Maths - Choosing Bases - 2D

As discussed on the previous page, an algebra based on 2D vectors is the smallest, non-trivial clifford algebra so this may be a good chance to experiment with what we can change.

For instance, we can try changing the following aspects of the algebra:

Do we choose e1^e2 or e2^e1 as the basis bivector?

The bases are the vectors e1 and e2, however the higher level products such as e1^e2 is also as a basis for the bivectors.

The question is what order do we choose for this (which is equivalent to saying what sign do we use since e1^e2=-e2^e1).

The methodology used for choosing the order of these indexes is explained here.

In the case of 2D multivectors there is not much of an issue since the only higher level product is the pseudoscalar e1^e2 or e2^e1 and since its the only one we might as well put the indexes in order as we do for all pseudoscalars so we use e1^e2.

Do the basis vectors square to +ve, -ve or zero?

Lets try out the following combinations:

category   isomorphic to
G 2,0,0 Both vectors square to +ve.  
G+2,0,0 Even subalgebra of G 2,0,0 complex numbers
G 0,2,0 Both vectors square to -ve. H = quaternions
G+ 3,0,0
G 1,1,0 One vector squares to +ve and the other squares to -ve.
 
G 0,0,2 Two vectors square to zero.  
G 1,0,1 One vector square to zero and the other to +ve.  
G 0,1,1 One vector square to zero and the other to -ve. dual complex numbers

The multiplication table for each of these is derived below:

Both vectors square to +ve: G 2,0,0

This corresponds to normal 2 dimensional space. It can be fully defined by the multiplication table as follows:

a*b b.e b.e1 b.e2 b.e12
a.e e e1 e2 e12
a.e1 e1 e e12 e2
a.e2 e2 -e12 e -e1
a.e12 e12 -e2 e1 -e

This table was generated using the program described on this page

As the above link explains, the table was generated by a computer program from the laws of vector algebra, that is: non-equal vector bases anti-commute and equal vector bases square to scalars (+,- or 0 as required). We could have done this manually as follows:

e1 e1 = 1
e2 e2 = 1

because we have chosen to let vectors square to a positive scalar value.

e1 e2 = e12
e2 e1 = -e12

because we need vector multiplication to anticommute so that the vectors will cancel out when a general vector is squared: (a e1 + b e2)2 = a2 + b2.
e1 e12 = e2
e2 e12 = -e2 e21 = -e1
e12 e1 = -e21 e1= -e2
e12 e2 = e1
these are derived from the results above using these rules.
e12 e12 = -e12 e21 = -1  

Even Sub-algebra of G 2,0,0 which gives: G+2,0,0

To take the even sub-algebra we take the even grades: 0=scalars, 2=bivectors and remove the other grades to give:

a*b b.e b.e12
a.e e e12
a.e12 e12 -e

This table was generated using the program described on this page

This is the algebra of complex numbers where:

Although a simpler way to generate complex number algebra is to use G 0,1,0.

Both vectors square to -ve: G 0,2,0

This is similar to normal 2 dimensional space above, in some ways it is arbitrary choice whether vectors square to +ve or -ve. it seems to swap over the signs of the anticommuting terms of e12 so perhaps it defines left or right handedness. It can be fully defined by the multiplication table as follows:

a*b b.e b.e1 b.e2 b.e12
a.e e e1 e2 e12
a.e1 e1 -e e12 -e2
a.e2 e2 -e12 -e e1
a.e12 e12 e2 -e1 -e

This table was generated using the program described on this page

By compareing this to we quaternion table:

a*b
b.1 b.i b.j b.k
a.1 1 i j k
a.i i -1 k -j
a.j j -k -1 i
a.k k j -i -1

we can see that G 0,20, is isomorphic to H.

As the above link explains, the table was generated by a computer program from the laws of vector algebra, that is: non-equal vector bases anti-commute and equal vector bases square to scalars (+,- or 0 as required). We could have done this manually as follows:

e1 e1 = -1
e2 e2 = -1

because we have chosen to let vectors square to a negative scalar value.

e1 e2 = e12
e2 e1 = -e12

as in above case except (a e1 + b e2)2 = -a2 - b2.
e1 e12 = -e2
e2 e12 = -e2 e21 = e1
e12 e1 = -e21 e1= e2
e12 e2 = -e1
these are derived from the results above using these rules.
e12 e12 = -e12 e21 = -1  

One vector squares to +ve and the other squares to -ve:
G 1,1,0

This is like one dimension of time and one dimension of space, I'm not sure if this is consistent, please let me know of any issues:

a*b b.e b.e1 b.e2 b.e12
a.e e e1 e2 e12
a.e1 e1 -e e12 -e2
a.e2 e2 -e12 e -e1
a.e12 e12 e2 e1 e

As the above link explains, the table was generated by a computer program from the laws of vector algebra, that is: non-equal vector bases anti-commute and equal vector bases square to scalars (+,- or 0 as required). We could have done this manually as follows:

e1 e1 = -1
e2 e2 = 1

because we have chosen to let vectors square to these values.

e1 e2 = e12
e2 e1 = -e12

because we need vector multiplication to anticommute so that the vectors will cancel out when a general vector is squared: (a e1 + b e2)2 = a2 - b2.
e1 e12 = e2
e2 e12 = -e2 e21 = -e1
e12 e1 = -e21 e1= e2
e12 e2 = e1
these are derived from the results above using these rules.
e12 e12 = -e12 e21 = 1  

Note: this changes the sign of the square of the bivector.

Two vectors square to zero: G 0,0,2

Any term with two or more copies of e1 or e2 is zero:

a*b
b.e b.e1 b.e2 b.e12
a.e 1 e1 e2 e12
a.e1 e1 0 e12 0
a.e2 e2 -e12 0 0
a.e12 e12 0 0 0

As the above link explains, the table was generated by a computer program from the laws of vector algebra, that is: non-equal vector bases anti-commute and equal vector bases square to scalars (+,- or 0 as required). We could have done this manually as follows:

e1 e1 = 0
e2 e2 = 0

because we have chosen to let vectors square to these values.

e1 e2 = e12
e2 e1 = -e12

as before lets make vector multiplication anticommute so that the vectors will cancel out when a general vector is squared: (a e1 + b e2)2 = 0.
e1 e12 = 0
e2 e12 = -e2 e21 = 0
e12 e1 = -e21 e1= 0
e12 e2 =0
these are derived from the results above using these rules. Any term with two or more e2 terms is zero.
e12 e12 = -e12 e21 = 0  

One vector square to zero and the other to +ve: G 1,0,1

Lets say e2 squares to zero :

a*b
b.e b.e1 b.e2 b.e12
a.e 1 e1 e2 e12
a.e1 e1 1 e12 e2
a.e2 e2 -e12 0 0
a.e12 e12 -e2 0 0

As the above link explains, the table was generated by a computer program from the laws of vector algebra, that is: non-equal vector bases anti-commute and equal vector bases square to scalars (+,- or 0 as required). We could have done this manually as follows:

e1 e1 = 1
e2 e2 = 0

because we have chosen to let vectors square to these values.

e1 e2 = e12
e2 e1 = -e12

as before lets make vector multiplication anticommute so that the vectors will cancel out when a general vector is squared: (a e1 + b e2)2 = a2.
e1 e12 = e2
e2 e12 = -e2 e21 = 0
e12 e1 = -e21 e1= -e2
e12 e2 =0
these are derived from the results above using these rules. Any term with two or more e2 terms is zero.
e12 e12 = -e12 e21 = 0  

One vector square to zero and the other to -ve: G 1,1,0

Lets say e2 squares to zero :

a*b
b.e b.e1 b.e2 b.e12
a.e 1 e1 e2 e12
a.e1 e1 -1 e12 -e2
a.e2 e2 -e12 0 0
a.e12 e12 e2 0 0

As the above link explains, the table was generated by a computer program from the laws of vector algebra, that is: non-equal vector bases anti-commute and equal vector bases square to scalars (+,- or 0 as required). We could have done this manually as follows:

e1 e1 = 1
e2 e2 = 0

because we have chosen to let vectors square to these values.

e1 e2 = e12
e2 e1 = -e12

as before lets make vector multiplication anticommute so that the vectors will cancel out when a general vector is squared: (a e1 + b e2)2 = -a2.
e1 e12 = -e2
e2 e12 = -e2 e21 = 0
e12 e1 = -e21 e1= e2
e12 e2 =0
these are derived from the results above using these rules. Any term with two or more e2 terms is zero.
e12 e12 = -e12 e21 = 0  

This is equivalent to dual complex numbers as described on this page.

where the following are equivalent:

G 1,1,0
dual complex
e 1
e1 i
a.e2 ε
a.e12

Nullvectors non-Orthogonal Basis

If all our basis vectors square to +ve and anticommute, then we can choose any mutually perpendicular right-handed basis vectors and the algebra will be the same. Simarly if all the vectors square to -ve, but what if some of the vectors square to +ve and others square to -ve as in G 1,1,0 ? Somewhere between where they square to +ve and -ve they must square to zero: We can generate a Geometric algebra from these null vectors, e0 and eY, as follows:

a*b
b.e b.e0 b.eY b.eY0
a.e 1 e0 eY eY0
a.e0 e0 0 -1+ eY0 e0
a.eY eY -1- eY0 0 -eY
a.eY0 eY0 -e0 eY 1

Can anyone tell me how to derive this?

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