By Bunge M.
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Extra resources for A Ghost - Free Axiomatization of Quantum Mechanics
J=1 The analogous statement for charge currents is proved in a similar way. By (38), M Jj = j=1 d t τ (NS )|t=0 , dt λ and so for any τλ -invariant state η one has M η(Jj ) = 0. 3 The Equivalent Free Fermi Gas In this subsection we shall show how to use the exponential law for fermionic systems to map the SEBB model to a free Fermi gas. Let ⎞ ⎛ h ≡ C ⊕ hR = C ⊕ ⎝ M hRj ⎠ , ˜ ≡ CAR(h), O h0 ≡ ε0 ⊕ hR , j=1 and, with a slight abuse of notation, denote by 1, f1 , · · · , fM the elements of h canonically associated with 1 ∈ C and fj ∈ hRj .
Its true cornerstones are the Onsager reciprocity relations (ORR), the Kubo fluctuationdissipation formula (KF) and the Central Limit Theorem (CLT). All three of them deal with the kinetic coefficients. The Onsager reciprocity relations assert that the matrix Lji of a time reversal invariant (TRI) system is symmetric, Lji = Lij . (17) For non-TRI systems, similar relations hold between the transport coefficients of the system and those of the time reversed one. For example, if time reversal invariance is broken by the action of an external magnetic field B, then the Onsager-Casimir relations Lji (B) = Lij (−B), hold.
We will only consider Lρ (Φj , Φk ), the other cases are completely similar. Using the CAR, Formula (29) and the fact that ωρ+ (Φl ) = 0, one easily shows that ωρ+ (τλt (Φj )Φk ) = Tr (T+ eithλ ϕj e−ithλ (I − T+ )ϕk ). Since d ithλ e hRj e−ithλ , dt the integration can be explicitly performed and we have eithλ ϕj e−ithλ = − Lρ (Φj , Φk ) = − lim T →∞ 1 Tr (T+ eithλ hRj e−ithλ (I − T+ )ϕk ) 2 T . −T Writing eithλ hRj e−ithλ = eithλ e−ith0 hRj eith0 e−ithλ and using the fact that ϕk is finite rank, we see that the limit exists and can be expressed in terms of the wave operators W± as Lρ (Φj , Φk ) = 1 Tr (T+ W−∗ hRj W− (I − T+ )ϕk ) 2 − Tr (T+ W+∗ hRj W+ (I − T+ )ϕk ) .
A Ghost - Free Axiomatization of Quantum Mechanics by Bunge M.