Physics derived from logic alone
Originally posted
2/5/08
Document history

 

ABSTRACT

Starting from the premise that all facts are logically consistent with each other, reality can be defined as the logical conjunction of all facts,

R =

This conjunction implies that each fact proves every other, since



The conjunction of all these implications between states can be arranged into a disjunction of every possible "paths" from one state to another, where each path consists of steps from start to finish. For example, one path might be

 

It is possible to view a set of propositions as elements in set theory. The disjunction of paths can be viewed as a union of paths through a sample space. And measure theory prescribes that the measure of a union of subsets is the addition of the measures of the individual subsets. And a measure for implication can be derived from set theory to be the Dirac delta function. So the disjunction of paths can be replaces with an infinite number of integrations of an infinite number of delta functions.

When the measure for implication is taken to be

this can be substituted for the . And this can be manipulated into the Feynman path integral formulation of a free particle. And it is interesting to note that the path integral can be derived from nothing more than the definition of the Dirac delta function.

INTRODUCTION

THE PATHS OF LOGIC

Historically, students of physics are completely mystified by how quantum mechanics is possible. Where does the wave-function come from? How can the imaginary square-root of a probability have anything to do with reality? Some complain that it is completely counter-intuitive and even illogical. But the goal of this article is to prove that quantum mechanics and eventually all of physics can be derived from classical logic.

The most basic requirement we expect of reality is that no fact of reality ever proves the non-existence of another fact of reality. It is never the case that one fact proves false another fact. In symbols this is written as

  [1]

 where is logical negation and is the material implication of propositional logic, q₁ and q₂ are proposition which have values of either true or false. In other words, we insist that each part of reality is at least consistent with every other part of reality. But it should be noted that

[2]

where   is conjunction, or the logical AND function. This can easily be proved with a simple truth table. And if this is true between any and every two facts considered, then reality is a conjunction of all the facts, or 

[3]

where n could be infinite. We naturally expect all the facts, or physical states, to coexist together as a unified whole. Just looking around we see that the chair we are sitting on exists AND the floor holding up the chair exists AND the computer screen we are reading exists AND the room we are in exists AND the walls exist AND the doors of the room exist, etc, etc, etc. We presume this coexistence between facts at every level of existence down to the microscopic level even though it is not observable with our eyes. For if this much were not true, it would be impossible to say anything about reality since it might not hold in the next moment or the next room.

To be more precise, if we let be the set of all facts in reality, then

[4]

And saying that some fact "exists" means that it is true that it is included in the set of all facts in reality. So I think nothing is lost by shortening the notation to

[5]

Where the symbol used here is the logical conjunction of  n  statements.

That isn't to say we know what all the facts are, but whatever laws of physics there are, they must not contradict this conjunction of facts.

And it should be realized that 

[6]

which again can be proved by a simple truth table. And what this means for the whole conjunction of reality is

[7]

This is an implication, not an equivalence. Though the conjunction on the left hand side is false when all q_i are false, all the implications on the right hand side are true even when all the q_i are false. But if it is safe to at least assume that something in the set of all reality actually exists, then it becomes an effective equality. For then there will be an implication somewhere on the right hand side with a true premise but false conclusion, which would make the right hand side false just as the left hand side would be. So since we know it is true that something exists, the above is practically an equivalence.

The next thing to realize is that the conjunction of implications above can be rearranged into a conjunction of "paths", where each path consists of a series of implications that step from one element/fact/state (whatever you wish to call the q_i) to another from a starting state to a final state. Each path would be of the form,

[8]

where the conclusion of one implication is the premise of the next. There would of course have to be paths in the reverse direction for every path in the forward direction since a conjunction implies an implication both ways as in [6] above. I believe this is a fair representation of a path, for it describes a sequence of steps from start to finish. Material implication, , describes the "if...,then..." conditional statements of logic. And what is a path except to say that if you are at this point, then the next point will be here, and if you are at that point, then the next point will be there, and if you are at that point, then the next point will be here, etc. Such a path can be parameterized by a discrete variable t, where successive t enable successive steps.

If we are interested in finding any relationships between the facts, then we have to start with the relationship between two arbitrarily chosen facts. We require that the laws that are developed should be the same for any pair of facts one might choose. So we construct our paths with end points on these two arbitrarily chosen facts. Let the starting fact be labeled q₁, and let our ending fact be labeled q_n. But then any series of steps we use from start to finish can only be arbitrary. So in order to eliminate any preference in how paths are constructed, it is necessary to construct every possible path from start to finish. This means that some parts of each path will be transverse many times since some paths will differ only slightly from other paths. But we can include a series of steps as many times as we like since we know that a proposition is logically equal to the conjunction of that proposition with itself as many times as you like,

  [9]

So far we have

[10]

where  paths_j are defined similarly to [8] above, n  is the number of possible states, and  p  accounts for every possible path from q₁ to q_n in conjunction with every possible reverse path from q_n to q₁.

Yet it is easy to prove by truth table that

[11]

where   represents disjunction, or the logical OR function. Applying this to the above gives us


[12]

Where the    symbol used here is the logical disjunction of  p  terms.

But now, how does math enter into these concerns? How does one assign a mathematical representation to these logical statements and connectives in order to obtain the path integral formulation of physics? 

Propositional logic, predicate logic, set theory, and probability theory can all be described as different features of a sample space. A sample space is place where all the possible elements to be considered exist; each element is a sample in the space. It's possible to impose a coordinate grid over the sample space and specify particular coordinates for each element or sample.  Set theory explicitly lists the samples that belong to a region in the space where the samples exist. Predicate logic considers whether there may exist some, none, or all samples in a region. Propositional logic is only concerned with the overlap of the regions where the sets are defined. And probabilities compare the relative number of samples in various regions of a sample space.. (Details here).

Individual coordinates, x_i = (x,y,z,...) can be assigned to each of the propositions of reality, the q_i. This turns the set of all propositions into a collection of points on a graph with a coordinate system imposed. Assigning coordinates to propositions is not usually done explicitly. What is usually done is to express the logical AND function in terms of the intersection of sets and to express the logical OR function in terms of the union of sets. This is done, for example, to explain DeMorgan's Theorems. And coordinates are assigned to the elements of sets in order to establish a metric between points, and the measure of sets. This is typically done in the study of Topology and Measure Theory. So we can consider each proposition, q_i , to be an element of a set whose only element is q_i . Then a measure can be established between the q_i  in the same way that a measure is established for any other set. 

Probabilities are just a special kind of measure. Measure theory is used to assign a size to a set. The more elements a set has, the greater the size of that set. The more elements a set has, the greater the measure it has. And larger sets are made of the union of smaller subsets. So the measure of a union of disjoint subsets is the sum of the measure of the individual subsets. etc. And a probability is a measure that is normalized so that the measure of the whole space is 1. In a sample space, a probability is assigned to each subset. The probability of obtaining two independent events (non-intersecting regions) is equal to the probability of getting one event times the probability of also getting the second event. And the probability of getting either one event or the other is the probability of getting one event plus the probability of getting the second event. Conjunction of events results in multiplying the probabilities, and a disjunction results in adding the probabilities. So we can replace   with     and    with  ,  where   is some measure (maybe the probability) of  . Then [12] becomes,

[13]

Now we can defined every possible path through the samples/elements/facts/states by creating a path and incrementing in a sum only one step in the path at a time as shown below in [14].

[14]

where is a yet unidentified measure.

Here,

[15]

If coordinates, x_i, are assigned to the individual statements, q_i , then all the become,

[16]

Then as is typically done in calculus, as n becomes infinite, the become differentials, dx_i , so that

[17]

And the discrete sums become integrals so that [14] above becomes

[18]

 

Now if the measure of implication can be equated to 

[19]

then [18] becomes the Feynman path integral of physics and this is what will be shown below.

 

 

THE MEASURE OF IMPLICATION

To find a measure for implication consider how implication is expressed in set theory. If there exists a set, then that certainly implies that its elements exist; for a set is made up of its elements. So if the set B is a subset of A, symbolized as , then the set B consists only of the elements of A, and so the existence of A implies the existence of B, . We can symbolize the fact that B is a subset of A as . This says that A is a superset of B, which is the same as saying that B is a subset of A. Fig 1 below shows that B is a subset of A in terms of Venn diagrams. Fig 2 below shows that when B is not a subset of A, then A does not imply B.

                             

Typically sets are specified by listing their elements in parentheses. But sets can also be represented graphically as points in a region as shown in Fig 3 below. Here elements are seen as points labeled as 1, 2, 3, etc. Any set can be represented in this way, with the individual elements of the original set identified with differently numbered points in a region of space. For example, the original set A may be A={a,b,c,d,e,f}. Then it is true that (aA)(bA)(cA)(dA)(eA)(fA). And it is true that A={a}{b}{c}{d}{e}{f}. And we can map this set to a series of numbers A={0,2,3,6,8,9} as long as we maintain the correlation between a=0, b=2, c=3, d=6, e=8, and f=9. Drawing a circle around these points is just an alternative to parentheses enclosing a list. 

                   

Then a coordinate system can be imposed on this space as shown in Fig 4 above. Each element is assigned numbers to designate its place in the coordinate system. For example, x₄ would be assigned the (x,y) coordinates, (1,3). Then the set A is represented as the area inside the ellipse as shown. If any x is in the area A, , then A implies x, . There is not an element, x, at every pair of coordinates. But when the coordinates do match places where elements are inside A, then we want to count them as part of the set A.

The Kronecker delta function can be used as a means to count an element as part of a set. It is the most basic measure of the size of a set. The Kronecker delta function is defined as   if     ,and     if  . So if we sum the Kronecker delta function over all the coordinates in the space, it will add up to 1 if we also count the place where the element is. Or in math symbols,

[20]

if the range of i includes 0 in this case. We can include a Kronecker delta for every element in the set A of Fig 4. The summation would then be

[21]

and is equal to the number of elements in the set A, in this case 6, which give us a measure of the size of A.

As the number of elements becomes infinite, there becomes a continuous distribution of elements, perhaps a higher density of elements in some areas more than other areas. And in the continuous case, the summation in [20] becomes an integral over the area for which the set is defined. And the measure changes from the Kronecker delta function to the Dirac delta function, , so that

[22]

This essential characteristic of the Dirac delta function needs to be preserved no matter how the region A is defined so that the element at x₀ is counted only once. The dA is a differential piece of area which shrinks to zero. And since the region A could become very, very small, the function will have to tend towards infinity at x₀ so that when multiplied by dA the integral remains 1 for very small regions A. This means that   when   , and     if   .

Implication is shown above in set theory as requiring the conclusion to be the subset of the premise. But in the introduction, the development of the path integral relied on all the q_i being on an equal basis. One was not the subset of another. All of them were propositions that were sometimes a premise and sometimes a conclusion of implication. So we want to show how implication can relate one element of a set to another. This can be done by shrinking to a point the region that defines a set; so that the set becomes synonymous with the proposition which is the only element of that set.

With coordinates imposed, a set is seen as an area in the space. And an element is seen as an element of a set if the element's coordinates lie inside the area that defines a set. For example Fig 5 below shows that x₀ is an element of A, therefore, A implies x₀. And x₀ remains an element of the set no matter how small the region of the set becomes as seen in Fig 6.

                             

In Fig 6 above, the region defining the set shrinks from A to B and then to C. And in Fig 7 the region A shrinks to the point x₀. Here A still implies x₀ even though A has shrunk to the point x₀, so the delta function is infinite. In Fig 8 A has shrunk to x₁ which does not include x₀; so the delta function is zero. The point is that the Dirac delta function represents implication between elements.

                                                                  

Now there are many different functions that can mathematically represent the Dirac delta function. The one chosen here is

[23]

It has the property that as the constant  approaches zero, the delta function becomes infinite in such a way that the integral is one. All mathematical representations of the Dirac delta function have this parameter that approaches zero. The integral of [22] with equation [23] as its integrand is a little tricky to prove and is done in most books in quantum mechanics that cover the path integral formulation.

It doesn't matter what the constant is; it can just as easily be an imaginary number. Let

[24]

At this point m and h-bar are arbitrary constants here. And this is not an attempt to covertly introduce physics. I only use the labels m and h-bar because with the constant labeled as such they can serve the same function in my derivation of the path integral as mass and Planck's constant serve in the Feynman path integral of physics. At this point in the derivation, however, no special status should be assigned to these constants. Without loss of generality I could have just as easily scaled m to 2 and h-bar to 1. Then , and there would be no objection that physics is introduced. And the quantum mechanics would still result as surely as setting m to 2 and h-bar to 1 in Schrodinger's equation. But if we let equal the above as in [24], then [23] becomes

[25]

which is equal to 

[26]

When equation [26] is substituted for in [18], then [18] becomes

[27]

where the limits of the integrals are typically from minus infinity to plus infinity, and the delta-t are allowed to approach 0 as in the limits of equation [26]. When the terms are gathered, [27] becomes

[28]

Notice that the exponential term looks like the the Action integral for the kinetic energy of a particle. Here m is only a constant used as a conversion factor to cancel out the velocity squared term. And h-bar is a constant used to cancel out the units of the integral so that the exponent is dimensionless and can be evaluated. Equation [28] can be recognized as the Feynman path integral for the wave function of a free particle. The limits of the delta-t approaching zero is implied by the notation of dt and delta-t. 

It remains to be seen, but it is expected that the still missing potential energy term of the action integral will come from a non-constant distribution in the function that describes the density of elements in the sample space. And it is expected that quantum field theory will come from the density function changing with time.

 

Now it is possible to derive the path integral from the definition of the Dirac delta function alone. And this in itself provokes some interesting philosophical contemplations.

The Dirac delta is defined as

[29]

And for any function f(x), we get

[30]

But if f(x) is the Dirac delta function itself, then

[31]

This is a transitive property that can be applied as many time as desired. Applying it again gives

[32]

And applying it an infinite number of times gives

[33]

And then if we let

[34]

with

[35]

then rearranging terms as before

[36]

which equals

[37]

we see that

[38]

is equal to

[39]

And this is equal to

[40]

as before, where delta-t approaches zero, and where n approaches infinity.

This derivation of the path integral from the Dirac delta function is provide on a separate webpage for anyone wishing to link only to that derivation. That webpage is provided here.