Causality and the Quaternion Derivative

[sweetser]

Hello:

Causality in classical physics is not the same as causality in quantum mechanics. Classically, A causes B causes C, a totally ordered set of Rube Goldberg interactions. In quantum mechanics, one must accurately account for all possible histories: everything that could happen must be included and given the relevant weight. A happens a%, B happens b% and C happens c%, so one measures aA + bB + cC.

In this post I will provide a new reason why causality is different between classical and quantum physics. The key to the idea is a collision between the two most important mathematical developments in the four hundred year history of physics. Newton's calculus launched this area of science we now call physics. Calculus is the study of change. For a large number of systems, if the relevant differential equation is known, the future of the system can be calculated.

The second most important math step was Einstein's introduction of spacetime, the ability of measurements in time to mix with space. The four dimensional manifolds that Minkowski developed applied not only to measurements of time and 3-space, but to energy and 3-momentum (I wish I had a more extensive list, I know there is one for angular momentum for example). This was a significant development because it provide a bridge between things that are measurably different, such as time and space.

My thesis is that calculus done correctly in spacetime is the entire cause of the difference in causality between classical and quantum physics.

What we do today is work with 4-vectors, that can be added, subtracted or multiplied by a scalar. Recall the basic definition of a derivative:

df(q)/dq = lim dq->0 (f(q+dq) -f(q))/dq

Since a 4-vector does not come equipped with division, there is no way to take the derivative of a 4-vector function with respect to a 4-vector. 4-vectors are inadequte to study change in spacetime.

A quaternion is a collection of 4 numbers that together behave much like the real and complex numbers. Addition, subtraction, multiplication, and division are all defined. First spotted by Gauss, too much was dreamed by Hamilton and his fans, too little was delivered. Quaternions play a bit role for 3D rotations and as the unit quaternion SU(2) found in the standard model.

Quaternions are 4-vectors, but they do come with division as part of the package. They have many of the same properties as real numbers: addition, subtraction, multiplication, division, the associative law, the distributive law. There are important differences. The real numbers are a totally ordered set: a real number is either greater than, less than, or equal to another real number. Like the complex numbers, quaternions are not totally ordered, there is no one rule to put them all in line.

I think of quaternions as a way to represent an event in spacetime:

q = (t, x, y, z) = (t, R)

To make things easier, I use a capital letter for the 3-vector. Let's form the product of two quaternions:

q q' = (t, R)(t', R') = (tt' - R.R', tR' + Rt' + RxR')

The rule for division is similar except there are more minus signs and a norm tossed in:

q/q' = (t, R)(t', R')*/(t'² + R'²)^1/2

= (tt' + R.R', -tR' + Rt' - RxR')/(t'² + R'²)^1/2

One well known property of quaternion multiplication and division is that it does not commute because of the cross product: q q' does not equal q' q. The norms of the products are equal, and that is important.

Here's Doug's Rule of Mathematical Consistency: if a quaternion definition of an operation is done correctly, then it will apply without any modification to the real and complex numbers.

The rules for addition, subtraction, multiplication and division all work this way. The definition for a quaternion derivative of a quaternion function does not. Having the differential element go to zero on the left is different from having the differential element go to zero on the right. There are some people who work with left and/or right quaternion differentials, but we will not do so. Except for a few special cases, Nature is indifferent to right versus left.

Do you recall L'Hospital's rule? There are some limit processes that do not work the first time, but if two limits are taken in a row, then the right answer appears. The reason quaternions do not commute in general is from the cross product which entangles two 3-vectors. If the differential 3-vector were to go to zero first, then the remaining differential scalar would commute, so it would not matter which side it was written on. Here is the dual limit definition of a quaternion
differential on a quaternion manifold:

df(q)/dq = lim dq->0 (lim (dq-dq*)->0 (f(q+dq) - f(q))/dq)

where dq-dq* = (0, 2 dR)

I have had two people doing reviews on quaternion calculus email me to say they really liked this definition, that it makes many proofs straightforward since it is effectively a directional derivative along the real line (a well understood animal in the math world). What I failed to convince them of was the link to physics since they were math wonks. I believe this is the kind of derivative used for classical causality.

What does it mean that dR < c dt? A photon will tell you: it means that change is happening at less than the speed of light. That is the realm of classical physics. A can lead to B can lead to C because a photon can go from A to B to C if changes are happening at less than the speed of light.

This leads to an obvious question: what happens if the order of the two limits are reversed, so the scalar goes to zero before the 3-vector? The derivative cannot be consistently defined. Now we use the little bread crumb left a few paragraphs ago: the norm of the derivative is the same no matter if the noncommuting differential is on the left or the right. We can define:

|df(q)/dq| = lim dq->0 (lim (dq+dq*)->0 (f(q+dq) - f(q))/dq)

where dq+dq* = (2dt, 0)

This is known as a normed derivative. The size of the change can be calculated precisely, but the direction is necessarily not knowable.

What does it mean that dR > c dt? A photon will tell you: there is no way the changes that happen at A, B, and C can be linked by photons. The changes are independent, they all can make a contribution. How much of a contribution depends on the process.

I do not believe this explanation of the difference in causality between classical physics and quantum mechanics has any testable consequences. Instead, this is progress on "why", a direct link between a math definition and whether photons may or may not participate in a measured change.

Correct me if I am wrong, but I have never heard anyone with an explanation anything like this. Fun, fun! I developed the idea on a trip to a quaternion conference in Rome in 1999. I would have gone back to the next meeting, but it was not held. Quaternions are that popular. There is an episode of The Stand-Up Physicist devoted to this topic which won the 2005 Berkeley Video and Film
Festival "Best in Show, Education Category" that you can view here: http://www.archive.org/details/WhyisQMWeird

doug

[papageno]

My thesis is that calculus done correctly in spacetime is the entire cause of the difference in causality between classical and quantum physics.

A professor of mine made a point of showing us that the fundamental mathematical structure of classical mechanics and quantum mechanics was the same.
The only difference was:

pq - qp = i hbar for QM (Heisenberg principle)

pq - qp = 0 for CM.

The keyword was C* algebra.

Since a 4-vector does not come equipped with division, there is no way to take the derivative of a 4-vector function with respect to a 4-vector. 4-vectors are inadequte to study change in spacetime.

How is ds² = dx² + dy² + dz² - (c dt)² defined then?

Of course you can have "derivatives" of function in 4 dimensions. They can be defined in any dimension; the keyword is differential:

df(x₁, x₂, x₃, x₄...) = (partial derivative of f with respect to x₁) dx₁ + (partial derivative of f with respect to x₂) dx₂ + ....

[Quellen]

One thing that has intrigued me is that quaternions can be thought of as a sort of complex number that consists of two complex numbers; i.e. a two-deep recursive complex number. The idea is called the Cayley-Dickson construction and it is pretty neat. It works like this

z=A+Bi.

Let A=a+bj and B=c+dj.

j is not the same as i.

Then z=(a+bj)+i*(c+dj)= a+ci+bj+d(ij).

Just postulate that ij is something new called k and its all downhill from there.
You just have to keep the definitions of the different conjugates separate. Pretty cool!

Hamilton was a pretty conceptual guy, and I wonder if that was the "Aha!" that he had when he scratched his famous message on the bridge abutment. He realized pretty quickly that i,j,k could represent space and that 1 could represent time. What if Hamilton had figured out spacetime and Maxwell's equations 100 years early?

You can do the recursion trick just one more time and make octonions, which are nicely normed and even weirder than quaternions. An octonion algebra has seven distinct quaternion sub-algebras. 7 spaces and one time.

Then you have the fact that a spacetime point is sort of like a quaternion but not exactly. For the norm to work as a Minkowski distance, time has to be pure imaginary. So maybe a spacetime point is really the four surviving non-zero values in an octonion.

[sweetser]

Hello papageno:

There are many ways to approach quantum mechanics. I agree with your professor who claimed:

> pq - qp = i hbar for QM (Heisenberg principle)
>
> pq - qp = 0 for CM.

With that one bit of math, everything else follows. The question is why the heck should Nature use this rule? Notice that is says zero about causality, what photons can and cannot do in describing change in a system.

I have a similar experience in my quantum class. The professor said the Uncertainty Principle can be derived from properties of complex numbers. It was an amazing lecture, the logic was solid. Still no reason why complex numbers, or more generally C* algebras, need to play a role.

The interval ds² gets defined if you add a metric. Math guys like this sort of exercise, adding this and that to make things work so they know all the parts that go into it. I prefer a machine that works out of the box, without me fiddling with it. Square a quaternion differential:

(c dt, dx, dy, dz)² = (c² dt² - dx² - dy² - dz², 2 c dt dx, 2 c dt dy, 2 c dt dz)

If you change from one inertial observer to another, then the first term of the square is a Lorentz invariant (doesn't change), and the final three terms are Lorentz covariants (we know how they change). In the recently closed thread, "GEM, a rank 1 unified field proposal", I showed how moving around in a gravitational field will make the first term covariant while the other three are invariants.

What people do is define the 4-vectors over the real number or the complex numbers. One can do calculus with real and complex numbers because mathematically they are a field (you'd have to look at a technical math book like Rudin for what math wonks mean by field). One cannot however in general multiply or divide these 4-vectors. I think that is a major technical mistake. Look how often multiplication and division get used in real analysis. It is at the foundations. Follow the well worn path if you must, but I am going for the high quality explosives myself.

Quaternions are also a mathematical field. One should be able to do calculus. One can, but the current definition, it doeth bite in my opinion.

I just read up a bit on C*-algebra: http://en.wikipedia.org/wiki/C*-algebra. I remember doing lots of this sort of thing for quaternions. One cool difference is that one conjugate is not enough for quaternions. I had to invent two others:

q -> q = (t, x, y, z) an automorphism
q -> q* = (t, -x, -y, -z) an involutive antiautomorphism
q -> q*1 = (iqi)* = (-t, x, -y, -z) another involutive antiautomorphism
q -> q*2 = (jqj)* = (-t, -x, y, -z) another involutive antiautomorphism

I did all kinds of proofs with the norms. Here's the thing for people with Banach algebra fetish: you get all these things for free just by working with quaternions.

doug

[sweetser]

Hello Quellen:

I was familiar with the Cayley-Dickson construction, although other readers may not be, so thanks for the explanation. I actually think of a quaternion as a set of 3 complex numbers that all share the same real number. This makes more sense to me in terms of the physics. The shared real is time, and then I get to choose basis vectors for the space measurements, which could be Euclidean or spherical, depending on my mood swing.

I have not been the best student of quaternion history because I do research with quaternions, trying to push the cart forward. Hamilton looked for a 3D division algebra from more than ten years. Gauss may have spent one morning on the issue (I am not familiar with what problem Gauss needed to solve, but it was in one of his notebooks, unpublished and not understood until many years after his death). Hamilton was thinking about time and space and complex numbers and apparently he lost it. His books on quaternions are not understandable.

Then you have the fact that a spacetime point is sort of like a quaternion but not exactly. For the norm to work as a Minkowski distance, time has to be pure imaginary.

This does not make sense to me. There is the origin, (0,0,0,0), and an event, just to be concrete and arbitrary, at (5, 1, 2, 3). We can calculate the interval between the origin and the event:

dq dq = (25-1-4-9, 10, 20, 30) = (11, 10, 20, 30)

Now calculate the norm:

dq dq* = (39, 0, 0, 0)

These are both fine quaternions, (11, 10, 20, 30) and (39, 0, 0, 0), made from the same initial quaternion, (5, 1, 2, 3), but we happened to have formed different products, one using a conjugate operator, the other one not. I have seen other people say a similar thing, and I think it is silly. Quaternions are an algebraic field which means if you are at one place on this manifold, and there is some odd path to another place on the manifold, there is necessarily a way to get there from here. There is NO need for another imaginary unit when we already have three imaginary unit vectors on the playing field. Don't doubt the power of an algebraic field.

doug

[publius]

Of course you can have "derivatives" of function in 4 dimensions. They can be defined in any dimension; the keyword is differential:

df(x₁, x₂, x₃, x₄...) = (partial derivative of f with respect to x₁) dx₁ + (partial derivative of f with respect to x₂) dx₂ + ....

What (I think) Doug means is this. Consider complex numbers, where we can have f(z). df/dz is the "derivative". But note that dz is really two dimensional dz = dx + i dy. You can define a sort of derivative of a general
N-D function in terms of a Jacobian matrix of partials, which is something like a rank-2 tensor. So df = Jacobian * dr, where dr = (dx1, dx2, ......)

So there's something sort of special about complex functions where a df/dz can be a "number" and not some general matrix. I forget all the details and the terminology for this.

In 2 dimensions, you can do this, and that's the complex numbers and their functions. However, in 3 dimensions, this doesn't work. For example, consider the Jacobian matrix for a complex function. Now, the Cauchy-Riemann condition, where when you have f(z) = u(x, y) + i v(x, y), you must have (using '@' for the partial derivative symbol), @u/@x = @v/@y, and
@u/@y = -@v/@x for it "to work".

That means that Jacobian matrix must have the form

| a -b |
| b a |

which is the matrix representation of a complex number (a + ib), of course. That works out very neatly, and means df/dz can be a "number", just like z. And that's related to be able to divide by a two dimensional entity -- well, there's some complication about division anyway.

Now, in 3D you can't make anything like that work, as Hamilton found out But you can in 4D. And those are quaternions. You can only do that in powers of 2 dimensions, IIRC, 2, 4, 8, 16, etc dimensions. The 8D case are the "octonions". There's some complication about division, and you loose stuff with each increase in dimension. You loose commutivity with quaternions, and even loose associativity with octonions.

Anyway, the point is there's something special about a derivative that can be the same type of entity as the function and its domain itself.

-Richard

[papageno]

There are many ways to approach quantum mechanics. I agree with your professor who claimed:

> pq - qp = i hbar for QM (Heisenberg principle)
>
> pq - qp = 0 for CM.

With that one bit of math, everything else follows. The question is why the heck should Nature use this rule?

But that is beyond the scope of Quantum Mechanics.

Notice that is says zero about causality, what photons can and cannot do in describing change in a system.

That's because there is no problem with causality in QM.

The time-evolution of the state-vectors is deterministic. You are confusing the problem of measurement with a problem of causality.

The interval ds² gets defined if you add a metric.

The same goes for the derivative as you used it: dq -> 0 requires a way to measure distances.

Math guys like this sort of exercise, adding this and that to make things work so they know all the parts that go into it. I prefer a machine that works out of the box, without me fiddling with it. Square a quaternion differential:

(c dt, dx, dy, dz)² = (c² dt² - dx² - dy² - dz², 2 c dt dx, 2 c dt dy, 2 c dt dz)

And how is this any different from the usual 4-vectors?

What people do is define the 4-vectors over the real number or the complex numbers. One can do calculus with real and complex numbers because mathematically they are a field (you'd have to look at a technical math book like Rudin for what math wonks mean by field). One cannot however in general multiply or divide these 4-vectors. I think that is a major technical mistake. Look how often multiplication and division get used in real analysis. It is at the foundations. Follow the well worn path if you must, but I am going for the high quality explosives myself.

I have seen Maxwell cultists go on and on about quaternions, but they have never been able to show that quaternions are a better way nor that they provide more information than the usual 4-vectors.

But I still don't see anything relating the "causality problem" or Quantum Mechanics.

[sweetser]

Hello papageno:

The question is why the heck should Nature use this rule?

But that is beyond the scope of Quantum Mechanics.

That would be sad, such a shallow understanding of Nature. It is our current state: accept the role of C*-algebra and eat your cold porridge. There is a great reason to accept this approach: you can sleep well at night. It is the issue which upset Einstein for about thirty years.

What I have seen so far are references to how you understand quantum mechanics. It would appear that you think the main issue is about the problem of measurement. That is a perfectly respectable position. There are lots of papers, a number of books, and conferences that probe the issue of measurement. One can become an expert in the topic. I have seen zero indication that you understood my dual limit definition of a quaternion derivative. You have defended your homeland where you are familiar, and avoided the new idea here.

You do not appreciate the limitation on 4-vectors, and how those limitations are not there for quaternions. With a 4-vector, you can add, subtract or multiply it by a scalar. That is it, unless you want to add more stuff. If you add a metric, then you can calculate the interval (presuming a Minkowski metric for simplicity):

g_uvA^uA^v = a₀² - a₁² - a₂² - a₃²

Feed a metric 2 4-vectors, and you get a number back, not another 4-vector. You cannot just tack on the other 3 terms (dt dx, dt dy, dt dz) because those can involve cross products if the two 4-vectors point in different directions. Square a quaternion, and you get a quaternion back every time, guaranteed.

Let's make a chess analogy. You get to play with only 3 pieces: a king, a rook, or - you get to choose - a pawn or a queen. Pawns are more limited, but you can add cool rules like en passant to give them more power. I'll take the lady that can kill from a far along any diagonal or horizontal lines.

I have seen Maxwell cultists go on and on about quaternions

The red warning sign I have seen are the alleged "200 equations" people have claimed were in the first edition of Maxwell's treatise. I am NOT one of those guys, honest! Since I run quaternions.com, I have probably seen the claim more often than you. I did look into the first edition, and your assessment is correct. It is also not relevant to the discussion at hand.

> But I still don't see anything relating the "causality problem" or Quantum Mechanics.

Bummer. Only post #1 has the info. The others were dealing with quaternions as a math tool, not the specifics of the derivative of a quaternion which is what relates to causality. Let's ask a more precise question: how can one approach this issue of causality? One way is to focus on photons, those things that travel at dR/dt = c. One event could cause another if dR/dt = c or dR/dt < c. If two events have a spacelike separation, then one cannot cause the other. That is all I mean by causality: can photons play in the game? That is a precise definition.

The quaternion derivative as defined in post #1 falls into two classes. In the first class, the differential element has the property that dR/dt < c. That means one could have real particles bopping into each other. This is the kind of derivative everyone is used to. The more technical math wonks would call it a directional derivative along the real number line. It is the derivative used in classical physics.

The second class, the differential element has the property that dR/dt > c. This does NOT mean that there are tachyons (and I will always hate Star Trek because of all their yabber about tachyons). What it means is that you cannot measure a standard derivative. EVER. Respect the speed of light law and eat your porridge. What you can measure is the norm of a derivative, |df(q)/dq|. This is like saying you cannot measure an amplitude, but you can measure the square of an amplitude. This is the domain of quantum mechanics, where you are always limited to measuring |df(q)/dq|, the normed derivative. Now I have a "why", it is how a derivative can be defined using quaternions in spacetime.

doug

[Celestial Mechanic]

When you perform the division in (f(x+dx)-f(x))/dx, is that a pre-division or a post-division by dx? Will they have the same limit? I'll just bet someone can come up with a counterexample where the two limits (one using pre-division and one using post-division) are not equal.

This problem does not occur for functions of complex numbers since multiplication and division are commutative.

[sweetser]

Hello Mr. Good Quantum Wrench:

The way you wrote it, little to no effort is required to find an example where:

(f(q+dq)-f(q)) dq^-1 does not equal dq^-1(f(q+dq)-f(q)) dq

Notice you did not include the limit process, which is where the rabbit lives in the magic hat. If I were to use one limit process, this thread would be silly. I used a 2 limit process. If dq = (c dt, dR), and dR < c dt (the change is happening at less than the speed of light c), then dR goes to zero before dt goes to zero. After dR has gone to zero, one is left with just dt which will commute with any quaternion, and for that reason it does not matter if the differential element is pre- or post-division, the limits are identical. I am arguing that this case applies for classical physics.

The other case would be if dR > c dt in the differential element. Again the order will not matter only for the norm of the derivative, |df(q)/dq|. This is all that can be measured, not df(q)/dq. I hope people can see this as the domain of quantum mechanics.

For clarity, I probably should have written it one way or the other, my bad.

doug

[papageno]

But that is beyond the scope of Quantum Mechanics.

That would be sad, such a shallow understanding of Nature.

I never said that research stopped with Quantum Mechanics.

It is our current state: accept the role of C*-algebra and eat your cold porridge. There is a great reason to accept this approach: you can sleep well at night. It is the issue which upset Einstein for about thirty years.

The issue that upset Einstein are the implications of the measurement problem in Quantum Mechanics.

What I have seen so far are references to how you understand quantum mechanics. It would appear that you think the main issue is about the problem of measurement. That is a perfectly respectable position. There are lots of papers, a number of books, and conferences that probe the issue of measurement. One can become an expert in the topic. I have seen zero indication that you understood my dual limit definition of a quaternion derivative. You have defended your homeland where you are familiar, and avoided the new idea here.

From your first post:

Causality in classical physics is not the same as causality in quantum mechanics. Classically, A causes B causes C, a totally ordered set of Rube Goldberg interactions. In quantum mechanics, one must accurately account for all possible histories: everything that could happen must be included and given the relevant weight. A happens a%, B happens b% and C happens c%, so one measures aA + bB + cC.

The last part, which you present as a causality different from the classical one, is the measurement problem in Quantum Mechanics.
You are considering the probabilistic nature of the measurement outcomes as a causality problem.

I objected that the time-evolution of the state-vectors in QM is deterministic, just as in Classical Mechanics the time-evolution of the state of system is deterministic.

You do not appreciate the limitation on 4-vectors, and how those limitations are not there for quaternions.

I am waiting for people to show that these non-limitations of the quaternions give us more physical information.

I have seen Maxwell cultists go on and on about quaternions

The red warning sign I have seen are the alleged "200 equations" people have claimed were in the first edition of Maxwell's treatise. I am NOT one of those guys, honest! Since I run quaternions.com, I have probably seen the claim more often than you. I did look into the first edition, and your assessment is correct. It is also not relevant to the discussion at hand.

But my point is the same: do quaternions gives us better physical information than four-vectors?

But I still don't see anything relating the "causality problem" or Quantum Mechanics.

Bummer. Only post #1 has the info.

There is actually a typo. That should be:
"causality problem" of Quantum Mechanics.

The others were dealing with quaternions as a math tool, not the specifics of the derivative of a quaternion which is what relates to causality. Let's ask a more precise question: how can one approach this issue of causality? One way is to focus on photons, those things that travel at dR/dt = c. One event could cause another if dR/dt = c or dR/dt < c. If two events have a spacelike separation, then one cannot cause the other. That is all I mean by causality: can photons play in the game? That is a precise definition.

If you want to deal with photons, then you'll have to deal with Quantum Electrodynamics, not Quantum Mechanics.

But if you want to go the road of "spooky action at a distance", modern experiments are points toward non locality of wavefunctions, but there does not seem to be a violation of causality per your definition.

About the rest of your post, it is too late today for me to think about quaternions.

[sweetser]

Hello papageno:

I don't care what label we happen to give the difference between classical and quantum mechanics. Let's call it "the measurement problem in quantum mechanics". Actually, I don't like that label. There is no "problem", we can make measurements just fine. The measurements have different qualities than measurements that are made in classical physics. We understand darn well what those qualitative differences are. One class are the uncertainty relations, another is the sum of all histories. All these issues are linked via C* algebra. I will call it "the measurement problem in quantum mechanics" with the understanding that we know a heck of a lot about measurements in quantum mechanics.

You are considering the probabilistic nature of the measurement outcomes as a causality problem.

We may agree on the sort of data one can collect, and have these trivial discussions of semantics. I measure the energy of a system which can get contributions from A, B, and C. The measurement was aA + bB + cC. I'm guessing you might not like saying that part of the measurement was caused by aA. Fine, I don't care about the language, I care about the math. The number aA went into the total, as did bB, as did cC. This is not the way calculations are done in classical physics.

I objected that the time-evolution of the state-vectors in QM is deterministic

It sounds like you are carrying out an old debate, certainly not one I was making. A quick scan for that word "determ" indicates it appeared only in your comments, not mine.

I am waiting for people to show that these non-limitations of the quaternions give us more physical information.

I don't overplay my hand. Let us not discuss what "people" have claimed about quaternions in the past, the history is not pretty and is not interesting to me. You and I have wasted time on such efforts.

What I am trying, and failing, to have a discussion about is a "quaternion derivative". Those two words have as yet to appear side-by-side in your comments. That means we are not discussing the topic presented in post #1.

If you want to deal with photons, then you'll have to deal with Quantum Electrodynamics, not Quantum Mechanics.

Here's a counter-example. In special relativity, you draw a light cone, a pair of lines running at +/- 45 degrees. Quanta run along those lines, but QED is irrelevant to the exercise. I want to think about describing changes that lie inside a future light cone (classical physics I maintain), and to think about describing changes that lie outside the future light cone (quantum physics I assert). Let's visualize this:

The observer sits at (0, 0, 0, 0), with the 3-vector R bundled together making the graph 2D. The lightcone is in blue, no QED needed. The solid red line is the world line of some classical process. One could use lots of photons and collect information on every little curve and turn made. There are no limits to how many photons that can be tossed at the solid red line to collect detailed information about that worldline.

There are also a few dots outside the light cone. They certainly can exist, but the observer at (0, 0, 0, 0) cannot go and collect information about them like the solid red line inside the future lightcone. The observer can send no particles whatsoever to collect information about any of the dots. Bummer. Sounds like measurements will have to be fundamentally different.

This proposal is subtle, in a measurable way. My claim was that classical physics is about this quaternion derivative definition:

df(q)/dq = lim dq->0 (lim (dq-dq*)->0 (f(q+dq) - f(q)) dq^-1)

51 characters in all. On the other hand, quantum mechanics is about this quaternion derivative definition:

|df(q)/dq| = lim dq->0 (lim (dq+dq*)->0 (f(q+dq) - f(q)) dq^-1)

53 characters in all. A minus sign went to a plus, and the absolute value signs are necessary around the derivative.

There were two people who were doing reviews of quaternion derivatives who thought this definition was kind of cute (these were unsolicited comments), so if there was a technical problem, I think they would have brought it up. I know it feels solid to me. But I have yet to get someone steeped in physics to give the idea quiet reflection time.

doug

[Quellen]

QM can be considered to be a strictly spacelike phenomenon in a sense. The Born interpretation requires that the integral over some space-like surface of the PDF of an observable is exactly 1. But then if you transform to another frame with a different plane of simultaneity, the integral still has to be exactly 1, even though the whole store of possible intermediate events that led up to the PDF can be completely different.

Ah well ... somewhere around I have a quote from Schrodinger from around 1940 or so that states this idea a little better than I just did.

[papageno]

I don't care what label we happen to give the difference between classical and quantum mechanics. Let's call it "the measurement problem in quantum mechanics". Actually, I don't like that label. There is no "problem", we can make measurements just fine.

The problem is the interpretation of the measurements postulates.
It's the whole business of "wavefunction collapse" whose consequences led to the phrase "spooky action at a distance" and the so-called "EPR paradox".

The measurements have different qualities than measurements that are made in classical physics.

The difference is how much the system undergoing the measurement is perturbed by the measurement.

I would not call that "qualitative difference" since we can recover classical mechanics by going pq-qp -> 0 or via the Ehrenfest theorem.

We understand darn well what those qualitative differences are. One class are the uncertainty relations, another is the sum of all histories.

But the "sum over all histories" is basically a computational tool.
The difference of principle lies in the uncertainty relations.

You are considering the probabilistic nature of the measurement outcomes as a causality problem.

We may agree on the sort of data one can collect, and have these trivial discussions of semantics. I measure the energy of a system which can get contributions from A, B, and C. The measurement was aA + bB + cC.

The outcome of the measurement depends on how the observable affects the A, B and C parts of the state-vector.
If A, B and C are eigen-vectors of the energy that give the full state, the outcome of the measurement depend on how the corresponding observable commutes with the energ.
The uncertainty relations are such commutation relations.

I'm guessing you might not like saying that part of the measurement was caused by aA. Fine, I don't care about the language, I care about the math. The number aA went into the total, as did bB, as did cC. This is not the way calculations are done in classical physics.

Sorry, but you have given a confusing picture of the math of a measurement in QM.

The calculations in QM are based on Hamiltonians and Lagrangians, which were both developed for Classical Mechanics.

I objected that the time-evolution of the state-vectors in QM is deterministic

It sounds like you are carrying out an old debate, certainly not one I was making. A quick scan for that word "determ" indicates it appeared only in your comments, not mine.

You said:

For a large number of systems, if the relevant differential equation is known, the future of the system can be calculated.

The same works for the time-evolution of the state-vector: the differential equation being the Schroedinger equation in QM, and the equivalents in QED and QFT.

The only thing that messes up this deterministic time-evolution is the measurement and the corresponding collapse of the wavefunction.

I am waiting for people to show that these non-limitations of the quaternions give us more physical information.

I don't overplay my hand. Let us not discuss what "people" have claimed about quaternions in the past, the history is not pretty and is not interesting to me. You and I have wasted time on such efforts.

What I am trying, and failing, to have a discussion about is a "quaternion derivative". Those two words have as yet to appear side-by-side in your comments. That means we are not discussing the topic presented in post #1.

I was focusing on the "causality problem".
My point is, the only "problem" relates to the measurement problem.

If you want to deal with photons, then you'll have to deal with Quantum Electrodynamics, not Quantum Mechanics.

Here's a counter-example.

There is no counter-example. The electromagnetic field is not quantized in Quantum Mechanics, and therefore there are no photons in QM.
That's why I pointed you to QED and QFT, which include special relativity, unlike QM.

[sweetser]

Hello Quellen:

Interesting points. If you take another look at the red dots in the spacetime graph, you might notices that they have the same basic shape as the solid red line (not an accident). If the origin where shifted to the left so that all the isolated dots are now in the future lightcone of the observer, then the observer could shower those events with photons and learn all kinds of detailed information. Now shifted, the observer could not receive the same sort of information about the solid red worldline.

Cool.
doug

[sweetser]

Hello papageno:

You certainly have remained consistent. It appears difficult to discuss the problem of measurement in quantum mechanics with such precision as to please you. I think big and simple: classical physics is not like quantum physics, so, like, why? That is the big picture.

You have provided one math approach:

> pq - qp = i hbar for QM
> pq - qp = 0 for CM

I hope it is clear I accept this bit of math and every consequence it has. I am not going to go through a point-by-point analysis of your post because we agree on far too much (yes, I do understand how the calculations depend on Lagrange densities).

I will have to give you failing marks on reading comprehension. The main topic of this ATM thread is:

A. The measurement problem in quantum mechanics
B. The postulates behind measurements
C. The weather in Bali
D. A new definition of a quaternion function derivative on a quaternion manifold and the consequences for physics

A casual observer of our exchanges would think the answer is A and/or B. It is my hope to play with the line in post #12 that had 51 characters and compare it to the one that had 53 characters. If you don't wish to discuss those lines, please say so explicitly. We can then go elsewhere to discuss the riddles of measurement theory in quantum mechanics. In this thread, there is a specific, mathematical ATM approach to physics on the table.

doug

[papageno]

I will have to give you failing marks on reading comprehension. The main topic of this ATM thread is:

A. The measurement problem in quantum mechanics
B. The postulates behind measurements
C. The weather in Bali
D. A new definition of a quaternion function derivative on a quaternion manifold and the consequences for physics

A casual observer of our exchanges would think the answer is A and/or B. It is my hope to play with the line in post #12 that had 51 characters and compare it to the one that had 53 characters. If you don't wish to discuss those lines, please say so explicitly. We can then go elsewhere to discuss the riddles of measurement theory in quantum mechanics. In this thread, there is a specific, mathematical ATM approach to physics on the table.

In your opening post you said:

My thesis is that calculus done correctly in spacetime is the entire cause of the difference in causality between classical and quantum physics.

My point is that there is no difference in causality between Classical and Quantum Mechanics, except for problematic interpretations of the measurement postulates.

Before proposing a solution, you have to establish what the problem is.

[sweetser]

Hello papageno:

This assertion,

there is no difference in causality between Classical and Quantum Mechanics, except for problematic interpretations of the measurement postulates

indicates a bias about what sort of answers are worthy of discussion, namely those that deal with measurement postulates (answer B in the list). I understand why you think this is true, the story does fit together nicely.

Would this allow a discussion of the topic at hand?

My problem statement: "classical physics is not like quantum physics, so, like, why?" I apologize if that is not erudite.

My thesis: calculus done correctly in spacetime is the entire cause of the difference in measurement between classical and quantum physics.

A reason this might be considered an improvement is that measurement has long been a topic of professional discussions. If we were ever to address the new definition of a derivative of a quaternion function on a quaternion manifold, the issue of causality would naturally arise since we would be talking about limitation on particles participating in change.

doug

[papageno]

This assertion,

there is no difference in causality between Classical and Quantum Mechanics, except for problematic interpretations of the measurement postulates

indicates a bias about what sort of answers are worthy of discussion, namely those that deal with measurement postulates (answer B in the list). I understand why you think this is true, the story does fit together nicely.

If you don't agree with it, feel free to explain your reasons.

Would this allow a discussion of the topic at hand?

My problem statement: "classical physics is not like quantum physics, so, like, why?" I apologize if that is not erudite.

My thesis: calculus done correctly in spacetime is the entire cause of the difference in measurement between classical and quantum physics.

There is no difference in the math. That's the whole point of bringing up:

pq - qp = i hbar for QM
pq - qp = 0 for CM.

The only difference is the value of a parameter, not the mathematical formalism for state, observable and measurement.

The whole point of the C*-algebra story, is to point out that the mathematical objects describing the states of a physical system and the mathematical objects corresponding to the physical observables have the same nature in CM and QM.
And the same goes for the measurement: the application of the observable to the state.

A reason this might be considered an improvement is that measurement has long been a topic of professional discussions. If we were ever to address the new definition of a derivative of a quaternion function on a quaternion manifold, the issue of causality would naturally arise since we would be talking about limitation on particles participating in change.

My point is that you have not established yet that there is an issue to be resolved.
And if your definition gives rise to an issue which is not there in the traditional formulation, what is the point?

[sweetser]

Hello papageno:

Discussions of quantum mechanics can be as odd as quantum mechanics itself. Let's start simple:

> "classical physics is not like quantum physics, so, like, why?"

There are many different schools of thought about the why. There are folks who think about measurements, others who worry all about the wave function, there are many worlds... I cannot list all the different approaches.

papageno is trying to make an important point that some folks miss: the math tools one has developed in classical physics do apply to doing work in quantum mechanics. That is a profound insight, and not inconsistent with my own efforts.

pq - qp = i hbar for QM
pq - qp = 0 for CM.

The only difference is the value of a parameter, not the mathematical formalism for state, observable and measurement.

So, like, why is this parameter zero for classical physics but not for quantum mechanics? How can quantum systems avoid making a mistake and having it equal zero once in a while? Why should C*-algebra's have any relevance to describing Nature? Those are open questions.

Look at what we have now: 4-vectors. The boys who sit in the front of the classroom can tell you all the wonderful properties they have: they work for any coordinate system! can be any dimension so our great buddies can use them for string in eleven dimensions!

I sit in the back row myself. All day long I take one number and multiply or divide it by another - that's math. Adding, subtracting, multiplying, and dividing, the four amigos. Let's say I have one 4-vector, and I feel like dividing it by another 4-vector. Too bad. With 4-vectors I can add it to another 4-vector or multiply it by a scalar, and that's it. Wow, now my 4-vector is 3 times bigger. Pretty clucky tools (where "clucky" is jargon for "lacking in sublime sophistication").

Now pretend, just for fun, to be the kind of person who enjoys yanking the teacher's chain (OK, I don't have to pretend). Wherever they write a 4-vector which can be added or subtracted or multiplied by a scalar, you swap in a quaternion. Cool thing is that no new marks have to be made on the paper or in the book (being lazy is part of the back-row mentality). You can do stuff that is not on the teacher's radar screen. If the teacher decides to carefully introduce the notion of C*-algebras, you can play your gameboy because that is built into the quaternions. Teachers can be so slow, but we are used to that back here.

The most important tool in all of physics, the one that started out this branch of science, is calculus. The limit definition requires division. Works like a charm on the real line. Works like a charm for complex functions in the complex plane. Minkowski's spacetime, that requires the power tools of quaternions. I'll just rob the bank by myself. Folks in the back of the class cannot take the critique of men who wear fuchsia too seriously. Instead we like to do actual calculations.

Here's your doodlework (both harder and more interesting than the assigned work). Calculate:

d(a + b)^2/da

where a and b are arbitrary quaternions using both derivative definitions in post #1. Make sure all dot and cross products are included. It might take a half hour.

doug

[papageno]

> "classical physics is not like quantum physics, so, like, why?"

There are many different schools of thought about the why. There are folks who think about measurements, others who worry all about the wave function, there are many worlds... I cannot list all the different approaches.

papageno is trying to make an important point that some folks miss: the math tools one has developed in classical physics do apply to doing work in quantum mechanics. That is a profound insight, and not inconsistent with my own efforts.

This is not a particular brilliant insight.
Physics students have to go through Classical Analytical Mechanics (in the three flavors: Newtonian, Lagrangian and Hamiltonian formulation) before tackling Quantum Mechanics. And at that point the similarity of mathematical formalism is obvious.

pq - qp = i hbar for QM
pq - qp = 0 for CM.

The only difference is the value of a parameter, not the mathematical formalism for state, observable and measurement.

So, like, why is this parameter zero for classical physics but not for quantum mechanics?

We don't know.
We don't even know if there is a reason, or if it just happened.

Maybe one day students will get as a homework exercise "Derive the uncertainty relations from first principles".

How can quantum systems avoid making a mistake and having it equal zero once in a while?

Mistake?

Why should C*-algebra's have any relevance to describing Nature? Those are open questions.

And theorists are working on it.

The most important tool in all of physics, the one that started out this branch of science, is calculus. The limit definition requires division.

The definition of "limit" does not need division, nor does the definition of "differential".

[sweetser]

Hello papageno:

Looks like we are making progress, slowly, but yes, progress.

We don't know.
We don't even know if there is a reason, or if it just happened.
Maybe one day students will get as a homework exercise "Derive the uncertainty relations from first principles".
...And theorists are working on it.

So there are valid issues to think about. It is the issue I am concerned with.

The definition of "limit" does not need division, nor does the definition of "differential".

Again you get failing marks for reading comprehension. I was referring to the limit definition of a derivative, which is:

df/dx = lim dq->0 (f(x+dq) - f(x)) / dq

One might complain that I need to toss in the word derivative, but that is what a good reader will do - realize the context. The thread is about a dual limit definition of a quaternion function derivative on a quaternion manifold. The boys in the back of the class will be amused for a good long time ("What does that / mean?").

I said to the misses that any gentleman who has reached such an august station in academia to require wearing a funny hat would surely do an assigned problem, so I told her I'd flip her $20 if you did the work. I see no evidence to that effect, so the twenty stays in my pocket. I don't ask people to do a calculation I would not do, that is not fair. The snide folks in the back do have an ethic, not unlike pirates. The only way one can actually understand this proposal is by doing calculation such as this. It is the admission price. [note: a = (a, A), b = (b, B), dq = (dt, dR), and you will need to tell the difference between the lower letters by context].


Bunch of fun stuff going on. The most important is that this should look familiar according to Doug's Rule of Mathematical Consistency. It works for quaternions, and it would work fine for reals.

doug

[papageno]

Again you get failing marks for reading comprehension. I was referring to the limit definition of a derivative, which is:

df/dx = lim dq->0 (f(x+dq) - f(x)) / dq

And I pointed out that the definition of differential does not require division.

One might complain that I need to toss in the word derivative, but that is what a good reader will do - realize the context.

I rather be boring but unambiguous.

The thread is about a dual limit definition of a quaternion function derivative on a quaternion manifold. The boys in the back of the class will be amused for a good long time ("What does that / mean?").

I thought that the thread started about justifying the "causality difference" between Classical Mechanics and Quantum Mechanics.
Has the topic shifted now?

I said to the misses that any gentleman who has reached such an august station in academia to require wearing a funny hat would surely do an assigned problem, so I told her I'd flip her $20 if you did the work.

I was waiting for you to establish the existence of the problem whose solution you are proposing.

[sweetser]

Hello papageno:

It would appear that admitting error, large or small, is not your style. While true that a differential does not require division, neither does addition, or an orange. The list is rather extensive. For the most important tool in the physics tool chest, the derivative, division is required, ergo the shortcomings of 4-vectors might be of concern.

The topic is "Causality and the Quaternion Derivative". I am trying to discuss the quaternion derivative first so one could see the implications to measurement and causality. You have been clear that it is your practice to only consider measurement and its postulates to be central.

You are waiting? The bus has left. There is that famous story about guys in funny hats who refused to look into a telescope because they wanted to know how everything fit together first. So for those who like to do calculate, let's me point out a fun fact from #22. In the first calculation where dR->0 before dt->0 so all changes have a timelike relationship and thus can have a causal link, the next to last line read:

d(a+b)²/dt = lim dt->0 (2a + 2 b + dt, 2A + 2B)

The remaining differential element is in the scalar. Look at the second to last line in the next calculation where dt->0 first so change cannot happen by particles interacting:

|d(a+b)²/dt| = lim dR->0 (2a + 2 b + dt, -dR.dR dR -2A - 2B)

The remaining differential element is in the 3-vector. It is a small observation, but worth the effort.

I have fond memories of learning the limit definition of a derivative so many years ago, and being able to do simple polynomials with it. It is cool that the ratio dR/dt has something to do with it.

doug

[papageno]

It would appear that admitting error, large or small, is not your style. While true that a differential does not require division, neither does addition, or an orange. The list is rather extensive. For the most important tool in the physics tool chest, the derivative, division is required, ergo the shortcomings of 4-vectors might be of concern.

I opened chapter 25 in Volume 2 of Feynman's Lectures, which is about four-vectors, and I can see the differential operator. In chapter 2 the differential operator is defined in terms of the differential, whose definition, as we know from calculus, does not require division.

Can you point out the short-comings of the four-vectors?

The topic is "Causality and the Quaternion Derivative". I am trying to discuss the quaternion derivative first so one could see the implications to measurement and causality. You have been clear that it is your practice to only consider measurement and its postulates to be central.

Your opening post starts with this:

Causality in classical physics is not the same as causality in quantum mechanics.

and your thesis is that:

calculus done correctly in spacetime is the entire cause of the difference in causality between classical and quantum physics.

Now, I objected there is no substantial difference in causality between Classical and Quantum Mechanics, and gave my reasons.

Are you going to explain where am I wrong?
Or will you keep avoiding the issue of proving the existence of the problem?

[sweetser]

Hello papageno:

This is getting embarrassing. It would appear you didn't notice the limit definition of a derivative that was provided several times:

> df(x)/dx = lim dx->0 (f(x+dx) - f(x)) / dx

When one puts the differential to work, to take the derivative of a function, then this is when division by the differential element necessarily comes into play.

Feynman's book is about physics, not math, so many of the details are missing. There is an important issue going on concerning manifolds which casual students of math miss (http://en.wikipedia.org/wiki/Manifold). The idea is not hard, but the importance is often overlooked. One needs a differential manifold to do calculus, a means of taking the limit defined above. The only sort of manifold beginning students seam to believe exists are R¹ for doing Newtonian physics, R² for working with complex numbers, R⁴ for spacetime, and R^N for anything else. All the dimensions are independent from each other, a separate bucket for every bin. Some people can have entire professional careers without using any other sort of manifold. It is a logical thing to do.

For complex analysis, things like C*-algebra, these separate players do not work. Instead one often works with C¹. The notion of the C¹ manifold is very difficult to teach. People invariably believe there are 2 reals sitting inside the one complex number. That is not the way it works. There are only other complex numbers next to other complex number. The manifold is all about complex numbers being next to complex numbers. What statements that can be made about the complex numbers are independent of how the complex number is represented. For example, all kinds of things can be proved about z and z*. People think, z* = (x, -y), but that is only true for one of an infinite number of representations of a complex number (it would be wrong in polar coordinates, since r = (x² + y²)^1/2, tan A = y/x so y>-y does note do what you would expect). Working with the manifold C¹ is a generalization of the complex numbers used in grade school.

Many of the most important results in complex analysis rely on working on the manifold C¹. I think of it as the manifold having some of the properties of the complex numbers themselves, so the analysis ends up being a bit richer. So the difference comes down to working with R² versus C¹. One can have arguments about whether that should make any difference, but such debates only means one party does not understand C¹.

In Feynman, the manifold is R⁴. That means each of the 4 functions are being treated separately. When it comes to evaluation the differential element, one needs to divide by the differential element, which in this case is a single real number.

I am trying to work with H¹, the one dimensional differential manifold of quaternions. In order to communicate to my fellow man, I do make it look more familiar, like R⁴, but that is to avoid frightening people. The differential element can be a quaternion on the H¹ manifold. The impact of changing the manifold takes careful thought, all of which is tied up with the definition I provided that you have decided not look into.

Now, I objected there is no substantial difference in causality between Classical and Quantum Mechanics, and gave my reasons.

This does not sound like an accurate assessment. What you did was describe the problems with quantum mechanics as being about measurement and the postulates for measurement. I agreed that was a fine position to take. I argued with no apparent success that this is not the only position to take. You feel like the measurement problem is a "trump" problem, that everything can be recast that way. I don't think that is the case. Richard Feynman in "The Character of physical law" argued quite effectively that the sum of all possible histories was not just some computational tool. To get the path the photon follows, all complex amplitudes must be summed, then squared. It turns out that the classical ones contribute nearly all of it, but the others must be included. Feynman does as good a job at explaining that as is possible.

The issue of causality can be discussed a using Minkowski spacetime diagram.

The stuff in the future light cone, the upper and lower triagles, can have a timelike relationship to the observer at the origin, so there can be a causal connection between the events. One event could possibly cause the other. This is what I mean by causality. For the events outside, to the left and right, the observer cannot send a particle to those events. These two events are necessarily independent of each other because it is not possible to send a photon between the two, one event cannot cause the other. This is a precise notion of causality, can two events be linked by a 45 degree worldline?

I don't expect you to see the connection between this standard observation of special relativity and quantum mechanics. I do hope you finally get the limit definition of a derivative and the quaternion manifold. That would be progress.

doug

[papageno]

This is getting embarrassing. It would appear you didn't notice the limit definition of a derivative that was provided several times:

> df(x)/dx = lim dx->0 (f(x+dx) - f(x)) / dx

When one puts the differential to work, to take the derivative of a function, then this is when division by the differential element necessarily comes into play.

Do you mean, like this:

df(x₁, x₂, x₃, ...) = a₁dx₁ + a₂dx₂ + ... ?

Feynman's book is about physics, not math, so many of the details are missing.

Isn't this thread about physics?

Now, I objected there is no substantial difference in causality between Classical and Quantum Mechanics, and gave my reasons.

This does not sound like an accurate assessment. What you did was describe the problems with quantum mechanics as being about measurement and the postulates for measurement.

This is what I said:
"My point is that there is no difference in causality between Classical and Quantum Mechanics, except for problematic interpretations of the measurement postulates."

I also pointed out the similarity of the mathematical formalism between Classical and Quantum Mechanics, in particular concerning the time-evolution of the state.

I agreed that was a fine position to take. I argued with no apparent success that this is not the only position to take.

Where did you show that there is a causality difference between CM and QM, and that it involves calculus?

You gave a confusing picture of the measurement formalism in QM, complained about the shortcomings of four-vectors and then you went on to talk at length about quaternions.

The impression I get is that your alternative position is "I prefer quaternions".

You feel like the measurement problem is a "trump" problem, that everything can be recast that way.

No, I explained that the only thing that can be described as a "causality difference" between CM and QM are the problems of interpretation of the measurement postulates, which have nothing to do with calculus or four-vectors.

I also pointed out that the experiments show no sign of a "causality difference".

Richard Feynman in "The Character of physical law" argued quite effectively that the sum of all possible histories was not just some computational tool. To get the path the photon follows, all complex amplitudes must be summed, then squared. It turns out that the classical ones contribute nearly all of it, but the others must be included. Feynman does as good a job at explaining that as is possible.

Then you should have realized the path-integral method only brings the formalism of quantum theories closer to the formalism of Classical Mechanics (for example, principle of least action).

Now, where is this "causality difference"?

I don't expect you to see the connection between this standard observation of special relativity and quantum mechanics.

I already pointed out that Quantum Electrodynamics and Quantum Field Theory include Special Relativity.

[sweetser]

Hello papageno:

The silliness continues. If I meant "df(x1, x2, x3, ...) = a1dx1 + a2dx2 + ... ?" I would have written it. Here in the states we write division using the slash symbol, "/". I thought that was a global convension. The differential element can be a real number, it can be a complex number, a quaternion, but not a 4-vector. I hope I don't have to rub your nose in this again. A careful reading of post #1 would have made it clear, or any other time I wrote the limit definition of a derivative.

I put a considerable effort into discussing manifolds. Your deep reflection on that section was: "Isn't this thread about physics?" Math and physics are very tightly related. You were the one who brought up C*-algebras which sounds like pure math to me. I do mathematical physics if you want to point out the obvious. I have seen no evidence that you understand manifolds, and I don't give you the benefit of the doubt give your track record on reading comprehension, knowing what a "/" is, or the way you skip assigned problems because you need to wait.

I also pointed out the similarity of the mathematical formalism between Classical and Quantum Mechanics, in particular concerning the time-evolution of the state.

papageno is trying to make an important point that some folks miss: the math tools one has developed in classical physics do apply to doing work in quantum mechanics. That is a profound insight, and not inconsistent with my own efforts.

This is not a particular brilliant insight.

The way you say the same thing sounds so much better, I guess everything besides the time evolution of the state is trivial.

I also pointed out that the experiments show no sign of a "causality difference".

Either you haven't read Feynman, or you didn't get it. Yes, the principle of least action applies for both classical mechanics and quantum mechanics. The result for quantum mechanics with its "measurement problem" is the sum of all histories is used to calculate what is measured. What that means is that ever single possible path makes a material contribution to what is observed. If you want an experiment to prove a difference in causality, read how Feynman discusses the two slit experiment. In that book, he talks directly about the difference between bullets (classical physics) and the wave function.

Again I have no data that you understand what causality means on a Minkowski spacetime graph. You point out that special relativity is included in QED and QFT which demonstrates you know how to make points that are not germane.

For those of you who are watching this cat fight from the sidelines, here is a way to see Doug's Rule of Mathematical Consistency in action (if a quaternion definition of an operation is done correctly, then it will apply without any modification to the real and complex numbers).

Here are my two definitions of a derivative of a function on the manifolds R¹, C¹, or H¹

df(q)/dq = lim dq->0 (lim (dq-dq*)->0 (f(q+dq) - f(q)) dq^-1)
|df(q)/dq| = lim dq->0 (lim (dq+dq*)->0 (f(q+dq) - f(q)) dq^-1)

The first reaction might be fear: shouldn't there be only 1 derivative??? It doesn't matter if there are two that are always identical. On the real manifold, there is no dq*, so the dual limit reduces to a single limit process, just like it is defined in Thomas and Finney. The second derivative is the same wether the differential is on the left or the right, so the absolute value sign is dropped.

On the complex manifold, both limits are necessary for the calculation. Because complex numbers commute, there is no need to take the norm of the derivative, the derivative is the same if the differential element is written on the left or the right. The derivative of the complex function will be identical whether the real goes to zero then the imaginary, or the imaginary goes to zero first then the real. What this means physically is everything points in the same line in spacetime.

For those in the audience that program, it would be nice to work with such consistent math tools. Sure, while it may make a derivative on a real manifold appear more bulky, the code for doing so would not be one iota different from that needed by the quaternions. Solve once, apply everywhere.

doug

[papageno]

The silliness continues. If I meant "df(x1, x2, x3, ...) = a1dx1 + a2dx2 + ... ?" I would have written it. Here in the states we write division using the slash symbol, "/". I thought that was a global convension. The differential element can be a real number, it can be a complex number, a quaternion, but not a 4-vector. I hope I don't have to rub your nose in this again. A careful reading of post #1 would have made it clear, or any other time I wrote the limit definition of a derivative.

And I pointed out in post #2 the use of the differential, instead of the derivative of a four-vector with respect to a four-vector.

A careful reading of post #1 also gives this:

Causality in classical physics is not the same as causality in quantum mechanics.

which you still have to prove.

I put a considerable effort into discussing manifolds.

You should put more effort into proving the existence of the issue you propose to resolve.

Your deep reflection on that section was: "Isn't this thread about physics?" Math and physics are very tightly related. You were the one who brought up C*-algebras which sounds like pure math to me.

And from the context it is clear that I brought them up to explain that the mathematical formalisms in Classical and Quantum Mechanics are essentially the same. And it is relevant, since your thesis is that:

calculus done correctly in spacetime is the entire cause of the difference in causality between classical and quantum physics.

which seems to imply that the mathematical formalism is at the origin of this supposed causality difference.

I do mathematical physics if you want to point out the obvious. I have seen no evidence that you understand manifolds, and I don't give you the benefit of the doubt give your track record on reading comprehension, knowing what a "/" is, or the way you skip assigned problems because you need to wait.

Did I misunderstand who is proposing an ATM idea and is supposed to address critiques?

How many times do I need to ask you to prove that there is infact a "difference in causality between classical and quantum physics"?

The way you say the same thing sounds so much better, I guess everything besides the time evolution of the state is trivial.

The insight that "the math tools one has developed in classical physics do apply to doing work in quantum mechanics" is trivial for the people who have studied both.

I also pointed out that the experiments show no sign of a "causality difference".

Either you haven't read Feynman, or you didn't get it. Yes, the principle of least action applies for both classical mechanics and quantum mechanics. The result for quantum mechanics with its "measurement problem" is the sum of all histories is used to calculate what is measured. What that means is that ever single possible path makes a material contribution to what is observed. If you want an experiment to prove a difference in causality, read how Feynman discusses the two slit experiment. In that book, he talks directly about the difference between bullets (classical physics) and the wave function.

Did you forget to post the part where you explain that the difference between classical particles and quantum particles involves a causality difference?

Again I have no data that you understand what causality means on a Minkowski spacetime graph. You point out that special relativity is included in QED and QFT which demonstrates you know how to make points that are not germane.

You insist on using special-relativistic spacetime with QM. I simply pointed you to the theories that do include Special Relativity.

Actually it is not even clear why you insist on spacetime, since the issue is supposed to be a difference between Classical Mechanics and Quantum Mechanics.

For those of you who are watching this cat fight from the sidelines, here is a way to see Doug's Rule of Mathematical Consistency in action (if a quaternion definition of an operation is done correctly, then it will apply without any modification to the real and complex numbers).

I would suggest, instead, that the lurkers go to library and read on how Classical Mechanics, Quantum Mechanics and four-vectors actually are used in Physics.

[sweetser]

Hello Lurkers:

Could I have an independent judgment. He asks:

How many times do I need to ask you to prove that there is in fact a "difference in causality between classical and quantum physics"?

This question strikes me as so inane that it is hard to formulate a reply, particularly with a guy who doesn't know what division means. If causality is the same between classical and quantum physics, then the measurements we see in classical and quantum physics will also be the same because the have exactly the same cause. The whole point of thinking about causality is because it has consequences that can be measured. No difference in causality translates to no difference in measurements.

doug