Position and momentum spaces

Summary

In physics and geometry, there are two closely related vector spaces, usually three-dimensional but in general of any finite dimension. Position space (also real space or coordinate space) is the set of all position vectors r in Euclidean space, and has dimensions of length; a position vector defines a point in space. (If the position vector of a point particle varies with time, it will trace out a path, the trajectory of a particle.) Momentum space is the set of all momentum vectors p a physical system can have; the momentum vector of a particle corresponds to its motion, with units of [mass][length][time]−1. Mathematically, the duality between position and momentum is an example of Pontryagin duality. In particular, if a function is given in position space, f(r), then its Fourier transform obtains the function in momentum space, φ(p). Conversely, the inverse Fourier transform of a momentum space function is a position space function. These quantities and ideas transcend all of cl

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We compute a family of scalar loop diagrams in AdS. We use the spectral representation to derive various bulk vertex/propagator identities, and these identities enable to reduce certain loop bubble diagrams to lower loop diagrams, and often to tree- level exchange or contact diagrams. An important example is the computation of the finite coupling 4-point function of the large-N conformal O(N ) model on AdS(3). Remarkably, the re-summation of bubble diagrams is equal to a certain contact diagram: the D1,1,32,32zz function. Another example is a scalar with phi (4) or phi (3) coupling in AdS(3): we compute various 4-point (and higher point) loop bubble diagrams with alternating integer and half- integer scaling dimensions in terms of a finite sum of contact diagrams and tree-level exchange diagrams. The 4-point function with external scaling dimensions differences obeying (12) = 0 and (34) = 1 enjoys significant simplicity which enables us to compute in quite generality. For integer or half-integer scaling dimensions, we show that the M -loop bubble diagram can be written in terms of Lerch transcendent functions of the cross- ratios z and z. Finally, we compute 2-point bulk bubble diagrams with endpoints in the bulk, and the result can be written in terms of Lerch transcendent functions of the AdS chordal distance. We show that the similarity of the latter two computations is not a coincidence, but arises from a vertex identity between the bulk 2-point function and the double-discontinuity of the boundary 4-point function.

2020

Evolution of the Valley Position in Bulk Transition-Metal Chalcogenides and Their Monolayer Limit

Dumitru Dumcenco, Zahid Hussain, Gang Xu, Kai Yan, Yi Zhang, Bo Zhou

Layered transition metal chalcogenides with large spin orbit coupling have recently sparked much interest due to their potential applications for electronic, optoelectronic, spintronics, and valleytronics. However, most current understanding of the electronic structure near band valleys in momentum space is based on either theoretical investigations or optical measurements, leaving the detailed band structure elusive. For example, the exact position of the conduction band valley of bulk MoS2 remains controversial. Here, using angle resolved photoemission spectroscopy with submicron spatial resolution (micro-ARPES), we systematically imaged the conduction/valence band structure evolution across representative chalcogenides MoS2, WS2, and WSe2, well as the thickness dependent electronic structure from bulk to the monolayer limit. These results establish a solid basis to understand the underlying valley physics of these materials, and also provide a link between chalcogenide electronic band structure and their physical properties for potential valleytronics applications.

American Chemical Society2016

Conformal 3-Point Functions and the Lorentzian OPE in Momentum Space

Marc Gillioz

In conformal field theory in Minkowski momentum space, the 3-point correlation functions of local operators are completely fixed by symmetry. Using Ward identities together with the existence of a Lorentzian operator product expansion (OPE), we show that the Wightman function of three scalar operators is a double hypergeometric series of the Appell F-4 type. We extend this simple closed-form expression to the case of two scalar operators and one traceless symmetric tensor with arbitrary spin. Time-ordered and partially-time-ordered products are constructed in a similar fashion and their relation with the Wightman function is discussed.

SPRINGER2020