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It has been mentioned in Topic 12 that the transition from classical to quantum mechanics can be accomplished either by the path integral method or by the more ad hoc "canonical quantization" such that the momentum p and position q are no longer mere numbers but are operator satisfying the commutative relation: pq - qp ~ . The Schrodinger equation was developed and applied to the atomic and molecular system with great success. This formalism becomes increasingly inaccurate for phenomena in smaller domain where energy can manifest itself in a variety of ways, e.g., pair creation or particle moving at relativistic speed. Therefore, the Schrodinger equation has to be replaced by field equations such as the Klein-Gordon equation (for particle with no spin) or Dirac equation (for particle with spin 1/2). These field equations are invariant (unchanged) under a change of the space-time coordinate system (Lorentz transformation). It is referred to as relativistic invariance, which ensures the validity of the field equation at relativistic speed. To account for the creation/annihilation of particles in high energy interaction, the field is considered to be an operator. It is expanded into Fourier series in terms of harmonic functions and coefficients. These coefficients are then subjected to some quantization rules. Depending on whether the particle has integer or half integer spin, these operators satisfy the commutation or anti-commutation relations (for example: ab + ba = 1; the Pauli exclusion principle is guaranteed by quantization with the anti-commutation relations for spin 1/2 particles). They are the creation and annihilation operators, which operate on state vectors describing the number of particles in different states. This is called the second quantization, which endows particle property to the field (field + second quantization = quantum field). Thus, in quantum field theory the particles are just bundles of energy and momentum of the fields, which constitute the basic ingredient. A more mathematically oriented description on Quantum Field Theory can be found in the appendix.
Before performing the second quantization, a field equation has to be available to describe the dynamic of the field. It is found that the field equation can be derived by minimizing the "Action", which is a function of the field and its first derivative. Since there is an infinite choice for the form of the "Action", some conditions are imposed to limit the arbitrariness. For example, the "Action" should be invariant (unchanged) under the operation of translation, rotation, and time progression (these kinds of symmetry imply the conservation of momentum, angular momentum, and mass-energy respectively). However, the symmetry of the "Action" or the field equation does not guarantee the same for its solution. For example, the Schrodinger Equation (mentioned in topic 12) have rotational symmetry for the hydrogen atom, and yet only the wave function corresponding to zero angular momentum possesses a spherical configuration. (See Figure 12-07.)
Note that not all the fields in quantum theory are interpreted in the same way. While the wave function in the Schrodinger Equation (in non-relativistic quantum mechanics and not involved in 2nd quantization) has been considered as the probability amplitude of finding the particle at certain space and time, and the electromagnetic field is the expectation value corresponding to a certain state (in quantum field theory), there are no measurable identity for the other kind of quantum fields (such as the spinor in the Dirac Equantion) upon quantization.