## Vacuum Energy in φ4 Theory

Let us return to a result derived in the previous chapter: the vacuum state energy density of the free field is expressed by a divergent integral (4.64).

(5.1)

where w(p) = ViP + m 2 (5.2)

As was discussed in the previous chapter in section 4.5, the vacuum energy Eo is not an observable quantity. We can detect only deviations of the field energy from Eo. Usually, such deviations are due to particles (the field excitations in the background of the vacuum) which carry a finite energy. Another way of observing variations of Eo is to change slightly the function w(p) (5.2). Such a variation can result from, for example, changed boundary conditions or switching on an external field. This does cause finite observable energy variations. Famous examples of such phenomena are the Casimir effect and the Lamb shift 1. In this chapter, we shall consider the Casimir effect in a simplified model. In the second part of this chapter, we shall calculate the energy of the spatially homogenous field configuration, the so-called effective potential of the <p4 model. These are the first examples of the calculation program formulated in section 4.5. We shall restrict the integration (summation) in momentum space in expression (5.1) and then remove this restriction

when physically meaningful quantities are obtained. As we shall show below the resultant values remain finite. It should be emphasized that in this introductory chapter we try to simplify calculation techniques as much as possible, sometimes applying common sense instead of rigorous procedures. We shall give a more solid justification for the method used here in the next chapters.