Relativistic Classical and Quantum Mechanics

Abstract

To develop the foundations of a manifestly covariant mechanics, we must first examine the Einstein notion of time and its physical meaning. We will then be in a position to introduce the relativistic quantum theory developed by Stueckelberg (1941) and Horwitz and Piron (1973). We describe in this chapter a simple and conceptual understanding of the Newton-Wigner problem (Newton 1949) presented above, a rigorous basis for the energy time uncertainty relation, as well as a simple explanation of the Landau-Peierls (Landau 1931) uncertainty relation between momentum and time. These applications provide a good basis for understanding the basic ideas of the relativistic quantum theory. Schieve and Trump (1999) have discussed at some length the associated manifestly covariant classical theory, but some basic aspects will be discussed here as well.

Appendix A

We describe here the basic ideas of the so-called constraint theory formulation of a many particle (many event) relativistic mechanics . In this theory, describing the positions $\{ x^\mu _i\}$, and momenta $\{p^\mu _i\}$ for $i=1,2,3,\ldots ,N$ of the particles, a constraint is defined for each of the particles of the form (we use the metric $(-,+,+,+)$)

$$\begin{aligned} K_i = {p^\mu }_i {p_\mu }_i +m_i^2 + \phi _i(x,p), \end{aligned}$$

(2.67)

where, on the constraint hypersurface $K_i \approx 0$, the $\phi _i(x,p)$ are functions of all the x’s and p’s, and the $\{m_i\}$ are the given masses of the particles. This set of N constraints restricts the motion to an N dimensional hypersurface in the 8N dimensional phase space.

The “first class” constraints $K_i$ may act as generators of motion under Poisson bracket action (e.g. Itzyson 1980), thus defining the infinitesimal variations with respect to the corresponding parameters $\tau _i$ of the infinitesimal transformations of the coordinates and momenta by

$$\begin{aligned} \begin{aligned} {dx_i \over d \tau _i}&= i \{ K_i, x_i\}_{PB} \\ {dp_i \over d \tau _i}&= i \{ K_i, p_i\}_{PB}, \end{aligned} \end{aligned}$$

(2.68)

providing a set of first order equations describing the motion on this hypersurface. This manifestly covariant formalism has the advantage that one may assume the interaction terms $\phi _i$ vanish asymptotically when the particles are far apart; the constraint conditions then enforce the particles to lie on mass shell (${p^\mu }_i {p_\mu }_i +m_i^2 =0$).

In order to construct a world line for the system on the range of these motions, one generally introduces another set of $N-1$ constraints, called second class constraints, forming surfaces with intersection along a line on the N dimensional hypersurface, and an Nth constraint which cuts this line and is a function of a single parameter $\tau$, thus describing motion along this world line (Sudarshan 1981a). It is possible, however, to define these constraints in another way, by constructing a Hamiltonian of the form (Rohrlich 1981)

$$\begin{aligned} K = \Sigma _i \omega _i (x,p) K_i. \end{aligned}$$

(2.69)

The Poisson bracket of this Hamiltonian with any observable $\mathcal{O}( x,p)$ then forms a linear combination

$$\begin{aligned} {d\mathcal{O} \over d\tau } = \Sigma _i \omega _i {d \mathcal{O} \over d\tau _i}, \end{aligned}$$

(2.70)

where we have taken into account that the $K_i$ vanish on the constraint hypersurface; the $\omega _i$ are then identified with $d\tau _i /d\tau$, with the $\tau _i$ considered as functions of the overall evolution parameter $\tau$.

Although this approach is very elegant on a classical level, there are some difficulties in passing to the quantum theory. The condition $K_i = 0$ poses a difficult problem since, in general, the $K_i$ have continuous spectrum, and the eigenstates would lie outside the Hilbert space. This problem can be treated by defining N Schrödinger type equations of the form (as for the treatment of cases with states in the continuous spectrum in the nonrelativistic theory)

$$\begin{aligned} i {\partial \psi _{\tau _1,\tau _2, \dots } \over \partial \tau _i} = K_i \psi _{\tau _1,\tau _2, \dots } \end{aligned}$$

(2.71)

but the combination $\Sigma _i {\omega _i} (x,p) K_i$ would, in general, not be Hermitian. The symmetric product with the $\omega _i$’s would not be useful, since the functions $\omega _i$ have no well-defined action on $\psi _{\tau _1,\tau _2, \dots }$. Nevertheless, Rohrlich and the author succeeded in formulating a viable scattering theory in this framework (see references under Llosa 1982).