Well, you have to make sure the bacteria that does the actual fixing is present (this is why you often innoculate bean and pea seeds before planting) because it isn't the legume itself that does the fixing but a symbiotic bacteria.
I expect the experiments required to measure if beans or peas were fixing nitrogen in an aquaponics system would be really challenging to manage on a home scale. Sounds more like a lab type experiment to me.
However, if you were to have a system with low nitrate readings and you were to plant some beans and see that they were doing poorly too then plant some innoculated beans and see that they are doing better in might be an indication that nitrogen fixing works in an AP system.
By the way, the nitrogen fixing benefits of legumes are usually not available to other plants until after the legume crops have been turned under and allowed to compost a bit so I kinda doubt you will see a measurable increase in nitrates in an aquaponics system just because you have beans or peas growing well in it but perhaps after the legume crops is done providing it's beans or peas, you could cut it at gravel level and leave the roots to decompose in the bed to release some of the nitrogen to the system.
I'm not sure but personally I don't think it is applicable in a true AP system. But you can definitely see results in a hybrid wick system.
TC: You have a good head on your shoulders packed full of wonderful knowledge; a true credit to our cause. Keep up the good work! But as far as your suggestion about leaving anything in the system to decompose? I disagree. I believe this will cause eventually result in head, heart and back aches in the long run (unless over wintering).
I guess it's kind of a moot point anyway. If there is ever a shortage of nutrients, you simply add more fish or feed them more, as long as the tank is large enough. Thanks for the input.
I agree that there could be danger to leaving things in a system to decompose. However, when you pull a plant out, do you really get all the roots or do lots of them break off to decompose in the gravel and be eaten/processed by the heteratrophic (sp?) bacteria and worms? I have to admit that I don't go completely digging though my grow beds to try to get all the roots out when I pull a plant, that would be too hard on any plants still living in the bed and I don't think I've every completely pulled everything in a bed all at the same time.
Now while I don't do this myself, I know many people who do dig kitchen scraps into their grow beds to feed the worms and this is essentially leaving things in the system to decompose. This would not be a good idea in a system too heavily loaded with fish but I'm not going to argue with something that seems to work so well for so many people. I just caution that a hearty population of worms and very good aeration is necessary to keep this from causing a problem.
As to leaving stumps of plants in, I should have explained better. I'll often say cut off a tomato or pepper plant and perhaps leave the stump for a few days and then pull it out later. This seems to allow for more root die off and make the stump easier to pull out without disturbing the rest of the bed as much. I guess it does mean that I am leaving much more roots in the system to decompose and be processed by the bacteria and worms but so far I haven't seen problems with this in my system over the years. I am running twice as much gravel bed volume as I have fish tank volume though which means I do have plenty of space for the bacteria and worms to do their work. I've really only seen major clogging issues from the roots of still living and growing plants (beware the banana beast, Mint monster, and I think Lufa may soon join the club.)
Composting worms are best. Here in FL red wigglers can be found in the ground and they are composting worms but in much of the continent you may find earthworms or night crawlers and some of them might be good for the grow beds while others may tend to escape to find habitat they find more comforting to them. One handy way to catch local worms is to wet down a patch of ground and wet down some corrugated cardboard and lay it on the ground and keep it moist for several days, if you have a good population or worms you should be able to flip up the carboard and collect a bunch to put in the worm bin. I know my composting worms think cardboard is one to the best materials on earth. You may find more worms inside the corrugations of the cardboard when you peal it open. You can buy composting worms that people sell for worms bins if your local worms are not cooperative.