Fermion, definition

In formulating the second-quantized description of a system of noninteracting fermions, we shall, therefore, have to introduce distinct creation and annihilation operators for particle and antiparticle. Furthermore, since all the fermions that have been discovered thus far obey the Pauli Exclusion principle we shall have to make sure that the formalism describes a many particle system in terms of properly antisymmetrized amplitudes so that the particles obey Fermi-Dirac statistics. For definiteness, we shall in the present section consider only the negaton-positon system, and call the negaton the particle and the positon the antiparticle. 

The Wick chronological operator T is, therefore, defined in the same way as the P operator previously introduced, except that the T operator includes in its definition the sign of the permutation of the fermion factors. 

Let us briefly mention some formal aspects of the above-introduced formalism, which have been discussed in detail by Blaizot and Marshalek [218]. First, it is noted that the both the Schwinger and the Holstein-Primakoff representations are not unitary transformations in the usual sense. Nevertheless, a transformation may be defined in terms of a formal mapping operator acting in the fermionic-bosonic product Hilbert space. Furthermore, the interrelation of the Schwinger representation and the Holstein-Primakoff representation has been investigated in the context of quantization of time-dependent self-consistent fields. It has been shown that the representations are related to each other by a nonunitary transformation. This lack of unitarity is a consequence of the nonexistence of a unitary polar decomposition of the creation and annihilation operators a and at [221] and the resulting difficulties in the definition of a proper phase operator in quantum optics [222]. 

The definition of the pair part or, equivalently the no-pair part of Hmat is not unique. The precise meaning of no-pair implicitly depends on the choice of external potential, so that the operator Hm t depends implicitly on the external potential, whereas the sum Hmat Hm t + Hm t is independent of the choice of external potential. Since the no-pair part conserves the number of particles (electrons, positrons and photons) we can look for eigenstates of Hm j in the sector of Fock space with N fermions and no photons or positrons. Following Sucher [18,26,28], the resulting no-pair Hamiltonian in configuration space can be written as... 

M. Nakata, H. Nakatsuji, M. Ehara, M. Fukuda, K. Nakata, and K. Fujisawa, Variational calculations of fermion second-order reduced density matrices by semi-definite programming algorithm. J. Chem. Phys. 114, 8282-8202 (2001). 

A -representability conditions [28]. Let us start this description by focusing on the RDM s properties, which may be deduced from their definition as expectation values of density fermion operators. Thus the ROMs are Hermitian, are positive semidefinite, and contract to finite values that depend on the number of electrons, N, and in the case of the HRDMs on the size of the one-electron basis of representation, 2/C. Thus... 

We have utilized the fermion creation and annihilation operators denoted a a, and apa, respectively. These operators act on the electron in the pth orbital with the projected spin a. The set 0p(r)) represents the molecular orbitals and the last term, hauc, in Eq- (13-3) is the nuclear repulsion energy. We use the following definitions of the one-electron excitation operator... 

The definition of the no-sea approximation for is not completely unambiguous. As discussed in Appendix B we define it through neglect of all vacuum fermion loops in the derivation of an approximate [/]. Alternatively, one could project out all negative energy states, thus generating a direct equivalent of the standard no-pair approximation. As one would expect the differences between these two schemes to be small, we do not differentiate between these approximations here. 

By definition, the Hamiltonian of a system of identical particles is invariant under the interchange of all the coordinates of any two particles. The wave function describing the system must be either symmetric or antisymmetric under this interchange. If the particles have integer spin, the wavefunction is symmetric and the particles are called bosons if they have half-integer spins, the wavefunction is antisymmetric and the particles are fermions. Our discussion will be restricted to electrons, which are fermions. 

Using Eq. (30), the definition of the antisymmetrized two-electron integral, and the anticommutation relation of fermion annihilation operators, Eq. (33) may be written... 

Having established that creation and annihilation operators are rank 1 covariant and contravariant tensors, respectively, with respect to the operator ( )L,S, we can define an rath-rank boson operator as consisting of a like number of fermion creation and annihilation operators. Then the normal product of an rath-rank boson operator is a natural definition for the irreducible tensor. 

Eq. (58) represents the starting point for all approximate propagator methods. Even though in the derivation we only discussed the linear response functions or polarization propagators, a similar equation holds for the electron propagator. The equation for this propagator has the same form but there are differences in the choice of h and in the definition of the binary product (Eq. (52)), which for non-number-conserving, fermion-like operators should be... 

Example Choosing both reference states ip) and v ) equal to the N particle ground state q of a fermionic system, the definition of the so-called extended particle-hole states becomes... 

Definition The p-product between two states F), Z) e Y is given by the matrix element Y p Z) where p, = diag(l, — 1,1, 1) is called the metric operator. Again, here and in the following, the upper sign refers to the fermionic case and the lower to the bosonic case. For extended states of the form (1) we introduce the following shorthand notation for the /i-product ... 

A preliminary justification for the classification given for the Green s functions and extended states is provided by the fact that the first or physical component of the defining vector of the extended state ] A, B) is related to the state AB ip) [see definition (1)]. In this paper we will choose the reference state ijj) to describe the exact ground state J ) of an interacting iV-fermion system. The physical component of the extended states... 

We will review here some definitions and properties of reduced density matrices for fermionic systems. The focus will be on the general density operator p and the corresponding Liouville equation... 

Note that the definition (A2) normalizes p = to the number of pairings of N fermions [13], Other normalizations [14,15] also exists, i.e. 

The atomic theory of matter, which was conjectured on qualitative empirical grounds as early as the sixth century BC, was shown to be consistent with increasing experimental and theoretical developments since the seventeenth century AD, and definitely proven by the quantitative explanation of the Brownian motion by Einstein and Perrin early in the twentieth century [1], It then took no more than a century between the first measurements of the electron properties in 1896 and of the proton properties in 1919 and the explosion of the number of so-called elementary particles - and their antiparticles - observed in modern accelerators to several hundred (most of which are very short lived and some, not even isolated). Today, the standard model assumes all particles to be built from three groups of four basic fermions - some endowed with exotic characteristics - interacting through four basic forces mediated by bosons - usually with zero charge and mass and with integer spin [2],... 

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