Download e-book for iPad: A Measure Theoretical Approach to Quantum Stochastic by Wilhelm Waldenfels

By Wilhelm Waldenfels

ISBN-10: 3642450814

ISBN-13: 9783642450815

ISBN-10: 3642450822

ISBN-13: 9783642450822

This monograph takes as start line that summary quantum stochastic approaches should be understood as a quantum box concept in a single house and in a single time coordinate. accordingly it truly is acceptable to symbolize operators as strength sequence of construction and annihilation operators in normal-ordered shape, which might be completed utilizing classical degree theory.

Considering intimately 4 easy 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 issues contain the idea of the Hudson-Parthasarathy equation and the amplified oscillator challenge. consequently, a bankruptcy on white noise calculus has additionally been included.

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Additional info for A Measure Theoretical Approach to Quantum Stochastic Processes

Example text

3 Consider a family of operators T (t), t ∈ R, with T (0) = 1 and t → T (t)f norm continuous for f ∈ V , and T (t) ≤ const ert ; then T (t) is a oneparameter group if and only if R(z) satisfies the resolvent equation for | Im z| > r. Proof Assume to begin with that Im z1 > r and Im z2 > r; then ∞ −(z2 − z1 ) 0 ∞ 0 eiz1 tt +iz2 t2 T (t1 + t2 ) − T (t1 )T (t2 ) dt1 dt2 42 3 ∞ = −(z2 − z1 ) t dt1 eiz1 t1 +iz2 (t−t1 ) T (t) − (z2 − z1 )R(z1 )R(z2 ) dt 0 One-Parameter Groups 0 = R(z1 ) − R(z2 ) − (z2 − z1 )R(z1 )R(z2 ).

Xn−1 ). Then ∞ ax |f = n=0 1 n! f (ξ )ax |ξ = |ax f , ξ ∈X n 22 1 ax+ |f = ∞ n=0 1 n! Weyl Algebras f (ξ )a + |ξ = ax+ f . ξ ∈X n Proof We have f (x1 , . . , xn )ax |x1 , . . , xn n 1 n! x f (x1 , . . , xn ) δx,x1 |x2 , . . , xn + · · · + δx,xn |x1 , . . , xn−1 n 1 n! x f (x, x2 , . . , xn )|x2 , . . , xn = |ax f . n n n! ,xn For ax there is a similar calculation. We use the notation of Sect. 5. If α is a finite set and xα ∈ X α , then axα = ax+α = axc ; c∈α ax+c ; |xα = ax+α Φ. c∈α For c ∈ / α we have ax+c |xα = |xα+c , where we have used the shorthand α + c = α + {c}.

Similarly if Ψ is the measure defined by Ψ (f ) = Ψ |f = f (∅), then a(ϕ)Ψ = 0. We have therefore Ψ |Φ = 1. We define the mapping μ ∈ M (X) → μ(Φ) and use the notation for it μ(Φ) = Φ(μ) = Φ|μ . 3 Creation and Annihilation Operators on Locally Compact Spaces a(ϕ), a + (ν) = 33 ν(dx)ϕ(x) = ϕ|ν . We define μ|f = Δw μ(dw)f (w) = Δα μ(dxα )f (xα ), α X f |μ = μ|f . 1 We have a + (ν)μ|f = μ|a(ν)f , a(ϕ)μ|f = μ|a(ϕ)+ f or Δw a + (ν)μ (dw)f (w) = Δw a(ϕ)μ (dw)f (w) = Δw μ(dw) a(ν)f (w), Δw μ(dw) a + (ϕ)f (w).

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A Measure Theoretical Approach to Quantum Stochastic Processes by Wilhelm Waldenfels

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