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ULTIMATE SYMMETRY:

Fractal Complex-Time and Quantum Gravity

by Mohamed Haj Yousef



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I.1.2 Symmetry in Electromagnetism


As the name implies, electromagnetism is unified theory of electric and magnetic fields, including light and other forms of electromagnetic waves. In the 18th century, the concept of field was suggested by Faraday to explain how electric and magnetic forces act through space. This concept proved to be one of the most powerful theoretical tools of modern physics, and it was generalized later in many ways in Relativity and Quantum Field Theory. Using a set of four equations, Maxwell was able to describe the electromagnetic field using a set of differential equations that describe how electric charges and currents create electromagnetic fields, as well as describing how an electric field can generate a magnetic field, and vice versa:

 

 (1.1)

 (1.2)

 (1.3)

 (1.4)

 

The first two equations are Gauss s law for electricity and magnetism, which allow the calculation of the electric and magnetic fields, respectively, where:  is the divergence operator,  is the curl operator,  is the electric field in units of volt per meter,  is the electric displacement field in coulomb per square meter,  is the magnetic flux density in Tesla,  is the free electric charge density in coulomb per cubic meter.

The other two equations describe how fields circulate around their sources, and these are Ampere s law with Maxwell s correction, and Faraday s law, respectively. The magnetic fields circulates around electric currents and time varying electric fields, and the electric fields circulate around time varying magnetic fields, where:  is the magnetic field strength in ampere per meter, and  is the free current density in ampere per square meter.

Originally, Ampere s law states that magnetic fields can be generated by electric current, but Maxwell added the displacement current , and this is particularly important because it makes the set of equations mathematically consistent for non static fields, without changing the laws of Ampere and Gauss for static fields. Consequently, this it predicted the fact that a changing magnetic field induces an electric field and vice versa, which allow self-sustaining electromagnetic waves to travel through empty space. The speed calculated for electromagnetic waves, which could be predicted from experiments on charges and currents, exactly matches the speed of light, which proved that light is one form of electromagnetic radiation, thereby unifying the theories of electromagnetism and optics.

In general, fields are something which associate a quantity with each point in space and time. In electromagnetism this quantity is a vector describing the electric and magnetic strength and direction, which require six numbers to describe them. However, Maxwell also noticed that electric and magnetic fields are intimately linked, and it was later shown that both phenomena can be associated with a single particle, the photon, which require only two numbers to be characterized at each point in space and time. This implied that there are some symmetries linking between the six numbers characterizing electric and magnetic fields relations.

For example, if someone is walking along the circumference of a circle, we can either describe the position by the height and width from the center of the circle, or we can use the angle around the circle s circumference, because changing the angle will change both height and width simultaneously. This connection comes from the fact that the circle is rotationally symmetric. This symmetry is called a gauge symmetry, because it provides a new mathematical description, without influencing what we can measure. This will be described further in section II.1.1 when we discuss symmetry in Quantum Mechanics below, although Gauge symmetry in Electromagnetism was recognized before the advent of Quantum Mechanics.

Based on this concept of symmetry, Maxwell saw the need for the extra term of displacement current  in the equation that relates current density to a resulting magnetic field. This generalization of the laws of electrodynamics revealed the connection between these equations and optical phenomena, which lead to the prediction of the whole spectrum of radiation, such as radio waves, X-rays, infrared and gamma rays.

After introducing Special Relativity, Einstein recognized that Maxwell s equations are covariant with respect to the Lorentz transformations between inertial frames of reference. This means that the set of transformations of the space-time coordinates that project the laws of electrodynamics from any observer s reference frame to any other inertial frame, such that the laws remain unchanged, is the symmetry group called after Poincar group. As we shall discuss further in section III.1.1, the theory of Special Relativity asserted that not only the laws of electrodynamics and optics, but all of the laws of nature must be covariant under such transformations. Einstein then generalized this symmetry to assert that all of the laws of nature must remain covariant with respect to transformations between frames of reference that are in arbitrary types of relative motion, which is the theory of General Relativity that led to a new explanation of gravity where space-time geometry plays an essential role in the mathematical representation of the laws of nature.

This indicated that all the laws of nature must be field laws that are mapped in Riemannian curved space-time. Einstein later concluded that every attempt to establish a unified field theory must start from the group of transformations which is no less general than that of the continuous transformations of the four coordinates. This covariance in terms of a continuous group does not admit the discrete reflections in space or time. This was also reflected in Noether s theorem which prescribes that the transformations that define the covariance in relativity theory must be non-singular everywhere. Such groups of continuous transformations are covered by Lie algebra.

Therefore, the continuous group that underlies Special and General Relativity implies that all discrete symmetries must be excluded from the laws of nature. However, as we shall see in section II.2.2 below, it was discovered in 1957 that the weak interaction violates spatial reflection, called parity, so if there is any unified field theory, in which all forces of nature are manifestations of a single force field, then this means that the electromagnetic and nuclear forces must also violate parity and time reversal symmetry.

 



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