For the first posting in this series we are going to go ahead and dive right in to some heady yet interesting terrain; the principle of gauge invariance. Our current understanding of the way the universe is put together is based on an idea known as the gauge principle. Put simply, the way the universe behaves should not depend on the way we look at it. A basic version of this principle has been known for some time in the form of space-time invariance. By space-time invariance we mean to say that the laws of physics should not change simply by moving around in space or time: if you are listening to your local rockin' band playing live and then decide to move to a different location in the room or perhaps wait for the inevitable mind-bending refrain one minute later, the way the sound waves propogate (for example) should not correspondingly change. You might be hearing different frequencies due to the acoustics in the room or in that the refrain has a different set of chord progressions, but the fundamental physics behind the sound waves does not change. This may seem obvious, but the best ideas in science are those obvious ideas that have profound consequences. In our example, as a consequence of the physics not changing in time we can actually prove the law of conservation of energy. That's right, this simple idea is the origin of the world's energy crisis... talk about profound.
Space-time invariance has been well known since Galileo (and likely significantly earlier), but another kind of symmetry has popped up more recently which is gauge invariance. To understand this we must review some quantum field theory (it's okay, just keep petting your fish) which says that all kinds of matter can be described by a field that exists everywhere in space, and the properties of that field will dictate how different kinds of matter will interact with each other. This idea is itself nothing new. Photons (the particles that make up light) are described by electromagnetic fields for example, which has been known for over a century. But, our new theory predicts that everything; electrons, protons, neutrinos, etc. have an associated field and that oscillations of that field produce the particles in question. Pretty neat, eh?
Let us focus on electrons, those damn things that shock you when moonwalking too much on your grandma's shag carpet. The field associated with electrons is known as a Dirac field. Now, it turns out that if we take a Dirac field and multiply it by a complex number, the physical quantities that we can measure from the field do not change. Therefore we also expect that the equations that dictate the behavior of the Dirac field should also not change. This process of multiplying a Dirac field by a complex number is known as a guage transformation, and the expectation that the behavior of the field should remain unchanged is known as gauge invariance (or the gauge principle). Now here is the punchline: it turns out that as a by-product of imposing the gauge invariance restriction on the equation that dictates the behavior of the field (known as the Lagrangian), we have to by hand put terms into this equation, and those terms predict the interaction between electrons and photons! Yes indeed. Before we had to just assume that an electron and a photon interacted in a particular way (our cell phones work, so we're pretty sure they interact), but now we actually have a physical principle that predicts this interaction. This theory is known as quantum electro-dynamics, or QED. One can take this principle furthur to tie together QED with other particles, leading in the end to a unified view of the universe that connects together all the known forces... except for gravity, but that's a different story.
So, here is the challenge: the Lagrangian which describes the Dirac field is given by
Here the psi symbol is the Dirac field (which is a column vector), the gamma symbol is a constant matrix, and m is the mass of the electron. The bar over the psi refers to the vector that has been transposed, complex-conjugated, and multiplied by another constant matrix, and the derivative (the delta symbol) is with respect to position. If we make the transformation on our Dirac field
where theta is a general function of position x, then we will also transform the Lagrangian into a new function, and this new function had better have the same form as the old one. Show that by taking the derivative of the Dirac field transformation we get two terms: one that is proportional to the derivative of the field, and another that is proportional to the field itself.
This second term is what must be cancelled in the Lagrangian equation, thus leading to the inclusion by hand of a cancelling term which coincidentally predicts electromagnetic interaction. Holy cow!
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