Is it possible that the Dirac factorization of the Klein-Gordon operator just by chance adds the necessary degree of freedom that can be used to describe the spin 1/2 of the electron?
Does the Dirac derivation of his equation predict the spin 1/2?
The Klein-Gordon equation is Lorentz covariant, and Dirac derives his own equation in a way which makes it Lorentz covariant, too.
But what properties of the electron does the Dirac equation actually predict?
The eigenfunctions of the equation can be defined as a standard plane wave times a 4-component Dirac spinor.
We then note that if we define an operator S in a way similar to the Pauli equation, S obeys rules which emulate the simple commutation rules we expect from a particle of spin 1/2. That is, if we know the spin projection in the z direction, we know nothing of the projection in the x direction, and so on.
Did the Dirac equation predict the spin 1/2? One might say: no - we just picked an arbitrary operator S which emulates the known rules of spin 1/2. But the Dirac equation made it possible to use a very simple operator S. At least in that sense, the Dirac equation predicted the spin 1/2.
The Dirac hamiltonian H commutes with L + S, where L is the usual orbital angular momentum operator. That fact suggests that S really describes some kind of angular momentum.
But H does not commute with S if the electron is relativistic. Does that make sense? If we have a free electron, why would its spin change during its flight?
Does the Dirac equation predict the gyromagnetic ratio 2?
The Pauli equation is the non-relativistic limit of the Dirac equation.
In the general form of the Pauli equation, the interaction with a magnetic field B is hidden in the minimal coupling
(σ • (p - qA))^2
kinetic term.
But in the standard form, the interaction is shown explicitly as
σ • B.
The Dirac equation does predict the correct interaction strength and the gyromagnetic ratio 2.
Is it possible that the Dirac equation by chance gets the ratio 2 right, even if the equation does not "really" describe the physical system? That looks unlikely.
The Dirac factorization of the Klein-Gordon operator is a general way to add a spin degree of freedom?
The Klein-Gordon equation with the minimal coupling describes the behavior of a spinless electrically charged massive point particle.
The Dirac factorization trick of the Klein-Gordon operator adds a spin degree of freedom, and furthermore, the minimal coupling gives the right interaction strength with a magnetic field for the spin, too. The minimal coupling term was designed to describe the behavior of a spinless point particle, but it magically produces the right interaction also for the spin angular momentum.
Is it a general rule that factorization of an operator adds a spin-like degree of freedom? If yes, why?
Does the factorization always give a sensible strength for the interaction of the spin with an external field?
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