The mathematical foundations of quantum theory

rpenner

28th January 2007 - 01:40 AM

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

jal

28th January 2007 - 02:59 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

jal

28th January 2007 - 05:25 PM

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

Confused2

28th January 2007 - 10:41 PM

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.

Confused2

29th January 2007 - 12:52 AM

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.

rpenner

29th January 2007 - 01:29 AM

QUOTE (Confused2+Jan 29 2007, 12:52 AM)

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)?

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

jal

29th January 2007 - 04:01 PM

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

Confused2

29th January 2007 - 11:39 PM

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

QUOTE

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

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.

jal

30th January 2007 - 12:13 AM

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

Solid State Universe

30th January 2007 - 06:29 AM

QUOTE

"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."

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.

Confused2

30th January 2007 - 09:04 AM

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.

jal

30th January 2007 - 04:17 PM

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 ...

QUOTE

. It explains many hitherto unexplained features, but it is perhaps oversimplified and wrong in details, and lacking in quantitative analysis.
We can say with some finality, however, that the Big Bang and the Standard Model are to the physics of the future as Phlogiston is to modern chemistry.

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

jal

30th January 2007 - 07:56 PM

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

Confused2

30th January 2007 - 10:34 PM

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.

Solid State Universe

30th January 2007 - 10:39 PM

How about (p!)?

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

Where:

!p + p! = 2

rpenner

30th January 2007 - 11:10 PM

In standard logic.

not-not-p = ¬¬p = p http://us.metamath.org/mpegif/pm4.13.html
not-p and p = ¬p∧p = p∧¬p = false, so ¬(¬p∧p) is always true, it is impossible for the same well-formed expression to be both true and false http://us.metamath.org/mpegif/pm3.24.html
not-p or p = ¬p∨p = p∨¬p = true, which is the law of the excluded middle, a well-formed expression is either true or false http://us.metamath.org/mpegif/exmid.html

Confused2

30th January 2007 - 11:55 PM

Hi SSU,

Hopefully it is already clear that I am operating outside my field of competence however I can clarify my intentions ..

We may have fallen into a universe where the probability of an event happening may (formally) be less than zero. As a result I feel the need to offer my monkeys a chance to press a key that is less than 'no key'. For example :- I define a red 'b' to have the quality of negating a black 'b'. The use of the '!' is borrowed from logic where it is an operator with the meaning of 'NOT'. In binary logic I would (normally) use the not (!) in the context of !1 = 0. In binary logic the other operators used are '+' for OR and '.' for AND. Within that notation 1 AND 0 = 0 is equivalent to 1.0 = 0 Another example would be 1 OR 0 = 1 .. which I might write as 1 + 0 = 1

Hopefully this short summary of logic enables us answer the old question

2b + !(2b)

This aside .. we need to keep our eyes on the trick rpenner is pulling. He's making some sort of claim about a number at right angles to the way we normally look at numbers. He got us into this so I think he should be the one to get us out of it. Meanwhile, have a look at http://en.wikipedia.org/wiki/Complex_number but don't let it frighten you. After a while it all seems as natural as feeding a banana to a monkey .. which is why I'm trying to put the whole thing into a nice simple bananas and monkeys sort of context. BUT .. quantum mechanics is rpenners world .. not mine. If we can explain what we can't see it might help him to explain what he can see. At least, that is the plan.

Best wishes,

-C2.

Confused2

31st January 2007 - 12:00 AM

I see rpenner has posted while I was preparing my response.

Mine remains unedited.

jal

31st January 2007 - 12:51 AM

We are not the only ones looking. See this thread. http://www.physicsforums.com/showthread.ph...535#post1228535
Converge on new QG formulation

I'll shut up and listen.
jal

Confused2

31st January 2007 - 11:04 AM

Hi rpenner,jal, Solid State Universe,

I don't know if it's the monkeys causing the problem.

Belatedly, I'll abandon the ! notation for the red letters. Not for the first time .. finding a notation looks like being a large part of solving the problem.

I'll try p (press) for a key because we don't want to get muddled up with K which is the number of degrees of freedom ..so we get p_b .. p_z.. and if the red shift is pressed at the same time then we have -p_a and so on.

I still don't know what a pure state is .. but maybe rpenner (or stronger medication ) will sort this out.

The question? Ah .. yes .. the axiom
A composite system consisting of subsystems A and B satisfies N = N_A N_B and also K = K_A K_B
One at a time ..

QUOTE (definition+)

The dimension, N, is defined as the maximum number of states that can be reliably distinguished from one another in a single shot measurement.

I suspect the clue is in the word 'subsystem' .. one monkey presses a key eg p_a and then another (subsystem) monkey presses a key eg p_b . To describe what they've done we have to write something like [p_a,p_b] . If one monkey (subsystem A) has 20 keys to play with and the other monkey (subsystem B ) has 10 keys to play with ... we need an oblong thing with 20 boxes (say horizontally) for what the first monkey pressed and then 10 boxes (vertically) for what the second monkey pressed. Multiplying 20 by 10 we get 200. It looks like this could be the meaning of 'dimension N ', the dimensions of the subsystems were just the number of keys each monkey was allowed to play with..
Hopefully I now know the meaning of
N = N_A N_B
"The dimension N of a system is equal to the product of the dimensions of of the subsystems."
This passes my test of being 'simple' (though not immediately obvious). If all is well so far, and everybody is happy, I'll take a few shots at the meaning of 'K'. If not, one bullet should suffice.
Best wishes,
-C2.

With hindsight it might have been better to have used p_a for an 'a' pressed by the first monkey (subsystem A) and q_b for a key pressed by the second monkey.

At this rate it could be several weeks before I am quite ready to derive Dirac's equations .. but hopefully the foundations are being laid.

Confused2

31st January 2007 - 10:52 PM

The second definition
"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."
We have advance notice of ..

QUOTE (rpenner+)

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

I am not convinced that these monkeys with typewriters are a good model. On the assumption that their output is purely random I don't think they have a 'state' that can be determined by any number of measurements. Without a suitable model the axioms make no sense and until the axioms make sense I am unable to select a suitable model.
I think I need some help here,please, if possible.
-C2.

Guest_SteveA

31st January 2007 - 11:36 PM

QUOTE (Confused2+Jan 30 2007, 11:55 PM)

Physical observations are differential. There are two components to an observation - the observer and the observee.

I don't believe something can have a negative probability of occuring, but of course you can have a negative bias relative to some non-negative value.

If we made the assumption that everything has some probability of occuring, then it would only be negative biases to this that make them not occur. An observer is limited in what they can coherently detect and so this would naturally provide a negative bias in the probability of observing something. I haven't followed the thread, though I love the subject and want to read this more closely (when I'm not at work

).

But I'd assume the probability of witnessing an event would be a product of both the likelyhood of the event occuring, as well as the likelyhood of it being coherently witnessed by the observer.

It's not immediately obvious how you could convert this multiplication into an addition, but I wanted to point out that the appearance of negative probabilities could occur when an expected event goes unnoticed by an observer (though again, I can't see how a negative probability could physically exist except as some abstract mathematical expression).

(I'll have to look at this thread again when I have more time. It's a very interesting subject)

Confused2

1st February 2007 - 12:30 AM

Hi StevenA et al,

Please disregard the 2b + !2b as no more than a red herring . 2b .. to be + (or) ! (not) 2b .. it is of no significance.

The rest of my posts are a serious attempt to deal with the proposition that this is a simple theory. I suspect the problem lies in the way I am trying to look at it (and/or lack of IQ) rather than that the proposition is false.

Best wishes,

-C2.

rpenner

1st February 2007 - 06:41 AM

QUOTE (Confused2+Feb 1 2007, 12:30 AM)

The rest of my posts are a serious attempt to deal with the proposition that this is a simple theory.

Let's see where the problem is. Lucian Hardy states what his intended plan of proof is on pages 2-3

QUOTE

The basic idea of the proof is simple. First we show how the state can be described by a real vector, p, whose entries are probabilities and that the probability associated with an arbitrary measurement is given by a linear function, r · p, of this vector (the vector r is associated with the measurement). Then we show that we must have K = N^r where r is a positive integer and that it follows from the simplicity axiom that r = 2 (the r = 1 case being ruled out by Axiom 5). We consider the N = 2, K = 4 case and recover quantum theory for a two dimensional Hilbert space. The subspace axiom is then used to construct quantum theory for general N. We also obtain the most general evolution of the state consistent with the axioms and show that the state of a composite system can be represented by a positive operator on the tensor product of the Hilbert spaces of the subsystems. Finally, we show obtain the rules for updating the state after a measurement.

But to actually follow along in the discussion, you need to have taken a college level course in linear algebra. It's a bit confusing when the author jumps between classical probability, standard quantum mechanics where Hermitian matricies of complex number are user, and the equivalent formulation in terms of a real vector.

One of the tools they use is the bra-ket formulation of linear algebra, where |a> is a column vector "ket", <a| is the dual (complex conjugate row vector) "bra", and <b|a> is the sum (with i over all the dimensions of the vecotor) of conjugate(b_i) times a_i, and |a><b| is the "projection" operator, which is a matrix where each element M_ij = a_i time conjugate(b_j). Since for any non-zero vector a, <a|a> is a real number > 0, and <Na|b> = conjugate(N)<a|b> and <a|Nb> = N <a|b>, then <a|b> has the properties of an inner product.

This relates to the quantum concept of measurement, because the measurements operations are of the form |a><a|, which operate on the state |b> like |a><a|b>, ands since <a|b> is just a number, M, then measurement takes the state from |b> to M times |a>. Measurement changes the state.

// Edit. What about Scott Aaronson's lecture?

StevenA

1st February 2007 - 05:38 PM

If everything is occuring, but an observer only detects events whenever they create a detectable state change for the observer, then events could be seen to have an associated rate, instead of a probability.

The average rate of an event occur could be greater than once per observation.

For example, when you see the hand of a clock pass by some number for an "instant" and then continue on, it wouldn't matter whether it remained for 1 "instant" or 1,000, it's only the visible changes in position that create your perception of time.

So imagine an observer interacting with the environment. Most changes in the environment would go undetected by the observer and not create any event perceivable as time. The observer could only detected the limited number of events that create a detectable internal state change for the observer (a small subset of the total change occuring).

So if you're watching a clock hand pass by some location but you're only seeing it in specific "snapshotted instants" of when internal changes occur for you, variations in time or clock rates between the two systems can occur (like time dilation when the rate of common time defining events is reduced between the two systems).

Anyway, so an observer could be limited to observing a compressed view of things as discrete events with specific probabilities of them occuring but those could be due to the observer sampling a system that possesses a much higher rate of interactions that could be well above the observed rate.

The question would be whether matter, like conscious perceptions, is capable of "overlooking" some events.

If you take a look at the quantization of energy that an atomic orbital responds to, you'll find thresholds. The photon receptors of your eye, or the color of light reflected off some material and the term "quanta" itself all indicate that detectable events are tuned to specific wavelengths/energies/frequencies. In quantum mechanics, the rotation of an object is not linear and continuous but instead stochastic and discrete - quantized.

So imagine instead that all possible events occur (and nothing is impossible

), of which an observer is limited to making one observation at a time and this naturally imposes observations of probabilities whereas the underlying reality is rates, which can extend well beyond 100% and be 200% ... etc.

When measuring these as probabilities though this extended range is compressed into a range of 0-100%, because of the bias induced by the observer making limited samples. Each such sample is a unit of time and has it's own associated rate, but truly this rate would also appear to be set physically relative to the rate of some other event transpiring.

So the probability of seeing a specific measurement result is determined by the underlying relative rate at which that class of event (as determined by whatever is making the measurements) versus the underlying rate at which other classes of time defining events occur and the only events capable of creating time for an observer are those that interact with it. Despite popular misperception, you can't freeze time in the environment and then proceed to walk around interacting with time frozen photons or air molecules etc. or you're altering them and violating the idea of them being frozen in time, and of course freezing time for the observer doesn't allow them to witness the world zipping around them in an instant because they aren't experiencing any internal changes that would allow this perception to occur.

I believe the compression from a continuous and potentially infinite value for rates, into discrete events with probabilities would operate much like a half-life or exponential decay where multiple identical results are compressed into a perception of only a single event occuring. The probability of seeing that event is determined by the probability of a sequence of events transpiring that includes it.

User posted image

The above image could be seen to predict the probability of some event occuring at some relative rate to another reference event. A simple way of seeing this is to flip the graph upside down (using 1-y instead of y as the vertical axis) and looking at the probability of NOT seeing an event as its rate of occurance is increased.

This formula generates an exponential of the form:

1-(e^-rt) for the probability of seeing an event occuring stochastically at a rate r or
1-(1-(e^-rt)) = e^-rt for the probability of NOT seeing that event occur.

If r is a complex value then you can can have periodic events. I'd assume this isn't externally imposed but internally imposed by an observer once you consider the effects that the requirement for memory has on what types of observations are coherently detectable. (Again, seeing an identical event occur more than once makes the duplicate occurances undetectable and so the appearance of virtual cancellations can subjectively appear and also for something to be seen as an objective and comprehensible class of event, it needs to possess a repeative and predictable component that make it definable ... of course this is also observer dependent and relies upon a memory as well)

To tie this in some with relativity consider that seeing a strong bias toward one type of event occuring, slows the perception of time because you're saturated with observing identical events, which appear merged as a single event.

For example, if took a very simple example of some external state being 50% likely to be in one state and 50% likely to be in a complimentary state, then perceived 'ticks' of time would only be available on the transitions between states and not during multiple identical states. So you'd always have a 50% chance of toggling to the other state and so your clock would be running, on average, at half the maximum (light speed related?) rate.

But let's say that something imposed a large bias to these probabilities and that you have a 90% chance of observing one state versus a 10% chance of seeing the alternate state (in this specific case, time would have no directionality and going "forward" or "backward" in time would have no meaning - every transition is forward or could be viewed as moving in a random direction throught time, depending upon what class of system you're trying to use it as a reference for).

The you'd experience transitions between these two states at a rate of 90% * 10% + 10% * 90% = 18%, instead of a 50% rate. It's x*(1-x) or x-x^2.

If these small scale motions through time were not directly correlated with macroscopic units, then the age of a system as determined by making a statistical measurement of the overall change in internal state, which would experience cancelations but retain an overall forward diffusion in a single direction through time proportional to the square root of the exposure time (brownian motion through time isn't directional with reference to individual units, but instead experiences a macroscopic diffusion outward or forward in time once relative measurements of distances are made between individual units - that's one manner in which time possesses a direction and this also creates the appearance of flows between pressure differentials and can give many fluid characteristics as well as creating the appearance of an invisible attractive force when you consider and observer is diffusing through space as well - if light was entirely consistant, and travelled in straight lines with constant velocity, then we shouldn't be limited to having statistical descriptions of when/where/what it is - imagine instead a pressure differential diffusing outward at some approximate rate, with more error at smaller scales and being detected when this difference passes by a detector and then disappearing once a new steady state condition is reached - you might not even detect the differential/photon if the detector wasn't in a state capable of detecting at that moment, so a virtualy sea of photons could exist beyond what's observed - detectors have a range of energies over which they detect quantum units)

Oh well, I ramble as usual and have to get to work but I couldn't resist posting on this thread because I love the subject.

Aerohead

1st February 2007 - 06:00 PM

Hi StevenA,

As usual, I like your "ramblings." Actually they're not that at all, IMO.

Interesting how we actually need "comparative state changes" to actually sense time, i.e., one "system" in which we're highly confident that no event aliasing occurs to compare against another system in which other (seemingly) unrelated "events" are of interest.

Yeah, I've got to get back to work too! Interesting thread here... ~Jim

StevenA

1st February 2007 - 06:46 PM

QUOTE (Aerohead+Feb 1 2007, 06:00 PM)

Thanks much. Yes, this is the stuff I've been thinking about for quite a while - how to derive properties that appear to agree with characteristics of the physical universe from basic logic and ideas in information theory (taking some hints from physical limits observed in the universe ... or in the mind).

Actually, to really do it right, you need at least 3 states and a memory. Being frozen witnessing a single state obviously doesn't allow for much to happen, but even witnessing the repeative cycling between two states (with multiple repetitions of each state being unobservable) isn't enough to do much of anything either. It seems for any useful physical reality to occur you need at least 3 possible fundamental states to be detectable (so no matter what the last state was you have 2 other states to select from for the next transition) and you also need to have a way to built a memory of these transitions in order to decode larger structures for space. For example, try to imagine how it would be possible to recognize and understand even something as fundamental as gravity without some ability to correlate relationships between experiences from moment to moment. It would be like to try to learn something with Alzheimers - the properties of space itself are learned and you couldn't communicate intelligently with anything that didn't learn a compatible representation of space because you wouldn't even be using the same physical structures to do it.

Here's something else to consider for the appearance of virtual cancellations - physical measurements are all relative. You need a reference in order to create a relationship with something else - a voltmeter has two inputs and it measures the difference between them, day only has a physical realization when there's a night to compare it with etc.

We normally think in logical terms of AND and OR but there's also the EXCLUSIVE OR operation that is true when either of the events is true, but not both.

Let's say for a second that a photon is detected whenever an energy differential exists, and is observed, between two points.

The wave function has both a negative and positive component but we only physically measure the square of this intensity (which interestingly happens to appear as having half the wavelength, but that's not important right now).

So let's say some probability wave was passing by some point in one direction, while an identical wave was passing in the opposite direction past this point.

In this case, the probability of both ends of this detector would increase symmetrically and no detectable differential would exist, but instead the detector would be riding at a higher baseline for that probability (that would be an invisible complex component or state, not physically detectable, yet potentially statistically measurable by comparing other observations elsewhere).

If you convert these boolean relationships into algebraic ones, you get:

prob(A AND B) = prob(A)*prob(B)

prob(A OR B) = prob(A)+prob(B) - prob(A)*prob(B)
(notice that for rare events the prob(A)*prob(B) term could easily go unnoticed because it's a second order effect, so it can easily appear linear)

prob(A XOR B) = prob(A)+prob(B) - 2*prob(A)*prob(B)
Here you can actually have both events occur simultaneously but not see either.

If you have A and B take the range of +/-1 instead of 0 to 1, then the XOR operation simplifies to pure multiplication.

Anyway, the virtual cancellations would seem likely to arise from the opposite direction and from the detector/observer itself and the resulting observations and rates of times would occur as these internal differences are detected and equalized (for every action there's an equal and opposite reaction, except for creative events) and no more information/energy is available to flow. (That may be why the perception of a will is associated with consciousness as well, because something needs to be able to generate changes so that the flow of these can be sensed, or at least have a manner in which an observer can be internally altered in order to "absorb" an event ... I'm just thinking out loud, trying to piece the picture together in a way that makes sense though maybe some of this will 'click' for someone else)

But let's say that an electron is orbitting a nucleus and is effectively generating it's own wavefunction. If a wave passes by that alters the background in the same phase as the electron, then no differential force necessarily occurs and they don't interact, whereas when this wave is out of phase, a strong differential exists between the two and an "observer" is very likely to measure a variety of local states - and seeing this variety would allow for time defining observations to occur.

If we look at the intensity, it's proportional to the square of wave function. Let's say we were observing a single point in space over two slices in time. If the baseline probability of observing some event at a specific moment was p, then probability of seeing it twice in a row would be p^2. If it takes two values in order for a physical differential measurement to occur, then "observers" could be seen as not making measures over some vague measure of "local space" but instead over some distance in time at a point. The average equivalent light speed distance this would be similar to would depend upon the overall density of local transitions between space (if space was thoroughly innudated with event class A only, then time stops and all observers are experiencing a moment witnessing class A ... it could last eons but it would appear as a single event, this automatically provides a renormalization that makes infinities appear as nothing because it's only the contrasts that are detected and as soon as you have even a single class B event around, then that infinitesmal event can create true eons of subjective time).

Dang, I have a lot of fun musing over these ideas but my wife is going to be giving me dirty looks if I don't head off to work LOL!

Confused2

2nd February 2007 - 12:43 PM

QUOTE (rpenner+)

Edit. What about Scott Aaronson's lecture?

Sorry about the delay.. IMHO it (the lecture) seems inspired and inspiring ..given a second chance I would have started with it (rather than the axioms). I think Stephen Hawking drew attention to the rule that you lose half your readers every time you include an equation and this might be what has happened here (7+ equations in the lecture). For me, and I suspect a few others, a little elementary linear algebra would work wonders. I suspect I (we, the 'others') need (at least) to get pretty confident about the meaning of the complex conjugate of a row vector. See Wiki ( http://en.wikipedia.org/wiki/Complex_conjugate ) . As a result of many years spent changing light bulbs and connecting electrical things together I am pretty sure that we don't need all of the stuff in the Wiki article to get our feet onto the first rung of the ladder. A lot of it looks pretty much like the sort of phase and power calculations I am used to .. and mathematics for engineers generally neither assumes nor requires very much beyond monkey see monkey do. IF I am right (?) then anyone who can wire up a plug ought to be able to get through most of Aaronson's lecture.. if they wanted to. I could (of course) be completely wrong. Meanwhile .. a physical significance for the complex conjugate ?

I suspect .. if it were a rotating vector.. rotatiing the other way. Impedance mismatches ... hm...

rpenner

2nd February 2007 - 07:12 PM

Complex Numbers
The Complex numbers are just the Real numbers along with i, a mysterious entity which is to be treated like a number. It obeys i² = -1.
Obviously 0 + i = i + 0 = i and 1 × i = i × 1 = i and -1 × i = i × -1 = -i and 0 × i = i × 0 = 0 = i + -i = -i + i and i + i = 2i, etc. So it follows all Complex numbers can be written as a + ib, where a and b are any Real numbers.

The basic rules of addition apply, allowing us to write.
( a + ib ) + ( c + id ) = ( a + c ) + i (b + d).
Multiplication is only slightly more tricky because i² = -1.
( a + ib ) × ( c + id )
= a × ( c + id ) + ib × ( c + id )
= a × c + i ( a × d ) + i( b × c ) + i² ( b × d )
= a × c + i ( a × d + b × c ) - b × d
= ( a × c - b × d ) + i ( a × d + b × c )

Complex Conjugate
But, (and this is important) the equation x² = -1 has two solutions over the Complex numbers. Both i and -i will solve it. In fact for any polynomial with only Real coefficient, if a + ib is a root of this polynomial, so is a - ib. The mapping between a + ib and a - ib is obviously a well-defined one-to-one function which happens to be its own inverse. Let us designate this operation (complex conjugation) as conj(x) or abbreviate the complex conjugate as x* = conj(x).

Then we can do things like:

Extract the real part of x. Re(x) = (x + x*) / 2.
Extract the imaginary part of x. Im(x) = (x - x*) / 2.
Prove exp(x*) = exp(x)* MetaMath: efcj
Compute the magnitude of x, |x| = √(x* × x )

Lets go into that last one a bit more.
|x| = √(x* × x )
so
|a + ib|
= √(( a + ib )* × ( a + ib ))
= √(( a - ib ) × ( a + ib ))
= √( a² + aib - aib - i²b² )
= √( a² - i²b² )
= √( a² - (-1)b² )
= √( a² + b² ) Since a and b are both real numbers, this means x* × x is always a real number
= √( r² )
= r -- the Euclidean (Pythagorean, 2-norm, etc) distance between 0 and x.

It obeys |x × y| = |x| × |y| which gives us our final formula:

The distance-angle representation of complex numbers

For each complex number, x there exists a real numbers r and θ such that x = r × exp(iθ) which is useful since x × y = r_x exp(iθ_x) r_y exp(iθ_y) = (r_x r_y) × exp(i (θ_x + θ_y)).

Also, (r × exp(iθ))* = (r × exp(iθ)*) = (r × exp((iθ*))) = r × exp(-iθ). So complex conjugation is simple in this representation of the complex numbers, also.

For complex multiplication, the distances to zero multiply and the angles add.

Back on track

But in order to create expressions like √(x* × x ), we need the complex conjugation function (or operator) which is not definable in terms of operations like +, -, ×, etc.

Dual vectors

If |x> is a complex column vector, it is much like a n×1 matrix. We define its dual, <x|, as the transpose and complex conjugate of |x>. So <x| is a row vector, 1×n, and all of its elements are complex conjugates of those in |x>.
So when we do the product <x|x> we are doing matrix multiplication to get a 1x1 matrix, and if x_i are all the elements of |x>, then the result is a real number which is Σ x_i* × x_i = Σ |x_i|² = Σ (a_i)² + Σ (b_i)² = Σ (r_i)² which is the square of the "Euclidean distance" which separates |x> from the zero vector.

Confused2

2nd February 2007 - 10:51 PM

I have asked rpenner for clarification and he's given it .. I guess the same rules apply to anyone else .. please don't feel you're holding me up if you want to understand this and would like help.
Complex Numbers
Happy.
Complex Conjugate
Flip the sign of the complex part .. the power of this is shown in the examples.
I've done my own test of exp(x*) = exp(x)* (see at the end) .. hopefully MetaMath (!!!) is a unnecessarily rigorous for present purposes.
The distance-angle representation of complex numbers
Happy
Back on track
I see the potential usefulness of flipping the sign of the imaginary part
Dual vectors
As long as I'm not expected to transpose an mxn matrix this looks fine.
C2.

Show (loosely!) exp(x*) = exp(x)*
In words " e to the complex conjugate of x = complex conjugate of e to the x "

1/ Let x be a + ib
2/ conjugate is a - ib
3/ e^(a+ib) = (e^a) (e^ib) <- the original
4/ e^(a-ib) = (e^a)(e^-ib) <- the test conjugate
5/ Coffee arrived
6/We know e^ix = cos(x) + isin(x) (Euler http://en.wikipedia.org/wiki/Euler%27s_formula ) and e^-ix = cos(x) - isin(x)
7/call e^a = k because its a real number and its just a factor
8/ from 3 (the original) we get
9/ k(cosb + isinb)
And from 4/ (the conjugate test) we get
10/ k(cosb - isinb)
From 9/ and 10/ we see the imaginary part has flipped sign .. which is (by definition) the complex conjugate.
Criticism welcome.

Solid State Universe

2nd February 2007 - 11:33 PM

If one were to construct a 'complex' magnetic field using a gyroscope... similar to the one built in the movie Contact (but on a much smaller scale), would the resultant magnetic field be said to have an 'imaginary' component in it's complexity?

rpenner

3rd February 2007 - 12:03 AM

QUOTE (Solid State Universe+Feb 2 2007, 11:33 PM)

You are using the work "complex" to mean nested or non-trivial, which is not the same sense of the meaning of the term as in "Complex numbers" where it is frequently capitalized because it is the name of a unique object, and therefore a proper name.
The magenetic field is a function of time and position. It is a real vector value, not a Complex number. Therefore it has no imaginary component.
If you use the tensor-notation formulation of Maxwell's equations, the electromagnetic field is everywhere real-valued and also has no imaginary component.
Quantum physics restricts us to measuring only real-based eigenvalues of various Hermitian operators. The result is quantum physics predicts no meausured quantity would have a complex measurement. This is good because as I indicated above there is no mathematical reason to a distinction between i and -i. If we could distinguish between them in a physical measurement, say be making a measurement with a Complex value, then the result would be inconsistant with the Complex numbers. It would be a different math.

Solid State Universe

3rd February 2007 - 12:17 AM

Well, this is what I'm wondering.

If the magnetic field generated by each of the spinning planes of the gyroscope propagates at a 90 degree angle to each of the gimbals as they spin and rotate, operating the gyroscope along three seperate axis should produce a magnetic field that propagates in a 4th dimension, 90 degrees away from the three being manipulated by the spinning gyroscope.

That's what I meant by complex.

rpenner

3rd February 2007 - 01:14 AM

Maxwell would not agree with you. He would say each of the three rings is a solenoid, which can be approximated by a simple current loop. Each current loop by itself has a well-defined magnetic dipole field. The total field is just the sum of three magnetic dipole fields, which at distances large relative to the outermost ring can be approximated by a single dipole. The fields literally sum like vectors.

At the exact center of the three, nested current loops, the field is

B = (μ_0/2) (I_x/R_x,I_y/R_y,I_z/R_z)

where I_w is the current in the loop perpendicular to the w-axis and R_w is it's radius.

http://hyperphysics.phy-astr.gsu.edu/hbase...tic/curloo.html
http://www.netdenizen.com/emagnet/offaxis/iloopoffaxis.htm

Confused2

3rd February 2007 - 02:08 PM

The proposal to be tested is whether or not the mathematical foundations of QT are sufficiently simple that it can reasonably be described as a 'simple theory'. Whether it works or whether or not we like it should not affect our judgment of the mathematical simplicity.
http://www.scottaaronson.com/democritus/lec9.html

QUOTE

The second way to teach quantum mechanics .. starts directly from the conceptual core -- namely, a certain generalization of probability theory to allow minus signs. Once you know what the theory is actually about, you can then sprinkle in physics to taste, and calculate the spectrum of whatever atom you want.This second approach is the one I'll be following here.

QUOTE (->

QUOTE

I suggest we add one point for every bit that everybody understands, a bonus point for every (relevant) question that is answered to the satisfaction of the questioner, and subtract one point for every (genuine) claim .. "I don't understand".
Why are we doing this? .. Schrodinger's cat might help http://en.wikipedia.org/wiki/Schr%C3%B6dinger's_cat ?
So far rpenner has successfully clarified some of my problems with the maths and there are no complaints so that's 1 to the proposition.
From the lecture ( still with http://www.scottaaronson.com/democritus/lec9.html )
1/ We can express the probabilities of those events by a vector of N real numbers: (p1,....,pN)
Well, the probabilities had better be nonnegative, and they'd better sum to 1. We can express the latter fact by saying that the 1-norm of the probability vector has to be 1. (The 1-norm just means the sum of the absolute values of the entries.)

[C2].. obviously he's talking about something with a total probability of 100% .. I wish I could think of of something more cheerful than death. If we listed the probability of each cause of death then obviously it would add up to 100%. If we made a list of probable causes of death we'd get a (p1,..pn) type thing. OK so far.
With some lack of continuity the lecturer discards the (p1..pn) vector and restricts discussion to a True/False situation... I think we've moved on to dead or alive cats. If the probability of it being alive is p then the probability of it being dead must be 1-p , can't get much simpler than that. The nice man tells the monkey to replace p with a^2 and (1-p) with b^2 so instead of p+1-p = 1 we get sqrt(a^2+b^2) = 1. On the assumption that we have virtually no memory (reasonable) we are reminded that a^2 is the probability of a 0 (dead cat?) and b^2 is the probability of a 1 (live cat?). I assume we retain the option of writing a^2 and b^2 as a column vector .. but we're going to do something else now.

QUOTE

This "2-norm bit" that we've defined has a name, which as you know is qubit. Physicists like to represent qubits using what they call "Dirac ket notation," in which the vector (α,β) becomes a|0> + b|1> α is the amplitude of outcome 0, and β is the amplitude of outcome 1.

Unless there are any problems, simple theory scores about 3 with nothing difficult so far.
Is this good/useful/bad ? .. comments welcome.
-C2.

Confused2

4th February 2007 - 12:01 AM

Continuing with http://www.scottaaronson.com/democritus/lec9.html

I assume we're all happy to let physicists use pointy brackets and call things qubits if they want to.

QUOTE

So given a qubit, we can transform it by applying any 2-by-2 unitary matrix -- and that leads already to the famous effect of quantum interference. For example, consider the unitary matrix [ this is an attempt to reproduce a graphic from the lecture ]
[ 1/sqrt(2),-1/sqrt(2) ]
[ 1/sqrt(2),1/sqrt(2) ]

Unfortunately I can't really remember how to multiply matrices .. clearly we want to multiply the [a,b] by this unitary matrix.
My first attempt at this would give the result ..
[ a/sqrt(2),-b/sqrt(2) ]
[ a/sqrt(2),b/sqrt(2) ]
If a's and b's were the amplitudes for dead and live cats it looks like I've spread my cats out .. and I don't know how to collect them back up into a qubit. I could try a thing called Cramer's rule ( http://en.wikipedia.org/wiki/Cramer's_rule ) but this might be a cruel and unnatural thing to do to a cat.
Assistance please .. to carry out the above matrix multiplication and (I assume) end up with a new qubit. .
Please note ..the simplest applicable rule will suffice.
-C2.

Edit ..
The 'Cover up rule' ?
http://www.engin.brown.edu/courses/en3/Not...s9/vectors9.htm .. monkey see monkey do. All we need is approval that it gives the right answer. (Never try to calculate anything unless you already know the answer - Wheeler?)

Solid State Universe

4th February 2007 - 12:05 AM

QUOTE (rpenner+Feb 3 2007, 01:14 AM)

Ummm... this 'w' axis... I'm assuming it's perpendicular to the x,y and z axis in our discussion?

Confused2

4th February 2007 - 12:18 AM

QUOTE (SSU+)

Ummm... this 'w' axis... I'm assuming it's perpendicular to the x,y and z axis in our discussion?

Wouldn't the vector addition of any vectors in three dimensions give a resultant vector (still) in three dimensions? The right (or left) hand rule is a three dimensional item .. takes two orthogonal fingers and gives you a third finger at right angles to both but still within three dimensions.. unless you've got very unusual hands.

-C2.

Solid State Universe

4th February 2007 - 12:21 AM

I'm not sure. Thats why I'm asking.

I'm just wondering... a gyroscope of electromagnetic fields would be pouring energy into the center of the gyroscope. Where does that energy go?

I'm curious because I've been looking at the magnetic traps used in BEC formation and wondered what would happen if you placed a BEC in the center of a magnetic gyroscope.

AlphaNumeric

4th February 2007 - 12:24 AM

QUOTE (Solid State Universe+Feb 4 2007, 01:05 AM)

Ummm... this 'w' axis... I'm assuming it's perpendicular to the x,y and z axis in our discussion?

No, he means w is either x, y or z.

Just because you have 'another' dimension doesn't mean it's automatically related to 'i' where i^2 = -1. Vast chunks of differential geometry are dedicated to the study of R^n, that being n dimensional real spaces. You have e_1, e_2, ...., e_n basis vectors (if n=3 you might be more familiar with x,y,z or i,j,k).

Adding another dimension to give you e_(n+1) doesn't mean e_(n+1) is complex, it's utterly equivalent to e_1, ..., e_n. Infact, a simple change of basic transformation makes them all inter equivalent in R^(n+1).

As I told you before, the extra dimensions in string theory are not 'complex' in the sense they have a basis system involving the square root of -1. Relativity is a Pseudo-Riemannian manifold based theory over a Real manifold so both SR and GR work with 'Real' dimensions, therefore quantum field theory (and Dirac's work) work over the same Real dimensions. There was no complexifying or 'imaginifying' (it's not a word but you see what I mean by it) of dimensions by Dirac or other QFT authors. They went from considering time and space on a Euclidean metric to a Lorentzian one.

There is also plenty of work done on complex valued basis systems. I recommend Nakahara - "Geometry, Topology and Physics", Chapter 8. It covers hermitian manifolds and Kahler geometries in a very good way. I've spent the last week or so reading through much of the book and it gives plenty of explaination.

Though somehow I doubt you'll bother trying to find it or anything similar....

rpenner

4th February 2007 - 04:37 AM

QUOTE (Confused2+Feb 4 2007, 12:01 AM)

Close, you just need to sum up some values.
Let M =
[ a b ;
c d ], which is a n×m (2×2) matrix. Let |V> be
[ e ;
f ] which is a m×o (2×1) column vector, then the rule for computing M|V> is to create a n×o matrix or vector where every element of the matrix is a "dot product" between a row of the first matrix and a column of the second matrix. (This forces the "m" dimension, which is 2 here, to match, otherwise the multiplication is not allowed.)

So M|V> is a 2×1 column vecotor which is
[ a×e + b×f ;
c×e + d×f ]

For your case,
M = [ 1/sqrt(2) -1/sqrt(2) ;
1/sqrt(2) 1/sqrt(2) ]
and
|V>= [a;
b]
so the result is
[ a/sqrt(2) -b/sqrt(2);
a/sqrt(2) + b/sqrt(2) ]
= (sqrt(2)/2) times
[ a - b;
a + b ]

Note that M² is
[ 0 -1;
1 0 ] (Prove it!)
and since M(M|V>) = (MM)|V> = M²|V>, applying M twice turns
|V> =
[ a ;
b]
into
M²|V> = [
-b ;
a ]

In this semi-colon notation borrowed from MatLab and the like, <V| (a row vector)
is [ a* b* ] so following exactly the same matrix multiplication rule, <V|V> is
a*×a + b*×b = |a|² + |b|²

QUOTE (Solid State Universe+Feb 4 2007, 12:05 AM)

Ummm... this 'w' axis... I'm assuming it's perpendicular to the x,y and z axis in our discussion?

Nope. w is just a label used to explain what the various I's and R's mean.

QUOTE (Confused2+Feb 4 2007, 12:18 AM)

Correct. Although there is a Science Fiction novel by Robert Heinlein called The Number of the Beast which makes a similar confusion with the cross product and gyroscopes.

QUOTE (Solid State Universe+Feb 4 2007, 12:21 AM)

I'm just wondering... a gyroscope of electromagnetic fields would be pouring energy into the center of the gyroscope. Where does that energy go?

Energy "going" somewhere is called "Power" not Energy. An analysis of the rings shows that if they remain mutually perpendicular, no energy changes hands. This of course neglects mechanical friction and electrical resistance.

Solid State Universe

4th February 2007 - 08:52 AM

I'm not talking about a gyroscope where the rings remain perpendicular. I've read Heinlein, I know what you're talking about.

Did you see the movie Contact? That was a Carl Sagan book.

User posted image

What kind of magnetic field would a superconducting array like that make if each ring were spinning like that?

Would that qualify as a 'complex' magnetic field?

AlphaNumeric

4th February 2007 - 09:48 AM

It qualifies as a complicated field, not a 'complex' as in "Square root of -1" field.

In some cases, particularly electrical engineering applications, the complex component of the plane wave e^(i[vt-wt]) is considered but not as a literal physical thing, but simply as the orthogonal part of a 2d wave form.

Solid State Universe

4th February 2007 - 10:06 AM

QUOTE

In some cases, particularly electrical engineering applications, the complex component of the plane wave e^(i[vt-wt]) is considered but not as a literal physical thing, but simply as the orthogonal part of a 2d wave form.

Now that was news to me. Thank you kindly for that one, Alpha.

Bit off topic... but would you care to speculate as to the effects one might produce within a condensate if one were to subject one to such a 'complicated' field?

Just speculation, mind you... but it would be a curious experiment.

Confused2

4th February 2007 - 12:28 PM

Rpenner,
Many thanks for your assistance and extraordinary patience. A third of the way through the lecture (by my reckoning) the score stands at about 6 ordinary and 2 bonus points on the simple side , 0 on the difficult. I have to do some work (new kitchen) for She Who Must Be Obeyed but will return to the evaluation exercise as soon as possible. I suspect we have reached the stage when it might be better to use heads and tails (coins) rather than cats. Anyone with a comparable complete absence of ability and suitable qualifications is invited to continue the evaluation in my absence.
-C2.

Confused2

4th February 2007 - 08:06 PM

Belatedly I find http://www.sosmath.com/matrix/matrix1/matrix1.html ( SOS .. that's me)

(There might be a better notation at http://mathworld.wolfram.com/MatrixMultiplication.html but I'd rather stay with the simple stuff as long as possible. )

Your post was not wasted because we have a notation for these columns and rows and now I have to prove the M^2 result

adopting ';' as meaning 'next row' . Checking that 0| denotes a row vector and |0 denotes a column vector.

Find M^2 where
M = [ 1/sqrt(2) -1/sqrt(2) ; 1/sqrt(2) 1/sqrt(2) ]

From http://www.sosmath.com/matrix/matrix1/matrix1.html ..
MxN = [a b; c d] x [ e f;g h] = [ae+bg af+bh; ce+dg cf+dh]

so .. temporarily using a for 1/sqrt(2)

M = [ a -a ; a a]
----- a' b c d <- cheat!
-------e f g h
M^2 = [ a -a; a a] x [ a -a; a a]
= [ aa-aa a(-)a+(-)aa ; aa+aa a(-a)+aa]
= [0 -1; 1 0] (

)

-----------------------
For M x [a;b]
you say
(Eq 1) (sqrt(2)/2) times [ a - b; a + b ]
sqrt(2)/2 !

Testing this operation on a [1;0] vector ..
Hopefully (?) with the significance that the amplitude of a 0 (say tails) outcome is 1 and that of a 1 (heads) outcome is zero.
from your Eq 1 we get
1/sqrt(2) [ 1 ;1 ]
(Eq 2). = [1/sqrt(2) ; 1/sqrt(2) ] .. which looks very 45 degrees ish
We're pure real ( a = a* ) giving the probability of a 0 (or tail) = 1/2 .. being (a* times a)
Likewise the probability of a 1 (or head) = 1/2 .. being (b* times B )
applying M to Eq 2 ..(enthusiasm) we get
(Eq 3) 1/sqrt(2) [ 0 ; 2/sqrt(2) ] = [0;1] .. which is 90 degrees ish
applying M to Eq.3 ..(more enthusiasm) we get
(Eq 4) [-1/sqrt(2) ; 1/sqrt(2) ] 135degrees .. negative amplitude for 0 (tails) but positive probability.( 1/2 )
So far I haven't actually seen any complex numbers in the lecture and the use of the '<' has been trivial .. I hope I haven't missed something.

Anybody who is not happy (especially rpenner but not She Who Must Be Obeyed ) .. comments most welcome.
-C2.

rpenner

5th February 2007 - 07:47 AM

It's astute of you to identify this operation with rotation.

Begin optional exposition

The complete world of 2×2 complex matrices with non-zero determinant form a representation of the General Linear group in 2 dimensions, called GL(2,C).

There is a subgroup of GL(2,C) which has the property that for every member, A, the transpose of the conjugate is also the matrix inverse ( transpose(conjugate(A)) = transpose(conjugate(A)) = A†, and A† A = A A† = [1 0; 0 1]. It's a subgroup because if A and B have this property, then so does AB. This subgroup is called the Unitary group in two dimensions. U(2).

Unitary groups are especially important, because any finite algebraic group can be represented as a subgroup of a Unitary group.

Closely related to U(2) is the subgroup of U(2) where the determinant is exactly 1. This is the Special Unitary group, SU(2).

Also closely related is the Orthogonal group, O(2,C), where for each element, A, the transpose is also the matrix inverse ( transpose(A) A = A transpose(A) = [1 0; 0 1]. ).

Finally, there is the Special Orthogonal group over to the reals, SO(2,R).

So, Special means Determinant(A) = 1, Orthogonal means transpose(A) = inverse(A), and Unitary means conjugate(transpose(A)) = inverse(A).

End exposition

Every member of SO(2,R) can be written as
[ cos(a) -sin(a); sin(a) cos(a)] which is exactly what we think of a Euclidean rotation in two dimensions.

But over the complex numbers, U(1,C) can be parameterized as [ e^ia ] which has the same multiplication rule as SO(2,R). Thus U(1,C) is isomorphic to SO(2,R).

Confused2

5th February 2007 - 03:11 PM

A mind-biggling post. A while back Good Elf gave a link to the Standard Model with SU's etc. written in .. I can't find it quickly (that damn kitchen) .. it might be interesting to see it (or something like it) again so I can decide whether or not you've just zapped me with what I think you've just zapped me with. Since e^ix is cyclic .. that would suggest U(1,C) is 'closed' (if that is the right word) .. hence also SO(2,R) .. yes? Leaving .. how can I possibly be expected to work on a kitchen?
-C2.

rpenner

5th February 2007 - 06:04 PM

All groups are "closed," which just means that if A and B are in the group then so is C = A B.

I think you mean "cyclic" as all the finite cyclic groups Z_2, Z_3, Z_4,... Z_N .... are isomorphic to a subgroup of U(1,C), with the value of the map of an element k is e^2πik/N.

To see this for Z_12 (clock arithmetic), List the elements of Z_12
0,1,2,3,4,5,6,7,8,9,10,11
and the law of composition (in this case, written with a plus sign)
example: 7 + 9 = 4
and the mapping to U(1,C)
e^2πi7/12 × e^2πi9/12 = e^(2πi7/12 + 2πi9/12) = e^2πi16/12 = e^2πi4/12
and the mapping to SO(2,R)
[ cos(2πi7/12) -sin(2πi7/12) ; sin(2πi7/12) cos(2πi7/12) ] [ cos(2πi9/12) -sin(2πi9/12) ; sin(2πi9/12) cos(2πi9/12) ]
= [ cos(2πi7/12)cos(2πi9/12)-sin(2πi7/12)sin(2πi9/12) -(cos(2πi7/12)sin(2πi9/12)+sin(2πi7/12)cos(2πi9/12)) ; sin(2πi7/12)cos(2πi9/12)+cos(2πi7/12)sin(2πi9/12) cos(2πi7/12)cos(2πi9/12)-sin(2πi7/12)sin(2πi9/12) ]
= [ cos(2πi4/12) -sin(2πi4/12) ; sin(2πi4/12) cos(2πi4/12) ]

http://www.sosmath.com/trig/Trig5/trig5/trig5.html

Confused2

6th February 2007 - 04:39 PM

OK.
What now? Consolidation? Challenge? New territory?

I think engineers generally like to have something solid to hang on to (a bar is ideal) .. anything too abstract and they lose concentration. If there are inputs then it's nice to have them connected to something ('driven'), same goes for outputs.. they like to have an oscilloscope on the outputs so they can play about with the inputs and watch what happens.

It might be safe to assume that e^ix is a little oasis of competence in an otherwise barren desert.

biggling=boggling

-C2

Confused2

7th February 2007 - 10:53 PM

Surely there must be others willing to test the claim that quantum theory is 'simple'. Not only but also .. I seem to be the only one who regards this thread as an opportunity that is too good to miss. It is perhaps unfortunate that there were a few equations in the original paper and on this thread rpenner has clarified a few definitions. Since there would be little point in continuing if those definitions were not understood rpenner (quite reasonably) suggested trying them out .. hence a bit more arithmetic. Hopefully anyone who was not happy with those steps would have posted 'help!' just as I did. It is in the nature of 'simple' that you can do it if you want to .. but it might only seem simple after you've made a tiny effort.

In the UK 'matrices' were introduced into the school curriculum a few years after I left .. there is every chance that a ten year old would be better qualified than I am to continue with this test of simplicity however ..
.
From the optional exposition..

QUOTE (rpenner+)

The complete world of 2×2 complex matrices with non-zero determinant form a representation of the General Linear <http://en.wikipedia.org/wiki/General_linear_group> group in 2 dimensions, called GL(2,C).

In my limited experience of matrices I think they have always been what I would call operators .. they map one point (or vector) to another. It seems they can also lead independent lives .. unless anyone else wishes to pursue this option then I would prefer to leave it for the present.
.
From the non-optional exposition

QUOTE (rpenner+)

Every member of SO(2,R) can be written as
[ cos(a) -sin(a); sin(a) cos(a)] which is exactly what we think of a Euclidean rotation in two dimensions.

But over the complex numbers, U(1,C) can be parameterized as [ e^ia ] which has the same multiplication rule as SO(2,R). Thus U(1,C) is isomorphic to SO(2,R).

Guessing that U[1,C] is a one dimensional complex matrix the claim would seem to be that
(1) We have [e^ia] which is sneakily equivalent to [Re(e^ia),Im(e^ia)]
see http://en.wikipedia.org/wiki/E_%28mathematical_constant%29
(2) since e^ia = cos(x) + i sin(x)
(3,4) we see [e^ia] = [cos(x),isin(x)] or possibly [cos(x); isin(x)]
If we took the magnitude of this we'd get
(4a) cos^2(x) + sin^2(x) .. which we all know is 1. I forget how I know that but I do.
incidentally from the wiki entry .. e^ia ..described by Richard Feynman as "[...] the most remarkable formula in mathematics [...], our jewel". Even before seeing that I was considering including an aside to the effect that 'Lord of the Rings' (Tolkien) ( http://en.wikipedia.org/wiki/Lord_of_the_rings ) was an allegorical account of an engineer (Gollum) who had 'lost' e^ia .

We're testing (1) against 'SO(2,R)' .. a two dimensional real matrix of the form
given by rpenner as
(5) [ cos(a) -sin(a); sin(a) cos(a)] .
SO means Special Orthogonal
To be special it satisfies the condition that Determinant(A) = 1. see http://www.sosmath.com/matrix/determ0/determ0.html
Evaluating (2) (ad-bc) gives
(6) cos(a)cos(a) + sin(a)sin(a) or sin^2(a) + cos^2(a) .. which hopefully we know is 1 (ask if not sure)
To be orthogonal it satisfies the condition that ..
(7) The matrix times it's transpose is equal to the unitary matrix. I'll let that one go for the present.

QUOTE

But over the complex numbers, U(1,C) can be parameterized as [ e^ia ] which has the same multiplication rule as SO(2,R). Thus U(1,C) is isomorphic to SO(2,R).

Incoming .. multiplying any matrix is pretty tricky (for me) .. multiplication by a complex matrix is 'unclear'.
-C2

AlphaNumeric

7th February 2007 - 11:26 PM

QUOTE (Confused2+Feb 7 2007, 11:53 PM)

This thread is one of the most mathsy I've ever seen on this forum. Given the higher than typical ratio of cranks on this physics site, I'm not suprised this thread gets little attention, despite being of a very interesting topic. Most cranks run at the thought of anything more complex than trigonometry. Introductory group theory is another few levels above that!

QUOTE (Confused2+Feb 7 2007, 11:53 PM)

Matrices and their extensions (since they, along with vectors, are just specific forms of tensors) make up vast areas of maths and you can't do theoretical physics without them (at least if you're mainstream, cranks get by often without calculus!).

Just as vectors can form 'vector spaces', you can do the same with many groups of matrices. Add two matrices together, you get another matrix. Multiply a matrix by a scalar, you get another matrix.

They are operators in the sense they generate linear maps from a vector space (of vectors or even matrices) to another space. The most interesting (and physically meaningful) ones are those which map a space to itself, like rotations.

I could (literally) go on for hours about their applications and intricacies. Though I initially cared little for group theory, having seen just how much linear algebra there is in physics and that it's actually much more elegant than I initially thought, I now find it a fascinating topic (call me sad...).

QUOTE (Confused2+Feb 7 2007, 11:53 PM)

(1) We have [e^ia] which is sneakily equivalent to [Re(e^ia),Im(e^ia)]

A 1x1 martrix is simply a number. You also have to be careful when moving from a real to a complex representation and back. You go from a complex vector space [e^ia] to a real one [Re(e^ia),Im(e^ia)] without making the distinction in your notion. It's obvious from your context but if you were doing a lot of algebra, it might be easy to mix. Just having subscripts R or C is enough, so if you see [X]_R, you know that X is real, not complex.

QUOTE (Confused2+Feb 7 2007, 11:53 PM)

Incoming .. multiplying any matrix is pretty tricky (for me) .. multiplication by a complex matrix is 'unclear'.

It's plain to see that both SO(2) and U(1) (dropped the R and C because they aren't neeed, U means complex, O means real) are both parameterised by a single variable, a. If you can show their group structure is the same, then they are isomorphic.

Consider two elments of U(1). e^(ia) and e^(ib) Compose them to give you a tird element, e^(ic) = e^(ia)e^(ib) = e^(i[a+b]).

Taking real and imaginary parts of the equation e^(ia)e^(ib) = e^(i[a+b]) gives you

cos(a+b) = cos(a)cos(b)-sin(a)sin(b)
sin(a+b) = cos(a)sin(b)+sin(a)cos(b)

Now consider two elements of SO(2). You have the property that

[ cos(a) -sin(a); sin(a) cos(a)] [ cos(b) -sin(b); sin(b) cos(b)] = [ cos(a+b) -sin(a+b); sin(a+b) cos(a+b)]

Multiply out the left hand side and you end up showing again that

cos(a+b) = cos(a)cos(b)-sin(a)sin(b)
sin(a+b) = cos(a)sin(b)+sin(a)cos(b)

by comparing matrix entries.

You also have that e^(ia)e^(ib) = e^(ib)e^(ia) and that
[ cos(a) -sin(a); sin(a) cos(a)] [ cos(b) -sin(b); sin(b) cos(b)] =
[ cos(b) -sin(b); sin(b) cos(b)] [ cos(a) -sin(a); sin(a) cos(a)] (this basically says rotating by an angle a then by an angle b is the same as doing b than a) and that |e^ia| = 1 and det [ cos(b) -sin(b); sin(b) cos(b)] = 1.

Therefore, the groups share all the same defining properties therefore they ARE the same group, up to how you describe to represent it. There's tons of work done on how you represent some groups, because as with a lot of things in physics, your choice of description can sometimes turn a horribly complicated system into a simple one, just as polar coordinates are nicer for circular motion than cartesians.

rpenner

8th February 2007 - 06:33 AM

QUOTE (Confused2+Feb 7 2007, 10:53 PM)

Guessing that U[1,C] is a one dimensional complex matrix the claim would seem to be that
(1) We have [e^ia] which is sneakily equivalent to [Re(e^ia),Im(e^ia)]
see Wikipedia: E (mathematical constant)
(2) since e^ia = cos(x) + i sin(x)
(3,4) we see [e^ia] = [cos(x),isin(x)] or possibly [cos(x); isin(x)]

1,3 and 4 confuse the concepts of Complex Numbers, which are supposed to be treated like numbers with matrices. Perhaps this is my fault, since I depicted a 1×1 Complex matrix. e^ia = cos(a)+isin(a) -- which is a statement about 1 real number (a) and 2 Complex numbers (e^ia) and (cos(a)+isin(a)).
The Complex Number post
None of [Re(e^ia),Im(e^ia)], [cos(x),isin(x)], (1×2 matrices) or [cos(x); isin(x)] (a 2×1 matrix) can fill in for a Complex number. No matrix can, because matrices have to agree in (one) dimension for multiplication to work. You can't multiply two 1×2 matrices.
So (1) I would re-write as [e^ia] = [Re(e^ia) + i Im(e^ia)] with a note that a is Real, not complex.
but (3) [e^ia] = [cos(a) + i sin(a)] just to be a reminder.

QUOTE (Confused2+Feb 7 2007, 10:53 PM)

If we took the magnitude of this we'd get
(4a) cos^2(x) + sin^2(x) .. which we all know is 1. I forget how I know that but I do.

Because the geometric interpretation of sine and cosine are the sides of a right-triangle. And Pythagoras applies.
Wikipedia: Trigonometric function
Wikipedia: Pythagorean trigonometric identity

QUOTE (Confused2+Feb 7 2007, 10:53 PM)

incidentally from the wiki entry .. e^ia ..described by Richard Feynman as "[...] the most remarkable formula in mathematics [...], our jewel". Even before seeing that I was considering including an aside to the effect that 'Lord of the Rings' (Tolkien) ( http://en.wikipedia.org/wiki/Lord_of_the_rings ) was an allegorical account of an engineer (Gollum) who had 'lost' e^ia .

e^iπ + 1 = 0 -- Five numbers which you should know. Wikipedia: Euler's identity

QUOTE (Confused2+Feb 7 2007, 10:53 PM)

We're testing (1) against 'SO(2,R)' .. a two dimensional real matrix of the form
given by rpenner as
(5) [ cos(a) -sin(a); sin(a) cos(a)] .
SO means Special Orthogonal
To be special it satisfies the condition that Determinant(A) = 1. see http://www.sosmath.com/matrix/determ0/determ0.html
Evaluating (2) (ad-bc) gives
(6) cos(a)cos(a) + sin(a)sin(a) or sin^2(a) + cos^2(a) .. which hopefully we know is 1 (ask if not sure)
To be orthogonal it satisfies the condition that ..
(7) The matrix times it's transpose is equal to the unitary matrix. I'll let that one go for the present.

Identity matrix [1 0;0 1] (in 2 dimension), not Unitary matrix.

I claim M=[cos(θ)e^iA sin(θ)e^iB;sin(θ)e^iC cos(θ)e^iD] is a member of U(2,C), where θ,A,B,C, and D are all Reals. Also D= B+C+π-A. You can test that it's Unitary by computing
M† = [cos(θ)e^-iA sin(θ)e^-iC;sin(θ)e^-iB cos(θ)e^-iD]
M†M = [cos(θ)cos(θ)(e^-iA)(e^iA)+sin(θ)sin(θ)(e^-iC)(e^iC) cos(θ)sin(θ)(e^-iA)(e^iB )+sin(θ)cos(θ)e^(-iC)(e^iD); sin(θ)cos(θ)(e^-iB )(e^iA)+cos(θ)sin(θ)(e^-iD)(e^iC) sin(θ)sin(θ)(e^-iB )(e^iB )+cos(θ)cos(θ)(e^-iD)(e^iD)]
= [cos²(θ)+sin²(θ) cos(θ)sin(θ)((e^-iA)(e^iB )+e^(-iC)(e^i(B+C+π-A))); sin(θ)cos(θ)((e^-iB )(e^iA)+(e^-i(B+C+π-A))(e^iC)) sin²(θ)+cos²(θ)]
= [1 cos(θ)sin(θ)((e^i(B-A))-(e^i(B-A))); sin(θ)cos(θ)((e^i(A-B ))-(e^i(A-B ))) 1]
= [1 0; 0 1]
MM† = [cos(θ)cos(θ)(e^iA)(e^-iA)+sin(θ)sin(θ)(e^-iB )(e^iB ) cos(θ)sin(θ)(e^iA)(e^-iC)+sin(θ)cos(θ)(e^iB )e^(-iD); cos(θ)sin(θ)(e^-iA)(e^iC)+sin(θ)cos(θ)(e^-iB )e^(iD) sin(θ)sin(θ)(e^iC)(e^-iC)+cos(θ)cos(θ)(e^iD)(e^-iD)]
= [1 cos(θ)sin(θ)( (e^iA)(e^-iC)+(e^iB )(e^-i(B+C+π-A)) ); cos(θ)sin(θ)( (e^-iA)(e^iC)+(e^-iB )e^(i(B+C+π-A)) ) 1]
= [1 cos(θ)sin(θ)( (e^i(A-C))-(e^i(A-C)) ); cos(θ)sin(θ)( (e^i(C-A))-e^(i(C-A)) ) 1]
= [ 1 0 ; 0 1 ]
So the Unitary group is pretty Complicated.

This example was to show that complex matrix multiplication is the same process as real matrix multiplication, but you use complex numbers.

Likewise, the rule for the determinant is the same, so
| M | = cos(θ)(e^iA)cos(θ)(e^iD) - sin(θ)(e^iB )sin(θ)(e^iC)
= cos²(θ)(e^i(B+C+π)) - sin²(θ)(e^i(B+C))
= cos²(θ)(e^i(B+C+π)) + sin²(θ)(e^i(B+C+π))
= e^i(B+C+π)
So M is part of SU(2,C) if B = π-C, But if B = π-C, then D = 2π-A = -A (modulo 2π)
So I claim that M is in SU(2,C) if it is in the form:
M=[cos(θ)e^iA -sin(θ)e^-iC;sin(θ)e^iC cos(θ)e^i(-A)]

P.S. I cheated a bit, because I experimented in trying to find the conditions for a Unitary matrix in a+ib format first, where I discovered some facts which got me going on this particular form for M.

Janus

9th February 2007 - 12:13 AM

Hi All

Us cranks are reading ... but we are pissing ourselves with laughter.

Talk about about complicating simple issues ... long live the monster.

Oh well ... some people think they can push themselves forward by patting themselves on the back.

Cheers

Janus

AlphaNumeric

9th February 2007 - 12:31 AM

QUOTE (Janus+Feb 9 2007, 01:13 AM)

Talk about about complicating simple issues ... long live the monster.

So I assume you've got a working replacement for quantum mechanics if it's actually quite simple? Oh yeah, you don't

These things in this thread are actually quite simple, though the lack of maths type doesn't help in making the equations look nice here (LaTeX support would be nice). Showing U(1) and SO(2) are isomorphic is the kind of thing 1st year maths students do in about 30 seconds. But then you and I know you have a bit of trouble with group theory don't we? How many identity elements does a group have again?

QUOTE (Janus+Feb 9 2007, 01:13 AM)

Oh well ... some people think they can push themselves forward by patting themselves on the back.

The success of quantum mechanics speaks for itself. Also, this kind of material is worked on extensively just from a maths point of view (pursuit of knowledge and all that).

You appear to think you can push things forward by saying everyone else is wrong and that it's all so simple to you.

Shame you've nothing to actually show for it other than demonstrating you don't understand maths

Confused2

9th February 2007 - 02:11 PM

I have a suggestion that you are probably going to really hate..

We seem to be in two dimensions at the moment and we have two arms... (this is just to establish the principle).

Waves left arm frantically .. then waves right arm frantically .. then waves both arms in a fairly normal fashion (in phase or out of phase?). Mentions that the special property of A is that it constrains the amount thissa and thatta to follow the rule X.

Holds arms at right angles .. weaves about with dreamy expression on face. Announces that the special property of B is that it does THIS (more weaving) to THESE (flaps hands to draw attention to arms in case virtual audience is missing the significance of the demonstration).

Maybe one of these constrained by that (possibly plus more bits yet to be revealed) has no choice but to be THIS

.

Using imaginery legs and carefully chosen poses it might be possible to show that this and this (there has been a change of pose) are both THIS

.

To the best of my knowledge I have never, ever (not even once) managed to teach anything to anyone.. but it has not been for lack of trying.

-C2.

Confused2

11th February 2007 - 01:00 AM

The claim is 'simple'.

QUOTE (rpenner+)

I claim M=[cos(θ)e^iA sin(θ)e^iB;sin(θ)e^iC cos(θ)e^iD] is a member of U(2,C), where θ,A,B,C, and D are all Reals. Also D= B+C+π-A. You can test that it's Unitary by computing

Due to the elegance of your computation the first part is accepted as 'simple'.. it would have been even easier to follow if you had added a comma as well as a space between items because my browser seems to like to run things together as much as possible. Of D= B+C+pi-A :- there are clearly an infinite number of M's that are not U(2,C) .. if this is an important result (rather than a fun result) then I'd be pleased to give it further consideration... please clarify the importance of this result.

QUOTE (rpenner+)

Likewise, the rule for the determinant is the same, so ..
So M is part of SU(2,C) if B = π-C, But if B = π-C, then D = 2π-A = -A (modulo 2π)
So I claim that M is in SU(2,C) if it is in the form:
M=[cos(θ)e^iA -sin(θ)e^-iC;sin(θ)e^iC cos(θ)e^i(-A)]

At this stage I would claim that the conditions for generating a Unitary matrix are 'non-obvious'. My experience of natural phenomena suggests that they will either emerge naturally or will require us to compute them within constraints that reduce the infinite set of M to a sensible number, I could well be wrong, of course.
I apologise for the way I continue to have the educational handicap of a nine-year-old who has not yet been introduced to matrices.
Despite the fact that you have not cooperated by suggesting suitable choreography for my imaginary legs dance I look forward to your next post.
-C2.

rpenner

11th February 2007 - 05:29 AM

QUOTE (Confused2+Feb 11 2007, 01:00 AM)

Nope, it was just fun for me.
When I explored this, I started with M = [w x; y z] (all of which are Complex) and computed both M†M and MM† and if these are both equal to I, then w*w + x*x = w*w + y*y = z*z + x*x = z*z + y*y = 1, which means |w| = |z| and |x| = |y| and |w|² + |x|² = 1, which is the relation between sin and cos.
Thus w = cos(θ)e^iA, x=sin(θ)e^iB, y=sin(θ)e^iC, and z=cos(θ)e^iD,
since |e^iR| = 1 (with R being Real)
But in addition to the elements of I which are 1, others are 0. These constrain D = B+C+π-A, so instead of 5 free parameters, there are only 4.
Another way to write this is z=cos(θ)(e^-iA)(e^iB)(e^iC)(e^iπ) = -cos(θ)(e^-iA)(e^iB)(e^iC), then all the M's are clearly in U(2,C).

Probably the second deepest thing about any Unitary representation (matrix) I can say is that the definition M† = M^-1 replaces a "hard" matrix operation (taking the inverse) with an easy operation (taking the conjugate of the transpose).

QUOTE (Confused2+Feb 11 2007, 01:00 AM)

There's actually a concrete way to generate (heh, heh, inside joke) a fully faithful representation of SU(N,C) for any N. Probably is one for U(N,C) also.
AlphaNumeric knows that this forum is not going to be one where I explain this process, it's just too hard to write down the math.
U(1,C) has 1 free parameter.
U(2,C) has 4 free parameters, and SU(2,C) has 3 free parameters.
U(3,C) has 9 free parameters, and SU(3,C) has 8 free parameters.
In the standard model, this is very closely related to the 1 photon, the W+, W- and Z bosons of the weak force and the eight gluons.

QUOTE (Confused2+Feb 11 2007, 01:00 AM)

I apologise for the way I continue to have the educational handicap of a nine-year-old who has not yet been introduced to matrices.

No need to apologize. We're all friends here, except those who aren't.

Janus

11th February 2007 - 11:44 PM

Hi All,

Sorry to digress.

I’m just asking for intellectual seriousness and innovative thinking … anything else … I can get out of a book … it’s the crank ideas that make this forum and some of their ideas can be fruitful … not necessarily their dream … but it might spark off someone in a direction they might not have thought of going.
To me I don’t care if its drivel … there’s always something to be gained … even if only a chuckle.
As for the 5-pips next to your name types … well they should have a closed thread if they just want to discuss book stuff amongst themselves.
Oh for more experienced professors on this forum … to bring some balance … but then again they’d probably be shat on if they spoke their minds.

Quote Alphanumeric:
Showing U(1) and SO(2) are isomorphic is the kind of thing 1st year maths students do in about 30 seconds.

My point exactly.

Quote Alphanumeric:
But then you and I know you have a bit of trouble with group theory don't we? How many identity elements does a group have again?

Definition:

“In abstract algebra, a group is a set with a binary operation that satisfies certain axioms.”

Look it’s not my fault that mathematics doesn’t know how to divide by zero or that everything multiplied by zero becomes zero.
I’ts also not my fault that that there is no multiplicative inverse to zero.

But I clearly showed why there should be an additive inverse to zero.

You know I was trying to introduce the concept of a negative zero as an axiom (but not in abstract algebra) … and also why … and you after several failed attempts decided to introduce the concept of groups to win the case … this is your usual tactic … shift the ground and state the person is an idiot … brilliant … you should be a spin doctor.

You at times are just a prattling placebo affecting very few positively or negatively.
Too few times do your explanations increase the readers understanding of the concept.

Now mathematics/physics or anything … will include many many concepts real and abstract … its like a sweet shop you can pick-and-mix … whose to say what is right or wrong … that’s what being an adult means … the right to choose … I can’t wait for some people to grow-up.

Think on this:
Lockwood told us … that for all the empirical and theoretical successes of the sciences, the nature of matter is no less enigmatic than that of consciousness.
Can you really be sure that you understand what the equations of mathematics are telling you?
Are you not fooling yourself with contrived levels of mathematical depiction?

And on this:
If I asked you to prove that the earth is round and not flat mathematically … you could not do it … try it … I await your results?
Unfair?
prove 1 + 0 = 1 then … or even … just explain what 1 and zero is?

And this:
"Mathematics may be defined as the subject where we never know
what we are talking about, nor whether what we are saying is true."
Bertrand Russell

Quote: Confused2
To the best of my knowledge I have never, ever (not even once) managed to teach anything to anyone.. but it has not been for lack of trying.

If you want to teach adults … then you have to teach something they want to learn … give them a reason to learn … answer their questions … their problems … not your own.
For example a person in prison on a long sentence who cannot read or write can be taught to read by telling him he’ll be able to read porn books instead of just looking at pictures … wont take him long to learn to read and then maybe he might read something else to his benefit.

Cheers

Janus

AlphaNumeric

12th February 2007 - 12:08 AM