for a measure. What exactly is a measure? Is it counting samples for which a
proposition is true? Does it only apply to operations (or formulas) on an
underlying set? Is it the information or entropy of an operation on a set?Record of Document Changes
for physics from logic effort.
Last modified this page: 8/7/08
by adding 25) of the TO-DO LIST
The article at this link attempts to derive all of physics from the principles of logic alone. And the page here comments on this effort. I expect this will be an ongoing effort, and changes will be made to these pages with further developments. Therefore, this page documents the changes made to those two pages. That way it will not be necessary to download the main page again unless significant changes have been made. I also intend to document a to-do list of areas that need more development. This will serve as a reminder to myself and as direction for anyone who wishes to help.
If you have any constructive comments or suggestions that might help in this effort, please feel free to contact me through email at: mjake (at_) sirus (_dot) com.
DOCUMENT CHANGES:
4/17/07 - first mention of this effort was posted . I will keep this page intact as part of the record. It tries to make use of the square root as an expression of implication. But this was mostly based on intuition and a more rigorous derivation was desired.
11/4/07 - first posted the comments page.
2/5/08 - The next effort was posted. This effort is more rigorously developed. Implication is instead represented by the Dirac delta function, and reasons for this are developed from first principles.
2/8/08 - Added hyperlink to Comments page at bottom of main article.
2/8/08 - Posted comments page to Web.
2/10/08 - Tried to improve the ABSTRACT mention of measure theory (second to last paragraph).
2/13/08 - Used clearer language to explain the arbitrary use of m and h-bar in the delta of the gaussian in equation [18] so that it does not appear to be covertly introducing physics. Thanks for discussion on alt.sci.physics.
6/2/08 - Simplified notation of infinite multiple integral to single integral sign. Added remarks to show that the gaussian form of the Dirac delta with complex exponent was no special choice in order to force the formulation to conform to physics. Renumbered formulas.
TO-DO LIST:
(ideas that need development)
1) Include potential term in the action integral. This is most likely a result of a non-constant density-field of elements.
2) Show that quantum field theory is a result of the density-field changing with time.
3) I might wish to explain how classical lagrangian physics is derived from the path integral - that the contribution of far flung paths cancel each other out to leave the path of classical mechanics as the greatest contributor.
4) I might wish to show how Schrodinger's equation is derived from the path integral in the usual way.
5) I might wish to show how special relativity and the uncertainty principle come from this.
6) Include more about measure theory
requiring addition for a union of sets and especially multiplication for an intersection of
sets. And adopt the more commonly used notation of
for a measure. What exactly is a measure? Is it counting samples for which a
proposition is true? Does it only apply to operations (or formulas) on an
underlying set? Is it the information or entropy of an operation on a set?
7) Make truth table that shows how a disjunction of every possible conjunction of implications becomes more likely equal to the implication between start and finish as more propositions are added to the truth table.
8) Show how the Born probability rule is obtained from all this. If the wavefunction is the Dirac delta, then somehow the delta times its complex conjugate must be finite and real. Perhaps this is done by dividing out the infinities when normalizing the probability. Can measurement be interpreted as the measure of the implication from start to finish times the measure of the implication from finish to start.
9) Show that since the limits of the integral of the Dirac delta are undefined, the integral is 1 no matter what the size of the interval, this acts as an indefinite integral so that when one takes the derivative of that integral wrt space, the Dirac delta function results.
10) If we recognize in the gaussian form of the Dirac delta the inner product of the momentum and velocity in the exponential, then m can be seen as the "mass metric". And the considerations of a manifold - configuration space and its dual phase space - enter the picture. This inner product has extensions into Special Relativity and perhaps General Relativity. The Dirac delta is both a metric and a measure. Could it be a measure for QM and a metric for GR? See also 19) and 24) below.
11) Is the Big Bang a representation of the Dirac delta from which all of physics follows? The Big Bang seems to be a singularity with infinite energy at an infinitesimal point in space, like the Dirac delta function.
12) Is the QFT calculation of infinite absolute energy (ignore by only considering differences) caused by the fact that there is a Dirac delta everywhere in spacetime?
13) Is the Dirac delta responsible for the need to renormalize the anomalous infinities observed in QFT?
14) Is the derivation from a conjunction of facts to the disjunction of paths unique? Or is there some other formula (physics) that can be derived from the conjunction of facts? Is it true that in order to get any measure there must be a disjunction (from this conjunction of facts)?
15) If all of physics is derived from the Dirac delta, and since the Dirac delta is a distribution, is it possible to calculate the entropy or information of the Dirac delta? Is this a constant independent of space time? Does this prove a conservation of information for the universe as a whole? All of physics is not derived from the Dirac delta unless particle interactions are available from it as well. So far I have only the free particle in flat space. I need to find interactions - possibly as an alternative interpretation of a potential term.
16) Add comments about the relevance of Godel's Incompleteness Theorem to the effort to derive physics from logic. Nothing in Godel's theorem contradicts the methods used in my derivation.
17) Show that the Standard Model can be achieved as a result of the path integral being invariant wrt other observables. Observables must be a result of the invariance of the formulation with respect to other observables (such as space and time), since in order to measure or observe any quantity, it must not change as you are measuring it. Is the Dirac delta related to variational principles, and if so how?
18) Show how this all is related to Hilbert spaces. This might be done by noting that the Dirac delta is also an inner product of states in ... Hilbert space.
19) Can Quantum Gravity be realized by this formulation? If m is the metric between the configuration space and its dual phase space, then m can be a metric function g(x). If this is the metric of GR, then can it be treated as the field in the action integral instead of (or perhaps in addition to) the kinetic or potential terms? And if this metric is in both the exponential and the leading factor, does this provide a non-perturbative approach of solving the path integral? See also 10) and 24).
20) The Dirac delta function can be represented as a gaussian distribution. And a gaussian distribution represents the bell curve of random processes. Perhaps we should start with totally random processes because this assumes no other more basic structure. We are looking to develop structure from scratch and not assuming any mathematical structure to begin with. Structure (laws of physics) are developed from scratch (total randomness)
21) Perhaps a little more explanation is needed in how one goes from the discrete to the continuous, from the sum to the integral, how the deltas in [14] go to the differentials in [15] as n goes to infinity.
22) For those who like to think in metaphysical terms, perhaps the Dirac delta fulfills the expectation of a God with infinite power who created all things and exists everywhere, making sure all things are just and reasonable. Perhaps the Dirac delta is the "mustard seed" from which the "Kingdom of God" grows with all its "branches".
23) Maybe some connection with the "cosmological argument" can be made since the Dirac delta function is a mathematical representation of material implication which symbolizes cause and effect. So the question as to whether the cosmos can come from something or nothing can be answered by this, and perhaps even issues of ultimate cause (as in cause and effect, a.k.a. implication).
24) Non-relativistic Quantum Mechanics was derived from the definition of the Dirac delta function. Can General Relativity and/or Quantum Gravity be developed from this formulation by using the definition of the Dirac delta function on a generally curved spacetime of a Riemannian manifold? See also 10) and 19) above. The volume for an arbitrary region, R, of dimension n, of a Riemannian manifold is
.
This is the integral of a function which equals 1. And of course we could multiply this by the number c for the integral of a function f(x)=c. But I'm still not sure what the equivalent of a Dirac delta function would look like in a generalized curved space. Does the fact that any Riemannian manifold is locally Euclidean mean that the Dirac delta does not change since the effective region of concern shrinks to zero? Or does the difference in the exponent of the gaussian integral require a special metric for Riemannian spaces?
25) Include the following in the narrative: A sequence of one event leading to another event leading to the next event is an ancient concept. And the history of argument has also shown that other people describe alternative sequences of events to explain how we got from one fact to another. So naturally we end up consider how probable each path of events is. All this argues for the real truth being somehow derived from the disjunction of all these sequences of events. So it should come as no surprise that the laws of physics could be derived from such a formulation.