Quantum Field Theory I

44 CHAPTER 1. INTRODUCTIONS
In this way, one is lead to some specific differential equation for ϕ(x), e.g. to
the equation
−i∂ t ϕ(x) = √ m 2 −∂ i ∂ i ϕ(x)
Because of the square root, however, this is not very convenient equation to
work with. First of all, it is not straightforward to check, if this operator obeys
all the commutation relations of the Poincaré algebra. Second, after the Taylor
expansion of the square root one gets infinite number of derivatives, which
corresponds to a non-local theory (which is usually quite non-trivial to be put
in accordwith specialrelativity). Another uglyfeature ofthe proposedequation
isthatittreatedthetimeandspacederivativesinverydifferentmanner, whichis
atleast strangein a would-berelativistictheory. The awkwardnessofthe square
rootbecomesevenmoreapparentoncethe interactionwithelectromagneticfield
is considered, but we are not going to penetrate in such details here.
For all these reasons it is a common habit to abandon the above equation
and rather to consider the closely related so-called Klein–Gordon equation
(
∂µ ∂ µ +m 2) ϕ(x) = 0
as a kind of a relativistic version of the Schrödinger equation (even if the order
of the Schrödinger and Klein–Gordon equations are different).
Note, however, that the Klein–Gordon equation is only related, but not
equivalent to the equation with the square root. One of the consequences of this
non-equivalence is that the solutions of the Klein-Gordon equation may have
both positive and negative energies. This does not pose an immediate problem,
since the negative energy solutions can be simply ignored, but it becomes really
puzzling, once the electromagnetic interactions are switched on.
Another unpleasant feature is that one cannot interpret |ϕ(x)| 2 as a probability
density, because this quantity is not conserved. For the Schrödinger
equation one was able to derive the continuity equation for the density |ϕ(x)| 2
and the corresponding current, but for the Klein–Gordon equation the quantity
|ϕ(x)| 2 does not obey the continuity equation any more. One can, however,
perform with the Klein–Gordon equation a simple massageanalogousto the one
knownfromthetreatmentoftheSchrödingerequation,togetanothercontinuity
equation with the density ϕ ∗ ∂ 0 ϕ−ϕ∂ 0 ϕ ∗ . But this density has its own drawback
— it can be negative. It cannot play, therefore the role of the probability
density.
All this was well known to the pioneers of the quantum theory and eventually
led to rejection ofwave-functioninterpretation ofϕ(x) in the Klein–Gordon
equation. Strangely enough, the field ϕ(x) equation remained one of the cornerstonesof
the quantum field theory. The reasonis that it was not the function
and the equation which were rejected, but rather only their wave-function interpretation.
The function ϕ(x) satisfying the Klein–Gordon is very important — it becomes
the starting point of what we call the field-focused approach to the quantum
field theory. In this approachthe function ϕ(x) is treated asaclassicalfield
(transforming according to the considered representationof the Poincarégroup)
and starting from it one develops step by step the corresponding quantum theory.
The whole procedure is discussed in quite some detail in the following
chapters.