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Subject: Principia Physica

Hi Ark,

I would be curious as to what you think about such a radical reassessment of quantum theory and nonlocality. Thanks!

Gary B.


(Note: please refer to original paper for mathematical notation which is lost/distorted by conversion from PDF format to plain text)

Principia Physica

Mukul Patel

Southwestern College

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 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 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.


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 ;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 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 theory. 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 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.

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]

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 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.

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.


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 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.

[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 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.

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 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.

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.

From: Arkadiusz Jadczyk

To: "Gary S. B."

Subject: Re: Principia Physica

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



On 27 Feb 00, at 21:47, G.B. wrote:

> Hi Ark,


> I would be curious as to what you think about such a radical

> reassessment of quantum theory and nonlocality.


> Thanks!


> Gary B.

Hi Gary,

The author seems to be a mathematician, but he does not write as a

mathematician. He is vague at important point. Therefore I can not

proceed linearly.... I have to study his "theorems" to see what are

his assumptions and definitions. I do not like being forced to do it, but

that how it is. Will tell you when getting at some clear assesment of the

value of the paper's claims.



P.S. The very idea of a "nonlocal connection" is not a new one.

Physicists were studying "bilocal field theories" fifty years ago!


From: Arkadiusz Jadczyk

To: "Mukul Patel


Subject: gr-qc/0002012

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

Dear Collegue,

I am reading your paper. The following term is unclear to me:

"affine derivative".

If I have a differentiable map f:U --> V, f(x)=y, then how you define

"affine derivative" of f at x? I know what is "ordinary derivative" of f,

but what is "affine derivative"?



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

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

From: Mukul Patel <>

Subject: Re: gr-qc/0002012


Dear Collegue,

thanks for your kind interest in the paper.

if we had required that f(x) = y, then the natural candidate would be the

"ordinary derivative"....i.e. a LINEAR map from tangent space at x to that

at y. But as things stand f(x) maybe a point differnt than y. so we have a

linear map (the derivative) from tangent space at x to that at f(x)

we apply our principle to the pair f(x) and y....again there is a

derivative map form tangent space at f(x) to that at f(f(x)), and we

appply the prinicple to the pair of paoints f(f(x)) and y....we repeat the

process and get successive linear maps from f^n(x) to f^(n+1)(x)...for

each n we have a compisiton from tangent space at x to that of f^n(x)

can be show that the sequence of points f^n(x) converges to y and the

wbove compistion converges to a linear map from tnagent space at x to that

at y. But we have an extra pice of information...the sequence of points

f^n(x)....this determines a vector at y. so what we have is a Linear map

from x to y AND vector at y. In short, we have an AFFINE map form x to y

(tangent spaces at these)....this map is what i calle "roughly speaking,

an 'affine derivative of f'"....i hope this clarifies the vague allusion

to the term affine. If you have any further questions in this regard,

please feel free to write. Sincerely, Mukul Patel


From: Arkadiusz Jadczyk

To: "Gary B."

Subject: Re: Principia Physica

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

He is using undefined terms. I have to inquire him directly about his

definitions. I am getting skeptical.




From: Arkadiusz Jadczyk

To: Mukul Patel <>

Subject: Re: gr-qc/0002012

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



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

> If you have any further questions in this regard,

> please feel free to write. Sincerely, Mukul Patel

You write:


Now we formulate the nonlocality principle more


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.


Yet, it is not a precise mathematical definition.

So, please, let me know what PRECISELY are the

assumptions here (when I say "precisely", I mean

according to the standards where the order of

"for every", "there exists", "such that" are

important. Please let me know formal mathematical

assumption here. Like:

"We assume that for for every two points x,y,

x different from y, and for every two neighbourhoods

U and V of x and y resp. there exists etc..."


"We assume that for for every two points x,y,

x not necessarily different from y, there exists

neighborhoods U,V and map ...."

"The family of maps f[x,y,U,V] is assumed to

satisfy the following ...."


So, please let me know the precise assumptions

according to the above standards.



From: Arkadiusz Jadczyk
To: Mukul Patel <>
Subject: Re: gr-qc/0002012
On 28 Feb 00, at 13:02, Mukul Patel wrote:

> If you have any further questions in this regard,
> please feel free to write. Sincerely,

Mukul Patel


I have also two other questions:
In formula (1.1) F is assumed to be an arbitrary tensor field, right?

Question 1: 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?

Question 2: What 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?


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.



Last modified on: June 27, 2005.