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Laplace transforms of Heisenberg operators in finite-dimensional spaces

Ф. С. Джепаров

Published May 1, 1977• 0 citations
OpenAlex

Abstract

The integral X (lambda) =..integral../sup infinity//sub 0/ dt exp(-(lambda-iF) t)X exp(-iHt) and the solution to the matrix equation (lambda-iF) X (lambda)+iX (lambda) H=X, where F and H are Hermitian matrices of dimensionality M and N, respectively, are written as polynomials of order N-1 in H and M-1 in F. The coefficients are determined from the characteristic polynomials of the matrices F and H.

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Supporting material and practice exercises

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First prove that fµν(−ω) = fνµ(ω). To see this consider φµν(t) =∫ β 0 dλ〈Jν(−i~λ)Jµ(t)〉 or φµν(−t) = ∫ β 0 dλ〈Jν(−i~λ)Jµ(−t)〉 = ∫ β 0 dλtr(Jµ(−t)e−βHJν(−i~λ)) = dλtr(e−iHt/~Jµ(0)e−βH+λHJν(0)e−λH), (8.1) where the cyclic property of the trace operation is used, further, the inte- grand can be written as tr(e−βHe(β−λ)HJµ(0)e−(β−λ)HeiHt/~Jν(0)e−iHt/~) or 〈Jµ(−i~(β − λ))Jν(t)〉. Thus, φµν (−t) = ∫ β 0 dλ〈Jµ(−i~(β − λ))Jν(t)〉. (8.2) Shift the integration variable from λ to λ′ = β − λ φµν(−t) = ∫ β 0 dλ′〈Jµ(−i~λ)Jν(t)〉 = φνµ(t). (8.3) Now, in the definition of the Fourier transform of the response function (Equa- tion (3.53)) change ω to −ω and then shift the integration variable from t to −t fµν(−ω) = ∫ +∞ −∞ dtφµν(−t)eiωt = ∫ +∞ −∞ dtφνµ(t)e iωt = fνµ(ω). QED.

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The integral X (lambda) =..integral../sup infinity//sub 0/ dt exp(-(lambda-iF) t)X exp(-iHt) and the solution to the matrix equation (lambda-iF) X (lambda)+iX (lambda) H=X, where F and H are Hermitian matrices of dimensionality M and N, respectively, are written as polynomials of order N-1 in H and M-1 in F. The coefficients are determined from the characteristic polynomials of the matrices F and H.
The purpose of this note is to investigate dynamical features of the 2x2 matrix approach contained in Einstein´s 1905 energy momentum equation E2 = p2 + m2 (1) In particular, we wish to investigate the existence of physical oscillations consistent with the model. The model was based on conversion of (1) into a 2x2 matrix eigenvalue system with various matrices as operators. The matrices and eigenvectors obey quantum mechanical math with exp(-iHt) as a time evolution operator. It is thus possible to calculate O(t)=exp(iHt) O exp(-iHt) where O is an operator. This leads to oscillatory motion along and perpendicular to the x axis in the one dimensional approach used. This oscillatory motion is considered in sme detail and comparisons with transformations of light from a stationary to moving frame are made.
Some results of the the dependent formalism in scattering theory are presented. The asymptotic condition with an observable algebra A acting on a small Hilbert space, h, is imposed. H is given by H = H/sub 0/ + V, where H/sub 0/ is the free Hamiltonian on h and V the potential present. The demonstrations given are based on the convergence properties of the spectral measurements associated with the vectors, exp(-iHt) psi, psi being the scattering state. It is shown thnt the OMEGA /sup plus or minus / operators plus or minus infinity), where g belongs to a group of dense h vectors and where T/sub t/ is a family of limited operators in A. A necessary and sufficient condition is given for OMEGA /sup plus or minus / to exist and to permit a choice of T/sub t/ unitary operators to be made. (auth)
First prove that fµν(−ω) = fνµ(ω). To see this consider φµν(t) =∫ β 0 dλ〈Jν(−i~λ)Jµ(t)〉 or φµν(−t) = ∫ β 0 dλ〈Jν(−i~λ)Jµ(−t)〉 = ∫ β 0 dλtr(Jµ(−t)e−βHJν(−i~λ)) = dλtr(e−iHt/~Jµ(0)e−βH+λHJν(0)e−λH), (8.1) where the cyclic property of the trace operation is used, further, the inte- grand can be written as tr(e−βHe(β−λ)HJµ(0)e−(β−λ)HeiHt/~Jν(0)e−iHt/~) or 〈Jµ(−i~(β − λ))Jν(t)〉. Thus, φµν (−t) = ∫ β 0 dλ〈Jµ(−i~(β − λ))Jν(t)〉. (8.2) Shift the integration variable from λ to λ′ = β − λ φµν(−t) = ∫ β 0 dλ′〈Jµ(−i~λ)Jν(t)〉 = φνµ(t). (8.3) Now, in the definition of the Fourier transform of the response function (Equa- tion (3.53)) change ω to −ω and then shift the integration variable from t to −t fµν(−ω) = ∫ +∞ −∞ dtφµν(−t)eiωt = ∫ +∞ −∞ dtφνµ(t)e iωt = fνµ(ω). QED.