In noncommutative geometry the paradigm of a geometric space is given in spectral terms, by a Hilbert space H in which both the algebra A of coordinates and the analogue of the inverse line element ds?1are represented, the latter being embodied by an unbounded self-adjoint operator D whichplaystheroleoftheDiracoperator. The local geometric invariants such as the Riemannian curvature are extracted from the functionals de?ned by the coefficients of heat kernel expansion Tr(ae~(-tD~2))～t↘∑n≥0an(a,D~2)t-d+n/2,a∈A, where d is the dimension of the geometry. Equivalently, one may consider special values of the corresponding zeta functions. Thus, it is the high frequency behavior of the spectrum of D coupled with the action of the algebra A in H which detects the local curvature of the geometry.
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机译：在非可交换几何中，几何空间的范式由光谱形式给出，由希尔伯特空间H给出，在该希尔伯特空间H中，坐标的代数A和反线元素ds？1的类似物均被表示，后者由无界的自播放Diracoperator的角色的伴随运算符D。从热核膨胀系数Tr（ae〜（-tD〜2））〜t↘∑n≥0an（a，D〜2）t定义的函数中提取局部几何不变量，如黎曼曲率-d + n / 2，a∈A，其中d是几何尺寸。同样，可以考虑相应zeta函数的特殊值。因此，正是D频谱的高频行为与H中的代数A的作用相结合才能检测出几何形状的局部曲率。
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