For Fqxi Essay Contest 2016-17 Physical Mathematics Explaining The Physics of Ten Dimensions
For Fqxi Essay Contest 2016-17 Physical Mathematics Explaining The Physics of Ten Dimensions
For Fqxi Essay Contest 2016-17 Physical Mathematics Explaining The Physics of Ten Dimensions
PHYSICAL MATHEMATICS
EXPLAINING THE PHYSICS OF TEN DIMENSIONS.
basudeba mishra
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.
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.
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.
DEFINING DIMENSION
Dimensions is the interface between the internal structural space and the external relational
space of an object depicted by the necessary parameters. 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.
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.
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?
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.
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. Sound travels between
bodies separated only by a medium – like gravitational interaction. Smell replicates strong
interaction. Taste replicates beta decay component of weak interaction. Touch replicates the rest
of weak interaction – like alpha decay.
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. Others say: certain animals,
like the mantis shrimp, see different colors than we do. These are capacity to see different
wavelengths and not a different sense. Could there be dimensions that no organism, terrestrial for
otherwise, could perceive? Whether it’s an issue of size or our limited senses, 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.
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
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.
BIBLIOGRAPHY