Vacuum expectation value fields

Similar results can be obtained for the vacuum expectation value of electromagnetic field operators. The spectral representation takes the form... 

The second Higgs field acts in such a way that if the vacuum expectation value is zero, ( ) = 0, then the symmetry breaking mechanism effectively collapses to the Higgs mechanism of the standard SU(2) x U(l) electroweak theory. The result is a vector electromagnetic gauge theory 0(3)/> and a broken chiral SU(2) weak interaction theory. The mass of the vector boson sector is in the A(3) boson plus the W and Z° particles. 

Due to the fact that the Lagrangian incorporates the creation and destruction of field quanta, not even the time-development of a single particle is a simple matter. The time development can be expressed in terms of the electron (fermion) and photon propagators, which are defined as the vacuum expectation values of the time-ordered product of field operators. For the fermions one has... 

The last necessary ingredient toward a well-defined and consistent quantum field theory of radiation interacting with a fermionic matter field is the introduction of normal-ordered products of field operators. The necessity for this step is easily realized by consideration of the vacuum expectation value of Gauss law. 

Recall that, in the SM, a natural scale is set by the vacuum expectation value of the Higgs field v, related to the parameters of the Higgs potential, and it has a rather precise value (4.2.38)... 

In [8] Witten was able to obtain the Jones polynomial within the context of the topological quantum field theory of the Chern-Simons kind. Nearly immediately other polynomials were obtained such as the HOMFLY and the Kauffman polynomials [9]. The main idea consists in that the observables (vacuum expectation values) of the TQFT s are by itself topological invariants for knots, links, tangles and three-manifolds. 

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... 

If the coupling constant Y is comparable to the coupling constant g, then the Fermi expectation energies of the Fermions occur at the mean value for the Higgs field (4>0)- In this case the vacuum expectation of the vacuum is proportional to the identity matrix. This means that the masses acquired by the right chiral plus left chiral gauge bosons A + /iM are zero, while the left chiral minus right chiral... 

Finally, it is worth remarking that retarded van der Waals dispersion potentials between molecules in ground and excited electronic states may also be calculated [12,51] using the fluctuating moment method. Because dispersion forces arise from the perturbation induced by the zero-point energy associated with the vacuum electromagnetic field, the expectation value of... 

If we assume that initially the field is in the vacuum state, then the free-field part Eq+) (R, t) does not contribute to the expectation values of the normally ordered field operators, and we obtain the following expressions for the first-and second-order correlation functions... 

Our "solution value is closer to the 60° mininimum from the statics study. Forcefields such as used by Mardsen et al. are solution equivalent (i.e. contain information on the interactions in aqueous solution). Therefore, their results are not for a truly isolated molecule, but might be expected to be equivalent to our solution model. To allow determinations of conformation in vacuum and other solvents, water information should not appear in the basic potentials. The presence of water information in force fields is a common problem. 

The gauge potentials and are 2x2 Hermitian traceless matrices, and the Higgs fields and % are also 2x2 matrices. These expectations are real-valued, and the nonzero contributions of the Higgs field on the physical vacuum are given by the diagonal matrix entries [95] ... 

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