Quantum Theory Derived from Logic

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  ABSTRACT

After a very brief introduction to logic, I show how a type of path integral can be constructed in terms of propositional logic. This logic is then transformed into the Feynman Path Integral of quantum mechanics using general techniques.

Reality can be defined as the conjunction of all the facts we observe. Even space itself consists of a conjunction of different points that we label with different coordinates. A conjunction of points, however, means that every point in fact logically implies every other point. And we can use the conjunction of all points to show that an implication between any two points equates to the disjunction of every possible sequence of implications from one point to the another. Each sequence of implications is a path consisting of a conjunction of implications from the first point to the second, and from the second point to the third, and from the third to the forth, etc. I then show that an implication can be represented mathematically by a Dirac delta function, and that disjunction is represented by addition, and that conjunction is represented by multiplication. When the exponential gaussian function is used to represent the delta function for each implication, the conjunction of implications for any one path become the multiplication of exponential functions. The exponents then add up to form an Action integral, and the disjunction of every possible path forms the Feynman path integral of quantum mechanics. This is 1st quantization. The wave function is the mathematical representation of logical implication. This process can be iterated to give us the quantum field theory of 2nd quantization. And the process can be iterated again to even give us 3rd quantization if needed. I also show where the Born Rule comes from to give us probabilities from the square modulus of the wave function. And finally, I give some reason to expect that these iterations prescribe that the complex numbers iterate to quaternions and octonions, which are believed to be responsible for the U(1)SU(2)SU(3) symmetry of the Standard Model.

 

Section 1
INTRODUCTION

Historically, quantum mechanics was developed in a rather ad-hoc manner, using trial and error to find some mathematics that eventually proved useful in making predictions. But the ultimate reasons why nature operates according to the equations of quantum mechanics has remained elusive. And some students of physics are mystified to the point of distraction by quantum mechanics because there does not seem to be any underlying principle for it that they can understand. 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 counter-intuitive and even illogical. But the goal of this article is to prove that quantum mechanics can be derived from classical logic without any physical assumptions.

Those most interested in foundational issues are those exposed to the subject for the first time. It's usually easier to accept more complicated implications of a theory when the basic premises of it are well understood. Therefore, in order to broaden the audience, I include about a page worth of paragraphs briefly describing the basic introductory definitions in logic. And I include about a page worth of introduction to the integration process of calculus. The fundamentals in a subject should be relatively easy, so my intention is to keep this article under a sophomore college level. It is hoped that the ease of this material will be appreciated. The web pages I link to should contain a bibliography for those interested in further reading. Advanced readers can skip to the next section if they are familiar with the symbols used in logic.

Anyone can make claims about any subject they like, but that only brings up questions as to what evidence there is to support those claims and what those claims imply. And some may like to think they are being reasonable in what they believe. But how can we know that the conclusions they reach are correctly derived in a reasonable way? Logic is the study of correct argumentation. Given facts in relation to each other, logic is a tool to help us determine what other truths these facts equate to or imply. In this section I briefly touch on three topics in logic: Propositional Logic, Set theory, and Predicate logic.

Propositional logic studies how the truth or falsity of statements effect the truth or falsity of other statements. Propositions are the same thing as statements or facts or claims which can either be true or they can be false, but they cannot be neither true nor false, and they cannot be both true and false at the same time. Propositional logic does not consider what the statements are about; it does not consider whether the statements are about abstract concepts such as math, or about physical facts or about feeling, emotions, or beauty. All Propositional logic does is label different statements with different letters such as a, b, c, etc. and treats them as variables whose values can be either true or false. I will use T for true and F for False. Compound statements can be constructed from simple statements using connectives such as AND and OR and IMPLIES and NOT. And the truth of the compound statement depends on how the simple statements are connected. Symbols are used for these connectives. I will use for AND (=conjunction), for OR (=disjunction), and for IMPLIES (=material implication), and for NOT (=negation). In Table 1 below is a truth table that shows the effect of these connectives on two statements, p and q.

 

1 2 3 4 5
p q p pq pq pq
F F T F F T
F T T F T T
T F F F T F
T T F T T T

Column 1 in the table lists every possible combination of T and F that p and q can have. Column 2 shows that the operation of negation (NOT) has the effect of reversing the truth-value of p. If p is T, then p is F, and visa versa. Column 3 shows that the statement "p AND q" is T only when both p is T and q is T. Column 4 shows that "p OR q" is T whenever either p is T or q is T or when both are T. Material implication is the IF, THEN function of logic. If p implies q, this means if p is true, then q is true. To say that p implies q is the same thing as saying if p then q, or p proves q, or p therefore q, or p results in q, or p causes q, etc. Here the first operand, p, is called the premise, and the second operand, q, is called the consequence. Column 5 shows the relationship of material implication. It is true that p proves q for any truth-values of p and q except when p is T but q is F. The consequences might still be true regardless of the premises on which they're based, but you can not have premises that are true without consequences; that would mean there is not an implication between them. Some things to note are that conjunction (AND) is commutative, which means you can reverse the order of the operands, p and q, so you get pq=qp. It is also true that disjunction (OR) is commutative. But implication () is not commutative, pq is not equal to qp.

Set Theory constructs lists of objects called elements. For example, the set S whose elements are objects labeled a and b and c and d is written as S={a,b,c,d}, where aS symbolizes that a is an element of the set S. Then Set theory examines how differing sets can be combined. You can combine sets by considering the union of the two sets, or the intersection between them or the complement of a set. For example, if you have only two sets, A={a,b,c,d,e,f} and B={d,e,f,g,h}, then the union between them is AB={a,b,c,d,e,f,g,h}. The intersection of the two sets is AB={d,e,f}. And if these two sets contain all the possible elements in the universe of our discourse, then the complement of B is B={a,b,c}. It is also possible to have sets which are subsets of other sets. For example, the set C={a,c,e} is a subset of A, symbolized as CA or as AC which says A is a superset of C, same thing as saying C is a subset of A.

Many times propositions can be described as objects with a particular property. In Predicate logic, if a specific object labeled q has the property labeled P, then Pq is the common notation for saying it is true that q has the property P. The extension of a predicate, P, here labeled P, is the set of all those specific objects which have the property P. In other words, P={q1,q2,q3,q4}, where it is true that Pq1 and Pq2 and Pq3 and Pq4. The expansion of the predicate P is a proposition, here labeled P, which is the conjunction of statements of all those objects that have the property P. In symbols, P = Pq1^Pq2^Pq3^Pq4. If it is understood that q1, q2, q3, and q4 are each propositions such that q1=Pq1, q2=Pq2, q3=Pq3, and q4=Pq4, then we can shorten the notation to P = q1^q2^q3^q4. And we can consider the consistency between all the statements in the set.

 

Section 2

THE PATHS OF LOGIC

Consistency among statements in a theory means that nothing in the theory will prove that the statement is both true and false. And this means, of course, that no statement in the theory will prove itself false. So if we are given a set of statements that are asserted to be true, then consistency requires that no statement in the set will ever prove false any other statement in that set. Or in symbols, if q1 and q2 are asserted to coexist as true statements of the theory, then

(q1q2)[1]

But it should be noted that

(q1q2) = q1^q2 [2]

This can easily be proved with a simple truth table. And if this is true between any two statements in the set, then a consistent set can be seen as the conjunction of all its statements:

q1^q2^q3^...^qn = qi[3]

where all the qi belong to the same set, and where n could be infinite, and where the symbol used here is the logical conjunction of  n statements.

To apply these ideas to nature, we can say that reality consists of all the objects within it. We can use the letter U to symbolize the property of belonging to the universe, and symbols such as q1, q2,q3, q4, etc. to represent various kinds of objects. We write Uq1, Uq2, Uq3, etc. to represent the statements that those objects have the property of actually existing in the universe. We can abbreviate those statements as q1, q2, q3, etc., which means q1=Uq1, q2=Uq2, q3=Uq3, etc., and they describe facts in the universe in terms of propositions that can be considered either true or false. The extension of the property U would be the set U={q1, q2, q3,...}, and the expansion of U would be the proposition U=q1^q2^q3^... And we would say that the universe consists of all the facts in reality coexisting in conjunction with each other.

It may be that some of the facts, qi, might be broken down into a conjunction of even more propositions which represent even smaller objects that have differing properties. And it may be that still other facts, qj, may share some of these differing properties in common. But it's still clear that the extension of these differing properties are subsets of the universal set, U. And the expansion of these properties only contribute propositions that exist in conjunction with everything else. So we can ultimately describe the universe as consisting of a conjunction of all the facts that describe all the parts of the universe. We use propositions to describe individual facts in reality all the time. For we describe situations in nature with propositions - this physical situation has this or that property, it's made of these parts, it's located at this place at this time. And we often argue about whether a statement about reality is actually true. We use the word "true" for those propositions that do describe what's real and "false" for those propositions that do not describe what's real. Larger physical systems are described with smaller physical subsystems. And we strive to find the smallest constituents of reality which will themselves always end up being described with one statement or another whose truth-value we argue about until we are completely satisfied.

So nature can be considered a consistent set of statements. And we expect that no fact in reality will ever contradict any other fact in reality. 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 AND the atoms they are made of exist, etc, etc, ad infinitum. We presume this coexistence between facts at every level of existence down to the most microscopic level even though it is not observable with our eyes. For if this much were not true, I don't suppose we would be able to describe anything in reality. So in the most general sense, I think it's fair to describe reality at the smallest possible level as consisting of a consistent set of propositions. That isn't to say we know what all the facts are or what properties they have, but whatever laws of physics there are, they must not contradict this conjunction of facts.

Continuing from equation [3], it should be realized that

q1^q2 (q1 q2)^(q2 q1)[4]

which again can be proved by a simple truth table. And this would be the case between any two facts in the universe. So what this means for the whole conjunction of reality is

qi (qi qj) = (qi qj) [5]

This conjunction would include factors such as (qi qi) which are true by the definition of material implication. And such factors do not change the conjunction since p=pT for any proposition p. You can always factor in a truth in a conjunction.

The conjunction on the left hand side (LHS) of equation [5] only implies the right hand side (RHS); it is not an equivalence. When all the qi are T, the LHS equals the RHS, and both sides are T. If there is a mixture of T and F for the qi, where some of the qi are T, but other qi are F, then the LHS will be F since there is an F in a conjunction. On the RHS for this same combination of T and F for the qi, there will be a factor of the form (F T)=T, but when that same factor is reversed elsewhere in that conjunction, there will be a factor of the form (T F)=F, making the whole conjunction on the RHS false just as it was on the LHS for that mixed combinations of T and F. The only difference, therefore, between the LHS and the RHS is when all the qi are F. Though the conjunction on the LHS is false when all qi are false, all the implications on the RHS are T when all the qi are F. This is because (F F)=T is a true statement by definition of implication. But if it is safe to at least assume that something in the set is true, then equation [5] becomes an effective equality. For then there will be an implication somewhere on the RHS of the form (T F)=F, which would make the conjunction of the RHS false just as the LHS would be. And in the case of reality, it's probably safe to assume that there must be something that truly exists. For we can at least say that the universe exists.

So how are paths constructed? Consider the following:

(qs qf) = (qs qf) [(qs q1)(q1 qf)]

where qs is the starting state, qf is the final state, and q1 is an intermediate state. The last term, (qs q1)(q1 qf), indicates the conjunction between (qs q1) and (q1 qf). This conjunction forms a two step "path" between qs and qf . I call it a path because it has an intermediate step of q1 between qs and qf. There's no value of q1, T or F, that can negate the equality. If the LHS is false, this only happens when qs=T and qf =F, and then if q1=T, that would make the factor (q1 qf)=(T F)=F making the conjunction term false, which in disjunction with the first term, (qs qf)=F, makes the RHS to be F, just as the LHS would be. Or, if q1=F, that would make the factor (qs q1)=(T F)=F, again making the conjunction term and thus the RHS false, just as the LHS is still false.

Now consider when the LHS is true, (qsqf)=T, then we also have that true term on the RHS already ORed in so that it does not matter what the last path term is since T=T (qsq1)(q1qf).

More intermediate states can be used to OR in more paths to get

(qs qf) = (qs qj)(qj qf)[6]

This is n parallel paths of two steps each. The index j cycles through all n propositions in the universal set so that qj acts like a variable taking on the value of various propositions. And so j will eventually take on the value of  s, and there will be a term on the RHS of equation [6] of the form; (qsqs)(qsqf)=(qsqf) that will be ORed in with the rest of the paths. And for the same reasons as stated before, there is no values of the rest of the qj's that can make any of these paths negate the equality. Note, however, that j will eventually also take on the value of f, and this will give us another term of the form (qsqf)(qfqf)=(qsqf) on the RHS. But this is totally acceptable in logic since p=pp for any proposition p. This may work in logic, but we'll have to be careful with the range of the index j when we get to the math. Note that since qj is the only variable in equation [6]. The factors (qsqj) and (qjqf) can be thought of as functions of the single variable qj, with qs and qf being held constant. Then equation [6] can be thought of as a type of mathematical expansion in terms of other functions.

This procedure can be applied again, and intermediate states can be inserted between, say qi1 and qf, to get

(qs qf) = (qs qi1)(qi1 qi2)(qi2 qf)

And applying the procedure m times,

(qs qf) = (qs qi1)(qi1 qi2)...(qim qf)[7]

In this case, factors like (qi1qi2) are functions of two variables, since both qi1 and qi2 act like variables which cycle through various propositions. If m=n, so that the i's range through every possible state in the universal set, then equation [7] is the combination of every possible path through the universal set. Already we can see this is setting us up to derive Feynman's path integral.

It might be interesting to consider that equation [7] could have been anticipated long ago. For it seems to represent every disagreement we have. We might agree about the state of affairs at some point in the past, and we might agree about some other point after that. But we might disagree about what sequence of events got us from the first point to the second point. One party proposes one sequence of event. The other party proposes a different sequence of events. And we are left considering the alternative sequences of events. For example, a man on trial for murder. Both parties agree that the victim was alive at some point and then was found dead at another point. Prosecution will argue that a series of events happened to prove that the accused committed the crime. Whereas, the Defense will argue a different sequence of events in which the man is innocent. The jury ends up considering alternative sequences of events.

So putting together what we have so far, we get

qi (qi qj)[8]

           = ((qi qi1)(qi1 qi2) ...(qim qj))

Generally we can't be expected to know what all the facts, qi, are. And we certainly cannot measure every possible thing in the universe. But typically we want to know how strong the relationship is between two facts called cause and effect, qs and qf. And besides, we need to know how to solve equation [7] before we can even consider equation [8].

A possible term in the disjunction of equation [7] is

(qs q3)(q3 q7)(q7 q21)(q21 q5)(q5 qf)

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 propositional 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.

Section 3

THE MEASURE OF IMPLICATION

Expressions of Propositional logic use connectives like AND and OR and NOT that operate on statements whose values are true or false. But the laws of physics are expressed in terms of mathematical operations that operate on variables that have numerical values. So if we are going to go from logic to math, we need a way to assign mathematical operations to logical connectives and to give numerical value to propositions, implication, and to paths.

In propositional logic, statements are either T or F. In math we must be able to count objects from 0 to 1 to 2 to 3, etc. The ability to count from 0 to 1 is the most basic operation of math; counting to higher numbers is just an iteration of this basic ability. So when mapping propositional logic to mathematics, we need to know how to go from F and T to 0 and 1. How does a proposition get mapped to a number? It's by counting it.

In Predicate logic, we represented a proposition as an object with a particular property q=Pq, where Pq is a statement that is true when q has the property P. And this meant that q P, where the set P was the extension of the property P. I suppose it's always possible to assign an proposition to any object that has a certain property and belongs to a certain set. For we can at least associate a true proposition q to an object q that has the property T of being a true proposition, which would mean that it belongs to the set T of all true propositions. Or, we could stipulate that we can always create a property Q with extension Q such that q=(q Q) = Qq, perhaps even Q has only one element, Q={q}, so that its property Q only assigns its one object q to the proposition q.

The most basic nature of counting is to scan the area of interest, and if you encounter an object of concern then you count one. In other words, if x is the object of concern and A is the area of interest, then we count 1 if x A; otherwise we count 0 if x A. And we usually limit our area of interest to a subset of the universe. We don't scan the sky for stones on the ground. So we need a function to accomplish the operation of counting 1 for set membership. The Dirac measure accomplishes this task. See here for the Dirac measure. The Dirac measure is denoted x(A) and is defined such that,

x(A) = 1 if x A
0   if x A

If the proposition x = (x A) is true, then the Dirac measure maps x to the value of 1. And if x = (x A) is a false proposition, then the Dirac measure maps x to the value of 0. So the Dirac measure maps T to 1 and F to 0. Notice that the delta symbol used here, , is in italic bold font, indicating that its inputs are elements and sets. Later when the Dirac delta function of coordinates is used, , the italic not bold font will be used to indicate that its input is coordinate numbers, not elements or sets.

When we scan the area A for the object x, we may notice that there are other members of A which are not the object x. We may list all the objects in A and get A={a,b,c,d,x,e,f,g,h}, for example. And scanning a region A may be as simple as taking notice of each of the elements in the list in turn until you encounter an x, or not. But if the set has been defined, then we can take the expansion of it to get, A = a^b^c^d^x^e^f ^g^h, where a = (a A), b = (b A), etc. And we can consider the truth and falsity of A and x independently from one another and ask how the truth-value of each are related.

Since true propositions are defined in terms of set membership, (a is true if (a A), for example), an expansion of a set is always true because each proposition in the conjunction is based on an element which is guaranteed to be in the set. Remember, here we are considering only one set, typically A, or in physics U, which is being considered constant, and propositions are being defined in terms of membership in that set. This allowed us to use the Dirac measure to map propositions to numbers. So A is always true, and the only thing to consider is whether an element is a member of A or not.

If x = (x A) is true, then note that A x. For if a conjunction is true, then so are all of its statements. But notice that x does not prove A, xA; for even though A is always true, x need not be an element of A, so that x is false. Thus, if x = (x A) is true, then A x is true. And if x = (x A) is false, then A x is false. This means that (x A)=(A x). So if x(A) represents the set inclusion x A, then x(A) also represents the implication A x. The Dirac measure is a measure on implication in this special case where (x A)=(A x). But is it possible that the Dirac measure could be a measure on implication in the general case between two propositions, say x and y for which {x,y}A?

In the notation of x(A) notice that x is an element and A is a set and not an element. Yet, we need a math operation for the implication between one element and another element. For paths were constructed above with the material implication between propositions, where each proposition relates to an element in U. So we need to manipulate x(A) to be more of the form x({y}), which would mathematically represent more closely the implication between two propositions.

To accomplish this, note that the set A in the notation of x(A) is a set whose number of elements is not specified. We expect (x (A)=(A x) to be true no matter the size of A as long as x remains an element of A. So we should still have x(A) representing implication even if A is shrunk down to the size of an element. Let A shrink down to a variable element, call it y, then we have A={y}, where y is a place holder for any one of the elements in the universe of discourse, then we can write

x(A) = yx = 1 if y = x, where A={y}.
0 if y x

because we know that if y=x, then x{y} would equal x{x}, which is an inherently true statement that gets mapped to 1. Otherwise, if y x, then y{x}, where {x} is every element other than x. And we know that x{x} is an inherently false statement that gets mapped to 0.

Previously when we considered A={a,b,c,d,x,e,f,g,h}, the expansion was A= a^b^^d^x^e^f ^g^h. But now, when we think of A={y}, the expansion is A = y. So Ax becomes y x, and x({y}) is a mathematical representation of y x, the implication of one proposition with another, which is what is needed for the conjunction of implications in a path. I labeled x(A) above as yx to remind us that A={y}. I call yx the point-to-point Dirac measure. It's not the Kronecker delta function because the input for the function here is still elements, not numbers.

Of course, for larger sets with more elements, these can be equated to the union of sets, each consisting of one element of the larger set. For example, if A={a,b,c,d,x,e,f,g,h}, then A={a}{b}{c}{d}{x}{e}{f}{g}{h}. Then it is true that

x(A)    =   yx    = 1     if     x A
0     if     x A

The y A under the symbol means that there is a numerical sum of the yx terms, where each term is evaluated with a different value of y which cycles one at a time through every element in A. Eventually y will equal x, if x A, and then yx will equal 1 there. All the rest of the terms will be 0. So the total sum will be 1. Note that y is the only thing varying, and since x is being held constant, yx can be treated as a function of the one variable element y. If x were allowed to vary as well, then in that case, yx would have to be seen as a function of two variable elements.

Now we're in a position to develop a mathematical representation of conjunction and disjunction, implication and paths. The rest of this article is basically only concerned with the algebra.

Notice that,

(qi qj)  = T ,   if   qj A,   or 1 j n[9]
F ,   if   qj A,   or j < 1 or n < j ,

where qi is a variable proposition based on qi which is a variable element that cycles through every element of A as i cycles from 1 to n. This is because as i cycles through all the qi in A, then eventually i will equal j, if qjA, and there will be a term of the form (qj qj) which is identically T since qj{qj}. This one term being T will be ORed in with the rest and makes the whole disjunction T. But suppose qjA, then qi{qj} will never be true, qi will never equal qj, (qi qj) will always be F, making every term and thus the whole disjunction F. And again, since qi is the only thing that varies in equation [9], we can look on (qi qj) as a function of one variable, qi, in this case, since qj is held constant here. It will be interesting to compare equation [9] with its mathematical counterpart that will be a useful completeness relation.

When we start using the form of implication between propositions, (qi qj), the truth of it is determined by whether qj{qi} and not by whether qjA. In other words, qj = (qj{qi}), which is inherently T when i=j, and F otherwise. So we lose track of whether any of the qj are an element of A or not, and we cannot say it is true or false that qjA. Therefore, when we map (qi qj) to the point-to-point Dirac measure, qiqj, it essentially becomes a function of just the single index variable i. If i were plotted on a number line, then i could be considered to be a coordinate that keeps track of where the elements are in that coordinate system. So qiqj becomes a function of coordinates, (i,j), where j is held constant. And when i are discrete whole numbers, (i,j) is usually labeled as a Kronecker delta, ij. Note the use of the italic not bold font for the delta to indicate that it is a function of coordinates or indices, not elements. So with T mapped to 1 and F mapped to 0 and (qi qj) mapped to ij, equation [9] can be mapped to

ij  = 1   if   1 j n[10]
0   if   if not

where ij is seen here to be a function of the one variable i, with j held constant.

Section 4

THE MATH OF IMPLICATIONS

In equation [10] above I just assumed that disjunction, , is mathematically represented by addition. This is mostly out of intuition to make equation [10] appear to be the mathematical representation of equation [9]. But is there any way of proving this? And what math operation would we use for conjunction, ? There may be other logical operations that can be mapped to math operations such as negation. But in order to arrive at the path integral all we need is a map for conjunction, disjunction and implication to map equation [7] to Feynman's path integral.

Logic has an algebra of ANDs and ORs that operate on proposition with values of T or F. But math has an algebra of plus and minus and multiplication and division that operates on variables with numeric value. So when we consider how to map the algebra of logic to the algebra of math, logic operators need to be mapped to mathematical operators, and logic variables need to be mapped to mathematical variables in order to preserve the algebra. Otherwise, if a math variable did not change with a logic variable, then you could not invent any rules to correlate any expression in logic to some expression in math. And we also need operators that are commutative in logic to map to commutative operators in math to maintain the equality in both logic and math if the variable values should be interchanged. Since disjunction, , and conjunction, , are commutative, we will need to use a commutative math operation for each. And since we are considering basic counting operations, the obvious choices are +, -, , and /. But - and / are not commutative, since (a-b)(b-a) and since a/bb/a. So we are left with + and .

To find a math operation for disjunction, , we can consider the disjunction of equation [9] with n=2,

(q1 qj) (q2 qj)   = T ,        if  qj A , or 1 j 2
F ,        if not

The map from logic to math that we are sure of so far is T 1,  F 0, and (qi qj) ij. And let's map to some as yet unknown math operation, call it for now. Then the last equation above gets mapped to,

1j 2j  = 1     if   1 j 2
0     if not

Since disjunction is commutative, (q1 qj)(q2 qj) = (q2 qj)(q1 qj), we need to have to be commutative as well, 1j 2j = 2j 1j. If j < 1 or 2 < j, then 1j = 2j = 0, and 1j 2j = 0. But if 1 j 2, then either 1j = 1 or 2j = 1, but it's never the case that both 1j and 2j are 1 at the same time. So we have the following table,

Table 1

(q1qj)

(q2qj)

1j  

2j  

(q1qj)(q2qj)

1j 2j

condition

F F 0 0 F  0   j < 1 or 2 < j
F T 0 1 T  1 j = 2
T F 1 0 T  1 j = 1

And the math operation that gives 00=0, 01=1, and 10=1 would be addition, +, as originally suspected. It cannot be multiplication since there is a 0 for every condition, and anything times 0 is 0, and we'd never have a 1 as needed. So the mathematical map for equation [9] is

{   (qi qj) = T,    eq[9]  }      {   ij = 1,   eq[10]  }.

Even if n were very large in equation [10], there would still only be one term that is 1; the rest would be 0, so that the total would always be 1. But let's look again at the situation. Equation [9] describes the disjunction of every possible alternative, only one of which turns out to be the case. And Kronecker deltas in equation [10] assigned a value of 1 to only one choice with the rest being 0. This can be viewed as the most basic of probability distributions with only one of the alternatives being possible. But there is no reason not to replace the simplest distribution, ij, with a more complicated probability distribution, p(i), that can assign a nonzero number to each of the alternatives. The rules of commutation from logic to math still apply, along with the rule that alternatives that are assigned 0 cannot make the whole mapping 0. So the disjunction of alternatives still maps to addition with the added requirement that the probabilities are assigned so that the addition is always 1. This is the Sum rule for alternative probabilities. We will see the Product rule for a sequence of events emerge shortly.

Next, let's find a math operator for conjunction, . Consider equation [6] above with n = 1,

(qs qf) = (qs q1)(q1 qf)

where each of the numbers s and f may or may not be 1. If this equation were mapped to mathematical terms, we would get

sf  = s1 1f

where is the as yet unknown math operation for conjunction. Since we are allowed to consider arbitrary values of s and f, the last equation shows that sf is 1only when s=f=1. And we have the following table for the math operation for conjunction,

Table 2

(qsq1)

(q1qf)

s1  

1f  

(qsq1)(q1qf)

sf = s1 1f

condition

F F 0 0 F           0 s1, f 1
F T 0 1 F           0 s 1, f =1
T F 1 0 F           0 s=1, f 1
T T 1 1          1 s=1, f =1

From Table 2, the math operator, , must fulfill the requirement that 00 = 0, 01 = 0, 10 = 0, and 11 = 1. Clearly, must be multiplication, , so that we have the map,

[11]

{   (qs qf) = (qs qj)(qj qf)     [6]  }      {  sf sjjf    }

And as before, the Kronecker delta can be replaced with a more general probability distribution. Replace the deltas with the probability distribution p( sf ) which means the probability of going from state s to another state f. And suppose n=1 in equation [11]. Then equation [11] tells us that p( sf ) = p( sj ) p( jf ), or the probability of a series of events is the multiplication of the probabilities of each step in the sequence. This is the Product rule for a series of possibilities.

Section 5

INTEGRAL CALCULUS

This section is a brief introduction to the definition of integration as studied in Calculus. If you are already familiar with calculus, you can skip to the next section.

If we were to graph the Kronecker delta function, ij, the value of i would be plotted along the horizontal axis and the numeric value of ij would be plotted on the vertical axis as shown in Fig 1 below. Here,  j = 4, and is held constant. Then the graph shows that when i = j = 4, then ij = 1, but is 0 for every other value of i.

And a more general version of a discrete probability distribution might look like that in Fig 2 below, where the probability of the i th alternative is labeled p(i).

Notice that all the points are well below 1 since we need the sum of all the values for the probability distribution to  equal 1,

p(i)  =  1.[12]

But equation [12] can also be written as,

1  =  ( p(i) i )[13] 

where i = 1. Equation [13] can be seen as a sum of areas each with a width of i and a height of p(i) at various i, as is shown in Fig 3 below.

The total area after summing these up is an approximation of the area between the i-axis and the curve represented by the function p(i) from imin = 1 to imax = 7. More generally, however, we can make i = (imax - imin) / (n -1), where in Fig 3, imin = 1, imax = 7, and n = 7, so that i = (7 - 1) / (7 - 1) = 1. When i takes on successive whole numbers on the i-axis, i will always be 1 and is usually omitted.

However, what happens when we want to divide the interval, imin i imax, by a larger number of sub-intervals? This would give us a closer approximation to the area under the p(i) curve. In that case, equation [13] can be written as

1   =     p( imin + [ j-1] i  i [14]

Here n does not necessarily represent the number of whole number steps from imin to imax as before. The number n could be very large in which case i = (imax - imin)/n  and can become arbitrarily small as n increases. As j steps from 1 to n,  p( imin + [ j-1] i ) is evaluated in increments of i along the i-axis. With arbitrarily large values of n,  p(i) could be evaluated at any real value of i, not just whole numbers.  And p(i) will have to be a continuous function with a corresponding value for every real number of i for which p(i) is evaluated.

So we must consider what happens as we let the discrete variable i become a continuous variable. When i become continuous, it's customary to label the i-axis as the x-axis, where x can take on any real value. Then p(i) becomes p(x) and must be a continuous function. The interval, imin to imax, becomes xmin to xmax, and i becomes x = (xmax -xmin) / (n -1), and imin + [ j -1] i becomes xj = xmin + [ j -1] x, where j still takes on values from 1 to n.

The process of integration found in the study of calculus is to let n increase without bound in equation [14]. We say "in the limit as n approaches infinity" and write in formulae and more simply n in text. And so the process of integration applied to equation [14] would be written,

p(xj) x = 1[15]

Since x = (xmax - xmin) / (n -1), as n approaches infinity, n , x approaches zero, x 0. But n never actually reaches infinity since that number is really not defined. And so x never actually reaches zero, but it is increasingly small. The notation of x 0 is usually shortened to dx and is referred to as "differential x" meaning that it is increasingly small. And the function p(x) in equation [15] no longer assigns a probability for each discrete alternative as p(i) did in equation [12]. In equation [15], p(x) is a probability density, assigning a probability for events to happen between x and x+dx. The notation is a little cumbersome to write, so it is usually shortened to where xmin is called the lower limit of integration and xmax is called the upper limit of integration. So changing to this notation equation [15] becomes

p(x)dx = 1[16]

And it is call the integral of p(x) from xmin to xmax that is set equal to 1.

 

Section 6

LOGIC OF DIRAC DELTAS

Now let's convert the summations of equations [10] and [11] to integrals. This becomes necessary when the density of propositions in the coordinate system becomes so dense that there is a continuous distribution of them. Using the techniques of the previous section, the continuous version of equation [10] becomes the integral

(x-x0)dx       = 1,    if   x0 R[17]
0,    if   x0 R

where R is some interval on the x-axis. The function (x-x0) is called the Dirac delta function, which is the continuous version of the Kronecker delta function ij. Note that equation [17] is a continuous version of x(A) = yx of Section 3, where x0 in equation [17] is the coordinates of the element x, and x is the coordinates of element y, and R is the region in the coordinate system that the set A occupies.

But now we need to understand the characteristics of the Dirac delta function (x-x0). Previously, ij was interpreted as a probability distribution which assigns a probability of 1 to the j th alternative and 0 to the others. This was generalized to a probability distribution p(i) that assigned various probabilities to various alternatives where the total must still equal 1. In these discrete distributions, p(i) can be seen as a probability for each alternative separated by a distance of 1 between successive i. In other word, p(i) is the probability per i = 1. And so p(i) can be written p(i)i. But as i becomes a continuous variable, we label the i-axis as the x-axis, p(i) becomes a continuous function of x, labeled p(x), which now becomes a probability density function, and i becomes x. Then when n, we get x0, usually written dx, so that p(i)i, becomes

p(x)x  =  p(x)dx.

From this notation one can see that since x 0, there is approaching a 0 probability for any particular event at p(x). The only way to get any meaningful number in the continuous case is to integrate p(x) between some limits.

Likewise, since (x-x0) is inside an integral, (x-x0) is a density function giving a number per unit of the x-axis. And since it is multiplied by dx, it cannot be just one specific number at x0 and zero everywhere else like ij. For that one number at x0 would be multiplied by dx which is vanishingly small and would give a zero result. The integral must have some limits to it, and (x-x0) must be a continuous function that has a value for every x on the x-axis between those limits.

The Dirac delta function (x-x0) can be derived from the Dirac measure x(A). In Section 3 the Dirac measure was defined as x(A) = 1, if  x A, and x(A) = 0, if x A, which has to be the case no matter how large or small the set A is. And when we put this in terms of a coordinate system, x0 becomes the coordinate of the element x, and R becomes the region in x-space that occupies the set A. And in the notation (x-x0), x is allowed to vary anywhere on the x-axis. But since (x-x0)dx = 1, if x0 R, even when R becomes a very, very small region, (x-x0) will have to become very, very large so that the integration of it still produces 1 even when the integration interval is very small. So this specifies another limiting process such that (x-x0) at x = x0, and (x-x0)0 for x x0. The limiting process of (x-x0) is controlled by aparameter, . I'll call it cap-delta, such that as 0, (x-x0). This is a different limiting process than the n limit process involved in integration. One has to hold at some finite value and then do the integration on the continuous Dirac delta function (x-x0), and then after integration require 0. For it would not be possible to do the integration if one were to allow (x-x0) to approach infinity first.

In the literature the region R in equation [17] is usually the entire real line, - x +, but this does not necessarily have to be the case. Yet if R in equation [17] were the entire real line, then x0 would certainly be included in it, and we get,

(x-x0)dx = 1[18]

which is mapped from the logical equation [9], and the Kronecker delta equation [10].

Next consider what effect equation [17] would have on an arbitrary function f (x),

f (x)(x-x0)dx.

Since the function (x-x0) is practically 0 far from x0 and very large near x0, we have that f (x)(x-x0) is practically 0 far from x0 and large near x0. This means we can restrict the interval of integration, R, to a very small interval that includes x0. And when R becomes a very small, f (x) will essentially be f (x0) if R is a small enough interval around x0, and the above equation becomes

f (x)(x-x0)dx = f (x0) near xo (x-x0)dx = f (x0) 1 = f (x0).

So that we have,

f (x)(x-x0)dx = f (x0)    for x0 R[19]

But if the interval, R, does not include x0, or x0 R, then x0 will be far away from x on the entire interval, R, and (x-x0) will essentially be 0 throughout the integration interval. Thus we have,

f (x)(x-x0)dx = 0        for x0 R[20a]

Now let's change the integration variable in equation [20a] from x to x1. Then f (x) becomes f (x1), and (x-x0) becomes (x1-x0), and dx becomes dx1, and equation [20a] becomes

f (x1)(x1-x0)dx1 = 0        for x0 R.

But if f (x1) were to be a Dirac delta function itself, (x-x1), we get

(x-x1)(x1-x0)dx1    =  (x-x0)       for x0R[20b]
0                 for x0R

Of course, in equation [20b], it would be just as easy to let f (x1) = (x1-x0) instead of (x-x1), and then multiply this f (x1) by the Dirac delta (x-x1). This new integral would result in the same (x-x0) as before, but now with the condition that x R. And as before the integral would be 0 for x R. So the two possible deltas for f (x1) together give,

(x-x1)(x1-x0)dx1    = (x-x0)           for {x, x0}R[21]
for {x, x0}R

It is interesting to note that equation [21] is a continuous math representation of the logical equation [11]. Equation [21] is also a recursion relation for the Dirac delta function which we can iterate again to get,

[22]

(x-x2)(x2-x1)(x1-x0)dx2dx1 = (x-x1)(x1-x0)dx1 = (x-x0)

And iterating an infinite number of times we get,

[23]

(x-xn)(xn-xn-1)(x1-x0)dxndxn-1dx1 = (x-x0)

for {x, x0}R. And the integral is 0 for {x,x0}R. Obviously the x1, x2,..., xn are each within the interval of R since we are integrating with respect to those variable within R. And equation [23] can be seen as a continuous math representation of the logical equation [7].

To sum up, the progression has been to go from logical equations to discrete summations to integrals in the continuous case. The disjunction of implications in equation [9] was mapped to a discrete summation of Kronecker deltas in equation [10] that became the integral of equation [17] in the continuous case. Or,

(qiqj) = T                ij = 1                  (x-x0)dx = 1

This was possible because implications, (qiqj), were able to be counted or not using the Kronecker delta function, ij, which was derived from the Dirac measure. This enabled us with the help of Table 1 to discern that disjunction must be represented by addition.

Table 1

(q1qj)

(q2qj)

1j  

2j  

(q1qj)(q2qj)

1j 2j

condition

F F 0 0 F  0   j < 1 or 2 < j
F T 0 1 T  1 j = 2
T F 1 0 T  1 j = 1

Then equation [9] was inserted into an implication as though it were an identity which resulted in equation [6].The Kronecker delta version of this was equation [11], and the Dirac delta version was equation [21]. Or,

(qsqf)    =    (qsqj)(qjqf)

                 sf   =    sjjf            (x-x0)  =  (x-x1)(x1-x0)dx1

In order to satisfy the Kronecker delta version, equation [11], it was required that conjunction be mapped to multiplication as seen in Table 2.

Table 2

(qsq1)

(q1qf)

s1  

1f  

(qsq1)(q1qf)

sf = s1 1f

condition

F F 0 0 F           0 s1, f 1
F T 0 1 F           0 s 1, f =1
T F 1 0 F           0 s=1, f 1
T T 1 1          1 s=1, f =1

The math representation of addition and multiplication for disjunction and conjunction is an easy exercise when dealing with the Kronecker delta in a discrete system. There you have ij equal to 1 or 0 which makes the math easier. But when you have  (x-x0) at x = x0 as 0, and (x-x0)0 elsewhere, it becomes conceptually less clear how multiplication continues to represent conjunction. For example, what is (x-x1)(x1-x0) when (x-x1) but (x1-x0)0? And when you replace discrete summations with integration, it becomes less obvious how addition represents disjunction. How do you add alternatives when the alternatives are very close together and not well distinguished form each other. So one wonders how Tables 1 and 2 would be affected by the use of the Dirac delta and the integration process.

In the development of equation [17] for the continuous case, disjunction is still mapped to addition; there is just an infinite number of propositions infinitesimally apart in the coordinate space. And instead of a difference of a unit distance multiplying the Kronecker delta in each term of the sum of n terms, there is a vanishingly small differential, dx, multiplying the Dirac delta in the integral of the continuous case. Since we do the integration before allowing (x-x0) at x = x0, we have that (x-x0) is a finite number being multiplied by dx which approaches 0. So each term approaches zero in the infinite sum of the integration process. This means you can't consider one or two terms in isolation in the infinite sum of the integral. One must integrate between some finite interval, R, to get any meaningful number.

And thus in Table 1, instead of two implications being listed with every combination of T or F as in the first two columns, there would be an infinite number of implications that would need to be listed. And since that's impractical, those columns are omitted. And the number of Dirac deltas for those implications would also be infinite and impractical; so those are omitted too. Instead of the disjunction of just two implication (q1qj)(q2qj), we would need to have (qiqj), with n = , since we cannot consider a few terms in isolation when there is a continuous distribution of them. And instead of a column for 1j 2j we'd have an infinite sum and write (x-x0)dx. For the conditions column it would be impractical to write out the infinite number of possibilities for j along a continuum. So we would write instead just two possibilities, whether x0R, or x0R. And in essence, equation [17] captures the logic of Table 1 in the continuous case.

(x-x0)dx       = 1,    if   x0 R[17]  
0,    if   x0 R

And again in Table 2, the columns that list each implication and corresponding Kronecker delta to be used would require an infinite number in a continuum which is not practical and are omitted. The column listing the conjunction of two implication, (qsq1)(q1qf), would become (qsqj)(qjqf) with n= since the continuum of states prevents us from considering a finite number of terms in isolation. And the column listing sf  =  s1 1f  would become (x-x0) = (x-x1)(x1-x0)dx1 instead. The conditions in Table 2 lists whether s and/or f is within the range of n, in that case n=1. So the corresponding conditions in the continuous case would list whether x and x0 would be within the range of R which is covered by x1. If x R, then (x-x1) is guaranteed to be zero in the range of integration, R, making the integral zero. Likewise, if x0 R, then (x1-x0) will be zero in integration range, R, making the integral zero in that case too. It's only when {x,x0}R that we have (x-x0) = (x-x1)(x1-x0)dx1. And so equation [21] captures the logic of Table 2 in the continuous case.

(x-x1)(x1-x0)dx1     = (x-x0)           for {x, x0}R[21]
0                     for {x, x0}

Now the question remains as to what function should be used to represent (xn-xn-1) in equation [17]. Note that the Dirac delta functions in equation [21] are functions of one variable since x and x0 are being held constant and x1 varies across R. But in equation [23], the Dirac delta functions are functions of two variables since now, nothing is being held constant and both its variables vary across R. So the Dirac delta functions will have to be functions of two variables, x1 and x0.

There may be many functions that could be used to represent the Dirac delta function. And the functions of interest will have to satisfy equations [17], [21], and [23]. One such function is the gaussian form of the Dirac delta,

δ(x x 0 )= lim Δ0 1 (π Δ 2 ) 1 2 e (x x 0 ) 2 / Δ 2 [24]

It has the property that as approaches zero, the delta function becomes infinite in such a way that the integral of equation [18] remains one. The integration of the gaussian Dirac delta is a little tricky to prove and is done in many books on quantum mechanics that cover the path integral. (No physics is necessary in the proof.)

For any non-zero value of , equation [24] represents a gaussian distribution of any measurement across many samples. The gaussian distribution is also called a normal distribution and represents completely random processes where no external forces or intelligence influences the measurements. It represents the minimal amount of information necessary to produce the result. There is no other structure in the distribution that needs to be explained; there is nothing biasing the samples that requires investigation. Then, as approaches zero, the distribution becomes more and more representative of perfect process, where there is no uncertainty that every measurement will be exactly the same as the next.

And it seems an unbiased, random distribution would have to be the starting point on which to build a fundamental theory. For otherwise it would not be fundamental because biased samples need further explanation and points to mysterious causes having some effect. So in this respect the gaussian distribution recommends itself as the mathematical representation of the Dirac delta function on which to build a fundamental theory.

The gaussian Dirac delta function of equation [24] also satisfies the recursion relation of equation [21] since,

[25]

+ ( λ 2π(t t 1 ) ) 1 2 e λ (x x 1 ) 2 2(t t 1 ) ( λ 2π( t 1 t 0 ) ) 1 2 e λ ( x 1 x 0 ) 2 2( t 1 t 0 ) d x 1 = ( λ 2π(t t 0 ) ) 1 2 e λ (x x 0 ) 2 2(t t 0 )

 

where (t-t1) and (t1-t0) both act like the previous and approach zero as (t-t0) approaches zero. This equation is called a Chapman-Kolmogorov equation and is proved in The Feynman Integral and Feynman's Operational Calculus, by Gerald W. Johnson and Michael L. Lapidus, page 37, eq 3.2.8. I don't know what other functions would solve the Chapman-Kolmogorov equation. But if it turns out that the gaussian is the only function that does, then this would prove that the only representation for the Dirac delta function would be the gaussian distribution. Or if every mathematical representation of the Dirac delta function is essentially equivalent, then it is fair you use the gaussian distribution.

If the gaussian form of the Dirac delta function is to continue to be used, then notice that as stated in equation [24] that this function makes (xn-xn-1) = (xn-1-xn). But we need a modification to this since the Dirac delta is supposed to represent implication. And since (p q)(qp), we need to have  (xn-xn-1)(xn-1-xn). The only parameter left to manipulate in equation [24] is ; we need to have depend on whether we use xn-xn-1 or xn-1-xn in the Dirac delta function. For example, let's start by trying the simple substitution 2=(tn-tn-1), where we let x be a function of a parameter called t, or x = x(t), such that xn = x(tn) and xn-1 = x(tn-1), etc. Successive t would then mark off successive steps along a path. Then the exponent in equation [24] will be either positive or negative depending on whether we use tn-tn-1 for xn-xn-1 or tn-1-tn for xn-1-xn. And the denominator will be a complex number when we take the square-root of a negative number. But we will have (xn-xn-1)(xn-1-xn) required since (pq)(qp).

However, there are a couple of problems with this choice of 2=(tn-tn-1). As tntn-1 so that 20 in equation [24], the exponent of -(xn-xn-1)2/(tn-tn-1) could approach - if tn >tn-1 as tn tn-1, or the exponent could approach + if tn <tn-1 as tntn-1. This means that the exponential term, exp[-(xn-xn-1)2/(tn-tn-1)] could approach infinity or zero depending on whether tn approached tn-1 from above or from below. There would be a discontinuity in the distribution. Also, the denominator of equation [24], (2)1/2 = 1/2(tn-tn-1)1/2, would suddenly change from a real number to an imaginary number as tn-tn-1 changed from a positive to a negative number near 0.

So in order to eliminate discontinuities and jumps into pure imaginary numbers, let's modify 2 and make it 2=i(tn-tn-1), where i=. Then the only difference between (xn-xn-1) and (xn-1-xn) is a phase shift. And the complex number would result in the denominator for both positive and negative tn-tn-1 which would be multiplied by the real and complex numbers from the imaginary exponent since,

eix=cos(x) + i sin(x).

This give us both a real and imaginary number for (xn-xn-1) in all cases so at least the nature of (xn-xn-1) doesn't abruptly change as tn >tn-1 goes to tn <tn-1. This indicates that the way to get an always real number would be by multiplying (xn-xn-1) by its complex conjugate (xn-1-xn) by interchanging tn and tn-1. I explore more about what this mean in Section 8 below.

 

Section 7

PATH INTEGRATION

So let us make the following substitution in equation [24],

[25]

where m and are arbitrary constants for the purposes here, and , then we can rearrange equation [24] to get,

which equals

[26]

Using m and above is not an attempt to covertly introduce physics. I only use the labels m and because with the constants labeled this way they can serve the same uses in this derivation of the path integral as mass and Planck's constant serve in the Feynman path integral of physics.

And inserting equation [26] into equation [23], we get

with the appropriate limits implied, and where the R in the integrals of equation [23] is the entire real line. By gathering terms, this is equal to

[27]

Notice that the exponential term looks like 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 is a constant used to cancel out the units of the integral so that the exponent is dimensionless and can be evaluated. Equation [27] can be recognized as the Feynman Path Integral for the propagator of the wave function for a free particle in quantum mechanics. The limits of the t approaching zero is implied by the notation of dt and t.

The development so far was to insert the complex gaussian form of the Dirac delta function into equation [23] many times. And then the variance of equation [25] was inserted into all those complex gaussians. This allows the ability of gathering all the exponents into a complex action as required by the path integral. The Dirac delta function was used because it closely parallels the math of the Kronecker delta which mimics the logic. But the problem is that the integration of all those Dirac delta functions in equation [23] resulted in another Dirac delta function. And the Dirac delta function is very sharply spiked and has limited use. We used ± as the limits in the path integral and not simply R. So how can a gaussian with a wide variance be justified, and what justifies the use of ±   instead of R?

Equation [23] only used Dirac delta functions to get a Dirac delta function. But the Chapman-Kolmogorov equation [25] could have just as easily been used, where only gaussians were used to obtain a wider gaussian. If equation [25] were iterated many more times by integrating even more gaussian exponential functions, then all the ( t n t n1 ) factors in the denominators would become smaller for each new exponential that's integrated in. In the limit of an infinite number of integrations (as in the path integral), the ( t n t n1 ) terms approach zero just as they do in the limit of equation [24] which defines the Dirac delta function with a gaussain. The Chapman-Kolmogorov equation used limits of ± , but as ( t n t n1 ) approaches zero, we could just as easily used the limits defined by R. Then all the considerations of x 0 R or not so that the integral is equal to 1 or 0 in equations [17] and [21] are still valid. The reason that the limits of ± are used is because we are only concerned about what's in the universe, where x 0 R and not anything outside the universe where x 0 R . So we let R be the interval from to + which insures that x 0 R .

 

Section 8

THE POTENTIAL IMPLICATIONS

Now what happens if each of the Dirac delta functions is weighted by a function, ? This would suggest that some implications are stronger and have more of an effect than others. Or might be viewed as a density function, and this might be another way of saying that some regions have more implications than others. Why not?

Then equation [23] becomes

[28]

And equation [26] becomes

[29]

But since in such a way so that , we can write , where , and where leaves the implication not weighted.

Then equation [24] becomes

and equation [27] becomes

which is the Feynman path integral for a particle in a potential which is called the wave function labeled, .

Section 9

THE BORN RULE OF PROBABILITIES

The Born rule tell us, at least in part, that the probability density, p(x), for finding a particle between x and x+dx with wave function, , is equal to the wave function times the complex conjugate of the wave function. Or in symbols,

This can be explained in the context of these efforts as follows: Equation [4] is

q1^q2 (q1q2)^(q2q1)

which is an equality if at least one of q1 or q2 is true. When we map this in mathematical terms, each of q1 or q2 is a proposition mapped to a value between 0 and 1 depending on how likely it is. So, for example, q1 maps to a number that behaves as the probability that the proposition q1 is true. And factors like (q1q2) generate the path integral which is another way of describing a wave function, . We learned that (q1q2) maps to a complex number so that (q2q1) must be its complex conjugate. And ^ maps to multiplication. So q1^q2 maps to a probability of finding q1 time the probability of finding q2, or p(q1)p(q2).

The physical interpretation of (q1q2) is that the state described by a proposition q1 leads to the state described by proposition q2. In terms of an experiment, q1 would be the setup of the experiment and q2 would be the measured result. Now, experiments are set up in a known state with certainty so that the results can be repeated. That means here that p(q1) would be 1. So what we have left is p(q2) equal to a wave function representing (q1q2) times the complex conjugate of the wave function representing (q2q1). If we let q2 be located at x, then p(q2) is replaced by p(x), and (q1q2) is represented by , and (q2q1) is represented by to get the Born rule:

where must be interpreted as the square root of a probability.

This means that the wave function expresses how one fact implies another. It does not give enough information to predict a measurement because the measurement of an experiment assumes you know both the setup and the result. You must know that the setup and the result both exist in conjunction. Otherwise you cannot form a correlation between cause and effect if you don't know what caused your effect or if you don't know what effect your cause had. So the wave function tells us what effect a cause will have, and the conjugate wave function tells us what caused an effect. And together you know both cause and effect and you can calculate the relationship (probability) between them.

And it seems only intelligence is concerned with calculating the probability between cause and effect. A screen hit by an electron doesn't care where it came from; it could come from anywhere and have the same effect. And an atom emitting a photon doesn't care what effect the photon has on any screen. Physical events don't care what the probabilities are; they simply respond to stimuli. But conscious beings with intelligence calculate probabilities so they can make intelligent decisions.

 

Section 10

THE LARGER IMPLICATIONS

The quantum mechanics of the wave function/path integral obtained above is usually called 1st quantization. Functions are obtained with this procedure. There is also a branch of quantum physics called quantum field theory which is sometimes called 2nd quantization. It takes the fields obtained in 1st quantization and plugs them into a very similar quantization procedure to get 2nd quantization. Again, it seems like there is little justification for further quantizing the fields other than it just so happens to produce correct results. It occurs to me, however, that quantum field theory comes naturally to the procedure I describe here.

We started with the fact that

qi (qiqj)[5]

which is an equality if at least one of the qi is true. And so it became necessary to evaluate

(qiqj) = (qiqi1)(qi1qi2)...(qimqj)

which when represented in mathematical form became the path integral of 1st quantization.

But there is no reason not to apply equation [5] again to get

qi (qi qj) ((qi qj) (qk ql))

in which the last conjunction is an equality if at least one of the (qiqj) is true which will be the case if at least one of the qi is true. And if we let qij = (qiqj), then we have

qi (qij qkl)

which would necessitate the evaluation of

(qij qkl) = (qij qi1j1)(qi1j1 qi2j2)...(qimjm qkl)

In this case the mathematical representation of (qi1j1 qi2j2) would be

( i1j1(x,t) i2j2(x,t))

where i1j1(x,t) is the wavefunction of 1st quantization and is the mathematical representation of qi1j1 = (qi1qj1). The delta here would be expected to still be the complex gaussian with i1j1(x,t) replacing xi in the exponential. And would replace dx in the integrals to finally get

which is the path integral of 2nd quantization used in quantum field theory.

Some of the details may need further attention. I'm not sure what the double subscripts imply. Maybe they can be treated as spinors that result in antimatter.

And I don't see why the same procedure can't be used to get 3rd quantization except that keeping track of the indices might be tedious. Yet it might be worth the effort. For just as 2nd quantization gives the particles used in 1st quantization, 3rd quantization might give us the fields used in 2nd quantization. This method will probably not give us the charge and mass of particles since logic is not concerned with our arbitrary units of measure. But it might give us a way to derive a ratio of one field's values to other field's values so that only one measurement is needed to deduce everything else. Would this be a non-perturbative approach to QFT? I wonder.

Previously the complex numbers were used in the wave function of 1st quantization. And the complex numbers establish the U(1) symmetry of QED. I have to wonder if a similar effort for the four numbers associated with the (qi1j1qi2j2) of second quantization or the eight numbers associated with third quantization might establish the quaternions or octonions used in the quaternionic representation of Isospin SU(2) or the octonionic formulation of SU(3) used in particle physics. I am by no means an expert in these matters. I only noticed their use in my reading, and now it seems they may become relevant to this effort. John Baez has a brief introduction to quaternions and octonions here. There the iteration from complex numbers to quaternions to octonions is very similar to the iteration from first to second to third quantization here and suggests their use. Further references on quaternions and octonions of symmetry groups in physics are here and here.

The real numbers, complex numbers, quaternions, and octonions are specific examples of the larger Clifford algebra as explained here. And Clifford algebra has also been used as an alternative description of differential geometry that is used to formulate the curvature equation of General Relativity as explained here. So I have to wonder, if the quaternions and octonions are justified by principle alone, as I suspect, then do they put a constraint on the Clifford algebra used in differential geometry to produce General Relativity? If this turns out to be the case, then we may have a means of deriving both the Standard Model and General Relativity from logic alone. Obviously, more study is needed to confirm these suspicions.

Section 11

DISCLAIMER

Having noticed a parallel between paths constructed from logical implication and paths constructed of particle trajectories, I extended that analogy to reconstruct Feynman's Path Integral from simple logic. The conversion is achieved by representing the material implication of logic with the Dirac delta function and then using the complex gaussian form of the Dirac delta. However, at this point my derivation has not been reviewed by reputable sources. It has yet to pass inspection by mathematical logicians. Until that time, this effort should be considered preliminary.

I may not have given a full account of all of the quantum mechanical formalism yet. I've not derived Schrodinger's equation, eigenvalues and eigenvectors, Hilbert or Fock space, or Heisenberg's uncertainty principle, for example. But I suspect that the rest may be implied by the wave function that I have derived. For example, the Schrodinger equation is derived from the path integral in many quantum mechanics text.

Keep in mind that I'm not claiming to have derived all of physics from logic. In order to claim a logical derivation of physics, one would have to derive physical quantities such as some of the 20 or so constants of nature or the principles of General Relativity. So I will keep an eye on such efforts. And I'll try to include more as time and insight allow.

However, this does open an intriguing possibility for deriving the laws of nature. Typically physicists use trial and error methods for finding mathematics that describe the data of observation in very clever ways. These theories are then used to make predictions that experiment may confirm or falsify. When very many observations are consistent with the equations, we have confidence that the theory is correct. However, such theories can never be proven correct and are always contingent on future observations confirming them. But we can never say they are completely proven true. For we don't know whether some observation in the future may falsify the theory. Now, however, there may be the possibility that physical theory can be derived from logical considerations alone. Such a theory would in essence be a tautology and proved true by derivation. We would have to check our math against observation, of course. But if even one observation was consistent with such a theory, how could we say that other observations would not be? Can we expect that some parts of nature are logical but others are not when they coexist in the same universe? 

We may not have any choice but to derive physics from logic since the ability to confirm ever deeper theories will require energies that are beyond our abilities to control. After all, we cannot recreate the universe from scratch many times over in order to confirm some proposed theory of everything. So we may be forced to rely on logical consistency alone. And I think I have a start in that direction.

Now, having derived the transition amplitudes of a particle from logic alone, I use these transition amplitudes in a description of virtual particle pairs. These virtual particle pairs come directly from the conjunction of points on a manifold and can be used to describe many of the phenomena we see in nature, perhaps all. For more details see this article.

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Thank you.