by Steve Bryson
The current success of quantum field theories describing virtually all known elementary particle interactions is based on a particular type of field theory call a gauge theory.
A gauge theory is based on the recognition of a certain fact that is true in any quantum theory. This is that the only way that a wave function can be observed is through the probability that the wave function predicts, which is given by the strength of the wave. Up until now we have been thinking of the wave function as a number at each point in space. In actuality, the wave function is somewhat more complicated. In the case of an electron, the wave function is given in terms of two numbers (for you techies out there the wave is complex valued). Now these two numbers can be thought of as an arrow, not in our normal spacetime, but in another space that we envision attached to spacetime everywhere. The buzzword for this is 'fiber bundle', so I will call this new space the bundle space. Then the strength of the wave function becomes the length of the arrow. Now the crucial observation of gauge theories is that in quantum mechanics it is only the strength of the wave that counts (it gives the probabilities). If the wave is actually described by an arrow (as it is here), then it is only the length of the arrow that counts and not its direction. In other words, we can never know what direction the arrow is pointing. Physicists call this a 'gauge symmetry', as at each point of space the theory is symmetric under rotation of the arrow.
According to the mathematics of the theory, however, the wave function really is given by an arrow and so we cannot ignore its direction even though the direction cannot be observed. In particular, the arrow may be pointed in one direction at one point of normal space and if you move to another point in normal space the arrow may rotate to point in another direction. Even though at each point you cannot observe the direction of the arrow, you do observe this rotation. It turns out that when you create a quantum field theory of just electrons (no electromagnetic force yet) that respects and describes this rotating of arrows this rotating of the arrows plays a role that is identical to the role of the electromagnetic field in QED! Thus the electromagnetic force arises from the rotation of the electron's wave function arrow as you move around in space.
I'll say this again a couple of different ways. According to the natural quantum field theory of electrons, the electron's wave function is a little arrow (pointing into the bundle space) at each point of spacetime. Say that we move from one point to another in spacetime. If the electron's wave function arrow rotated between the points we would experience that rotation as an electromagnetic force in spacetime. If we traveled to another point such that the arrows did not rotate, then we would experience no electromagnetic force between those two points:
A more abstract and geometrical approach is the following. Instead of presuming that for some reason the electron's wave function arrows were not all aligned, make the equivalent presumption that the bundle space where the arrows point is curved. This means that when all the arrows are aligned in the curved bundle space, they will appear to be rotated from the perspective of observers in spacetime. Then in the same way as above we would experience that rotation as an electromagnetic force. The beauty of this view is that if one considers an electron moving in the total space (spacetime + bundle space), then one can show that if the bundle space is curved an electron moving in a straight line in the total space appears to move in a curved path in spacetime. The equations for this curved path that you get from this approach are exactly the equations describing the path of a particle under the influence of a classical electromagnetic force. In this way we see the electromagnetic force as not a force at all, but as the manifestation of 'straightest line' motion in a curved total space. (In this way electromagnetism becomes precisely analogous to gravity as the curving of spacetime.)
Thus QED is the simplest gauge theory, and in fact this gauge aspect of QED is crucial in proving that QED is renormalizable.
Once you understand electron wave functions as arrows in a two dimensional bundle space (that is described by two numbers), you can ask if you can have wave functions in bundle spaces of higher dimensions! It turns out that quarks are correctly described by a wave function that is described by an arrow that points into a six-dimensional bundle space (that is described by six numbers). It is said that quarks come in three 'colors', each color labeling two of the six directions, and the resulting gauge theory is called Quantum Chromodynamics or QCD. (The three colors are not to be confused with the flavors of quarks, called up, down, etc.. Each flavor comes in three colors, that is each flavor is described by a six dimensional arrow.) So physicists play the same game that they did for QED as a gauge theory except now the bundle space is six dimensional rather than two. By presuming that the direction of the six dimensional quark wave arrow is unobservable while the rotations of that arrow are observable, we derive a QCD force between quarks (and only between quarks) that can be viewed as the curvature of the six dimensional bundle space. The rotation of the six dimensional arrow is characterized by eight quantities called gluons.
It turns out that because this bundle space is six dimensional, using renormalization techniques we discover that the strength of the force found in this way between quarks depends on the separation of the quarks. Two quarks very close together act as if there is no force at all between them while if the quarks try to move apart the force gets stronger and stronger. Thus two quarks can never move far apart as the force between them gets infinitely strong as they try to move apart. Thus the only things that can move apart from one another are things that are not represented by arrows in this six dimensional bundle space and so do not generate a force. Such objects are electron type particles and certain groups of quarks (where the quark arrows in some sense cancel). Thus quarks can only exist in the world as certain groups (which we call protons, neutrons, etc.) and never by themselves. This is called 'quark confinement' and is currently understood as coming from the fast that this six dimensional bundle force gets stronger with increasing distance (called ''asymptotic freedom' due to the fact that short distance corresponds to high energy). While it turns out that it is very hard to make specific predictions in QCD, those that have been made match very well with observations, especially the actual observation of asymptotic freedom at SLAC in Stanford in 1969.
While these gauge theories were successful in explaining the interaction of matter and light (QED) and the behavior of quarks (QCD), there was still one force that acts on elementary particles that did not fit into a symmetric gauge theory. This is the weak nuclear force which actually does not look much like a force at all but shows up in the radioactive decay of atomic nuclei. To create a gauge theory of the weak force, physicists had to drop the assumption that it is only strength of the wave function arrow describing a particle that counts. They had to allow the direction of the wave arrow to become directly observable. In addition, to make the theory work, the physicists had to incorporate electromagnetism with the weak force in a single gauge theory. What resulted was the first unified quantum filed theory that both describes electromagnetism and the weak nuclear force. This theory is called 'the electroweak theory' or the Glashow-Weinberg-Salam model after its prime inventors.
This theory has several remarkable features that I can only briefly mention here. First, the wave function arrow for particles in this theory is described by four numbers and so is in a four dimensional bundle space. Now, however, the direction of the arrow is directly observable. Two of the directions in the bundle space correspond to the two directions of the electron wave function, and the other two correspond to the two directions of the wave function for a particle called a neutrino. Thus when the wave function points in one direction we may see an electron while if it points in another direction we would see a neutrino. The rotating of this four dimensional wave arrow from point to point in normal space is experienced as a combination of the weak and electromagnetic force. This rotation of the arrows is described by four quantities called the photon (for the electromagnetic part), W+, W-, and Z0 (for the weak part). The force of this theory gets stronger with distance (like QCD), but not as strong as QCD.
This theory actually starts with all particles massless. When building the theory you assume that electrons, quarks, and so on do not have any mass. The mass then arises by the mechanism that allows the direction of the wave arrow to become detectable. Though in the current form you have to be very clever and construct your theory just right to explain all the masses, we have for the first time some explanation for the existence of mass. This explanation is, however, at a very primitive and unnatural level.
What is predicted, however, is that the W and Z quantities behave like particles with a certain mass, and they were observed to actually have that mass at CERN in Europe in 1983. This is a spectacular confirmation of the electroweak theory.
The electroweak theory is founded on a mathematical construction called spontaneous symmetry breaking or SSB. This is a rather artificial and ad hoc mechanism, however. Yet in using it we are certainly doing something right. Taken at face value, the form of the SSB that works predicts an as yet unseen massive particle called the Higgs boson which plays a major role in generating the SSB. The fact that this particle has not yet been seen is not a problem, as the electroweak theory makes no predictions whatsoever about the properties of the Higgs. In particular it may be so heavy that we have not been able to produce it yet. There are also theoretical reasons to think that the Higgs does not actually exist and the mechanism of SSB is not yet understood completely. There has been considerable effort to understand SSB in a more natural way (called technicolor theories), but so far there has been no success.
Inspired by the success of the electroweak model, since 1974 several physicists were lead to postulate that the wave functions of particles actually pointed into a higher dimensional bundle space and all the particles we see correspond to different directions in that space. Several theories were written down, all of then using one form or another of the spontaneous symmetry breaking mechanism. Currently there is no observational evidence favoring or discrediting any one over the others, but here I will describe the simplest one so you can get some feel for what these theories look like.
This theory is called 'The minimal SU(5) model'. Here it is postulated that the wave function arrow points into a 10 dimensional bundle space, where six of the directions correspond to the quark color directions, two directions correspond to the electron directions, and the last two correspond to the neutrino directions. As the gauge symmetry is broken, these directions actually look different to observers in spacetime, and that difference appears as quarks vs. electrons vs. neutrinos. The rotations of this ten dimensional wave arrow are described by 24 quantities, 8 of which are the gluons, three are the W+, W-, and Z0, one is the photon, six of the new ones are labels X and the last six are labeled Y.
This theory has reproduced all the predictions of the electroweak theory and QCD. In addition, this theory predicts that the X and Y quantities act like particles of a very large mass. Also, because now the ten dimensional wave can rotate from pointing in a quark direction to pointing in an electron direction (for example), we can have quarks turning into electrons. Thus as a proton is made out of quarks, we expect to see protons decaying into other particles. This theory predicts that a proton should decay, on the average, in about 1031 years. This has not yet been seen, which is something of a problem for the SU(5) model, but has not yet ruled it out. This theory also predicts that the electrical charge of quarks should be 1/3 that of the electron, in accord with observation. There are also some technical predictions that match well with observation.
There are many other models built along these lines, but it is currently beyond the range of particle accelerators to test their special predictions.
Another approach to the wave function:
The Feynman Path Integral
There is a new approach to quantum mechanics due to Richard Feynman that is rapidly taking over in research in quantum field theory. Here I want to give a brief and simple description of this approach.
Feynman starts by saying that there is no wave function, rather what is strange about the world and gives rise to the wave aspects of nature is that motion is not what you think it is. Consider an electron moving from one end of the room to another, saying that it started at some particular place and ended at another place. Label the starting point A and the ending point B. Classically, in the absence of any forces, the electron would travel in a straight line, the straight line from A to B:
Feynman says "no, actually the electron traveled along every conceivable path starting at A and ending at B at the same time." That is, the electron did not travel in any sense along any particular path, but rather was somehow everywhere at once (even though it is still a particle):
Then Feynman says that there is associated with this particle a quantity called an amplitude which is given by adding up all the particle paths (whatever that means) with each term in the sum of particle paths multiplied by a weight function (a mathematical term). Thus each term in the sum looks like
(weight function) times (path)
And the amplitude of the particle looks like
amplitude = sum of [(weight function) times (path)] added up over all paths
This amplitude is called the Feynman path integral.
All the physics is contained in the weight function, and it is designed so that the paths that are radically different from the classical straight line will, on the average, cancel each other out in the sum that gives the amplitude. This cancellation becomes less and less likely for paths closer and closer to the classical straight line path.
Then Feynman discovered that this amplitude with this special weight function behaves exactly like the wave function of orthodox quantum mechanics. In this way Feynman derived the wave function of quantum mechanics by taking a radically different view of motion.
The significance of this amplitude is that when we look to find the electron at a particular point, the probability that we will find the electron at that point is given by the magnitude of the amplitude. Thus in the above example where the electron is moving from point A to point B, the amplitude along the classical straight line is very large so it is likely that we will find the particle traveling along the classical straight line. Far from the classical path, the amplitude is small (though not zero) due to the average cancellation, so it is unlikely that we will see the particle there (though there is some chance):
While all this is very pretty and somewhat compelling as an understanding of how quantum phenomena arise, it should be kept in mind that the construction of the weight function is very artificial and ad hoc (though not any more so than orthodox quantum theory). It would be very nice to motivate this weight function as then we would have some understanding of why quantum mechanics is true. Currently, the closest we are to this is the observation that the Feynman path integral arises naturally when asking questions about random processes in general mathematics (the field of stochastic calculus). Perhaps this implies that at some level the universe is truly random. This is a very new area of research.
The Path Integral in Quantum Field Theory
In describing the collision of two electrons, one starts with the two electrons widely separated and therefore not (for practical purposes) interacting and asks what are the chances that after the collision one will find two electrons widely separated and not interacting. In terms of the wave function, one asks how one performs a transition from one wave describing two electrons through a wave describing a collision to a wave describing two electrons. Feynman says that one should consider all possible waves that look like the wave of two electrons in the distant past and also look like the wave for two electrons in the distant future, and can look like anything in between. Then the amplitude for getting two electrons, widely separated, out of the collision is given by the weighted sum (or Feynman path integral) over all these waves. Thus the probability for such a process is 'simply' given by calculating this sum. You can also ask what is the probability of starting with two electrons and ending up with two muons (or two quarks or anything else allowed by certain conservation laws). Then you would look at a wave that starts as a wave for two electrons and ends as a wave for two muons. Then the sum in the amplitude for getting the muons out. The problem is that no one knows how to figure out what this sum is exactly. The best anyone can currently do is use an approximation scheme.
This is where the difficulties of quantum field theory arise. In this approximation scheme, one expresses the Feynman path integral as new sum reflecting better and better approximations to the actual answer. It looks something like this:
Feynman sum is approximately = A + B + C + ...
Here the letters stand for the approximate terms. It turns out that these approximate terms can be expressed as diagrams of lines that can be usefully thought of as representing particles and particle processes. These are the Feynman Diagrams discussed in last week's notes. I feel, however, that as these diagrams are expressions for a mathematical approximation series they should not be taken as representing anything in the structure of the world.
Now the problem is that the first few approximation terms (A and B and sometimes C) turn out to be infinite when you calculate them. This is bad as no probability can be infinite (when something happens with certainty we say that it has a probability = 1 and something cannot have a higher probability than that!). It turns out that it is possible to eliminate these infinities by a clever redefinition of the weight function of the Feynman sum. This is called renormalization.
Many physicists are uncomfortable with renormalization as it involves cavalierly throwing about infinities as if they were normal numbers. In the case of the quantum field theory of the interaction of electromagnetism and matter (Quantum Electrodynamics or QED), however, the answers that you get after renormalizing match observations to the current limit of accuracy (currently about one part in a million!). Thus at least in QED we must be doing something right. Also, recent work in simpler nonphysical models of field theories have shown that even though the approximation scheme in these models give infinities (just like in QED), the actual total sum is finite. In other words if we could calculate what the sum is we would get a finite answer. Thus the infinities arising out of the approximation scheme came from the approximation scheme and not from the Feynman sum, at least in these simpler models. The hope is that this is true for all actual quantum field theories.
Even though this approximation scheme and its required renormalization are somewhat funky, they do predict a rather surprising result. That is that the strength of the forces described by the theories is not constant but is a function of the energy of the system. In QED, for example, the strength of the electromagnetic force actually increases with energy very slowly. Thus a magnet at very high temperatures (higher than the center of the sun to be noticeable) is stronger than a magnet in our room. In the case of the strong nuclear force, the force actually gets weaker with higher energies, and seems to get infinitely strong (a real infinity this time) for lower and lower energies. This is called 'asymptotic freedom' as it predicts that for very high energies we should see something like free quarks (this was observed at SLAC in 1969) and is used to explain why you can never see free quarks at everyday energies. This is a spectacular vindication of the quantum field theory approach.