TYPES OF ENTANGLEMENT (आरादुपकारक & संनिपत्योपकारक) – 6- Shri Basudeba Mishra
PHYSICAL MATHEMATICS OR MATHEMATICAL PHYSICS?
Since the wave function has no physical meaning, and only its square has meaning, it is necessary to discuss the relationship between physics and mathematics. 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), which is 6(C+H2O). 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.
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 (गणितम्), which explains the accumulation and reduction of numbers linearly or non-linearly of confined or discrete objects (गणँ स॒ङ्ख्याने॑). Number is that which universally differentiates one from the others (भेदाभेद विभागोहि लोके सङ्ख्या निबन्धन). The differentiation is done step by step or unit by unit. This unit is one (एक इता संख्या । इता अनुगता सर्वत्र या संख्या सा एक), which is universally perceived similarly in all things. Even analog fields are quantized based on this principle.
Number (स॒ङ्ख्या) 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, which leads to validation of predictions (प्रमाणतोऽर्थप्रतिपत्तौ प्रवृत्तिसामर्थ्यादर्थवत् प्रमाणम्). Logic is the special proof necessary for knowing the unknown aspects of something generally known (तर्को न प्रमाणसंगृहीतो न प्रमाणान्तरं, प्रमाणानामनुग्राहकस्तत्त्वज्ञानाय कल्पते). Thus, the validity of a mathematical statement rests with its logical consistency.
Differentiation is related to perception by dissection (अवच्छेदः). Perception is taking note of the result of measurement (मानः व्यवहारः). Measurement is a process of comparison between similars based on previous data to identify similarity (चित्तसमुन्नतिरक्षुद्रता). 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 by which, after detailed analysis, we arrive at a conclusion about the invariant nature of something (तत्त्वम् – तनोति सर्वमिदं – तनुँ श्रद्धोपकर॒णयोः तनुँ॑ विस्ता॒रे च). We have concept about something. The objects or subjects may differ, but their fundamental “concepts” 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 (भाषा – स्वहृदयस्थो भावो यया परहृदये समुन्नीयते). 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 (तत्थ्यम्) is gathered, one can achieve a “God’s eye view”. Data is valid in the specific context (तथा साधु) and is not synonymous with knowledge (ज्ञानम्). Knowledge is the concepts stored in memory based on previous sensory experience (स्मृतिपूर्वानुभूतार्थविषयं ज्ञानमुच्यते). 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! Space and time are related to everything in the universe in the same way (अमूर्तः) – they are intervals between objects and events (कालात् क्रियाबिभज्यन्ते आकाशात् सर्वमूर्तयः) and are the universal base for everything – thus analog and infinite (आधारशक्तिप्रथमा सर्वसंयोगिनां मता) – the more you measure, still it is there. Everything we see is related to limited objects and events (मूर्तः) and discreet (अवच्छिन्न). Hence they can’t be unified except as container-contained (आधाराधेयभावः). Just like we can’t do mathematics with apples and oranges, we can’t do mathematics with space/time and other objects. We can only measure the intervals between objects or events – points in space and time, and do mathematics with them.
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).
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.