Quaternion Quantum Mechanics

Notation Summary

Start with the Pauli Quaternions:

(1)

This time, instead of the basis s₀,s₁,s₂,s₃ use I,s₁,s₂,s_{3 .} Define:

Also define the surface bivector units:

(3)

It is possible to avoid the imaginary unit i in all equations including the Dirac equation with the help of these bivector units. The imaginary unit is mathematically convenient however, and most people are used to seeing it. For that reason, what I’ve done is to group i with the appropriate line vector. When you see i ez for example, mentally replace it with

Let be quaternions (the coefficients may be complex) then define the vector conjugate

(4)

and Hermitian conjugate

(5)

In matrix form we can write:

(6)

Scalar Products

There is a four-vector dot product and a space-time scalar product:

(7)

Resolve a Quaternion into Components

Given q=q₀I+q_xe_x+q_ye_y+q_ze_z to find q₀,q_x,q_y,q_z use:

(8)

Quaternion Exponential and Related Functions

Magnitude of a quaternion, angle of a quaternion and two types of ‘unitizing’ operators U and U₂

(9)

In U₂ , carg is the argument of a complex number.

Note that the magnitude of a quaternion is not necessarily real or positive. The first unitizing operator U converts all quaternion to a quaternion with magnitude equal to positive one. U2 preserves the sign (and complex argument) of the original magnitude.

The quaternion exponential :

(10)

If the quaternion is written as a 2x2 matrix then the following equivalent form is computationally useful:

(11)

Also useful are the logarithm and the square root:

(12)

(13)

Quaternions and Space-Time Geometry

A location r in space-time is represented by a quaternion with real coefficients:

(14)

r is Hermitian r^†=r. (because the coefficients x,y,z,t are real) r is said to be time-like, light-like or space-like depending on whether |r|²=det(r) is >0 ,=0 or <0.

Consider a Quaternion L of unit norm: , Then a Lorentz transform of an Hermitian quaternion r is

(15)

Eq (15) represents only homogenous Lorentz transforms that are proper(preserves spatial parity or ‘handness’), and orthochronous(preserves time direction)). It’s obvious that multiplying L by a complex number of the form

won’t change the Lorentz transform at all. It is most convenient to choose the parameters so that |L|=1 and the scalar part of L is pure real. Or det(L)=+1 and Im(Trace(L))=0.

If L happens to be Hermitian: then L effects a pure velocity transform If then we have a pure 3D rotation (see below) .

The Lorentz transform preserves the space-time norm of any quaternion:

(16)

3d rotation

To rotate thru the angle about a unit axis

(note that , alternatively one could use the axis that would remove the factor of i in the rotation formula and introduce a factor of i in the Lorentz formula but that would spoil the geometric interpretation. Note that

i a is the unit bivector defining the plane of rotation)

use the quaternion:

Then the rotation is

(Right hand rule: point thumb along a curl fingers in direction of positive)

Lorentz Transform

To effect a Lorentz transform of speed v along the unit axis:

(That is: transform to a frame where an object that was at rest, in the initial reference frame now appears to move at velocity v in the new reference frame)

First, let , then use the quaternion

The Lorentz transform of q is

For the space-time event , the coordinates will transform to

(For a velocity in the opposite direction (-v/c) replace a with –a in the expression for L)

Find the Lorentz transform that ‘rotates’ space-time event A into B. That is for unit Hermitian quaternions A and B find L such that:

Then

(Note that A and B must be Hermitian and time-like)

Canonical form Lorentz Transform

As mentioned det(L)= +1 and the scalar part of L is pure real. With these restrictions, the most general form of L is

(17)

where a_L and a_R are real (Hermitian) unit three-vector quaternions of the form

and are real numbers. These six independent parameters together with det(L)=+1 and Im (Tr(L))=0 determine the eight components of L, Use the quaternion logarithm function (12) to find these parameters. The usual rule for the product of exponentials generally does not work for quaternion exponentiation, however through Singular Value Decomposition L can be factored into:

(18)

That is any Lorentz transform can be resolved into a rotation followed by a boost along the z axis followed by another rotation. Formulae for the parameters in (18) are exceedingly complicated and won’t be written here. They can be obtained from any good symbolic algebra program—just apply SVD to a general complex 2x2 matrix.

So to get a product of Lorentz quaternions L₁ L₂..L_n into the form (17), use the logarithm function. To get the canonical form

(18) use singular value decomposition.

Quaternion Gradients

(19)

Meaning of Quaternion Gradient

Let S(t,x,y,z) be a scalar valued function of the coordinates that is, some field takes on the value S(t,x,y,z) at the space time point . The gradient operator connects the values at neighboring points. Let be the usual small displacement from r then in terms of the ordinary gradient we write:

(20)

To get the same expression in terms of the quaternion gradients of (19) we use the scalar dot product (7a)

(21)

(The second line is a notation convention)

Next, consider an Hermitian quaternion-valued field Q(r):

(22)

Let a be the three vector with real components (a_x,a_y,a_z),The quaternion gradient combines the gradient, divergence, curl :

(23)

For a anti-hermitian quaternion

(24)

(25)

The quaternion gradient can be defined as a volume integral:

(26)

Here, dx,dy,dz,dt are quaternions understood to be dx e_xwith dx a scalar and so forth.

Quaternion Dirac Equation

Reference: J Scott: Complex four-vectors and the Dirac equation

A very good paper. Some of his terminology is a little different from mine. For example when his term quaternion corresponds to Hermitian quaternion.

In the original form as described in the Principles of Quantum Mechanics, Dirac defined the following 4x4 matrices, Portray them here in block-partitioned format.

(27)

The four-vector wave function evolves according to:

(28)

We can also write this as:

(29)

Solutions

Let

(30)

Define:

(31)

Then

(32)

Quaternion form

Define the following quaternion:

(33)

Then either (28) or (29) is equivalent to:

(34)

Direct Form

To solve (34), apply the Lorentz transform to the particle at rest solution:

(35)

(E,p,w are defined in(30))

(This normalization makes ).

(This normalization makes the probability density r=1. see below).

There is also a time reversed negative energy solution to (34).

(36)

Also note that
(37)

Through simple algebra, we get the following variations on (34).

(38)

Furthermore, let

(39)

Eq (34), (38b) and definitions (39) can be combined into:

(40)

The advantage of the last form is that it avoids the conjugate of the wave function. It doubles the number of equations and might be explored as a generalization of the Dirac equation. Eq (39) and 35 (35) are special solutions. These equations were first discovered by Lanczos and Proca in the 1930’s, later rediscovered by others.

Another advantage of (40) is that we can most easily relate it to the QRW process.