Quantum Mechanics from logic alone
Originally posted
4/21/07

 

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,



This conjunction implies that each fact proves every other, since



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



In a sample space, the probability of a conjunction results from multiplying the probabilities of each simple event. And if  conjunction, , maps to multiplication, then implication, ,  should map to a square root of a probability, an amplitude. The square root requires that the number introduced by implication is complex. The exponential factors, , for each step in a path are multiplied together; the product of exponentials equals an exponential of a sum of the phases. This sum resembles an action integral, . When this exponential of the complex action is added together for every possible path, the result is a path integral formulation of quantum mechanics.


INTRODUCTION

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 I'd like to suggest that quantum mechanics is derived from classical logic only, without imposing any physical considerations.

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 negation and is the material implication of propositional logic. 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 infinity. 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 must not contradict this conjunction. It is also interesting to note that one could say this about any topology - that every point of the topology is an element of that topology. And perhaps these ideas can be applied more broadly to any manifold whatsoever.

But it should be realized that 

[6]

which again can be proved by 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 nothing exists, all the implications on the right hand side are true even when all the facts involved 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 whole conjunction of implications false just as it would be on the left hand side. So since we know 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 them) 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..." conditionals 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. Implication seems to be working here as a function mapping points to other points. It seems to be working like the successor function in topology that maps one point to the next point along a path. Such a path can be parameterized by a discrete variable t_j, where successive j enable successive steps.

If we are interested in finding any relationships between the facts, then we have to start with the relationship between two facts. So we construct our paths with end points on 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 others. 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  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.

But it is easy to prove by truth table that

[11]

where the    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. Probabilities compare the number of samples in various regions of a sample space. Set theory explicitly lists the samples that belong to a region. Predicate logic considers whether there may exist some, none, or all samples in a region. And Propositional logic is only concerned with whether the regions themselves exist or not. (Detail here)

In a sample space, a probability is assigned to each event. Since the samples can be counted, it is natural to form a probability by counting the relative number of samples. The probability of obtaining two independent events 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 results in multiplying the probabilities, and disjunction results in adding the probabilities.

So when the mathematics of sample spaces is applied to the facts of reality, each fact is a sample in the space of all possible facts whether they are observed, measured, or not. And there is a probability associated with obtaining a new fact when it is measured. Measurement or observation is the act of conjoining a new fact. It states that the new fact exist in conjunction with the first facts. It is not enough to know that the first facts prove the new fact. For the first facts might also prove many other facts as well. And the new fact may not in turn prove the first. But if the new fact is confirmed to coexist with the others, then it must be that the first facts prove the new fact AND the new fact proves the first facts.

So if a conjunction of states results in a multiplication of the associated probabilities, and disjunction results in the addition of probabilities, then what operation does implication result in? Conjoining a new fact with a measurement means multiplying by a number, a probability. But we have , where we assume one of the states is the initial state that exists with absolute certainty because we prepared that state. This makes the conjunction practically equal the two implications. And then conjoining the next state means multiplying by one number, a probability. There is one conjunction symbol but two implication symbols. So implication must represent an operation that gives a number that when multiplied by a number gained by the reverse implication results in a probability. The operation represented by will be used in constructing paths over the entire space, so it must have a similar form independent of which states it acts on. It cannot prefer the order of the operands, but it must give a real number when multiplied by its inverse. So can only be the square root of a number that when multiplied by the square root of a number obtained from gives a real probability. cannot give a number that is the reciprocal of that obtained by , for when multiplied together that could only give 1. And probabilities are usually different than 1. And cannot give the negative of , for when multiplied together that would be negative. And probabilities are always positive. So has to be the square-root of a probability, which we will call an amplitude.

Since introduces the square root of an unknown function of the states q₁ and q₂, that introduces the possibility of imaginary numbers, since that unknown function might be negative. So in general results in a complex number,

[13]

where controls the phase of the complex number, whether it is positive, negative, or imaginary. And must satisfy the following:

, for any pair of q_i or q_j.[14]

, if and only if q_i=q_j.[15]

And since must be real, it must be that 

[16]

so that .

It must also be the case that 

[17]

so that the square root associated with is the same as the square root associated with so that there is no preference of the order of operands.

Then the conjunction of steps in a single path from q₁ to q_n will map to a product of complex numbers,

[18]

which is only meant to indicate that the starting state is q₁ and the final state is q_n. This is not meant to state how many steps there are from start to finish, nor is it meant to say what path is taken. But this can be broken down to

[19]

And when all the paths are summed up, we get:

[20]

This looks very much like Richard Feynman's path integral formulation of quantum mechanics.

In order to finish, we can switch over to the notation used in quantum mechanics. No generality is lost in doing so; this will not impose any physical constraints or considerations. The derivation still proceeds from logic alone. Only the notation will change.

Instead of writing , this can be written as, , which is called the probability amplitude of going from the state q_j to the state q_k. The reverse implication of is the complex conjugate of . In other words, , where is the complex conjugate of .

And instead of writing , this is written as,

[21]

It represents the probability of getting the state q_k from the known state q_j. And is called the modulus of a complex number .

Now there doesn't seem to be any alternative but to label various states with different numbers. This means that different states occupy different points in a topological space. And since probabilities can never be more than 1, it must be that

[22]

For adding two amplitudes might be more than one. But no probability amplitude can be more than one. And so the above inequality along with the previous requirement for means that is a metric, and the states occupy a metric space. If the states occupy a Euclidean space, then

[23]

And if the space is so dense that the paths can be considered continuous, then,

[24]

But in general, if the space is not Euclidean, then

[25]


If the exponential

[26]

has any meaning in the continuum limit, then

[27]

has to be finite as n approaches infinity. This means that must be a differential. So setting , where H is some function yet to be determined and t parameterizes the steps along the path, then

[28]

And as the number of states becomes a continuum, the number of paths from the starting state to the final state also becomes infinite. Each path becomes a infinite product of infinitesimal steps, which forms in itself an infinitesimal. So the initial sum becomes an integral:

[29]

where the last integral is usually written more briefly as

[30]

Putting it all together, the path integral:

[31]

becomes:

[32]

The only thing that remains is to show how H in the exponential can be related to another function L in the same way that the Hamiltonian relates to the Lagrangian functions of physics.

If L is related to H by

[33]

where the dot over x represents time differentiation and where , for some constant m, then the above path integral becomes

[34]

And if it can be shown that

[35]

is a path independent constant, then it can be pulled out of the path integral and treated as a scale factor in the conversion between H and L. Some help here would be appreciated. Then the path integral becomes the familiar form of physics:

[36]

This is Richard Feynman's path integral formulation of quantum mechanics. And from here the Schroedinger wave equation proceeds as usual as the differential form of the above.

Comments and Concerns