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The set () of smooth vector fields along is a vector space under pointwise vector addition and scalar multiplication. [18] One can also pointwise multiply a smooth vector field along γ {\displaystyle \gamma } by a smooth function f : [ 0 , 1 ] → R {\displaystyle f:[0,1]\to \mathbb {R} } :
More precisely, the Hartle-Hawking state is a hypothetical vector in the Hilbert space of a theory of quantum gravity that describes the wave function of the universe.. It is a functional of the metric tensor defined at a (D − 1)-dimensional compact surface, the universe, where D is the spacetime dimension.
Note: This page uses common physics notation for spherical coordinates, in which is the angle between the z axis and the radius vector connecting the origin to the point in question, while is the angle between the projection of the radius vector onto the x-y plane and the x axis. Several other definitions are in use, and so care must be taken ...
In coordinate-free language, any vector space over complex numbers may be thought of as a real vector space of twice as many dimensions, where a complex structure is specified by a linear operator J (such that J 2 = −I) which defines multiplication by the imaginary unit i. Any such space, as a real space, is oriented.
The concept of universal wavefunction was introduced by Hugh Everett in his 1956 PhD thesis draft The Theory of the Universal Wave Function. [8] It later received investigation from James Hartle and Stephen Hawking [9] who derived the Hartle–Hawking solution to the Wheeler–DeWitt equation to explain the initial conditions of the Big Bang ...
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, inner product, norm, or topology) and the linear functions defined on these spaces and suitably respecting these structures.
In the natural sciences, a vector quantity (also known as a vector physical quantity, physical vector, or simply vector) is a vector-valued physical quantity. [ 9 ] [ 10 ] It is typically formulated as the product of a unit of measurement and a vector numerical value ( unitless ), often a Euclidean vector with magnitude and direction .
In the natural sciences, a vector quantity (also known as a vector physical quantity, physical vector, or simply vector) is a vector-valued physical quantity. [ 1 ] [ 2 ] It is typically formulated as the product of a unit of measurement and a vector numerical value ( unitless ), often a Euclidean vector with magnitude and direction .