## Thursday, 2 April 2015

### Joy Christian for dummies (like me)

Over the last decade, Joy Christian has published many articles on his theory that involves a local realistic solution for the EPR type experiment, that is intended to demonstrate entanglement. This solution heavily relies on geometric algebra (GA), a mathematical framework that is not so commonly known. The reason however for using this is that it seems to handle rotations in a more natural way. Furthermore the framework defines a product that does not commute (two objects A and B for which A x B is unequal to B x A), which Joy uses to describe the strong quantum correlations found in EPR type experiments in a local realistic way.

## Geometric Algebra (GA)

Geometric algebra is said to extend the concept of vector algebra. It starts with the definition of orthogonal base vectors, usually called e1, e2 and e3 (here used in 3 dimensions, similar to calling them x, y and z). So any vector can be defined as ae1+be2+ce3, where a, b and c are real numbers. Such a vector can of course as well be written as (a, b, c), which means a times in the e1 direction, b times in the e2 direction and c times in the e3 direction.

### Bivector

Then GA goes on extending the possible objects. Next in line is a bivector, which can somewhat be seen as a combination of two vectors. Is is an oriented part of a plane, and in the same plane as the vectors. The size or weight of the bivector can be calculated as the surface of the parallelogram spanned by the vectors.

The sequence of the vectors indicate the rotational orientation. Therefore the bivector below is different, although constructed from the same vectors.

However a bivector is not a parallelogram. It can as well be drawn as a circle with the same surface. The rotational orientation is now indicated by the lines sticking out from the circle:

To get the bivector spanned by two vectors the wedge product (^) must be used. So when having two vector a and b, a bivector v can be a^b. When the vectors a and b are written in components as

a1e1 + a2e2+a3e3

and

b1e1 + b2e2+b3e3

then the wedge product is

(a1e1 + a2e2+a3e3)^(b1e1 + b2e2+b3e3)

and can be expressed as fractions of the bivectors between the base vectors:

(a1b1-a2b2)(e1^e2) + (a3b1-a1b3)(e3^e1)+(a2b3-a3b2)(e2^e3)

### Trivector

The bivector is a plane, the trivector a volume. It is made of the wedge product between all three of the base vectors: e1^e2^e3.  Now the size of the volume spanned by the three vectors is called the weight. Like the bivecor, the trivector has also rotational orientation (handedness), which is either clockwise or anti-clockwise.

Again it is of no importance how the trivector is drawn. There is no such thing like for example a trivector multiplied in the e1 direction, because one can only multiply its weight.
The same trivector can thus be drawn as sphere, where lines pointing in or outwards indicate the sense of rotational orientation:

The unit trivector, having a weight of 1, is often called 'I' in GA.

### Multivector and operations

In a way, the objects in GA look like calculating with complex numbers or quaternions. A complex number like a+bi, where a and b are real numbers, cannot further be simplified. However one can add two complex numbers by adding their components: (a+bi) + (c+di)= (a+c)+(b+d)i;
The real number, the vector, the bivector and the trivector can all be seen as special cases of a multivector (now a1 to a8 are real numbers):

a1 + a2e1 + a3e2 + a4e3 + a5e1^e2 + a6e2^e3 + a7e3^e1 + a8e1^e2^e3

Another special multivector is the rotor (which is similar to the quaternion):

a1 + a5e1^e2 + a6e2^e3 + a7e3^e1

Adding or subtracting multivectors is by adding or subtracting its components, just like we did with the imaginary numbers. Also its worth remembering that a normal multiplication of one of the base vectors e1, e2 or e3 with itself results in -1, just as  i-squared is in complex numbers.

But in GA there are several types of multiplication between its objects defined.  Here I give an explanation for two ordinary vectors. The product definitions however are usually extended to the complete algebra (1).

Wedge product (a^b):
(a1b1-a2b2)(e1^e2) + (a3b1-a1b3)(e3^e1)+(a2b3-a3b2)(e2^e3)
Outer or cross product (a X b):
(a2b3-a3b2)(e1) + (a1b3-a3b1)(e2)+(a1b2-a2b1)(e3)
Inner product (a.b):
a1b1 + a2b2 +a3b3
Geometric product (ab):
(a1 * b1 + a2 * b2 + a3 * b3)+
(a1 * b2 - a2 * b1)e1^e2 +
(a2 * b3 - a3 * b2)e2^e3 -
(a1 * b3 - a3 * b1)e3^e1

For multivectors some product definitions can be expressed in terms of other product definitions, like:
a b = a . b + a ^ b (for vectors)

Dependent on the grade of the multivector (scalar is grade 0, vector grade 1, bivector grade 2 etc.) the geometric product may or may not be commutative.In the previous post one can see that the non-commutativity between bivectors is responsable for the effect Joys describes in his paper.

I like to conclude by referring again to the program GAViewer (3), which was used to create all the images in this post.