Wilhelm Waldenfels's A Measure Theoretical Approach to Quantum Stochastic PDF

By Wilhelm Waldenfels

ISBN-10: 3642450814

ISBN-13: 9783642450815

ISBN-10: 3642450822

ISBN-13: 9783642450822

This monograph takes as place to begin that summary quantum stochastic methods should be understood as a quantum box concept in a single area and in a single time coordinate. for this reason it truly is acceptable to symbolize operators as energy sequence of production and annihilation operators in normal-ordered shape, that are completed utilizing classical degree theory.

Considering intimately 4 uncomplicated examples (e.g. a two-level atom coupled to a warmth tub of oscillators), in each one case the Hamiltonian of the linked one-parameter strongly non-stop team is decided and the spectral decomposition is explicitly calculated within the type of generalized eigen-vectors.

Advanced themes contain the idea of the Hudson-Parthasarathy equation and the amplified oscillator challenge. thus, a bankruptcy on white noise calculus has additionally been included.

Show description

Read Online or Download A Measure Theoretical Approach to Quantum Stochastic Processes PDF

Best quantum theory books

Quanta, logic, and spacetime : variations on Finkelstein's - download pdf or read online

For researchers in nonlinear technology, this paintings comprises insurance of linear platforms, balance of options, periodic and nearly periodic impulsive structures, critical units of impulsive platforms, optimum regulate in impulsive structures, and extra 1. Foundations -- 2. good judgment and Set idea -- three. crew Duality, Coherence and Cyclic activities -- four.

Read e-book online Mathematical Feynman Path Integrals and Their Applications PDF

Even if greater than 60 years have handed when you consider that their first visual appeal, Feynman direction integrals haven't begun to lose their fascination and luster. they don't seem to be just a ambitious tool of theoretical physics, but in addition a mathematical problem; actually, numerous mathematicians within the final forty years have committed their efforts to the rigorous mathematical definition of Feynman's principles.

Additional resources for A Measure Theoretical Approach to Quantum Stochastic Processes

Example text

As the values of a function at a given point are generally not defined, we cannot define ax and ax+ for a given x ∈ X. But the definitions at the end of Sect. 7 can be generalized. Define for f ∈ L(n + 1) and g ∈ L(1) a(g)f (x1 , . . xn ) = λ(dx0 )g(x0 )f (x0 , x1 , . . , xn ) and for f ∈ L(n − 1) a + (g)f (x1 , . . , xn ) = g(xc )f (x[1,n]\{c} ). c∈[1,n] One obtains in the usual way a(g) Γ + a (g) √ n+1 g Γ f √ ≤ n g Γ f Γ ≤ Γ Γ, with, of course, g 2 Γ = 2 λ(dx) g(x) . The mappings a(g) and a + (g) can be extended to operators Γfin (X, λ) → Γfin (X, λ), and one has f |a(g)h = a + (g)f |h and the commutator a(f ), a + (g) = λ(dx)f (x)g(x).

Dx1 ) · · · λ(dxn ) fn (x1 , . . , xn ) 2 for f = f0 ⊕ f1 ⊕ f2 ⊕ · · · with fn ∈ L(n), and g accordingly. So f is in Γ , if only and if f Γ < ∞. We define the subspace Γfin ⊂ Γ of those f such that fn = 0 for n sufficiently large. W. 1007/978-3-642-45082-2_2, © Springer-Verlag Berlin Heidelberg 2014 25 26 2 Continuous Sets of Creation and Annihilation Operators Recall the definition of X = {∅} + X + X 2 + · · · and provide X with the measure ∞ eˆ (λ)(f ) = f (∅) + n=1 1 n! λ(dx1 ) · · · λ(dxn )fn (x1 , .

If α = (α0 , α1 , α2 , . , we have |f = (Δα)f (xα )|xα , α f |g = (Δα)f (xα )g(xα ). α One obtains for an additional index c (axc f )(xα ) = f (xα+c ) and for xc ∈ X ax+c f (xα ) = δxc ,xb f (xα\c ). 7 Finite Sets of Creation and Annihilation Operators 23 If g : X → C is a function, then define a + (g) = g(x)ax ; a(g) = x∈X g(x)ax+ . x∈X We obtain for f ∈ Ks (X) a(g)f (xα ) = g(xc )f (xα+c ), xc ∈X a + (g)f (xα ) = g(xc )f (xα\c ). c∈α One has also for the commutator a(g), a + (h) = g|h . Chapter 2 Continuous Sets of Creation and Annihilation Operators Abstract We define first the operators a(ϕ) and a + (ϕ) on the usual Fock space.

Download PDF sample

A Measure Theoretical Approach to Quantum Stochastic Processes by Wilhelm Waldenfels

by Daniel

Rated 4.17 of 5 – based on 7 votes