Second quantization, Introduction to Malliavin calculus and analysis in Wiener space

The states of a quantum system form a Hilbert space.  In elementary quantum mechanics, this is usually the space of square-integrable functions on a Euclidean space of the appropriate dimension, or its slightly modified version---to account for additional degrees of freedom.  To describe a system with a variable particle number, or a quantum field, a more complicated Hilbert space is needed - a Fock space.  To define it, one starts from a Hilbert space  H - the space of one-particle states.  The n-th tensor power of H is the space of n-particle states.  The direct sum of all tensor powers of  H is called the Fock space over H and denoted F(H).  Its subspace F₊(H), consisting of vectors invariant under permutations of factors, is called the symmetric (or  bosonic) Fock space over H.  It describes the states of a system of identical particles, satisfying the Bose-Einstein statistics.  An important example is the electromagnetic field whose quanta---photons---are bosons.  

 

 

Quite unexpectedly, and very interestingly, the structure of a bosonic Fock space is hidden in a space which appears in the theory of stochastic processes---the space L²(P_W) of square-integrable functionals of a Wiener process.  This was first discovered by Norbert Wiener, and later studied and applied by several mathematicians, in particular Irving Segal, who realized the potential of this idea in mathematical physics. In an independent development, Paul Malliavin created a calculus of Wiener functionals, introducing the Malliavin derivative operator D and its dual---the divergence operator \delta.  These tools turned out to be very powerful, in particular leading to a probabilistic proof of the celebrated Hörmander Hypoellipticity Theorem.  As we are going to see, they are an incarnation of the creation and annihilation operators, fundamental for quantum mechanics of many-body systems.

 

The probabilistic representation of the bosonic Fock space has far-reaching consequences.  In particular, it allows to apply methods of stochastic analysis to study open quantum systems.  This series of lectures will be an introduction to these fascinating topics, important for both mathematics and physics.