Hamiltonian vacuum expectation values

Note that the first and second terms on the right-hand side of this equation are simply the spin-orbital Fock operator (in normal-ordered form), and the last two terms are the Hartree-Fock energy (i.e., the Fermi vacuum expectation value of the Hamiltonian). Thus, we may write... 

Due to the redefinition of the vacuum (<4 0s) = 0) the Hamiltonian is still not bounded from below. This property must be implemented by a renormalization of the energy scale, i.e. by subtraction of the vacuum expectation value of Ha,... 

To illustrate the use of the Goldstone program, let us consider the wave operator and correlation energies of some atom or molecule. For close-shell systems, of course, we can choose always a 1-dimensional model space, 0c), which coincides with the reference state total energy of the system up to first order is equal to the vacuum expectation value of the normal-ordered Hamiltonian... 

There are several points to be noted about this operator. First, the second term creates an electron-positron pair, and the third term annihilates an electron-positron pair. This means that the Hamiltonian connects states with different particle numbers, that is, particle number is not conserved, though charge is. The existence of these terms embodies the idea of an infinitely-many-body problem that arose from the filling of the negative-energy states in Dirac s interpretation. Second, the order of the operators in the fourth term means that the vacuum expectation value of this operator is not zero, but... 

To avoid the negative infinite vacuum energy, the vacuum expectation value is subtracted from the Hamiltonian to define a new, QED Hamiltonian ... 

The QED Hamiltonian has the features we were looking for above. Its vacuum expectation value is zero (by construction), that is, the vacuum has zero energy. The positronic term now has a negative sign, and the energy of a positron state ) =... 

The subtraction of the vacuum expectation value from the Hamiltonian makes the operator vacuum-dependent. The argument for this proposition may be developed as follows. 

From the above mentioned relations it is easy to see that the vacuum expectation value of the electronic Hamiltonian (3.4) is zero. The particle-hole formalism implies a redefinition of the vacuum state. Since correlation energy is defined with respect to the Hartree-Fock energy, we redefine the vacuum state as being the occupation-number vector corresponding to the converged HF determinant, the Fermi vacuum. This leads to a redefinition of creation... 

In order to interpret the above results, consider the expectation value of the total energy density in the vacuum state, i.e., of the hamiltonian density, Eq. (10-12). There is a contribution J u(x)Al(x) from the external field and a contribution m<0 j (a ) 0)ln 4 (a ) from the induced current, hence to lowest order... 

The conclusion of this analysis is that the normal-ordered QED approach as presented here, with a floating vacuum, is equivalent to the empty Dirac approach. It appears that the reinterpretation of the negative-energy states as positron states has no influence on the combination of matrix elements that results from the commutator. Normal ordering then only affects the terms involving the positron operators, and at least for the one-electron Hamiltonian this means that the reference energy will be identical in both the empty Dirac and the QED approaches, since they only have occupied electron states, and the terms that survive in the refCTence expectation value are identical in the two approaches. 

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