Principia Physica

Mukul Patel

Southwestern College

Department of Mathematics

http://xxx.lanl.gov/abs/gr-qc/0002012

Abstract

A comprehensive physical theory explains all aspects of the physical

universe, including

quantum aspects, classical aspects, relativistic aspects, their

relationships, and

unification. The central nonlocality principle leads to a nonlocal

geometry that ex-plains

entire quantum phenomenology, including two-slit experiment,

Aspect-type

ex-periments,

quantum randomness, tunneling etc. The infinitesimal aspect of this

geom-etry

is a usual (differential) geometry, various aspects of which are

energy-momentum,

spin-helicity, electric, color and flavor charges. Their interactions are

governed by a mathematically automatic field equation—also a grand

conservation principle. New predictions: a new particle property;

bending-of-light estimates refined over relativity’s; shape of the

universe; a no gravitational singularity theorem; etc. Nonlocal physics is

formulated using a nonlocal calculus and nonlocal differential equations,

replacing inadequate local concepts of Newton’s calculus and partial

differential equations. Usual quantum formalisms followfrom our theory—the

latter doesn’t rest on the former.

1.1 Introduction

The primary purpose of a scientific theory is to understand complex

phenomena with

the aid of simpler and readily comprehensible concepts.

Modern physics is faced with

a great challenge—quantum phenomena. These still lack consistent

rational explanation

after a hundred years since their discovery. Most of the paradoxes of the

quantum theory are paradoxes of the theory rather than those of observed

phenomena. Clearly, a fresh consistent set of concepts are needed to

actually understand the seemingly bizarre quantum phenomena. Thus, we

abandon the entire opaque machinery of quantum mathematics and all its

interpretations. Aspect-type experiments reveal the inherent nonlocality

of the physical world. Hence, we also abandon the very basis of classi-cal

physics—the tacit assumption that phenomena are governed by local

mechanisms. Instead, we propose a nonlocal physics, constructed from

scratch. This physics also lays bare the integrated reality which

underlies all the myriad fields, particles, their properties, and their

fields.

Fortunately, this entire new conceptual framework can be deduced

from

one physical

principle, the nonlocality principle. The latter leads to a nonlocal

geometry, which

is specified by a nonlocal connection (as opposed to classical, local

connection in the sense of differential geometry). This nonlocal

connection explains entire quantum phe-nomenology on one hand while its

local aspect, a classical connection, is the universal field which yields

an integrated geometric description of all the forces, particles, fields,

energy-momentum, charges, and other quantum numbers.

Due to nonlocality, partial differential equations are inadequate to

predict events.

Thus, the local concepts of Newton’s calculus are inadequate to

completely describe

the physical world. While the classical physics is encoded in terms of

relationships among (local) rates of change of physical quantities, the

crucial concepts of nonlocal physics involve the way physical quantities

are related to each other nonlocally. Thus, the new laws should be

formulated as statements of these nonlocal relationships. For these

reasons, we devise a new nonlocal calculus and nonlocal differential

equations. Although this calculus correctly, and completely, encodes

nonlocal dependences among fields, the fields are still local in that they

are defined point-wise. The nonlocal connection mentioned above is not a

field defined point-wise. It is defined at ordered pairs of points; and

its value at a pair relates vectors and tensors at one point with those at

the other. To analyze this essentially nonlocal field and its various

aspects, we devise a calculus of nonlocal fields, along with a nonlocal

differential geometry. Then we write down the field equation, in terms of

concepts of this calculus. This equation governs all aspects of the

physical world.

We make no attempt to explain any of the current quantum formalisms,

nor

is any

consideration given to the paradoxes arising out of these formalisms. We

only explain observed physical phenomena. Our theory is formulated

completely using real numbers—noncommutative variables are not needed.

Besides explaining a great many unexplained phenomena, several

new

predictions

are deduced.

It can’t be over-emphasized that, while current theoretical trends in

science and philosophy actively shun determinism and rationality, our

theory brings us back to the realm of classical logic, and to a

determinism stronger than that of Newton’s. Ironically enough, this form

of determinism has plenty of room for ‘free will’, and it also causes the

apparent quantum randomness.

This report consists of seven sections, divided into to three chapters.

Chapter 1 introduces the nonlocality principle and explains quantum

phenomena. Chapter 2 ex-amine the fundamental field, which follows from

the nonlocality principle. Since this field is a nonlocal object and has a

local, infinitesimal aspect, the analysis takes three forms. The first

analyzes the local aspect using infinitesimal methods; i.e., using

New-ton’s calculus. Here, various aspects of the local aspect will be

identified with various properties of particles. Using these fields we can

build various particles, and the field equation mandates that they should

interact. As the information extracted from this analysis is inadequate to

exactly predict events, we next analyze the local aspect using a nonlocal

analysis. This yields a more natural set of equations encoding complete

information on the local aspect. We are still left with an unknown—the

nonlocal con-nection itself. This being a truly nonlocal object, we devise

a calculus for such objects. For more details, see the table of contents.

Chapter 3 lists several new predictions. We have systematically suppressed

much technical details and all proofs. An exhaustive discussion of

concepts and technical details, can be found in the forthcoming research

monograph, Principia Physica, by the author.

1.2 Nonlocal field

1.2.1 Nonlocality Principle

Classically, it is conceived that an individual event affects events

only in its immediate

vicinity, and this effect travels from point to point with definite

speed. The discovery

of a series of quantum effects, which culminated in Aspect-type

experiments, forces

us to abandon this classical, local, way of describing the physical

world. It has been

evident for at least fifteen years that the quantum world is ruled by

essentially nonlocal mechanisms, and there is no way to reduce this

nonlocality to classical local objects. We propose that this fact of the

physical world be adopted as a fundamental physical principle.

Thus, we propose a fundamentally different mechanism of how events

affect each

other. While the classical viewpoint is essentially local, we propose that

any two events (points) reflect events in each-other’s vicinity, and this

is an immediate reflection with-out any notion of a signal traveling from

one point to another. This may sound absurd at first, but its full

implications are very naturally intuitive and consistent with observation.

This is because at any given point x, the reflections from other points

add up to describe the events in the immediate neighborhood of x. For

example, the value of a field at a point is the sum of the values

reflected from all the other points of spacetime. Thus, even though each

point reflects events everywhere else, all points do not look the same.

Also, as we look at successive space-like sections, we perceive some

effects to be moving from point to point and with definite speeds. Thus,

although our hypothesis asserts a strong action-at-a-distance, its

cumulative effects may appear to be traveling from point to point with

definite speeds; consequently, the classical viewpoint is not

contradicted, but is supplemented at a more fundamental level. We call

this hypothesis of events being reflected in different neighborhoods the

nonlocality principle. This principle can be refined using mathematical

language. For this we need to define two basic concepts. Following

Einstein, we think of the universe as the set of events: each event

corresponding to a mathematical point in spacetime 1 ;butwe call them

point events instead, to underscore their exact conceptual content. Now we

define an event to be any set of point events. E.g., entire spacetime is

an event, and so is a single point event. Also, the trajectory (or part

thereof) of an electron is an event.

Now we formulate the nonlocality principle more precisely:

Spacetime, X; the set of point events, is a four-dimensional manifold,

such that every pair of point events, (x; y); is nonlocally connected in

the following sense:Each one of the neighborhoods Uand Vof point events x

and y; respectively, reflects events in the other: these reflections are

described by smooth maps between neighborhoods, and are asymptotically

exact; i.e. as the neighborhoods become smaller, the reflections converge

to inverse, one-to-one, and onto reflections. ones.

The seminal consequence of the principle, mentioned below, will

follow

regardless

of how we choose to formalize the asymtotic convergence (there are

several ways); so

we can afford to be vague about the latter—at least for the time being. We

can deduce entire physics from this single hypothesis. In particular, we

propose that nothing exists but this scheme of reflections between pairs

of points. Thus, our theory does not even assume the existence of matter,

energy, fields, particles etc.; rather, we deduce all these from the

nonlocality principle as formulated above.

The seminal consequence of the principle is that it implies a nonlocal

connection on spacetime. We use the word ‘connection’ in the sense that it

lets us compare vectors at any pair of points in spacetime. The classical

connection, as conceived in differential geometry, is local in the sense

that it is essentially a way of relating vectors at any point with those

in its immediate (infinitesimal) vicinity. A posteriori, it allows us to

compare...

[1 Actually it is possible to formulate the principle—without any

reference to a pre-existing spacetime—in

the mathematical language of categories; essentially by formalizing

the

way one arrives at the concept of

a point from that of a neighborhood. It is not clear whether this will add

to our understanding of physical phenomena, though.]

...vectors at distinct points through parallel transport along paths. We

note that this way of comparing vectors at distinct points depends on the

path along which one transports the vectors. This again points out the

fact that the classical connection is essentially an infinitesimal object

whose integral is the classical parallel transport. As opposed to that,

our nonlocal connection is nonlocal in the sense that it provides a means

of comparing vectors at any two distinct points directly, without any

primary notion of infinitesimal transport of vectors or the accompanying

path-dependant parallel transport. We will also show that this nonlocal

connection has an infinitesimal (local) aspect which is nothing but a

classical (local) connection. Our theory accomplishes three important

objectives:

(1) The nonlocal connection explains all quantum aspects of the

physical

world.

(2) The local, infinitesimal, aspect of this connection unifies all the

fields, particles, quantum numbers, charges, mass, energy, momentum, etc.

(3) It is devoid of singularities.

We observe that the quantum aspects are more apparent at smaller

scales

because

the nonlocal reflections grow more and more accurate as the

neighborhoods grow

smaller. The ‘rate’ at which these convergence to perfect accuracy

occurs depends

on the metric, and it will be shown that the latter is an arbitrary

choice in our the-ory.

Thus, the Planck’s constant, which seems very intimately related to this

rate, also seems to be dependant on our choice of the metric. We will not

go into this any further because we can formulate our entire theory

without any reference to it, or any other constants—dimensional or

dimensionless.

If an intuitive picture of the universe is sought, we can say that it is a

giant kalei-doscope, each point being an infinitesimal mirror reflecting

all other mirrors. Another metaphor would be a cross-section of a bundle

of optic fibers, which are fused together at one end into a single point.

Yet another visualization of these nonlocal connec-tions is the image of a

telephone exchange, where each point of spacetime corresponds to a

telephone; each phone being in direct instantaneous communication with all

the other phones. Then, cumulative information at each point may appear to

travel at finite speeds despite the underlying instantaneous communication

among the phones.

1.2.2 Preliminary consequences of the principle

Nonlocal affine connection

As a consequence of the principle, there exists a one-to-one

correspondence between

vectors at x and vectors at y. It is easy to visualize this

correspondence. Every vector

can be thought of as a tangent to a particle trajectory. Since events,

such as particle trajectories, are reflected between pair of

neighborhoods, we see that this induces a correspondence of vectors at

points in these neighborhoods. Mathematically, this correspondence is an

affine isomorphism from the tangent vector-space Tx at x to the tangent

vector-space Ty at y; roughly speaking, this isomorphism, say !xy; is the

‘affine derivative’ of the correspondence referred to in the nonlocality

principle. Thus, we have, for every pair of points in spacetime, a way to

compare vectors at one of the points with those at the other. This is

reminiscent of the notion of connection from differential geometry, which

lets us compare vectors at a point with those at points in its

infinitesimal neighborhood. This, the classical kind of connection, is

consequently a local connection. As opposed to that, what we have above is

best described as a non-local connection, say !, whose value at an ordered

pair of points (x; y) is the affine isomorphism !xy. Note that !xy and !yx

are inverse isomorphisms. Note that !xy naturally extends to the whole

tensor algebra at x:

Sum of reflections

Consider a fixed point x in spacetime. For any other point y, events

around y will be

reflected in events around x. For example, if an elementary physical

field F takes the

value F (y) at y, then it will contribute !yx(F (y)) to the value of the

field F at x. Here !yx(F (y)) is the value of the vector F (y) under the

map !yx. Thus, the field value at x is the sum of all these contributions

as y ranges over entire spacetime. This sum is described mathematically by

an integral 2 :

F (x) =Z !yx(F (y))dy (1.1)

Note that the integrand is a function on X with values in the

vector-space Tx.

We see that, elementary physical fields are extremely nonlocal objects in

the sense that values at each point depend on values at all other

points—not just nearby points.

1.3 Quantum consequences

1.3.1 Two-slit experiment

Because of equation (1.1) we can view an elementary particle as a

field

which may

appear localized in a portion of spacetime and yet be spread-out over

entire spacetime; e.g., we can visualize an electron as a very intense

region of a field. Now, since this part of the field is the sum-total of

the field everywhere else (see equation 1.1), it can also be viewed as

spread-out over entire spacetime. Reciprocally, this nonlocal summation

can give rise to intense regions in the field, which are dependant on the

values of the field everywhere else. Thus, discreteness and contiguousness

exist simultaneously, and yet in a non-contradictory way. Also, more

localized the particle-like phenomenon is, more it comes under the purview

of nonlocal connection, and more it manifests its wave-particle duality.

This is the basic picture to keep in mind when trying to understand

quantum phenomena. The two-slit experiment becomes immediately

comprehensible from this picture. As an aside, we mention that since

clumpiness naturally arises from the nonlocal character of spacetime, it

may explain COBE-type data and distribution of galaxies.

2 This integral is defined using a volume form on X; integrating

vector-valued functions component-wise.

The volume form is determined only up to a scalar multiple, but the

class of the permissible fields is

determined uniquely by this integral. Also, the physics is independent of

the choice of this form because the fields and equations are invariant

under this choice. If spacetime is compact (see Sec. 3.1.3), then there is

a unique volume form with integral 1:

1.3.2 Quantum randomness

Consider the history of an observer in time as a one-parameter family of

space-like sections of spacetime. Then, given a field, the nonlocality

principle implies that the observer will not be able to predict exactly

how the field will change over his own history. This is because the data

he has on the field is from the past and the present. He has no data on

the future slices. Since the value of the field at any point in spacetime

is directly dependant on that at every point, it is not possible in

general to predict the exact value of the field at any point using partial

differential equations and partial data. Since we recognize fields and

particles as the same entity, we see that it is not possible to predict

any event exactly as conceived in classical physics. Thus, the observer is

left with the feeling that the events are purely random, and he is led to

believe that physical objects such as particles don’t have physical

properties until they are observed. All the famous paradoxes of quantum

theory are based on this assumption and on the undue significance that the

process of measurement receives due to it.

1.3.3 Aspect-type experiments

The basic picture mentioned above also makes Aspect-type results

transparent, lending

a solid physical explanation for the violation of Bell’s inequality.

1.3.4 Quantum tunneling

This is just a manifestation of the apparent randomness and

unconnectedness of two

events: vanishing of a particle at one point, and its reappearance

elsewhere. The point

is that a particle doesn’t have to go through a wall to appear on the

other side. The field configuration over the whole spacetime, when viewed

as space-like slices, appears to evolve in such a way that it exhibits

local effects, such as presence of its particle on one side of the wall in

one instance, and on the other side in the next instance.

1.3.5 Deterministic choice

We have already noticed that the nonlocality principle is an expression of

an extreme form of determinism. Despite this, there is considerable room

for choice in this the-ory. Consider the case of an elementary field being

monitored by an observer. At any instance in time according to his frame

of reference, the field configuration in his past is already determined.

Taking into account the total nonlocal dependence of the field, one would

think that the field configuration in the future, too is completely

determined. This is not the case: Since the value of the field at any

point is given by an integral over X; there can be infinitely many

configurations, each differing another on at the most a set of measure

zero. Consequently, the future of the configuration has a fair degree of

freedom without the need to alter the past.

[mathematical analysis deleted due to cut and paste incompatibility]

http://xxx.lanl.gov/abs/gr-qc/0002012

Structure of particles

Basic constructions

We propose that the observed particles/fields are nothing but the

manifestations of the

field

with varying intensity of the constituent fields. For instance,

electron is a

field/particle whose only nonzero components are (i) negative charge

field (ii) energy-momentum

field (iii) spin-helicity field. Similarly, we view leptons, quarks, and

so-called gauge bosons as particles/fields with various combinations of

nonzero field components.

Interactions

All the elementary fields interact because they have to satisfy equation

(2.5). All the interactions are built into this equation.

Conservation and source-supply symmetry

We have already mentioned the tentative modified conservation laws.

We

also know

that under this scheme, every field splits into a source field, and the

corresponding supply field (traceless part), and because of the

conservation laws, the source can con-vert into supply and vice versa.

Thus, under right circumstances, we should be able to observe electric

charge convert into electromagnetic field, and energy-momentum convert

into pure gravitation. More generally, an electric charge may convert into

pure gravitation or vice versa.

2.2 Local Aspect: Nonlocal analysis

2.2.1 Inadequacy of partial differential equations

Since the value of an elementary field at a point depends on its value at

every other point, the system of equations we have are completely

over-determined. It is obvious that because of the nonlocal

interdependence of values of the fields, the partial differ-ential

equations can not predict the exact events unless we know the field at

(almost) all points—in which case, equations are not needed! Thus, it

would be impossible to solve any initial-value or boundary-value problems.

It follows that the nonlocal char-acter of the universe forces us to

abandon description in terms of partial differential equations and forces

us to adopt some nonlocal concepts for description, so we would be able to

compute empirical predictions. Furthermore, in a nonlocal universe, it is

only natural that laws of physics are best formulated in terms of nonlocal

concepts. For these reasons, we introduce nonlocal calculus and nonlocal

differential equations in the next section. We then formulate the

fundamental nonlocal field equation. This equation contains lot more

information on our fields than does the local equation. Note that fields

analyzed here are local but the analysis is nonlocal. In the following

sections we will analyze nonlocal fields.

2.2.2 Nonlocal calculus of fields

[See original paper for mathematical analysis]

2.3 Nonlocal aspect: Nonlocal analysis

The most fundamental structure on spacetime is the nonlocal

connection

!: This is an

example of what we will call a (nonlocal) form. Unlike the usual fields,

this field is defined at an ordered pair of points, and is an isomorphism

from the tangent space of the first point to that of the other. In order

to analyze !; we develop an analysis of such forms.

Chapter 3

THE CONCLUSION

3.1 Predictions

3.1.1 Rotation of polarization of Light

The new field spun-heluxicity, the traceless part accompanying the

field

spin-helicity

(2.1.3), would rotate the polarization of light 1 (and other such

‘directed’ properties)

just like curvature bends light.

3.1.2 Bending of light near magnetars

Since electromagnetism is a non-metric aspect of the universal

connection, given a

stellar object, such a magnetar, with an immense magnetic field that is

comparable in curvature to its gravitational field, bending of light

should be significantly different than that predicted by general

relativity.

3.1.3 Causality and compactness

Since a metric is just a convention of a frame of reference, there is no

absolute concept of causal character of vectors. Consequently, the concept

of a closed time-like curve has no absolute meaning. Indeed, given a

closed curve, we can always find a metric with respect to which it is

time-like. Thus, it is not possible to avoid closed-loop con-tradictions

(such as an individual killing his own parents before he was conceived)

even if the spacetime is non-compact. In short, the mathematical deduction

that spacetime must be non-compact (if causal), has no absolute meaning

and is not very useful. If we were to preserve this kind of causality,

instead of restricting ourselves to non-compact spacetime, we should

identify frames of references ( which includes a choice of met-ric) that

prohibit closed time-like curves; and then declare these the only

physically permissible ones. A similar discussion holds for other

causality conditions such as strong causality, etc.

[1 Just before the submission of this paper, it was pointed out to the

author that such phenomena have already been observed.]

3.1.4 No-singularities and big-bang

The general conservation principle (2.2) implies that if the gravitation

part of curva-ture increases than the non-gravitational part will

compensate for it. Consequently, black-hole type situation can’t lead to

singularities. This mechanism that prevents sin-gularities can be

interpreted as a sort of anti-gravity, which is not an extra force of

nature, but is built into our theory by virtue of the grand conservation

law (2.1). The lack of absolute metric implies that there are no absolute

notions of expansion and contraction of space. Thus, expansion is not an

absolute feature of the universe; big-bang is an actual point of

spacetime, and is no different than any other point in the spacetime.

3.1.5 Parallelizability and shape of spacetime

Nonlocality principle implies that spacetime is parallelizable and

hence

orientable, and

this removes the possibility of Moebius-strip type circumstances from our

physical uni-verse. Also, it follows from last two sections, that the

hypothesis of complete (‘without holes’) and compact spacetime is viable;

and this combined with parallelizability can restrict the shape of

spacetime in very significant ways—if it is boundary-less. Also, the above

discussion can be brought to bear on the recent cosmological observations

of Class 1-a supernovae. Currently, these observations are interpreted by

saying that the spacetime has negative large-scale curvature. But this

interpretation assumes that the connection of the universe is the

Levi-Civita connection of a metric. Our theory implies that the connection

of spacetime has non-metric components as well; and it is this connection

whose curvature is negative with respect to the metric g. Thus, the

gravitation part of the connection need not have negative curvature. This

removes the restriction on the shape of the (space-like sections of)

universe that it must be a hy-perbolic 3-manifold, and this in turn saves

us from the implication that spacetime (if compact) should be a multiply

connected manifold.

3.1.6 Micro predictions

Corresponding to the field spun-heluxity, there is a new particle

property, which should

be inferable from observation of rotation mentioned in section (3.1.1) in

particle inter-actions, as well.

Our viewpoint also validates particles of other fields such as sound

and

heat when

these are determined at micro-scales, e.g. in solid-state. More

generally, we predict

anyons corresponding to any conceivable physical field determined at

extremely small

scale.

http://xxx.lanl.gov/abs/gr-qc/0002012

To: "Gary S. B."

Date sent: Mon, 28 Feb 2000 08:27:27 -0400

From: Arkadiusz Jadczyk

Date sent: Mon, 28 Feb 2000 09:53:59 -0400

Date sent: Mon, 28 Feb 2000 13:02:02 -0800

To: Arkadiusz Jadczyk (by way of Mehri Arfaei <arfaei@jinx.sck>)

From: Mukul Patel <mukulp_@ca.freei.net>

From: Arkadiusz Jadczyk

To: "Gary B."

Date sent: Mon, 28 Feb 2000 09:55:37 -0400

From: Arkadiusz Jadczyk

To: Mukul Patel <mukulp@ca.freei.net>

Date sent: Mon, 28 Feb 2000 13:16:10 -0400

From: Arkadiusz Jadczyk

To: Mukul Patel <mukulp@ca.freei.net>

On 28 Feb 00, at 13:02, Mukul Patel wrote:

Forwarded to: Date forwarded: Tue, 29 Feb 2000 18:16:33 -0400 From: "Arkadiusz Jadczyk" To: Mukul Patel Date sent: Tue, 29 Feb 2000 12:06:22 -0400 Subject: Re: gr-qc/0002012 Priority: normal On 29 Feb 00, at 11:50, Mukul Patel wrote: > >If omega_{yx} is an affine map between vector spaces, how do > >you extend it to tensors? I know how to extend a linear map, but > >how you extend an affine map? > > > replace the word "bilinear" by "bi-affine" It does not work. Give me precise definition of your extension. > >Question 2: > > > >Whats is the meaning of the symbol "dy" in (1.1). Integration > >needs measure. Or it needs volume form. How you integrate > >without mesure and without volume form? > > > please look at the footenote on that page. You say that volume form is defined up to a scalar multiple. But this scalar multiple is an arbitrary FUNCTION. If manifold compact - this arbitrariness holds as well. So, Which volume form do you choose in (1.1)? Notice that if (1.1) holds for one volume form, it will not hold for another. ark

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