Particle-hole duality

A quantum system of N particles may also be interpreted as a system of (r — N) holes, where r is the rank of the one-particle basis set. The complementary nature of these two perspectives is known as the particle-hole duality [13, 44, 45]. Even though we treated only the iV-representability for the particles in the formal solution, any p-hole RDM must also be derivable from an (r — A)-hole density matrix. While the development of the formal solution in the literature only considers the particle reduced Hamiltonian, both the particle and the hole representations for the reduced Hamiltonian are critical in the practical solution of N-representability problem for the 1-RDM [6, 7]. The hole definitions for the sets and are analogous to the definitions for particles except that the number (r — N) of holes is substituted for the number of particles. In defining the hole RDMs, we assume that the rank r of the one-particle basis set is finite, which is reasonable for practical calculations, but the case of infinite r may be considered through the limiting process as r —> oo. 

By particle-hole duality, the same block structure appears in the spin-adapted two-electron RDM. The four blocks of the 2-RDM have the following traces [57] ... 

The CSE allows us to recast A-representability as a reconstruction problem. If we knew how to build from the 2-RDM to the 4-RDM, the CSE in Eq. (12) furnishes us with enough equations to solve iteratively for the 2-RDM. Two approaches for reconstruction have been explored in previous work on the CSE (i) the explicit representation of the 3- and 4-RDMs as functionals of the 2-RDM [17, 18, 20, 21, 29], and (ii) the construction of a family of higher 4-RDMs from the 2-RDM by imposing ensemble representability conditions [20]. After justifying reconstmction from the 2-RDM by Rosina s theorem, we develop in Sections III.B and III.C the functional approach to the CSE from two different perspectives—the particle-hole duality and the theory of cumulants. 

Many-body problems in quantum mechanics are usually described by the number of particles N in the system and the probabilities of finding those particles at different locations in space. If the rank of the one-particle basis is a finite number r, an equally valid description of the system may be given by specifying the number of holes r N in the system and the probabilities of finding these holes at different locations in space. This possibility for an equivalent representation of the system by particles or holes is known as the particle-hole duality. By using the fermion anticommutation relation... 

The reconstruction functionals, derived in the previous section through the particle-hole duality, may also be produced through the theory of cumulants [21,22,24,26,39,55-57]. We begin by constructing a functional whose derivatives with respect to probe variables generate the reduced density matrices in second quantization. Because we require that additional derivatives increase the number of second quantization operators, we are led to the following exponential form ... 

In the Af-representability literature these positivity conditions are known as the D- and the g-conditions [5, 7, 63]. The two-particle RDM and the two-hole RDM are linearly related via the particle-hole duality,... 

This duality of holes and particles allows us to write... 

Thus, the wavelength-frequency relation (2.1) implies the Compton-effect formula (2.10). The best we can do is to describe the phenomena constituting the wave-particle duality. There is no widely accepted explanation in terms of everyday experience and common sense. Feynman referred to the experiment with two holes as the central mystery of quantum mechanics. It should be mentioned that a number of models have been proposed over the years to rationalize these quantum mysteries. Bohm proposed that there might exist hidden variables whieh would make the behavior of each photon deterministic, i.e., particle-like. Everett and Wheeler proposed the many worlds interpretation of quantum mechanics in which each random event causes the splitting of the entire universe into disconnected parallel universes in whieh eaeh possibility becomes the reality. 

It is essential that under duality transformation (A9) the holes and particles in Eq. (C2) change place, so identity (A 10) satisfies automatically for D =D p. The appropriate q-extended ( > 1) hole-particle indices can be cast explicitly... 

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