short description of Dirac trick

diractrick.hs

-- The situation Dirac faced was (and we are still) faced with is this:

psi  = c_lower ((> 0) energy) + c_lower ((< 0) energy)
psiT = c_raise ((> 0) energy) + c_raise ((< 0) energy)

c_lower = undefined -- a fermion annihilation operator
c_raise = undefined -- a fermion creation operator
                    --
                    -- and I have deliberately avoided using pluses and minuses
                    -- except where we we are actually summing over all possible states

--
-- The trick that Dirac profited from was the fact that:
--
--    The removal: c_lower ((< 0) energy)
-- 
--        of a particle with negative energy is exactly the same as:
--
--    The creation: c_raise (anti $ (> 0) energy)
--
--        of a (anti-) particle with positive energy



-- Ultimately we transform this:
psi  = c_lower ((> 0) energy) + c_lower ((< 0) energy)
psiT = c_raise ((> 0) energy) + c_raise ((< 0) energy)

-- By a relabeling to this: 

psi  = c_lower ((> 0) energy) + c_raise (anti $ (> 0) energy)
psiT = c_raise ((> 0) energy) + c_lower (anti $ (> 0) energy)

-- Which is the same as:

psi  = c_lower ((> 0) energy) + c_antip_raise ((> 0) energy)
psiT = c_raise ((> 0) energy) + c_antip_lower ((> 0) energy)