The mathematical foundations of quantum theory, Perhaps it's a simple theory after all

Easy: Scott Aaronson's lecture on Quantum Physics

"Quantum mechanics is what you would inevitably come up with if you started from probability theory, and then said, let's try to generalize it so that the "probabilities" can be negative numbers. As such, the theory could have been invented by mathematicians in the 19th century without any input from experiment. It wasn't, but it could have been."

Probability vector: Constant 1-norm, "a stochastic matrix is the most general sort of matrix that always maps a probability vector to another probability vector."

Quantum Mechanics vector: Constant 2-norm, a unitary matrix is "the most general sort of matrix that always maps a unit vector in the 2-norm to another unit vector in the 2-norm" Cancellation between positive and negative amplitudes can be seen as the source of all "quantum weirdness" -- the one thing that makes quantum mechanics different from classical probability theory.

Advanced: Lucien Hardy "Quantum Theory From Five Reasonable Axioms" quant-ph/0101012

The state associated with a particular preparation is defined to be (that thing represented by) any mathematical object that can be used to determine the probability associated with the outcomes of any measurement that may be performed on a system prepared by the given preparation.

Central to the axioms are two integers K and N which characterize the type of system being considered.

• The number of degrees of freedom, K, is defined as the minimum number of probability measurements needed to determine the state, or, more roughly, as the number of real parameters required to specify the state.
• The dimension, N, is defined as the maximum number of states that can be reliably distinguished from one another in a single shot measurement.

Axiom 1 Probabilities. Relative frequencies (measured by taking the proportion of times a particular outcome is observed) tend to the same value (which we call the probability) for any case where a given measurement is performed on a ensemble of n systems prepared by some given preparation in the limit as n becomes infinite.

Axiom 2 Simplicity. K is determined by a function of N (i.e. K = K(N)) where N = 1, 2, . . . and where, for each given N, K takes the minimum value consistent with the axioms.

Axiom 3 Subspaces. A system whose state is constrained to belong to an M dimensional subspace (i.e. have support on only M of a set of N possible distinguishable states) behaves like a system of dimension M.

Axiom 4 Composite systems. A composite system consisting of subsystems A and B satisfies N = N_A N_B and K = K_A K_B

Axiom 5 Continuity. There exists a continuous reversible transformation on a system between any two pure states of that system.

Surreal: Bob Coecke "Kindergarten Quantum Mechanics" quant-ph/0510032

This post has been edited by rpenner on Jan 28 2007, 02:27 AM

Hi rpenner!
I wish I had more education.
The paper
Advanced: Lucien Hardy "Quantum Theory From Five Reasonable Axioms" http://www.arxiv.org/PS_cache/quant-ph/pdf/0101/0101012.pdf
Was the most understandable to me.
As far as I could see my model does not violate the Axioms. It is a pure state that I have added "packing" in order to obtain the required symmetry in 3D.
If I knew more, I would be able to cite this paper to demonstrate that I am using a valid approach.

jal

Here a some quotes
p.9
The surface of the set of normalized states must therefore be N2−2 dimensional. This means that, in general, the pure states are of lower dimension than the surface of the convex set of normalized states. The only exception to this is the case N = 2 when the surface of the convex set is 2-dimensional and the pure states are specified by two real parameters. This case is illustrated by the Bloch sphere. Points on the surface of the Bloch sphere correspond to pure states.

p.16
6. We show that the N = 2 case corresponds to the Bloch sphere and hence we obtain quantum theory for the N = 2 case.

p.20
8.6 The Bloch sphere
We are left with K = N2 (since K = N has been ruled out by Axiom 5). Consider the simplest nontrivial case N = 2 and K = 4. Normalized states are contained in a K−1 = 3 dimensional convex set. The surface of this set is two-dimensional. All pure states correspond to points on this surface. The four fiducial states can all be taken to be pure. They correspond to a linearly independent set. The reversible transformations that can act on the states form a compact Lie Group. The Lie dimension (number of generators) of this group of reversible transformations cannot be equal to one since, if it were, it could not transform between the fiducial states. This is because, under a change of basis, a compact Lie group can be represented by orthogonal matrices [21]. If there is only one Lie generator then it will generate pure states on a circle. But the end points of four linearly independent vectors cannot lie on a circle since this is embedded in a two-dimensional subspace. Hence, the Lie dimension must be equal to two. The pure states are represented by points on the two-dimensional surface.
(See my example using the orange to see the minimum size of the points. http://forum.physorg.com/index.php?showtopic=5203&st=45)
Furthermore, since the Lie dimension of the group of reversible transformations is equal to two it must be possible to transform a given pure state to any point on this surface. If we can find this surface then we know the pure states for N = 2. This surface must be convex since all points on it are extremal. We will use this property to show that the surface is ellipsoidal and that, with appropriate choice of fiducial states, it can be made spherical (this is the Bloch sphere).

p. 21
(84) This equation defines a two dimensional surface T embedded in three dimensions.

p.22
Therefore, we have obtained quantum theory from the axioms for the special case N = 2.

Since we have now reproduced quantum theory for the N = 2 case we can say that
• Pure states can be represented by |ψihψ| where |ψi = u|1i+v|2i and where u and v are complex numbers satisfying |u|2 + |v|2 = 1.
• The reversible transformations which can transform one pure state to another can be seen as rotations of the Bloch sphere, or as the effect of a unitary operator ˆU in SU(2).
This second observation will be especially useful when we generalize to any N.

However, each two-dimensional fiducial subspace must, by Axiom 3, behave as a system of dimension 2. Hence, if we take those elements of D which correspond to an N = 2 fiducial subspace they must have the form given in equation (87). We can then calculate that for N = 3

p.26
While continuous dimensional spaces play a role in some applications of quantum theory it is worth asking whether we expect continuous dimensional spaces to appear in a truly fundamental physical theory of nature. Considerations from quantum gravity suggest that space is not continuous at the planck scale and that the amount of information inside any finite volume is finite implying that the number of distinguishable states is countable. (see my orange example)
Given the mathematical difficulties that appear with continuous dimensional Hilbert spaces it is also natural to ask what our motivation for considering such spaces was in the first place.

Would this paper be a suitable citation for my model?
jal

Hi rpenner,jal et al,

Many thanks for starting this thread .. I'll maybe have some questions when I've read all and absorbed as much as I can.

Meanwhile..

Wavelength ... looks so much like rotation of that qubit and it would be so easy to say that it was but nobody does, there must be a reason .. would it be possible to give a hand-waving or direction pointing type of explanation?

Best wishes,

-C2.

Hi rpenner,

On axiom 4 .. I am 'notationally challenged' .. notationally defeated might be closer to the truth.

Advanced version ( http://www.arxiv.org/PS_cache/quant-ph/pdf/0101/0101012.pdf )

N= N_A N_B , K= K_A K_B

Is this simply the normal arithmetic product of the terms (eg K= K_A times K_B)?

Best wishes,

-C2.

This post has been edited by Confused2 on Jan 29 2007, 12:59 AM

It is indeed a simple product of integers.

Since K is a function of N, K(N) = K(N_A N_B) = K(N_A) K(N_B) which turns out to be important to the proof that K(N) = N^2

This post has been edited by rpenner on Jan 29 2007, 01:33 AM

Good day!

Lucien Hardy’s next paper examines the challenges of mathematically analyzing a dynamic system and comes up with an approach.
This is where everybody is stuck.

http://arxiv.org/PS_cache/gr-qc/pdf/0509/0509120.pdf
Probability Theories with Dynamic Causal Structure: A New Framework for Quantum Gravity
Lucien Hardy
Perimeter Institute,
31 Caroline Street North,
Waterloo, Ontario N2L 2Y5, Canada
May 18, 2006
Hence, we require a mathematical framework for physical theories with the following properties:
1. It is probabilistic.
2. It admits dynamic causal structure.
The approach taken in this paper is operational. We define an operational notion of space-time consisting of elementary regions Rx. An arbitrary region R1 may consist of many elementary regions. In region R1 we may perform some action which we denote by FR1 (for example we may set a Stern-Gerlach apparatus to measure spin along a certain direction) and observe something
XR1 (the outcome of the spin measurement for example).
p. 16
We apply the group of continuous reversible transformations (implied by the continuity postulate) to show that the set of states must live inside a ball (with pure states on the surface). This is the Bloch ball of quantum theory for a two dimensional Hilbert space. Thus, we get the correct space of states for two dimensional Hilbert space. We now apply the information postulate to the general N case to impose that the state restricted to any two dimensional Hilbert space behaves as a state for Hilbert space of dimension 2. By this method we can construct the space of states for general N. Various considerations give us the correct space of measurements and transformations and the tensor product rule and, thereby, we reconstruct quantum theory for finite N.
p.18
To this end we will give a framework (which admits a formulation of quantum theory) which does not take as fundamental the notion of an evolving state. The framework will, though, allow us to construct states evolving through a sequence of surfaces. However, these surfaces need not be space-like (indeed, there may not even be a useful notion of space-like)
p. 39
Up till now everything we have done has been quite general. In particular, all this works for any choice of nested regions R(t) or, equivalently, for any choice of disjoint elementary time-slices Rt. To deal with (ii) we need to add spatial structure which we will deal with later.
p.51
The causaloid is a fixed object. Yet at the same time we have not assumed any fixed causal structure in deriving the causaloid formalism. That is to say we have not specified any particular causal ordering between the elementary regions. In this sense we must have allowed the possibility of dynamic causal structure. It interesting to see a little more explicitly how this can work in the causaloid formalism.
p.52
Dynamic causal structure is likely to be quite generic for causaloids. However, it is unlikely to be as clear cut as the hypothetical example we just discussed. In general we cannot expect the sort of clear cut causal structure we see evident in the causaloid diagrams of Fig. 9. In general, the causal relationship between nodes may be more complicated than can be represented by pairwise links. Thus, when we speak of “causal structure” we do not necessarily intend to imply that we have well defined causal structure of the type that allows us to determine whether two nodes are separated by a time-like or a space-like interval.
p.56
Thus, we see that the causaloid formalism provides us with a new calculus capable of dealing with situations where Newton’s differential calculus would be inappropriate. The advantage of differential calculus and the implied ontology is that, where it works, it affords a simple picture of reality which allows significant symmetries to be applied. We can hope that increased familiarity with the causaloid approach may achieve something similar.
p.57
It is often stated that experiments to test a theory of QG will involve probing nature at the Planck scale. It is no coincidence that apparatuses we might construct to do this would have to be very big. As illustrated above, postulated variation at a small scale shows up at a large scale and we might even doubt that there is any ontological meaning to talking about what is happening on this
small scale. The fiducial measurements in the causaloid formulation for such an experiment will, we expect, be at a much larger scale than the Planck scale. ( see my orange example)
p.63
The causaloid formalism deals with matrices between elementary regions. In the case that there exist RULES we may only need to specify local lambda matrices and lambda matrices for pairs of regions (as in QT). This is closer to Einstein’s original approach than providing an amplitude for an entire history is.
p.64
One problem which is common to most approaches which start with a Planck scale picture is that it is difficult to account for the four dimensional appearance of our world at a macroscopic level (Smolin calls this the “inverse problem” [33]). Since the approach in this paper starts at the macroscopic level, it may allow us to circumvent this problem in the same way Einstein does in GR. Thus, we would not attempt to prove that space-time is four dimensional at the macroscopic level but put this in by hand. This is not an option in Planck scale approaches to QG because the constraint that a four dimensional world emerges at the macroscopic scale has no obvious expression at the Plank scale. The best approach, however, may be to combine an approach which posits some properties at a Planck scale with the causaloid approach. By working in both directions we might hope to constrain the theory in enough different ways that it becomes unique.
p.65
The approach taken here attempts to combine the early operational philosophy of Einstein as applied to GR with the operationalism of Bohr as applied to QT (see [35] for a discussion of how Einstein and Bohr might have engaged in a more constructive debate). We do this primarily for methodological reasons to obtain a mathematical framework which might be suitable for a theory of QG without committing ourselves to operationalism as a philosophy of physics. In fact it is interesting just how close this early philosophy of Einstein is to the later philosophy of Bohr.

His analysis will find opposition from people who have theories that fail his reasoning and will get support from people who have theories that are within his description.
I find that he has gone too far in the philosophy direction to be able to produce a model.
For a comparison read.
http://arxiv.org/PS_cache/gr-qc/pdf/0701/0701142.pdf
Quantum gravity and cosmological observations
Martin Bojowald
26 Jan. 2007
p. 2
There is an additional expectation from quantum gravity, namely that space has a
discrete structure on very small scales. One can think of this structure as an irregular lattice whose typical plaquette size p is close to ℓ2 P. But unlike the Planck length, this is a geometrical parameter or field specifying the quantum gravity state and can thus be dynamical. This parameter brings in crucial information from quantum gravity, unlike ℓP which is determined simply by parameters of quantum mechanics and classical gravity.

Everyone is still looking.
Jal

Hi rpenner,jal,

I tried out axioms [1..4] on an ensemble of monkeys with keyboards. My monkeys were perfectly happy with keyboards and even reduced character set keyboards .. I just had a look at what they'd typed from time to time. I was getting nice numbers like

K_totalkeys = (N_totalkeys - N_alphakeys) (N_nonalphakeys)

But then it went horribly wrong ..

Any chance of a useful definition (or example) of a pure state (definition?) of a type that exhibits reversible transformations. I can't see how to get pure monkeys and keyboards and besides they don't look reversible .. I seem to be missing something .
Best wishes,
-C2.

This post has been edited by Confused2 on Jan 29 2007, 11:44 PM

Hi Confused2!
Lot's of people reading ... not too many wanting to adventure any answers.

I'll add this bit of info.

causaloid…. Plaquette ….. SPOT …. What’s in a name?
If you want to produce a model, then you should also be familiar with the following information.
http://arxiv.org/PS_cache/gr-qc/pdf/0601/0601097.pdf
Planck-scale physics: facts and beliefs
Diego Meschini∗
Department of Physics, University of Jyv¨askyl¨a,
PL 35 (YFL), FI-40014 Jyv¨askyl¨a, Finland.
January 23, 2006
The relevance of the Planck scale to a theory of quantum gravity has become a worryingly little examined assumption that goes unchallenged in the majority of research in this area. However, in all scientific honesty, the significance of Planck’s natural units in a future physical theory of spacetime is only a plausible, yet by no means certain, assumption. The purpose of this article is to clearly separate fact from belief in this connection.

We will argue that quantum gravity scholars, eager to embark on the details of their investigations, overlook the question of the likelihood of their assumptions regarding the Planck scale—thus creating seemingly indubitable facts out of merely plausible beliefs.

p. 5
Also Baez (2000) made welcome critical observations against the hypothetical relevance of the Planck length in a theory of quantum gravity. Firstly, he mentioned that the dimensionless factor (here denoted Kl) might in fact turn out to be very large or very small, which means that the order of magnitude of the Planck length as is normally understood (i.e. with Kl = 1) need not be meaningful at all.
More interestingly, Baez also recognized that “a theory of quantum gravity might involve physical constants other than c, G, and hbar.”
[see my orange for an example for a definition of what c as a constant implies)

p.8
The second alternative takes for granted that at least all three constants G, h, and c must play a role in quantum gravity. Although this is a seemingly sensible expectation, it need not hold true either, for a theory of quantum gravity may also be understood in less conventional ways. For example, not as a quantum-mechanical theory of (general-relativistic) gravity but as a quantum mechanical theory of empty spacetime, as we explain below.
p. 9
In view of the repeated difficulties and uncertainties encountered so far in attempts to uncover gravity’s quantum mechanical aspects, one may wonder whether the issue might not rather be whether spacetime beyond its metric field—i.e. empty spacetime as characterized by its bare points—may have quantum-mechanical aspects.
(see my orange example for a definition of points)
p.13
Further, we argued that the physical meaning of the Planck units could only be known after the successful equations of the theory which assumes them—quantum gravity—were known. To achieve this, however, the recognition and observation of some phenomenological effects genuinely related to spacetime are essential.

(Like the fact that the speed of light is constant.)
You cannot get the right answers if you do not have the right model. Now…. Go look at my model and then try to make a better model. Who knows? Your model might make it possible to reach the next level of technical innovations and get a better understanding of the universe.
jal

It's been done.

It's called the Dirac Relativistic Wave Equation.

Look up D.L. Hotson's reinterpretation of this equation and you should find everything you're looking for.

Hi Solid State Universe,jal,rpenner,

I think we may have to accept that the point of the thread is to show that the foundations of quantum theory are 'simple'. Once we have (hopefully) established the simplicity of the axioms we should be in a better position to understand the nature of the problem to which quantum mechanics is a solution.

Meanwhile .. these monkeys .. one has got as far as ..

To be or not to be,
That is the fumfumsnakker

Best wishes,

-C2.

Good Day Solid State Universe, Confused2, rpenner, ...
I am aware of his presentation.
He uses a fixed time interval {This minimum time appears to be 2e2/3mc3, or 6.26 x 10-24 seconds) to develop his approach.
He concludes ...

I agree .... those are good enough reasons to keep looking.
jal

Good Day!
There is no harm in looking at “simple” models to get intuition. I am still looking.
The following links are also of people who have done some looking.
They might think that they have found the final answer…. I don’t!
However, they are looking at interesting options.
These links are similar to my approach and, I expect, TRoc’s approach.
We are all looking to apply “simple axioms” .
http://www.blazelabs.com/f-p-develop.asp
The Particle
A proper model has to be compliant with experimental evidence and so be in perfect agreement with the spectral data for each atom.
If quantum numbers are unique, it then follows from our knowledge about the 6 unique basic platonics (5+dual tetra), that all basic elements can be described by no more than 6 pricipal quantum numbers.

http://www.21stcenturysciencetech.com/articles/moon_nuc.html
Advances in Developing the Moon Nuclear Model In the atomic nuclear structure hypothesized by Dr. Robert J. Moon1 in 1986, protons are considered to be located at the vertices of a nested structure of four of the five Platonic solids (Figure 1).
http://www.21stcenturysciencetech.com/Arti...Periodicity.pdf
The Geometry of the Nucleus

I repeat,
the challenges of mathematically analyzing a dynamic system and comes up with an approach.
This is where everybody is stuck.
This is where, I believe, that the next answers will come from.

rpenner You are awful quite. I would like to hear what you have found and what are your thoughts.
jal

This has happened before.. I find myself isolated .. last time it was jumping mice .. now it's monkeys..

I'll just carry on with my monkeys if that's OK with everybody. If anybody spots anything sensible from either me or the monkeys then please let me know.

I think my monkeys need a shift bar so they can type in two colours (like in the old days) red p's and black p's.. Defining a red p to be a not 'p' thus ..

p + !p = 0
!p + p = 0
p = 1
!p = -1

We train our monkeys to press a key (maybe including the red shift key) whenever a small tambourine in struck.

Either ten monkeys and one tambourine tap or one monkey and ten tambourine taps .. I don't know. This all seemed so fresh and exciting when I started.

Giving the monkeys various keys to press may not be quite the same as giving them a full degree of freedom .. but I don't really want them wandering about too much. Maybe it's enough .. maybe it isn't. If rpenner doesn't come to the rescue (heeelp!) then I'll try out the K's and N's on them if and when I feel a bit more confident.

-C2.

How about (p!)?

p = 1
!p = -1
p! = +1

Where:

!p + p! = 2