The Principle of Constructive Interference

The principle that gives rise to the above-mentioned new phase space coordinates is the principle of constructive interference according to which the spacetime trajectory of a particle is the locus of events where the semi-classical wavefunction

$$\psi \sim \int_{-\infty}^\infty \int_{-\infty}^\infty \int_{-\in... ...f(P_\tau, P_v, P_x, P_y)\mathcal{A}(x^\alpha)e^{iS'/\hbar}dP_\tau dP_vdP_xdP_y$$

has maximum modulus.

The mathematical formulation of this principle is based on the evaluation of the superposition expressed by this integral whenever the exponential phase factor is a function varying rapidly compared to the slowly varying amplitude $\mathcal{A}$. Based on a Gaussian weight factor $f$, the superposition integral is a Gaussian also. Its maximum is located at those events which satisfy

$$\left. \begin{array}{c} \displaystyle\frac{\partial S'}{\partial ... ...}=Q^x\\ \\ \displaystyle\frac{\partial S}{\partial P_y}=Q^y \end{array} \right.$$ (22)

The left column are the conditions for constructive interference. They comprise in Minkowski spacetime the particle's world line

$$\{x^0(\tau),x^1(\tau),x^2(\tau),x^3(\tau) \}=\{u,v,x,y\}$$

as obtained from

$$\frac{m}{2P_v}u+ (\tau-\tau_0) \equiv Q^\tau=\frac{m}{2P_v}u_0$$ (23)

$$v-\frac{1}{4P_v^2} \int^u_{u_0} \left[(P_x-qA_x(u))^2+(P_y-qA_y(u))^2\right. \left.+2mP_\tau\right]du$$

$$\equiv Q^v=v_0$$ (24)

$$x+\frac{1}{2P_v}\int^u_{u_0} \left( P_x-qA_x(u) \right)du \equiv Q^x=x_0$$ (25)

$$y+\frac{1}{2P_v}\int^u_{u_0} \left( P_y-qA_y(u) \right)du \equiv Q^y=y_0~.$$ (26)

The capitalized $P$'s and $Q$'s refer to the initial value data of the world line at $\tau=\tau_0$.

The pictorial representation of the principle of constructive interference consists of the intersection of the isograms of two slightly different solutions to the H-J equation. The points of intersection are where constructive interference takes place. The particle worldline is understood to pass through these successive points. Figure 5 and 6 illustrate this process for free charge and and for a charge driven by the e.m. field of a plane wave. Figure 7 illustrates it for an e.m. pulse with a finite number of oscillations. Note that, once suitably averaged, its spacetime region acts a refractive medium for the particle world line.

Problem 3: a) For Figure 7 formulate what in Euclidean space corresponds to Snell's law.

b) Can one identify a refractive index for the laser pulse history? If so, what is it?

Figure 5: Constructive interference between two sets of wave front histories, the solid isograms of $S(t,z;P_v)$ and dashed isograms of $S(t,z;P_v+\Delta P_v)$. These two sets of wave front histories are the familiar relativistic De Broglie matter waves. The particle is understood to be located at that event where an isogram of one intersects with an equal-value isogram of the other. The fact that the heavy world line and the intersecting isograms are straight is a reflection of the fact that the charged particle is free: there is no e.m. field.

Figure 6: Same as Figure 5 except that the charged particle is driven by the periodic e.m. field of a travelling wave. In such a circumstance the oscillating e.m. field distorts the De Broglie wave front histories so that constructive interference results in an oscillating particle trajectory.

Figure 7: Relativistic De Broglie wave front histories (solid and dashed isograms) distorted by a three cycle laser pulse. Its spacetime history occupies the diagonal 45$^o$ swath. The constructive interference expresses the circumstance where the particle with slight negative z-velocity passes through the +z travelling pulse, gets jiggled three times, before it emerges with its original velocity from the back of the pulse. The net effect of this process is that the particle gets shifted by an amount proportional to the duration of the pulse. If one ignores the oscillatory effect on the De Broglie waves by averaging over each of their histories, then the spacetime history of the laser pulse acts as a refractive medium for the averaged particle world line.