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