Document Changes to Physics from Logic

Record of Document Changes
for physics from logic effort.

Last modified this page: 11/27/09
added derivation of least action page
also added intro to functional calculus page
added 34) to 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.

11/25/09 - Added link in main page to newly created Developments page.

11/25/09 - Added page where least action principle is derived (here)

11/25/09 - Added intro to functional calculus page

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. [See 27) and 28) below]

3) I might wish to explain how classical Euler-Lagrange equation of physics is derived from this path integral - that the contribution of far flung paths cancel each other out to leave the path of classical mechanics as the greatest contributor, so that the least action principle is necessary in order for the evaluate the path integral. I wonder if there is any mathematical proof that the path integral cannot be evaluated unless there is zero to functional derivative of the action.

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. This seems to be what is stated by eq[33] if the Dirac delta function represents implication.

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. Godel's incompleteness theorem states that some things can be true in an axiomatic system but not provable by that system. But this is nothing more than saying that some things are implied. For implication includes the possibility of a false premise nevertheless resulting in a true conclusion, in other words, a statement can be true but not proved.

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". Along these lines, it's interesting to note that it is said that "God is one", and we also have that the integral of the delta function is 1.

Einstein once said, "God does not play dice with the universe". This indicates God in scientific terms is considered to be an underlying controlling mechanism that maintains the consistency of all facts so that nothing happens without cause. Another interesting connection is that in ancient Hebrew language in which the bible was written, just about every sentence begins with the letter "vav" which stands for the word "and". It's as though it were written, " and this and that and this too and that, etc." All of which seems to be written to show how God maintains cause and effect; sin results in punishment, virtue results in rewards, etc. So perhaps my derivation here can be considered to be biblical physics.

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. The main reason for understanding the universe and searching for a TOE is to assure ourselves that everything is consistent. So it is satisfying to see that consistency produces the laws of physics.

26) Can the dimensionality of classical spacetime be derived by this method as well? It seems the space dimensions which labels propositions is arbitrary. It is just as easy to label each proposition with a different number in one dimension as it is to label them with different set of numbers in multiple dimensions, as long as each fact is labeled uniquely whether with one number or a set of numbers. Since a proposition is logically equivalent to a conjunction of any number of the same proposition, R=R^R^R..., paths from one fact to another now include going from one dimension to other dimensions for various points to others and even from that one point in one dimension to the same point in other dimensions. I wonder if this transdimensional path integral has a classical path in the 3 space dimensions we're familiar with. I also wonder if this means that particles have equivalent descriptions in other dimensions, for example as strings or other branes. Although I don't know how one would add length to area or volume. Maybe the other dimensions are just set to zero. Or maybe there is a common function that works in any number of dimensions, such as the metric or the curvature, which would work as a Lagrangian in that transdimensional path integral.

Then, perhaps as a dimension is varied in the process of integration, the higher the dimensionality of a contribution the less it contributes to the overall sum because now the same function is described with many dimensions, only one of which is being varied, which means the function is not changing as much and cancels out with other values of the dimension as that integration variable is varied. This would have a tendency to favor the lower dimensions, and perhaps lead to the classical observation of 3 dimensions of space.

Then perhaps the number of classical dimensions would change as the universe of integration gets smaller.

27) Instead of a kinetic energy term, I wonder what kind of restrictions there must be in the kinds of functions used in the exponents so that it remains a gaussian form of the Dirac delta function? For example what if "x" in eq[23], or in eq[28] where a general field function, say Phi , would this be the way to obtain QFT? Are there any symmetries necessary so it remains a Dirac delta? Perhaps this is why gauge symmetries are relevant, since they leave the Lagrangian unchanged. Does this mean that every function that satisfies these conditions has necessarily a physical meaning?

28) If terms are added in the "Lagrangian", this is the same as multiplying exponentials with those terms to the original exponentials. As long as these multiplied exponentials also form a gaussian Dirac delta, this is still multiplying an infinite number of Dirac deltas as in the original formulation, eq[33]. The delta that consists of the exponent with the new term still has an infinite number of them multiplied together with the same limits as with the other integrals... So does this mean we can still aggregate the new term as is done from eq[27] to eq[28] and put the new term under the same integral sign of the action?

29) Perhaps the reason that the exponent in the gaussian delta function is complex is in order to maintain translational invariance. If the exponent is not complex, then translating the function to a different spacetime coordinate will give a different value. But if the exponent is complex, then the modulus will remain 1 for translational transformations. But this would also allow ANY function whatsoever in the exponent, positive or negative, since the modulus remains 1, so I'm not sure about this idea.

30) Taking the derivative of the field function in the "Lagrangian" in the "action integral" in the exponent and then squaring it is a type of functional, a function whose input is a function. And 27) above suggest we really don't know how to restrict the type of functions, , that apply. Perhaps we should think in terms of what kind of functionals are allowed in the exponent. Maybe this formulation may restrict the functional to the traditional Lagrangian of physics.

31) The idea in 27) above suggests that any added terms must be square integrable. The idea in 28) above suggests that added functions or functionals must form a linear space. Do these together form a requirement for a Hilbert space? I wonder what the inner product of this space would necessarily mean. I find a curious familiarity between functionals operating on functions and operators acting on vectors. I wish I could find reference that explores this similarity more completely. Perhaps operator math on vectors in Hilbert space is just a different notation from that of functional calculus on function space with the added restrictions of linearity.

32) It might be desirable to show the usual derivation of the path integral from the identity. And then work backwards and show how the inner product describes an implication from starting state to final state (mention Ariel Caticha's work). Then show that if multiplication is replaced with conjunction and addition is replaced with disjunction as is done with probabilities, then this can be shown to be the disjunction of a conjunction of implications as I derive from only a conjunction (conjunction implies cross implications). It is then a much more easy leap of faith to see how this is implied by a conjunction of all things. Along these lines note how states are labeled with space and time coordinates. Note also how implication is described by the delta function. Pehaps my ideas are not so fringe afterall.

33) Show how the least action principle can be derived by integrating the path integral then taking the first variation. Since the path integral is a Dirac delta function, its integration is equal to 1. Then the variation of this, which is a constant, is equal to zero. But the process of functionally differentiating the path integral forces a functional derivative factor next to the usual exponent. And the only way this can be zero is if the variation of the action is zero. This is a little confusing because the finite difference is also zero for a constant. And the finite difference can be expressed as a taylor series using functional derivatives. This would mean that all higher order variation would also have to be zero. Yet I see higher order variation of the action being used in QFT books.

34) The path integral can be derived by inserting the identity an infinite number of time and then substituting the gaussian form of the Dirac delta for each of the inner products <x_i|x_j>. Typically the gaussian form is justified through a convoluted reasoning using the Schrodinger's equation. But I wonder if one could more directly infer the use of a gaussian form for the <x_i|x_j>. Then the path integral would be derived entirely on principle alone. See 20) and 32) above.

Is there any merit to this reasoning for the use of the gaussian form of the dirac delta function for each of the <x_i|x_j>? Since the path integral includes every product, <x_i|x_j>, with equal weight, then singling out any one for special consideration must have been an arbitrary and thus totally random choice. Does this mean that the inner product <x_i|x_j>, which is a distribution, must be represented by a random process itself, by a gaussian distribution? What else could <x_i|x_j> mean if not the probability of obtaining it? That's how we are accustomed to thinking about <x_i|x_j> in QM. But more generally, is that how we should consider and inner product, as a amplitude or probability of obtaining it? It seems to me that <x_i|x_j> has no meaning apart from the context of all the other vectors in the space that |x_i> and |x_j> are a part of. In fact it seems necessary to consider all the rest of the |x_k> in order to even calculate <x_i|x_j>. For we have that <x_i|x_j> = 0 if i not equal j and <x_i|x_j>=1 if i=j. So it seems that the evaluation of <x_i|x_j> depends on the context of choicing it from all the other possible <x_i|x_j>. Any further thoughts out there? Maybe I'm touching on some mathematics that I'm not familiar with.