass to electron charge? If not, what is the right Lagrangian formulation? For that matter, what is the statement of the Lorentz force equation in terms of J?

I feel guilty about composing a post which is all questions but no answers. In partial atonement, here is a list of resources that I found helpful in getting me this far. For a nice introduction to Lagrangians, see the lecture on the “Principle of Least Action” in Volume II of Feynman’s lectures. To learn more about Lagrangians by working examples, see David Morin’s notes. (I am still working from the physical copies that Morin gave me nine years ago, but I think my link goes to identical files.) To convince yourself that those pretty path integrals you’ve heard talked about really do lead to Schroedinger’s equation, see John Baez’s notes from the Quantization and Cohomology Seminar. To convince yourself that solutions of Schroedinger’s equation do (sometimes) look like particles in motion, see John Baez’s Photons, Schmotons page. This final one may only be useful to people working from Zee’s book specifically — Zee starts by presenting a scalar field theory where the force carrying particles are massive but don’t interact with each other. I found it very helpful to discover that the classical analog of this theory is discussed at the end of the lecture on “Electromagnetic Mass” in volume II of Feynman’s lectures (the section entitled “The Nuclear Force Field”).

Thanks in advance for any hints!

## 18 thoughts on “Help David learn Quantum Field Theory”

1. A.J. Tolland says:

Question 1: Neither true nor false as stated (which I guess is normally what happens when one is confused).

The state of a single particle is described, as you say, by a wavefunction on spacetime.

The state of a quantum field is described by a wave function on the space of solutions to the classical field equations.

The subtlety is this: When we quantize _free_ fields on flat spacetime, we Fourier decompose the field into various modes of oscillations. These individual field modes behave like particles. More mathematically, you might say that the state space for free fields is the Fock space built from the state space for a particle.

Things can get much more complicated in interacting theories, but if you’re reading Zee, you’re really trying to learn _perturbative_ QFT, and in perturbative QFT we cheat and think of our Hilbert space as a the Hilbert space of a free theory.

Question 2: The field whose excitations are electrons is the “electron field”, usually written $\psi$. Unlike the gravitational and electromagnetic fields, it wasn’t studied by classical physicists. They didn’t have the language. $\psi$ is a spinor-valued fermionic field. This means, amongst other things, that it’s nilpotent.

Classical physicists studied instead the electron current, which is the vector field $J^\mu = \psi^*_a (\gamma^0)^a_b \gamma^{\mu,b}_c \psi^c$, where the $\gamma$ are the Dirac matrices (which you should think of as intertwining three representations of the Lorenz group: the vector, the spinor, and the dual spinor).

OK, I’m going back to work. I’ll write more later, if no one else does first.

2. matteo says:

I’m not a great physicist but I would like to try give some hints: in between QM and QFT Dirac tried to treat “ordinary particles” as relativistic quantum mechanical particles, and in this early work the particle is described by a wave function on spacetime. But this was not correct and effects such as Lamb shift couldn’t be accounted for, you had to quantize the wave function (classical field) itself, that is to promote it to an operator on some Hilbert space (Fock space): the state of the system is described by a vector in that space, which is not the space of classical solutions to the fields equations. But anyway sometimes (most times) you can treat the electron as a quantum wave-function, so therefrom comes the confusion. On the other side force-carriers were already classically described as fields so you simply had to quantize them.
As to the lagrangian formulation, the current is not the right field for the electron since it’s a vector and you can easily see that the electron wave functions doesn’t transform as a vector, since its Lorentz group representation is doubly covered by rotations: 2\pi rotations around a fixed axis invert the elicity orientation. So you need to use spinors to describe it. Its lagrangian is some sort of square root of that of a scalar field.

3. “Classical physicists studied instead the electron current, which is the vector field… ”

Oh, thank you! That helps a lot. (I should have guessed that J was some second order expression in $\psi$, since densities are like squares of wave functions, but I didn’t.) I also need to go back to work, I’ll write some more this evening if I have time.

4. reager says:

Question 1: Neither. In QFT the path integral and canonical quantization are two complementary approaches. Zee mostly focuses on the path-integral side of the story. Particles satisfying $-k^2 + \omega^2 = -m^2$ (the Fourier transform of the classical equations of motion) are said to be “on mass-shell.” Particles that don’t satisfy this constraint are called “virtual particles” and are off-shell. For a nice illustration that virtual particles have physical consequences see the Breit-Wigner distribution for $e^{+}e^{-} \rightarrow \text{hadrons}$ sharply peaked around the Z-boson mass in Peskin and Schroeder 20.2. I also highly recommend Mark Srednicki’s “Quantum Field Theory.” http://www.physics.ucsb.edu/%7Emark/qft.html

5. A.J. Tolland says:

Reager: I’m not quite sure that’s an answer to a question David was asking…

But it’s worth asking: David, what did you mean by force carrier particle? A virtual particle appearing inside a Feynman diagram? Or a gauge (or Higgs) field? I assumed the latter. In which case, the Hilbert space formalism doesn’t really distinguish between matter fields (like electrons) and gauge fields. The distinction is between particles (which have wave functions that live on spacetime and take values in a Lorenz group representation) and fields (which have wave functions that live on the space of classical fields).

One more thing: The second quantization formalism matteo mentions never actually made very much sense to me. I’ve never understood why we should be quantizing particle wave functions.

The physically sensible formalism goes in more or less the other direction: We start with a quantum theory of fields on Minkowski spacetime. This has a Hilbert space, which is a representation of the Poincare group. We can try to break this space up into irreducible representations; individual irreps correspond to particles, some of which are “fundamental” and some of which are composites of fundamental fields. When we think about particles in the field theory context, what we’re doing is focusing our attention on the way these irreps behave. You can actually develop perturbation theory just by thinking of these irreps and some combinatorial data, but the full Standard Model (in particular the Higgs potential and the anomalous chiral symmetry in QCD) seem to require fields. There’s more to field theory than particles…

6. matteo says:

I was just mentioning second quantization because I think David got confused by the fact that sometimes you treat quantum fields as they were wave-functions (on spacetime) of a particle, and this is not correct from the viewpoint of canonical quantization, and sometimes as they were wavefunctions on the space of classical fields. The path integral formulation cannot generate confusion since you have fields from the beginning.
I admit that the second quantization procedure is kind of wierd if you think of the wavefunction as a wavefunctions, since it’s easier to think about particles as particles, but it makes much more sense if you think about the wave side of the wave-particle duality of QM, then the wavefunctions is a classical field and it is almost natural to quantize it.

7. David Speyer says:

Uggh, I spent a while writing this up last night, and now I see new things to respond to in AJ’s reply. Still, I don’t want to hold this up forever.

———–

Regarding force carrying particles: here is what I was thinking, which may well be wrong. In the first few chapters of Zee, here are the particles which are analyzed from the perspective of wave functions on space of solutions to the field equations : photons, pions, gravitons. Here are the particles which are analyzed in terms of wave functions on spacetime: electrons, neutrinos, protons. Notice that every particle on the first list is associated with a force (electro-magnetism, strong force, gravity) while the particles in the second list are not. I was generalizing from this pattern. (I am being unfair to Zee here, since last night I realized that he does discuss electrons from both perspectives, just in a section which I don’t understand yet.) Is that what you mean the contrast between particles and gauge fields?

I think that the confusion that I am having is indeed related to something one might reasonably call second quantization and to some of the issues that matteo raises. Sticking to just E&M to avoid confusion, it seems that there is a table we can compile

Level 0: Positions of photons?    Positions of Electrons
Level 1: The potential A    The Dirac field psi
Level 2: Functions on the    “Superfunctions” on the
space of possible A’s    space of possible psi’s

(Since I am still confused by any sentence which contains the word “canonical”, I think that it is safe to assume that I am using the path integral perspective here.) Earlier I had thought that J was what belonged on the right hand column at level 1, so thank you for clearing that up. It would be mathematically consistent (although false) to imagine a world which just operated on Level 0 or which just operated on level 1.

The thing that seems so odd is that the Dirac field psi “feels” quantum while A “feels” classical. More precisely (1) the Dirac equation contains complex numbers in a pretty fundamental way, while I don’t think that Maxwell’s equations do (2) I can’t think of any sort of “wave function collapse” phenomenon for A. I have some vague ideas about how to explain this difference, but I’m going to stop and see if someone who actually knows what they are talking about has something to say first.

8. Hi David,

There is a “wave function collapse” phenomenon for the A field as well. In fact it is basically the same as for the psi field. What I’m thinking of here is one of the classic eperiments that tells us that classical physics really fails to describe what’s going on in the real world: The double slit experiment.

Everyone Interested in the real world should know this real world experiment. In it what you do is you set two walls, one with two very thin vertical slits and the other with detectors on it. Then you shoot a “beam of particles” through the slits towards the detectors and you measure the pattern you see. For example this can be a beam of electrons (like the ones used in older, non-LCD TVs) (this would be the psi field) or it can be from light source like a laser or a light buld. You need the beam to be somewhat homogeneous e.g. the electrons should all have about the same energy or the light should be the same frequency/color. Then, assuming that you have a lot of electrons/particles what you see is striped interference pattern, there will be stripes where the detectors detect a lot and stripes where they detect almost none.

If you are trying to model the physics you say Aha! This is acting like a wave! In fact E&M already describes the way these waves move in the case of light, and calculate various things like the spacing of the stripes (note: Here the “wave function” is actually a vector valued function). You can get a similar thing for the electrons, but to get it to match up with the real life physic we see, you need to use spinor valued wave functions. This is why (well one reason) there are those pestky complex numbers; Spin(3) = SU(2) so we use complex numbers to simplify things. As an aside, you could just as well use a quaternions since Spin(3) = Sp(1) as well.

Now for the “wave function collapse” part: let’s turn the intensity of the beam way down. What happens? Well, if you lived a little over a hundred years ago it’d be kinda a shock. What happens is that the detectors (if they’re sensative enough) detect individual particles. This probably matches your intuition for the electrons, but it also works for light. As you dim the light way down, the detectors will detect nothing, then nothing, then pow! a burst of energy concentrated in a single precise location (e.g. on just one of you detectors in the array), then nothing again until the next “particle”.

So this is why we say there is a wave/particle duality thing going on. On the one hand, at high energies we get totally wave like behavoir = interference patterns, but at low energies we get individual particles being detected.

What’s even wierder, is the following. Let’s keep the beams at low intensity so that only one particle is going through the slits at a time, but let’s keep track of where the particles hit on our grid of detectors. After a ling time we will have a bunch of dots on our grid and guess what? They form the same interference pattern we saw with the high intensity beams! This is really wierd ’cause it’s like a single particle is “interfering with itself”.

No one really understands what’s going on or why this happens, but we can model it. You can play with the wave functions and calculate there “amplitude” but at the end of the day you still interpret it as some sort of probability. Or you can try to make sense out of the path-integral mojo and think of the particle as “traveling along all paths at once”.

All of this is at “level 1” of your little table. The thing that goes wrong with this wave function stuff is that it doesn’t accurately model particles interacting. You need ot have paricles that can smash into other particles and which can create new particles. This is the whole reason for “level 2” and second quantization. It gives us a way to deal with particle interactions.

9. A.J. Tolland says:

1) We should probably drop the issue for now about force-carrier particles vs matter particles. It’s not relevant to the question about Hilbert spaces. Although for the record, when I say matter or force carrier, I’m usually thinking about the Standard Model. Matter means spin 1/2 fermions, and force carrier means spin 0 Higgs or spin 1 gauge boson. Gravity not included, but obviously a force carrier. Pions are composite particles, so I’m not including them in my list. (For the record, pions are associated with the residual strong force left over after the extremely strong color force has confined itself. The actual strong force is mediated by gluons.)

2) Your table is basically correct.

Level 0 would be a classical theory describing photons and electrons. We’d need to make up some ad-hoc rules to describe their interactions, but there are physical situations where this is good enough, extremely high energy photons behave very much like particles.

Level 1 is the quantum mechanics of photons and electrons. Again, we need some ad-hoc rules to describe their interactions, but there are situations where you can get away with this. Most of the quantum mechanics pioneers were thinking in these terms.

Level 2 is the quantum mechanics of fields. It’s a bit of a leap to get here from Level 1, and it looks in some ways like the leap from Level 0 to Level 1, so people call it 2nd quantization. As I’ve mentioned, I object to this terminology.

The transition is in some ways clearer when we think about Hilbert spaces and linear operators. At level 1, we have a Hilbert space H_particle and operators on it that tell us about the positions and momenta of particles. At level 2, we have a different Hilbert space, and now our operators measure the strengths of fields at various positions in spacetime.

This is pretty intuitive for the EM field, because you’re used to the idea of measuring EM field strengths. And I agree that it’s confusing to think of Dirac fields classically. There are two slightly entangled reason for this: 1) As I mentioned above, the Dirac fields commutation relations become a nilpotency condition as $\hbar \to 0$. So $\psi$ itself is hard to measure. But you can measure composite operators like the current J. And J does behave like a classical field as long as you’re studying phenomena which are large-scale relative to the mass of electrons. However, and here’s point 2), if you look at J carefully enough, you will notice that it’s somewhat discrete, because those electron currents are made up of huge numbers of electrons. Electron field excitations (i.e. electrons) have a minimum energy $m_\psi \simeq .5 MeV$, and consequently electron fields are more easily seen to be coarse. The EM field on the other hand has massless quanta, so you can have as many quanta as you like; it’s harder to see the “quantum coarseness”.

And as Chris said: If you want to handle interactions, especially the sorts of interactions that change the number particles in the system, without resorting to ad-hoc rules, then you really should think in terms of fields.

10. matteo says:

I would like to rewrite your scheme this way:

Level 1: (particles) photons – electrons
Level 2: (waves) EM field A – electron field \psi
Level 3: (second quant.) operator A – operator \psi
Level 4: (path integral) functional of field A – functional of field \psi

Classical physics takes EM from level 2 and particles from level 1, first quantization returns photons to level 1 and electron waves to level 2 (double slit experiment), “second quantization” interprets A and \psi as operators on some Fock state space, path integral formulation, which is equivalent to second quantization, uses a functional over the space of classical fields (in the sense that they are not operators as in second quantization!). Note that these fields need not be classical solutions to lagrange equations since they not necessarily minimize the action.

The wierd thing is that crossing connection between classical and quantum interpretation of fields and particles.

I also point out that despite being awkward and inelegant, the “second quantization” procedure deserves attention since it consists of imposing (anti)commutation relations on your objects (and therefore you need to interpret them as operators), which is the usual way used in QM to define “quantization”. In path integral formulation you might not really see where the quantization procedure takes place (you can actually detect it by comparison with classical Hamilton-Jacobi equation, not the easiest thing to do and interpret)

11. A.J. Tolland wrote: “The second quantization formalism matteo mentions never actually made very much sense to me. I’ve never understood why we should be quantizing particle wave functions.”

We’re not. We’re quantizing fields. Particle wave functions only emerge as an approximate concept in the nonrelativistic limit.
(That’s why you never hear about wave functions for photons …)

12. A.J. Tolland says:

Mark Srednicki wrote: “We’re not. We’re quantizing fields.”

That’s why I was suggesting that David ignore the second quantization stuff.

Thanks for the correction about the non-existence of photon wave-functions, though. I was trying to cheat there, and probably causing more confusion than I resolved.

So, for David: You can sometimes pretend that a classical EM field is the wavefunction of a photon. The math works out OK if you think of a plane wave as the wave function for a single photon, when you’re trying to calculate the probability of a detector going “bing”, as probabilities and radiation intensities are both quadratic in the field.

But classical fields aren’t really single photon states, and the field A(x,t) isn’t actually a probability amplitude for any single particles. In fact, classical fields correspond to coherent states, which have non-zero inner product with any n-photon state.

Where are we now? Are you unconfused or more confused?

13. mo says:

A.J. Tolland wrote: “The second quantization formalism matteo mentions never actually made very much sense to me. I’ve never understood why we should be quantizing particle wave functions.”

Mark Srednicki responded: We’re not. We’re quantizing fields. Particle wave functions only emerge as an approximate concept in the nonrelativistic limit.
(That’s why you never hear about wave functions for photons …)

mo to A.J. Toland: According to S. Schweber (An Introd. to Relativistic Quantum Field Theory, p. 122), “The importance of [second quantization] derives from the fact that it permits performing calculations which automatically take into account the combinatorial aspects arising from the particular statistics (Bose-Einstein or Fermi-Dirac) obeyed by the particles. Secondly, it allows an extension of ordinary nonrelativistic quantum mechanics to systems [of identical particles] for which the number of particles is no longer a constant of the motion. Such an extension is in fact necessary to describe the physical phenomena encountered in the relativistic domain.”

mo to Mark Srednicki: The second quantization has been extensively used in old books, and what people sometimes quantized were wave functions indeed. See e.g. L. Schiff’s Quantum Mechanics, section 55 “Quantization of the nonrelativistic Schrodinger equation,” or Landau-Lifschitz’s Quantum Mechanics, sections 64 and 65. As you know, in this formalism wave functions psi(x) are replaced by psi-operators psi^(x) obeying the same Schrodinger equation, that’s why it has been called the second quantization.

The reference to photons is not a good example. It is well know that Maxwell equations for fields E(x,t), H(x,t) or potentials A(x,t) can be reinterpreted as quantum-mechanical equations for the same fields or potentials (in the momentum space) but considered now as one-particle photon wave functions. These can then be quantized, etc. For details, refer to Akhiezer and Berestetsky’s Quantum Electrodynamics. These authors do use such concept as photon wave function. Landau and Peierls made it clear that photon wave functions don’t make much sense in the coordinate space, but are meaningful in the momentum space.

For a good discussion, see Iwo Bialynicki-Birula, Photon wave function, quant-ph/0508202. Here is an abstract:

“Photon wave function is a controversial concept. Controversies stem from the fact that photon wave functions can not have all the properties of the Schroedinger wave functions of nonrelativistic wave mechanics. Insistence on those properties that, owing to peculiarities of photon dynamics, cannot be rendered, led some physicists to the extreme opinion that the photon wave function does not exist. I reject such a fundamentalist point of view in favor of a more pragmatic approach. In my view, the photon wave function exists as long as it can be precisely defined and made useful.”

And finally, why “Particle wave functions only emerge as an approximate concept in the nonrelativistic limit”? In many QFT books, one can find such things as exact one-particle, two-particle, etc., wave functions. I found them even in your own QFT book, e.g., “a state of n particles with momenta k1, …, kn” (Problem 3.3, quoting from a free electronic version of the text). To be sure, these are undressed states, but, by all means, go ahead and dress them (by applying an appropriate operator)–and you get exact realistic n-particle wave functions.

14. mo says:

I wrote: I found them even in your own QFT book, e.g., “a state of n particles with momenta k1, …, kn” (Problem 3.3, quoting from a free electronic version of the text). To be sure, these are undressed states, but, by all means, go ahead and dress them (by applying an appropriate operator)–and you get exact realistic n-particle wave functions.

Correction: Please omit “realistic” above. The states obtained by the dressing of bare states of the form a+(k1)…a+(kn)|0> are exact, but not realistic as they are not normalizable. One has to build wave packets from plane-wave states.

15. I was implicitly defining “wave function” as a function psi(x,t) such that |psi(x,t)|^2 dV is the probability to find the particle in a volume dV centered on x at time t, and such that psi(x,t) obeys some sort of local, linear, differential equation. You can of course choose to relax one or more of these criteria.

16. carlbrannen says:

The review article of comment 13 is inadequate in that it gets the question of the existence of the photon position (coordinate) wave function completely wrong. It is true that it was believed for many years that a photon wave function was impossible and this is still generally believed.

The contrary was shown by Margaret Hawton. See Phys. Rev. A 59, 954 (1999) or read the articles available on the web: Hawton on arXiv.

Fortunately, for the vast majority of working physicists, it really doesn’t matter one way or another. It matters to me because on an ontological basis, I would prefer to see QM written in position space rather than momentum space.

It’s been a few years since I thought about this. My memory is that the secret of getting the photon wave function is that it requires that the wave be split into left and right handed chiral waves. I think that this is very important physically, as the same sort of thing appears in the standard model of particle physics. The natural particles seem to be the left and right handed components (paradoxical though that may be for many reasons), and to understand mass we need to understand the interaction between those chiral states.

By the way, great blog. I can’t believe I didn’t find it before now.

17. Scott Carnahan says:

Just a side note: David’s use of levels is similar to the complexity hierarchy described in Tao’s blog, from Borcherds’ lecture. It seems to be one of those remarkably useful conceptual tools that never get mentioned.