## Tuesday, February 13, 2018

### HYSICAL MATHEMATICS EXPLAINING THE PHYSICS OF TEN DIMENSIONS.

PHYSICAL MATHEMATICS
EXPLAINING THE PHYSICS OF TEN DIMENSIONS.

The goal of physics is to analyze and understand natural phenomena of the universe - properties of matter, energy, their interaction, and consciousness/observer. Random occurrences are not encountered by chance wandering. There is a causal law putting restrictions on these. The validity of a physical statement rests with its correspondence to reality. The validity of a mathematical statement rests with its logical consistency. Mathematical laws of dynamics can be valid physical statements, as long as they correspond to reality. Dynamics is more than action of forces moment by moment or calculated over the particle’s entire path throughout time. The changeover from LHS to RHS in an equation is not automatic. The sign = or → is not an arithmetic total, but signifies special conditions like dynamical variables or transition states, etc.

The reaction 2H2+ O2 → 2H2O is not automatic - they must be ignited to explode. The ratio of hydrogen to oxygen is 2:1, the ratio of hydrogen to water is 1:1 and the ratio of oxygen to water is 1:2. Water molecule is like H-O-H. So in the reverse reaction, the bonds between the two atoms of each of the gaseous molecules of H2 and O2 must break, which requires energy. Once the atoms recombine to form water, the net energy in the hydrogen bonds in the molecules is much lower than what was there in the individual molecular bonds of gaseous hydrogen and oxygen. So the end result is surplus energy - to the tune of 286 Kilo Joules per mole. Thus, the correct equation is: 2H2+ O2 → 2H2O + Energy. The equations simply do not add up. The → sign indicates the requirement of energy to be added to the reactants as a catalyst. Presence of catalysts lower the thermal barrier changing the variables. But it does not show up in the equation and is not mathematically derived - it must be physically measured. In nature, plants use chlorophyll and energy from the Sun to decompose water. The reaction produces diatomic oxygen. Hydrogen released from water is used for the formation of glucose (C6H12O6). But the equations only shows: C6H12O6 + O2 = H2O + CO2.

Hydrogen is a nontoxic, nonmetallic, odorless, tasteless, colorless, and highly combustible diatomic gas. Oxygen is a colorless, odorless, tasteless diatomic gas of the chalcogen group on the periodic table and is a highly reactive nonmetallic element. It readily forms compounds (notably oxides) with almost all other elements, second only to fluorine. Water is attractive to polar molecules, has high-specific heat, high heat of vaporization, the lower density of ice, and high polarity. Hydrogen and oxygen are gases, but water is fluid at NTP. It brings down temperature. From the equation 2H2+ O2 → 2H2O, can we find these properties? No. Equations do not explain the difference in the properties of water from its constituents. It is true in all reactions. Thus, equations do not give complete information.

MATHEMATICAL PHYSICS VS PHYSICAL MATHEMATICS

Wigner defined mathematics as “the science of skillful operations with concepts and rules invented just for this purpose”. This is too open-ended. What is skillful operation? What are the concepts and Rules? Who invented them? What is the purpose? Do all concepts and rules have to be mathematical only? Wigner says: “The great mathematician fully, almost ruthlessly, exploits the domain of permissible reasoning and skirts the impermissible”, but leaves out what is permissible and what is not; leaving scope for manipulation – create a problem through reductionism and then solve it through manipulation! Finally call it unreasonable effectiveness of mathematics and incompleteness theorem!

One reason for the incompleteness of equations is the nature of mathematics, which explains the accumulation and reduction of numbers linearly or non-linearly of confined or discrete objects. Even analog fields are quantized. Number is a quality of objects by which we differentiate between similars. If there are no similars, it is 1. If there are similars, it is many, which can be 2, 3, 4,…..n, depending upon the sequential perception of ‘one’s in any base. Accumulation or reduction is possible only in specific quantized ways and not in an arbitrary manner (even fractions or decimals are quantized). Proof is the concept, whose effect remain invariant under laboratory conditions. Logic is the special proof necessary for knowing the unknown aspects of something generally known. Thus, the validity of a mathematical statement rests with its logical consistency.

Differentiation is related to perception by dissection. Perception is taking note of the result of measurement. Measurement is a process of comparison between similars. Hence result of measurement is always a scalar quantity. Without the concept of units, it has no meaning. Concept is an intelligent process universally applicable to all subjects or objects. We have concept about something. The objects or subjects may differ, but their “conception” in our memory or CPU remains same – only their detailed descriptions differ. This brings in the Observer, who must differentiate between the objects or subjects and determine which concept is applicable in a given context. Concepts are expressed in a language.

Language is the transposition of some information/command on the mind/CPU of another person/operating system. Mathematics tells us how much a system changes in the right hand side, when the parameters of the left hand side change. This information is universal and invariant in cognition. To that extent, mathematics is a language of physics. But it does not describe what, why, when, where, or how about the parameters or the system. It gives partial information. Generalizing such partial information misleads. Thus, it cannot be the only language of Nature. There is physics beyond mathematics. There is no equation for the observer. Yet, the observer has an important role in physics. No equation can describe the smile on the lips of the beloved. It is not the same as curvature of the lips. Detaching physics from equations is misleading interpretation in quantum physics - it is not weird.

The technological advancements in various sectors has led to data-driven discoveries in the belief that if enough data is gathered, one can achieve a “God’s eye view”. Data is not synonymous with knowledge. Knowledge is the concepts stored in memory. By combining lots of data, we generate something big and different, but unless we have knowledge about the physical mixing procedure to generate the desired effect, it may create the Frankenstein’s monster - a tale of unintended consequences. Already physics is struggling with misguided concepts like extra-dimensions, gravitons, strings, Axions, bare mass, bare charge, etc. that are yet to be discovered. If we re-envision classical and quantum observations as macroscopic overlap of quantum effects, we may solve most problems.

Scientists blindly accepts rigid, linear ideas about the nature of space, time, dimension, etc. These theories provide conceptual convenience and attractive simplicity for pattern analysis, but at the cost of ignoring equally-plausible alternative interpretations of observed phenomena that could possibly have explained the universe better. And sometimes they misguide!

What is the basic difference between quantum physics and classical physics? Notices of the American Mathematical Society Volume 52, Number 9 published a paper which shows that the theory of dynamical systems used to design trajectories of space flights and the theory of transition states in chemical reactions share the same set of mathematics. Our ancients considered the difference as that of the individual and the universal. Moving from individual to universal involves energy. Further, the tiny quantum mass is more susceptible to interference – noise from the environment. This makes the linear interaction to become non-linear.

In the Standard Model, which is not as successful as it is made out to be, we deal with quarks and leptons individually. In classical physics, we deal with their combinations collectively. The observable universe is explained by QED (photons exchange energy with electrons etc.) The rest of the SM deal with strong/weak interaction and yet to be incorporated gravity. We can model the interaction across all scales. We can add certain frequencies, phase them together like in holography - there are macro equivalents of all micro particles. Planet Jupiter is a macro equivalent of protons. Earth is a macro equivalent of neutrons. Our galaxy is a miniature universe, which is spinning around its axis like everything else in the universe. This will explain many observations, without invoking any novel phenomena.

String theory, which was developed with a view to harmonize General Relativity (GR) with Quantum theory, is said to be a high order theory where other models, such as super-gravity and quantum gravity appear as approximations. Unlike super-gravity, string theory is said to be a consistent and well-defined theory of quantum gravity, and therefore calculating the value of the cosmological constant from it should, at least in principle, be possible. On the other hand, the number of vacuum states associated with it seems to be quite large, and none of these features three large spatial dimensions, broken super-symmetry, and a small cosmological constant. The features of string theory which are at least potentially testable - such as the existence of super-symmetry and cosmic strings - are not specific to string theory. The features that are specific to string theory - the existence of strings - either do not lead to precise predictions or lead to predictions that are impossible to test with current levels of technology.

Apart from no evidence in support of existence of strings, there are many unexplained questions relating to its concept. Given the measurement problem of quantum mechanics, what happens when a string is measured? Does the uncertainty principle apply to the whole string? Or does it apply only to some section of the string being measured? Does string theory modify the uncertainty principle? If we measure its position, do we get only the average position of the string? If the position of a string is measured with arbitrarily high accuracy, what happens to the momentum of the string? Does the momentum become undefined as opposed to simply unknown? What about the location of an end-point? If the measurement returns an end-point, then which end-point? Does the measurement return the position of some point along the string? The string is said to be a Two dimensional object extended in space. Hence its position cannot be described by a finite set of numbers and thus, cannot be described by a finite set of measurements. How do the Bell’s inequalities apply to string theory? No answer.

String theories require 26 or 11 dimensions. M-theory requires 10 dimensions. But scientists have no idea about what these mathematical dimensions are. The strings are said to be excitations in hyperspace in 26 or 11 dimensions of a particle with zero mass and two units of spin. The extra dimensions are thought to be compactified or curled up into tiny pockets inside observable space. The particular vibrations of the strings within a multidimensional hyperspace are thought to correspond to particles that form the basis of all matter and energy. No one knows whether such hyperspace or strings or compactified dimensions exist. Time has come to switch over to physical mathematics. We will show the 10 dimensions in observable space.

DEFINING DIMENSION (परिमाण)

Dimension is a structural attribute (विस्तार), a measurable extent of spread in a given direction: length, breadth, depth, or height - the space an object takes up. In physics, dimension is considered as an expression of the character of a derived quantity in relation to fundamental quantities, without regard for its numerical value. In all measurements, unit is considered fundamental and the result is derived from it by comparison. For quantum particles like quarks, we cannot measure their extent and compare with others – they are indiscernible. Mesons, though composite quarks, are highly unstable. Even in the discernible macro world, the same object may be perceived differently from different angles or different distances. In both cases, they may be stable or unstable. Thus, we have to choose a precise description to cover these aspects: discreet/indiscreet (नित्य-अनित्य) and unit/quantity (अणु-महत्).

Dimensions is the interface (प्रचय) between the internal structural space and the external relational space (परिमाण) of an object depicted by the necessary parameters (संख्या). In visual perception, where the medium is electromagnetic radiation, we need three mutually perpendicular dimensions corresponding to the electric field, the magnetic field and their direction of motion. Measurement shows the relationship of dimension with numbers in a universalized manner. In the case of number, it is one or the totality of ‘one’s. But dimension is not the same as measurement of length or breadth or height – it is the constant in all three – spread (विस्तारस्य यथैवार्थ आयामेन प्रकाशित । तथारोहसमुच्छ्रायौ पर्यायवाचिनौ मतौ । - विश्वकर्मा).

Some claim that if there is some observable phenomena that we can measure by defining units of measure and counting the quantity of these units, then there is an associated dimension which is not unit based but the units reside within it or are composed of the dimension being measured. They posit, number of dimensions are not limited to the dimensions of space and time but include all manner of observable phenomena which can be quantified and measured. Thus dimension should include time-duration, electric current, thermodynamic temperature, amount of substance and luminous intensity. In the case of indiscernible, the concept of dimension is different than that of discernible. Let us examine their view.

OF VECTORS SPACES, LINEAR ALGEBRA & FIELDS

Some people claim that if V is a vector space, then its dimension is the cardinality of a minimal spanning set or maximal linearly independent set of vectors. What this is for infinite dimensional vector spaces depends on whether we want a Hamel basis, i.e. do we allow or disallow infinite direct sums. But physically, what does it mean? A vector space is said to be a space consisting of vectors, together with the associative and commutative operations of vectors and the associative and distributive operation of multiplication of vectors by scalars. For a general vector space, the scalars are members of a field F, in which case V is called a vector space over F. This is a statement and not a precise definition, as it uses the term ‘space’ without defining it precisely and showing whether such definition applies to the term Vector space. Also, how different is vector space from observed space.

Both space and time arise from our concepts of sequence and interval. When objects are arranged in an ordered sequence, the interval between them is called space. The same concept involving events is called time. We describe objects only with specific markers. Since intervals have no markers, they cannot be described. Thus, we use alternative symbolism to define space and time by using the limiting conditions, i.e., by the limiting objects and events. Space is described as the interval between limiting objects and time as the interval between limiting events.

A vector in physics is a quantity having direction as well as magnitude, especially as determining the position of one point in space relative to another. Movements are related to shifting mass. Even a wave, which passes on momentum, involves mass, as momentum itself is mass x velocity. All movements occur in space in some direction. There is no space, which is empty. Vector addition and multiplications are related to use of different forces to move mass in different directions in the same space. Intervals are not described by their mass. Then how does vector space differ from ordinary space?

Linear algebra deals with linear equations. When plotted, a linear equation gives rise to a line. Most of linear algebra takes place in the so-called vector spaces. It takes place over structures called field, which is a set (often denoted F) which has two binary operations +F (addition) and ·F (multiplication) defined on it. Thus, for any a, b F, a +F b and a ·F b are elements of F. They must satisfy certain rules. A nonempty subset W of a vector space V that is closed under addition and scalar multiplication (and therefore contains the 0-vector of V) is called a linear subspace of V, or simply a subspace of V, when the ambient space is unambiguously a vector space. This is not mathematics, but politics, where problems multiply by division. What does it physically mean?

Some people use the term ‘quantity of dimension one’ to reflect the convention in which the symbolic representation of the dimension for such quantities (like linear strain, friction factor, refractive index, mass fraction, Mach number, Reynolds number, degeneracy in quantum mechanics, number of turns in a coil, number of molecules, etc.) is the symbol 1. But they cannot define the ‘quantity of dimension one’ and how it is determined to be a dimension. Dimension is not a scalar quantity and a number has no physical meaning unless it is associated with some discrete object. Moreover, two lengths cannot be added or subtracted if they are perpendicular to each other, even though both have length.

A field is a region of space, upon entering which we experience a force. By convention, depending upon the nature of the force, we designate the field as electric field, magnetic field etc. Why complicate it with unnecessary details which has no physical meaning; like complex numbers?

WHAT IS NOT A DIMENSION

Some say: we can specify the time and place of an event in the universe by using three Cartesian coordinates for space and another number for time. This makes space-time four-dimensional. It shows that we can specify time using a number. An object remain invariant under mutual transformation of the dimensions: like rotating length to breadth or height, even though the measured value of the new axes change. Time does not fulfill these criteria. Further, we can change our directions in space, but not in time. We can measure both sides of our position in space and remember the result of measurement. But we cannot remember future. Hence time is not a dimension, though it is intricately linked to space due to the following reason.

Earlier, we have defined number as a universal quality of all substances by which we differentiate between similars. Zero is that which is not present at here-now, but is present elsewhere. Elsewhere we have proved mathematically that division of a number by zero is not infinity, but it leaves the number unchanged. Infinity is like one – without similars, with one exception. While the dimensions of one are discrete – hence clearly perceived, the dimensions of infinity are analog and not clearly perceived. Space, time, coordinates and Consciousness are the only infinities. We use their digital segments like buckets of water from ocean. Infinities do not interact as interaction involves change of position, which is possible only in discrete objects. Infinities can coexist. Thus, space and time coexist to appear as spacetime.

Some hold that the dimension of a physical quantity is defined as the power to which the fundamental quantities are raised to express the physical quantity. Suppose there is a geometric shape with some associated quantity and we scale up the lengths of all sides of the shape by 2. If the associated quantity scales 2d, then d is the dimension. For example, take a plane polygon on a graph. If we double its side-lengths, we multiply it by 22 – change in area. For a polyhedron, doubling the sides gives a factor of 23 - change in volume. But these changes have other known geometrical properties also. When we take higher values like 4 or n, can these values be derived like length, area or volume for dimensions 1, 2, and 3 respectively? There is no higher dimension with similarly increasing geometrical properties. Why should we presume higher dimensions?

Can luminous intensity be a dimension? No, because dimension is a fixed quality that depicts invariant extent in a given direction, but intensity is neither invariant nor has a direction. It is uniform within its spread area. Is the mass or the amount of substance a dimension? No, because mass is defined as a dimensionless quantity representing the amount of matter in a particle. Can an effectively ‘dimensionless dimension of one’ be defined such that it is derived as a ratio of dimensions of the same type: as in deriving angle? No, because the statement is self-contradictory.

Can the measurement change the phenomenon, body, or substance under study in such a way that the quantity actually measured differs from the measurand: like the potential difference between the terminals of a battery may decrease when using a voltmeter with a significant internal conductance to perform the measurement? No; it is a difference of intensity – not dimension. For the same reason, thermal temperature is not a dimension. The open-circuit potential difference can be calculated from the internal resistances of the battery and the voltmeter. Further, this definition differs from that in VIM, 2nd Edition, Item 2.6, and some other vocabularies, that define the measurand as the quantity subject to measurement. The description of a measurand requires specification of the state of the phenomenon, body, or substance under study. In chemistry, the measurand can be a biological activity.

Do the number of dimensions we see is limited by our senses that define our perceptions? Are sight, sound, taste, smell, and touch the only senses an organism can have? Yes; they replicate the fundamental forces of Nature. Eyes use only electromagnetic radiation (उपयाम). Sound travels between bodies separated only by a medium – like gravitational interaction (उद्याम). Smell replicates strong interaction (अन्तर्याम). Taste replicates beta decay component of weak interaction (वहिर्याम). Touch replicates the rest of weak interaction – like alpha decay (यातयाम).

Some say birds have another sense – they can perceive and navigate by the Earth’s magnetic fields. This is not a different sense, but one aspect of touch (स्पर्श). Others say: certain animals, like the mantis shrimp, see different colors than we do. These are capacity to see different wavelengths (रूप) and not a different sense. Could there be dimensions that no organism, terrestrial or otherwise, could perceive (अतीन्द्रिय)? Whether it is an issue of size (अणुपरिमाण) or our limited senses (सङ्कुचितशक्ति), could extra-dimensions be reason for science to turn to mathematics as a means of advanced exploration? No. Speculation is not science.

Some say: dimension of a physical quantity is the index of each of the fundamental quantity (Length, mass, time,) which express that quantity. The dimension of mass, length and time are represented as [M], [L] and [T] respectively. For example, the dimension of speed can be derived as: Speed= distance/time = length/time = L/T = L.T-1

In the above expression, there is no mention of mass, current or temperature because they do not play any role in defining this quantity. Or the dimension of mass, current, luminous intensity, temperature in this expression is zero. This is the brute force approach. A system consists of several necessary parameters. By arbitrarily reducing these parameters to zero, the system no longer remains as it is. Thus, it is a wrong description.

According to the principle of homogeneity of dimensional equations, the dimensions of fundamental quantities on LHS of an equation must be equal to the dimensions of the fundamental quantities on the RHS of that equation. The famous equation e = mc2 fails this test. Let us consider three quantities A, B and C such that C = A + B. According to this principle, the dimensions of C are equal to the dimensions of A and B. For example: we can write the dimensional first equation of kinematics:  v = u + at as: [M0 L T-1] = [M0 L T-1] + [M0 L T-1] X [M0 L0 T] = [M0 L T-1].

Apart from the fact that mass and time are not dimensions as shown above (also being variables or emergent properties), the equation does not give information about the dimensional constant common to all parameters like mass, length and time. If a quantity depends on more than three factors having dimension, the formula cannot be derived. From the above equation, we cannot derive the formulae containing trigonometric function, exponential functions, logarithmic function, etc. The exact form of relation cannot be developed when there are more than one part in any relation. It gives no information whether a physical quantity is scalar or vector.

Others say: high-dimensional abstract spaces (independent of the physical space we live in) like parameter spaces or configuration spaces such as in Lagrangian or Hamiltonian mechanics exist. This implies that position coordinates are not the only dimensions. For example, if a system consisting of homogenous ideal gas particles following the postulates of Kinetic Theory of Gases contained in an ideal confinement, the Pressure P; Volume V; Temperature T; and amount of gas i.e. no. of moles n, are the only required dimensions to state all the properties of that system. These are mere words. What is the proof in support of this argument? Has these spaces been discovered?

Some say: dimension is basically a number needed to specify something. For example the surface of a sheet of paper is two-dimensional because we can specify a point on the sheet of paper using the Cartesian coordinate system. But a graph is not the same as the real object it represents. The paper itself is three dimensional with varying thickness. We use one of its surfaces for plotting the graph. The real object that the graph represents has three dimensions. The graph gives only partial information. Further, what we “see” is the radiation emitted by a body – not the body proper. What we touch is the body proper and not the radiation emitted by it. Thus, both give incomplete information, which needs to be mixed to get a complete picture. For this reason, we have two eyes.

Dimension is not a sequence of addresses existing at different address locations along the street at different years. A fixed physical address and time does uniquely identify a specific house, but that is an arbitrary nomenclature – not a universal rule to qualify as dimension.

THE 10 DIMENSIONS

Dimension is an existential description. Change in dimension changes the existential description of the body irrespective of time and space. It never remains the same thereafter. Since everything is in a state of motion with reference to everything else at different rates of displacement, these displacements could not be put into any universal equation. Any motion of a body can be described only with reference to another body. Poincare and other have shown that even three body equations cannot be solved precisely. Our everyday experience shows that the motion of a body with reference to other bodies can measure different distances over the same time interval and same distance over different time intervals. Hence any standard equation for motion including time variables for all bodies or a class of bodies is totally absurd.

Dimension is generally understood as the number of independent coordinates needed to specify any point in a given space. For describing the size of an object, we use three numbers: length, breadth and elevation. For describing any position on Earth, we use three numbers: longitude, latitude and elevation, which also express the same information for a spherical structure. Photon and other radiation that travel at uniform velocity, are massless or without a fixed background structure – hence, strictly, are not “bodies”.

The three or six dimensions (including their negative directions from the origin) are not absolute terms, but are related to the order of placement of the object in the coordinate system of the field in which the object is placed. Since
1.      dimension of an object (वयुन) is related to the spread of the object, i.e., the relationship between its “confined structural inner space” and its “outer space” through which it is related to others (प्रचय संयोग),
2.      the outer space (वयोनाध) is infinite,
3.      the outer space does not affect inner space without breaking the dimension (वय),
the three or six dimensions remain invariant under mutual transformation of the axes (पर्यायवाची). If we rotate the object so that x-axis changes to the y-axis or z-axis, there is no effect on the structure (spread - विस्तार) of the object, i.e. the relative positions between different points on the body and their relationship to the space external to it remain invariant.

Based on the positive and negative directions (spreading out from or contracting towards) the origin, these describe six unique functions of position, i.e. (x,0,0), (-x,0,0), (0,y,0), (0,-y,0), (0,0,z), (0,0,-z), that remain invariant under mutual transformation. Besides these, there are four more unique positions, namely (x, y), (-x, y), (-x, -y) and (x, -y) where x = y for any value of x and y, which also remain invariant under mutual transformation. These are the ten dimensions and not the so-called “mathematical structures”. Since time does not fit in this description, it is not a dimension.

Our ancients named these 10 dimensions as: 1) Maahendree (माहेन्द्री), 2) Vaishwaanaree (वैश्वानरी), 3) Yaamyaa (याम्या), 4) Nairhtee (नैऋती), 5) Vaarunee (वारुणी), 6) Vaayavee (वायवी), 7) Kouveree (कौवेरी), 8) Aishaani (ऐशानी), 9) Braahmee (ब्राह्मी) and 10) Naagee (नागी). The nomenclature indicates their confining character (संस्त्यान).

We will discuss strings physically in another paper.

BIBLIOGRAPHY

1)      VAISHESHIKA SOOTRA (वैशेषिकसूत्रम्) by KANADA (कणादः)
2)      COMPENDIUM ON PROPERTIES OF MATTER (पदार्थधर्मसंग्रह) by PRASHASTAPADA (प्रशस्तपादः)