# SPECIAL SYMBOLS and their physical/mathematical meanings.

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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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