Explicit construction of a unitary double product integral

• Autorzy: R. L. Hudson , Paul Jones
• Języki publikacji: EN

Abstrakty

EN

In analogy with earlier work on the forward-backward case, we consider an explicit construction of the forward-forward double stochastic product integral $∏^{→→}(1 + dr)$ with generator $dr = λ(dA^† ⊗ dA - dA ⊗ dA^†)$. The method of construction is to approximate the product integral by a discrete double product
$∏^{→→}_{(j,k)∈ℕ_m×ℕₙ} Γ(R_{m,n}^{(j,k)}) = Γ(∏^{→→}_{(j,k)∈ℕ_m×ℕₙ} (R_{m,n}^{(j,k)}))$
of second quantised rotations $R_{m,n}^{(j,k)}$ in different planes using the embedding of $ℂ^m ⊕ ℂⁿ$ into L²(ℝ) ⊕ L²(ℝ) in which the standard orthonormal bases of $ℂ^m$ and ℂⁿ are mapped to the orthonormal sets consisting of normalised indicator functions of equipartitions of finite subintervals of ℝ. The limits as m,n ⟶ ∞ of such double products of rotations are constructed heuristically by a new method, and are shown rigorously to be unitary operators. Finally it is shown that the second quantisations of these unitary operators do indeed satisfy the quantum stochastic differential equations defining the double product integral.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Banach Center Publications
• Rocznik: 2011
• Tom: 96
• Numer: 1
• Strony: 215-236

Daty

wydano

2011

Twórcy

autor

R. L. Hudson

• Mathematics Department, Loughborough University, Loughborough, Leicestershire LE11 3TU, Great Britain

autor

Paul Jones

• Mathematics Department, Loughborough University, Loughborough, Leicestershire LE11 3TU, Great Britain

Identyfikatory

DOI

10.4064/bc96-0-14