CHAPTER 1
Statistical Mechanics and Quantum
Field Theory
1.1 Quantum Particles
We have no better way of describing elementary particles than quantum field theory. A quantum field in general is an assembly of an infinite
number of interacting harmonic oscillators. Excitations of such oscillators are associated with particles. The special importance of the
harmonic oscillator follows from the fact that its excitation spectrum is
additive, i.e. if
and £2 ^re energy levels above the ground state then
£ 1 + £2 will be an energy level as well. It is precisely this property that
we expect to be true for a system of elementary particles. Therefore we
attempt to identify the Hamiltonian of the particles with the Hamiltonian of coupled oscillators (there is a familiar example from solid state
physics: the excitations of a crystal lattice can be interpreted as
particles—phonons). All this has the fiavour of the XIX century, when
people tried to construct mechanical models for all phenomena. I see
nothing wrong with it because any nontrivial idea is in a certain sense
correct. The garbage of the past often becomes the treasure of the
present (and vice versa). For this reason we shall boldly investigate all
possible analogies together with our main problem.
A very important analogy, which will be extensively used below, is
the one between the quantum mechanics of a ^-dimensional system
and the classical statistical mechanics of a ^ + 1-dimensional system.
Let us demonstrate it in the simplest case of the ^ = 1 quantum
mechanics of one particle. According to the Feynman principle, the
transition amplitude £ from the point x to the point x' is given by the
sum over all possible trajectories connecting points x and x', each
trajectory entering with the weight exp((i/ft)S[x(i)]) where 5[x(i)] is the
1
DOI: 10.1201/9780203755082-1
Statistical Mechanics and Quantum
Field Theory
1.1 Quantum Particles
We have no better way of describing elementary particles than quantum field theory. A quantum field in general is an assembly of an infinite
number of interacting harmonic oscillators. Excitations of such oscillators are associated with particles. The special importance of the
harmonic oscillator follows from the fact that its excitation spectrum is
additive, i.e. if
and £2 ^re energy levels above the ground state then
£ 1 + £2 will be an energy level as well. It is precisely this property that
we expect to be true for a system of elementary particles. Therefore we
attempt to identify the Hamiltonian of the particles with the Hamiltonian of coupled oscillators (there is a familiar example from solid state
physics: the excitations of a crystal lattice can be interpreted as
particles—phonons). All this has the fiavour of the XIX century, when
people tried to construct mechanical models for all phenomena. I see
nothing wrong with it because any nontrivial idea is in a certain sense
correct. The garbage of the past often becomes the treasure of the
present (and vice versa). For this reason we shall boldly investigate all
possible analogies together with our main problem.
A very important analogy, which will be extensively used below, is
the one between the quantum mechanics of a ^-dimensional system
and the classical statistical mechanics of a ^ + 1-dimensional system.
Let us demonstrate it in the simplest case of the ^ = 1 quantum
mechanics of one particle. According to the Feynman principle, the
transition amplitude £ from the point x to the point x' is given by the
sum over all possible trajectories connecting points x and x', each
trajectory entering with the weight exp((i/ft)S[x(i)]) where 5[x(i)] is the
1
DOI: 10.1201/9780203755082-1