A second quantization procedure is formulated in the framework of Nelson’s stochastic mechanics. It is shown that the annihilation operator is given by an integration over a Gaussian white noise Ḃ(s) with the kernel exp(iω(t−s)) and the creation operator is given by an integration over a time reversed Gaussian white noise Ḃ*(s) with the kernel exp(−iω(t−s)). The eigenfunctions on which the operators act are shown to be Gaussian random variables whose collection is dense in a Hilbert space. Employing the above operators, we can construct {N}-representation in the stochastic mechanics.
This is a self-contained and hopefully readable account on the method of creation and annihilation operators (also known as the Fock space representation or the "second quantization" formalism) for non-relativistic quantum mechanics of many particles. Assuming knowledge only on conventional quantum mechanics in the wave function formalism, we define the creation and annihilation operators, discuss their properties, and introduce corresponding representations of states and operators of many-particle systems. As the title of the note suggests, we cover most topics treated in sections 6.7 and 6.8 of Feynman's "Statistical Mechanics: A Set of Lectures". As a preliminary, we also carefully discuss the symmetry of wave functions describing indistinguishable particles. We note that all the contents of the present note are completely standard, and the definitions and the derivations presented here have been known to many. Although the style of the present note may be slightly more mathematical than standard physics literatures, we do not try to achieve full mathematical rigor.(Note to experts: In particular we here DERIVE the (anti)commutation relations of the creation and annihilation operators, rather than simply declaring them. In this sense our approach is quite close to that of Feynman's. But we here focus on the action of creation/annihilation operators on general
N body wave functions, while Feynman makes a heavy use of Slater-determinant-type states from the beginning. We hope that our presentation provides a better perspective on the formalism.)
In this paper we investigate a quantum stochastic calculus build of creation, annihilation and number of particles operators which fulfill some deformed commutation relations. Namely, we introduce a deformation of a number of particles operator which has simple commutation relations with well known q-deformed creation and annihilation operators. Since all operators considered in this theory are bounded we do not deal with some difficulties of a non deformed theory of Hudson and Parthasarathy. We define stochastic integrals and estimate them in the operator norm. We prove Ito's formula as well.
IntroductionLet
at and
at∗ be respectively annihilation and creation operators at apoint
t∈R .In the theory of white noise operators [23][24], it is known that
at is acontinuous linear operator from
Γu(Ec) , the (Fock) space of test functions over acomplexified nuclear space
Ec , into itself, and at is acontinuous linear operator from
Γu(EC)∗ , the dual space of
Γu(EC) , into itself.In paticular,
at+at∗ and
at+at∗+at∗at+I are cffied the quantum Gaussian white noise and the quantum Poisson white noise, respectively (cf.[11] [24]).These are known to be elemental generalzed qunatum stochastic processes.The main purpose of this work is to examine relationships between classical and quantum white noises from the point of the classical white noise theory of Gaussian [19][20] and Poisson types [13].Then it will be clarified that quantum Gaussian and Poisson white noises are the Fock space realzations of classical Gaussian and Poisson white noises, respectively.We shall discuss them in Section 4. As amatter of fact, one can see that such facts can be proved not only through
J -transform given by (2.4) and the holomorphy, but also through Sx-transform gllven by (3.3) depending on the exponential function
ϕξX .The function
ϕξX determines aunitary isomorphism between Boson Fock space and
L2(E∗,μx) ,
X=G,
P , where
μG and
μP are a Gaussian measure and aPoisson measure on
E∗ , respectively.As is pointed out in Remark 4.3, the choice of
ϕξX (kernel function of
SX -transform)is the essential part to represent the above different quantum noises acting on the same space
Γu(Ec)∗ .In Appendix, some connections between
[6][7] and [10]
wiU be noted.
The basic integrator processes of quantum stochastic calculus, namely, creation, conservation, and annihilation, are introduced in the Hilbert space of square integrable Brownian functionals. Stochastic integrals with respect to these processes and a quantum Itô’s formula are described. As an application two examples of quantum stochastic differential equations are discussed. A continuous time version of Stinespring’s theorem on completely positive maps in C *-algebras is exploited to formulate the notion of a quantum Markov process and indicate how classical Markov processes are woven into the fabric of the Schodinger-Heisenberg dynamics of quantum theory.
A review of the second quantization formulation of quantum mechanics is given based on the idea of a graded Hilbert space whose elements are represented by column matrices containing zero, one, two, ···, particle functions. The creation and annihilation operators are defined in terms of intuitive ideas about adding a particle to or removing a particle from the system. The (anti)commutation relations are shown to follow from the (anti)symmetry of the wavefunctions. The field operators which describe the creation or annihilation of a particle at a point are then obtained and their properties discussed. These operators are then used to express general operators in second quantization. Finally, a basis is introduced and the equations of occupation-number space are obtained.