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The Glashow–Weinberg–Salam Theory of the Electroweak Interaction
Published in K Grotz, H V Klapdor, S S Wilson, The Weak Interaction in Nuclear, Particle and Astrophysics, 2020
K Grotz, H V Klapdor, S S Wilson
We have already touched upon mass generation via an SU(2)L Higgs field Φ in Chapter 4. For the vacuum expectation value () 〈Φ〉=12(0v)
Variational Methods and Functional Legendre Transforms
Published in A.N. Vasiliev, Patricia A. Millard, Functional Methods in Quantum Field Theory and Statistical Physics, 2019
where |0) is the ground state of the full Hamiltonian of the theory with action S = S′ + S″ and φ^ is the field operator of this theory. The vacuum expectation value (6.5) is actually independent of the time, so the field operator can be used in any representation. In quantum statistics the average would be obtained rather than the vacuum expectation value (6.5).
Frozen-density embedding employing configuration interaction as a subsystem method
Published in Molecular Physics, 2020
Nils Schieschke, Tilmann Bodenstein, Sebastian Höfener
While in case of ‘pure’ wavefunction methods, contributions to the energy are formulated in terms of operators, this is not the case for contributions based on density-functional theory due to non-linear contributions in the non-additive terms. This holds in particular for exchange-correlation contributions for which the energy must be computed via the energy density and not the exchange-correlation potential which is included in the Kohn–Sham equations. The total ground-state energy of an embedded system using the FDE is therefore calculated as The subsystem energy is computed as an expectation value of the vacuum Hamilton operator without the embedding contribution since all interaction contributions are included in the interaction energy . Note that the vacuum expectation value, , contains implicitly embedding contributions as the CI coefficients are obtained from Equation (29), i.e. in the presence of the embedding potential. This is in analogy to model A of References [33,45], where the ground state density is used for computation of the interaction energy for all states k.
Systematically improvable excitonic Hamiltonians for electronic structure theory
Published in Molecular Physics, 2019
Using the same conventions as above, but applied to the orbitals , we have the set , whose members satisfy where the operator is built from annihilation operators (with indices in descending order). We can quickly verify that is the set of biorthogonal complements to by using the aforementioned anticommutation relationships where the angle brackets denote a vacuum expectation value. This makes use of the fact that, since tuples are always in ascending order, if , then they are different in composition.
Diagonal Born–Oppenheimer correction for coupled-cluster wave-functions
Published in Molecular Physics, 2018
There are two ways to evaluate the CC norms. In the algebraic-diagrammatic, field-theoretical framework based on the Wicks theorem [1,50], the norm is viewed as a vacuum expectation value involving all possible contractions of powers of and powers of . This expression can be seen to involve an infinite number of connected and disconnected diagrams resulting from the underlying exponential ansatz. Such a scheme of evaluation also contains the exclusion-principle violating (EPV) terms [4]. Another method is to introduce a resolution-of-identity in (29a) leading to The above non-diagrammatic form explicitly removes all the EPV terms and yields a finite series suitable for evaluation. In the conference talk, it was incorrectly claimed that (29a) and (31) are different [51]. Using the Wicks theorem, it is possible to verify that both are equivalent when all terms are included. In alternative CC schemes, the diagrammatic form of norm is used for formal manipulations to obtain connected expressions for energy and molecular properties [49]. In these cases, the resulting expressions are infinite series and need to be suitably truncated [4].