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
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)
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