I guess your spacing didn’t come out right; feel free to try again.

Euclidean versus Wick space depends on a factor of “i” and therefore a rotation of 90 degrees through a complex plane. I really hadn’t thought about it in those terms, so let me tell you what I do have in mind and maybe you can make some sense of it.

Insofar as Figure 1, I think of the baryon as a “finite state machine.” (In fact, just this past week, I watched some of Feynman’s QED old lectures, and if you look at him go through Feynman diagrams, he also takes the approach that nature is in one state, then a photon is exchanged, then it is in the next state, then another photon exchange to the next state, etc. And that everying we observe macroscopically (and classically) is built up out of trillions of these individual, finite state, photon exchanges.) So, you can think of Figure 1 as iterating perhaps 10^20 times per second or more from subscripted states 1 to 2 to 3 and then, via the dashed lines, back to 1. The temporal component is in the iteration over and over. Three quarks. Then gluon exchanges. Back to three quarks. Then gluon exchanges. And so on. The Dashed lines are meant to schematically indicate that state 3 is really just recycled to state 1 once again. That is, 1 –> 2 –> 3/1 –>2 –> 3/1, etc. THE DASHED LINES CAN BE REMOVED IF YOU WISH.

Another way to think about Figure 1, which lays the foundation for Figure 3, is to take away all dashed lines, since they are meant to be iterative anyway. From Figure 2, right side, you can draw a meson as either two quark lines going in “opposite” directions (upper right), or as one quark and one antiquark line going in the same direction (lower right). In either case, in these figures, time is an independent line flowing from right to left.

So, if you take the dashed lines out of Figure 1, you see that the only thing that flows “out,” or “in,” are quarks and antiquarks in pairs. Equivalently and alternatively, for every quark that flows in, another quark flows out. Think about it: that is exactly what we know about baryons — their only way of INTERacting strongly (versus INTRAacting strongly), is by emitting quark-anti-quark pairs, or equivalently and alternatively, absorbing one quark for every emitted one quark. The baryon can emit a meson, and then reabsorb it immediately. Or, the meson can vanish entirely with a quark/antiquark anihillation — these are short-lived, after all. When the dashed lines are retained in Figure 1, that is showing the INTRAactions of the baryon.

Now go to Figure 3. Remove, it you wish, all four of the dashed “recycling” lines, and turn the lines connecting the two baryons from dashed to solid. Now, we see a quark sent from the left baryon “simultaneously” with a quark being sent from the right baryon to the left baryon. Equivalently and alternatively, we see a meson emitted from the left baryon and absorbed by the right baryon. Or, if you move the time arrow the other way, there is a meson emitted from the right baryon and being absorbed by the left baryon. This is the most interesting of phenomenon, because this is the nuclear force in action.

This final thing to keep in mind regarding the flow of time, which is perplexing you, is this: When I put arrows on the diagrams, it is so as to think of this in terms of s,t,u channel processes, bringing in all the rules and techniques for doing scattering calculations in this context. Thus, the diagram for each baryon is really a closed, three-node Mandelstam diagram, and one can use that formalism in place of having to think explicitly about time. Thus, in each baryon, with an s or a t or a u possible at each of three nodes, one gets 3^3=27 possible combinations, though some are redundant. On top of this, one can play with the vertices by doing various “crossings,” because the quarks and gluons can take various INTRAbaryon paths. For example, look at all the various crossing diagrams I have posted on my web site http://home.nycap.rr.com/jry/FermionMass.htm , about 3/4 of the way down the page. I believe this is an exhaustive first order group of diagrams. Once the gluons start to beget more gluons, or quark loops, then you are into second order and higher.

I hope that fills in a bit more.

Best as always,

Jay.

There are a bunch of resonances going on between this and the stuff I’m working on. Of course I think that these sorts of diagrams are also useful for preons where the quarks and leptons are composed of three preons each. And in addition, the use of matrices here somehow reminds me of Margaret Hawton’s photon density wave function, but I can’t put my finger on it. In her case, it was spin-1/2 case of SU(2) that was simple (in that angular momentum commutes with density, which allows density to be treated as a scalar), while the spin-1 and larger cases of SU(2) for massless particles end up without the ability to define a complete(!) set of angular momentum operators (as spin-1 massless has 3 degrees of freedom, but only 2 are populated, and which two depends on orientation). But in your case the simplicity of SU(2) is trumped by the complexity of SU(N>2).

The other thing I was wondering about were the Feynman diagrams. As you know, I think bound states of quarks (as well as preons) should be represented by things that look very similar to what you’ve got here. However, I always want to keep one end of my Feynman diagrams in the past and the other end in the future, so that the propation is from old to new.

In your Figure 1 Feynman diagrams, it seems that this sort of thing could only be done by breaking the dashed lines. Are you working in Euclidean or Wick-rotated spacetime so that time is no longer the usual? Then I guess I understand these, at least superficially.

Basically, when you un-Wick rotate Figure 1, you end up with three fermions propagating from the past into the future, with each exchanging gauge bosons by pairs. After cutting the dashed lines, you can choose either end to be the initial or final state, and you get various permutations of the order in which the gauge bosons are created or annihilated.

Finally, Figure 3 kind of bothers me. If you are using Wick rotated Feynman diagrams, what happens when I un rotate it? I don’t know what to do when a fermion loop has two dashed lines in it. I can only cut one of them I think. But to get two baryons, I really must make 6 cuts, so that means I have to cut that one fermion loop twice. Then it seems like what is going on is some sort of vertex exchange…

I guess I’d prefer to see a meson exchange between two baryons that leaves the structure of the two baryons almost unchanged. So the baryon would interact with itself almost entirely, except for every now and then having a much rarer interaction with a fermion loop that is the meson. I.e., putting the bound quark on the left and the fermion loop / meson on the right, (I’ll add periods to prevent WordPress from killing spaces):

+.+~+………………..
+.+.+………………..
+~~~+………………..
+.+.+………………..
+.+~+………………..
+~+.+…..++++++>+++…..
+.+.+~~~~+……….+~~~~
+.+.+…..++++++>+++…..
+.+~+………………..
+~+.+………………..

Carl