List of Symbols

List of Symbols

APPENDIX LIST OF SYMBOLS n the following table of symbols, the many transitory symbols that I have no application beyond their immediate relevance to the deri­ vation involved have not been included for the sake of clarity. Even so, many symbols have had to be used for more than one purpose. Where possible I have kept to the standard usage, but many of the symbols still need to be taken in the context of the equation. For example, Z is used for both the ionic charge number and the renormalization constant, but since the former always appears with the electronic charge e, the danger of confusion is small. As an indication the chapter number has been given if a symbol has a definition restricted to that chapter. Throughout, the following standard notation has been used to indicate the possible meanings of a symbol (0). o An operator. o The matrix form of a function. o A vector. 0+ The Hermitian conjugate. 0-1 The inverse. 6 The time derivative. (0) The expectation value. °lm The matrix element (1101 m). Heisenberg to Schrodinger representation transformation operator. A(k,w) Spectral weight function. A(1,2) The two-particle propagator [G(1,2)G(1,2)]. An(x) A known function of x (Chap. 5). Ax(t) Fourier coefficient of the electromagnetic potential for the wave vector kx and polarization A. A(r,t) Electromagnetic vector potential. a Interatomic separation of atoms on the linear chain. 311 APPENDIX lth components of the expansion of the Landau parameters in Legendre polynomials. Creation and annihilation (boson) operators for the normal mode k. Creation and annihilation operators in the "thermal" Heisenberg representation. Annihilation and creation operators corresponding to the condensate state. Coefficient of the quasiparticle state of wave vector k, band index n. Landau interaction parameters. Spectral weight function. Annihilation and creation operators for the state j. Annihilation and creation operators for the plasmon excitation of wave vector k. Velocity of light. Fermion annihilation and creation operators for the state m. D(rl ,tl ,r2,t2) Phonon Green's function (propagator). Do(rv t 1,r2, t2) Bare phonon Green's function. E Energy of a system. Ek Energy of the jth state of the M-particle system (j = 0 corresponds to the ground state). Energy of the state n. Eigenvalue of the particle i in the state ~~(r) (Chap. 1). Real and imaginary parts of the energy (Chap. 5). Et/> Energy of the N-particle system for a given external potential ¢(r). (Chap. 12). Ground-state energy of the interacting system (Chap. 5). E(k) Energy of the isolated Landau quasiparticle state k in the presence of other quasiparticle (Chap. 5). E(k) Energy of the state k. 312 LIST OF SYMBOLS E(k) Energy eigenvalue for the state corresponding to the transformed operators &jJk (Chap. 13). Energy of the isolated Landau quasiparticle state k (Chaps. 5 and 9). E(v) Energy of the N-particle system as a function of the velocity of the observer (Chap. 5). E(r,t) Electric field. E[n] The energy as a functional of the electron density. ExJn] The exchange and correlation energy functional. e Electronic charge. e Polarization vector. F Number of closed fermion loops in a diagram (Chap. 8). F Free energy (Chaps. 10, 12). F(rI1Tl1r 2,T2) Anomalous Green's function (Chap. 13). f(k,k') Landau quasiparticle interaction function. r(kF,8),f(kF,8) The direct and exchange parts of the quasiparticle interaction. f(r) Arbitrary function (Chap. 2). G Reciprocal lattice vector (Chap. 9). Expansion coefficients for the Green's function (Chap. 2). Anomalous Green's functions. The normal (Le., nonpairing) part of the two- particle Green's function (Chap. 13). G(k,w) Momentum-energy dependent Green's function. G(r,r',t,t') Time-dependent Green's function (Chap. 2). G(r,r',E) Energy-dependent Green's function (Chap. 2). G(x,x',w) Energy-dependent Green's function. G[n] The functional which contains the kinetic, exchange, and correlation contributions to the total energy (Chap. 12). Gc(X l1 t 1,X2l t2) Condensate part of the Green's function. Gix1,tI1 X2,t2) Real-time thermal Green's function. G:(X1,tI1 X2l t2) Advanced real-time thermal Green's function. G:(Xl1 t 1,X2,t2) Retarded real-time thermal Green's function. GO(Xl,tI1X2,t2) Noninteracting Green's function. 313 APPENDIX Green's function corresponding to the single­ particle Hamiltonian without the Hartree potential (Chap. 7). nth order approximation to the single-particle Green's function. G2( xI,t I,X2f t2,X3, t 3,X4, t4) Two-particle Green's function. G±(r,r,t - t') Advanced (+) and retarded (-) Green's function (Chap. 2). Noncondensate part of the Green's function (Chap. 11). G-I(Xl1tl,X2ft2) Inverse of the single-particle Green's function. G'(XI,tl,X2,t2) The matrix Green's function for a system with a condensate (Chap. 11). G(k,w n) Momentum-energy dependent thermal Green's function. g (X, r ,x',r') Thermal Green's function. go(x,r,x',T') Noninteracting thermal Green's function. go(XI ,X2fWn) Noninteracting energy-dependent thermal Green's function. The spin-up electron propagator (Chap. 13). g The strength of the effective electron-electron interaction (Chap. 13). Energy functions (not functionals) of the electron density. The energy density functional. Hamiltonian. Hamiltonian describing the background neutralizing charge (Chap. 11). Hamiltonian containing the coulomb repulsion (Chap. 12). Hel Electron Hamiltonian (Chaps. 11 and 12). Hel-ion Electron-ion interaction Hamiltonian (Chap. 11). HINT Hamiltonian describing the interaction between the N-particle system and the added electron (Chap. 11). Electron-phonon interaction Hamiltonian (Chap. 12). Ion Hamiltonian. 314 LIST OF SYMBOLS Coupling Hamiltonian between localized and de localized states (Chap. 12). Hamiltonian for the N-particle system (Chap. 11). Hamiltonian for the delocalized states (Chap. 12). HPh Phonon Hamiltonian (Chap. 12). Ho Single-particle Hamiltonian. HI Hamiltonian for one particle (Chap. 11). Ho(xl) Single-particle Hamiltonian. H(r,t) Magnetic field (Chap. 3). H[n1 Hamiltonian as a functional of the electron density. H[cf>1 Hamiltonian as a function of the external potential. 71 Hamiltonian density. 71 Magnetic field (Chap. 12). 71/ An effective internal magnetic field (Chap. 12). 71 0 External magnetic field (Chap. 12). h Plank's constant. 1(1,2,3,4) T-matrix for electron-electron scattering. i(x) Single-particle operator. Jo(r) Zero-order Bessel function (Chap. 9). d(x) Operator density. K Effective Spring constant for internuclear forces (Chap. 3). Grand canonical Hamiltonian. Soluble part of the grand canonical hamiltonian. Eigenvalue of the grand canonical hamiltonian for the state n. The normal (i.e., nonpairing) part of the grand canonical Hamiltonian (Chap. 13). Variable conjugate to x [= (k, spin)]. Boltzmann constant. Fermi wave vector, i.e., wave vector of state at the Fermi level. k Wave vector of a plane-wave state. L Lagrangian. .L Lagrangian denSity. 315 APPENDIX M Atomic mass. M Occupation number of the q = 2kF phonon in the one-dimensional solid (Chap. 12). M(k,j,q) Matrix element appearing in the response function of an insulator. m Electron mass. m* Effective m'ass of an electron. N Number of atoms in the linear chain (Chap. 3). N Electron density (Chap. 11). N(E) Density of states at energy E. N. Number eigenvalue of the state I n) (Chap. 10). n Number of particles not in the condensate (Chap. 11). n Order number of a diagram (Chap. 8). n(k) Number of particles in the eigenstate k. n(r) Density of electrons. fl(r) Density operator. n[¢] The electron density as a functional of the external potential. Number operator for the normal mode of wave vector k. Number operator for the state 1,0". no(k) Ground-state occupation factor (Chap. 5). o General operator. OH(q,t) Heisenberg representation operator. O[(q,t) Interaction representation operator. OK(q,T) Operator in the "thermal" Heisenberg representation. Os(q,t) Schrodinger representation operator. P(q,w) Energy-momentum dependent polarization propagator. P( X II t I1X21 t2) Polarization propagator. P(Xl1TllX21T2) Thermal response function. P{} Permutation operator which produces all possible permutation of the labels; thus, P{1/Ia1/l~1/I.,} = 1/Ia1/l~1/I., + 1/Ia1/l.,1/I~ + 1/I.,1/Ia1/l~ + t/;'Yt/;~t/;a + t/;~t/;'Yt/;a + t/;~t/;at/;'Y' Probability of finding a system in the state In} (Chap. 10). 316 LIST OF SYMBOLS PO(X1,t1,X2,t2) First-order polarization propagator. p Momentum operator. P Momentum operator (single-particle). Momentum of ith atom (Chap. 3). Fourier transform of Pi and momentum coordinate of the normal mode of wave vector k (Chap. 3). Particle momentum. Momentum of the ith atom (Chap. 11). Momentum coordinate for the normal mode (k,A). Q(k,k') The transformed interaction matrix element in a paired electron system (Chap. 13). q Generalized position variable. qi Shift of ith atom from its equilibrium position (Chap. 3). Position variable of the normal mode of wave vector k. The Fourier transform of qi (Chap. 3). Position coordinate for the normal mode (k,A) (Chap. 11). q Wave vector variable. R(k,k') As Q(k,k') (Chap. 13). R(r1,t1,r2,t2) The plasmon Green's function (or propagator). RA Operator occurring in the electron electromagnetic field interaction (Chap. 4). Atomic position vector. Position vector describing the equilibrium position of the ith atom in a lattice. Electron position vector for the nth electron. 5 Entropy of a system (Chap.

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