Explicit construction of a unitary double product integral

R. L. Hudson, and Paul Jones. "Explicit construction of a unitary double product integral." Banach Center Publications 96.1 (2011): 215-236. <http://eudml.org/doc/282100>.

@article{R2011,
abstract = {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.},
author = {R. L. Hudson, Paul Jones},
journal = {Banach Center Publications},
keywords = {double product integrals},
language = {eng},
number = {1},
pages = {215-236},
title = {Explicit construction of a unitary double product integral},
url = {http://eudml.org/doc/282100},
volume = {96},
year = {2011},
}

TY - JOUR
AU - R. L. Hudson
AU - Paul Jones
TI - Explicit construction of a unitary double product integral
JO - Banach Center Publications
PY - 2011
VL - 96
IS - 1
SP - 215
EP - 236
AB - 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.
LA - eng
KW - double product integrals
UR - http://eudml.org/doc/282100
ER -