A generalization of the path integral formulation of quantum theory which
suggests the existence of multiple universes
zerologics@gmail.com
Keywords: Path Integral Formulation, Multiverse, Anthropic Principle, Time Slicing
Abstract:
In the present work, we describe a generalization of the time-slicing approach to
path integration that suggests the existence of many different universes corresponding
to alien laws of physics. Our generalization consists of allowing the number of slices in
the time-discretization procedure to be complex. As there exist many different ways in
which a complex number may approach infinity in the plane, this apparently leads to the
existence of many different laws of motion. We speculate that these correspond to
different universes; after developing the necessary mathematical mechanisms, we
calculate how these universes behave in the case of free particles.
1.
A Brief Introduction
A very simple generalization of the path integral formulation of quantum theory
may lead to some new developments in particle theory. If we fully acknowledge the
existence of complex numbers -- by infusing them into the time-slicing procedure of path
integration -- a description of many parallel universes results. These universes have
differing physical laws, and -- in particular -- in these different universes conservation of
energy is violated. This application of complex numbers to Feynman’s formulation may
lead to some startling new discoveries in the field of subatomic physics.
In this paper I shall describe this generalization of the path integration procedure.
It would be helpful for the reader to be familiar with the work Quantum Mechanics and
Path Integrals (by Richard Feynman) so that she may understand some of the basic
theory foundational to this work. However, at least a brief explanation of Feynman’s
formulation would be useful to provide.
1.1 The Time-Discretization Approach to Path Integration
The basic idea behind Feynman’s path integral approach to quantum mechanics
is simple and elegant. To calculate the probability amplitude that any event will occur,
we add up the probability amplitudes for each way in which the event may occur. In
particular, to calculate the probability amplitude that a particle may move from one
location to another, we add up the probability amplitudes corresponding to each path
the particle may travel along. Feynman gave the contribution each path makes to the
total probability amplitude, and the only problem that remains is calculating a sum of all
the contributions of the paths.
Feynman describes in Quantum Mechanics and Path Integrals a very simple
approach to this problem1. First of all, it is important to note two facts: (1) that we are
considering paths through space-time, and not only through space, and (2) that we are
considering only one spatial dimension, or two dimensions when time is included.
Feynman began by considering only a discrete number of times and corresponding
positions of the particle.
Xn (xb ,tb)
X3
X2
X1
X0 (xa, ta)
Figure 1-The time-slicing approach to path integration.
Figure 1 shows the situation: a particle moves from xa at time ta to xb at time tb.
To calculate the associated probability amplitude, we consider n discrete times and
corresponding positions of the particle. We imagine that the particle moves along
straight lines between these times.
Next, Feynman incorporates the fundamental idea of calculus – that of
approximation. We may approximate a general curved path with a path composed of
straight lines. As we increase the number of straight lines, the approximation becomes
more and more accurate. So, as we increase n in our discrete paths, they do a better
and better job of approximating a general curved path.
To carry this out, we must know the form of contribution each path makes to the
whole probability amplitude. To agree with experiments, Feynman suggested a
contribution of the form:
𝑖
exp {ℏ 𝑠}
where s is the action of the path.
It is important to note that S is defined as
∫
𝑚 2
𝑥′ − 𝑉 𝑑𝑡
2
In which V is the potential acting on the particle and x’ is defined, for a single leg
of our discrete path, to be
𝑥𝑖+1 − 𝑥𝑖
𝜀
(the xi and xi+1 are the two endpoints of the leg of our path). Here,  measures the
relative magnitude, or size, of each and every time slice. In our case, it is easy to see
that , the size of each stretch of time, can be defined as
𝑇
𝑛
In which T is the total stretch of time across all the slices.
Now, suppose in the straight-line paths that we label the positions of the
particle x0, x1, … xn (see Figure 1). To add up the contributions from straight-line paths
with given n, we merely integrate the contribution formula over all possible x1…xn-1. By
our previous observation, as we increase n, this answer will approach the total
probability amplitude in which all paths are taken into account. Thus, we are led to the
equation:
𝑖
𝑃 = lim ∬ … ∫ exp { 𝑠} 𝑑𝑥1 … 𝑑𝑥𝑛−1
𝑛→∞
ℏ
for the total probability amplitude of a particle moving from one location to
another.
This equation is incorrect in one very important respect.
Of course, any reader of Quantum Mechanics and Path Integrals should know
that this limit, as it stands, is utterly divergent. To force this limit to converge, and to
normalize the result (in other words, to make the integral of the probability amplitude
over all xb equal to 1), Feynman introduced a normalization constant, a function of n that
multiplies the multiple integral within the limit. Taking this important function into
account, we have:
(1)
𝑖
𝑃 = lim 𝐴(𝑛) ∫ ∫ … ∫ exp {ℏ 𝑠} 𝑑𝑥1 … 𝑑𝑥𝑛−1
𝑛→∞
It is important to note here that the integrals extend from -∞ to +∞.
1.2 An Introduction to the Generalization
The generalization I wish to propose in the present work is very simple.
Feynman only considered an integer number of points of time in the straight-line paths.
However, if we allow for the existence of a complex number of “time slices” in
Feynman’s path integral expression, we are led to a profound and very startling
conclusion. Figure 2 illustrates that there are many ways in which it is possible to
approach infinity in the complex plane; thus, if n in Equation 1 is allowed to take on a
complex value, many different probability amplitudes exist for the particle’s motion, each
corresponding to a different way in which n may approach infinity in the complex plane.
These different physical laws seemingly correspond to different universes, each with its
own laws of motion. The different ways in which n may approach infinity would then
correspond to different universes.
Figure 2-Some ways in which a complex number may approach infinity in
the plane (a very small sample!)
Now, it is a mathematical tradition to denote integers by n and non-integers
perhaps by q, so we will replace Equation 1 by:
(2)
𝑖
𝑃 = lim 𝐴(𝑞) ∫ ∫ … ∫ exp {ℏ 𝑠} 𝑑𝑥1 … 𝑑𝑥𝑞
𝑞→∞
where q is complex. This describes the infinity of possible universes according to
this generalization.
It is important that, in order for this generalization to work, the limit in equation (2)
must not exist. That is, in order for the different universes to be separate from one
another,the limit in equation (2) must not have a single value but rather must have a
multitude of values, each corresponding to a different way in which q may approach
infinity. It is certainly true that this is the case with a variety of functions of q (for
example, exp{-q}, which approaches zero on the real line and oscillates on the
imaginary axis), and we hope that this is the case for equation (2).
Indeed, to ensure that each and every limit yields a different result, we shall
redefine the action S to introduce an absolute value term. As often occurs in complex
analysis, the introduction of an absolute value allows each different limit in the complex
plane to yield a different result (for example, |q|/q approaches different limits in the
complex plane). Thus, we shall still define S as
∫
𝑚 2
𝑥′ − 𝑉 𝑑𝑡
2
But in this case our  is now defined as
𝑇
|𝑞 + 1|
While this may seem an arbitrary attempt to secure the theoretical edifice of the
theory, it is actually, it seems, a natural way to define the relative size or magnitude of
each time slice. After all, the relative magnitude of a complex number is described by an
absolute value. So, this simple and natural re-definition of the action for different limits in
the complex plane not only ensures the existence of different limits, but comes from a
simple definition of , which describes the relative size or magnitude of the time-slices.
Now, the skeptic might ask, What does it even mean to integrate over a complex
number of different variables? In other words, how do we even define the expression we
wrote above? It is true that we do need to develop some sort of definition of the process
of integrating over a complex number of variables in order to give meaning to (2). But it
is still certainly possible, at least in principle, to give such a meaning, as witness for
example the fractional calculus, which defines derivatives with a complex order by
utilizing properties kept by derivatives of an integer order2. Indeed, we shall develop a
meaning for (2) along these lines later in this paper.
To formalize the result, we note that each of the paths in Figure 2 may be
described by a parametric formula of the form 𝑥 = 𝑓(𝑡), 𝑦 = 𝑔(𝑡). Either 𝑓 or 𝑔
approaches infinity in the examples. Thus, we have a more formal version of Equation
2:
(3)
𝑃=
lim
𝑓 𝑜𝑟 𝑔 𝑜𝑟 𝑏𝑜𝑡ℎ→∞
𝑖
𝐴(𝑓(𝑡) + 𝑖𝑔(𝑡)) ∫ ∫ … ∫ exp {ℏ 𝑠} 𝑑𝑥1 … 𝑑𝑥𝑓(𝑡)+𝑖𝑔(𝑡)
A number of seemingly insurmountable difficulties confront us. First of all, what
definitions shall we use to integrate over a complex number of variables? And second
of all, how would we ever test this generalization experimentally? And why should we
consider this idea, anyway? It is impossible to divide a time interval into a complex
number of pieces, isn’t it?
I shall consider some of these questions later in this paper. However, I shall now
address the last one. History has shown, again and again, that generalizing the
concept of number has proved vital in the progression of science. While complex
numbers themselves were once viewed as a numerical monstrosity, their application to
physics was essential to quantum theory. Another prime example was the concept of
negative energy which, while seemingly absurd, was utilized by Dirac to develop the
very notion of antimatter. Additionally, the fractional calculus, which defines derivatives
of a complex order, is becoming essential to many practical applications2.Following
these historical guides, we note that another generalization to complex numbers may
prove useful to physics. Even if the theory itself proves incorrect, these results may
become useful in other areas of science.
With that, we should study the foundational mathematics of the theory.
2. Integration Over a Complex Number of Variables
We shall base the mathematics of this theory, which involves multiple integration
over a complex number of variables, on two fairly simple axioms:
1)
Separation. This axiom will be quite critical in what follows. In fact, this
axiom has two parts: that of sums and that of products. The first part is
simple. We merely generalize the sum rule from integration, and claim that
it holds for integrals with a complex number of variables (examples will
follow in another section). The second part to this axiom is almost as
simple.
Consider the multiple integral
𝑏
𝑏
𝑏
∫ ∫ … ∫ 𝑓1 (𝑥1 )𝑓2 (𝑥2 ) … 𝑓𝑛 (𝑥𝑛 ) 𝑑𝑥1 … 𝑑𝑥𝑛
𝑎
𝑎
𝑎
Quite obviously, by the very nature of a multiple integral, this may be
written as
𝑏
𝑏
𝑏
∫ 𝑓1 (𝑥1 )𝑑𝑥1 · ∫ 𝑓2 (𝑥2 )𝑑𝑥2 … · ∫ 𝑓𝑛 (𝑥𝑛 )𝑑𝑥𝑛
𝑎
𝑎
𝑎
If all these integrals are the same, we may simply calculate a power to
get our result.
We merely generalize this law to the case of multiple integrals with a
complex number of variables. These two parts to postulate 1 will be very
important to the theory, as they will allow us to integrate a vast variety of
functions, some of which that will arise in physical situations.
2)
Substitution. Many problems that arise in the context of path integration
are not, at least immediately, of a form that separation can be directly
applied to. Our strategy shall be to reduce the integrand to a simpler
expression, one that can be separated. Our strategy shall utilize
substitution.
Of course, integral substitution is a fundamental technique of
integration, and is taught in virtually all calculus courses that cover
integrals. The Jacobian becomes important when one studies usubstitution with multiple integrals, and with each substitution, the
integrand is multiplied by the Jacobian function.
Consider the integral
∫ … ∫ 𝑓(𝑥1 … 𝑥2 ) 𝑑𝑥1 … 𝑑𝑥2
Now, suppose we make the substitution
𝑧1 = 𝑥1 − 𝑎 (a is constant)
𝑧2 = 𝑥2 − 𝑥1
⋮
𝑧𝑛 = 𝑥𝑛 − 𝑥𝑛−1
(The reason why we consider this type of substitution shall be
considered later). We postulate (or, rather, define) that this substitution
corresponds to a Jacobian of 1, even when integrals are considered over
a complex number of variables. This seems quite clear, and it is certainly
true for an integer number of variables; thus, it makes sense to generalize
this to the case of a complex number of variables.
These postulates must be followed with a little explanation. Frequently, we shall
use these postulates to simplify expressions involving sums over a complex number of
terms. These sums are meaningless entities in and of themselves; one cannot add up
the first i integers as one might sum the first 6 square numbers. However, when placed
within a multiple integral, as the examples that follow will show, these sums do gain a
meaning. This illustrates an important lesson: the integrands of our bizarre integrals are
meaningless, but the integrals have an actual value. This is not altogether different from
the situation with differentials, which while meaningless in and of themselves, gain
meaning when placed within an integral.
Now, we come to the critical question: Why consider these axioms and none
others? It is because these axioms, dealing with the fundamental properties of
separation and substitution, are really the simplest way to define these integrals in a
way relevant to path integration. Those three properties are really the foundational
pillars of integration and sums, and thus it makes sense to use them to extend the
definition of integration.
However, we must wonder, Why would these definitions be chosen by nature
above others? History shows us that often the simplest definitions are the ones most
pertinent to nature. For example, negative numbers were defined to operate in very
simple ways so as to preserve the properties of positive numbers, and their application
to energy levels by Dirac was successful. Following these historical examples, we
define these integrals in a simple way, so as to preserve the most fundamental
properties of integrals and sums.
Incidentally, when we refer to 𝑞 𝑟 , with q and r complex, we refer to the principal
value of that quantity. This is quite clear, as this is how the integrals are defined for a
real number of variables; thus, to preserve the properties held by normal multiple
integrals, we shall use this convention.
3. Notation and Terminology
Our notation shall be quick and efficient. Let us denote the multiple integral
𝑏
𝑏
𝑏
∫ ∫ … ∫ 𝑓(𝑥1 … 𝑥𝑞 ) 𝑑𝑥1 … 𝑑𝑥𝑞
𝑎
𝑎
𝑎
as
∫
𝑏
𝑎𝐶𝑞
𝑓(𝑥1 … 𝑥𝑞 ) 𝑑𝑖𝑣
Where we contract the original equation into a much simpler form.
I call a multiple integral over a complex number of variables a “complintegral”, for
“complex” and “integral”. Therefore I shall call this theory “Quantum
Complintegrodynamics”, as it is a quantum theory of motion.
Before we get too far into the theory, I must clear up some confusion that might
have befuddled the reader: our x1…xq take on only real values. One never measures a
particle having a “complex” position, like being i units to the left of the origin. In fact, this
theory should result in all real observable quantities. Indeed, the wavefunction is
designed so that it always gives a real answer for the probability that a particle will be
found in a given region, regardless of how “imaginary” the original wavefunction is. We
are, in fact, using complex numbers intermediately to give interesting information about
our “real” universe. Indeed, this goes along with the well-known mathematical proverb,
“The shortest path between two points in the real domain goes through the complex.”
4. Examples of complintegral calculation
Here, I will provide some examples of how the axioms above may be used to
calculate a variety of complintegrals.
Example 1. To calculate the integral
𝑞
∫
1𝐶
0 𝑞
∑ 𝑥𝑛 𝑑𝑖𝑣
𝑛=1
This is excruciatingly simple. Just apply the first part of axiom 1 to take the sum
sign out of the integral sign, yielding
𝑞
∑ ∫ 𝑥𝑛 𝑑𝑖𝑣
1
𝑛=1 0𝐶𝑞
𝑏
Now, ∫𝑏𝐶 𝑓(𝑥𝑛 )𝑑𝑖𝑣 is simply (𝑏 − 𝑎)𝑞−1 · ∫𝑎 𝑓(𝑥)𝑑𝑥 (which comes from multiplying
𝑎 𝑞
𝑓(𝑥𝑛 ) by 1𝑞−1 and applying separation), and thus, the answer is simply q.
Example 2. To calculate
𝑞
∫ ∏ 𝑥𝑛 𝑑𝑖𝑣
2𝐶
0 𝑞
𝑛=1
This is even simpler than the last problem. By axiom 2, this is simply 2 q.
This example brings up an important point. How do we know that we can just go
around, pulling products and sums out of integral signs, without worrying about
convergence? Well, notice: for real q, we may pull the product sign out of the
integral. Thus, it seems only appropriate that we may define our complintegrals
to conserve that property when generalizing to the complex (indeed, we did:
that’s what axiom 2 was about!)
Example 3. To calculate the complintegral
𝑞
∫ ∞𝐶 ∑𝑛=0 𝑒𝑥𝑝(−[𝑥𝑛+1 − 𝑥𝑛 ]2 ) 𝑑𝑖𝑣 (where 𝑥𝑞+1 is some constant, along with 𝑥0 )
−∞ 𝑞
We shall illustrate axiom 2 by using that principle to simplify this expression. We
substitute 𝑧𝑛 for 𝑥𝑛+1 − 𝑥𝑛 and obtain
𝑞−1
∫
1𝐶
0 𝑞
2
∑ 𝑒𝑥𝑝(−(𝑧𝑛 )2 ) + 𝑒𝑥𝑝 (−[𝑥𝑞+1 − 𝑥𝑞 ] ) 𝑑𝑖𝑣
𝑛=0
Now,𝑥𝑞 = 𝑧𝑞−1 + 𝑥𝑞−1 = 𝑧𝑞−1 + 𝑧𝑞−2 + 𝑥𝑞−2 = ⋯ = 𝑧𝑞−1 + 𝑧𝑞−2 … + 𝑧0 + 𝑥0 . Thus,
we have
𝑞−1
∫
1𝐶
0 𝑞
𝑞
2
∑ 𝑒𝑥𝑝(−(𝑧𝑛 )2 ) + 𝑒𝑥𝑝 (− [𝑥𝑞−1 − ∑ 𝑧𝑞−𝑛 − 𝑥0 ] ) 𝑑𝑖𝑣
𝑛=0
𝑛=1
This expression, as it turns out, may be simplified using axiom 1, by expanding
the second term in a Fourier Integral of easier-to-deal-with functions and then applying
separation. I have described this general process in detail in the section “The
Complintegrodynamical Problem of a Free Particle”, where a very similar problem
appears. Thus, I shall not waste several pages of my paper explaining a long process
that I shall describe later. Suffice it to say that axiom 2 allows us to manipulate
expressions, and bring them to a form that separation may be applied to. Of course,
separation could be applied in the very beginning of this problem; but the purpose was
not to solve it the most efficient way possible, but to illustrate axiom 2; similar somewhat
to the way classes in differential equations sometimes solve simple equations by
complex methods just to illustrate those procedures.
Now that we have completed the mathematical theory and illustrated the basic
postulates, the next step is to turn to physics. We now turn to the problem of the free
particle.
5. The Quantum Complintegrodynamical Problem of a Free Particle
In Feynman’s book, one of the first situations considered is that of a free particle.
This elementary problem serves to illustrate many of the ideas Feynman developed
earlier. So, certainly, the most natural place for us to begin is with the free particle; this
will not only illustrate the theory, but help us develop a profound and amazing
conclusion later in this paper.
First, we begin with equation (2),
𝑖
𝑒𝑥𝑝 ( 𝑆) 𝑑𝑖𝑣
ћ
∞
−∞𝐶𝑞
lim 𝐴(𝑞) ∫
𝑞⟶∞
Now, we must express S (the action) in terms of 𝑥1 … 𝑥𝑞 (𝑥0 and 𝑥𝑞+1 are the fixed
endpoints, and so they will be considered constants). The action of a path, of course, is
defined by
𝑡𝑏
(3) ∫
𝑡𝑎
𝑚
(𝑥′)2 𝑑𝑡
2
for a free particle, in which 𝑡𝑎 and 𝑡𝑏 are the start and end times of the path in
interest, and 𝑥′ is the rate of change of 𝑥 with respect to time. Equation (3) is written in
a rather awkward way; to apply it to our discrete paths, we must break it up into several
integrals for each straight line our path is composed of. Carrying this process out, we
obtain
𝑞
𝑡𝑛+1
∑∫
𝑛=0 𝑡𝑛
𝑚 (𝑥𝑛+1 − 𝑥𝑛 )2
𝑑𝑡
2
𝜀2
Where 𝑡0 … 𝑡𝑞+1 are the times associated with 𝑥0 … 𝑥𝑞+1 , and ε is again the
magnitude of the time interval between 𝑡𝑛 and 𝑡𝑛+1 . Now, calculating the integral, we
have
𝑞
𝑞
𝑛=0
𝑛=0
𝑚 (𝑥𝑛+1 − 𝑥𝑛 )2
𝑇
𝑚 (𝑥𝑛+1 − 𝑥𝑛 )2
∑
·
=∑
2
𝜀2
𝑞+1
2
𝜀′
In which ’ is 𝜀 2
𝑞+1
𝑇
. Now, we have written the action of the discrete paths in
terms of 𝑥0 … 𝑥𝑞+1. So, we may insert this expression for the action into equation (2),
yielding
𝑞
𝑖𝑚
lim 𝐴(𝑞) ∫
𝑒𝑥𝑝 (
∑(𝑥𝑛+1 − 𝑥𝑛 )2 ) 𝑑𝑖𝑣
𝑛→∞
2ћ𝜀′
∞
−∞𝐶𝑞
𝑛=0
Now we have a standard complintegral. Noticing the 𝑥𝑛+1 − 𝑥𝑛 term, we use our
substitution axiom (this is why we took special note of this particular substitution, so that
axiom 2 is now tailor-made to this problem). Thus, we substitute 𝑧𝑛+1 for 𝑥𝑛+1 − 𝑥𝑛 ,
yielding
𝑞−1
𝑖𝑚
𝑖𝑚
2
lim 𝐴(𝑞) ∫
𝑒𝑥𝑝 (
∑(𝑧𝑛+1 )2 +
(𝑥𝑞+1 − 𝑥𝑞 ) ) 𝑑𝑖𝑣
𝑞→∞
2ћ𝜀′
2ћ𝜀′
∞
−∞𝐶𝑞
𝑛=0
Now, using the same reasoning as in example 3, we note that
𝑥𝑞 = 𝑧𝑞 + 𝑥𝑞−1 = 𝑧𝑞 + 𝑧𝑞−1 + 𝑥𝑞−2 = ⋯ = 𝑧𝑞 + 𝑧𝑞−1 + 𝑧𝑞−2 … + 𝑥0
Thus, we have the complintegral
𝑞−1
𝑞−1
𝑛=0
𝑛=0
2
𝑖𝑚
𝑖𝑚
lim 𝐴(𝑞) ∫
𝑒𝑥𝑝 (
∑(𝑧𝑛+1 )2 +
[𝑥
− ∑ 𝑧𝑞−𝑛 − 𝑥0 ] ) 𝑑𝑖𝑣
𝑞→∞
2ћ𝜀′
2ћ𝜀′ 𝑞+1
∞
−∞𝐶𝑞
Now, we may break up the exponential, yielding
𝑞−1
𝑞−1
𝑛=0
𝑛=0
2
𝑖𝑚
𝑖𝑚
𝑒𝑥𝑝 (
∑(𝑧𝑛+1 )2 ) 𝑒𝑥𝑝 (
[𝑥
− ∑ 𝑧𝑞−𝑛 − 𝑥0 ] ) 𝑑𝑖𝑣
2ћ𝜀′
2ћ𝜀′ 𝑞+1
∞
−∞𝐶𝑞
lim 𝐴(𝑞) ∫
𝑞→∞
This brings up a very important point: How do we know that these bizarre
complintegral exponentials behave like their real counterparts? The answer is simple:
we define them to be so. After all, we are free to choose how these exponentials act; so
let us define them in a way so as to preserve the properties of the real exponentials.
Here will be our strategy: we shall expand the rightmost exponential into a
Fourier Integral (eliminating the troublesome squared term) and apply separation. This
will tell us, eventually, how complintegrodynamical considerations affect the problem of
the free particle.
Carrying out this strategy, we write
𝑞−1
2
𝑖𝑚
𝑒𝑥𝑝 (
[𝑥
− ∑ 𝑧𝑞−𝑛 − 𝑥0 ] ) =
2ћ𝜀′ 𝑞+1
𝑛=0
1
√2𝜋
𝑞−1
∞
∫ 𝐹(𝑘)𝑒𝑥𝑝 (𝑖𝑘 [𝑥𝑞+1 − ∑ 𝑧𝑞−𝑛 − 𝑥0 ]) 𝑑𝑘
−∞
𝑛=0
where
𝐹(𝑘) =
1
√2𝜋
∞
∫ 𝑒𝑥𝑝 (
−∞
𝑖𝑚 2
𝑥 ) 𝑒𝑥𝑝(−𝑖𝑘𝑥)𝑑𝑥
2ћ𝜀′
Now, we may carry out this integral, yielding
(−𝑖𝑘)2
𝜋
1
𝑒𝑥𝑝
(−
)
·
√ 𝑖𝑚
𝑖𝑚
√2𝜋
−
4
2ћ𝜀′
2ћ𝜀′
Simplifying, we calculate
√𝑖
2ћ𝜀′𝜋
ћ𝜀′𝑘 2
1
𝑒𝑥𝑝 (−𝑖
)·
𝑚
2𝑚
√2𝜋
Now that we have our Fourier Integral, we may carry out the plan. This yields
∞
ћ𝜀′𝑘 2
2ћ𝜀′𝜋
𝑒𝑥𝑝 (−𝑖 2𝑚 ) √𝑖 𝑚
−∞
√2𝜋
(4) lim 𝐴(𝑞) ∫
𝑞→∞
𝑞−1
𝑖𝑚
∫
𝑒𝑥𝑝 (
∑(𝑧𝑛+1 )2 )
2ћ𝜀′
∞𝐶
−∞ 𝑞
𝑛=0
𝑞−1
· 𝑒𝑥𝑝 (𝑖𝑘 [𝑥𝑞+1 − ∑ 𝑧𝑞−𝑛 − 𝑥0 ]) 𝑑𝑖𝑣 𝑑𝑘
𝑛=0
In which we have taken the Fourier Integral out of the complintegral. To calculate
the complintegral within,
𝑞−1
𝑞−1
𝑛=0
𝑛=0
𝑖𝑚
∫
𝑒𝑥𝑝 (
∑(𝑧𝑛+1 )2 ) · 𝑒𝑥𝑝 (𝑖𝑘 [𝑥𝑞+1 − ∑ 𝑧𝑞−𝑛 − 𝑥0 ]) 𝑑𝑖𝑣
2ћ𝜀′
∞𝐶
−∞ 𝑞
we shall utilize axiom 1. First, we combine the two exponentials, yielding
𝑞−1
𝑞−1
𝑛=0
𝑛=0
𝑖𝑚
∫
𝑒𝑥𝑝 (
∑(𝑧𝑛+1 )2 + 𝑖𝑘 (𝑥𝑞+1 − ∑ 𝑧𝑞−𝑛 − 𝑥0 )) 𝑑𝑖𝑣
2ћ𝜀′
∞𝐶
−∞ 𝑞
and utilize separation. To this effect, let us combine the two sums,
𝑞−1
∫
∞
−∞𝐶𝑞
𝑒𝑥𝑝 (∑ [
𝑛=0
𝑖𝑚
(𝑧 )2 − 𝑖𝑘𝑧𝑞−𝑛 ] + 𝑖𝑘𝑥𝑞+1 − 𝑖𝑘𝑥0 ) 𝑑𝑖𝑣
2ћ𝜀′ 𝑛+1
Obviously, this is equivalent to
𝑞−1
∫
𝑒𝑥𝑝 (∑ [
∞
−∞𝐶𝑞
𝑛=0
𝑖𝑚
(𝑧 )2 − 𝑖𝑘𝑧𝑛+1 ] + 𝑖𝑘(𝑥𝑞+1 − 𝑥0 )) 𝑑𝑖𝑣
2ћ𝜀′ 𝑛+1
Taking the sum out of the exponential term, we have
𝑞−1
𝑖𝑚
(𝑧𝑛+1 )2 − 𝑖𝑘𝑧𝑛+1 ) 𝑑𝑖𝑣
∏ 𝑒𝑥𝑝 (
2ћ𝜀′
∞𝐶
−∞ 𝑞
𝑒𝑥𝑝(𝑖𝑘[𝑥𝑞+1 − 𝑥0 ]) ∫
𝑛=0
Applying separation, we produce
𝑞
∞
𝑖𝑚 2
𝑒𝑥𝑝(𝑖𝑘[𝑥𝑞+1 − 𝑥0 ]) (∫ 𝑒𝑥𝑝 (
𝑥 − 𝑖𝑘𝑥) 𝑑𝑥)
2ћ𝜀′
−∞
Now, evaluating the integral, we have
𝑞
(−𝑖𝑘)2
𝜋
1
𝑒𝑥𝑝(𝑖𝑘[𝑥𝑞+1 − 𝑥0 ]) (√
𝑒𝑥𝑝 (−
)·
)
𝑖𝑚
𝑖𝑚
√2𝜋
−
4
2ћ𝜀′
2ћ𝜀′
This allows us to “simplify” equation 4 to
∞
1
lim 𝐴(𝑞) ∫
𝑞→∞
−∞ √2𝜋
·√
2𝑖ћ𝜀′𝜋
𝑚
𝑞
𝜋
𝑖𝑘 2 ћ𝜀′(𝑞 + 1)
· (√
) 𝑒𝑥𝑝 (𝑖𝑘(𝑥𝑞+1 − 𝑥0 )) 𝑒𝑥𝑝 (−
) 𝑑𝑘
𝑖𝑚
2𝑚
−
2ћ𝜀′
𝑞
2𝑖ћ𝜀′𝜋
Now, since√
𝑚
· (√
𝜋
−
𝑖𝑚
2ћ𝜀′
) does not depend on k, it may be removed from the
integral. Now, notice this: in Quantum Theory, of course, a constant factor in a kernel
(as Feynman calls this probability amplitude; see his book) may be disregarded. So
we may immediately disregard those expressions mentioned just above. Now, what
really matters in the kernel is the dependence on the end points. This involves the
𝑒𝑥𝑝(𝑖𝑘[𝑥𝑞+1 − 𝑥𝑞 ]) expression. Now, the other exponential factor is “intertwined”, so
to speak, within the integral with the spacial dependence factor, so we can not ignore
it so quickly. Indeed, we may calculate the integral, and the result we obtain is
2
2𝑚𝑋 2 |𝑞 + 1|
lim 𝐴(𝑞)exp (𝑖
(
) )
𝑞→∞
4ℏ𝑇 𝑞 + 1
In which X is the difference in the end point positions and T is the analogous
expression for the times. This result is thoroughly fascinating; it really illustrates why
we introduced the absolute value term in the new generalized definition for the action.
It is interesting to note that conservation of energy is violated in some of these other
universes; when we define the expectation value of the Hamiltonian in the way
described in normal quantum mechanics (see “Comments about the Theory”), we will
see that the quantity actually changes from time to time in these various parallel
realities.
It is furthermore also interesting that the probability amplitudes may not be
normalized. Indeed, the integral of the absolute square of the probability amplitude
over all space changes from time to time in many of these parallel universes. We
interpret such seemingly anomalous probabilities in the following manner: an integral
of probability over all space being less than one indicates a chance that the particle
may vanish or disappear entirely. Similarly, we may interpret such a quantity being
greater than one as indicating the creation of new particles. Indeed, for example, if
100 particles are set off independently in an experimental apparatus and there is a
200 percent probability that they will be found in a given location, we interpret this as
implying that 200 particles will be found at that location. This is yet another example
of the violation of the laws of conservation of charge and conservation of energy in
Quantum Complintegrodynamics. We note that, in the free particle case at least, and
for a particle that starts at a specific position, these effects seem to be concentrated
for small t (close to the initial creation of the particle) and seem to dissipate for large t.
We further note that the kernel, in this case, satisfies the differential equation
−ħ2
𝜕
(𝑎 + 𝑏𝑖)∇2 𝛹 = 𝑖ħ 𝛹
2𝑚
𝜕𝑡
In which a+bi is a complex number on the unit circle. We note that this equation
is a sort of combination of the diffusion equation and the familiar Schrodinger
equation. With this generalized wave equation, we may now solve the problem of a
free particle that starts with a specific momentum. We find that such a particle has the
wavefunction
𝑒𝑥𝑝 [
𝑖𝑝𝑥
𝑝2 (𝑐 + 𝑑𝑖)𝑡
] 𝑒𝑥𝑝 [
]
ħ
2𝑚ħ
In which c+di is again a complex number appearing on the unit circle. The c
parameter indexes the various universes, and this formula delineates the behavior of
a free particle starting with a specific momentum in these different universes.
6. The Quantum Complintegrodynamical Problem of a Particle in a
Potential
We shall find that utilizing a perturbation expansion will be fundamental to
calculating the way in which a particle in a potential behaves in Quantum
Complintegrodynamics. Indeed, in Feynman’s own text, an infinite series or perturbation
expansion was essential to solving the problem of a particle in a potential, and we shall
use a similar method in this section of the paper.
We shall begin by writing the formula that delineates the behavior of the various
complintegrodynamical universes:
𝑖
𝑒𝑥𝑝 ( 𝑆) 𝑑𝑖𝑣
ħ
∞
−∞𝐶𝑞
lim 𝐴(𝑞) ∫
𝑞→∞
In which S, of course, describes the (generalized) action of the path in question.
𝑡
Note that the action S may be written S[0] -∫𝑡 𝑏 𝑉(𝑥(𝑠), 𝑠)𝑑𝑠, which describes a
𝑎
breakdown into free and potential components. Thus, we are led to the expression
𝑖
𝑖 𝑡𝑏
lim 𝐴(𝑞) ∫
𝑒𝑥𝑝 ( S[0]) 𝑒𝑥𝑝 (− ∫ 𝑉(𝑥(𝑠), 𝑠)𝑑𝑠) 𝑑𝑖𝑣
𝑞→∞
ħ
ħ 𝑡𝑎
∞
−∞𝐶𝑞
Now, note that we may apply the infinite series expansion of the exponential to
rewrite this equation as
𝑡𝑏
𝑖
𝑒𝑥𝑝 ( S[0]) ∫ 𝑉(𝑥(𝑠), 𝑠)𝑑𝑠 𝑑𝑖𝑣 + ⋯
ħ
∞
𝑡𝑎
−∞𝐶𝑞
lim 𝐴(𝑞) ∫
𝑞→∞
In which the ellipsis delineates the infinite series corresponding to other terms.
Now, reversing the order of integration and rewriting the integrals, we obtain
𝑞−1
𝑡𝑛+1
lim 𝐴(𝑞) ∑
𝑞→∞
∫
𝑖
𝑒𝑥𝑝 ( S[0]) 𝑉(𝑥(𝑠), 𝑠)𝑑𝑖𝑣 𝑑𝑠 + ⋯
ħ
∞
−∞𝐶𝑞
∫
𝑡𝑛
𝑛=0
Now, x(s) is clearly equal to xn + (s-tn)(xn+1-xn)/(tn+1-tn) for a single leg of our
discrete path; that is, within the domain of one of the integrals within the above sum.
Now, here comes the critical link in our chain of reasoning: we may set xn equal to xn+1.
That is, we may allow one leg of our discrete path (as illustrated in Figure 1) to be
“vertically” aligned. After all, such a discrete path (with xn equal to xn+1) preforms equally
well at approximating an arbitrary curved path in the limit as q approaches infinity as a
normal discrete path with xn not necessarily equal to xn+1.
Thus, x(s) then becomes xn, and our expression becomes
𝑞−1
lim 𝐴(𝑞) ∑
𝑞→∞
𝑡𝑛+1
∫
𝑖
𝑒𝑥𝑝 ( S[0]) 𝑑𝑥1 … 𝑑𝑥𝑛−1 𝑑𝑥𝑛+1 … 𝑑𝑥𝑞 𝑑𝑥𝑛 𝑑𝑠 + ⋯
ħ
∞
−∞𝐶𝑞
∫
𝑡𝑛
𝑛=0
∞
𝑉(𝑥𝑛 , 𝑠) ∫
−∞
Which comes from reversing the order of integration. Now, our experience with
the free particle tells us that this is equivalent to
𝑞−1
lim 𝐴(𝑞) ∑
𝑞→∞
𝑡𝑛+1
∫
∫ 𝐾(𝑥𝑎 , 𝑡𝑎 ; 𝑥𝑛 , 𝑠)𝐾(𝑥𝑛 , 𝑠; 𝑥𝑏 , 𝑡𝑏 ) 𝑉(𝑥𝑛 , 𝑠)𝑑𝑥𝑛 𝑑𝑠 + ⋯
𝑡𝑛
𝑛=0
∞
−∞
In the limit as q goes to infinity, where K is the probability amplitude
corresponding to a free particle. Now, we may join together the sum and integral
portions, yielding
𝑡𝑏
∞
lim 𝐴(𝑞) ∫ ∫ 𝐾(𝑥𝑎 , 𝑡𝑎 ; 𝑥𝑛 , 𝑠)𝐾(𝑥𝑛 , 𝑠; 𝑥𝑏 , 𝑡𝑏 ) 𝑉(𝑥𝑛 , 𝑠)𝑑𝑥𝑛 𝑑𝑠 + ⋯
𝑞→∞
𝑡𝑎
−∞
This same process may be applied to the other terms in the expansion, with
similar results. This is a very interesting expression; it illustrates that without the
presence of an absolute value in the generalized definition of the action, not only would
the free particle kernel fail to vary from universe to universe, but so would the probability
amplitude corresponding to a particle in a potential. Furthermore, it is interesting to note
that the kernel corresponding to real q (that is, in non-complintegrodynamical physics)
but with the contribution formula
1
𝑒𝑥𝑝 ( (𝑎 + 𝑏𝑖) 𝑆)
ħ
In which a+bi is a complex number on the unit circle, and S is the action
corresponding to the potential V/(a+bi), gives the same result; thus, the different parallel
realities in Quantum Complintegrodynamics can also be viewed as corresponding to
different contribution formulae, rather than different limits in the complex plane.
7. Comments About the Solution
First of all, we make special note of the fact that the very bedrock of physical law
varied from universe to universe (most particularly conservation of energy) when we
studied the case of a free particle. Thus, it seems highly likely that even more bizarre
gems lie in our alien landscape of parallel worlds.
How might conservation of energy vary specifically from universe to universe?
Conservation of energy finds its place in quantum mechanics when we imagine the
universe as a multi-particle system, and we say that the total of all the expectation
values of the energy of all particles is exactly constant. This involves a critical
balance of all the probability amplitudes in the universe, and it seems that
complintegrodynamical considerations disrupt this balance.
To make this quantitative, let us consider the expectation value of the energy of a
particle in quantum mechanics. It is given by the formula3
∞
∫  𝐻  𝑑𝑥
−∞
Where H is the Hamiltonian Operator and  is the wavefunction of the particle.
Since conservation of energy is defined in quantum theory in terms of the expectation
value of the energy3, it seems quite clear that complintegrodynamical effects, that
change the wavefunction of the particle from its description in ordinary quantum
theory, will disrupt conservation of energy. And indeed, this is precisely the result we
found with the free particle, when we observed that it did not obey the laws of
conservation of matter and energy in a complintegrodynamical setting.
Perhaps one of the concepts most commonly associated with the idea of multiple
universes is the anthropic principle. This principle states that the physical properties
of the universe should conform to allowing the eventual origin of intelligent life
capable of observing the cosmos. While we only provide a rough sketch of how the
anthropic principle may be applied to Complintegrodynamics to determine the specific
properties of our universe, it is clear that the existence of intelligent life places
constraints on how our universe behaves in the setting of quantum
complintegrodynamics.
We take particular note of the final solution in Section 5, in which we computed
how a free particle starting with specific momentum behaves in the various universes
described by Quantum Complintegrodynamics. Observe that the formula predicts, in
lieu of our interpretation of non-normalized probabilities, a deluge of such particles
set off independently with a specific momentum will either diminish or increase in
number. We also note that the evolution of intelligent life (on Earth at least) required
approximately four billion years and a terrestrial planet with a moderately pressurized
atmosphere. If complintegrodynamical effects are too extreme, atoms and particles
within planets will disappear or increase in number over time, changing the mass of
the planets, disrupting the environments compatible with life and preventing the
evolution of intelligence.
It is clear that the status of a planet’s atmosphere (and therefore its ability to bear
life) depends critically on its mass. Quantum Complintegrodynamical effects, which
change the mass of objects over time due to the increase or decrease in the number
of their constituent particles, could disrupt such life-bearing environments.
In our very rough sketch of an anthropic argument we shall pay attention merely
to the case of negative c in the momentum solution of Section 5. We further suppose
that no terrestrial planets exist with over one hundred times the mass of the Earth, a
conservative assumption that has been borne out through astronomical observations.
We also postulate that all intelligent life evolves on terrestrial planets of approximately
Earth mass over a time scale of roughly four billion years. These last two
assumptions are somewhat unjustified, but they should suffice for a rough sketch of
an anthropic argument. We also speculate that the average momentum of atoms
within the cores of terrestrial planets, in standard atomic units, is approximately
100,000,000 (which comes from a temperature of roughly 50,000 degrees Fahrenheit
(typical of the cores of terrestrial planets) and a mass of 50 times that of hydrogen
(typical of the iron atoms that make up much of the cores of such planets)). Now, we
may apply the momentum solution of section 5 to determine how the sizes of such
planets diminish or increase over large time scales.
Inputting the relevant parameters, we find that such planets increase or decrease
in size by the factor
𝑒𝑥𝑝(𝑐(100000000000)𝑡)
In the appropriate atomic units, with c, again, taking on a negative value. Utilizing
our previous postulates and standard atomic units, we find that c must be at most
2 x 10-38
In order for the sustainment of terrestrial planets and therefore of life. Due to the
inaccuracies of our rough sketch this quantity could conceivably be off by factors of
hundreds in either direction. However, it is clear that restrictions must be placed on
the veracity of complintegrodynamical effects in order to ensure the eventual
evolution of intelligent life.
Conclusion
A very simple generalization of the path integral approach to quantum
mechanics-in which we allow a time interval to be divided into a complex number of
“slices”-predicts a vast complex of different universes, or entirely separate realities
corresponding to different laws of nature. Although we have not proven this idea
rigorously here, it seems perfectly possible that these universes do indeed have
different and alien physical laws, and may behave in ways we never thought possible.
However, we did prove that the scenario of a free particle varies across all the
universes, involving the introduction of many new, bizarre laws of nature, including
the violation of the law of conservation of energy and matter, particularly in small
intervals of time.
We provided a rough sketch of an anthropic argument that suggested that very
few of these universes will ever produce intelligent life. This argument focused on the
stability of terrestrial planets, and how complintegrodynamical effects could disrupt
those environments that cradle the evolution of intelligent life.
Thus, this theory not only naturally follows from previous historical mathematical
advances from the real to the complex, but also impacts our understanding of the big
bang (because the difference between universes becomes manifest at high
potentials, it seems), might be tested with observations of high potential phenomena,
and is an easy way to apply the anthropic principle. All around, it seems to be a
promising proposal and to be quite worthy of future scientific research.
Time and time again, the generalization of the concept of Number has proven
vital in the progression of science. If we work to solve equation (5), and apply this to
the anthropic principle to determine how our universe behaves, we may have access
to some new domains of particle physics.
Acknowledgements
The author would like to thank Dr. Michael Carchidi of Drexel University for
invaluable tutoring on various important theories of subatomic physics.
References
1
Richard Feynman, Quantum Mechanics and Path Integrals (McGraw-Hill, New
York, 1965) pp.32-33.
2Keith
B. Oldham and Jerome Spanier, The Fractional Calculus (Academic, New
York, 1974)
3David
J. Griffiths, Introduction to Quantum Mechanics (Pearson Prentice Hall,
Upper Saddle River, 2005) pp.17, 37.