3 Quantisation of Scalar Field

N. Panchapakesan

1.Learning Outcomes

Learn the way time dependence in the formalism is shifted from state vectors to operators in the different pictures.
Learn how commutators of operators bring in quantisation and prevent possibility of simultaneous measurement.
Learn to quantize the simplest field with one component, the scalar field.
Learn to go to the complementary momentum space and interpret the Hamiltonian in terms of creation and annihilation operators
Notice the symmetry of the operators when commuted and the characteristic of Bose-Einstein statistics.

2. INTRODUCTION

In quantizing particles in Quantum mechanics the Schrodinger picture was used where the wave function (or more precisely the state vector) carried the time dependence and the operators were time independent. The observations are given by the matrix elements of the operators or alternatively the integrals of operators between the wave function and its conjugate. The wave function has a time dependence obtained by solving the Schrodinger equation . This way of looking at the time evolution of a quantum system was referred to by Dirac as using the Schrodinger picture. We can use another approach in which the operator carries the time dependence instead of the state vector . The time evolution is carried by the operators and the State vector or wave function is independent of time. This way of proceeding is referred to as working in the Heisenberg picture. When using perturbation theory it is convenient to use a third picture called the “interaction picture”. In this picture the time dependence due to unperturbed Hamiltonian is carried by the state vector and the evolution due to the perturbed part is carried by the operators. We describe the first two pictures below and discuss the third , the interaction picture , later when using perturbation theory.

3. Schrodinger & Heisenberg Picture:

Schrodinger Picture

In this picture the state vector denoted by |ψS(t)〉, depends on time, while the operator, along with other operators, is independent of time. The equation of motion is the Schroedinger equation in the form

as it should be. This ensures that both pictures lead to the same predictions (matrix elements) and are hence equivalent.

4 Quantisation of Scalar Field

Quantization: Imposing non commutativity

The main difference when going to quantum mechanics is the inability to simultaneously measure two physical variables like position and momentum. Mathematically this happens as the two variables (often called conjugate) do not commute. So to go to quantum mechanics or quantum field theory from classical physics, we have to impose non commutativity on conjugate variables. In Schrodinger picture the operators do not depend on time. In Heisenberg picture this implies , imposing commutation relation on the field operator and its conjugate at the same time.

Quantum Oscillators in momentum space.

It is much more convenient to go to momentum space (sometimes called Fock space) and deal with creation and annihilation operators. The field then behaves like a collection of quantized harmonic oscillators.

We expand the field as a Fourier integral over plane wave solutions:

The quantisation of the field, thus leads to states of particles or excitations.

5. Hamiltonian in terms of harmonic oscillator creation and annihilation operators ?_?, ?_?⁺.

The eigen states have fixed number of particles given by the eigen value ?_?. The creation operators ?_?⁺ raise the number of particles by one and annihilation operators decrease the number by one. So they are often also called ladder operators, taking one from one state to a state with higher number of particles.

6 Two (Many)-particle states and Statistics

Bose-Einstein Statistics

wave function or state vector is symmetric under the interchange of any two particles, it is said to obey Bose-Einstein statistics. The particles , quanta or excitations are called Bosons.

1. Learning Outcomes (Times New Roman , size 14)

After studying this module, you shall be able to (Times New Roman Font, size 11,)

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