Feynman's Non-relativistic QM

Feynman's Space-Time Approach to Non-Relativistic Quantum Mechanics

Notes by Jack Sarfatti

Version 0.2

Dec 4, 1996

Details of Feyman’s geometry of paths.

“... to define a probability amplitude for a particular completely specified spacetime path ... we shall limit ourselves to a one-dimensional problem ... Assume that we have a particle which can take up various values of a coordinate x.”

This is consistent with Bohm’s hidden-variable/pilot-wave ontological interpretation of quantum mechanics but not with Bohr’s Copenhagen interpretation -- which Bohr correctly recognized immediately upon hearing Feynman for the first time on this topic.

“Imagine that we make an enormous number of successive position measurements, let us say separated by a small time interval epsilon..”

What about the quantum Zeno effect here? This will be important when the time epsilon is smaller than the time tau during which the localized particle wave packet spreads a distance greater than the statistically averaged distance between successive position measurements.

“Then a succession of measurements such as A, B, C, D, ... might be a succession of measurements of the coordinate x at successive times t1, t2, t3, ... where ti+1 = ti + epsilon. Let the value, which might result from measurement of the coordinate at time ti be xi. ... From a classical point of view, the successive values, x1, x2, x3, .... of the coordinates practically define a path x(t). Eventually, we expect to go to the limit epsilon -> 0.”

The probability amplitude is a multiple integral, one integration for each xi. This algorithm takes into account all possible paths connecting the same starting point x0 at t0 and xf = xn+1 at tf where n = (tf - t0)/epsilon. Therefore, holding to and tf fixed, and letting epsilon squeeze down toward 0 pushes n toward infinity in the sense of the theory of limits in differential calculus. At some point it appears that we will run into the quantum Zeno effect which freezes the eigenvalues xi, so there is a practical resolution limit depending on the inertial mass of the particle. Feynman did not take this into account. But I may be wrong about this. I need to look in more detail at the Zeno effect.

“I. If an ideal measurement is performed to determine whether a particle has a path lying in a region R of spacetime, then the probability that the result will be affirmative is the absolute square of a sum of complex contributions, one from each path in that region. ... A path is first defined by the positions xi through which it [the particle] goes at a sequence of equally spaced times ti = ti-1 + epsilon. Then all values of the coordinates within R have an equal weight. The actual magnitude of the weight depends upon epsilon and can be so chosen that the probability of an event which is certain shall be normalized to unity. ...

“The contribution from a single path is postulated to be an exponential whose (imaginary) phase is the classical action (in units of hbar) for the path in question. The total contribution from all paths reaching x,t from the past is the wave function psi(x,t).”

But he also says that the complex-conjugate, chi*(x,t) is from all paths reaching x,t from the future. This is generally ignored!

Click your mouse on the above image for more images.

The classical action is “the time integral of the Lagrangian taken along the path”.

“If we suppose it to be a quadratic function of the velocities, we can show the mathematical equivalence of the postulates here and the more usual formulation of quantum mechanics.”

In general for many degrees of freedom x(t) the system point is in classical comfiguration space. Note in the pre-Feynman second-quantized notation, the number of degrees of freedom of configuration space is proportional to the number of creation operators acting on the vacuum. Also we have the new nonclassical feature of coherent superposition of classical configuration spaces of different dimensions as in Glauber coherent and squeezed boson states, for example.

Feynman shows that the Lagrangian cannot depend on time derivatives of position x higher than the first. If it did “the end points are not sufficient to define the classical path”. Well what about the radiation reaction force which depends on the third time derivative of the position suggesting a second time derivative of position in the Lagrangian L. Consider the resulting equation of motion from the universal Action Principle.

Variation of Action = 0

dL/dx - (d/dt)dL/d(dx/dt) = 0

Suppose a term proportional to (dx/dt)(d^2x/dt^2), i.e., product of velocity and acceleration, in the Lagrangian. This gives a radiation reaction force term proportional to d^3x/dt^3. This means that the end points do not provide enough information to define the classical path. Feynman avoided this by using one half the sum of advanced and retarded potentials from all the other particles with no self-action.

Imagine a present moment t with R’ a spacetime region in its past and R” a region in its future. We are doing nonrelativistic geometry so there is no elsewhere outside the light cone. In other words, the speed of light is effectively infinite in this approximation of Galilean relativity. Feynman shows that the ordinary quantum wave function at here-now event (x,t) is from the sum of all paths that come from R’ and end at (x,t). That is, the ordinary wave function is the “retarded” sum of the phase factors of all past paths that end at (x,t). Similarly, the complex conjugate wave function is from all future paths from R” backward in time to (x,t). That is, the complex conjugate is an advanced waveform.

Feynman defines the, usually off-diagonal, transition amplitude to go from past R’ to future R”. That is, we coarse grain so we do not need to know a single path. We are only interested in a bundle of paths determined by the resolution of our detectors which must always average of spacetime regions.

= Integral over spacelike surface at t of advanced complex conjugate wave function chi*(x,t) multiplied by retarded wave function psi(x,t).

This is Feynman’s equations (13) and (14). When the complex conjugate chi*(x,t) is the same function as the wavefunction psi(x,t), I.e., in the diagonal case where chi* = psi*, we get the Born probability density. Thus the quantum probability density for the wavefunction not to quantum jump to a different wavefunction is seen to be from all the past influences from the past to the present event at (x,t) modulated by all the future teleological or destined influences projecting backward in time from the future to that same present event (x,t).

“This split is possible because ... the Lagrangian is a function only of positions and velocities. ... Thus, with psi and chi* we have separated the past history from the future experiences of the system. This permits us to speak of the relation of past and future in the conventional manner.”

Not quite I think. Feynman is doing a shell game here. Yes, if you ask for transition amplitude we can speak of past and future in the conventional manner. But if you ask for the Born probability density at a single point (x,t) you have both advanced and retarded influences at (x,t). There is an alternative interpretation here due to Cramer, but it deviates from Feynman's actual argument. Cramer imagines that psi* includes the same paths from R' but leaving (x,t). The difference is shown in the following diagram.

“Likewise, the function chi*(x,t) characterizes the experience, or , let us say, experiment to which the system is to be subjected.”

So, the inner product in Hilbert space (R”|R’) is really an integral of past causes modulated by future causes of all of the present events (x,t). That is, the integral is over all possible x at fixed t. If we look at just one here-now event (x,t), then we are interested in (R”|x,t)(x,t|R’) which is the Born probability density. These past and future causes of the actual historical event here-now can be considered to be in a different level of quantum reality called “potentia” in Stapp’s Heisenberg-James ontology that he defines in his book, Matter, Mind and Quantum Mechanics. Note that the transition probability from R’ to R” is |(R”|R’)|^2 which involves all x at a fixed t. In contrast, simply (R’|x,t)(x,t|R”) is the probability density for the single event (x,t) to actualize when (R”| and |R’) are duals of the same projective ray in Hilbert space.

Derivation of Schrodinger equation.

not finished -- under construction

BACK