For this reason is often referred to as the " vacuum field ", although it is of course a Heisenberg-picture operator acting on whatever state of the field happens to be appropriate at 0 } } . is the source field, the field generated by the dipole and acting on the dipole.
32.
In the current example the coupling of a dipole oscillator to the electromagnetic field has a dissipative component, in the form of the zero-point ( vacuum ) field; given the existence of radiation reaction, the vacuum field must also exist in order to preserve the canonical commutation rule and all it entails.
33.
The fact that the canonical commutation relation for a harmonic oscillator coupled to the vacuum field is preserved implies that the zero-point energy of the oscillator is preserved . it is easy to show that after a few damping times the zero-point motion of the oscillator is in fact sustained by the driving zero-point field.
34.
In fact we can show that the vacuum field is essential for the preservation of the commutators and the formal consistent of QED . When we calculate the field energy we obtain not only a contribution from particles and forces that may be present but also a contribution from the vacuum field itself i . e . the zero-point field energy.
35.
In fact we can show that the vacuum field is essential for the preservation of the commutators and the formal consistent of QED . When we calculate the field energy we obtain not only a contribution from particles and forces that may be present but also a contribution from the vacuum field itself i . e . the zero-point field energy.
36.
Variation of the first term of the action with respect to the tetrad e ^ { \ alpha } _ { \ I } gives the ( mixed index ) Einstein tensor and variation of the second term with respect to the tetrad gives a quantity that vanishes by symmetries of the Riemann tensor ( specifically the first Bianchi identity ), together these imply Einstein's vacuum field equations hold.
37.
Nethertheless as Jaffe himself notes in his paper " no one has shown that source theory or another S-matrix based approach can provide a complete description of QED to all orders . " Furthermore, Milonni has shown the necessity of the vacuum field for the formal consistency of QED . In QCD, color confinement has led physicists to abandon the source theory or S-matrix based approach for the strong interactions.
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