Quantization and Field Equations
Klein-Gordon Equation of the Scalar Field
A relativistic theory still uses the same kind the canonical quantization rules as outlined in Eqs.(1) - (4). But the energy equation is now replaced by the one from the theory of special relativity, i.e.,
E2 = (mc2)2 + (pc)2 ---------- (15)
In this formulation, (px, py, pz, iE/c) together form a 4-momentum, which transforms like a 4-vector under the Lorentz transformation. The probability density has also to be redefined as the 4th (time) component of the flux density :
which satisfies the continuity equation :
| ---------- (16)|
where the Greek indices run from 1 to 4. By substituting Eqs.(3a) and (3b) into the energy equation Eq.(15) (ignoring the y, z components again) we obtain the Klein-Gordon Equation :
| ---------- (17)|
2[c2 - ] = m2c4 ---------- (18).
The natural units are often adopted in particle physics with c = = 1 by changing the unit of length to [1/(3x1010)]sec, and the unit of time to 1/(1.054x10-27erg), then the unit of the 4-momentum, energy and mass becomes the inverse of length, while the charge, velocity, and angular momentum are dimensionless. The original value of c and can always be re-introduced by dimensional consideration (see more in Natural Units). Thus, Eq.(18) can be re-written as :
[ - ] = m2, ---------- (19)
the solution of which is :
= A e-i(t-kx) ---------- (20)
where =E, and k2=E2-m2. Thus the 4-momentum can be written in the form : (kx, ky, kz, iE). By re-arranging the energy equation into the form : E2=m2+k2 and taking the square root :
E = (m2+k2)1/2 ---------- (21).
It is immediately clear that we have a problem with the negative energy (as mentioned earlier). Another troublesome formula comes from the definition of probability density (via Eq.(16)) :
P = is4 = 2AE ---------- (22)
which becomes negative for negative energy. But there is no such thing as negative probability density. The solution is to re-interpret P as the charge density, then particle with different sign of energy would associate with different charge leading to the particle/anti-particle duality. This logical step was not taken until 1934 by Pauli and Weisskoff after the discovery of positron in 1932.
It was later shown in addition that the particle/anti-particle pair is represented by the and * fields respectively.
The Klein-Gordon equation was first considered as a quantum wave equation by Schrodinger in his search for an equation to describe the de Broglie waves. The problem with negative energy and the wrong prediction on the fine structure of the hydrogen atom leaded him to introduce the Schrodinger equation instead (omitting the fine structure). The equation was proposed by Klein and Gordon a year later in 1927 for a relativistic electron. It is now recognized that this scalar field equation describes a spin 0 particle such as the pion or the proposed Higgs particle. The fine structure of the hydrogen atom will be determined correctly by the Dirac equation for spin 1/2 particle.