Confused by Feynman sum over histories

  • #1
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Main Question or Discussion Point

The Feynman formulation of Quantum Mechanics builds three central ideas from the de Broglie hypothesis into the computation of quantum amplitudes: the probabilistic aspect of nature, superposition, and the classical limit. This is done by making the following three three postulates:

1. Events in nature are probabilistic with predictable probabilities P.
2. The probability P for an event to occur is given by the square of the complex magnitude of a quantum amplitude for the event, Q. The quantum amplitude Q associated with an event is the sum of the amplitudes associated with every history leading to the event.
3. The quantum amplitude associated with a given history is the product of the amplitudes associated with each fundamental process in the history.
This seems to contradict the very tools we use to perform the scientific method. Immediately after performing any experiment, the measurement becomes a historical event, and this says that we cannot say that any historical event has actually occurred. Thus, I cannot actually say that I performed an experiment and obtained a measurement....and if you cannot obtain concrete evidence, then you cannot prove scientific theories! In fact, if you cannot say that any past event has actually occurred, then you cannot use logic to describe the universe, because in logic a statement is true or false but cannot be "true with some probability"...and mathematics is derived from logic, so mathematics also breaks down!
 

Answers and Replies

  • #2
Events are probabilistic in the sense that our experiments repeat probabilistically. That is, try as we might to do the exact same thing every time we still get variations. The distribution of the variations is repeatable, so the experiment is repeatable in that sense.

Once a measurement is made then probability drops out of the picture. A measurement is a measurement. We use these measurements to get a probability distribution.

So I guess an "event" would be a measurement that has not yet actually been made.
 
  • #3
epenguin
Homework Helper
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Immediately after performing any experiment, the measurement becomes a historical event, and this says that we cannot say that any historical event has actually occurred.
You must have abbreviated your argument, missing out some essential premises that would make it comprehensible.

For instance on the face of it, more plausible than "we cannot say that any historical event has actually occurred" seems that we cannot say anything else about the physical world. Almost.

Also what you present as an objection to quantum mechanics, or maybe a particular formulation or interpretation of it, seems to apply to all Science.
 
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  • #4
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apparently your interpretation is one view: Feynman seems to have shown that his approach is equivalent to a classical one:

http://en.wikipedia.org/wiki/Sum_over_histories

Feynman showed that this formulation of quantum mechanics is equivalent to the canonical approach to quantum mechanics, when the Hamiltonian is quadratic in the momentum. An amplitude computed according to Feynman's principles will also obey the Schrödinger equation for the Hamiltonian corresponding to the given action.

Classical action principles are puzzling because of their seemingly teleological quality: given a set of initial and final conditions one is able to find a unique path connecting them, as if the system somehow knows where it's going to end up and how it's going to get there. The path integral explains why this works in terms of quantum superposition. The system doesn't have to know in advance where it's going or what path it'll take: the path integral simply calculates the sum of the probability amplitudes for every possible path to any possible endpoint. After a long enough time, interference effects guarantee that only the contributions from the stationary points of the action give histories with appreciable probabilities
Roger Penrose in THE ROAD TO REALITY, pages 666-669 discusses interpretations of Feynman's path integral/sum over histories approach...in a general way....a continuous infinity of classical alternatives, infinite dimensional configuration space, etc, but beginning on page 670 discusses DIVERGENT PATH INTEGRALS: FEYNMAN'S RESPONSE where Penrose discusses the Feynman propagator and some mathematical properties associated with his approach....I'm not qualified to really understand the math he references.
 
  • #5
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I have no issue with summing over all possible paths in order to compute the probability of measuring a particular outcome in an experiment...this prevents us from making exact predictions about what the outcome will be, but it does not prevent us from making definitive statements such as "I performed an experiment."

The issue with summing over all possible histories from the origin of the big bang singularity is that there are some histories in which we do not even exist and never performed an experiment. If this is the case then we cannot actually say "I performed an event" or "I exist" we can only say "With some probability, I exist.." but then this confuses the very notion of probability, which is a description of the odds of making some measurement under the assumption that definitive measurements can be made.
 
  • #6
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Nevermind, I figured it out.
 

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