On Thursday, April 16, 2020 at 3:54:54 AM UTC-5, Thomas Koenig wrote:
Stephen Wolfram [...] physics based on graph theory.
[...]
I wonder what the physics community thinks of his approach.
Well, naturally, to answer that question you go over to ArXiv and do a
search! Like this
https://arxiv.org/abs/2004.14810
It's little more than another crusade to try to force yet another
narrative or framework out of the blue onto facts. Theories are supposed
to be led by and follow the facts, not the other way around.
I played that game a long time ago (a thesis titled "Finite
Electromagnetism"); something I outgrew more than a generation back.
But this much I will say: note the mention of "updating events" in
"causal invariant graphs" in the reference. What, exactly, does
"updating" mean in the setting of quantum theory ... particularly if you
are in the Heisenberg Picture?
And it's here that we shift it from the forced Wolfram narrative back to matters more of concern to Physicists - the Born Rule.
The standard rule used to establish a correspondence between Quantum
Theory and the real world (and to provide a interpretation thereof) is
the Born Rule. What does the Born Rule look like in the Heisenberg
Picture? What's being updated?
Perhaps you'll find a correspondence in the following to some of the
details laid out in the above reference, and see if you don't find any correspondence. But better grounded.
The Born Rule has a causal dependency. The input states for two or more measurements depends on the causal order of those measurements. Thus, in
order to even be able to *define* a Born Rule, you need to *first* have
a causal frame in place that gives you a causal ordering between the measurements.
That's part of the background!
Thus, a Heisenberg Picture version of the Born Rule contains a large
amount of hidden infrastructure that would otherwise be behind the
scenes if you had used the Schroedinger picture to frame the Born Rule
inside of. It consists of - at a minimum - the following:
(a) A set C of measurements (which you're going to be applying the Born
Rule to all of).
(b) A causal ordering on the set C.
(c) A FAMILY of Heisenberg states - one state for each partition of the
set into "before/after" subsets of C such that
(c1) If B and A are respectively before and after subsets then no
measurement in A causally precedes any measurement in B; no measurement
in B causally follows any measurement in A.
(c2) A and B, together, exhaust all of C
(c3) the intersection of A and B is empty - it is a partition of C.
So for each such partition C like so, there is one Heisenberg state
associated with it.
(d) For any two partitions that agree except for one measurement; i.e.
A1 = A0 union {c}, B0 = B1 union {c}, the Heisenberg state associated
with (A1,B1) is obtained from the Heisenberg state associated with
(A0,B0) by applying the Born rule to measurement (c).
To get the Born Rule, you map the time coordinate (t) so that the t = 0
surface separates B1 from A0 with t < 0 for A0, t > 0 for B1 and has measurement (c) at t = 0; and then transform to the Schroedinger Picture
using this as the reference time. Apply the Born Rule for measurement
(c) at t = 0, then transform back to the Heisenberg Picture. The
Heisenberg state derived from the t > 0 Schroedinger state is then the
state associated with (A1,B1), while the Heisenberg state associated
with the t < 0 Schroedinger state is the state that is associated with
(A0,B0).
When you apply the Born Rule to the (A0,B0) state, you get a mixed state
for (A1,B1) in the first stage of the Born Rule, and then this further
reduces to one of the pure state components (presumably, one of the eigenvectors of the matrix version of the mixed state); the resulting
being a stochastic transformation from the (A0,B0) to (A1,B1) state. So,
the whole graph C is threaded by a family of Heisenberg states that are connected to one another by stochastic updates. It is, in effect, a
stochastic finite Heisenberg-state automaton.
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