I agree about the 'not wrong'. My objection is rather of aesthetical nature.

Since the same phenomenon can be explained by a variety of ways, why not to pick up the most elegant one? Let's add the criterion of aesthetical completeness in our analysis...

If we are able to do such an enormous JUMP, then I think that our modern theories are just for those who are irrelevant and can be impressed. Otherwise, we are doing (silently) next trick:

1)we start from a well accepted theoretical result like Schroedinger's equation (S.E.)

2)we remember the least action principle that almost 'always works'

3)we choose such a function that after extremizing the path action can lead to S.E. by taking the linear term in Taylor series expansion.

Sorry, but this reminds me some incredible proofs in Calculus:

Just to lough with it I give you one such 'Proof':

Example.

=======

Let the function f:R-->R with

f(x)=x^2*sin(1/x), if x#0

f(x)=0,if x=0

Show that f is continuous in R.

'Proof'

=====

Indeed, let x0#0 and epsilon>0. We choose

delta=1/2*min{1,|x0|/2,epsilon/(2|x0|+3)}

Then... (I omit the proof)

End of 'proof'.

As you saw the 'Big' Professor of Calculus had the divine effulgence to choose from the beginning such a suitable value for delta! Let's glory him!

By the same meaning, glory to Feynman and Hibbs!

I agree about the 'not wrong'. My objection is rather of aesthetical nature.

Since the same phenomenon can be explained by a variety of ways, why not to pick up the most elegant one? Let's add the criterion of aesthetical completeness in our analysis...

A probably, but unrevealed from the writers, first legitimation is this:

Since the wave function psi is a complex one then any multiplication by a term of the form exp(i*phi) leaves it invariant. So, why not to use action S as a phase exponent? But, again the division by hbar is still non-justified.

A second justification could be the exponential infinitesimal generator, but the relationship is not so obvious like the 1st one.

Who knows exactly why they introduced the term phi[x(t)]=const exp((i/hbar) S[x(t)]) ?

Good discussion, folks !

Let's perform our familiar linearisation process and take the linear approximation of wave function phi (see attachment previously pdf). Then the squared amplitude (=probability) is just:

P=k^2+(k^2/hbar^2)*S^2

i.e. a quadrature term for probability. What does such a value mean in reality?

We recall that we must achieve the minimum action S for the most probable path, so in the above linear case the probability P is becoming also minimum!

How do you find this result?

Please do not object about linearisation if you don't read first here:

https://www.researchgate.net/post/Are_you_satisfied_with_our_linear_science

Another possible explanation:

Every path from a to b is represented by a complex wave function phi[a,b] with the same amplitude, say k, but different phase, say theta. So we have that:

phi[a,b]=k*exp(i*theta)

Feynman & Hibbs in order to argue that when we are far away from the optimal action path S* our solution has to highly oscillates so the contribution actually vanishes, they chose as theta=S/hbar, a dimensionless quantity which when S>>hbar oscillates around zero very fast. But, the legitimation is still unjustified, since they could choose another quantity that could also fast oscillate.

The Feynman action S is naturally normalized by \hbar because \hbar is the quantum mechanical minimum action. \hbar is just a unit action in a discrete quantum action space.

I recommend to consult

R. P. Feynman, Space-Time Approach to Non-Relativistic Quantum Mechanics, Rev. Mod. Phys. 20 (1948) 367-387; reprint in: J. Schwinger (Ed.), Selected Papers on Quantum Electrodynamics, 1958, No. 27 (here, also Dirac's precursor paper is reprinted)

Feynman correctly observes that the Schrödinger transition probabilities, P_ab obey the Chapman-Kolmogorov eq.,

P_ac = Sum(b) P_ab P_bc

My only question is, how mathematically well founded is the limit procedure leading to the path integral?

Hope this helps,

Peter

Peter,

the limit (as defined by Feynman) leading to the path integral as Riemannian integral holds only for quadratic Lagrangians. If Lagrangian were of higher order than quadratic, there occur problems (the trajectories can exhibit singularities) .

I remember a recent very nice paper by J. Bohacik and P. Presnajder in J.Math. Physics where the problem has been solved in an elegant way. At the moment I do not have the exact reference, I can send it later

I am adding the promised reference:

J. Bohacik and P. Presnajder, J . Math. Physics 49, Iss.11, 13505, 2008

It is but a rephasing of quantum mechanics. Schrödinger got his equation just with the converse route, using the Hamilton-Jacobi equation. There are actually two derivations, the first from a variational principle, and the second heuristical.

Aestetically, it is just a consequence of the linearity of the wave equation. Every point is a source that radiates in every direction, and all that is superposed, so that it can be viewed as if all paths are traversed, but it is only one of the infinite representations possible, that has no special ontological status. For a wave travelling in a one dimensional space, it is easy to see that the phase is proportional to the action, the constant of proportionality having no special ontological status either, it only depends on the units. The Newton equations are already a hint of the wave nature of matter.

The wave nature is just an experimental fact. The superposition principle in the absence of interaction is just the definition of an interaction. It seems that interaction has no special ontological status either, since this defintion leads to parodoxes.

There are experimental observations of the *nature* of wave, which is interference phenomena. A wave isn't a physical entity, it is a theoretical concept. There is nothing in the real world that is identical to this concept, a particle also has properties of a corpuscle along with some of a wave.

When there is no interaction, there is no observation. As long as there is a measurement, there is the measurement problem.

There is no probability with a corpuscle, alone it is necessarily deterministic. Once you have a probability theory and a law of evolution in time, you can express it through a normed wave function. The normed condition is equivalent to the sum of the probabilities being equal to one, and the modulus squared ensures that the probability is positive. Probability due to a lack of knowledge isn't probability, but just lack of knowledge. Consequently, probability *is* nature of wave.

The evolution law of the probability must be invariant by translation and rotation. The only way to write it is with a wave equation. (the only other alternative is a diffusion equation, but it doesn't show interference phenomena, and is then excluded by experiment.) The evolution of the wave function is deterministic, it is the reason why it is able to describe the undeterministic motion of a corpuscle. That's not old notion, that's logics.

Even in causal interpretations like the one of Böhm, the motion of the corpuscle is deterministic, we only don't know their initial conditions. And that doesn't work without a weird "quantum potential" that is non local, otherwise there is no interference.

If the law of evolution of the probability (of whatever) is covariant, then there are only two possibilities: a diffusion like equation, or a wave like equation. Therefore, only experiment can decide on the wave nature, it is false to say that the interference phenomena can be derived just from the constraints of probability theory. That doesn't mean that all particles are waves, but that means that none particle is a corpuscle.

The definition of an interaction is the same for waves and for corpuscles, since a set of non interacting corpuscles also satisfies the principle of superposition. The equation of evolution then must be linear, and there are again two possibilities: diffusion or wave, which correponds to hyperbolic or elliptic. A diffusion equation is necessarily probabilistic, so it is natural that a wave equation be probabilistic too.

We can as well derive a diffusion equation, there is no contradiction.

A theorem is not only an argument, it is also a set of hypothesis. In the real world, there are phenomena described by a diffusion equation, like heat conduction, that are random in nature. Some of the hypothesis must therefore be false.

Two no interacting corpuscles can be describe as the superposition of two systems of one corpuscle. Applying the equation of motion to any system gives exactely the same result. The density of a system of a great number of corpuscles with random initial conditions is described on average by a linear diffusion equation. That's fully analogous to the superposition of waves. Wave equations and diffusion equations differ only by the imaginary unit. They are essentially the same, and there is no wonder they can be proven together through some theorem.

The Böhm theory precisely talks about only one corpuscle with a diffusion equation. Because when the corpuscles are non interacting, it makes no difference whether there is only one or many of them, if the initial condition is not known. That's the principle of a thermodynamical ensemble.