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Ideas

What is science?

• Science is the quest to capture the processes of nature in formal mathematical representations

So “math is the blueprint of reality” in the sense that formal systems are the foundation of science.
In a nutshell:

• Natural systems are a subset of reality, i.e., the observable universe
• Guided by thought, observation and measurement natural systems are “encoded” into formal systems
• Using logic (rules of inference) in the formal system, predictions about the natural system can be made (decoding)
• Checking the predictions with the experimental outcome gives the validity of the formal system as a model for the natural system

Physics can be viewed as dealing with the fundamental interactions of inanimate matter.

For a technical overview, go to the here.

Paradigm

• Mathematical models of reality are independent of their formal representation

This leads to the notions of symmetry and invariance. Basically, this requirement gives rise to nearly all of physics.

Classical Mechanics

Symmetry, understood as the invariance of the equations under temporal and spacial transformations, gives rise to the conservation laws of energy, momentum and angular momentum.

In layman terms this means that the outcome of an experiment is unchanged by the time and location of the experiment and the motion of the experimental apparatus. Just common sense…

Mathematics of Symmetry

The intuitive notion of symmetry has been rigorously defined in the mathematical terms of group theory.

Physics of Non-Gravitational Forces

The three non-gravitational forces are described in terms of quantum field theories. These in turn can be expressed as gauge theories, where the parameters of the gauge transformations are local, i.e., differ from point to point in space-time.

The Standard Model of elementary particle physics unites the quantum field theories describing the fundamental interactions of particles in terms of their (gauge) symmetries.

Physics of Gravity

Gravity is the only force that can’t be expressed as a quantum field theory.

Its symmetry principle is called covariance, meaning that in the geometric language of the theory describing gravity (general relativity) the physical content of the equations is unchanged by the choice of the coordinate system used to represent the geometrical entities.

To illustrate, imagine an arrow located in space. It has a length and an orientation. In geometric terms this is a vector, lets call it a. If I want to compute the length of this arrow, I need to choose a coordinate system, which gives me the x-, y- and z-axes components of the vector, e.g., a = (3, 5, 1). So starting from the origin of my coordinate system (0, 0, 0), if I move 3 units in the x direction (left-right), 5 units in the y-direction (forwards-backwards) and 1 unit in the z direction (up-down), I reach the end of my arrow. The problem is now, that depending on the choice of coordinate system - meaning the orientation and the size of the units - the same arrow can look very different: a = (3, 5, 1) = (0, 23.34, -17). However, everytime I compute the length of the arrow in meters, I get the same number independent of the chosen representation.

In general relativity the vectors are somewhat like multidimensional equivalents called tensors and the commonsense requirement, that the calculations involving tensor do not depend on how I represent the tensors in space-time, is covariance.

It is quite amazing, but there is only one more ingredient needed in order to construct one of the most estethic and accurate theories in physics. It is called the equivalence principle and states that the gravitational force is equivalent to the forces experienced during acceleration. This may sound trivial, has however very deep implications.

Physics of Condensed Matter

This branch of physics, also called solid-state physics, deals with the macroscopic physical properties of matter. It is one of physics first ventures into many-body problems in quantum theory. Although the employed notions of symmetry do not act at such a fundamental level as in the above mentioned theories, they are a cornerstone of the theory. Namely the complexity of the problems can be reduced using symmetry in order for analytical solutions to be found. Technically, the symmetry groups are boundary conditions of the Schrödinger equation. This leads to the theoretical framework describing, for example, semiconductors and quasi-crystals (interestingly, they have fractal properties!). In the superconducting phase, the wave function becomes symmetric.

Conclusion

The Success

It is somewhat of a miracle, that the formal systems the human brain discovers/devises find their match in the workings of nature. In fact, there is no reason for this to be the case, other than that it is the way things are.

The following two examples should underline the power of this fact, where new features of reality where discovered solely on the requirements of the mathematical model:

• In order to unify electromagnetism with the weak force (two of the three non-gravitational forces), the theory postulated two new elementary particles: the W and Z bosons. Needless to say, these particles where hitherto unknown and it took 10 years for technology to advance sufficiently in order to allow their discovery.
• The fusion of quantum mechanics and special relativity lead to the Dirac equation which demands the existence of an, up to then, unknown flavor of matter: antimatter. Four years after the formulation of the theory, antimatter was experimentally discovered.

The Future…

Albeit the success, modern physics is still far from being a unified, paradox-free formalism describing all of the observable universe. Perhaps the biggest obstacles lies in the last missing step to unification. In a series of successes, forces appearing as being independent phenomena, turned out to be facets of the same formalism: electricity and magnetism was united in the four Maxwell equations; as mentioned above, electromagnetism and the weak force were merged into the electroweak force; and finally, the electroweak and strong force were united in the framework of the standard model of particle physics. These four forces are all expressed as quantum (field) theories. There is only one observable force left: gravity.
The efforts to quantize gravity and devise a unified theory, have taken a strange turn in the last 20 years. The problem is still unsolved, however, the mathematical formalisms engineered for this quest - namely string/M-theory and loop quantum gravity - have had a twofold impact:

• A new level in the application of formal systems is reached. Whereas before, physics relied on mathematical branches that where developed independently from any physical application (e.g., differential geometry, group theory), string/M-theory is actually spawning new fields of mathematics (namely in topology).
• These theories tell us very strange things about reality:
  • Time does not exist on a fundamental level
  • Space and time per se become quantized
  • Space has more than three dimensions
  • Another breed of fundamental particles is needed: supersymmetric matter

Unfortunately no one knowns if these theories are hinting at a greater reality behind the observable world, or if they are “just” math. The main problem being the fact that any kind of experiment to verify the claims appears to be out of reach of our technology…

From: http://j-node.homeip.net/knowledgebase/overview/

tags: science, fundamental, analytical, reality, mathematical models

This entry was posted on Wednesday, March 28th, 2007 at 12:16 pm and is filed under This'n'that, science overview. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.