SPECIAL SYMBOLS
and their physical/mathematical meanings.
Alg( Hilb ) Algebra of bounded operators on Hilb
Alg( Hilb(n) ) Algebra of bounded operators on Hilb(n)
Alg(n) = Alg( Hilb(n) )
B QM annihilation operator
B! QM creation operator
B(n) Truncated annihilation operator
B!(n) Truncated creation operator
~B(n) Skew truncated annihilation operator
~B!(n) Skew truncated creation operator
C(n) C!(n) Cyclic matrices
C The complex field
CHAH_k(z) k-th Chebyshev polynomial od the first kind
D_k(z) Parabolic Cylinder functions
~D_k(z) Normalized Parabolic Cylinder functions
d_k Column normalizing factor of XI(n)
delta_kj Kronecker delta
delta_(kj) Kronecker delta
epsilon_ijk Permutation symbol (Epsilon density)
epsilon_(ijk) Permutation symbol (Epsilon density)
F(n) Generator for C(n)
Fr(n) Q-P finite Fourier transform
G(n) Fundamental Indefinite form on Hilb(n)
Hilb A separable infinite dimensional Hilbert space
Hilb(n) An n-dimensional Hilbert space
Hilb_+(n) G(n)-positive part of Hilb(n)
Hilb_-(n) G(n)-negative part of Hilb(n)
Hilb_0(n) G(n)-null part of Hilb(n)
H(n) Radial operator in pseudo-polar form for
B(n) and B!(n)
H_k(z) Unnormalized k-th Hermite polynomial
~H_k(z) Normalized k-th Hermite polynomial
h-bar Reduced Plank constant
I Identity in Alg(Hilb)
I(n) nXn identity
L A Lie algebra
L* A complexified Lie algebra
lambda_k(n) The (3) dimensional IRREP of su(3)
M(n)
N QM Number operator
N(n) Truncated Number operator
N'(n) Truncated Number operator
N_-(n) Truncated Number operator with negative
eigenvalues
P QM Momentum operator
P(n) Truncated "Momentum" operator
P'(n) Truncated "Momentum" operator
PHI Abstract Fourier transform
PI(n) Diagonalizer of P(n)
Q QM Position operator
Q(n) Truncated "Position" operator
Q'(n) Truncated "Position" operator
rho( A ) Spectral radius of operator A
S_a(n) Generators for n dimensional IRREP of su(2)
Sp( A ) Spectrum of operator A
sigma_k Normalized Pauli spin matrices
SHA Shift operator on a one-side infinitely
Hilb
SHA(n) SHA!(n) Nilpotent shift operators on Hilb(n)
t(n), t_(+|-)(n) Time operators
Tr( A ) Trace of operator A
U_g(n) Kinematical invariance group of
canonical transformations
U(G, n) Kinematical invariance group of
canonical transformations
UPSILON(n) N-F-M finite Fourier transform
and diagonalizer of C(n)
X_a(n) 2 dimensional IRREP of su(2)
XI(n) Diagonalizer of Q(n)
Y_a(n) Canonical bilinears
Z Dense invariant domain in Hilb for B and B!
|n, k> Eigenvectors of N(n)
|q(n, k)> Eigenvectors of Q(n)
|p(n, k)> Eigenvectors of P(n)
|mu(n, k)> Eigenvectors of C(n)
! Complex conjugate transpose
* Complex conjugate
CM Classical Mechanics
CCR Canonical Commutation Relation
CAR Canonical Anticommutation Relation
EM Classical Electromagnetic theory
FCCR Finite Canonical Commutation Relation
GR General Relativity
IRREP Irreducible Representation
LGH Left G(n)-Hermitean
LGS Left G(n)-skew Hermitean
QFT Quantum Field Theory
QG Quantum Gravity
QST Quantum Space-Time
QM Standard Quantum Mechanics
RGH Right G(n)-Hermitean
RGS Right G(n)-skew Hermitean
RQM Relativistic Quantum Mechanics
SR Special Relativity
UIRREP Unitary Irreducible Representation
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http://graham.main.nc.us/~bhammel/FCCR/symbs.html
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