By Chern S.S., Li P., Cheng S.Y., Tian G. (eds.)

Those chosen papers of S.S. Chern speak about issues equivalent to critical geometry in Klein areas, a theorem on orientable surfaces in 4-dimensional area, and transgression in linked bundles

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1 Partition function in path integral formalism This paragraph discusses the scalar field which is described by the Schrodinger field operator ϕ(x) ˆ where x is the spatial coordinate. 1) where ϕ(x) is a c-number function. Let π ˆ (x) be its conjugate momentum operator. 2) where δ-is the Dirac delta function. 3) with eigenvalies π(x). 6) Suppose that a system is in state |ϕa > at a time t = 0. After time tf it will evolve into exp(−iHtf )|ϕa >. 7) For statistical mechanics purposes, consider the case in which the system returns to its original state after time tf .

6 Green’s function of fermi field The finite temperature fermionic Green’s function (or fermionic propagator) may be introduced in the similar way to scalar field. 79) with k µ = (ωn , k), ωn = (2π/β)(n + 1/2). 81) where {¯ γµ } is the set of Euclidean gamma matrices. 7 Notation As in the case of the scalar field one can work with a quantum model in Minkowski space-time, restoring the time variable t = −iτ defined in the interval [0, −iβ]. 83) with ωn = (2πi/β)(n + 1/2). 84) Chapter 3 THERMODYNAMICS OF QUANTUM GASES AND GREEN’S FUNCTIONS This section develops a method for calculation of thermodynamic potentials directly from finite temperature Green’s functions.

21). 64) ∂ψ − H(ψ + (x, t), ψ(x, t)) ∂t For the time variable τ , get Zβ = N exp ′ β dτ 0 d3 xψ + µ − idψ + dψ ∂ + iγ 0 γ · ∇ − mγ 0 ψ ∂τ The quantization of a Fermi system can be obtained as a result of integration over the space of anticommuting functions ψ(x, τ ) (x ∈ V, τ ∈ [0, β]), which are the elements of an infinite Grassman algebra. 65) are ψ(x, τ ) = (βV )−1/2 exp(i[ωτ + kx])ψn (k) n k ψ + (x, τ ) = (βV )−1/2 exp(−i[ωτ + kx])ψn+ (k) n k where ψn (k) and ψn+ (k) are the generators of Grassmann algebra.