I don’t know whether you have read my recent messages at Kibitzer’s Café or not and so I am posting it here too. On Monday of last week I have become proud father of (the second) daughter named Aneta. You can see her picture on the site of regional newspaper “Kladensky denik” (Kladno’s Daily) at http://kladensky.denik.cz/miminka/m... where is just now till this Friday (12:00 A.M. CET) running a polling contest for the most sympathetic baby born last week in regional maternity hospitals organized by the newspaper. Personally I am not much interested in such a kind of competitions but my wife likes it very much and she would be much happy to see our little girl on the top. Right now we are on the second place in very tense and close race with two other contenders and so every additional vote is very important for the final outcome. If you would like to help me to make my wife even happier than she already is now, just click on the link above, flag “Aneta Cervenkova, Stredokluky” (the first name in gray box on the right side on the screen) and hit the “hlasovat” button below. It is possible to vote repeatedly always after 60 minutes from one IP address... :-D

<I want to start with the antiparticles and describe why there must be antiparticles if you try combine quantum mechanics with relativity. It also permits us to solve another problem which was very mysterious pre-relativistic times, one of the grand mysteries of the world, the Pauli Exclusion Principle. (That when you exchange two particles, you get... um... you put in a minus sign.)

It's easy to demonstrate that in non-relativist mechanics that if Nature started that way, it would be that way all the time. And so the problem would be pushed back to creation, and God only knows how that was done.

<laughter>

But with the existence of antiparticles we make new pairs, and therefore new electrons, and the mystery now is, why does the new electron that's just been made have to be antisymmetric with respect to the others? That it can't get into the same state as the others which are already there?

And so the existence of particles and artiparticles permits us to ask the question in a practical way, suppose I make two new pairs with two electrons and I compare the amplitudes for when they annihilate directly, or when they exchange before they annihilate. Why is there a minus sign?

All these things have been solved in a beautiful way in the spirit of Dirac, with lots of symbols and operators and so on, but I am going back to Maxwell's "gear wheels" to try to explain to you, as best I can, these things in such a way that they appear not so mysterious. I am adding nothing to what is known before, this is purely exposition.

And so, here we go, as to how things work, and why there must be anti-particles.>

Quite a nice lead-in don't you think? So far not much is over my head, I even understand the reference to Maxwell's gear wheels. I also *think* I understand this minus sign that Feynman regards as the core of the mystery, but perhaps it's best for me to explain it to my ability, so that if I'm hung up on the introduction we can clear the air immediately.

I didn't have it memorized but a quick google search gave me the formula behind Pauli's idea (for fermions)

ψ = ψ1(a)*ψ2(b) - ψ1(b)*ψ2(a)

where ψ (psi) is the probability of electrons 1 & 2 being in quantum states a & b. ψ1 (really ψ sub 1, but my keyboard is limited) is the probability amplitude for electron 1, and ψ2 is the probability amplitude for electron 2. Ergo, if a=b (both electrons are in the same quantum state) the probability will be zero, which is to say, that aint' happenin'.

I used google to refresh my memory on what a "quantum state" is and learned it's defined as a combination of 1) energy, 2) angular momentum magnitude, 3) angular momentum orientation, and 4) orientation of intrinsic spin.

Here my confusion starts already. Did that web page forget to add "time" to the list? Or is that implicit? Obviously an electron on the tip of my nose might be in the same state as some electron that existed on an oak tree a million years ago, right? I have this desire to add "5) time" to that list, but that's just my own notion.

I would say you got a bad definition of a quantum state! :)

A quantum state is, as it sounds, just a state that a quantum system may be in. This state will have defining characteristics, and usually one is interested in determining the minimum number of variables that can be used to define that state. As an analogy, suppose we have a staircase with n steps, and my 'state' is defined by which step I'm on. In this case we can easily define my state using a single variable - the value of n. But now suppose my state is defined by both which step I'm on, and which foot I'm standing on. Now for each value of n, we have both a left foot state, and a right foot state, so it's necessary to define my state in terms of two variables.

I guess what you're saying with your staircase example is, "it depends" ... the variables you use to describe a photon would not be the same ones that describe a neutron. (Or are they?) And as you add more particles to the system it gets hairier and hairier, I imagine.

So let's take a simple example, a single electron, like one that comes out of the CRT inside my television. What variables would we pick to define its state?

Well, a TV is actually a pretty complicated system! So let's say we're just talking about an electron traveling through space, minding its own business. In that case, the electron's state would be well described by its energy (or momentum) and its spin - 2 variables.

To take things a little further, the quantum states of a system correspond to the solutions of the Schrodinger equation, which you can see here: http://en.wikipedia.org/wiki/Schr%C...

Our example of an electron moving through space interacting with nothing else is a special case of V = 0. In this case, the solutions are waves, with the frequency/wavelength determined by energy/momentum.

<Funny side-note: I was doing a google search for "strangeness spin and charm" and I got a news article about Barak Obama. ROFL>

It's a lot like the newtonian physics I did in college, where you come up with functions that define the location of an object with respect to time.

Exactly, the key difference is that:

In Newtownian mechanics you define the position of each particle as a function of time, x(t). The velocity is then dx/dt, and the momentum is m*dx/dt.

I understand that, but it definitely is worthy of stating and restating.

It's important to emphasize that electrons are not like bullets or baseballs. From time to time you may hear me talk about the location of an electron but that's just because I'm not educated to use the right terminology. What I really mean is, the probability field that dictates its possible locations.

One of the traps to understanding physics is an over reliance on metaphors. There's nothing wrong with thinking in metaphors--the smartest people in the world do it. And so, it's very tempting to think of an electron like a bullet, or a wave, or a cloud, or a string, or something that is familiar to us in our day-to-day lives. But if you take that approach, you're doomed from the start, because nature on that scale is so far detached from our normal experience that there is nothing that we can compare it to.

Well, I should say "nothing except for one thing", and that one thing is mathematics.

And this is a very important thing, and therefore must be the result of some sort of theorem, and so I made up a mathematical theorem, the proof of which I don't know, but I'm sure it's right. >

[roaring laughter from audience]

< Uh, in THIS application.

That is, if we put together a function only with positive frequencies to find this way (the integral over positive or negative) with any weights whatsoever, to produce a function f(t), then that function CANNOT BE ZERO for a finite interval of time.

>

[ blank stares from audience ]

< Oh, I... um... you see, ordinarily you're a little bit surprised there, because you know that you can take a piecewise function that's zero over a range and Fourier analyze it, but then you'll get both positive and negative frequencies. And I'm insisting that there only be positive frequencies, we've got half as much data to work with and I have to have two functions, the real and the imaginary part of f, both be zero.

Now maybe that some clever mathematician cook up a some way that can almost happen, with some kind of "e to the 1/x" or some terror, but I want it to be zero over an interval of time, you see, inside the light cone. Then if I change x (which changes all the phases any everything else) it's still zero. I don't think you can make it work, zero outside the light cone.

So the point is, it's definitely so this function cannot be zero outside the light cone. In other words, there is an amplitude for particles apparently to propagate faster than the speed of light, and no arrangement of superposition can get around that.

Intuitively one might expect all of these symmetry relations to be true, and that seems to be the case for gravity, electromagnetism and the strong nuclear force. But weirdly enough, violations in C, P and T have all been exhibited by the weak nuclear force.

But, if you invert C, P, and T together, the resultant universe seems to be indistinguishable from our own - this is known as CPT symmetry (at least there haven't been any violations seen yet, and if any are seen it will have serious theoretical ramifications).

<Now here's a surprise: if we put together these waves with some amplitude that's any function of p whatsoever, this amplitude to get from 1 to 2 is NOT--and in fact it cannot be--zero, when 2 is outside of the light cone of 1. And that's a shock to anybody that doesn't know that: that if you started a series of waves out, they can't be confined in the light cone! If the energy is always positive.>

Surely he's not saying that waves actually escape their own light cones--that's ridiculous. I get impression that he is trying to prove the existence of antiparticles using a kind of proof by contradiction, and the notion of waves not contained in their light-cones is the contradiction that proves that the premise is false. Sort of like in geometry, when after much effort you finally get to say "...and therefore the parallel lines A and B meet at point P, which is of course absurd, therefore the premise is false, qed."

That is, if you insist that the wave equations be relativistically invariant, then you have to believe that the wave goes outside of the light cone. But you know that is absurd, so faced with a contradiction you have to "fix" your theory somehow, and the way to fix it is by adding antiparticles. To Dirac that was as simple as employing a minus sign, but the chalk on the blackboard had far-reaching implications for the real world.