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## F=ma Classical Mechanics Quantum Mechanics

Esquivel, Teresa, Managing Editor has reference to this Academic Journal, PHwiki organized this Journal Quantum Geometry: A reunion of math in addition to physics Physics in addition to Math are quite different: Physics Math Although to an uninitiated eye they may appear indistinguishable

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Math: deals with abstract ideas which exist independently of us, our practice, or our world (Plato) Physics: the study of the most fundamental properties of the real world, especially motion in addition to change (Aristotle) Mathematicians prove theorems in addition to value rigorous proofs. E.g. Jordan curve theorem: Every closed non-self-intersecting curve on a plane has an inside in addition to an outside. Seems evident but is not easy to prove. Physicists are more relaxed about rigor.

Since the times of Isaac Newton, physics is impossible without math: Laws of Nature are most usefully expressed in mathematical as long as m. In a sense, physics is applied math.

Most of the time, physicists are consumers of math: They do not invent new mathematical concepts. And mathematicians usually do not need physics. But once in a while physicists have to invent new math concepts to describe what they see around them. Isaac Newton had to invent calculus to be able to as long as mulate laws of motion. F=ma Here a is time derivative of velocity. The invention of calculus was a revolution in mathematics. Relativity theory of Einstein did not lead to a mathematical revolution. It used the tools which were already available: The geometry of curved space created by Riemann.

But quantum mechanics does require radically new mathematical tools. Some of these have been invented by mathematicians inspired by physical problems. Some were intuited by physicists. Some remain to be discovered. What sort of math does one need as long as Quantum Physics Classical Mechanics Observables (things we can measure) are real numbers Determinism Positions in addition to velocities are all we need to know

Quantum Mechanics Observables are not numbers: they do not have particular values until we measure them. Outcomes are inherently uncertain, physical theory can only predict probabilities of various outcomes. Cannot measure positions in addition to velocities at the same time (Heisenberg’s uncertainty principle). Heisenberg’s Uncertainty Principle x is the uncertainty of position p is the uncertainty of momentum (p=mv) =6.626 10-34 kgm2/sec is Planck’s constant The better you know the position of a particle, the less you know about its momentum. And vice versa:

How can we describe this strange property mathematically The answer is surprising: Quantum position in addition to quantum momentum are entities which violate a basic rule of elementary math: commutativity of multiplication X P P X Recall that ordinary multiplication of numbers is commutative: a b=b a in addition to associative: a (b c)=(a b) c One can often define multiplication of other entities. It is usually associative, but in many cases fails to be commutative. Which other entities can be multiplied Example 1: functions on a set X. A function f attaches a number f(x) to every element x of the set X. The product of functions f in addition to g is a function which attaches the number f(x) g(x) to x. This multiplication is commutative in addition to associative.

Example 2: rotations in space. Multiplying two rotations is the same as doing them in turn. One can show that the result is again a rotation. This operation is associative but not commutative. Another difference between the two examples is that functions on a set X can be both added in addition to multiplied, but rotations can be only multiplied. When some entities can be both added in addition to multiplied, in addition to all the usual rules hold, mathematicians say these entities as long as m a commutative algebra. Functions on a set X as long as m a commutative algebra. When all rules hold, except commutativity, mathematicians say the entities as long as m a non-commutative algebra. Quantum observables as long as m a non-commutative algebra! This is a mathematical reflection of the Heisenberg Uncertainty Principle.

But there are many more non-commutative algebras than commutative ones. Just like there are more not-bananas than bananas. Bananas Not bananas There are many special cases, where we know the answer. Say, as long as a particle moving on a line, we have position X in addition to momentum P. XP-PX=i where i is the imaginary unit, i2 =-1. Their algebra is determined by the following commutation relation But how do we find suitable multiplication rules in other situations To find the right algebra, we can try to use the Correspondence Principle of Niels Bohr: Quantum physics should become approximately classical as becomes very small.

Slight difficulty: has a particular value, how can one make it smaller or larger But this is easy: imagine you are a god in addition to can choose the value of when creating the Universe. A Universe with a smaller will be more classical. A Universe with a larger will be more quantum. Tuning to zero will make the Universe completely classical. Conversely, we can try to start with a classical system in addition to turn it into a quantum one, by cranking up . correspondence quantization This is called quantization. Classical Quantum Let’s recap. To describe a quantum system mathematically, we need to find the right non-commutative algebra. We can start with the mathematical description of a classical system in addition to try to quantize it by cranking up . This is called quantization.

Varieties of Quantum Geometry Non-commutative geometry: Quantization of phase space Hidden dimensions may be non-commutative (A. Connes) Stringy geometry Mirror symmetry Hidden dimensions in non-perturbative string theory (M-theory, strings in low dimensions) Quantized space-time The idea that physical space-time should be quantized in addition to perhaps non-commutative is attractive. Motivation: in quantum gravity, one cannot measure distances shorter than some minimal length. Reason: achieving a very accurate length measurement requires a lot of energy, which may curve the space-time in addition to distort the result. What is the structure of space-time at very short length in addition to time scales Is it non-commutative Is it stringy It is a safe bet that answering these physical questions will require entirely new math.

## Esquivel, Teresa Managing Editor

Esquivel, Teresa is from United States and they belong to Food Product Design and they are from Phoenix, United States got related to this Particular Journal. and Esquivel, Teresa deal with the subjects like Beverages; Food

## Journal Ratings by Colby College

This Particular Journal got reviewed and rated by Colby College and short form of this particular Institution is US and gave this Journal an Excellent Rating.