Originally Posted by Fortis

However,

ds = ct.e₀ + i.(x.e₁ + y.e₂ +z.e₃)

is a 4-d vector (with some imaginary components), where I have used bold to indicate a vector quantity, and e_i would be the ith basis vector in an orthonormal basis set.

The dot product,
ds.ds = c²t²-(x²+y²+z²)

which is just the usual interval of SR.

There is nothing really stopping you from making space imaginary and time real. As I have said before, the key thing is that the interval is invariant under a coordinate transformation from one inertial frame to another. At times (real ones ) I wonder if you are constructing a strawman version of SR.

See the above. If I stick an "i" in front of the spatial 3-vector, instead of the time scalar, in the 4-vector, it all works equally fine.

So in that sense you don't have a problem with imaginary space, though you seem to have a problem with imaginary time. This doesn't seem to be very consistent.

What is the operation II, or IJ?
Is it the dot product, the vector product, or the dydic product?
its multiplication using Hamilton's Rules you don't have to have dot or cross product, just multiplication by ijk= i^2 = j^2 = k^2 = -1! II=-1 IJ=K...
Why does the coordinatization of space time have to be associative? This seems to be a big issue for you, but I have never felt the need to multiply Alice's location by Bob's location, by Charlie's location.

Originally Posted by yawyaw

No. Do you?

Originally Posted by Fortis Just double checking, because I want to be sure about what it is that you are claiming. If I carry out a coordinate transformation from one inertial reference frame to another, p2 = (c2t2-(x2+y2+z2)) + 2.c.t.(i.x.+j.y+k.z) will be the same in the transformed coordinates. Is that correct? Or are you just suggesting that it is invariant only in some qualitative sense, not in a quantitative sense?

Originally Posted by yawyaw

(II)J = + J could be turn left and I(IJ)= -J turn right. In a control system, one operator writes (II)J and the other I(IJ) two different results unless manual specifies which way to write it, added confusion. But enough said on that.

Originally Posted by Fortis

I had thought that we were talking about position coordinates in spacetime. You are quite correct about rotations, but you started this discussion by taking the product of two quaternions describing position, i.e.

P1 = ct1 + ix1 + jy1 + kz1

and

P2 = ct2 + ix2 + jy2 + ky2

Am I missing something?

Originally Posted by Fortis

I'm happy with the first term being invariant, but you had said,

Based on this, I assumed that you believed that all of the quaternion product will be invariant, not just the real part.

Please can you be explicit and write down what it is that you believe to be invariant.

Originally Posted by yawyaw

I believe all of the quaternion product will be invariant, the real and the vector part, to changes of reference frames.

Originally Posted by Fortis

So why do you disagree that each of the individual components, i.e. the real and the vector part, must also be invariant?

Originally Posted by yawyaw

There may be rotations that switch the axis so that the value on the axis may be differently ordered but still the vector be the same in size but differently ordered, e.g i + oj + k j + > io + j + k.