2 Corrected typo in commutation relation

## Background

Suppose we have a finite-dimensional Hilbert space $H = \mathbb{C}^s$ (for a natural number s) and we construct the symmetric (or bosonic) Fock space built from it: $$F(H):= \mathbb{C} \oplus H \oplus S(H \otimes H) \oplus S(H \otimes H \otimes H) \oplus \ldots$$ where S is the symmetrising operator.

Vectors in F are sequences of vectors $\psi = (\psi_0, \psi_1,\psi_2,\ldots)$ such that $\psi_0 \in \mathbb{C}$, $\psi_1 \in H$, $\psi_2 \in S(H \otimes H)$ etc such that $\sum_{n=0}^\infty ||\psi_n||_n^2 < \infty$ where || ||n denotes the appropriate norm.

For any vector f $\in$ H we can define a pair of unbounded densely defined operators $a^\dagger(f)$ and $a(f)$ acting on F. These are called the "creation and annihilation operators". They are mutually adjoint and satisfy a commutation relation of the form: $$a^\dagger(f) a(ga(f) a^\dagger(g) - a(g) a^\dagger(g) a(f) = \langle f, g\rangle$$ where $\langle f, g\rangle$ is the inner-product of f, g $\in$ H.

The best reference for all this is M. Reed, B. Simon, "Methods of Mathematical Physics, Vol 2", section X.7 p207-212. This is partially available on Google books here: http://books.google.co.uk/books?id=Kz7s7bgVe8gC&lpg=PA141&dq=reed%20and%20simon%20x.7&client=firefox-a&pg=PA210#v=onepage&q=&f=false

The sum $\phi(f) = a(f) + a^\dagger(f)$ is self-adjoint (more properly the closure of their sum is self-adjoint) and is called the Segal quantisation of f (up to a factor of $\sqrt{2}$).

Since $\phi(f)$ is self-adjoint we can apply the spectrum theorem to it. The question is, what is its spectral decomposition? Or more loosely, what are its eigenvalues and eigenvectors? or what can we tell from about its spectral decomposition?

1

# Spectral theory for self-adjoint field operators on a symmetric Fock space

## Background

Suppose we have a finite-dimensional Hilbert space $H = \mathbb{C}^s$ (for a natural number s) and we construct the symmetric (or bosonic) Fock space built from it: $$F(H):= \mathbb{C} \oplus H \oplus S(H \otimes H) \oplus S(H \otimes H \otimes H) \oplus \ldots$$ where S is the symmetrising operator.

Vectors in F are sequences of vectors $\psi = (\psi_0, \psi_1,\psi_2,\ldots)$ such that $\psi_0 \in \mathbb{C}$, $\psi_1 \in H$, $\psi_2 \in S(H \otimes H)$ etc such that $\sum_{n=0}^\infty ||\psi_n||_n^2 < \infty$ where || ||n denotes the appropriate norm.

For any vector f $\in$ H we can define a pair of unbounded densely defined operators $a^\dagger(f)$ and $a(f)$ acting on F. These are called the "creation and annihilation operators". They are mutually adjoint and satisfy a commutation relation of the form: $$a^\dagger(f) a(g) - a(g) a^\dagger(g) = \langle f, g\rangle$$ where $\langle f, g\rangle$ is the inner-product of f, g $\in$ H.

The best reference for all this is M. Reed, B. Simon, "Methods of Mathematical Physics, Vol 2", section X.7 p207-212. This is partially available on Google books here: http://books.google.co.uk/books?id=Kz7s7bgVe8gC&lpg=PA141&dq=reed%20and%20simon%20x.7&client=firefox-a&pg=PA210#v=onepage&q=&f=false

The sum $\phi(f) = a(f) + a^\dagger(f)$ is self-adjoint (more properly the closure of their sum is self-adjoint) and is called the Segal quantisation of f (up to a factor of $\sqrt{2}$).

Since $\phi(f)$ is self-adjoint we can apply the spectrum theorem to it. The question is, what is its spectral decomposition? Or more loosely, what are its eigenvalues and eigenvectors? or what can we tell from about its spectral decomposition?