Confusion about How Dirac discovered Dirac's Equation

[petergreat]

When I learned about Dirac's Equation, textbooks usually say that the earlier Klein-Gordon equation isn't linear in time derivative, contrary to what we expect from the time-dependent Schrödinger equation, therefore Dirac had to come up with a version that's linear. However, I think this doesn't really make sense. Klein-Gordon equation is perfectly acceptable if electrons were bosons. The only justification for Dirac's equation is the fermionic nature of electrons.

The "linear time derivative" argument just seems to be some irrelevant out-dated intuition from the structure of non-relativistic QM, and its only benefit is that we preserve the form of Schrödinger's equation and we can still talk about "Hamiltonian" and "energy levels" in a rather similar manner, e.g. in atomic physics, without going into quantum field theory. In contrast, for the Klein-Gordon equation you must treat it as a quantum field to recover the concept of Hamiltonian (now a field Hamiltonian) and a time-dependent Schrödinger equation which by definition is linear in time derivative. In short, the "linear time derivative" property just makes semi-classical treatment easy, and doesn't really have physical content.

Does what I say make sense? Or am I confused about something?

 

[petergreat]

I'll summary my point again. Dirac's equation, with a suitable choice of representation for the gamma matrices, happens to look like a time-dependent Schrödinger equation, while the Klein-Gordon equation lacks this property. However in view of QFT, this fact is not meaningful, because both of the are just classical field equations.

 

[GeorgeRaetz]

...Dirac had to come up with a version that's linear. However, I think this doesn't really make sense. Klein-Gordon equation is perfectly acceptable if electrons were bosons. ...?

The "linear time derivative" (meaning the differential equation is first order in time) reflects a belief in strict deterministic time evolution of the wave function. The state of the system at one instant of time depends entirely on the state at the previous instant.
A system that evolves according to a DE that is second order in time needs state information at two previous instants(or state plus derivative) implying the state needs some sort of 'memory' to evolve in time.
A 2nd order system also generates those awkward advanced solutions.
At least that's the way I understand it.
I quote from section 27 of "Princples of Quantum Mechanics": Dirac postulates: "...the state at one time determines the state at another time.." .