The Lorentz Transform in modified complex algebra

 Author Carl Borrell e-mail carlborrell@btinternet.com

The Lorentz transform in one spatial dimension:

w' = w.cosh(a) - x.sinh(a)

x' = x.cosh(a) - w.sinh(a)

e = w + ix

e' = w' + ix'

e' = L(e) = [w.cosh(a) - x.sinh(a)] + i[x.cosh(a) - w.sinh(a)]

[Where tanh(a) = V/c; sinh(a) = 1/(1 - V2/c2)1/2 ; w = ct; c = speed of light ; V = relative velocity]

cannot be expressed as a function of a complex variable because the Cauchy-Reimann conditions are not satisfied. However if we change the product rule for complex multiplication:

a = a0 + ia1

b = b0 + ib1

a.b = (a0.b0 - a1b1) + i(a0b1 - b0a1)

We can still regard i as (-1)1/2 because,

(0 + i).(0 + i) = (0.0 - i.i) + i(0.i - i.0) = -1

The algebra is non-commutative:

a.b = (b.a)*

The square of any complex variable is always a purely real number in this algebra:

a2 = (a0a0 - a1a1) + i(a0a1 - a0a1) = a02 - a12

So the square is defined unambiguously even though the algebra is non-commutative.

This leads to a definition of the reciprocal function as:

1/a = a/a2 = (a0 + ia1) / (a02 - a12)

The Lorentz transform can be expressed as multiplication by a complex constant:

L = cosh(a) + isinh(a)

e' = L.e = [w.cosh(a) - x.sinh(a)] + i[x.cosh(a) - w.sinh(a)]

This suggests the following proof of Lorentz invariance:

e'2 = (L.e)2

= L2.e2

w'2 - x'2 = (cosh2(a) - sinh2(a)).(w2 - x2)

= w2 - x2 = e2

[Because cosh2(a) - sinh2(a) = 1 for all a].

This shows the invariance of the interval w2 - x2 under this transform. In fact all purely real numbers have an infinite number of square roots corresponding to the space-time co-ordinates of the same event viewed from each of the infinite number of inertial frames in standard configuration.

In terms of the velocity vector v = (c + iV) the Lorentz factor can be expressed as follows:

L = 1/(1 - V2/c2)1/2 + i(V/c(1 - V2/c2)1/2)

= c/(c2 - V2)1/2 + i(V/(c2 - V2)1/2)

= (c + iV)/(c2 - V2)1/2

= v/(v2)1/2

Now v2 is purely real therefore (v2)1/2 has an infinite number of solutions. In each case L is analogous to unity and L2 = 1.

Consequences for the algebra

Some other consequences for the algebra are as follows:

Theorem: Zero factorisation:

0 = (a + ia).(b + ib)

In terms of special relativity where ct = x the interval corresponds to two events joined by a light pulse.

Theorem: All even powers are purely real:

a = w + ix

a2 = (w2 - x2)

a2n = (w2 - x2)n

Theorem: Two definitions for odd powers:

a = w + ix

a(2n + 1) = (w2 - x2)n . a or

a(2n + 1) = a.(w2 - x2)n

These two possibilities are complex conjugates of each other, the time component is the same but the space component is reversed. Perhaps this is related to chirality?

Binomial

(a + b)2 = a2 + a.b + b.a + b2

We cannot collect the cross products because the algebra is non-commutative, therefore

(a + b)2 = a2 + a.b + (a.b)* + b2

= a2 + 2Re(a.b) + b2

So (a + b)2 is purely real as required.

Derivatives and partial derivatives

Let x = x0 + ix1 ; f(x) = f0 (x0,x1) + if1(x0,x1)

What is the relationship between the derivative df(x)/dx

and the partial derivatives d

[i.e Cauchy-Reimann conditions do not seem to be satisfied there appears to be a relationship with the curl and divergence operators]

Path Integrals

What are the consequences of the zero factorisation?

Taylor Series

Let s = x2

Then for infinitely differentiable function p(x),

p(x) = a0 + a1.x + a2.x2/2! + a3.x3 /3! ...

= f(s) + x.g(s)

Quaternions

A similar change can be made to the four-dimensional algebra defined by Sir W.R. Hamilton in the early 1840's.

[Hamilton's original papers are available at the history of mathematics web site, Trinity College, Dublin ref.

In Hamilton's quaternion algebra the product is defined as

i2 = j2 = k2 = -1

i.j = k

j.k = i

k.i = j

j.i = -k

k.j = -i

i.k = -j

a = a0 + a1i + a2j + a3k

b = b0 + b1i + b2j + b3k

a.b = (a0b0 + a1b1 + a2b2 + a3b3) +

(a0b1 + a1b0 + a2b3 - a3b2)i +

(a0b2 + a2b0 + a3b1 - a1b3)j +

(a0b3 + a3b0 + a1b2 - a2b1)k

Which is the origin of the more familiar vector algebra due to Gibbs.

The quaternion product can be expressed in vector algebra, if we regard a quaternion as the sum of a vector and a scalar quantity

a = a0 + v ; where v = a1i + a2j + a3k

b = b0 + u ; where u = b1i + b2j + b3k

Then a.b = a0b0 + v.u + a0u + b0v + v X u

The quaternion algebra can be modified in the same way as the complex algebra as follows:

a.b = (a0b0 + a1b1 + a2b2 + a3b3) +

(a0b1 - a1b0 + a2b3 - a3b2)i +

(a0b2 - a2b0 + a3b1 - a1b3)j +

(a0b3 - a3b0 + a1b2 - a2b1)k

[In vector algebra, if

a = a0 + v ; where v = a1i + a2j + a3k

b = b0 + u ; where u = b1i + b2j + b3k

Then a.b = a0b0 + v.u + a0u - b0v + v X u]

which gives

a2 = (a02 - a12 - a22 - a32)

i.e. the Lorentz invariant interval in four dimensions.

Field equations

Is it possible to express the laws of electromagnetism in terms of field equations in this 4-d algebra?

What is the relationship between the derivative and the curl and divergence operators?

Can the equations be transformed as follows?

f(s) = [p(x)]2 ; s = x2

This give a scalar function of a scalar variable in terms of Lorentz invariant quantities.