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.

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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.

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A Measure Theoretical Approach to Quantum Stochastic Processes by Wilhelm Waldenfels


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