QUANTUM ANALYSIS AND OPERATOR CALCULUS

Quantum Operator Calculus (QOC)

goes back to the Feynman's famous rules for calculation of the variation of the operator exponential function in the form of a perturbation series. Since then many authors made attempts to clarify these mysterious rules and put them on a solid mathematical bases, the most notorious among them was Maslov's Operator Calculus. My own explicit triangular-matrix formula
MATH
is the simplest algorithmic form of the operator chain rule which gives an alternative approach to QOC. It was discovered as a particular case of the quantum functional "Itô formula" [1, 2] which is based on the canonical triangular-matrix representation for Quantum-Itô variations MATH of four basic QS processes of exchange MATH, creation $A_{\circ }^{+}$, annihilations $A_{-}^{\circ }$ and preservation $A_{-}^{+}$. As this formula suggests, finding a variation of a function of operator $X$ corresponding to its noncommutative variation $\QTR{rm}{d}X$ is reduced to the evaluation of the upper right element MATH of the block-triangular operator MATH, the only nonzero element of the difference MATH. Thus the noncommutative chain rule is defined as soon as the function $f$ of the block-triangular operator MATH, where is the $2\times 2$-ampliation MATH of $X$, and MATH is its $2\times 2$-nilpotent variation MATH, makes sense.

Quantum Stochastic Calculus (QSC)

explores the HP (Hudson-Parthasarathy) "Itô table" which has simple multiplication structure
MATH
of the matrix product for the triangular matrices MATH and MATH having zero elements $B_{\nu }^{\mu }=0$ if $\mu =+$ or $\nu =-$ and indefinite-metric $\star $-involution MATH defined by the reflection MATH in the Hermitian conjugate matrix MATH. I discovered this $\star $-algebraic structure of the Quantum Stochastic Calculus (QSC) in 1987 while developing nondemolition stochastic calculus for solving quantum time-continuous measurement problem and filtering [3,4]. Using this underlying 'Heaven structure' I extended the HP Itô multiplication rule to quantum functional and nonadapted Itô formula [5] presented for the adapted QS operator differentials MATH in [1,2] as the four differences sumMATH
over $\mu =-,\circ $ and $\nu =\circ ,+$. It was obtained as a down-to-Earth projection of the finite difference chain rule in a 'Heaven calaculus' underlying the Quantum Stochastic Analysis developed in [5-8].

Quantum Stochastic Analysis (QSA)

is a development of Quantum Stochastic Calculus (QSC) which started in the early 80's with the work of Hudson and Parthasarathy (HP) on coherent vectors in Fock space. My approach, based on Fock scale representation of the basic processes MATH, which I defined in [3] as bounded operators on a natural projective limit domain in this scale, extended the classical white noise calculus from the generalized functionals to generalized operators (kernels) on the Fock projective limit. This work lead to a new 'Heaven calculus' which unifies the HP approach with Maassen-Meyer kernel calculus and extends the HP QS integrals from the exponential to the natural kernel domain of the Guichardet Fock and pseudo-Fock 'Heaven space'. The explicit operator definition of the QS integrals, introduced in [4], was based on the discovered in [3] canonical triangular-matrix representation of the QS differentials. This was immediately generalized to the non-adapted QS processes in my QSA paper [5] by the analogy with Boson Wick calculus [6]. Thus the quantum analog of Skorohod integral and Malliavin calculus based on four fundamental QS point derivatives MATH instead of a single one as in the classical stochastic calculus was introduced. The non-adapted quantum Itô formula
derived in [5], which has the same formMATH
as in the adapted case but is given by nondiagonal $3\times 3$-ampliation MATH of the operator curve MATH with $X_{\nu }^{\mu }=0$ if $\mu =+$ or $\nu =-$ and four quantum "Malliavin derivatives" MATH , enabled the QS differential formulation of the unitarity and homomorphic conditions for non-adapted QS differential equations and flows in [5,6]. The infinite dimensional QSA with respect to the arbitrary quantum white noise was developed in [7,8], the Quantum Stochastic Calculus (QSC) of for quantum filtering -- in [9], and the question of finding the QS integrants for Fock space operators was studied in [10]. Attal and Lindsay recently developed the adapted version of the QSA on the natural domain of the Guichardet Fock space.

My 10 Relevant Publications:
  1. V. P. Belavkin: The Unified Ito Formula Has Pseudo-Poisson Structure. Journal of Mathematical Physics 34 (4) 1508--1518 (1993).
  2. V. P. Belavkin: Quantum Functional Ito Formula. In Quantum Probability and Related Topics 8 81--85 World Scientific, Singapore, 1993.
  3. V. P. Belavkin: A New Form and *-Algebraic Structure of Quantum Stochastic Integrals in Fock Space. Rendiconti del Seminario Matematico e Fisico di Milano LVIII 177--193 (1988).
  4. V. P. Belavkin: A Quantum Stochastic Calculus in Fock Space of Input and Output Non-Demolition Processes. In: Quantum Probability and Applications LNM 1442 99--125 Springer Verlag, Berlin 1990.
  5. V. P. Belavkin: A Quantum Nonadapted Ito Formula and Stochastic Analysis in Fock Scale. J of Funct Analysis 102 2 414--447 (1991). math-ph/0512076, PDF.
  6. V. P. Belavkin: A Boson Field Multiple Integrals and Quantum Filtering Equations in Fock Scale. Centro Matimatico V Volterra, Universita'degli studi di Roma II 80 1--26 (1991). math.PR/0512362, PDF.
  7. V. P. Belavkin: A Non-Adapted Stochastic Calculus and Non-Stationary Quantum Evolution in Fock Space. In Quantum Probability and Related Topics 6 137--179 World Scientific, Singapore 1991. math.PR/0512509, PDF.
  8. V. P. Belavkin: Stochastic Calculus and Filtering in Noncommutative Probability Theory. DSci Thesis. Steklov Mathematical Institute AN USSR, Moscow 1991. See also math.PR/0512265, PDF of the paper [5] on the QCC page.
  9. V. P. Belavkin: Quantum Stochastic Calculus and Quantum Nonlinear Filtering. Journal of Multivariate Analysis 42 (2) 171--201 (1992). math.PR/0512362, PDF.
  10. V. P. Belavkin and J. M. Lindsay: The Kernel of a Fock Space Operator. In Quantum Probability and Related Topics 8 87--94 World Scientific, Singapore 1993.