The Concept of Energy

One of the most important universal concepts of physics is the concept of energy, which we did not touch up to now in our technical development. In this chapter we will correct this state of affairs, introducing and exploring the concept of energy, thus continuing our probe into the fundamental structure of the theory, and extending it in this important new direction. This will require the redefinition of the theory in the context of the canonical formalism, which we will develop entirely on the Euclidean lattice. We will see that it is possible to do this without any problems, and that the usual familiar results are recovered in the $d=1$ case of quantum mechanics. However, significant differences with respect to the results of the traditional formalism will be found in the case of quantum field theory, that is, for the cases $d\geq 2$.

At first the exploration of the concept of energy will be limited to the study of the vacuum state of the theory, which has already been defined, but one is quickly led to consider other states, resulting in the definition and exploration of particle states, that is, states with energies and momenta corresponding to multiples of some fundamental quanta. These particle states will be associated to the modes of the $d$-dimensional cavity containing the physical system, in momentum space, an association which is of great physical importance, since it can be realized experimentally in the non-relativistic limit. The important on-shell condition characterizing the physical states of relativistic particles will appear naturally from the resulting structure when one considers the continuum limit.

We will see that the introduction of the particle states permits a deeper and more direct probe into the structure of the theory, which is closer to the observability aspects of the physical structure. However, some difficulties of a very fundamental nature will also be found, when we try to make closer contact with the traditional formalism involving state-vectors and operators in a Hilbert space. These difficulties do not appear in the definition and calculation of the correlation-function aspects of the structure, but only when one considers the issue of the definition of the energy and of particle states. The results of the last section of this chapter will lead us to depart even further from the traditional approach to the subject.