scholarly journals On the construction of invariant measure over the orthogonal group on the Hilbert space by the method of Cayley transformation

1974 ◽  
Vol 10 (2) ◽  
pp. 413-424 ◽  
Author(s):  
Hiroaki Shimomura
Keyword(s):  
Hilbert Space ◽  
Invariant Measure ◽  
Orthogonal Group ◽  
Cayley Transformation

scholarly journals HAMILTONIAN FORMALISM FOR BLACK HOLES AND QUANTIZATION II

1996 ◽  
Vol 05 (03) ◽  
pp. 227-250 ◽  
Author(s):  
MARCO CAVAGLIÀ ◽  
VITTORIO DE ALFARO ◽  
ALEXANDRE T. FILIPPOV
Keyword(s):  
Hilbert Space ◽  
Black Holes ◽  
Black Hole ◽  
Invariant Measure ◽  
Hamiltonian Formalism ◽  
Quantization Method ◽  
Schwarzschild Black Hole ◽  
Canonical Formulation ◽  
Gauge Fixing ◽  
Rigid Transformations

We quantize by the Dirac-Wheeler-DeWitt method the canonical formulation of the Schwarzschild black hole developed in a previous paper. We investigate the properties of the operators that generate rigid symmetries of the Hamiltonian, establish the form of the invariant measure under the rigid transformations, and determine the gauge fixed Hilbert space of states. We also prove that the reduced quantization method leads to the same Hilbert space for a suitable gauge fixing.

Spinors in Hilbert space

1976 ◽  
Vol 80 (2) ◽  
pp. 337-347 ◽  
Author(s):  
R. J. Plymen
Keyword(s):  
Hilbert Space ◽  
Vector Space ◽  
Orthogonal Group ◽  
Double Cover ◽  
Projective Representation ◽  
Complex Vector Space ◽  
Complex Vector ◽  
True Representation ◽  
Dimension 2 ◽  
Image Position

In 1913, É. Cartan discovered that the special orthogonal groupSO(k) has a ‘two-valued’ representation (i.e. a projective representation) on a complex vector spaceSof dimension 2n, wherek= 2nor 2n+ 1. The projective representation in question lifts to a true representation of the double cover Spin (k) ofSO(k). We restrict attention to the casek= 2n. Under the action of Spin (2n),Sbreaks up into 2 irreducible subspaces:The vectors inSare calledspinors(relative toSO(2n)), those inS+orS−are calledhalf-spinors(4).

SYMPLECTIC SPACE OR ORTHOGONAL SPACE OF n QUBITS

2011 ◽  
Vol 09 (06) ◽  
pp. 1449-1457
Author(s):  
JIAN-WEI XU
Keyword(s):  
Hilbert Space ◽  
Orthogonal Group ◽  
Symplectic Group ◽  
Orthonormal Basis ◽  
Mathematical Structure ◽  
Symplectic Space ◽  
Orthogonal Space ◽  
Spin Flip

In Hilbert space of n qubits, we introduce symplectic space (n odd) or orthogonal space (n even) via the spin-flip operator. Under this mathematical structure we discuss some properties of n qubits, including homomorphically mapping local operations of n qubits into symplectic group or orthogonal group, and proving that the generalized "magic basis" is just the biorthonormal basis (i.e. the orthonormal basis of both Hilbert space and the orthogonal space). Finally, a demonstrated example is given to discuss the application in physics of this mathematical structure.

scholarly journals TOPOLOGICAL MEASURE AND GRAPH-DIFFERENTIAL GEOMETRY ON THE QUOTIENT SPACE OF CONNECTIONS

1994 ◽  
Vol 03 (01) ◽  
pp. 207-210 ◽  
Author(s):  
JERZY LEWANDOWSKI
Keyword(s):  
Differential Geometry ◽  
Hilbert Space ◽  
Invariant Measure ◽  
Quotient Space ◽  
Vector Spaces ◽  
Integral Calculus ◽  
Topological Vector Spaces ◽  
Topological Measure ◽  
Non Linear ◽  
Densely Defined

Integral calculus on the space [Formula: see text] of gauge equivalent connections is developed. By carring out a non-linear generalization of the theory of cylindrical measures on topological vector spaces, a faithfull, diffeomorphism invariant measure is introduced on a suitable completion of [Formula: see text]. The strip (i.e. momentum) operators are densely-defined in the resulting Hilbert space and interact with the measure correctly

A GENERALISATION OF THE FROBENIUS RECIPROCITY THEOREM

2019 ◽  
Vol 100 (2) ◽  
pp. 317-322
Author(s):  
H. KUMUDINI DHARMADASA ◽  
WILLIAM MORAN
Keyword(s):  
Hilbert Space ◽  
Invariant Measure ◽  
Banach Spaces ◽  
Closed Subgroup ◽  
General Setting ◽  
Locally Compact Group ◽  
Reciprocity Theorem ◽  
Representation Space ◽  
Locally Compact ◽  
Frobenius Reciprocity

Let $G$ be a locally compact group and $K$ a closed subgroup of $G$. Let $\unicode[STIX]{x1D6FE},$$\unicode[STIX]{x1D70B}$ be representations of $K$ and $G$ respectively. Moore’s version of the Frobenius reciprocity theorem was established under the strong conditions that the underlying homogeneous space $G/K$ possesses a right-invariant measure and the representation space $H(\unicode[STIX]{x1D6FE})$ of the representation $\unicode[STIX]{x1D6FE}$ of $K$ is a Hilbert space. Here, the theorem is proved in a more general setting assuming only the existence of a quasi-invariant measure on $G/K$ and that the representation spaces $\mathfrak{B}(\unicode[STIX]{x1D6FE})$ and $\mathfrak{B}(\unicode[STIX]{x1D70B})$ are Banach spaces with $\mathfrak{B}(\unicode[STIX]{x1D70B})$ being reflexive. This result was originally established by Kleppner but the version of the proof given here is simpler and more transparent.

scholarly journals The orthogonal group of a Form Hilbert space

2007 ◽  
Vol 14 (5) ◽  
pp. 937-946 ◽  
Author(s):  
Hans A. Keller ◽  
Herminia Ochsenius
Keyword(s):  
Hilbert Space ◽  
Orthogonal Group

SOBOLEV REGULARITY OF INVARIANT MEASURES FOR GENERALIZED ORNSTEIN–UHLENBECK OPERATORS

2006 ◽  
Vol 09 (04) ◽  
pp. 595-606 ◽  
Author(s):  
ABDELHADI ES-SARHIR
Keyword(s):  
Hilbert Space ◽  
Invariant Measure ◽  
Sobolev Space ◽  
Invariant Measures ◽  
Separable Hilbert Space ◽  
Square Root ◽  
Absolutely Continuous ◽  
Sobolev Regularity ◽  
Reference Measure ◽  
Nikodym Derivative

This paper deals with the regularity of an invariant measure μ associated to a class of generalized Ornstein–Uhlenbeck operators. Regularity here means that μ is absolutely continuous with respect to a properly chosen Gaussian reference measure σ on a separable Hilbert space H. Moreover, the square root of its Radon–Nikodym derivative ρ should belong to some directional Sobolev space [Formula: see text].

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