Lecture 2. Contents. 1 Fermi s Method 2. 2 Lattice Oscillators 3. 3 The Sine-Gordon Equation 8. Wednesday, August 28

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1 Lecture 2 Wednesday, August 28 Contents 1 Fermi s Method 2 2 Lattice Oscillators 3 3 The Sine-Gordon Equation 8 1

2 1 Fermi s Method Feynman s Quantum Electrodynamics refers on the first page of the first lecture to Fermi s method, and calls it one of the simplest formulations of quantum field theory. The following description is from Fermi s 1932 review[1] of Dirac s Theory of Radiation: 1. Fundamental concept Dirac s theory of radiation is based on a very simple idea; instead of considering an atom and the radiation field with which it interacts as two distinct systems, he treats them as a single system whose energy is the sum of three terms; one representing the energy of the atom, a second representing the electromagnetic energy of the radiation field, and a small term representing the coupling energy of the atom and the radiation field. If we neglect this last term, the atom and the field could not affect each other in any way; that is, no radiation energy could be either emitted or absorbed by the atom. A very simple example will explain these relations. Let us consider a pendulum which corresponds to the atom, and an oscillating string in the neighborhood of the pendulum which represents the radiation field. If there is no connection between the pendulum and the string, the two systems vibrate quite independently of each other; the energy is in this case simply the sum of the energy of the pendulum and the energy of the string with no interaction term. To obtain the mechanical representation of this term, let us tie the mass M of the pendulum to a point A of the string by means of a very thin elastic thread a. The effect of the thread is to perturb slightly the motion of the string and of the pendulum. Let us suppose for instance that at the time t = 0, the string is in vibration and the pendulum is at rest. Through the elastic thread a the oscillating string transmits to the pendulum very slight forces having the same periods as the vibrations of the string. If these periods are different from the period of the pendulum, the amplitude of its vibrations remains always exceedingly small; but if a period of the string is equal to the period of the pendulum, there is a resonance and the amplitude of vibration of the pendulum becomes considerable after a certain time. This process corresponds to the absorption of radiation by the atom. If we suppose, on the contrary, that at the time t = 0 the pendulum is oscillating and the string is at rest, the inverse phenomenon occurs. The forces transmitted through the elastic thread from the pendulum to the string put the string in vibration; but only the harmonics of the string, whose frequencies are very near the frequency of the pendulum reach a considerable amplitude. This process corresponds to the emission of radiation be the atom. 2

3 2 Lattice Oscillators The harmonic oscillator is perhaps the simplest model with periodic motion in classical mechanics F = m d2 x dt 2 = mω2 x, x(t) = x(0) cos(ωt) + v(0) ω sin(ωt), E = m 2 ( v 2 + ω 2 x 2). The complete set of energy eigenvalues and eigenfunctions of the quantum harmonic oscillator can be obtained using ladder operators ( mω â = ˆx + iˆp ) ( mω, â = ˆx iˆp ) [â,, ] ( â = 1, Ĥ = ω â â + 1 ), 2 mω 2 mω 2 which is the most convenient formalism for generalizing to quantum field theory. The statistical mechanics of anensemble of quantum oscillators can also be solved exactly. For example, the partition function of a canonical ensemble at temperature T is ( ) n Z = Tre βĥ = ρ n e E n/(k B T ) = e ω/(2k BT ) e ω/(k BT ) e ω/(2k BT ) = 1 e. ω/(k BT ) n=0 The classical canonical partition function can be obtained as the high temperature limit, or by integrating over the classical phase space Z classical = dγe βh = 1 dp dx e (p2 +m 2 ω 2 x 2 )/(2mk B T ) = k BT 2π ω. 3 n=0

4 The Classical Harmonic Chain Altland-Simons Section 1.1 considers the low energy spectrum of a one-dimensional lattice of ions with mass M and equally spaced equilibrium positions R I = Ia, I = 1,..., N. The many-body Coulomb forces are approximated by classical elastic springs with Hooke s Law constant k s connecting neighboring ions. The Hamiltonian and Lagrangian functions are H = N I=1 [ P 2 I 2M + k ] s 2 (R I+1 R I a) 2, L = T V = N I=1 [ M 2 Ṙ2 I k ] s 2 (R I+1 R I a) 2 with periodic boundary conditions R I+N = R I. This model can also be solved exactly by transforming to normal coordinates., 4

5 Continuum Limit and Sound Waves Taking the limits N and a 0 with the length L = Na of the chain held fixed, and defining a function φ(t, x) by Ia x, a N I=1 L 0 dx, R I R I a φ(x), the Lagrangian can be written as the integral of a Lagrangian density L[φ] = L 0 dx L(φ, φ, x φ), L = M 2 φ 2 k sa 2 R I+1 R I a a 3/2 2 ( xφ) 2. φ x, The notation L[φ] indicates that the Lagrangian is a function of a continuously infinite number of variables, i.e., the infinite number of oscillator displacements in the continuum limit. Action and Hamilton s Principle The function φ(t, x) is an example of a classical field. A field φ is a smooth mapping φ : M T of a base manifold M to a target manifold T. In this instance, M is the set of points z = (t, x) in the two-dimensional plane R 2 of time and x-coordinate values, and T is the one-dimensional real line R of values φ(t, x). 5

6 The action functional of the field associated with a finite time interval t i < t < t f is defined by S[φ] = tf t f dt L[φ] = tf t i dt L 0 dx L(φ, t φ, x φ). The Euler-Lagrange equations of motion for the field φ(t, x) are determined by Hamilton s Principle, which states that the physical trajectories of the system with given initial configuration at time t i and final configuration at time t f are extrema of the action functional on the interval [t i, t f ]. The methods of multi-variable calculus can be generalized to derive the Euler-Lagrange equations for the field: L φ L µ µ φ = L φ L t t φ L x x φ = 0. 6

7 The field equation is the one-dimensional wave equation ) (M 2 t k sa 2 2 φ(t, x) = 0, 2 x 2 and the solutions are non-dispersing sound waves φ(t, x) = φ + (x vt) + φ (x + vt), v = a ks M. Functional Calculus The following table (Altland-Simons page 14) summarizes the generalization of multi-variable calculus to continuum calculus: 7

8 3 The Sine-Gordon Equation Consider a chain of physical pendulums rotating under gravity in the vertical plane, with neighboring masses connected by elastic Hooke s Law springs. The base manifold is x R. The target manifold is the circular trajectory of the pendulum bob 0 bφ < 2π. The gravitational potential energy of the mass φ(x) at point x is V g (φ) = Mgl [1 cos (bφ(x))] measured relative to the equilibrium position, where l is the length of each physical pendulum, and b is a constant with dimension [L] 1/2. Adding this potential to the oscillator chain Lagrangian gives the equation of motion ) (M 2 t k sa 2 2 φ(t, x) + Mglb sin(bφ) = 0, 2 x 2 8

9 which is called the Sine-Gordon Equation[2]. The quantized version of the Sine-Gordon field can be solved exactly, see for example Coleman s well-know article[3], and illustrates many remarkable properties of quantum field models in 2-dimensional spacetime. In the limit of small oscillations of the physical pendula, the Sine-Gordon equation reduces to ) (M 2 t k sa x + 2 Mglb2 φ(t, x) = 0, which has the same form as the relativistic Klein-Gordon equation ( 1 2 c 2 t m2 c 2 ) φ(t, r) = 0, 2 which describes relativistic scalar particles with rest mass m. 9

10 References [1] E. Fermi, Quantum Theory of Radiation, Rev. Mod. Phys. 4, (1932). [2] For explanation and references on the Sine-Gordon Equation see Wikipedia and Wolfram Mathworld. [3] S. Coleman, Quantum sine-gordon equation as the massive Thirring model, Phys. Rev. D 11, (1975). 10