Department of Mathematics
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
explains entire quantum phenomenology, including two-slit experiment,
Aspect-type ex-periments, quantum randomness, tunneling etc. The infinitesimal
aspect of this geometry 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.
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 classical 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
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 examine 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.
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 ;but we 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
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.
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...
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
(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 theory. Thus, the Planck’s constant, which seems very intimately
related to this rate, also seems to be dependant on our choice of the
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 kaleidoscope,
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 connections 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.
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
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
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]
of particles Basic constructions
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.
and source-supply symmetry
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
Inadequacy of partial differential equations
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
character 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.
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
Nonlocal calculus of fields
original paper for mathematical analysis]
Nonlocal aspect: Nonlocal analysis
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
3 THE CONCLUSION
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.
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.
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 contradictions (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.
Just before the submission of this paper, it was pointed out to the
author that such phenomena have already been observed.]
No-singularities and big-bang
general conservation principle (2.2) implies that if the gravitation
part of curvature 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.
Parallelizability and shape of spacetime
principle implies that spacetime is parallelizable and hence orientable,
and this removes the possibility of Moebius-strip type circumstances
from our physical universe. 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.
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