This is the first in a series of lecture notes for Physics 380. These notes are meant to provide a qualitative summary of each course lecture. Their goal is to prevent students, in their rush to take notes on every detail of the lectures, from losing sight of the broader picture: how topics fit together, what examples or particular topics show, and the motivations and justifications for steps taken in class.

Lecture 1:

Handouts:

1. Syllabus

2. Web Resources

First, introductions and discussion of the syllabus: grading policies, motivations for studying quantum field theory, and outline of course content. These are all written out in Handout 1.

Then, to physics: we first discussed units. When we study a physical system, we are free to choose the scales by which we measure mass, length, and time in a way that reflects the typical dimensions of the problem. Poor choices complicate our work by making us keep track of conversion ratios -- for example, in a derivation involving length, we might initially convert an input length from miles to meters; if we think of the input as always being in miles, we would always carry a constant factor (the meters/mile conversion ratio) through whatever algebraic steps our derivation requires.

Since this course concerns relativistic quantum theory, we will choose units most suited for such problems: that is, units that eliminate the conversion constants that always crop up in problems involving quantum mechanics and relativity. In our units, called "natural units", both of those constants, and c, are set to 1. (To completely specify the units, there is still one more degree of freedom, which is chosen so that Boltzmann's constant k -- which will play no role in this course -- is also 1).

These units are a little counterintuitive at first. Since c is 1, velocities are dimensionless -- they are just fractional numbers, which indicate at what fraction of the speed of light an object is traveling. There is only one dimensionful scale in our scheme: with the same units, we measure mass, energy, inverse time, and inverse length. That is,

where [x] means "the dimension of x". This formalizes relationships you recall from quantum mechanics: a particle's mass is inversely related to its deBroglie (also called Compton) wavelength; and an energy gap between states is inversely related to the lifetime of the higher energy state.

Some numbers were given to develop a sense of scale:

This means that an energy of 1 GeV (a typical particle physics energy, equal to times the energy required to accelerate an electron through a 1 Volt potential difference) corresponds to a length of , or about angstro ms -- where an angstrom is a typical atomic size.

A hierarchy of particle masses was given:

 
photon 				 massless

electron 			 .5 MeV   		(M means   )

proton, neutron 		 938 MeV  		(roughly the scale of the strong interactions)

W, Z				 85 GeV   		(the scale of the weak interactions).

Finally, we discussed some heuristic arguments about why constructing a quantum field theory -- that is, a theory that both explains quantum effects and obeys causality plus locality -- is a nontrivial extension of quantum mechanics. These were, in short:

1) Perturbation theory in quantum mechanics makes us expect that multiparticle intermediate states will have significant effects in the relativistic limit. Specifically, perturbation theory tells us that, in the presence of a perturbing (small) interaction, energy levels are shifted. That shift is affected by all possible intermediate states, although contributions are suppressed by the energy gap required to reach such states. At relativistic energies, the energy shift due to multiparticle intermediate states is only suppressed by a factor -- comparable to any relativistic effect.

2) a) The evolution of a localized quantum mechanical particle violates causality.

That is, start with a position eigenstate . As time goes on, it obeys the time-dependent Schrodinger equation

so it becomes . (H is the Hamiltonian operator; if we write the state as a linear combination of energy eigenstates, each eigenstate picks up a phase .)

Consider the amplitude to observe the particle at time t at position x': . If we calculate this, we find a small but nonzero probability to observe the particle outside its "forward light cone". That is, causality says that a particle can propagate no further than a distance ct in time t; but our quantum mechanical calculation shows a nonzero probability to do just that. This nonzero probability is a very serious problem: Lorentz invariance relates it to a probability to travel backward in time, with all the attendant paradoxes.

2b) This problem really comes about because we attempt to localize the particle. In fact, it only arises when we localize the particle within a Compton wavelength -- a situation that doesn't arise in atomic physics.

What happens as we try to localize the particle? The Heisenberg Uncertainty Principle tells us that, to localize the particle within a length scale L, we introduce uncertainty in the momentum of order 1/L, and thus in the energy, also of order 1/L in natural units. As we decrease L, grows, so that inside the box we may have energies up to 1/L. If 1/L becomes much larger than the particle mass m, energy barriers between single particle states and multiparticle states become negligible -- the energy can be carried by one particle, or many (charge conservation provides some constraint, but within the box we could have 1 electron, with rest mass m, or 2 electrons and 1 positron, with rest mass 3m, etc). Thus as we localize the particle the number of particles we have confined to the box becomes undetermined.

Again, then, we find multiparticle states become important. The hope is that, by including such states, we can restore causality to the theory. This hope will turn out to be true -- we can reconcile quantum mechanics with causality by building multiparticle quantum field theories.

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