2 Some Remarks about Path Integration

In this section we review briefly the definition of phase space path integral. The main purpose here is twofold: first, we wish to present a self-contained exposition for those readers who may not be acquainted with the idea of ``sum over histories'' proposed by Feynman[6,10]; second, we wish to isolate those fundamental properties of Quantum Mechanics, encoded in the path integral approach, which are relevant to the subsequent discussion of our own computational method .
One such fundamental property is the distinction between the square of the absolute value of the wave function, or probability density, and the wave function itself, or probability amplitude. With that distinction in mind, the peculiarity of the quantum mechanical world is that, in contrast to the classical rules of conditional probability, one computes probability amplitudes for the paths, and then sum the amplitudes; when amplitudes are superimposed, and then squared, an interference pattern is induced in the probability density. The famed ``particle-wave duality'' of the quantum world stems essentially from that new computational rule. According to Dirac and Feynman [11], the basic difference between classical and quantum mechanics is that the former selects, through the stationary action principle, a single trajectory connecting the initial and final particle position, while the latter assumes that a particle ``moves simultaneously along all possible trajectories'' connecting the initial and final end points. Of course, not all trajectories are equal, even if they correspond to wave functions having the same absolute value: the action $S\left[ {\vec{\mbox{\boldmath {$y$}}}}(t) \right]$ of a particular path connecting the two fixed end points determines the phase of the propagation amplitude associated with that path. The phase factor corresponding to any individual path can be written in the form $${ \exp \left\{ i S \left[ {\vec{\mbox{\boldmath {$... ...t) , {\vec{\mbox{\boldmath {$p$}}}}(t) \right] / \hbar \right\} }$$, and a very effective way to describe the quantum mechanical behavior of a particle is through the sum over histories proposed by Feynman. Symbolically, the amplitude to evolve from ${\vec{\mbox{\boldmath {$x$}}}}_0$ to ${\vec{\mbox{\boldmath {$x$}}}}$ is

$$A \left( {\vec{\mbox{\boldmath {$x$}}}}_0 , {\vec{\mbox... ...}(t) , {\vec{\mbox{\boldmath {$p$}}}}(t) \right] \quad ,$$ (1)

where

$$\Phi \left[ {\vec{\mbox{\boldmath {$y$}}}}(t) , {\vec... ... {\vec{\mbox{\boldmath {$p$}}}}(t) \right] \right\} \quad .$$ (2)

A sum of this type, defined over a space of functions, i.e., ${\vec{\mbox{\boldmath {$y$}}}}(t)$, and ${\vec{\mbox{\boldmath {$p$}}}}(t)$, is a functional integral, and the basic problem of the Feynman formulation of quantum mechanics is to determine what that formal sum means.
In order to construct the sum over all paths in phase space, one may follow the evolution of the system in a total time interval $[ t ,t' ]$ by dividing such interval in $N+1$ subintervals, with end points labelled by $t _{1}$, $t _{2}$, ..., $t _{i}$ , ..., $t _{N+1}$ so that

$$t = t _{0}$$

$$t' = t _{N+1} \quad .$$

For convenience, one may also choose each of the above intervals to be of equal length $\epsilon$,

$$t _{i+1} - t _{i} = \epsilon \quad ,$$ (3)

in such a way that

$$t _{k} = t + k \epsilon \quad , \qquad k = 0 , \dots , N+1 \quad .$$ (4)

Next, one has to determine an equation for the partial amplitude

$$A \left( q _{i+1} , t _{i+1} ; q _{i} , t _{i} \ri... ...+1} , t _{i+1} \vert q _{i} , t _{i} \right \rangle$$ (5)

that the system evolves from a position with coordinate $q _{i}$ at time $t _{i}$ to a position with coordinate $q _{i+1}$ at time $t _{i+1}$. Using the bracket notation, the amplitude for the first subinterval takes the form

$$\left \langle \tilde{q} , \epsilon \vert \bar{q} , 0 \right \rangle = \left \langle \tilde{q} \vert \exp \left( - i H \epsilon \right) \vert \bar{q} \right \rangle$$

$$= \delta \left( \tilde{q} - \bar{q} \right) - i \epsilon \left... ...ert H \vert \bar{q} \right \rangle + {\mathcal{O}} \left( \epsilon ^{2} \right)$$

$$= \int \frac{dp}{2 \pi} e ^{i p \left( \tilde{q} - \bar{q} \right)}... ...ac{\tilde{q} + \bar{q}}{2} \right) + {\mathcal{O}} \left( \epsilon ^{2} \right)$$

$$= \int \frac{dp}{2 \pi} e ^{i p \left( \tilde{q} - \bar{q} \right)}... ...{q} + \bar{q}}{2} \right) \right ] + {\mathcal{O}} \left( \epsilon ^{2} \right)$$

$$= \int \frac{dp}{2 \pi} \exp \left\{ i \left [ p \left( \tilde{q} -... ...on H \left( p , \frac{\tilde{q} + \bar{q}}{2} \right) \right ] \right\} \quad .$$ (6)

Generalizing the above expression to a generic subinterval, we find

$$\left \langle q _{i+1} , t _{i+1} \vert q _{i} , t _{i} \... ... \left( t _{i+1} - t _{i}\right) \right ] \right\} \quad .$$ (7)

Approximating the path from $q$ at time $t$ to $q'$ at time $t'$ with a finer subdivision of the total time interval, and in view of the fact that there are no predetermined conditions on the coordinate variable at intermediate times, the total amplitude can be written as the product of the amplitudes associated with each subinterval, integrated over all possible intermediate positions:

$$A \left( q' , t' ; q , t \right) = \lim _{N \to... ...} , t _{i} ; q _{i - 1} , t _{i - 1} \right \rangle \quad .$$ (8)

Combining the two equations above, we are led to the final step of the discretization procedure

$$A \left( q' , t' ; q , t \right) = \lim _{N \to + \infty} \int d q _{1} \int d q _{2} \int \dots \int d q _{N} \cdot$$

$$\qquad \cdot \int \frac{d p _{1}}{2 \pi} \int \frac{d p _{2}}{2 \pi} \int \dots \int \frac{d p _{N+1}}{2 \pi} \cdot$$

$$\qquad \qquad \cdot e ^{i \left[ p _{1} \left( q _{1} - q _{... ...ht) H \left( p _{1} , \frac{q _{1} + q _{0}}{2} \right) \right] } \cdot$$

$$\qquad \qquad \qquad \cdot e ^{i \left[ p _{2} \left( q _{2} ... ...t( p _{2} , \frac{q _{2} + q _{1}}{2} \right) \right] } \cdot \dots \cdot$$

$$\qquad \qquad \qquad \qquad \cdot e ^{i \left [ p _{N+1} \left(... ...\right) H \left( p _{N+1} , \frac{q _{N+1} + q _{N}}{2} \right) \right] }$$

$$= \lim _{N \to + \infty} \left( \prod _{i} ^{1,N} \int d q _{i} \right) \left( \prod _{j} ^{1,N+1} \int d p _{j} \right) \cdot$$

$$\qquad \cdot e ^{i \left \{ \sum _{k} ^{1,N+1} \left[ p _{k} \lef... ...ight) H \left( p _{k} , \frac{q _{k} + q _{k-1}}{2} \right) \right ] \right\} }$$

$$= \lim _{N \to + \infty} \left( \prod _{i} ^{1,N} \int d q _{i} \right) \left( \prod _{j} ^{1,N+1} \int d p _{j} \right) \cdot$$

$$\qquad \cdot e ^{i \left \{ \sum _{k} ^{1,N+1} \left[ p _{k} \fra... ..._{k-1}}{2} \right) \right ] \left( t _{k} - t _{k-1} \right) \right\} } \quad .$$

The expression in the last equality represents the discrete version of the path integral in phase space

$$\int \left[ {\mathcal{D}} q \right] \int \left[ {\mathcal{... ...t{q} - H \left( p , q\right) \right ] dt \right\} \quad .$$ (9)

which is the starting point for computing the sum over histories for the majority of the integrable systems encountered in the literature. Equation (9) is also the starting point of our own approach. However, in terms of the discretization procedure outlined above, a further simple elaboration of the phase space path integral leads to a computational method which seems mathematically more efficient and physically more enlightening, at least for some complex systems encountered in high energy physics.
Consider the first term in the expression (9), namely

$$\int p \dot{q} dt = \int p dq \quad .$$ (10)

This term corresponds to the discrete sum

$$\sum _{k} ^{1,N+1} p _{k} \left( q _{k} - q _{k-1} \right)$$ (11)

which we can rewrite as follows:

$$\sum _{k} ^{1,N+1} p _{k} \left( q _{k} - q _{k-1} \right) =$$

$$= \sum _{k} ^{1,N+1} p _{k} q _{k} - \sum _{k} ^{1,N+1} p _{k} q _{k-1}$$

$$= \sum _{k} ^{1,N+1} p _{k} q _{k} - \sum _{k} ^{1,N+1} p _{k} q ... ...} - \sum _{k} ^{1,N+1} p _{k-1} q _{k-1} + \sum _{k} ^{1,N+1} p _{k-1} q _{k-1}$$

$$= \sum _{k} ^{1,N+1} \left( p _{k} q _{k} - p _{k-1} q _{k-1} \right) - \sum _{k} ^{1,N+1} \left( p _{k} - p _{k-1} \right) q _{k-1}$$

$$= p _{N+1} q _{N+1} - p _{0} q _{0} - \sum _{k} ^{1,N} \left( p _{k} - p _{k-1} \right) q _{k-1} \quad .$$ (12)

The first two terms in the above expression represent a boundary contribution to the path integral, whereas the remaining sum corresponds to the following integral

$$\sum _{i} ^{1,N} \left( p _{k} - p _{k-1} \right) q _{k-1}... ...grightarrow \int q dp = \int q \dot{p} dt \quad .$$ (13)

Next, in what follows we make use of the following equality

$$\int d q _{k-1} \exp \left\{ i \left( p _{k} - p _{k-1} \right) q _{k-1} \right\} = \delta \left( p _{k} - p _{k-1} \right)$$ (14)

which gives the well known representation of the Dirac delta function as Fourier transform of the imaginary exponential. Thus, the net result of the above rearrangement of terms is encoded in the following correspondence

$$\int \left [ {\mathcal{D}} q \right] \exp \left\{ i \int d... ... \frac{p _{k} - p _{k-1}}{t _{k} - t _{k-1}} \right) \quad .$$ (15)

which we take as a definition of the ``functional Dirac delta distribution''.
In the next section, we apply this mathematical rearrangement of terms in the phase space path integral to compute the propagation kernel of a non relativistic particle. As we shall see, the physical payoff is a novel interpretation of the sum over histories in the sense that it underscores the special role played by the entire family of trajectories which are solutions of the classical equations of motion.