3.1 Path Integral and Propagation Kernel

The classical dynamics of a non-relativistic point-particle with mass $m$ is encoded into the Lagrange function, or in its corresponding Hamiltonian

$$L\left( {\vec{\mbox{\boldmath {$y$}}}}(t) , {\dot{\vec{\m... ...) ; t \right) ={1\over 2} m {\dot{\vec{\mbox{\boldmath {$y$}}}}}{}^2$$ (16)

$$H\left( {\vec{\mbox{\boldmath {$y$}}}}(t) , {\vec{\mbox{\... ...t) ; t \right)={1\over 2 m} {\vec{\mbox{\boldmath {$p$}}}}{}^2 \quad .$$ (17)

From these the classical equations of motion follow:

$${d{\vec{\mbox{\boldmath {$p$}}}}\over dt}=0 \quad \Rightarrow \quad {\vec{\mbox{\boldmath {$p$}}}}={\rm cost.}\equiv {\vec{\mbox{\boldmath {$q$}}}}$$ (18)

$${dH\over dt}=0 \quad \Rightarrow \quad H={\rm cost.}\equiv E$$ (19)

$$m{d{\vec{\mbox{\boldmath {$y$}}}}\over dt}={\vec{\mbox{\boldmath {$q$}}}}\quad \Rightarrow \quad {\vec{\mbox{\boldmath {$y$}}}}(t)={1\over m} {\vec{\mbox{\boldmath {$q$}}}} t+{\vec{\mbox{\boldmath {$x$}}}}_0 \quad ,$$ (20)

where we have taken into account the boundary conditions

$${\vec{\mbox{\boldmath {$y$}}}}(0)={\vec{\mbox{\boldmath {$x$... ...ox{\boldmath {$x$}}}}-{\vec{\mbox{\boldmath {$x$}}}}_0\over T}$$ (21)

so that

$${\vec{\mbox{\boldmath {$y$}}}}(t)={{\vec{\mbox{\boldmath {$x... ...$x$}}}}_0\over T} t+{\vec{\mbox{\boldmath {$x$}}}}_0 \quad .$$ (22)

Furthermore, for later reference, we recall that the classical action

$$S_{\mathrm{cl.}}( {\vec{\mbox{\boldmath {$x$}}}} , {\vec{\mbox{\boldmath {$x$}}}}_0 ; T ) = {1\over 2}m\int_0^T dt \left( {{\vec{\mbox{\boldmath {$x$}}}}-{\vec{\mbox{\boldmath {$x$}}}}_0\over T} \right)^2$$

$$= {m\over 2T}\vert {\vec{\mbox{\boldmath {$x$}}}}-{\vec{\mbox{\boldmath {$x$}}}}_0 \vert^2$$

is a solution of the Jacobi equation

$${\partial S_{\mathrm{cl.}}\over\partial T} + {1 \over 2m} \v... ...la S_{\mathrm{cl.}}\cdot\vec\nabla S_{\mathrm{cl.}}= 0 \quad .$$ (23)

Equivalently, one can follow the particle evolution by determining the propagation amplitude $K( {\vec{\mbox{\boldmath {$x$}}}}-{\vec{\mbox{\boldmath {$x$}}}}_0 ; T )$ that a particle propagates from an initial position ${\vec{\mbox{\boldmath {$x$}}}}_0$ to a final position ${\vec{\mbox{\boldmath {$x$}}}}$. As we have shown in the previous section, the amplitude is given by a formal sum over all phase space paths connecting ${\vec{\mbox{\boldmath {$x$}}}}_0$ to ${\vec{\mbox{\boldmath {$x$}}}}$ in a total time $T$, each path carrying a weight given by the phase factor $\exp\left( i S[ \hbox{path} ]/\hbar \right)$. We have also reviewed the standard method that gives meaning to the sum over histories: it goes through a discretization procedure which consists in subdividing the total time lapse $T$ into a number of infinitesimal intervals, thereby approximating the smooth phase space trajectory followed by a classical particle with a succession of ``jagged'' paths. It seems worth emphasizing at this point, that in the above implementation of the path integral, the classical trajectory followed by a particle is uniquely specified by Newtonian mechanics. Furthermore, we hasten to say that the sum over histories, so defined, is a sum over trajectories which are nowhere differentiable. As a matter of fact, a post modern interpretation of Feynman's discussion is that the quantum mechanical path of a particle is inherently fractal<sup>4</sup>. The gist of the argument is that, when a particle is more and more precisely located in space, its trajectory becomes more and more erratic as a consequence of Heisenberg's principle. In other words, it is the addition of quantum fluctuations around the classical trajectory that gives meaning to the idea that ``a particle moves along all possible paths'' connecting the initial and final configurations. Be that as it may, the discretization procedure turns the functional integral into an infinite product of ordinary integrals [13]. For some elementary systems, the integration over the momenta can be carried out, albeit with some efforts, and the final result for the amplitude is the ``Lagrangian path integral'':

$$K( {\vec{\mbox{\boldmath {$x$}}}} , {\vec{\mbox{\bold... ...,{\dot{\vec{\mbox{\boldmath {$y$}}}}}(t) ; t ] dt \quad .$$ (24)

For more complex systems, such as the relativistic extended systems encountered in contemporary high energy physics, the expression (24), may not tell the full story, and we have found it advantageous to start with a sum over histories in phase space. Thus, in order to illustrate our computational method, we start directly from the canonical phase space path integral for a free, non-relativistic particle

$$K( {\vec{\mbox{\boldmath {$x$}}}} , {\vec{\mbox{\bol... ...}}}}(t)- H_0({\vec{\mbox{\boldmath {$p$}}}}) \right] \quad ,$$ (25)

where

$$H_0({\vec{\mbox{\boldmath {$p$}}}})={{\vec{\mbox{\boldmath {$p$}}}}{}^2\over 2m}$$ (26)

and $N$ is a normalization constant to be fixed later on.
The focal point of our approach is the recognition that the path integral automatically assigns a special role to the whole family of trajectories which are solutions of the classical equation of motion. The precise meaning of the above statement is illustrated by implementing the computational steps following Eq.(9). Thus, we write the first term in the action as

$$\int_0^T dt {\vec{\mbox{\boldmath {$p$}}}}\cdot{\dot{\vec{\... ...ath {$y$}}}}\cdot {\dot{\vec{\mbox{\boldmath {$p$}}}}} \quad .$$ (27)

Accordingly, the path integral reads

$$K(\, {\vec{\mbox{\boldmath {$x$}}}}\, , \, {\vec{\mbox{\bold... ...nt_0^T dt\, H_0({\vec{\mbox{\boldmath {$p$}}}})\right]\right\}$$ (28)

and we note that, while the first term in the above expression is the integral of a total differential, and therefore independent of the path connecting the two points $\left( {\vec{\mbox{\boldmath {$x$}}}}_0 , {\vec{\mbox{\boldmath {$x$}}}} \right)$, the second term depends linearly on the spatial trajectory ${\vec{\mbox{\boldmath {$y$}}}}(t)$. A closer look at this term shows that it is a (functional) Dirac-delta distribution represented as a ``Fourier integral'' over the functions ${\vec{\mbox{\boldmath {$y$}}}}(t)$

$$\int_{{\vec{\mbox{\boldmath {$x$}}}}_0}^{{\vec{\mbox{\boldma... ...[ {d{\vec{\mbox{\boldmath {$p$}}}}\over dt} \right] \quad .$$ (29)

The Dirac-delta in equation (29) is non-vanishing only when its argument is zero, i.e., when the momentum ${\vec{\mbox{\boldmath {$p$}}}}$ solves the classical equation of motion

$${d{\vec{\mbox{\boldmath {$p$}}}}\over dt}=0 \quad \Rightarro... ...}(t)={\rm const.}\equiv {\vec{\mbox{\boldmath {$q$}}}} \quad .$$ (30)

This is our first, and central result which applies equally well, ``mutatis mutandis'', to relativistic point-particles and relativistic extended objects: once the spatial trajectories ${\vec{\mbox{\boldmath {$y$}}}}(t)$ are integrated out, the resulting path integral is non-vanishing only when the surviving integration variable, namely the three-momentum vector along the trajectory, is constrained to satisfy equation (30). This information is encoded in the following expression for the propagation amplitude

$$K( {\vec{\mbox{\boldmath {$x$}}}} , {\vec{\mbox{\bold... ...\! \! dt H_0({\vec{\mbox{\boldmath {$p$}}}}) \right\} \quad ,$$ (31)

where only the restricted family of constant (i.e., time independent) ${\vec{\mbox{\boldmath {$p$}}}}$-trajectories contributes to the path integral. Note that the value of the three-momentum along a classical trajectory is a fixed, even though arbitrary, number. Thus, on the mathematical side, the pay-off of our procedure is that we have traded the original path integral, i.e., a functional integral, with a single ordinary integral over the constant, but numerically arbitrary, components of the three-momentum.
On the physical side, our method underscores a conceptual difference between the mechanism of propagation envisaged here and the conventional one. By this we mean that in the conventional interpretation of the path integral, the sum over histories is obtained by summing over all possible quantum fluctuations around a single classical trajectory which is uniquely defined because both extremal position and momentum are preassigned. In contrast, in our interpretation of sum over histories, only the extremal coordinates of the particle are precisely specified by the boundary conditions, whereas the corresponding value of the momentum at the end points is constant but arbitrary, in consistency with the uncertainty principle. Since the momentum is a vector, it follows that all paths in configuration space contribute to the evolution of the wave function. In the above sense, it seems to us that the complementarity between particle and wave behavior and the role of the uncertainty principle are especially evident in our approach.
To summarize our discussion so far, ``the sum over histories'' of a free particle between two fixed end points, can be translated into the integration over all possible values of the linear momentum. Put briefly,

$$\int [{\mathcal{D}}p(t)] \delta\left[d{\vec{\mbox{\boldmath {$p$}}}}/dt\right]\left(\dots\right) = {\rm sum over the classical momenta}$$

$$= {\rm sum over constant momentum trajectories}$$

$$= \int d^3 q \left(\dots\right) \equiv {\rm ordinary momentum integral} \quad .$$

Next, if the momentum is constant, then the Hamiltonian is constant as well, and we can write

$$\int d^3 q \exp\left[ {i\over\hbar}{\vec{\mbox{\boldmath {... ...}}}}-{\vec{\mbox{\boldmath {$x$}}}}_0 \vert^2\right] \quad ,$$ (32)

where we have used the identity

$$\displaystyle {\int_{{\vec{\mbox{\boldmath {$x$}}}}_0}^{\vec... ...vec{\mbox{\boldmath {$x$}}}}-{\vec{\mbox{\boldmath {$x$}}}}_0}$$ (33)

and the formula for the multidimensional Gaussian integral listed in Appendix B. As a result, we obtain

$$K( {\vec{\mbox{\boldmath {$x$}}}} , {\vec{\mbox{\bold... ...}}}}-{\vec{\mbox{\boldmath {$x$}}}}_0 \vert^2\right] \quad .$$ (34)

Finally, the normalization constant $N$ can be fixed by the condition that, in the limit of a vanishing lapse of time, the particle is bound to be in its initial position. In other words,

$$\lim_{T\rightarrow 0} K( {\vec{\mbox{\boldmath {$x$}}}}- {... ... \quad N=\left( {1\over 2\pi\hbar^2} \right)^{3/2} \quad .$$ (35)

Hence,

$$K( {\vec{\mbox{\boldmath {$x$}}}}-{\vec{\mbox{\boldmath {$x$}}}}_0 ; T ) = \left({m\over 2i\pi\hbar T }\right)^{3/2} \exp\left[{im\over 2\hb... ...\vec{\mbox{\boldmath {$x$}}}}-{\vec{\mbox{\boldmath {$x$}}}}_0 \vert^2\right]$$

$$= \left({m\over 2i\pi\hbar T }\right)^{3/2} \exp\left[ i S_{\ma... ...}- {\vec{\mbox{\boldmath {$x$}}}}_0 ; T \right)/\hbar \right] \quad ,$$ (36)

where the phase factor is just the classical action measured in $\hbar$ units

$$S_{\mathrm{cl.}}\left( {\vec{\mbox{\boldmath {$x$}}}}- {\v... ...th {$x$}}}}-{\vec{\mbox{\boldmath {$x$}}}}_0 \vert^2 \quad .$$ (37)

The above expression for the propagation kernel is a well known result which tests the consistency of our approach: as Feynman and Hibbs demonstrated, for any lagrangian quadratic in the position and velocity variables, the corresponding propagation amplitude is given by a pre-factor, which is a function of the evolution parameter lapse, multiplied by the exponential of ``i'' times the classical action in $\hbar$ units.