The sense of achievement and closure for theoretical physics that came with the brilliant success of the classical field theory of electromagnetism was short lived. The new technology invented out of the mathematical unification of electricity with magnetism produced copious data about the nature of matter and light that snapped all of the mathematical threads that physicists had just succeeded in tying down.
   And after this new data was unraveled and understood and explained using mathematics, the unified worldview of classical theoretical physics became split into two very different views of the universe -- the particle view and the geometric view.

Particles and waves

   The first sign of trouble was when J.J. Thomson discovered the electron in 1897. Experimentalists began to see data that suggested a model of the atom with negatively charged particles orbiting around a positively charged core. But according to Maxwell's equations, such a system should be physically unstable. Classical field theory was unable to explain or describe the emerging data on atomic structure.
   Another big mystery that came out of Maxwell's equations was the thermal behavior of light. Hot objects, like a hot coal, glow by emitting light and that light is observed to consist of a distribution of waves of different frequencies. But physicists who tried to explain the observed distribution of frequencies using light waves as described by Maxwell's equations met with continued failure.
    Then as the new 20th century was beginning, a young German physicist, in an "act of despair" over the gaps in the understanding of thermal radiation, made a guess called the Quantum Hypothesis, which explained the observed thermal spectrum of light as coming from a collection of identical discrete quanta of energy. His formula worked, but he didn't know why.
   This was the beginning of the idea known as particle-wave duality, and the field of quantum mechanics.
   Einstein used Planck's idea to explain the newly-observed photoelectric effect. Einstein proposed that light was emitted or absorbed by an excited electron in discrete quanta called photons whose energy was proportional to the frequency of the light according to the relation

,

where h is a number called Planck's constant, determined by measurement to be 6.6 x 10^-34 joule seconds.
   If a light wave could behave like a particle, then could a particle behave like a wave of some kind? In 1923, French aristocrat Louis de Broglie put forward the idea that an electron traveling with some momentum p could act like a continuous wave with wavelength l according to the relation

   When the dust was settled, the new quantum theory described a given physical system not in terms of the path of a particle or the strength of a field, but as the probability amplitude for a given system to be in a given quantum state. This probability amplitude is the square of a function called the wave function Y(x,t), which is a solution to the Schrodinger equation

   Solutions to Schödinger equation for more then one identical particle have an interesting symmetry. For example, let's consider a two particle system and exchange the two particles. The wave function will obey the relation

In the plus case, the two particles are what we call bosons. Two bosons can occupy the same quantum state at the same time.
   In the minus case, the two particles are what we call fermions. Two fermions cannot occupy the same quantum state at the same time. This effect is called Pauli repulsion, and Pauli repulsion explains the structure of the periodic table of elements and the stability of atoms, and hence of all matter.

Relativity and geometry

   The radical new idea of the quantum physics of atoms and light marked one direction of departure from the comforting sureness of 19th century classical field theory. The other big surprise of the 20th century came with the astounding observation in an experiment by Michelson and Morley that the speed of light was independent of the motion of the observer.
   Now normally one would think that is a person were capable of throwing a javelin at 5 miles per hour while standing still, that same person, when running across the ground at 10 miles per hour, would be capable of making the javelin travel across the ground at a speed of 15 miles per hour.
   But according to the data from the Michelson-Morley experiment, if one uses a laser instead of a javelin, then whether the person is sanding still or running 60 miles per hour or in a rocket traveling near the speed of light -- the light from the laser still travels the same speed!
   This was an astounding result! How could it be explained using physics? Einstein came up with a powerful, simple theory, called the Special Theory of Relativity. Einstein used the geometric notion of a metric. The most familiar metric is just the Pythagorean Rule, which in three space dimensions in differential form looks like

This formula has the special property that it is invariant under rotations. In other words, the length of a straight line does not change when you rotate the line in space. In the Special Theory of Relativity the idea of a metric is extended to include time, with a very crucial minus sign:

Like the space metric, the spacetime is invariant under rotations in space. But now there is a new twist -- the spacetime metric is also invariant under a kind of rotation of space and time called a Lorentz transformation, and this transformation tells us how different observers who are moving with some constant velocity relative to one another see the world.
   And under a Lorentz transformation, the speed of light always stays the same, which is consistent with the shocking Michelson-Morley experiment.
   Einstein's next target of revision was Newton's Universal Law of Gravitation. In Newton's formula the gravitational force F₁₂ between two planets of masses m₁ and m₂ as depending on the inverse square of the distance r₁₂ between the planets

G_N is called Newton's constant and is measured to be 6.7x10-8 cm³ /(gm sec²).
   Newton's Law was extremely successful at explaining the observed motions of the planets around the Sun, and of the moon around the Earth, and easily extendible through the techniques of classical field theory to continuous systems.
   However, there was no hint in Newton's theory as to how a gravitational field would change in time, especially not in a manner that was consistent with the new understanding in Special Relativity that nothing can travel faster than the speed of light.
   Einstein took a very bold step, and reached out to some radical new mathematics called non-Euclidean geometry, where the Pythagorean rule is generalized to include metrics with coefficients that depend on the spacetime coordinates in the form

where repeated indices imply a sum over all space and time directions in the chosen coordinate system. Einstein extended the idea of Lorentz invariance to general coordinate invariance, proposing that the values of physical observables should be independent of a choice of coordinate system used to chart points in spacetime. He called this new theory the General Theory of Relativity.
   In Einstein's new theory, spacetime can have curvature, like the surface of a beach ball has curvature, compared to the flat top of a table, which doesn't. The curvature is a function of the metric g_ab and its first and second derivatives. In the Einstein equation

the spacetime curvature (represented by R_mn and R) is determined by the total energy and momentum T_mn of the "stuff" in the spacetime like the planets, stars, radiation, interstellar dust and gas, black holes, etc.
    The Einstein equation is not strictly a departure from classical field theory, and the Einstein equation can be derived as the solution to Euler-Lagrange equations that represent the stationary point, or extremum, of the action

Two views of the world

   Using quantum mechanics, the typical questions that can be answered concern the types of quantum states and allowed transitions in a system that features one or more particles that has some type of potential energy represented by the potential V(x). A typical method of working is to take some given V(x) and use the Schrödinger equation find the wave function, the energies of the quantum states of the system, and the allowed transitions between those states.
   In general relativity, things are very different. One performs calculations that compute the evolution and structure of an entire universe at a time. A typical way of working is to propose some particular collection of energy and matter in the universe,to provide the T_mn. Given a particular T_mn, the Einstein equation turns into a system of second order nonlinear differential equations whose solutions give us the metric of spacetime, g_mn, which holds all the information about the structure and evolution of a universe with that given T_mn.
   Given the difference in the fundamental questions and methodologies used in quantum mechanics and in general relativity, it seems hardy surprising that uniting quantum physics with gravity, for a theory of quantum gravity, would prove to be a very tough challenge.