The Feynman Path Integra from First Principles

[Majik Won]

It now appears that the Feynman Path Integral of Quantum Mechanics can
be derived from logical considerations alone. This makes QM a branch of
mathematics. The details are at: http://logictophysics.com/QMlogic.html

The math is relatively easy; it's designed at the freshman/sophomore
level. The website is designed with references visible on mouseovers so
you don't have to scroll back and lose your place. There is even a
truth-table calculator to confirm logic equations without doing the work
yourself.

The derivation may seem a bit coincidental, but the framework developed
here also allows the particle content of the Standard Model to be
derived. The electron and positron appear to be fundamental, but the
other particle seem to be composites. The particles are derived at:
http://logictophysics.com/StandardModel.html

If the framework developed here is valid (please check), then we have a
physics derived from a limited number of mathematical axioms. We might
then be able to check for completion and consistency. And we might be
looking at a theory of everything. See:
http://logictophysics.com/ToE%20and%20ML.html

If anyone would like to discuss this, I'd be happy to debate the issues
here.

[Majik Won]

Some of the innovations that you may wish to review are the following:

I identified each point in space with a proposition. For each point in
space can be described by its unique coordinates, and a description is a
kind of proposition. This way it's possible to use logic to space, by
saying that it's true or false that each point exists or not as part of
the universe.

Then I used the Dirac measure to go from logic to math. The Dirac
Measure assigns a value of 1 if a particular element is included inside
a specific set; it's value is 0 if the element is not in that set. (See
https://en.wikipedia.org/wiki/Dirac_measure)

And set inclusion can be used to represent the material implication of
propositional logic. If the set exist, then so does its elements. So if
the element does not exist, then neither does the set. But just because
an element exists does not mean the set exists. This represents the
relationship of premise to conclusion, with the premise being the set,
and the conclusion being the element.

I also had to shrink the set of the Dirac measure so that it could
include or not only one particular point. This way I could associate the
Dirac set with a particular point. And it then becomes possible to
equate the Dirac measure with the Kronecker delta function. If the
associated point assigned to the set is the same as the point assigned
to the element, then the value is 1 as the Kronecker delta would be.

This reduction of the Dirac measure to the Kronecker delta allows me to
map the disjunction of logic to the addition of math. It allows me to
map conjunction to multiplication. Implication gets mapped to the
Kronecker delta function.

When we go to the continuum limit, the Kronecker delta goes to the Dirac
delta function, and the infinite sum goes to an integral. I then use the
Gaussian exponential function to represent the Dirac delta function. I
make the Gaussian complex to insure no preference in value is assigned
to any particular implication between points which are treated as
propositions; each implication only gets assigned a different phase.

With all these considerations, the Feynman Path Integral easily emerges
from the logic. In essence the wave function of quantum mechanics is a
mathematical representation of logical implication.

The details are at: http://logictophysics.com/QMlogic.html

[Majik Won]

Physics is described by particles moving along a path and what
influences the shape of that path. But a path can be formulated as
a series of steps through space and time. Each step follows from
another; the previous step is necessary before the next step can
occur. If you are at this point, then the next point will be here;
then if you are at that point, then the next point will be there,
etc. So each step can be viewed as an implication of logic, where
the next step necessarily follows from the previous step. So a path
is conjunction of implications. And if we map conjunction with
multiplication and implication with the Dirac delta function in the
continuum limit, then a path gets mapped to a multiplication of
exponential Gaussian functions. The exponents in all the Gaussian
functions for all of the steps gets added together. And what results
seems to be an action integral of a Lagrangian. Now we know where
the Lagrangian comes from.

What I've shown is that a material implication of logic can be
equated to an infinite disjunction of an infinite conjunction of
implications. And when mapped to the math, this becomes an infinite
number of integrations of an exponential of an action integral;
this is equal to the Feynman Path Integral.

[Majik Won]

The logical equations I use are rather easy. And once I starting using
the Dirac delta function, the math is also relatively
straightforward. So I think the most relevant question is on the
validity from going from logic to math. How legitimate is it to use the
Dirac measure to represent the material implication of propositional
logic? Is it legitimate to shrink the set of the Dirac measure to the
size where it can be represented by a point?

I'm not an expert in math. I simply noticed these connections. And they
seem to produce physics. So I could use some expert math opinions. But
given what's at stake (a theory of everything and the completion of
physics), one should be prepared to rigorously defend their views.

So I wonder where else is the Dirac measure used? What else is it used
to prove? Is it used anywhere is physics? Or is it a purely mathematical
construct? It is ever used to derive the Dirac delta function?

According to the Wikipedia article at:
https://en.wikipedia.org/wiki/Dirac_measure , the Dirac measure is an
indicator function, and it is also considered to be a probability
measure. So I have to wonder if there is any information associated with
this distribution. The value of the Dirac measure is either 1 or 0
depending on whether the element is in the set or not, so it seems there
is 1 bit of information associated with the Dirac measure. I wonder if
this may lead to a means of calculating the information content of
various regions of spacetime with energy, maybe even a means to
calculate a holographic principle or even a conservation of information
principle.

Shrinking the set of the Dirac measure so that it surrounds just a point
seems reminiscent of the Hausdorff property of a manifold, where a
manifold is specified by the fact that each point has a neighborhood (a
set) that contains the point but separates it from any other point no
matter how close. So it seem my construction specifies a manifold as
well. And isn't this the same thing done in calculus, where a point
within in an infinitesimal region is used to represent the infinitesimal
region? Is this the axiom of choice a requirement to to this?

[b***@hotmail.com]

The website at
http://www.logictophysics.com/welcome.html
seems to be relevant.

I am an amateur physicist and this approach seemed interesting to
me, though I am not convinced that inserting Dirac deltas into the
logic equations comes naturally. That said, the only time I met
'implicates' was in reading this work to try to understand it. I
too like the drawing of the net around a set of points to find the
implicate. Like wave function collapse.

Physicists may not like it though.
See http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=33&hilit=born+rule

and https://groups.google.com/forum/#!searchin/sci.physics.foundations/implicate/sci.physics.foundations/BPWsDJ8Bcc8/3LMC4y03_mIJ

Best wishes

[[Mod. note --
It seems to me that this approach is fundamentally flawed
because it's based on a sort of "ontological type error". That is, this
approach is trying to use *mathematics* (which is ultimately about the
logical consequences of various axioms, without reference to what those
axioms might have to do with real world) to make inferences about the
real world (which is what physics is about).

Consider a mathematical theorem such as, say, the uniform limit theorem,
https://en.wikipedia.org/wiki/Uniform_limit_theorem
That theorem follows as a mathematical consequence of a set of axioms
(e.g., the ZFC axioms of set theory) + the usual definition of integers
and integer arithmetic, then the definition of rational numbers & rational
arithmetic in terms of integer arithmetic, then real numbers (as either
Dedikind cuts or equivalence classes of Cauchy sequences) in terms of
rational numbers, thens the notion of continuity, convergence, and uniform
convergence in terms of epsilon-delta analysis. Each step in this process
is mathematically valid and can be (and is, in many standard textbooks)
written out in sufficient detail to convince any reasonable person that
it does indeed follow from the stated axioms. (This level of detail is
typical for how professional mathematicians write mathematics.) Indeed,
I suspect that some or all of these steps have in fact now been formally
verified as following from the stated axioms. So, we know that the
uniform limit theorem is a logical consequence of our original axioms.

*BUT* nothing in this mathematical analysis tells us anything at all
about the real world. Nothing in this analysis tells us that there are
any things in the real world for which the mathematical concept of (say)
"real number" or "continuous function" is a good model. *That* would
require actual observation of the real world (which of course is what
physics and physicists do all the time).

And if we do that... all we can really learn is that there are things
in the real world which can be well-modelled by (say) continuous functions.
But that doesn't tell us that those things actually obey the mathematical
properties of continuous functions. After all, if something "looks like
it's continuous", but it's really discontinuous by 1 part in 10^100, we're
unlikely to notice that experimentally... but that's still enough to break
the mathematical proof, because that "arbitrary $\epsilon > 0$" in the
proof is allowed to be (much) smaller that 1/10^100.

So, I do not think that mathematical analysis alone can tell me what path
integral actually describes quantum mechanics in the real world. It may
well be that a mathematical analysis can show that Feynman's form has
various nice properties and consequences, but without experiment we have
no way to know that the path integrals correctly describe the observed
behavior of real-world systems.
-- jt]]

[Jos Bergervoet]

Post by b***@hotmail.com
The website at
http://www.logictophysics.com/welcome.html
seems to be relevant.
I am an amateur physicist and this approach seemed interesting to
me, though I am not convinced that inserting Dirac deltas into the
logic equations comes naturally. That said, the only time I met
'implicates' was in reading this work to try to understand it. I
too like the drawing of the net around a set of points to find the
implicate. Like wave function collapse.
Physicists may not like it though.
See http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=33&hilit=born+rule
and https://groups.google.com/forum/#!searchin/sci.physics.foundations/implicate/sci.physics.foundations/BPWsDJ8Bcc8/3LMC4y03_mIJ
Best wishes
[[Mod. note --
It seems to me that this approach is fundamentally flawed
because ...

...

Post by b***@hotmail.com
So, I do not think that mathematical analysis alone can tell me what path
integral actually describes quantum mechanics in the real world. It may
well be that a mathematical analysis can show that Feynman's form has
various nice properties and consequences, but without experiment we have
no way to know that the path integrals correctly describe the observed
behavior of real-world systems.

But what question is there about them? Aren't they just derived
from the choice of field theory?

I'd think that the path integral merely is a way to compute the
time evolution from a state [*] at t1, to a new state at t2. And
it does this by summing over all paths weighed by the complex
exponential of the action, which by itself is derived from the
Lagrangian density, which is fixed by our choice of field theory.

So it adds nothing more than a mathematical tool, comparable to
doing a lattice calculation, which would also just have to stick
to whatever field theory we choose to describe the universe..

[*] Actually, the 'state' in general can be the wave functional
of a quantum field, or the much simpler wave function of a
Schroedinger equation, or the position of a classical point
particle, etc. The tool is quite versatile, but computes just
what could also be computed in other ways.

[Majik Won]

[Moderator's note: I have re-formatted this post. It arrived with each
paragraph as one long line. While YOUR software might insert line
breaks, not everyone's does. Please format the post as it should look
without relying on some softare to do so, i.e. write a text file with
explicit line breaks then send that "as is". --P.H.]

Post by b***@hotmail.com
[[Mod. note --
It seems to me that this approach is fundamentally flawed
because it's based on a sort of "ontological type error". That is, this
approach is trying to use *mathematics* (which is ultimately about the
logical consequences of various axioms, without reference to what those
axioms might have to do with real world) to make inferences about the
real world (which is what physics is about).

"fundamentally flawed" can not be asserted without proof. Show me
mathematical proof that I made a mistake, and I'll shut up and go back
to the drawing board. If my approach is mathematically correct, then the
question is whether it is unique.

If physics can not be derived from logic, then what you seem to be
suggesting is that physics might be something that is true though not
provable by logic. I can't imagine what observation would prove this,
can you?

Whatever the ontology of reality, it seems inescapable that it is a
collection of things that all coexist in logical conjunction. I start
with this assumption and assume a conjunction of things down to the
smallest conceivable things, that every point in space exists in
conjunction with every other point in space. That this conjunction means
every point implies every other is a simple logical consequence. That
set inclusion also specifies implication is also obvious. That the Dirac
measure is about set inclusion is also obvious. All this leads to my
approach to deriving quantum theory.

Post by b***@hotmail.com
*BUT* nothing in this mathematical analysis tells us anything at all
about the real world. Nothing in this analysis tells us that there are
any things in the real world for which the mathematical concept of (say)
"real number" or "continuous function" is a good model. *That* would
require actual observation of the real world (which of course is what
physics and physicists do all the time).

So you seem to be asking what logic has to do with existence. To me it
seems we can not escape it. Ontology has to do with what exists. But
existence is a binary concept such that binary logic applies. Either
something exists or it does not. There is not middle ground. So we are
perfectly justified in using propositions which are either true or false
to describe what exists or not. This is just the correspondence
principle between what is true and what exists. Physicists use
propositions to describe what exists without exception. So it is
unavoidable that logic should be applied to reality.

Post by b***@hotmail.com
... but without experiment we have
no way to know that the path integrals correctly describe the observed
behavior of real-world systems.
-- jt]]

The "experiment" is the observation that all the various parts of
reality coexist in logical conjunction. Surely you've noticed this. Then
in order to even talk about the world or theorize about reality, we have
to assume we can use proposition which are either true or false to
describe various parts of reality which either exist or not. Are you
assuming that there is some scale of reality that does not exist in
conjunction?

[Jos Bergervoet]

[Moderator's note: New text moved to bottom and quoted text trimmed
somewhat; top-posting means that one has to look to the bottom first for
context. -P.H.]

Post by Majik Won
Show me
mathematical proof that I made a mistake, and I'll shut up and go back
to the drawing board. If my approach is mathematically correct, then the
question is whether it is unique.
If physics can not be derived from logic, then what you seem to be
suggesting is that physics might be something that is true though not
provable by logic. I can't imagine what observation would prove this,
can you?
Whatever the ontology of reality, it seems inescapable that it is a
collection of things that all coexist in logical conjunction. I start
with this assumption and assume a conjunction of things down to the
smallest conceivable things, that every point in space exists in
conjunction with every other point in space. That this conjunction means
every point implies every other is a simple logical consequence. That
set inclusion also specifies implication is also obvious. That the Dirac
measure is about set inclusion is also obvious. All this leads to my
approach to deriving quantum theory.

Post by b***@hotmail.com
*BUT* nothing in this mathematical analysis tells us anything at all
about the real world. Nothing in this analysis tells us that there are
any things in the real world for which the mathematical concept of (say)
"real number" or "continuous function" is a good model. *That* would
require actual observation of the real world (which of course is what
physics and physicists do all the time).

So you seem to be asking what logic has to do with existence. To me it
seems we can not escape it. Ontology has to do with what exists. But
existence is a binary concept such that binary logic applies. Either
something exists or it does not. There is not middle ground. So we are
perfectly justified in using propositions which are either true or false
to describe what exists or not. This is just the correspondence
principle between what is true and what exists. Physicists use
propositions to describe what exists without exception. So it is
unavoidable that logic should be applied to reality.

Post by b***@hotmail.com
... but without experiment we have
no way to know that the path integrals correctly describe the observed
behavior of real-world systems.
-- jt]]

The "experiment" is the observation that all the various parts of
reality coexist in logical conjunction. Surely you've noticed this. Then
in order to even talk about the world or theorize about reality, we have
to assume we can use proposition which are either true or false to
describe various parts of reality which either exist or not. Are you
assuming that there is some scale of reality that does not exist in
conjunction?

Post by b***@hotmail.com
[[Mod. note --
It seems to me that this approach is fundamentally flawed
because it's based on a sort of "ontological type error". That is, this
approach is trying to use *mathematics* (which is ultimately about the
logical consequences of various axioms, without reference to what those
axioms might have to do with real world) to make inferences about the
real world (which is what physics is about).

"fundamentally flawed" can not be asserted without proof.

Why do you believe that? If someone says:
"I know 2+2=4 because I had a revelation telling me so"
is that fundamentally flawed or not? Can you prove it?

What you seem to say is: (admittedly not the same as above)
"Feynmans path integral is the correct description of the universe
because certain mathematical equations that I have derived are
consistent with it."
That does not sound as a proof, not even as something interesting.

I have only glanced through your paper and, like most other
physicists, will not do more than that unless you first tell
us why it is relevant. Basically your abstract has to explain
this, it doesn't do a good enough job right now, I'm afraid..

We already know that the path integral describes the time evolution
of quantum mechanical amplitudes [*], and does so in a logically
consistent way (that's actually why states become *entangled*
the way they do).

Is your approach simply restating this with a new notation?
Or are you offering something new? If the latter, then please
answer: "what do we know by using your approach, that we would
_not_ know without it?"

[*] Here I'm assuming that this is the path integral application
you mean, the mathematics of it can most likely be used for many
other things like finance, biological evolution, etc.

Regards,

[Majik Won]

[Moderator's note: I have re-formatted this post. It arrived with each
paragraph as one long line. While YOUR software might insert line
breaks, not everyone's does. Please format the post as it should look
without relying on some softare to do so, i.e. write a text file with
explicit line breaks then send that "as is". --P.H.]

Post by b***@hotmail.com
I am an amateur physicist and this approach seemed interesting to
me, though I am not convinced that inserting Dirac deltas into the
logic equations comes naturally.

It did take me a few years to find how the Dirac measure can be used.
But now it seem unavoidable. The Dirac measure is about set inclusion
which can also be used to represent logical implication. This seems to
be the most direct and obvious way to get from the most basic property
of numbers, 1 and 0, to the most basic property of logic, true and
false. How did we miss this? Isn't this just your basic property of
cardinality? Isn't this the most basic property of probability and
implication?

I recently found a website stating that the Dirac delta function is the
same as the Dirac measure. It's at:

https://www.sciencedirect.com/topics/engineering/dirac-measure

Search for: the Gaussian distribution is degenerated: its probability
density f is a Dirac measure f(x) = Î'(x â' m)

Although I don't know exactly what math is involved when they go from
the measure to the delta function. But it at least shows that I'm not
the only one doing this.

What got me started was to recognize that the iterative property of
implication, equation [6] at: http://logictophysics.com/QMlogic.html is
mimicked or mapped to the iterative property of the Dirac delta
function, equation [20], which is mapped to the iterative property of
the Gaussian function, equation [24]. As far as I know, the Gaussian
function is the only equation I know that has this iterative property.
This is called the Chapman-Kolmogorov equation. And I've not seen it
worked out for any function other than the exponential Gaussian.

[Majik Won]

Post by Jos Bergervoet
But what question is there about them? Aren't they just derived
from the choice of field theory?
I'd think that the path integral merely is a way to compute the
time evolution from a state [*] at t1, to a new state at t2. And
it does this by summing over all paths weighed by the complex
exponential of the action, which by itself is derived from the
Lagrangian density, which is fixed by our choice of field theory.
So it adds nothing more than a mathematical tool, comparable to
doing a lattice calculation, which would also just have to stick
to whatever field theory we choose to describe the universe..
Jos

But this all begs the question as to where the fields come from to
begin with or why the Lagrangian or where the path integral comes
from. I think my construction derives all these things. Accordingly,
the quantum field arise directly from spacetime itself, precisely
because the points of spacetime coexist in conjunction, leading to
implication between all points. And number of ways these implications
can be combined and how the implication between implications, etc,
can be combined is what forms the various kinds of particles of
physics. I show this at:
http://logictophysics.com/StandardModel.html

[Jos Bergervoet]

Post by Majik Won

Post by Jos Bergervoet
But what question is there about them? Aren't they just derived
from the choice of field theory?
I'd think that the path integral merely is a way to compute the
time evolution from a state [*] at t1, to a new state at t2. And
it does this by summing over all paths weighed by the complex
exponential of the action, which by itself is derived from the
Lagrangian density, which is fixed by our choice of field theory.
So it adds nothing more than a mathematical tool, comparable to
doing a lattice calculation, which would also just have to stick
to whatever field theory we choose to describe the universe..
Jos

But this all begs the question as to where the fields come from to
begin with or why the Lagrangian or where the path integral comes
from.

Exactly! None of the mathematical equation handling (yours included)
can answer that question. Of course the same holds for Newton's
classical point-particle world. Classical physics does not explain
why there should be point particles in the first place, or why there
should be forces between them.

Post by Majik Won
I think my construction derives all these things. Accordingly,
the quantum field arise directly from spacetime itself,

Why? You definitely do not prove that any 4-dimensional manifold
must inevitably have a set of quantum fields living inside it.

Post by Majik Won
... precisely
because the points of spacetime coexist in conjunction, leading to
implication between all points. And number of ways these implications
can be combined and how the implication between implications, etc,
can be combined is what forms the various kinds of particles of
physics.

What does this mean? Points in a 4 dimensional space always
"coexist in conjunction" but how does this prove that quantum
fields are present?

Post by Majik Won
http://logictophysics.com/StandardModel.html

You go even further: there you write that even the number of particle
species, charge ratios and the presence of three particle generations
are proven by your theory. So then you not only claim to prove that
the standard model of physics must exist, but that it is the only
possible set of quantum fields that can exist.

From the mere fact that points in spacetime always coexist in
conjunction, you would be able to prove all this?!

[NB: I'm still trying to find out what exactly you are claiming, and
what are the postulates about spacetime that you use as a starting
point to get there! At the moment it seems you derive something from
nothing.]

[Majik Won]

Post by Jos Bergervoet

Post by Majik Won

Post by Jos Bergervoet
But what question is there about them? Aren't they just derived
from the choice of field theory?
I'd think that the path integral merely is a way to compute the
time evolution from a state [*] at t1, to a new state at t2. And
it does this by summing over all paths weighed by the complex
exponential of the action, which by itself is derived from the
Lagrangian density, which is fixed by our choice of field theory.
So it adds nothing more than a mathematical tool, comparable to
doing a lattice calculation, which would also just have to stick
to whatever field theory we choose to describe the universe..
Jos

But this all begs the question as to where the fields come from to
begin with or why the Lagrangian or where the path integral comes
from.

Exactly! None of the mathematical equation handling (yours included)
can answer that question. Of course the same holds for Newton's
classical point-particle world. Classical physics does not explain
why there should be point particles in the first place, or why there
should be forces between them.

Post by Majik Won
I think my construction derives all these things. Accordingly,
the quantum field arise directly from spacetime itself,

Why? You definitely do not prove that any 4-dimensional manifold
must inevitably have a set of quantum fields living inside it.

Post by Majik Won
... precisely
because the points of spacetime coexist in conjunction, leading to
implication between all points. And number of ways these implications
can be combined and how the implication between implications, etc,
can be combined is what forms the various kinds of particles of
physics.

What does this mean? Points in a 4 dimensional space always
"coexist in conjunction" but how does this prove that quantum
fields are present?

Post by Majik Won
http://logictophysics.com/StandardModel.html

You go even further: there you write that even the number of particle
species, charge ratios and the presence of three particle generations
are proven by your theory. So then you not only claim to prove that
the standard model of physics must exist, but that it is the only
possible set of quantum fields that can exist.
From the mere fact that points in spacetime always coexist in
conjunction, you would be able to prove all this?!
[NB: I'm still trying to find out what exactly you are claiming, and
what are the postulates about spacetime that you use as a starting
point to get there! At the moment it seems you derive something from
nothing.]
--
Jos

No, I don't start from nothing. I assume that a background spacetime
exists. It's still a mystery to me how spacetime expands, especially
from a singularity. The key to my developments is that conjunction
(of any two points in space) leads to an implication. And through
the use of Dirac measure, implication leads to complex Gaussian
which leads to a wavefunction. Since a conjunction gives rise to
an implication both ways, there is an wavefunction for both directions
in time. This accounts for a particle and an antiparticle traveling
synchronously as an entangled virtual pair. Or in other words, we
see here that points of spacetime are entangled. And this begins
to look like the efforts to find quantum gravity by means of the
entanglement of various points of space.