(13/Sep/2011 Revision notes moved to bottom, some formatting fixed)
London, UK 11/11/2010
Postulates of Quantum Mechanics J.B. Gallagher
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Abstract
========
We propose a set of postulates for Quantum Mechanics.
The Postulates
==============
0) A universe, U, was created. (inflation?)
1) U consists of a finite number n, of "states" s, which are complex numbers.
i) |s| > 0 if s represents a quantum.
ii) |s| = 0 if s does not represent a quantum.
2) (POSTULATE OF RANDOMNESS)
Any state can change its (complex) phase randomly.
3) (POSTULATE OF DISCRETE TIME EVOLUTION)
When a single state changes its phase the universe evolves in time tdelta > 0,
according to the LINEAR SCHROEDINGER EVOLUTION RULE:
U(t+tdelta) = M.U(t)
where M is an n x n complex valued matrix, such that |Ms| = |s| for all states s. (unitary?)
(Exact form to be determined, must encode the Standard Model at least)
(M is exp(L) - I for some anti-hermitian matrix L (lagrangian) and identity matrix I)
Corollaries
===========
a) There are no "elements of reality" (Einstein) in the universe. A state must be measured*
to determine its value. The universe exists ontologically for "fleeting" time intervals, tdelta.
*The concept of a "measurement" is discussed below.
b) (quantum foam) The entire universe updates after a single state change. There is no way to
know which state changed. The entire universe is (effectively) randomly updated.
c) (wave-particle duality, complimentarity) When a state is measured its phase is fixed.
A state is a "particle" when measured and a "wave" otherwise.
d) (Uncertainty Principle for position/momentum) To measure momentum you must measure a state
for successive time intervals tdelta, the more intervals the more precise direction and speed
are known, but less precise is position. (Large mass means large number of states, so position
is not well-defined in this case) NOTE this is "hand-waving"
e) (Speed of Light limit) There is a maximum (and invariant) speed a state can propagate in the
universe due to finite evolution time, tdelta, and the linear evolution rule.
f) (quanta conservation) |Ms| = |s| for all states s.
g) U has an "Arrow of Time"
h) No wave-function collapse or decoherence mechanism is required (but decoherence mechanism can
be observed) The cat is dead or alive, but we have to "look" to know which one.
ie "wave function collapse" occurs at every evolution step,
and the universe is in a superposition of ALL possibilities before each evolution step.
Conjectures
===========
Pauli Exclusion principle is due to constraint on state occupation ("constraints" encoded in matrix M)
Gravity is described by M (linear graviton model),
otherwise it is a statistical effect, entropic a la Verlinde?
Phases are discrete values, multiples of 2*PI/m for some integer m. (18/11/2010)
No, this this is unlikely I think, phases are surely continuous (as is Time) (21/8/2011)
For more structure (eg degrees of freedom), space of states may be CxC, CxCxC etc (19/11/2010)
NOTE: "Space" and geometry are not defined here, they will be emergent properties encoded in M.
But Time is absolute (and continuous), relativistic physics follows from discrete time evolution.
--> invariant and maximum "speed of light"
(everything moves at ~"speed of light" at microscopic level)
Continuous rotation and translation symmetries are emergent from discrete QM superpositions
(eg hep-th/0707.4568 )
Holographic principle urges that fundamental space should be encoded on a 2D surface,
and 3D space is emergent.
MEASUREMENT AND CONSCIOUS FREE-WILL (24/11/2010)
===================================
(This may be considered speculative and ignored, it is independent of the above theory)
The above postulates present a mechanical "interpretation" of QM fully consistent with current experimental observations.
However some may protest "what about the quantum zeno effect?". Well I would rather protest "how can I walk back and forth
in my kitchen AT WILL?" Surely the zeno effect or my apparent free choice to move in a direction is not encoded in the
above postulates. Statistically speaking there is a small probability that the observations in the zeno effect and my
movements are explainable by randomness of states and linear schroedinger evolution, but the probability is so vanishingly
small as to be not a scientifically acceptable explanation.
What is a measurement? At a microscopic level (of states) the universe U exists ontologically for fleeting moments tdelta,
if we know the value of a state at time t we do not know its value at time t+tdelta or any previous or future time.
(We may obtain excellent probabilistic predictions by a feynman path integral calculation over ALL possibilities) But the
linear schroedinger rule allows that MACROSCOPIC structures may persist in some stable manner, although consisting of
underlying probabilistic microstates. Furthermore these macroscopic structures cannot propagate in the universe faster than
some upper speed limit, which we may identify as the "speed of light", due to finite evolution time tdelta and evolution rule
given by matrix M. It is these stable macroscopic structures that we can identify with a "measurement" of the universe, they
give us some probabilistic information indicating the random evolutionary "path" the universe has taken in its past.
We know there were dinosaurs because the persisent macroscopic structures of fossils are still "existing". We know of our
past through photographs and other persistent "records". However, the unfortunate inevitable fate of all these records seems
to be dissipation in eventual "heat death" of the universe U, IF the universe evolves purely according to the above postulates.
But the universe is surely not evolving purely according to the above postulates. I can walk back and forth in my kitchen,
we can observe the quantum zeno effect, the Apollo 11 mission to the moon is so unlikely to be encoded in matrix M evolution
and randomness that we must accept a further mechanism governs the behaviour of humankind.
I propose that we can make CHOICES, exercise FREE-WILL, we can modify microstates s to our choosing, or perhaps "load the dice"
on a subset of states to probabilistically ensure a macroscopic effect, such as walking one way or the other. In quantum zeno
experiments WE CHOOSE to make a measurement, or we choose to program a macroscopic device to do the measuring.
We record measurements as macroscopically persistent objects, such as writing on paper, or digital records in computer memory,
we cannot transmit the information faster than the speed of light. The apparent "non-locality" of the linear schroedinger
evolution rule is completely useless for transmitting information, as microstates s are fundamentally random.
Whether there may be some clever and subtle way of transmitting a macroscopic collection of states (in some reasonably
stable manner) faster than light will require exact determination of the form of matrix M. But for now we must accept it is
unlikely to be possible.
[
Speculating informally, we may amend the linear evolution rule to
U(t+tdelta) = M.U(t) + V(t)
where V(t) is a global potential which our consciousness accesses to alter values of microstates.
]
We are unique (as far as we know) in the universe, so it does not seem unreasonable that a unique theory must apply to us.
Before we evolved here on Earth, the universe evolved according to the above postulates, but once we gained consciousness
we were able to "observe" the fleeting moments of existence tdelta and use free-will to propagate our observations to
macroscopically stable records. Before that we have only the records that persist "by chance" to tell us what happened.
Rather worryingly, it's only our free-will versus the almighty matrix M and randomness that can prevent "heat death" in the
far future. So we must take our responsibility due to possessing conscious free-will rather seriously. A lot is at stake!
(Note: these ideas are speculative and I may substantially edit the whole section in the near future. I also realise that
my prose skills are not great, and may sadly fail to get across what I think is an immensely important point)
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Notes/revisions
===============
(17/Nov/2010) Rev 1, Don't need to specify what type of energy quanta represent
(17/Nov/2010) Rev 2, M need not be a permutation (duh!)
(24/Nov/2010) Rev 3, Added a (controversial) discussion of "measurement"
(02/Dec/2010) Rev 4, Added explanatory note that "space" and geometry are emergent concepts.
(11/Feb/2011) Rev 5, U is C^n or "grid" of n x C^n vectors, 3D space is emergent (cf holographic principle).
Gravity due to either emergence from linear gravitons or statistics.
Lorentz invariance due to "speed of light" invariance,
continuous (Poincare) symmetries emergent from discrete QM superpositions.
(21/Mar/2011) Conjecture M is a (product of) rotation matrix(es) (sufficiently complex to encode the Standard Model)
followed by a transitive permutation (which may be as simple as a cyclic shift of the states) to get propagation
throughout the universe. We need the random phase change to propagate throughout the universe after one or a small
number of evolution steps, perhaps just have a permutation or mapping of the PHASES of the states only.
(14/Apr/2011) Suppose we construct a complex permutation matrix M which encodes the linear groups of the SM (at minimum),
this will have many zero entries, and phases are not redistributed globally at each evolution step,
so now we PERTURB the matrix in U(n) to get small but non-zero entries almost everywhere - we now have to ask:
"how do we choose the correct perturbation?" and may end up in anthropic arguments...
(18/May/2011) Trying to figure out SUSY compatibility/plausibility.
(11/Jul/2011) SUSY is too hard to work out right now :-), but another idea - supose M^4 = M, then 3 dimensions of space
might arise from cyclic evolution, does this impose a "handedness" on nature?
Are we fooled by our dumb minds which have just evolved to decode the cyclic structure (M^4 = M) as spatial dimensions?
(I really prefer/want SU2/2 ~= SO3 to be explanation for our dumb minds "seeing" 3 dimensions.)
(19/Jul/2011) Or perhaps rather M^4 ~= M
(M^4 is similar to M on a macroscopic scale but there may be more intricate evoluton on a microscopic scale)
(22/Jul/2011) "Expansion of Universe" is rather slowing of evolution time - increasing tdelta.
(tdelta is increasing, perhaps due to increasing entropy the evolution law "takes longer to calculate")
(21/Aug/2011) Looking for a hierarchy of "mixing" matrices, ie matrices within matrices
(obviously M can't be a just disjoint product of small matrices, since nothing could move across the universe)
I don't want this to be anthropically defined though (ie matrix coefficients for the mixing matrices are not just determined
by experiment), I want some (reasonably) simple mathematical construction that will appear "natural"
NB Need to decide if gravity is from gravitons or pure statistics (entropic), and how a graviton model might be modelled by M.
(13/Sep/2011) Note that initial state of universe could just be one single state s with |s| > 0 (so can say |s|=1 to normalise),
There will be mostly zero states but entropy will increase VERY quickly initially.
(13/Sep/2011) The postulates are consistent with Feynman's path integral formulation of quantum mechanics, (indeed,
this was a major motivation) and in fact explain why the path integral approach works.
At each evolution step the entire universe is in a (quantum) superposition of all possible configurations of the states
under evolution by matrix M given ANY (continuous) phase change of ANY state.
"Wave function collapse" (urgh!) occurs at every evolution step.
Conjecture that tdelta ~= planck time (10^-43 secs), but may not be constant
(average value of tdelta ~= planck time, but this average may be increasing)
(14/Sep/2011) Born Rule derivation - Gleason's Theorem? Or a simple frequentist explanation?
See also "Quantum Mechanics as a Theory of Probability" - Itamar Pitowsky http://arxiv.org/abs/quant-ph/0510095
Need to start running some computer simulations here, problem is the evolution rule requires a computer more powerful
than exists (or ever will) - especially if M is some unnatural perturbation in U(n).
(Need to experiment with small universes, look for interesting qualitative behaviour, construct matrices exhibiting
simple conservation rules etc)
(18/Sep/2011) I know of no mathematical/computational studies applicable to this model.
Considering all the crap mathematicians have studied in the last century you'd think a simple model like this might have
at least been considered by some small group - but no - we are in unchartered territory here -
amazing really since it's the simplest model that might describe Nature.
(21/Sep/2011) I can only test up to about 10000 x 10000 matrices, with the 2GB Core2 Duo I have. Don't need to multiply two
matrices, just a matrix * vector (complex-valued) so can be quite fast, eg the Armadillo library http://arma.sourceforge.net/
does it in ~0.4 secs (and takes just over 6 secs to create the (double precision) random complex-valued matrix).
I guess the first thing to try is looking at random 10000 x 10000 UNITARY matrices (google for an algorithm to create them)
and running a graphical simulation of the evolution in complex space, see if anything interesting appears.
This is obviously a bit desperate, so will need to think about the form of the matrix more cleverly. :-)
(eg unitary matrices which are small pertubations of (complex) permutation matrices)
Here's an example of multiplying a complex vector by a complex matrix if you install armadillo libraries
( you can then compile this with g++ -larmadillo -O2 complex.cpp -o complex, execute with ./complex )
// complex.cpp
#include
#include
#include "armadillo"
using namespace arma;
using namespace std;
int main(int argc, char** argv)
{
int size = 1000; // default matrix size is 1000x1000
wall_clock timer;
// use any size value supplied on the command line instead of the default
if ( argc == 2 ) size = atoi( argv[1] );
// create a size x size complex-valued matrix with random moduli uniformly distributed in [0,1]
srand(time(0));
cx_mat M = randu(size,size);
// create a random complex-valued vector with n elements
cx_vec U1 = randu(size);
// multiply it by M and time how long this takes
timer.tic();
cx_vec U2 = M*U1;
cout << "time taken = " << timer.toc() << " secs" << endl;
return 0;
}
(21/Sep/2011) If speed is an issue then could go with something like CUDA since I have an nvidia gpu, but it's much
more important here to get the qualitative structure of M correct, and then see if the simulation exhibits anything
like what the vacuum is supposed to do, with particle creation/annihilation etc.
(22/Sep/2011) Well actually speed is an issue here since, assuming tdelta ~10^-43 secs, we'll only be
simulating events that can occur in small multiples of planck time, which probably won't be very interesting.
Interesting events may be emergent in a complex/statistical way from the exact linearity of the evolution rule.
Have to use CUDA and smaller universe, maybe 5000 states, but this is probably not big enough to model any
elementary particle interactions, let alone see non-linear gravitation emerge!
Simulation may be hopeless, maybe better to prove that the Standard Model particles and forces could be modelled
(without actually finding the exact model) and prove that non-linear gravity could emerge.
(05/Oct/2011) Recent Nobel Awards are interesting. To explain the accelerating "expansion of the universe" will
probably involve something more subtle than varying tdelta due to global entropy increase. This will be tricky to
see in a computer simulation, since floating point mutiplication/addition of zeros probably takes the same number
of cpu cycles as any other mutiplication/addition (whereas it might not in nature, if you get what I mean)
The quasicrystal award for chemistry gives me a neat idea, what about if matrix M has a quasicrystal structure
(irregular tiling a la Penrose) - that might solve transitivity problem for the evolution (such a matrix will
propagate a single state throughout the entire universe, eventually). Problem is that the phases are then locally
propagated and I want global propagation of phases - or ... maybe I don't :-) ;-)
(05/Oct/2011) I noticed Steven Weinberg has just posted a paper on a proposed stochastic evolution model for QM
('Collapse of the State Vector' arxiv.org/abs/1109.6462 ) James Hartle was consulted, so I guess Weinberg is talking
about the 'wavefunction of the universe' (like here) but seems reluctant to say so - he refers to 'the state vector
of any system, large or small'. Very happy to read that he will be publishing a book on quantum mechanics
('Lectures on Quantum Mechanics' is mentioned in note [1] in the paper).
(30/Jan/2012) Ok, I realised what we actually need for M is a type of (complex) Adjacency Matrix or variations thereof,
ironically this is complementary to a (complex) permutation matrix, a term that I confused it with, but that's because
I never did any/much graph theory, so didn't know the terminology, http://en.wikipedia.org/wiki/Adjacency_matrix :
M = [ 0 w w w w ... w ] or with permuted rows, eg [ w w w 0 w ... w ]
[ w 0 w w w ... w ] [ w 0 w w w ... w ]
[ w w 0 w w ... w ] [ w w w w w ... 0 ]
[ w w w 0 w ... w ] [ w w w w w ..0.. w ]
[ w w w w 0 ... w ] [ 0 w w w w ... w ]
[ ... ... w ] [ ... ... ]
[ ... ... w ] [ ... ... ]
[ w w w w ... 0 w ] [ w w w w w ..0.. w ]
[ w w w w w ... 0 ] [ w w w w w ..0.. w ]
where w is complex with |w| = 1 (the ws are not all the same value, but to simplify and save space I did not label the
entries as w_ij), probably M needs to be symmetric or hermitian
Also we need state space to be at least CxC and random phase change to "balance" in each state "doublet" to maintain some
kind of symmetry eg if state = (z1,z2) and "random phase change" is pi in z1 then we also need phase change of pi in z2.
Furthermore, as originally conjectured, we probably need to consider larger states in C^r for some +ve integer r, and the
more sophisticated symmetries that this allows, maybe we should call them "strings" ( lol ;-) )
Now, quanta conservation requires some structure in the distribution of states (can't be arbitrary) depending on exact
structure of M. eg in simplest case, if universe = [ +1 -1 0 0 -1 +1 ] then M can be trivial adjacency matrix with 0s
on the diagonal and 1s everywhere else. This will result in a very simple dynamical evolution of the universe where the
signs are permuted at each step. But note that a "random phase change" will have to permute a PAIR of signs to maintain
conservation of quanta in any case (if we randomly flip just one sign, then the evolution under matrix M will map some
of the 1s to 2s and the 0s to +/-1)
Of course, numerically, for large matrices, this is a nightmare, because of the "numerical sign problem", I guess Nature
doesn't really care about that though, http://en.wikipedia.org/wiki/Numerical_sign_problem
(31/Jan/2012) We may need to multiply M by a small numerical factor ~ 1/n or 1/sqrt(n), which would mitigate the
requirement for the random phase change to be "balanced" by other elements of the same state.
( eg w = i/sqrt(n) everywhere in M except for 0s on the diagonal ) ( sqrt(n) because 2d random walk distance ~sqrt(n) )
Note that in this model, space and geometry are emergent from the dynamics generated by evolution under M, and bear in
mind that, in appropriate units, length = time, so distances are just the TIME it takes for a signal to propagate. The
"speed of light" limit is bounded by the evolution time tdelta, but I guess it might be possible for neutrinos to move
slightly faster than photons if their interactions are more feeble - this may be a statistical effect, or an exact one
encoded in the dynamics of neutrinos described by M (this may appear to be "movement" in small extra dimensions in an
alternative mathematical description)
But the point is, that in this new paradigm, nothing has to physically "move" to move, we just need the phases to
redistribute. A signal can propagate as a phase pattern moving amongst the states. But the states can also move
around under the evolution law, but this doesn't translate to "movement" in our observed 3D world in a simple way.
The evolution rate is bounded by tdelta, hence a cosmic speed limit, and in fact everything moves at "the speed of
light" relative to everything else at the microscopic level of states, but as we become more macroscopic, we have
complex statistical and emergent patterns which will propagate less rapidly than this cosmic speed limit.
Incidentally, the adjaceny matrix is natural when one considers that everything in the universe is connected, this
was the original inspiration I had for the idea, as I wanted a random phase change to influence the entire universe
via the schroedinger evolution, I just fumbled the implementation initially, pity someone helpful was not around to
point out that a "complex permutation which linearly redistributes phases" could be implemented by this kind of matrix.
(01/Feb/2012) so,btw, M is not unitary but rather hermitian, probably. :-) (unless we have the weird permuted rows)
(20/Feb/2012) After some experimentation/investigation it looks like we need an anti-symmetric or anti-hermitian matrix,
so we can interpret M as an adjacency matrix of a directed graph. (Remarkably there is some established theory of these
"quivers" in mathematical physics, but I hope I don't end up in combinatorial braneworld madness or similar :-)
see http://en.wikipedia.org/wiki/Quiver_(mathematics) ). In fact, even the simple matrix M(i,j) = sign(i-j) (sign(0)=0)
gives some non-trivial dynamics acting on C^n , note the max eigenvalue of such a matrix -> n*2i/pi as n->infinity so we
should multiply the matrix by pi/2n for bounded dynamics.
(27/Feb/2012) Looks like we don't need to worry about ftl neutrinos now that the cables have been tightened :-)
OK, this damned matrix, we're looking for a matrix consistent with Feynman's path integral formalism - so why not just
dumbly take the same steps as Feynman took when interpreting Dirac's lagrangian formulation of QM,
ie U(t+tdelta) = exp(i.Lagrangian).U(t), and assume nature's Lagrangian has some simple matrix form:
(anti-)symmetrical/hermitian maybe composed of many identically repeating submatrices taken from a basis of some
suitable lie algebra. Then construct the matrix from the first few terms of the exp expansion 1+x+x^2/2+....
(28/Feb/2012) So M is unitary after all ( M = exp(lagrangian) where lagrangian is anti-hermitian )
A linux program to experiment with graphically iterating dynamics using such a matrix:
http://jbg.f2s.com/exphermitian_plot.cpp
You need armadillo matrix library installed, usage and compilation notes are included at top of the program code.
in fedora linux you need to do 'sudo yum install armadillo armadillo-devel gcc-c++ libX11-devel'
in ubuntu you need to do 'sudo apt-get install libarmadillo0 libarmadillo-dev libboost-dev libx11-dev g++'
(03/Mar/2012) After correcting the code I'm noticing period 3 cycles when the state vector is projected into a 2D view,
(regardless of the phase angles chosen to contsruct the random anti-hermitian matrix, ie not just for pi/3 phases)
so maybe, as mentioned above 11/Jul/2011, M^3 ~ I (ie M^4 is approximately M - with a small rotation), and we need to
project each cycle into a separate dimension to get 3D space - ie this is the reason we appear to live in 3-D "space".
(04/Mar/2012) Period 3-cycles are in dynamics generated by (e(L) - I) not e(L)
(06/Mar/2012) I'm convinced the period 3 dynamics are real, and is the explanation for 3D space. I need to correct the c++
code to definitely show this, the version linked to above is not correct, but I have an almost correct version which I am
checking does not show period 3 for some trivial reason. This makes sense in this new paradigm, remember that length=time
in appropriate units, distances are just the time it takes a signal to propagate. We have incorrectly been trying to think
of time as an additional spatial coordinate when in fact it is the 3 spatial coords that are additional time coordinates!
ie 3D space is literally a dynamical phenomena. (exp(L) - I)U(t) gives us the CHANGE in the universe after an iteration by
Schroedinger evolution (L is an anti-hermitian matrix, the lagrangian if you like, so exp(L) is a unitary matrix)
And our conscious perception of this dynamical period 3 evolution manifests itself as 3 degrees of freedom in space.
(07/Mar/2012) New code, http://jbg.f2s.com/exp_antihermitian_plot.cpp
I've increased the exp taylor approximation to the 10th power, so it takes a little longer to create the initial matrix.
With the default parameters there may also be a longer 2-cycle noticeable within the 3-cycles as an expansion/contraction,
since I create the random anti-hermitian matrix using 2x2 sub-matrices. But no obvious explanation for the 3-cycles yet.
It may be a computational quirk with armadillo matrix library, or may be due to rounding, or a quirk of the gnu rand()
function generating random numbers.
(08/Mar/2012) Looks OK, there were minor bugs which I corrected and added several checks to rule out rounding errors etc.
The dynamics are VERY sensitive to the scaling factor lambda that I multiply the anti-hermitian (lagrangian) matrix by.
To find lambda for a given matrix size and random distribution of complex values you must execute the iteration with
autozoom OFF, and find a value of lambda which stabilises the dynamics. A small change in lambda will cause the dynamics
to blow up or shrink to zero. lamdba seems to vary as ~1/sqrt(matrix size), which is promising, since we want something
like M = exp(i.h.L), where h ~ 1/(sqrt(#states_in_universe)) and matrices L,M are n x n where n = #states_in_universe.
So we might have found origin of speed of light (tdelta) , planck's constant and 3 dimensions of "space".
(09/Mar/2012) The computer simulations suggest that I need to prove the following result:
Given any n x n anti-hermitian matrix L of the form,
L = [ 0 w12 w13 w14 w15 ... w1n ]
[ z21 0 w23 w24 w25 ... w2n ]
[ z31 z32 0 w34 w35 ... w3n ]
[ z41 z42 z43 0 w45 ... w4n ]
[ z51 z52 z53 z54 0 ... w5n ]
[ ... ... . ]
[ ... ... . ]
[ zn1 zn2 zn3 zn4 zn5 ... 0 ]
where zji = -wij* (* denotes complex conjugate), (includes zji = wij = 0)
then there exists a positive real number h = h(L) ~ O(1/sqrt(n)) (for |wij| ~ |zji| ~ O(1) ) such that the matrix,
M = exp(hL) - I
(where I denotes nxn identity matrix, and exp(hL) is the exponential matrix I + hL + (hL)^2/2! + (hL)^3/3! + ... )
has at least one (non-trivial) set of globally attractive period-3 fixed points {R1, R2, R3} in C^n.
ie M^3.Ri = Ri for i=1,2,3 and |M^(3k).U - Rj| --> 0 (some j=1,2,3) as k --> infinity for all non-zero U in C^n.
(probably need contraction mapping theorem here)
Then 3D "space" is our conscious perception of the "period-3" dynamical system Delta(t) = U(t+tdelta) - U(t)
under Schroedinger evolution U(t+tdelta) = exp(hL).U(t) of the universe U(t) in C^n (the evolution is seeded by
random phase jumps in U(t) to give quantum superpositions, QM is not a simple deterministic dynamical system)
(11/Mar/2012) A new version of the code http://jbg.f2s.com/unitary3d.cpp (TODO: auto-searching for lambda)
I'm gonna suggest something more radical, the evolution equation of the universe is actually
U(t+tdelta) = exp(hL).U(t) - U(t) ( so M = exp(hL) - I )
ie the ONLY thing remaining after time tdelta is the CHANGE in the universe from the previous time step,
the past is GONE FOREVER, and thus CANNOT BE CHANGED - there is no possibility of causal paradoxes in nature,
and no time-travel to the past. The only thing existing is the inexorable flow of probability states - yes indeed
Heraclitus, "everything flows" (at the "speed of light" at the microscopic level of probability states)
I'll probably publish this with computer code for exhaustively checking the result for n up to one or two thousand,
and work out the mathematical proof later. ( that's ok in physics :-) )
(12/mar/2012) I submitted a brief note of the result to vixra.org - http://jbg.f2s.com/"Everything Flows.pdf"
I hope people like the cute title. Now hopefully they don't reject it and let's see if anyone can debunk the thing.
(My worries are that this is a known mathematical result and I have overlooked something, but best to find out now rather
than later)
final version of code with auto-search for lambda - http://jbg.f2s.com/unitary3d.cpp.final
http://vixra.org/abs/1203.0039
(16/Mar/2012) The more I think about it the more beautiful it appears, the simple step of "subtracting the universe"
makes it all work - this is what feynman et al discovered with renormalisation but couldn't justify subtracting the
large numbers.
(20/Mar/2012) I may be over-interpreting the result slightly :-)
Anyway, it's not a too bad mathematical result to think about
A full mathematical proof is in preparation, but the essential idea is that the max eigenvalue s of L is
pure imaginary, thus exp(hs) - 1 lies in the lhs of the complex plane. Now we adjust h until a periodic
point is found which can only be for a 3-cycle (can't be a 2 cycle since im(exp(hs)-1) != 0) corresponding
to angle 2pi/3 (4 cycle would be on pure imaginary axis, 5-cycles and higher would be in rhs of the complex
plane). And we then get period-3 convergence since s is the max eigenvalue all the other components will
converge to zero.
(22/Mar/2012) No, I changed my mind, I'm interpreting just fine :-)
Seems justification for 3 dimensions of "space", renormalisation, speed of light, size of planck's constant and
a sane explanation of QM superpositions isn't enough for the community (so far). It's probably difficult for
people to adjust to this model where 3d "space" is redefined as a dynamical phenomena so I need a concrete
calculation of some particles - but this will be incredibly difficult since all phases in the universe are
updated every evolution step - I mean things are blimin' complicated down there at microscopic level! However
we must have the linear group theory amongst all the stuff, so need to look at matrix coefficients with some sane
kind of distribution (not just randomly distributed in the unit ball or similar)
(03/Apr/2012) The three fixed points R1, R2, R3 of U(t+dt) = exp(hL).U(t) - U(t) are 2pi/3 phase shifts of a
single point R in C^n ie R1=R, R2=e^i(2pi/3)R, R3=e^i(4pi/3)R , this reminds one of drawing 3d axes on paper:
## Y
##
##
##
##
##
## ##
## ##
## ##
Z ## ## X
The existence of fixed points is trivial except in the case where matrix L has multiple eigenvalues of max modulus,
then you will get some aperiodic trajectories on a subset of C^n in the eigenspace of the other eigenvalues. Are these
rare cases just a mathematical curiosity or do they have any physical significance?
(29/May/2012) I'm working out emergent General Relativity as a linear approximation. Also thinking about 2 spin degrees
of freedom, perhaps these are emergent simply as the period-6 cycle you get multiplying the period 3 cycle by exp(hL)
(rather than by exp(hL) - I , ie "spin" degrees of freedom are constantly being swapped - they alternately disappear
into the past universe along with everything else) but unlike the 3 "spatial" degrees of freedom they are not observable
in the "present" defined by the evolution equation U(t+tdelta) = exp(hL).U(t) - U(t), they are a sort of "virtual"
degree of freedom.
I created openGL version of the code using glut, this speeds up graphics on some hardware (a bit too fast to see the
emergence of the 3-cycle) but might slow things down with non-accelerated graphics - http://jbg.f2s.com/glunitary3d.cpp
And something speculative ( :-) ) - when I awoke this morning I was reminded that our dreams often lack colour, we seem
to dream in shades of grey, I think there is a simple explanation for this. Colour is not created by our brains but is
objectively existing as frequencies in the evolution equation - it is not an easy job for our brains to recreate this
phenomena - a complicated statistical superposition of many frequencies. Our brains create geometrical shapes and forms,
(in 3D because the brain has only ever observed 3 "dimensions" emergent from the evolution equation) and shades of grey,
or light and dark contrast - but not colour, at least not easily. I only recall very vivid single colours appearing -
perhaps once the impression is made very strongly in the memory it can be recalled in a dream.
I also think this is why we often fly or experience strange forces in dreams - our brains can't accurately copy the
emergent forces (gravity) in the evolution equation - it is too complicated, so it has a very poor approximate model
which results in flying and lack of inertia when running etc.
A correction to the mathematical argument above (20/Mar/2012) - we choose eigenvalue s so that e(s)-1 is maximum (not
max eigenvalue of L) and then adjust h until e(hs)-1 has modulus 1, which corresponds to a period 3 cyclic point
(10/Jun/2012) I'm troubled by the dynamical nature of the evolution and interpretation of this wrt spacetime and
relativity, we must have mathematically sound arguments for the flow of the quantum evolution - not just hand-wavy shit.
However I expect this will be possible, just require debunking of a whole load of crap generated by ~100 years of
GR theory.
(24/Jul/2012) Oh, I remember now why I put zeroes on the diagonal - I wanted the evolution equation to contain only
real-valued quantities. In retrospect this is a bit silly, since I assume the state vector has complex valued elements.
(imaginary diagonal values were added to the computer code)
jbgallagher2000@yahoo.co.uk