This was posted to JayZero in my journal, for all you techy folks.
Yeah, that's right - lumens are candela-steradians which is a measure of light times a unit of solid angle. If you double the number of lights you double the number of candelas, but you're essentially using the same (approximate) solid angle, so you have to account for area (which is a square unit) and so you have to take the square root. Lumens can be a poor measure for plants anyway, depending on the spectrum of the lamp. Lumens are really a measure of how bright something looks to the human eye and not how bright something looks to chlorophyll. If your sources all have a similar specral distribution then then it's a decent comparison, but 10,000 lumens of CFL doesn't necessarily equal 10,000 lumens of MH or HPS, for example.
^ if you look here, you see the absorption spectrum of chlorophyll A and B. From 400nm to 500nm is "blue" light, from 500-570 is "green", 580-600 is "yellow" , and 600-700 is "red" and beyond that is infrared.
During veg it's the blue peak that is most important and during flowering it's the red peak on the right. You see that there is almost no absorption by the leaf in the middle, which is green (and which is why plants look green!).
Lumens, however, measure light as seen by the human eye, which sees most dominantly in the green portion of the spectrum. Thus it's possible to have a very green light with a high luminous intensity but which doesn't have anything that a plant could ever use. It could almost use a new unit of measurement like plant-lumens, or even red-plant-lumens and blue-plant lumens - a number that tells you how bright a light looks to a plant, not to a human.
This is why you see some of these new LED grows which don't seem to have very bright sources, but which work really well because you can get LEDs tuned exactly to what chlorophyll wants - that's to say that the LED lamp spectrum looks much more like the chlorophyll absorption spectrum. It's all enough to twist your head in knots...