Carrying capacity

"Carrying capacity, at its most basic level, is about organisms and food supply, where "X" amount of humans need "Y" amount of food to survive. If the humans neither gain or lose weight in the long run, the calculation is fairly accurate." - Wikipedia

Because I adore this kind of calculation, here is the theoretical maximum capacity of Earth, if everybody became vegan:

(1.74000 × (10^17) W * 0.05) / (2000 (kilocalories per day)) = 8.98279159 × 10^13

First number is the total power from sunlight. 5% is the efficiency of photosynthesis for growing plants. 2000 Calories a day is how much energy each person eats. Well over ten trillion people.

If we discount any genetic engineered plants that can grow in the oceans (which would require massive ecosystem changes) and only count arable land (41.4 million square kilometers):

(((((1.36600 kilowatts) per square meter) * (41.4 million square kilometers)) /
2) * 0.05) / (2000 kilocalories per day) = 1.45976558 × 10^13

The first number is the solar constant (amount of light per sq meter). Divide arable land mass by 2 to account for day/night cycle. Rest of calculation is the same. Still about 10 trillion people.

I've totally ignored factors like the supply of fresh water that's required to grow crops. The water factor places a hard limit on the amount of plants that can grow, unless:
(a) Genetically modify plants to use salt water.
(b) Cheaper desalinization.
(c) Create more rainfall.
There's significant research into both approaches.