From Osher Doctorow
Einstein's tensors have finite matrix representations (finite size or
rank matrices), while Heisenberg's quantum matrices have countably
infinite size, so the question is: what is an uncountably infinite
matrix?
To find an uncountably infinite matrix is simply a generalization of
finding an uncountable number of vectors with tails at a particular
point, for example the origin. But for unit vectors, these represent
precisely the directions in which vectors point from the origin, which
in turn can be represented in terms of the uncountably infinite angles
with and/or in some arbitrary plane(s).
Of course, we can still visualize the countably infinite matrix as one
with integer rows and columns (or for three-dimensional matrices, rows
and columns and depth-number), and then visualize additional rows and
columns as being associated with first rational and then irrational
numbers interpolating between the integers.
It would be ironic if quantum gravity is not merely a combination of
Heisenberg's countably infinite matrices and Einstein's finite tensors
but a further "phase" beyond them, namely an uncountably infinite
tensor.
In addition, notice what happens in the complex representation:
1) z = r exp(i theta)
When r is a function of time t, as the argument of the Riccati
Differential Equation, then (1) violates the Principle which I
discussed earlier in this thread of observables not involving
multiplication of two variables (two observables), unless one of them
is held constant. But holding theta or A (angle) constant is
precisely what we have been considering. If, on the other hand, r
is held constant, then theta or A can vary. In the latter case, the
Universe can trace out spatial structure and spatial surfaces at a
particular time.
Osher Doctorow
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