ABSTRACT
After a very brief introduction to logic in order to establish
language and notation, I show how a
type of path integral can be constructed in terms of propositional logic that can then be mapped
to the mathematical expression of Feynman's Path Integral, which is the
wavefunction of quantum mechanics.
In logic
an implication can
be equated to the
disjunction of every possible path, where a path is defined as a conjunction
of many implications in which the consequence of one implication becomes the
premise of the next implication.
I show
how implication can be mathematically represented as a Dirac
delta function, how disjunction is mapped to addition, and how conjunction is mapped to multiplication. It will be shown that the complex gaussian form of the Dirac
delta function will be required to represent implications in logic. When the gaussian delta functions are multiplied
together as required by the conjunction of implications in each path, the
exponents of the gaussian delta functions can be added to form an Action
integral with a lagrangian of kinetic energy. And when the paths are added as
required by the disjunction of paths, the Feynman path integral is formed.
Potential energy can be added in the lagrangian by considering a weighting
function on each implication. This gives us the wavefunction
of 1st
quantization.
The process can be iterated to give a prescription for the
quantum field theory
of 2nd
quantization. And the process can be iterated
to
provide a 3rd,
or
even a 4th
quantization procedure, if needed. Finally, I give some reason to expect that these
iterations prescribe that the complex numbers iterate to quaternions
in 2nd
quantization and to octonions
in the 3rd
quantization, which are believed to
be responsible for the
U(1)SU(2)SU(3)
symmetry of the Standard Model.
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. And physics would then be a result of
applying this tool to the facts of reality.
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 distributed throughout this article briefly
describing the basic introductory definitions in logic. 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 and they cannot be both. 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),
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.
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 4
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 object 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
a
S
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
A
B={a,b,c,d,e,f,g,h}.
The intersection of the two sets is
A
B={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
C
A
or as
A
C
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.
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
[1]
But it should be noted that
[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:
[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 are 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, qi,
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
[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
[5]
This conjunction would include factors such as 
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
(FT)=T,
but when that same factor is reversed elsewhere in that conjunction, there
will be a factor of the form
(TF)=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
(FF)=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
(TF)=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:
where
q1 is an intermediate state, and
where the last term,
,
indicates the conjunction between
(qiq1)
and
(q1qf).
This conjunction forms a two step "path"
between qi and qf .
I call it
a path because it has an intermediate step
of q1
between qi and qf
.
There's no value
of q1, T of F,
that can negate
the equality. If the LHS is false, this only happens when
qi=T
and
qf =F,
and then if
q1=T,
that would make the factor
(q1qf)=(TF)=F
making the conjunction term false, which in disjunction with the first term,
(qiqf)=F,
makes the RHS
to be F,
just as the LHS would be. Or, if
q1=F,
that
would make the factor
(qiq1)=(TF)=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,
(qiqf)=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 (qiq1)(q1qf).
More intermediate states can be
used to OR in more paths to get
[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 i,
and there will be a term on the RHS of
equation [6]
of the form;
(qiqi)(qiqf)=(qiqf)
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 qi'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
(qiqf)(qfqf)=(qiqf)
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
(qiqj)
and
(qjqf)
can be thought of as functions of the single
variable
qj,
with qi and qf being held constant. Then
equation [6]
can be thought of as a
type of expansion in terms of other functions.
This
procedure can be applied again, and intermediate states can be inserted
between,
say 
and 
,
to get
And
applying the
procedure m
times,
[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 antisipated 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
[8]
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,
qi 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
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.
THE MEASURE OF IMPLICATION
Expressions
of Propositional logic have connectives of AND and OR and NOT that operate on values of true
or false. But real world applications usually involve calculating numerical
values. So we need a way to assign mathematical operations to logical
connectives and to give measurability 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 qP,
where
the set P
was the extension of the
property P.
I suppose
it's
always possible to assign an object to any proposition 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=(qQ)=
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 add one. In other words,
if x
is the
object of concern
and A
is the area of interest, then
we count 1
if xA;
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
adding 1
for set membership. The Dirac measure
accomplishes this task. See here
for the Dirac measure. The Dirac measure is
denoted 
and is defined such that,
So
if x is a proposition such that x=xA,
then the Dirac measure
maps F to 0
and T to 1
for the
proposition x.
When we scan the area A for the
object x,
we may notice that there are other members
of A
which may not
be an
x.
We may list all the objects
in A
and get
A={a,b,c,d,x,e,f,g,h}.
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.
Then
since A
is a
proposition, notice that
Ax;
if a conjunction is true, then so are all of its statements. But notice that
x
A;
for an element tells us nothing about which set it may belong to. Thus, if
xA
is true, then
Ax
is true. And if
xA
is false, then
Ax
is false. This means that
(xA)=(Ax).
So if
represents the set inclusion
xA,
then
also represents implication Ax.
The Dirac measure is a measure on implication.
Now
notice in the notation
of
that x is an element on equal footing with all the rest of the elements
in A.
But A
is a set whose number of elements is not specified. But we expect
(xA)=(Ax)
to be true no matter
the size of A
as long as x remains an
element of A.
So we
should still
have
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
Y={y},
where
y
stands for one of the elements in the universe of discourse, then we can
write
This
is the Kronecker delta function; it is a
point-to-point
Dirac measure. And since each element in the universal
set has a corresponding proposition, we can
use it to represent the implication of one proposition to another.
We have
that 
is a mathematical representation of
(yx).
This is what we want to use since the paths in the logical equations were
based on the implication of one proposition with another proposition, and
not on a proposition with an expansion.
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
The
yA
under the 
symbol means that there is a sum
of the
terms, where each term is evaluated with a different value
of y
which cycles
through every
element in A.
If y cycles through all the
elements of A
in this
fashion, then
eventually y
will
equal x,
if
xA,
and
will
equal 1.
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,
can be treated as a function of the one
variable y.
Later,
when x
is allowed
to vary as well, then in that case,
would have to be seen as a function of two variables.
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.
As was just
developed,
(q2q1)
gets mapped mathematically
to 
.
Or since all the propositions involved
are a q
to some subscript,
we can
drop the q
and
write 
to get,
(q2q1)
.
And
notice that
if i
cycles through every element
in the set A,
then
(qiqj)
= T , if qjA
, or 1 
j 
n
[9]
= F , if qjA
, or j < 1 or n < j ,
where the elements of A
are labeled with the
subscript i
which 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,
and if
qjA,
there will be a term of the form
(qjqj)
which is
identically T
which is ORed in and makes the whole
expression T.
If
qjA,
then
qj=F,
but all the rest of
the qi are T
since the rest of
the
qiA.
This makes all the terms of the form
(TF)=F,
and thus the whole disjunction
is F.
And again,
since qi
is the only thing that varies in
equation [9],
we look on
(qiqj)
as a function of one variable in this case,
since qj
is held constant here.
So with T mapped to 1 and
F mapped to 0
and
(qiqj)
mapped
to 
,
equation [9]
gets mapped to
[10]
where
is seen here to be a function of the
one variable i,
with j held
constant.
Logical
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. So we need to use a
commutative math operation for the
logical
operation since it's commutative. 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 .
If we
were to use ,
there would always be a factor
somewhere that is 0,
which would multiply with
all the rest of the factors to make the
whole mapping 0
for
any disjunction ().
So our only choice is to map
to +
to get the map:
(qiqj)
.
It's
interesting to consider that
equation [10]
acts like a type of completeness
relation for the Kronecker delta, and so likewise
equation [9]
looks like a
completeness relation for material implication.
Let's
look again at
equation [10].
It states that for alternative choices, one of
the choices can be assigned a value
of 1
with the others
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, 
,
with a more complicated probability
distribution, 
,
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.
For now, let's
insert
equation [9],
the identity of the completion relation for implication,
into
(qhqj),
to get
(qhqj)
=
(qhqi)(qiqj)
[11]
Remember,
the requirement from
equation [9]
was that
1 j
n
in order to ensure that
qjA.
No mention was made that qhA,
and so there is no requirement that
1 h
n.
And
as i
cycles
from 1 to n,
eventually i
should take on the
value of
j.
Then the RHS will have a term in it of
the form
(qhqj)(qjqj)
=
(qhqj)T
=
(qhqj),
making the
RHS=LHS
since the rest of the terms will have no effect (see discussion about
equation [6]
above). If we make the now required substitutions to map it from logic to math,
+, and
, and (qhqj)
, and (qhqi)
, and (qiqj)
,
equation [11]
becomes
= *
[12]
The question becomes with what mathematical operation do
we replace
the * symbol.
The
math operation
of *
has to be commutative since
the conjunction it replaces is commutative. So the possibilities
are + and .
When i=j,
=1,
and
if * were +,
then 1
would be
added to ,
with the other terms being zero. That would make the RHS one more
than the LHS and thus not equal to the LHS.
But if * were
multiplication,
then
multiplying
by
=1
leaves the RHS equal to the LHS. So
the * symbol
must be
multiplication, and conjunction maps to multiplication.
Now
let's replace the deltas with a more complicated probability
distribution p(ij)
which means the probability of going from
state i
to another
state j.
And
suppose n=1
in
equation [12].
Then
equation [12]
tells us that
p(hj)=p(hi)p(ij),
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.
If we were to graph the
Kronecker delta function, the value
of i
would be plotted along the
horizontal axis and the numeric value
of
would be plotted on the vertical axis as shown in
Fig 1
below. Here,
j=4,
and the graph shows that when
i=j=4,
then
=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. Notice that all the points are well below 1
since we need the sum of all the values to
equal 1.
Let's change the i axis to the more traditional
x
axis.
Then
would change
to 
,
which would be a function of the single
variable x,
and where the
constant j
has been changed
to x0.
And the discrete
probability
distribution
would change
to 
.
And
consider what happens as we let the
discrete x
become a continuous
variable. The result of going from a discrete to a continuous variable on the
Kronecker delta
function
is to change it to its continuous version
(x-x0),
called the
Dirac delta
function.
The requirement of a discrete sum in
equation [10]
becomes an integral relation as the number of
states, n,
goes to
infinity. Or,
[13]
where
R is some region of the
x axis.
And this defining characteristic of the Dirac delta function needs to be
preserved no matter how large or small the
region R
becomes so that the element
at x0
is counted only once if it's within the
region R.
The dx
is a differential
piece of length which shrinks to zero. And since the
region R
could be very,
very small, the function
(x-x0),
will
have to tend towards infinity
at x0 so that when multiplied
by dx
in this small region, the integral remains equal
to 1.
This
means that
(x-x0)
as xx0,
and
(x-x0)0
anywhere away
from x0.
And, of course, if we were to
integrate everywhere,
then x0
would certainly be included.
Or,
[14]
which
is mapped from the logical
equation [9],
and the Kronecker delta
equation [10].
The Dirac delta function must also satisfy the
following requirement,
[15]
So
that if 
were to be the Dirac delta function itself,
(x-x1),
we get
[16]
which
can be mapped from the logical
equation [11],
and the Kronecker delta
equation [12].
This is a recursion relation for the Dirac delta function.
We can iterate
equation [16]
again to get,
And
iterating an infinite number of times we get,
[17]
which
can be mapped from the logical
equation [7].
Now the question remains as to what function should
be used to represent the
(xn-xn-1)
in
equation [17].
Note that the Dirac delta functions in
equation [16]
are functions of one variable since
x and x0
are
being held constant
and x1
varies
from -
to +.
But in
equation [17],
the Dirac delta functions are functions of two variables since
now, nothing is being held constant and both its variables vary
from -
to +.
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 [14], [16],
and [17].
One such function is the gaussian form of the Dirac
delta,
[18]
It has the property that
as 
approaches zero, the
delta function becomes infinite in such a way that the integral of
equation [14]
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 in necessary in the
proof.)
For any
non-zero
value
of 
,
equation [18]
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 [18]
also satisfies the recursion relation of
equation
[16]
since,
[19]
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 it fair you use the gaussian distribution.
And
there seems to be some reason to expect
(xn-xn-1)
to be a complex number when used in the general case of
equation [17].
For there are now two variables in
(xn-xn-1), one for the premise and another for the conclusion. But we cannot
have
(xn-xn-1)=(xn-1-xn),
since
(pq)(qp).
We also cannot have
(xn-xn-1)=
(xn-1-xn),
since implication between any two propositions was defined as only
adding 1.
And we cannot have
(xn-xn-1)
= 1/(xn-1-xn),
since the reciprocal of
equation [18]
does not integrate
to 1 as approaches zero. So we are left with having to specify two numbers for the
representation of material implication, one number
for p
and a second
for q.
And the order of the numbers would depend on whether
(pq)
or
(qp).
So we are forced to treat
(xn-xn-1)
like a vector of two numbers.
In the paper, "Origin of Complex
Quantum Amplitudes and Feynman's Rules", arXiv:0907.0909v3,
the authors prove that if a pair of real numbers are assigned to outcomes of
experimental quantum setups and outcomes, and if those setups and outcomes can be added in
series and parallel combinations similar to using Feynman's rules, then the
algebra involved requires the number pairs to form a complex numbers. What
is interesting is that setup and outcomes of experiment in their paper can be
viewed as a type of premise and conclusion, the premise being the setup and
the conclusion being the outcome. And the Feynman rules they use for series
and parallel experiments behaves in the same way that conjunction and
disjunction are used on the implications developed here. They even use the
symbols of logical conjunction and disjunction for series and parallel
experiments. All this proves that the Dirac delta function needs to
be a complex number. And perhaps this justifies the use of a complex gaussian
function for the Dirac delta since by Euler's forumula,
which
would give us the complex number we need.
So let us make
the following substitution in
equation [18],
[20]
where m and
h-bar
are arbitrary
constants for the purposes here, and 
, then we can rearrange
equation [18]
to get,
which equals
[21]
Using m and
h-bar
above is not an attempt to covertly introduce physics.
I
only
use the
labels m
and
h-bar
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 [21]
into
equation [17],
we get
with the appropriate limits implied. And by gathering terms, this is equal to
[22]
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 [22]
can be recognized as the Feynman path integral for the wavefunction
of a free particle. The limits of the
delta-t
approaching zero is implied by
the notation
of dt
and
delta-t.
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 [17]
becomes
[23]
And
equation [21]
becomes
[24]
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 [22]
becomes
which
is the Feynman path integral for a particle in a potential which is called the
wavefunction
labeled, 
.
THE
BORN RULE OF PROBABILITIES
The Born rule tell us,
at least in part, that the probability
density, p(x),
for finding a particle
at x
that has a
wavefunction 
is
equal to the wavefunction times the complex conjugate of the wavefunction. Or
in symbols,
This
can be explained in the context of these efforts as follows:
Equation [4]
is
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 
generate the path integral which is another way of describing a
wavefunction, .
We learned that maps to a
complex number so that 
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
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 wavefunction representing
times the complex conjugate of the wavefunction representing .
If we
let q2
be located
at x,
then p(q2)
is replaced
by p(x),
and is represented
by ,
and is represented
by 
to get the Born rule:
where
must be interpreted as the square root of a probability.
This
means that the wavefunction 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 wavefunction
tells us what effect a cause will have, and the conjugate wavefunction 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.
THE
LARGER IMPLICATIONS
The quantum mechanics of the
wavefunction/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 above.
We started with the
fact that
[5]
which is an equality if at least one of
the qi
is true. And so it became necessary to evaluate
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
in
which the last conjunction is an equality if at least one of the 
is true which will be the case if at least one of
the 
is true. And if we let 
,
then we have
which
would necessitate the evaluation of
In
this case the mathematical representation of 
would be
where 
is the wavefunction of 1st
quantization and is the mathematical representation of 
.
The delta here would be expected to still be the complex gaussian
with 
replacing 
in the
exponential.
And 
would
replace 
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 fields' values so that only one measurement is needed to
deduce everything else. Would this be a
non-perturbative
approach to QFT? I
wonder.
The authors of the paper, "Origin of Complex
Quantum Amplitudes and Feynman's Rules", show that the complex numbers
arise in the algebra when a pair of numbers are used for entities that obey
the Feynman Rules which seem to behave similarly to conjunction and
disjunction on paths of implication for the purposes here. 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
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 conjectures.
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 wavefunction 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 cleaver 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.
I hope this has
encouraged some confidence that nature is run by perfectly logical principles
after all. And perhaps I've inspired some to look further into these efforts.
If you feel you
have some helpful comments on content or presentation, please email me at:
m_a_j_i_k_1
at c_h_a_r_t_e_r dot n_e_t.
(Remove
the underscores and spaces and use the correct dot and at symbols)
You may address me as Michael.
Thank you.