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Wiki - Field Theory of Macroeconomics [fundamentals]

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(the following is an extract of some articles from http://en.wikipedia.org ) =====Field theory===== Field theory usually refers to a construction of the dynamics of a field, i.e. a specification of how a field changes with time or with respect to other components of the field. Usually this is done by writing a [[http://en.wikipedia.org/wiki/Lagrangian|Lagrangian]] of the field, and treating it as a system with an infinite number of degrees of freedom. The resulting field theories are referred to as classical or quantum field theories. In modern physics, the most often studied fields are those that model the four fundamental forces which one day may lead to the Unified Field Theory. ==Classical fields== There are several examples of classical fields. The dynamics of a classical field are usually specified by the Lagrangian density in terms of the field components; the dynamics can be obtained by using the action principle. Michael Faraday first realized the importance of a field as a physical object, during his investigations into magnetism. He realized that electric and magnetic fields are not only fields of force which dictate the motion of particles, but also have an independent physical reality because they carry energy. These ideas eventually led to the creation, by James Clerk Maxwell, of the first unified field theory in physics with the introduction of equations for the electromagnetic field (Maxwell's equations). At the end of the 19th century, the electromagnetic field was understood as a collection of two vector fields in space. Nowadays, one recognizes this as a single antisymmetric 2nd-rank tensor field in spacetime. Einstein's theory of gravity, called general relativity, is another example of a field theory. Here the principal field is the metric tensor, a symmetric 2nd-rank tensor field in spacetime. ==Quantum fields== It is now believed that quantum mechanics should underlie all physical phenomena, so that a classical field theory should, at least in principle, permit a recasting in quantum mechanical terms; success yields the corresponding quantum field theory. For example, quantizing classical electrodynamics gives quantum electrodynamics. Quantum electrodynamics is arguably the most successful scientific theory; experimental data confirm its predictions to a higher precision than any other theory. General relativity, the classical field theory of gravity, has yet to be successfully quantized. Classical field theories remain useful wherever quantum properties do not arise, and can be active areas of research. Elasticity of materials, fluid dynamics and Maxwell's equations are cases in point. =====Symmetries of fields===== A convenient way of classifying a field (classical or quantum) is by the symmetries it possesses. Physical symmetries are usually of two types: ==Spacetime symmetries== Fields are often classified by their behaviour under transformations of spacetime. The terms used in this classification are: * scalar fields (such as temperature) whose values are given by a single variable at each point of space. This value does not change under transformations of space. * vector fields (such as the magnitude and direction of the force at each point in a magnetic field) which are specified by attaching a vector to each point of space. The components of this vector transform between themselves as usual under rotations in space. * tensor fields, (such as the stress tensor of a crystal) specified by a tensor at each point of space. The components of the tensor transform between themselves as usual under rotations in space. * spinor fields are useful in quantum field theory. ==Internal symmetries== Fields may have internal symmetries in addition to spacetime symmetries. For example, in many situations one needs fields which are a list of space-time scalars: (φ<sub>1</sub>,φ<sub>2</sub>...φ<sub>N</sub>). For example, in weather prediction these may be temperature, pressure, humidity, etc. If there is a symmetry of the problem, not involving spacetime, under which these components transform into each other, then this set of symmetries is called an '''internal symmetry'''. One may also make a classification of the charges of the fields under internal symmetries. ==Symmetry as an invariance== Invariance is specified mathematically by transformations that leave some quantity unchanged. This idea can apply to basic real-world observations. For example, temperature may be constant throughout a room. Since the temperature is independent of position within the room, in this special case the temperature is ''invariant'' under a shift in the measurer's position. Similarly, a uniform sphere rotated about its center will appear exactly as it did before the rotation. The sphere is said to exhibit spherical symmetry. A rotation about any axis of the sphere will preserve how the sphere "looks". == Invariance in force == The above ideas lead to the useful idea of ''invariance'' when discussing observed physical symmetry; this can be applied to symmetries in forces as well. For example, an electric field due to a wire is said to exhibit rotational symmetry. Rotating the wire about its own axis does not change its position or charge density, hence it will preserve the field. In Newton's theory of mechanics, given two bodies, each with mass ''m'', starting from rest at the origin and moving along the ''x''-axis in opposite directions, one with speed ''v''<sub>1</sub> and the other with speed ''v''<sub>2</sub> the total kinetic energy of the system (as calculated from an observer at the origin) is 1⁄2 m(v<sub>1</sub><sup>2</sup> + v<sub>2</sub><sup>2</sup>) and remains the same if the velocities are interchanged. The total kinetic energy is preserved under a reflection in the ''y''-axis. The last example above illustrates another way of expressing symmetries, namely through the equations that describe some aspect of the physical system. The above example shows that the total kinetic energy will be the same if ''v''<sub>1</sub> and ''v''<sub>2</sub> are interchanged. =====Conservation laws and symmetry===== The symmetry properties of a physical system are intimately related to the conservation laws characterizing that system. **Noether's theorem gives a precise description of this relation.** The theorem states that each continuous symmetry of a physical system implies that some physical property of that system is conserved. Conversely, each conserved quantity has a corresponding symmetry. For example, the isometry of space gives rise to conservation of (linear) momentum, and isometry of time gives rise to conservation of energy. ====Noether's theorem==== Noether's (first) theorem states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. The theorem was proved by German mathematician Emmy Noether in 1915 and published in 1918. The action of a physical system is the integral over time of a Lagrangian function (which may or may not be an integral over space of a Lagrangian density function), from which the system's behavior can be determined by the principle of least action. **Noether's theorem has become a fundamental tool of modern theoretical physics and the calculus of variations.** For illustration, if a physical system behaves the same regardless of how it is oriented in space, its Lagrangian is rotationally symmetric; from this symmetry, Noether's theorem shows the angular momentum of the system must be conserved. The physical system itself need not be symmetric; a jagged asteroid tumbling in space conserves angular momentum despite its asymmetry — it is the laws of motion that are symmetric. As another example, if a physical experiment has the same outcome regardless of place or time (having the same outcome, say, somewhere in Asia on a Tuesday or in America on a Wednesday), then its Lagrangian is symmetric under continuous translations in space and time; by Noether's theorem, these symmetries account for the conservation laws of linear momentum and energy within this system, respectively. **Noether's theorem is important, both because of the insight it gives into conservation laws, and also as a practical calculational tool. It allows researchers to determine the conserved quantities from the observed symmetries of a physical system.**

 

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fundamentals.txt · Zuletzt geändert: 2012/10/05 01:23 (Externe Bearbeitung)
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