By Robert Carroll

The publication surveys mathematical kin among classical and quantum mechanics, gravity, time and thermodynamics from quite a few issues of view and plenty of resources (with acceptable attribution). The emergence subject is constructed with an emphasis at the that means through arithmetic. A historical past subject of Bohemian mechanics and connections to the quantum equivalence precept of Matone et al. can be built in nice aspect. a few unique paintings pertaining to the quantum power and Ricci movement can be integrated.

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In [177] we considered given a function Q ∈ L∞ (Ω) (for Ω a bounded domain) and looked for R ∈ H01 (Ω) satisfying Q = −( 2 /2m)(∆R/R) ≡ ∆R + (2m/ 2 )QR = 0. We showed that if Q < 0 (β = (2m/ 2 )) then there is a unique solution and if 0 is not in the countable spectrum of ∆R + βQR then 37 38 2. SOME GEOMETRIC ASPECTS ∆R + βQR = 0 has a unique solution for any Q ∈ L∞ . The corresponding HJ equation (1F) ∂t S + (1/2m)(∇S)2 + Q + V = 0 and the continuity equation (1G) ∂t R2 + (1/m)∇(R2 ∇S) = 0 must then be solved to obtain some sort of generalized quantum theory.

54)-2. 54)-2). 2. ON THE GUTZWILLER TRACE FORMULA 25 Next one shows that the Mehlig-Wilkinson operators coincide with the metaplectic operators SˆW,m when S = SW and will determine the correct choice for ν (which is related to the usual Maslov index in [376]). 56) det(SW − I) = det(B)det(B −1 A + DB −1 − B −1 − (B T )−1 Thus when S is written as in ( ) then (3S) det(SW − I) = det(L−1 )det(P + T Q − L − L ). 57) A−I C B D−I 0 I = C − (D − I)B −1 (A − I) B −1 (A − I) B D−I 0 I hence (3T) det(SW −I) = det(B)det[C −(D−I)B −1 (A−I)].

3) For χ a smooth function with compact support contained in ]E −δE, E + δE[, equal to 1 in a neighborhood of E, assume well defined the “reguˆ Aˆχ (H)g[(E ˆ ˆ larized density of states” ρA (E) = T r[χ(H) − H)/ ]. 1 implies the spectrum of H is purely discrete in a neighborhood of E so that ρA (E) is well defined. 1. 5) for g. 69) ρA (E) ≡ π −n/2 gˆ(0) + γ∈(ΓE )T −(n−1) A(α)dσE (α) + ΣE   ei[(Sγ / )+σγ π/2)] (2π)(n/2)−1 gˆ(Tγ )  |det(I − Pγ |1/2 ck (ˆ g) k k≥−n+2 Tγ∗ dγj (ˆ g) A(αs )ds + 0 j≥1 j    ˆ Tγ∗ is the primitive period of γ, σγ is where A(α) is the classical Weyl symbol of A, the Maslov index of γ (σγ ∈ Z), Sγ = γ pdq is the classical action along γ, ck (ˆ g) are distributions in gˆ with support in {0}, dγj are distributions in gˆ with support {Tγ } and dσE is the Liouville measure on ΣE , namely dσE = dΣE /|∇H| (where dΣE is Euclidean measure).

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