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Title:  Unitary double products as implementors of bogolubov transformations 
Authors:  Jones, Paul 
Keywords:  Quantum stochastic differential equations Quantum probability Bogolubov transformation Double product integrals 
Issue Date:  2013 
Publisher:  © Paul Jones 
Abstract:  This thesis is about double product integrals with pseudo rotational generator, and
aims to exhibit them as unitary implementors of Bogolubov transformations. We
further introduce these concepts in this abstract and describe their roles in the
thesis's chapters. The notion of product integral, (simple product integral, not
double) is not a new one, but is unfamiliar to many a mathematician. Product
integrals were first investigated by Volterra in the nineteenth century. Though often
regarded as merely a notation for solutions of differential equations, they provide
a priori a multiplicative analogue of the additive integration theories of Riemann,
Stieltjes and Lebesgue. See Slavik [2007] for a historical overview of the subject.
Extensions of the theory of product integrals to multiplicative versions of Ito and
especially quantum Ito calculus were first studied by Hudson, Ion and Parthasarathy
in the 1980's, Hudson et al. [1982]. The first developments of double product integrals
was a theory of an algebraic kind developed by Hudson and Pulmannova
motivated by the study of the solution of the quantum YangBaxter equation by
the construction of quantum groups, see Hudson and Pulmaanova [2005]. This
was a purely algebraic theory based on formal power series in a formal parameter.
However, there also exists a developing analytic theory of double product integral.
This thesis contributes to this analytic theory. The first papers in that direction are
Hudson [2005b] and Hudson and Jones [2012]. Other motivations include quantum
extension of Girsanov's theorem and hence a quantum version of the BlackScholes
model in finance. They may also provide a general model for causal interactions in
noisy environments in quantum physics. From a different direction "causal" double
products, (see Hudson [2005b]), have become of interest in connection with quantum
versions of the Levy area, and in particular quantum Levy area formula (Hudson
[2011] and Chen and Hudson [2013]) for its characteristic function. There is a close
association of causal double products with the double products of rectangular type
(Hudson and Jones [2012] pp 3). For this reason it is of interest to study "forwardforward"
rectangular double products.
In the first chapter we give our notation which will be used in the following chapters
and we introduce some simple double products and show heuristically that they
are the solution of two different quantum stochastic differential equations. For each
example the order in which the products are taken is shown to be unimportant;
either calculation gives the same answer. This is in fact a consequence of the so
called multiplicative Fubini Theorem Hudson and Pulmaanova [2005].
In Chapter two we formally introduce the notion of product integral as a solution
of two particular quantum stochastic differential equations.
In Chapter three we introduce the Fock representation of the canonical commutation
relations, and discuss the Stonevon Neumann uniqueness theorem. We define
the notion of Bogolubov transformation (often called a symplectic automorphism,
see Parthasarathy [1992] for example), implementation of these transformations by
an implementor (a unitary operator) and introduce Shale's theorem which will be
relevant to the following chapters. For an alternative coverage of Shale's Theorem,
symplectic automorphism and their implementors see Derezinski [2003].
In Chapter four we study double product integrals of the pseudo rotational type.
This is in contrast to double product integrals of the rotational type that have been
studied in (Hudson and Jones [2012] and Hudson [2005b]). The notation of the
product integral is suggestive of a natural discretisation scheme where the infinitesimals
are replaced by discrete increments i.e. discretised creation and annihilation
operators of quantum mechanics. Because of a weak commutativity condition, between
the discretised creation and annihilation operators corresponding on different
subintervals of R, the order of the factors of the product are unimportant (Hudson
[2005a]), and hence the discrete product is well defined; we call this result the discrete
multiplicative Fubini Theorem. It is also the case that the order in which the
products are taken in the continuous (nondiscretised case) does not matter (Hudson
[2005a], Hudson and Jones [2012]). The resulting discrete double product is shown
to be the implementor (a unitary operator) of a Bogolubov transformation acting
on discretised creation and annihilation operators (Bogolubov transformations are
invertible real linear operators on a Hilbert space that preserve the imaginary part
of the inner product, but here we may regard them equivalently as liner transformations
acting directly on creation and annihilations operators but preserving adjointness
and commutation relations). Unitary operators on the same Hilbert space
are a subgroup of the group of Bogolubov transformations. Essentially Bogolubov
transformations are used to construct new canonical pairs from old ones (In the
literature Bogolubov transformations are often called symplectic automorphisms).
The aforementioned Bogolubov transformation (acting on the discretised creation
and annihilation operators) can be embedded into the space L2(R+) L2(R+) and
limits can be taken resulting in a limiting Bogolubov transformation in the space
L2(R+) L2(R+). It has also been shown that the resulting family of Bogolubov
transformation has three important properties, namely bievolution, shift covariance
and timereversal covariance, see (Hudson [2007]) for a detailed description of these
properties.
Subsequently we show rigorously that this transformation really is a Bogolubov
transformation. We remark that these transformations are HilbertSchmidt perturbations
of the identity map and satisfy a criterion specified by Shale's theorem. By
Shale's theorem we then know that each Bogolubov transformation is implemented
in the Fock representation of the CCR. We also compute the constituent kernels
of the integral operators making up the HilbertSchmidt operators involved in the
Bogolubov transformations, and show that the order in which the approximating
discrete products are taken has no bearing on the final Bogolubov transformation
got by the limiting procedure, as would be expected from the multiplicative Fubini
Theorem.
In Chapter five we generalise the canonical form of the double product studied in
Chapter four by the use of gauge transformations. We show that all the theory
of Chapter four carries over to these generalised double product integrals. This
is because there is unitary equivalence between the Bogolubov transformation got
from the generalised canonical form of the double product and the corresponding
original one.
In Chapter six we make progress towards showing that a system of implementors
of this family of Bogolubov transformations can be found which inherits properties
of the original family such as being a bievolution and being covariant under shifts.
We make use of Shales theorem (Parthasarathy [1992] and Derezinski [2003]). More
specifically, Shale's theorem ensures that each Bogolubov transformation of our
system is implemented by a unitary operator which is unique to with multiplicaiton
by a scalar of modulus 1. We expect that there is a unique system of implementors,
which is a bievolution, shift covariant, and time reversal covariant (i.e. which
inherits the properties of the corresponding system of Bogolubov transformation).
This is partly ongoing research. We also expect the implementor of the Bogolubov
transformation to be the original double product. In Evans [1988], Evan's showed
that the the implementor of a Bogolubov transformation in the simple product case
is indeed the simple product. If given more time it might be possible to adapt
Evan's result to the double product case.
In Hudson et al. [1984] the three properties of bievolution, shift covariant, and time
reversal covariant (in only one degree of freedom) were used to show uniqueness of
the so called timeorthogonal unitary dilation" in the case of nonFock quantum
stochastic calculus. The double" case in this thesis is much harder. I show using
the bievolutionarity and shiftcovariance that there exists an unique system of implementors
up to multiplication by a family of scalars
{ e^{i(ba)(ts)} }_{(0<a<b;0<<st)}
for some real parameter. It is expected that the parameter can be uniquely specified
using the time reversal property. It is not clear, as of yet, how this should be
done, even though it is quite simple in the similar case of Hudson et al. [1984]. 
Description:  A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University. 
URI:  https://dspace.lboro.ac.uk/2134/14306 
Appears in Collections:  PhD Theses (Maths)

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