What is Theoretical Physics

[MjolnirH]

Basic

Theoretical physicists use mathematics to describe certain aspects of Nature. Sir Isaac Newton was the first theoretical physicist, although in his own time his profession was called "natural philosophy".

By Newton's era people had already used algebra and geometry to build marvelous works of architecture, including the great cathedrals of Europe, but algebra and geometry only describe things that are sitting still. In order to describe things that are moving or changing in some way, Newton invented calculus.

The most puzzling and intriguing moving things visible to humans have always been been the sun, the moon, the planets and the stars we can see in the night sky. Newton's new calculus, combined with his "Laws of Motion", made a mathematical model for the force of gravity that not only described the observed motions of planets and stars in the night sky, but also of swinging weights and flying cannonballs in England.

Today's theoretical physicists are often working on the boundaries of known mathematics, sometimes inventing new mathematics as they need it, like Newton did with calculus.

Newton was both a theorist and an experimentalist. He spent many many long hours, to the point of neglecting his health, observing the way Nature behaved so that he might describe it better. The so-called "Newton's Laws of Motion" are not abstract laws that Nature is somehow forced to obey, but the observed behavior of Nature that is described in the language of mathematics. In Newton's time, theory and experiment went together.

Today the functions of theory and observation are divided into two distinct communities in physics. Both experiments and theories are much more complex than back in Newton's time. Theorists are exploring areas of Nature in mathematics that technology so far does not allow us to observe in experiments. Many of the theoretical physicists who are alive today may not live to see how the real Nature compares with her mathematical description in their work. Today's theorists have to learn to live with ambiguity and uncertainty in their mission to describe Nature using math.

[MjolnirH]

Advanced

Theoretical physicists today still use a core technology that was developed in the 18th century out of the calculus pioneered by Isaac Newton and Gottfried von Leibniz.
Isaac Newton derived his three Laws of Motion through close, almost obsessive observation and experimentation, as well as mathematical reasoning. The relationship he discovered between force and acceleration, which he expressed in his own arcane notation of fluxions, has had the most impact on the world in the differential notation used by his professional rival, Wilhelm von Leibniz, as the familiar differential equation from freshman physics:




After Newton accused Leibniz of plagiarism in the discovery of calculus, Leibniz' vastly more convenient and intuitive differential and integral notation failed to become popular in England, and so the majority of advances in the development of calculus in the next century took place in France and Germany.

At the University of Basel, the multitalented Leonhard Euler began to develop the calculus of variations that was to become the most important tool in the tool kit of the theoretical physicist. The calculus of variations was useful for finding curves that were the maximal or minimal length given some set of conditions.
Joseph-Louis Lagrange took Euler's results and applied them to Newtonian mechanics. The general principle that emerged from the work of Euler and Lagrange is now called the Principle of Least Action, which could be called the core technology of modern theoretical physics.

In the Principal of Least Action, the differential equations of motion of a given physical system are derived by minimizing the action of the system in question. For a finite system of objects, the action S is an integral over time of a function called the Lagrange function or Lagrangian L(q, dq/dt), which depends on the set of generalized coordinates and velocities (q, dq/dt) of the system in question.



The differential equations that describe the motion of the system are found by demanding that the action be at its minimum (or maximum) value, where the functional differential of the action vanishes:



This condition gives rise to the Euler-Lagrange equations



which, when applied to the Lagrangian of the system in question, gives the equations of motion for the system.
As an example, take the system of a single massive particle with space coordinate x (in zero gravity). The Lagrangian is just the kinetic energy, and the action is the energy integrated over time:



The Euler-Lagrange equations that minimize the action just reproduce Newton's equation of motion for a free particle with no external forces:



The set of mathematical methods described above are collectively known as the Lagrangian formalism of mechanics. In 1834, Dublin mathematician William Rowan Hamilton applied his work on characteristic functions in optics to Newtonian mechanics, and what is now called the Hamiltonian formalism of mechanics was born.

The idea that Hamilton borrowed from optics was the concept of a function whose value remains constant along any path in the configuration space of the system, unless the final and initial points are varied. This function in mechanics is now called the Hamiltonian and represents the total energy of the system. The Hamiltonian formalism is related to the Lagrangian formalism by a transformation, called a Legendre transformation, from coordinates and velocities (q, dq/dt) to coordinates and momenta (q,p):



The equations of motions are derived from the Hamiltonian through the Hamiltonian equivalent of the Euler-Lagrange equations:



For a massive particle in zero gravity moving in one dimension, the Hamiltonian is just the kinetic energy, which in terms of momentum, not velocity, is just:



If the coordinate q is just the position of the particle along the x axis then the equations of motion become:



which is equivalent to the answer derived from the Lagrangian formalism.

Classical mechanics would have had a brief history if only the motion of finite objects such as cannonballs and planets could be studied. But the Lagrangian formalism and the method of differential equations proved well adaptable to the study of continuous media, including the flows of fluids and vibrations of continuous n-dimensional objects such as one-dimensional strings and two-dimensional membranes.

The Lagrangian formalism is extended to continuous systems by the use of a Lagrangian density integrated over time and the D-dimensional spatial volume of the system, instead of a Lagrange function integrated just over time. The generalized coordinates q are now the fields q(x) distributed over space, and we have made a transition from classical mechanics to classical field theory. The action is now written:



Here the coordinate xa refers to both time and space, and repetition implies a sum over all D+1 dimensions of space and time.
For continuous media the Euler-Lagrange equations become



with functional differentiation of the Lagrange density replacing ordinary differentiation of the Lagrange function.

What is the meaning of the abstract symbol q(x)? This type of function in physics that depends on space and time is called a field, and the physics of fields is called, of course, field theory.

The first important classical field theory was Newton's Law of Gravitation, where the gravitational force between two particles of masses m1 and m2 can be written as:



The gravitation force F can be seen as deriving from a gravitational field G, which if we set x1=0 and x2=x, can be written as:



Newton's Law of Gravitation was the beginning of classical field theory. But the greatest achievement of classical field theory came 200 years later and gave birth to the modern era of telecommunications.

Physicists and mathematicians in the 19th century were intensely occupied with understanding electricity and magnetism. In the late 19th century, James Clerk Maxwell found unified equations of motion of the electric and magnetic fields, now known as Maxwell's equations. The Maxwell equations in the absence of any charges or currents are:



Maxwell discovered that there exist electromagnetic traveling wave solutions to these equations, which can be rewritten as



and in 1873 he postulated that these electromagnetic waves solved the ongoing question as to the nature of light.

The greatest year in classical field theory came in 1884 when Heinrich Hertz generated and studied the first radio waves in his laboratory. Hertz confirmed Maxwell's prediction and changed the world, and physics, forever.

Maxwell's theoretical unification of electricity and magnetism was engineered into the modern human power to communicate across space at the speed of light. This was a stunning and powerful achievement for theporetical physics, one that shaped the face of coming 20th century as the century of global telecommunications.
But this was just the beginning. In the century that was just arriving, the power of theoretical physics would grow to question the very nature of reality, space and time, and the technological consequences would be even bigger.

[Kaber]

Ok so what about the Casimir effect?

What happens if you take two mirrors and arrange them so that they are facing each other in empty space? Your first reaction might be "nothing at all". In fact, both mirrors are mutually attracted to each other by the simple presence of the vacuum. This startling phenomenon was first predicted in 1948 by the Dutch theoretical physicist Hendrik Casimir while he was working at Philips Research Laboratories in Eindhoven on - of all things - colloidal solutions. The phenomenon is now dubbed the Casimir effect, while the force between the mirrors is known as the Casimir force.

For many years the Casimir effect was little more than a theoretical curiosity. But interest in the phenomenon has blossomed in recent years. Experimental physicists have realized that the Casimir force affects the workings of micromachined devices, while advances in instrumentation have enabled the force to be measured with ever-greater accuracy.

The new enthusiasm has also been fired by fundamental physics. Many theorists have predicted the existence of "large" extra dimensions in 10- and 11-dimensional unified field theories of the fundamental forces. These dimensions, they say, could modify classical Newtonian gravitation at sub-millimetre distances. Measuring the Casimir effect could therefore help physicists to test the validity of such radical ideas.

Unlike what would be expected from elementary electromagnetism, two parallel, neutral planes at a given distance do exert a force upon one another. This effect, referred to as the Casimir effect, is often introduced as a purely quantum electrodynamical effect in the published literature. Such is not the case, however, as this surface force can be understood as a completely classical phenomenon present in acoustics and observed in the past centuries between ships at sea under particular conditions. In the electromagnetic case, even space completely devoid of matter contains energy in the form of random fluctuations of the electric and magnetic fields. Such fluctuations, which are collectively referred to as the zero-point field, can be described classically by arbitrarily including random fields into Maxwell's equations, or quantum mechanically, as a natural consequence of Heisenberg's uncertainty principle.


1. A Mysterious Force

In order to appreciate the physical principles, it is useful to first focus on a particular manifestation of so called dispersion forces, a phenomenon referred to as the Casimir effect. In its typical manifestation, the Casimir effect is a force between two infinite, parallel planes of given optical properties at a given distance from one another- usually on the order of a micrometer or less (one micrometer is equal to one millionth of a meter). For instance, we can consider two highly conducting metal plates, or any other combination of two conducting, insulating, or magnetic plates facing one another.

It is an experimental fact that such two surfaces, although completely neutral, will exert a force upon one another. This force is in addition to, and completely different from, the expected gravitational attraction between them. It is also quite unexpected. We are taught in basic physics classes that any two objects, such as plates, particles, or planets, can only interact because of one or more of the four fundamental interactions of nature: (1) gravitation, (2) the weak force, (3) electromagnetism, and (4) the strong force, listed in order of increasing intensity. Of these, gravitation and electromagnetism are referred to as long-range forces, whereas the weak and the strong force are referred to as short-range forces.

We are all of course familiar with gravitation, which is the force that dominates all aspects of our Earth-bound lives. In the description given to us by Sir Isaac Newton, the gravitational force between (for instance) two homogenous, spherical objects is proportional to the product of their masses and inversely proportional to the distance between their centers squared. Albert Einstein of course radically changed this view of gravitation, from that of a force between objects to the effect that their mass and energy has on the space-time where other objects move. In other words, whereas Newton would have said that his famous apple was being attracted by the Earth all the way down from the apple tree, Einstein would have envisioned an apple in free-fall under the action of absolutely no force, but in a space-time curved by the presence of Earth's mass.

The weak force is a much more exotic interaction than the gravitation we are familiar with. For instance, it is ultimately responsible for some very important radioactive processes, such as the beta-decay of the neutron, which causes this particle to be unstable. Interestingly, in recent years a unified theory of electromagnetism and of the weak force has been developed and many of its predictions have been experimentally verified. This theory of the electro-weak force, referred to as the Standard Model, is one of the most successful theories of the fundamental nature of the Universe ever developed.

The first step towards this process of force unification was taken by James Clerk Maxwell in the 19th century, when he introduced his famous set of Maxwell equations, which showed that electricity and magnetism are not separate phenomena, but are actually deeply intertwined as different manifestations of the same electromagnetic interaction.

In some way, electromagnetism is also a force with which we have familiarity. It is clearly around us, although it does not appear to dominate our interaction with our surroundings as decisively as gravitation does. In fact, electromagnetism is the strongest long-range force in the Universe. The reason that gravitation appears to play a more important role in our daily lives is that, most often, different regions of the Universe are neutral. However, whenever objects around us become electrically charged or an electric current flows, electromagnetism easily displays its immense superiority to gravitation - for instance when an entire car is lifted from the ground with an electromagnet or lightning briefly turns the night into day. The importance of this interaction is further understood if we recall that light itself is an entirely electromagnetic phenomenon

The strong force is responsible for the very existence of atomic nuclei, as it is the "glue" that binds protons and neutrons together. As one would expect, a force capable to bring together protons within distances that are extremely small fractions of the size of one atom must indeed be incredibly intense, so as to overwhelm the electric repulsion between the positively charged protons. However, the range of action of the strong force is not much larger than the size of an atomic nucleus itself. Therefore, although matter as we know it could not be possible without the existence of the strong force, we never directly experience the most intense interaction in the Universe.

If we now return to our neutral parallel plates, it is easy to see that the mysterious force between them shown by the experiments does not immediately appear to be related to any of the four fundamental forces of nature. This interaction between the plates does not fit the description of a short-range force, it is not of a gravitational nature, and it does not appear to be electromagnetic because the surfaces are uncharged. This would appear to leave us only with the option to consider a new, previously unknown force to explain why our neutral planes exert a measurable force upon one another, so giving rise to the Casimir effect


Information was provided by the following URL's

www.interstellartechcorp.com/phyTheoretical.html?PHPSESSID=6b9da1513185013c675422e40358eada

and

physicsweb.org/articles/world/15/9/6