Heisenberg Equation of Motion Yields Field Equation

Background math needed for delta function relation

From Arfken and Weber, Mathematical Methods for Physicists, 4^th ed, pg 85,

,

(1)

where in our case we will have

,

(2)

so that (1) becomes

.

(3)

Deriving the Scalar Field Equation

,

(4)

and for a complex scalar field, this is

.

(5)

Thus,

(6)

where the quantities inside the integral are all functions of x^/ and t.  Since (x,t)  is not a function of x^/, we can evaluate the commutator inside the integral.  The second and third terms inside the integral of (6) commute with , and thus drop out.  Writing out the independent variable dependence only when needed for clarity and using the field commutation relations for  and π in the second line below, we find (6) becomes

(7)

Next,

(8)

Noting that , and using (3) in the third line below, the second term in (8) becomes

(9)

The reader can verify that evaluation of the third term in (8) leads to

(10)

Combining (7) and (10), one ends up with the Klein-Gordon equation

.

(11)

 

Return to Unifying Chart 3 Part 2.