Friday, April 3, 2015

Quantionic Quantum Mechanics and Dirac's Theory of the Electron

Now we will present the relationship between quantions and spinors. They are basically two methods of taking the square root of the d'Alembertian: while spinors work in any dimensions, quantions are related to Hodge decomposition and this works only in 4 dimensions because of the interplay between 2-forms and their dual.

But let's start with the beginning. Nikola Zovko from the Ruder Boskovic institute in Croatia was following Emile Grgin's work very closely and he wanted to relate this work with known physics. To this aim he proposed an interpretation for quantionic quantum mechanics, an interpretation which generalizes Born rule: instead of probabilities the inner product will produce a 4-vector probability density current. Grgin calls this "the Zovko interpretation" and everything follows from it. In the regular complex number quantum mechanics, the wavefunction of say the electron in the hydrogen atom attaches to each point in space a complex number. Now in quantionic quantum mechanics each point in space time has attached a quantion and we know from last time that \(q^{\dagger} q\) (the "algebraic norm") is a future-oriented 4-vector. Summing over all complex number or quantion algebraic norms over the entire space yield either a positive scalar or a future oriented 4-vector and this is the Born rule. For quantions if  \(q^{\dagger} q = j\) is a 4-vector current then we must have an equation of continuity:

\({\partial}_{\mu} j^{\mu} = {\partial}_{\mu} (q^{\dagger} q)^{\mu} = 0\)

So now suppose we have a "quantionic field": \(q(x) = (q_1 (x), q_2 (x), q_3 (x), q_4 (x))\) with x the usual 4-vector in relativity. Then the continuity equation can be written as:

\({\partial}_{\mu} j^{\mu} = \frac{1}{2} [q^{\dagger} D(q) + {D(q)}^{\dagger} q ]= 0\)


\( D = \left( \begin{array}{cc} \partial_0 + \partial_3 & \partial_1 + i \partial_2 \\ \partial_1 - i \partial_2 &  \partial_0 - \partial_3 \end{array}\right)\)

and so the real part of \( q^{\dagger} D q\) must vanish. If we split \(D q\) into:

\(D q = i H q + i A q\)

with H hermitian and A anti-hermitian matrices and we interpret H as outside potential, for a free particle we have: D q = -iAq and "A" can be expressed as:

\(A = m e^{i\psi} [cos \theta \gamma^1 + sin \theta cos \phi \gamma^3 + i sin \theta sin \phi \gamma^0 \gamma^5]\)

 This is more generic than the usual Dirac's equation because quantionic quantum mechanics describe a SU(2)xU(1) gauge theory. If we restrict however to the case of \(A = m \gamma^1\) we recover completely Dirac's theory. In this case there is a one-to-one correspondence between the 4 quantionic components \(q\)s and Dirac's spinors \(\Psi\)s:

\( q = \left( \begin{array}{c} q_1 \\ q_2 \\ q_3 \\ q_4  \end{array}\right) = \sqrt{2} \left( \begin{array}{c} -\Psi_2 \\ {\Psi}_3^{*} \\ \Psi_1 \\{\Psi}_4^{*} \end{array}\right)\)

\( \Psi = \left( \begin{array}{c} \Psi_1 \\ \Psi_2 \\ \Psi_3 \\ \Psi_4  \end{array}\right) = \frac{1}{\sqrt{2}} \left( \begin{array}{c} q_3 \\ -q_1 \\ q_2^* \\q_4^* \end{array}\right)\)

and the quantionic current is Dirac's current:

\(j^{\mu} = {(q^{\dagger} q)}^{\mu} = \Psi^{\dagger} \gamma^0 \gamma^{\mu} \Psi\)

But how come nobody else noticed an SU(2)xU(1) gauge theory before? Actually... this was discovered independently by David Hestenes. 

David Hestenes

He calls it: the spacetime algebra. Quantionic algebra is nothing but the spacetime algebra. Next time we'll talk about the physics of quantionic quantum mechanics and see to what degree it can represent nature.

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