Problem 16: Coherent states

Friedrich-Schiller-Universität Jena Summer Term 2020 Prof. Andreas Wipf

M.Sc. Michael Mandl

Problems Quantum Field Theory

Sheet 5

Problem 16: Coherent states

Consider a hermitian scalar eld φ(x) given by φ(x) =

Z

dµ(p)

a p u p (x) + a ^† _p u ^∗ _p (x)

,

where we use notation established on previous exercise sheets. Now consider the coher- ent state |ηi , dened by

|ηi = exp Z

dµ(p)η(p)a ^† _p

|0i ,

where η(p) is a normalizable function dened on the mass shell p ² = m ² , p ⁰ > 0 : Z

dµ(p)|η(p)| ² < ∞ .

1. Show that |ηi is an eigenstate of the annihilation operator a(p) and therefore diagonalizes the annihilation part of φ(x) .

2. Show that coherent states |η ₁ i and |η ₂ i are non-orthogonal for η 1 (p) 6= η 2 (p) . Are they normalized?

3. Let |{n, p}i = |n ₁ , p 1 ; n 2 , p 2 ; ...; n N , p N i denote a state in Fock space with n 1

particles of momentum p 1 , n 2 particles of momentum p 2 , etc. Show that |ηi can be used as a generating function for |{n, p}i .

4. Now dene φ f as the smearing of φ(x) (which is an operator-valued distribution) with some suitable test function f(x) :

φ _f = Z

d ³ xf (x)φ(x) .

One may study the probability distribution of φ f in a state |{n, p}i , dened as ρ _{n,p} (α) = h{n, p}|δ(φ _f − α)|{n, p}i =

Z ds

2π e ^−iαs h{n, p}|e ^isφ

|{n, p}i . Since |ηi generates the |{n, p}i , the problem of nding ρ _{n,p} (α) reduces to that of nding hη ₁ |e ^isφ

|η ₂ i . Thus, compute hη ₁ |e ^isφ

|η ₂ i . Then, show explicitly that the vacuum distribution ρ ₀ (α) = h0|δ(φ _f − α)|0i is a Gaussian distribution centered around a = 0 with a variance σ ² = R

dµ(p)| f ˜ (p)| ² , where f ˜ (p) =

Z

d ³ x u p (x)f (x)

is the Fourier transform of f(x) .

Problem 17: Heisenberg picture

Consider a charged scalar eld in the Schrödinger picture:

φ(x) = Z

dµ(p)

a p u p (x) + b ^† _p u ^∗ _p (x)

.

Now transition to the Heisenberg picture by introducing time dependent operators a p (t) = e ^iHt a p e ^−iHt ,

etc., where H denotes the Hamiltonian of the theory. Show that, in the Heisenberg picture,

φ(x) = 1 (2π) ^3/2

Z

dµ(p)

a p e ^−ipx + b ^† _p e ^ipx

, where px = x _µ p ^µ .

Problem 18: Helicity

The angular momentum of the radiation eld is given by J =

Z

d ³ x x ∧ (E ∧ B) .

Dene the corresponding operator. Then, consider the component of J in the direction of propagation, i.e., the projection

J · ˆ k , where ˆ k denotes the normalized vector. For a state

|k, i = (α a ^† _k,1 + β a ^† _k,2 )|0i ,

determine the coecients α und β , such that the (normalized) state is an eigen-

state of J · ˆ k. (Here, the expansion of the eld in terms of momentum modes and

the operators a ^† _k,1/2 are dened as in the lecture).